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Dynamic marketing decisions in the presence of perishable demand Swami, Sanjeev 1998

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D Y N A M I C M A R K E T I N G D E C I S I O N S IN T H E P R E S E N C E O F P E R I S H A B L E D E M A N D by Sanjeev Swami B. Engg. (Production and Industrial) University of Allahabad M . Tech. (Industrial Management) I.I.T. Kanpur A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S F A C U L T Y O F C O M M E R C E A N D B U S I N E S S A D M I N I S T R A T I O N We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A August 1998 © Sanjeev Swami, 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Date DE-6 (2/88) Abstract This thesis seeks to advance our understanding of how quantitative models can be de-veloped and applied to marketing in complex dynamic environments characterized by demand perishability. Specifically, it involves three essays on the dynamic shelf-space management of movies. The problem is particularly complex for exhibitors - the re-tailers in the motion picture supply chain - given the short life cycles of movies, their perishable and uncertain demand, and complicated contracts. Our objective is to un-derstand, formalize, and develop optimal normative policies for such decision making situations. Essay 1 considers this problem from a theoretical standpoint by addressing the stochas-tic aspects of movie replacement, which is analogous to equipment replacement in main-tenance theory. We formulate this problem as a Markov decision process model. A scenario analysis reveals that the exhibitor is better off when shelf-space becomes scarcer for the distributors. A smart exhibitor associates a cost with contract parameters and bears it if it makes economic sense. The results underscore the importance of precise information for making smart replacement decisions. The optimal policy under special conditions resembles a control limit policy, which is easy to implement and compute. Essay 2 applies the theoretical concepts developed in Essay 1 to a special case of the movie replacement problem. The output of this essay is SilverScreener, which is a decision support model for movie exhibitors. The model helps the exhibitors both select (which) and schedule (when, how long) the movies. The model is readily implementable and appears to lead to considerable improvement in profitability in different comparative cases. The general nature of optimal policy emerges as: choose fewer "right" movies and i i run them longer. The robustness of the results is shown through sensitivity analyses. Essay 3 proposes a two-tier application of the SilverScreener model to show its effec-tiveness as a managerial aid. The master plan helps the manager in bidding and planning for movies before a season. The rolling horizon approach is useful for weekly replace-ment decisions during the season. Our results show that SilverScreener can improve the manager's profitability and promises to be an effective scheduling and planning tool. ii i Table of Contents Abstract ii List of Tables vii List of Figures ix Acknowledgments x 1 Introduction 1 1.1 Nature and Scope of This Research 1 1.2 Organization of this Thesis 6 1.3 Summary of the Three Essays 7 1.3.1 Essay 1: Stochastic Modeling of Replacement of Movies on Screens 7 1.3.2 Essay 2: SilverScreener: A Decision Support Model for Movie Screens Management 9 1.3.3 Essay 3: Approaches for Estimating Managerial Gains from Silver-Screener 11 2 Stochastic Modeling of Replacement of Movies on Screens 14 2.1 Overview 14 2.2 Modeling Framework 20 2.2.1 Machine Deterioration and the Movie Replacement Problem . . . 20 2.2.2 Statement of the M D P Model 22 2.2.3 A n Example Problem 28 iv 2.3 Optimality Criterion 33 2.3.1 Finite-Horizon Policy Evaluation 35 2.4 Scenario Analysis 37 2.4.1 Parameter Setup 38 2.4.2 Problems Motivation, Design and Analysis Results 41 2.5 Rank-based Optimal Policies 57 2.5.1 Motivation 57 2.5.2 Problem Design 58 2.5.3 Results 59 2.5.4 Managerial Insights 61 2.6 Conclusion 62 2.6.1 Contributions : 62 2.6.2 Limitations and Future Research 65 3 SilverScreener: A Decis ion Support M o d e l 91 3.1 Overview 91 3.2 Problem Formulation and Modeling Approach 95 3.2.1 The Exhibitor Problem 95 3.2.2 SilverScreener-DP: The Dynamic Programming Approach 98 3.2.3 SilverScreener-IP: Integer Programming Approach 105 3.3 Normative vs. Actual Decision Making: A Case Study of 84th St. Sixplex in New York 109 3.3.1 Data Description 109 3.3.2 Comparison Approaches 113 3.3.3 SilverScreener's Normative Results 115 3.3.4 Sensitivity Analyses 120 v 3.4 Conclusion 124 3.4.1 Contributions 124 3.4.2 Limitations and Future Research. 125 4 Approaches to Estimating Managerial Gains from SilverScreener 138 4.1 Overview 138 4.1.1 Decision Support Models in Marketing 138 4.1.2 Implementation Examples from Previous Research 140 4.1.3 Types of Potential Gains . . 144 4.2 Implementation Scenarios: Simulating a Manager 149 4.2.1 Master Plan Development 150 4.2.2 Rolling Horizon Approach 163 4.2.3 Heuristics . ' 167 4.3 Conclusion 171 4.3.1 Contributions 171 4.3.2 Limitations and Future Research 173 5 Conclusion 185 5.1 Discussion 185 5.2 Future Research 189 Bibliography 197 Appendix 204 A Programming Code in C Language for MDP Algorithm 204 vi List of Tables 2.1 Expected Net Revenue Values for Example Problem in Section 2.2 71 2.2 Summary of Scenario Analysis Results 72 2.3 Expected Net Revenue Values for Scenario Analyses 73 2.4 Initial Probability Matrices for Scenario Analyses 74 2.5 Transition Probability Matrices for Scenario Analyses 75 2.6 Supply Conditions Analyses: Design and Results 76 2.7 Results of Obligation Period Analyses 77 2.8 Results of Value of Information Analyses 78 2.9 A Hypothetical Example of a Rank-Based Control Limit Policy for the Movie Replacement Problem 79 2.10 Results of Rank-Based Optimal Policy Analyses . 80 3.11 Parameter Values of Example Problem in Section 3.2 128 3.12 Restricted Consideration Set 129 3.13 Expanded Consideration Set 130 3.14 Actual Schedule 131 3.15 Optimal Schedule Using Restricted Set Data 132 3.16 Optimal Schedule Using Expanded Set Data 133 3.17 Characterization of Solution 134 3.18 Sensitivity Analysis (Cost Data) 135 3.19 Sensitivity Analysis (Revenue and Obligation Period) 136 vii 3.20 Sensitivity Analysis (Revenue Sharing Terms) 137 4.21 Master Plan Schedule (Using ex ante Data for Revenue Prediction and Optimization Approach for Schedule Generation) 176 4.22 Bidding Solutions 177 4.23 Characterization of Solution 178 4.24 Actual Implied Schedule Obtained Using Rolling Horizon Ap-proach 179 4.25 Characterization of Rolling Horizon Results 180 4.26 Schedule Generated Using Distributors' Pressure Heuristic and ex ante Data for Revenue Estimates 181 4.27 Schedule Generated Using Distributors' Pressure Heuristic and ex post Data for Revenue Estimates 182 4.28 Schedule Generated Using Rank-Based Heuristic and ex ante Data for Revenue Estimates 183 4.29 Schedule Generated Using Rank-Based Heuristic and ex post Data for Revenue Estimates 184 viii List of Figures 2.1 The Supply Chain in the Motion Picture Industry 83 2.2 Timing of Events in the Movie Replacement Problem 84 2.3 Release Scenario for the Example Problem for Explanation of MDP Model 84 2.4 Expected Revenue Decay Patterns of Different Clusters (Movie Types) 85 2.5 Results of Supply Conditions Analyses 86 2.6 Release Situation for Obligation Period Analyses 87 2.7 Results of Obligation Period Analyses 88 2.8 Release Situation for Value of Information Analyses 89 2.9 Release Situation for Rank-Based Optimal Policy Analyses . . . 90 ix Acknowledgments First of all, I express my sincere gratitude to the Lord Almighty for providing His inces-sant grace and support in my efforts towards completing this thesis. In this endeavor, I have also benefited from the guidance and help of many individuals. Foremost is my supervisor, Dr. Charles Weinberg (or Chuck, as he likes to be called), who has been an incredible teacher and guide throughout my association with him during the thesis process. Chuck has always led by example and I can only hope to emulate him in the years to come. I will continue to envy his skill and capacity to be "on top of everything," even in the busiest times. It is with great pleasure that I express my gratefulness for his alacrity and willingness to provide continuous advice, insights, creative ideas, and support (including financial!). I also wish to thank the members of my thesis committee. Dr. Martin (Marty) Put-erman for providing critical feedback and showing enthusiasm in my research questions. I was very fortunate to have taken his marquee course, Markov Decision Processes. The theoretical foundation of this thesis has benefited a lot from this course, and I thank Marty for that. I also thank Dr. Shelby Brumelle for always being available to discuss research ideas and challenging me to think "beyond the box." He has shown me what a true academic should be. Last, but not the least, I thank Dr. Eunkyu (Kyu) Lee for providing useful comments on my ideas. I thank him for the SilverScreener brand name for my second essay. Being the youngest member of my committee, K y u has also been my "houng nim" (elder brother in Korean)! He also taught me how to teach. I would also like to thank Dr. Jehoshua (Josh) Eliashberg of Wharton Business School for his enthusiasm about my research questions. It was in his seminar course at U B C in x the summer of 1995 that we generated seminal ideas for this thesis. Though he is not a member of my thesis committee, he has shown a keen interest in my thesis. Perhaps most importantly of all, I thank both Josh and Chuck for expressing their confidence in my abilities. Then, there are people who have provided me their friendship, advice, help, love and support. The list is long, but some names deserve special mention - Michael, Murray, Moren, Jas, David, Darren, K P (from the senior PhD batches), Don, Jennifer, Dora (my cohorts), Lorenzo, Srinivas and Joshua (from the junior batches). I would also like to thank the other doctoral students and members of the marketing group at U B C . Special thanks are due to Ms. Rosalea Dennie, the secretary of marketing division, and Ms. Ruby Visser, the coordinator of the PhD program, for their support throughout my PhD program. I thank the satsangee families in the Vancouver and Surrey areas for making my stay here a comfortable and memorable one. Finally, how could I ever thank my family members, especially my parents, for their unflinching support during the last four years! They always encouraged me to "keep it up" when the goings were tough. They were there when I needed their help. I thank them for providing me that extra push in the times when I felt that I will never be able to do it! x i Chapter 1 Introduction 1.1 Nature and Scope of This Research Quantitative modeling in the field of marketing is progressing rapidly. These promis-ing analytic and data-based findings are important to the development of a marketing theory that enables marketers to understand and manage markets more effectively in an increasingly dynamic and complex environment. In particular, there has been a rapid development in the research addressing the issues related to dynamic decision making. Examples can be found in areas such as advertising (Erickson 1995), brand choice (Erdem and Keane 1996), new product development (Bayus 1995), channel coordination (Chin-tagunta and Jain 1992), consumer behavior (Krishna 1994), market research (Bockholt and Dillon 1997), pricing (Raman and Chatterjee 1995), promotion (Papatla and Kr-ishnamurthi 1996), and retailing (Krider and Weinberg 1997). Most previous marketing research on dynamic decision making has been concerned with consumer durable goods, not with services or intangibles. This thesis seeks to advance our understanding of how quantitative model building can improve marketing decision making in complex dynamic environments by focusing on a perishable entertainment product. In particular, it in-volves three essays concentrating on the development and application of marketing and management science and operations research methods to dynamic retail management problems of products characterized by demand perishability. Previous researchers have recognized the interface between marketing and operations 1 Chapter 1. Introduction 2 as an important research domain. In the operations management literature, Acquilano and Chase (1991, p. 17) mention that marketing specialists "need an understanding of what the factory can do relative to meeting customer due dates, product customization, and new product innovation. In service industries, marketing and production often take place simultaneously, so a natural mutuality of interest should arise between marketing and O M [operations management]." Eliashberg and Steinberg (1993) also suggest that such an integrative viewpoint has the potential to build on synergy between the two functions. More recently, Karmarkar (1996) stresses the need to do more integrative research between marketing and operations management. This thesis addresses such needs and uses both marketing and operations management tools in solving management problems. In particular, we apply scheduling and replacement methodologies from oper-ations management literature to the general area of shelf space (display) management of marketing. Shelf (display) space management is an area of important concern to retailers. First, consumers choose from the products that are displayed on the shelf. In this sense, shelf space management determines the ultimate profitability of the retailer. Second, most retailers have limited shelf space to display their products. Therefore, the choice of which products to display becomes an important retailing decision. The decision is particularly complex in those industries in which many new products are continually introduced in market. Finally, anticipating and adapting to dynamic changes in consumers tastes and demand are key concerns to most retailers. Moreover, such decisions have to be made for several product categories. Though previous researchers (Corstjens and Doyle 1983; Bultez and Naert 1988; Borin, Farris, and Freeland 1994) have addressed some of these issues, they have generally been in the context of packaged goods in a supermarket chain setting. However, other industry settings, such as the motion picture (or movie) industry considered in this thesis, pose different challenges and intriguing problems. Chapter 1. Introduction 3 A movie is an interesting product for several reasons. Most movies are separate entities and can be considered an innovation because of the new features (e.g., actors, storyline, music, etc. ) usually included in them. These new movies are released every week throughout the year. Hollywood's major studios produce over 450 feature films every year. The product life cycle of movies is relatively short and is measured in weeks. Second, the movie industry is a big business and the industry's revenues for attendance at theaters were between five and six billion dollars in the 1990's. The product is seasonal in nature and the two prominent seasons are Summer and Christmas. Finally, one of the most interesting attribute of movies from a research standpoint, and a major com-ponent of the complexity associated with the problems addressed in this thesis, is their perishability. Typically, perishability is thought of in terms of physical deterioration of a product such as a grocery item with an expiry date. In some sense, this view comes from the supply side of the product. In contrast, in this thesis we adopt a demand side view in the context of perishability. Thus, the physical product in the problems considered remains the same, but its demand perishes over time 1. In recent years considerable work has been done on the treatment of perishability in inventory control (e.g., Abad 1996, Jain and Silver 1994). In inventory control, per-ishability refers to the physical deterioration of units of a product. For the movies, we analogously define perishability as the decrease in the value/appeal of a movie with the passage of time. As mentioned earlier, previous research in shelf space management and dynamic decision making has mainly focused on non-perishable goods. In the context of movies, the complexity introduced by perishability in dynamic shelf space management 1The notion of perishability adopted in this thesis refers mainly to gradual perishability. Another type of perishability, sudden perishability, has been addressed in the area of yield management (Little-wood 1982; Belobaba 1987; Brumelle et al. 1990), which is also referred to as perishable-asset revenue management (PARM) (Weatherford and Bodily 1992). Sudden perishability implies that the product is available only for a finite period of time, after which the unsold stock has no (or limited) value (e.g., fashion goods, cruise lines, airline seats booking). Chapter 1. Introduction 4 problem can be summarized as: which motion pictures to choose to show each week and for how long to play them. This decision making problem is further complicated by some other factors specific to the movies context. We discuss such factors in detail in the later chapters to follow. Though we focus on movies in this thesis, our methodology and results can readily be generalized to other entertainment products (e.g., performing arts, books, video games), travel services, fashion goods, and educational programs. In response to the dynamics and challenges posed by the above characteristics of movies, a stream of research, particularly addressing the marketing of movies, has con-tributed to the marketing literature. At the consumer behavior level, some of the re-search has questioned the relevance of the traditional information-seeking framework for studying the consumption of movies (e.g., Hirschman and Holbrook, 1982; Holbrook and Hirschman 1982). Another stream of research has focused on forecasting the enjoyment of movies at the individual level (Eliashberg and Sawhney 1994) as well as forecasting the commercial success of movies at the aggregate level (Smith and Smith 1986; Austin and Gordon 1987; Dodds and Holbrook 1988; Sawhney and Eliashberg 1996; Eliashberg and Shugan 1997). Additionally, some research has begun to emerge addressing diffusion (Mahajan, Muller, and Kerin 1984; Jones and Ritz 1991), seasonality (Radas and Shugan 1995), release timing (Krider and Weinberg 1998), clustering (Jedidi, Krider, and Wein-berg 1998), sequential products (Lehmann and Weinberg 1998; Prasad, Mahajan, and Bronnenberg 1998), contract design (Swami, Lee, and Weinberg 1998), and the impact of advertising (Zufryden 1996). This thesis advances the above stream of research. We now briefly discuss the specific modeling techniques used in the current research. In this thesis, we focus on a special type of dynamic decision making situation - that requiring a sequence of interrelated decisions to be made. These are usually formulated as a sequential decision model (Puterman 1994). In these problems, at a specified point in time, a decision maker, or controller, observes the state of a system. Based on this Chapter 1. Introduction 5 state, the decision maker chooses an action. The action choice produces two results: the decision maker receives an immediate reward, and the system evolves to a new state at a subsequent point in time according to an equation of motion determined by the action choice. At this subsequent point in time, the decision maker faces a similar problem, but now the system may be in a different state and there may be a different set of actions to choose from. The objective of the decision maker is to choose a sequence of actions which causes the system to perform optimally with respect to some predetermined performance criterion. When the decision is assumed to be made at discrete points in time and the equation of motion is assumed to be deterministic, the problem is usually modeled as a (deter-ministic) dynamic programming problem (Bellman 1957). The idea behind the dynamic programming approach is to breakdown a large problem into several smaller stages -called separability - to devise a sequence of simpler optimal solutions that lead to the overall optimal solution. A discrete time problem that allows for stochasticity in the equation of motion is usually modeled as a Markov decision process (MDP) problem (Puterman 1994), also known as a stochastic dynamic programming problem. In these problems, the system makes transitions from a discrete point in time to another accord-ing to a transition probability function. Finally, it is also possible to write an equivalent linear program for a finite state deterministic dynamic programming problem (Bertsekas 1987; Ahuja, Magnanti, and Orlin 1993). This equivalence between dynamic and linear programming allows for the problems involving binary decision variables to be formulated as integer programs. In this thesis, therefore, we use the following quantitative modeling techniques: dynamic programming, Markov decision processes, and integer programming. A l l of these techniques have been used extensively in management science and operations research literature for solving scheduling and replacement problems. To summarize, our objective in this thesis is to understand and formalize, using Chapter 1. Introduction 6 quantitative modeling techniques, certain dynamic decision making situations involving shelf space management of perishable products. The understanding and formal structure help us in developing optimal normative policies for these problems. We present three essays in the next three chapters towards this end. 1.2 Organizat ion of this Thesis The rest of this document is organized as follows. In the first essay (presented in Chapter 2), our objective is to formalize the problem of dynamic movie replacement on theater screens. In doing so, we propose a theoretical model of the movie replacement phe-nomenon. This formal modeling and the resulting normative policies seek to provide the decision maker with conceptual insights regarding the general direction of the optimal policy for complex real problems. The second essay (presented in Chapter 3) applies the theoretical concepts developed in the first essay to a special (deterministic) case of the general movie replacement problem. The normative optimal policy in this essay results in a readily implementable decision support model, SilverScreener. In the third essay (presented in Chapter 4), we discuss the types of potential managerial gains that result from a decision support model. We provide various methods for estimating those gains in the context of SilverScreener, the model developed in the second essay. Therefore, we begin from the theoretical treatment of the general problem in Essay 1 and address increasingly applied issues as we approach Essay 3. We conclude in Chapter 5 by sum-marizing and interpreting the findings of the current research, and noting the directions for future research. We now provide a summary of the three essays contained in this thesis. Chapter 1. Introduction 7 1.3 Summary of the Three Essays 1.3.1 Essay 1: Stochastic Modeling of Replacement of Movies on Screens Managing the allocation of shelf space for new products is a problem of significant impor-tance for retailers. The problem is particularly complex for the exhibitors - the retailers in the motion picture supply chain - because they face dynamic challenges, given the short life cycles of movies, the exponentially decaying level of demand over time, and the com-plex revenue sharing contract between the exhibitor and the distributor. The dynamic environment of this problem gives rise to the notions of decay and aging of the movies. Decay is the intrinsic weekly decline in the box-office attraction of a movie playing at a theater (Krider and Weinberg 1998). Aging is the decline in the box-office attraction of a movie from an exhibitor's perspective if there is a delay (e.g., by a week) in exhibiting the movie at the theater. Aging, therefore, results in an opportunity cost of not being able to play a particular movie. The demand for a movie is stochastic in nature. Movies decay and age in an uncertain fashion. Another source of demand uncertainty is the opening strength of the movie. Essay 1 (Chapter 2) considers this general problem from a theo-retical standpoint by addressing the stochastic aspects associated with movie scheduling and replacement. We formulate this problem as a Markov decision process (MDP) model. The problem is analogous in important ways to the equipment replacement problem in maintenance theory. The decay of a movie playing at a screen is considered equivalent to the deterioration of an equipment in the maintenance framework. The objectives of this essay are as follows. First, we introduce the theoretically appealing and conceptually interesting M D P modeling framework to understand and formalize movie replacement phenomenon. In doing so, we define a "smart" (exhibition) manager who takes decision according to the normative optimal policy recommended by Chapter 1. Introduction 8 the solution of the M D P problem, that is, a decision maker who takes into account un-certainty in the environment and adopts a long-term view in his/her decisions. Second, we investigate optimal movie replacement policies to be followed by the exhibitor under different setting in a stochastic environment. These normative policies suggest how a smart manager should behave under different settings. The third objective, which is re-lated to the second one, is to characterize the optimal policies that emerge. In particular, we explore the possibility that the optimal policy possesses a simple structure for the movie replacement problem. We investigate various decision making settings by simulating three different scenarios. Our results from supply conditions analyses show that the exhibitor is better-off in those situations in which his/her shelf-space becomes a scarcer commodity for the upstream channel members (e.g., producers, distributors). We find that it is the interaction of greater quantity and better quality of the movies that brings increased value to the exhibitor. The quality of a movie is defined in terms of its opening box-office strength and its weekly decay rate. In this analysis, we also find that a smart exhibitor would prefer the movies with slow decay rate with an early release. The obligation period (a contract parameter) analysis indicate that a smart exhibitor associates a "cost" with different parameters of the contract provided by the distributor. He/she is willing to bear that cost if it makes economic sense according to the chosen optimality criterion. Together, the supply conditions and obligation period analyses have important implication for the release strategies to be adopted by the distributors. The value of information analysis stresses the importance of having the right information for making smart decisions in the exhibition business. We find that as the quality of information about a forthcoming movie deteriorates, the exhibitor makes sub-optimal movie scheduling decisions and retains existing movies longer than desirable. The sub-optimality is characterized as a cost (in terms of decrease in value function) the exhibitor bears as a result of making bad choices. Chapter 1. Introduction 9 Finally, we explored the possibility of characterizing the structure of the optimal policy for the movie replacement problem in a similar way to a control limit policy for equipment replacement problem. Such policies are easier to implement because they dras-tically reduce the computational effort required to solve the decision problem optimally. Due to some fundamental differences between the two problem settings, we do not find the optimal policy has a strict control limit policy form. However, when characterized on a reduced state variable comprising only of ranks, the optimal policy resembles a control limit policy. We discuss a heuristic based on this rank-based optimal policy, which is used and tested in Essay 3 (Chapter 4). 1.3.2 Essay 2: SilverScreener: A Decis ion Support M o d e l for M o v i e Screens Management Essay 2 applies the theoretical concepts developed in Essay 1 to a special case of the general movie replacement problem. The output of this essay is SilverScreener, which is a decision support model for movie exhibitors, the retailers of the motion picture industry. This essay advances the marketing research stream that focuses on the development of decision support models to help retailers improve their decisions making (e.g., Bultez and Naert's (1988) S H A R P model, and Abraham and Lodish's (1993) P R O M O T I O N S C A N system). More specifically, the objective of SilverScreener is to help the movie exhibitors decide which movies to show on the multiple screens of their theaters. In the U.S. and Canada, the total number of screens has remained relatively constant in the last five years. However, the total number of mass market movies released by the major studios seems to be rising steadily. Combined with the stable pool of moviegoers, this trend suggests that a major exhibition chain with an objective of effective screens management faces a complex scenario. The complexity comes from various sources. First, the increased supply of high-budget movies increases the difficulty of deciding Chapter 1. Introduction 10 which movies to play. It also implies that the ensuing scarcity of "shelf space" would require special attention in managing the screens effectively and profitably. The shelf space scarcity phenomenon is aptly summarized in an Apri l , 1995 headline in Variety, "So Many Pix, So Few Screens." The dynamic decision making scenario thus induced is further complicated by the distributor-exhibitor contract, which is unique to the motion picture industry. A typical exhibition contract states a fixed obligation period and a differential revenue sharing scheme in different weeks between the distributor and the exhibitor. The obligation period limits the ability of an exhibitor to replace a "rapid decay" movie while the replacement movies "age". The manner in which the revenue sharing scheme is designed favors the distributor in the first few weeks and the exhibitor later on. Distributors thus have a strong incentive to promote the movies in their initial play period. On the other hand, the longer the exhibitor plays a movie, the larger becomes his/her share of the (declining) box office receipts. In the face of this complexity, our model provides a structure to analyze the problems of above nature. We use a deterministic dynamic programming approach to formulate the problem. Then, treating the multiple screens as parallel machines and the movies as jobs to be scheduled, we provide an analogy of the exhibitor problem to the parallel machine scheduling problem (Baker 1993; Pinedo 1995). The resulting problem is for-mulated as an integer program, which makes it convenient to use standard mathematical programming algorithms to solve large size problems generally encountered in practice. The developed model is readily implementable, as we demonstrate in an illustrative ex-ample, and appears to lead to improved profitability in two different comparative cases. In the first case, restricted set analysis, the algorithm developed considers only those movies that were shown by the theater chosen for the analysis. In the second panded set analysis, the algorithm also considers, in addition to the restricted set movies, the movies that appeared in the "Top 50" list of the movie trade magazine Variety but Chapter 1. Introduction 11 were not shown by the theater. The improvement over actual decisions in terms of prof-itability appears to result from a combination of both better selection and scheduling of the movies. The general nature of the optimal policy emerges as: choose fewer "right" movies and run them longer. Through sensitivity analysis, we demonstrate that these results are robust to various parameters of the problem. 1.3.3 Essay 3: Approaches for Estimating Managerial Gains from Silver-Screener Montgomery and Weinberg (1973) note in the context of marketing decision support models: I n 1967, when one of the authors was asked by a Coca-Co la vice president at the beginning of a model ing project to " t e l l us a l l about the successful applications of market ing models ," the answer d i d not take long. W h e n the same question was asked by another executive last month , the answer ended only w h e n the mode l builder was reminded that he h a d exceeded the t ime available. This trend has continued and a number of successful applications of marketing decision support models can be found in the literature. As Rangaswamy (1993) states, "Decision models are playing an increasingly important role in supporting management decision making." The major objective of this essay is to show that SilverScreener, the model developed in Essay 2, can achieve the same level of success as similar previous modeling applications. In particular, the objectives of this essay are as follows. First, we identify the types of potential managerial gains that result from a decision support model, such as SilverScreener. Next, we provide methods for estimating those gains. Finally, we show that the implementation of SilverScreener model may improve managerial decision making. Chapter 1. Introduction 12 We propose a two-tier application of SilverScreener. The first tier involves develop-ment of a master plan that would help the manager plan a season before the start of that season. A n ex ante revenue prediction scheme is developed for this application which uses the attributes of a movie such as genre, M P A A Rating, stars, distributor, and so on. The master plan could assist the manager to decide, before the season, whether to bid for a movie or not. It can also help the exhibitor decide on some of the contract terms for the movies chosen to bid. Our analysis shows some of the ways in which a master plan could be developed. The results show that, despite a less than perfect revenue prediction scheme, the master planning exercise can help the exhibitor make effective planning and bidding decisions. The second tier, rolling horizon approach, can be used for weekly decisions during the season, possibly after the development of the master plan. In the beginning, this approach requires an estimate of the opening strength and decay rates for the set of available movies. Subsequently, it would involve would involve "rolling" decisions and updating data, from one time window to another. To operationalize this approach, we use the one-week ahead forecasts of movies' box-office revenue that is pub-lished in Variety every week. Our results show that the rolling horizon approach, using these non-hindsight revenue forecasts, can achieve an improvement (in profit terms) over actual schedule of the same order as that achieved by a perfect information based one-shot optimization approach. In addition to the two-tier application, we propose two alternative decision rules (heuristics) that provide a comparison criterion for the SilverScreener model. We compare the performance of SilverScreener model with the heuristics under two different levels of information availability: ex ante revenue prediction and ex post perfect information. SilverScreener outperforms the two heuristics under both levels of information availability. The results of distributor pressure heuristic show that the exhibitor could lose money if he/she tries to please all the distributors in the market. The results of rank-based Chapter 1. Introduction 13 heuristic show that it performs reasonably well. This heuristic is a simple version of the rank-based optimal policy (as explained above) of Essay 1. Its good performance in a multiple screen setting using restricted set data and a longer planning horizon suggests that some reasonable rank-based heuristic could be derived for the exhibitor problem. Collectively, the results of this essay show that the SilverScreener model has potential for improving profitability of exhibitors, and can provide several related benefits such as an aid in bidding, weekly planning and control, scenario analysis, and synthesis of marketing information on an ongoing basis. C h a p t e r 2 S t o c h a s t i c M o d e l i n g o f R e p l a c e m e n t o f M o v i e s o n S c r e e n s 2.1 O v e r v i e w Every week, motion picture exhibitors have to make an important decision regarding the replacement of the movies playing at the screens in their theaters. Figure 2.1 describes the typical supply chain in the motion picture industry. Movies are, in general, pro-duced and distributed by the major studios and played for public consumption by an exhibitor. Movies are a seasonal product, and the two prominent seasons are Summer and Christmas. In either of these seasons, a number of movies are released every week by the studios/distributors. The dynamic competitive environment thus induced gives rise to the notions of decay and aging of movies. Decay is the intrinsic weekly decline in the box office attraction and gross revenues (grosses in industry jargon) of a movie playing at a theater (Krider and Weinberg 1998). Aging is the decline in the value, that is, gross generating power, of a movie from an exhibitor's perspective if there is a delay (e.g., by a week) in exhibiting the movie at the theater. Aging, therefore, results in an opportunity cost of not being able to play a particular movie. The replacement of movies on theater screens is a complex task. The complexity comes from various sources. First, the managerial decision making involved is dynamic and long-term in nature. It is dynamic since the replacement (or continuation) decisions have to be made every week because of the weekly releases of various movies. The weekly availability of movies and a specialized contract between a distributor and exhibitor (to 14 Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 15 be described later) imply that the decisions taken at a point in time affect the immediate profits and the future decisions and corresponding long-term profits. Second, movies decay and age in an uncertain fashion. This uncertainty in demand for movies makes the task of their replacement a difficult one. For example, suppose the exhibitor decides to replace an existing movie with a new one. Because of the uncertainty in demand, he/she may be worse off than before if the new movie decays very rapidly. In an opposite situation, suppose he/she decides to continue with the existing movie. In this case, he/she may be better off replacing the existing movie if the new movie is a good one, and the existing movie decays rapidly. This complexity is further increased because the uncertainty has to be considered for a number of movies. Another source of demand uncertainty is the opening strength of the movie. Finally, the nature of the distributor-exhibitor contract in the motion picture industry is unique. In signing a contract to play a film in its theaters, the exhibitor becomes obligated to play the film for a certain period of time, even when consumer demand is weak. This minimum obligation period (playtime), which is negotiated between the two parties, may vary by movie as well as by studio. The financial arrangements between studios and exhibitors are also unique to the motion picture industry. The box-office grosses are split between the exhibitors and distributors of motion pictures. The manner in which the box-office grosses are split favors the studio (distributors) in the first few weeks of the movie playing, but shifts to the exhibitor's favor later on. Distributors thus have a strong incentive to promote the movies intensively in their initial play period. On the other hand, the longer the exhibitor plays the movie, the larger becomes his/her share of the box-office receipts1. At the same time, theater attendance for a movie typically declines the longer a movie 1For a detailed explanation of the nature of the contract in the movie industry, the readers are referred to the introductory sections of Chapter 3. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 16 plays2. The above explanation represents the key elements of the movie replacement problem faced by the managers in the motion picture exhibition industry. The complexity of the movie replacement phenomenon poses interesting challenges for its quantitative modeling. The insights to be drawn from modeling will be useful for several reasons. First, a formal model would provide a conceptual and theoretical framework for the complex problem. This framework will provide a better understanding of the problem and would permit subsequent testing of various example situations. Second, by optimally solving the resulting dynamic model with a long-term objective, we will be able to prescribe normative strategies that would suggest how an optimally acting manager should behave in various problem settings. Accordingly, for the rest of this chapter, we define a smart manager as the one who optimally solves his/her movie replacement problem by taking demand uncertainty into account and adopting a dynamic and long-term approach in decision making. Finally, the model of movie replacement, and its solution, will help us characterize the general nature of normative strategies that emerge. The major portion of the complexity in the movie replacement problem, as explained above, comes from the stochasticity of demand. Accordingly, our emphasis in this chap-ter is on the issues associated with stochasticity of demand for a movie and their relation with the movie replacement problem. The insights from the treatment of stochasticity may be useful because they provide a richer description of reality. It has been noted previously that "Demand for a new movie is highly uncertain, since movies are "expe-riential" products " (Sawhney and Eliashberg 1996). In an uncertain environment, 2There are a few other elements of the general movie exhibition problem that bring additional com-plexity in the overall management of theater screens by a major exhibition chain. These features are the availability of multiple screens in a theater, increased supply of movies by the studios and the resulting shelf-space scarcity. As we discuss later, our emphasis in this chapter is on the aspects related to the stochasticity of demand from a theoretical standpoint. We address these other elements in Chapters 3 and 4. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 17 therefore, the exhibitor may know the revenue potential of a movie only probabilistically. The incorporation of uncertainty increases the number of possibilities to be considered for the future revenue streams. A l l such possibilities must be included in the model because the decision at different points in time are inter-related, and together affect the long-term objective of the exhibitor. This implies that one of the most desirable features for a model for the movie replacement problem is that its solution recommends a decision corresponding to every possible state of the realization of uncertainty. The mathematical model presented in the following sections is aimed at fulfilling this objective. The weekly nature of the movie replacement problem makes the dynamic program-ming approach a good candidate to model this situation if we could visualize the weekly transition in a revenue stream as a discrete time Markov chain. Indeed, as we show in the next section, such a Markov chain structure is achieved by modeling the revenue variable in a special way. The resulting dynamic programming model is termed a discrete time stochastic dynamic program, or Markov Decision Process (MDP) model. The M D P model falls in the general area of the probabilistic sequential decision model. The usual setting of these problems is as follows: A decision maker, agent, or controller is faced with the problem, or some might say, the opportunity, of influencing the behavior of a probabilistic sys-tem as it evolves through time. He/she does this by making decisions or choosing actions. The action choice produces two results: the decision maker receives an immediate reward, and the system evolves to a new state at a subsequent point in time according to a probability distribution determined by the action choice. His/her goal is to choose a sequence of actions which causes the system to perform optimally with respect to some predetermined performance criterion. Since the system we model is ongoing, the state of Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 18 the system prior to tomorrow's decision depends on today's decision. Con-sequently, the decision maker must not make decisions myopically, but must anticipate the opportunities and costs (or rewards) associated with future system states. (Puterman 1994) This simple probabilistic sequential decision model has generated a rich mathematical theory over the last four decades in the fields of management science and operations research. For a complete discussion of the theory of the stochastic aspects of the dynamic programming problems, readers are referred to the excellent texts by Puterman (1994) and Bertsekas (1987). In a M D P problem, at each decision epoch, the decision maker chooses an action in the state occupied by the system at that time. A decision rule3, dt(s), specifies the action to be chosen at time t given state s. A policy is a sequence of decision rules. A n attractive feature of the M D P model is that its solution provides the decision maker with an optimal policy, or strategy, that is, the prescription for choosing the optimal action in any possible future state. M D P models are, therefore, a useful tool in modeling a variety of situations involving sequential decisions. In a series of papers, White (1985, 1988, and 1993) surveys a wide range of applications of MDP's in the areas of fisheries, forestry, water resources, airline booking, vehicle replacement, inventory management, and so on. Although a few modeling efforts have appeared in the field of marketing science that use dynamic programming (e.g., Zufryden 1986; Horsky and Mate 1988; Krishna 1994), most of those have not considered stochastic elements. Recently, however, Bitran and Mondschein (1996) apply the M D P framework in the area of direct marketing. Their study examines optimal policies in the catalog sales industry where there is limited access to capital. Our work in this paper is similar to such efforts as Bitran and Mondschein's in 3Deterministic, Markovian decision rule. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 19 modeling and solving the sequential decision making problems in marketing that involve uncertain scenarios. To best of our knowledge, ours is the first attempt to address the stochastic aspects associated with the replacement decisions in movie exhibition. A few earlier attempts at capturing the stochastic elements in the context of movies have been mainly concerned with forecasting. Eliashberg and Sawhney (1994) focus on forecasting the enjoyment of movies at the individual level, while Sawhney and Eliashberg (1996) focus on forecasting the commercial success of movies at the aggregate market level. Their approach needs at least three weeks of data for estimation. This requirement makes it unsuitable for the current problem in which the exhibitor must have, even though probabilistically, an estimate of the revenue potential "before" the release of a movie. De Vany and Walls (1996) apply chaos theory to present a mathematical model of how moviegoers behave. They propose that information cascades generate box office hits and flops. They focus on the distribution function at the market level in the movie industry, and do not specifically address the replacement issues related with the exhibition function. The objectives of this research are the following. First, we introduce the theoretically appealing and conceptually interesting framework of Markov Decision Process (MDP) methodology in solving a complex replacement problem. In doing so, we define a "smart" manager who takes into account uncertainty in the environment and adopts a long-term view in his/her decisions. Second, we investigate optimal movie replacement policies to be followed by the exhibitor under different setting in a stochastic environment. We investigate the different settings in various scenario analyses designed to address key aspects of movie replacement phenomenon. The third objective, which is related to the second one, is to characterize the optimal policies that emerge. In particular, we explore the possibility that the optimal policy possesses a simple structure for the movie replacement problem. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 20 The first set of the scenario analyses considers the effect of changing supply conditions on the manager's replacement policies. We manipulate these conditions by varying the quantity and quality (to be defined later) of the movies available in the market for exhi-bition. The second problem investigates the effect of a contract parameter (an obligation period imposed by distributor) on the exhibitor's replacement policies. The third prob-lem examines the value of having the precise information about movies' revenue potential on making smart replacement decisions. The rest of this chapter is organized as follows. Section 2.2 presents our modeling framework. In Section 2.3, we discuss the optimality criterion we use to evaluate different policies. We present the policies that emerge in various scenario analyses in Section 2.4. In Section 2.5, we explore the possibility of characterizing the optimal policy obtained for movie replacement problem as a simple control limit policy. We conclude in Section 2.6 with discussion on limitations and future research. 2.2 Modeling Framework 2.2.1 Machine Deterioration and the Movie Replacement Problem The classical machine (or equipment) replacement problem introduced by Derman (1963) is one of the standard applications of MDP's . It assumes that a machine is observed at discrete intervals and takes values in a finite state space = {1, • • •, z}, where 1 represents the best state (condition) and z, the worst. If the machine is in state i at the beginning of a period and is not replaced, then the probability of its being in state j at the beginning of the next period is pij (i.e., the machine deteriorates according to a Markov chain with stationary transition matrix P = [ptj]). Since the seminal work by Derman (1963), there has been a good amount of research on the topic of equipment replacement. Recent work on this topic includes papers by Hopp and Nair (1994), and Abdel-Hameed (1995). In Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 21 this type of problem, at each decision epoch, the decision maker observes the current state of the system and decides whether to continue or replace the machine. A control limit policy has been found to be optimal for such problems under certain conditions. That is, for each time period there exists a control limit with the property that if the current state of the system exceeds the control limit, the optimal decision is to replace the machine, otherwise operate it for another period. We consider the movie replacement problem for a single screen theater as similar in structure to the above machine deterioration and replacement problem. We visualize the movie playing on a screen as the deteriorating machine. Lehmann and Weinberg (1998) found by conducting empirical analyses on a database of movies that the exponentially declining model fits the revenue pattern of most mainstream movies well. It is therefore reasonable to assume equivalence between the exponential decay of a movie at a screen and the deterioration of a machine. In addition to the movie playing at a screen, we con-sider the aging of all other available movies' revenues also according to the exponentially declining model. To model the deterioration of a movie at a screen, we introduce a parameter, rank, of a movie. We define rank of a movie as its relative position with respect to other movies in terms of the box-office gross revenue. We assume that in any week, a movie can be in one of the ranks from the set {1, • • •, z}, where 1 denotes the highest rank and z, the lowest. We further assume that the knowledge of the rank of a movie provides information about its expected gross revenue. In other words, different movies with the same rank have the same expected gross revenue. These assumptions are not unrealistic because Variety publishes rank data every week for the "Top 50" movies of the week in terms of their box-office gross collection for the week. Moreover, some evidence suggests that ideas akin to a rank structure are being used in the movie industry. For example, Barry Reardon, president of Warner Brothers Distributing Corporation, states: Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 22 What follows is one of the three scenarios. If the picture opens strongly,.... If it opens moderately or poorly,.... (Squire 1992, p. 317) In our proposed framework of ranks, these comments suggest a three-rank structure - High, Average, and Low. The transitions of a movie between different ranks can be modeled according to a Markov chain. The transition matrix of the Markov chain can be designed appropriately to mimic the exponentially declining pattern of a movie. Previous researchers have also used the notion of ranks to characterize performance of a movie. For example, De Vany and Walls (1997) analyze a large sample of movies as an evolving rank tournament of survival and death. They analyze a 50-rank data set based on data from Variety. Their results also find support for an exponential decay of a movie. We now present the M D P formulation of the movie replacement problem. 2 . 2 . 2 S t a t e m e n t o f t h e M D P M o d e l We model the movie replacement problem of the previous section as a discrete time finite-horizon M D P problem. To simplify the analysis, we make the following assumptions. 1. The theater has only one screen on which to show movies, 2. The replacement decisions are made on a weekly basis. Further, the replacement decision for the coming weekend is made on the previous Monday, and delivery of the replacement movie occurs instantaneously, 3. The probability of the transitions of a movie between various ranks is stationary over time and independent of the other movies, 4. The release dates of the movies considered during the planning horizon are deter-ministic and known in advance, Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 23 5. Movies once replaced at a theater are not available for screening at the same theater, and 6. The theater is a monopoly theater in its geographic location. Assumptions 2, 4, and 5 are consistent with industry practice. The rest of the as-sumptions have been made for the tractability of the analysis. Assumption 3 is quite limiting because of the following reasons. First, the independence between the transition probabilities of movies may be questionable in some cases, because decay of a movie may affect that of another if they are released close to each other. Second, the transition of the movie between the ranks may vary depending on whether the movie is in the early or later part of its run. We make this assumption to keep the analysis simple at this level of generality. The resulting description is parsimonious, yet it accommodates a wide range of interesting issues. The relaxation of some of these assumptions will provide attractive future research opportunities. We comment on these issues in the final section. Assumption 2 implies the timing of events in this problem, which we present in Figure 2.2. The timing-of-events diagrams are helpful in understanding the points in time when the relevant information becomes available and when the crucial decisions are made on a periodic basis. As shown in the figure, the movie m t is playing at the beginning of week t. The exhibitor assesses its weekend box-office performance on Monday. We assume that this assessment of the movie's weekend performance gives the exhibitor enough information to determine the rank of the movie for the upcoming decision epoch4. 4 As long as instantaneous delivery (Assumption 2) can be assumed, there is no difference between a Monday and the following Friday from a decision making standpoint. This suggests another alternative for the timing of events. In this case, we could assume that both the availability of information (about a movie's rank) and decision making occur on the Friday of a week. Thus, the exhibitor assesses the movie's performance over the last week (not just the weekend), which gives him/her the rank of the movie. If the exhibitor continues the existing movie, he/she incorporates the transition probability into his/her decisions to predict which rank the movie will be in the following Friday. If the exhibtior decides to replace, then the delivery of the replacement movie occurs instantaneously on the current Friday. In Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 24 This is reasonable to assume because the weekend is the period of higher demand in a movie's weekly performance. This concurs with practice in the movie industry (Verter and McGahan 1997). Given this assumption, since a movie's rank for a given week is known on a Monday, the exhibitor incorporates transitions of the movie from the current rank to the lower ranks next Monday through a transition probability matrix (to be discussed later) and makes the replacement decision for the coming Friday. Thus, the decisions are made on a weekly basis, and the decision epochs are the Monday mornings of every week. If the decision is to continue the current movie, mt+\ is the same as mt, otherwise a different movie is chosen from the set of the available movies. If the decision is to replace the current movie by an already released movie, then the timing of events essentially operates in a similar fashion as the current movie. That is, the rank of such a replacement is known on a Monday for the replacement decision for the coming Friday. The exhibitor could have this knowledge of rank because either an additional print of the movie might be playing at a theater in some other location or its nationwide performance might give an indication of its rank. Let us consider the case of a new movie now. Since the replacement decision is made on the Monday preceding the Friday when a movie is released, we assume that the opening rank of a new movie is known on that Monday. Sources of information for the exhibitor to be confident of a movie's opening include advertising levels, word-of-mouth, private market research, and so forth. Moving earlier in time, the exhibitor incorporates the initial probability (to be described later) of a movie opening in a particular rank a week before the Monday before the movie's release. case of a new movie, the following sequence of events will emerge. The movie is released (from distributor to exhibitor) on a Friday, the exhibitor knows its rank instantaneously, and the delivery also occurs on the same Friday. The exhibitor invokes the initial probability of a movie's opening rank the preceding Friday into his/her decisions. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 25 We now define the parameters of the model development. W - length of planning horizon in weeks, say summer season5, M - set of movies considered during the planning horizon, M = { m i , 7712, m^, • • • mjv}, N - total number of movies considered during the planning horizon, N = \M\, R - set of ranks of a movie, R = {0,1, • • •, z, z + l } 6 , RN - set of all possible combinations of ranks of the N movies in the set M 7 , OPD - the length of the obligation period (assumed equal for all movies), m - index number of the movie playing at a decision epoch, m G M , n - total number of weeks the movie m has played before a decision epoch, Z - vector of ranks, at a decision epoch, of the movies considered during the planning horizon 8, E[NR(s, a)] - expected net revenue to the exhibitor at a decision epoch given state s and action a, Pj(^Ki) _ probability of transition of movie j, when it is available, from the rank i\ to i2, h,i2 e 5Generally considered from the last week of May (Memorial Day in the U.S.) to the first week of September (Labor Day in the U.S.) every year in North America. 6 A n available movie can assume any rank from 1 to z, where 1 is the best rank and z, the worst. We augment these ranks by two levels, 0 and z + 1, to arrive at set R. This enables the state variable to convey information about the movies that are not available. Thus, if a movie has not been released, its rank will be set to 0, and if it has been replaced after having played for some weeks, then its rank will be set to z + 1. 7The variable denotes the cross-product of the vector R for N times with itself to generate N-tuples of rank combinations. 8We consider a movie currently playing as "available," therefore we do not need to include its rank as a separate component in the state variable. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 26 P}n(0 _ the initial probability that movie j is in rank i (at its first week release) from rank 0 (i.e., a week before release), i € {1, • • •, z}. *mi> • • • i tmN - release dates of the movies in M during the planning horizon, We propose the M D P model in terms of its five basic components (Puterman 1994): Decision Epochs, States, Actions, Rewards, and Probability Transition Function. We explain this apparently complex model afterwards by means of an example problem. Decis ion Epochs (beginning of every week) 9. States (the movie playing, its play length, and the ranks of all the movies at a decision epoch): ( S — {(m, n, Z), m E M, n € {—OPD, —OPD -f-1, • • • , 0,1, 2, • • •}, Z € RN}. Act ions (the movie which might be scheduled at a decision epoch)1 0: For s = (m, n, Z), m, if n < 0, or Zj E {0, z + 1} V j 6 M - {m}, As={ j, otherwise, V j € M , Zj £ {0, z + 1} 9 W e assume that the last decision is m a d e at decision e p o c h W a n d no decision is m a d e at W + 1. 1 0 T h e o r e t i c a l l y , it is possible for a n exhibitor to show no movies at all . However, we make a n i m p l i c i t a s s u m p t i o n here that at least one movie is always available that yields positive net revenue to the exhibitor. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 27 Rewards (expected net revenue to exhibitor in week w): rw(s,a) = E[NR(s,a)], w = 1, • • • , W - 1, s € S, a € As rw(s) = 0, s € S. Trans i t ion Probabi l i t ies : pw(s2\s1, a) = { for UjPj(Zj,2\Zjti) if (a = m 1 , n 2 = n i + 1, Zjt2 > Zjti > 0), or (a ^ m i , n 2 - - O P D , Z m i , 2 = 2 + 1, Z i f 2 > Zj-i > 0); j € M , 0 otherwise, si , 52 G 5, si = ( m i . n i , Zi), 52 = (m2,n2, Z 2 ) ; a G A s ; to = 1, • • •, W - 1, where, and for k G {1, P i ( Z A 2 = 0|Z i , i = 0) = < 1 if u> < i,- — 1, 0 otherwise (2.1) Pi,«,(^,2 = k\Zjtl = 0) = . V f (*) , w = tj-li Zj7^0Vw> tj - 1, pj(Zjji = z + l\Zj,1 = z + l) = l. (2.2) Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 28 2.2.3 A n Example P r o b l e m We first present a hypothetical scenario of movie releases, given in Figure 2.3, for a small size problem which we find useful in explaining each component of the above model. The figure shows that Movie a is released, and available in the first week. Movies c and b are available subsequently in Weeks 4 and 5 respectively, that is, ta = 1, t\, = 4, and tc = 5. We choose the following values for the parameters: W = 8, M = {a, b, c}, N = 3, R = {0,1,2,3}, OPD = 2, and P o ( * 2 | * l ) = 0.3 0.7 0 1 Pb(h\h) = Pc{h\h) 0.5 0.5 0 1 = (0.2 0.8),rt*(t) = (0.7 0.3),p*(i) = (0.8 0.2), where i, ix, i<2 € {1, 2}. In this problem, the planning horizon is eight weeks long. There are three movies, a, b, and c, to be considered during the planning horizon. Each of them has an obligation period of 2 weeks. These movies can be in the ranks 1 or 2 when they are available; 1 being the "high" rank and 2, the "low" rank. Rank 0 implies that the movie has not been released and Rank 3 implies that the movie has been replaced after having played. The transition and the initial probabilities are defined only in terms of the ranks of availability, 1 or 2. These probabilities imply that Movie a is the weakest of the three movies. This is because its initial probability of opening in Rank 1, the better rank, is Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 29 low (0.2). Moreover, even if it opens in Rank 1, its transition probability matrix suggests that it is highly likely (with a probability of 0.7) to decay to Rank 2. Movie b is stronger than a, but weaker than c in terms of the likelihood of its opening. However, it is likely to retain itself better than c, which decays.rapidly. The expected net revenue generated by a movie at a decision epoch depends on its rank and the number of weeks it has played before that epoch. The expected net revenue is equal for the three movies in different weeks of their playing as long as they have the same ranks and the number of weeks played. The time-varying expected net revenue, presented in Table 2.1, have been generated by assuming common hypothetical contract terms. According to these terms, the exhibitor will earn the same net revenue from a movie in any week as long as it remains in the higher rank. However, if the movie decays to Rank 2, then the net revenue the exhibitor earns will depend on the week of a movie's run. The exhibitor earns a higher net revenue the longer a movie plays. This specification of weekly net revenues is quite representative of the way the contract terms work in this industry. We explain these terms in a greater detail in the next chapter. A n explanation of the M D P model follows. We consider decision epochs as the begin-ning of every week of the planning horizon. This follows directly from Assumption 2 and the timing of events given in Figure 2.2. Therefore, we use weeks and decision epochs interchangeably for the remainder of this chapter. The state variable in Week w, sw, is a set of three elements. The first element, mw, denotes the movie playing at the theater in week w. We identify each movie by a unique index number. The second element, nw, denotes the play length of the movie mw. A negative value of the variable nw denotes that the movie is still in its obligation period 1 1 . The third (vector) element, Zw, denotes 1 1 A similar notation has been used in the MDP formulation of the "Blast Furnace (Maintenance) Problem" (Stengos and Thomas, 1980). In this problem, the blast furnace needs maintenance from time to time, either because it has failed during the operation, or to prevent such a failure. The only action available during this period is to continue the maintenance routine. It is clear that the state of a movie in its obligation period in the movie replacement problem is analogous to the state of the blast furnace Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 30 the vector of ranks, at decision epoch w, of all the movies considered in the planning horizon. A movie is assumed to be available from its release date until the end of the planning horizon. However, movies replaced in a week are assumed to be unavailable in subsequent weeks for screening at the same theater. The ranks 0 and z + 1 convey the information of unavailability of a movie. In our example problem, suppose that we observe the system in Week 6 and find that Movie c has been playing for two weeks after replacing Movie a in Week 4. Therefore, me = {c}, and n$ = -2 + 2 = 0, because the obligation period is two weeks. Further, suppose that the ranks of Movies c and b are 1 and 2 respectively in Week 6, which implies that Ze — (3, 2, l ) 1 2 . Thus, the state variable, SQ, is equal to {c,0,(3,2,l)}. The action variable determines the continuation or replacement of a movie playing at the theater. This decision depends on the movies' availability, and the number of weeks that the movie currently playing has run with respect to its obligation period. Thus, the only action is to continue playing the current movie if either there is no movie available for replacement, or the current movie is in its obligation period. If both of these conditions are false, then the action set consists of both continue and replace decisions. In the example problem, the only feasible action in the first three weeks is {a}, because no other movie is available. In Week 5, on the other hand, if Movie a is still playing, then the set of feasible actions is {a, b, c}, because Movie a has played longer than its obligation period and the other movies are available. However, if Movie c is playing in Week 5, then the only feasible action is {c}, because Movie c is still in its obligation period. The reward earned by the exhibitor at decision epoch w depends on both the state variable and the action taken at decision epoch w. In the proposed model, the reward is under maintenance. 1 2 The rank of Movie o is 3, because it has been replaced. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 31 the one-period expected net revenue received by the exhibitor by taking action aw in state sw. For the sake of simplicity, we do not provide the reward function in this chapter, but simply note that it depends on the contract terms, expected gross revenue, and expected concession profits. The exact expression of the reward function is similar to the one in the SilverScreener-DP model of Chapter 3 1 3 . We assume that, given the rank of a movie, the exhibitor knows its expected gross revenue deterministically. We set the reward at period W (i.e., salvage value) as zero. A n appropriate salvage value can, however, be chosen depending on a particular situation. For our example problem, suppose Movie a is playing in Week 5 (i.e., it has played for a total of four weeks). Further suppose that the ranks of Movies a, b and c are 1, 1, and 2 respectively. Thus, 5 5 = {a, 2, (1,1, 2)}. Then, if either of the Movies a or b is chosen, the one-period immediate reward to the exhibitor, given our parameter values, is $290. On the other hand, if Movie c is chosen, it yields $60 to the exhibitor. The probability transition function, pw(s2\si, a), specifies the probability that the system will be in the state s2 = (m2, " 2 , Z2) in the next decision epoch given the current state of the system is si = (mi, n i , Z\) and action a is taken. In the movie replacement problem, the "system" consists of all the movies making transitions between ranks. We assume that the transitions of different movies between various ranks are independent of each other. Thus, the joint probability of transition is simply the product of their individual probabilities. The action possibilities suggest two different conditions in which this joint probability is operational. If the action is to continue with the current movie, then the movie index (i.e., the first element of the state variable) remains the same in 1 3 To keep our analysis at a general level in this chapter, we do not include in reward some specific cost-related elements such as fixed and variable costs. In the analyses performed in this chapter, therefore, we consider the expected reward as expected net revenue, and not expected profits. We use the expected profit as the reward criterion in the next chapter. However, the form of the contract terms is the same in both the chapters. Moreover, a preliminary analysis suggested that the two cost elements do not affect the policy results. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 32 the next period. The play length of the movie increases by one period. A l l the movies that have been released either stay in the same rank or make transitions to a lower rank according to their Markov chain probabilities. In the second condition, however, the action is to replace the current movie by another available movie. The first element of the state variable in the next period becomes the movie index of the new movie. The play length is set to —OPD in the next period, that is, the movie undergoes its obligation period. The rank of the current movie mx is set to z + 1 in the next period. The rest of the available movies makes transitions according to their respective stationary Markov chains1 4. Some movies may not be available at a particular decision epoch because either they have not been released by that week or they have been replaced after having played. The transition probability expression for these two cases are given by Equations 2.1 and 2.2 respectively. Equation 2.1 implies that if a movie has not been released (i.e., its rank is equal to zero), then its rank remains zero with certainty in the next period until a period before its release. At one period before its release, the movie has an initial probability of opening in the next week in any of the ranks from 1 to z. We assume that this initial probability is known to the exhibitor based on the genre, actors, director, distributor, and critics' reviews of the movie. Equation 2.2 implies that once a movie is replaced (i.e., its rank is equal to z + 1), it is never available for screening again, that is, its rank 1 4This operationalization of the action variable is similar to a bandit model (Puterman 1994, pp. 57-62), which suggests another possibility of formulating this problem. A colorful description of a typical one-armed bandit problem is as follows. A gambler in a smoke-filled casino may either pay c units and pull the lever on a slot machine that pays one unit with probability q and zero units with probability 1 -q, or decide not to play. Unfortunately, the gambler does not know q; instead she summarizes her beliefs regarding its values through a probability density function /(g), on [0, 1]. By playing the game several times, the gambler acquires information about the distribution of q and revises her assessed probability density accordingly. The action of "pulling the lever" in this problem is similar to the replace action in the movie replacement problem, and the decision of not to play is similar to continuing with the current movie in the movie replacement problem. The bandit problem is usually modeled in a Bayesian framework. In the last section of this chapter, we discuss some other formulations that also adopt the Bayesian techniques. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 33 remains equal to z + 1 with certainty. Notice that this is consistent with Assumption 5. We now demonstrate for our example problem the calculation of the transition prob-ability in two different cases. As a first case, suppose Movie a is playing in Week 6, that is, all the three movies are available. Further, suppose that the ranks of Movies a, b, and c are 1, 1, and 1 respectively. Therefore, Si = {a, 3, (1,1,1)}. If we choose action {b}, that is, replace Movie a by Movie b, then the probability of system making transition to s 2 = {&> — L (3,1, 2)} (i.e., Movie a is not available for replacement, Movie b remains in the same rank and Movie c decays to Rank 2) is 1 x 0.5 x 0.9 = 0.45. In the second case, suppose Movie a is playing in Week 3 and its rank equals 1. Therefore, = {a, 0, (1, 0, 0)}. In the next week, however, Movie c is released and becomes avail-able. The probability of transition from sx to s2 = {a, 1, (1, 0,1)}, therefore, involves the initial probability of Movie c opening in Rank 1, and equals 0.3 x 1 x 0.8 = 0.24. 2.3 O p t i m a l i t y C r i t e r i o n We use the expected total reward as the criterion for comparing various policies to derive an optimal policy. We will use this criterion in the scenario analyses to be presented in the next section. Let v^(s) represent the expected total reward over the decision making horizon (of length W) if policy n is used and the system is in state s at the first decision epoch. For the most general class of M D P problems, we define expected total reward by w v^(s) = E*3{Y,rw{XW) Yw) + rw+1(Xw+1)} (2.3) where r represents rewards, and Xw and Yw are random variables which take values in the set of states, S, and, the actions set, A, respectively. In the following sections, we restrict n to the class of policies that are Markovian Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 34 deterministic, that is, 7r G J[MDl5. The decision rules in such policies choose actions de-terministically (i.e., It = ^t(st)-)> a n d the immediate rewards and transition probabilities depend on the past only through the current state of the system (i.e., the Markovian property). Puterman (1994, pp. 89-90) states that by restricting attention to Markovian deterministic policies, which are simple to implement and evaluate, we may achieve as large an expected total reward as we might using some other more complicated policies (e.g., randomized history-dependent policies).1 6 M D P theory and algorithms for finite-horizon problems using the above criterion primarily concern determining a policy 7T* G UMD with the largest expected total reward. That is, we seek a policy 7r* for which v^(s)>v^(s), seS, neUMD (2.4) Such a policy is referred to as optimal policy 17. In other words, we seek to characterize the value of the M D P in question, uj^, defined by vw( s) =  vw( s) = max Vyy(s), s G S (2.5) TT€IIMD A n intuitive interpretation of the value of an M D P is in order here. We earlier mentioned that an attractive feature of the M D P model is that its solution provides the decision maker with an optimal policy, or strategy, that is, the prescription for choosing the optimal action in any possible future state. The corresponding value of the M D P , 1 5 J J M D denotes the set of all policies of class Markovian deterministic (MD), 1 6For a complete theoretical treatment of this topic, see Puterman (1994). 1 7We assume that an optimal policy for the problems considered is attainable, and therefore we do not address issues regarding e-optimality. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 35 therefore, provides a quantifiable summary measure of such a prescription in any state at the first decision epoch 1 8. In the scenario analyses that follow, we will use this notion of value of an M D P to compare and evaluate various alternative scenarios. Recall from the overview section that we define a smart manager as the one who optimally solves his/her movie replacement problem by taking demand uncertainty into account and adopting a dynamic and long-term approach in decision making. A l l the requirements for this definition are met by the value criterion. Thus, the value criterion provides us a way of specifying under which scenario an optimally behaving manager is better off. 2.3.1 Finite-Horizon Policy Evaluation M D P theory and computation are based on using backward induction to evaluate ex-pected rewards recursively. In this section, we present the fundamental method to eval-uate the expected total reward of a Markovian deterministic policy, n € HMD. Let denote the expected total reward obtained by using policy 7r at decision epochs w, w - f-1, . . . , W . Then u7^ is defined by w n=w where r, X, and Y are defined in the same fashion as that for Equation 2.3 above. Puterman (1994, pp. 80-82) shows how to compute the value of a policy, v\\-, by inductively evaluating The difference between v\\ and tt£, is that Vyy includes reward over the entire future, while v?w only incorporates rewards from decision epoch w onward. Thus, the value of a policy f^(s) = ux (s) for all s € S. In the scenario analysis section that follows, we use the backward induction and invoke the above relationship between 1 8 We consider problems in which the decision maker wishes to choose an optimal policy for all possible initial-system states. In practice, all that might be needed is an optimal policy for a specified initial state. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 36 v^r(s) and « i ( s ) to calculate the value of a policy. To calculate u£, recursively, we use backward induction algorithm for solving finite-horizon discrete-time MDP's . The algorithm has as its theoretical foundation "The Principle of Optimality," aptly verbalised A n o p t i m a l pol icy has the property that whatever the i n i t i a l state and i n i t i a l de-cisions are, the remaining decisions must constitute an op t ima l pol icy w i t h regard to the state result ing f rom the first decision. Based on the M D P model presented earlier, the backward induction algorithm operates as follows. by Bellman (1957, p. 83). The Backward Induct ion A l g o r i t h m 1. Set w = W + 1 and «{V+i( s w+i) =  rw+i(s\v+i), sw+i £ S (2.6) 2. Substitute w — 1 for w and compute u^s^j) for each sw € S by (2.7) Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 37 Set 1 9 A*sw,w = axZ m a x a e A „ i rw( sw,a)+ Y2pw(j\sw,a)u*w+1(j)}. (2.8) jes 3. If w=l , stop. Otherwise, return to step 2. The above backward induction algorithm can be used to determine the policy 7r*, as discussed earlier, which yields the largest expected total reward. 2.4 Scenario Analys i s We now present a scenario analysis on some key parameters of the problem introduced in the earlier sections. At a broad level, the analysis is aimed at accomplishing two different but related objectives. First, it shows that, faced with different decision situations that emerge depending on various values the parameters assume, how a smart manager should behave. This will provide normative guidelines to the managers in the exhibition industry. Second, which is related to the first objective, the scenario analysis shows that how a smart manager would behave under different decision situations. This will offer important learning about the movie replacement phenomenon and implications for the other players in the industry who deal with such a manager. The characterization of various decision situations helps us examine some important issues related with the movie replacement problem. First, we investigate the impact on the exhibitor's decision making and overall value if the supply conditions change in the market. We consider the changes in supply conditions brought by varying the quality, 1 9 We define arg max in the following way. Let X be an arbitrary set, and g(X) is a real-valued function on X. Then arg msxx€Xg(x) = {x € X : g(x ) > g(x) V x e X} In the current context, therefore, arg max is the set of (optimal) actions that yield the maximum in Equation 2.7. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 38 quantity, and release sequence of movies released. Second, we choose one of the contract parameters, obligation period, and examine its impact on an optimally acting manager's movie replacement policy. Next, we estimate the value of information in making smart replacement decisions. We vary the quality of information about weekly decay of box-office revenue from a perfect information case to gradually deteriorating information conditions. These problems, their objectives, main result, and managerial implications are summarized in Table 2.2. In order to construct test problems representing these situations, we need to make some mild assumptions about the general parameter setup of the problem. 2.4.1 Parameter Setup We refer to Jedidi, Krider and Weinberg's (1998) clustering study for the four types of classes in which most mainstream movies generally tend to fall. They fit a two-parameter exponential demand model of the following form for weekly market share data for 102 major motion pictures released between December 1990 and Apri l 1992. E[Revw] = ea+/hu (2.9) where E [Revw] denotes expected box-office gross revenue in Week w of the movie's run (w = 0,1,2, . . . ) . Their results show that the a and /3's for the four clusters are as follows: Cluster 1 2 3 4 a -1.484 -1.614 -2.52 -2.255 -0.224 -0.11 -0.439 -0.258 Jedidi, Krider and Weinberg (1998) label these clusters as follows: Cluster 1: "Holly-wood Heroes" (movies that open strongly but decay sharply), Cluster 2: "Mega Movies" . Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 39 (movies that open reasonably strongly and retain themselves well, that is, decay slowly), Cluster 3: "Fast Fades" (movies that open weakly and decay very sharply), and Cluster 4: "Fair Flicks" (movies that open weakly and decay sharply). The decay patterns of the four movie types are shown in Figure 2.4. R e v e n u e D a t a We assume that once we know the rank of a movie for a week, we know its correspond-ing expected box-office gross revenue. The different problems that were investigated in this chapter involved 2, 3, 4, or 5 ranks. For various problems investigated, the expected box-office gross revenue was varied from 100 to 500 by dividing this range in approx-imately equal intervals corresponding to the expected revenue of each successive rank. For example, in a three-rank case, the expected box-office gross revenue was assumed 100, 300, and 500 in ranks 1,2, and 3 respectively. In order to calculate the exhibitor's share of the box-office revenue generated by any movie in a given rank and week, we contacted a major theater chain for a sample of their revenue sharing terms. We chose the modal contract term 2 0 out of the different types of terms used by that theater as the representative contract term for our calculation purposes. We generated the exhibitor's share of the box-office gross revenue using the above scheme (refer Table 2.3). These data were generated for the four different cases of ranks, where ER(x) represents ex-pected box-office gross revenue in Rank x. I n i t i a l a n d T r a n s i t i o n P r o b a b i l i t i e s We assume that a movie's type (i.e., its cluster type) conveys the information about its initial and transition probabilities. We assign initial and transition probability values 2 0This contract term format was used by that theater chain for more than 60% of its movies. We do not elaborate on the specifics of the contract terms in this chapter. However, we use this contract term again in Chapter 3. Interested readers are referred to Type 1 contract terms in Table 3.20 of Chapter 3. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 40 to different movie types in such a way that they mimic the general exponential decay pattern of the four movie types of Jedidi, Krider and Weinberg's (1998) study. Thus, these matrices emerge as a general form of the exponential decay curve. Consistent with an exponential decay pattern, these matrices do not allow transitions from a lower rank to higher. However, they are more general in that they also permit the movies to stay in the same rank or even make transitions to more than one lower rank. The initial and transition probability matrices corresponding to the four cases of ranks are provided in Tables 2.4 and 2.5 respectively. We choose values for the probability matrices to keep them as close to the exponential curves of the four clusters found by Jedidi, Krider and Weinberg (1998). Let us illustrate this for a two-rank case. For example, Type 1 movies open strongest among the movies of all four clusters. Therefore, their initial probability of opening in Rank 1 (i.e., the high rank) is set to 0.8 (refer the first column values of Table 2.4). Accordingly, the probability of opening in the low rank is 0.2. The corresponding values for transition probabilities of Type 1 movies, because of their faster decay rate, are 0.4 and 0.6 respectively. The probability values for the other movie types are set in a similar fashion. The probability values for the greater number of ranks (3, 4, or 5) are set analogously by redistributing the original values in a two-rank case21. The specific probability values for the different movie types was chosen to reflect the nature of their curves from Equation 2.9, and their respective a and /?'s. This specification provides us a meaningful way to operationalize the availability of various movies types in the market. Moreover, we find in some test cases that minor variations in these figures only alter the (numerical) value, but do not affect the policy results significantly for the various problems investigated. 2 1 The rationale behind the redistribution is that as the number of ranks increase, the probability assignments to the new ranks should come at the cost of the probability assignments to respective ranks in the previous case. Tables 2.4 and 2.5 show that this redistribution preserves the original relative positions of the different movie types in terms of their opening strengths and decay rates. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 41 2.4.2 Problems Motivation, Design and Analysis Results We conducted scenario analyses on a variety of parameters. In this section, we explain the objectives of each of the analyses, design of the various problems constructed to achieve those objectives, and the results of the ensuing investigations. We focus on single screen settings in investigating these problems. Unless specified otherwise, we consider an eight-week planning horizon. We choose this horizon mainly due to the complexity associated with the problem of a larger size. However, an eight-week length reasonably represents two months of Summer season, or an entire Christmas season. Finally, noting that Types 3 and 4 are much weaker than Types 1 and 2 in terms of both opening strength and retention rates, a pretest was conducted which showed that there was not much difference between Types 3 and 4 as far as policy implications are concerned22. Therefore, we only used Type 3 for further analysis. We use a backward induction algorithm to calculate the values of various policies in different problems. The algorithm was implemented by writing a programming code in C language on H P U N I X SunOS 4.1.3 system. The code involves a data structure that was designed to facilitate the tabular calculations typically used in M D P calculations. The inputs to the code are net revenue, initial and transition probabilities, release dates and obligation periods of various movies. Two subroutines of the code check state vari-ables' and actions' feasibility, respectively. Various combinations of the ranks of different movies were generated by a matrix generator, which is essentially a series of nested loop statements. The program is initialized by assigning zero salvage values at the end of the horizon. The algorithm then proceeds using backward induction algorithm. The inter-ested readers may refer to the copy of the code provided as an appendix to this thesis for additional details. The solution obtained from the algorithm was verified by comparing 2 2Both of the Types 3 and 4 get almost similarly dominated by Types 1 and 2 in a typical test scenario. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 42 it against some example problems solved manually. For the size of problems considered in the current analysis, the algorithm took only a few seconds to compute the solution. However, as the number of movies and ranks increases, the size of the matrix for the ranks of different movies expanded considerably and it was not feasible for the computer to solve such problems2 3. Supply Conditions Analysis Motivation In this analysis, our objective is to investigate the effect on exhibitor's value under different scenarios of movies' supply, in terms of quality, quantity, and release sequence variables. Since movie types 1 and 2 are better than Type 3 both in terms of opening strength and decay rates, a scenario in which we observe a greater proportion of Type 1 and 2 movies than Type 3 movies is characterized as a high quality scenario. A scenario in which we observe a greater total number of movies is characterized as a high quantity scenario. The other variable investigated is the release sequence of the movies within a season. That is, to what extent and under what conditions do exhibitors prefer the earlier release of high quality movies? These variables also have implications for other related phenomena present in the market, such as, availability, variety, substitutability, season effects, and relative bargaining powers of the upstream and downstream channel members. We examine the extent to which the exhibitor is better in one scenario versus another by comparing value of the optimal policies in the respective cases. Problem Design The analysis was conducted under two different problem settings, which we label as 2 3 The sizes (in terms of its parameters) of some representative larger problems we could solve using this algorithm are: 8-week horizon, (6 movies, 4 ranks); (7 movies, 3 ranks); (5 movies, 5 ranks); and similar other combinations. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 43 Problem A and Problem B respectively. The only difference between the two problem types lies in manipulation of quality; Problem A involves movie types 1 and 3, while Problem B involves movie types 2 and 3. Though both Types 1 and 2 represent high quality movies, Type 1 characterizes movies that open strongly but decay rapidly, while Type 2 movies open moderately but decay more slowly. Thus, the two problem settings let us examine the quality effects of both Types 1 and 2 movies. Type 3 represents a low quality movie in the two problems. In both of the problems, the following general parameters were chosen: single screen, planning horizon = 9 weeks, number of ranks = 3, obligation period = 2 weeks for all movies. A 2 X 4 matrix design is used to examine the issues in these problems (refer Table 2.6). As shown in the table, we manipulate both quantity and quality of movies, along with their release patterns. In Problem A , for example, there are two movies in the first italicized row as compared to four in the second row. Within a row, the quality and release pattern manipulations are done. For example, in the first row (i.e., the low quantity case), the case 33 denotes that two Type 3 movies are released during the planning horizon, representing a low quality scenario. Similarly, though both of the cases 31 and 13 represent medium quality scenarios, they represent different release sequence24. R e s u l t s Table 2.6 and Figure 2.5 show the results for the various analyses of Problems A and B. These results have been obtained using the optimal policy as described in the previous sections. The results suggest the following2 5. •^The movies in all the cases are released at equal intervals during the horizon. For example, in case 31, Type 3 movie is released at Week 1 and Type 1 at Week 5. 2 5 A brief explanation on how to interpret these and subsequent results is in order here. The values in Table 2.6 and y-axis of Figure 2.5 denote the value the decision maker expects to achieve at the beginning of the planning horizon considering all future possibilities, which is our comparison criterion for various cases. This value is shown for three cases of starting ranks for the movie (Type 3) playing at start of the horizon. For example, in the case 33, the exhibitor can expect to achieve $660 over a period of 8 Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 44 • The effect of increased quality on exhibitor's value is generally much greater than the increase of quantity. Also, higher opening rank (another indicator of quality) can substantially improve exhibitor's value. For example, compare the increase in values from case 33 to 11 (51%, 38% and 33% increases in value in the Ranks 1, 2, and 3 respectively) with that from 33 to 3333 (3%, 3%, and 4% increases). • The increased quantity of low quality movies has little effect on value. For example, compare the increase in values from case 33 to 3333 (3%, 3%, and 4% increase in value over the Ranks 1, 2, and 3 respectively) with that from other cases such as 11 to 1111 (20%, 31%, and 39% increases). • The results show that the exhibitor is substantially better-off in Problem B cases (involving Type 2 movies) as compared to the corresponding cases in Problem A (Type 1). Although Type 1 movies are slightly stronger than Type 2 movies in terms of their opening strength, they are much weaker than Type 2 movies in retention strength. • The policy results from the two problems (considering different cases such as 31, 13, 32, 23, and so forth) suggest that if a movie retains itself long enough in the higher ranks, then it is optimal to continue with it even if there is a stronger movie available later on in the horizon. This point is also related to the way contract terms operate in this industry. The policy results also suggest that if a Type 2 movie is available early on in the horizon, then the exhibitor does not consider the movies released later (irrespective of their type) for replacement unless the first movie has decayed to the lower ranks. • When quantity is kept constant, exhibitors generally prefer that the season open weeks, if the first Type 3 movie opens in Rank 1. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 45 with a high quality movie (Type 1 or 2) in the highest rank than with the low quality movie (Type 3). For example, in Problem A , the values in cases 13 and 1313, Rank 1 (842, 1012) are higher than those in cases 31 and 3131, Rank 1 (829, 997) respectively. Interestingly, however, case 31 is preferred to 13 if Type 1 movie does not open strongly (i.e., opens in Rank 2 or 3) in the latter case. .We explain this inversion in two steps. If movie 1 in case 13 opens in Rank 3, then case 13 reduces to case 33 (because of the way transition probabilities are designed), which is worse than case 31. If movie 1 in case 13 opens in Rank 2, then it is interesting to note that case 31 is still preferred to case 13. This probably happens because of the rapid decay property of the movie type 1. Case 13 becomes less attractive later on in the horizon because of this rapid decay of movie 1, while case 31 still has a new movie type 1 to be released later on. In contrast, these inversions do not take place in Rank 2 for the corresponding cases of Problem B because of the better retention strength of Type 2 movies. Thus, it is better from the exhibitor's perspective to have Type 2 movies released early on in the horizon. Manager i a l Implicat ions At a first glance, it appears that the exhibitor should be better-off in a high quantity than a low quantity scenario. The reason is that a high quantity scenario offers the ex-hibitor increased flexibility in scheduling movies. However, this only helps if the increase is accompanied by a corresponding increase in quality. On the other hand, movies that sustain themselves well (Type 2 vs. Type 1) lessen the need for a quantity of movies. In general, therefore, the exhibitor should be more vigilant of the availability of the movies with strong retention strength than other movies, say Type 1 (which have a strong open-ing but low retention). We find this result because the decision maker in our model adopts a long-term view and accounts for future uncertainties. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 46 It is also advisable that the exhibitor consider the high quality movies with either strong opening or retention strength (preferably both) that are released early in the horizon so that they could schedule them more profitably during the planning horizon. In general, the policy results suggest that if an existing movie has run long enough to reap benefits of more favorable contract terms, then it is advisable to continue playing it as long as it maintains itself in at least a moderate rank (say, Rank 2). This result concurs with and is a precursor to our broader policy result in Chapter 3. It also suggests that the distributors of the strong movies should aim at releasing their movies earlier in the season so that the exhibitor has an incentive to play them longer. This is consistent with Krider and Weinberg's (1998) study on release timings. Contractual Obligation Period Motivation As discussed earlier, movie contracts specify that once the exhibitor decides to play a movie, he/she is obligated to run it for a minimum number of weeks known as obligation period. The contract also specifies a sliding percentage scale for sharing the box-office gross revenues between the distributor and exhibitor. The scale favors the distributor in the first few weeks of the movie's run, and the exhibitor later on. However, the appeal of most movies is highest in its first few weeks. Moreover, distributors invest a lot in advertising, especially at the national level, before a movie is released to build a high early demand. In order to "squeeze the juice" from this early demand and to cover their already incurred costs, the distributors specify an obligation period so that their movie is not replaced for a minimum period of time. The above discussion about the nature of the contract implies that probably the ex-hibitor would prefer to run a movie longer to take advantage of the increased sharing percentage in his/her favor in the later part of the run of the movie. However, two factors Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 47 inhibit the exhibitor's choice to run a movie longer. First, as previously mentioned, most movies (with the exception of Type 2 movies, whose proportion is very low as compared to the other movies) do not retain themselves well at the box-office in the later part of their run. Second, there are many movies released during a peak season such as Summer or Christmas. Ideally, the exhibitor would like to play only those movies longer whose retention power is high and he/she is quite sure about it. A n optimally acting exhibitor would make such replacements as soon as it makes "economic sense" to do so 2 6. The obligation period limits the exhibitor's ability to schedule other, possibly high quality, movies released during the same period. In certain sense, the exhibitor faces an oppor-tunity cost (of not being able to show other movies) associated with obligation period, which is revealed by the decrease in his/her value. In order to investigate various effects of the obligation period parameter on the exhibitor's decision making, we analyze differ-ent problem situations as described below. P r o b l e m Design A relatively straightforward question of interest is the effect an increase in obligation period of a particular movie will have on the value of an optimally acting exhibitor. In Problem C, we present a scenario in which a weak movie (Type 3) is currently playing at the theater and a strong movie (Type 2) is to be released two weeks later (refer Figure 2.6, Problem C). We vary the obligation period of Type 3 movie from 1 to 4 weeks. In general, we would expect the exhibitor to be worse-off with an increase in obligation period of a particular movie. The major variable of interest in this investigation is the extent of such a decrease in the exhibitor's value. The second major question of interest is the effect of increase in obligation period on 2 6This economic sense depends on the interplay of several factors such as revenue figures and sharing percentages (depending on the week of movies' run) of many movies. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 48 the exhibitor's replacement strategy. In this case, we consider variation in the obligation period of a forthcoming movie, and not of a currently playing movie. We choose three different situations that let us investigate various issues. In Problem D, we examine a situation in which the obligation period of a forthcoming weak movie (Type 3) is var-ied, and a strong movie (Type 2) is available to be released behind it (refer Figure 2.6, Problem D). W i l l an optimally acting exhibitor ever schedule the weaker movie, and if so, under which values of its obligation period? In Problem E, we examine a situation in which both of the forthcoming movies are strong, but differ in terms of their quality parameters. The movie whose obligation period is varied is Type 1 and the other strong movie is Type 2, which is released later (refer Figure 2.6, Problem E). In Problem F, we examine the opposite situation of Problem E, that is, the movie which is released earlier, and whose obligation period is varied, is Type 2, and the second movie is Type 1 (refer Figure 2.6, Problem F) . The following general problem parameters were chosen: single screen, planning horizon = 8 weeks, number of ranks = 3. We explain the results obtained from the analysis of these problems below. Results The results of this analysis are shown in Table 2.7 and Figure 2.7, which suggest the following. • In all cases, as expected, the exhibitor's value decreases with an increase in obliga-tion period. The figures in parentheses of Table 2.7 show that the biggest drop (if it occurs) in the value occurs for the increase in the smaller values of the obligation period. This suggests that as the obligation period increases, the cost associated with lost opportunity becomes less significant because the alternative movie also loses its appeal, or it ages. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 49 • The analysis of Problem D shows that the value decreases with an increase in obligation period from one week to two weeks. This is because if the exhibitor schedules the second Type 3 movie in Week 3, then he/she is obligated for at least two weeks, and misses the opportunity of adopting a much stronger movie (Type 2) in Week 4. The opportunity cost of being obligated with a weak movie becomes so high that the policy never recommends to replace the first Type 3 movie by the second one at Week 3. This is in anticipation of a strong movie (i.e., Type 2) to • be released later. At Week 4 or later, it is optimal to replace by the second Type 3 movie only if Type 2 movie opens rather poorly. These results hold, and the value does not change, when obligation period of the second Type 3 movie is increased beyond two weeks. • The analyses of Problems E and F show that, as better quality movies become available (Types 1 and 2), the impact of obligation period on exhibitor's value decreases. As the values in Table 2.7 show, the exhibitor is better off than the corresponding cases of the Problems C and D, because he/she has two better movies as replacements. This corroborates the quality related results discussed in the earlier section. In Problem E, unlike Problem D, the exhibitor does not always wait for Type 2 movie to be released, when the obligation period of Type 1 movie is increased from 1 to 2 weeks. This is because a better movie (Type 1) is available in this problem than the one (Type 3) in Problem D. The replacement depends on the relative attractiveness of the two movies in terms of ranks. The results of Problem F corroborate the earlier results that the exhibitor does not put too much weight on Type 1 movie's (to be released later) opening strength when Type 2 movie with strong retention strength is available early. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 50 Manager i a l Implicat ions In general, an increase in the obligation period tends to decrease the value that an exhibitor achieves in a given scenario. We recommend that the exhibitor consider this decrease as an opportunity cost of not being able to play other movies. However, the extent of such decrease is quite situation specific and disappears in some cases. The above test problems quite effectively demonstrate such differences. Under some conditions, the exhibitor finds it optimal to forego the opportunity of showing a new movie in anticipation of a stronger movie to be released later on in the horizon. This suggests that a smart exhibitor should rather wait for a stronger movie to be released later than "lock up" his/her display space in a binding commitment with a low quality movie. A strong movie, on the other hand, might be worth playing despite the obligation period attached to it. These results have important implications for the release strategies of the distributors of weaker movies. The significant drop in the exhibitor's value for an increase in obligation period of the weak movies suggests that distributors of weak movies should take care in deciding on the obligation period of their movies. The decision would primarily depend on the "spacing" available from the stronger movies' releases. The distributors of a weak movie should either keep enough space between the release of their movie, or be willing to decrease the obligation period for his/her movie. These results also concur with Krider and Weinberg's (1998) study on release timings of movies. The methodology adopted in analyzing the effect of obligation period can also be extended to other contract parameters. For example, the effects of different sliding per-centage scales, that divide the box-office revenues between the distributor and exhibitor, can also be analyzed under the current framework. Swami, Lee, and Weinberg (1998) analyze related issues in their study. The exhibitor should quantify the effects of various parameters as a cost in conducting such analyses. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 51 Value of Information Motivation In this analysis, we examine the effect of variation in the exhibitor's estimates of a movie's decay pattern on his/her replacement decision rules. The policy the exhibitor recommends for a particular scenario could be sub-optimal if the estimates about the movies' behavior are incorrect. In the context of our model, the question becomes whether or not the exhibitor precisely knows the structure of the Markov chain of a movie's decay pattern. If the quality of information about a movie's behavior deteriorates, and consequently the exhibitor's estimates are imprecise, then that "error" might be reflected in his/her decision rules. For example, if the exhibitor expects a forthcoming movie to decay rapidly (i.e., the exhibitor attaches high probabilities of decay from the higher to lower ranks), he/she may recommend not adopting that movie or may suggest its quick replacement if it is adopted. However, if the exhibitor's estimates were wrong, and the movie turned out to "actually" decay slowly, then his/her recommended policy might be sub-optimal. What is the value (or the loss of it) associated with the quality of information a decision maker has? How much deterioration in information quality alters the exhibitor's optimal policies? We now discuss a framework to simulate the aforementioned errors. We begin with a simple case in which the exhibitor knows a movie's decay pattern correctly. In this perfect information world, for example, the exhibitor may know that if the movie is in Rank 1 in any given week, it stays in that rank with certainty in the next week; if it is in Rank 2, it stays in Rank 2, and so on. When the quality of information deteriorates, the exhibitor's confidence in his/her transition probability estimates also decreases. For example, he/she may only be, say, 80% confident that the movie will stay in a given rank (i.e., he/she expects a 20% chance of it decaying to one of the lower Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 52 ranks.) We would expect this change in exhibitor's confidence in the transition probabil-ity estimates to produce changes in the replacement strategy. Another issue of interest is the nature of such changes. That is, would the exhibitor become more conservative after his/her confidence decreases in the estimates due to poor information, and as a result continue playing the existing movies? Or, given the exhibitor's reduced confidence about whether a movie will stay in a higher rank, a "grass-is-greener-on-the-other-side" effect will operate on the decisions, and the exhibitor will replace movies sooner? P r o b l e m Design The design of the problems in this analysis is different from the rest of the analyses. We consider a simple 2-movie, 8-week, 3-rank scenario, as shown in Figure 2.8. That is, our regular Type 3 movie (with obligation period of two weeks) is playing at the beginning of the horizon. However, though the initial probability vector of the second movie, say Type X , is assumed the same as Type l's, we design three different decay patterns for this movie as follows. Vc{h\h) = ( 1 0 0 ^ 0 1 0 0 0 1 P m ( * 2 | * i ) = / 0.8 0.1 0.1 ^ 0 0.8 0.2 0 0 1 PU2(*2|*l) = 0.5 0.25 0.25 0 0.5 0.5 0 0 1 Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 53 The first transition matrix represents a case in which the exhibitor is certain (hence, the subscript C) about how the movie is going to "decay" from one rank, say ii, to another, say i2. In the second C£lS6j £tS the information quality deteriorates; the exhibitor's confidence in his/her decay estimates decreases. For example, now he/she is only 80% certain that if the movie opens in Rank 1, it will stay in that rank. We use the subscript U l to denote that this is the first case involving errors in estimates, or the incorporation of uncertainty. As shown, these errors increases even further in the third case, U2. We propose the following scheme to compare the cases C, U l , and U2. First, we get policy results for the three cases using their respective transition probability matrices for Type X . Though the policies can be compared on the basis of their recommendations, their value are not comparable because they use different transition probability data. Therefore, we need to construct specific instances of realization of state variable, and compare the value based on one set of transition probability data. We assume that the "actual world" behaves more in accordance with Case C than with the other cases, that is, once Type X movie becomes available in a particular rank, it remains in that rank for the rest of the horizon. The optimal policies for the three cases, C, U l , and U2, would tell us their recommended decisions at each decision epoch for the specific instance of state variable. Then, for the uncertainty cases, U l and U2, we will infer the value achieved at each decision epoch by their policies calculated according to Case C's data. We create two instances of the realization of the state variable, presented as Problems G and H . We label the first (Type 3) and second (Type X) movies released in the horizon as a and b respectively. In Problem G, Movie a, as long as it is available to be played, remains in Rank 1. When Movie b becomes available, in Week 3, it is also available in Rank 1. In Problem H , Movie a does not retain itself in Rank 1 after Week 2 and decays to Rank 3. Movie b opens in Rank 2 and remains in that rank throughout the planning horizon. The state variables corresponding to the eight weeks of the planning horizon Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 54 are shown in the second columns of Table 2.8. In Problem G, the state variable at Week 1 is {a,-2,(l,0)}, denoting that Movie a is playing, has just entered its obligation period (indicated by -2), the rank of Movie a is 1, and that of Movie 6 is 0 (because it is yet to be released). In Week 4, there are two possibilities, corresponding to whether Movie a is still playing or was replaced by Movie b in Week 3. In the first case, the state variable is simply {a,l ,(l , l)} . In the second case, the state variable is {b,-l,(4,l)}, indicating that Movie b is playing, it is in its obligation period, Movie a has been replaced2 7 and Movie b is available in Rank 1. Results The optimal decision rules for Problems G and H for the perfect information case, C, and the two cases of imperfect information, U l and U2, are shown in Table 2.8. The corresponding values at each decision epoch are provided in the parentheses28. Since the results from the two problems are quite similar, we will focus on Problem G in the following explanation. • The optimal decision rules of the perfect information case, C, show that it is optimal to replace Movie a by Movie b at Week 3, that is, when Movie b is released. It is optimal to schedule Movie b for both cases of state variable in Week 4. At Week 5, however, it is interesting to note that it is optimal to continue playing a, if it has not been replaced so far. In other words, if the decision maker has made a mistake at some point in the horizon, the policy recommends, that point onwards, what the optimal decision is at a particular decision epoch taking future possibilities into account. 2 7 Reca l l that this is a three-rank case and we have augmented the rank set by 0 and 4, where 4 indicates that the movie has been replaced. 2 8 A l l the values in the parentheses are calculated using Case C's data. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 55 • The cost of delay in playing a movie is reflected by comparing the associated values in the parentheses. For example, in Case C, if Movie a is still playing at Week 4, the optimal action b gets the exhibitor a value of 850, which is less than 890, the value he/she would have received had Movie a been replaced by b in Week 3. • In the first imperfect information condition U l , we find a difference from Case C's recommendations at Week 4 as the policy based on Case U l ' s data suggests continuing with a. This results in a slight drop in value from 850 in Case C to 841 in Case U l . As the quality of information deteriorates further in Case U2, the exhibitor's confidence in Movie &'s retaining itself in higher ranks decreases. Thus, the decisions rules for U2 suggest not to replace by Movie b at all. It appears that since the exhibitor does not have much confidence in Movie b and Movie a retains itself in high ranks, this makes Movie a more attractive in the initial weeks of Movie &'s release. After that, as explained above, it is too late in the horizon to replace by b and consequently, the exhibitor finds it optimal to continue with the already playing movie. The values in the parentheses reflect the loss associated with such mistakes in decision making as compared to the certainty case. Manager i a l Implicat ions Our results show that the exhibitor can make wrong replacement decisions in situa-tions characterized by the poor quality of available information. Though the magnitude of the effects of such errors in our small-size test problems is moderate, in practice these resulting losses could be substantial. We also found that there are some costs associated with the opportunity foregone to show a movie after its release. Using the value terms, we were able to quantify such "losses." Thus, having the right information is crucial in making the optimal replacement decisions on an on-going basis. We note that the exhibitors tend to continue with a currently running movies if Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 56 the information is imperfect because their confidence about the future performance of a forthcoming movie is reduced. In other words, they become overly conservative and impose some form of penalty on the attractiveness of the forthcoming movie. Therefore, we recommend that the distributors in this industry provide additional information to the exhibitors to reduce the uncertainty about their movies' performance. Another interesting result suggests that by using recommended policy, it is possible to find optimal decisions for the remainder of the horizon even after making a mistake earlier on in the horizon. A n analogy of this situation can be drawn with the notion of off-equilibrium path in game theory. By making a wrong choice earlier in the horizon, the decision maker is off the equilibrium. However, he/she can still do best for the sub-game that he/she faces after the mistake. This further recommends that the exhibitors adopt a long-term thinking in decision making, and use the models, such as the one in this and the following chapters, on a rolling basis. That is, even if they make a mistake at a decision epoch, they detect that mistake, discover the additional information, and take optimal actions in the future. In conclusion of scenario analysis, we investigated some key issues of the movie re-placement problem. The results from these analyses have some important managerial implications that are summarized at the end of each analysis. A summary of the re-sults obtained from these analyses is also presented in Table 2.2. In the next section, we explore the possibility of characterizing a rank-based optimal policy for the movie replacement problem. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 57 2.5 Rank-based O p t i m a l Policies 2.5.1 M o t i v a t i o n In this section, we address the decision-support aspects of our modeling framework. One of the principal uses of M D P methods is to establish the existence of optimal policies with a special structure. Such structured policies are useful because of their appeal to decision makers: their ease in implementation and commercialization, and their enabling efficient computation. Examples of policies with a simple structure include (s, 5) policies in inventory control models2 9, and control limit (or critical number) policies in queuing control or equipment replacement models. Under some conditions on the parameters (such as costs) of the model (for example, see Puterman 1994, p. 108), a control limit policy is composed of decision rules of the form:-where ax and a 2 are distinct actions, and S* is a control limit. That is, such a policy divides the state-space of the problem under consideration by a critical limit in such a way that it is optimal to take one action below and another action above that limit. If one establishes that such policies are optimal for a problem, the task of finding an optimal policy reduces to that of determining S*. Since the movie replacement problem is similar in structure to the machine deterioration problem, we expect the optimal structured policy for the movie replacement problem, if one exists, to be of a form similar to that for the machine deterioration problem. For example, the simplest form of such a policy could be: In a given week, continue a currently playing movie if either it is in its obligation 2 9 A n example of such a rule is: Order sufficient stock to raise the inventory to S units whenever the inventory level at the beginning of a period is less than s units. When the inventory level at the beginning of a period is s units or greater, do not place an order. o i , s < S* (2.10) a 2, s > S* Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 58 period or its rank is below30 a critical number, otherwise replace it by a new movie. A pictorial representation of such a policy, assuming a five-rank case and that the existing movie has already played beyond its obligation period, is as shown in Table 2.9. As shown in the table, this simple policy recommends (after the existing movie's obligation period is over) that continue playing the existing movie if it sustains itself in top two ranks, otherwise replace it. In this section, we examine whether the optimal policies for the movie replacement problem have a special structure that can be established in terms of difference in ranks of the movie playing and those available. 2.5.2 P r o b l e m Design We designed three problems towards this end using the scenario analysis data. To keep the analysis simple, we consider the case of two movies3 1; the first one (a) is playing at the beginning of the horizon, and the other one (b) becomes available at Week 3 (see Figure 2.9). In Problem I, Movie a is Type 1 and Movie b is Type 2. The objective of this problem is to investigate the optimal replacement policy when the first movie has a strong opening, but decays rapidly, and another movie with strong retention becomes available later. The objective is opposite in Problem J, in which Movie a is Type 2 and Movie b is Type 1. In Problem K , Movie a is Type 1 and Movie b is also Type 1. Since both existing and replacement movies are similar types, this case is a step closer to a typical machine replacement scenario. The following parameters were chosen in each sub-problem: single screen, planning horizon = 8 weeks, obligation period = 2 weeks. To examine rank-based implications of policy in greater detail, we increased the number of ranks to five in these problems. ^Recall that a higher rank number of a movie indicates a worse position. 3 1 We also restrict our analysis to better quality movies, Types 1 and 2 , because they dominate the other types on both quality parameters. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 59 2.5.3 Results The results of these analyses are presented graphically as an optimal action space in Table 2.10. Since the replacement movie is available from Week 3 onwards, the action spaces are shown starting from Week 3 in these figures. The interpretation of these results is as follows. For example, consider the optimal action space of Problem I for Week 3. The figure suggests that if Movie a is in Rank 1 or 2, then it is optimal to continue playing it (shown as Retain space) no matter which rank Movie b opens in. However, if Movie a has deteriorated to Rank 3, then it is optimal to replace it (shown as shaded Replace space) if Movie b opens in the top 2 ranks. The other figures can also be interpreted similarly. The results of the three problems suggest the following. • As mentioned above, in Week 3 of Problem I, the optimal policy does not recom-mend replacement of Type 1 movie as long as it maintains itself in the top 2 ranks. This is because the Type 1 movie retains itself well against its expected rapid decay. This increases the attractiveness of Type 1 movies in the higher ranks. • In Problem I, at Week 4, the optimal rules for the ranks 1, 2, 4, or 5 of Movie a remain the same as those at Week 3. For Rank 3 of Movie a, we find a difference in that Movie b can be ho worse than Rank 1 in order for it to replace a. That is, with a lapse of one week, the optimal policy imposes a more stringent criterion on Movie b than what it was earlier. The gradual decrease in the size of the replacement area through the horizon in all the problems shows that the optimal policy imposes increasingly stringent criterion for the movie available for replacement. This phe-nomenon concurs with our earlier results in the obligation period analysis that as a movie ages, it loses its appeal. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 60 • In Problem J, since the existing movie is Type 2 (with better holding strength) which is available early in the horizon, the optimal policy requires the replacement movie (Type 1) to be highly attractive in order to enforce replacement. Therefore, a more stringent criterion is imposed on the replacement movie in this problem as compared to Problem I. This is shown by the smaller size of the respective replacement areas of Problem J than Problem Fs. • In all of the problems, similar to a control limit policy, the optimal policy partitions the state-space (or rank-space) into two portions such that the "Retain" option is optimal in one portion, and the "Replace" option is optimal in the other. However, as discussed below, the multi-dimensional nature of the state space in our problem poses problems for the optimal policy to be specified as a control limit policy. We find different optimal policy results for the different problem settings. Similar to control limit rules in machine or equipment replacement problems, we find that the optimal policy divides the rank-space into two distinct portions such that it is optimal to take "Retain" action in one of the portions and "Replace" in the other one. However, the state space in our problem highlights some important differences between the two problems. First, we show Retain and Replace areas in terms of one of the components of the state space, the rank of the movie. The other components of the state variable also affect the policy. Given the other components of the state variable, we could treat the boundary between the Replace and Retain areas as a control limit in our two-dimensional rank (action) space. The challenge in this case is to define the functional form of that boundary. Our results show that the optimal policy results are problem and decision epoch specific, therefore it is quite difficult to define the boundary between Retain and Replace regions. Second, even if the other components in our state variable are held constant, the rank vector in our state variable involves ranks of all the other movies. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 61 This is limiting for the cases involving more than two movies. Moreover, there are some fundamental differences between our problem and a typical replacement problem. In our problem, the new movie's initial state is uncertain. The new movie is also subject to deterioration (i.e., aging) if replacement does not occur immediately. Given these differences, our results do not have a strict control limit form. When conditioned on a reduced state variable, however, the optimal policy resembles a control limit policy. For example, in Problem I (refer Table 2.10), given that movie o is playing at decision epoch 3 (the first two-dimensional action space), that is, it has played for two weeks, and the replacement movie b is available in Rank 1, then the optimal policy recommends that it is optimal to continue a if it sustains itself in top two ranks. Recall that this is the simple form of control limit policy we explained earlier in Table 2.9. It is interesting to note that since this is a two movie case, the optimal policy can also be conditioned on the rank of the replacement movie. For example, another form of the same policy as discussed above could be: given that movie a is playing at decision epoch 3, that is, it has played for two weeks, and the existing movie a is in Rank 3, then the optimal policy recommends that it is optimal to replace a if the replacement movie is available in top two ranks. Thus, if we could keep enough elements of the state variable fixed, then the optimal policy could have a control limit form. However, this reduction in the state variable would involve a greater number of elements as the number of replacement movies increases. 2.5.4 Manager i a l Insights The optimal policies, and the corresponding action spaces, provide some interesting man-agerial insights. First, different problem situations (at least for a simple two-movie case) can be compared according to the sizes of replacement areas. The relative replacement areas would further suggest which situations require quicker replacements of the existing Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 62 movies than the others. Within a given problem, the variation of the replacement area at different decision epochs determine the attractiveness of an alternative movie. We also find that the optimal policy imposes increasingly stringent criterion on the movie available for replacement with the progress of the horizon. These results suggest that the manager cannot use the same replacement rule for a given setting in the later part of the horizon as in the earlier part. A n observation consistent across all the results is that as long as an existing movie is in a rank better than the replacement movie, the optimal policy recommends continuing with the existing movie. This suggests a simple, and probably profitable, decision making heuristic: Each week, if a new movie becomes available, replace the existing movie if it is not in its obligation periods and the rank of the new movie is better than the one being considered for replacement, otherwise do not replace. In Chapter 4, we propose a version of this heuristics and examine its performance against other decision making approaches. 2.6 Conclus ion 2.6.1 Contr ibut ions In this chapter, we propose a modeling approach to analyze an interesting problem re-garding the replacement of motion pictures on theater screens. The contributions of this chapter are as follows. First, we propose an M D P model to formulate the movie replacement problem. The problem is complex because it involves stochastic elements in a dynamic decision making setting. The M D P formulation helps us model this complex problem by breaking it into separate components. These components are easier to explain in part than for the whole. Indeed, one of the major strengths of the M D P formulation is that its componential structure helps explain a complex phenomenon that would oth-erwise be hard to interpret. Next, we provide analogy of the movie replacement problem Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 63 with a machine/equipment problem and build upon the theoretical foundations of this stream of literature. However, we also note the differences between the setting of our problem and that of a standard replacement problem's. The resulting model provides theoretical and conceptual insights into the replacement phenomenon of perishable prod-ucts. In particular, we use a rank-based procedure to model the deterioration of a movie. Such schemes have been used by previous researchers and can be used in future work to model the problems related with the other perishable products. Third, the modeling of this problem helps us design some typical decision situations faced by the exhibitors in the motion picture industry. We investigate these situations by designing different scenarios. Finally, using expected total reward as the optimality criterion, we defined a "smart" manager as the one who accounts for uncertainty in the environment and adopts a long-term view in his/her actions. This definition provides normative guidelines for the managers in the exhibition industry, which we discuss in various scenario analyses. The analysis of supply conditions suggest that the exhibitor is better-off in a market situation in which his/her shelf-space becomes a scarcer commodity for upstream channel members (i.e., producers or distributors). The shelf-space scarcity at the exhibition level, and consequently increased value to the exhibitor, can come from an interaction of quantity and quality with release timing of the movies. The prescription to the managers is that they develop a mechanism to discriminate between the quality of various movie types. They should also prefer the movies that have strong retention strength and are available early in the horizon. The analysis of obligation period reveal that there is a "cost" associated with a certain obligation period. The smart manager is willing to bear that cost for a movie if it makes economic sense to do so according to the chosen optimality criterion. On the other hand, if too high an obligation period is attached with a less attractive (in expected revenue terms) movie, then the manager will not play that movie, especially when there is a more Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 64 attractive movie to be released in the future. Thus, one of the important messages of this analysis is that the producers/distributors of the movies have to choose the obligation period carefully in designing the contracts they offer to a smart exhibitor. The availability of other movies influences how sensitive a manager is to the obligation period. Another important implication of the obligation period analyses for the producers/distributors of the movies is regarding their release strategies. When faced with smart downstream channel members, they must release their movies as far away from the stronger movies as possible. This result concurs with earlier studies such as Krider and Weinberg (1998), which focused on distributors. The value of information analysis stresses the importance of having the right informa-tion for making smart decisions in the exhibition business. As the quality of information deteriorates, the exhibitor's confidence in a movie's performance decreases. Consequently, he/she becomes more conservative and continues with the existing movie and imposes a sort of penalty on the forthcoming movie. This suggests that the upstream channel member (e.g., producer, distributor) needs to send enough signals in the market for the downstream channel members (e.g., exhibitors, audience) in order to reduce ambiguity regarding quality of his/her movies. Some such signals could be upfront advertising, having big stars, favorable early reviews, and so forth. The statements to this effect have been made earlier. For example, Arthur DeVany, an industrial economist at the University of Southern California, Irvine, states, " A bankable star gets the exhibitor to take the film, so you get lots of screens." (Cassidy 1997, p. 37). Similarly, B i l l Mechanic, chairman of Twentieth Century Fox Studio, comments: "Even with the crazy salaries, it makes sense to put the right star in the right vehicle." (Cassidy 1997, p. 42). In the context of the advertising of the movie Independence Day, Cassidy (1997, p. 42) notes: "Somebody in Fox's marketing department had an idea: Why not broadcast a trailer from the film during the Super Bowl . . . If it had worked for Budweiser and Coca-Cola, Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 65 why not do it with a film, even if the release date was more than six months away? The trailer, which showed the White House being blown up, instantly turned Independence Day into a media phenomenon and staked its claim for the July 4th weekend, one of the most valuable release dates in the calendar." We also investigated the possibility of characterizing the structure of rank-based opti-mal policies as a control limit policy under different conditions. Since our model is similar to a machine replacement problem, we expected to find the structure of the optimal poli-cies to be similar to a control limit policy. However, some basic differences in the settings between the movie replacement and typical machine replacement problems cause some important differences to emerge. Though the optimal policies results are not exactly like control limit rules, we find some interesting results in this analysis. Our policy results depend on the state of both the current and replacement movies. This creates a two (or multi-)-dimensional decision space whose area depends on the relative attractiveness of the current and replacement movies, and the stage of the planning horizon. Our results show that the optimal policy imposes increasingly severe rank-based criterion on the re-placement movie to replace the existing movie as the horizon progresses. We propose a simple heuristic based on the results of this section, which is used and tested in Chapter 4. Therefore, another contribution of this chapter is that it lays the foundation for the later chapters. 2.6.2 L imi ta t ions and Future Research We now discuss some of the limitations of our study in this chapter. First of all, we chose a rank-based approach to operationalize the revenue variable. This operational-ization helps us model the movie replacement problem as a M D P problem. However, there are other ways in which we could have formulated this problem. To explain those, Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 66 we must first recognize that there are two major parts to the movie replacement prob-lem. One is revenue estimation, and the other is scheduling (i.e., replacement). Various methods emerge according to how we treat these two parts. In the current rank-based scheme, with some mild assumptions about rank-based structure, we treat both the parts simultaneously32. However, other methods could emerge depending on the treatment of these two parts. For example, one of the ways is to first use Bayesian forecasting for rev-enue prediction. Similar approaches are used in state-space modeling using Kalman filter in time-series analysis (Chatfield 1989), or in dynamic generalized linear models (West, Harrison, and Migon 1985; Fahrmeir and Tutz 1994). The next step would be to develop a deterministic space allocation optimization scheme for scheduling. The problem with such approaches is that they dissociate the two parts^ revenue prediction and scheduling, and therefore, the resulting approach might be sub-optimal. A n improved possibility would be to use a fully Bayesian approach in an M D P framework, that is, simultaneous optimization and revenue prediction without using the rank-based Markov chain frame-work. In other words, the approach would assume some priors for all the movies in the movie consideration set, arrive at their revenue estimates based on posteriors as the data become available, consider all such possibilities according to the assumed distributions, and then produce an optimal replacement plan simultaneously33. It is obvious that the complexity associated with problem of this nature would increase enormously with prob-lem size and would not permit investigation of reasonable size scenarios. Moreover, the objective in this chapter is to provide a conceptual framework and derive useful insights for the movie replacement problem an exhibitor faces. We deal with decision support aspects of this problem in detail in Chapter 3. 3 2There are some other approaches to model the movie replacement problem such as partially-observed Markov decision processes (POMDP's) (Zilla 1993), but they are similar in spirit to the rank-based approach adopted in this chapter. 3 3 We earlier alluded to one of the Bayesian approaches, bandit model, in our modeling section. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 67 The second limitation is regarding the assumed stationarity of probability transition matrices. In other words, our model assumes that the exhibitor's confidence about a movie's performance remains the same no matter how long it has been available in the market. In practice, this confidence would change with time as the movie does better or worse than expected. Clearly, we made this assumption for the tractability of the analysis. However, in future research, it would be interesting to examine'how non-stationarity of transition matrices would affect optimal policy; We also assumed that the exhibitor owns a monopoly theater in its geographical area. In reality, however, the demand for a movie would also be affected by the movies playing in the other nearby theaters. Our model does not account for such effects. We discuss this assumption further in later chapters. Suffice to note here that the inclusion of the competitive effects would take us away from the major objectives of this study. However, the modeling of competitive effects suggests attractive future research opportunities. Another issues concerns the length of the planning horizon in our example problems. The crucial factors in this context are starting- and end-of-horizon effects. In our exam-ples, we began the horizon with a weak movie to ensure the possibility of a replacement occurring, given our short horizon. However, one may argue that certain seasons begin with a strong movie already playing at the theater. One way to incorporate such cases in a large-scale simulation set-up would be to randomize the occurrence of an opening movie of a season. This could be done using a uniform random variable and the proportions of the movie types in a large-size database such as Jedidi, Krider, and Weinberg's (1998). The end-of-horizon effects, on the other hand, suggest that a season that ends with a strong movie is better for the exhibitor from a long-term perspective. This end effect is usually summarized in dynamic programming techniques by a salvage value at the end. Though we use zero salvage value in our example problems for the sake of simplicity of exposition, the methodology can be extended to include a reasonable salvage value in the Chapter 2. Stochastic.Modeling of Replacement of Movies on Screens 68 following way. The extreme situation to consider while incorporating such effects is that a movie is chosen to play, and hence enters its obligation period, at the last decision epoch. Then, assuming that this movie would have played for at least its obligation period (say, two weeks) into the next season, we can calculate its salvage value as its expected revenue for those two weeks. We could similarly summarize such two-week expected values for all the movies as their salvage values. Finally, let us consider how we could extend the current single screen M D P model to a multiple screen case. Instead of presenting the mathematical details of the model, we discuss the changes that would be required for the multiple screen model. The decision epochs in the multiple screen problem would remain the same as before. The state variable would have to be expanded to include the information about movies playing at each screen. The movie index and the play length elements of the state variable will be changed to vectors of the size of number of screens. The rank vector would remain the same since it depends on the number of movies. The action variable would also become a screen specific vector. It will consider two different cases. The first one checks a condition whether any movie in the current movie set (a vector in the state variable) is still in its obligation period, in which case that movie would have to be scheduled on one of the screens. If the condition in the first case is false, then other movies could be scheduled which could be either from the current movie set or from the other available movies. The reward variable will also be changed to be screen specific. This leads to the screen capacity issue, that is, the maximum revenue a movie can generate at a screen is the amount the screen yields if it is sold out. The reward variable can account for this condition because movies are screen specific. Finally, the transition probability expressions would remain essentially the same as long as we assume independence between different movies. The only change from the current model will be that the probability expressions would have to check additional conditions whether a movie belongs to the current movie set at the Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 69 current and the next decision epochs. The above operationalization raises some interesting issues which we now discuss. The first issue concerns the different capacity of different screens. This will be a criti-cal variable to the formulation only if the demand for a movie exceeds screen capacity. Indeed, the primary reason for the exhibitor to have different screen sizes is that the at-tendance varies for different movies and having different screen sizes increases scheduling flexibility. The main decision variables that seem to be affected by different screen sizes are as follows. The first variable is the multiple screenings of a movie, that is, whether to play the same movie on more than one screen of a theater simultaneously. This is usually done for the blockbuster movies whose demand is expected to exceed the capacity of the highest capacity screen. However, such movies are rare in a season. The second variable is which movie to allocate on which screen in the beginning and switching of the movies between screens in the later weeks. A movie may be switched from a higher to a lower capacity screen because of its depleting demand and the availability of a more attractive movie for the higher capacity screen. However, the opposite case may also occur sometimes when a movie's demand builds on word-of-mouth in the later weeks34. In practice, it appears that the switching between screens is mainly decided on the basis of the rank-ordering of the movies. Thus, the different screen capacity related decisions seem to follow the decision "which-movies-to-pick." The second issue that the multiple screen model raises is regarding the direction of optimal policy results. That is, whether the results would be similar to the single screen case as investigated in this chapter. We speculate that the broad policy results will be similar, although there will be some changes specific to the multiple screen scenario. ^These phenomena raise another interesting research question. Given that the movies make tran-sitions stochastically, when should the exhibitor switch a movie from one screen to another (or drop it altogether), or start showing a movie on one screen from two screens. These questions can also be addressed by formulating them as optimal stopping problems (Puterman 1994, pp. 47-50). Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 70 For example, the exhibitor might still prefer the movies with strong retention strength early in the horizon, but the optimal policy might schedule some movies for a longer period because of the increased flexibility in screening a movie and possibility of multiple screenings of a movie. The above issues raise interesting research questions, that we believe deserve separate modeling efforts. It is evident that the proposed multiple screen formulation would increase the model complexity at this level of generality. In Chapter 3, we examine a simplified version of the general multiple screens problem as proposed above. One of the major objectives of Chapter 3 is to solve bigger size problems and develop an implementable decision support model. The treatment of these issues comes at the expense of making some simplifying assumptions. In particular, we examine the case of a multiple screen theater but assume a deterministic setting in the movie replacement phenomenon. The assumption of deter-ministic knowledge of revenues is relaxed in a later analysis in Chapter 4, which estimates potential gains from the decision support model developed in Chapter 3. Moreover, as we explain above, the different screen capacity raises other modeling issues which are not the focus of the current study. Therefore, we restrict ourselves to an equal capacity mul-tiple screens setting and focus on "which-movies-to-pick" and "how-long-to-play-them" decisions in Chapter 3. To summarize, our main objective in this chapter was to provide a theoretically appealing and conceptually interesting framework for the movie replacement problem faced by the exhibitors. Building on the dynamic nature of this problem, .we present M D P methodology to solve it. Besides providing some interesting results and managerial insights, M D P approach naturally leads us into Chapter 3 because of its dynamic nature. Moreover, we use the insights from the rank-based optimal policy analysis in this chapter to generate an efficient heuristic in Chapter 4. Thus, the current chapter sets the stage for the following chapters. Chapter 2. Stochastic Modeling of Replacement ofMovies on Screens 71 Table 2.1: Expected Net Revenue Values for Example Problem in Section 2.2 Rank\Week 1 2 3 4 5 6 7 8 1 290 290 290 290 290 290 290 290 2 60 70 80 90 90 90 90 90 Chapter 2. Stochastic Modeling of Replacement of Movies on Screens Table 2.2: Summary of Scenario Analysis Results 72 Analysis and Objective Problems Description Main Results Managerial Summary Supply Conditions • To investigate impact of quantity, quality and release sequence of movies on exhibitor's value A. Involves various cases of high quality Type 1 and low quality Type 3 movies B. High quality Type 2 movies and low quality Type 3 movies /") Exhibitor prefers greater quantity of higher quality movies, //') Increased quantity of low quality movies has minimal impact, ///) Early availability of high quality movies, especially Type 2, improves exhibitor's value, iv) Exhibitor achieves higher values in Problem B than A because of strong retention strength of Type 2 movies Use increased quantity to find movies that retain well and stay with them, earlier is better Obligation Period -To investigate impact of obligation period (minimum play length associated with a movie in its contract) on exhibitor's value and replacement decisions C. Type 2 movie to released after existing Type 3 movie, whose obligation period is varied. D, E & F. Three movies cases. Type 3 movie currently playing. Two forthcoming movies, obligation period of first one varied. D. Type 2 movie released after another Type 3 movie, E. Type 2 movie to be released after Type 1 movie, F. Type 1 released after Type 2 movie C. Exhibitor's value decreases with increase in obligation period, D. j) With increase in obligation period, exhibitor's value first decreases, then stabilizes, ii) Exhibitor finds it optimal to "wait" for high quality movie than replace with low quality movie earlier, E & F. /) Exhibitor's value decreases with increase in obligation period, //) Type 2 more attractive in F since available earlier in the horizon, /'//') Replacement depends on the difference in ranks of the two replacement movies, iv) Overall, exhibitor better off than in C or D Increased obligation periods for weak movies substantially decreases value, anticipate and wait for better replacements Value of Information -To examine impact of variation of information about a movie's performance on exhibitor's value and replacement decisions Two movies case. First movie Type 3. Variable information about second movie. G. Instance involves first movie retains itself in top rank. H. Instance involves first movie decays to a lower rank. With deterioration in information about a forthcoming movie, exhibitor, /) makes sub-optimal replacement decisions, ii) continues existing movies longer, in) incurs opportunity cost of not being able to show a better movie Low quality information leads to lower value and resistance to switch existing movies Chapter 2. Stochastic Modeling of Replacement ofMovies on Screens 73 Table 2.3: Expected Net Revenue Values for Scenario Analyses Case 1 - 2-ranks, ER(1)=500, ER(2)=100 RankAWeek 1 2 3 4 5 6 7 8 1 150 200 230 230 230 230 230 230 2 30 40 50 60 65 65 65 65 Case 2 - 3-ranks, ER(1)=500, ER(2)=300, ER(3)=100 Rank\Week 1 2 3 4 5 6 7 8 1 150 200 230 230 230 230 230 230 2 90 120 150 180 195 195 195 195 3 30 40 50 60 65 65 65 65 Case 3 - 4-ranks, ER(1)=500, ER(2)=400, ER(3)=250, ER(4)= =100 Rank\Week 1 2 3 4 5 6 7 8 1 150 ; 200 230 230 230 230 230 230 2 120 " 160 200 220 220 220 220 220 3 75 100 125 150 163 163 163 163 4 30 40 50 60 65 65 65 65 Case 4 - 5-ranks, ER(1)=500, ER(2)=400, ER(3)=300, ER(4)=200, ER(5)=100 Rank\Week 1 2 3 4 5 6 7 8 1 150 200 230 230 230 230 230 230 2 120 160 200 220 220 220 220 220 3 90 120 150 180 195 195 195 195 4 60 80 100 120 130 130 130 130 5 30 40 50 60 65 65 65 65 Chapter 2. Stochastic Modeling of Replacement ofMovies on Screens 74 Table 2.4: Initial Probability Matrices for Scenario Analyses Movie Type\ Number of Ranks 2 3 4 5 1 2 3 4 (0.8 0.2) (0.6 0.4) (0.2 0.8) (0.3 0.7) (0.7 0.2 0.1) (0.5 0.3 0.2) (0.1 0.1 0.8) (0.1 0.2 0.7) (0.6 0.2 0.1 0.1) (0.5 0.2 0.2 0.1) (0.1 0.1 0.1 0.7) (0.1 0.2 0.3 0.4) (0.6 0.1 0.1 0.1 0.1) (0.5 0.2 0.1 0.1 0.1) (0.1 0.1 0.1 0.1 0.6) (0.1 0.1 0.2 0.2 0.4) Chapter 2. Stochastic Modeling of Replacement ofMovies on Screens 75 Table 2.5: Transition Probability Matrices for Scenario Analyses Movie Type\ Number of Ranks 2 3 4 5 1 '03 0.6 O.f 0 0.4 0.6 ,0 0 1, ^0.2 0.6 0.1 O.f 0 0.3 0.6 0.1 0 0 0.4 0.6 0) 0 0 1, ^0.1 0.6 0.1 0.1 O.f 0 0.2 0.6 0.1 0.1 0 0 0.3 0.6 0.1 0 0 0 0.4 0.6 ,0 0 0 0 1, 2 '0.7 0.2 O.f 0 0.8 0.2 ,0 0 \j '0.6 0.2 0.1 O.f 0 0.7 0.2 0.1 0 0 0.8 0.2 0 0 1, '0.5 0.2 0.1 0.1 O.f 0 0.6 0.2 0.1 0.1 0 0 0.7 0.2 0.1 0 0 0 0.8 0.2 K0 0 0 0 1, 3 (0.1 0.9) (o 1 J '0.10.1 0.8N 0 0.1 0.9 ,0 0 1, '0.1 0.1 0.1 0.7> 0 0.1 0.1 0.8 0 0 0.1 0.9 ,0 0 0 1; '0.1 0.1 0.1 0.1 0.6"> 0 0.1 0.1 0.1 0.7 0 0 0.1 0.1 0.8 0 0 0 0.1 0.9 ,0 0 0 0 \j 4 ( 0 , 0.S) '0.10.2 0.7> 0 0.2 0.8 ,0 0 1; '0.1 0.1 0.2 0.6^ 0 0,1 0.2 0.7 0 0 0.2 0.8 ,0 0 0 \j '0.1 0.1 0.1 0.2 0.5^  0 0.1 0.1 0.2 0.6 0 0 0.1 0.2 0.7 0 0 0 0.2 0.8 <0 0 0 0 \j Chapter 2. Stochastic Modeling of Replacement ofMovies on Screens 76 Table 2.6: Supply Conditions Analyses: Design and Results Problem A Case 33 31 13 11 Rank 1 2 3 1 2 3 1 2 3 1 2 3 Value* 660 581 512 829 750 681 842 633 512 997 799 681 Case 3333 3131 1313 1111 Rank 1 2 3 1 2 3 1 2 3 1 2 3 Value* 680 601 532 997 920 851 1012 819 704 1199 1047 949 Problem B Case 33 32 23 22 Rank 1 2 3 1 2 3 1 2 3 1 2 3 Value* 660 581 512 955 875 806 1268 938 512 1423 1138 681 Case 3333 3232 2323 2222 Rank 1 2 3 1 2 3 1 2 3 1 2 3 Value* 680 601 532 1172 1095 1027 1428 1146 825 1554 1320 1128 * - Represents the value the exhibitor achieves if the movie opens in a respective rank at the first decision epoch. Note: The values have been rounded-off. Chapter 2. Stochastic Modeling of Replacement ofMovies on Screens 11 Table 2.7: Results of Obligation Period Analyses Problem C Obligation Period 1 2 3 4 Rank 1 2 3 1 2 3 1 2 3 1 2 3 Value 950 873 805 950 873 805 (0) (0) (0) 831 752 683 (13) (14) (15) 141 661 592 (22) (24) (26) Problem D Obligation Period 1 2 3 4 Rank 1 2 3 1 2 3 1 2 3 1 2 3 Value* 909 830 761 893 814 745 (2) (2) (2) 893 814 745 (2) (2) (2) 893 814 745 (2) (2) (2) Problem £ Obligation Period 1 2 3 4 Rank 1 2 3 1 2 3 1 2 3 1 2 3 Value* 1007 930 861 969 891 823 (4) (4) (4) 932 854 785 (8) (8) (9) 912 833 764 (9) (10) (11) Problem F Obligation Period 1 2 3 4 / Rank 1 2 3 1 2 3 1 2 3 1 2 3 Value* 1021 944 876 1006 929 861 (1) (2) (2) 995 918 849 (3) (3) (3) 990 913 844 (3) (3) (4) * - Represents the value the exhibitor achieves if the movie opens in a respective rank at the first decision epoch. The figures in parentheses represent the percentage drop in value from the value corresponding to obligation period of one week. Note: The values have been rounded-off. Chapter 2. Stochastic Modeling of Replacement ofMovies on Screens 78 Table 2.8: Results of Value of Information Analyses Problem G Decision epoch (w) State Variable ( S w ) Optimal decisions under various conditions, C, U l , and U2 (associated values in parentheses) d * c ( Sw ) d*ui(Sw) d*U2(Sw) 1 (a,-2,(l,0)} a ( 1 234 ) a ( 1 234 ) a ( 1 2 3 4 ) 2 (8,-1,(1,0)} a ( 1136 ) a ( 1136 ) a ( 1136 ) 3 (a,0,(l,l)} b ( 1040 ) b ( 1 040 ) a ( 1 020 ) 4 (a,l,(l,l)} b ( 850 ) a ( 841 ) a ( 841 ) (b,-l,(4,l)} b ( 890 ) b ( 890 ) * 5 {a,2,(l,l)} a ( 6 7 2 ) * * a ( 672 ) a ( 672 ) (b,0,(4,l)} b ( 730 ) b ( 730 ) * 6 (a,3,(l,l)} a ( 5 0 4 ) * * a ( 504 ) a ( 504 ) (b,l,(4,l)} b (560) b ( 560 ) * 7 (a,4,(l,l)} a ( 3 4 5 ) * * a ( 345 ) a ( 345 ) (b,2,(4,l)} b ( 380 ) b ( 380 ) * 8 (a,5,(l,l)} a ( 1 9 0 ) * * a ( 190 ) a ( 190 ) (b.3,(4.1)} b (190) b ( 190 ) * - T h e c o r r e s p o n d i n g case does no t o c cu r . ** - T h e c o r r e s p o n d i n g v a l u e s i f a c t i o n b w e r e t a k e n are 6 60 , 4 8 0 , 3 10 , a n d 150 respec t i ve l y . Problem H Decision epoch (w) State Variable ( S w ) Optimal decisions under various conditions, C, U l , and U2 (associated values in parentheses) d'c (Sw) d*ui(s«) d*U2(Sw) 1 (8,-2,(1,0)} a ( 1234 ) a ( 1234 ) a ( 1234 ) 2 (8,-1,(1,0)} a ( 1136 ) a ( 1136 ) a ( 1 136 ) 3 (8,0,(3,2)} b (860) b ( 860 ) a ( 810 ) 4 (8,1,(3,2)} b ( 700 ) b ( 700 ) a ( 660 ) (b,-l,(4,2)} b ( 740 ) b ( 740 ) * 5 (8,2,(3,2)} b (540) a ( 520 ) a ( 520 ) (b,0,(4,2)} b (610) b ( 610 ) * 6 (8,3,(3,2)} a ( 390 ) a (390) a ( 390 ) (b, 1,(4,2)} b ( 470 ) b ( 470 ) * 7 (8,4.(3,2)} a ( 250 ) a ( 250 ) a ( 250 ) (b,2,(4,2)} b (320) b ( 320 ) * 8 (8,5,(3,2)} a ( 130 ) a ( 130 ) a ( 130 ) (b,3,(4,2)} b ( 160 ) b ( 160 ) * - T h e c o r r e s p o n d i n g case does no t o c cu r . Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 79 Table 2.9: A Hypothetical Example of a Rank-Based Control Limit Policy for the Movie Replacement Problem Actions Retain Replace Rank Action 1 2 3 4 5 Chapter 2. Stochastic Modeling of Replacement of Movies on Screens 80 Table 2.10: Results of Rank-Based Optimal Policy Analyses Problem I M o v i e c u r r e n t l y p l a y i n g = a ( T y p e 1) M o v i e a v a i l a b l e f o r r e p l a c e m e n t = b ( T y p e 2 ) A c t i o n s R e t a i n a R e p l a c e a D e c i s i o n e p o c h O p t i m a l A c t i o n s S p a c e 3 4 N u m b e r o f w e e k s M o v i e a p l a y e d * 0 R a n k s o f M o v i e a\b 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 D e c i s i o n e p o c h N u m b e r o f w e e k s M o v i e a p l a y e d * R a n k s o f m o v i e a\b 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 B e y o n d i t s o b l i g a t i o n p e r i o d o f t w o w e e k s . Chapter 2. Stochastic Modeling of Replacement of Movies on Screens Table 2.10: Results of Rank-Based Optimal Policy Analyses (Continued) Problem J M o v i e c u r r e n t l y p l a y i n g = a ( T y p e 2 ) M o v i e a v a i l a b l e f o r r e p l a c e m e n t = b ( T y p e 1) A c t i o n s B R e t a i n a R e p l a c e a O p t i m a l A c t i o n s S p a c e D e c i s i o n e p o c h 3 4 5 N u m b e r o f w e e k s M o v i e a p l a y e d * 0 1 2 R a n k s o f M o v i e a\b 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 D e c i s i o n e p o c h 6 7 8 N u m b e r o f w e e k s M o v i e a p l a y e d * 3 4 5 R a n k s o f m o v i e a\b 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 B e y o n d i t s o b l i g a t i o n p e r i o d o f t w o w e e k s . Chapter 2. Stochastic Modeling of Replacement of Movies on Screens Table 2.10: Results of Rank-Based Optimal Policy Analyses (Continued) Problem K M o v i e c u r r e n t l y p l a y i n g = a ( T y p e 1) M o v i e a v a i l a b l e f o r r e p l a c e m e n t = b ( T y p e 1) A c t i o n s B R e t a i n a R e p l a c e a O p t i m a l A c t i o n s S p a c e D e c i s i o n e p o c h 3 4 5 N u m b e r o f w e e k s M o v i e a p l a y e d * 0 1 2 R a n k s o f M o v i e a\b 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 D e c i s i o n e p o c h N u m b e r o f w e e k s M o v i e a p l a y e d * R a n k s o f m o v i e a\b 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 * - B e y o n d i t s o b l i g a t i o n p e r i o d o f t w o w e e k s . Chapter2. Stochastic Modeling of Replacement of Movies on Screens Figure 2.1: The Supply Chain in the Motion Picture Industry 83 DISTRIBUTOR (e.g., Warner Bros.) AUDIENCE ftHftft Chapter 2. Stochastic Modeling of Replacement of Movies on Screens Figure 2.2: Timing of Events in the Movie Replacement Problem 84 Start of Start of Wee c t Weekend demand Week days demand Wee FRIDAY-Movie nit playing MONDAY-Movie rrit playing -Decide movie m t + i for the coming Friday FRIDAY -Movie mt+i playing-Figure 2.3: Release Scenario for the Example Problem for Explanation of M D P Model Movies Released a c b 1 2 3 4 5 6 7 8 Decision Epoch (Week) Chapter 2. Stochastic Modeling of Replacement ofMovies on Screens 8 5 Figure 2.4: Expected Revenue Decay Patterns of Different Clusters (Movie Types) Note: The expected revenue figures for the four clusters are according to a common scale. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens Figure 2.5: Results of Supply Conditions Analyses 86 Problem B 400 f,vm-,v,v,^^ $ 8 R N Case Chapter 2. Stochastic Modeling of Replacement ofMovies on Screens 87 Figure 2.6: Release Situations for Obligation Period Analyses Problem C M o v i e s R e l e a s e d Type 3' T y p e 2 1 2 3 4 5 6 7 8 D e c i s i o n E p o c h ( W e e k ) Problem D M o v i e s R e l e a s e d T y p e 3 Type 3' T y p e 2 1 2 3 4 5 6 7 8 D e c i s i o n E p o c h ( W e e k ) Problem £ M o v i e s R e l e a s e d T y p e 3 Typel' T y p e 2 1 2 3 4 5 6 7 8 D e c i s i o n E p o c h ( W e e k ) Problem F M o v i e s R e l e a s e d T y p e 3 Type 2' T y p e 1 1 2 3 4 5 6 7 8 D e c i s i o n E p o c h ( W e e k ) * - Represents the movie whose obligation period is varied. Chapter 2. Stochastic Modeling of Replacement of Movies on Screens Figure 2.7: Results of Obligation Period Analyses 88 Problem C Problem D § 950 5 900 2 * 750 700 A;,; I D ± A T X""' .... ' " " ' " O ! I M n " " " " o " 1— X • 1 — i 2 3 Obligation Period Problem E § 1100 5 ,_ 1000 &t  900 S 700 4 ™ 600 > 1 — • — Rank 1 s —A—Rank 2 X —X—Rank 3 Obligation Period Problem F o 1050 | 1000 950 900 850 800 Si 6 _ —' n m i l " 11 o x — ft "" — A T .VnV.Y,Y,V,V«.V.Tf.T.T.Tm —~<—x ••YMYIYMM ,(,YMY MY.Y.Y, i x — — i 2 3 Obligation Period o—Rankl •A—Rank 2 I—X—Rank 3 Chapter 2. Stochastic Modeling of Replacement ofMovies on Screens 8 9 Figure 2.8: Release Situation for Value of Information Analyses Movies Released a (Type 3) b (TypeX) 1 2 3 4 5 6 7 8 Decision Epoch (Week) Chapter 2. Stochastic Modeling of Replacement ofMovies on Screens 90 Figure 2.9: Release Situation for Rank-Based Optimal Policy Analyses Movies Released a b 1 2 3 4 5 6 7 8 Decision Epoch (Week) Chapter 3 SilverScreener: A Decision Support Mode l for Movie Screens Management 3.1 Overview In Chapter 2, we introduced the general problem of movie replacement at theater screens. In this chapter, we consider a special case of the general movie replacement problem and apply it to the shelf-space management problems faced by exhibitors, the retailers of the motion picture industry. Retailing is becoming an increasingly important area of management attention and academic research. Much of the research has focused on the strategic aspects of retail management — both within the distribution channel and between retailers (or retailer-manufacturer systems) — but another important area has been the development of decision support models to help retailers improve their decision-making. For example, Bultez and Naert's (1988) S H A R P model helps retailers decide on shelf space allocations and Abraham and Lodish's (1993) P R O M O T I O N S C A N system helps managers in developing and evaluating short-term retail promotions. Our work is in a similar spirit to these models, its aim is to help retailers improve their decision making. Most of the published research in this area focuses on consumer packaged goods sold through supermarkets. However, other retail formats and industry settings pose different challenges and intriguing problems. We focus on products which have relatively short life cycles, so that effective retail management requires regular attention to the issue of which products should be stocked. In particular, we concentrate on the motion picture industry, 91 Chapter 3. SilverScreener: A Decision Support Model 92 though other entertainment (e.g., books, video games) and fashion goods industries have similar characteristics. The motion picture industry is emerging as an area of increased interest to marketing scholars and researchers. The industry is undergoing several revolutionary changes -due mainly to advances in technologies and marketing strategies - that are likely to change the manner in which movie watchers will consume entertainment products and the way firms conduct their businesses. Hollywood's major studios produce over 450 feature films each year. These films are exhibited on more than 27,000 theater screens in the U.S. and Canada. Movie making is an expensive and risky business. The average movie costs $40 million to produce and an additional $20 million to market. While some movies do become "blockbusters," more than half of the movies produced by Hollywood do not recoup their investment even after their release to foreign markets, cable, and broadcasting T .V . (Vogel 1994). In response to these dynamics and challenges, a stream of research, addressing var-ious aspects related to the marketing of movies, has begun to emerge. At the con-sumer behavior level, some of the research has questioned the relevance of the traditional information-seeking framework for studying the consumption of movies (e.g., Hirschman and Holbrook, 1982; Holbrook and Hirschman 1982). Another stream of research has focused on forecasting the enjoyment of movies at the individual level (Eliashberg and Sawhney 1994) as well as forecasting the commercial success of movies at the aggregate level (Smith and Smith 1986; Austin and Gordon 1987; Dodds and Holbrook 1988; Sawh-ney and Eliashberg 1996; Eliashberg and Shugan 1997). Additionally, some research has begun to emerge addressing diffusion (Mahajan, Muller, and Kerin 1984; Jones and Ritz 1991), seasonality (Radas and Shugan 1995), release timing (Krider and Weinberg 1998), clustering (Jedidi, Krider, and Weinberg 1998), sequential products (Lehmann and Wein-berg 1998; Prasad, Mahajan, and Bronnenberg 1998), contract design (Swami, Lee, and Chapter 3. SilverScreener: A Decision Support Model 93 Weinberg 1998), and the impact of advertising (Zufryden 1996), all in the context of motion pictures. There are more than 300 exhibitors in the U.S. and Canada. Though the total number of theater screens owned by these exhibitors has remained relatively constant in the last five years, the total number of mass market movies released by the major studios seems to be rising steadily. Combined with a fairly stable pool of moviegoers, this trend suggests that movies' distributors (i.e., studios) will face limited screen availability for their films, while the exhibitors will have to manage their bookings and screens very effectively to maintain and improve profitability. Especially during the peak summer season, the synchronization of movies supply and theater screens may represent a serious challenge. In reviewing the situation for the summer of 1997, a Wall Street Journal article comments: After an epic binge on production of big-budget movies, the film industry now faces a glut of expensive "event" pictures and too few summer weekend slots to release them all. With more than $1 billion wrapped up in 12 mega-budget films spread over 10 weeks this summer, it's statistically impossible for every movie to recoup its investment. (Bannon 1997, p. B l ) In the summer of 1997, the major studios planned to open 59 movies between May and Labor Day, up from 49 in 1996. However, . . . there aren't enough movie screens to keep big films running for the months they need to recoup their costs. By July 4th, 1997, many are predicting gridlock. Films that are performing only moderately well will likely be scaled back in favor of the new releases. (Bannon 1997) Similar observations have been noted in the recent past. For example, L. Klady (1995) reports in Variety : Chapter 3. SilverScreener: A Decision Support Model 94 Summer 1995 almost certainly will be another box office record-breaker, but for distribs [distributors] it could be a summer of sturm und drang. Competition to secure screens will be tough. Keeping those screens will be even tougher as the majors fight the bloody battle for market share. . . . there will be 19% more movies. A major exhibition chain with an objective of effective screens management thus faces a complex scenario. The complexity comes from various sources1. First, the increased supply of movies by various studios increases the difficulty of deciding which movie to play. This decision is further complicated because it is to be made for a number of screens in a multiple screens theater (i.e., a multiplex). Second, an additional supply of movies brings more pressure from the studios to guarantee sufficient playtime for their movies. Relationship management in the motion picture industry is considered by many as very crucial. On the other hand, the scarcity of "shelf space" requires special attention in managing the screens effectively and profitably. Third, the nature of the distributor-exhibitor contract in the motion picture industry is unique. In signing a contract to play a film in its theaters, the exhibitor becomes obligated to play the film for a certain period of time, even when consumer demand is weak. This minimum obligation period (playtime), which is negotiated between the two parties, may vary by movie as well as by studio. The financial arrangements between studios and exhibitors are also apparently unique to the motion picture industry. Unlike wholesale/retail pricing practices commonly employed in the consumer goods industry, box-office grosses are split between the exhibitors and distributors of motion pictures. The manner in which the box-office grosses are split favors the studio (distributors) in the first few weeks of the movie playing, but shifts to the exhibitor's favor later on. Distributors thus have a strong incentive to promote the movies intensively in their initial play period. On the other hand, the longer the exhibitor plays the movie, the larger becomes his/her share of the box-office receipts. At the same 1We briefly mentioned some of these sources in Chapter 2. Chapter 3. SilverScreener: A Decision Support Model 95 time, theater attendance for a movie typically declines the longer a movie plays. In the face of this complexity, the aim of current research is to provide a structure for analyzing management problems of exhibitors, the retailers of the movie industry. Using a dynamic programming approach and a fast, but readily accessible algorithm, we propose a decision support model to help the exhibitor make effective and timely decisions regarding theater screens management. The major objective of this modeling exercise is to help the exhibitor both select and schedule movies at his/her theater. The developed model is readily implementable, as we demonstrate in an illustrative example, and appears to lead to improved profitability. Through sensitivity analysis, these results are shown to be robust to various parameters values. The rest of this chapter is organized as follows. The screens management model, SilverScreener, its properties, and solution procedures are presented in the next section. A n illustrative application of the model, implications for screens management policy, and the results of sensitivity analyses are given in Section 3.3. We provide concluding remarks and suggest directions for further research in Section 3.4. 3.2 Problem Formulation and Modeling Approach 3.2.1 The Exhibitor Problem In Chapter 2, we introduced the movie replacement problem. This replacement phe-nomenon provides the, foundation of the decision problem faced by the exhibitor. We revisit the salient features of this problem in this section. Every week, motion picture exhibitors have to make an important decision regarding the replacement of the movies playing at the screens in their theaters. Movies are a seasonal product, and the two prominent seasons are Summer and Christmas. In either of these seasons, a number of Chapter 3. SilverScreener: A Decision Support Model 96 movies are released every week by their distributors. The dynamic competitive envi-ronment thus induced gives rise to the notions of decay and aging of movies. Decay is the intrinsic weekly decline in the box office attraction and gross revenues (grosses in industry jargon) of a movie playing at a theater (Krider and Weinberg 1998). Aging is the decline in the value, that is, gross generating power, of a movie from an exhibitor's perspective if there is a delay (by week) in exhibiting the movie at the theater. Aging, therefore, results in an opportunity cost of not being able to play a particular movie. The above scenario is further complicated by the nature of the contract between the distributor and the exhibitor. The basic structure of the contract is fairly standard between different distributor-exhibitor pairs although the individual terms may vary de-pending on the relationship between the two parties. A typical exhibition contract states a fixed obligation period and a differential revenue sharing scheme in different weeks between the distributor and the exhibitor. The obligation period limits the ability of an exhibitor to replace a movie with less than satisfactory box-office performance in the initial weeks after its release.2 In a given week, the revenue sharing scheme splits the gross of a movie between a distributor and exhibitor by one of the two rules: a) 90%/10% over house nut 3 , or b) minimum gross percentage. If the 90%/10% over house nut rule operates, then the distributor receives 90% of the gross after the exhibitor has deducted and retained the house nut amount. Accordingly, under this rule the exhibitor keeps 10% of the gross over house nut plus the house nut amount. The exhibition contract also contains minimum percentage figures ("floors") as specified by the distributor for every week of the expected play length of a movie. These figures will be used if the minimum gross percentage rule 2The obligation period may range from two to ten weeks depending on the respective bargaining power of the distributor and exhibitor and the marketability of a particular movie. 3House nut is a small negotiated amount which the exhibitor receives from the distributor. It does not necessarily bear any relationship to the theater's actual expenses, and is only meant to allow for some cushion in the exhibitor's profit margins. Chapter 3. SilverScreener: A Decision Support Model 97 is invoked for revenue sharing. Under this rule, the whole gross amount (without house nut deduction) is split according to the specified minimum percentage for that week. A n illustrative example may clarify how these financial terms apply. The splitting terms (in favor of distributor and exhibitor, respectively) specified by the distributor under a typical contract may appear as follows (see, for example, Squire (1992), p. 315 for the movie Batman): 90%/10% over approved house allowance with minimums of: First week at 70%/30% Next two weeks at 60%/40% Next week at 50%/50% Next week at 40%/60% Balance at 35%/65% Suppose for instance that the movie grosses in its second week $10,000 and that the house nut is $5,000. The distributor will evaluate first the "90%/10% over house nut" option: 90% of (10,000 - 5,000) = $4,500. Next the distributor will evaluate the "mini-mum" option: 60% of $10,000 = $6,000. Since $6,000 > $4,500, the "minimum" option prevails, yielding $6,000 and $4,000 to the distributor and the exhibitor respectively. In another instance suppose that the movie grosses $50,000 for that week. Since 90% of the $(50,000-5,000) > 60% of $50,000 (under the "minimum" option), the "90%/10%" option prevails. In this case the distributor and the exhibitor receive $40,500 and $9,500, respectively. In addition to the revenue earned from the box office gross, the exhibitor also earns some income from concession sales such as popcorn, candies, and soft drinks. The concession sales, however, depend on individual demands generated by the movies playing at the theater. The exhibitor does not share the concession income with the distributor. Chapter 3. SilverScreener: A Decision Support Model 98 A majority of theaters nowadays have multiple movies playing at their multiple screens. Given the complexity of the revenue sharing scheme and the dynamics of movie availability and decision making, the managers of such multiplexes are faced with non-trivial decisions of selecting and scheduling movies on different screens in a fixed planning horizon. The exhibitor problem can be summarized as: Schedule the movies i n such a way that the e x h i b i t o r ' s c u m u l a t i v e profi t ( i n c l u d i n g concession profits) f r o m the chosen movies is m a x i m i z e d over a fixed p l a n n i n g h o r i z o n , subject to the constraints: 1. o b l i g a t i o n per iods for a l l the movies are met , 2. t o t a l n u m b e r of movies scheduled i n a week < t o t a l n u m b e r of screens i n the m u l t i p l e x In the next sub-section, we present a dynamic programming-based model, SilverScreener-DP, which aims at helping managers address the above problem. 3.2.2 S i l v e r S c r e e n e r - D P : T h e D y n a m i c P r o g r a m m i n g A p p r o a c h We formulate the exhibitor problem of the previous section as a finite-horizon determinis-tic dynamic programming (DP) problem. In order to simplify the exposition, we assume the following: 1. The availability of the movies to be released during the planning horizon is known in advance, 2. The weekly revenues to be generated by the movies considered during the planning horizon can be estimated in advance, 3 . The replacement decisions are made on a weekly basis, Chapter 3. SilverScreener: A Decision Support Model 99 4. A l l the screens in the multiplex are of equal capacity, 5. There is no time lag between placing an order for a new movie and its arrival, 6. The theater is a monopoly theater in its geographic location. Assumptions 1, 3 and 5 are not limiting and, in fact reflect the industry current and future practice. Assumption 2 is a strong assumption about a priori knowledge of movie revenues. However, we relax this assumption partially in the empirical analysis section. Moreover, data collected from Variety suggest that managers have a reasonable estimate of box-office gross revenue of a movie. We relax this assumption in Chapter 4 and show how the forecast data can be incorporated in our optimization model using an adaptive approach. In Chapter 4, we also discuss an ex ante revenue prediction scheme that can be used in conjunction with our optimization model. In the conclusion of Chapter 2, we discussed that the different screen capacity problem does not fit the objectives of our study. We therefore assume equal capacity screens for the current analysis (Assump-tions 4), which addresses the questions regarding which movies to pick and how long to show them. The assumption keeps the current analysis tractable, and can be relaxed in future work. While zones exclusivity provides the local exhibitor with some degree of monopolistic power, a complete relaxation of Assumption 6, would require the analysis of strategic replacement decisions by two (or more) competing theaters in vicinity of each other. Since this would require a major shift from the objectives of the proposed research, we refrain from the treatment of competitive aspects in this chapter. The model proposed in this section is a dynamic programming (DP) model based on the M D P model of the previous chapter. However, there are several differences between the two models. First, the D P model is for a multiple screens (equal capacity) case. Therefore, the movie and play length indices in the state variable and the actions are sets instead of single elements. Second, since we assume knowledge of deterministic Chapter 3. SilverScreener: A Decision Support Model 100 revenues (Assumption 2), the DP model does not involve the probability transitions and the notion of rank of a movie. We could treat each movie as having two ranks, 0 or 1, where 0 indicates that a movie is available and 1 indicates that it is unavailable4. Given that a movie is available, we know its revenue in a week deterministically. Third, we consider profits as rewards in this model. Since the major objective of this chapter concerns decision support aspects, we include specific elements of the contract terms and costs (fixed and variable) in the rewards expression. Finally, in this model we allow for the obligation periods to be movie specific, which helps in the empirical applications presented later. We now define the parameters used in the model. W - length of planning horizon, H - number of screens in the multiplex, M - set of all the movies considered during the planning horizon, N - total number of movies considered during the planning horizon, N = \M\, mi - index number of the movie playing at screen i at a decision epoch, ni - total play length of movie rrii at a decision epoch, rh,n - sets of rrii and respectively of the movies playing at a decision epoch at the H screens, Z - set of binary variables (for each movie) indicating whether a movie is available for replacement at a decision epoch, ai - movie scheduled on screen i at a decision epoch, 4Similar to Chapter 2, we assume that the release dates of different movies, ti, • • • , ijv, a r e known in advance (Assumption 1). This is an input data and gives us a knowledge of whether a movie is available at a particular decision epoch. However, at this level of specification of the DP model, we simply specify the model in terms of 0-1 variables, and do not explicitly include release date variables. Chapter 3. SilverScreener: A Decision Support Model 101 a - set of aj of the movies chosen for screening on H screens, GROSSjW - box-office gross revenue potentially generated by movie j in week w, POPjW - concession profit potentially generated by movie j in week w, VCjW - variable cost due to movie j in week io, FCW - fixed cost in week w, FLOORjW - "minimum" gross percentage (in exhibitor's favor) for movie j in week w, OPDj - obligation period of movie j, C - house nut. We formulate our model SilverScreener-DP in terms of the four basic components of a deterministic dynamic program (Puterman 1994)5 : Decision Epochs, States, Actions, Rewards. Decis ion Epochs (Monday morning of every week): T = { l , 2 , . . . , « v l W } > W<oo. States (set of movies playing, their play lengths and movie availability set at a decision epoch): S = {m, n, Z}. Act ions (the movies to be scheduled at a decision epoch): For s € S, As = {5} = {(alt - • • ,aH)}. 5Since we address the case of a deterministic dynamic program, we do not provide an expression for the transition probability matrices. However, the dynamic program presented here can be considered a special case of the MDP model presented in the previous chapter if the transition probabilities are modeled appropriately to reflect deterministic replacement actions. Chapter 3. SilverScreener: A Decision Support Model 102 m«, if rii < OP Dm. or Zj = 0 V j 6 M - {m*} g, otherwise, Zq = 1 t = 1, • • •, H. Rewards (profit to exhibitor in week w): -FCW + J2(POPjw- VCjw) + I0iw * {0.1 * {GROSSjw - C) + C} + (1 - Iejw) * FLOORjw * GROSSjw, s e S,a E A3, w = 1, • • •, W - 1, rw(s) = 0, s € S, where 9jW is a logical condition given by 9jw = (0.9 * (GROSSjw - C) > (1 -FLOORjw) * GROSSjw), We consider decision epochs as the Monday morning of every week of the planning horizon, which is consistent with the timing of events discussed in Chapter 2. The state variable is a set of three vectors. The first of these vectors, m, denotes the movies playing at each screen of the theater in a week. We identify each movie by a unique index number. The second vector, h, denotes the play lengths of the corresponding movies in rh. The third vector, Z, gives the information about the availability of all the movies considered in the planning horizon at a particular decision epoch. A movie is assumed to be available from its release date until the end of the planning horizon. However, movies replaced in a week are assumed to be unavailable in subsequent weeks for screening at the same and 1 if X = TRUE, 0 otherwise. Chapter 3. SilverScreener: A Decision Support Model 103 theater6. The action variable determines the continuation or replacement of a movie playing at a screen. This decision depends on the movies' availability, and the number of weeks that the movies currently playing have run with respect to their obligation periods. The reward function specifies the expression for the profit to the exhibitor in two different conditions of the contract terms. It follows directly from the revenue sharing terms described in the previous section. In addition, there are variables for fixed and variable costs. Fixed costs are shown to vary only by week, not by movies. In practice, they need not even vary by week. These are the costs that the exhibitor has to incur irrespective of the traffic generated by the movies playing in a particular week. It may involve costs like rent, lights, weekly maintenance costs, and so forth. The other type of cost, variable cost, varies by movies as well as weeks. These may include such costs as salaries of the temporary staff hired for a particular movie, or other general and administrative expenses. We use the following two-parameter exponentially declining function to estimate box-office gross revenue. Lehmann and Weinberg (1998) found this function to be a reasonable model of the revenue pattern for most major movies (see also, Sawhney and Eliashberg 6In a strict sense, these changes in the availability vector should be specified by a (probability) transition function, which takes on a value 1 for the feasible actions of continue or replacement of a movie, and 0 otherwise. The release dates data, ti,"-,tjv, would also be needed for the correct specification of such a transition function. To keep the exposition of the DP model simple, we do not specify the transition function in this chapter. However, it will use the concepts similar to the transition probability function of the MDP model of Chapter 2. Chapter 3. SilverScreener: A Decision Support Model 104 1996, Krider and Weinberg 1998)7. GROSSjw = aje^w+£ (3.11) where ctj > 0 and Pj < 0 are opening and decay factors respectively of movie j and e ~ normal (0, c r 2 ) . We set the value of the reward function arbitrarily to zero for the weeks following the planning horizon. However, a suitable salvage value could be chosen as discussed in the final section of Chapter 2. Finally, the value and the associated action set of the optimal policy can be found by the classical backward induction algorithm (Bellman 1957) as follows: V*(s) = map {r w(s, a) + V^s)}, (3.12) A*a,w = 'Kg m a x a G A s { r „ ( s , a) + K * + 1 ( s ) } , (3.13) w = l,---,W, seS. The value, V^, is defined based on the concepts developed in Chapter 2. It denotes the cumulative reward (or maximum profit) the decision maker obtains by taking optimal actions from Week w to the end of the horizon. The state variable at Week w, sw, is related to the state variable at Week w + 1, s^+i, by the action taken at the Week w. If action is to continue playing a movie, m ^ , on Screen i in Week w, then mitW+\ is the same as m^w, its play length, niiW, increases by one week (i.e., riitW+1 = riitW +1), and the movie remains available in the set Zw+\. If action is to replace the first movie by 7The exponential demand function used in the current research is a special case of the general demand function introduced by Sawhney and Eliashberg (1996). Their approach starts from a stochastic model of movie-goers' behavior. They model time to adopt a movie (by a movie-goer) as a three-parameter generalized gamma distribution from which they derive expressions for expected demand. In certain conditions of two of its parameters, their model reduces in the case of most major movies to a two-parameter demand function similar to the one used in this research. We find that in our empirical application (to be presented later) also, Sawhney and Eliashberg's (1996) model reduces to the two-parameter model in most cases. Therefore, we propose this simple yet reasonable function to model demand for a movie. Chapter 3. SilverScreener: A Decision Support Model 105 a new movie, then mijW+i (the new movie) is different from m,itW, the play length of the new movie is set to 1 week in Week w 4-1 (i.e., riiiW+\ = 1), and the movie m,iiW becomes unavailable in the set Zw+\. With its dynamic nature, the dynamic programming formulation appears to be a natural framework for the exhibitor problem. The conceptual insights drawn from the dynamic programming formulation and its theoretical appeal are attractive features for appreciating the underlying decision problem. The implementation of the above dynamic program would further require development of an optimization routine in a programming language such as C. However, one of our major objectives is to present our model in a framework as close to user-friendly implementation as possible. Toward this end, it would be beneficial to use the specialized computer codes such as those available for solving large-size linear programming problems. Moreover, it should be possible, at least theoretically, to write a linear program equivalent to a deterministic dynamic program. More specifically, the exhibitor problem has a special structure, which we explore in the next section, that helps us develop an integer linear programming formulation of this problem.8 3.2.3 SilverScreener-IP: Integer Programming Approach The formulation of the exhibitor problem in this section builds upon the general area of parallel machine scheduling problems (Baker 1993; Pinedo 1995). The solution ap-proaches for these problems are based on integer programs involving binary decision variables. The usual setting of these problems is as follows. A set of N jobs has to be scheduled on H parallel machines. Each job j, (j = 1, • • •, AT), must be processed without interruption during a period of length OPDj. A machine can handle no more than one job at a time, and is continuously available from 8We thank David Williams for his comments on this point. Chapter 3. SilverScreener: A Decision Support Model 106 time zero onwards. Each job j has a release date rj, and a due date dj, the time by which it should ideally be completed.9 We are asked to find an optimal feasible schedule, that is, a set of start times such that the capacities, availability and time limit constraints are met, and a given objective function is optimized. A special case of these problems is when all of the parallel machines are identical in terms of their capacities, and the processing period is restricted to be at least OPDj. When applied to the exhibitor problem, the analogy of this case is obvious: the (assumed) equal capacity screens are machines and movies are jobs. Each job (i.e., movie) has its own release date rj. Except for a few special cases, we assume that all the movies have a common due date, end-of-the-horizon, W. However, the framework is flexible to incorporate movie specific due dates. This flexibility allows the exhibitor to book certain slots for pre-commitments and still arrive at an optimal feasible schedule for the rest of the planning horizon. However, we do not explicitly model such effects in this paper. The difference between the exhibitor problem and machine scheduling problem is that all movies do not have to be played in the exhibitor problem, while all jobs have to be scheduled in the machine scheduling problems. Accordingly, the machine scheduling problems also have different objectives, such as, minimum tardiness, lateness, and so forth. We now introduce the time-indexed formulation of the scheduling problems that is particularly useful for solving the exhibitor problem (Sousa and Wolsey 1992, Williams 1997). This formulation is based on the idea of dividing the planning horizon [0, • • •, W] into W discrete intervals of unit length. It introduces binary variables XjW, where XjW = 1 if job j starts in period w, and 0 otherwise. The time-indexed formulation can also be generalized to encompass such extensions as screen capacities, precedence constraints between jobs, and movie specific due dates. 9 T h e n o t i o n of due date is i m p o r t a n t i n the cases w h e n customer places a n order before the c o m p l e t i o n of a job, or the machine has to b e m a d e available for a m o r e c r u c i a l j o b at a specified date. Chapter 3. SilverScreener: A Decision Support Model 107 To model the screen management problem, we define a binary variable, XjiW, which equals 1 if movie j is shown for i weeks beyond its obligation period starting in week w of the planning horizon, and 0 otherwise. Notice that the obligation period constraint is included in the definition of Xjiw itself. For example, if obligation period of Movie Number 3 is 2 weeks, then x^oi = 1 implies that it is shown for 2 weeks starting in week 1. The definition of XjiW also suggests that the variables corresponding to the scheduling of the same movie for different play lengths and different start times represent different jobs. We define PjiW, the total profit corresponding to each XjiW, as: P- — w+SCRji-1 X ; (-FCU + POPju - VCju) + hiu * {0.1 * {GROSSju - C) + C} u=w + (1 - J„.J *FLOORju * GROSSju, j = 1, • • • , N, i = 0, • • • , kj, w = rj, • • •, dj — SCRji + 1. where, kj = dj — Tj — OPD j-\- \ = maximum possible number of weeks movie j can be shown beyond its obligation period starting in rj or any feasible week thereafter, SCRji = OPDj '+ i = total screening period for movie j if it is shown for i weeks beyond its obligation period, where i = 0, • • • , kj. The variables kj and SCRji h e lp us "cover" all feasible (P,x) pairs for a movie. For example, suppose movie j has parameters: OPDj = 2,r,- = 1, and dj = 4. Thus, if the movie is scheduled in week 1, then it can be shown for 2, 3 or 4 weeks. Similarly, if it is scheduled in week 2, then it can be shown for 2 or 3 weeks. Finally, if it is scheduled in week 3, then it can be shown only for 2 weeks. Table 3.11, constructed for the above example, shows how the variables kj and SCRji parsimoniously capture the desired effect. Chapter 3. SilverScreener: A Decision Support Model 108 The expression for profit above is based upon the one-period reward function pre-sented in the previous section. This operationalization of P ^ ' s simplifies the solution procedure considerably since they can now be computed independently of the optimiza-tion routine. The time-indexed formulation of the exhibitor problem is as follows: N kj dj-SCRji+1 m a x E E E PjiwXjiw (3.14) j=l i=0 VJ=Tj subject to kj dj-SCRji+1 E E i = l , - . . , i V (3.15) t=0 W=Tj N kj w E E E ^j<H, w = l,...,W,- (3.16) j=l i=0 qj=w-SCRji+l rj <qj <dj- SCRjt + 1, j = l,...,N; t = 0 , . . . , fy , (3.17) xjiwe{0,l} (3.18) In the above model, Constraint 3.15 ensures that a movie is played in only consecutive weeks. The next constraint restricts the total number of movies scheduled in any week of the planning horizon to the total number of screens in the multiplex. In doing so, it sums up all the movies which are released earlier than or in the week under consideration. The set of inequalities 3.17 is an indexing constraint, which restricts the variable qj in Constraint 3.16 to feasible values. Constraint 3.18 defines XjiW to be a binary variable. We coded the above model in A M P L (Fourer, Gay, and Kernighan 1993), a modeling language for mathematical programming. A M P L uses a mixed integer programming code C P L E X as a solver. Our empirical analysis results, which we will present in the next section, have been obtained using the code in A M P L . The analysis was conducted on an Intel Pentium class computer. A typical small size application involved 6117 variables and 65 constraints, after the presolver of the code eliminated a few redundant variables and constraints. The time taken to solve such problems was of the order of a few seconds. Chapter 3. SilverScreener: A Decision Support Model 109 3.3 Normative vs. Actual Decision Making: A Case Study of 84th St. Sixplex in New York 3.3.1 Data Description The 84th St. Sixplex is a six screen theater, located at 84th St. and Broadway in New York City's Upper West Side. We collected the empirical data for this theater from Variety for the summer of 1989. We chose this theater for the following reasons. It is a reasonable size theater, and usually plays first run movies. It can be considered a monopoly theater in its vicinity. 1 0 Moreover, it is one of the few theaters in New York whose data for the year 1989 are publicly available in Variety. The year 1989 was chosen because the contract terms of a major movie in 1989, Batman, are available from Squire (1992). Since Batman was played by this theater, we use its contract terms as representative of those of the other movies. This assumption is not limiting since, as we will show in our sensitivity analysis section, variations in contract terms do not materially affect our results. The following information is available to the exhibitor for making scheduling deci-sions. The exhibitors knows the revenue data for all the movies that have played at his/her theater. Variety regularly published until 1989 a theater-specific weekly box-office revenue data for a sample of theaters from New York and Los Angeles, and a few other major cities in the U.S. Besides the revenue data for the movies that have already played, the exhibitor may also have an estimate or forecast for the movies to be released in the future. Variety also publishes a one-week ahead box-office revenue forecast of the movie the theater is planning to show. The exhibitor receives a formal letter in advance from the distributor of a movie which specifies the contract terms of a movie. Variety publishes screen specific house nut amount (one of the contract term parameters) for 10Except for a small theater with two screens and less than 20% of the Sixplex's capacity, there were no other first run theaters north of 70th St. on Manhattan's West Side. Chapter 3. SilverScreener: A Decision Support Model 110 the theaters such as the 84th St. Sixplex. Finally, the exhibitor may also have access to the box-office performance of a movie nationally. Such data are published by Variety on a weekly basis. The "national" is defined in terms of a sample of 2500-3000 screens. It reports Top 50 movies in terms of their weekend box-office revenue. It also reports average revenue per screen for such movies. Additionally, the exhibitor may also have some in-house data sources. Specifically, the data items used in the following analyses and their respective sources are as follows: 1. Schedule of the movies actually played by the theater (Source: "Domestic Box Office" data from Variety) 2. Gross revenues generated by the movies actually played at the theater (Source: "Domestic Box Office" data from Variety) 3. One-week ahead gross revenue forecasts for the movies actually played at the theater (Source: "Domestic Box Office" data from Variety) 4. Availability and sample average gross revenue data for the movies not played by the theater (Source: "50 Top Grossing Films" data from Variety) 5. House nut amounts (Source: "Domestic Box Office" data from Variety) 6. Costs (variable and fixed) and concession profits (Source: "Financial Statements" of major theater chains from Security and Exchange Commission (SEC) filings on the Internet) Table 3.12 shows the list of the movies shown at the theater. We term it the "restricted (consideration) set" of movies since we restrict ourselves to only the movies actually chosen by the exhibitor. Some movies released during the summer of 1989 appeared in Chapter 3. SilverScreener: A Decision Support Model 111 the "Top 50 List of Movies" in Variety, but were not chosen by the exhibitor. We term the set of movies obtained by appending these additional movies to the restricted set the "expanded (consideration) set." In other words, this is the "full" consideration set the exhibitor would have used in developing the restricted set choices. The expanded set is shown in Table 3.13. There are 43 movies in the restricted set, and 87 movies in the expanded set. We represent all the movies in Table 3.13 by suitable acronyms for ease of presentation. The actual schedule followed by the exhibitor for the 27 weeks of the planning horizon is given in Table 3.14. As shown in the table, the theater showed a total of 43 different movies. There are a number of movies with play length of one week or two weeks11. In addition, three movies, Indiana Jones and the Last Crusade (IJ), Star Trek V (ST5), and Batman (B), were double-booked. That is, the same movie played simultaneously on two screens for some period of time in its run. In such cases, the algorithm treats this as if there were two different movies. There is only one instance of "cycling"- a movie returning on one of the screens after being replaced once - in the case of Dead Poets Society (DPS). We treat it as a special case and assume the second run of DPS to be a different movie. The three double booked movies and the second run of DPS increase the number of movies in both the restricted and expanded sets by four. Thus we have 47 movies in the restricted set, and 91 movies in the expanded set for data analysis. In the analysis that follows, we compare the actual schedule to the schedules based on our algorithm using restricted and expanded consideration sets respectively. We assume the obligation period to be two weeks for all the movies in these consideration sets. This is the minimum of the play lengths of most movies in the actual schedule12. The revenue 1 1 Generally, the one-week movies are relatively low revenue movies, and may have been used by the manager as "filler" movies. Therefore, in our algorithm's application, we use a common obligation period of two weeks, the minimum play length of the rest of the movies. 1 2Though the actual case schedules a number of movies for only one week, we proceed with a two-week obligation period for the sake of consistency. It is clear that this extra restriction works against Chapter 3. SilverScreener: A Decision Support Model 112 sharing terms for the movie Batman in 1989 were as specified in the previous section. The house nut amount for all the screens of 84th St. Sixplex was $14,500 in 1989 (see Variety 1989). We have no public information available on the variable and fixed costs, and concession profits for this theater for the year 1989. Therefore, we used the financial statements of the four major theatrical chains, A M C , Cineplex Odeon, United Artists, and Carmike, to examine how their cost data vary with box-office revenue. The data are available on the Internet and have been collected from the Security and Exchange Commission (SEC) filings of these theatrical companies. We find that, over a number of financial statements (both quarterly and annual) of these companies, the operating costs vary from 56% to 66% of box-office revenue. The concession profits vary from 30% to 40% of box-office revenue. These ranges conform with the industry standards available from other sources such as Squire (1992). To present results for a representative case in the comparative analysis, we assume operating costs of 66% and concession profits of 40% 1 3. The break-up of operating costs into fixed and variable costs is not available from the financial statements. We make an assumption that a half of the operating costs is fixed cost and the other half is variable cost. Thus, for the representative case, fixed and variable costs are each 33% of the box-office revenue. While the variable cost varies with weekly revenues of different movies, the fixed cost has been assumed to stay constant from week to week. Since one of the our major criteria of comparison is cumulative profit over the season, we generate cumulative net revenues for various comparison cases which include variables that vary with weekly revenues and movies, that is, concession profit, variable cost, share of the box-office revenue, and house nut. We then subtract from the cumulative net revenue the cumulative fixed cost over the season which is common across our comparison approaches, and our profit results will be conservative to this extent. 1 3In the sensitivity analysis section to be presented later, we examine several other values of the operating costs and concession profits ranges to check the robustness of our results. Chapter 3. SilverScreener: A Decision Support Model 113 all the comparison cases. To calculate the cumulative fixed cost, we use the cumulative box-office revenue for the actual case of all the movies that played at the theater during the season. 3.3.2 Comparison Approaches As mentioned above, we compare the actual schedule with the schedules recommended by our model using the restricted and expanded data sets separately. Accordingly, we have two different cases for comparison, which we now explain. Restricted Set Approach In this approach, we consider only those movies that were actually played at the theater, that is, the restricted set movies. In order to compare the schedule produced by our model with the actual decisions of the exhibitor, we need to predict box-office gross revenues for the weeks beyond the actual play length of a movie. We fit the linear regression version of the demand model introduced in Equation 3.11 of the previous section after log-transforming their gross revenues. After deleting two "outlier" movies, an average adjusted R was found to be 0.92 for the movies tested.1 4 The revenue of the two outlier movies first increases for a brief period and then exponentially decreases. We could possibly use a three-parameter Generalized Gamma distribution to model the revenue data for this type of movies (Sawhney and Eliashberg 1996). However, it is not necessary for our purposes since the increase in revenue occurs in the first two weeks when these movies are still in their obligation period. The revenue is exponentially declining by the end of their play length. Therefore, we simply fit the two-parameter model for the exponentially declining portion of the revenue function of such movies. It is obvious 1 4 The average adjusted R2 does not include the two outlier movies or the movies that played only for 1 week. It also does not include the movies that played for exactly 2 weeks, since their R? value is equal to one and would innate the average R2. Chapter 3. SilverScreener: A Decision Support Model 114 that this approach is applicable only to the restricted set movies that played longer than one week. To generate revenue estimate for the later weeks of the movies that actually played only for one week at the theater, we use a median decay rate across all the movies in the restricted set. From here onward, we use the term "optimal restricted set schedule (approach)" to denote the schedule produced by our model using the revenue data generated in the above fashion. Expanded Set A p p r o a c h In this approach, we consider the movies from the expanded set, that is, the movies that were played at the theater as well as other movies that were released during the summer of 1989 but were not played at the theater. In the case of the expanded set, we need revenue predictions for the weeks beyond the play length of a movie that was shown at the theater, and for all the weeks of a movie not shown at the theater. In the case of the movies that were shown at the theater, similar to the restricted set approach, we use the two-parameter exponential model for estimation. However, this model cannot be applied to those movies of the expanded set that were never scheduled by the theater. For such movies, we collected their "Average Per Screen" gross revenue data from the "Top 50 List" published by Variety every week. This average is calculated on a sample of 2,500 to 3,000 screens (thereby 10-12 % of the screens in North America) in 17 to 20 major and medium metropolitan domestic market areas. Since the 84th St. Sixplex is fairly representative of such theaters, we expect that, in any given week, the box-office gross of a movie there should be highly correlated with the corresponding Average Per Screen data. Moreover, analysts at Standard Data Corporation, New York, which compiles this data, suggested that "• • -the "Average Per Screen" column often can be more meaningful than aggregate b.o. [box office] dollar total for "This Week" [another entry in the data] • • •". Accordingly, we pool the weekly revenue ( W K R E V ) for all the weeks of the restricted set movies and Chapter 3. SilverScreener: A Decision Support Model 115 regress it on the corresponding "Average Per Screen" revenue (APSREV) . After deleting one outlier observation15, we have 161 observations16. Simple linear regression resulted in an adjusted R2 of 0.80. We, therefore, used the regression-based estimate as an estimate of the revenue of a movie not shown at the 84th St. Sixplex theater. The estimated regression equation was WKREV = 10117.3 + 2.22 APSREV t = (10.113) (25.57) Our comparative analysis approaches from the above discussion can be summarized as follows: • Restricted Set Approach - use actual revenue data for the weeks of a movie that is shown at the theater. For the weeks beyond, generate the revenue estimates using the two-parameter exponential model. • Expanded Set Approach - use actual revenue data for the weeks a movie that is shown at the theater. For the weeks beyond of such movies, generate the revenue estimates using the two-parameter exponential model. Use the national box-office based regression estimates for the movies that are not shown at the theater. 3.3.3 SilverScreener's Normative Results The results presented in this section were obtained by the implementation of SilverScreener-IP algorithm on the empirical data collected. The optimal schedules generated by SilverScreener-IP for the restricted and expanded sets are given in Tables 3.15 and 3.16 1 5 The outlier observation corresponds to the first week observation of the movie DPS (Dead Poets Society), which was released only on 8 screens in its first week. Therefore, its first-week average per screen revenue was much more than the national figures on average. 1 6Total observations (including the outlier movie) = 27 weeks * 6 screens = 162. Chapter 3. SilverScreener: A Decision Support Model 116 respectively. As shown, the output of the algorithm is a set of movies that the manager should adopt for each week of the planning horizon. The algorithm also provides the profitability associated with the recommended schedule. A summary characterization of the results for the two approaches is provided in Table 3.17. Restr ic ted Set A p p r o a c h The results presented in Tables 3.15 and 3.17 suggest the following: • The use of the SilverScreener optimization approach results in the exhibitor earning a higher cumulative profit than for the actual schedule, • The percentage improvement for the exhibitor in profit terms is 37.7% in this case. The improvement figure compare favorably with similar recent studies in marketing such as Reddy, Aronson, and Stam (1998) which reports a 2% improvement in overall profits in the context of T .V . spot scheduling. Moreover, our profitability improvements are achieved despite the fact that the restricted set-based analysis is the most stringent test, given the data available, of our optimization approach. Recall, we use a reduced consideration set, which includes sub-optimal choices of movies made by the exhibitor. We only allow double-booking for the three movies that the theater double-booked. Moreover, we use the exponentially declining model for the extrapolation of revenue streams, which always forecasts decreased revenues in future weeks. A n improvement of 37.7% in the restricted set case, therefore, attests to the potential effectiveness of our approach in rather unconstrained cases. • In the restricted (and also expanded set) approach, an improvement over actual decisions is achieved from two sources. One is better selection, that is, choosing the "right" movie to show. The other source of improvement is better scheduling, that Chapter 3. SilverScreener: A Decision Support Model 117 is, deciding the "right" 1 7 run-length of the movie chosen. It is clear from the results of both approaches that it is a combination of both better scheduling and selection, and not one of them alone, that drives the results. In the case of the restricted set, the improvement from better selections is bounded by the set of choices made by the exhibitor. • As shown in Table 3.17, the nature of the optimal policy is to choose fewer "right" movies and run them longer. Expanded Set Results The results presented in Tables 3.16 and 3.17 suggest the following: • The use of the SilverScreener optimization approach in expanded set approach results in the exhibitor earning a higher cumulative profit than for both actual and restricted set based schedules. The percentage improvement for the exhibitor in profit terms is 121.2% in the expanded set case. • As mentioned above, the improvement results from the algorithm's ability to both select and schedule the right movies. The nature of the optimal policy is also preserved in this case, that is, to choose fewer "right" movies and run them longer. A n interesting observation in this respect is that even though the expanded set approach considers 87 movies, the optimal expanded set approach schedules only 25 movies, even fewer than 27 scheduled by the optimal restricted set approach. • Taken as a set, the distributors' net revenue increase as compared to the actual returns by 20.5% in the case of the expanded set. Though we observe a slight drop in this figure in the case of restricted set, it is relatively insignificant (-0.27%). While 1 7 " R i g h t " is s t r i c t l y i n t h e sense o f o p t i m i z a t i o n c r i t e r i o n f o l l o w e d b y t h e m a n a g e r . Chapter 3. SilverScreener: A Decision Support Model 118 not all distributors gain individually from this one theater's improved scheduling, this implies that a possibility of "win-win" situation exists from an overall system standpoint. • The optimal expanded set schedule retains 15 out of 27 movies from the restricted set. Although there may be some external constraints that limit managerial flexibil-ity, an improvement in decision making is consistent with Behavioral Decision The-ory (BDT) (Slovic, Fischhoff and Lichtenstein 1977), which indicates that human judgments are subject to various types of biases under complex and/or uncertain scenarios18. Of course, the model should be best used in conjunction with man-agerial judgments, as happened with the ARTS P L A N model (Weinberg 1986) for planning a performing arts series. Huber (1975) also suggests that "bootstrapping" judgments are generally more accurate than the original judgments. Indeed, our model significantly improves upon actual decisions while retaining approximately 50% of managerial choices. Managerial decisions may gradually converge with the model's recommendation as the model is used over an extended period of time. Thus, our approach seems ideally suited to situations in which the model "evolves" with usage (Weinberg 1986). • When choices are limited, as in the restricted set, the optimal approach is to run the major double-booked movies for a longer period of time. Wi th more flexibility in the expanded set, however, the optimal schedule has reduced the double booking length. To summarize these points, the use of optimization approaches like SilverScreener holds promise to improve managerial decisions in the movie exhibition business. The 1 8Though it can be argued that some of the other movies scheduled in the expanded set approach may be playing in the other theaters, our model does not address such competitive effects. Chapter 3. SilverScreener: A Decision Support Model 119 proposed approach can be applied to realistic decision making in two different ways. The first is for developing a Master Plan. Conversations with theater owners indicated that exhibition executives often develop a master schedule several months before the start of the summer season. The master schedule helps them bid for some movies before the start of a season. It also helps them make weekly adjustments and scheduling decisions later on during the season. A one-shot application of SilverScreener, as presented in the previous sections, can help the exhibition executives develop a Master Plan before the start of a season. The exhibitor may refer to the historical data of the box-office grosses at his/her theater to generate forecasts for the movies yet to be released. In doing so, he/she can relate a particular movie to a previously played movie on the basis of perceived similarity between the two in terms of the actors, producer, director, movie genre, and so on. SilverScreener can then be easily applied by converting these perceptions into a relative measure of revenue potential with a common base. Alternatively, the exhibitor may use his/her own judgment to forecast the movie revenue. The weekly box office gross revenue prediction data published in Variety suggest that some exhibitors have fairly reasonable estimates of movie revenue potentials at their theaters. These one-week ahead forecast data are published weekly in Variety for a set of theaters in New York (including 84th St. Sixplex) and Los Angeles. This forecast is for every movie a theater is showing or is planning to show1 9. The second instance of SilverScreener's application can be for weekly decisions, possibly after the development of the Master Plan. We term this adaptive application the Rolling Horizon Approach since it involves "rolling" from one time window (i.e., a week) to another. Based on these ideas, we discuss a two-tier application of our model in Chapter 4. 19These one-week ahead forecasts appear to be quite accurate. We found an overall correlation between the actual and predicted data of 0.96 for the movies shown at 84th St. Sixplex in 1989. Chapter 3. SilverScreener: A Decision Support Model 120 3.3.4 Sensitivity Analyses The results obtained in the previous section depend on our assumptions about cost data, the estimates of the future revenue streams and certain contract terms. We perform sensitivity analysis on each of these parameters to check the robustness of our results. The results presented in this section are based on the restricted set data only; an analysis on the expanded set will preserve the general direction of these results. Fixed and Variable Costs, and Concession Profits We mentioned earlier that based on a number of financial statements (both quarterly and annual) of four major theatrical companies, we find that the operating costs vary from 56% to 66% of box-office revenue, and the concession profits vary from 30% to 40% of box-office revenue. Considering the extreme points, this gives us a 2 X 2 matrix to consider a range of the values of operating cost and concession profit. Since the financial statements do not supply the break-up of operating cost into fixed and variable costs, we consider three different cases. Case A assumes that the operating cost is totally variable. Case B assumes that 50% of the operating cost is fixed and 50% is variable. Finally, Case C assumes that the operating cost is totally fixed and there are no variable elements. These three different assumptions about the split of operating cost, and values for operating costs and concession profits, result in twelve different cases to be examined in this analysis. For each of these cases, we calculate the profit earned by the exhibitor in the actual case and using the optimal restricted set approach. The results are presented in Table 3.18. As the results show, over a range of assumptions about operating costs and con-cession profits, the optimal approach shows a wide range of improvement over actual profits. In only one case was the profit improvement slightly below 20%. The highest Chapter 3. SilverScreener: A Decision Support Model 121 (lowest) level of improvement results when the operating costs are at their higher (lower) level and totally variable, and the concession profits are at their lower (higher) level. In general, the fully variable cost cases show a greater variation in improvements than other cases. Gross Revenue A plausible way of conducting sensitivity analysis on the gross revenue would be to systematically increase or decrease it by a fixed percentage for all the movies. Since this systematic change would only rescale the objective function, it is obvious that policy recommendations will not change. Adding a reasonable random error term is another possibility for conducting sensitivity analysis of the gross revenue. Instead of using a standard stochastic error distribution, we use more realistic data, published in Variety, based on the exhibitor's explicit forecasts for box-office gross for the following week. Using these data, we can examine the impact of forecasting errors on the optimal policy and the improvement over actual decisions. The optimal schedule for this problem is quite similar to the one obtained in the restricted set case. Therefore, we simply report a summary of the results in Table 3.1920. The results of the actual schedule and optimal restricted set approach are also presented in Table 3.19 for comparison purposes. There is a slight decrease in the percentage improvement in profit terms over the actual case as compared to the restricted set approach. However, it is clear from the summary given in the table that policy results are similar to those reported in the previous section. Thus, our results are quite robust for such random variations in the revenue data as considered in this section. 2 0 The last column of Table 3.19 reports the results of obligation period sensitivity analysis, which we present later. Chapter 3. SilverScreener: A Decision Support Model 122 Obl igat ion P e r i o d In our earlier analyses, we assumed an obligation period of two weeks for all the movies in the consideration sets. However, it might be reasonable to expect that some stronger movies have a higher obligation period attached with them. To conduct sensitivity anal-ysis on obligation period based on this reasoning, we compute the cumulative box-office revenue generated over the play length of each movie that played at the theater. We treat the cumulative revenue of a movie as an indicator of its strength. We sort these data and median split the data into two halves. We assume the obligation period of those movies that have cumulative revenue below the median to be two weeks, and of those above the median to be four weeks. The results of this analysis are presented in the last column of Table 3.19. The results show that the added obligation period restriction causes slight decrease in profitability. It also causes the resulting schedule to select fewer (25) movies than those scheduled by optimal restricted set schedule. However, still 12 out of those 25 movies are scheduled for at least four weeks. Therefore, the results are quite comparable with optimal restricted set schedule in both profitability and policy terms 2 1. Revenue Sharing Terms The sensitivity analysis of revenue sharing terms is essentially equivalent to varying "minimum" percentage figures for the different weeks. This is because the "90%/10%" part of a contract is fixed across weeks. Moreover, the house nut amount of $14,500 is the same for all the screens at 84th St. Sixplex. We contacted a major theater chain for a sample of revenue sharing terms it used in 199622. We conducted sensitivity analysis on 2 1 We conducted another analysis on obligation period by varying its value from one week to five weeks for all movies. We do not present the results here, but they suggest that as the obligation period increases, the optimal schedule selects fewer movies and runs them longer. Since an increased obligation period implies reduced flexibility for the exhibitor, the profit is lowered. These results are useful in explaining the effect of obligation period on policy recommendation. 2 2For reasons of confidentiality, the name of the chain cannot be revealed. Chapter 3. SilverScreener: A Decision Support Model 123 the four major type of revenue sharing terms used by that theater, which covered 83% of the movies played in that theater during the summer of 1996. Table 3.20 shows the results of this analysis. In addition to the above four types of terms, we also include the results of the terms for the movie Batman which were used in the previous sections. The five different types of terms were used for calculating both the optimal and actual cumulative net revenues. The first two rows of Table 3.20 show the comparative results. The optimal approach does consistently better than actual decisions across all five contract terms. The percentage improvements in profitability are between 15% and 38% in all the cases. The results are quite comparable on other parameters. The main policy difference appears to be that if the exhibitor gets more favorable terms in the first few weeks, then he/she will schedule more movies for shorter runs. To summarize, the SilverScreener model leads to an improvement in profitability for the exhibitor under a broad range of parameter estimates, cost assumptions, contract terms, and decision-making structures. The prescriptive advice to the manager, as com-pared to current practice, is to concentrate on a smaller number of movies, which are selected to maximize profits, and consequently to play these movies for a longer period of time. Choice of which specific movies to play is, of course, a critical decision and the scenarios we investigate lead to recommendations of different sets of movies, played for different lengths of time. The results presented-along with the flexibility and speed of the algorithm developed here-however, suggest that the improvement to profit from using SilverScreener is readily attainable in a particular setting. In the next chapter, we show ways in which the model could be used by managers to improve decision making. Chapter 3. SilverScreener: A Decision Support Model 124 3.4 Conclus ion 3.4.1 Contr ibut ions Like other retail space allocation decisions, the choice of which movies to schedule in a theater is a complex one, involving trade-offs among numerous alternatives and with implications for both current and future profits. The movie industry poses some par-ticularly challenging opportunities for managers (and for modelers) as new products are introduced every week, and the attractiveness of existing products decays systematically and usually rapidly over time. Moreover, contract terms are complex and designed in ways to favor the distributors, making the choice of the most profitable set of movies to schedule difficult for the exhibitors. Using a dynamic programming model, the special structure of the underlying problem, an analogous production scheduling problem, and a powerful, but readily available, mathematical programming language, we are able to structure the exhibitor's problem and to derive optimal solutions which lead to substan-tially improved profits. Using the most representative data, SilverScreener leads to an 37.7% improvement in profit when the exhibitor is restricted to the movies actually scheduled and 121.2% when the exhibitor's consideration set includes movies that were nationally available, but not scheduled in the particular theater studied. Given the weekly publication of actual revenue data (as well as forecasts of revenue data) as well as both private and public data on contract terms, we are able to show that these improvements are quite robust. In particular, since weekly a priori forecasts of attendance by theater were reported in Variety for the period studied, the improvements appear to be attainable. Moreover, the algorithm can be solved in only a few seconds of C P U time, suggesting that SilverScreener can be incorporated into real-time decision-making situations. Chapter 3. SilverScreener: A Decision Support Model 125 3.4.2 L imi ta t ions and Future Research A n interesting line of future research concerns how the SilverScreener model would be used by managers and how its usage would change the decision-making environment. As reported in Weinberg's (1986) discussion of ARTS P L A N , the issue is not model vs. manager, but how model and manager interact. In the current work, we view the manager as using the model at two points in time. The first is before a season starts when the manager seeks to develop an overall schedule for (say) the summer season. This situation is quite analogous to the use of ARTS P L A N , as a scheduler before the season begins. Unlike ARTS P L A N , however, only some of the commitments are binding; the others are subject to change. This leads to the second proposed usage of SilverScreener, as a weekly planning and control device. In the movie industry, typically on Monday, the theater owner receives the weekend grosses for the theater and has to decide whether to retain the current movies (not in the obligation period) or drop them in favor of other movies. SilverScreener,can use both the objective and subjective data that the manager has available-as well as a forecasting routine—to suggest an optimal movie schedule for the upcoming week. These issues are considered in the next chapter2 3. Another area of future research concerns competition. A combination of historical relationships with movie distributors (although limited by antitrust law) as well as short term contracts may limit the availability of movies to theaters. In addition, attendance estimates may vary depending on which movies are scheduled at nearby theaters. These concerns, and similar ones, can be readily incorporated into the SilverScreener algorithm. More challenging, however, would be dealing with the strategic interplay of theaters each attempting to optimize its own revenues. To date, there are only limited results available on strategies available to multiproduct retailers in environments as complex as this, 2 3 We also discuss the relaxation of deterministic revenues in the next chapter. Chapter 3. SilverScreener: A Decision Support Model 126 so that considerable simplification would be required before analytic results could be obtained. A limitation of the current research is regarding the assumed equal capacity screens. In practice, the exhibitors have different capacity screens so that they can "switch" their movies from one screens to another. We do not model this aspect explicitly in this study because of the unavailability of the right kind of data in this regard. To elaborate on this point further, let us consider how we could model the "different capacity" case in the exhibitor problem. In the SilverScreener-IP problem, for example, we would define a new binary variable, Xji3W, which equals 1 if movie j is shown for i weeks on screen s starting in week to, and 0 otherwise. The corresponding profit, PjiSW, depends on GROSSjU variable, that is, the relevant box-office gross revenue, which would have to be modeled differently for the different capacity case. In this case, it would have to be recast as min(MAXs, GROSSjU), where MAXS is the maximum weekly revenue any movie can earn on screen s if that screening room is filled to its capacity 2 4. Such a reformulation of gross revenue would allow for a movie's revenue potential to be greater than the maximum revenue capacity of a screen, in which case it may be scheduled on a higher capacity screen. As in the current problem, this decision will depend on other factors also, such as, availability of more attractive movies that are better suited for the higher capacity screens. A related future research idea in this effort would be modeling of micro-scheduling issues such as scheduling different shows within a day on different capacity screens. While these are important issues, we find that the equal capacity screens assumption gives us a reasonable starting point. Finally, a third extension would be to consider an exhibitor who owns multiple the-aters. In fact, this is the common practice, and in any one regional area, typically most theater screens are controlled by a few chains. Such a setting may allow the theater ^In our empirical application, we do not have information available on the MAXS variable. Chapter 3. SilverScreener: A Decision Support Model 127 owner to optimize over multiple sites, so that, for example, movies with limited audi-ences could be profitability scheduled at only a small number of the chain's theaters. Similarly, policies concerning multiple booking of movies within a single theater and innovative booking policies might be more readily undertaken when a system-wide ap-proach is employed. To the extent that our algorithm was to encompass system-wide effects as compared to merely aggregating individual theaters, extensive changes in the model would be required to capture these effects. To the best of our knowledge, despite the dominance of national chains in a vast array of specific retail categories, there is very limited modeling available to guide managers when considering system-wide impacts of shelf space decisions in individual stores. Yet, clearly individual stores in a chain carry different product assortments. The movie industry would appear to be a promising area in which to begin such analyses. Chapter 3. SilverScreener: A Decision Support Model 128 Table 3.11: Parameter Values of Example Problem in Section 3.2 > I SCR„ w Corresponding (P,x) pairs 0 2 1, 2, 3 (PjOhXjoi), (Pj02,Xj02), (Pj03,Xj03) 1 3 1,2 (PjU.Xjll), (Pjl2,Xji2) 2 4 1 (P,2I,X,2l) * - The value of kj is 2. Therefore, / = 0,...,2. Chapter 3. SilverScreener: A Decision Support Model Table 3.12: Restricted Consideration Set 129 Movie Title Movie Title Number Number 1 New York Stories 2 Chances Are 3 Dangerous Liaisons 4 Lean on Me 5 Police Academy VI 6 Leviathan 7 Dead Bang 8 Heathers 9 Sing 10 Major League 11 Dead Calm 12 The Accused 13 Disorganized Crime 14 See You in the Morning 15 Lover Boy 16 Pet Semetary 17 Speed Zone 18 Scandal 19 Miss Firecracker 20 See No Evil, Hear No Evil 21 Earth Grids Are Easy 22 For Queen and Country 23 Pink Cadillac 24 Indiana Jones and the Last Crusade 25 Dead Poets Society 26 No Holds Barred 27 Star Trek V 28 Batman 29 Honey I Shrunk the Kids 30 Lethal Weapon II 31 The Package 32 Peter Pan 33 Night Game 34 License to Kill 35 Lock Up 36 Turner and Hooch 37 Friday the 13th VIII 38 A Nightmare on Elm Street V 39 UHF 40 Rude Awakening 41 Cookie 42 Let It Ride 43 Relentless Chapter 3. SilverScreener: A Decision Support Model Table 3.13: Expanded Consideration Set 130 Movie Title Movie Tide Number Number 1 New York Stories (NYS) 2 Chances Are (CA) 3 Dangerous Liaisons (DL) 4 Lean on Me (LOM) 5 Police Academy VI Q?A6) 6 Leviathan (LTHN) 7 Dead Bang (DB) 8 Heathers (H) 9 Sing (SING) 10 Major League (ML) 11 Dead Calm (DC) 12 The Accused (TA) 13 Disorganized Crime (DOC) 14 See You in the Morning (SYM) 15 Lover Boy (LB) 16 Pet Semetary (PS) 17 Speed Zone (SZ) 18 Scandal (SDL) 19 Miss Firecracker (MF) 20 See No Evil, Hear No Evil (SNEHNE) 21 Earth Girls Are Easy (EGE) 22 For Queen and Country (FQC) 23 Pink Cadillac (PC) 24 Indiana Jones and the Last Crusade (IJ) 25 Dead Poets Society (DPS) 26 No Holds Barred (NHB) 27 Star Trek V(ST5) 28 Batman (B) 29 Honey I Shrunk the Kids (HISK) 30 Lethal Weapon II (LW2) 31 The Package (TP) 32 Peter Pan (PP) 33 Night Game (NG) 34 License to Kill (LTK) 35 Lock Up (LU) 36 Turner and Hooch (TH) 37 Friday the 13th VIII (FTS) 38 A Nightmare on Elm Street V(NES5) , 39 UHF (UHF) 40 Rude Awakening (RA) 41 Cookie (C) 42 Let It Ride (LIR) 43 Relentless (RL) 44 Sea of Love (SOL) 45 Fletch Lives (FL) 46 Skin Deep (SD) 47 Rain Man (RM) 48 The Rescuers (TR) 49 Bill & Ted's Adventures Q3TA) 50 Roof Tops (RT) 51 Cousins (CSN) 52 Working Girl (WG) 53 Troop Beverly Hills (TBH) 54 The Dream Team (TDT) 55 Cyborg (CBRG) 56 Adventures of Baron Munchausen (ABM) 57 Say Anything (SA) 58 She's Out of Control (SOC) 59 Winter People 60 Red Scorpion (RS) 61 Field of Dreams (FD) 62 K-9 (K-9) 63 Criminal Law (CL) 64 The Horror Show (THS) 65 Listen to Me (LTM) 66 Lost Angels (LA) 67 Road House (RH) 68 How I Got Into College (HGC) 69 Miracle Mile (MM) 70 Renegades (RG) 71 Ghost Busters II (GB2) 72 The Karate Kid III (KK3) 73 Do the Right Thing (DRT) 74 Great Balls of Fire (GBF) 75 Weekend at Bernies (WB) 76 When Harry Met Sally (WHMS) 77 Shag (SHAG) 78 Parenthood (PH) 79 Young Einstein (YE) 80 Uncle Buck (UB) 81 Casualities of War (COW) 82 Cheetah (CH) 83 The Abyss (TA) 84 Sex, Lies and Videotape (SLV) 85 Wired (WIRED) 86 Adventure of Milo & Otis (AMO) 87 Kick Boxer (KB) Chapter 3. SilverScreener: A Decision Support Model Table 3.14: Actual Schedule WeekA Screen 1 2 3 4 5 6 1 NYS C A D L L O M PA6 L T H N 2 NYS C A D L L O M DB L T H N 3 D L NYS C A H DB SING 4 D L NYS M L H D C T A 5 DOC M L NYS H D C T A 6 S Y M M L DOC PS SZ H 7 S Y M LB M L SDL PS H 8 S Y M L B M L SDL PS M F 9 SNEHNE E G E M L SDL PS M F 10 SNEHNE E G E FQC SDL PS M F 11 E G E PC IJ IJ SDL SNEHNE 12 DPS NHB IJ IJ SNEHNE PC 13 DPS ST5 ST5 IJ IJ SNEHNE 14 DPS ST5 ST5 IJ IJ SNEHNE 15 IJ ST5 DPS B B HISK 16 IJ ST5 DPS B B HISK 17 IJ LW2 DPS B B HISK 18 L T K LW2 B IJ PP HISK 19 L T K LW2 B n UHF HISK 20 L T K LW2 B T H FT8 HISK 21 L T K LW2 B T H FT8 L U 22 L T K L U B T H LW2 NES5 23 R A LIR B L U LW2 NES5 24 LIR TP B L U LW2 C 25 DPS TP B RL LW2 C 26 DPS TP B RL LW2 C 27 R L TP B C LW2 N G Chapter 3. SilverScreener: A Decision Support Model Table 3.15: Optimal Schedule Using Restricted Set Data* 132 WeekY Screen 1 2 3 4 5 6 1 NYS C A D L L O M L T H N PA6 2 NYS C A D L L O M L T H N PA6 3 NYS C A D L L O M L T H N H 4 NYS D C D L T A M L H 5 NYS D C D L T A M L H 6 NYS D C PS T A M L H 7 NYS , D C PS T A M L H 8 SDL D C PS T A M L H 9 SDL D C M F T A E G E H 10 SDL D C M F T A E G E H 11 SDL D C M F IJ E G E IJ 12 SDL DPS MF IJ E G E IJ 13 SDL DPS ST5 IJ ST5 IJ 14 SDL DPS ST5 IJ ST5 U 15 B DPS B IJ HISK IJ 16 B DPS B IJ HISK U 17 B DPS B LW2 HISK IJ 18 B DPS B LW2 HISK L T K 19 B DPS B LW2 HISK L T K 20 B DPS B LW2 T H L T K 21 B DPS B LW2 T H L U 22 B DPS B LW2 NES5 L U 23 B DPS B LW2 NES5 L U 24 TP DPS B LW2 C L U 25 TP DPS B LW2 C RL 26 TP DPS B LW2 C RL 27 TP DPS B LW2 C R L * - Using the following cost assumptions - Variable costs are 33% of box-office revenue, concession profits are 40% of box-office revenue, fixed costs are 33% of the total box-office revenue in a week. Chapter 3. SilverScreener: A Decision Support Model Table 3.16: Optimal Schedule Using Expanded Set Data 133 Week\ Screen 1 2 3 4 5 6 1 NYS C A A B M R M FL L T H N 2 NYS C A A B M R M F L L T H N 3 NYS C A A B M R M F L H 4 NYS M L A B M R M FL H 5 NYS M L A B M R M SA H 6 FD PS A B M R M SA H 7 FD PS A B M R M SA SDL 8 FD PS A B M R M SA SDL 9 FD M F A B M R M SA SDL 10 FD M F A B M R M SA SDL 11 FD MF A B M IJ IJ SDL 12 FD DPS A B M U IJ SDL 13 FD DPS A B M D IJ ST5 14 FD DPS GB2 IJ IJ ST5 15 B DPS GB2 IJ FflSK B 16 B DPS DRT IJ HISK B 17 B DPS DRT IJ LW2 B 18 B DPS DRT WHMS LW2 L T K 19 B DPS DRT WHMS LW2 L T K 20 B DPS DRT WHMS LW2 L T K 21 PFI DPS DRT WHMS LW2 SLV 22 P H DPS DRT WHMS LW2 SLV 23 P H DPS DRT WHMS LW2 SLV 24 P H DPS DRT WHMS LW2 SLV 25 P H DPS DRT WHMS LW2 SLV 26 P H DPS DRT WHMS LW2 SLV 27 PH DPS DRT WHMS LW2 SLV * - Using the following cost assumptions - Variable costs are 33% of box-office revenue, concession profits are 40% of box-office revenue, fixed costs are 33% of the total box-office revenue in a week. Chapter 3. SilverScreener: A Decision Support Model Table 3.17: Characterization of Solution 134 Actual Optimal Optimal Schedule Restricted Set Expanded Set Schedule Schedule* Cumulative Profit ($'s) 585,175 805,988 1,294,408 Percentage improvement over . 37.7 121.2 actual Number of different movies 43 27 25 scheduled Average run length 3.8 6 6.5 (screening slots) Percentage scheduled for 1 week 18.6 (8/43) 0 (0/27) 0 (0/25) 2 weeks 27.9 (12/43) 11.1 (3/27) 20 (5/25) 3 weeks 14 (6/43) 22.2 (6/27) 16 (4/25) > 4 weeks 39.5 (17/43) 66.7(18/27) 64 (16/25) Double booking for Batman 3 weeks 9 3 Indiana Jones 4 6 4 Star Trek V 2 2 0 Screening slots for Batman 16 21 9 Indiana Jones 13 13 11 Lethal Weapon II 11 11 11 Dead Poets Society 8 16 16 New York Stories 5 6 5 Scandal 5 8 6 The Accused 2 7 -When Harry Met Sally - - 10 Adventures of Baron - - 13 Munchausen Field of Dreams - - 9 * - Using the following cost assumptions - Variable costs are 33% of box-office revenue, concession profits are 40% of box-office revenue, fixed costs are 33% of the total box-office revenue in a week. * * - (27 weeks * 6 screens) / 43 movies = 3.8 slots Chapter 3. SilverScreener: A Decision Support Model 135 Table 3.18: Sensitivity Analysis (Cost Data) _ * Case Operating Costs Concession Actual Profit Optimal Restricted Set (as % of gross Profit Profit (Percentage revenue) (as % of gross revenue) Improvement over actual) A l 56 30 551,857 768,508 (39.2%) A2 66 30 66,924 390,822 (483.9%) A3 56 40 1,036,790 1,225,613 (18.2%) A4 66 40 551,857 768,508 (39.3%) B l 56 30 580,124 791,662 (36.5%) B2 66 30 100,235 302,867 (202.2%) B3 56 40 1,065,054 1,295,802(21.7%) B4 66 40 585,175 805,988 (37.7%) CI 56 30 608,388 877,873 (44.3%) C2 66 30 133,550 403,035 (201.8%) C3 56 40 1,093,318 1,386,972 (26.9%) C4 66 40 618,480 912,134(47.5%) * - The figures for both operating costs and concession profit reflect the correlation with which they vary with the box-office revenue. The following assumptions have been made in the various cases for the percentage split of the operating costs in fixed and variable costs. Cases Fixed Variable A 0 100 B 50 50 C 100 0 For example, in Case B4, the variable costs vary as 33% of weekly box-office revenue, while fixed costs are 33% of the total box-office revenue received over a season. Chapter 3. SilverScreener: A Decision Support Model Table 3.19: Sensitivity Analysis (Revenue and Obligation Period) 136 Actual Schedule Optimal Restricted Set Schedule* Optimal Schedule* (Using Exhibitor's Forecasts from Variety) Optimal Schedule* (Using median split scheme for obligation period) Profit ($'s) 585,175 805,988 769,238 762,185 Percentage - 37.7 31.4 30.2 improvement over actual Number of different 43 27 31 25 movies scheduled Average run length 3.8 6 5.2 6.5 (screening slots) Percentage scheduled for 1 week 18.6 (8/43) 0 (0/27) 0(0/31) 0 (0/25) 2 weeks 27.9 (12/43) 11.1 (3/27) 25.7(8/31) 24 (6/25) 3 weeks 14.0 (6/43) 22.2 (6/27) 16.1 (5/31) 28 (7/25) > 4 weeks 39.5 (17/43) 66.7 (18/27) 58.2(18/31) 48 (12/25) Double booking for Batman 3 weeks 9 3 4 Indiana Jones 4 6 6 4 Star Trek V 2 2 2 0 Screening slots for Batman 16 21 16 15 Indiana Jones 13 13 16 9 Lethal Weapon II 11 11 11 9 Dead Poets Society 8 16 16 14 New York Stories 5 6 6 4 * - Using the following cost assumptions - Variable costs are 33% of box-office revenue, concession profits are 40% of box-office revenue, fixed costs are 33% of the total box-office revenue in a week. Chapter 3. SilverScreener: A Decision Support Model 137 Table 3.20: Sensitivity Analysis (Revenue Sharing Terms)* Sharing Terms Type** 1 2 3 4 5 Profit ($) - Actual 654,185 962,035 848,105 903,525 585,175 Profit ($) - Optimal restricted 880,722 1,112,591 1,043,204 1,092,753 805,988 (percentage (34.6%) (15.6%) (23%) (20.9%) (37.7%) improvement over actual) Movies scheduled 28 30 30 30 28 Average run length 5.8 5.4 5.4 5.4 5.8 (screening slots) Percentage of movies for 1 week 3.6 3.3 3.3 3.3 3.6 2 weeks 17.8 20 20 16.7 10.7 3 weeks 10.7 16.7 16.7 20 17.9 > 4 weeks 67.9 60 60 63.3 67.9 Double booking for Batman 6 6 6 6 9 Indiana Jones 6 6 6 6 6 Star Trek V 2 2 2 2 2 Screening slots for Batman 19 19 19 19 21 Indiana Jones 13 13 13 13 13 Dead Poets Society 16 16 16 16 16 * - Using the following cost assumptions - Variable costs are 33% of box-office revenue, concession profits are 40% of box-office revenue, fixed costs are 33% of the total box-office revenue in a week. * * - Different types of revenue sharing terms (minimum percentage in favor of exhibitor) are as follows. Type 1 - 30, 40, 50, 60, 65, 65,... Type 2 - 50, 60, 65, 65,... Type 3 - 40, 50, 60, 65, 65,... Type 4 - 40, 60, 65, 65,... Type 5 (Base case) - 30, 40, 40, 50, 60, 65, 65,... Chapter 4 Approaches to Estimating Managerial Gains from SilverScreener 4.1 Overview 4.1.1 Decision Support Models in Marketing Over the last three decades, decision support modeling has flourished in marketing. In 1967, as mentioned in the introduction of this thesis, very few successful applications of quantitative modeling could be reported in the field of marketing. The situation has changed dramatically since then, and today one can name many successfully imple-mented decision support models in marketing, such as P R O M O T I O N S C A N (Abraham and Lodish 1993), Rangaswamy, Sinha and Zoltners's (1990) model on sales force restruc-turing, S H A R P (Bultez and Naert 1988), ARTS P L A N (Weinberg and Shachmut 1972), B R A N D A I D (Little 1972),. C A L L - P L A N (Lodish 1971), and so on 1 . These models have addressed various aspects in marketing at different levels of a supply chain. For example, C A L L P L A N efficiently allocates salesforce to customers and products. ARTS P L A N helps manager plan a series of performing arts presentations. P R O M O T I O N S C A N helps manager in developing and evaluating short-term retail promotions. S H A R P model helps retailers decide on shelf-space allocations. Our model SilverScreener, presented in Chap-ter 3, is similar to P R O M O T I O N S C A N and S H A R P models, its aim is to help retailers (of the motion picture industry) improve their decision making. Marketing decision support systems (MDSS) (Little 1979, p. 9) are intended to assist •"^ Many commercial models based on the similar concepts are now also available in the market. 138 Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 139 decision makers in taking advantage of available information. Decision makers should benefit from the availability of more or better data by incorporating the information derived from these data into their decision processes (Blattberg and Hoch, 1990). For this purpose MDSS contain marketing models that make it possible to perform so-called "what-if analysis (Wierenga et al. 1994). Blattberg and Hoch (1990) report that a com-bination of model and manager often outperfroms either of these two alone. Hoch (1994) attributes this to the relative strengths of models, which compensates for the relative weaknesses of managers. In general, some of the ways in which the use of decision sup-port models can aid a marketing executive are the following (Montgomery and Weinberg 1973): • Helps to better utilize a manager's judgment, • Requires an explicit listing of input assumptions which leads to more informed discussion, • Provides a method for quick and convenient evaluation of the consequences of alternative plans. • Allows the emergence of unexpected solutions which open up new areas of problem-solving, • Expands the range of questions which can be answered by use of the notion of derived judgment, • Distills from available data relevant information as in new product forecasting, • Provides a basis for relating marketing inputs to market results and, hence, serves as basis for marketing planning, and Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 140 • Diagnoses, based on early data, the adequacy of a market plan and locates areas needing improvement. Inspite of these benefits, one must recognize the following important points about de-cision support models. First, models are an aid to the decision maker, not a replacement. Marketing models often can help the manager make a better decision, but models do not make executive decisions by themselves. For example, ARTS P L A N model allows the manager to revise the model's forecasts based on additional information or plans, such as special promotions, that the manager may have. Second, models should be proposed as useful tools to the end-user (i.e., manager), not as academic curiosities. Finally, models can be useful to managers in many different ways, and may offer opportunities for effi-ciency in a broad range of managerial activities. These observations drive our objectives in this chapter. We now refer to the previous research that has resulted in successful implementation of modeling exercise in the industry. 4.1.2 Implementation Examples from Previous Research We briefly review two successfully implemented decision support models from previous research, C A L L P L A N and ARTS P L A N , that provide us guidelines to characterize some potential managerial gains that may result from the implementation of our model, Silver-Screener. We choose these two application settings because, on the one hand, they bear some similarities with the proposed application of our model. On the other hand, they address quite different problems, and their implementation procedure is also different from each other. Together, these two application settings address a broad range of issues which would be useful in the further development of our yet-to-be implemented model. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 141 C A L L P L A N C A L L P L A N (Lodish, 1971) is an interactive salesperson's call-planning system. Its objective is to determine call-frequency norms for each client (current customers) and each prospect (account not currently buying from the salesperson). Call frequencies are the numbers of calls per effort period. The salesperson has a higher probability of getting business if he/she calls on a client than a prospect. Moreover, he/she is likely to spend less time convincing the clients than prospects in an equivalent sales effort. Thus, C A L L P L A N addresses the basic question: How can salespersons best utilize their time, allocating it between clients and prospects, and across products? The C A L L P L A N procedure has two phases. In the calibration phase, the expected profit associated with different call policies for each client and prospect is determined. Lodish uses a decision calculus2 approach for calibrating the sales response function with the salesperson's own best estimate of customer response to changes in call frequency. In the optimization phase, optimal allocation of time to clients and prospects is established. A dynamic programming-based routine is proposed for optimization. A heuristic is then proposed which provides very efficient solutions. The approach is usually interactive with sales-person putting in input values, looking at the results, modifying the input values, and so on. A controlled experiment of the C A L L P L A N approach was run at United Airlines. Twenty salespersons participated in the experiment; ten pairs of salesperson were matched by local management. Ten C A L L P L A N participants were chosen from each pair. The remaining salespersons comprised the control group. After six months the C A L L P L A N group had an 11.9% increase in sales from the previous year, while the control group had only a 3.8% increase3. The actual sales improvement over that of the control group for 2Little (1970) defines decision calculus as a model-based set of procedures for processing judgments and data to assist a manager in decision making. 3The difference was significantly different from 0 at 0.025 level. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 142 just these ten salespersons was "well into seven figures" (Fudge and Lodish, 1977, p. 104). A R T S P L A N ARTS P L A N (Weinberg and Shachmut 1978) is a user-oriented, decision support system designed to aid a performing arts manager in planning a schedule of perform-ing arts events. The model was first implemented in 1976 by the management of the "Lively Arts at Stanford" (LAS) program, a nonprofit organization. The system helps the manager achieve a variety of goals. As is typical in the nonprofit organizations, the goals conflicted to a certain extent. Moreover, given the "artistic" nature of the product concerned, only parts of management decision making were amenable to formal analytic procedures. Thus, this application describes and underscores the importance of handling the manager-model interface with care. The model consists of two major components. A forecasting system is first proposed to predict attendance at a scheduled event. A dummy variable regression analysis of historical data (three years) is performed to predict atten-dance. The model's prediction could be revised by the manager if he/she disagrees with the forecast. The second component of ARTS P L A N is an interactive planning model by which the manager can test the impact of scheduling different performing arts events on total attendance for the season. The planning model helps the manager determine whether a tentative or planned schedule will meet his/her attendance objectives4 for the year and what the impact of promoting certain events would be on the attendance predictions. The ARTS P L A N system has been used as an aid in the management of an on-going season and in the planning of a future season. In an interesting follow-up article on ARTS P L A N , Weinberg (1986) describes how ARTS P L A N model evolved after its 4 N o t i c e that the nonprofit organization has other objectives, i n this case attendance, t h a n profit m a x i m i z a t i o n i n case of businesses. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 143 introduction. Obviously, given the nature of a nonprofit organization's objectives, the usefulness of ARTS P L A N model could not be quantified in profit terms. The clearest evidence of its usefulness, therefore, lies in the fact that the model was enthusiastically embraced by L A S management and was used on an on-going basis. The management's enthusiasm is noted by Weinberg (1986) as follows: One year, in fact, logistical problems made it difficult for LAS management to run the model on a terminal so the computer forecasts were generated by hand. While there are many claims of computers replacing managers, this is probably one of the few instances of a manager replacing a computer. Some of the reasons for ARTS PLAN' s successful implementation could be as follows. One of the authors had collaborated with L A S management on some previous projects and that had build a feeling of trust. Second, the regression forecast were generally accu-rate in predicting attendance. Third, the system was so developed that the manager felt he/she had control over it. The final reason, which is related to the issue of control, was the way the managerial planning problem was handled itself. The ARTS P L A N system was deliberately designed to help the manager with a limited, but important, compo-nent of the process of planning a season. As mentioned earlier, recognizing the "artistic" nature of the performing arts, the ARTS P L A N analytical aid still left a considerable "room" for the manager to use his/her judgment when needed. Together, these two implementation cases give us a number of lessons. The first point that is highlighted by these examples is that the model can be a good aid to manager even if it addresses only a part (albeit significant) of the managerial problem. This underscores the importance of managing the manager-model interface carefully. Some relevant issues of concern are: how to build trust, how to demonstrate potential or resulting gains, how much control should manager have over model, and so forth. Second, Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 144 in both applications, the model emerges as a useful scenario (what-if) analysis tool. For example, C A L L P L A N allows the manager to re-allocate sales force after viewing the profit impact of a previous allocation. Similarly, ARTS P L A N allows the manager to revise the model's attendance prediction. This scenario analysis is time-efficient because it is done on a computer. The time efficiency in turn opens up new opportunities for the manager to examine various possibilities which would otherwise be impossible to generate by hand. Also, computer interfaces are important links to existing databases, which can be useful in the situations involving ongoing availability of data. Third, both applications include two components: prediction (or calibration) and planning. The two applications suggest that several possibilities exist as to how we could treat the prediction part. For example, C A L L P L A N uses managerial judgments as input values for the calibration of sales response function. ARTS P L A N , on the other hand, relies more on historical database which it uses in its regression model to predict attendance. It also presents another possibility that uses both historical database and managerial judgments. Finally, the implementation exercises of C A L L P L A N and ARTS P L A N conforms with our intuition in Chapter 3 that a decision-support model could be used in one of the following two major ways. A one-shot application of the model before the start of a season would give us a master planning application of the model. A n interactive and adaptive approach, on the other hand, offers a rolling horizon approach. Moreover, it could offer other benefits such as its availability as a "what-if analysis tool and marketing information system. 4.1.3 Types of Potential Gains Based on the above ideas, we now discuss the types of gains our model SilverScreener can potentially offer a manager of a major theater chain. SilverScreener can help an exhibition manager in the following ways. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 145 M a s t e r P l a n n i n g The master planning exercise would help an exhibitor design a tentative schedule for a forthcoming season. Our personal interviews with several exhibitors confirm that they prepare a master plan before a season. They also expressed keen interest in a mathematical model-based master plan because it would help them in their bidding and season planning decisions5. As mentioned in Chapter 3, exhibitors know the release dates for many major movies in advance. This is especially so if a distributor is aiming for a wide release of a movie. Approximately three to four months before the summer season, distributors screen their movies for exhibitors at a show in Las Vegas. After the screening, distributors send out bid solicitations to most exhibitors in the country. A typical bid invitation letter (see Squire 1992, p. 315 and pp. 344-345) contains the release date of the movie, the contract terms (i.e., obligation period and sharing terms) that vary by both movies and distributors, bid return deadline, and so on. A cover letter often accompanies the bid invitation letter. The cover letter, which might feature a brief synopsis of the movie, along with the name of the stars, director, producer and writer, would typically include an excerpt of the following nature6: Studio A is pleased to offer the release of Picture B, starring Star C in an action-adventure thriller directed by Director D . . . [italics added] (Squire 1992, p. 344) Using the information about the contract terms for various movies from their bid invi-tation letters, and a suitable revenue prediction scheme (which could either be historical-data based, or managerial judgment based), the exhibitor can then come up with a 5In fact, most exhibitors revealed that they develop a "sort-of-master-plan" before the start of a season. However, the plan optimization task may be too complicated to be done on a pen-and-paper level. 6We use the italicized terms in the master plan development scheme to be presented in the next section. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 146 tentative schedule for a multiplex using SilverScreener. The schedule can further help the exhibitor decide whether or not to bid for a particular movie. The decision would be based on the respective play-lengths recommended by SilverScreener. Such a decision might be quite different from the one the exhibitor arrives at using some analysis which works on a movie-by-movie basis, or which does not take into account the impact of a movie's profit potential on the exhibitor's overall (i.e., long-term for the entire season) profitability. The master plan would therefore reject some movies outright, either because their contract terms are too unattractive, or their estimated profit potential is not attrac-tive enough as compared to other movies. However, in a.spirit similar to C A L L P L A N and ARTS P L A N models, the exhibitor would have flexibility to override some of the model's recommendations (if it does not meet some feasibility criteria of the manager), and re-schedule the season. Another benefit that the master plan offers comes simply from its ability tox rec-ommend the play length for a movie. This may be beneficial in the cases when the distributor's bid invitation letter contains more complicated contract terms. For exam-ple, the contract terms for a movie may contain three different sets of obligation period and revenue sharing terms. The lowest value of obligation period may have the least at-tractive of the three sharing terms from an exhibitor's perspective. The next higher value of obligation period may have better sharing terms than the former, and so on. Based on SilverScreener's recommendations, the exhibitor can examine the impact of different sets of obligation periods and the associated sharing terms on his/her profitability, and can then decide whether or not to bid, and if chosen to bid, the set of contract terms for bidding. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 147 Weekly In-season Scheduling Based on the master plan, the exhibitor may choose to bid for some, but not neces-sarily all, of the movies three months in advance. Moreover, as mentioned earlier, not all the movies have wide release pattern. This leads to another proposed use of Silver-Screener, a rolling horizon approach, as a weekly planning and control device during a season7. Some commitments due to bidding done based on master plan might still be binding during the season. This suggests that once the season begins, the exhibitor might have "holes", or empty slots, to fill in his/her weekly schedule. The model is capable of optimizing the allocation of movies for the rest of the schedule keeping some allocations fixed. The exhibitor is likely to choose a shorter planning horizon (a time window) than the master plan for this application of the model because he/she will be relatively more certain of the availability and revenue potentials of various movies in that period. In the next section, we describe in detail how an exhibitor might make weekly decisions rolling from one time window to another. In doing so, we use minimal information that an exhibitor has about a movie's revenue potential before the start of the movie. After a movie plays at the theater (if it is chosen to play), and the corresponding actual revenue data becomes available, the exhibitor can update his revenue prediction model on the basis of the new data. If we stack the first week decisions of each time window, we end up with a quasi-master plan, which we term as actual implied schedule. In the next section, we compare the performance of this minimal information-based schedule with the one derived using ex post restricted set data in Chapter 3. Therefore, the rolling horizon application also serves as our method for examining the extent to which hindsight bias contributes to the profitability improvements claimed in Chapter 3. 7Though we do not explicitly show in this chapter, the rolling horizon approach of this chapter can also be generalized to an MDP framework of Chapter 2. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 148 Other Managerial Benefits SilverScreener can also be used as a scenario analysis tool. Using the model, the manager can examine the impact of different contract terms for the same movie. In the movie industry, the theater owner has an ongoing relationship with the distributors, and likely will not make decisions on a single transaction basis. On the one hand, the avail-ability of SilverScreener provides an optimal schedule. On the other hand, it provides an estimate of the value of honoring a relationship as opposed to making a more profitable, but perhaps short-term, decisions. A related managerial benefit from SilverScreener is its availability as a marketing information system. The model can easily be adapted to provide a summary of potential profits to be obtained from different distributors/movies. When compared with similar data over previous seasons, these summaries can provide a more concrete way of estimating the cost of honoring relationships with the distributors. It can also provide a measure of the attractiveness of the contract terms of some distrib-utors/movies. The exhibitor can also use this information in his/her negotiations with the respective distributors and might bargain for better terms in the future. The model's output can also provide a break-up of profits earned from the box-office gross revenue and the concessions. Since the exhibitors do not share the concession profits with the distributors, they may well find that, in certain cases, even though a movie's contract terms are not attractive, it provides more than proportionate profit from concessions because of increase in box-office appeal. A summary measure of such effects would be useful from management's perspective in movie exhibition. At a broader and more general level, we expect SilverScreener to provide managerial benefits in two different ways. The first, and more straightforward, benefit is that it provides the manager a more profitable way of scheduling. In the next section, we show the analyses which support our claim of profitability improvements. Second, the model, with the aid of computer, provides better efficiency to the manager in doing a complex Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 149 task8. If included on the back end of some automated forecasting system, the model could become part of a package that provides enormous efficiency benefits at a routine task. The major objective at this stage of our model development is that despite "not-being-there" (i.e., the model is yet to be implemented in the real-world), we must show that SilverScreener would be an effective decision aid to the manager. It would then be more meaningful to compare the model's output with actual decision making. In this chapter, therefore, we device ways to simulate an implementation scenario or "create" a manager. In the next section, we discuss various ways of implementing our model in the real-world setting. To summarize, the objectives of this chapter are as follows. First, we propose the types of potential managerial gains that can result from a decision support model, such as SilverScreener. Next, we provide methods for estimating those gains. Third, we show that the implementation of SilverScreener model may improve managerial decision making. 4.2 Implementation Scenarios: Simulating a Manager In this section, we discuss different ways of simulating an implementation situation for our model, SilverScreener, that was introduced in Chapter 3. We present the first two implementation scenarios as a two-tier application of the SilverScreener model. The first application discusses the master planning approach and the second application discusses the rolling horizon approach. We then discuss some heuristics which represent plausible simple decision rules that the manager might use in different situations. 8This benefit is quite independent of the profitability improvement. Thus, the model might provide benefits by automation of a time-consuming task even if it is accompanied with no improvements in profitability. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 150 4.2.1 Master Plan Development Revenue Prediction Scheme In this section, we present a model master plan development exercise. In doing so, we put ourselves in an exhibition manager's "shoes" and address the following question: how to develop a master plan before the season, and how to use it for bidding purposes? To develop a master plan a few months before the movies of a particular season are released, the two most important pieces of information the exhibitor needs are the release dates of the movies and estimates of their box-office revenue potentials. The release dates are more easily available and generally well-known in advance9. We now discuss the scheme to generate the estimates of the box-office revenue potentials of various movies. Description of Estimation Data Assuming that a typical manager has access to at least as much historical data as those that appear in Variety, we propose an ex ante revenue prediction scheme which uses previous years' data to generate revenue estimates of forthcoming movies. It is clear that this scheme does not suffer from any hindsight effects. We presented in Chapter 3 the optimal restricted set schedule for the 84th St. Sixplex theater for the year 1989. This schedule considers only the movies that the exhibitor showed at the theater. In the master plan application, we use the same movie consideration set as that for the optimal restricted set. In generating revenue estimates for 1989 in our master planning application, we use the data from 84th St. Sixplex for the year 1988 for the corresponding season of 27 weeks10. 9While distributors sometimes change the movie release dates due to strategic issues (Krider and Weinberg 1998) or production difficulties (for example, Titanic), these release dates are generally deter-mined several months before the start of a season. 1 0Though a few data used for estimation are from the years preceding 1988 (e.g., in case of sequels), most of the prediction data are used from the year 1988. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 151 Demand Model The model we use for revenue prediction is the exponential model used earlier in Chapter 3, namely GROSSjw = aje^w+e (4.19) where GROSSjW denotes box-office gross revenue of Movie j in Week w of the movie's run (w = 0,1, 2, . . . ) . In the context of this model, our use of the term revenue prediction implies estimation of a and pi for different movies. Alternatively, a manager can use some other revenue prediction model depending on the level of sophistication required for the manager's situation. A simple model could be in terms of percentage declines. For example, a manager might say, "I expect Movie X to open at a and decline at b% every week." It is clear that we capture open and decline by a and /?, respectively, in our estimation scheme. The revenue data of 1988 movies were collected to estimate a and /J's (using Equation 4.19 for the corresponding movies of the Summer season of 1989)11. Movies Classification Procedure We collect data on some key attributes (to be described later) of both 1988 and 1989 movies. The objective of this data collection procedure is to classify the movies according to these variables. We use a genre-based approach for the classification purposes. Once such a classification is done, we can predict the revenue for a 1989 movie with certain attributes by "matching" the movie with similar movies from 1988. By collecting the relevant data and matching it with the previous years' movies, we attempt to mimic the managerial behavior of the following type: When faced with a new movie, the manager says, "Last year, the movies of this type generated X dollars on average in their first n W e collected data on 44 movies of the 1988 season, similar to 43 from the 1989 season. In addition, as mentioned previously, we also collected data for a few movies from 1987. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 152 week, and dropped by Y percent every week on average. Therefore, I expect this movie to open at X and drop at the rate of Y every week."12 There is some evidence that such rules are used in the motion picture industry. In fact, some distributors use even more sophisticated rules than the one suggested above. For example, Barry Reardon, president of Warner Bros. Distributing, reports: Our sophisticated data base and in-house computer tracking system kicks in [after a movie's release], and executives are at their desks Saturday morning analyzing the grosses as they are received from EDI (Entertainment Data Inc.), a service that reports on every theatre opening our picture. Since our data base contains the grossing history of every theater in the country over ten years, we instantly know where we stand and can make reliable projections. Built into these projections are weather issues (heavy snow in Chicago), T V competition (a hot mini-series), current events (international unrest) or other influences. [Italics added] (Squire 1992, p. 317) We choose the following five attributes for the classification purposes: Genre, M P A A Rating, Sequel, Stars, and Distributor. These attributes are based on previous studies such as Sawhney and Eliashberg (1996), and Jedidi, Krider and Weinberg (1998). Pre-vious research (Wallace, Seigerman, and Holbrook 1993; Sawhney and Eliashberg 1996) has reported some success in forecasting revenues based on such attributes. We must point out, however, that we do not suggest that the overall appeal of a movie can be reduced to some function of its attributes. Instead, the objective of the proposed revenue prediction scheme is to help us present a sample application of our model, SilverScreener. Therefore, we do not rule out the possibility of better prediction schemes to be used in conjunction with SilverScreener in future. 12Notice that since the year-to-year variations would be fixed across all movies, they would not affect the schedule generation process. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 153 It should also be noted that the classification (or clustering) of the various movies could be done at several levels. For example, we could classify movies depending simply on whether they contain major stars or not. Notice that such a classification yields only two classes and would not provide a good discrimination scheme. We could generate many other classifications according to various combinations of the different attributes. However, we did not observe any similarities in the a. and /?'s of different movies when the classification is done simply on Stars and Distributor attributes1 3. We adopt a genre (and M P A A) based classification approach because the 1988 data reveal that movies of a similar genre and M P A A rating show greater similarity in terms of their a and 0's. We now explain each of the five attributes in detail. A t t r i b u t e s The first attribute is genre of the movie. We include the following genres in our analysis: Action, Comedy, Drama, Horror, Crime, and Science Fiction. The second at-tribute, M P A A Rating, captured the Motion Picture of America Association's (MPAA) classification of the movie as R, P G , or PG-13. The Sequel attribute (Yes/No variable) indicates whether the movie was a sequel or not. We used Blockbuster Entertainment Guide (Castell, 1996) to gather data on the above variables. The fourth attribute, Stars (Yes/No variable), indicates whether the movie included major stars (as classified by Quigley's Motion Picture Almanac, Klain, 1990) or not. Finally, the Distributor at-tribute (Yes/No variable) indicates whether the movie was being distributed by a major distributed or not. The following distributors are recognized as the major distribu-tors: Warner Brothers, TriStar, Touchstone, Buena Vista, Walt Disney, Paramount, and United Artists 1 4 . Notice that the above five attributes of a movie can be collected ex ante 1 3 The case of Sequel attribute is treated differently from the rest of the attributes, which we describe later. 1 4 We also collected data on another attribute, Director, but it was dropped later on. There is so much Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 154 and are assumed to be known to the exhibitor at the time of receiving the bid invitation letter. Estimation of Demand Parameters We estimate the two parameters of the demand model for the various movies of 1989 using the corresponding 1988 movies' estimates by the following scheme. All the movies except blockbuster and sequel movies The following scheme ap-plies to all the movies that are neither blockbusters (e.g., Indiana Jones) nor sequels (e.g., Lethal Weapon II). • If, for any 1989 movie, there is a unique corresponding 1988 movie (i.e., matches on all five attributes and is the only such movie), then we use the cv and /3 of that movie as estimates of the 1989 movie. • If there are multiple movies from 1988 that match on all five attributes, then we use the average of a and /?'s of all such movies as estimates of the 1989 movie. • If there is no movie from 1988 that matches the 1989 movie on all five at-tributes, then we examine whether there are movies in 1988 data that match on the next four attributes (including Genre and M P A A ) and use the relevant averages of their estimates. If not, then we match on the next three variables, and proceed similarly as before. We therefore adopt a stepped approach and match the movies next on Genre and M P A A , then on Genre alone, and use the relevant averages. • Finally, if there is no movie in 1988 data set that matches the 1989 movie even variation on this variable from one year to another that it does not classify the movies effectively. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 155 on genre, then we use averages of all the movies in 1988 data set . In this case, we envision the manager as saying, "Faced with a completely different movie about which I have no specific classifying information, I'll use the estimates that are based on all the movies in my last year's information set." B lockbuster movies We define those movies as blockbusters that involve huge pro-duction budgets, and are supported by heavy upfront advertising. In recent years, such movies as Jurrasic Park, Independence Day, or Titanic would qualify on this criterion. It is therefore easy to see that in the 1989 data set for the 84th St. Sixplex, only Batman and Indiana Jones can be treated as blockbusters. This is also reflected in their longer play lengths and double booking status in the actual schedule. For the blockbusters movies of 1989, we choose as their estimates the maximum a and minimum /3 (assuming a negative value of /?) of all the movies from 1988. In this case, therefore, the manager uses the following rule, "Since these are blockbuster movies, I expect them to open like the movie with the best opening last year, and sustain like the movie with the strongest legs (i.e., minimum decay rate) from last year." Sequel movies For a sequel movie, we use as its estimates the a and f5 of its immediate predecessor from the most recent previous year. The predecessors of Friday the 13th Part VIII and A Nightmare on E lm St. 5 are available from 1988 data. We use 1987 data for the estimates of License to K i l l and Lethal Weapon II. The estimates of some movies are not available under this scheme because either the movie was not played by the theater, or the predecessor was released more than two years ago. In such cases, their estimates are based on their Genre type attributes. 1 5This occurs for the movie SING that has Music as its Genre according to Blockbuster Entertainment Guide. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 156 We use the above scheme to generate the master schedule using the same cost and contract related assumptions as that for the optimal restricted set schedule (Table 3.5) in Chapter 3. In order to check the predictive ability of this revenue prediction scheme, we pooled all the actual revenue observations for the weeks that a movie played at the theater and their corresponding estimate predicted by our scheme. We then regressed the estimates obtained by the prediction scheme on the corresponding actual observations16. The resulting overall R2 value was found to be 0.28. The low R2 value reflects the difficulty associated with the a priori revenue prediction of new movies. Previous re-searchers have also met with only partial success in such efforts. For example, Sawhney and Eliashberg (1996) used a meta-analytic approach for a priori prediction of movie revenues. Their results show that they found acceptable (according to the chosen crite-rion) a priori predictions for only two out of the ten movies in their holdout sample. In light of the previous work, therefore, our scheme provides a reasonable starting point for master plan design. This is especially so because our major objective is not forecasting of box-office revenues, but to show how a master plan can be constructed. M a s t e r P l a n Results The master plan is presented in Table 4.21. The resulting schedule suggests the following. 1 6In this regression, we do not include the movies that played for only one week and the movie DPS (Dead Poets Society), an outlier. The one-week movies were not included because these movies are "filler" movies and the prediction scheme could not predict their revenue for just one week accurately. The movie DPS was not included because it was a "surprise" movie in 1989. The actual revenue of this movie at 84th St. Sixplex theater increased after the first few weeks of its release, possibly as a result of favorable word-of-mouth. We also checked the national release pattern of this movie. It followed a release pattern known as "platform" release in industry jargon. That is, it opened on very few screens (eight) in the first week of its release, and then gradually increased the number of screens it played on with the progress of time. It is clear that such a revenue pattern is very difficult to be captured by any ex ante prediction scheme.. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 157 Bidding Results As we discussed earlier in this chapter, one of the major uses of the master plan is in the bidding process. Let us explore how the master plan might suggest to the manager which movies to bid on (and possibly for how long a period). We have 43 different movies in the actual consideration set. Since the schedule generated by the restricted set approach is optimal for this data set, we assume that some movies out of the 27 movies scheduled by the optimal restricted set approach are good choices for bidding. The exact number of movies chosen for bidding depends on some appropriately chosen criterion. We wish to examine the number of movies that would be chosen for bidding by the master plan using the same criterion. To select a movie for bidding from master plan, we use a cut-off criterion in terms of the recommended play length by the master plan. For example, for a certain case, such a criterion could be: Select all the movies with recommended play lengths greater than or equal to X weeks. Such a criterion is reasonable since it captures a range of the manager's risk taking capacity. At a lower value of X , say two weeks, the manager is more risk prone in that he/she chooses a movie to bid as long as it appears for at least two weeks in his/her master plan. A more cautious (risk-averse) manager, on the other hand, might decide not to bid on a movie unless it appears for a reasonable period in the master plan, and bid on fewer movies. In other words, this cut-off procedure may be useful in not only deciding ultimately which movies to bid on, but also prioritizing the bidding choice based on the recommended play-lengths. Thus, it would also help the manager decide on which movies to bid first. A noteworthy point about the master plan approach is that its generation implicitly imposes a cut-off rule on the weaker choices. Thus, by choosing the movies to bid only from the master plan, the manager automatically rules out the weaker movies that the master plan never schedules. Notice that such a ruling-out procedure takes into Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 158 account the dynamics of expected revenues, release dates and contract terms. This in turn automates a task of sifting the good movies from the bad if the consideration set is large in size. In Table 4.22, we vary X from 2 to 6 weeks and report the movies that would be chosen for bidding by both the master plan and optimal restricted set approach. Our objective is to examine how similar the two sets of movies are, that is, how many common movies are chosen by the two approaches. If the master plan without the benefit of hindsight could recommend a similar set of movies for bidding as that recommended by the optimal restricted set approach, then it would corroborate the proposed usefulness of SilverScreener model. As shown in Table 4.22, at different values of the cut-off criterion, the master plan selects a similar set of movies for bidding as that selected by the optimal restricted set approach. As expected, the number of movies chosen for bidding decreases with an increase in the cut-off criterion in cases of both the master plan and optimal restricted set. At the cut-off value of 2 weeks, all the movies are chosen as scheduled by both the approaches17. The number of movies common to the two schedules is also highest in this case. The respective figures progressively decrease as the cut-off value increase. At the cut-off value of 6 weeks, the master plan recommends 7 movies, while the optimal restricted set approach recommends 10 movies for bidding. The number of the movies common to the schedules, 4, is the lowest for this case. The major difference in the recommendations from the two approaches results from the different play lengths of a few "surprise" movies, such as DPS (Dead Poets Soci-ety). It is a surprise movie because, as explained earlier, its demand "picks up" after the first few weeks of the movie's run and it retained itself well during the later part of its run. It is clearly difficult to capture this pattern at master planning level. Similarly, another movie, SDL (Scandal), showed strong retention power (in actual data) in the 1 7This is because we impose an obligation period of two weeks on all the movies in these analyses. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 1 5 9 later part of its run, quite contrary to its predicted decay pattern. The movie SDL is of Drama genre, has M P A A rating of R, is not a sequel, does not contain any major stars (John Hurt, Joanne Whalley-Kilmer, Bridget Fonda, Ian McKellen), and is distributed by Palace Pictures, which is not a major distributor. Its estimates are based on the relevant averages from (Drama, R) combination of (Genre, M P A A Rating). The average estimates from 1988 show that the movies of this type decay rather rapidly as compared to the other movie types. Therefore, it is difficult to predict a priori that SDL should be chosen to play longer by the master schedule. However, it opened and retained its box-office appeal well as its revenue data show1 8. The optimal restricted set approach captures this phenomenon through a combination of the information about a movie's actual revenue and a reasonable demand prediction (two-parameter exponential) model. Since the master plan is unable to capture such situations, it underscores the importance of the rolling horizon approach (described in the next section), an adaptive scheme which is the second tier of the proposed application of our model. Characterization of Overall Master Plan Results The characterization of the master plan results is presented in Table 4.231 9. In this table, we also present for comparison purposes the summary results of actual schedule and optimal restricted set from Chapter 3. The following salient points emerge from these results. • The master plan schedules fewer movies (with longer average run lengths) than the actual case. This is in the direction suggested by the optimal policy from Chapter 3. 18Blockbuster Entertainment Guide also gives Scandal 4 points in rating on a scale from 1 to 5. 1 9 As mentioned earlier, all the results presented in Table 4.23 have been obtained using restricted set data. Please ignore the last four columns of Table 4.23 at this point. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 160 • The master plan schedules 31 different movies, close to 27 scheduled by the opti-mal restricted set approach which has the benefit of ex post data. Out of the 31 movies scheduled by the master plan, 23 movies are also scheduled by the optimal restricted schedule. As discussed above, a majority of movies scheduled by the optimal restricted set approach are also selected by the master plan. • Some important differences exist between the master plan and the optimal restricted set schedule. A few of these differences are shown in Table 4.23. First, the master plan schedules 29% of its movies for 2 weeks. Though this percentage is less than 46.5% as scheduled in the actual case, it is substantially higher than the optimal restricted set case. The master plan schedules 48.5% of its movies 4 weeks or longer period, greater than 39.5% in actual case, but lower than 66.7% in the optimal restricted set case. These results could be attributed primarily to the availability of better information in the latter case. • The percentage change in profit figure shown in Table 4.23 indicates that the master plan shows a decrease of 29.4% in profit as compared to the actual case. This is also substantially lower than the level of profitability shown by the optimal restricted set approach. However, a clear description of this comparison criterion is in order here. First, the profit for the master schedule is calculated using the optimal restricted set data. The optimal restricted set uses ex post data for revenue prediction and enjoys a distinct advantage over the master schedule. Therefore, if our ex ante prediction scheme could generate the estimates close to the ex post estimates, the master schedule would show the same level of improvement over actual case as that shown by the optimal restricted set case. However, the estimates are not accurate for some key movies, as also reflected by an overall low B? value, which creates the major differences in profitability. Second, the manager in the actual case had Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 161 the advantage of making weekly adjustments with the availability of additional information as the season progressed. Indeed, as the revenue forecast data from Variety show, the forecasts are updated from week to week. Since the master plan is one-shot application of the model, it cannot compete with an adaptive decision making approach. Therefore, in some sense, the comparison criterion based on profit is "unfair" to the master planning approach2 0. We adopt this criterion because it provides us with a common comparison basis on which we can compare other approaches for schedule generation such as heuristics (to be discussed later). Finally, in generating the master schedule, we impose an obligation period of two weeks on all the movies. In the actual case, however, there are several movies that played only for one week. • The double booking and screening slots results from Table 4.23 show that the major difference in profit between the master plan schedule and the optimal restricted set schedule result from the altered play lengths of the surprise movies such as SDL and DPS (as we already discussed in the bidding results). On a smaller scale, some differences between the master plan and optimal restricted set schedules are also caused by the differences in the play lengths of the movies such as N Y S (New York Stories), LW2 (Lethal Weapon II), and IJ (Indiana Jones), which are either replaced too soon, if scheduled at all, or are scheduled for too long a period., The increase in run length of some movies compensates for the shortened lengths of the other movies, and results primarily because the prediction scheme expects these movies to do well at the box-office. This discussion concludes our example of master planning approach, the first tier of 2 0 W e m u s t r e m i n d t h e r e a d e r s h e r e t h a t t h e m a s t e r p l a n i s b a s e d o n t h e r e s t r i c t e d se t m o v i e s , t h a t i s , o n l y t h o s e m o v i e s t h a t t h e e x h i b i t o r s h o w e d . T h e p r o f i t r e s u l t s w o u l d p r o b a b l y b e m o r e g e n e r o u s t o w a r d t h e m a s t e r p l a n i f t h e a d d i t i o n a l e x p a n d e d se t m o v i e s a r e a l s o i n c l u d e d i n t h i s a n a l y s i s . Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 162 the proposed application of our decision-support model. Our results show that the model could provide useful guidelines to the manager for bidding purposes. Using an ex ante revenue prediction approach and a cut-off criterion in terms of play length of a movie for bid choice, we find that the master plan recommends a similar set of movies for bidding as that recommended by an optimal plan that has ex post information about revenues. This suggests that the proposed master planning approach would be even more useful if our scheduling model is used at the back end of a forecasting model that uses more sophisticated techniques than those used in our prediction scheme in this section. We also find that our approach shows some limitations in the case of those movies which perform substantially different from the expectation based on the information about their key attributes. To summarize, if the forthcoming season could be visualized as a two-dimensional (week by screen) "matrix," then that matrix contains empty "cells" before the master planning process. After the master plan is developed, the manager may choose to "f i l l " some of those cells by bidding. The number of cells thus filled would depend on the manager's attitude towards risk. If the manager is risk-prone, then he/she might choose to bid on a lot of movies and may get "locked" in several binding commitments, leaving relatively less decision making during the season. If the manager is risk-averse, then he/she might choose to bid on a relatively fewer number of movies, and rather wait for the season to commence for weekly negotiations. In either case, this reasoning suggests a second tier of the application of our model, a rolling horizon (adaptive) approach. We present the rolling horizon approach in the following section2 1. 2 1 In presenting the rolling horizon application, we start with an empty "matrix," that is, we proceed as if none of the "cells" are filled by a preceding master plan. This helps us explain the rolling horizon approach more completely. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 163 4.2.2 Rolling Horizon Approach Motivation In Chapter 3, we provided an introduction to the rolling horizon approach. The basic idea behind this application of SilverScreener is to use it every week of the season for a short planning horizon, say, six to ten weeks. It is reasonable to expect that the exhibitors are relatively more certain about the availability and revenue potential of various movies in a shorter horizon. If we demonstrate that our model is effective using minimal information that an exhibitor has at a decision epoch in a season, then it will further support the usefulness of our model and help establish the robustness of our scheduling model. The general approach is described as follows. Consider a theater manager who manages a multiplex. The manager may begin using SilverScreener with an eight-week time window in the first week of the season us-ing his/her judgment of revenue estimates. Depending on the model's recommendation, he/she arrives at the first week scheduling decision. In the second week (time window - second to ninth week), he/she may update the revenue estimates on the basis of the actual revenue received in the first week. Based on SilverScreener's recommendation for the second to ninth week, he/she makes the second week scheduling decision. From the third week onwards, the manager could use a combination of the two-parameter expo-nential model (for the movies that have played for two weeks) and managerial estimates for the prediction of revenues. The exhibitor could thus come up with weekly scheduling decisions by "rolling" from one time window to the next. Analysis and Results To analyze the scheduling decisions more in line with the actual decision-making, and to show the robustness of our profitability improvement results from the previous Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 164 chapter, we use minimal information in the rolling horizon application of our model. We assume that the only information available to the exhibitor about the revenue of a movie before its release is its first-week forecast. This forecast is in fact published in Variety (1989). To generate prediction for the later weeks, we use a median 2 2 exponential decay rate of -0.28 (calculated across the movies in the restricted set). The idea is that with no a priori information, the executive is likely to rely on an average across all movies or a similar historical summary measure. This is in the same spirit as using a prior distribution over the decay rate. After a movie has played for one week, we fit the two-parameter exponential model using the actual data from the first week and the second week's forecast to generate predictions for the later weeks. We fit the two-parameter exponential model to the actual data for the movies that played for at least two weeks. The obligation period variable is assumed to be movie specific to account for the beginning- and end-of-the-window effects. We follow the rolling horizon approach for the first 20 time windows, covering the entire summer season. For the sake of brevity, we report not the schedules obtained, but note some key characteristics of the weekly schedules. First, in each of the eight-week windows, the optimal (rolling horizon) approach schedules consistently fewer movies than it "considers" (the movies released during that time window plus all the movies released in previous time windows). Thus, even with a planning horizon as short as eight weeks, we observe a pattern that is consistent with the optimal policy obtained by the one-shot application of the model in the optimal restricted set case. Second, one can view the screening slots allocated to an unreleased movie as an indicator of the executive's "belief (following SilverScreener) in the strength of a movie before it is released. This belief, however, changes from window to window depending on the release dates and revenue estimates of other movies. We do not address the dynamics of such competitive effects 2 2 The distribution of decay rates is skewed to the right, therefore we used median instead of mean. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 165 in this thesis. The modeling of these effects, however, represents an attractive future research idea. Although a schedule is generated for all the weeks within a time window, it should only be viewed as the "expectation" of a schedule by the executive. It is the first week recom-mendation that is used for decision making. We term the schedule obtained by stacking the first week recommendation of all time windows as the actual implied schedule23. The actual implied schedule compiled from first 20 windows of the rolling horizon approach is given in Table 4.24. It is clear from Table 4.24 that, barring a few exceptions, the actual implied schedule derived in an adaptive manner is quite similar to the optimal restricted set schedule derived in a one-shot manner for the first 20 weeks. One of the differences is that there is an instance of "(job) splitting" in the case of the actual implied schedule, that is, the movie - Dead Calm (DC) returns in Week 7 after dropping out in Week 5. Also, there are a few movies that are scheduled for one week because we do not impose obligation period restriction from one week to another in constructing the actual implied schedule. Both of these differences are due to the stacking of the first week recommendation of each time window. Since there are only a few such instances and given that our major objective in presenting this application is for illustrative pur-poses, we did not apply various constraints such as obligation period or non-splitting from window-to-window. We compare the actual implied schedule (obtained using rolling horizon approach) with the optimal restricted set schedule on the basis of the data used for the optimal restricted set schedule. The results are shown in Table 4.25. For the first 20 weeks of the planning horizon, the actual implied schedule achieves more than 90% of the improvement obtained by the one-shot optimal restricted set schedule for the same period. Moreover, 2 3 We must emphasize here that this schedule is arrived at using an ex ante revenue prediction approach. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 166 the policy results are also quite similar for the two schedules24. This is to be expected since in the limit, the one-shot optimal restricted set schedule can be viewed as a single window of 27 weeks. The only difference between the two approaches is of information availability, which is greater in the optimal restricted set schedule. For our illustrative purposes, however, it is more important to note that policy recommendations from the minimal information rolling horizon approach mimic those from a hindsight based optimal restricted set approach. To summarize, in this section we present rolling horizon approach, the second tier of the two-tier application of the SilverScreener model. This adaptive approach uses minimal information that is readily available to an exhibitor at any decision epoch. It considers a shorter planning horizon than the master planning approach and incorporates the additional demand information as it becomes available with the progress of the season to revise its estimates. The approach itself is quite general to incorporate a sophisticated demand prediction model, managerial judgments, or a combination of both. We find that the policy results recommended by this approach are similar to those recommended by the one-shot optimal restricted set approach. We also find that this approach can achieve almost the same level of profitability as that of the optimal restricted set approach. This completes our example of the two-tier application of the SilverScreener model. In the next section, we examine some other decision rules (heuristics) that offer plausible scheduling alternatives for the exhibitor problem. •^The only difference appears to be in average play lengths of the two schedules. This is because there are some movies which enter the actual implied schedule with a one-week play length. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 167 4.2.3 Heuris t ics We consider two types of heuristics in this chapter. The first heuristic, which we call Distributors' Pressure Heuristic, considers the case of a manager who attempts to sched-ule all the movies that are released in the market. The second heuristic is a Rank-Based Heuristic, which we alluded to in Chapter 2. This heuristic is based on the ideas from the control-limit policies in machine replacement literature. We use two different revenue data sets to manipulate the level of information avail-ability to the manager who generates schedules based on these heuristics. At the first level (a low level) of information, we use the revenue data produced by the ex ante rev-enue prediction scheme used in the master planning approach. We treat it as a low information level because the expected revenue data generated by this approach relies only on the attributes of the movies that are known before a movie is released. The second level (a high level) of information uses the ex post data used in the restricted set approach. This level is representative of the cases in which the manager has access to a sophisticated system that produces "perfect" information about the revenue of a movie. Though we generate the two revenue data sets being used before arriving at the schedules, the heuristics are applied as if the manager knows the data corresponding to a decision epoch only at that decision epoch. Thus, these heuristics do not take into account future realizations of revenues. The two heuristics along with the two levels of information availability let us compare the scheduling recommendation made by alternative decision rules with those made by the master plan and optimal restricted set approaches25. The description of the two heuristics is given below. ^The basic difference between the master plan and optimal restricted set approach also lies in the extent of information availability. Since we solve the optimal restricted set also as a one-shot case, it could be considered a master plan application with ex post data. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 168 Distributors' Pressure Heuristic As mentioned above, this heuristic considers the case of a manager who acts under a lot of pressure from the distributors to schedule their movies, perhaps to maintain good relations with them. Let us assume that such a manager using the following decision rule: Each week, if a new movie becomes available, replace the weakest of the existing movies if it is not in its obligation period with the new movie. If more than one new movies become available in a week, then consider the next to the weakest existing movie applying the same criterion as before, and so on. Notice that the manager following this heuristics tries to accommodate the movies released by a l l 2 6 the distributors as long as he/she has "space" to show it as permitted by the obligation period commitments of the movies already being played in a given week. Though the manager tries to show as many movies as possible in the above in the above fashion, he/she replaces only his/her worst performing movies (in terms of expected revenue) at any decision epoch. Thus, there is some element of smartness in the manager's decisions under this heuristic. However, the exhibitor gets penalized for the attempt to show all the movies because even though the movie being replaced is the weakest among the ones playing, it could still be stronger than the new movie in terms of revenue. Rank-Based Heuristic In Chapter 2, we introduced the notion of rank-based optimal policy. We found that the optimal policy recommends the replacement of an existing movie with a new movie if the difference between their ranks exceeds a certain number depending upon the relative strengths of the two movies and the week of the planning horizon being considered. In general, we found that the new movie must open in a better rank than the existing for 2 6 "AH" could also be interpreted as "most favored" distributors. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 169 the replacement to take place. In this chapter, we introduce a rank-based heuristic which is based on a simplified version of the above rank-based optimal policy. First, we sort the weekly revenues for all the movies in a data set. This gives us a range in which the weekly revenues of different movies in the data set fall. We divide this range in equal intervals to develop a system of ranks. For example, the revenues for the ex post optimal restricted data set vary from a minimum of 63 to a maximum of 1006 (after some scaling). We divide this range in equal intervals of 100. Thus, we come up with an 11-rank system, where revenues greater than 1000 fall in Rank 1 and between 0 and 100 fall in Rank 11. We came up with a similar systems of ranks for the ex ante data set. We envision a manager following the rank-based heuristic as using the following decision rule: Each week, if a new movie becomes available, replace with the new movie the weakest of the existing movies if it is not in its obligation period, and if the rank of the new movie is better than the one being considered for replacement, otherwise do not replace. If more than one new movies become available in a week, then consider the next to the weakest existing movie applying the same criterion as before, and so on. Notice that a manager following this heuristic is not bound to replace an existing movie with a new one unless the rank criterion suggests so. Heur i s t ic s ' Results The schedules generated using the two heuristics using ex ante and ex post data respec-tively are presented in Tables 4.26 to 4.29. We characterize these results in the last four columns of Table 4.23. The following points emerge from these results. The manager following the distributors' pressure heuristic performs much worse on the profitability criterion (relative to the actual case) when using ex ante data 2 7 . This is despite the fact that it chooses such major movies as Batman and Indiana Jones for long 2 7 We remind the readers that the profit is calculated using optimal restricted set data. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 170 play lengths. However, the decrease in profit results generally from a huge proportion of the movies that are scheduled for shorter play lengths, that is, in the opposite direction as suggested by the optimal policy in the previous chapter. With the availability of better information in the ex post data, the manager improves upon this deficiency, and comes close to the actual case in profit terms. Given that the total number of movies scheduled under this heuristic are equal to those in actual case, these results are interesting because they provide an explanation for the short average play lengths by the manager in the actual case. In other words, possibly the 84th St. Sixplex manager was committed to show the 43 movies of all the distributors he/she had good relation with, and, therefore, quickly replaced the already playing movies in many instances. The manager following the rank-based heuristic performs much better than the dis-tributors' pressure heuristic in both cases of information availability. It performs remark-ably well with the ex post information. This heuristic improves by 25.8% in profitability over the actual schedule. It is inferior to the optimal restricted set approach. How-ever, the policy results of this heuristic are in the direction recommended by the optimal (restricted set) policy, that is, it schedules fewer movies for longer average run lengths. Wi th the ex ante data set, the performance of this heuristic deteriorates and it does worse than the actual case in profitability terms. However, it still performs better than the distributors' pressure heuristic. Moreover, the policy results are generally in a direc-tion recommended by the optimal policy. Thus, it appears that even with a simplified version of the rank-based policy, we achieve a level of improvement that is comparable to that achieved by the optimal approaches. If these results could be replicated under different cost and contract assumptions, and with different data sets, then they would suggest that some efficient heuristics could be developed for the exhibitor problem. In summary, our results show that the level of profitability that a manager might achieve in an application setting would depend on the quality of information he/she has Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 171 about the revenue potentials of various movies. In the case of good information (e.g., ex post data), the opportunity for increased profitability is enormous if used with a good scheduling tool, as demonstrated by the optimal restricted set and rank-based heuristic approaches. However, the availability of good information alone does not guarantee increased profits. If the replacement rule is myopic and is based on somewhat perverse, but understandable, managerial behavior, then even good information cannot bring extra value to the manager, as exemplified by the distributor pressure heuristic with ex post information. 4.3 Conclus ion 4.3.1 Contr ibut ions In this chapter, we presented a two-tier application of our decision support model Sil-verScreener. The first tier is a master planning exercise which the exhibition manager engages in some time before the season begins. The second tier is an adaptive rolling horizon approach that the manager could use once the season begins. Our major objec-tive in presenting these applications is to show that the implementation of SilverScreener model will improve managerial decision making. We also propose the types of potential managerial gains that result from our model. In doing so, we presented methods for estimating these managerial gains. We develop an ex ante revenue prediction scheme to show the implementation of the master plan. The prediction scheme is developed primarily to assist the description of how a manager could use our model in practice. Thus, its aim is not to provide a sophisticated revenue forecasting system. We used some key attributes of a movie for the prediction scheme. However, in a real world application, the system could benefit even more from additional inputs from an experienced manager. Using our revenue prediction Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 172 scheme, we show that the master plan can help the manager in deciding which movies should he/she bid on. The other benefit of using our model in this setting is that the manager could readily perform "what-if analyses to examine a range of scenarios to check his/her assumptions. The model is useful for this type of analysis because it automates a complicated task that involves the dynamics of release dates, varying revenue streams and different revenue sharing terms of various movies. As a result of these analyses and based upon his/her selection criterion, the manager may decide to bid on some, but not necessarily all of the movies. This possibility highlights the importance of the second tier of the application of our model. The second tier of our proposed application is a rolling horizon approach that would be used once the season begins. This is an adaptive approach in which the manager discovers demand as it becomes available with the progress of the season, incorporate it to revise his/her estimates of a movie's performance and comes up with the weekly scheduling decisions. To show the robustness of our claims of the benefits from our model, we use the minimal information in this application that a manager would have at any decision epoch. Our results from the rolling horizon approach show that it can achieve an improvement over actual schedule (in profit terms) of the same order as that achieved by a hindsight based one-shot application of the model. Our model is also capable of handling such constraints as would be imposed by the manager's precommitment to certain movies because of bidding in the first stage. Finally, we proposed some alternative decision making rules in the form of two heuris-tics under two extreme conditions of the quality of information about movies' perfor-mance. The results of the distributors' pressure heuristic show that no matter how good the information is, the exhibitor could lose money if he/she tries to please all the distrib-utors in the market. This lends further support to our normative policy results that the exhibitors need to be more selective about their scheduling decisions and should run their Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 173 movies longer instead of replacing them quickly. This result also makes intuitive sense given the way the contractual agreements in this industry turn to the exhibitors' favor only in the later part of a movie's run. The results of the second heuristic, rank-based heuristic, show that it performs quite well in profit terms. We use a simple version of the rank-based optimal policy we alluded to in Chapter 2. Still, its good performance in a multiple screen setting and a longer planning horizon suggests that some reasonable rank-based heuristics could be derived for the exhibitor problem. 4.3.2 L imi ta t ions and Future Research One of the obvious limitations of our analyses in this chapter and the previous chapters is that our model is yet to be implemented in a real-world setting. Though we attempt to show in this chapter that the profit levels resulting from the use of our model are quite attainable in a real-world setting, it is difficult to anticipate all the complications involved in the actual implementation process. The next step for future research in this effort is to arrive at an understanding regarding how a manager might use our model in his/her routinized activities. We speculate on some key areas of importance in this endeavor. A n important area of concern deals with the issues involved with the manager-model interface. How will the manager use the schedules suggested by the model and how will this change over time? In the case of ARTS P L A N , the manager extensively used the A R T S P L A N model as it reduced uncertainty and saved time, but the manager also made personal choices in scheduling and revised model forecasts. In the movie industry, the theater owner has ongoing relationships with the distributors, and likely will not make decisions on a single transaction basis. The availability of SilverScreener both provides an optimal schedule, and an estimate of the value of honoring a relationship as compared to making a more profitable, but perhaps short-term, decision. W i l l such information Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 174 change the nature of the relationship over time, and the way decisions are made? A n issue related with the above question is whether the manager using the model will become a better decision maker, or the availability of an automated system will make him lose his intuition. A recent study by Bruggen, Smidts, and Wierenga (1998) suggests that the use of a decision support model 2 8 increases the effectiveness of marketing decision makers. The reason they provide is that when managers are confronted with a complex decision environments, they may lapse into mental-effort-reducing heuristics such as anchoring and adjustment29. In an experimental study based on M A R K S T R A T simulation (Larreche and Gatignon 1990), Bruggen, Smidts, and Wierenga (1998) show that the decision makers using the model with an automated system are less susceptible to applying such heuristics, and, therefore, show more variation in their decisions in a complex and dynamic environment. A related issue of interest in future research is to consider the critical factors that might affect the implementation process. In other words, we need to address the following question: Under which conditions, or for which type of managers, is the use of our model (or decision support models in general) most recommended? Previous researchers (Chakravarti et al. 1979; Bruggen, Smidts, and Wierenga 1998) suggest two factors that are relevant to current study: the cognitive style of the manager, and time pressure faced by the manager. In general, their results suggest that low-analytical decision makers who do not operate under a lot of time pressure find the decision support models more useful. The design of our decision support system must include proper assessment of such factors in the implementation process. In conclusion, the major objective of this chapter was to show how the SilverScreener •^We use the term decision support model for marketing decision support system (MDSS) in Bruggen, Smidts, and Wierenga's (1998) study. •^Decision makers typically employ the adjustment from an anchor heuristic (Tversky and Kahneman 1974) in making judgments under uncertainty. Decisions are made by anchoring on the previous decision and then adjusting by a certain percentage. These adjustments are often nonoptimal since decisions will be biased towards their initial values, which may be insufficient for present market conditions. Chapter 4. Approaches to Estimating Managerial Gains from SilverScreener 175 model can be implemented in a realistic setting and achieve the level of improvement claimed in the earlier chapter. Towards this end, we proposed the types of managerial gains our model offers, and provided the methods for estimating those gains. The two-tier application of SilverScreener, involving master plan and rolling horizon approaches, shows how the model could assist managerial decision making in the exhibition business. At a broad level, our model would benefit managerial decision making in two different ways. One is by helping the managers improve their scheduling techniques, and consequently profitability. That is, the model could help the manager adopt a new decision making style which is based on a scientific approach. The second way is by helping the manager perform the same scheduling task in a more efficient way. That is, even if the manager is able to schedule the movies in an "optimal" fashion, the effort required to do so might be exorbitant. The availability of a computer-based algorithm will automate this process and help save management's time. This benefit would be especially valuable for the managers of theater chains who have to make such scheduling decisions for a number of theaters on a routinized basis. Chapter 4: Approaches to Estimating Managerial Gains from SilverScreener 176 Table 4.21: Master Plan Schedule (Using ex ante Data for Revenue Prediction and Optimization Approach for Schedule Generation)* Week\Screen 1 2 3 4 5 6 1 NYS C A D L L O M PA6 L T H N 2 NYS C A D L L O M PA6 L T H N 3 NYS C A D L L O M SING H 4 D C T A M L L O M SING H 5 D C T A M L L O M DOC H 6 D C T A M L S Y M DOC H 7 D C T A M L S Y M DOC H 8 D C T A M L MF DOC H 9 E G E SNEHNE M L M F DOC H 10 E G E SNEHNE M L M F DOC H 11 PC SNEHNE M L IJ DOC IJ 12 PC SNEHNE DPS IJ NHB IJ 13 ST5 SNEHNE DPS IJ NHB IJ 14 ST5 SNEHNE DPS IJ NHB IJ 15 B SNEHNE B IJ HISK IJ 16 B SNEHNE B IJ HISK U 17 B SNEHNE B IJ LW2 IJ 18 B L T K B IJ LW2 IJ 19 B L T K B U LW2 H 20 B L T K B IJ LW2 IJ 21 B L U B U LW2 IJ 22 B L U B IJ LW2 NES5 23 B R A B IJ LW2 NES5 24 B R A B IJ TP C 25 B R A RL IJ TP C 26 B R A R L IJ TP C 27 B R A RL IJ TP C * - Using the following cost assumptions - Variable costs are 33% of box-office revenue, concession profits are 40% of box-office revenue, fixed costs are 33% of the total box-office revenue in a week. Chapter 4: Approaches to Estimating Managerial Gains from SilverScreener 111 Table 4.22: Bidding Solutions Criterion Master Plan Optimal Restricted Set Approach > 2 weeks Number of movies chosen for bidding Number of common movies All 31 movies scheduled by the master plan 31 2 All 27 movies scheduled by the optimal restricted set 27 3 > 3 weeks Number of movies chosen for bidding Number of common movies NYS, CA, DL, L O M , H, DC, TA, M L , DOC, SNEHNE, MF, DPS, NHB, IJ*, B*, LTK, LW2, RA, RL, TP, C 21 1 NYS, CA, DL, L O M , H, DC, TA, M L , PC, SDL, MF, EGE, DPS, IJ*, B*, HISK, LTK, LW2, L U , RL, TP, C 22 7 > 4 weeks Number of movies chosen for bidding Number of common movies L O M , H, DC, TA, M L , DOC, SNEHNE, IJ*, B*, LW2, RA, TP, C 13 NYS, DL, H, DC, TA, M L , SDL, MF, E G E , DPS, IJ*,B*,HISK, LW2, L U , TP, C 17 9 > 5 weeks Number of movies chosen ibr bidding Number of common movies H, DC, TA, M L , DOC, SNEHNE, IJ*, B*, LW2, R A 10 NYS, DL, H , DC, TA, M L , SDL, DPS, IJ*, B*,HISK, LW2 12 7 > 6 weeks Number of movies chosen for bidding Number of common movies H, M L , DOC, SNEHNE, IJ*, B*, LW2 7 NYS, H, DC, TA, SDL, DPS, U * , B * , HISK, LW2 10 4 - On two screens. Chapter 4: Approaches to Estimating Managerial Gains from SilverScreener 178 Table 4.23: Characterization of Solution Actual Optimal Master Distributor Distributor Rank Rank Schedule Restricted Plan1 Pressure Pressure Based Based Set Heuristic1 Heuristic2 Heuristic1 Heuristic2 Schedule2 Profit ($'s) 585,175 805,988 413,328 329,525 571,990 388,220 736,138 Percentage change in 0 +37.7 -29.4 -43.7 -2.2 -33.7 +25.8 profit from actual Number of different 43 27 31 43 43 34 35 movies scheduled Average run length 3.8 6 5.2 3.8 3.8 4.8 4.6 (screening slots) -Percentage scheduled for <=2 weeks 46.5 11.1 29 46.5 48.8 23.5 25.7 3 weeks 14 22.2 22.5 30.2 20.9 38.2 28.6 > 4 weeks 39.5 66.7 48.5 23.3 30.3 38.3 45.7 Double booking for (weeks ) Batman 3 9 9 8 11 8 5 Indiana 4 6 11 7 6 9 6 Jones Star Trek V 2 2 0 2 2 2 2 Screening slots for Batman 16 21 21 21 22 20 18 Indiana 13 13 28 24 13 26 13 Jones Lethal 11 11 7 2 11 5 11 Weapon II Dead Poets 8 16 3 3 7 3 16 Society New York 5 7 3 3 3 3 5 Stories Scandal 5 7 0 2 5 0 6 1 - Using ex ante data, 2 - Using ex post data Note - The following cost assumptions are used for all the cases - Variable costs are 33% of box-office revenue, concession profits are 40% of box-office revenue, fixed costs are 33% of the total box-office revenue in a week. Chapter 4: Approaches to Estimating Managerial Gains from SilverScreener 179 Table 4.24: Actual Implied Schedule Obtained Using Rolling Horizon Approach WeekAScreen 1 2 3 4 5 6 1 NYS C A D L L O M PA6 L T H N 2 NYS C A D L L O M DB L T H N 3 NYS C A D L L O M SING H 4 NYS M L D L D C T A H 5 NYS M L DOC D C T A H 6 S Y M M L PS SZ T A H 7 SDL M L PS D C T A H 8 SDL M L PS MF T A H 9 SDL M L PS SNEHNE T A H 10 SDL FQC PS SNEHNE T A H 11 SDL E G E IJ SNEHNE T A IJ 12 SDL DPS IJ SNEHNE NHB IJ 13 SDL DPS IJ SNEHNE ST5 ST5 14 SDL DPS IJ SNEHNE ST5 IJ 15 B DPS IJ B HISK H 16 B DPS IJ B HISK U 17 B DPS IJ B HISK LW2 18 B DPS L T K B HISK LW2 19 B DPS L T K B HISK LW2 20 B DPS L T K B FT8 LW2 Chapter 4: Approaches to Estimating Managerial Gains from SilverScreener Table 4.25: Characterization of Rolling Horizon Results 180 Actual Optimal Restricted Actual Implied Schedule* Set Schedule* Schedule (rolling horizon approach)* Profit ($'s) 126,030 259,470 250,309 Percentage change in profit over actual 0 +105.9 +98.6 Number of different movies scheduled 35 22 31 Average run length 3.4 5.5 3.9 (screening slots) Percentage schec uled for <=2 weeks 48.5 9.1 38.7 3 weeks 14.3 18.2 12.9 > 4 weeks 37.2 72.7 48.4 Double booking for (weeks) Batman 3 6 6 Indiana Jones 4 6 5 Star Trek V 2 2 1 Screening slots for Batman 9 12 12 Indiana Jones 13 13 12 Lethal Weapon II 4 4 4 Dead Poets Society 6 9 9 New York Stories 5 7 5 Scandal 5 7 8 * - Over a period of 20 weeks. Note - The following cost assumptions are used for all the cases - Variable costs are 33% of box-office revenue, concession profits are 40% of box-office revenue, fixed costs are 33% of the total box-office revenue in a week. Chapter 4: Approaches to Estimating Managerial Gains from SilverScreener 181 Table 4.26: Schedule Generated Using Distributors' Pressure Heuristic and ex ante data for Revenue Estimates* Week\ Screen 1 2 3 4 5 6 1 NYS C A D L L O M PA6 L T H N 2 NYS C A D L L O M PA6 L T H N 3 NYS SING D L L O M H DB 4 M L SING D C T A H DB 5 M L SING DC T A H DOC 6 M L S Y M PS SZ H DOC 7 SDL S Y M PS SZ LB DOC 8 SDL S Y M PS MF LB DOC 9 SNEHNE S Y M E G E MF L B DOC 10 SNEHNE FQC E G E MF LB DOC 11 SNEHNE FQC PC IJ IJ DOC 12 SNEHNE DPS PC IJ IJ NHB 13 ST5 DPS ST5 IJ LT NHB 14 ST5 DPS ST5 IJ JJ NHB 15 B B HISK IJ LT NHB 16 B B HISK IJ IJ NHB 17 B B HISK IJ LI LW2 18 B B L T K IJ PP LW2 19 B B L T K IJ PP UHF 20 B B T H IJ FT8 UHF 21 B B T H IJ FT8 L U 22 B B T H IJ NESS L U 23 B R A LIR IJ NESS L U 24 B R A LIR IJ TP C 25 B R A RL IJ TP C 26 B R A RL IJ TP c 27 B R A RL IJ N G C * - Using the following cost assumptions - Variable costs are 33% of box-office revenue, concession profits are 40% of box-office revenue, fixed costs are 33% of the total box-office revenue in a week. Chapter 4: Approaches to Estimating Managerial Gains from SilverScreener 182 Table 4.27: Schedule Generated Using Distributors' Pressure Heuristic and ex post Data for Revenue Estimates* WeekA Screen 1 2 3 4 5 6 1 NYS C A D L L O M PA6 L T H N 2 NYS C A D L L O M PA6 L T H N 3 NYS C A D L DB SING H 4 M L D C T A DB SING H 5 M L D C T A DB DOC H 6 M L S Y M PS SZ DOC H 7 M L S Y M PS SZ SDL L B 8 M L S Y M PS MF SDL L B 9 M L SNEHNE PS MF SDL E G E 10 FQC SNEHNE PS MF SDL E G E 11 FQC SNEHNE IJ PC SDL IJ 12 NHB SNEHNE IJ PC DPS IJ 13 NHB ST5 IJ ST5 DPS IJ 14 NHB ST5 IJ ST5 DPS IJ 15 HISK B IJ B DPS IJ 16 HISK B IJ B DPS IJ 17 HISK B LW2 B DPS IJ 18 L T K B LW2 B DPS PP 19 L T K B LW2 B UHF PP 20 T H B LW2 B UHF FT8 21 T H B LW2 B L U FT8 22 T H B LW2 B L U NES5 23 R A B LW2 B LIR NES5 24 R A B LW2 TP LIR C 25 R A B LW2 TP R L C 26 R A B LW2 TP R L C 27 N G B LW2 TP RL C * - Using the following cost assumptions - Variable costs are 33% of box-office revenue, concession profits are 40% of box-office revenue, fixed costs are 33% of the total box-office revenue in a week. Chapter 4: Approaches to Estimating Managerial Gains from SilverScreener 183 Table 4.28: Schedule Generated Using Rank-Based Heuristic and ex ante Data for Revenue Estimates* WeekA Screen 1 2 3 4 5 6 1 NYS-. C A D L L O M PA6 L T H N 2 NYS C A D L L O M PA6 L T H N 3 NYS C A D L L O M H SING 4 M L D C T A L O M H SING 5 M L D C T A L O M H DOC 6 M L DC T A S Y M H DOC 7 M L LB T A S Y M H DOC 8 M L LB T A MF H DOC 9 M L E G E SNEHNE MF H DOC 10 M L E G E SNEHNE MF FQC DOC 11 IJ IJ SNEHNE PC FQC DOC 12 LT IJ SNEHNE PC DPS NHB 13 H IJ ST5 ST5 DPS NHB 14 IJ IJ ST5 ST5 DPS NHB 15 IJ IJ B B HISK NHB 16 IJ IJ B B HISK NHB 17 IJ IJ B B HISK LW2 18 LT IJ B B L T K LW2 19 IJ IJ B B L T K LW2 20 IJ T H B B L T K LW2 21 IJ T H B B L U LW2 22 IJ T H B B L U NES5 23 U R A B B L U NES5 24 IJ R A B TP C NES5 25 JJ R A B TP C RL 26 IJ R A B TP C R L 27 IJ R A B N G C R L * - Using the following cost assumptions - Variable costs are 33% of box-office revenue, concession profits are 40% of box-office revenue, fixed costs are 33% of the total box-office revenue in a week. Chapter 4: Approaches to Estimating Managerial Gains from SilverScreener 184 Table 4.29: Schedule Generated Using Rank-Based Heuristic and ex post Data for Revenue Estimates* Week\ Screen 1 2 3 4 5 6 1 NYS C A D L L O M PA6 L T H N 2 NYS C A D L L O M PA6 L T H N 3 NYS C A D L L O M DB H 4 NYS M L D C T A DB H 5 NYS M L D C T A DOC H 6 S Y M M L PS T A DOC H 7 S Y M M L PS T A SDL H 8 M F M L PS T A SDL H 9 M F SNEHNE PS T A SDL E G E 10 M F SNEHNE FQC T A SDL E G E 11 IJ SNEHNE FQC IJ SDL E G E 12 IJ SNEHNE DPS IJ SDL NHB 13 u ST5 DPS IJ ST5 NHB 14 IJ ST5 DPS IJ ST5 NHB 15 U B DPS IJ B HISK 16 IJ B DPS U B HISK 17 LW2 B DPS U B HISK 18 LW2 B DPS L T K B HISK 19 LW2 B DPS L T K B HISK 20 LW2 B DPS L T K T H FT8 21 LW2 B DPS L U T H FT8 22 LW2 B DPS L U T H NES5 23 LW2 B DPS L U LIR NES5 24 LW2 B DPS TP LIR C 25 LW2 B DPS TP R L C 26 LW2 B DPS TP RL c 27 LW2 B DPS TP RL c Using the following cost assumptions - Variable costs are 33% of box-office revenue, concession profits are 40% of box-office revenue, fixed costs are 33% of the total box-office revenue in a week. Chapter 5 Conclus ion 5.1 Discussion In this thesis we examined how quantitative model building can assist marketing decision making in complex dynamic environments involving perishable products. In particular, we applied marketing and management science (operations research) methods to dynamic retail management of movies. In the introduction we explained the nature and scope of this thesis. This thesis is positioned at the interface of marketing and operations, which has been recognized as an important research domain by previous researchers. It focuses on shelf (display) space management, an area of important concern to retailers. This thesis advances the shelf space management research by considering the case of a perishable product such as movies. The problem addressed is quite complex for several reasons. Some of these reasons are due to the factors specific to the movies context, which we have discussed in the earlier chapters. However, the major contribution to the complexity comes from two sources. The first is the perishable nature of the product considered, that is, the appeal of a movie declines every week. The second source, which is related to the first one, is the dynamic aspect of the decision making involved. Together, the two sources imply that the decision maker must have a long-term profitability objective in order to make smart exhibition decisions in the movie industry. That is, since an action taken today affects the future states of the nature, the decisions at each epoch must be made keeping 185 Chapter 5. Conclusion 186 in mind all possible future realizations of the states of the system. We formalize this dynamic decision making situation by means of a sequential decision model (Puterman 1994). In this thesis, we use two different versions of this model, the Markov decision process (MDP) (Puterman 1994) and (deterministic) dynamic programming (Bellman 1957) approaches. We also proposed an integer programming model, which is equivalent to the deterministic dynamic programming version of the model. Chapter 2 considers the general problem of movie replacement from a theoretical standpoint by addressing the stochastic aspects associated with movie scheduling and replacement. We formulate this problem as a Markov decision process (MDP) model. The problem is analogous in important ways to the equipment replacement problem in maintenance theory. Our analysis reveals that the exhibitor is better off in those situations in which his/her shelf-space becomes a scarcer commodity for the upstream channel members. The analysis of a contract parameter indicates that a smart exhibitor associates a "cost" with the parameters of the contract provided by the distributor. Our results also show that it is important to have the right information for making smart decisions in the exhibition business. Finally, the characterization of optimal policy results from various problems reveals that the optimal policies for the movie replacement problem are in the general direction of a control limit policy, when the state variable comprises of an element, rank of a movie, conditioned appropriately on the other elements. This rank-based characterization results in a simple heuristic, which is easier to implement and compute and is used in Chapter 4. Chapter 3 applies the theoretical concepts developed in Chapter 2 to a special case of the general movie replacement problem. The output of this essay is SilverScreener, which is a decision support model for movie exhibitors, the retailers of the motion picture industry who own multiple screen theaters. The model helps the exhibitors to both select (which movies) and schedule (how long, when) the movies at their theater. The developed Chapter 5. Conclusion 187 model is readily implementable and appears to lead to considerable improvement in profitability in two different comparative cases. The general nature of the optimal policy emerges as: choose fewer "right" movies and run them longer. This holds even in an expanded set analysis when the number of movies available almost doubles. Through sensitivity analysis, we demonstrate that these results are robust to various parameters of the problem. The major contribution of Chapter 4 is that it proposes a methodology to demonstrate effectiveness of a mathematical model without having the advantage of "being there," that is, the availability of a real application setting. We propose a two-tier application of SilverScreener. The first tier involves development of a m a s t e r plan that would help the manager plan a season before the start of that season. The master plan could assist the manager to decide, before the season, whether to bid for a movie or not. The results show that the master planning exercise can help the exhibitor make effective planning decisions. The second tier, rolling horizon approach, can be used for weekly decisions during the season, possibly after the development of the master plan. This application involves "rolling" and updating data, from one time window to another. Our results show that the rolling horizon approach can achieve an improvement (in profit terms) over actual schedule of the same order as that achieved by a perfect information based one-shot optimization approach. We also propose two alternative decision rules (heuristics) to compare the performance of SilverScreener model under two different levels of information availability. One of these heuristics is based on the ideas developed in Chapter 2, and performs reasonably well. However, SilverScreener outperforms the two heuristics under both levels of information availability. In this chapter, therefore, we show that application of the SilverScreener model can improve the manager's profitability and decision making, and promises to be an effective scheduling and planning tool. At a broad level, we interpret our results from the three chapters as providing three Chapter 5. Conclusion 188 important messages to the retailers of the perishable products. First, these results show that a complex problem can be modeled conceptually and resolved in a way that may lead to the implementation of its solution. The solution yields two types of results. At a broad level, the normative policy recommends having a bias against quick replacements. At a more specific level, the decision support model can help in up-front season planning, and scheduling every week later during the season. Second, the general recommendation to the retailers of perishable products such as movies is that they should pick the right product and "stay" with them longer instead of replacing too quickly. The recommenda-tion is somewhat surprising given that this industry is characterized by a wide range of innovations with short life cycles, which yield a major portion of their revenues early in their run. It would appear reasonable to have a bias towards quick replacements, which concurs with practice. However, the recommendation is based on the economics of the managerial situation considered from a long-term standpoint, which brings us to our last recommendation. Finally, we recommend that these retailers need to adopt a long-term view in their dynamic decisions. Though having a long-term view is advisable for almost any business situation, such a view becomes especially critical for the retailers of the perishable products. This is because not only does the existing product decay over time, but the replacement product may also loses its appeal every period. For example, in Chapter 2, we find, in the context of obligation period, that the smart exhibitor finds it optimal to wait for high quality later than replace with a low quality movie earlier that is available with a high obligation period. Similarly, Chapters 3 and 4 show that the improvements in profitability result when decisions are taken keeping in mind the future availability and appeal of movies. In fact, the long-term view forms the basis of part of Chapter 3's general policy to play the movies longer. Collectively, these results imply that the retailers need to develop precise methods to determine the appeal of the products available in the market. This was found especially Chapter 5. Conclusion 189 critical in Chapter 2's results on the supply conditions and the value of having precise information in this industry. The estimation methods could take form of either a fore-casting routine, or a discrimination mechanism between high and low quality products. Especially challenging in this endeavor would be a pre-launch assessment of the prod-uct's quality. Some early models to this effect are beginning to appear in the academic literature (Sawhney and Eliashberg 1996; Manceau, Eliashberg and Rao 1998). Since the focus of our research was on scheduling, we did not explore the forecasting area in detail. However, our results, especially from Chapters 3 and 4, show that the potential for profitability improvement could be enormous if the models such as SilverScreener are joined at the back end of an effective forecasting routine. 5.2 Future Research We now discuss the directions for related future research in the current area. Since we have noted the future research issues specific to a chapter in the respective chapter, we discuss some general implications and the areas of broad interest in this section. A n a l y t i c a l Developments The general model presented in Chapter 2 is quite complex, which does not allow its application to the large-size problems. This is due mainly to the fact that our model considers the probability transitions of the weekly revenue of each movie separately since this formulation considers each movie as an independent entity. However, studies like Jedidi, Krider and Weinberg (1998) show that movies tend to fall in distinct clusters in terms of their opening market shares and decay (or retain) rates. If we recognize these latent "classes" of movies, then the exhibitor problem can be reduced to a more manageable size. The basic premise in using such a notion of classes is that it would help Chapter 5. Conclusion 190 the exhibitor identify and characterize similar movies by a common parameter1. We propose two approaches for the allocation of different movies to classes: 1) attribute-based, and 2) historical data-based. In the attribute-based approach, we can classify movies by one or more of their attributes. One such classification scheme based on the genre of a movie could be: A) action/adventure/science fiction, B) kids/comedy, and C) emotional/romance. Notice that these classes could vary from theater to the-ater depending on the history of the performance of a number of movies at a particular theater. For example, an exhibitor may notice over time that Class A movies open very strongly, but decay rapidly. Class C movies, which primarily build on word-of-mouth, may open moderately, but retain themselves well. The historical data-based approach, instead of imposing a pre-defined classification structure, would try to "infer" the differ-ent classes of movies from the historical data based on the revenue patterns of various movies from the past data. Having inferred the classes from the past data, the exhibitor may then ascribe the classes certain labels. The optimal policy derived for the above operationalizations would be based on the "historically average behavior" of a number of movies within a class. A control limit or critical number policy in these cases will comprise decision rules that "detect" better or worse than average performances. For example, a policy for a three-class (say, X , Y and Z) case based on the above scheme could be2: If a Class X movie is playing currently, then continue if it is in its obligation period, or its rank is less than Sx,or no Class Y or Z movies are available in the ranks less than SY and respectively; else replace it by a Class Y or Z movie, whichever is available in a better rank. The critical numbers, Sx, SY, and S%, may be functions of the historical performance of movies within a class, and could be different for each class. The idea behind such a policy is that the exhibitor •'Though it has different objectives, we present a movie classification and revenue prediction approach based on similar ideas in Chapter 4. 2We remind the reader that a higher rank number represents a worse position of a movie. Chapter 5. Conclusion 191 continues to play an existing movie as long as it makes economic sense in view of the attractiveness of the alternative movies, otherwise he/she replaces it if the existing movie is not in its obligation period. The other future research ideas related to the modeling aspect of Chapter 2 could be the problem's formulation as a bandit problem and the incorporation of dynamic gener-alized models or Bayesian techniques in modeling. For example, in Bayesian frameowrk, the prior probability for a movie belonging to a certain class could be based on its char-acteristics, such as genre, actors, director, and so forth. Then, as more data becomes available, the Bayesian updating could assign a movie to a class with greater probability. M u l t i p l e x Environment The first extension we referred to in the earlier chapters is the treatment of capacity issues in a multiple screen theater case. This problem could be modeled separately, say as a two-screen case, and can address interesting issues such as: Should the exhibitor continue, switch to the lower capacity screen, or drop an existing movie in the wake of perishing demand and varying availability of alternative movies? The other extensions could be classified as micro-scheduling issues. One of these issues is the timing of differ-ent shows on different screens within a day. This may be important to exhibitor for two reasons. First, if different movies on all the screens are scheduled simultaneously, then the exhibitor may lose some concession customers, a major source of revenue, because of the traffic generated at the concession stands. A waiting-time simulation experiment could be coupled with a scheduling routine to generate useful insights into this issue. Second, given that some audiences may be late for a particular show, a proper timing of the other shows may make them stay for another movie, say, half an hour later. Notice that in addition to scheduling within a day, these issues are also related to important product portfolio issues. For example, should two films of the same genre be shown on Chapter 5. Conclusion 192 the same day, and if so, then how far apart should their show timings be. This is analo-> gous to increasing product variety for the customers such that if they cannot afford (are late for) one product, they still have another choice. Similarly, how far apart in time a kids/cartoon movie (which is likely to generate a lot of concession demand!) be shown from another movie. Implementat ion Issues The results from Chapters 3 and 4 show that the SilverScreener model holds promise to improve managerial decisions making by providing an effective scheduling tool. We must emphasize here that the improvement claims made in the previous chapters are based on the scheduling property of our models. This decision making aspect should not be confused with the issues related with revenue prediction or forecasting. This is because revenue estimates are inputs to our model. Thus, the first future research idea in implementation context is the development of a good forecasting routine. This is espe-cially important given our results that suggest that the manager could make sub-optimal if he/she does not have the right information available. There are several alternatives available towards this end. We used a version of Sawhney and Eliashberg's (1996) fore-casting model, which appears reasonable for our purposes. More sophisticated models could be developed, such as those for new product forecasting, depending on the specific implementation scenario. Alternatively, the manager could use judgments for prediction purposes. The challenge here would to design an easy to use procedure for eliciting man-agerial estimates. In either case, it is clear that a successful application would require an interaction of the manager and model. This suggests another interesting line of future consideration concerning how the model would be used by managers and how its usage would change the decision-making environment. In the current work, we noted that the manager could use the decision support model Chapter 5. Conclusion 193 in two different ways, master planning and rolling horizon. The master plan situation is quite analogous to the use of ARTS P L A N , as a scheduler before the season begins. The second usage of SilverScreener is as a weekly planning and control device once the season begins. It is difficult to predict at this time how the manager will use the sug-gested schedule and how this will change over time. In the movie industry, the ongoing relationships of the exhibitor with the distributors will be of paramount concern. The availability of the model provides an estimate of the value of honoring a relationship as compared to making a more profitable, but perhaps short-term, decision. This suggests the use of the model as a scenario analysis tool. It will be interesting to examine whether such information will change the nature of the relationship over time, and the way de-cisions are made3. Alternatively, the manager may choose to use a rank-based heuristic based on the ideas developed in Chapter 2. The exact form of such a heuristic would depend on the specific application context. The above discussion also suggests that the model will be best used if it evolves with managerial use as time progresses. For example, in the case of ARTS P L A N , the manager extensively used the ARTS P L A N model as it reduced uncertainty and saved time, but the manager also made personal choices in scheduling and revised model fore-casts. The evolutionary approach can also be interpreted in terms of the level of model sophistication. For example, Montgomery and Weinberg (1973) note in the context of implementation of marketing models that, "One way to overcome these difficulties [in implementing complex models] is to adopt an evolutionary approach which starts from a simple model and becomes increasingly sophisticated as greater understanding of the problem and rapport with management is developed. Urban's (1970) S P R I N T E R (Spec-ification of Profits with Interdependencies) is a new product model which exists in three 3Interestingly, a manager of one of the exhibition chains expressed interest in using the model after the season with ex post data. This would provide the manager a benchmark against which he/she could compare his/her decisions. Chapter 5. Conclusion 194 main evolutionary versions (Mod I, II, and III)." The decision support model developed in this thesis could be interpreted as a first version in the evolution of such models. More sophisticated models could be developed in future which handle greater complexity as introduced by the differential capacity constraints, joint revenue prediction and schedul-ing, and so on. D i s t r i b u t i o n C h a n n e l Issues Despite some evidence in our analysis suggesting a win-win situation for the entire system, the SilverScreener model provides in general more power to the exhibitor as com-pared to the distributor. With improved usage of information available to the exhibitor, he/she should be able to improve profitability. This is consistent with what appears to be a general trend of increased power to the retailer, and a growing interest by marketing academics in analyzing problems from the retailer's as compared to the manufacturer's perspective (Messinger and Narsimhan 1995). However, as in other areas of distribution channels, the retailers and manufacturers do not always have opposing interests, and a possibility of win-win situation might exist. Such a possibility will depend on the nature of the contract terms between the distributor-exhibitor pair. In this context, it would be intriguing to examine the impact of contract terms on scheduling in particular and industry profitability more generally. Present contracting practices were designed in an environment quite different from the one existing now. Early results from Swami, Lee, and Weinberg's (1998) study suggest that win-win situation could result if the current contractual practices were replaced with even simpler contract terms such as two-part tariffs, fixed sharing and slotting allowance schemes. Another extension in distribution channel context could be the consideration of the competitive effects between two the-aters. The competitive effects could cause some movies to be available to a theater, and not to another. Another dimension this "game" could affect is the bidding strategy of a Chapter 5. Conclusion 195 theater depending on the reactions of the other theater. Genera l izat ion Though we focus on movies in this thesis, our methodology and results can readily be generalized to other entertainment products (e.g., performing arts, books, video games), travel services, fashion goods, and educational programs. Not only are the scheduling problems studied here analogous to those in many areas of the performing arts, but there are analogies to a variety of entertainment related products. For example, the computer game market features virtually continuous new product introductions and rapid obso-lescence of existing products. Electronics Arts, for example, a producer of such leading games as N H L Hockey and Madden NFL , reports that more than 50% of its quarterly sales are from products that did not exist in the previous quarter. According to a Wall Street Journal article on the book industry (Knecht 1997), the growth of mega-stores has transformed the publishing industry into one similar to movies, in which a limited number of recent and popular books command a large portion of sales and get the prime shelf space. If the books do not sell, they are removed within a matter of weeks from the prime display areas. Interestingly, the contract terms in the book industry are such that unsold books can be returned by the retailer to the manufacturer for a refund. Wi th a reported 35% of a mass market book's print run being returned to the publisher on average, the opportunity for shelf space algorithms such as SilverScreener appears to be considerable. In conclusion, the objective of this thesis was to understand and formalize, using quantitative modeling techniques, certain dynamic decision making situations involving shelf space management of perishable products. The understanding and formal struc-ture helped us develop and characterize optimal normative policies for these problems. Chapter 5. Conclusion 196 Towards this end, we presented three essays in the three core chapters of this thesis. The wide range of future research ideas listed above suggests that this thesis opens a new arena of research. 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Huizingh, and P. A . F. M . van Campeh (1994), "Hierarchical Scaling of Marketing Decision Support Systems," Decision Support Systems, 12, 219-232. Williams, David N. (1997), "Time Indexed Formulation of Scheduling Problems," M.Sc. Thesis, University of British Columbia, Vancouver, Canada. Zilla, Sinuany-Stern (1993), "Replacement Policy Under Partially Observed Markov Decision Process," International Journal of Production Economics, 29, 159-166. Zufryden, Fred S. (1986), " A Dynamic Programming Approach for Product Se-lection and Supermarket Shelf-Space Allocation," Journal of the Operational Research Society, 37(4), 413-422. " (1996), "Linking Advertising to Box Office Performance of New Fi lm Releases-A Marketing Planning Model," Journal of Advertising Research, (July-August), 29-41. Appendix A Programming Code in C Language for MDP Algorithm The following code in C language was used to solve the scenario analysis problems in Chapter 2. The program is assembled from seven different files: two header files and five program files. The description of each file is given in the first line of the respective file. /* PROCEDRS.H - Header file for procedure templates and structure declarations of MDP.C Program */ void readatafj; void matj5en0; int chk_state_feasibility(int movie_playing, int runlength, int level_no, int stage_no); int chk_action_feasibility(int movie_playing, int runjength, int level_no, int stage_no, int action); float get_probab (int first, int second, int movie_playing, int action, int stage_no); struct level2 { float maxmoney; unsigned short maxindex; }; struct time2 { struct level2 level[NUM_LEVELS]; }; struct active2 { struct time2 time[NUM_WEEKS]; }; struct week { struct active2 active|MJMJviOVIES]; }; 204 Appendix 205 /* PRMTRS.H - Header file for parameters (for an example problem) of MDP.C Program */ # ifndef PRMTRS # define PRMTRS # define N U M W E E K S 9 # define N U M M O V I E S 4 # define N U M L E V E L S 625 /*from combination of number of ranks and movies*/ # define N U M R A N K S 5 float rev|NUMJl\NKS][>rUMJVEEKS], initialjrrNUM_MOVIES][NTJM_RANKS], ind_pr[NUM MOVIES] [NUM_RANKS] [NUMRANKS]; int rel_dt[NUM_MOVTES]; unsigned short nmt[NUM_LEVELS+l][NUM_MOVIES+l]; int OPD|NUM_MOvTES]; #endif /* MDP.C - The main driver of the program, which includes the main program for solving MDP problems of Chapter 2*1 #include <stdio.h> #include <math.h> #include <stdlib.h> #include "prmtrs.h" #include "procedrs.h" void mainO { int run, stage, mv, i , j , k, 1, m, t, state_feasible, action_feasible; float MAXIM, probab, tran_prb; struct week wk[NUM_WEEKS]; mat_gen0; readataO; BIG_NO=999; for (stage=NUM_WEEKS-1; stage>=0; stage-) { for (mv=0; mv<NUM MOVIES; mv++) { for (run=0; run<= (stage-reldt [mv]); run++) { for (i=0; i<NUM_LEVELS; i++) { wk[stage]. active[mv] .time[run]. level [i]. max_money=0; wk[stage] .active[mv] .time[run] .level[i] .rnax_index=BIG_NO; state_feasible=chk_state_feasibility(mv, run, i , stage); if (state_feasible>0) Appendix 206 for (1=0; KNUM_MOVIES; 1++) { MAXIM=0; action_feasible^hk_action_feasibility(mv, run, i, stage, 1); if (action_feasible>0) { if(l=mv) MAXIM=MAXIM+rev[mat[i] [1]] [run]; else MAXIM=MAXIM+rev[mat[i] [1]] [0]; if (stage < NUM_WEEKS-1) { if (l=mv) t=run+l; else t=l; for (k=0;k<NUM_LEVELS; k++) { tran_prb= get_probab (i, k, mv, 1, stage); ]V1AXIM=MAXIM+ tran_prb * wk[stage+l].active[l].time[t].level|li].max_rnoney; }; }; }; if (MAXIM>wk[stage] .active[mv] .time[run] .level[i] .max_money) { wk[stage] .active[mv] .time[run] .level[i] .max_money=MAXIM; wk[stage] .active[mv] .time[run] .level[i] max_index=l; }; }; / * for(l=0.... */ }; /*if(state_feasible */ }; /*for(i=0; .... */ };/*for(run */ }; / * for (mv=0; ....*/ }; / * for (stage=0; ....*/ } /*end of main */ Appendix 207 /* M A T G E N . C - Function that generates different combinations of the ranks of various movies */ #include <stdio.h> #include <math.h> #include <stdlib.h> #include "prmtrs.h" void mat^genO { int q, num_replcd,ij; int mO; int m l ; intm2; intm3; q=0; for (mO=0;mO<NUM_RANKS;mO++) for (ml=0;ml<NUM_RANKS;ml++) for (rn2=0;rn2<NUM_RANKS;m2++) for (m3=0;m3<NUM_RANKS;m3++) { mat[q][0]=m0; mat[q][l]=ml; mat[q][2]=m2; mat[q][3]=m3; num_replcd=0; q++; }; Appendix 208 /*: READ ATA. C - Function that reads the problem data */ #include <stdio.h> #include <math.h> #include <stdlib.h> #include "prmtrs.h" void readataO {FILE *infile; int j , k, 1; if((infile^open("example.(iat,,,,,r,,))=NULL) {printf(,,Could not open 'example.dat'"); exit(l); }; for (j=0J<NUM_RANKS for (k=0;k<NUM_WEEKS;k++) { fscaniTjnfiie, "%f, &rev[j][k]); }; for (j=0J<NUM_MOVIESJ-H-) for (k=0;k<NUM_RANKS ;k++) { fscanfXinfile, "%f\ &initial_pr[j][k]); }; for (j=0J<NUM_MOVIESiJ-H-) for (k=0;k<NXrM_RANKS;k++) for (1=0;1<NUM_RANKS;1++) { fscatif(infile, "%f, &ind_pr[j][k][l]); }; for 0=OJ<NUM_MOVIES J++) { fscanf(infile, "%d", &rel_dt[j]); }; for (k=0;k<NUM_MOVIES;k++) { fscanf(infile, "%d", &OPD[k]); }; fclose(infile); } Appendix 209 /* FEASIBLE. C- Set of two functions that check the feasibility of a given state and action variable, respectively. State variable is given as function argument in the first function, and both state and action variables are given in the second function. Both functions return 1 if feasible, 0 otherwise */ i^nclude <stdio.h> #include <math.h> #include <stdlib.h> #include "prmtrs.h" int chk_state_feasibility(int movie_playing, int runjength, int levelno, int stageno) { inti j , return_value,rl_datel,rl_date2,mvl; return_value= 1; for (j=0; j<NUM_MOVIES; j++) { if ((mat[level_no][j] = 0 && rel_dt[j]<stage_no) || (mat[level_no][j] = 0 && rel_dt[j]=stage_no) || (mat[level_no]D] ==NUM_RANKS-1 && rel_dt[j]=stage_no) || (mat[level_no]|j] > 0 && rel_dt[j]>stage_no) || (mat[level_no][moviejjlaymg]==(NUM_RANKS-l)) || (mat[level_no][movie_playing]=0) || (movie_playing=0 && runjength!=stage_no) || (movie_playing=0 && j!=movie_playing && mat[level_no][j]==NUM_RANKS-l) || (movie_playing!=0 && mat[level_no][0]!=NUM_RANKS-l)) return_value=0; }; return(returnvalue); } int chk_action_feasibility(int movie_playing, int runlength, int level_no, int stageno, int action) { int returnvalue; return_value=l; if ((mat[level_no][action]=0) || (rnat[level_no][action]==NUM_RANKS-l) || (run_length<OPD[movie_playing] && action! =movie_playing)) return_value=0; return(return_value); } Appendix 2 1 0 /*GET_PROB.C - Function that calculates transition probability for two given rank vectors. */ #include <stdio.h> #include <math.h> #include <stdlib.h> #include "prmtrs.h" float get_probab (int first, int second, int movie_playing, int action, int stage_no) { float probab; int m; probab=l; m=0; while (m<NUM_MOVTES & & probab>0.0) { i f (m=movie_playing) { i f (mat[first][m]>0 & & mat[first][m]<(NUM_RANKS-l) & & mat[second][m]==(NUM_RANKS-1) & & action!=movie_playing) probab=probab; else i f (mat[first][m]>0 & & mat[first][m]<(NUM_RANKS-l) & & mat[second][m]>0 & & mat[second][m]<(NUM_RANKS-l) & & action==movie_playing) probab=probab*ind_pr[m] [mat[first] [m]] [mat[second] [m]]; else probab=0.0; } else { i f ((mat[first][m]==0 & & mat[second][m]=0 & & stage_no<(rel_dt[m]-l)) || (mat[flrst][m]=NUM_RANKS-l & & mat[second][m]=NUM_RANKS-l)) probab=probab; else i f (mat[first][m]>0 & & mat[first][m]<{NUM_RANKS-l) & & mat[second][m]>0 & & mat[second][m]<(NUM_RANKS-l)) probab=probab*ind_pr[m] [mat[first] [m]] [matfsecond] [m]]; else i f (mat[first][m]==0 & & stage_no=(rel_dt[m]-l)) probab^robab*initial_pr[m] [mat[second] [m]]; else probab=0.0; }; m-t-+; }; return(probab); } 

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