International Construction Specialty Conference of the Canadian Society for Civil Engineering (ICSC) (5th : 2015)

Simulated schedule delay mitigation via float allocation Lucko, Gunnar; Thompson, Richard C. Jun 30, 2015

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5th International/11th Construction Specialty Conference 5e International/11e Conférence spécialisée sur la construction    Vancouver, British Columbia June 8 to June 10, 2015 / 8 juin au 10 juin 2015   SIMULATED SCHEDULE DELAY MITIGATION VIA FLOAT ALLOCATION Gunnar Lucko1 and Richard C. Thompson, Jr.1 1 Department of Civil Engineering, Catholic University of America, Washington, DC, USA Abstract: As delays in construction project schedules are widely documented, new approaches to manage this problem are needed. Previous studies have explored float as an inherent ability of schedules to absorb delays. However, all ignore the critical path; some recommendations as to float ownership directly contradict the inherently unfair ‘first come first serve’ principle; and none have derived testable methods to fairly allocate float to multiple participants. This study therefore employs the mathematical analogy of decision-making among a group of unequally sized individuals to explicitly allocate project float a priori to those most vulnerable – the critical path. While intuition might indicate allocation proportional to activity duration, or perhaps equal shares, it is demonstrated that neither is truly fair and a mathematical compromise can be found. The performance of this approach is tested quantitatively by simulating a case example of project schedules with and without such protection. Delays are modeled as probabilistic events that affect activities. It is found that even a relatively small amount of project float allocated along the critical path provides significant delay mitigation. 1 INTRODUCTION Construction project schedules in many countries unfortunately are frequently inundated with delays (Gündüz et al. 2013). Delays in this research are defined in a strictly temporal view, i.e. negative changes of start and/or finish of activities between the as-planned and as-updated project schedules. What makes delays particularly perilous is the fact that they are not an isolated local phenomenon within schedules, but may initiate a cascading set of ripple effects in later activities that depend on the earlier delayed one. Whatever the diverse causes of such delays are, which may stem from one or several risk factors that are materializing (González et al. 2013), is less relevant to this discussion than the unfortunate outcome, that it becomes likely that the contractually agreed turnover date will be met, i.e. the entire project will be late. Contractual effects of late completion typically are that the contractor is required to reimburse the owner for consequential cost (liquidated damages) of postponed occupancy of the new facility, which often leads to further side-effects, e.g. protracted court cases over liability that may damage the reputation and lose future clients. The central question remains not one of identifying and treating symptoms, but of curing the cause: What inherent defenses does a project schedule have against incurring delays? Such capabilities could take two forms, either preventive (lowering the probability of a delay occurring) – akin to inoculation; or reactive (gaining better means of alleviating a delay that occurs) – akin to therapeutics in medicine. To begin answering this question, this research will first review existing theory of project management regarding conceptual limitations of traditional scheduling and opportunities for protecting against delays, then explore potential theory in decision-making from which inspiration for such resilience will be gleaned. 025-1 2 LITERATURE REVIEW 2.1 Scheduling Assumptions Construction schedules are traditionally conceived and handled as networks, which mathematically are acyclic ‘graphs’ (Chen et al. 2001) that consist of nodes (activities) and arcs (links) between them, whose direction is dictated by the passing of time. Exploiting a rather simple and repetitive structure of relations of temporal data among activities led to the so-called critical path method (Kelley and Walker 1959) with which dates are sequentially calculated: Within activities, adding the duration to its start gives its finish (S + D = F). And between activities, all successors must finish as a condition for the successor to start (max {Fpred} = Ssucc). This forward-looking calculation assumes the best case that all activities start as early as possible at their earliest (E) dates. Its inverse, the backward-looking calculation assumes the worst case that they start as late as possible at latest (L) dates, so as to just not yet delay the project finish itself. Comparing these two scenarios gives information for each activity on how much it can be delayed without impacting (a) any successor (free float, FFpred = min{ESsucc} - EFpred), or ‘only’ (b) the project finish (total float, TF = LF - EF = LS - ES). Thus float is defined by an assumed delay. Due to the aforementioned vital importance of the project finish for contractual liability purposes, this research will focus on the total float. Float in the literature has been presented as the opposite of criticality, which has been defined as having “zero total float” (de la Garza et al. 2007, p. 836), and inspired the name for this scheduling technique. A subset of activities within any network schedule forms a continuous path – the eponymous critical path – between the first and last activity, which just like the overall sequential schedule may branch and merge. Hidden assumptions also lie in the inputs to such analysis, notably links and durations. Links (or schedule ‘logic’) are primarily determined based on technical reasons per laws of physics, but may also be modified for administrative reasons or even to artificially generate opportunities for success or failure (Zack 1992). This research assumes that links are fixed. Durations are determined based on work quantity as taken off plan drawings and specifications, combined with the crew choice and their unit productivity. However, one must consider that forecasting durations is always loaded with uncertainty. While studies have explored distributions to provide realistic durations for probabilistic scheduling (Khamooshi and Cioffi 2013), project managers in the construction industry have not been oblivious to the need for realistic duration estimates as opposed to overly optimistic ones. In practice, how well it is accomplished depends on the expertise of individual project managers, who determine ‘raw’ durations (from productivity data) and empirically extend them to ‘realistic’ ones by including a contingency (Barraza 2011). Contingency refers to both time or cost (Xie et al. 2012), but here it means additional time. This research assumes that raw durations are known to a user of this new approach, here presumed to be the general contractor. Studies have explored how contingency aids in company performance (Deng and Smyth 2013). It “has been used in project planning to cover activities delays, oversights, and unknowns and as a cushion for possible time-estimating errors” (Barraza 2011, p. 259). Since time contingency is essentially float, it will be referred to as such for clarity. The nature, quantity, and most beneficial distribution of such contingency, or float, will be explored next. 2.2 Criticality Paradox Whereas risk has been explored in detail, with many studies listing risk factors and how to rank or weigh them, often using survey methodologies (e.g. Tran and Molenaar 2012), its realization in form of delays has been studied primarily to create numerous competing or even contradictory approaches to distribute liability between owner and contractor (Yang and Kao 2012) after excusable delays have been forgiven, i.e. assigning blame, and pondering the vexing issue of concurrent delays (Ibbs et al. 2011). However, a posteriori analysis, while unfortunately necessary, is not even reactive, but merely inculpative. In other words, delay liability appears has been the focus rather than delay mitigation or, even better, avoidance. Rephrasing the definition of criticality as ‘an inability to absorb delays’ reveals that it is based upon an assumption itself, that activities in network schedules will necessarily have unequal capabilities to handle delays. However, schedulers control the manner in which activities are linked in network schedules (Zack 1992), so that criticality is essentially an artifact of the schedule itself. Criticality as it is currently defined 025-2 for critical path method is not an inherent quality of the activity based on its nature or risk, although this fact is often hidden, because long or difficult activities often actually turn out to be part of the critical path. Artificially dividing activities into being critical or not by whether or not they have total float in fact creates a paradox (Thompson and Lucko 2012, p. 488): “By definition, noncritical subcontractors have float, but the critical ones have none, even though they need it most.” Such two classes, of which critical activities are designed to be on the verge of failure, run counter to what should be the optimum goal of a scheduler: Creating a schedule that attempts to minimize the overall risk level (or probability of local delays affecting any successor) of its activities by giving each the amount of float that it actually needs. In other words, this approach will advocate for a schedule that does not contain a ‘critical path’ in the traditional definition. 2.3 Float Ownership Although a notion of overcoming the well-established concept of a critical path may sound ambitious, it is not out of line with project management in practice: Project managers already attempt to set contingency by considering which of the various activities ‘needs more time’. However, this is performed empirically by “subjective management of the project time contingency” (Barraza 2011, p. 260), influenced by various factors, including the contract type that is used (Smith and Bohn 1999), but lacking any rigorous scientific foundation. Thus realism and quality (in terms of predictive ability) of a schedule depend to a large degree on such an individual’s expertise. Quantifying and formalizing this wisdom is the purpose of this research. If float can mitigate delays by being consumed, then it becomes a commodity with potential value to the project participants (de la Garza et al. 1991). A question naturally arises, who owns float? In practice the ‘first come, first serve’ approach of using float appears to prevail (Pasiphol 1994), which underlines the urgent need of a mathematical approach that is derived from a testable hypothesis. A multitude of studies in the literature has discussed float ownership, which continues to be contentious and unresolved. Among the arguments were that float should belong to the owner, who pays for a project, or the contractor, who is responsible for its schedule (Al-Gahtani 2006), and suggestions to share or split it (Prateapusanond 2003), but without any details on how it could be fairly and equitably accomplished. Splitting this Gordian knot once again can be achieved by rephrasing the initial question to overcome its implicit assumption: Who among the project participants should be allocated float, and how much? The dichotomy of owner and general contractor is not central to resolving this question, because in many cases the individual activities are performed by specialty subcontractors. As Al-Gahtani and Mohan (2007 p. 33) supported, float should be allocated by “linking the ownership with the party who carries the risk”. Subcontractors, being productive agents who perform the schedule activities, will therefore be the focus of this research. 2.4 Fair Allocation The final step in the logical argument pertains to the question of how much? This is fundamentally an allocation problem, where float takes on the role of a valuable but limited commodity and subcontractors are competing parties who each desire to receive float to protect their individual activities against delays. Since such activities are very different, it is necessary to establish a quantitative measure of need for float (rather than binary criticality) by which the allocation can proceed to achieve a fair and equitable result. 3 RESEARCH METHODOLOGY This research employs an analogy-adapting methodology to create a functioning mathematical approach and test its hypothesis. An unexpected intriguing analogy is found in the area of social choice (Thompson and Lucko 2011), which will be described in the following section. Different from previous work by these authors that first presented the idea; this research will compare three different possible measures of float allocation and visualize their respective performances regarding the goal of a fair and equitable allocation. The quantitative measure of need for float will use the activity duration, i.e. it is simplifyingly assumed that long activities are at a higher risk of incurring delays. While this is certainly true, characterizing the risk of delays by duration alone in practice likely is less realistic. It is envisioned that a weighted combination of e.g. duration, cost, and any other quantitative measure of relevant risk factors may be used, which will be 025-3 investigated under future research. It is assumed for the new approach that initially all durations are raw durations and that a regular calculation per the critical path method has been completed to obtain the raw project finish and a list of activities that are deemed critical. The difference between calculated raw project finish and the known contract turnover date is called contract float (CF), which is “post-CPM duration float that relates to the overall completion of the project” (Thompson and Lucko 2011, p. CN-002-7). While this approach may superficially resemble critical chain project management (CCPM) from the manufacturing industry, which intentionally accumulates all intermediate buffers into a single large one at the project finish (Steyn 2000), important differences exist, which counteract the simplistic assumptions of CCPM: • CCPM does not explicitly distinguish raw from realistic durations, but simplistically assumes a generic 50% reduction of the planned duration to determine “the likely duration” (Zhao et al. 2010, p. 1056), which questionably assumes that on average all subcontractors inflate their estimates by as much; • CCPM radically removes all ‘buffers’ from a schedule, even from the currently non-critical activities, into a single cumulative ‘project buffer’ (Rand 2000) at the project end, thus changing schedule dates; • CCPM artificially and wilfully generates severe criticality across a schedule, because it assumes that the subcontractors will only work timely if they have the impression that they work with zero flexibility; • CCPM over-ambitiously seeks to minimize project duration (Herroelen and Leus 2001), but does not focus on reducing the project risk by increasing the consistency of achieving a realistic duration; • “CCPM does not assign safety time to each individual activity” (Ma et al. 2014, p. 1), which is inverse to the philosophy new approach, and prevents determining any mathematically fair float ownership; • CCPM reinserts ‘feeding’ buffers into non-critical paths (Rand 2000), changing schedule dates again; • CCPM continues to ignore the critical path, whose risk has increased when all ‘safety’ was removed. Besides such conceptual discrepancies, neither CCPM nor similar buffering approaches that were purely simulation-based (Gurgun et al. 2013, Barraza 2011) and essentially functioned by “taking off contingency buffers from individual activities and pooling them”, then “putting the resized or newly introduced buffer between activities” (Park and Peña-Mora 2004, p. 630), provided a rigorous scientific foundation for fair and equitable apportionment of buffers, or rather CF. They can therefore be considered to be rudimentary attempts of solving a related ambition, but differed from the analogy-based approach that is presented in this research by turning away from, instead of improving, the vital backbone of a project – its critical path. Note that this methodology and its assumptions and intended outcome – a project schedule ‘without’ any critical path – are not as radical as they may initially appear. Project managers already will ‘inflate’ the raw durations to what they consider realistic ones, but they do so in an individual and empirical manner. In the best case, this research will thus formalize and codify the best professional practice by providing it with a quantitative scientific foundation whose entire predictive and preventative approach can be validated. 3.1.1 Research Analogy Arrow (1964) distinguished decision-making approaches into being voting (which is used in democratic systems to reach political decisions), and marketplace mechanisms (which are used in capitalist systems to reach economic decisions). Under these approaches, persons or groups engage in decision-making in several ways; e.g. dictatorially or unilaterally, forming a coalition to compose a majority of similar-minded individuals, or collective bargaining (hierarchical negotiation). Such approaches are often refined by extra rules and constraints, e.g. veto power – an ability to overrule a decision, per its Latin meaning “I forbid.” Regardless of its detailed approach, any type of decision-making process seeks to reach a consensus by systematically aggregating many individual views. A rich body of research has examined decision-making under the heading of social choice; within it problems like fair division (of desirable assets or undesirable chores) and voting models (for parliamentary processes) have been explored mathematically. These offer analogies for the internal structure and operation of network schedule systems. Stated briefly, participants in such system (here: project) are voters (here: subcontractors), who jointly generate a decision (here: to expend float or not) by their actions (here: timely completion or delay) (Thompson and Lucko 2011). This conceptual analogy warrants investigation, especially exploring the applicability of their tenets to network schedule systems and project management in general for the problem at hand; namely float allocation. 025-4 A fundamental question emerges when participants are not of equal size (e.g. countries in the European Union), yet each of them should receive a fair weight within a decision-making process. To solve this vexing problem, two related mathematical voting models have emerged; the Penrose (1946) square root law and the Banzhaf (1965) power index. The former was inspired by the founding of the United Nations. It addressed problems of ‘resolute’ blocs that may dominate decisions and small nations, whose view either is irrelevant or who may become the tiebreaker. In other words, a truly fair balance is necessary to ensure that participants of small size are neither never nor always influencing the decision. Transferred newly to project management, Penrose (1946, p. 53) serves as inspiration for the new float allocation: In general, the power of the individual vote[r] can be measured by the amount by which his chance of being on the winning side exceeds one half. The power, thus defined, is the same as half the likelihood of a situation in which an individual vote can be decisive – that is to say, a situation in which the remaining votes are equally divided upon the issue at stake. The general formula for the probability of equal division of n  random votes, where n  is an even number, approaches πn2  when n  is large. It follows that the power of the individual vote[r] is inversely proportional to the square root of the number of people in the committee. He distinguished voting weight (which is a simple factor) from voting power (how often a participant sways a decision). Of course, all participants should receive equal power, regardless of their size. For countries with population n, a balance between the two unfair extremes of ‘one vote per country’ (proportional to n0, which gives small ones too much power) versus ‘one vote per person’ (proportional to n1, which gives small ones too little power) is required for a fair and equitable decision-making process (Pöppe 2007). He suggested that the newly formed General Assembly should allocate votes based on the square root (n0.5) of country population. This somewhat counterintuitive principle that using direct proportionality to number of voters in each country is not the fairest possible approach remains a central insight of these studies. Transferring this conceptual analogy to network schedules systems, political systems in representative democracies with two or more levels (federations like e.g. the United Nations, European Union, or United States) that make binary decisions provide the analogy, which correlates with critical subcontractors, who individually decide to consume their allocated float or not, and collaboratively build a project on time or not. Just as voters cast votes in an election (a decision forum), so subcontractors participate in a project with delay mitigation by deciding on the expenditure of float. “[V]oting typifies a possible determinant for the analogous behavior to those participating in large construction projects; and a potential mitigation approach to network schedule uncertainties and their impacts” (Thompson and Lucko 2011, p. CN-002-4). A question remains of whether n0.5 is the most appropriate value, i.e. a pure Penrose approach to float allocation, or if applying this analogy to construction projects warrants another value between n0 and n1. 3.1.2 Research Hypothesis It is hypothesized that allocating float to the activities on the critical path based on the square root of the measure of float need leads to less need to project float than an allocation based on order zero or one. 4 SIMULATION CALCULATIONS 4.1 Conceptual Approach To efficiently perform the simulation calculations for the case example and derive comparisons, the entire new scheduling approach has been implemented in commercial computer software. Coding has followed a modular approach: Inputs are provided as a file that contains an activity list with name identifier for each activity as well as duration (distributions with its respective shape parameters) and point-to-point relations between the predecessors and successors that compose a network schedule. The number of repetitions of the simulation (i.e. runs) are specified separately. Output is written to another file and post-processed for analysis and interpretation. During each run the probability distribution of every activity is randomly sampled to determine its respective duration instance for one particular realization of the entire project. 025-5 Repeated simulation of a model that contains individual probabilistic variables to observe the probabilistic behavior of the entire system is known as Monte Carlo method (Chantaravarapan et al. 2004). It is often employed if the system (here a network schedule) is sufficiently complex to preclude a direct prediction of how exactly its elements will interact and cannot be solved by direct mathematical calculation. A distant conceptual relation of the new approach of this research exists to the so-called criticality index (Tang et al. 2013, p. 3238), which “is defined as the probability of an activity becoming critical”. However, several fundamental differences must be understood to contrast the criticality index with the new float allocation: • A criticality index does not pre-allocate any contract float, neither to critical nor to any other activities; • A criticality index is based on duration distributions for realistic durations, not raw ones, even though the former are more difficult to determine than the latter, which can rely directly on productivity data; • A criticality index is performed within the calculated project duration, oblivious of its contract deadline; • A criticality index analysis, using a Monte Carlo approach, tracks whether or not the critical path will pass through a particular activity within a single randomized realization of a given network schedule; • A criticality index analysis is randomized across a single variable, the duration from the distributions; • A criticality index analysis generates a single quantitative values to indicate a form of ‘meta-criticality’; • The float allocation is randomized across two variables, durations and also increasing available CF; • The float allocation generates a curve for each activity that indicates at what total CF that is available to all critical activities within a schedule said activity ceases to exceed its own allocated portion of CF. For continuity, the case example is selected from Thompson and Lucko (2011). The left half of Table 1 lists its input with 15 activities that initially amount to a 72-day-long project from mobilization (Mob.) to turnover (T/O) per a CPM calculation of raw durations. Seven bold activities are deemed critical and will be the ones that receive float from the three allocation mechanisms. For simplicity, links have no leads or lags and no link types other than finish-to-start are used; these items will be explored in future research. In the subsequent simulation it is assumed that the duration distribution of activities has an asymmetric triangular shape with a minimum at 90% of the planned duration per Table 1 and a maximum of 125% to reflect that fact that activities rarely take shorter than planned, but often longer (Thompson 2012). Other probability distributions, e.g. beta (Fente et al. 2000), will be examined within the future research as well. The following approach was taken for simulation. The first module performs the forward pass, backward pass, and float calculations of traditional CPM, which identifies critical activities. Afterward, it replaced the fixed planned durations with aforesaid triangular distribution and performs numerous repetitions (here 1,000) in a Monte Carlo type simulation of the now probabilistic schedule. Data analysis was performed on the list of seven critical activities only. Their actual durations as sampled from the distribution were compared per Equation 1 with the float that had been allocated, called distributed float DF (Thompson and Lucko 2011) and is a portion of the contract float CF itself that is available to the entire critical path. [1] Actual duration – (planned duration + distributed float) = Float overrun Besides the actual duration of each activity, CF was the second variable in the simulation. It was varied in integer increments between CF = {1 to 40} days. Efficiently, it was not necessary to re-run the simulation to count all of the individual cases of float overrun (or have sufficient float, which is of course desirable), or even to repeat it 30 times, once for each scenario of CF. Rather, only ‘actual duration’ in Equation 1 is randomized. ‘Planned duration’ is fixed and known from Table 1. And ‘distributed float’ depends only on the mechanism of float allocation, planned duration, and CF, but not actual duration. The reason for this uncoupling is that the entire approach is designed to function a priori, before actual durations are known. Rounding presented an important challenge for float allocation, because it directly influenced how exactly the limited valuable CF became DF for each critical activity. It was rounded to integer days and any non-allocated single day of float was not explicitly assigned, but was left to ‘first-come, first-serve’. However, activities were rounded up so that each activity could only start at an integer time, to reflect the fact that if a subcontractor finishes before the end of a workday in construction practice, the next one will not arrive for merely a few hours, but on the morning of the next regular workday as Bashford et al. (2007) implied. 025-6 Table 1: Case Example Input and Float Allocation Output Traditional CPM Calculations Duration0.0 Duration0.5 Duration1.0 Act. Dur. Succ. ES LS EF LF TF FF L R m s L R m s L R m s Mob. 7 A, B, E 0 0 7 7 0 0 3 11 4.80 3.85 4 13 5.87 3.52 4 14 6.34 3.82 A 19 D, I, J 7 13 26 32 6 0 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A B 10 C 7 7 17 17 0 0 3 18 5.98 4.53 3 16 5.62 3.65 3 15 5.60 3.79 C 6 D, F, J 17 17 23 23 0 0 3 11 4.24 3.05 4 14 5.43 3.32 5 16 6.52 3.64 D 18 L 26 33 44 51 7 7 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A E 15 F, G 7 8 22 23 1 0 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A F 17 H, I, K 23 23 40 40 0 0 3 32 9.83 7.38 2 19 6.15 4.19 2 16 5.48 3.60 G 16 H, I, K 22 24 38 40 2 2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A H 6 M 40 53 46 59 13 0 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A I 11 L 40 40 51 51 0 0 3 18 6.79 5.21 3 15 6.06 3.72 3 14 5.85 3.72 J 19 L 26 32 45 51 6 6 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A K 15 T/O 40 54 55 69 14 14 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A L 18 T/O 51 51 69 69 0 0 3 32 10.69 8.36 2 18 6.60 4.75 1 15 5.17 4.04 M 10 T/O 46 59 56 69 13 13 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A T/O 3 N/A 69 69 72 72 0 0 3 4 3.50 0.00 6 9 7.56 0.37 9 12 11.07 0.69 Grand Average Across Activities 3.0 18.0 6.55 4.63 3.4 14.9 6.19 3.36 3.9 14.6 6.58 3.33 4.2 Results and Interpretation Float overrun was plotted as the response variable in comparison to the CF as the explanatory variable. Figure 1 shows side-by-side the three different mechanisms of float allocation: Equal shares (planned duration0.0), square root of duration (planned duration0.5), and directly proportional (planned duration1.0). The number of occurrences of float overrun was counted for each CF scenario and for every activity. Said counts were of course different under the three mechanisms. Shapes of the curves in Figure 1 represent the efficacy of each mechanism to protect the critical path against delays. For low CF the critical activities have not yet received sufficient DF to be individually protected. This explains the ‘plateau’ of their curves in the left half of the diagrams: While on this plateau, activities will consume more float than they own, or have not even received any DF yet. The latter is due to the fact that only integer days of CF become DF; and until the CF exceeds the number of critical activities, critical activities exist that have an allocation of zero. Next, activities experience a more or less steep drop-off or ‘cliff’, as float shows its powerful effect in absorbing any delays locally. If all activities will remain within their DF, the entire project will not exceed its planned raw duration plus CF. For each activity, a relatively narrow range exists, where DF changes from insufficient (on average) to sufficient (on average). After this cliff, a lower plateau is reached in the right half of the diagrams, where activities have excess DF and the project contains more CF than is needed. To capture the cliff behaviour, which is of vital interest for this analysis, these parameters were calculated from the data and listed in Table 1: Left and right cutoff L and R (the right edge of upper plateau and the left edge of lower plateau), and within this range a weighted mean m and weighted standard deviation s on the horizontal axis, which indicated where and how narrow the cliff was. For example, if an activity has a result of y = {10, 10, 1, 0, 0, 0} counts of overrun for x = {0, 1, 2, 3, 4, 5} days of CF, then L = 1; R = 3; m = [9 · (2 - 1) + 1 · (3 - 2)] / (9 + 1) = 1.6 (weighted by drops Δy within their integer ranges of x); and s = ( ) ( )[ ] ( )( )[ ]21210/6.15.216.15.19 22 −⋅−⋅+−⋅  = 0.42 (average weighted distance of the integer range midpoints to m). Note that both m and s are only calculated across a cliff itself, not the plateaus besides it. 025-7 The ranges of L and R and weighted grand averages and standard deviations across all critical provided a quantitative comparison of the efficacy of the three mechanisms. The equal shares (planned duration0.0) has a wide range {3.0 to 18.0} within which all activities exhibit their drop-off behaviour, a weighted mean of 6.55, and a weighted standard deviation of 4.3, which indicates a merely gradual saturation of critical activities with their urgently needed DF. This approach is not recommended, due to being oblivious to the relative risk of activities, which here is represented by the proxy of duration. More interesting is comparing square root (planned duration0.5) versus proportional shares (planned duration1.0) approaches of Figure 1. The former has a narrower average range {3.4 to 14.9} of width 11.5 for all critical activities per Table 1; the latter is even slightly narrower {3.9 to 14.6) of width 10.7. Weighted means and standard deviations give a more distinct view: The former has a weighted mean of 6.19 compared to 6.58 of the latter. Clearly the square root model on average provides all critical activities earlier with the DF that is needed to satisfy their need. In fact, m is almost identical for (a) and (c) in Figure 1, which supports the hypothesis that (b) is the balance approach between these extremes. Finally, weighted standard deviations compare nearly identical at 3.36 versus 3.33 per Table 1. Both are much lower than s = 4.63 for (a). Testing for statistical significance at an alpha-level of 0.05 finds the difference between means not significant; however, this is most likely due to the very small case example with only seven different activities. More experimentation is therefore needed to substantiate the positive indications of the square root approach for float allocation. 01002003004005006007008000 10 20 30 40 01002003004005006007008000 10 20 30 40 01002003004005006007008000 10 20 30 40 (a) Equal Shares (b) Square Root (c) Proportional Shares Figure 1: Case Example Performance for Three Mechanisms of Float Allocation 5 CONCLUSIONS AND RECOMMENDATIONS This research has presented a vision for overcoming the long-standing and deeply entrenched paradigm within construction project scheduling by uncovering its various implicit assumptions. They have included focusing excessively on dissecting risk factors on the one hand and delay liability on the other hand; not explicitly distinguishing raw from realistic durations and basing criticality assessment on the latter, not the former; and artificially creating a paradoxical two-class society of critical versus non-critical activities that actively contradicts any desire to minimize the overall risk level. It is thus understandable that previous approaches to float ownership failed to solve the true underlying issue – minimizing risk by mitigating delays. This research stipulated that all critical activities should receive float, not by directly inserting it from its location between calculated and contract finish to being distributed along the critical path within a network schedule itself, but rather by providing an unambiguous approach to calculate optimum portions of ownership. This new approach is based on the premise that a directly proportional allocation to ‘size’ of the critical subcontractors’ contributions to the project would not fulfill the overall goal of being fair and equitable. Rather, three cases have been tested for numerical comparison; an allocation based on equal shares (n0), the square root of size – or risk – (n0.5), or proportional shares (n1). Initial results are positive, but not statistically conclusive between the square root approach or proportional shares of float allocation. More experimentation with larger and more realistic examples of network schedules is recommended. Count of Overrun Contract Float Count of Overrun Contract Float Count of Overrun Contract Float 025-8 This research aids in fulfilling the unrealized vision of “Total Float Traded as Commodity” (de la Garza et al. 1991, p. 716). Only that it will not be the TF of non-critical activities, but CF, and that it will be designed to benefit critical activities. And that it will require having completed a pre-allocation of CF. Combined with the creative analogy from voting mathematics, which provides the seminal inspiration for how to achieve a provable fairness between large and small subcontractors who compete for CF, a market of opportunities can emerge. More research is necessary on how to determine dollar value, which should be listed in the contract to give all subcontractors certainty. Future research should investigate how the emphasis that is placed on measures of ‘size’ – whether it is duration, cost, risk (depending on type and nature of activity) – can be weighted and converted into a single numerical metric (Thompson and Lucko 2011). The current approach has merely assumed that a numeric value of risk (or size) is available as input. Closely related to this, it should also be examined how the potentially different subjective monetary valuations of CF by said contractors can be aggregated. Further analogies from areas of knowledge besides construction project management should be parsed to identify proven concepts that can handle subjective valuation. Acknowledgements The support of the National Science Foundation (Grant CMMI-1265989) for portions of the work presented here is gratefully acknowledged. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The computer programming by Mr. Yi Su, graduate research assistant and doctoral student, is gratefully acknowledged. References Al-Gahtani, K. S. 2006. A comprehensive construction delay analysis technique – enhanced with a float ownership concept. Dissertation, State University of New York at Buffalo, Buffalo, NY: 388 pp. Al-Gahtani, K. S., Mohan, S. B. 2007. Total float management for delay analysis. Cost Engineering, 49(2): 32-37. Barraza, G. A. 2011. Probabilistic estimation and allocation of project time contingency. Journal of Construction Engineering and Management, 137(4): 259-265. Banzhaf, J. 1965. Weighted voting doesn’t work. Rutgers Law Review 19(2): 317-343. Bashford, H. H., Sawhney, A., Felt, J., Koh. D. 2007. Benchmarking the contribution of labor in production homebuilding. Proceedings of the 2007 Construction Research Congress, Grand Bahama Island, Bahamas, May 6-8, 2007, American Society of Civil Engineers, Reston, VA, 2: 843-851. Chantaravarapan, S., Gunal, A., Williams, E. J. 2004. On using Monte Carlo methods for scheduling. Proceedings of the 36th Winter Simulation Conference, Washington, DC, December 5-8, 2004, Winter Simulation Conference Foundation, Catonsville, MD: 1870-1875. Chen, Y.-L., Rinks, D., Tang, K. 2001. The first K minimum cost paths in a time-schedule network. Journal of the Operational Research Society, 52(1): 102-108. de la Garza, J. M., Vorster, M. C., Parvin, C. M. 1991. Total float traded as commodity. Journal of Construction Engineering and Management, 117(4): 716-727. de la Garza, J. M., Prateapusanond, A., Ambani, N. 2007. Preallocation of total float in the application of a critical path method based construction contract. Journal of Construction Engineering and Management, 133(11): 836-845. Deng, F., Smyth, H. 2013. Contingency-based approach to firm performance in construction: Critical review of empirical research. Journal of Construction Engineering and Management, 0733-9364/04013004(14). Fente, J., Schexnayder, C. J., Knutson, K. 2000. Defining a probability distribution for construction simulation. Journal of Construction Engineering and Management 126(3): 234-241. González, P., González, V., Molenaar, K. R., Orozco, F. 2013. Analysis of Causes of Delay and Time Performance in Construction Projects. Journal of Construction Engineering and Management, 0733-9364/04013027(9). Gündüz, M., Nielsen, Y., Özdemir, M. 2013. Quantification of delay factors using the relative importance index method for construction projects in Turkey. Journal of Management in Engineering, 29(2): 133-139. 025-9 Gurgun, A. P., Zhang, Y., Touran, A. 2013. Schedule contingency analysis for transit projects using a simulation approach. Journal of Civil Engineering and Management, 19(4): 465-475. Herroelen, W., Leus, R. 2001. On the merits and pitfalls of critical chain scheduling. Journal of Operations Management 19(5): 559-577. Ibbs, C. W., Nguyen, L. D. 2011. Concurrent delays and apportionment of damages. Journal of Construction Engineering and Management, 137(2): 119-126. Kelley, J. E., Walker, M. R. 1959. Critical-path planning and scheduling. Proceedings of the Eastern Joint Computer Conference, ed. Heart, F. E., Boston, MA, December 1-3, 1959, National Joint Computer Committee, Association for Computing Machinery, New York City, NY: 160-173. Khamooshi, H., Cioffi, D. F. 2013. Uncertainty in task duration and cost estimates: Fusion of probabilistic forecasts and deterministic scheduling. Journal of Construction Engineering and Management, 139(5): 488-497. Ma, G., Wang, A., Li, N., Gu, L., Ai, Q. 2014. Improved critical chain project management framework for scheduling construction projects. Journal of Construction Engineering and Management 04014055(12). Pasiphol, S. 1994. New concepts for total float distribution in CPM project scheduling. Dissertation University of Texas, Austin, TX: 235 pp. Penrose, L. 1946. The elementary statistics of majority voting. Journal of the Royal Statistical Society 109(1): 53-57. Pöppe, C. 2007. Sterben für eine Quadratwurzel? Web publication, Spektrum der Wissenschaft June 16, 2007. Prateapusanond, A. 2003. A comprehensive practice of total float preallocation and management for the application of a CPM-based construction contract. Dissertation, Virginia Polytechnic Institute and State University, Blacksburg, VA: 230 pp. Rand, G. K. 2000. Critical chain: the theory of constraints applied to project management. International Journal of Project Management 18(3): 173-177. Smith, G. R., Bohn, C. M. 1998. Small to medium contractor contingency and assumption of risk. Journal of Construction Engineering and Management, 125(2): 101-108. Steyn, H. 2000. An investigation into the fundamentals of critical chain project scheduling. International Journal of Project Management, 19(): 363-369. Tang, P., Mukherjee, A., Onder, N. 2013. “Construction schedule simulation for improved project planning: Activity criticality index assessment.” Proceedings of the 45th Winter Simulation Conference, Washington, DC, December 8-11, 2013, Winter Simulation Conference Foundation, Catonsville, MD: 3237-3248. Thompson, R. C., Lucko, G. 2011. Modeling measures of float allocation for quantitative risk management of construction projects. Proceedings of the 2011 3rd International / 9th Construction Specialty Conference at the Canadian Society for Civil Engineering Annual Conference, Ottawa, Ontario, June 14-17, 2011, Canadian Society for Civil Engineering, Montréal, Quebec: CN-002-1 – CN-002-10. Thompson, R. C. 2012. Risk measurement, allocation, and pricing in network schedule systems. Dissertation, Catholic University of America, Washington, DC: 43 pp. Thompson, R. C., Lucko, G. 2012. Modeling measures of float monetization for quantitative risk management of construction projects.” Proceedings of the 2012 Construction Research Congress, West Lafayette, IN, May 21-23, 2012, American Society of Civil Engineers, Reston, VA: 485-494. Tran, D., Molenaar, K. R. 2012. Critical risk factors in project delivery method selection for highway projects. Proceedings of the Construction Research Congress, West Lafayette, IN, May 21-23, 2012, American Society of Civil Engineers, Reston, VA: 331-340. Xie, H., AbouRizk, S. M., Zou, J. (2012). Quantitative method for updating cost contingency throughout project execution. Journal of Construction Engineering and Management, 138(6): 759-766. Yang, J.-B., Kao, C.-K. 2012. Critical path effect based delay analysis method for construction projects. International Journal of Project Management 30(3): 385-397. Zack, J. G. 1992. Schedule ‘games’ people play, and some suggested remedies. Journal of Management in Engineering, 8(2): 138-152. Zhao, Z. Y., You, W. Y., Zuo, J. 2010. Application of innovative critical chain method for project planning and control under resource constraints and uncertainty. Journal of Construction Engineering and Management 136(9): 105-1060.  025-10  5th International/11th Construction Specialty Conference 5e International/11e Conférence spécialisée sur la construction    Vancouver, British Columbia June 8 to June 10, 2015 / 8 juin au 10 juin 2015   SIMULATED SCHEDULE DELAY MITIGATION VIA FLOAT ALLOCATION Gunnar Lucko1 and Richard C. Thompson, Jr.1 1 Department of Civil Engineering, Catholic University of America, Washington, DC, USA Abstract: As delays in construction project schedules are widely documented, new approaches to manage this problem are needed. Previous studies have explored float as an inherent ability of schedules to absorb delays. However, all ignore the critical path; some recommendations as to float ownership directly contradict the inherently unfair ‘first come first serve’ principle; and none have derived testable methods to fairly allocate float to multiple participants. This study therefore employs the mathematical analogy of decision-making among a group of unequally sized individuals to explicitly allocate project float a priori to those most vulnerable – the critical path. While intuition might indicate allocation proportional to activity duration, or perhaps equal shares, it is demonstrated that neither is truly fair and a mathematical compromise can be found. The performance of this approach is tested quantitatively by simulating a case example of project schedules with and without such protection. Delays are modeled as probabilistic events that affect activities. It is found that even a relatively small amount of project float allocated along the critical path provides significant delay mitigation. 1 INTRODUCTION Construction project schedules in many countries unfortunately are frequently inundated with delays (Gündüz et al. 2013). Delays in this research are defined in a strictly temporal view, i.e. negative changes of start and/or finish of activities between the as-planned and as-updated project schedules. What makes delays particularly perilous is the fact that they are not an isolated local phenomenon within schedules, but may initiate a cascading set of ripple effects in later activities that depend on the earlier delayed one. Whatever the diverse causes of such delays are, which may stem from one or several risk factors that are materializing (González et al. 2013), is less relevant to this discussion than the unfortunate outcome, that it becomes likely that the contractually agreed turnover date will be met, i.e. the entire project will be late. Contractual effects of late completion typically are that the contractor is required to reimburse the owner for consequential cost (liquidated damages) of postponed occupancy of the new facility, which often leads to further side-effects, e.g. protracted court cases over liability that may damage the reputation and lose future clients. The central question remains not one of identifying and treating symptoms, but of curing the cause: What inherent defenses does a project schedule have against incurring delays? Such capabilities could take two forms, either preventive (lowering the probability of a delay occurring) – akin to inoculation; or reactive (gaining better means of alleviating a delay that occurs) – akin to therapeutics in medicine. To begin answering this question, this research will first review existing theory of project management regarding conceptual limitations of traditional scheduling and opportunities for protecting against delays, then explore potential theory in decision-making from which inspiration for such resilience will be gleaned. 025-1 2 LITERATURE REVIEW 2.1 Scheduling Assumptions Construction schedules are traditionally conceived and handled as networks, which mathematically are acyclic ‘graphs’ (Chen et al. 2001) that consist of nodes (activities) and arcs (links) between them, whose direction is dictated by the passing of time. Exploiting a rather simple and repetitive structure of relations of temporal data among activities led to the so-called critical path method (Kelley and Walker 1959) with which dates are sequentially calculated: Within activities, adding the duration to its start gives its finish (S + D = F). And between activities, all successors must finish as a condition for the successor to start (max {Fpred} = Ssucc). This forward-looking calculation assumes the best case that all activities start as early as possible at their earliest (E) dates. Its inverse, the backward-looking calculation assumes the worst case that they start as late as possible at latest (L) dates, so as to just not yet delay the project finish itself. Comparing these two scenarios gives information for each activity on how much it can be delayed without impacting (a) any successor (free float, FFpred = min{ESsucc} - EFpred), or ‘only’ (b) the project finish (total float, TF = LF - EF = LS - ES). Thus float is defined by an assumed delay. Due to the aforementioned vital importance of the project finish for contractual liability purposes, this research will focus on the total float. Float in the literature has been presented as the opposite of criticality, which has been defined as having “zero total float” (de la Garza et al. 2007, p. 836), and inspired the name for this scheduling technique. A subset of activities within any network schedule forms a continuous path – the eponymous critical path – between the first and last activity, which just like the overall sequential schedule may branch and merge. Hidden assumptions also lie in the inputs to such analysis, notably links and durations. Links (or schedule ‘logic’) are primarily determined based on technical reasons per laws of physics, but may also be modified for administrative reasons or even to artificially generate opportunities for success or failure (Zack 1992). This research assumes that links are fixed. Durations are determined based on work quantity as taken off plan drawings and specifications, combined with the crew choice and their unit productivity. However, one must consider that forecasting durations is always loaded with uncertainty. While studies have explored distributions to provide realistic durations for probabilistic scheduling (Khamooshi and Cioffi 2013), project managers in the construction industry have not been oblivious to the need for realistic duration estimates as opposed to overly optimistic ones. In practice, how well it is accomplished depends on the expertise of individual project managers, who determine ‘raw’ durations (from productivity data) and empirically extend them to ‘realistic’ ones by including a contingency (Barraza 2011). Contingency refers to both time or cost (Xie et al. 2012), but here it means additional time. This research assumes that raw durations are known to a user of this new approach, here presumed to be the general contractor. Studies have explored how contingency aids in company performance (Deng and Smyth 2013). It “has been used in project planning to cover activities delays, oversights, and unknowns and as a cushion for possible time-estimating errors” (Barraza 2011, p. 259). Since time contingency is essentially float, it will be referred to as such for clarity. The nature, quantity, and most beneficial distribution of such contingency, or float, will be explored next. 2.2 Criticality Paradox Whereas risk has been explored in detail, with many studies listing risk factors and how to rank or weigh them, often using survey methodologies (e.g. Tran and Molenaar 2012), its realization in form of delays has been studied primarily to create numerous competing or even contradictory approaches to distribute liability between owner and contractor (Yang and Kao 2012) after excusable delays have been forgiven, i.e. assigning blame, and pondering the vexing issue of concurrent delays (Ibbs et al. 2011). However, a posteriori analysis, while unfortunately necessary, is not even reactive, but merely inculpative. In other words, delay liability appears has been the focus rather than delay mitigation or, even better, avoidance. Rephrasing the definition of criticality as ‘an inability to absorb delays’ reveals that it is based upon an assumption itself, that activities in network schedules will necessarily have unequal capabilities to handle delays. However, schedulers control the manner in which activities are linked in network schedules (Zack 1992), so that criticality is essentially an artifact of the schedule itself. Criticality as it is currently defined 025-2 for critical path method is not an inherent quality of the activity based on its nature or risk, although this fact is often hidden, because long or difficult activities often actually turn out to be part of the critical path. Artificially dividing activities into being critical or not by whether or not they have total float in fact creates a paradox (Thompson and Lucko 2012, p. 488): “By definition, noncritical subcontractors have float, but the critical ones have none, even though they need it most.” Such two classes, of which critical activities are designed to be on the verge of failure, run counter to what should be the optimum goal of a scheduler: Creating a schedule that attempts to minimize the overall risk level (or probability of local delays affecting any successor) of its activities by giving each the amount of float that it actually needs. In other words, this approach will advocate for a schedule that does not contain a ‘critical path’ in the traditional definition. 2.3 Float Ownership Although a notion of overcoming the well-established concept of a critical path may sound ambitious, it is not out of line with project management in practice: Project managers already attempt to set contingency by considering which of the various activities ‘needs more time’. However, this is performed empirically by “subjective management of the project time contingency” (Barraza 2011, p. 260), influenced by various factors, including the contract type that is used (Smith and Bohn 1999), but lacking any rigorous scientific foundation. Thus realism and quality (in terms of predictive ability) of a schedule depend to a large degree on such an individual’s expertise. Quantifying and formalizing this wisdom is the purpose of this research. If float can mitigate delays by being consumed, then it becomes a commodity with potential value to the project participants (de la Garza et al. 1991). A question naturally arises, who owns float? In practice the ‘first come, first serve’ approach of using float appears to prevail (Pasiphol 1994), which underlines the urgent need of a mathematical approach that is derived from a testable hypothesis. A multitude of studies in the literature has discussed float ownership, which continues to be contentious and unresolved. Among the arguments were that float should belong to the owner, who pays for a project, or the contractor, who is responsible for its schedule (Al-Gahtani 2006), and suggestions to share or split it (Prateapusanond 2003), but without any details on how it could be fairly and equitably accomplished. Splitting this Gordian knot once again can be achieved by rephrasing the initial question to overcome its implicit assumption: Who among the project participants should be allocated float, and how much? The dichotomy of owner and general contractor is not central to resolving this question, because in many cases the individual activities are performed by specialty subcontractors. As Al-Gahtani and Mohan (2007 p. 33) supported, float should be allocated by “linking the ownership with the party who carries the risk”. Subcontractors, being productive agents who perform the schedule activities, will therefore be the focus of this research. 2.4 Fair Allocation The final step in the logical argument pertains to the question of how much? This is fundamentally an allocation problem, where float takes on the role of a valuable but limited commodity and subcontractors are competing parties who each desire to receive float to protect their individual activities against delays. Since such activities are very different, it is necessary to establish a quantitative measure of need for float (rather than binary criticality) by which the allocation can proceed to achieve a fair and equitable result. 3 RESEARCH METHODOLOGY This research employs an analogy-adapting methodology to create a functioning mathematical approach and test its hypothesis. An unexpected intriguing analogy is found in the area of social choice (Thompson and Lucko 2011), which will be described in the following section. Different from previous work by these authors that first presented the idea; this research will compare three different possible measures of float allocation and visualize their respective performances regarding the goal of a fair and equitable allocation. The quantitative measure of need for float will use the activity duration, i.e. it is simplifyingly assumed that long activities are at a higher risk of incurring delays. While this is certainly true, characterizing the risk of delays by duration alone in practice likely is less realistic. It is envisioned that a weighted combination of e.g. duration, cost, and any other quantitative measure of relevant risk factors may be used, which will be 025-3 investigated under future research. It is assumed for the new approach that initially all durations are raw durations and that a regular calculation per the critical path method has been completed to obtain the raw project finish and a list of activities that are deemed critical. The difference between calculated raw project finish and the known contract turnover date is called contract float (CF), which is “post-CPM duration float that relates to the overall completion of the project” (Thompson and Lucko 2011, p. CN-002-7). While this approach may superficially resemble critical chain project management (CCPM) from the manufacturing industry, which intentionally accumulates all intermediate buffers into a single large one at the project finish (Steyn 2000), important differences exist, which counteract the simplistic assumptions of CCPM: • CCPM does not explicitly distinguish raw from realistic durations, but simplistically assumes a generic 50% reduction of the planned duration to determine “the likely duration” (Zhao et al. 2010, p. 1056), which questionably assumes that on average all subcontractors inflate their estimates by as much; • CCPM radically removes all ‘buffers’ from a schedule, even from the currently non-critical activities, into a single cumulative ‘project buffer’ (Rand 2000) at the project end, thus changing schedule dates; • CCPM artificially and wilfully generates severe criticality across a schedule, because it assumes that the subcontractors will only work timely if they have the impression that they work with zero flexibility; • CCPM over-ambitiously seeks to minimize project duration (Herroelen and Leus 2001), but does not focus on reducing the project risk by increasing the consistency of achieving a realistic duration; • “CCPM does not assign safety time to each individual activity” (Ma et al. 2014, p. 1), which is inverse to the philosophy new approach, and prevents determining any mathematically fair float ownership; • CCPM reinserts ‘feeding’ buffers into non-critical paths (Rand 2000), changing schedule dates again; • CCPM continues to ignore the critical path, whose risk has increased when all ‘safety’ was removed. Besides such conceptual discrepancies, neither CCPM nor similar buffering approaches that were purely simulation-based (Gurgun et al. 2013, Barraza 2011) and essentially functioned by “taking off contingency buffers from individual activities and pooling them”, then “putting the resized or newly introduced buffer between activities” (Park and Peña-Mora 2004, p. 630), provided a rigorous scientific foundation for fair and equitable apportionment of buffers, or rather CF. They can therefore be considered to be rudimentary attempts of solving a related ambition, but differed from the analogy-based approach that is presented in this research by turning away from, instead of improving, the vital backbone of a project – its critical path. Note that this methodology and its assumptions and intended outcome – a project schedule ‘without’ any critical path – are not as radical as they may initially appear. Project managers already will ‘inflate’ the raw durations to what they consider realistic ones, but they do so in an individual and empirical manner. In the best case, this research will thus formalize and codify the best professional practice by providing it with a quantitative scientific foundation whose entire predictive and preventative approach can be validated. 3.1.1 Research Analogy Arrow (1964) distinguished decision-making approaches into being voting (which is used in democratic systems to reach political decisions), and marketplace mechanisms (which are used in capitalist systems to reach economic decisions). Under these approaches, persons or groups engage in decision-making in several ways; e.g. dictatorially or unilaterally, forming a coalition to compose a majority of similar-minded individuals, or collective bargaining (hierarchical negotiation). Such approaches are often refined by extra rules and constraints, e.g. veto power – an ability to overrule a decision, per its Latin meaning “I forbid.” Regardless of its detailed approach, any type of decision-making process seeks to reach a consensus by systematically aggregating many individual views. A rich body of research has examined decision-making under the heading of social choice; within it problems like fair division (of desirable assets or undesirable chores) and voting models (for parliamentary processes) have been explored mathematically. These offer analogies for the internal structure and operation of network schedule systems. Stated briefly, participants in such system (here: project) are voters (here: subcontractors), who jointly generate a decision (here: to expend float or not) by their actions (here: timely completion or delay) (Thompson and Lucko 2011). This conceptual analogy warrants investigation, especially exploring the applicability of their tenets to network schedule systems and project management in general for the problem at hand; namely float allocation. 025-4 A fundamental question emerges when participants are not of equal size (e.g. countries in the European Union), yet each of them should receive a fair weight within a decision-making process. To solve this vexing problem, two related mathematical voting models have emerged; the Penrose (1946) square root law and the Banzhaf (1965) power index. The former was inspired by the founding of the United Nations. It addressed problems of ‘resolute’ blocs that may dominate decisions and small nations, whose view either is irrelevant or who may become the tiebreaker. In other words, a truly fair balance is necessary to ensure that participants of small size are neither never nor always influencing the decision. Transferred newly to project management, Penrose (1946, p. 53) serves as inspiration for the new float allocation: In general, the power of the individual vote[r] can be measured by the amount by which his chance of being on the winning side exceeds one half. The power, thus defined, is the same as half the likelihood of a situation in which an individual vote can be decisive – that is to say, a situation in which the remaining votes are equally divided upon the issue at stake. The general formula for the probability of equal division of n  random votes, where n  is an even number, approaches πn2  when n  is large. It follows that the power of the individual vote[r] is inversely proportional to the square root of the number of people in the committee. He distinguished voting weight (which is a simple factor) from voting power (how often a participant sways a decision). Of course, all participants should receive equal power, regardless of their size. For countries with population n, a balance between the two unfair extremes of ‘one vote per country’ (proportional to n0, which gives small ones too much power) versus ‘one vote per person’ (proportional to n1, which gives small ones too little power) is required for a fair and equitable decision-making process (Pöppe 2007). He suggested that the newly formed General Assembly should allocate votes based on the square root (n0.5) of country population. This somewhat counterintuitive principle that using direct proportionality to number of voters in each country is not the fairest possible approach remains a central insight of these studies. Transferring this conceptual analogy to network schedules systems, political systems in representative democracies with two or more levels (federations like e.g. the United Nations, European Union, or United States) that make binary decisions provide the analogy, which correlates with critical subcontractors, who individually decide to consume their allocated float or not, and collaboratively build a project on time or not. Just as voters cast votes in an election (a decision forum), so subcontractors participate in a project with delay mitigation by deciding on the expenditure of float. “[V]oting typifies a possible determinant for the analogous behavior to those participating in large construction projects; and a potential mitigation approach to network schedule uncertainties and their impacts” (Thompson and Lucko 2011, p. CN-002-4). A question remains of whether n0.5 is the most appropriate value, i.e. a pure Penrose approach to float allocation, or if applying this analogy to construction projects warrants another value between n0 and n1. 3.1.2 Research Hypothesis It is hypothesized that allocating float to the activities on the critical path based on the square root of the measure of float need leads to less need to project float than an allocation based on order zero or one. 4 SIMULATION CALCULATIONS 4.1 Conceptual Approach To efficiently perform the simulation calculations for the case example and derive comparisons, the entire new scheduling approach has been implemented in commercial computer software. Coding has followed a modular approach: Inputs are provided as a file that contains an activity list with name identifier for each activity as well as duration (distributions with its respective shape parameters) and point-to-point relations between the predecessors and successors that compose a network schedule. The number of repetitions of the simulation (i.e. runs) are specified separately. Output is written to another file and post-processed for analysis and interpretation. During each run the probability distribution of every activity is randomly sampled to determine its respective duration instance for one particular realization of the entire project. 025-5 Repeated simulation of a model that contains individual probabilistic variables to observe the probabilistic behavior of the entire system is known as Monte Carlo method (Chantaravarapan et al. 2004). It is often employed if the system (here a network schedule) is sufficiently complex to preclude a direct prediction of how exactly its elements will interact and cannot be solved by direct mathematical calculation. A distant conceptual relation of the new approach of this research exists to the so-called criticality index (Tang et al. 2013, p. 3238), which “is defined as the probability of an activity becoming critical”. However, several fundamental differences must be understood to contrast the criticality index with the new float allocation: • A criticality index does not pre-allocate any contract float, neither to critical nor to any other activities; • A criticality index is based on duration distributions for realistic durations, not raw ones, even though the former are more difficult to determine than the latter, which can rely directly on productivity data; • A criticality index is performed within the calculated project duration, oblivious of its contract deadline; • A criticality index analysis, using a Monte Carlo approach, tracks whether or not the critical path will pass through a particular activity within a single randomized realization of a given network schedule; • A criticality index analysis is randomized across a single variable, the duration from the distributions; • A criticality index analysis generates a single quantitative values to indicate a form of ‘meta-criticality’; • The float allocation is randomized across two variables, durations and also increasing available CF; • The float allocation generates a curve for each activity that indicates at what total CF that is available to all critical activities within a schedule said activity ceases to exceed its own allocated portion of CF. For continuity, the case example is selected from Thompson and Lucko (2011). The left half of Table 1 lists its input with 15 activities that initially amount to a 72-day-long project from mobilization (Mob.) to turnover (T/O) per a CPM calculation of raw durations. Seven bold activities are deemed critical and will be the ones that receive float from the three allocation mechanisms. For simplicity, links have no leads or lags and no link types other than finish-to-start are used; these items will be explored in future research. In the subsequent simulation it is assumed that the duration distribution of activities has an asymmetric triangular shape with a minimum at 90% of the planned duration per Table 1 and a maximum of 125% to reflect that fact that activities rarely take shorter than planned, but often longer (Thompson 2012). Other probability distributions, e.g. beta (Fente et al. 2000), will be examined within the future research as well. The following approach was taken for simulation. The first module performs the forward pass, backward pass, and float calculations of traditional CPM, which identifies critical activities. Afterward, it replaced the fixed planned durations with aforesaid triangular distribution and performs numerous repetitions (here 1,000) in a Monte Carlo type simulation of the now probabilistic schedule. Data analysis was performed on the list of seven critical activities only. Their actual durations as sampled from the distribution were compared per Equation 1 with the float that had been allocated, called distributed float DF (Thompson and Lucko 2011) and is a portion of the contract float CF itself that is available to the entire critical path. [1] Actual duration – (planned duration + distributed float) = Float overrun Besides the actual duration of each activity, CF was the second variable in the simulation. It was varied in integer increments between CF = {1 to 40} days. Efficiently, it was not necessary to re-run the simulation to count all of the individual cases of float overrun (or have sufficient float, which is of course desirable), or even to repeat it 30 times, once for each scenario of CF. Rather, only ‘actual duration’ in Equation 1 is randomized. ‘Planned duration’ is fixed and known from Table 1. And ‘distributed float’ depends only on the mechanism of float allocation, planned duration, and CF, but not actual duration. The reason for this uncoupling is that the entire approach is designed to function a priori, before actual durations are known. Rounding presented an important challenge for float allocation, because it directly influenced how exactly the limited valuable CF became DF for each critical activity. It was rounded to integer days and any non-allocated single day of float was not explicitly assigned, but was left to ‘first-come, first-serve’. However, activities were rounded up so that each activity could only start at an integer time, to reflect the fact that if a subcontractor finishes before the end of a workday in construction practice, the next one will not arrive for merely a few hours, but on the morning of the next regular workday as Bashford et al. (2007) implied. 025-6 Table 1: Case Example Input and Float Allocation Output Traditional CPM Calculations Duration0.0 Duration0.5 Duration1.0 Act. Dur. Succ. ES LS EF LF TF FF L R m s L R m s L R m s Mob. 7 A, B, E 0 0 7 7 0 0 3 11 4.80 3.85 4 13 5.87 3.52 4 14 6.34 3.82 A 19 D, I, J 7 13 26 32 6 0 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A B 10 C 7 7 17 17 0 0 3 18 5.98 4.53 3 16 5.62 3.65 3 15 5.60 3.79 C 6 D, F, J 17 17 23 23 0 0 3 11 4.24 3.05 4 14 5.43 3.32 5 16 6.52 3.64 D 18 L 26 33 44 51 7 7 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A E 15 F, G 7 8 22 23 1 0 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A F 17 H, I, K 23 23 40 40 0 0 3 32 9.83 7.38 2 19 6.15 4.19 2 16 5.48 3.60 G 16 H, I, K 22 24 38 40 2 2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A H 6 M 40 53 46 59 13 0 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A I 11 L 40 40 51 51 0 0 3 18 6.79 5.21 3 15 6.06 3.72 3 14 5.85 3.72 J 19 L 26 32 45 51 6 6 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A K 15 T/O 40 54 55 69 14 14 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A L 18 T/O 51 51 69 69 0 0 3 32 10.69 8.36 2 18 6.60 4.75 1 15 5.17 4.04 M 10 T/O 46 59 56 69 13 13 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A T/O 3 N/A 69 69 72 72 0 0 3 4 3.50 0.00 6 9 7.56 0.37 9 12 11.07 0.69 Grand Average Across Activities 3.0 18.0 6.55 4.63 3.4 14.9 6.19 3.36 3.9 14.6 6.58 3.33 4.2 Results and Interpretation Float overrun was plotted as the response variable in comparison to the CF as the explanatory variable. Figure 1 shows side-by-side the three different mechanisms of float allocation: Equal shares (planned duration0.0), square root of duration (planned duration0.5), and directly proportional (planned duration1.0). The number of occurrences of float overrun was counted for each CF scenario and for every activity. Said counts were of course different under the three mechanisms. Shapes of the curves in Figure 1 represent the efficacy of each mechanism to protect the critical path against delays. For low CF the critical activities have not yet received sufficient DF to be individually protected. This explains the ‘plateau’ of their curves in the left half of the diagrams: While on this plateau, activities will consume more float than they own, or have not even received any DF yet. The latter is due to the fact that only integer days of CF become DF; and until the CF exceeds the number of critical activities, critical activities exist that have an allocation of zero. Next, activities experience a more or less steep drop-off or ‘cliff’, as float shows its powerful effect in absorbing any delays locally. If all activities will remain within their DF, the entire project will not exceed its planned raw duration plus CF. For each activity, a relatively narrow range exists, where DF changes from insufficient (on average) to sufficient (on average). After this cliff, a lower plateau is reached in the right half of the diagrams, where activities have excess DF and the project contains more CF than is needed. To capture the cliff behaviour, which is of vital interest for this analysis, these parameters were calculated from the data and listed in Table 1: Left and right cutoff L and R (the right edge of upper plateau and the left edge of lower plateau), and within this range a weighted mean m and weighted standard deviation s on the horizontal axis, which indicated where and how narrow the cliff was. For example, if an activity has a result of y = {10, 10, 1, 0, 0, 0} counts of overrun for x = {0, 1, 2, 3, 4, 5} days of CF, then L = 1; R = 3; m = [9 · (2 - 1) + 1 · (3 - 2)] / (9 + 1) = 1.6 (weighted by drops Δy within their integer ranges of x); and s = ( ) ( )[ ] ( )( )[ ]21210/6.15.216.15.19 22 −⋅−⋅+−⋅  = 0.42 (average weighted distance of the integer range midpoints to m). Note that both m and s are only calculated across a cliff itself, not the plateaus besides it. 025-7 The ranges of L and R and weighted grand averages and standard deviations across all critical provided a quantitative comparison of the efficacy of the three mechanisms. The equal shares (planned duration0.0) has a wide range {3.0 to 18.0} within which all activities exhibit their drop-off behaviour, a weighted mean of 6.55, and a weighted standard deviation of 4.3, which indicates a merely gradual saturation of critical activities with their urgently needed DF. This approach is not recommended, due to being oblivious to the relative risk of activities, which here is represented by the proxy of duration. More interesting is comparing square root (planned duration0.5) versus proportional shares (planned duration1.0) approaches of Figure 1. The former has a narrower average range {3.4 to 14.9} of width 11.5 for all critical activities per Table 1; the latter is even slightly narrower {3.9 to 14.6) of width 10.7. Weighted means and standard deviations give a more distinct view: The former has a weighted mean of 6.19 compared to 6.58 of the latter. Clearly the square root model on average provides all critical activities earlier with the DF that is needed to satisfy their need. In fact, m is almost identical for (a) and (c) in Figure 1, which supports the hypothesis that (b) is the balance approach between these extremes. Finally, weighted standard deviations compare nearly identical at 3.36 versus 3.33 per Table 1. Both are much lower than s = 4.63 for (a). Testing for statistical significance at an alpha-level of 0.05 finds the difference between means not significant; however, this is most likely due to the very small case example with only seven different activities. More experimentation is therefore needed to substantiate the positive indications of the square root approach for float allocation. 01002003004005006007008000 10 20 30 40 01002003004005006007008000 10 20 30 40 01002003004005006007008000 10 20 30 40 (a) Equal Shares (b) Square Root (c) Proportional Shares Figure 1: Case Example Performance for Three Mechanisms of Float Allocation 5 CONCLUSIONS AND RECOMMENDATIONS This research has presented a vision for overcoming the long-standing and deeply entrenched paradigm within construction project scheduling by uncovering its various implicit assumptions. They have included focusing excessively on dissecting risk factors on the one hand and delay liability on the other hand; not explicitly distinguishing raw from realistic durations and basing criticality assessment on the latter, not the former; and artificially creating a paradoxical two-class society of critical versus non-critical activities that actively contradicts any desire to minimize the overall risk level. It is thus understandable that previous approaches to float ownership failed to solve the true underlying issue – minimizing risk by mitigating delays. This research stipulated that all critical activities should receive float, not by directly inserting it from its location between calculated and contract finish to being distributed along the critical path within a network schedule itself, but rather by providing an unambiguous approach to calculate optimum portions of ownership. This new approach is based on the premise that a directly proportional allocation to ‘size’ of the critical subcontractors’ contributions to the project would not fulfill the overall goal of being fair and equitable. Rather, three cases have been tested for numerical comparison; an allocation based on equal shares (n0), the square root of size – or risk – (n0.5), or proportional shares (n1). Initial results are positive, but not statistically conclusive between the square root approach or proportional shares of float allocation. More experimentation with larger and more realistic examples of network schedules is recommended. Count of Overrun Contract Float Count of Overrun Contract Float Count of Overrun Contract Float 025-8 This research aids in fulfilling the unrealized vision of “Total Float Traded as Commodity” (de la Garza et al. 1991, p. 716). Only that it will not be the TF of non-critical activities, but CF, and that it will be designed to benefit critical activities. And that it will require having completed a pre-allocation of CF. Combined with the creative analogy from voting mathematics, which provides the seminal inspiration for how to achieve a provable fairness between large and small subcontractors who compete for CF, a market of opportunities can emerge. More research is necessary on how to determine dollar value, which should be listed in the contract to give all subcontractors certainty. Future research should investigate how the emphasis that is placed on measures of ‘size’ – whether it is duration, cost, risk (depending on type and nature of activity) – can be weighted and converted into a single numerical metric (Thompson and Lucko 2011). The current approach has merely assumed that a numeric value of risk (or size) is available as input. Closely related to this, it should also be examined how the potentially different subjective monetary valuations of CF by said contractors can be aggregated. Further analogies from areas of knowledge besides construction project management should be parsed to identify proven concepts that can handle subjective valuation. Acknowledgements The support of the National Science Foundation (Grant CMMI-1265989) for portions of the work presented here is gratefully acknowledged. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The computer programming by Mr. Yi Su, graduate research assistant and doctoral student, is gratefully acknowledged. References Al-Gahtani, K. S. 2006. A comprehensive construction delay analysis technique – enhanced with a float ownership concept. Dissertation, State University of New York at Buffalo, Buffalo, NY: 388 pp. Al-Gahtani, K. S., Mohan, S. B. 2007. Total float management for delay analysis. Cost Engineering, 49(2): 32-37. Barraza, G. A. 2011. Probabilistic estimation and allocation of project time contingency. Journal of Construction Engineering and Management, 137(4): 259-265. Banzhaf, J. 1965. Weighted voting doesn’t work. Rutgers Law Review 19(2): 317-343. Bashford, H. H., Sawhney, A., Felt, J., Koh. D. 2007. Benchmarking the contribution of labor in production homebuilding. Proceedings of the 2007 Construction Research Congress, Grand Bahama Island, Bahamas, May 6-8, 2007, American Society of Civil Engineers, Reston, VA, 2: 843-851. Chantaravarapan, S., Gunal, A., Williams, E. J. 2004. On using Monte Carlo methods for scheduling. Proceedings of the 36th Winter Simulation Conference, Washington, DC, December 5-8, 2004, Winter Simulation Conference Foundation, Catonsville, MD: 1870-1875. Chen, Y.-L., Rinks, D., Tang, K. 2001. The first K minimum cost paths in a time-schedule network. Journal of the Operational Research Society, 52(1): 102-108. de la Garza, J. M., Vorster, M. C., Parvin, C. M. 1991. Total float traded as commodity. Journal of Construction Engineering and Management, 117(4): 716-727. de la Garza, J. M., Prateapusanond, A., Ambani, N. 2007. Preallocation of total float in the application of a critical path method based construction contract. Journal of Construction Engineering and Management, 133(11): 836-845. Deng, F., Smyth, H. 2013. Contingency-based approach to firm performance in construction: Critical review of empirical research. Journal of Construction Engineering and Management, 0733-9364/04013004(14). Fente, J., Schexnayder, C. J., Knutson, K. 2000. Defining a probability distribution for construction simulation. Journal of Construction Engineering and Management 126(3): 234-241. González, P., González, V., Molenaar, K. R., Orozco, F. 2013. Analysis of Causes of Delay and Time Performance in Construction Projects. Journal of Construction Engineering and Management, 0733-9364/04013027(9). Gündüz, M., Nielsen, Y., Özdemir, M. 2013. Quantification of delay factors using the relative importance index method for construction projects in Turkey. Journal of Management in Engineering, 29(2): 133-139. 025-9 Gurgun, A. P., Zhang, Y., Touran, A. 2013. Schedule contingency analysis for transit projects using a simulation approach. Journal of Civil Engineering and Management, 19(4): 465-475. Herroelen, W., Leus, R. 2001. On the merits and pitfalls of critical chain scheduling. Journal of Operations Management 19(5): 559-577. Ibbs, C. W., Nguyen, L. D. 2011. Concurrent delays and apportionment of damages. Journal of Construction Engineering and Management, 137(2): 119-126. Kelley, J. E., Walker, M. R. 1959. Critical-path planning and scheduling. Proceedings of the Eastern Joint Computer Conference, ed. Heart, F. E., Boston, MA, December 1-3, 1959, National Joint Computer Committee, Association for Computing Machinery, New York City, NY: 160-173. Khamooshi, H., Cioffi, D. F. 2013. Uncertainty in task duration and cost estimates: Fusion of probabilistic forecasts and deterministic scheduling. Journal of Construction Engineering and Management, 139(5): 488-497. Ma, G., Wang, A., Li, N., Gu, L., Ai, Q. 2014. Improved critical chain project management framework for scheduling construction projects. Journal of Construction Engineering and Management 04014055(12). Pasiphol, S. 1994. New concepts for total float distribution in CPM project scheduling. Dissertation University of Texas, Austin, TX: 235 pp. Penrose, L. 1946. The elementary statistics of majority voting. Journal of the Royal Statistical Society 109(1): 53-57. Pöppe, C. 2007. Sterben für eine Quadratwurzel? Web publication, Spektrum der Wissenschaft June 16, 2007. Prateapusanond, A. 2003. A comprehensive practice of total float preallocation and management for the application of a CPM-based construction contract. Dissertation, Virginia Polytechnic Institute and State University, Blacksburg, VA: 230 pp. Rand, G. K. 2000. Critical chain: the theory of constraints applied to project management. International Journal of Project Management 18(3): 173-177. Smith, G. R., Bohn, C. M. 1998. Small to medium contractor contingency and assumption of risk. Journal of Construction Engineering and Management, 125(2): 101-108. Steyn, H. 2000. An investigation into the fundamentals of critical chain project scheduling. International Journal of Project Management, 19(): 363-369. Tang, P., Mukherjee, A., Onder, N. 2013. “Construction schedule simulation for improved project planning: Activity criticality index assessment.” Proceedings of the 45th Winter Simulation Conference, Washington, DC, December 8-11, 2013, Winter Simulation Conference Foundation, Catonsville, MD: 3237-3248. Thompson, R. C., Lucko, G. 2011. Modeling measures of float allocation for quantitative risk management of construction projects. Proceedings of the 2011 3rd International / 9th Construction Specialty Conference at the Canadian Society for Civil Engineering Annual Conference, Ottawa, Ontario, June 14-17, 2011, Canadian Society for Civil Engineering, Montréal, Quebec: CN-002-1 – CN-002-10. Thompson, R. C. 2012. Risk measurement, allocation, and pricing in network schedule systems. Dissertation, Catholic University of America, Washington, DC: 43 pp. Thompson, R. C., Lucko, G. 2012. Modeling measures of float monetization for quantitative risk management of construction projects.” Proceedings of the 2012 Construction Research Congress, West Lafayette, IN, May 21-23, 2012, American Society of Civil Engineers, Reston, VA: 485-494. Tran, D., Molenaar, K. R. 2012. Critical risk factors in project delivery method selection for highway projects. Proceedings of the Construction Research Congress, West Lafayette, IN, May 21-23, 2012, American Society of Civil Engineers, Reston, VA: 331-340. Xie, H., AbouRizk, S. M., Zou, J. (2012). Quantitative method for updating cost contingency throughout project execution. Journal of Construction Engineering and Management, 138(6): 759-766. Yang, J.-B., Kao, C.-K. 2012. Critical path effect based delay analysis method for construction projects. International Journal of Project Management 30(3): 385-397. Zack, J. G. 1992. Schedule ‘games’ people play, and some suggested remedies. Journal of Management in Engineering, 8(2): 138-152. Zhao, Z. Y., You, W. Y., Zuo, J. 2010. Application of innovative critical chain method for project planning and control under resource constraints and uncertainty. Journal of Construction Engineering and Management 136(9): 105-1060.  025-10  Simulated Schedule Delay Mitigation via Float Allocation   by  Gunnar Lucko Richard C. Thompson  International Construction Specialty Conference June 9, 2015 Overview  •  Introduction  •  Literature Review §  Scheduling assumptions, criticality paradox §  Float ownership, fair allocation   •  Research Methodology §  Square-root analogy, hypothesis  •  Simulation Calculations §  Conceptual approach §  Results and interpretation  •  Conclusions and Recommendations Introduction  •  Delays inundate construction (Gündüz et al. 2013)  •  Act not just locally, but may have ripple effects  •  Caused by risk factor(s) that materialize (González et al. 2013) and have been amply studied already  •  Effects of late completion: Lost bonus, liquidated damages, claims, court cases, damaged reputation à Should cure cause (preventive), not symptoms: What inherent defense schedule has against delay? Scheduling Assumptions   •  Acyclic networks (Chen et al. 2001)  §  Critical path method (Kelley and Walker 1959) sequentially adds durations: max {Fpred} = Ssucc §  Best case if all activities start as soon as possible §  Compare with worst case: How much can it be delayed until…  Impact any succ.: FFpred = min{ESsucc}-EFpred)  Impact project: TF = LF-EF = LS-ES (used here)  •  Float is defined by assumed delay / opposite of criticality (de la Garza et al. 2007)  •  Focus should be on delay avoidance, not claims Hidden Assumptions   •  Links: For technical reasons, but also administrative reasons, or even to create or reduce float (Zack 1992) à Assume links as fixed  •  Durations: Determined from quantity take-off and resource unit productivity §  Distributions for probabilistic durations possible (Khamooshi & Cioffi 2013) §  In practice, ‘raw’ plus contingency (added float) à Assume durations as unmodified  •  Contingency “used in project planning to cover activities delays, oversights, and unknowns and as a cushion for possible time-estimating errors” (Barraza 2011, p. 259) Criticality Paradox  •  Definition of criticality as ‘inability to absorb delays’ §  Based on assumption that activities are unequal §  But schedulers create criticality and float as artifacts of chosen links and durations (Zack 1992) §  Not inherent to nature of activity or its unique risk   •  Artificial split: “By definition, noncritical subcontractors have float, but… critical ones have none, even though they need it most” (Thompson & Lucko 2012, p. 488) §  Counterintuitive: One activity type on verge of failure  •  Goal:  Create schedule that minimizes overall risk level (or probability of local delays affecting any successor) of all activities by allocating each float that it Float Ownership  •  Schedulers have to use “subjective management of the project time contingency” (Barraza 2011, p. 260) §  Empirical approach only §  Quantify and formalize float need for realism  •  Float mitigates delays §  Becomes valuable commodity (de la Garza et al. 1991)  •  Contentious unresolved issue: Who owns float? §  First-come, first-served? (Pasiphol 1994)? §  Owner or contractor? (Al-Gahtani 2006)? §  Split somehow (Prateapusanond 2003)?  à Should link “ownership with the party who carries the risk” (Al-Gahtani & Mohan 2007 p. 33) Fair Allocation  •  Who should own float, and how much? §  Activities on critical path in as-planned schedule §  Allocated from pool at project end, between sum of raw durations and contract deadline §  Inverse to critical chain project management (CCPM)  •  But: CCPM aggregates, not allocates §  Radically removes all buffers, simplistically reduces 50% “the likely duration” (Zhao et al. 2010, p. 1056) §  Creates project buffer (Rand 2000), changes dates §  Then reinserts ‘feeding buffers’ on non-critical paths §  Focus on duration, not risk (Herroelen & Leus 2001)  •  Fair quantitative approach is required a priori Methodology  •  Analogy-adapting to create mathematical approach  •  Inspired by social science (Thompson & Lucko 2011) §  Compare float allocations – fair and equitable? §  Buffer simulation not a priori (Park & Peña-Mora 2004)  •  Initial assumptions §  Raw durations w/o contingency, risk ~ duration §  Weighted combination of time, cost, other factors will be examined under future research  •  Definition: CF = contract finish – calculated finish §  Critical path from baseline as-planned schedule Square-Root Analogy  •  Decision-making (Arrow 1964): Voting versus markets §  Rules and hierarchies to reach a consensus §  Social choice problems: Fair division, voting models §  {System, voters, decision} ↔ {project, subs, delay}  •  Fairness challenge if participants have unequal ‘size’ •  Two related models: §  Penrose (1946) square root law (inspired by UN) §  Banzhaf (1965) power index  Both explore how to fairly balance ‘small’ vs. ‘large’ participants so that they neither never nor always dominate a decision (in majority or as tiebreaker) Inspiration  •  Penrose (1946, p. 53) inspires new float allocation:  [P]ower of the individual vote[r] can be measured by the amount by which his chance of being on the winning side exceeds one half. (…) [F]ormula for the probability of equal division of  random votes, where n is an even number, approaches           when  is large. It follows that the power of the individual vote[r] is inversely proportional to the square root of the number of people in the committee.  •  Distinguished weight (factor) from power (changing a decision) à All participants should have equal power  •  Population n of EU countries, unfair extremes: §  ‘One vote per country’ (~n0) §  ‘One vote per person’ (~n1) πn2Methodology  •  Transfer conceptual analogy to network schedules: §  Vote Y/N ↔ Use float (due to delay) or not §  Voters ↔ Subcontractors §  Election ↔ Project §  {System, voters, decision} ↔ {project, subs, delay}  •  Hypothesis  It is hypothesized that allocating float to the activities on the critical path based on the square root of the measure of float need leads to less need for project float than an allocation based on order zero or one. Simulation  •  Modular computer implementation: §  Input text file: Activities, links, duration distributions §  Output text file: Post-processed for analysis §  Each run samples distributions for unique project  •  Monte Carlo method (Chantaravarapan et al. 2004) •  Criticality index (CI, Tang et al. 2013, p. 3238) “is… the probability of an activity becoming critical”, BUT: §  CI based on (difficult) distr. of full duration, not raw §  CI exists within calculated project duration, no CF §  CI assumes critical path may vary (more complex) §  Float allocation sim. will vary and chart dur. and CF Simulation  •  Assume asymmetric triangular distr. 90%/100%/125%  •  Realistic because activities take longer more often  •  Future research will use e.g. beta (Fente et al. 2000) §  Forward / backward pass and float of baseline schedule with fixed durations à critical path §  Replace with distr. for 1000 repetitions, CF={1-40} §  Distributed float (DF) is a priori allocated part of CF §  Post-processing (1 run, not 40):  ActualD – (PlannedD + DF) = Float overrun  •  Rounding is future challenge: Limited valuable integer CF becomes integer DF for critical subcontractors Case Example Input and Allocation Traditional CPM Calculations Duration0.0 Duration0.5 Duration1.0 Act. Dur. Succ. ES LS EF LF TF FF L R m s L R m s L R m s Mob. 7 A, B, E 0 0 7 7 0 0 3 11 4.80 3.85 4 13 5.87 3.52 4 14 6.34 3.82 A 19 D, I, J 7 13 26 32 6 0 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A B 10 C 7 7 17 17 0 0 3 18 5.98 4.53 3 16 5.62 3.65 3 15 5.60 3.79 C 6 D, F, J 17 17 23 23 0 0 3 11 4.24 3.05 4 14 5.43 3.32 5 16 6.52 3.64 D 18 L 26 33 44 51 7 7 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A E 15 F, G 7 8 22 23 1 0 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A F 17 H, I, K 23 23 40 40 0 0 3 32 9.83 7.38 2 19 6.15 4.19 2 16 5.48 3.60 G 16 H, I, K 22 24 38 40 2 2 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A H 6 M 40 53 46 59 13 0 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A I 11 L 40 40 51 51 0 0 3 18 6.79 5.21 3 15 6.06 3.72 3 14 5.85 3.72 J 19 L 26 32 45 51 6 6 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A K 15 T/O 40 54 55 69 14 14 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A L 18 T/O 51 51 69 69 0 0 3 32 10.69 8.36 2 18 6.60 4.75 1 15 5.17 4.04 M 10 T/O 46 59 56 69 13 13 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A T/O 3 N/A 69 69 72 72 0 0 3 4 3.50 0.00 6 9 7.56 0.37 9 12 11.07 0.69 Grand Average Across Activities 3.0 18.0 6.55 4.63 3.4 14.9 6.19 3.36 3.9 14.6 6.58 3.33 Thompson and Lucko (2011) Performance for Three Allocation Mechanisms 01002003004005006007008000 10 20 30 4001002003004005006007008000 10 20 30 4001002003004005006007008000 10 20 30 40Count of Overrun Contract Float Count of Overrun Contract Float Count of Overrun Contract Float Results  •  Plotting float overrun over CF for three mechanisms  •  Shapes represent efficacy of delay mitigation: §  ‘Plateau’: Low CF, some critical activities consume more than they own or may not even have DF yet §  ‘Cliff’: Float shows powerful absorption of delays §  ‘Bottom’: Activities within their DF even if delayed; project has minimum prob. of exceeding deadline §  ‘Beyond’: Excess DF, project could be shorter   •  Describing ‘cliff’: L edge, R edge, mean m and standard deviation s (weighted as cliff is stepped) §  n0,n0.5,n1: {3.0-18.0, 3.4-14.9, 3.9-14.6} = {15, 11.5, 10.7} §  m = {6.55, 6.19, 6.58} not sign. diff., s = {4.63, 3.36, 3.33} Conclusions  •  Vision for overcoming paradigm of float ownership §  Targets critical activities, but DF remains at project end §  Unambiguous and preventive §  Fair and equitable – even risk §  à DF as tradable commodity (de la Garza et al. 1991)   •  Recommendations for future research §  Duration, cost, risk as combined input §  More large examples, other distributions §  Rounding challenge (different methods) §  Measure instances and severity of overrun §  Monetary valuation in market of opportunities Acknowledgement    The support of the National Science Foundation (Grant CMMI-1265989) for portions of the work presented here is gratefully acknowledged. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. Thank you! Do you have any questions? Contact Gunnar Lucko, Ph.D. Director, CE&M Program Associate Professor Department of Civil Engineering Catholic University of America 620 Michigan Avenue NE Washington, DC 20064  202-319-4381 lucko@cua.edu http://faculty.cua.edu/lucko Richard C. Thompson, Ph.D. Postdoctoral Research Associate       thompsrc@cua.edu 

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