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Quantification of risks during feasibility analysis for capital projects Ranasinghe, Kulatilaka Arthanayake Malik Kumar 1986

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QUANTIFICATION OF RISKS DURING FEASIBILITY ANALYSIS FOR CAPITAL PROJECTS by KULATILAKA ARTHANAYAKE MALIK KUMAR RANASINGHE A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CIVIL ENGINEERING We accept this thesis as conforming to the reguired standard THE UNIVERSITY OF BRITISH COLUMBIA 10 DECEMBER 1986 © Kulatilaka Arthanayake Malik Kumar Ranasinghe, 1986 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the The University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. DEPARTMENT OF CIVIL ENGINEERING The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: 10 DECEMBER 1986 ABSTRACT The purpose of this thesis is to propose a consistent theory and a model based on i t to estimate the uncertainty of project duration, cost, revenue, and net present value probabilistically. The model can be used to assist decision making on such strategic, feasibility analysis issues as contingency provision, r e l i a b i l i t y of an estimate for the "go-no go" decision, adopting phased or fast-track construction, etc. Project cost and revenue are evaluated in terms of current and discounted dollars, thereby emphasising the economic effect of time and inflation on net present value which is considered as the decision criterion. The model is derived mathematically by treating a l l the issues which effect the estimation of project cost, duration and revenue through the mechanism of linked work packages. Issues found to be significant in the evaluation of work package duration are: the scope of work, the productivity, and the labour usage. For work package cost they are: the duration and the starting time, unit rates for labour, equipment, and materials, labour and equipment usage, sub-contractor and indirect cost, inflation and interest rates. For revenue the issues are: the gross revenue, operating & maintenance cost, inflation rates, duration, and the starting time. Moments of work package cost, duration and revenue streams are f i r s t evaluated using subjective estimates of percentiles for the independent variables, deriving moment information from these estimates, and then processing this information using the expectation operator on the Taylor series expansion of the performance measure about the mean. These moments along with the Pearson family of distributions are used to quantify the uncertainity of project duration, cost, revenue, and net present value. The decision maker is provided with probabilistic estimates, of duration, cost and revenue at both the work package/revenue stream and project levels and of the net present value. i i A computer program is developed to implement the proposed theory and to organise and simplify the calculation process. Table of Contents ABSTRACT i i LIST OF' FIGURES v i i i LIST OF TABLES x ACKNOWLEDGEMENTS xi 1. INTRODUCTION 1 1 .1 Background for the Research 1 1.2 The Research Needs 3 1.3 Thesis Objectives 4 2. LITERATURE REVIEW 6 2.1 General 6 2.2 Probabilistic Estimation Models 6 2.2.1 Time Estimation Models 7 2.2.2 Cost Estimation Models 12 2.3 Assessment of Probabilities 18 2.4 Summary 2 3 3. MODEL CONCEPT I : THE APPROACH AND THE BASIC TOOLS ..25 3.1 Background for the Proposed Model 25 3.2 The Approach for the Proposed Model 26 3.3 The Primary Variable 26 3.3.1 Subjective Estimates for a Primary Variable 28 3.3.2 The Pearson Family of Distributions 28 3.3.3 The First Four Moments of a Primary Variable 34 3.4 The Dependent Variable 37 3.4.1 The Additive Form 38 3.4.2 The Subtractive Form 40 3.4.3 Scalar Performance Functional Form 41 i v 3.4.4 Derived Variable From Dependent Random Variables 44 3.5 Cumulative Distribution Function 46 4. MODEL CONCEPT II : THE MATHEMATICAL FRAMEWORK 48 4.1 General 48 4.2 Work Package Duration Model 48 4.3 Project Duration Model 50 4.3.1 Project Network 50 4.3.2 Early Start Time of a Work Package 50 4.3.3 Project Duration 52 4.4 Work Package Cost Model 53 4.4.1 Model Development 53 4.4.2 Escalation and Interest During Construction 55 4.4.3 Discounted Work Package Cost 57 4.5 Project Cost Model 62 4.6 Project Revenue Model 63 4.6.1 Net Revenue Stream 64 4.6.2 Net Present Value of Project Revenue 66 4.7 Net Present Value Model 67 4.8 Model Assumptions 68 4.8.1 General Assumptions 69 4.8.2 Assumptions for the Duration Models 70 4.8.3 Assumptions for the Cost Models 71 4.8.4 Assumptions for the Revenue Models 72 5. DEVELOPMENT OF A COMPUTER MODEL 74 5.1 General 74 5.2 The Input Data 76 v 5.2.1 Data Fil e for Control Unit 3 77 5.2.2 Data Fil e for Control Unit 4 79 5.2.3 Data F i l e for Control Unit -5 80 5.3 Step 1 : Moments of the Primary Variables 81 5.4 Step 2 : The Duration Models 82 5.4.1 Subroutine "WPDURA" 82 5.4.2 Subroutine "NETWRK" 83 5.5 Step 3 : The Cost Models 84 5.5.1 Subroutine "WPCOST" 84 5.6 Step 4 : Revenue Models 86 5.6.1 Subroutine "REVNUE" 86 5.7 Step 5 : Net Present Value Model 87 5.8 Cumulative Distribution Function 87 5.8.1 Subroutine "CDFUNC" 88 5.8.2 Subroutine "GRAPH" 88 6. APPLICATION OF THE MODEL : RESULTS AND DISCUSSION ...89 6.1 General 89 6.2 Example 1 : Model Behaviour 89 6.3 Example 2 : Model Application 99 6.3.1 Construction schedule and work package cost 101 6.3.2 Modified data for the model 101 6.3.3 Project Cost and Contingency Allocation ..104 6.3.4 Internal rate of return 110 7. CONCLUSIONS AND RECOMMENDATIONS 117 7.1 Conclusions 117 7.2 Recommendations for Future Research 118 7.2.1 The Theory 118 vi 7.2.2 The Model 119 7.2.3 Input data and the output 120 REFERENCES 122 APPENDIX A 1 28 APPENDIX B 151 v i i LIST OF FIGURES 2.1 Activity sharing between paths j & k 09 2.2 A Typical Work Package 23 3.1 Cumulative subjective function corresponding to the subjective estimtes 27 3.2 Diagram showing the areas and bounding curves associated with the different solutions of Karl Pearson's differential equation 29 3.3 Contours in the (1. , plane on which the 5.0% and 2.5% distances are constant for Pearson curves 32 4.1 General Cash Flow Diagram for a Construction Project 56 4.2 Cash Flow Diagram assumed for the Model 56 4.3 Discounted Work Package Cost Diagram 58 5.1 The structure and the major modules of the program 75 6.1 A network of work packages 90 6.2 The effect of uncertainty at primary variable level on the work package level for the total dollar cost estimate 92 6.3 The effect of uncertainty at primary variable level on the project level for the total dollar cost estimate 93 v i i i 6.4 The effect of uncertainty at work package level on the project level for the total dollar cost estimate 96 6.5 The effect of uncertainty of the total project cost on the net present value 98 6.6 Construction Schedule 100 6.7 Constant Dollar Cost Estimate 106 6.8 Current Dollar Cost Estimate 107 6.9 Total Dollar Cost Estimate 108 6.10 Discounted Project Cost 111 6.11 Discounted Project Revenue 112 6.12 Net Present Value Diagram 113 6.13 Net Present Value vs Discount Rate 115 ix L I S T O F T A B L E S 6.1 Deterministic (median) estimates for work package costs and durations 102 6.2 Probabilistic estimates for work package costs and durations 103 6.3 Comparision between the deterministic and probabilistic analyses for the project cost 105 x ACKNOWLEDGEMENTS I wish to express my sincere gratitude to Dr. Alan D. Russell, my supervisor for his valuable advice and guidance throughout my studies. I greatly appreciate his effort and time in reviewing this thesis and the valuable suggestions to improve the content. My thanks to Dr. W.F. Caselton for reviewing this thesis. Acknowledgement is most gratefully extended to the Canadian Commonwealth Scholarship and Fellowship Committee and the Association of Universities and Colleges of Canada who provided the scholarship which enabled me to pursue graduate studies in Canada. A special thank you to professors, friends and colleagues who gave valuable advice and support during this study. Finally, to my wife Deepthi, your patience, support and encouragement is most gratefully acknowledged. xi To my parents and to my grandmother f o r t h e i r guidance, support and encouragement in a l l my endeavours x i i 1. INTRODUCTION 1.1 BACKGROUND FOR THE RESEARCH The increase in demand for energy and raw materials with the advancement of technology in the recent past saw the emergence of large, complex projects. Typically these projects have long durations, high costs, multiple investors and are undertaken in highly uncertain environments. The high degree of uncertainty in duration, cost and scope of these projects creates complexities for the project management function. Further, the large investments required for such projects, have necessitated the adoption of concepts such as fast-track and phased construction. The very nature of these concepts coupled with the increasing size, complexity and riskiness of projects necessitates explicit treatment of risk in the decision making process, especially in its early stages. The successful completion of a project implies, among many things, delivering i t within a specified duration and budget. This achievement depends much on the decisions management takes during its f e a s i b i l i t y stage. A common mistake in those projects which have been financial failures is the inadequate effort devoted to their f e a s i b i l i t y study resulting in the setting of unrealistic targets for performance. 1 2 The.purpose of the fea s i b i l i t y analysis is to develop and evaluate alternatives so that the most desirable alternative can be selected and implemented. A project is economically feasible i f the net present value of the benefits generated from it exceeds the net present value of its cost at the minimum attractive rate of return (MARR). The net present value of a project at MARR and its internal rate of return are two parameters that guide the decisions on the economic fe a s i b i l i t y of a project. Generally speaking, the selected alternative should be, in management's view, the best in terms of technical, socio p o l i t i c a l , economic, and financial f e a s i b i l i t y . The importance of decision making on the overall project estimate and duration is maximum during this stage and gives management significant leeway to make changes in the scope of the project, either partially or totally, or even to cancel the project with minimum loss. S t i l l , the limited information available at this stage increases the uncertainty of such decisions. The a b i l i t y to quantify some of this uncertainty would improve the quality of these decisions significantly. Today, a need exists for a tool that could assist management in its important decisions at the feasibility stage , which not only evaluates the net present value of the project at MARR and its internal rate of return, but also quantifies its uncertainty. Such a tool would help considerably in the "go - no go" decisions, setting 3 p e r f o r m a n c e t a r g e t s , s e l e c t i n g s t r a t e g i e s s u c h a s f a s t - t r a c k a n d / o r p h a s e d c o n s t r u c t i o n , a n d d e t e r m i n i n g c o n t i n g e n c y a l l o w a n c e s r e f l e c t i n g t h e c l i e n t ' s v i e w p o i n t on a c c e p t a b l e p r o b a b i l i t i e s o f f u t u r e . 1.2 THE RESEARCH NEEDS A c o n s i s t e n t t h e o r y i s n e e d e d , t h a t t r e a t s t h e u n c e r a i n t y o f t i m e and c o s t o f a n e s t i m a t e , t h a t i n t e g r a t e s t h e t i m e a n d c o s t a s p e c t s , t h a t r e c o g n i s e s d a t a l i m i t a t i o n s b u t a l s o t h a t p r o v i d e s i n t e r m e d i a t e i n f o r m a t i o n on t h e r e l a t i v e c o n t r i b u t i o n s e a c h b a s i c p a r a m e t e r makes t o u n c e r t a i n t y . R e s e a r c h work i s r e q u i r e d t o o b t a i n an a p p r o a c h t h a t : 1 . M o d e l s t h e d u r a t i o n , c o s t , r e v e n u e , n e t p r e s e n t v a l u e a n d i n t e r n a l r a t e o f r e t u r n o f a p r o j e c t b o t h a c c u r a t e l y a n d r e a l i s t i c a l l y , a n d 2. Q u a n t i f i e s t h e u n c e r t a i n t y a s s o c i a t e d w i t h e a c h o f t h e m o d e l e d e s t i m a t e s . W h i l e t h e u l t i m a t e d e c i s i o n s on f e a s i b i l i t y a r e a t t h e o v e r a l l p r o j e c t l e v e l , t h e i m p o r t a n c e o f a two l e v e l d e s c r i p t i o n o f a p r o j e c t ( i . e a t p r o j e c t l e v e l a n d a t work p a c k a g e l e v e l ) f o r m o d e l l i n g i s r e c o g n i s e d . A s t u d y o f t h e b e s t l e v e l o f s e p a r a t i o n t o c o m p l e m e n t t h e a v a i l a b l e i n f o r m a t i o n w i t h t h e a b i l i t y t o m o d e l i s i m p o r t a n t . Once s e p a r a t e d , i t i s n e c e s s a r y t o f i n d ways t o i n t e g r a t e t h e p l a n n i n g p r o c e s s a t t h e two l e v e l s . F u r t h e r , a p r o b a b i l i s t i c 4 a p p r o a c h f o r t h e t r e a t m e n t o f c o s t a n d t i m e t h a t i s b o t h a c c u r a t e a n d m a t h e m a t i c a l l y c o n s i s t e n t i s r e q u i r e d . I d e a l l y , an a n a l y s t u s i n g t h i s t o o l s h o u l d be a b l e t o e x p l o r e a s many a l t e r n a t i v e s a s p o s s i b l e a n d e v a l u a t e them w i t h r e s p e c t t o v a r i o u s c r i t e r i a . The a l t e r n a t i v e s e l e c t e d a s a r e s u l t o f h i s s t u d y s h o u l d be t h e one t h a t i s most c o n s i s t e n t w i t h t h e c l i e n t o b j e c t i v e s . 1 . 3 T H E S I S O B J E C T I V E S T h i s t h e s i s d e s c r i b e s t h e d e v e l o p m e n t o f a c o n s i s t e n t t h e o r y a n d a c o m p u t e r m o d e l b a s e d on i t w h i c h i s s u i t a b l e f o r p r e l i m i n a r y p l a n n i n g p u r p o s e s . The m o d e l e s t i m a t e s t h e d u r a t i o n , c o s t , r e v e n u e , n e t p r e s e n t v a l u e , a n d t h e u n c e r t a i n t y a s s o c i a t e d w i t h e a c h o f t h e s e c r i t e r i a o f a p r o j e c t i m p l e m e n t e d i n t h e c o n v e n t i o n a l , f a s t - t r a c k , o r p h a s e d m ethod o f c o n s t r u c t i o n . C h a p t e r 2 c o n t a i n s a r e v i e w o f t h e l i t e r a t u r e on some o f t h e a v a i l a b l e p r o b a b i l i s t i c d e c i s i o n t o o l s a n d some a p p r o a c h e s t o o b t a i n i n g p r o b a b i l i s t i c d a t a . The r e v i e w e d d e c i s i o n m o d e l s a r e c a t e g o r i s e d a s p r o b a b i l i s t i c t i m e a n d c o s t m o d e l s t o f a c i l i t a t e a c o m p a r i s i o n o f t h e i r a b i l i t i e s a n d a s s u m p t i o n s . The k e y o b s e r v a t i o n o f t h e r e v i e w i s t h e l i t t l e work done t o d a t e t o i n t e g r a t e b o t h t i m e a nd c o s t a s p e c t s . C h a p t e r 3 c o n t a i n s t h e b a s i c t o o l s n e c e s s a r y t o d e v e l o p t h e m a t h e m a t i c a l f r a m e w o r k o f t h e m o d e l . T h e s e i n c l u d e t h e 5 p r o p o s e d method t o e v a l u a t e t h e moments o f a p r i m a r y v a r i a b l e , t h e o p e r a t i o n s on t h e d e p e n d e n t v a r i a b l e t o o b t a i n i t s moments and t h e m e t h o d t o o b t a i n t h e c u m u l a t i v e d i s t r i b u t i o n f u n c t i o n o f t h e d e p e n d e n t v a r i a b l e . S i m p l e e x a m p l e s a r e g i v e n t o i l l u s t r a t e t h e b a s i c r e l a t i o n s h i p s a n d t h e p r o c e d u r e s . I n C h a p t e r 4 t h e m a t h e m a t i c a l f r a m e w o r k f o r t h e p r o p o s e d m o d e l i s d e v e l o p e d . T h i s f r a m e w o r k i s b a s e d on t h e t o o l s d e v e l o p e d i n t h e p r e v i o u s c h a p t e r a n d s e r v e s a s t h e f o u n d a t i o n f o r t h e m o d e l l i n g d e s c r i b e d i n t h e f o l l o w i n g c h a p t e r . The a s s u m p t i o n s made i n d e v e l o p i n g t h e f r a m e w o r k a r e d i s c u s s e d a t t h e end o f t h e c h a p t e r . C h a p t e r 5 c o n t a i n s t h e m o d e l l i n g p r o c e s s . I t i n c l u d e s t h e m e t h o d o f p r o v i d i n g t h e d a t a r e q u i r e d by t h e m o d e l , t h e s t r u c t u r e o f t h e m o d e l , how i t i s o p e r a t e d , a n d t h e d e f a u l t s i t u a t i o n s . C h a p t e r 6 c o n t a i n s two e x a m p l e s t o d e m o n s t r a t e t h e a p p l i c a t i o n o f t h e m o d e l . The f i r s t d e a l s w i t h a s i m p l i f i e d e x a m p l e t o d e m o n s t r a t e t h e m o d e l b e h a v i o u r a n d t h e i n f l u e n c e o f v a r y i n g u n c e r t a i n t y a t t h e i n d e p e n d e n t p a r a m e t e r l e v e l on t h e p e r f o r m a n c e m e a s u r e s o f i n t e r e s t . The s e c o n d d e a l s w i t h d a t a o b t a i n e d f r o m an a c t u a l d e t e r m i n i s t i c f e a s i b i l i t y a n a l y s i s f o r a m i n i n g p r o j e c t . F i n a l l y , C h a p t e r 7 c o n t a i n s c o n c l u s i o n s a n d r e c o m m e n d a t i o n s f o r f u t u r e d e v e l o p m e n t s t o t h e c o m p u t e r m o d e l a n d r e s e a r c h work i n o r d e r t o p r o d u c e an e f f e c t i v e t o o l f o r p r o b a b i l i s t i c d e c i s i o n m a k i n g . 2 . LITERATURE REVIEW 2 . 1 GENERAL T r a d i t i o n a l l y , e s t i m a t e s f o r p r o j e c t c o s t a n d d u r a t i o n h a v e been t r e a t e d a s d e t e r m i n i s t i c , i g n o r i n g t h a t b o t h t h e c o s t a n d d u r a t i o n o f t h e i n d i v i d u a l work p a c k a g e a n d t h e p r o j e c t a r e random. A number o f a u t h o r s h a v e r e c o g n i s e d t h e p r o b a b i l i s t i c n a t u r e o f e s t i m a t e s i n d e v e l o p i n g t h e i r m o d e l s . I n t h i s c h a p t e r , a r e v i e w o f e x i s t i n g work on p r o b a b i l i s t i c e s t i m a t i o n m o d e l s a n d t h e i r a p p l i c a t i o n s a n d t h e a p p r o a c h e s t o a s s e s s i n g p r o b a b i l i t i e s i s d o n e . 2 . 2 PROBABILISTIC ESTIMATION MODELS P r o b a b i l i s t i c e s t i m a t i o n m o d e l s i n c o n s t r u c t i o n c a n be c l a s s i f i e d d e p e n d i n g on t h e i r a p p l i c a t i o n s i n t o , 1. Time E s t i m a t i o n M o d e l s : t h o s e w h i c h e s t i m a t e t h e d u r a t i o n o f t h e a c t i v i t i e s a n d t h e p r o j e c t , 2. C o s t E s t i m a t i o n M o d e l s : t h o s e w h i c h e s t i m a t e t h e c o s t o f a p r o j e c t . 6 7 2.2.1 T I M E E S T I M A T I O N M O D E L S Networks and network analyses have been used as a starting point for analysis and scheduling in time estimation models since the " C r i t i c a l Path Method (CPM)" was developed by Sperry Rand Corporation for Dupont and the "Program Evaluation and Review Technique (PERT)" was developed by the U.S.Navy Special Projects for use in the Polaris Missile Program. The C r i t i c a l Path Method is based on the assumption that the activity duration is deterministic, and accordingly the project duration is deterministic. Due to the uncertainty in construction, i t is d i f f i c u l t to predict the activity duration with certainty. Ang et al (1) state that activity durations should be modeled as random variables and the project duration evaluated as a problem of probabilistic network analysis. While PERT considers the activity durations as random variables, Ang et al (1) show that PERT invaribly underestimates the required completion time for a given network. Elmaghraby (13) studied PERT in depth and showed that, 1. the simplifying assumptions of PERT restrict the shape of the probability distribution of activity duration to only one of three, those of skewness 1 / V 2 , -1 / / 2 , or 0 ; 2 . PERT's estimate of the expected duration of the project is biased on the low side, i.e i t is optimistic ; 8 3 . PERT assumes that the mode is given, thus fixing the relationship between the two shape parameters of the beta distribution X1 and X2 ; 4. the range of uncertainty in the estimate of the variance of an activity is not reduced by assuming a beta distribution, mainly due to the other assumptions of the PERT model ; and 5 . PERT ignores paths other than the c r i t i c a l path. Therefore, the issue of correlation is ignored when evaluating the project duration variance. Ang et al (1) developed, the "Probabilistic Network Evaluation Technique (PNET)" to evaluate the project completion time probability. PNET models activity durations as random variables and assumes them to be s t a t i s t i c a l l y independent, ignoring the correlation brought about by the use of shared resources such as manpower, equipment, management, etc. Accordingly, for path j , the mean duration i s , n = L i=1 (2. 1) and its variance i s , 2 n = Z i = 1 2 (2. 2) 9 Figure 2.1 Activity sharing between paths j & k PNET assumes that there is correlation among paths arising from shared activities, and those paths highly correlated with a major path could be represented by that major path. The correlation between paths j and k because of shared activities, as shown in figure ( 2 . 1 ) , is defined as 10 Those paths with > 0.5 are represented by path j assuming that p = 0.5 represents the transition between high and low correlation. The project completion time probability p(t) is given approximately by the product p(t) - P(T 1<t).P(T 2<t) P(Tn<t) where P(T 1<t), P(T 2<t), , P(Tn<t) are the probability of the "representative" paths. Then assuming the "representative path" durations to be normally distributed the completion time probability of the project is approximated. The PNET model was validated by a Monte Carlo simulation. Crandall (7) identifies, the major problems of PNET as: 1. the selection of representative paths. This includes the number of paths to be considered and the inclusion of those paths with large variations ; 2. the selection of a correlation coefficient p to tune the model ; and 3. the possibility that the final project completion distribution may not be normally distributed i f a few act i v i t i e s dominate the distribution. 11 Crandall (8) compares PNET and Monte Carlo simulation as probabilistic techniques to reduce network information to an acceptable level while providing reliable statistics relating to the expected distribution for the project completion time. While he favors simulation as the more reliable method, the choice between the methods for the analysis of a network is left to the user. In a recent study, Crandall and Woolery (9) address the question of developing viable "milestone dates" from a Monte Carlo simulation network. A method was developed to produce a control schedule based on an acceptable level of risk without using the assumption of normality for node time distributions ("node point" refers to a milestone node in Arrow Network Convention or a milestone activity in Precedence Networking Convention). The method uti l i z e d the Edgeworth series to calculate the node point estimation at a desired percentile value (acceptable level of risk) given the f i r s t four moments about the mean for the distribution. They conclude that for those projects where the acceptable level of risk is very low (CDF > 0.95) the proposed Edgeworth series method provide a viable tool, and where the acceptable risk is higher (CDF < 0.9) Edgeworth series method is superior but the normal distribution can provide the basis to evaluate adequate point estimates. Pritsker, Happ and Whitehouse (26),(27) used Monte Carlo simulation to develop the "Graphical Evaluation and Review Technique, (GERT)" to analyse procedures for complex, 1 2 stochastic networks. GERT models activity durations and logic as stochastic variables and then uses Monte Carlo simulation to obtain the project completion time distribution. Kennedy and Thrall(20) modified GERT to develop the "Project Length Analysis and Evaluation Technique, (PLANET)" for the NASA space shuttle program. PLANET incorporates c r i t i c a l path and slack analysis, which GERT ignores, on every simulation step. PLANET allows the computation of the probability that an activity is c r i t i c a l , the construction of completion time and slack distributions for both events and a c t i v i t i e s , and the completion time distribution for the project duration from a Monte Carlo simulation. The information necessary for the simulation is provided at the input phase in the form of events, a c t i v i t i e s with in i t i a t i n g and terminating events and duration distributions, and special network data. 2.2.2 COST ESTIMATION MODELS The cost estimation models reviewed in this section are those which consider the probabilistic nature of cost. An important observation is that, when cost is treated probabilistically, only some models treat time as a probabilistic quantity but none treat the linkage between the time and cost of an estimate. 1 3 Hemphill (16) used the logarithmic normal distribution to describe an estimate of a cost element. The justification was that the lower bound of a cost estimate is zero and that they are skewed right. Standard normal distribution was substituted for the logarithmic normal when calculating the confidence interval, on the assumption that the low side error can be limited to 40% (i.e when the skewness of the confidence interval becomes significant). The model computes the expected accuracy of an estimate, knowing only the expected value and the confidence interval of each cost element in the estimate. H i l l i a r d and Leitch (17) used the bivariate log normal distribution to modify the Jaedicke and Robichek (1964) model for Cost - Volume - Profit (CVP), based on the same reasoning as above. After their study, which included the effect of correlation, i t was concluded that the log normal assumption was better than Jaedicke and Robichek's normal assumption. Johnson (18) has shown that the bivariate log normal distribution imposes a highly restrictive and unrealistic regressive relationship between the two dependent random variables. Kottas and Lau (22) state that i t is theoretically invalid to impose the bivariate log normal distribution to two dependent random variables with log normal marginals without proper justification. Spooner (32) developed a model to quantify the risk and the uncertainty associated with a bid estimate. Developed using the conventional unit cost, unit man-hour approach, i t 1 4 allows for the treatment of linear correlation between the input variables. A triangular distribution for the input quantities is assumed to determine the mean and the variance. A normal distribution is assumed for the output variable, provided the input s t a t i s t i c s do not vary. This assumption was verified with a Monte Carlo simulation. Shafer (30) based his model on the risk element approach. The risk element method is based on the thesis that a l l project/estimate risk is a result of elements which are c l a s s i f i e d into "Design", "Pricing", or "Contingency" groups. The most probable cost (MCP) and upper and lower limits for the project are derived from the addition of allowances for undefined cost to the defined project cost. The undefined cost allowances are derived from the risk elements. A probability density function for the project cost is developed by adjusting a normal distribution's left-hand side (i.e keeping the area constant but elevating the lower limit) with the MPC as the mode and median value, but not necessarily the mean value. This curve is used to develop a cumulative distribution function which relates the risk level of any amount which might be appropriated. Deshmukh (10) also used risk elements to develop a model called PAUS. The model assumed a triangular distribution for the random variables, with a correction for the subjective minimum and maximum values. A Monte Carlo simulation was then used to obtain the statistics of the cost estimate. 1 5 Bjornsson (4) developed a risk analysis model called RISK using simulation. The model, which functions in an interactive mode, requires subjective estimates and probability distributions selected from from several types, for the random variables. These input values are then used for a Monte Carlo simulation to obtain output variable s t a t i s t i c s . The model has the capacity to handle correlations i f the dependencies are specified subjectively. The results of the analysis are presented as a risk preference profile. Moeller (23) developed the "Venture Evaluation and Review Technique, (VERT)" for the assesment of risk of a new miltary or business venture. VERT provides for functional and stochastic modeling of time, cost and performance for each activity. The stochastic modeling is aided by ten input s t a t i s t i c a l distributions. Simulation is then used to obtain time, cost and performance stati s t i c s of the project. (If desired VERT could provide for discounted cost). These st a t i s t i c s are ,in the form of pictorial histogram approximations of the probability density function and the cumulative distribution function. Van Tetterode (34) developed a discounted cash flow model to obtain a probability distribution for the rates of return of a project. Every cost factor that is not fixed is defined as a random variable of the model. These random variables are assumed to have a double triangular distribution. A Monte Carlo simulation was then used to 1 6 b u i l d a p r o b a b i l i t y d i s t r i b u t i o n f o r t h e d i s c o u n t e d c a s h f l o w a n d h e n c e , t h e r a t e s o f r e t u r n . C o r r e l a t i o n s among c o s t f a c t o r s were t r e a t e d by m o d i f y i n g t h e random number s a m p l i n g p r o c e d u r e s w i t h a w e i g h t i n g f a c t o r . B e y , D o e r s c h a n d P a t t e r s o n (3) s t u d i e d t h e NPV a s a c r i t e r i o n t o s c h e d u l e a p r o j e c t . NPV c r i t e r i o n a l w a y s y i e l d s a s o l u t i o n w h i c h m a x i m i z e s t h e e x p e c t e d v a l u e o f t h e p r o j e c t , w h e r e a s s o l u t i o n c r i t e r i a s u c h a s m i n i m i z i n g t h e p r o j e c t d u r a t i o n a s u s e d by most s c h e d u l i n g t e c h n i q u e s , may n o t y i e l d t h e maximum e x p e c t e d p r o j e c t v a l u e . A v e r y i m p o r t a n t o b s e r v a t i o n t h a t comes o u t of t h e s t u d y i s t h e c r i t i c a l i t y o f t h e e c o n o m i c e f f e c t o f t i m e on an i n v e s t m e n t . D i e k mann (11) r e v i e w e d f o u r p r o b a b i l i s t i c e s t i m a t i n g p r o c e d u r e s ; 1. d i r e c t a n a l y t i c a l t e c h n i q u e s , 2. d i r e c t a n a l y t i c a l t e c h n i q u e s w i t h c e n t r a l l i m i t t h e o r e m , 3. a p p r o x i m a t e mean an d v a r i a n c e o f a g e n e r a l f u n c t i o n , 4. s i m u l a t i o n m e t h o d s , i n t h e c o n t e x t o f t h e i n p u t d a t a a v a i l a b i l i t y , d a t a c o r r e l a t i o n , u s e o f p r o b a b i l i s t i c c o s t e s t i m a t e s , f o r m o f t h e c o s t m o d e l , a n d t h e number o f e l e m e n t s i n t h e m o d e l . The r e v i e w f a v o u r s t h e f l e x i b i l i t y o f M o n t e C a r l o s i m u l a t i o n o v e r t h e o t h e r m e t h o d s , b u t n o t e s i t s s t r i n g e n t d a t a r e q u i r e m e n t s . K o t t a s and L a u (22) s t a t e t h a t w h i l e t h e " p e r f o r m a n c e c r i t e r i o n ( p . c ) " d e v e l o p e d f r o m a s i m u l a t i o n a r e e f f i c i e n t i n a n s w e r i n g q u e s t i o n s r e l a t i n g t o t h e p r o b a b i l i t y o f t h e p.c a t t a i n i n g a g i v e n t a r g e t l e v e l o r t h e l e v e l o f p.c a t t a i n a b l e w i t h a g i v e n d e g r e e o f c e r t a i n t y , t h e y a r e l e s s 1 7 effective in estimating the higher order moments of the p.c. This is because the sample estimates of higher order moments have very large standard errors. They argue that analysts often restrict the model's primary variables (random variables) to a "universal" distribution, as in the case of many models reviewed, to avoid excessive complications in subsequent derivation of the p.c's stochastic characteristics. Instead, the use of the shape characteristics of random variables, the location, dispersion, skewness, and kurtosis to evaluate the shape characteristics of the performance criterion is proposed. Kottas and Lau (21) demonstrated the use of shape characteristics in stochastic modeling by extending Starr and Tapiero's "linear break-even analysis model" to evaluate the f i r s t four moments of the performance criterion (the pro f i t ) . The f i r s t four moments are it ' s mean and i t ' s second to fourth central moments. A comparision with Starr and Tapiero's Chebyshev's Inequality approach demonstrated the superiority of the four moments approach. Moreover, i t is stated, if the value of the third and fourth central moments are required for u t i l i t y calculations, then the computed values are "exact", where as the central moments estimated through simulation have high standard errors and are much less reliable. 18 2.3 ASSESSMENT OF PROBABILITIES Reliability of an estimate relies heavily on the level of detail and the precision of that detail. Bjornsson (4) states that there are two basic ways in obtaining the detail required by a probabilistic estimation model. 1. Relative frequency : a measure of probability which can be used only if the situations on which the historical data are based are similar to the actual situation and i f i t can be assumed that the conditions in the future w i l l resemble those in the past. 2. Subjective probability : a measure of the confidence that an individual has in the translation of a belief in the occurrance of an uncertain event into a probabilistic quantity. Typically, use of subjective probabilities predominates because most of the variables of interest represent predictions of future events for which historical data has l i t t l e relevance. Winkler (35) investigated the suitability of some of the techniques for assessing prior distributions. While there is no "true" prior distribution, the assessor has certain prior knowledge which is not easy to quantify without careful thought. Therefore an eli c i t a t i o n technique should generate enough information to be able to write down a prior distribution which accurately reflects the assessor's prior knowledge and judgement. A questionnaire was developed to study four of the major e l i c i t a t i o n 19 techniques, these being the Cumulative Distribution Function (CDF) and the Probability Density Function (PDF) which were labelled as direct techniques as compared to the indirect techniques of Hypothetical Future Samples (HFS) and Equivalent Prior Sample Information (EPS). The study indicated that i t was feasible to question people about subjective distributions although this depends on the assessor and on the specific assessment technique(s) used. The distributions corresponding to the direct techniques tended to be more dispersed than those corresponding to the indirect techniques. The use of feedback sessions suggested that more instructions could lead to some reduction of the differences. The important product of the study was the revised questionnaire to assess the prior distribution from subjective probability. Winkler (36) investigated the development of methods involving payoff ( bets and penalty functions ) to help the assessor to determine probabilities which are in accordance with his judgement. The approach to assessing probabilities by direct interrogation has been c r i t i c i z e d on the grounds that the assessor has no incentive to make his assessments correspond with his judgement. The study however suggests that the methods involving direct questions and payoff complement each other. Direct interrogation provides for the quantification of judgement and payoff thereby giving an incentive for a quantification consistent with the assessor's judgement. The possibility that the presence of 20 the payoff defeating the purpose stated above, with risk-taker assessing a distribution which is highly concentrated about a single point and risk-avoider assessing a uniform distribution to assure himself of a certain minimum score is noted. Improved instructional procedures for the various techniques of direct interrogation, training and experience is believed to reduce the rate of occurance of inconsistencies. Smith (31) proposed a psychometric ranking technique to obtain subjective probabilities. The method divides the horizontal axis into a fixed number of intervals ( seven to ten intervals were proposed as a large number could be too cumbersome to handle ) . These intervals are then ranked in the ascending order according to the expected relative probabilities. The rankings are then quantified using the relative differences. The quantified rankings are converted into a probability distribution by a smooth curve passing through the subjectively derived histogram. The stated advantages of the method are that the derived distributions are not restricted to a given prior distribution nor are they required to be unimodal. Morrison (24) in a critique of the above technique states that the cumulative distribution function (CDF) approach which is much simpler, contains a l l the stated advantages, and is easier to apply. The wisdom of dividing into seven to ten intervals is questioned, with the possibility of smaller intervals becoming more significant 21 in deriving rankings. Green (14) shows that the same results could be obtained by a simpler approach than with the proposed elaborate mathematical approach. Winkler (37) studied the "consensus problem", which is defined as the development of a single distribution from a combination of a number of distributions to provide input to a formal Bayesian analysis. This distribution f(0) represents the consensus of the experts' judgement. A number of approaches, some mathematical and the others using feedback and/or group discussion were considered. While there was no "correct" f(0) and i t is impossible to select one "correct" method, i t is emphasized that the decision maker bore the final responsibility for arriving at an f(0). Thus the choice of an f(0) should be based on the decision maker's judgement and the methods proposed should be used only to the extent that they simplified his problem. Hampton, Moore, and Thomas (15) reviewed the literature on the measurement of subjective probabilities for both the individual and group consensus situation. After reviewing a number of methods of assessment, including empirical studies, scoring rules and psychological evidence, they recommended the use of direct fractile assessments for the individual decision maker and the Delphi technique for the group situation. Experience, good training and measurement techniques were recognized as essential ingredients for consistent assessments. 22 Chesley ( 6 ) reviewed the literature to obtain available evidence concerning the theory of how to e l i c i t subjective probabilities. E l i c i t a t i o n techniques examined included direct methods, indirect methods, hybrid methods, multiple assessments and the interviewer approach, the subject's knowledge and assessment results, conservatism and the Bayesian revision experiments, and the assessment of subjective probabilities by groups. Fractile assessment by direct questioning was recommended as the better technique for el i c i t a t i o n of individual subjective probabilities, with a caution regarding the ordering of the questions. The possibility of bunching answers against the extreme values for s t r i c t l y ascending or descending questions, and anchoring for questions starting with the best estimate were noted. Graphical techniques and indirect assessment techniques such as payoff were recommended as a check on responses from other techniques. While scoring rules as an incentive device were not recommended, the use of an interviewer was. An interesting conclusion was the superiority of the "nominal group" which uses discussion without group pressures over the popular "Delphi technique". But the limited research for the conclusion on the group assessment technique was noted. 23 2.4 SUMMARY It is essential to integrate the cost and time aspects of an estimate to study the economic effect of time on that estimate. While most of the models reviewed achieve most of their stated objectives, none integrates the cost and time aspects of an estimate. A DURATION (d) Time Figure 2.2 A Typical Work Package To integrate the cost and time aspects of a typical work package as shown in Fig (2.2) a model should be able to estimate the present value of the work package cost. Typically this integration is achieved by the early start time.. •(t .)•-• of the work package. For a stochastic model to achieve this integration i t should model the duration and the early start time as probabilisitic quantities by providing distribution functions for both. 24 W h i l e t he t e c h n i q u e s r e v i e w e d f o r a s s e s s i n g p r o b a b i l i t i e s s e p a r a t e the i n d i v i d u a l and the g roup as t h e i r s o u r c e , t he r i s k assessmen t t a s k i n t he e n g i n e e r i n g domain r e q u i r e s a c o m b i n a t i o n of i n d i v i d u a l e x p e r t o p i n i o n and g roup o p i n i o n . 3 . M O D E L C O N C E P T I : T H E A P P R O A C H A N D T H E B A S I C T O O L S 3 . 1 B A C K G R O U N D F O R T H E P R O P O S E D M O D E L The inspiration for the proposed model is the unified s t a t i s t i c a l framework for probabilistical planning models proposed by Kottas and Lau (22). In their framework, they propose the use of the f i r s t four moments of the primary variables to evaluate the f i r s t four moments of the performance criterion. The f i r s t four moments of a primary variable are derived by assuming the primary variable to be a member of one of the recommended families of distributions namely, beta distribution, Johnson's family, generalized Tukey's lamda distribution, or the family developed by Schmeiser and Deutsch. Then, using the assumption that the frequency distribution of the performance criterion can be approximated by the highly flexible Pearson's system of curves and the tables compiled by Johnson et a l (19) for the normalised Pearson variable, a cumulative distribution function for the performance criterion is developed. This cumulative distribution function is then used for probabilistic decision making. 25 26 3.2 THE APPROACH FOR THE PROPOSED MODEL The approach for the proposed model is based on the assumption that the frequency distributions of both the primary variables and the performance criterion can be approximated by the Pearson system of curves. This assumption permits the evaluation of the f i r s t four moments of a primary variable, i f the estimator's subjective probability distribution of that variable approximates to a Pearson type distribution. These moments combined with s t a t i s t i c a l operations on the dependent variable provide the basis for the evaluation of the f i r s t four moments of the performance criterion. Then, following the recommended approach the cumulative distribution function of the performance criterion is developed for probabilistic decision making. 3.3 THE PRIMARY VARIABLE The procedure to estimate the f i r s t four moments of a primary variable from a set of subjective estimates and the Pearson family of distributions to which the primary variable is assumed to approximate is described in this sect ion. 27 o Variable Estimates Figure 3.1 Cumulative distribution function corresponding to the subjective estimates. 28 3.3.1 SUBJECTIVE ESTIMATES FOR A PRIMARY VARIABLE In d e v e l o p i n g the proposed model i t has been assumed t h a t i t i s f e a s i b l e to q u e s t i o n people about s u b j e c t i v e d i s t r i b u t i o n s . T h i s assumpt ion i s based on past r e s e a r c h rev iewed in s e c t i o n 2.3 of the p r e v i o u s c h a p t e r . Whi le views are d i v e r s e r e g a r d i n g the best e l i c i t a t i o n t echn ique and q u a n t i f i c a t i o n of p r i o r knowledge, most a u t h o r s recommend f r a c t i l e assesment by d i r e c t q u e s t i o n i n g as the b e t t e r t e c h n i q u e for the e l i c i t a t i o n of i n d i v i d u a l s u b j e c t i v e p r o b a b i l i t i e s . I t i s l e f t to f u t u r e r e s e a r c h to deve lop se t s of c o n s i s t e n t q u e s t i o n s which would e l i c i t the i n d i v i d u a l s u b j e c t i v e d i s t r i b u t i o n s from f r a c t i l e assessment for each of the pr imary v a r i a b l e s of i n t e r e s t . For t h i s s tudy i t i s assumed that the e s t i m a t o r can p r o v i d e e s t i m a t e s of h i s s u b j e c t i v e d i s t r i b u t i o n of a p r i m a r y v a r i a b l e . These e s t i m a t e s c o r r e s p o n d to the 2 . 5 , 5 . 0 , 5 0 . 0 , 9 5 . 0 , and 97.5 percentage p o i n t s of the d i s t r i b u t i o n , (see f i g u r e 3.1) 3.3.2 THE PEARSON FAMILY OF DISTRIBUTIONS The Pearson f a m i l y of d i s t r i b u t i o n s has 12 d i s t i n c t members that cover the (v//3 1,j3 2) p lane (see f i g u r e 3 . 2 ) , a l l of which are d e r i v e d as s o l u t i o n s of a s i n g l e d i f f e r e n t i a l e q u a t i o n which , when the o r i g i n of x i s a t the mean, has the form, 29 Figure 3.2 : Diagram showing the areas and bounding curves associated with the different solutions of Karl Pearson's dif ferential • equation Source : Johnson, Nixon, and Amos (Reference No. **) Biometrika(1963), 50, 3 and 4, p.460 30 df(x) = - (x+b) . f(x) dx a+bx+cx where x i s the random v a r i a t e , and f(x) i s the p r o b a b i l i t y density function The parameters a, b, and c are functions of the centra l moments n (x) of f ( x ) , and may be expressed as, 4/32 - 3 ^ 1O02 - \2(i. -- 18 "1 (P2 + 3) 10^2 - 1 2P1 -- 18 2/32 - 301 " 6 1O02 - 1 2/3, -- 18 where 2 3 2 j31 = M 3 / M 2 and 0 2 = M 4 / M 2 and u 2 , M3 , and JI^ are the second, t h i r d , and fourth centra l moments of the primary random variable x. 31 Johnson, Nixon and Amos (19) compiled tables for the Pearson's system of curves. These tables, tabulate the standardized deviate for fifteen percentage points based on the values of v//31 and 0 2 . The fifteen percentage points are namely the median, upper and lower 0.25, 0.5, 1.0, 2.5, 5.0, 10.0, and 25.0 percentage points. Pearson and Tukey (25) developed an approximation to the mean and standard deviation for the Pearson system of curves based on the distances between percentage points of frequency curves. The approximation to the mean is The approximation for the standard deviation using an iteration scheme is M = [50%] + 0.185A (3. 1) where A = [95%] + [5%] - 2[50%]. (3. 2) a = max { a 0.05 a 0.025 } (3. 3) 32 0 0 1 0 2 0 3 0 AO fix Contours of 5 % distance in fiv fi. piano. 0 0 1 0 1 0 3 0 4-0 fix Contours of 2-5% distance in filt /?, plane. Figure 3.3 : Contours in the 0 ] f P2 plane on which the 5.0% and 2.5% distances are constant for Pearson curves Source : Pearson and Tukey (Reference No. **) Biometrika(1965), 52, 3 and 4, p.537 33 where * [95%] - [5%] ° o 05 = ? < 3 - 4 ) U , U D max { 3.29 - 0.1(A/a 0 > 0 5> , 3.08 } , and * [97.5%] - [2.5%] ° o 025 = 5 (3.5) u , u " max { 3.98 - 0.138(4/5 0 > 0 2 5 ) , 3.66 } . ^0.05 a n c^ ^ 0 025 a r e t l i e a P P r o x i m a t i ° n s t 0 t n e standard deviation from the previous iteration. For the f i r s t iteration 5Q Q 5 and o n Q 2 5 are defined on the basis of figure (3.3) as, and [95%] - [5%] s°-°6= ~ T 7 5 — <3-6) [97.5%] - [2.5%] "0.025 - — It is stated that the error in the estimation of the mean is not more than 0.1% for a large area of the , 0 2 ) plane (see figure 3.3), and not more than 0.5% for the rest. Similarly, the error for the standard deviation is less than 0.5% for a very large area of the ( / / J , , ^ ) plane. These 34 approximations are recommended for use in other families of distributions without much f a l l in accuracy. 3.3.3 THE FIRST FOUR MOMENTS OF A PRIMARY VARIABLE The f i r s t four moments of a continous random primary variable are i t ' s expected value and second to fourth central moments. The proposed method to estimate the f i r s t four moments of a primary variable is described in a step by step procedure with an example of the estimation of the f i r s t four moments of the quantity of a work package. Step 1 : S u b j e c t i v e E s t i m a t e s Obtain the estimates for the 2.5, 5.0, 50.0, 95.0, and 97.5 percentage points of the estimator's subjective distribution of a primary variable. The five estimates corresponding to estimator's subjective distribution for the work package quantity in cubes are 6000.0, 6200.0, 8000.0, 10000.0, and 10200.0. Step 2 : The E x p e c t e d V a l u e and the Sta n d a r d D e v i a t i o n The five subjective estimates from step 1 are used in equations (3.1) to (3.7) to approximate the expected value and the standard deviation of the primary variable. From equation (3.2) A is 200.0 and hence the expected value of the work package quantity from equation (3.1) is 8037 cubes. From equations (3.6) and (3.7) a n n R and a n n 9 R 3 5 are 1 1 6 9 . 2 3 and 1 0 7 1 . 4 3 for the f i r s t i t e r a t i o n . From * * i t e r a t i o n of equations ( 3 . 4 ) and ( 3 . 5 ) ^ Q 05 A N D ° 0 0 2 5 are 1 1 5 6 . 0 6 7 , and 1 0 5 6 . 5 8 9 . Hence, from equation ( 3 . 3 ) the standard deviation of the work package quantity i s 1 1 5 6 . 0 6 7 cubes. Step 3 : Normalise the S u b j e c t i v e Estimates Using the approximations for the expected value and the standard deviation of the primary variable from step 2, the five subjective estimates from step 1 are normalised by, XP = X p - M (3. 8) The normalised estimates of the work package quantity for the 2 . 5 , 5.0, 5 0 . 0 , 9 5 . 0 , and 97 . 5 percentage points from equation ( 3 . 8 ) are - 1 . 7 6 2 , - 1 . 5 8 9 , - 0 . 0 3 2 , 1.698, and 1 . 8 7 1 . Step 4 : y / /3 1 and 0 2 f o r the Primary V a r i a b l e The normalised estimates from step 3 are then compared with the 2 . 5 , 5.0, 5 0 . 0 , 9 5 . 0 , and 97 . 5 percentage points for the normalised Pearson variable tabulated by Johnson et a l , by the method of least squares, to obtain and / 3 2 that best approximate to the normalised estimates. The model permits a maximum cumulative error of 10% of the standard deviation for this approximation. The average 36 e r r o r o f an e s t i m a t e a t t h i s maximum c u m u l a t i v e e r r o r i s 4.47% of the s t a n d a r d d e v i a t i o n . I t i s p r o p o s e d t o p e r m i t t he u s e r t o s e t t he maximum c u m u l a t i v e e r r o r i n t he f u t u r e . The and 0 2 f o r the work package q u a n t i t y f rom the a p p r o x i m a t i o n a r e 0.1 and 2 . 2 . The v a l u e s o f the f i v e n o r m a l i s e d P e a r s o n v a r i a b l e p e r c e n t a g e p o i n t s c o r r e s p o n d i n g t o t he above v a l u e s of and 0 2 a r e - 1 . 7 6 2 5 , - 1 . 5 8 9 3 , - 0 . 0 2 6 3 , 1 .6793 , and 1 .8953 . Hence , the maximum c u m u l a t i v e e r r o r o f the a p p r o x i m a t i o n i s 3.12% of t he s t a n d a r d d e v i a t i o n . Step 5 : The Central Moments From / / J , and 0 2 f o r the a p p r o x i m a t e d d i s t r i b u t i o n f rom s t e p 4 and the s t a n d a r d d e v i a t i o n a p p r o x i m a t e d a t s t e p 2 t he s e c o n d , t h i r d and f o u r t h c e n t r a l moments o f t he p r i m a r y v a r i a b l e a r e e v a l u a t e d f r o m , M 0 ( x ) = a 2 (3. 9) M 3 ( x ) = i / p , * a 3 (3. 10) M 4 ( x ) = S32 * a (3. 11) 37 Using the approximated / j 3 1 and 0 2 of 0.1, and 2.2, and standard d e v i a t i o n of 1156.067, the second to f o u r t h c e n t r a l moments f o r the work package q u a n t i t y from equations (3.9), (3.10) and (3.11) are 1.3365xl0 6, 1.545xl0 8 and 3.9296x!0 1 2. 3 . 4 THE DEPENDENT VARIABLE The estimated moments of the primary v a r i a b l e s are then used to e v a l u a t e the f i r s t four moments of the dependent v a r i a b l e . The moments of the dependent v a r i a b l e depend on both the moments of the primary v a r i a b l e and the f u n c t i o n a l r e l a t i o n s h i p between the primary v a r i a b l e s and the dependent v a r i a b l e . These r e l a t i o n s h i p s c o u l d be from simple a d d i t i o n to c o m p l i c a t e d f u n c t i o n a l forms. The proposed model has three b a s i c f u n c t i o n a l forms, namely, 1. the l i n e a r a d d i t i v e form 2. the l i n e a r s u b t r a c t i v e form 3. s c a l a r performance f u n c t i o n a l form ( i . e not l i n e a r a d d i t i v e , s u b t r a c t i v e , or m u l t i p l i c a t i v e ) In d e v e l o p i n g the r e l a t i o n s h i p s f o r the moments of the dependent v a r i a b l e i t has been assumed that the primary v a r i a b l e s are s t a t i s t i c a l l y independent. T h i s assumption of independence i s maintained throughout the development of the mathematical framework f o r the proposed model. A b r i e f example i s given at the end of t h i s s e c t i o n to demonstrate the p o t e n t i a l importance of c o r r e l a t i o n among both primary and dependent v a r i a b l e s . I t s d e t a i l e d treatment i s l e f t f o r 38 f u t u r e w o r k . 3.4.1 THE ADDITIVE FORM The moments o f a d e p e n d e n t v a r i a b l e f r o m an a d d i t i o n o f two random v a r i a b l e s o f t h e f o r m , u = x + y c a n be e v a l u a t e d f r o m e q u a t i o n s ( 5 ) a n d (7) f r o m K o t t a s a n d L a u ( 2 2 ) . The e x p e c t e d v a l u e f r o m e q u a t i o n ( 5 ) i s , j u ( u ) = M ( x ) + j u ( y ) a n d t h e c e n t r a l moments f r o m e q u a t i o n ( 7 ) a r e , M n ( u ) = j 0 { ( n C i > t " i , n - i < x ' ^ * where n C i = n! / [ i ! ( n - i ) ! ] I f x a n d y a r e s t a t i s t i c a l l y i n d e p e n d e n t , t h e n M m n ( x ' y ) = * i m ( x ) ^ n ( y ) 39 Now consider the moments of a dependent variable from an addition of n s t a t i s t i c a l l y independent random variables of the form n Y = 2 X . . i = 1 1 Then, from above reasoning, the expected value i s , n E [ Y ] = L v . (3. 12) i = 1 X l and second to fourth central moments are, M , ( Y ) = 2 M , ( X . ) (3. 13) * i = 1 z 1 n M , ( Y ) = L M o ( X . ) (3. 14) J 1=1 J 1 n n n UA(Y) = I u , ( X . ) + 6 L Z M,(X. ) ./u~(X.) (3.15) 4 i=1 q 1 i=ij=i+i ^ 1 J 40 3.4.2 THE SUBTRACTIVE FORM The f i r s t four moments of a dependent variable from a subtraction of two independent random variables of the form, u = x - y can be evaluated from equations (8) to (11) from Kottas and Lau (21). The expected value of the dependent variable from equation (8) i s , M(U) = M(X) - M ( y ) (3.16) and the second to fourth central moments from equations (9), (10), and (11) are, M2(u) = M 2(X) + M 2 ( y ) (3.17) ixAu) = Mo(x) - M o ( y ) (3. 18) M 4(U) = M 4(X) + M 4 ( y ) + 6 . M 2 ( X ) . M 2 ( y ) (3. 19) 41 3.4.3 S C A L A R P E R F O R M A N C E F U N C T I O N A L F O R M A l l functional forms other than direct addition, subtraction, or linear multiplication between the primary variables are defined as the scalar performance functional form. The moments of the dependent variable of such a function are evaluated by an approximate method. First, the function is expanded in a Taylor series about the mean values of the primary variables. Then, using the expansion with the definition of the moments, the moments of the dependent variable are approximated. If Y is a function of several random variables such that, then the expansion of g(x.) in a Taylor series about the Y = 9 (X., X 2' mean values u X1' MX2' Xn is given by n 3g Y = g(M x 1, M X2' ' "Xn> \Z=] ( X i - " x l * + J_ 2 42 where the derivatives are evaluated at p. XT M2' Xn* The Taylor series is then truncated at the second order such that the truncation error of the approximation i s , The second order approximation provides reasonable mathematical ease for the st a t i s t i c a l analysis. A third or higher order approximation would give more.accurate results, but mathematical complexity prohibits its use. The expected value of Y from the second order approximation is | R 2 | = J _ L 2 L ( X . - M X I ) ( X - - M X - J ) ( X.-M X K ^ J f i = i j = i k = 1 i X i 3 X ] n n n E[Y] = g(M X1 X2' covCX^X.) and the central moments are M n(Y) = E[ (Y - E[Y]) n ] 43 When the random variables are assumed to be st a t i s t i c a l l y independent the expected value from equation ( 1 5 . 1 2 ) from Bury (5) i s , E [ Y ] = g U x 1 , M X 2 , M X n ) n 3 2a + 1 Z 9 Mo (X . ) (3. 20) 1 i = 1 3 X : 2 2 1 and the second to fourth central moments from equations ( 1 5 . 1 3 ) . , ( 1 5 . 1 4 ) , and ( 1 5 . 1 5 ) from Bury (5) are, •- -2 M , ( Y ) = Z (l^L)2. M , ( X . ) + Z f-L.JLl. M , ( X . ) 2 i - 1 3X. 2 1 i - 1 3 X i 9x . 2 3 1 M . ( Y ) = Z ( 1 ? _ ) 3 . Mo(X. ) (3. 22) 6 i=1 3 X i 6 1 HAY). = Z ( 1 9 _ ) 4 . M , ( X . ) 4 i=1 3 X i 4 1 + 6 Z Z n ( ! i _ ) 2 . M , ( X . ) . ( ! 9 _ ) 2 . M , ( X . ) f J . 2 3 / i = 1j = i + 1 3 X . ^ 9 X _ . 44 3 . 4 . 4 DERIVED VARIABLE FROM DEPENDENT RANDOM VARIABLES A brief example is presented for the linear addition of dependent random variables to show the potential importance of correlation on the derived variable. The variable Y is derived from the addition of n dependent random variables of the form, n Y = 2 X. i=1 1 The expected value of the derived variable Y i s , n E[Y] = 2 » . i = 1 X 1 The second central moment of Y is n n n M,(Y) = 2 n0 (X.) + 2 2 2 P--.O..Q. * i=1 c 1 i = 1 j = i + 1 1 - ' 1 1 1 where p^j is the correlation coefficient between variables i and j , and a- and CT. the standard deviation of variables i and j . 45 The third central moment of the derived variable Y is n Mo(Y) = Z M , (X. ) J i = 1 J "•±1, j§,lc§i E [ U i " M x i ) ( x j " * X j ) ( X k " " X k ) ] when i , j and k are not a l l equal at the same time. The fourth central moment of the derived variable Y is n n n n.(Y) = Z n, ( X . ) + 6 Z Z M , ( X . ) M 0 ( X . ) * i=1 * 1 i = i j = i+i l 2 V A i y M 2 V A j + J i E [ ( x i - * i x i ) ( x i - * x i ) ( x k - « x k ) ( x i - M X I > 3 •i * X i ' V A j X j ' V A  M X k when i , j , k and 1 are not a l l equal or not equal in pairs at the same time. When these equations for the f i r s t four moments of the derived variable are compared with equations ( 3 . 1 2 ) to ( 3 . 1 5 ) , ( i.e the f i r s t four moments of the derived variable when the random variables are assumed to be independent) the 46 terms that are neglected are evident. These neglected terms are the effect of the correlation on the moments. While i t is possible to obtain estimates for the correlation coefficient necessary for the estimation of the second central moment, obtaining reliable information on the cross moments for the third and fourth central moments is almost, if not completely, impossible. The treatment of correlation becomes more complicating when faced with scalar performance functional forms. Further research in the future is proposed to develop an approach by which the information necessary for the cross moments can be obtained such that the correlation between the variables can be treated effectively. 3.5 CUMULATIVE DISTRIBUTION FUNCTION The estimated moments of the dependent variable can then be used to quantify the uncertainty associated with i t . To do so i t is assumed that the dependent variable can be described by the Pearson family of distributions. The distribution chosen is the one which corresponds most closely to the dependent variables's shape characteristics, skewness and kurtosis. They are evaluated from its moments as : skewness: = * 3 ( Y ) U2(Y)] * 5 (3. 24) 47 M 4(Y) kurtosis: 0O = sr O . 25; M 2 ( Y ) 2 With the evaluation of the shape characteristics, the basis for the development of the cumulative distribution function is laid. The development of the cumulative distribution function and its use for probabilistic decision making is discussed later. 4. MODEL CONCEPT I I : THE MATHEMATICAL FRAMEWORK 4.1 GENERAL The model is derived mathematically by integrating a l l the issues which affect the project cost and duration of distinct work packages. The f i r s t four moments (i.e the expected value and second to fourth central moments) for the cost and duration of these work packages are evaluated using the moment analysis approach. These moments are then used to quantify the uncertainty of project cost and duration. Net present value is developed as the decision criterion for the model, emphasising the economic effect of time and inflation on the cost. The frequency distributions of the primary variables are assumed to be approximated by the highly flexible Pearson's system of curves. The effect of correlation between primary variables is not treated in the development of the model. 4.2 WORK PACKAGE DURATION MODEL A model to evaluate the duration of the a work package in terms of i t s f i r s t four moments is developed with the simple relationship, 48 49 T. = — i (4.1) P L i L i The primary variables , PLi» a n d of the model are respectively, the quantity descriptor, the labour productivity rate and labour usage profile of the i* " * 1 work package (the latter two variables are assumed to be invariant). From moment analysis, the expected value and second to fourth central moments of the work package duration using, equations (3.20),(3.21),(3.22) and (3.23) are, X* 3 2 E[T.] = l i + 1 Z i Z i . M 7 ( X . ) (4.2) X 2 i x 3 i 2 J - i 3 X j 2 3 3 3T. 9 3 3T. 32T. M,(T. ) = 2 ( i l l ) 2 . M 9 ( X , ) + I I i i . M o ( X . ) r^. 5; 2 1 3=1 3Xj 2 3 j-1 3 X j 3 x . 2 3 3 M , (T. ) = z (01±)3. MO(X.) r*. J j=1 3Xj J 3 M,(T. ) = z (Hi)4. nAZ.) 4 j=1 3Xj 4 3 + 6 z z 3 (Hi)2.M,(x.). (Hi)2.M9(x,) j=ik=j+i 3 X z 3 9 X k where X1 = Q i ('X 2 = P L i, and X 3 = 1^. 50 The f i r s t f o u r moments o f t h e p r i m a r y v a r i a b l e s a r e o b t a i n e d f r o m t h e p r o c e d u r e d e s c r i b e d i n s e c t i o n 3.3.3. The e v a l u a t e d f i r s t f o u r moments o f t h e work p a c k a g e d u r a t i o n a r e t h e n u s e d f o r t h e e v a l u a t i o n o f t h e f i r s t f o u r moments o f t h e work p a c k a g e s t a r t t i m e s , t h e work p a c k a g e c o s t a n d t h e p r o j e c t d u r a t i o n . 4 . 3 P R O J E C T D U R A T I O N M O D E L 4 . 3 . 1 P R O J E C T N E T W O R K A p r e c e d e n c e n e t w o r k t o s e q u e n c e a l l o f t h e work p a c k a g e s i n a p r o j e c t i s d e v e l o p e d f r o m t h e l o g i c ( t r e a t e d a s d e t e r m i n i s t i c ) p r o v i d e d by t h e e s t i m a t o r . The f i r s t f o u r moments f o r e a c h o f t h e work p a c k a g e d u r a t i o n s a r e c o m b i n e d w i t h t h i s n e t w o r k t o o b t a i n t h e f i r s t f o u r moments o f , 1. t h e e a r l y s t a r t t i m e o f a work p a c k a g e a n d 2. t h e p a t h d u r a t i o n s . 4 . 3 . 2 E A R L Y S T A R T T I M E O F A W O R K P A C K A G E The e a r l y s t a r t t i m e o f a work p a c k a g e i s t h e e a r l i e s t p o i n t i n t i m e t h a t a work p a c k a g e c a n b e g i n . I t i s E d e t e r m i n e d by t h e l o n g e s t p a t h t o a work p a c k a g e . I f T. i s t h E t h e e a r l y s t a r t t i m e o f t h e i work p a c k a g e , T^ i s t h e e a r l y s t a r t t i m e , a n d T^ i s t h e d u r a t i o n , o f t h e p r e c e d i n g h work p a c k a g e , t h e e a r l y s t a r t t i m e o f t h e i work 51 package is defined by, = Max V-h ( T h E + T h) (4. 6) where Max Vh implies that the maximization is to be over a l l the links "h to i " terminating at the i f c ^ work package. The f i r s t four moments of the early start time of the i f ck work package are evaluated from the generalised additive equations based on equations (3.12), (3.13), (3.14), and (3.15), and assuming the work package durations to be st a t i s t i c a l l y independent. For, where there are n work packages in the longest path to the i f c ^ work package, the expected value of the early start time i s , T. l E n = Z T . (4. 7) j = 1 E n (4. 8) 52 a n d t h e s e c o n d t o f o u r t h c e n t r a l moments a r e , n L (4. 9) j= 1 n L j = 1 (4. 10) n L n n (4. 11) j = 1 I n t h i s e v a l u a t i o n , o n l y t h e s t a t i s t i c s o f t h e l o n g e s t p a t h t o t h e work p a c k a g e a r e c o n s i d e r e d . W h i l e t h i s a p p r o a c h p r o v i d e s a method t o i n t e g r a t e t i m e a n d c o s t a s p e c t s i t t e n d s t o i g n o r e s h o r t e r b u t more u n c e r t a i n p a t h s . I t i s p r o p o s e d i n t h e f u t u r e t o d e v e l o p a c o n s i s t e n t a p p r o a c h b a s e d on t h e PNET a l g o r i t h m t o o b t a i n t h e s t a t i s t i c s o f t h e e a r l y s t a r t t i m e s w h i c h w o u l d a c c o u n t f o r t h e c o r r e l a t i o n a r i s i n g f r o m s h a r e d work p a c k a g e s a n d m u l t i p l e p a t h s . 4 . 3 . 3 PROJECT DURATION The c o m b i n a t i o n o f t h e p r e c e d e n c e n e t w o r k , t h e work p a c k a g e d u r a t i o n s , a n d t h e l a g t i m e s , p r o v i d e s t h e c o m p l e t i o n d u r a t i o n s o f i n d i v i d u a l p a t h s . The l o n g e s t p a t h d u r a t i o n i s g e n e r a l l y c o n s i d e r e d t o be t h e p r o j e c t d u r a t i o n . The u n c e r t a i n t y a s s o c i a t e d w i t h t h i s e s t i m a t e d p r o j e c t 53 duration is essential for decision making. The model at present u t i l i z e s the statis t i c s of the longest path as those of the project duration but does not develop a completion time probability function for the project duration. It is proposed in the future to use the PNET algorithm developed by Ang et al (1) to evaluate the completion time probability of the project duration. The PNET algorithm was discussed in detail in section 2 . 2 . 1 . 4.4 WORK PACKAGE COST MODEL 4.4.1 MODEL DEVELOPMENT The work package cost model is developed by integrating a l l the variables which affect the estimation of the project cost into i t . The variables treated are: * duration and the early start time of the work package; * unit and inflation rates for labour, equipment, and materials; * productivity and usage profiles for labour and equipment; * sub contractor and indirect cost; * and interest rates. The constant dollar cost model for the i* " * 1 work package in terms of these variables i s , C o i = [ CL' 2 + CM'Q + CE' 2 ]i + Cj.T. PL PE (4. 12) 54 where CL' CM' a n d CE a r e fc^e u n^- t r a t e s f° r labour, materials and equipment, Q is the quantity descriptor, P L and P E the productivity rates , for labour and equipment, S and Cj are the sub contractor and indirect cost, C o i a n d T i t h e w o r k package constant dollar cost and durat ion. From the fundamental relationship for the work package duration in equation (4.1), the constant dollar cash flow model can be modified to C o i ( t ) = C L i ' L i + C E i - E i + C M i - P L i ' Li + h + C I ( 4 - l 3 ) T. l where and E^ are the labour and equipment usage profiles. In developing these relationships i t has been assumed that resources are consumed at a constant rate. The validity of this assumption is discussed at the end of the chapter. 55 4 . 4 . 2 ESCALATION AND INTEREST DURING CONSTRUCTION The generalised cash flow diagram for a construction project is shown in figure (4.1). For the development of this model a more simplified cash flow diagram as shown in figure (4.2) is assumed. In this simplified scenario, the expenditure for design and construction comes from a combination of equity and borrowed funds. The borrowed funds are assumed due at the end of the project. Due to the numerous financing alternatives available in the market, no attempt has been made to model the permanent financing. External economic variables have a strong influence on the construction cost estimate. Escalation primarily due to inflation and interest payments for the construction loan can be a significant portion of the estimate. Hence i t is essential to model these variables as r e a l i s t i c a l l y as possible. In treating escalation during construction, the proposed model deviates from the popular but often incorrect assumption of a common rate for inflation, by allowing different rates for different categories of cost. In providing this f a c i l i t y , i t is assumed that these different rates for inflation in a work package are constant over the duration of the project. Similarly, for the financing of a work package, a constant interest rate over the project duration is assumed. $^$ DRAW DOWN OF BORROWED FUNDS lPERMA^ REVENUE SALVAGE VALUE EXPENDITURE FOR DESIGN & CONSTRUCTION REPAYMENT FOR PERMANENT FINANCING TIME DISCHARE OF LOAN \ ^ BALANCE REPAYMENT OF CONSTRUCTION LOAN f Figure 4.1 General cash flow diagram for a construction project $$ ^ ^ ^ ^ ^ DRAW DOWN OF REVENUE BORROWED FUNDS EXPENDITURE FOR TIME DESIGN & CONSTRUCTION REPAYMENT OF CONSTRUCTION LOAN > Figure 4.2 Cash flow diagram assumed for the model 57 4.4 .3 D I S C O U N T E D WORK P A C K A G E C O S T A generalised discounted cost model for a work package as shown in figure (4.3), incorporating the equity fraction of the investment, escalation and interest during construction, can be represented by, PWC = / f f.C Q(t).exp 8 t. exp" y t.dt t s + (l-f).exp-y TP. / f C f t ( t ) . e x p e t . e x p r ( T P - t ) . (4.14) fcs where PWC is the discounted work package cost, C Q(t) is the constant dollar work package cost, 6, r and y are the inflation, interest, and discount rates, t g , and t^ the work package start and finish times, f the equity fraction, and Tp the time at which the construction loan is due. Applying the generalised discounted cost equation (4.14) to the modified constant dollar cost equation (4.13), the discounted cost model for the i* " * 1 work package is developed. The discounted cost model for the i f c ^ work package i s , 58 CONSTANT $$ COST t s DURATION t u r e 4.3 D i s c o u n t e d Work P a c k a g e C o s t D i a g r a m 59 PWC. = f { C L..L.. g [ e x p ( f l L ) ] + C E..B..g[exp(e E)] + C ^ . P ^ . L . . g [ e x p ( * M ) ] + Cj ,g[exp(ei) ] + S.. g[exp ( 0 s ) ] } T i + ( l - f ).exp ( r"y ) TP { C L..L.. g[dexp(0L)] + C E i.E..g[dexp(e E)] + C M..P L..L.. g[dexp ( e y ] + Cj.g[dexp( t9 j ) ] + s. . g[dexp(0g)] } (4. 15) T • where (t9.-y)(t .+T.) t .(e.-y) g[exp (c9j ) ] = [exp 3 S 1 1 - exp S 1 3 ] (4.16) and (0.-r)(t .+T. ) t .(fl.-r) g[dexp ( 0 . ) ] = [exp 3 S 1 1 - exp S 1 3 ] (4.17) (d.-r) where j = L, E, M, I, and S, PWC^  is the discounted cost of the i t h work package, 60 #L, 0 M , Sg, and 6^ are the inflation rates for the labour, materials, equipment, sub contractor, and indirect cost, and can be expressed in terms of Q, PL^ and L^. Equation (4.15) which incoporates a l l the issues found to be significant in the estimating of the project cost is referred to herein as the fundamental equation of the proposed work package cost model. The f i r s t four moments of the work package cost are evaluated using the fundamental equation and equations ( 3 . 2 0 ) , ( 3 . 2 1 ) , ( 3 . 2 2 ) , and ( 3 . 2 3 ) . The expected value of the discounted work package cost i s , E [ P W C i ] = PWC. [ T . , T p f C L i , C M i , C E i , P L i , C l , L^, E^ , ,r,t s i ' e 0 Q , e T ] + 1 . z s 1 2 j=1 7 32PWC-i . M 2 ( X j ) (4. 18) and second to fourth central moments are, M 2(PWC i) = 17 Z j = 1 9PWC• 9 2PWC. (4. 19) 61 17 3PWC. 1 Mo (PWC.) = I r ^ I i ) 3 . Mo(X.) (4.20) 6 j = 1 3Xj 3 1 1 M 4 (PWC.) = 2 7 ( 3 P ! ! £ i ) 4 . M , ( X . ) 4 j=1 SXj 4 3 . . I7 _ 1 7 ,3PWC-,2 x /9PWC-.2 ,„ x „ + 6 2 I ( 1 ) . M 0 ( X . ) ( 1 ) , M 0 ( X . ) (4.21) j=1 k=j+1 3 X 3 X 2 k J K w h e r e X 1 = C L i ' X 2 = C M i ' X 3 = C E i ' X 4 = P L i ' X 5 = S L ' X 6 = 5 M ' X 7 = ^ E ' X 8 = 0 S ' X 9 = e i ' X 1 0 = r ' X 1 1 = L i ' X 1 2 = E i ' X 1 3 = S i ' X 1 4 = C I ' X 1 5 = T i ' X 1 6 = t s i ' a n d X 1 7 = T p -I n d e v e l o p i n g t h e e q u a t i o n s f o r t h e f i r s t f o u r moments o f t h e d i s c o u n t e d work p a c k a g e c o s t i t h a s been assumed t h a t , * t h e p r i m a r y v a r i a b l e s a r e a l l i n d e p e n d e n t , a n d * t h e a p p r o x i m a t i o n s o b t a i n e d by n e g l e c t i n g t h e h i g h e r o r d e r t e r m s o f a s e c o n d o r d e r T a y l o r s e r i e s e x p a n s i o n g i v e s u f f i c i e n t l y a c c u r a t e v a l u e s t o r e p r e s e n t t h e f i r s t f o u r moments o f t h e d e p e n d e n t v a r i a b l e . 62 4 . 5 PROJECT COST MODEL The present worth of project cost is simply the sum of a l l the discounted work package costs. The project cost model, for n work packages in the project i s , The project cost is estimated primarily in terms of discounted dollars, the decision criterion for the model. If desired, the project cost can be estimated in current dollars by setting the discount rate to zero and equity fraction to one in the work package cost model. Similarly, by setting equity fraction to one and discount and inflation rates to zero, the project cost can be estimated in constant dollars. The f i r s t four moments of the discounted project cost are evaluated from the generalised additive equations, (3.12), (3.13), (3.14) and (3.15). For a project with n work packages the expected value i s , n P.C = I PWC i = 1 1 (4. 22) n E [ P . C ] = L E[PWC-] i — 1 i i=1 (4. 23) and second to fourth central moments are, 6 3 n M 2 ( P . C ) = 2 M 2 ( P W C I ) i=1 (4. 24) n M 3 ( P . C ) = 2 M 3 ( P W C 1 ) i=l (4. 25) n H.(P.C) = 2 j u A ( P W C . ) n n + 6 . 2 2 i = 1 j = i + 1 . M 2 ( P W C 1 ) . M 2 ( P W C • ) (4. 26) I n d e v e l o p i n g t h e f i r s t f o u r moments f o r t h e d i s c o u n t e d p r o j e c t c o s t i t h a s b e e n a s s u m e d t h a t t h e work p a c k a g e c o s t s a r e s t a t i s t i c a l l y i n d e p e n d e n t . 4.6 PROJECT REVENUE MODEL The g e n e r a t i o n o f r e v e n u e i s t h e f u n d a m e n t a l o b j e c t i v e o f an i n v e s t m e n t i n a p r o j e c t . The i n i t i a l d e c i s i o n t o i n v e s t , t h e r e f o r e , i s b a s e d on t h e a b i l i t y o f t h e p r o j e c t t o g e n e r a t e a r e v e n u e t h a t w o u l d j u s t i f y t h e i n v e s t m e n t . The c r i t e r i o n f o r t h i s d e c i s i o n i n t h e p r o p o s e d m o del i s n e t p r e s e n t v a l u e o f b e n e f i t s m i n u s c o s t s . T h e r e f o r e , t h e a b i l i t y t o s t u d y t h e e c o n o m i c e f f e c t o f t h e g e n e r a t e d r e v e n u e w i t h r e s p e c t t o t i m e i s e s s e n t i a l . 64 The possibility of a project generating a number of revenue streams at different points in time is typical of large projects. The proposed model therefore provides the f a c i l i t y to model a number of revenue streams starting at different points in time. Each revenue stream can be described in terms of i t s f i r s t four moments. Total discounted project revenue is simply the sum of a l l the discounted revenue streams. The moments of total discounted project revenue are obtained from equations (3.12), (3.13) (3.14) and (3.15). 4.6.1 NET REVENUE STREAM The net revenue stream is defined as the difference between the gross revenue and its operation and maintenance cost. Both, the gross revenue and the operation and maintenance cost are inflated with different rates and discounted. The proposed discounted net revenue for the i * " * 1 stream i s , T NR. = / f R. .exp ( 9R"y ) t : .dt T s T, - / O&M..exp(eo y ) t -dt T s (4. 27) 65 where R^  is the gross revenue, O&M^  is the operation and maintenance cost, (9R and 6Q inflation rates for revenue and operation and maintenance cost, T s and T^ the start and finish time of the revenue stream as random variables. The f i r s t four moments of the i f c * * net revenue stream are evaluated from the f i r s t four moments of its gross revenue and operation and maintenance cost. In so doing, i t has been assumed that both the constant dollar gross revenue and operation and maintenance costs are uniformly distributed over the duration of the revenue stream and that they are independent of each other. These simplifing assumptions are discussed later in section 4.8.4. From equation (3.16) the expected value .of the net revenue of the i f c ^ stream i s , E[NR^] = EtRj] - E[O&M^] (4.28) and the second to fourth central moments from equations (3. 17), (3. 18) , and (3.19) are, 66 M 2(NR i) = M 2(R i) + M2(0&Mi) (4. 29) M 3(NR i) = M 3(R i) " M3(0&Mi) (4. 30) M 4(NR i) = M 4(R i) + M4(0&Mi) + 6.M 2(R i). M^O&N^) (4.31) 4 . 6 . 2 N E T P R E S E N T V A L U E O F P R O J E C T R E V E N U E The discounted net project revenue i s the sum of a l l the discounted net revenue streams. The functions for the f i r s t four moments of the net project revenue are developed from the additive equations based on equations (3.12), (3.13), (3.14) and (3.15). The expected value of the net project revenue from n net revenue streams i s , n E[NPR] = L E[NR.] i=1 1 (4. 32) and the second to fourth central moments are, 67 n M , ( N P R ) = Z M , ( N R . ) (4. 33) * i=1 z 1 Mo(NPR) = Z M o ( N R . ) (4. 34) J i=1 * 1 n n n M A ( N P R ) = Z uA(NR• ) + 6 . Z Z Mo(NR. ) . M o ( N R . ) (4. 35) * i=1 4 1 i = i j = i + i ^ l ^ ] 4 .7 NET PRESENT VALUE MODEL The n e t p r e s e n t v a l u e i s t h e d e c i s i o n c r i t e r i o n o f t h e p r o p o s e d m o d e l . By d e f i n i t i o n , t h e n e t p r e s e n t v a l u e i s t h e d i f f e r e n c e b e t w e e n t h e d i s c o u n t e d r e v e n u e a n d c o s t . The m o d e l l i n g o f t h e p r o j e c t n e t r e v e n u e a n d t h e p r o j e c t c o s t i n t e r m s o f d i s c o u n t e d d o l l a r s a l l o w s f o r t h e n e t p r e s e n t v a l u e m o d e l t o be d e v e l o p e d a s a d i f f e r e n c e b e t w e e n two s t a t i s t i c a l l y i n d e p e n d e n t v a r i a b l e s . The f i r s t f o u r moments o f t h e n e t p r e s e n t v a l u e a r e d e v e l o p e d f r o m e q u a t i o n s ( 3 . 1 6 ) , ( 3 . 1 7 ) , ( 3 . 1 8 ) , a n d ( 3 . 1 9 ) . The e x p e c t e d v a l u e i s , E [ N P V ] = E [ N P R ] - E [ P C ] (4. 36) 68 a n d t h e s e c o n d t o f o u r t h c e n t r a l moments a r e , M 2 ( N P V ) = M 2 ( N P R ) + M 2 ( P C ) (4. 37) M 3 ( N P V ) = M 3 ( N P R ) - M 3 ( P C ) (4.38) M 4 ( N P V ) = M 4(N P R) + M 4 ( P C ) + 6 . M 2 ( N P R ) . M 2 ( P C ) (4.39) 4 . 8 MODEL ASSUMPTIONS The s t r e n g t h o f a m o d e l i s i t s c o n s i s t e n c y w i t h r e s p e c t t o i t s a s s u m p t i o n s . The key a s s u m p t i o n s o f t h e p r o p o s e d m o d e l h a v e b e e n n o t e d w i t h t h e d e v e l o p m e n t o f t h e m a t h e m a t i c a l f r a m e w o r k . I n t h i s s e c t i o n , t h e n e e d f o r t h o s e a s s u m p t i o n s , a n d t h e c o r r e s p o n d i n g c o m p l e x i t i e s w h i c h w o u l d o t h e r w i s e p r e v a i l , i s d i s c u s s e d . 69 4 . 8 . 1 G E N E R A L A S S U M P T I O N S Two k e y a s s u m p t i o n s a r e u s e d t o d e v e l o p t h e m a t h e m a t i c a l f r a m e w o r k . 1. The e s t i m a t o r i s c a p a b l e o f p r o v i d i n g s u b j e c t i v e e s t i m a t e s f o r p r i m a r y v a r i a b l e s , a n d t h e c u m u l a t i v e d i s t r i b u t i o n f u n c t i o n s d e v e l o p e d f r o m t h o s e e s t i m a t e s c a n be a p p r o x i m a t e d by t h e h i g h l y f l e x i b l e P e a r s o n ' s s y s t e m o f c u r v e s . 2. The p r i m a r y v a r i a b l e s a r e i n d e p e n d e n t , a n d h e n c e t h e r e i s no c o r r e l a t i o n b e t w e e n d e p e n d e n t v a r i a b l e s . The f i r s t g e n e r a l a s s u m p t i o n i s i m p l e m e n t e d w i t h t h e m o d e l a c c e p t i n g o n l y a s e t o f e s t i m a t e s w h i c h f i t a P e a r s o n t y p e d i s t r i b u t i o n . I t i s i m p o s s i b l e t o f i t e v e r y s e t o f e s t i m a t e s t o a d i s t r i b u t i o n . The r e q u i r e m e n t t h a t a s e t o f e s t i m a t e s f i t a P e a r s o n d i s t r i b u t i o n i s n o t v i e w e d a s o v e r l y r e s t r i c t i v e s i n c e t h e P e a r s o n ' s s y s t e m o f c u r v e s c o v e r s a r a n g e o f d i s t r i b u t i o n s . The s e c o n d g e n e r a l a s s u m p t i o n i s made t o s i m p l i f y t h e f r a m e w o r k . When c o r r e l a t i o n i s n e g l e c t e d , i t c a n r e s u l t i n an u n d e r e s t i m a t i o n o f t h e r i s k ( s i n c e i f v a r i a b l e s a r e c o r r e l a t e d t h e y t e n d t o be p o s i t i v e l y c o r r e l a t e d ) . V e r y s i g n i f i c a n t m e asurement a n d m a t h e m a t i c a l d i f f i c u l t i e s a r e e n c o u n t e r e d when one a t t e m p t s t o t r e a t c o r r e l a t i o n . D e t a i l e d t r e a t m e n t o f t h i s t o p i c i s recommended f o r f u t u r e r e s e a r c h . 70 4.8.2 ASSUMPTIONS FOR THE DURATION MODELS The work package duration model is based, in addition to the general assumptions, on the fundamental relationship between the quantity, productivity, labour usage, and duration. In most instances, this relationship provides a resonable approximation, but for work packages where the duration cannot be defined in terms of these fundamental variables, the model allows the duration to be specified subject ively. The s t a t i s t i c s of the early start time of a work package are obtained from those of the longest path to the work package. This assumption ignores shorter but more uncertain paths. It is proposed to develop an approach based on the PNET algorithm to obtain the stati s t i c s of the early start time which would account for the correlation arising from shared work packages and multiple paths to a work package. The s t a t i s t i c s of the project duration are obtained by assuming those of the longest path. Incorporating the PNET algorithm at the project level would f a c i l i t a t e the development of the completion time probability function for the project duration. This function would account for the correlation arising from shared work packages and multiple paths and hence improve the current assumption. 71 4 .8 .3 ASSUMPTIONS FOR THE COST MODELS I n t h e d e v e l o p m e n t o f t h e work p a c k a g e c o s t m o d e l , t h e p r i m a r y v a r i a b l e s a r e assumed t o be s t a t i o n a r y o v e r t h e d u r a t i o n o f t h e work p a c k a g e . I n r e a l i t y , a l l o f t h e p r i m a r y v a r i a b l e s a r e t i m e d e p e n d e n t ( i . e l a b o u r u s a g e , p r o d u c t i v i t y , i n f l a t i o n , i n t e r e s t r a t e s , e t c cha n g e w i t h t i m e ) . As an e x a m p l e , w i t h o u t t h e t h i r d a s s u m p t i o n t h e l a b o u r component o f t h e work p a c k a g e c o s t model i n d i s c o u n t e d d o l l a r s i s , t / e T ( r ) . d r • t f O L . L.C = f J C , . L ( t ) . e x p . e x p y t . d t t T / e L ( r ) . d T / r ( r ) . d r -vT t f O t + ( ! - f ) . e x p 1 J C T . L ( t ) .exp .exp . d t fcs When t h e p r i m a r y v a r i a b l e s a r e assumed t o be u n i f o r m l y d i s t r i b u t e d o v e r t h e d u r a t i o n o f t h e work p a c k a g e , t h e l a b o u r component i n d i s c o u n t e d d o l l a r s i s , 8 - +• L.C = f / C L . L . e x p L .exp y t . d t + ( ! - f ) . e x p y T ; t f C L . L . e x p 8 L . e x p r ( T fc).dt A s i m i l a r m a t h e m a t i c a l s i m p l i f i c a t i o n i s u s e d f o r t h e o t h e r c o m p o n e n t s o f t h e work p a c k a g e c o s t m o d e l . 72 Justification for this simplification comes in part from the following observations. F i r s t , unless feedback data from previous projects are readily available, insufficient information exits in the f e a s i b i l i t y phase with which to assess time varying profiles for work package primary variables. Second, i t can be shown that total discounted cost is relatively insensitive to time variations of resource usage within a work package. This assumption, by providing a trade-off between complexity and the ab i l i t y to model, becomes the backbone to the development of a simplified but r e a l i s t i c work package cost model. This is essential as i t facil i t a t e s the speedy formulation of a number of planning alternatives. It is through the evaluation of several alternatives that performance is optimized. The costs associated with a work package are assumed to be s t a t i s t i c a l l y independent of a l l other work package costs. This assumption is made for mathematical ease. 4.8.4 ASSUMPTIONS FOR THE REVENUE MODELS In the development of the net revenue stream, the primary variables are assumed to be uniformly distributed over the duration of the revenue stream. This assumption was made solely for convenience and could easily be relaxed. Incorporation of time varying revenue functions (which could be important for natural resource projects) is outside the 73 scope of the present study. The net revenue streams are assumed to be s t a t i s t i c a l l y independent in the development of the discounted net project revenue. The reasonableness of this assumption is not known and i t is left for future work to expand the model to treat correlation. 5. DEVELOPMENT OF A COMPUTER MODEL 5.1 GENERAL Development of the computer model based on the mathematical framework described previously is discussed in this chapter. The program is structured around a five step evaluation process. The five steps comprising the model are: Step 1 : Evaluation of the moments of the primary variables for work package duration, cost and revenue models, Step 2 : Evaluation of work package and project duration models, Step 3 : Evaluation of work package and project cost models, Step 4 : Evaluation of net and project revenue models, and Step 5 : Evaluation of net present value model. Shown in figure (5.1) are the major modules of the program with supporting subroutines and data f i l e s . The computer program was written in Fortran 77 and run on the AMDHAL 470 V/6, Model II computer. 74 75 DATA F I L E ( i ) WPSUM DATA F I L E ( i i i ) \ / VARBLE / WPDURA f NET WRK DATA F I L E ( i v ) WPCOST MAIN p r o j e c t c o s t DATA F I L E (v) DATA FI L E ( i i ) REVNUE MAIN p r o j e c t revenue MAIN net p r e s e n t v a l u e CDFUNC GRAPH Figure 5.1 The structure and major modules of the program 76 At present the program is limited in its sophistication especially in the area of user friendliness. To date, the emphasis has been placed on evaluating the proposed analytical approach. Generalization of the model and enhancement of the user-machine interface are l e f t for future work. To function e f f i c i e n t l y the model requires a substantial amount of data. This requirement forces the user to be f a i r l y sophisticated in the f i e l d of construction estimation. 5.2 THE INPUT DATA Data needed for model evaluation are stored in five f i l e s and are accessed from five control units. The f i l e s are: (i) a data base consisting of 2.5, 5.0, 50.0, 95.0, 97.5 percentage points, and 0 2 from the tables compiled by Johnson et al (1963) for the Pearson family of distributions called "TABSORT" accessed at unit 1; ( i i ) a data base of the complete table of percentage points for the Pearson family of distributions compiled by Johnson et al (1963) called "PEARSON" accessed at unit 2; ( i i i ) subjective estimates for the primary variables of the duration and cost models of the work packages accessed at unit 3; 77 (iv) logic relationships of the work packages accessed at unit 4; and (v) subjective estimates for the primary variables of the revenue models accessed at unit 5. Data f i l e s (i) and ( i i ) are self explanatory. The remaining three f i l e s are described in more detail in the following subsections. 5 .2 .1 DATA F I L E FOR CONTROL UNIT 3 The data f i l e for control unit 3 contains a l l the data necessary to evaluate the duration and the cost of a work package. The f i r s t line of this data f i l e contains the number of actual work packages plus two which the model considers as the total number thereafter, the starting and finishing discount rate for the search of project internal rate of return, the minimum attractive rate of return, and the equity fraction in that order. The two work packages are added in order to provide a start and finish work package for the project network. A maximum of ninety eight (98) work packages for a project are allowed. The rest of the f i l e contains work package identity codes and five percentage point estimates for each of the primary variables used to determine the work package duration and cost. The work package duration and cost models have fifteen primary variables. Thus each work packge is 78 described by sixteen data lines. The f i r s t data line for a work package is its identity code described by an integer. The codes are : Code 0 : to evaluate a work package duration from the fundamental relationship; Code 1 : to evaluate a work package duration specified subject ively; Code 2 : to evaluate a work package cost as a lump sum from subjective estimates; code 3 : to evaluate a work package consisting only of a duration. The data lines of subjective estimates for the primary variables have to be entered in the following order : 1. the quantity descriptor of the work package (Q^), 2. the productivity rate for labour ( P L i ^ ' 3. the labour usage profile ( L ^ ) , 4. the equipment usage profile (E^), 5. the subcontractor cost (S^), 6. the unit rate for indirect cost (Cj), 7. the unit rate for labour (C L^), 8. the unit rate for materials (C.,.), M l 9. the unit rate for equipment (C_,), t i I 10. the interest rate for the construction loan (r), 11. the inflation rate for labour ( 0 ^ ) , 12. the inflation rate for materials (0w-), M l 13. the inflation rate for equipment {$„•), 79 14. the inflation rate for subcontractor cost (0_.), O 1 15. the inflation rate for indirect cost (QQ^), A deterministic estimate for a variable is accepted when the deterministic value is specified as the median and the other percentage point values as zero. The only requirement with respect to units is that a l l of the estimates must be consistent with the unit for time. If the required time unit of the estimate is in months, then the estimates of such variables as usage profiles, interest and inflation rates, etc, should be with respect to a month as the time unit. Since continuous compounding is assumed, a l l interest and inflation rates are nominal ones. 5.2.2 DATA F I L E FOR CONTROL UNIT 4 The data f i l e for control unit 4 contains a l l the logic relationships between the work packages which are necessary to develop the project network. This combined with the previous f i l e provide a l l the data required by the program to evaluate the project duration, cost and their moments. The f i r s t line of this data f i l e contains the total number of work packages (i.e the actual number of work packages plus two), and the projected start and finish dates, in that order. Thereafter the work package relationships are entered using two lines of the data f i l e for each work package. 80 The f i r s t line of the work package data contains the number, the number of dependencies and the alphanumeric description for that work package. The second line contains the work package numbers of its predecessors. The model permits up to forty (40) predecessors for a work package. Only one work package can be without a predecessor, the start work package and only one without a successor, the finish work package. If this condition is not satisfied the model terminates, requesting the user to check his data f i l e at unit 4. To avoid this inconvenience, the model is developed with a built in start and finish work package, which always have 1 and n the total number as their work package numbers. 5 .2 .3 DATA F I L E FOR CONTROL UNIT 5 The data f i l e for control unit 5 contains a l l the data necessary to evaluate the project revenue. The f i r s t line contains the total number of revenue streams and the rest of the f i l e contains data describing each of the revenue streams. Each revenue stream occupies six lines of the data f i l e . The f i r s t data line for each revenue stream contains a work package number, and a fraction of a work package duration. The number is of the work package that links the revenue stream to the project network, and the fraction is that of the linking work package duration, after which the 81 revenue stream w i l l be operative. The other five lines contain five percentage point estimates for each of the primary variables of the revenue model. These estimates must be entered in the following order : 1. the projected revenue per time period (R^), 2. the operation and maintenance cost (O&M^), 3. the projected duration of the revenue stream (T^), 4. the inflation rate for revenue ( ^ R ^ ) f 5. the inflation rate for operation and maintenance cost (0Qi)* The requirement with respect to units for the primary variables of the revenue model is that they should be consistent with the unit for time. The inflation rates are nominal rates from time zero. 5.3 STEP 1 : MOMENTS OF THE PRIMARY VARIABLES The f i r s t step of the program is to estimate the f i r s t four moments of the the primary variables used to define the duration, cost and revenue variables. This estimation is done with the subroutine "VARBLE". The subroutine "VARBLE" uses the five subjective estimates for the 2.5, 5.0, 50.0, 95.0, and 97.5 percentage points for a variable along with equations (3.1) to (3.5) to estimate the expected value and the standard deviation of that variable. These subjective estimates are then 82 normalised using equation (3.8). The normalised estimates are compared with the data base "TABSORT" by the method of least squares to obtain the "best f i t " distribution. The shape characteristics of this distribution are used along with equations (3.9), (3.10), and (3.11) to estimate the central moments of the variable. 5.4 STEP 2 ; THE DURATION MODELS The duration models are developed at both the work package and project levels. The stat i s t i c s of the work package duration are estimated by the subroutine "WPDURA" while the subroutine "NETWRK" estimates the statistics of work package early start time and of the project duration. It is proposed to use the PNET algorithm to estimate the project completion time probability as a future development to the model. Currently the longest path (PERT) approach is used. 5.4.1 SUBROUTINE "WPDURA" The subroutine "WPDURA" which incorporates equations (4.2) to (4.5) is used to estimate the f i r s t four moments of a work package duration. This subroutine is called by the main program for a l l work packages numbered 2 to (n-1). 83 5.4.2 SUBROUTINE "NETWRK" The subroutine "NETWRK" estimates the f i r s t four moments of the early start time of a work package and of the project duration. This subroutine is based on the program "NETWORK" developed by Thomas Y. Tong (30). The program "NETWORK" is an interactive precedence network processor using CPM techniques. The structure developed in "NETWORK" is maintained in the subroutine "NETWRK". A few modifications necessary to accomodate the computer language and probabilistic estimation were done to Tong's program. The subroutine uses equations (4.8) to (4.11) to estimate the f i r s t four moments of the early start time of a work package. These statistics are very important as they link the cost and time models by considering the economic effect of time on the cost. The f i r s t four moments of the early start time of the n work package, the finish work package corresponds to the stat i s t i c s for the project duration. At present only finish to start relationships with no lag time (i.e FS = 0) are permitted in the network model. It is proposed that overlapping networks with work packages having different precedence link relationships and lag times be treated in future work. 84 5.5 STEP 3 : THE COST MODELS Cost models are developed at both the work package and project levels. The statistics of the work package cost are estimated by the subroutine "WPCOST" while the statistics of the project cost are estimated in the main program. The f i r s t four moments for the project cost are based on equations (4.23) to (4.26). Subroutine "WPSUM" estimates the stat i s t i c s of the work package cost when a work package cost is specified as a lump sum. The importance of current dollar and constant dollar estimates for possible decision purposes is recognised. The model estimates the cost in current dollars when the discount rate is set to zero in the data f i l e at control unit 3, and in constant dollars when the discount and the inflation rates are set to zero. 5.5.1 SUBROUTINE "WPCOST" Subroutine "WPCOST" is based on equations (4.18) to (4.21). The function for the work package cost, and its seventeen variables make this the longest routine of the model. This subroutine is called for a l l the work packages numbered 2 to (n-1). Subroutine "WPCOST" contains a default situation in order to treat the case of the inflation rate equaling the 85 discount rate. Consider, for example the labour component of the equity portion of the i f c ^ work package cost. From equation (4.15), . f c T r u L - y ) ( t s i + T i } t s i ^ L - y * . i = * Li* i * e x p ~ exp ] (8T-y) If the inflation rate for labour (0^) = discount rate (y), then A i = 0 / 0 Applying L'Hospital's rule, one obtains A i = f ' C L i ' L i - T i ^ • f . c L i . L i . [ ( t s i + T , ) 2 - t s i 2 ] 3X L  ! S - f . c L i . L i . [ ( t s i + y 3 - t s i 3 ] 3 X L ^ where X L = (8L - y) 86 The default situation is evoked for a difference of less than 0.001 between the discount rate and an inflation rate. 5.6 STEP 4 : REVENUE MODELS Revenue models are developed at both the revenue stream and project levels. The st a t i s t i c s of the revenue stream are estimated by the subroutine "REVNUE" while the statistics of the net project revenue are estimated in the main program. The f i r s t four moments of the net project revenue are based on equations (4.32) to (4.35) and the statistics of the net revenue streams are estimated in the subroutine "REVNUE". 5.6.1 SUBROUTINE "REVNUE" Subroutine "REVNUE" is based on equations (4.28) to (4.31). It estimates the f i r s t four moments of a net revenue stream based on the assumption that the revenue and the operation and maintenance cost streams are s t a t i s t i c a l l y independent. This subroutine also contains the default situation developed for the subroutine "WPCOST", when the discount rate equals the inflation rate. 87 5 .7 STEP 5 : NET PRESENT VALUE MODEL The net present value model is evaluated in the main program. The f i r s t four moments of the net present value are based on equations (4.36) to (4.39) and the estimated sta t i s t i c s for the project cost and revenue. The model estimates the net present value of the project, for the specified range of discount values in steps of 0.5%. This development is to f a c i l i t a t e the estimation of the statistics for the internal rate of return. It is l e f t to future work to develop an approach in order to obtain the sta t i s t i c s of the internal rate of return. 5 . 8 CUMULATIVE DISTRIBUTION FUNCTION The model generates plots for the cumulative distribution function for the project cost, revenue, and net present value at the specified minimum attractive rate of return (MARR). If the cumulative distribution functions of the decision parameters are required at different discount rates, they can be obtained by changing the MARR in the data f i l e for control unit 3. The model uses two subroutines, "CDFUNC" and "GRAPH" to generate a plot of the cumulative distribution function. 88 5 . 8 . 1 SUBROUTINE "CDFUNC" Subroutine "CDFUNC" estimates the percentage values of the cumulative distribution function for the desired dependent variable. This subroutine extracts the normalised percentage points for the shape characteristics of the variable, from the data base "PEARSON" accessed from control unit 2. It does an interpolation between the tabulated y/$. and $2 values to obtain accurate normalised percentage points for those shape characteristic values. Then using the equation, x_ = X .a + u P P i t estimates the percentage values of the cumulative distribution function. 5 . 8 . 2 SUBROUTINE "GRAPH" Based on the estimated percentage values of the cumulative distribution function from the subroutine "CDFUNC", subroutine "GRAPH" generates a plot for that cumulative distribution function. Subroutine "GRAPH" uses DISSPLA 9.0, a proprietary software product of ISSCO, San Diego, CA, available in the system to generate this plot. 6. APPLICATION OF THE MODEL : RESULTS AND DISCUSSION 6.1 GENERAL This chapter contains two examples to demonstrate application of the model. First a simplified example is examined to demonstrate the model behaviour and the influence of varying uncertainty at the independent parameter level on performance measures of interest. Second, the model is applied to data from an actual deterministic fea s i b i l i t y analysis for a mining project. The results obtained are compared with the deterministic analysis and used to answer some of the questions arising at the feasibility stage. 6.2 EXAMPLE 1 : MODEL BEHAVIOUR A project consisting of eight identical work packages (ten in total including the start and finish work packages) logically related as shown in figure ( 6 . 1 ) is used to demonstrate model behaviour. Identical work packages provide a basis from which to study the influence of one level of the project on the other. 89 W P 1 • W P 4 • W P 7 Figure 6.1 A network of work packages o 91 Coefficient of variation, a non-dimensional measure, is used to study the influence of varying uncertainty at the independent parameter level on the work package and project level performance measures. The total dollar cost is the performance measure of interest for this study. Two skewness scenarios for the independent primary variable level were examined, namely: 1. when the skewness is less than or equal to |0.1| and 2. when the skewness is greater than or equal to |1.6|. For both cases the work packages are kept identical for each application of the model but the coefficient of variation of the primary variables are changed from application to application. Figure (6.2) demonstrates that uncertainty at the primary variable level is magnified by nearly 100% to the work package level. This magnification is due to the functional form of the work package model as shown by equation (4.15). An interesting observation is that the third central moment term in equation (4.19) is not significant even for highly skewed variables in this magnification. The magnitudes of the derivatives of this term could account for the small impact of skewness. Figure (6.3) demonstrates that the overall variation of the total dollar cost estimate for the project is less than that of the primary variables. This implies a diversification of uncertainty from the primary variable level to the project level. 92 UORK PACKAGE COV VS PRIMARY V A R I A B L E COV o o o SKEUNESS > 1.6 H • SKEUNESS < 0.1 "1 i i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 PRinflRY VARIABLE COV Figure 6.2 The effect of uncertainty at primary variable l e v e l on the work package level for the t o t a l do l lar cost estimate. 93 PROJECT COST COV VS PRIMARY V A R I A B L E COV in n o e> SKEUNESS > 1.6 -« SKEUNESS < 0.1 > 1/1 o2-o C J (M o UJ -—> O in T -1 r 0.45 o.o i r 0.05 r 0.1 i r 0.15 i 0.2 I 0.2S 0.3 0.35 PRiriRRY VARIABLE COV 0.4 0.5 Figure 6.3 The effect of uncertainty at primary variable level on the project level for the total dollar cost estimate. 94 While the relationship between the primary variable and the project level does not provide a direct reason for the sharing of uncertainty the relationship between the work package and the project level does. Figure (6.4) demonstrates a significant reduction of the variation when the work package costs are summed to obtain the total dollar cost estimate for the project. This is due to the diversification of risk between the work package costs when they are summed up. This effect can be demonstrated mathematically for the special case considered in this example. When a l l the work packages are identical then the coefficient of variation for the total dollar cost estimate of the project cost, the derived variable from the addition of n work package costs of the form n Y = I X . i = 1 1 can be shown to be n n 0.5 V = V [ n + 2 Z Z p. • ] Y _JL i = 1j = i + 1 ^ where V y is the coefficient of variation of the project cost, V x is the coefficient of variation of an identical work package, 95 p^j is the correlation coefficient between dependent t h t h i and j work packages, and n is the number of work packages. When the work packages are s t a t i s t i c a l l y independent (p—=0), the coefficient of variation of the project cost is V y = V x / ,/n The above relationship for s t a t i s t i c a l l y independent work packages demonstrates that as the network gets larger there is more sharing of the risk and regardless of the skewness of the work packages i t is true if the work packages are identical. This is because the skewness of variables in linear additive form does not contribute towards the variation of the derived variable (skewness only marginally affects the variance at the work package level, see figure (6.2)). For the case of total dependancy (p^j=l), i t can be shown that the coefficient of variation of the project cost is V = V Y X 96 -> in 8 -C O o L U —> O ir> 8: 5 PROJECT COST COV VS WORK PflCKnGE COV O © SKEUNESS > 1.6 4 >- SKEUNESS < 0.1 0.0 0.1 i 1 1 1 1 1 1 1 1 i r i - i — 0 2 0.3 0.4 0.5 0.6 0.7 UORK PACKAGE COV i 1 1 r 0.8 0.9 1.0 F i g u r e 6.4 The e f f e c t of u n c e r t a i n t y a t work package l e v e l on the p r o j e c t l e v e l f o r the t o t a l d o l l a r c o s t e s t i m a t e 97 A n o t h e r i m p o r t a n t o b s e r v a t i o n f r o m t h e r e s u l t s f r o m t h e a p p l i c a t i o n o f t h e model ( s e e A p p e n d i x A) i s t h e t e n d e n c y o f t h e . s k e w n e s s o f t h e t o t a l d o l l a r c o s t e s t i m a t e f o r t h e p r o j e c t c o s t t o a p p r o a c h z e r o . T h i s o b s e r v a t i o n i s i n a g r e e m e n t w i t h t h e a p p l i c a t i o n o f t h e c e n t r a l l i m i t t h e o r e m f o r v a r i a b l e s t h a t a r e i n d e p e n d e n t a n d i d e n t i c a l l y d i s t r i b u t e d . F i g u r e 6.5 i m p l i e s t h a t t h e s k e w n e s s a t t h e p r i m a r y v a r i a b l e l e v e l a f f e c t s t h e u n c e r t a i n t y o f t h e p r o j e c t n e t p r e s e n t v a l u e s i g n i f i c a n t l y . W h i l e t h e r e i s a s m a l l e f f e c t c a r r i e d o v e r f r o m t h e p r o j e c t c o s t v a r i a t i o n t h e r e a l c a u s e i s t h e f u n c t i o n a l f o r m o f t h e n e t p r e s e n t v a l u e . The s u b t r a c t i v e f u n c t i o n a l f o r m c a n c a u s e s t a t i s t i c a l i n s t a b i l i t i e s r e s u l t i n g i n a m i s l e a d i n g i n t e r p r e t a t i o n o f t h e e s t i m a t e . F o r e x a m p l e when t h e d i s c o u n t r a t e a p p r o a c h e s t h e i n t e r n a l r a t e o f r e t u r n t h e c o e f f i c i e n t o f v a r i a t i o n o f t h e n e t p r e s e n t v a l u e i n c r e a s e s s i g n i f i c a n t l y . ( I n f a c t , when t h e d i s c o u n t r a t e e q u a l s t h e i n t e r n a l r a t e o f r e t u r n t h e c o e f f i c i e n t o f v a r i a t i o n i s i n f i n i t e ) . T h i s phenomenon i s c a u s e d by t h e s u b t r a c t i v e f u n c t i o n a l f o r m o f t h e n e t p r e s e n t v a l u e a n d d o e s n o t i m p l y u n r e l i a b i l i t y o f t h e e s t i m a t e . F o r t h i s r e a s o n , i t i s recommended t h a t i n t e r n a l r a t e o f r e t u r n be u s e d a s an a d d i t i o n a l d e c i s i o n c r i t e r i o n i n o r d e r t o g a u g e t h e s i g n i f i c a n c e o f t h e u n c e r t a i n t y m e a s u r e d w i t h t h e n e t p r e s e n t v a l u e c r i t e r i o n . 98 NET PRESENT VALUE COV VS PROJECT COST COV CM o CC > i — s -UJ CD U l cr Q_ oo UJ o co Q o SKEUNESS > 1.6 + SKEUNESS < 0.1 0.0 —I— 0.15 0.2 0.25 PROJECT COST COV 0.35 0.4 Figure 6.5 The effect of uncertainty of the t o t a l project cost on the net present value. 99 6 . 3 E X A M P L E 2 : M O D E L A P P L I C A T I O N The model is used to estimate the decision parameters for the economic f e a s i b i l i t y of a copper mine. Data was obtained from an actual deterministic feasibility analysis for a mining project in South America. The results obtained from the study are compared with the deterministic analysis and used to answer some of the questions which must be addressed at the fe a s i b i l i t y stage, these questions being the uncertainty of project performance measures, provision of contingency, fe a s i b i l i t y at minimum attractive rate of return, etc. The deterministic f e a s i b i l i t y analysis contained a construction schedule, estimates for cost centers (detailed analysis for some and lump sums for others), projected annual revenue and operating costs for a study period of fifteen years. An inflation rate of 6% was estimated for costs, while a financing rate of 9% and an equity fraction of 15% was assumed. From the deterministic analysis for the pretax base case an internal rate of return of 36% was reported. In the original feasibility analysis, inflation of the sales revenue and manufacturing costs were neglected on the assumption that any inflationary increases in cost would be offset by corresponding increases in sales price. 100 START WORK PACKAGE W P 1 ENGINEERING 4 MOBILIZATION W P 2 CONSTRUCTION OP A TEMPORARY FUEL TANK W P 3 BOAO & R A L FOR EOUFM6NT TRANSFER W P 4 CAMP EXPANSION W P 7 MINE AUXILIARY 3UIDNGS WP8 TOWNSITE -PHASE 1 W P 9 POWER HOUSE CONSTRUCTION W P 5 ROAOS FOR CONSTRUCTION REQUIREMENTS RAINY SEASON 3 MONTH DOWNTIME - YEAR 2 -W P 6 WATER SUPPLY SCHEME W P 1 0 OFFICE. CHANGEHOUSE i LABORATORY FOR PLANT W P 1 1 ROAO/RAIL/PORT TRANSFER FACILITIES FOR PRODUCE W P 1 2 CONSTRUCTION OF PROCESS PLANT W P 1 3 TAILINGS 0AM W P 1 4 TOWNSITE - PHASE 2 W P 1 5 POWER PLANT, SUPPLY ANO DISTRIBUTION W P 1 6 ROAOS FOR OPERATIONAL REQUIREMENTS - PHASE 1 RAINY SEASON 3 MONTH OOWNTIME - YEAR 3 -W P 2 2 TOWNSITE - PHASE 3 W P 2 3 TAILINGS PIPELINE PHASE 2 W P 2 4 TAILINGS THICKNER PHASE 2 y y P 2 5 EQUIPMENT ANO INSTALLATION OF PROCESS PLANT W P 2 6 ROAOS FOR OPERATIONAL REQUIREMENTS-PHASE 2 W P 1 7 CONSTRUCTION OF A PERMANENT FUEL SYSTEM W P 1 8 TAILINGS PIPELINE - PHASE 1 W P 2 1 TAILINGS THICKNER PHASE 1 W P 2 7 RECLAIM WATER SYSTEM W P 1 9 PLANT SHOP 1 WAREHOUSE START UP W P 2 0 PRE - PRODUCTION FINISH WORK PACKAGE W P 2 8 PROJECT MANAGEMENT. LOCAL ORGANISATIONAL EXPENSES ANO TECHNOLOGY WPORT TAX Figure 6.6 Construction Schedule 101 6 . 3 . 1 C O N S T R U C T I O N S C H E D U L E A N D W O R K P A C K A G E C O S T For study purposes herein, the original construction schedule was simplified to contain only start to finish relationships with a zero lag as shown in figure (6.6). The logic of the original network was maintained throughout the modification of the schedule. As the deterministic analysis did not contain basic productivity information, the starting point for this study was at the work package level. The work packages were developed to correspond to the modified construction schedule. From the estimates for cost centers of the deterministic analysis the work package costs were estimated such that the sum of the work package cost in constant dollars was equivalent to constant dollar cost estimate of the deterministic analysis. Table (6.1) contains the deterministic estimates for cost and duration of work packages. 6 . 3 . 2 M O D I F I E D D A T A F O R T H E M O D E L The data obtained from the deterministic analysis were modified to a probabilistic form acceptable to the model. For this modification, the deterministic estimates for the respective variables (i.e work package duration and cost, annual revenue and operating costs, inflation and financing rates) were assumed to be the median value of their respective frequency distributions. 102 W.P i WORK PACKAGE DESCRIPTION DURA W.P COST Mdl S h i mths $$ 01 — S t a r t work package 02 01 E n g i n e e r i n g & m o b i l i s a t i o n 4 2800000 03 02 C o n s t r u c t i o n of a temporary f u e l tank 3 200000 04 03 Road & r a i l f o r equipment t r a n s f e r 3 2520900 05 04 Camp expansion 3 2620000 06 05 Roads f o r c o n s t r u c t i o n r e q u i r e m e n t s 8 2400000 07 06 Water s u p p l y scheme 1 1 2501100 08 07 Mine a u x i l i a r y b u i l d i n g s 11 4233800 09 08 Tow n s i t e Phase 1 8 3552200 10 09 Power house c o n s t r u c t i o n 5 865800 11 Rainy season - year 2 : Downtime 3 12 1 0 O f f i c e , changehouse, & l a b o r a t o r y f o r p l a n t 10 2497200 13 1 1 R o a d / r a i l / p o r t t r a n s f e r f a c i l i t i e s f o r produce 10 4198000 14 12 C o n s t r u c t i o n of p r o c e s s p l a n t 8 4996300 15 13 T a i l i n g s dam 8 3980000 16 14 Townsite Phase 2 8 4000000 17 15 Power p l a n t , s u p p l y and d i s t r i b u t i o n 13 6958300 18 16 Roads f o r o p e r a t i o n a l r e q u i r e m e n t s Phase 1 9 3500000 19 17 C o n s t r u c t i o n of permanent f u e l system 9 743900 20 18 T a i l i n g s p i p e l i n e Phase 1 4 550000 21 19 P l a n t shop & warehouse 13 1513600 22 20 P r e p r o d u c t i o n 21 33047700 23 — R a i n y season - year 3 : Downtime 3 24 21 T a i l i n g s t h i c k n e r Phase 1 5 440000 25 22 Townsite Phase 3 6 2000000 26 23 T a i l i n g s p i p e l i n e Phase 2 5 682500 27 24 T a i l i n g s t h i c k n e r Phase 2 4 346000 28 25 Equipment & i n s t a l a t i o n of p r o c e s s p l a n t 6 1 1853700 29 26 Roads f o r o p e r a t i o n a l r e q u i r e m e n t s Phase 2 6 1475000 30 27 R e c l a i m water system 9 1356100 31 — S t a r t up 3 600000 32 28 P r o j e c t management, l o c a l o r g expenses and t e c h n o l o g y i m p o r t t a x 36 18018000 33 — F i n i s h work package (Revenue 15ys p e r i o d ) T o t a l Base E s t i m a t e 3 y r s 124450100 Table 6.1 Deterministic (median) estimates for work package costs and durations W.P f CONSTANT DOLLAR COST DURATION MODEL EXP VALUE STD DEV SKW EX.VL ST.DV SKW $$ mths mths 01 0 0 0 0 0 0 02 2836998 913186 0.1 3.98 0.51 0.4 03 203700 67051 0.2 3.02 0.27 0.7 04 2549598 912707 0. 1 2.98 0.33 -0.3 05 2649598 912707 0.1 2.98 0.33 -0.3 06 2418497 881804 0.1 7.98 0.91 0.2 07 2537628 913159 0.1 1 1 .04 0.84 0.1 08 4258292 1337783 0.1 11 .02 0.45 0.5 09 3579133 1140381 0.1 7.98 0.91 0.2 10 859450 273656 0.0 5.01 0.44 0.1 1 1 0 0 0 3.02 0.27 0.7 12 2535108 913267 0.1 10.03 0.58 0.2 13 4198738 1276595 0.0 10.05 0.55 0.3 14 5034666 1581373 0.1 7.93 0.27 0.4 15 4042898 1370344 0.1 7.94 0.33 0.4 16 4073997 1341011 0.1 7.93 0.27 0.4 17 7029227 2220856 0.1 13.01 1.06 0.0 18 3536998 1095333 0.1 9.02 0.33 0.4 19 746157 237101 0.0 9.02 0.33 0.4 20 555550 191632 0.1 3.98 0.51 0.4 21 1517816 471158 0.1 13.01 0.55 0.1 22 33400032 10952315 0.1 21 .02 1.06 0.1 23 0 0 0 3.02 0.27 0.7 24 445550 149120 0.1 4.98 0.40 0.0 25 2018498 638774 0.2 6.02 0.40 0.2 26 688975 219015 0.1 5.01 0.30 0.4 27 349330 118624 0.1 3.96 0.51 0.0 28 12074329 3992589 0.2 6.02 0.55 0.1 29 1502748 518038 0.2 6.02 0.48 0.2 30 1390841 458266 0.3 9.02 0.33 0.4 31 607400 194779 0.1 3.02 0.33 0.4 32 18751312 6156602 0.5 36.02 1.18 -0.1 33 0 0 0 1 5 y r s 0 0 Table 6.2 P r o b a b i l i s t i c estimates for work package costs and durations 1 04 From this basic assumption values for the 2.5, 5.0, 95.0, and 97.5 percentage points of the distribution were subjectively estimated. From past experience the estimates for annual revenue were selected with a negative skew while the estimates for the other variables were selected with a positive skew. The coefficients of variation for work packages were assumed to be over 30% in the selection of estimates. This is to account for the magnification of the uncertainty at the work package level when the costs are estimated from the primary variables. This phenomenon was demonstrated in the previous example. Table(6.2) contains the expected value, standard deviation and skewness of the work package duration and constant dollar cost used in this study. 6.3.3 P R O J E C T C O S T A N D C O N T I N G E N C Y A L L O C A T I O N The project cost from the deterministic analysis was presented in base (constant) dollars with estimated provisions for the escalation during construction (EDC) and the interest during construction (IDC). For a comparision with the deterministic analysis the output from the model should contain a constant dollar cost estimate, a current dollar cost estimate and a total dollar cost estimate for the project. The constant dollar cost estimate is the estimate of the project in some base year dollars. It is obtained by 105 setting the equity fraction to 1.0, the inflation rate for work package costs and the minimum attractive rate of return to zero. A summary result for the constant dollar estimate is shown in figure (6.7) and the detailed results are contained in Appendix (B-1). The current dollar cost estimate is the base estimate for the project plus the escalation during construction. It is obtained by setting the equity fraction to 1.0 and the minimum attractive rate of return to zero. A summary result for the current dollar estimate is shown in figure (6.8) and the detailed results are contained in Appendix (B-2). The total dollar cost estimate is the base estimate for the project plus the escalation during construction and the interest during construction. Assuming 15% equity input and interest paid at the end of the project a summary result for the total dollar cost estimate is shown in figure (6.9) and the detailed results are contained in Appendix (B-3). DETERMINISTIC PROBABILISTIC EXP.VAL STD.DV Constant $$ E s t i m a t e 124450100 126392880 14041892 EDC ( i n c i n t e r e s t ) 15522000 EDC (exc i n t e r e s t ) 13472976 IDC (w/o e s c a l a t i o n ) 1 1600000 IDC (wth e s c a l a t i o n ) 13896848 T o t a l d o l l a r c o s t 151572100 153762704 17029992 Table 6.3 Comparision between the deterministic and probabilistic analyses for the project cost 106 C . D . F O F T H E P R O J E C T C O S T Figure 6.7 Constant Dollar Cost Estimate C.D.F OF THE PROJECT COST 107 Figure 6.8 Current Dol lar Cost Estimate 108 C.D.F OF THE PROJECT COST F i g u r e 6 . 9 T o t a l D o l l a r Cos t E s t i m a t e 109 Table (6.3) shows a comparision between the results from the deterministic analysis and the probabilistic analysis performed by the model. The most significant improvement of the probabilistic analysis over the deterministic analysis is that while giving an estimate for performance measures of interest i t also quantifies the uncertainty associated with that estimate. The quantification of uncertainty achieves one of the important objectives of this thesis, a rational basis on which to allocate the contingency provision. The contingency allocation is an important risk mitigating strategy and its provision is a key decision to be made at the f e a s i b i l i t y stage. It should be neither too large thereby tying up funds unnecessarily nor too small thereby carrying a greater risk of overrunning the budget. For example, the authors of the deterministic analysis provided 10% of the constant dollar capital cost estimate as a contingency provision. While this may be an accepted factor of safety from experience, i t provides no other information on its goodness as a risk mitigating stategy. In contrast, the developed cumulative distribution function for the total project cost provides the guidance to set up a contingency allocation to a desired probability of success. If the desired probability of success is 90%, then the estimate for the desired probability of success is the 90fc^ percentile value of the cumulative distribution function for the project cost. Hence the contingency provision is the 1 10 d e s i r e d p e r c e n t i l e v a l u e m i n u s t h e e x p e c t e d v a l u e o f t h e p r o j e c t c o s t . ( I t i s n o t e d t h a t a t a r g e t b u d g e t , e x c l u s i v e o f c o n t i n g e n c y c o r r e s p o n d i n g t o a v a l u e o t h e r t h a n t h e e x p e c t e d v a l u e may be more a p p r o p r i a t e . ) To be c o n s i s t e n t w i t h t h e d e t e r m i n i s t i c a n a l y s i s t h e 9 0 f c ^ p e r c e n t i l e v a l u e f o r t h e c o n s t a n t d o l l a r c o s t e s t i m a t e f r o m f i g u r e ( 6 . 7 ) i s 145039144.0. Hence t h e c o n t i n g e n c y p r o v i s i o n f o r a 90% p r o b a b i l i t y o f s u c c e s s i s 1 8 6 4 6 2 6 4 . 0 . T h i s i s a b o u t 14.75% o f t h e c o n s t a n t d o l l a r c o s t e s t i m a t e . 6 . 3 . 4 I N T E R N A L R A T E O F R E T U R N The i n t e r n a l r a t e o f r e t u r n f o r a p r o j e c t i s t h e d i s c o u n t r a t e a t w h i c h t h e d i s c o u n t e d r e v e n u e i s e q u a l t o t h e d i s c o u n t e d c o s t . I n o t h e r w o r d s , t h e d i s c o u n t r a t e a t w h i c h t h e n e t p r e s e n t v a l u e i s z e r o . The d e t e r m i n i s t i c a n a l y s i s r e p o r t e d a 36% p r e t a x i n t e r n a l r a t e o f r e t u r n . The p r o b a b i l i s t i c a n a l y s i s assumes c o n t i n u o u s c o m p o u n d i n g . Hence f o r a c o m p a r i s i o n w i t h t h e d e t e r m i n i s t i c a n a l y s i s t h e p r o b a b i l i s t i c a n a l y s i s s h o u l d be a t t h e e q u i v a l e n t n o m i n a l r a t e . A n o m i n a l r a t e r f o r c o n t i n u o u s c o m p o u n d i n g e q u i v a l e n t t o an e f f e c t i v e r a t e o f i i s e s t i m a t e d f r o m r e x p = (1 + i ) 111 C.D.F OF THE PROJECT COST F i g u r e 6.10 D i s c o u n t e d P r o j e c t C o s t 1 12 C.D.F OF THE NET PROJECT REVENUE Figure 6.11 Discounted Project Revenue 113 C.D.F OF THE PROJECT NET PRESENT VALUE Figure 6.12 Net Present Value Diagram 1 14 Summary results of the analysis for discounted project cost, discounted revenue and the net present value for a discount rate of 30.7%, the equivalent nominal rate for an effective rate of 36% are given in figures (6.10), (6.11) and (6.12) while the detailed results are given in Appendix(B-4). The analysis shows a positive net present value at the nominal discount rate of 30.7%. The coefficient of variation of the net present value at this discount rate is around 170%. The high variability implies that the discount rate is close to the internal rate of return. While the model has the capability to estimate the net present value as a probabilistic parameter i t does not have the capability to estimate the internal rate of return as a probabilistic quantity. The internal rate of return is an implicit function of the net present value and hence does not provide a direct functional form for s t a t i s t i c a l operations. The importance of the internal rate of return as the decision criterion for economic f e a s i b i l i t y was demonstrated in the previous example. While some of the possible approaches are discussed below, i t is le f t for future research to develop a consistent approach to estimate the internal rate of return as a probabilistic parameter. NPV f Figure 6.13 Net Present Value vs Discount Rate The typical relationship between the net present value, the discount rate and the internal rate of return (IRR) as shown in figure (6.13) could be used to develop an interpolation approach to determine the statistical properties of the internal rate of return. The model at present estimates the f i r s t four moments of the net present value for a deterministic discount rate. Hence using the sta t i s t i c a l properties of negative and positive net present values, the sta t i s t i c a l properties of the internal rate of return could be estimated from either a linear or a quadratic polynomial interpolation. 1 1 6 The i n i t i a l studies done for this approach show, f i r s t l y , the need to treat correlation between net present values as they are perfectly correlated since the only difference in the set is the discount rate. Secondly, because of the high variability of the net present value as i t approaches zero, higher order moments become very significant in the equations for the s t a t i s t i c a l parameters. It must be stressed that the high va r i a b i l i t y of the net present value as i t approaches zero is caused by its subtractive functional form and does not imply the unreliability of the estimate. Two other approaches are suggested with a caution that their f e a s i b i l i t y has not been investigated as at present. The f i r s t is the use of the r e l i a b i l i t y function and the Rackwitz - Fiessler algorithm (28) from structural r e l i a b i l i t y to develop a probability density function for the internal rate of return. The second is by using the approach proposed for "implicitly given mappings of uncertain quantities" by Ove Ditlevsen (12) as the internal rate of return is an implicit function of the net present value. 7. CONCLUSIONS AND RECOMMENDATIONS 7.1 CONCLUSIONS The o b j e c t i v e o f t h i s t h e s i s was t o d e v e l o p a c o n s i s t e n t t h e o r y w h i c h c o u l d a s s i s t i n s t r a t e g i c d e c i s i o n m a k i n g a t t h e f e a s i b i l i t y s t a g e o f a p r o j e c t . The t h e o r y a n d t h e c o m p u t e r m o d e l d e v e l o p e d a c h i e v e s t h i s o b j e c t i v e by o v e r c o m i n g t h e c o n s t r a i n t s i d e n t i f i e d i n s e c t i o n 1 . 2 . I n p a r t i c u l a r : 1. The m o d e l r e l i e s on s u b j e c t i v e p r o b a b i l i t i e s t o o b t a i n d a t a r e c o g n i s i n g t h e d a t a l i m i t a t i o n s i n t h e c o n s t r u c t i o n i n d u s t r y ; 2 . I t i n t e g r a t e s t h e t i m e and c o s t a s p e c t s o f a work p a c k a g e by c o m b i n i n g t h e s t a t i s t i c a l p r o p e r t i e s o f t h e e a r l y s t a r t t i m e o f t h e work p a c k a g e s w i t h work p a c k a g e t i m e a n d c o s t e s t i m a t i o n ; 3. I t e s t i m a t e s t h e u n c e r t a i n t y o f t h e p r o j e c t p e r f o r m a n c e p a r a m e t e r s ( i . e p r o j e c t c o s t , p r o j e c t r e v e n u e , n e t p r e s e n t v a l u e ) by u t i l i z i n g t h e moment a n a l y s i s a p p r o a c h a n d t h e P e a r s o n s y s t e m o f c u r v e s c o n s i s t e n t l y ; 4. The work p a c k a g e c o n c e p t i s u t i l i z e d a s t h e a p p r o a c h t o o b t a i n i n t e r m e d i a t e i n f o r m a t i o n on t h e r e l a t i v e c o n t r i b u t i o n s t o u n c e r t a i n t y ; 5 . The q u a n t i f i c a t i o n o f u n c e r t a i n t y p r o v i d e s a b a s i s t o a n s w e r s u c h s t r a t e g i c q u e s t i o n s a s s e t t i n g up o f t h e 1 1 7 118 contingency provision and the r e l i a b i l i t y of an estimate for the "go - no go" decision; and 6. Above a l l the theory develops a consistent analytical approach to obtain the s t a t i s t i c s of a dependent variable from a set of random primary variables without the use of simulation. 7 . 2 R E C O M M E N D A T I O N S F O R F U T U R E R E S E A R C H 7 . 2 . 1 T H E T H E O R Y The theory developed in this thesis assumes that the primary variables of the work package and revenue models, and the work packages and the revenue models themselves, are s t a t i s t i c a l l y independent. It has been emphasized throughout the thesis that this assumption underestimates the risks. A consistent approach to estimate the effect of correlation among the variables from available and obtainable information should be developed. This should be given the highest priority in the future work. The internal rate of return has been shown to be an important decision criterion in the probabilistic evaluation of the economic fe a s i b i l i t y of a project. The d i f f i c u l t y in obtaining s t a t i s t i c s for the internal rate of return and some of the possible approaches to developing a consistent method were discussed in section 6.3.4. Based on the recommendations in section 6.3.4, a consistent approach to 1 19 obtain the statistics of the internal rate of return should be developed. This is an essential improvement for the completion of this theory. The model at present uses a PERT approach to estimate the st a t i s t i c s of the early start time of a work package. A method based on the PNET algorithm and the Pearson curves is recommended for this estimation. It is proposed to combine the cumulative distribution function developed from the PNET algorithm with the Pearson curves to obtain statistics for the early start time. Such an approach would overcome some of the limitations of PERT as the PNET algorithm considers the correlation arising from shared work packages and multiple paths. The completion time probability function for the project duration could then be developed similarly by incorporating the PNET algorithm and the Pearson curves at the project level. 7.2.2 THE MODEL The model at present lacks sophistication especially in the area of user friendliness. Improvement in this area is necessary for the model to become a practical tool. Primarily i t should be interactive. Secondly i t should contain f a c i l i t i e s that makes i t easy to use. These f a c i l i t i e s should include default situations and the a b i l i t y to process overlapping networks with the work packages 1 20 having different precedence link relationships and lead times. Another important function with respect to the model is that i t should be validated. The validation could be by comparing the information obtained from the model with those from a detailed simulation. 7.2.3 INPUT DATA AND THE OUTPUT The theory at present assumes that the estimator can provide the estimates for the necessary percentage points of his subjective distribution for a variable. Therefore consistent sets of questions to obtain the estimates for the desired variables should be developed. The basis for the development of these sets of questions can be obtained from the available literature in the f i e l d of subjective probabilities. When providing answers to probability related questions i t is important to gauge the r e l i a b i l i t y of them. This r e l i a b i l i t y depends much on the accuracy of the input data and the analytical approach. Therefore, i t would be useful to study the r e l i a b i l i t y of the results obtained from the model. A study of"the amount of useful information that could be obtained at the project and intermediate levels and how i t would best be represented would help considerably in reducing information overload to management. The d i f f i c u l t y 121 of such a study is to generalise the needs of different clients with different objectives. Nevertheless i t s t i l l would give a basis for presenting information required by management for the strategic decisions that should be made at the feasibility stage. REFERENCES 01. ANG, A.H.S., ABDELNOUR, J . , and CHAKER, A.A., "ANALYSIS OF ACTIVITY NETWORKS UNDER UNCERTAINTY", JOURNAL OF THE ENGINEERING MECHANICS DIVISION, A.S.C.E., VOL 101, NO.EM4, AUGUST 1975, pp.373~387. 02. BENJAMIN, J.R., and CORNELL, C.A., PROBABILITY,  STATISTICS, AND DECISION FOR CIVIL ENGINEERS, McGRAW-HILL BOOK COMPANY, NEW YORK, 1970. 03. BEY, R.B., DOERSCH, R.H., and PATTERSON, J.H., "THE NET PRESENT VALUE CRITERION: ITS IMPACT ON PROJECT SCHEDULING", PROJECT MANAGEMENT QUARTERLY, JUNE 1981, pp.35-45. 04. BJORNSSON, H.C., "RISK ANALYSIS OF CONSTRUCTION ESTIMATES", TRANSACTIONS OF THE AMERICAN ASSOCIATION  OF COST ENGINEERS, MILWAUKEE, WISCONSIN, 1977, pp.182-189. 05. BURY, K.E., STATISTICAL MODELS IN APPLIED SCIENCE, JOHN WILEY & SONS, INC, NEW YORK, 1975. 06. CHESLEY, G.R., "ELICITATION OF SUBJECTIVE PROBABILITIES: A REVIEW", THE ACCOUNTING REVIEW, VOL 50, APRIL 1975, pp.325-337. 07. CRANDALL, K.C., "PROBABILISTIC TIME SCHEDULING", JOURNAL OF THE CONSTRUCTION DIVISION, A.S.C.E., VOL 102, NO.C03, SEPTEMBER 1976, pp.415-423. 1 22 08. CRANDALL, K.C., "ANALYSIS OF SCHEDULE SIMULATIONS", JOURNAL OF THE CONSTRUCTION DIVISION, A.S.C.E., VOL 103, NO.C03, SEPTEMBER 1977, pp.387-394. 09. CRANDALL, K.C. , and WOOLERY, J.C, "SCHEDULE DEVELOPMENT UNDER STOCHASTIC SCHEDULING", JOURNAL OF  THE CONSTRUCTION DIVISION, A.S.C.E., VOL 108, No.C02, JUNE 1982, pp 321-329. 10. DESHMUKH, S.S., "RISK ANALYSIS", TRANSACTIONS OF THE  AMERICAN ASSOCIATION OF COST ENGINEERS, BOSTON, MASSACHUSETTS, 1976, pp.118-121. 11. DIEKMANN, J.E., "PROBABILISTIC ESTIMATING: MATHEMATICS AND APPLICATIONS", JOURNAL OF  CONSTRUCTION ENGINEERING AND MANAGEMENT, A.S.C.E., VOL 109,No.3, SEPTEMBER 1983, pp.297-307. 12. DITLEVSEN, O., UNCERTAINTY MODELLING WITH  APPLICATIONS TO MULTIDIMENSIONAL CIVIL ENGINEERING  SYSTEMS, McGRAW - HILL INC., 1981 13. ELMAGHRABY, S.E., "PROBABILISTIC ACTIVITY NETWORKS (PANS) : A CRITICAL EVALUATION AND EXTENSION OF THE PERT MODEL", ACTIVITY NETWORKS: PROJECT PLANNING AND  CONTROL BY NETWORK MODELS, JOHN WILEY & SONS, 1977, pp.228-320. 14. GREEN, P.E., "CRITIQUE OF: RANKING PROCEDURES AND SUBJECTIVE PROBABILITY DISTRIBUTIONS", MANAGEMENT  SCIENCE , VOL 14, No.4, DECEMBER 1967, pp.B250-B252. 124 15. HAMPTON, J.M., MOORE, P.G., and THOMAS, H., "SUBJECTIVE PROBABILITY AND ITS MEASUREMENT", JOURNAL  OF THE ROYAL STATISTICAL SOCIETY ASSOCIATION, VOL 136, PART 1, 1973, pp.21-42. 16. HEMPHILL, R.B., "A METHOD FOR PREDICTING THE ACCURACY OF A CONSTRUCTION COST ESTIMATE", TRANSACTIONS OF THE  AMERICAN ASSOCIATION OF COST ENGINEERS, 1 2 t h NATIONAL MEETING, HOUSTON, TEXAS, 1968, pp.20-1to20-18 . 17. HILLIARD, J.E., and LEITCH, R.A., "COST-VOLUME -PROFIT ANALYSIS UNDER UNCERTAINTY: A LOG NORMAL APPROACH", THE ACCOUNTING REVIEW, JANUARY 1975, pp.69-80. 18. JOHNSON, N.L., "BIVARIATE DISTRIBUTIONS BASED ON SIMPLE TRANSLATION SYSTEMS", BIOMETRIKA, VOL 36, 1949, pp.297-304. 19. JOHNSON, N.L., NIXON, E., and AMOS, D.E., "TABLE OF PERCENTAGE POINTS OF PEARSON CURVES, FOR GIVEN */$ . AND 0 2 , EXPRESSED IN STANDARD MEASURE", BIOMETRIKA, VOL 50, 1963, pp.459-498. 20. KENNEDY, K.W., and THRALL, R.M., "PLANET: A SIMULATION APPROACH TO PERT", COMPUTER AND OPERATION  RESEARCH , VOL 3, 1976, pp.313-325. 21. KOTTAS, J.H., and LAU, H.S., "STOCHASTIC BREAKEVEN ANALYSIS", JOURNAL OF THE OPERATIONS RESEARCH  SOCIETY, VOL 29, 1978, pp.251-257. 22. KOTTAS, J.H., and LAU, H.S., "A REVIEW OF THE STATISTICAL FOUNDATIONS OF A CLASS OF PROBABILISTIC PLANNING MODELS", COMPUTER AND OPERATIONS RESEARCH, VOL 7, 1980, pp.227-284. 23. MOELLER, G.L., "VERT - A TOOL TO ASSESS RISK", TECHNICAL PAPERS, A.I.I.E., MAY-JUNE 1972, pp.211-222. 24. MORRISON, D.G., "CRITIQUE OF: RANKING PROCEDURES AND SUBJECTIVE PROBABILITY DISTRIBUTIONS", MANAGEMENT  SCIENCE, VOL 14, No.4, DECEMBER 1967, pp.B253-B254. 25. PEARSON, E.S., and TUKEY, J.W., "APPROXIMATE MEANS AND STANDARD DEVIATIONS BASED ON DISTANCES BETWEEN PERCENTAGE POINTS OF FREQUENCY CURVES", BIOMETRIKA, VOL 52, 1965, pp.533-546. 26. PRITSKER, A.A.B., and HAPP, W.W., "GERT : GRAPHICAL EVALUATION AND REVIEW TECHNIQUE, PART I: FUNDAMENTALS" JOURNAL OF INDUSTRIAL ENGINEERING, VOL 17, MAY 1966, pp.267-274. 27. PRITSKER, A.A.B., and WHITEHOUSE, G.E., "GERT : GRAPHICAL EVALUATION AND REVIEW TECHNIQUE, PART II: PROBABILISTIC AND INDUSTRIAL ENGINEERING APPLICATIONS", JOURNAL OF INDUSTRIAL ENGINEERING, VOL 17, JUNE 1966, pp.293-301. 28. RACKWITZ, R., and FIESSLER, B., "STRUCTURAL RELIABILITY UNDER COMBINED RANDOM LOAD SEQUENCES", COMPUTERS & STRUCTURES, VOL.9, 1978. 126 29. SAVAGE, L.J., "ELICITATION OF PERSONAL PROBABILITIES AND EXPECTATIONS", JOURNAL OF THE AMERICAN  STATISTICAL ASSOCIATION, VOL 66, No.336, DECEMBER 1971, pp.783-801. 30. SHAFER, S.L., "RISK ANALYSIS FOR CAPITAL PROJECTS USING RISK ELEMENTS", TRANSACTIONS OF THE AMERICAN  ASSOCIATION OF OF COST ENGINEERS, LOS ANGELES, CALIFORNIA, 1974, pp.218-223. 31. SMITH, L.H., "RANKING PROCEDURES AND SUBJECTIVE PROBABILITY DISTRIBUTIONS", MANAGEMENT SCIENCE, VOL 14, No.4, DECEMBER 1967, pp.B236-B249. 32. SPOONER, J.E., "PROBABILISTIC ESTIMATING", JOURNAL OF  THE CONSTRUCTION DIVISION, A.S.C.E., VOL 100, No.COl, MARCH 1974, pp.65-77. 33. TONG, T.Y., "PRECEDENCE NETWORK PROCESSOR", A PROJECT REPORT PRESENTED FOR M.Eng.(Building), CONCORDIA UNIVERSITY, MONTREAL, QUEBEC, JULY 1982. 34. VAN TETTERODE, L.M., "RISK ANALYSIS OR RUSSIAN ROULETTE?", TRANSACTIONS OF THE AMERICAN ASSOCIATION  OF COST ENGINEERS, MONTREAL, CANADA, 1971, pp.124-129. 35. WINKLER, R.L., "THE ASSESSMENT OF PRIOR DISTRIBUTIONS IN BAYESIAN ANALYSIS", JOURNAL OF THE AMERICAN  STATISTICAL ASSOCIATION, VOL 62, 1967, pp.776-800. 1 27 36. WINKLER, R . L . , "THE QUANTIFICATION OF JUDGEMENT : SOME METHODOLOGICAL SUGGESTIONS", JOURNAL OF THE  AMERICAN STATISTICAL ASSOCIATION, VOL 62, 1967, pp.1105-1120 . 37. WINKLER, R . L . , "THE CONSENSUS OF SUBJECTIVE PROBABILITY DISTRIBUTIONS", MANAGEMENT SCIENCE, VOL 15, N o . 2 , OCTOBER 1968, p p . B 6 l ~ B 7 5 . APPENDIX A EXAMPLE 1 : MODEL BEHAVIOUR 1 28 1 29 APPENDIX A-1 SKEWNESS OF PRIMARY VARIABLES < 0.1 CORRELATION OF PRIMARY VARIABLES = 0.07 130 • W.P 0 • 1 • W.P 0 • 2 • W.P # • 3 • W.P 0 • 4 • W.P * • 5 • W.P 0 • 6 • W.P * • 7 • W.P » • a • W.P 0 • 9 • W.P 0 • 10 • W.P 0 • 1 • W.P 0 • 2 • W.P 0 * 3 • W.P * * 4 * • EXPECTEO VALUE * • 0 .0 » • EXPECTED VALUE • * 0.5370 • • EXPECTED VALUE • • 0.5370 EXPECTED VALUE • • 0.5370 • • EXPECTED VALUE • * 0.5370 * « EXPECTED VALUE 0.5370 * • EXPECTED VALUE 0.5370 * * EXPECTED VALUE 0.5370 * * EXPECTED V A L U E - * * 0.5370 • * EXPECTED VALUE « * 1.0000 * • • EXPECTED VALUE *« 0 .0 * • EXPECTED VALUE * • 0 .0 * • EXPECTED VALUE « • 0 .0 • * EXPECTED VALUE • • 0 .0 WORK PACKAGE DURATION MODEL « « « • • • * * STANDARD DEVIATION • • • SKEWNESS * • * 0 .0 0 .0 • * STANDARD DEVIATION * • * * * SKEWNESS • * • 0.0G05 -0.02 «• STANDARD DEVIATION *« * • « SKEWNESS * * * 0.0605 0.060S 0.0605 * * STANDARD DEVIATION * • 0.0605 * • STANDARD DEVIATION • * 0.0605 0.0605 0.0 -0.02 * • STANDARD DEVIATION • * * • * SKEWNESS • • * -0.02 * • STANDARD DEVIATION • * *«« SKEWNESS * * • -0.02 SKEWNESS * * • -0 .02 SKEWNESS * * * -0.02 * * STANDARD DEVIATION *« * * * SKEWNESS • * * -0 .02 •« STANOARD DEVIATION • * * * • SKEWNESS • * * 0.0605 * • STANOARD OEVIATION • * • * « SKEWNESS * * * 0.0 STATISTICS OF EARLY START TIME OF A WORK PACKAGE • * STANDARD DEVIATION • * * * * SKEWNESS • • • 0 .0 0 .0 • * STANDARD DEVIATION • * • SKEWNESS • • • 0 .0 0.0 • • STANDARD DEVIATION • * * • * SKEWNESS * * * 0 .0 0.0 • * STANDARD DEVIATION * • * SKEWNESS * • * 0 .0 0.0 • • COEFF' OF VARIATION • • 0 .0 * * COEFF OF VARIATION * * O. 1 1 * • COEFF OF VARIATION • • 0. 11 • « COEFF OF VARIATION » • O. 11 • * COEFF OF VARIATION * * O. 11 * * COEFF OF VARIATION • • O. 11 * * COEFF OF VARIATION • • • O. 1 1 • * COEFF OF VARIATION * * 0.11 »• COEFF OF VARIATION •*•• O. 1 1 • • COEFF OF VARIATION * • 0 .0 * • COEFF OF VARIATION • • 0 .0 * * COEFF OF VARIATION « * O.O COEFF OF VARIATION «* 0 .0 * * COEFF OF VARIATION *« 0 .0 131 • w .p 0 • • • EXPECTED VALUE STANDARD OEVIATION « • • SKEWNESS ••• • • COEFF OF VARIATION • • s 0.5370 0.0605 -0.02 0. ,11 • V .p 0 • EXPECTED VALUE *• STANDARD DEVIATION ... SKEWNESS • • COEFF OF VARIATION ** 6 0.5370 0.0605 -0.02 0. . 1 1 • w .P 0 * • • EXPECTED VALUE ** • • STANDARD DEVIATION * * * SKEWNESS ••* * • COEFF OF VARIATION • * 7 0.5370 0.0605 -0.02 0. , 1 1 • w .P 0 • • * EXPECTED VALUE STANDARD DEVIATION •* • •* SKEWNESS *** • « COEFF OF VARIATION • • 8 1.0741 0.0856 -0.01 0. .08 • w .P 0 • * * EXPECTED VALUE * • STANDARD DEVIATION •* * » * SKEWNESS ••• »* COEFF OF VARIATION * * 9 1.0741 0.0856 -0.01 0 .08 * w .P 0 • EXPECTED VALUE STANDARD DEVIATION ... SKEWNESS ••* .. COEFF OF VARIATION • * 10 1.6111 0.1048 -0.01 0 .07 •*•••* PROJECT DURATION MODEL »••••• ........ PROJECT DURATION ........ 1.61 1 14 **• FIRST FOUR MOMENTS OF PROJECT DURATION ••• 1.61114 0.01099 -0.00001 0.00011 «... CURRENT S$ WORK PACKAGE COST MODEL *•** * W.P * * EXPECTED VALUE ** *• STANDARD DEVIATION •* SKEWNESS •** *• COEFF OF VARIATION •* 2 1046915. 145704. 0.00 0.14 * W.P 0 • ** EXPECTED VALUE *• STANDARD DEVIATION •* *•« SKEWNESS ••• COEFF OF VARIATION 3 1046915. 145704. 0.00 0.14 * W.P 0 • •• EXPECTED VALUE •• *• STANDARD DEVIATION **• SKEWNESS ••• COEFF OF VARIATION 4 1046915. 145704. 0.00 0.14 * W.P 0 * •* EXPECTED VALUE *• *• STANDARD DEVIATION *• ••* SKEWNESS ••• ** COEFF OF VARIATION »• 5 1031561. 143633. 0.00 0.14 132 * W.P # • E X P E C T E D VALUE 6 1 0 3 1 5 6 1 . * W.P » • •• E X P E C T E D VALUE •• 7 1 0 3 1 5 6 1 . * W.P 0 * •* E X P E C T E D V A L U E ** 8 1 0 1 7 7 1 9 . * W.P # • E X P E C T E D VALUE •* 9 1 0 1 7 7 1 9 . * W.P # * *• E X P E C T E D VALUE »• 10 0 . ** STANDARD D E V I A T I O N 1 4 3 6 3 3 . •• STANOARD D E V I A T I O N 1 4 3 6 3 3 . STANDARD D E V I A T I O N ** 1 4 1 9 4 8 . STANDARD D E V I A T I O N *• 1 4 1 9 4 8 . »* STANDARD D E V I A T I O N ** O. .••• SKEWNESS ••• 0 . 0 0 SKEWNESS •** 0 . 0 0 •** SKEWNESS **• 0 . 0 0 ••• SKEWNESS *•* 0 . 0 0 **• SKEWNESS •** 0 . 0 COEFF OF V A R I A T I O N 0 . 14 •• COEFF OF V A R I A T I O N ** 0 . 14 •• COEFF OF V A R I A T I O N 0 . 14 COEFF OF V A R I A T I O N 0 . 14 ** COEFF OF V A R I A T I O N *• 0 . 0 CURRENT St PROJECT COST MODEL ** EXPECTED V A L U E »• * STANDARD DEVIATION * *** SKEWNESS *•• *•• KURTOSIS *•* 8 2 7 0 8 6 4 . 4 0 7 2 8 3 . 0 . 0 0 1 2 . 9 8 1 CURRENT S$ NET REVENUE MODEL •••••• R . S » • •* E X P E C T E D VALUE •* ** STANDARD D E V I A T I O N *• ••• SKEWNESS •** ** COEFF OF V A R I A T I O N 1 1 7 5 6 0 3 8 4 . 5 6 8 8 1 8 7 . 0 . 1 4 0 . 3 2 • » • • • CURRENT $t PROJECT REVENUE MODEL *•** E X P E C T E D VALUE *• • STANDARD DEVIATION • SKEWNESS *** •*• KURTOSIS ••• 1 7 5 6 0 3 8 4 . 5 6 8 8 1 8 8 . 0 . 1 4 1 4 . 3 6 3 NET PRESENT VALUE MODEL E X P E C T E D VALUE •* • STANDARD DEVIATION • 9 2 8 9 5 2 0 . 5 7 0 2 7 5 0 . SKEWNESS 0 . 140 ••• KURTOSIS ••• 4 . 3 4 9 133 APPENDIX A-2 SKEWNESS OF PRIMARY VARIABLES < 0.1 CORRELATION OF PRIMARY VARIABLES = 0.15 134 •••• CURRENT tt WORK PACKAGE COST MODEL •••• EXPECTED VALUE •• 1 0 4 7 0 8 2 . EXPECTED VALUE •• 1 0 4 7 0 8 2 . *• EXPECTED VALUE •* 1 0 4 7 0 8 2 . •* EXPECTED VALUE 1 0 3 1 7 5 4 . *• EXPECTED VALUE 1 0 3 1 7 5 4 . *• EXPECTED VALUE ** 1 0 3 1 7 5 4 . *• EXPECTED VALUE •* 1 0 1 8 0 7 3 . EXPECTED VALUE •• 1 0 1 8 0 7 3 . ** E X P E C T E D VALUE •* 0 . STANDARD D E V I A T I O N ••• SKEWNESS *•* 3 3 2 6 0 8 . 3 3 2 6 0 8 . 3 3 2 6 0 8 . 3 2 7 9 1 7 . 3 2 7 9 1 7 . ** STANDARD D E V I A T I O N •« 3 2 7 9 1 7 . 3 2 4 0 9 9 . 3 2 4 0 9 9 . 0 . 0 0 «• STANOARO D E V I A T I O N ••• SKEWNESS ••« 0 . 0 0 STANDARD D E V I A T I O N •*• SKEWNESS ••* 0 . 0 0 *« STANOARD D E V I A T I O N •* SKEWNESS ••• 0 . 0 0 STANDARD D E V I A T I O N *« • • ' SKEWNESS «•• 0 . 0 0 SKEWNESS *•• 0 . 0 0 »« STANDARD D E V I A T I O N ••* SKEWNESS ••• 0 . 0 0 STANDARD D E V I A T I O N **» SKEWNESS ••• 0 . 0 0 STANOARD D E V I A T I O N •» ••* SKEWNESS •*« 0 . 0 •* COEFF OF V A R I A T I O N 0 . 3 2 COEFF OF V A R I A T I O N 0 . 3 2 •« COEFF OF V A R I A T I O N •* 0 . 3 2 •« COEFF OF V A R I A T I O N •• 0 . 3 2 ** COEFF OF V A R I A T I O N •• 0 . 3 2 *« C O E F F OF V A R I A T I O N «• 0 . 3 2 ** COEFF OF V A R I A T I O N *• 0 . 3 2 ** COEFF OF V A R I A T I O N -« 0 . 3 2 ** COEFF OF V A R I A T I O N *• 0 . 0 CURRENT tt PROJECT COST MODEL EXPECTED VALUE • STANDARD D E V I A T I O N * 8 2 7 2 6 5 1 . 9 2 9 8 1 4 . «*• SKEWNESS ••• 0 . 0 0 2 '•• KURTOSIS *•* 2 . 9 7 9 CURRENT tt NET REVENUE MODEL EXPECTED VALUE •* 1 7 5 6 0 1 9 2 . STANDARD D E V I A T I O N 5 6 9 3 0 6 2 . ••• SKEWNESS ••• 0 . 14 COEFF OF V A R I A T I O N 0 . 3 2 . . . . . CURRENT tt PROJECT REVENUE MODEL «*•• EXPECTED VALUE •• 1 7 5 6 0 1 9 2 . • STANDARD D E V I A T I O N • 5 6 9 3 0 6 3 . ••• SKEWNESS ••• O. 140 KURTOSIS ••• 4 . 3 5 8 NET PRESENT VALUE MODEL EXPECTED VALUE 9 2 8 7 5 4 1 . STANDARD D E V I A T I O N • 5 7 6 8 4 9 3 . ••• SKEWNESS ••• 0 . 135 KURTOSIS ••* 4 . 2 8 9 1 35 APPENDIX A-3 SKEWNESS OF PRIMARY VARIABLES < 0.1 CORRELATION OF PRIMARY VARIABLES = 0.24 1 36 •**• CURRENT t t WORK PACKAGE COST MODEL •••• EXPECTED VALUE •* 1 0 4 7 3 4 1 . »• EXPECTED VALUE 1 0 4 7 3 4 1 . *• EXPECTED VALUE •* 104734 1. ** EXPECTED VALUE •• 1 0 3 1 8 2 9 . EXPECTED VALUE ** 1 0 3 1 8 2 9 . •* EXPECTED VALUE •* 1 0 3 1 8 2 9 . «« EXPECTED VALUE ** 1 0 1 8 2 5 1 . •• EXPECTED VALUE •• 1 0 1 8 2 5 1 . *• EXPECTED VALUE *• 0 . STANDARD D E V I A T I O N •• ••• SKEWNESS ••* S 2 6 0 0 9 . STANDARD D E V I A T I O N •• 5 2 6 0 0 9 . 5 2 6 0 0 9 . 5 1 8 5 6 2 . •• STANOARD D E V I A T I O N 5 1 8 5 6 2 . ** STANDARD D E V I A T I O N •• 5 1 8 5 6 2 . 5 1 2 5 1 1 . »* STANDARD D E V I A T I O N *• 5 1 2 5 1 1 . 0 . 0 0 SKEWNESS 0 . 0 0 *• STANDARD D E V I A T I O N SKEWNESS •*• 0 . 0 0 •* STANDARD D E V I A T I O N •* *•• SKEWNESS «** 0 . 0 0 SKEWNESS 0 . 0 0 SKEWNESS 0.00 STANOARD O E V I A T I O N «* *•* SKEWNESS *** 0 . 0 0 SKEWNESS 0.00 «» STANDARD O E V I A T I O N •»» SKEWNESS •** 0 . 0 . 0 •• COEFF OF V A R I A T I O N *• 0 . 5 0 •* COEFF OF V A R I A T I O N •• 0 . 5 0 •• COEFF OF V A R I A T I O N *• 0 . 5 0 •• COEFF OF V A R I A T I O N •• 0 . 5 0 •• COEFF OF V A R I A T I O N *• 0 . 5 0 *• COEFF OF V A R I A T I O N 0 . 5 0 COEFF OF V A R I A T I O N •* 0 . 5 0 COEFF OF V A R I A T I O N 0 . 5 0 *• COEFF OF V A R I A T I O N ** 0 . 0 CURRENT t t PROJECT COST MODEL EXPECTED VALUE *- • STANDARD D E V I A T I O N • 8 2 7 4 0 0 7 . 1 4 7 0 4 1 1 . SKEWNESS *•• 0 . 0 0 1 KURTOSIS *** 2 . 9 8 1 CURRENT t t NET REVENUE MODEL •»•••• •« EXPECTED VALUE •* STANDARO D E V I A T I O N ••• SKEWNESS ••• 1 7 5 5 8 8 8 0 . 5 7 0 1 6 7 1 . 0 . 1 4 COEFF OF V A R I A T I O N *• 0 . 3 2 CURRENT t t PROJECT REVENUE MODEL •••• EXPECTED VALUE ** • STANOARD D E V I A T I O N * 1 7 5 5 8 8 8 0 . 5 7 0 1 6 7 2 . SKEWNESS ••• 0 . 140 KURTOSIS **• 4 . 3 4 9 NET PRESENT VALUE MODEL EXPECTED VALUE •• • STANDARO DEVIATION • 9 2 8 4 8 7 3 . 5 8 8 8 2 2 2 . ••• SKEWNESS **• 0 . 127 •• KURTOSIS 4 . 186 137 APPENDIX A-4 SKEWNESS OF PRIMARY VARIABLES < 0.1 CORRELATION OF PRIMARY VARIABLES = 0.34 138 •••• CURRENT tt WORK PACKAGE COST MODEL •••• W.P # • •• EXPECTED VALUE *• STANDARD DEVIATION •*• SKEWNESS *•• 2 1 0 6 1 1 0 0 . 7 7 4 4 4 2 . 0 . 0 0 W.P * • •* EXPECTED VALUE •• •* STANDARD D E V I A T I O N ** *•* SKEWNESS *•• 3 1 0 6 1 1 0 0 . 7 7 4 4 4 2 . 0 . 0 0 W.P * • •* EXPECTED VALUE •* •* STANDARD D E V I A T I O N *•• SKEWNESS *** 4 1 0 6 1 1 0 0 . 7 7 4 4 4 2 . 0 . 0 0 W.P * • •• EXPECTED VALUE •* *• STANDARD O E V I A T I O N *• ••• SKEWNESS 5 1 0 4 5 1 0 0 . 7 6 3 4 8 5 . 0 . 0 0 W.P » ' " E X P E C T E D VALUE •* STANDARD D E V I A T I O N ••• SKEWNESS ••• 6 1 0 4 5 1 0 0 . 7 6 3 4 8 5 . 0 . 0 0 W.P H • EXPECTED VALUE ** STANDARD O E V I A T I O N •* SKEWNESS **• 7 1 0 4 5 1 0 0 . 7 6 3 4 8 5 . 0 . 0 0 W.P * • EXPECTED VALUE •• STANDARD D E V I A T I O N *» ••* SKEWNESS ••* 8 1 0 3 1 5 6 1 . 7 5 4 5 7 9 . 0 . 0 0 W . P * • *• EXPECTED VALUE •* *• STANOARD D E V I A T I O N **• SKEWNESS **• 9 1 0 3 1 5 6 1 . 7 5 4 5 7 9 . 0 . 0 0 W.P # • ** EXPECTED VALUE »• STANDARD D E V I A T I O N ** SKEWNESS ••* 10 O . 0 . 0 . 0 . . . . . . CURRENT $$ PROJECT COST MODEL •••*•• •* EXPECTED VALUE ** * STANDARD D E V I A T I O N * ••• SKEWNESS ••• 8 3 8 1 7 1 7 . 2 1 6 4 8 9 9 . 0 . 0 0 2 . . . . . . CURRENT »$ NET REVENUE MODEL »•••»• R . S H ' E X P E C T E D VALUE *• STANDARD D E V I A T I O N ** ••• SKEWNESS *** 1 1 7 5 5 1 9 2 0 . 5 7 1 5 3 3 2 . 0 . 1 4 . . . . . CURRENT tt PROJECT REVENUE MODEL ••*• •• EXPECTED VALUE •« • STANDARD D E V I A T I O N • •«* SKEWNESS •*• 1 7 5 5 1 9 2 0 . 5 7 1 5 3 3 2 . 0 . 1 3 8 • NET PRESENT VALUE MODEL *• EXPECTED VALUE * STANDARD D E V I A T I O N • •*• SKEWNESS •*» 9 1 7 0 2 0 3 . 6 1 1 1 6 1 2 . 0 . 1 1 3 COEFF OF VARIATI ON 0 . 7 3 •• COEFF OF VARIATI ON •• 0 . 7 3 ** COEFF OF V A R I A T I O N 0 . 7 3 COEFF OF VARIATION *• 0 . 7 3 COEFF OF VARIATI ON 0 . 7 3 COEFF OF V A R I A T I O N 0 . 7 3 *• COEFF OF V A R I A T I O N •« 0 . 7 3 •• COEFF OF V A R I A T I O N *• 0 . 7 3 COEFF OF V A R I A T I O N •* 0 . 0 KURTOSIS *"* 2 . 9 7 9 COEFF OF VARIATI ON -« 0 . 3 3 •• KURTOSIS 4 . 333 •• KURTOSIS 4 . 0 1 9 1 39 APPENDIX A-5 SKEWNESS OF PRIMARY VARIABLES < 0.1 CORRELATION OF PRIMARY VARIABLES = 0.42 140 *• CURRENT tt WORK PACKAGE COST MODEL E X P E C T E D VALUE •• 1 0 5 1 4 8 1 . *• E X P E C T E D VALUE 1 0 5 1 4 8 1 . ** E X P E C T E D VALUE ** 1 0 5 1 4 8 1 . •* E X P E C T E D VALUE •* 1 0 3 5 3 1 9 . E X P E C T E D VALUE 1 0 3 5 3 1 9 . E X P E C T E D VALUE 1 0 3 5 3 1 9 . •* EXPECTED VALUE •» 1 0 2 2 0 4 3 . E X P E C T E D V A L U E *• 1 0 2 2 0 4 3 . ** E X P E C T E D VALUE •• 0 . STANDARD DEVIATION •* 9 4 0 6 7 4 . •» STANDARD DEVIATION 9 4 0 6 7 4 . ** STANDARD DEVIATION •* 9 4 0 6 7 4 . *• STANDARD DEVIATION •• 9 2 7 3 0 1 . ** STANDARD D E V I A T I O N 9 2 7 3 0 1 . STANDARD D E V I A T I O N 9 2 7 3 0 1 . •• STANDARD DEVIATION 9 1 6 3 9 0 . *• STANDARD DEVIATION •• 9 1 6 3 9 0 . •* STANDARD DEVIATION •* 0 . SKEWNESS ••• 0 . 0 0 SKEWNESS ••• 0 . 0 0 SKEWNESS •*• 0 . 0 0 SKEWNESS ••• 0 . 0 0 SKEWNESS **• 0 . 0 0 SKEWNESS ••• 0 . 0 0 SKEWNESS •*• 0 . 0 0 SKEWNESS **• 0 . 0 0 SKEWNESS 0 . 0 •• COEFF OF V A R I A T I O N •* 0 . 8 9 •• COEFF OF V A R I A T I O N 0 . 8 9 ** COEFF OF V A R I A T I O N •* 0 . 8 9 COEFF OF V A R I A T I O N *• 0 . 9 0 COEFF OF V A R I A T I O N »• 0 . 9 0 COEFF OF V A R I A T I O N 0 . 9 0 *• COEFF OF V A R I A T I O N 0 . 9 0 *• COEFF OF V A R I A T I O N 0 . 9 0 *• COEFF OF V A R I A T I O N 0 . 0 CURRENT tt PROJECT COST MODEL E X P E C T E D VALUE • STANDARD OEVIATION • 8 3 0 4 4 8 1 . 2 6 2 9 4 1 1 . SKEWNESS *•* 0 . 0 0 2 •*• KURTOSIS *** 2 . 9 7 9 CURRENT St NET REVENUE MODEL »• E X P E C T E D VALUE *• •* STANDARD DEVIATION ••• SKEWNESS *•• 1 7 5 4 7 4 7 2 . 5 7 2 8 9 8 3 . 0 . 1 4 CURRENT tt PROJECT REVENUE MODEL *••• •* COEFF OF V A R I A T I O N •• 0 . 3 3 E X P E C T E D VALUE *• • STANDARD OEVIATION • 1 7 5 4 7 4 7 2 . 5 7 2 8 9 8 3 . >•• SKEWNESS 0 . 1 3 7 KURTOSIS *** 4 . 3 1 B NET PRESENT VALUE MODEL E X P E C T E D VALUE ** • STANDARD DEVIATION • 9 2 4 2 9 9 1 . 6303574 ••• SKEWNESS ••* 0 103 ••- KURTOSIS *** 3 898 141 APPENDIX A-6 SKEWNESS OF PRIMARY VARIABLES > 1.6 CORRELATION OF PRIMARY VARIABLES = 0.07 142 CURRENT $$ WORK PACKAGE COST MODEL •••• E X P E C T E D VALUE •• 1 0 7 6 6 1 0 . ** E X P E C T E D VALUE ** 1 0 7 8 6 1 0 . E X P E C T E D VALUE 1 0 7 8 6 1 0 . *• E X P E C T E D VALUE •• 1 0 6 2 8 9 9 . ** E X P E C T E D VALUE »* 1 0 6 2 8 9 9 . *• E X P E C T E D VALUE »• 1 0 6 2 8 9 9 . E X P E C T E D VALUE •* 1 0 4 8 7 4 4 . »* E X P E C T E D VALUE ** 1 0 4 8 7 4 4 . *• E X P E C T E D VALUE ** 0 . •• STANDARD O E V I A T I O N ••• SKEWNESS ••• 1 4 9 4 1 4 . STANDARD D E V I A T I O N •• 1 4 9 4 1 4 . 1 4 9 4 1 4 . •• STANDARD D E V I A T I O N 147292 . •• STANDARD D E V I A T I O N •* 147292 . 1 4 7 2 9 2 . 1 4 5 5 7 5 . STANDARO D E V I A T I O N •• 1 4 5 5 7 5 . 0 . 0 7 SKEWNESS 0 . 0 7 •• STANDARD D E V I A T I O N ••* SKEWNESS *** 0 . 0 7 ••* SKEWNESS **• 0 . 0 7 *•* SKEWNESS *** 0 . 0 7 •• STANDARD D E V I A T I O N •• •*• SKEWNESS *** 0 . 0 7 ** STANDARD D E V I A T I O N •* •*• SKEWNESS •** 0 . 0 7 SKEWNESS *•• 0 . 0 7 STANDARD D E V I A T I O N »• *•* SKEWNESS •** 0 . 0 • • • C O E F F OF V A R I A T I O N •• 0 . 14 •• COEFF OF V A R I A T I O N 0 . 14 *• COEFF OF V A R I A T I O N 0 . 14 •• COEFF OF V A R I A T I O N •• 0 . 1 4 •• COEFF OF V A R I A T I O N 0 . 14 *• COEFF OF V A R I A T I O N •• O . 14 •• COEFF OF V A R I A T I O N •• 0 . 14 •• COEFF OF V A R I A T I O N 0 . 14 •• COEFF OF V A R I A T I O N 0 . 0 CURRENT S* PROJECT COST MODEL E X P E C T E D VALUE *• • STANDARD D E V I A T I O N ' 8 5 2 2 0 1 5 . 4 1 7 6 6 3 . ••• SKEWNESS ••• 0 . 0 2 5 «•• KURTOSIS ••* 3 . 0 6 2 CURRENT tt NET REVENUE MODEL E X P E C T E D VALUE •• 1 7 5 3 6 3 0 4 . •• STANDARD D E V I A T I O N *• 5 6 7 8 4 4 4 . *** SKEWNESS ••• 0 . 14 CURRENT St PROJECT REVENUE MODEL •••• COEFF OF V A R I A T I O N •» 0 . 3 2 E X P E C T E D VALUE •• • STANDARD D E V I A T I O N • 1 7 5 3 6 3 0 4 . 5 6 7 8 4 4 5 . ••• SKEWNESS ••• 0 . 141 NET PRESENT VALUE MODEL E X P E C T E D VALUE •• • STANDARD D E V I A T I O N • 9 0 1 4 2 8 9 . 5 6 9 3 7 8 3 . ••• SKEWNESS ••• 0 . 140 ••• KURTOSIS •** 4 . 3 6 5 ••• KURTOSIS •*• 4 . 3 5 0 1 43 APPENDIX A-7 SKEWNESS OF PRIMARY V A R I A B L E S Z 1.6 CORRELATION OF PRIMARY V A R I A B L E S = 0.16 144 CURRENT J * WORK PACKAGE COST MOOEL •••• •• EXPECTED VALUE •* t ( 0 4 3 6 3 . *• EXPECTED VALUE •* 1 1 0 4 3 6 3 . EXPECTED VALUE 1 1 0 4 3 6 3 . EXPECTED VALUE *• 1 0 8 8 2 1 9 . ** EXPECTED VALUE •« 1 0 8 8 2 1 9 . EXPECTED VALUE ** 1 0 8 8 2 1 9 . *• EXPECTED VALUE *• 1 0 7 3 8 4 0 . •• EXPECTED VALUE 1 0 7 3 8 4 0 . ** E X P E C T E D VALUE •* 0 . *• STANDARD D E V I A T I O N ••• SKEWNESS 3 7 4 4 S 1 . STANDARD D E V I A T I O N •• SKEWNESS 3 7 4 4 5 1 . STANDARD O E V I A T I O N •• 3 7 4 4 5 1 . 3 6 9 1 2 2 . 3 6 9 1 2 2 . •« STANOARD D E V I A T I O N *• 3 6 9 1 2 2 . 3 6 4 8 2 0 . "* STANOARD O E V I A T I O N 3 6 4 8 2 0 . 0 . 0 7 SKEWNESS *• STANDARD D E V I A T I O N •• «•• SKEWNESS ••• 0 . 0 7 •* STANDARD D E V I A T I O N •*• SKEWNESS *** 0 . 0 7 SKEWNESS 0 . 0 7 STANDARD D E V I A T I O N •* *•• SKEWNESS SKEWNESS 0 . 0 7 STANDARD D E V I A T I O N *• ••* SKEWNESS ••• 0 . 0 . 0 ** COEFF OF V A R I A T I O N *• 0 . 3 4 *• COEFF OF V A R I A T I O N *• 0 . 3 4 *• COEFF OF V A R I A T I O N •• 0 . 3 4 *• COEFF OF V A R I A T I O N •• 0 . 3 4 *• COEFF OF V A R I A T I O N •«. 0 . 3 4 COEFF OF V A R I A T I O N *• 0 . 3 4 ** COEFF OF V A R I A T I O N 0 . 3 4 •* C O E F F OF V A R I A T I O N 0 . 3 4 COEFF OF V A R I A T I O N *• 0 . 0 CURRENT »$ PROJECT COST MODEL EXPECTED VALUE ** * STANDARD O E V I A T I O N • 8 7 2 5 4 2 6 . 1 0 4 6 7 0 0 . •** SKEWNESS *** 0 . 0 2 5 CURRENT $$ NET REVENUE MODEL •* EXPECTED VALUE •* STANDARD D E V I A T I O N ••* SKEWNESS ••* 1 7 5 1 9 0 2 4 . 5 6 7 6 8 2 5 . 0 . 1 4 . . . . . CURRENT %% PROJECT REVENUE MODEL •*•• EXPECTED VALUE *• * STANDARD D E V I A T I O N • 1 7 5 1 9 0 2 4 . 5 6 7 6 8 2 5 . •*• SKEWNESS ••• 0 . 1 4 0 NET PRESENT VALUE MODEL EXPECTED VALUE ** • STANDARD D E V I A T I O N • 8 7 9 3 5 9 8 . 5 7 7 2 5 1 4 . SKEWNESS 0 . 1 3 3 KURTOSIS *"* 3 . 113 COEFF OF V A R I A T I O N •• 0 . 3 2 ""* KURTOSIS 4 . 3 6 0 •« KURTOSIS •** 4 . 2 7 2 1 45 APPENDIX A-8 SKEWNESS OF PRIMARY VARIABLES > 1.6 CORRELATION OF PRIMARY VARIABLES = 0.24 146 CURRENT-$$<» WORK PACKAGE COST MODEL E X P E C T E D VALUE 1 2 3 2 0 7 7 . •• EXPECTED VALUE «• 1 2 3 2 0 7 7 . EXPECTED VALUE 1 2 3 2 0 7 7 . E X P E C T E D VALUE •• 1 2 1 4 3 7 2 . •* E X P E C T E D VALUE ** 1 2 1 4 3 7 2 . •« E X P E C T E D VALUE ** 1 2 1 4 3 7 2 . E X P E C T E D V A L U E *• 1 1 9 8 9 1 1 . EXPECTED VALUE •* 1 1 9 8 9 1 1 . •* E X P E C T E D VALUE *« O . STANDARO D E V I A T I O N ••" SKEWNESS *•• 6 3 0 4 8 2 . 6 3 0 4 8 2 . 6 3 0 4 8 2 . •« STANDARO O E V I A T I O N •* 6 2 1 7 2 2 . STANDARO O E V I A T I O N •• 6 2 1 7 2 2 . 6 2 1 7 2 2 . 6 1 4 6 9 1 . STANDARD D E V I A T I O N *• 6 1 4 6 9 1 . 0 . 0 8 STANOARD D E V I A T I O N •• ••• SKEWNESS •*• 0 . 0 8 ** STANDARD D E V I A T I O N •• •*• SKEWNESS ••• 0 . 0 8 SKEWNESS SKEWNESS 0 . 0 8 «• STANDARD D E V I A T I O N •* ••• SKEWNESS ••« 0 . 0 8 ** STANDARD D E V I A T I O N SKEWNESS ••• 0 . 0 9 SKEWNESS «* STANDARD D E V I A T I O N ••* SKEWNESS *** O. 0 . 0 COEFF OF V A R I A T I O N •• 0 . S 1 COEFF OF V A R I A T I O N 0 . 5 1 *• COEFF OF V A R I A T I O N 0 . 5 1 COEFF OF V A R I A T I O N 0 . 5 1 •« COEFF OF V A R I A T I O N •• 0 . 5 1 •* COEFF OF V A R I A T I O N 0 . 5 1 COEFF OF V A R I A T I O N •* 0 . 5 1 •« COEFF OF V A R I A T I O N •• 0 . 5 1 •* COEFF OF V A R I A T I O N •* 0 . 0 CURRENT SS PROJECT COST MODEL EXPECTED V A L U E • STANDARD D E V I A T I O N • 9 7 3 7 1 6 9 . 1 7 6 2 9 0 3 . *•• SKEWNESS *** 0 . 0 3 0 KURTOSIS 3 . 0 4 6 . . . . . . CURRENT $$ NET REVENUE MODEL E X P E C T E D VALUE ** STANDARO O E V I A T I O N «*• SKEWNESS ••* 1 7 4 3 4 9 7 6 . 5 6 4 9 0 5 2 . 0 . 1 4 CURRENT SS PROJECT REVENUE MODEL •*** COEFF OF V A R I A T I O N 0 . 3 2 EXPECTED VALUE *• • STANDARD D E V I A T I O N • 1 7 4 3 4 9 7 6 . 5 6 4 9 0 5 2 . '*• SKEWNESS ••• 0 . 140 KURTOSIS *•* 4 . 3 5 9 NET PRESENT VALUE MODEL EXPECTED VALUE •* • STANDARD O E V I A T I O N • 7 6 9 7 8 0 7 . 5 9 1 7 7 3 6 . ••* SKEWNESS *•• 0 . 121 KURTOSIS •** 4 . 129 147 APPENDIX A-9 SKEWNESS OF PRIMARY VARIABLES > 1.6 CORRELATION OF PRIMARY VARIABLES = 0.33 1 4 8 CURRENT t$ WORK PACKAGE COST MODEL •••• •* E X P E C T E D VALUE •« 1 3 0 9 6 0 8 . EXPECTED VALUE 1 3 0 9 6 0 8 . •• EXPECTED VALUE •* 1 3 0 9 6 0 8 . *• EXPECTED VALUE 1 2 9 0 6 7 3 . •• EXPECTED VALUE *• 1 2 9 0 6 7 3 . EXPECTED VALUE 1 2 9 0 6 7 3 . ** E X P E C T E D VALUE •« 1 2 7 4 6 5 2 . EXPECTED VALUE *• 1 2 7 4 6 5 2 . ** EXPECTED V A L U E 0 . STANDARD D E V I A T I O N 9 3 5 3 9 2 . ** STANDARD O E V I A T I O N •• 9 3 5 3 9 2 . ** STANDARD O E V I A T I O N «• 9 3 5 3 9 2 . •• STANDARD O E V I A T I O N •• 9 2 2 4 3 8 . STANDARD D E V I A T I O N 9 2 2 4 3 8 . •* STANDARD D E V I A T I O N *• 9 2 2 4 3 8 . •* STANDARD D E V I A T I O N *• 9 1 2 1 0 9 . *• STANDARD D E V I A T I O N 9 1 2 1 0 9 . •* STANDARD D E V I A T I O N •* O . ••• SKEWNESS ••• 0 . 0 9 ••• SKEWNESS ••• 0 . 0 9 SKEWNESS •** 0 . 0 9 ••• SKEWNESS 0 . 0 9 **• SKEWNESS *** 0 . 0 9 «*• SKEWNESS •*• 0 . 0 9 •*• SKEWNESS ••• 0 . 0 9 *•* SKEWNESS ••• 0 . 0 9 *•• SKEWNESS ••• 0 . 0 *• COEFF OF V A R I A T I O N •* 0 . 7 1 C O E F F OF V A R I A T I O N 0 . 7 1 C O E F F OF V A R I A T I O N 0 . 7 1 •• COEFF OF V A R I A T I O N •* 0 . 7 1 C O E F F OF V A R I A T I O N 0 . 7 1 •* C O E F F OF V A R I A T I O N *• 0 . 7 1 •• C O E F F OF V A R I A T I O N *• 0 . 7 2 •• C O E F F OF V A R I A T I O N 0 . 7 2 •* COEFF OF V A R I A T I O N •* 0 . 0 CURRENT tt PROJECT COST MODEL EXPECTED VALUE • STANDARD D E V I A T I O N • 1 0 3 5 0 1 4 7 . 2 6 1 5 6 1 3 . **• SKEWNESS •"* 0 . 0 3 1 KURTOSIS •** 3 . 0 4 5 CURRENT tt NET REVENUE MODEL «* EXPECTED VALUE •• •• STANDARD O E V I A T I O N ••• SKEWNESS *•• 1 7 3 9 7 7 2 8 . 5 6 4 4 0 5 6 . 0 . 1 4 CURRENT tt PROJECT REVENUE MODEL *••• C O E F F OF V A R I A T I O N *• 0 . 3 2 EXPECTED VALUE •* • STANDARO D E V I A T I O N • 1 7 3 9 7 7 2 8 . 5 6 4 4 0 5 6 . ••• SKEWNESS **• 0 . 139 NET PRESENT VALUE MODEL KURTOSIS •** 4 . 3 5 1 EXPECTED VALUE ** » STANDARD D E V I A T I O N • 7 0 4 7 5 8 1 . 6 2 2 0 6 7 5 . *•• SKEWNESS O. 102 KURTOSIS 3 . 9 1 7 1 49 APPENDIX A-10 SKEWNESS OF PRIMARY VARIABLES > 1.6 CORRELATION OF PRIMARY VARIABLES = 0.42 150 •••• CURRENT tt WORK PACKAGE COST MODEL •••• ** E X P E C T E D VALUE •« 1 4 3 7 1 9 7 . *• E X P E C T E D VALUE *• 1 4 3 7 1 9 7 . •* E X P E C T E D VALUE •• 1 4 3 7 1 9 7 . ** E X P E C T E D VALUE •• 1 4 1 6 3 6 4 . ** E X P E C T E D VALUE •• 1 4 1 6 3 6 4 . •* E X P E C T E D VALUE •* 1 4 1 6 3 6 4 . E X P E C T E D VALUE ** 1 3 9 9 4 5 4 . •• E X P E C T E D VALUE ** 1 3 9 9 4 5 4 . *« E X P E C T E D VALUE •* 0 . •• STANDARO DEVIATION •• 1308209 . STANDARD DEVIATION 1 3 0 8 2 0 9 . •• STANDARO DEVIATION "« 1 3 0 8 2 0 9 . «• STANDARD DEVIATION •• 1 2 9 0 3 2 2 . STANDARD DEVIATION 1 2 9 0 3 2 2 . •» STANDARD DEVIATION 1 2 9 0 3 2 2 . •* STANDARO DEVIATION ** 1 2 7 6 1 5 5 . STANDARD DEVIATION *• 1 2 7 6 1 5 5 . •* STANDARD DEVIATION «• 0 . SKEWNESS *•* 0 . 0 9 ••• SKEWNESS 0 . 0 9 •«• SKEWNESS *•• 0 . 0 9 «*• SKEWNESS ••• O . 10 **« SKEWNESS 0 . 10 •*« SKEWNESS 0 . 10 SKEWNESS *** 0 . 1 0 *•• SKEWNESS ••• 0 . 10 *** SKEWNESS •** 0 . 0 COEFF OF V A R I A T I O N 0 . 9 1 •• COEFF OF V A R I A T I O N *• 0 . 9 1 •• COEFF OF V A R I A T I O N 0 . 9 1 »• COEFF OF V A R I A T I O N *• 0 . 9 1 •• COEFF OF V A R I A T I O N 0 . 9 1 ** C O E F F OF V A R I A T I O N «« 0 . 9 1 *" COEFF OF V A R I A T I O N •• 0 . 9 1 COEFF OF V A R I A T I O N •• 0 . 9 1 ** COEFF OF V A R I A T I O N 0 . 0 . . . . . . CURRENT tt PROJECT COST MODEL E X P E C T E D VALUE ** * STANOARD D E V I A T I O N • 1 1 3 5 9 5 9 1 . 3 6 5 8 7 1 3 . SKEWNESS *•• 0 . 0 3 4 ••• KURTOSIS •** 3 . 0 4 5 CURRENT tt NET REVENUE MODEL •* E X P E C T E D VALUE •* •* STANDARO DEVIATION ••• SKEWNESS 1 7 3 3 9 7 7 6 . 5 6 3 0 4 2 5 . 0 . 1 4 CURRENT tt PROJECT REVENUE MODEL •••• COEFF OF V A R I A T I O N •• 0 . 3 2 E X P E C T E D VALUE *• • STANDARD D E V I A T I O N • 1 7 3 3 9 7 7 6 . 5 6 3 0 4 2 5 . SKEWNESS ••• 0 . 139 KURTOSIS •** 4 . 3 4 4 NET PRESENT VALUE MODEL E X P E C T E D VALUE * STANDARD D E V I A T I O N * 5 9 8 0 1 8 5 . 6 7 1 4 7 5 0 . SKEWNESS 0 . 0 7 6 *** KURTOSIS •*• 3 . 6 6 8 A P P E N D I X B E X A M P L E 2 : M O D E L A P P L I C A T I O N 1 51 1 52 APPENDIX B-1 CONSTANT DOLLAR COST ESTIMATE •»•* CURRENT $$ WORK PACKAGE COST MODEL •»»« • W.P 0 • 2 • W.P 0 » 3 • W.P 0 * 4 » W.P 0 • 5 • W.P 0 * 6 • W.P 0 • 7 • W.P 0 a • w .p * • 9 • W.P 0 ' 10 • W.P 0 • 11 • W.P 0 • 12 • W.P * • 1 3 • W.P 0 * 14 • W.P # • 15 EXPECTED VALUE ** STANDARO DEVIATION •* *•• SKEWNESS,»•• EXPECTED VALUE STANDARD DEVIATION *• *•• SKEWNESS **• 203700. 67051. 2549598. 912707. 2649598. 912707. 24 18497. 881804. 2537628. 1337783. 3579133. 1 140381 . 859450. 273656. 0.20 ** EXPECTED VALUE »• »* STANDARD DEVIATION •« «•» SKEWNESS *»• O. 10 EXPECTED VALUE STANDARD DEVIATION •* SKEWNESS 0. 10 EXPECTED VALUE »» »» STANDARD DEVIATION *« *** SKEWNESS *»• 0. 10 •• EXPECTED VALUE •* STANDARD DEVIATION SKEWNESS •** O. 10 « EXPECTED VALUE •* •* STANDARD DEVIATION «« •** SKEWNESS *** O. IO EXPECTED VALUE *• *• STANDARD DEVIATION *• ••* SKEWNESS *•• »» EXPECTED VALUE •» STANDARD DEVIATION *• SKEWNESS »** 0.0 *• EXPECTED VALUE »• ** STANOARD OEVIATION ** *** SKEWNESS *** O. O. 0.0 EXPECTED VALUE •* STANDARD DEVIATION •» SKEWNESS **• 2535108. 913267. 0. 10 EXPECTED VALUE •* STANDARO DEVIATION •» »•* SKEWNESS *•* 4 1 9 8 7 3 8 . 0.0 *• EXPECTED VALUE »• STANDARD DEVIATION •* *•» SKEWNESS 5034666. 1581373. O. 10 *« EXPECTED VALUE •* »• STANDARD DEVIATION *** SKEWNESS 4042898. 1370344. O. 10 •« COEFF OF VARIATION O. 32 •« COEFF OF VARIATION O. 33 COEFF OF VARIATION *• 0.36 COEFF OF VARIATION ** 0.34 »» COEFF OF VARIATION »» 0.36 COEFF OF VARIATION 0.36 «* COEFF OF VARIATION »• O. 31 •» COEFF OF VARIATION 0.32 •* COEFF OF VARIATION •• 0.32 •* COEFF OF VARIATION •* 0.0 «• COEFF OF VARIATION 0.36 »• COEFF OF VARIATION O. 30 COEFF OF VARIATION •* 0.31 COEFF OF VARIATION *• 0.34 CO • W.P 0 • •» EXPECTED VALUE •» 16 4073997. • W.P 0 • EXPECTED VALUE 17 7029227. • W.P * • EXPECTED VALUE •* 18 3536998. • W.P * • » • EXPECTED VALUE 19 746157. • W.P 0 * • • EXPECTED VALUE 20 555550. • W.P 0 • EXPECTED VALUE 21 1517816. » W.P 0 • EXPECTED VALUE »• 22 33400032. • W.P * • « • EXPECTED VALUE 23 0 . • W.P 0 • EXPECTED VALUE 24 445550. • W.P 0 • « • EXPECTED VALUE •» 25 2018498. • W.P 0 • • * EXPECTED VALUE 26 688975. • W.P 0 • EXPECTED VALUE 27 349330. • W.P 0 • » • EXPECTED VALUE *« 28 12074329. • W.P * • »« EXPECTED VALUE *• 29 1502748. • W.P * • *« EXPECTED VALUE »• 30 1390841. »• STANDARD DEVIATION *» 1341011. *• STANDARD DEVIATION •» 2220856. •* STANOARD OEVIATION »* 1095333. » • STANDARD DEVIATION »» 237101. •» STANDARD DEVIATION »* 191632. •* STANDARD DEVIATION » • 471158. *• STANOARD DEVIATION »« 10952315. STANDARD DEVIATION •» O. STANDARD DEVIATION ** 149120. »» STANDARD DEVIATION • * 638774. STANDARD DEVIATION »* 219015. • • STANDARD DEVIATION »» 118624. •* STANDARD. DEVIATION ** 3992589. STANDARD DEVIATION •» 518038. STANDARO DEVIATION 458266. »*» SKEWNESS »*» O. 10 »•» SKEWNESS 0 . 10 » * • SKEWNESS • • • O. 10 • * » SKEWNESS 0 . 0 SKEWNESS • • • 0 . 10 »** SKEWNESS * •» 0 . 10 * • • SKEWNESS • • • O. 10 SKEWNESS • * • O.O • * • SKEWNESS • * * 0 . 10 •*» SKEWNESS »•» O. 20 *»» SKEWNESS • * • 0 . 10 ' * * • SKEWNESS * • * 0 . 10 * * • SKEWNESS «*« 0 . 2 0 SKEWNESS • • • 0 . 2 0 • • * SKEWNESS 0 . 3 0 * • COEFF OF VARIATION O. 33 COEFF OF VARIATION 0 .32 • • COEFF OF VARIATION 0 .31 *• COEFF OF VARIATION 0 . 3 2 COEFF OF VARIATION »• 0 . 34 COEFF OF VARIATION 0 .31 ** COEFF OF VARIATION • • 0 . 3 3 • • COEFF OF VARIATION O.O • • COEFF OF VARIATION •« 0 . 3 3 »* COEFF OF VARIATION •» O. 32 COEFF OF VARIATION 0 . 3 2 ** COEFF OF VARIATION *• O. 34 ** COEFF OF VARIATION O. 33 COEFF OF VARIATION O. 34 • • COEFF OF VARIATION O. 33 if* 1 « W.P 0 • EXPECTED VALUE »• 31 607400. • W.P 0 * EXPECTEO VALUE 32 18751312. • W.P 0 » EXPECTED VALUE * • 33 O. STANDARD DEVIATION *» 194779. *• STANDARD OEVIATION •» 6156602. •* STANDARD DEVIATION *• 0 . * • * SKEWNESS **» O. 10 • » • SKEWNESS.•»* 0 . 5 0 • • • SKEWNESS • * * 0 . 0 • * COEFF OF VARIATION ••• 0 . 3 2 •» COEFF OF VARIATION O. 33 •« COEFF OF VARIATION 0 . 0 . . . . . . CURRENT $$ PROJECT COST MODEL * * • * « » •« EXPECTED VALUE *• 126392880. « STANDARD DEVIATION * 14041892. • » • SKEWNESS **» 0 .095 *** KURTOSIS • * * 2 .607 . . . . . PERCENTAGE VALUES OF THE CUMULATIVE DISTRIBUTION FUNCTION OF THE PROJECT COST . . . . . . . . . . 0.25% VALUE . . . . . 92435960. « * * » . 0.50% VALUE . . . . . 94309352. * » . » * 1.0% VALUE » . » » . 96538504. • • » * » 2.5% VALUE * • » ' • 100198536. * * • • * 5.0% VALUE • • » » • 103703368. . . . . . 10.0% VALUE . . . . . 108124792. . . . . . 25.0% VALUE * « • « • 116283704. . . . . . 50.0% VALUE • » » » * 126123480. ..... 75.0% VALUE » • « • • 136224376. . . . . . 90 .OX VALUE . . . . . 145039144. ..... 95.0% VALUE . . . . . 149797400. ..... 97.5% VALUE . . . . . 154068552. ..... 99.0% VALUE . . . . . 158436552. ..... 99.5% VALUE . . . . . 161175176. ..... 99.75% VALUE «««»« 163531944. END OF OISSPLA 9 . 0 - - 7309 VECTORS GENERATED IN 1 PLOT FRAMES. PROPRIETARY SOFTWARE PRODUCT OF ISSCO. SAN DIEGO, CA. 2258 VIRTUAL STORAGE REFERENCES: 7 READS; O WRITES. APPENDIX B-2 CURRENT DOLLAR COST ESTIMATE E X P E C T E D V A L U E •* 2 8 6 4 6 0 0 . E X P E C T E D V A L U E •» 2 0 9 2 1 6 . E X P E C T E D V A L U E *• 2 6 1 8 4 0 6 . E X P E C T E D V A L U E 2 7 2 1 1 0 6 . E X P E C T E D V A L U E 2 5 1 4 3 3 9 . E X P E C T E D V A L U E «• 2 6 5 8 0 1 1 . E X P E C T E D V A L U E ••' 4 5 2 6 1 3 3 . E X P E C T E D V A L U E • • 3 7 7 5 2 8 4 . E X P E C T E D V A L U E «• 9 0 0 1 6 2 . E X P E C T E D V A L U E *• 0 . E X P E C T E D V A L U E •• 2 8 3 6 1 2 4 . E X P E C T E D V A L U E ' • 4 6 2 8 7 5 5 . E X P E C T E D V A L U E •• 5 5 2 1 4 6 7 . E X P E C T E D V A L U E •• 4 4 3 3 S 9 5 . C U R R E N T $ $ WORK P A C K A G E C O S T •* S T A N O A R D D E V I A T I O N »• 9 2 2 0 7 7 . •« S T A N D A R D D E V I A T I O N ** 6 8 8 6 9 . S T A N D A R D D E V I A T I O N 9 3 7 3 7 1 . •* S T A N D A R D D E V I A T I O N *• 9 3 7 3 7 4 . ** S T A N D A R O D E V I A T I O N 9 1 6 8 0 1 . »* S T A N D A R O O E V I A T I O N 9 5 6 5 4 3 . S T A N D A R D D E V I A T I O N 1 4 2 2 0 9 7 . *» S T A N D A R D D E V I A T I O N 1 2 0 3 0 1 9 . *• S T A N D A R D O E V I A T I O N •« 2 8 6 6 4 3 . ** S T A N D A R D D E V I A T I O N «» O . ** S T A N D A R D O E V I A T I O N »* 1 0 2 1 9 5 5 . •* S T A N O A R D D E V I A T I O N «« 1 4 0 7 8 4 3 . S T A N D A R D D E V I A T I O N *• 1 7 3 4 8 0 8 . *• S T A N D A R D D E V I A T I O N »« 1 5 0 3 2 9 6 . •*• S K E W N E S S •** O . 10 »*» S K E W N E S S * » * 0 . 2 0 **» S K E W N E S S *•* O . 10 *»* S K E W N E S S ••* O . 10 •»» S K E W N E S S O . 10 »«« S K E W N E S S O . 10 »** S K E W N E S S **• O . 10 S K E W N E S S •»• O . 10 *•* S K E W N E S S **• O . O O •»» S K E W N E S S 0 . 0 *«• S K E W N E S S O . 10 ••* S K E W N E S S 0 . 0 0 **• S K E W N E S S **• O . 10 ••• S K E W N E S S O . 10 C O E F F OF V A R I A T I O N O . 32 C O E F F OF V A R I A T I O N 0 . 3 3 »» C O E F F OF V A R I A T I O N 0 . 3 6 C O E F F OF V A R I A T I O N O . 34 C O E F F OF V A R I A T I O N O . 36 *• C O E F F OF V A R I A T I O N <* O . 3 6 *• C O E F F OF V A R I A T I O N 0 . 3 1 *• C O E F F OF V A R I A T I O N 0 . 3 2 C O E F F OF V A R I A T I O N O . 32 •* C O E F F OF V A R I A T I O N 0 . 0 *• C O E F F OF V A R I A T I O N O . 3 6 C O E F F OF V A R I A T I O N O . SO •* C O E F F OF V A R I A T I O N 0 . 3 1 C O E F F OF V A R I A T I O N O . 34 W.P * • •• E X P E C T E D V A L U E •• 1G 4 4 6 7 9 1 0 . W . P * • •* E X P E C T E D V A L U E •* 17 7 8 0 5 5 3 3 . W.P # • E X P E C T E D V A L U E ** 18 3 8 8 9 4 1 7 . W.P D • ' • E X P E C T E D V A L U E •* 19 8 2 0 5 0 2 . W.P # • •• E X P E C T E D V A L U E »» 2 0 6 0 3 5 0 1 . W . P # • •• E X P E C T E O V A L U E 2 1 1 6 8 5 4 3 8 . W.P * < E X P E C T E D V A L U E 22 3 7 8 2 9 8 5 6 . W.P # ' •• E X P E C T E D V A L U E 23 0 . W.P # • E X P E C T E D V A L U E •* 24 4 9 4 7 0 3 . W.P * • •• E X P E C T E D V A L U E ** 25 2 3 3 6 4 2 6 . W.P # • •• E X P E C T E D V A L U E 26 7 9 5 5 3 1 . W.P * • •• E X P E C T E D V A L U E •• 27 4 0 2 3 2 5 . W.P # • •• E X P E C T E D V A L U E " 28 1 3 9 7 6 1 4 0 . W P » • •• E X P E C T E D V A L U E •• 2 9 1 7 3 9 4 4 4 . W . P * • *• E X P E C T E D V A L U E *« 3 0 1 5 9 7 9 1 9 . S T A N D A R D D E V I A T I O N 1 4 7 1 0 8 0 . •* S T A N D A R D D E V I A T I O N 2 4 6 7 0 6 6 . «• S T A N D A R D O E V I A T I O N »* 1 2 0 4 8 6 6 . *• S T A N D A R D D E V I A T I O N 2 6 0 8 0 6 . »• S T A N D A R D D E V I A T I O N •» 2 0 8 2 1 5 . S T A N D A R O D E V I A T I O N *• 5 2 3 3 8 7 . •» S T A N D A R D D E V I A T I O N 1 2 4 1 0 3 4 6 . »* S T A N D A R D D E V I A T I O N •» 0 . •« S T A N O A R D D E V I A T I O N »• 1 6 5 6 2 3 . *• S T A N D A R D D E V I A T I O N «» 7 3 9 8 2 1 . •* S T A N D A R D D E V I A T I O N 2 5 3 0 3 1 . *« S T A N D A R D D E V I A T I O N »» 1 3 6 6 8 5 . •* S T A N D A R D D E V I A T I O N 4 6 2 3 9 3 9 . •• S T A N D A R D D E V I A T I O N 5 9 9 9 2 5 . S T A N D A R D D E V I A T I O N *• 5 2 6 7 6 0 . S K E W N E S S 0 . 10 S K E W N E S S *•* O . 10 S K E W N E S S ••• O . 10 S K E W N E S S 0 . 0 0 S K E W N E S S • » * O . 10 S K E W N E S S •** 0 . 10 S K E W N E S S ••* O . 10 S K E W N E S S *•» 0 . 0 S K E W N E S S »•• 0 . 10 S K E W N E S S *** 0 . 2 0 S K E W N E S S O . 10 S K E W N E S S *•* O . 10 S K E W N E S S *•* 0 . 2 0 S K E W N E S S •»• 0 . 2 0 S K E W N E S S ••* 0 . 3 0 •* C O E F F OF V A R I A T I O N 0 . 3 3 »• C O E F F OF V A R I A T I O N O . 32 «* C O E F F OF V A R I A T I O N O . 3 1 C O E F F OF V A R I A T I O N •• O . 32 ** C O E F F OF V A R I A T I O N •* 0 . 3 5 C O E F F OF V A R I A T I O N •• 0 . 3 1 *• C O E F F OF V A R I A T I O N O . 3 3 C O E F F OF V A R I A T I O N •• 0 . 0 *• C O E F F OF V A R I A T I O N O . 3 3 C O E F F OF V A R I A T I O N 0 . 3 2 *• C O E F F OF V A R I A T I O N O . 32 C O E F F OF V A R I A T I O N O . 34 ** C O E F F OF V A R I A T I O N •• O . 33 C O E F F OF V A R I A T I O N •* O . 34 *• C O E F F OF V A R I A T I O N •* W.P # 31 W . P * 32 W.P # 33 • • EXPECTED VALUE 718705. EXPECTED VALUE 20495088. " EXPECTED VALUE • * 0 . >• STANDARD OEVIATION <• j . 2 3 0 6 3 9 . >• STANDARD DEVIATION * * 6730359. STANDARD DEVIATION »• O. * » » SKEWNESS • • • O. 10 • • • SKEWNESS • • • 0 . 5 0 * * * SKEWNESS * • * 0 . 0 * • COEFF OF VARIATION • • O. 32 * * COEFF OF VARIATION O. 33 * * COEFF OF VARIATION 0.0 « • • • • • CURRENT $$ PROJECT COST MODEL « » « • » • EXPECTED VALUE • STANDARD OEVIATION • 139865856. 15759003. • • • SKEWNESS • * * 0 .094 • * • KURTOSIS * • • 2 . 594 PERCENTAGE VALUES OF THE CUMULATIVE DISTRIBUTION FUNCTION OF THE PROJECT COST * » • » • • • • • • 0.25% VALUE 101888648. * • • « • 0.50% VALUE • » » * * 103954792. • • * « « 1.0% VALUE • • • 106421944. >•• 2.5% VALUE • « • » » 110488840. * • • • » 5.0% VALUE » • » * • 114397832. • • • • • 10.0% VALUE • • • • • 1193440O8. 25.0% VALUE • * • « * 128500200. » • • * • 50.0% VALUE 139478440. • • • • • 75.0% VALUE 150923576. 90.0% VALUE • • » • < 1G080960B. « * * » * 95.0% VALUE • • • • • 166357 176. 97.5% VALUE 170887368. 99.0% VALUE » « • • « 175735224. * * * * * 99.5% VALUE • * * • » 178762328. • • » • • 99.75% VALUE 181358136. END OF DISSPIA 9 . 0 - - 7 162 VECTORS GENERATED IN 1 PLOT FRAMES. PROPRIE TARY SOFTWARE PRODUCT OF ISSCO, SAN DIEGO. CA. 2268 VIRTUAL STORAGE REFERENCES; 7 READS; O WRITES. tn APPENDIX B-3 TOTAL DOLLAR COST ESTIMATE W.P # • E X P E C T E D V A L U E •• 2 3 5 4 1 9 3 6 . W.P * * E X P E C T E D V A L U E 3 2 5 3 0 2 3 . W.P * • *• E X P E C T E D V A L U E 4 3 1 6 7 0 6 6 . W.P # » »• E X P E C T E D V A L U E •* 5 3 2 9 1 2 8 4 . W.P * • *• E X P E C T E D V A L U E »• 6 2 9 9 3 4 6 8 . W.P # • E X P E C T E D V A L U E *• 7 3 1 3 4 1 5 4 . W.P # • *• E X P E C T E D V A L U E «• 8 5 2 3 6 9 9 8 . W.P * • •• E X P E C T E D V A L U E 9 4 4 1 1 3 1 6 . W.P * • '»• E X P E C T E D V A L U E 10 1 0 6 1 4 3 0 . W.P * • * ' E X P E C T E D V A L U E •« 1 1 0 . W . P 0 • E X P E C T E D V A L U E t2 3 0 7 3 1 9 5 . W.P * • •» E X P E C T E D V A L U E »• 13 5 1 1 0 6 4 5 . W.P * • E X P E C T E D V A L U E *• 14 6 1 3 6 8 4 5 . W . P 0 ' «• E X P E C T E D V A L U E 15 4 9 2 7 9 3 3 . • C U R R E N T $ $ WORK P A C K A G E C O S T »• S T A N D A R O O E V I A T I O N *» 114 1 9 0 8 . »» S T A N D A R D D E V I A T I O N •• 8 3 3 9 2 . S T A N D A R D D E V I A T I O N 1 1 3 4 9 3 8 . »• S T A N D A R D D E V I A T I O N 1 1 3 5 0 4 9 . S T A N D A R D D E V I A T I O N »• 1 0 9 2 4 1 3 . •* S T A N D A R D D E V I A T I O N •» 1 1 2 8 7 7 8 . *• S T A N D A R D D E V I A T I O N *• 1 6 4 6 9 2 4 . S T A N D A R O D E V I A T I O N 1 4 0 7 0 1 6 . •• S T A N D A R D D E V I A T I O N •» 3 3 8 3 5 1 . S T A N D A R D D E V I A T I O N «• O . *» S T A N D A R D D E V I A T I O N •* 1 1 0 7 7 0 9 . S T A N D A R D D E V I A T I O N *• 1 5 5 5 1 6 6 . S T A N D A R D D E V I A T I O N •» 1 9 2 9 1 1 9 . •» S T A N D A R D D E V I A T I O N »* 1 6 7 1 4 5 9 . ••• S K E W N E S S •*» O . 10 S K E W N E S S ••• O . 2 0 S K E W N E S S O . 10 ••• S K E W N E S S ••• O . 10 S K E W N E S S ••• 0 . 10 »** S K E W N E S S ••• O . 10 ••• S K E W N E S S O . 10 ••• S K E W N E S S O . 10 •** S K E W N E S S »»• 0 . 0 0 •«» S K E W N E S S »*• 0 . 0 *•» S K E W N E S S •** O . 10 •*« S K E W N E S S 0 . 0 0 »*• S K E W N E S S ••• O . 10 S K E W N E S S O . 10 C O E F F OF V A R I A T I O N O . 32 *• C O E F F OF V A R I A T I O N <• O . 33 C O E F F OF V A R I A T I O N •• O . 36 «» C O E F F OF V A R I A T I O N •• O . 34 C O E F F OF V A R I A T I O N •• O . 3 6 «• C O E F F OF V A R I A T I O N •«. 0 . 3 6 C O E F F OF V A R I A T I O N 0 . 3 1 *» C O E F F OF V A R I A T I O N 0 . 3 2 •• C O E F F OF V A R I A T I O N •» O . 32 »• C O E F F OF V A R I A T I O N 0 . 0 •* C O E F F OF V A R I A T I O N O . 3 6 C O E F F OF V A R I A T I O N •• O . 3 0 C O E F F OF V A R I A T I O N 0 . 3 1 C O E F F OF V A R I A T I O N O . 34 W.P # • •• E X P E C T E D V A L U E •• 16 4 9 6 5 8 6 9 . W.P * • E X P E C T E D V A L U E 17 8 5 3 9 0 2 8 . W.P * • E X P E C T E D V A L U E •• 18 4 3 0 8 1 5 0 . W.P * • •• E X P E C T E D V A L U E «• 19 9 0 8 8 3 7 . W.P # • E X P E C T E D V A L U E 2 0 6 7 8 9 5 5 . W . P * • E X P E C T E D V A L U E •• 2 1 1 8 4 3 8 2 4 . W.P # < •• E X P E C T E D V A L U E •* 22 4 0 3 6 2 0 1 6 . W.P # • •• E X P E C T E D V A L U E 23 0 . W.P * • E X P E C T E O V A L U E 24 5 4 1 2 2 0 . W P * • E X P E C T E D V A L U E 25 2 4 2 4 5 0 9 . W.P # • •• E X P E C T E D V A L U E •* 26 8 2 8 1 0 0 . W.P » • E X P E C T E D V A L U E 27 4 2 0 1 5 5 . W.P * " »• E X P E C T E D V A L U E 28 1 4 5 0 3 0 3 1 . W.P * • E X P E C T E O V A L U E 2 9 1 8 0 5 0 1 9 . W.P * • E X P E C T E D V A L U E ** 3 0 1 6 7 3 9 6 3 . S T A N D A R D D E V I A T I O N 1 6 3 5 7 6 8 . S T A N D A R D D E V I A T I O N *• 2 6 9 9 8 9 8 . •» S T A N O A R D D E V I A T I O N 1 3 3 5 2 3 7 . S T A N D A R D D E V I A T I O N »• 2 8 9 0 1 8 . *• S T A N D A R D D E V I A T I O N •* 2 3 4 3 6 1 . S T A N D A R D D E V I A T I O N »• 5 7 2 8 0 4 . •* S T A N D A R D D E V I A T I O N *• 1 3 2 4 3 9 9 2 . »• S T A N D A R D D E V I A T I O N »• O . *• S T A N D A R D O E V I A T I O N 1 8 1 2 5 9 . S T A N D A R D D E V I A T I O N »* 7 6 7 8 4 2 . *• S T A N D A R D D E V I A T I O N •* 2 6 3 4 3 6 . S T A N D A R D D E V I A T I O N 1 4 2 7 6 5 . »• S T A N D A R D D E V I A T I O N ** 4 7 9 8 9 7 4 . S T A N D A R D O E V I A T I O N 6 2 2 6 2 7 . *• S T A N D A R D D E V I A T I O N «» 5 5 1 9 2 3 . S K E W N E S S ••• O . 10 S K E W N E S S ••• O . 10 S K E W N E S S ••• 0 . 10 S K E W N E S S ••• 0 . 0 0 S K E W N E S S *•* O . 10 S K E W N E S S ••• O . 10 S K E W N E S S »»• O . 10 S K E W N E S S ••* 0 . 0 S K E W N E S S **• O . 10 S K E W N E S S O . 2 0 S K E W N E S S O . 10 S K E W N E S S ••• 0 . 10 S K E W N E S S •»* 0 . 2 0 S K E W N E S S O . 2 0 S K E W N E S S **• 0 . 3 0 »» C O E F F OF V A R I A T I O N O . 3 3 •• C O E F F OF V A R I A T I O N O . 32 •• C O E F F OF V A R I A T I O N 0 . 3 1 •• C O E F F OF V A R I A T I O N •• O . 32 C O E F F OF V A R I A T I O N O . 3 5 C O E F F OF V A R I A T I O N 0 . 3 1 »* C O E F F OF V A R I A T I O N O 3 3 »• C O E F F OF V A R I A T I O N »• 0 . 0 »• C O E F F OF V A R I A T I O N •• O . 3 3 C O E F F OF V A R I A T I O N «• O . 32 *• C O E F F OF V A R I A T I O N 0 . 3 2 C O E F F OF V A R I A T I O N O . 34 C O E F F OF V A R I A T I O N O . 3 3 »• C O E F F OF V A R I A T I O N O . 34 C O E F F OF V A R I A T I O N •• O . 3 3 • W.P n • E X P E C T E D V A L U E 31 7 2 S 3 8 2 . * W.P * ». •• E X P E C T E O V A L U E »• 32 2 2 8 9 5 5 2 0 . • W.P * • •• E X P E C T E O V A L U E 33 O . »• S T A N D A R D D E V I A T I O N •• 2 3 2 8 0 7 . »• S T A N D A R D D E V I A T I O N 7 5 2 2 7 5 9 . *• S T A N D A R D D E V I A T I O N •» O . . •»• S K E W N E S S ••* O . 10 *»* S K E W N E S S ••» 0 . 5 0 »•• S K E W N E S S * » • O . O •* C O E F F OF V A R I A T I O N O . 32 C O E F F OF V A R I A T I O N 0 . 3 3 C O E F F OF V A R I A T I O N •» 0.0 . . . . . . C U R R E N T $ $ P R O J E C T C O S T MODEL «••••« E X P E C T E D V A L U E • S T A N D A R O D E V I A T I O N * • « * S K E W N E S S **» K U R T O S I S ••• 1 5 3 7 6 2 7 0 4 . 1 7 0 2 9 9 9 2 . 0 . 0 9 6 2 . 6 1 2 . . . . . P E R C E N T A G E V A L U E S OF THE C U M U L A T I V E D I S T R I B U T I O N F U N C T I O N OF T H E P R O J E C T C O S T ••••• 0 . 2 5 % V A L U E »•••» . . . . . 0 . 5 0 % V A L U E »*»•• • • • • • 1 0 % V A L U E «•••• 1 1 2 5 1 7 4 8 0 . 1 1 4 8 0 6 7 2 8 . 1 1 7 5 2 G 2 3 2 . . . . . 2.5% V A L U E . . » » . . . . . . 5.0% V A L U E . . . . . . . . . . 1 0 . 0 % V A L U E • • • » • 1 2 1 9 8 3 5 1 2 . 1 2 6 2 4 5 1 2 8 . 1 3 1 6 1 4 6 8 0 . • • • « . 2 5 . 0 % V A L U E . . . . . • • • • • 5 0 . 0 % V A L U E » . . . . • • • • • 7 5 . 0 % V A L U E 1 4 1 5 1 1 0 0 0 . 1 5 3 4 3 6 0 5 6 . 1 6 5 6 7 7 4 9 6 . 9 0 . 0 % V A L U E . . . . . . . . . . 95.0% V A L U E . . . . . . . . . . 97.5*/. V A L U E »•*•« 1 7 6 3 6 8 7 4 4 . 182 1 6 5 3 6 8 . 1 8 7 3 3 8 7 4 4 . . . . . . 99.0% V A L U E •••»» . . . . . 99.5% V A L U E . . . . . . . . . . g g . 7 5 % V A L U E ••••• 1 9 2 6 5 6 6 0 0 . 1 9 5 9 9 6 1 6 8 . 1 9 8 8 7 4 4 4 0 . END OF O I S S P L A 9 . 0 - - 7 4 4 3 V E C T O R S G E N E R A T E D IN 1 P L O T F R A M E S . P R O P R I E T A R Y SOFTWARE PRODUCT OF I S S C O . S A N D I E G O . C A . 2 2 9 0 V I R T U A L S T O R A G E R E F E R E N C E S ; 7 R E A D S ; 0 W R I T E S . APPENDIX B-4 PROJECT DURATION, COST, REVENUE & NET PRESENT VALUE ESTIMATES FOR A NOMINAL DISCOUNT RATE OF 30. . . . . . . W 0 R K PACKAGE DURATION MODEL • * » * * * • W.P 0 • 1 • W.P 0 • 2 • W.P * • 3 • W.P 0 * 4 • W.P * • s • W.P » • 6 • W.P 0 • 7 • W.P 0 • 8 • W.P * * 9 • W.P 0 ' 10 • W.P 0 • 11 • W.P 0 • 12 • W.P 0 • 13 • W.P 0 * 14 • W.P 0 • •* EXPECTED VALUE •* 0 . 0 EXPECTED VALUE 0 .3318 « • EXPECTED VALUE •* 0 .2519 EXPECTED VALUE 0.2481 EXPECTED VALUE O.2481 *• EXPECTED VALUE 0 .6655 * • EXPECTED V A L U E . • • 0 .9200 EXPECTED VALUE 0.9181 •» EXPECTED VALUE •* 0 .6655 • * EXPECTED VALUE 0 .417S EXPECTED VALUE 0 .2519 EXPECTEO VALUE • • 0 .8356 EXPECTED VALUE 0 .8374 » • EXPECTED VALUE *• 0 .6609 EXPECTED VALUE •* STANDARD DEVIATION ** ; o.o STANDARD DEVIATION *» 0 .0428 »• STANDARD DEVIATION •* 0 .0228 STANOARD DEVIATION 0 .0278 STANDARD DEVIATION ** 0 .0278 STANDARD DEVIATION •* 0 .0763 STANDARD DEVIATION • * 0 .0700 STANDARD DEVIATION •* O.0377 ** STANDARD DEVIATION • * 0 .0763 »• STANDARD DEVIATION •* 0 .0365 •* STANDARD DEVIATION ** 0 .0228 STANDARD DEVIATION •* O.0488 STANDARD DEVIATION *• 0 .0460 STANDARD DEVIATION ** 0 .0228 STANDARD DEVIATION ** » • * SKEWNESS »»* 0 . 0 • » * SKEWNESS • * • O. 40 »»* SKEWNESS • * • 0 . 7 0 • • • SKEWNESS - O . 30 »*• SKEWNESS • • * - 0 . 3 0 SKEWNESS • • • O. 20 **• SKEWNESS *** O. 10 • « * SKEWNESS * • » 0 . 5 0 *** SKEWNESS *** 0 . 2 0 * • * SKEWNESS * • * O. 10 **• SKEWNESS *** 0 . 7 0 *»* SKEWNESS • * • 0 . 2 0 * * • SKEWNESS « * • 0 . 3 0 • • • SKEWNESS • • • 0 . 4 0 • * • SKEWNESS *** •» COEFF OF VARIATION 0 . 0 ** COEFF OF VARIATION « • O. 13 ** COEFF OF VARIATION 0 . 0 9 •» COEFF OF VARIATION •» 0 .11 COEFF OF VARIATION O. 1 1 COEFF OF VARIATION 0 .11 * • COEFF OF VARIATION 0 . 0 8 *• COEFF OF VARIATION • * 0 .04 ** COEFF OF VARIATION * • O. 1 1 ** COEFF OF VARIATION *• 0 . 0 9 COEFF OF VARIATION • * 0 . 0 9 ** COEFF OF VARIATION 0 . 0 6 •« COEFF OF VARIATION 0 . 0 5 COEFF OF VARIATION 0 . 0 3 COEFF OF VARIATION CTl U l 15 0 .6618 • W.P 0 • EXPECTED VALUE 16 0 .6609 * W.P 0 * •» EXPECTED VALUE ** 17 1.0837 * W.P 0 * • * EXPECTEO VALUE 18 0 .7518 • W.P 0 • • * EXPECTED VALUE 19 0 .7518 • W.P * • EXPECTED VALUE ** 20 0 .3318 • W.P * • •» EXPECTED VALUE •* 2 1 1.0837 • W.P 0 * *« EXPECTED VALUE 22 1.7518 ' • W.P 0 * * • EXPECTEO VALUE 23 0 .2519 • W.P 0 * * • EXPECTED VALUE *« 24 0 .4156 • W.P * • *» EXPECTEO VALUE •» 25 0 .5018 • W.P * • ** EXPECTED VALUE *• 26 0 .4175 • W.P * • •* EXPECTED VALUE *• 27 0 .3300 • W.P 0 • • • EXPECTED VALUE •* 28 0 .5018 • W.P 0 • EXPECTED VALUE 29 0 .5018 • W.P * • EXPECTED VALUE •* 0 .0278 • • STANDARD DEVIATION *• 0 .0228 ** STANDARD DEVIATION ** 0 .0882 i • »• STANDARD DEVIATION *• 0 .0278 STANDARD DEVIATION *• 0 .0278 STANDARD DEVIATION ** 0 .0428 STANDARD DEVIATION ** 0 .0456 STANDARD DEVIATION 0 .0882 • • STANDARD DEVIATION 0.0228 STANDARD DEVIATION •* 0.0334 STANDARD OEVIATION * • 0 .0335 •* STANDARD DEVIATION ** 0 .0252 •* STANDARD DEVIATION •* 0 .0426 ** STANDARD DEVIATION *» 0 .0457 STANDARD DEVIATION »• 0 .0396 »• STANDARD DEVIATION * • 0 . 4 0 * • * SKEWNESS * * • 0 . 4 0 * • * SKEWNESS • * * 0 . 0 * • • SKEWNESS • • * 0 . 4 0 • • • SKEWNESS * • • 0 . 4 0 *** SKEWNESS • * * 0 . 4 0 •** SKEWNESS * * • O. 10 *** SKEWNESS * * • O. 10 • • • SKEWNESS * * • 0 . 7 0 * • • SKEWNESS • * * O.O • » • SKEWNESS • • * 0 . 2 0 • • * SKEWNESS * • * 0 . 4 0 *** SKEWNESS *** O.O • • * SKEWNESS O. 10 *•« SKEWNESS * * • O. 20 * • * SKEWNESS *** 0 .04 COEFF OF VARIATION *« O.03 ** COEFF OF VARIATION ** O.OS •* COEFF OF VARIATION «* 0.O4 •* COEFF OF VARIATION 0 . 0 4 •» COEFF OF VARIATION O. 13 •* COEFF OF VARIATION • • 0 .04 » • COEFF OF VARIATION O.05 •* COEFF OF VARIATION 0 . 0 9 • * COEFF OF VARIATION O.OB *« COEFF OF VARIATION » • 0 .07 ** COEFF OF VARIATION ** 0 . 0 6 ** COEFF OF VARIATION *• O. 13 COEFF OF VARIATION *• 0 . 0 9 ** COEFF OF VARIATION 0 . 0 8 ** COEFF OF VARIATION * • 30 0 .7518 W.P * * EXPECTED VALUE *• 31 0 .2518 W.P * « EXPECTED VALUE •* 32 3.O018 W.P * * *• EXPECTED VALUE 33 15.0000 » W.P # • EXPECTED VALUE •* 1 0 . 0 W.P # • * • EXPECTED VALUE 2 0 . 0 W.P * • EXPECTED VALUE 3 0 .3318 W P * • EXPECTED VALUE 4 0 .3318 W.P # • EXPECTED VALUE *• 5 0 .3318 W.P * • EXPECTED VALUE •* 6 0 .3318 W P # * *• EXPECTED VALUE *• 7 0 .3318 W.P * * EXPECTEO VALUE » • 8 0 .5837 W.P # • EXPECTED VALUE 9 0 . 5 8 0 0 W.P # • •« EXPECTED VALUE • * 10 0 .5837 W.P * • •« EXPECTEO VALUE •* 11 0 .9974 0.0278 0 . 4 0 •« STANDARD DEVIATION •* SKEWNESS • * • 0 .0278 0 . 4 0 STANDARD DEVIATION ** SKEWNESS *** 0 .0980 - O . 1 0 *• STANDARD DEVIATION »• SKEWNESS • * • 0 . 0 0 . 0 STATISTICS OF EARLY START TIME OF A WORK PACKAGE STANDARD DEVIATION •« SKEWNESS •*» ' 0 . 0 0 . 0 STANDARD DEVIATION •* » • • SKEWNESS * * • 0 . 0 0 . 0 STANDARD DEVIATION *«* SKEWNESS **• 0 .0428 0 . 4 0 ** STANDARD DEVIATION »• *** SKEWNESS «** 0 .0428 0 . 4 0 •* STANDARD DEVIATION •» SKEWNESS * • • 0 .0428 0 . 4 0 STANDARD DEVIATION » * • SKEWNESS **• •* 0 .0428 0 . 4 0 STANDARO DEVIATION ** SKEWNESS * •* O.0428 0 . 4 0 •« STANDARO DEVIATION •« SKEWNESS «** 0 .0485 0 .35 •* STANOARO DEVIATION •** SKEWNESS •** 0 .0510 O. 19 *• STANOARD DEVIATION •* *** SKEWNESS * •* 0 .0485 0 .35 STANDARO DEVIATION »• * » • SKEWNESS *** *« 0 .0875 O. 18 0 .04 COEFF OF VARIATION • • 0 .11 COEFF OF VARIATION 0 .03 • • COEFF OF VARIATION 0 . 0 • COEFF OF VARIATION • • 0 . 0 • COEFF OF VARIATION 0 . 0 • COEFF OF VARIATION *« 0 . 13 • COEFF OF VARIATION O. 13 COEFF OF VARIATION O. 13 COEFF OF VARIATION O. 13 COEFF OF VARIATION • • O. 13 COEFF OF VARIATION 0 .08 COEFF OF VARIATION 0 . 0 9 COEFF OF VARIATION 0 .08 COEFF OF VARIATION •« 0 . 0 9 EXPECTED VALUE ** 1.5018 EXPECTED VALUE •« 1.2492 EXPECTED VALUE 1.2492 EXPECTED VALUE 1.2492 EXPECTED VALUE 1.2492 EXPECTED VALUE ** 1.2492 EXPECTED VALUE 1.2492 EXPECTED VALUE *• 1.2492 EXPECTEO VALUE 1.2518 EXPECTEO VALUE »* 1.2492 EXPECTED VALUE ** 1.2492 EXPECTED VALUE 2.0011 EXPECTED VALUE *» 1.5837 EXPECTEO VALUE *• 2.2529 EXPECTED VALUE 2 .2529 STANDARD DEVIATION »* 0.0614 STANOARD DEVIATION O.O904 STANDARD DEVIATION O.0904 •* STANDARD DEVIATION *• 0 .0904 •* STANDARD DEVIATION *• O.0904 STANDARD DEVIATION •* 0.0904 »» STANDARD DEVIATION ** 0.0904 STANDARD DEVIATION ** O.0904 STANOARD DEVIATION 0.0821 STANDARD DEVIATION ** O.0904 •* STANDARD DEVIATION •* O.0904 STANOARD DEVIATION 0.0946 • • STANDARD DEVIATION 0 .0925 STANDARD DEVIATION *• 0 .0973 *• STANDARD DEVIATION *» 0 .0973 * • * SKEWNESS • * * ** COEFF OF VARIATION 0 . 2 9 0 .04 »*« SKEWNESS *** ** COEFF OF VARIATION •* 0 .17 0 .07 * • * SKEWNESS •* COEFF OF VARIATION 0 .17 O.07 » » • SKEWNESS • * * * • COEFF OF VARIATION 0 .17 0 . 0 7 * » • SKEWNESS ** COEFF OF VARIATION 0 .17 0 . 0 7 *** SKEWNESS • * * * • COEFF OF VARIATION • * O.17 0 .07 SKEWNESS * •» ** COEFF OF VARIATION * • 0 . 17 0 . 07 *«* SKEWNESS • * * ** COEFF OF VARIATION ** 0 .17 0 .07 • • • SKEWNESS • * * ** COEFF OF VARIATION • * 0 . 1 2 0 . 0 7 »*• SKEWNESS * * • ** COEFF OF VARIATION • * 0 .17 0 .07 »*• SKEWNESS *** ** COEFF OF VARIATION ** O. 17 0 .07 •** SKEWNESS * • • ** COEFF OF VARIATION ** O. 16 0 .05 SKEWNESS • * * ** COEFF OF VARIATION O. 12 0 . 0 6 » • • SKEWNESS • * * «» COEFF OF VARIATION * • 0 . 1 6 0 .04 •*« SKEWNESS *** ** COEFF OF VARIATION 0 .16 0 .04 CTl CO / f • p 0 * • • EXPECTED VALUE • * STANDARD DEVIATION * • *** SKEWNESS * • * • * COEFF OF VARIATION • • 27 2.2529 0 .0973 0 . 16 0 .04 W.P 0 • * « EXPECTED VALUE * • * * STANDARD DEVIATION •* • • * SKEWNESS * * COEFF OF VARIATION • * 28 2 .2529 0 .0973 0 . 16 0 .04 W.P 0 • • • EXPECTED VALUE •* • * STANDARD DEVIATION * * * SKEWNESS **• * * COEFF OF VARIATION * • 29 2.2529 0 .0973 O. 16 0 .04 W.P 0 * * * EXPECTED VALUE • * STANDARO OEVIATION • • *** SKEWNESS •** * • COEFF OF VARIATION * * 30 1.9993 0.0984 0 . 10 0 . 05 W.P 0 * * + EXPECTED VALUE *« * * STANDARD DEVIATION *» * + • SKEWNESS • • * • • COEFF OF VARIATION • • 31 2.7548 0 .1029 0 . 14 0 . 04 W.P 0 * * • EXPECTEO VALUE * • * • STANDARD OEVIATION •« * * * SKEWNESS * • * * * COEFF OF VARIATION • * 32 0 . 0 0 . 0 0 . 0 0 . O W.P 0 * EXPECTED VALUE • * * » STANDARD DEVIATION »• * * * SKEWNESS « * • * * COEFF OF VARIATION * • 33 3.0066 0 .1066 0 . 13 0 . 04 • • • » • « PROJECT DURATION MODEL • * • * » * . . . . . . . . PROJECT DURATION * • » • « « * * 3.00665 FIRST FOUR MOMENTS OF PROJECT DURATION * * • 3.00665 0.01136 0 .00016 0.0O011 • » • • DISCOUNTED WORK PACKAGE COST MODEL • * • • V p * • *• EXPECTED VALUE STANDARD DEVIATION ** * * * SKEWNESS *** COEFF OF VARIATION • « 2 1645290. 530781. 0 . 10 0 . 32 w. P 0 * EXPECTED VALUE • * STANDARD DEVIATION * * * SKEWNESS • • * COEFF OF VARIATION • • 3 115355. 38049. 0 . 2 0 0 . 33 w. P 0 • EXPECTED VALUE • • STANDARD DEVIATION *• *** SKEWNESS *** COEFF OF VARIATION • • 4 1444048. 517817. 0 . 10 0 . 36 CTl LO • W.P * • 5 • W.P * • 6 • W.P 0 • 7 • W.P * • 8 • W.P 0 • 9 • W.P 0 • 10 • W.P * • 11 • W.P # • 12 • W.P 0 • 13 •W.P** 14 •W.P** 15 • W.P * • 16 •W.P** 17 • W.P * • 18 • W.P * • 19 •• EXPECTED VALUE •* 1500686. *• EXPECTED VALUE *• 1347638. ** EXPECTED VALUE •* 14O0414. •* EXPECTEO VALUE *« 2305380. •* EXPECTED VALUE *• 1956441. •* EXPECTED VALUE ** 474077. ** EXPECTEO VALUE *• O. ** EXPECTED VALUE *• 1288644. •* EXPECTED VALUE •* 2171950. ** EXPECTEO VALUE *« 2620282. •• EXPECTED VALUE *• 2104051. •• EXPECTED VALUE ** 2120304. *• EXPECTEO VALUE *• 3605884. •• EXPECTED VALUE •• 1835032. •• EXPECTED VALUE *• 387114. *• STANDARD DEVIATION ** 5)7902. •* STANDARD DEVIATION *• 492141. *• STANOARD DEVIATION «• 504748. •* STANDARD DEVIATION ** 725821 . »• STANDARD DEVIATION •• 624697. *• STANDARD DEVIATION ** 151275. *• STANDARD DEVIATION •• 0. •• STANDARD DEVIATION •« 464954. »* STANDARD DEVIATION *• 661941. ** STANDARD DEVIATION •* 824843. *• STANDARD OEVIATION ** 714491. *• STANDARD DEVIATION •* 699311. *• STANOARD DEVIATION •• 1141774. •• STANDARD OEVIATION •« 569568. *« STANDARD DEVIATION *• 123275. »*• SKEWNESS *** ** COEFF OF VARIATION •* 0.10 0.33 *** SKEWNESS *** ** COEFF OF VARIATION •• O.IO 0.37 *•• SKEWNESS *•* •* COEFF OF VARIATION ** 0.10 0.36 **• SKEWNESS *** ** COEFF OF VARIATION «• 0.10 0.31 *•* SKEWNESS ••• *• COEFF OF VARIATION •• 0.10 0.32 •*» SKEWNESS *** ** COEFF OF VARIATION ** -0.00 0.32 *** SKEWNESS •*• ** COEFF OF VARIATION •* 0.0 0.0 *** SKEWNESS *** ** COEFF OF VARIATION *« 0.10 0.36 •** SKEWNESS *** ** COEFF OF VARIATION «* -0.00 0.30 *** SKEWNESS *•* ** COEFF OF VARIATION ** 0.10 0.31 **• SKEWNESS *** ** COEFF OF VARIATION •« 0.10 0.34 *«* SKEWNESS •** •* COEFF OF VARIATION ** 0.10 0.33 *•• SKEWNESS ••* •• COEFF OF VARIATION •• O.IO 0.32 •*• SKEWNESS ••• •* COEFF OF VARIATION •• 0.10 0.31 **» SKEWNESS »•• ** COEFF OF VARIATION -0.00 0.32 O • W.P 0 • 20 • W.P 0 • 21 • W.P 0 • 22 • W.P * * 23 • W.P 0 • 24 • W.P 0 ' 25 • W.P * * 2E • W.P * • 27 • W.P 0 • 28 • W.P * • 29 • W.P * • 30 • W.P 0 • 31 • W.P 0 • 32 • W.P 0 • 33 EXPECTED VALUE 292452. • • EXPECTEO VALUE *• 778608. ** EXPECTEO VALUE ** 16763104. EXPECTED VALUE ** O. •* EXPECTED VALUE ** 228435. • • EXPECTEO VALUE 986859. EXPECTED VALUE • * 337750. *• EXPECTED VALUE •* 171730. »» EXPECTED VALUE 5903238. EXPECTED VALUE •« 734705. *• EXPECTED VALUE 685779. EXPECTED VALUE 289967. •* EXPECTED VALUE ** 9859165. •* EXPECTEO VALUE STANDARD DEVIATION *• 101057. •* STANDARD DEVIATION 242244. • * STANOARD DEVIATION *« 5508044. •* STANDARD DEVIATION ** O. ' * STANDARO DEVIATION *• 76601. ** STANOARD OEVIATION *• 313023. »• STANDARD DEVIATION *• 107610. STANDARD DEVIATION * • 58429. STANDARD DEVIATION *• 1956090. »* STANOARD DEVIATION * • 253753. ** STANDARD DEVIATION »* 226425. *• STANDARD DEVIATION « • 93208. •* STANOARD OEVIATION •« 3243038. ** STANDARD DEVIATION O. • • • SKEWNESS • * * ** COEFF OF VARIATION 0 . 1 0 0 .35 • * * SKEWNESS • • * * • COEFF OF VARIATION 0 . 1 0 0 .31 *** SKEWNESS • * * *» COEFF OF VARIATION 0 . 10 0 . 3 3 «** SKEWNESS *** » • COEFF OF VARIATION 0 . 0 0 . 0 • * * SKEWNESS • • * *• COEFF OF VARIATION •* 0 . 1 0 0 .34 SKEWNESS * * • *• COEFF OF VARIATION 0 . 2 0 0 . 3 2 **• SKEWNESS •** •« COEFF OF VARIATION *• 0 . 1 0 0 . 3 2 « • * SKEWNESS *** ** COEFF OF VARIATION • * 0 . 1 0 0 .34 * • * SKEWNESS * • • ** COEFF OF VARIATION •* 0 . 2 0 0 . 3 3 • * * SKEWNESS • • • COEFF OF VARIATION •* 0 . 2 0 0 . 3 5 *»» SKEWNESS * • • *» COEFF OF VARIATION 0 . 3 0 0 .33 SKEWNESS •* COEFF OF VARIATION 0 . 1 0 0 .32 SKEWNESS * • • ** COEFF OF VARIATION 0 .5O 0 .33 • * • SKEWNESS • • * »* COEFF OF VARIATION 0 . 0 0 . 0 ...... DISCOUNTED PROJECT COST MODEL «•*••• EXPECTED VALUE *• • STANDARD DEVIATION * *** SKEWNESS *** ••* KURTOSIS *** 65354256. 7157379. 0.097 2.629 *•**• PERCENTAGE VALUES OF THE CUMULATIVE DISTRIBUTION FUNCTION OF THE PROJECT COST •*••• *»•*• 0.25% VALUE »•*•* ..... 0.50% VALUE ••••• ..... «.o% VALUE •»••« 47943192. 48926968. 50O90OO8. ••••• 2.5% VALUE •*••• ..... 5.0% VALUE ***** ..... 10.0% VALUE ••«*• 51986248. 53790760. 56056136. ..... 25.0% VALUE ..... ..... 50.0% VALUE ..... ..... 75.0% VALUE ..... 60216024. 65216328. 70349912. •»••• 90.0% VALUE'***** ***.. 95.0% VALUE «**•• ..... 97.5% VALUE ••»*• 74845064. 77325112. 79482040. ..... 99.0% VALUE •»»•• ..... 99.554 VALUE «*««« ..... 99.75% VALUE •*••• 81744568. 83172552. 84408920. ENO OF DISSPLA 9.0 -- 7467 VECTORS GENERATED IN 1 PLOT FRAMES. PROPRIETARY SOFTWARE PROOUCT OF ISSCO, SAN DIEGO, CA. 2116 VIRTUAL STORAGE REFERENCES: 7 READS: 0 WRITES. ...... DISCOUNTED NET REVENUE MODEL ...... * R S H « EXPECTED VALUE •* *• STANDARD DEVIATION •• *** SKEWNESS ••* *« COEFF OF VARIATION *• 1 14865696. 3539280. -0.16 0.24 • R S * • •• EXPECTED VALUE •* •• STANDARD DEVIATION ** *•* SKEWNESS ••* «* COEFF OF VARIATION •• 2 13742297. 2437553. -0.08 0.18 • R.S * • ** EXPECTED VALUE *• •* STANDARD DEVIATION *• •*• SKEWNESS *•* ** COEFF OF VARIATION •* 3 10462015. 2559987. -0.21 0.24 * R S » * ** EXPECTEO VALUE «• *• STANOARD DEVIATION «• **« SKEWNESS •** *• COEFF OF VARIATION •* ro 4 1009695G. 1874154. - 0 . 0 9 0 . 19 • R .S * • * * EXPECTED VALUE •* • * STANDARD DEVIATION *• * * * SKEWNESS * * • * • COEFF OF VARIATION * • s 5297495. 1404062. - 0 . 1 7 0. ,27 • R s 0 * * • EXPECTED VALUE * * STANDARD DEVIATION ** ... SKEWNESS • * * • • COEFF OF VARIATION • • 6 7312847. 2083295. - 0 . 17 0 . 28 * R . S 0 • • * EXPECTED VALUE «• * • STANDARD DEVIATION • * • SKEWNESS • * • • • COEFF OF VARIATION • » 7 6965347. 1177514. 0 . 0 3 0 . 17 • R. S 0 » • * EXPECTED VALUE • • * * STANDARD DEVIATION ... SKEWNESS • * • * * COEFF OF VARIATION * * 8 2126687. 912861. - 0 . 16 0 . 43 ***** DISCOUNTED PROJECT REVENUE MODEL ***• *« EXPECTED VALUE ** * STANOARD DEVIATION • «*• SKEWNESS • • * • • * KURTOSIS • • • 70869264. 6088144. - 0 . 0 6 3 2 .936 • • * * • PERCENTAGE VALUES OF THE CUMULATIVE DISTRIBUTION FUNCTION OF THE PROJECT REVENUE • • • • • . . . . . o.25% VALUE • * * • » » « • * * 0.50% VALUE • • * « * « « » » « 1.0% VALUE • • • • • 53614408. 55016024. 56537960. * * « • * 2.5% VALUE • • * * • * * * • • 5.0% VALUE • * » • • * . . . » 10.0% VALUE ***** 58788904. 60736856. 62991224. ***** 25.0% VALUE ***** . . . . . 50.0% VALUE ***** . . . . . 75.0% VALUE • • * • • 66764456. 70935224. 75045256. * • • • * 90.0% VALUE • • • • * * * * * « 9 5 . 0 % VALUE « * « * • * * * * * 9 7 . 5 % VALUE * » • • « 78661304. 80370664. 82577448. . . . . . gg.0% VALUE ***** . . . . . gg.5% VALUE • • • * • . . . . . 99.75% VALUE • • • * • 84627688. 85995256. 8724 1336. END OF DISSPLA 9 . 0 - - 7557 VECTORS GENERATED IN 1 PLOT FRAMES. PROPRIETARY SOFTWARE PRODUCT OF ISSCO, SAN DIEGO, CA. CO • • • • • « • NET PRESENT VALUE MODEL • • « * « « • ' • EXPECTED VALUE • • 5515008. • STANDARD DEVIATION • 9396466. SKEWNESS • • • - 0 . 0 6 0 • • * KURTOSIS *** 2.864 • • « « « PERCENTAGE VALUES OF THE CUMULATIVE DISTRIBUTION FUNCTION OF ***** 0.25% VALUE ***** • • » • • 0.50% VALUE ***** -20670168. -18642744. * • • * • 2.5% VALUE « • • • • . . . . . 5.0% VALUE * • • * • -13067681. -10131872. 25.0% VALUE . « . . • . . . . . 50.0% VALUE • • • • • -875323. 5615141. . . . . . 90.0% VALUE • « » » • . . . . . 95.0% VALUE * « • • « 17593432. 19552120. ..... 99.0% VALUE ***** . . . . . 99.5% VALUE ***** 26583848. 28587992. ENO OF DISSPLA 9 . 0 - - 7319 VECTORS GENERATED IN 1 PLOT FRAMES. PROPRIETARY SOFTWARE PRODUCT OF ISSCO. SAN DIEGO, CA. 2452 VIRTUAL STORAGE REFERENCES; 6 REAOS: O WRITES. • * * * INTERNAL RATE OF RETURN MOOEL **** ** DISCOUNT RATE *• . . . . . EXPECTED VALUE * • • • • 0 .2870 11608128. ** DISCOUNT RATE ** . . . . . EXPECTED VALUE • « • * • 0 .2970 8398560. • • DISCOUNT RATE *• . . . . . EXPECTED VALUE ***** 0 .3070 5515008. •* OISCOUNT RATE • • . . . . . EXPECTED VALUE • « • * • 0 .3170 2925584. •* OISCOUNT RATE *• . . . . . EXPECTED VALUE • • « « * 0 .3270 602256. ** OISCOUNT RATE • • . . . . . EXPECTED VALUE ***** 0-3370 -1480800. THE NET PRESENT VALUE ***** ***** 1.0% VALUE ***** -16412977 . » • * * • 10.0% VALUE «*«•« -6695873 . ***** 75.0% VALUE • • • • * 12012370. ..... 97.5% VALUE • » • « • 23535256. ..... 99.75% VALUE ***** 30390472. * * • * STANDARD DEVIATION •* 10192215. * • * * STANDARD DEVIATION «* 9780798. * • • • STANDARD DEVIATION • • 9396465. * * « • STANDARO DEVIATION • • 9036279. • • • • STANDARD DEVIATION •* 8697739. « • * * STANOARD DEVIATION • • 8378696. 

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