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## Material Information- Title:
- Efficiency/equity analysis of water resources problems--a game theoretic approach
- Creator:
- Ng, Elliot Kin, 1950-
- Publication Date:
- 1985
- Language:
- English
- Physical Description:
- xi, 160 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Cost allocation ( jstor )
Cost efficiency ( jstor ) Cost functions ( jstor ) Cost of equity ( jstor ) Cost savings ( jstor ) Game theory ( jstor ) Linear programming ( jstor ) Minimization of cost ( jstor ) Water resources ( jstor ) Water use efficiency ( jstor ) Dissertations, Academic -- Environmental Engineering Sciences -- UF Environmental Engineering Sciences thesis Ph. D Information storage and retrieval systems -- Water resources development ( lcsh ) Water resources development -- Cost effectiveness -- Data processing ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1985.
- Bibliography:
- Bibliography: leaves 150-158.
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- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Elliot Kin Ng.
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- University of Florida
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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- 029549216 ( ALEPH )
14706818 ( OCLC ) AEH3714 ( NOTIS ) AA00004882_00001 ( sobekcm )
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EFFICIENCY/EQUITY ANALYSIS OF WATER RESOURCES PROBLEMSA GAME THEORETIC APPROACH By ELLIOT KIN NG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1985 To my parents and my wife, Eileen, and children, Matthew, Michelle, Michael ACKNOWLEDGMENTS I would like to thank my chairman, Dr. James P. Heaney, for the many hours spent guiding this research. His encouragement, support, and friendship during my three years at the University of Florida have been invaluable. I would also like to thank the other members of my supervisory committee, Dr. Sanford V. Berg, Dr. Donald J. Elzinga, Dr. Wayne C. Huber, and Dr. Warren Viessman, for their time and support. In addition, I wish to thank the U.S. Air Force for giving me the opportunity to pursue the Ph.D. degree. Thanks are also due to several fellow students who have made my program enjoyable and memorable. In particu lar, I wish to thank Mr. N. Devadoss, Mr. Mun-Fong Lee, and Mr. Robert Ryczak. I would also like to give special thanks to Mr. Robert Dickinson for keeping an extra copy of the LP-80 and Mrs. Barbara Smerage for doing such an excellent job typing this manuscript. I am extremely grateful to my parents for instilling in me a desire to seek further eduction. Furthermore, I am especially thankful to my wife, Eileen, for typing initial drafts of this manuscript and for her love, encouragement, and sacrifices throughout my program. We will miss the iii croissants, pizzas, and hoagies that supplemented my late night studies. Finally, I wish to thank my children, Matthew, Michelle, and Michael, for their love and under standing during the countless times I have chased them out of my study. IV TABLE OF CONTENTS Page ACKNOWLEDGMENTS iii LIST OF TABLES vii LIST OF FIGURES ix ABSTRACT x CHAPTER 1 INTRODUCTION 1 2 LITERATURE REVIEW 4 Efficiency Analysis 4 Equity Analysis 5 Conclusions 8 3 EFFICIENCY ANALYSIS 10 Introduction 10 Partial Enumeration Techniques 12 Total Enumeration Techniques 15 Modeling Network Problems as Digraphs 16 The Total Enumeration Procedure 21 Computational Considerations 30 Summary 3 8 4 EQUITY ANALYSIS 39 Introduction 39 Cost Allocation for Regional Water Networks 4 0 Criteria for Selecting a Cost Allocation Method 45 Ad Hoc Methods 4 8 Defining Identifiable Costs as Zero... 49 Defining Identifiable Costs as Direct Costs 54 v Cooperative Game Theory 64 Concepts of Cooperative Game Theory... 65 Unique Solution Concepts 75 Empty Core Solution Concepts 87 Cost Allocation in the Water Resources Field 88 Separable Costs, Remaining Benefits Method 9 0 Minimum Costs, Remaining Savings Method 9 5 Allocating Cost Using Game Theory Concepts. 99 The k Best System 99 The Dummy Player 108 Comparing Methods 115 Summary 119 5 EFFICIENCY/EQUITY ANALYSIS 120 Introduction 120 Maximum Cost 122 Minimum Cost 129 Fairness Criteria 132 Summary 13 3 6 CONCLUSIONS AND RECOMMENDATIONS 135 APPENDIX EFFICIENCY/EQUITY ANALYSIS OF A THREE- COUNTY REGIONAL WATER NETWORK WITH NONLINEAR COST FUNCTION 142 REFERENCES 15 0 BIOGRAPHICAL SKETCH 159 vi LIST OF TABLES Table Page 3-1 Example of Total Enumeration Procedure for 3-Node Digraph 27 3-2 The Number of Independent Calculations to Find the Costs of Spanning Directed Trees for All Possible Subdigraphs 31 3-3 Summary of Computational Effort for Digraphs Shown in Figure 3-4 34 3-4 Efficiency Analysis of a Three-User Water Supply Network with Nonlinear Cost Function Using Lotus 1-2-3 37 4-1 Projected Population Growth and Projected Average Per Capita Demand 41 4-2 The Costs and Percent Savings for All Options 44 4-3 Cost Allocation Matrix 50 4-4 Cost Allocation of Optimal Network Based on Population 52 4-5 Cost Allocation of Optimal Network Based on Demand 5 3 4-6 Cost Allocation of Optimal Network with Use of Facilities Method 56 4-7 Cost Allocation for the Use of Facilities Method 5 7 4-8 Cost Allocation of Optimal Network with Direct Costing/Equal Apportionment of Remaining Costs Method 60 4-9 Cost Allocation for Direct Costing/Equal Apportionment of Remaining Costs Method 61 vii 4-10 Core Geometry for Three-Person Cost Game Example 7 3 4-11 Cost Allocation for Three-County Example Using the Shapley Value 80 4-12 Cost Allocation for Three-County Example Using the Nucleolus 85 4-13 Empty Core Solution Methods 89 4-14 Cost Allocation for Three-County Example Using the SCRB Method 94 4-15 Cost Allocation for Three-County Example Using the MCRS Method 98 4-16 Nominal Versus Actual Core Bounds for Optimal Network Game 100 4-17 Cost Allocations for the Optimal Network and the Second Best Network ($) 103 4-18 Cost Allocation for Option 3 as a Two-Person Game Using the SCRB Method 10 9 4-19 Comparing Cost Allocations for Option 3 as Two-Person Game and Three-Person Game Using the SCRB and MCRS Methods Ill 4-20 Core Bounds for Option 3 as a Three-Person Game 112 4-21 Core Bounds for Option 3 as a Three-Person Game with County 1 as a Dummy Player 114 4-22 Core Bounds for Option 3 as a Two-Person Game 116 4-23 Comparison of Methods Discussed for Allocating Costs of Water Resources Projects. 117 5-1 Using Independent Calculations from the Total Enumeration Procedure to Find c(i), c(S), and c(N) for the Three-County Regional Water Network Problem 121 5-2 Efficiency/Equity Analysis of the Optimal Network 12 6 viii LIST OF FIGURES Figure Page 3-1 Types of Cost Functions 13 3-2 Example Digraph Representing a Regional Water Network Problem for Three Users 18 3-3 Flow Diagram of Total Enumeration Procedure for n-Node Digraph 23 3-4 Examples of 3,4,5-Node Digraphs 33 4-1 Lengths of Interconnecting Pipelines 43 4-2 Geometry of Core Conditions for Three-Person Cost Game Example 71 4-3 Core for the Optimal Network Game (C (N ) = $4,556,409 ) 101 4-4 Core for the Second Best Network Game ( C (N) = $4,556,826 ) 102 4-5 Reduction in. Core as c(N) Increases from c X(N) to c (N) 107 IX Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy EFFICIENCY/EQUITY ANALYSIS OF WATER RESOURCES PROBLEMSA GAME THEORETIC APPROACH By Elliot Kin Ng August, 1985 Chairman: James P. Heaney Major Department: Environmental Engineering Sciences Successful regional water resources planning involves an efficiency analysis to find the optimal system that maxi mizes benefits minus costs, and an equity analysis to appor tion project costs. Traditionally, these two problems have been treated separately. This dissertation incorporates efficiency analysis and equity analysis into a single regional water resources planning model. A reliable total enumeration procedure is used to find the optimal system for regional water network problems. This procedure is easy to understand and can be implemented using readily available computer software. Furthermore, the engineer can use realistic cost functions or perform detailed cost analysis and, also, examine good suboptimal systems. In addition, this procedure finds the optimal system for each individual and each subgroup of individuals; hence, an equity x analysis can be accomplished using the theory of the core from cooperative n-person game theory. Game theory concepts are used to perform an equity analysis on the optimal system as well as good suboptimal systems. For any system, an equitable cost allocation exists if a core exists. However, if a game is not properly defined, even a cost allocation in the core may be inequitable. A rigorous procedure using core conditions and linear programming is described to determine the core bounds. An individual's lower core bound and upper core bound unambigu ously measure the individual's minimum cost and maximum cost, respectively. Traditional approaches for quantifying minimum cost and maximum cost assume that either a regional system involving the grand coalition is built or all the individuals will go-it-alone. However, this rigorous procedure accounts for the possibility that a relatively attractive system involving subgroups may form. Furthermore, this rigorous procedure gives a general quantitative definition of marginal cost and opportunity cost. Once the minimum cost and maximum cost for each individual are determined, a basis for equitable cost allocation is available. Finally, efficiency analysis and equity analysis are not separable problems but are related by the economics of all the opportunities available to all individuals in a project. xi CHAPTER 1 INTRODUCTION In situations where multiple purposes and groups can take advantage of economies of scale in production and/or distribution costs, a regional water resources system is an attractive alternative to separate systems for each purpose and each group. However, a regional system imposes complex economic, financial, legal, socio-political, and organiza tional problems for the water resources professionals. This dissertation examines two problems associated with regional water resources planning that are typically treated separately, yet are closely related. The first problem involves performing an efficiency analysis to determine the economically efficient or optimal regional system that maximizes benefits minus costs. Once the optimal regional system is determined, a major task still remains to allocate project costs; therefore, an equity analysis must be performed to apportion project costs in an equitable manner. This second problem is viewed from the perspective of each purpose and each group because they must each be convinced that the optimal regional system is their best alternative; otherwise, voluntary participation will be difficult. No doubt, each purpose's and each 1 2 group's decision to participate in the optimal regional system depends on its allocated cost, and not necessarily on what is best for the region. The prevailing belief is that efficiency analysis and equity analysis are separate problems and, therefore, research has either focused entirely on efficiency analysis or equity analysis. Research on efficiency analysis has mainly been on the application of partial enumeration tech niques to find optimal regional systems, while research on equity analysis has continued to explore the application of concepts from cooperative game theory to allocate project costs. The purpose of this dissertation is to integrate efficiency analysis and equity analysis into a single regional water resources planning model characterized by economies of scale. The model to be presented incorporates a total enumeration procedure along with concepts from cooperative game theory for efficiency/equity analysis. The specific application is to determine the least cost regional water supply network and to determine a "fair" allocation of costs among the multiple users. Chapter 2 reviews selected works on efficiency analysis and equity analysis of water resources problems. Chapter 3 presents a reliable total enumeration procedure for effi ciency analysis of regional water supply network problems. However, unlike traditional partial enumeration techniques used for efficiency analysis that give only the optimal 3 solution, this procedure also gives all the optimal solu tions for each user and each subgroup of users which are necessary information to perform an equity analysis using concepts from cooperative game theory. In addition, this procedure gives all the suboptimal solutions. Chapter 4 shows how the information from the total enumeration procedure is used to perform an equity analysis of not only the optimal solution, but also "good" suboptimal solutions. Chapter 5 reveals how efficiency analysis and equity analysis are related. Finally, Chapter 6 summarizes the results and conclusions. CHAPTER 2 LITERATURE REVIEW Efficiency Analysis During the past two decades, the problem of finding the economically efficient or optimal regional water system has been extensively modeled as a mathematical optimization problem. A review of selected works on efficiency analysis of regional water systems that includes Converse (1972), Graves et al. (1972), McConagha and Converse (1973), Yao (1973), Joeres et al. (1974), Bishop et al. (1975), Jarvis et al. (1978), Whitlatch and ReVelle (1976), Brill and Nakamura (1978), and Phillips et al. (1982) indicates a variety of partial enumeration techniques, e.g., nonlinear programming, for finding optimal regional systems. These optimal regional systems can be a least cost system or a system that maximizes benefits minus costs. Generally, regional water resources planning problems exhibit economies of scale in cost and, therefore, involve nonlinear concave cost functions. Consequently, to a great extent, the selection of the partial enumeration optimization technique to apply to a particular problem depends on the characteri zation of the nonlinear concave cost functions. For instance, linear programming can be applied if the nonlinear 4 concave cost functions are represented by linear approximations. 5 Equity Analysis Unfortunately, successful regional planning is not merely knowing the optimal regional system but must also include an equity analysis to find an acceptable allocation of costs among the participants. Otherwise, the optimal system will be difficult to implement. Of the publications cited in the preceding paragraph, only McConagha and Converse (1973) dealt with both efficiency and equity in regional water planning. In addition to presenting a heuristic procedure for finding the least cost regional wastewater treatment facility for seven cities, they evalu ated the equity of several cost allocation procedures. Although they recognized that an equitable cost allocation should not charge any city or subgroup of cities more than the cost of an individual treatment facility, they did not include the possibility of subgroup formation in their analysis. Giglio and Wrightington (1972) introduced concepts from cooperative game theory as a way to consider the possibility of subgroup formation in allocating costs of water projects. However, their treatment of cooperative game theory was incomplete. Therefore, they concluded that the game theory approach rarely yields a unique cost allocation 6 and proceeded to recommend the separable costs, remaining benefits (SCRB) method or methods based on measure of pollu tion. Shortly thereafter, several researchers applied popular unique solution concepts from game theory like the Shapley value and the nucleolus to allocate the costs of regional water systems. Heaney et al. (1975) applied the Shapley value to find an equitable cost allocation of common storage units for storm drainage for pollution control among competing users. Suzuki and Nakayama (1976) applied the nucleolus to assign costs for a water resources development along Japan's Sakawa and Sagami Rivers. Loehman et al. (1979) used a generalization of the Shapley value to allocate the costs of a regional wastewater system involving eight dischargers along the lower Meramec River near St. Louis, Missouri. Subsequently, Heaney (1979) established that the fair ness criteria used for allocating costs in the water resources field and the concepts used in cooperative game theory are equivalent. Moreover, Straffin and Heaney (1981) showed that a conventional method for allocating costs used by water resources engineers is identical to a unique solu tion concept used by game theorists. More recently, Young et al. (1982) compared proportionality methods, game theoretic methods, and the SCRB method for allocating cost and concluded that the game theoretic methods may be too complicated while the SCRB method may give inequitable cost 7 allocations. Meanwhile, Heaney and Dickinson (1982) revealed why the SCRB method may fail to give equitable cost allocations and proposed a modification of the SCRB method that uses game theory concepts along with linear programming to insure an equitable cost allocation can be found if one exists. The possibilities of using concepts from cooperative game theory as a basis for allocating costs of water projects continue to develop. In fact, concepts from coop erative game theory are gaining acceptance in other fields as well. Researchers in accounting are looking toward coop erative game theory as a possible solution to the arguments by Thomas (1969, 1974) that any cost allocation scheme in accounting is arbitrary and hence not fully defensible. Recent works by Jensen (1977), Hamlen et al. (1977, 1980), Callen (1978), and Balachandran and Ramakrishnan (1981) applied concepts from cooperative game theory to evaluate the equity of existing and proposed cost allocation schemes in accounting. Meanwhile, in economics, concepts from cooperative game theory are frequently used as a basis for evaluating subsidy-free and sustainable pricing policies for decreasing cost industries, e.g., the work of Loehman and Whinston (1971, 1974), Faulhaber (1975), Sorenson et al. (1976, 1978), Zajac (1978), Panzar and Willig (1977), Faulhaber and Levinson (1981), and Sharkey (1982b). 8 Conclusions Three conclusions can be made from reviewing the literature on efficiency analysis and equity analysis of regional water resources planning. First, there is a gap in the research to jointly examine efficiency and equity in regional water resources planning. In spite of a continual effort to find economically efficient regional water systems and equitable cost allocation procedures, no published work incorporates both efficiency analysis and equity analysis in a single regional water resources planning model using realistic cost functions. Heaney et al. (1975) and Suzuki and Nakayama (1976) used linear cost models while Loehman et al. (1979) used conventional cost curves. Secondly, the cost allocation literature in the water resources field has consistently allocated the costs of treatment and piping together even though federal guidelines suggest that piping cost be allocated separately from treatment cost to the responsible users (Loehman et al., 1979; U.S. Environmental Protection Agency, 1976). Finally, the cost allocation literature has dealt with allocating the cost of the optimal system. However, situations in practice may require that "good" suboptimal systems be considered; therefore, an acceptable cost allocation procedure should be able to allocate the costs of several systems under consideration in an equitable manner. These three conclusions formed the basis for the research undertaken in this dissertation. Chapter 3 begins integrating efficiency analysis and equity analysis by searching for a computational procedure 9 to simultaneously perform an efficiency analysis and calculate all the necessary information to perform an equity analysis using concepts from cooperative game theory. CHAPTER 3 EFFICIENCY ANALYSIS Introduction The importance of both efficiency analysis and equity analysis in planning regional water resources systems is well recognized. Over the years, researchers have applied methods ranging from simple cost-benefit analysis to sophis ticated mathematical programming techniques to search for economically efficient or optimal regional water resources systems. Yet, the implementation of regional systems is difficult unless an equitable financial arrangement is found to allocate project costs among individuals (or partici pants) in a project. Until recently, a theoretically sound basis for allocating costs has eluded the water resources professional. However, there is increasing interest in using the theory of the core from cooperative n-person game theory as a basis for allocating costs, e.g., see Suzuki and Nakayama (1976), Bogardi and Szidarovsky (1976), Loehman et al. (1979), Heaney and Dickinson (1982), and Young et al. (1982). The theory of the core is based on principles of individual, subgroup, and group rationality. This means that no individual or subgroup of individuals should be allocated a cost in excess of the cost of nonparticipation, 10 11 while total cost must be apportioned among all individuals. The cost of nonparticipation is simply the cost that each individual and each subgroup of individuals must pay to independently acquire the same level of service by the most economically efficient means. As a result, to evaluate efficiency/equity for a regional system with n individuals, it is necessary to determine 2n-l optimal solutions. Although the close association between efficiency analysis and equity analysis is recognized, there have been few attempts to incorporate these two analyses in regional water resources planning. A typical efficiency analysis usually ends with determining the optimal solution for a problem without addressing cost allocation, and a typical equity analysis begins by assuming the 2n-l optimal solu tions are available to accomplish the cost allocation. This disjointed approach to efficiency/equity analysis is fostered by a belief that these two problems are independent (James and Lee, 1971; Loughlin, 1977). Furthermore, reliable techniques for finding the 2n-l optimal solutions to accomplish an efficiency/equity analysis of most problems encountered in actual practice are unavailable. This chapter begins by evaluating the applicability of partial and total enumeration techniques for finding the 2n-l optimal solutions for problems with different types of cost functions. Subsequently, a computational procedure is described to examine a regional water supply network problem 12 wherein we need to find the economic optimum and a "fair" allocation of costs among the individuals in the project. In order to do the cost allocation we need to find the costs of the optimal systems for each individual and each subgroup of individuals since these costs are going to be the basis for cost allocation. Partial Enumeration Techniques The difficulty of finding the optimal solution for a particular problem depends on the nature of the cost func tions. Generally, a cost function can be classified as either linear, convex, concave, S-shape, or irregular (see Figure 3-1). To find the optimal solution for problems with either linear or convex cost functions is straightforward using readily available and reliable linear programming codes. Accordingly, a vast body of overlapping theoretical results is available from classical economics and operations research, e.g., convex programming, for finding the optimal solution to problems with convex cost functions. However, problems with linear and convex cost functions are unable to characterize the economies of scale in cost typically encountered in regional water resources planning. The concave cost function is generally used to represent economies of scale, and several partial enumera tion techniques are available for dealing with this cost 13 Figure 3-1. Types of Cost Functions. 14 function. One approach surveyed by Mandl (1981) is separable programming which takes advantage of readily available linear programming codes by using a piecewise linear approximation of the concave cost function. Unfortunately, this approach is rather tedious to use and guarantees only a local optimal solution. A second approach is to retain the natural concave cost function and apply a general nonlinear programming code. However, according to surveys by Waren and Lasdon (1979) and Hock and Schittkowski (1983), general nonlinear programming codes may converge to local optima and may be subject to other failures, e.g., termination of code. A final approach used by Joeres et al. (1974) and Jarvis et al. (1978) is to approximate the concave cost function with several fixed-charge cost functions and apply a mixed-integer programming code. This approach guarantees a globally optimal solution, but standard mixed-integer programming codes are expensive to use. More importantly, unresolved problems remain as to how to properly define a fixed charge problem. If the fixed charge formulation is used because it is computationally expedient, then the resulting cost estimates may distort the cost allocation procedure. Given the current status of partial enumeration techniques for finding the optimal solutions to perform efficiency/equity analysis for problems with concave cost functions, one can conclude that other methods must be used. Obviously, this conclusion applies 15 to problems with S-shape and irregular cost functions as well. Total Enumeration Techniques Total enumeration techniques can be used to find the optimal solution for a problem regardless of the types of cost functions involved. The ability to handle irregular cost functions is especially important because this type of cost function is frequently used by state-of-the-art cost estimating models like CAPDET, i.e., Computer Assisted Procedure for Design and Evaluation of Wastewater Treatment Systems (U. S. Army Corps of Engineers, 1978) and MAPS, i.e., Methodology for Areawide Planning Studies (U. S. Army Corps of Engineers, 1980). For example, in MAPS, the cost function for constructing a force main is composed of separate cost functions for pipes, excavation, appurten ances, and terrain. Furthermore, each of these cost func tions is based on site-specific conditions. For instance, the cost function for pipe includes the cost of purchasing, hauling, and laying the pipe and depends on the material, diameter, length, and maximum pressure. No doubt, the composite site-specific cost function for a force main may be nonlinear, nonconvex, multimodel, and discontinuous. Another advantage with a total enumeration technique is that it presents and ranks all of the alternative solu tions. Unlike partial enumeration techniques which only 16 present the optimal solution for consideration, total enumeration techniques allow examination of suboptimal solutions which may be preferable when factors other than cost are considered. For example, proven engineering design or socio-political values are difficult to incorporate into an optimization model even if the problem is well defined, so the optimal solution may be so unrealistic that another solution must be selected. Depending on the size of the problem, a possible drawback with total enumeration techniques may be the compu tational effort to enumerate all possible solutions. However, for some problems, total enumeration may be the only meaningful approach. For these problems, the challenge with using a total enumeration approach is to find ways to reduce the computational effort by applying mathematical techniques or engineering considerations. After a discus sion on modeling network problems as digraphs, a total enumeration procedure that does not require extensive compu tational effort to find the least cost network for each individual and each group of individuals is presented. Modeling Network Problems as Digraphs Consider a situation wherein an existing water supply source, S, is going to serve n users with demands of Q^, Q2, . . Q respectively. Assume that the water source is able to supply the total demand by the n users without 17 facility expansion except for a new regional water network. Furthermore, consider a particular system with three users that can be served directly by the source, and engineering considerations, e.g., gravity flow, have determined that it is feasible to send water from user 1 to both user 2 and user 3, and from user 2 to user 3. For this particular system, assume the total cost function for constructing a pipeline is rather simple. From Sample (1983), the total cost function for constructing a pipeline is characterized by economies of scale and can be expressed as a linear function of distance and a nonlinear function of flow; or C = aQbL (3-1) where C = total cost of pipeline, dollars Q = quantity of flow, mgd L = length of pipeline, feet, and a, b = parameters, 0
Given this situation, the objective of the regional waterauthority is to determine the least cost water network for each user and each group of users in order to perform efficiency/equity analysis. This problem can be modeled as a digraph or directed graph (see Figure 3-2) consisting of nodes to represent the source and users, and directed arcs to represent all 18 Figure 3-2. Example Digraph Representing a Regional Water Network Problem for Three Users. 19 possible interconnecting pipelines. If water can be sent in either direction between two users, then the pipeline is represented by two oppositely directed arcs. Consequently, any regional water network problem can be modeled by a digraph. Before continuing, a few brief definitions and concepts are necessary since the nomenclature used in the network and graph theory literature is not standardized. A digraph or directed graph, D(X,A), consists of a finite set of nodes, X, and a finite set of directed arcs, A. A directed arc is denoted by (i,j) where the direction of the arc (shown by an arrow) is from node i to node j; node i is called the initial node and node j is called the terminal node. A subdigraph of D(X,A) has a set of nodes that is a subset of X but contains all the arcs whose initial and terminal nodes are both within this subset. A path from node i to node j is simply a sequence of directed arcs from node i to node j. An elementary path is a path that does not use the same node more than once. A circuit is an elementary path with the same initial and terminal node. A directed tree or an arborescence is a digraph without a circuit for which every node, except the node called the root, has one arc directed into it while the root node has no arc directed into it. A spanning directed tree of a digraph is a directed tree that includes every node in the digraph. If a cost, C(i,j) is associated with every arc 20 (i,j) of a digraph, then the cost of a directed tree is defined as the sum of the costs of the arcs in the directed tree. Finally, a minimum spanning directed tree of a digraph is the spanning directed tree of the digraph with the least cost. For the reader desiring more information regarding networks and graphs, numerous texts are available, e.g., Christofides (1975), Minieka (1978), and Robinson and Foulds (1980 ) . The problem of finding the least cost water network for each user and each group of users is the same as finding the minimum spanning directed tree rooted at node S for all possible subdigraphs as well as the digraph shown in Figure 3-2. In general, not every digraph has a spanning directed tree; however, for a realistic problem one can assume a pipeline is available to serve all individuals participating in a regional system. Thus, a spanning directed tree exists for digraphs representing realistic regional water network problems. Although algorithms are found in Gabow (1977) and Camerini et al. (1980a, 1980b) for finding the minimum spanning directed tree or the K best spanning directed trees, these algorithms assume a linear cost model in which the cost on each arc is given prior to initiating the algorithm. As a result, these algorithms are not applicable to problems with nonlinear costs on each arc. That is, the cost along each arc cannot be determined in advance 21 because the cost is a function of the quantity of flow along the arc; yet, the quantity of flow along the arc is a function of the path in which the arc belongs. The Total Enumeration Procedure The procedure for enumerating and calculating the costs of all the spanning directed trees for all possible sub digraphs as well as the digraph is based on recognizing that a large number of spanning directed trees of a digraph can be constructed from specific spanning directed trees of subdigraphs. These specific spanning directed trees are characterized by one arc emanating from the root node and are referred to as "essential spanning directed trees." In contrast, "inessential spanning directed trees" are charac terized by more than one arc emanating from the root node. The procedure sequentially calculates the costs of essential spanning directed trees for subdigraphs with increasing number of nodes, until the costs of essential spanning directed trees are calculated for all possible subdigraphs and for the digraph. Meanwhile, the cost of each inessen tial spanning directed tree for all possible subdigraphs as well as the digraph is calculated simply by summing the costs of essential spanning directed trees of subdigraphs that are associated with each arc emanating from the root node of the inessential spanning directed tree. That is, each arc emanating from the root node belongs to an 22 essential spanning directed tree of a subdigraph. By apply ing this procedure the costs of all the spanning directed trees can be systematically enumerated for all possible subdigraphs as well as the costs of all the spanning directed trees for the digraph. As a result, the least cost network for each user and each group of users is found. In the following discussion, "n-node" means the number of nodes, not including the root node, is n; e.g., an i-node digraph or subdigraph consists of i+1 nodes if the root node is counted. The total enumeration procedure for the n-node digraph is summarized by the flow diagram shown in Figure 3-3. Step 1 begins the procedure for evaluating all subdi graphs consisting of the root node and one other node, i.e., the 1-node subdigraphs. Step 2 initializes a count of the number of combina tions of i-node subdigraphs evaluated. Step 3 generates all possible combinations of i-node subdigraphs from the n-node digraph. The number of possible combinations is (^). For example, the 3-node digraph shown 3 in Figure 3-2 has (or three possible 2-node subdigraphs, i.e., subdigraphs consisting of the following sets of nodes {S,1,2 }, S,1,3), and (S,2,3>. Step 4 selects one i-node subdigraph not previously selected and enumerates all of its spanning directed trees. A spanning directed tree may not exist in a case where a path does not exist from the root node to every node in the 23 Figure 3-3. Flow Diagram of Total Enumeration Procedure for n-Node Digraph not every node in the i-node i-node subdigraph, i.e., subdigraph has an arc directed into it. Actually, only the essential spanning directed trees need to be enumerated. The enumeration of inessential spanning directed trees is simply done by finding all possible combinations of i-node digraphs from the entire set of essential spanning directed trees enumerated previously, i.e., all essential spanning directed trees for all possible subdigraphs of the i-node subdigraph. This process substantially reduces the effort involved in enumerating all the spanning directed trees for an i-node subdigraph because a large number of spanning directed trees are inessential. If the i-node subdigraph is unusually large and dense, algorithms are available in Chen and Li (1973), Christofides (1975), and Minieka (1978) for generating spanning directed trees. If necessary, a procedure in Chen (1976) can be used to compute the number of spanning directed trees of an i-node subdigraph or an n-node digraph. A directed tree matrix, M, is defined for a digraph, where equals the number of arcs directed into node i and nm ^ is equal to the negative of the number of arcs in parallel from node i to node j. The number of spanning directed trees rooted at node S for the digraph defined by M is given by the determinant of the minor submatrix resulting from deleting the Sth row and 25 column of M. Applying this procedure to the 3-node digraph in Figure 3-2 gives the following directed tree matrix. S 1 2 3 S 1 2 3 0 -1 -1 -1 0 1-1-1 0 0 2 -1 0 0 0 3 The determinant of the minor submatrix resulting from delet ing the Sth row and column is six, so there are six spanning directed trees rooted at node S for this digraph. Step 5 calculates the cost of each spanning directed tree enumerated in Step 4. The cost for each essential spanning directed tree is calculated independently. How ever, the cost for each inessential spanning directed tree is simply calculated by summing the costs of essential spanning directed trees of subdigraphs calculated previously that are associated with the arcs emanating from the root node. For inessential spanning directed trees the costs can be calculated along with the enumeration process described in Step 4. Step 6 ranks all the spanning directed trees for the i-node subdigraph according to cost. The minimum spanning directed tree is the least cost network for the users associated with the set of nodes for the i-node subdigraph. 26 Step 7 checks the counter to see if all possible combinations of i-node subdigraphs have been evaluated. If not, Step 8 advances the counter by one before returning to Step 4 to evaluate another i-node subdigraph. If all of the possible combinations of i-node subdigraphs have been evaluated, the procedure goes to Step 9 and begins the evaluation of subdigraphs with i+1 nodes. Step 10 checks if the n-node digraph has been evalu ated. If not, the procedure returns to Step 2 and proceeds to evaluate the subdigraphs with i+1 nodes; otherwise, the procedure terminates. The total enumeration procedure is illustrated in Table 3-1 using the regional water network problem modeled by the 3-node digraph shown in Figure 3-2. During the first iteration all combinations of 1-node subdigraphs are evaluated. For this simple case 3 three combinations, i.e., (^) = 3, are evaluated. Further more, each combination has only one spanning directed tree, and the one spanning directed tree is essential. As a result, the cost of the spanning directed tree for each combination must be calculated. Obviously each spanning directed tree is the least cost network for the associated user. During the second iteration, three combinations, 3 i.e., (2) =3, of 2-node subdigraphs are evaluated. In this case, each combination has two spanning directed trees, but the cost of only one spanning directed tree needs 27 Table 3-1. Example of Total Enumeration Procedure for 3-Node Digraph Iteration i-Node i Subdigraphs Spanning Directed Trees for i-Node Subdigraph Are Spanning Directed Trees Essential? i = l (S, 1} - Yes (S, 2} - (S, 3} i=2 {S,l,2} (D-^ {S, 1,3 } No Yes {S 2,3 } No Yes No 28 Table 3.1. Continued. Iteration i-Node i Subdigraphs Spanning Directed Are Spanning Trees for i-Node Directed Trees Subdigraph Essential? 29 to be calculated. The cost of the inessential spanning directed tree is simply found by summing the costs of the corresponding essential spanning directed trees calculated during the first iteration. The minimum spanning directed tree for each combination is the least cost network for the associated group of users. Finally, for the third itera tion, i.e., i=n, the 3-node digraph is being evaluated. This 3-node digraph has six spanning directed trees, and these six spanning directed trees can be enumerated by inspection. The four inessential spanning directed trees can be enumerated by simply finding all possible combinations of 3-node digraphs from the essential spanning directed trees generated during the first and second iterations. Thus, only two independent calculations are necessary to find the costs of the essential spanning directed trees. Mean while, the cost of the four inessential spanning directed trees is calculated simply by summing the costs of essential spanning directed trees for subdigraphs previously calcu lated during the first two iterations. For example, in Table 3-1, the cost for the inessential spanning directed tree consisting of the set of arcs {(S,3), (S,l), (1,2)} is determined by summing the costs of the two essential spanning directed trees consisting of the sets of arcs {(S,3) } and {(S,l), (1,2)} associated with the two sub digraphs consisting of the sets of nodes S,3} and {S,l,2}, respectively. Therefore, eight independent calculations are 30 necessary to find the costs of the six spanning directed trees for the digraph, and only two of the six spanning directed trees are essential. In fact, the eight indepen dent calculations enable us to find all 2n-l or seven optimal solutions necessary to perform efficiency/ equity analysis. Table 3-2 shows that the number of independent calculations necessary to find the cost of all the spanning directed trees for all possible subdigraphs is simply equal to the number of independent calculations to find the cost of all the spanning directed trees for the digraph less the number of essential spanning directed trees for the digraph. Consequently, for our 3-node digraph, six independent calculations are necessary to find the optimal solution for each user and each subgroup of users. For the balance of this chapter, the optimal solution for each user and each subgroup of users will be referred to as the 2n-2 optimal solutions. Finally, all suboptimal solutions are enumerated for all possible subdigraphs as well as for the digraph. Computational Considerations Although the number of independent calculations neces sary to find the costs of all the spanning directed trees for all possible subdigraphs as well as the digraph is uniquely determined by the configuration of the digraph, we can get a sense of the computational effort by examining the 31 Table 3-2. The Number of Independent Calculations to Find the Costs of Spanning Directed Trees for All Possible Subdigraphs. Independent Calculation Is Independent Calcu lation Used to Find the Costs of Spanning Directed Trees for the Digraph? Is Independent Calcu lation Used to Find the Costs of Spanning Directed Trees for All Possible Subdigraphs? -* Yes Yes 0H0) Yes Yes (D-KD Yes Yes Yes Yes Yes Yes -- Yes Yes dX Yes No Ql v ) Yes No Total Number of Yes 8 6 32 three digraphs shown in Figure 3-4. For the 3-node digraph, six independent calculations are necessary to find the costs of the four spanning directed trees for the digraph, and only one of the four spanning directed trees is essential. More importantly, 12 calculations are necessary to find the seven optimal solutions, but only 6 of the 12 calculations (50%) are independent. Furthermore, only five independent calculations are necessary to find the 2n-2 optimal solu tions. For the 4-node digraph, 10 independent calculations are necessary to find the cost of the eight spanning directed trees for the digraph, and only one of the eight spanning directed trees is essential. For this digraph, 33 calculations are necessary to find the 15 optimal solutions, but only 10 of the 33 calculations (30%) are independent. Moreover, only nine independent calculations are necessary to find the 2n-2 optimal solutions. Finally, for the 5-node digraph, 19 independent calculations are necessary to find the costs of the 24 spanning directed trees for the digraph, but only 2 of the 24 spanning directed trees are essential. In this case, 109 calculations are necessary to find the 31 optimal solutions, but only 19 of the 109 calculations (17%) are independent. From these 19 independent calculations, only 17 are necessary to find the 2n-2 optimal solutions. As we can see, summarized in Table 3-3, a large number of the spanning directed trees of a digraph are inessential. 33 gur 3~4 Â£* aPles f 3 ,4,5~Uocl e Di 9riPhi Table 3- 3. Summary of Computational Effort for Digraphs Shown in Figure 3-4. Digraph 2n-l Optimal Solutions Number of Spanning Directed Trees Number of Inessential Spanning Directed Trees Number of Calculations to Find 2n-l Optimal Solutions Number of Independent Calculations to Find 2n-l Optimal Solutions (%) Number of Independent Calculations to Find 2n-2 Optimal Solutions 3-node 7 4 3 12 6 (50%) 5 4-node 15 8 7 33 10 (30%) 9 5-node 31 24 22 109 19 (17%) 17 u> 35 Also, the percentage of independent calculations decreases as the number of nodes for a digraph increases. The 5-node digraph in Figure 3-4 shows that the actual number of independent calculations necessary to determine the 31 optimal solutions to perform efficiency/equity analy sis of a regional water network problem involving five users is rather small. In fact, a regional water network serving five users may be considered a fairly large network. As larger systems form, increases in transactions costs because of multiple political jurisdictions, growing administrative complexity, etc., may eventually offset the gains from a regional system. In any case, real regional water network problems probably involve fairly small and sparse networks. That is, large networks can usually be broken down into smaller networks for analysis based on natural geographical and hydrological features, political boundaries, etc. Also, in actual problems there may not be that many choices for routing pipelines. Thus, the number of independent calculations necessary to calculate the 2n-l optimal solutions for a realistic regional water network should not be unreasonable. One of the advantages of using this total enumeration procedure is that it can be accomplished on a personal computer using readily available software. Thus, decision makers involved with planning and negotiating a regional water network can have easy access to information to aid the 36 decision-making process. For instance, the procedure can be implemented using the extremely "user friendly" Lotus 1-2-3 spreadsheet software package. Lotus 1-2-3 has the mathematical functions to handle calculations involving nonlinear cost functions or involving detailed cost analysis. A sample Lotus 1-2-3 printout is shown in Table 3-4 for a hypothetical water network problem modeled by the 3-node digraph shown in Figure 3-2. This printout should be self-explanatory. The top portion of the printout contains the data for the problem, and the bottom portion is the calculations associated with the total enumeration pro cedure. The sorting capabilities of Lotus 1-2-3 allow automatic ranking of all the feasible solutions according to cost. Moreover, the Lotus 1-2-3 electronic spreadsheet automatically recalculates all values associated with a formula whenever a new value is entered or an existing value is changed. This automatically gives the total enumeration procedure the capability for sensitivity analysis. For example, the set of all feasible solutions ranked according to cost can be evaluated as the economies of scale, as represented by the value of b in equation (3-1), is varied over a specific range of values. Thus, for a regional network problem of realistic size, all the feasible solutions can be enumerated using a spreadsheet software package. 37 Table 3-4. Efficiency Analysis of a Three-User Water Supply Network with Nonlinear Cost Function Using Lotus 1-2-3. Sara Distance : L(i,j) is the distance in feet from i to j L (S, 1) = L(S,2) = 17320 L(S,3)= 26000 L(1,2)= 3025k L (1,3) = 13130 L (2,3) = 19673 15500 Demand : Q(i) is the demand in mgd for user i Q(1)= 1 Q(2) = Cost Function: a(Q~b)L a= 5 Q (3) = 38 b= 3.51 Calculations With C(i..j)[x]= Cost of network [x] for Total Enumeration i..j ; C(i..j)= L Procedure east cost C(1)[S1]= 646000 C(2)[S2]=2463343. C(3)[S3]=2012935. C(12)[SI,12]= 2984140. C (12)[SI;S2]= 3109348. C(12)= 2934140. C(13)[SI,13]= 2618975. C(13)[S1;S3]= 2658986. C(13)= 2618975. C(23) [S2,23]= 4061294. C(23)[32;S3]= 4476835. C(23)= 4061294. C(123)[SI,12,23]= 4648439. C (123)[SI,12;S3]= 4997126. C(123) [SI,12,13]= 4640756. C (123)[SI,13;S2]= 5032324. C (123) [SI;S2,23]= 4737294. C(123)[SI;S2;S3]= 5122335. C(123)= 4640756. Sort C(123) in ascending order Paths Cost C(123)[SI,12,13]= 4643755. C (123) [SI,12,23]= 4548439. C(123)[S1;S2,23]= 4707294. C (123)[SI,12;S3]= 4997126. C(123)[S1,13;S2]= 5082824. C(123) [S1;S2;S3]= 5122835. BEST C(123)= 4540756 38 Summary A total enumeration procedure for finding the optimal solutions necessary for efficiency/equity analysis of realistic regional water network problems is presented. The procedure can be easily understood and applied by engineers with little knowledge or experience in operations research techniques. Furthermore, the procedure allows the engineers to handle all problems regardless of the types of cost function involved or to perform detailed cost analysis. Finally, if the optimal solution is impractical for implementation, all suboptimal solutions ranked according to cost are readily available for consideration. CHAPTER 4 EQUITY ANALYSIS Introduction Proposed regional water resources systems involve multiple purposes and groups who must somehow share the cost of the entire project. The project may focus on construc tion of a large dam which serves numerous purposes such as water supply, flood control, and recreation. Also, canals from the dam direct the water to nearby users. A signifi cant portion of the total cost of this project may involve elements which serve more than one purpose and/or group. These costs are referred to as joint or common costs. In such cases, it is possible to find the optimal or the most economically efficient regional system, i.e., the one that maximizes benefits minus costs. However, a major effort remains to somehow apportion the project cost in an equitable manner. In fact, the importance of the financial analysis to apportion project cost is not limited to the optimal system but includes any other integrated systems being considered for implementation as well. This chapter examines principles of cost allocation using concepts from cooperative n-person game theory. An 39 40 example regional water network is used to illustrate these principles. Cost Allocation for Regional Water Networks A hypothetical situation similar to options contained in the West Coast Regional Water Supply Authority's master plan for Hillsborough, Pasco, and Pinellas counties in Florida (Ross et al., 1978) is now considered. Phase I (1980-1985) of the plan recommends the use of groundwater from existing and newly developed well fields to satisfy water demands in the tri-county area. For this hypothetical problem, assume that an existing well field is the most high quality and cost effective water supply source (S) available for three counties (1, 2, and 3) with projected demands of 1, 6, and 3 million gallons per day (mgd), respectively. The demand for each county is based on projected population growth and average per capita demand over a period of 5 years (see Table 4-1). Assume that the existing well field is currently operating below its capacity of 20 mgd and can satisfy the additional 10 mgd demanded by the three counties. In addition, assume that no facility expansion is required except for a new regional water network. Further more, each county can be served directly by the well field, and engineering considerations, e.g., gravity flow, have determined that water can be sent from county 1 to both county 2 and county 3, and from county 2 to county 3. The 41 Table 4-1. Projected Population Growth and Projected Average Per Capita Demand. County Projected Population Growth Projected Average Per Capita Demand (gal/cap-day) Projected Additional Demand (mgd) 1 8,000 125 1 2 40,000 150 6 3 18,750 160 3 Total 66,750 10 Weighted Average ... 150 .. 42 lengths of all possible interconnecting pipelines are shown in Figure 4-1. For our hypothetical problem, assume that the total cost of constructing a pipeline has strong economies of scale and is C = 38Q'^L, where C is total cost of pipeline in dollars, Q is quantity of flow in mgd, and L is the length of pipeline in feet. Given the problem just described, the cost of a pipe line serving county 1 alone is $646,000; the cost of a pipeline serving county 2 alone is $2,420,095; and the cost of a pipeline serving county 3 alone is $1,990,992. The total cost for three individual pipelines is $5,057,087. However, when the costs for all the options available to these three counties are enumerated using the procedure outlined in the preceding chapter, we see that the counties can do better by cooperating (see calculations in Appendix A using Lotus 1-2-3). There may be a slight difference between the numbers used in the text and the numbers in Appendix A because of rounding off. Also, cost data are only significant to the nearest thousand dollars. If the three counties cooperate, they can construct the least cost or optimal network consisting of pipelines from the well field to county 1, from county 1 to county 2, and from county 2 to county 3 (see Table 4-2). This optimal network costs $4,556,409 and represents a savings of 9.9% or $500,678 when compared with the cost for three individual pipelines. Obviously, constructing the optimal network is & 3 TC^ -V l>e ,rv<3 f te* c ,t^' ec O- y<5 f i-Q ue 44 Table 4-2. The Costs and Percent Savings for All Options. Option (Rank) Cost ($) Savings (%) 1 4,556,409 9.90 2 4,556,826 9.89 3 4,630,177 8.44 4 4,919,503 2.72 5 5,006,734 1. 00 6 5,057,087 0 45 in the best interest of the three counties, but to implement this least cost network, an equitable way to allocate the cost among the three counties must be found. This financial problem is known as a cost allocation problem. The complex ity is introduced because the counties share common pipes. Criteria for Selecting a Cost Allocation Method Several sets of criteria for selecting a cost alloca tion method are found in the literature. For the water resources field, criteria for allocating costs date back to the Tennessee Valley Authority (TVA) project in 1935 when prominent authorities were brought together to address the cost allocation problem. They developed the following set of criteria for allocating costs (Ransmeier, 1942, pp. 220-221) : 1. The method should have a reasonable logical basis. It should not result in charging any objective with a greater investment than the fair capitalized value of the annual benefit of this objective to the consumer. It should not result in charging any objective with a greater invest ment than would suffice for its development at an alternate single purpose site. Finally, it should not charge any two or more objectives with a greater investment than would suffice for alternate dual purpose or multiple purpose improvement. 2. The method should not be unduly complex. 3. The method should be workable. 4. The method should be flexible. 5. The method should apportion to all purposes present at a multiple purpose enterprise a share in the overall economy of the operation. 46 This set of criteria developed for the water resources field is similar to the following set of criteria proposed by Claus and Kleitman (1973) for allocating the cost of a network: 1. The method must be easy to use and under standable to users. They must be able to predict the effects of changes in their service demands. 2. The method must have stability against system breakup. It should not be an advantage to one or more users to secede from the system. Thus, there are limits to which a method can subsidize one user or class of user at the expense of others. 3. It is desirable, though not necessary, that the costing be stable under evolutionary changes in the system or under mergers of users. 4. It is again desirable that the method should preserve the substance and appearance of non discrimination among users. 5. If the method represents a change from present usage it is desirable that transition to the new method be easy. From these two sets of criteria, the most important criterion for selecting a method to allocate the cost of a regional water network is the method's ability to ensure stability or prevent breakup of the network. That is, the method should not allocate cost in a manner whereby an individual or a subgroup of individuals can acquire the same level of service by a less expensive alternative. Other wise, the individual or subgroup of individuals will con sider their allocated cost inequitable or unfair and secede from the regional network for a less expensive alternative. Heaney (1979) has expressed these fairness criteria for an equitable cost allocation mathematically as follows: 1) x(i) < minimum [b(i), c(i)] VieN (4-1) where x (i) cost allocated to individual i b (i ) benefit of individual i c(i) = the alternative cost to individual i of independent action, and N set of all individuals; i.e., N = {1,2 . ,n }. r r This criterion simply means that individual i should not be charged a cost greater than the minimum of individual i's benefit and alternative cost for independent action. 2) Z x(i) _< minimum [b(S), c(S)] V Scn i eS (4-2) where c(S) = alternative cost to subgroup S of independent action, and b(S) = benefit of subgroup S. This second criterion extends the first criterion to include subgroup of individuals as well. These two fairness criteria are now used to evaluate some simple and seemingly fair cost allocation schemes for our regional water network problem. Throughout this chapter, we will assume for our regional water network problem that each county's and each 48 subgroup of counties1 alternative cost of independent action is less than or equal to each county's and each subgroup of counties' benefits, respectively; i.e., c(i) = minimum [b(i), c(i)] V ieN, and (4-3) c(S) = minimum [b(S), c(S)] V ScN. Ad Hoc Methods Over the years, many ad hoc methods have been proposed or used to apportion the costs of water resources projects (Goodman, 1984). In general, ad hoc methods used in the water resources field for allocating costs can be described as follows: allocate certain costs that are considered identifiable to an individual directly and prorate the remaining costs, i.e., total project cost less the sum of all identifiable costs, among all the individuals in the project by some physical or nonphysical criterion. Mathe matically, this can be expressed as follows: x ( i ) = x(i)id + iMi)*rc (4-4) where x (i) x (i) id ip (i) rc cost allocated to individual i, costs identifiable to individual i, prorating factor for individual i, and remaining costs, i.e., c(N) 49 Furthermore, the requirement that Z iM i) = 1.0 should be ieN obvious. James and Lee (1971) summarize 18 ways for allocating the costs of water projects depending on the definition of identifiable costs and the basis for prorating the remaining costs (see Table 4-3). Basically, the differences among these 18 methods are the following three ways of defining identifiable costs: 1) zero, 2) direct or assignable costs, or 3) separable costs; and the following six ways of prorating remaining costs: 1) equal, 2) unit of use, 3) priority of use, 4) net benefit, 5) alternative cost, or 6) the smaller of net benefit or alternative cost. The next two sections analyze the effects of defining identifiable costs as either zero or direct costs. A detailed treatment of separable costs, i.e., the difference between total project costs with and without an individual, is given in the section on the separable costs, remaining benefits method. Defining Identifiable Costs as Zero The simplest way to allocate costs is to define identi fiable costs as equal to zero and prorate total project cost by some physical or nonphysical criterion. For example, population and demand are two ways to prorate total project 50 Table 4-3. Cost Allocation Matrix. Definition of Identifiable Cost Basis for Prorating Remaining Costs A. Zero B. Direct Cost C. Separable Cost a. Equal Aa Ba Ca b. Unit of Use Ab Bb Cb c. Priority of Use Ac Be Cc d. Net Benefit Ad Bd Cd e. Alternative Cost Ae Be Ce f . Smaller of d. or e. Af Bf Cf Source: Modified from James and Lee, 1971, p. 533. 51 cost (Young et al., 1982). Using these two ways to prorate the cost of the optimal network for our regional water network problem gives the following cost allocations (see calculations in Table 4-4 and Table 4-5): Proportional to Population County 1 $ 546,769 County 2 2,733,845 County 3 1,275,795 $4,556,409 Proportional to Demand County 1 $ 455,641 County 2 2,733,845 County 3 1,366,923 $4,556,409 Although these cost allocations are simple to calculate and easy to understand, they fail to implement the optimal network because county 2 considers these cost allocations unfair. In contrast to counties 1 and 3, county 2 loses money by being allocated a cost in excess of its go-it-alone costs using either of these two methods. Consequently, county 2 would rather acquire a pipeline by itself than cooperate with counties 1 and 3 to construct the optimal network. The principal failure with these proportionality Table 4-4 Cost Allocation of Optimal Network Based on Population. County i Population Percent of Total Population Allocated Cost ($) x (i) Go-It-Alone Cost ($) c (i ) Is x ( i ) < c ( i ) ? 1 8,000 12 546,769 646,000 Yes 2 40,000 60 2,733,845 2,420,095 No 3 18,750 28 1,275,795 1,990,992 Yes Total 66,750 100 4,556,409 5,057,087 Ln N) Table 4-5. Cost Allocation of Optimal Network Based on Demand County i Demand (mgd) Percent of Total Demand Allocated Cost ($) x (i) Go-It-Alone Cost ($) c (i ) Is x(i) < c(i)? 1 1 10 455,641 646,000 Yes 2 6 60 2,733,845 2,420,095 No 3 3 30 1,366,923 1,990,992 Yes Total 10 100 4,556,409 5,057,087 U1 OJ 54 methods is that they do not recognize explicitly each individual's contribution to total project cost. Defining Identifiable Costs as Direct Costs A way to recognize each individual's contribution to total project cost is by defining identifiable costs as those costs that can be directly assigned, and prorating the remaining costs by some physical or nonphysical criterion such as use or number of individuals; i.e., x (i) = x(i)direct + 4(i)-re (4-5) where x(i)-,. = direct cost or assignable cost direct -7 to individual 1. Although this direct costing approach intuitively seems fair, inequitable and unpredictable cost allocations can result. To illustrate, two direct costing methods are applied to our regional water network problem. A common approach to allocating remaining costs is by some physical measure of each individual's use of the common facilities; this method is generally referred to as the use of facilities method (Loughlin, 1977; Goodman, 1984). This traditional method is easy to understand and apply because quantitative information on a physical measure of use is generally available. In the water resources field, use can be measured in terms of the storage capacity and/or the 55 quantity of water flow provided by the common facilities. For our regional water network problem, the flow to each county is the obvious measure of use to apportion the costs of common pipelines since the assumed cost function depends on the flow. In the case of the optimal network, the only direct cost is the cost of the pipeline from county 2 to county 3 serving county 3, and the use of facilities method gives the following cost allocation (see calculations in Table 4-6). $ 204,283 2,221,299 2,130,827 $4,556,409 County 1 County 2 County 3 Total Unfortunately, this cost allocation does not implement the optimal network because county 3 can do substantially better by going alone, i.e., $1,990,992 versus paying $2,130,827. In addition to giving an inequitable cost allocation for the optimal network, the use of facilities method can promote noncooperation if other networks are also being considered. Table 4-7 shows the cost allocations for all possible options available to the three counties using the use of facilities method. Suppose the "second best" network or option 2 is also being considered by the counties. The second best network consists of the pipelines from the well field to county 1, from county 1 to county 2, and from county 1 to county 3. This second best network costs Table 4-6. Cost Allocation of Optimal Network with Use of Facilities Method Pipeline S-l 1-2 2-3 Total Cost ($) Go-It-Alone Cost ($) c (i ) Length (f t) 17,000 13,100 15,500 Q (mgd) 10 9 3 Pipeline Cost ($) 2,042,832 1,493,400 1,020,177 4,556,409 Cost for County 1 ($) Q=1 mgd 204,283 0 0 204,283 646,000 Cost for County 2 (?) Q=6 mgd 1,225,699 995,600 0 2,221,299 2,420,095 Cost for County 3 ($) Q=3 mgd 612,850 497,800 1,020,177 2,130,827 1,990,992 Ln 57 Table 4-7. Cost Allocation for the Use of Facilities Method. Cost Allocation to ($) County i Zx (i) Is Cost Allocation Option (Rank) County 1 x(l) County 2 x (2 ) County 3 x (3 ) ($) Equitable? 1 204,283 2,221,299 2,130,827 4,556,409 No x(3)>c(3 ) 2 204,283 2,445,055 1,907,488 4,556,826 No x(2)>c(2 ) 3 646,000 1,976,000 2,008,177 4,630,177 No x(3)>c(3 ) 4 244,165 2,684,346 1,990,992 4,919,503 No x(2)>c(2) 5 323,000 2,420,095 2,263,639 5,006,734 No x(3)>c(3) 6 646,000 2,420,095 1,990,992 5,057,087 (4) (5) (6) 58 $4,556,826 or $417 more than the optimal network; so, both networks are essentially comparable in cost, and either network might be considered the least cost network. In fact, the second best network becomes the optimal network if the economies of scale or the value of b in the cost function is .51 instead of .50 (see Table 3-4). Never theless, applying the use of facilities method to this second best network gives the following cost allocation. $ 204,283 2,445,055 1,907,4 88 $4,556,826 County 1 County 2 County 3 In this case, the cost allocation fails to implement the second best network because county 2 is better off going alone, i.e, paying $2,420,095 rather than $2,445,055. Furthermore, if we examine the cost allocation for the optimal network and the second best network, another problem is evident. Although the costs for the two networks are $417 apart, the difference in costs between the two networks for county 2 and county 3 is enormous. Consequently, this cost allocation method imposes another obstacle for the counties to cooperate and implement either one of the two networks. County 2 strongly opposes the second best network because of its substantially higher cost while county 3 strongly opposes the optimal network for the same reason. This problem is even more serious when more options are considered by the counties. Table 4-7 indicates tremendous 59 differences in allocated cost for each county depending on the network, thereby making cooperation very difficult. This situation shows the danger for individuals to simply accept the least cost network without carefully examining all of their options if the use of facilities method for allocating costs is chosen. Another simple way to prorate the remaining costs is to divide it equally among the individuals associated with the common facilities (see calculations for optimal network in Table 4-8). Table 4-9 shows the cost allocations using this egalitarian approach and indicates that none of the cost allocations for options with savings are equitable. At first glance, the cost allocation for option 5 appears equitable because each county is charged a cost less than or equal to its go-it-alone cost. However, closer examina tion reveals that counties 1 and 2 can do better as a coalition. They can construct a pipeline from the well field to county 1 and from county 1 to county 2, i.e., option 4, for $2,928,511 rather than pay the sum of their costs for option 5, i.e., $3,066,095. Unfortunately, a transition from option 5 to option 4 causes county 1 to lose money, i.e., $854,577 for option 4 versus $646,000 for option 5. To further complicate matters, option 5 only gives a 1% savings and requires county 1 to cooperate with county 3 to build a pipeline without getting any savings. Table 4-8. Cost Allocation of Optimal Network with Direct Costing/Equal Apportionment of Remaining Costs Method. Pipeline S-l 1-2 2-3 Total Cost ($) Go-It-Alone Cost($) c (i ) Length (ft) 17,000 13,100 15,500 Q (mgd) 10 9 3 Pipeline Cost ($) 2,042,832 1,493,400 1,020,177 4,556,409 Cost for County 1 ($) Q=1 mgd 680,944 0 0 680,944 646,000 Cost for County 2 ($) Q=6 mgd 680,944 746,700 0 1,427,644 2,420,095 Cost for County 3 ($) Q=3 mgd 680,944 746,700 1,020,177 2,447,821 1,990,992 o Table 4-9. Cost Allocation for Direct Costing/Equal Apportionment of Remaining Costs Method 61 Cost Allocation to County i Option (Rank) County 1 x (1) ($) County 2 x (2 ) County 3 x (3) f x ( i ) ($) Is Cost Allocation Equitable? 1 680,944 1,427,644 2,447,821 4,556,409 No X(1)>c(1) x(3)>c(3) 2 680,944 1,900,300 1,975,582 4,556,826 No x(l)>c(l) 3 646,000 1,482,000 2,502,177 4,630,177 No x(3)>c(3 ) 4 854,577 2,073,934 1,990,992 4,919,503 No X(1)>c(1) 5 646,000 2,420,095 1,940,639 5,006,734 No x(1)+x(2)> c (12 ) 6 646,000 2,420,095 1,990,992 5,057,087 (4) (5) (6) 62 Given these observations, the stability of option 5 as a regional water network is at best questionable. Again, if the allocated costs for counties 2 and 3 for the optimal network are compared to the second best network, a similar situation like the one discussed for the use of facilities method exists. That is, counties 2 and 3 face substantially different costs for these two networks with comparable costs. Thus, assigning direct costs does not help eliminate inequitable cost allocations. In fact, direct costing methods can impose additional obstacles to cooperation. This occurs because the assignment of direct costs depends on the configuration of the facilities. For instance, the cost of the pipeline from county 2 to county 3 for our regional water network problem can be a direct cost or a joint cost depending on the network. The cost of the pipe line is a direct cost for county 3 if the second best network, i.e., option 2, is being considered; yet, the cost of the pipeline is a joint cost for counties 2 and 3 if the optimal network, i.e., option 1, is being considered. These changes in the cost classification for the pipeline from county 2 to county 3 contribute to the tremendous difference in the cost allocations for counties 2 and 3 for the two comparable cost networks. This situation indicates an additional criterion not addressed by Claus and Kleitman (1973) for selecting a procedure to allocate network cost. 63 The cost allocation procedure should be independent of network configuration; otherwise, the cost allocation pro cedure can promote noncooperation if more than one network is being considered. In summary, two approaches for allocating costs in the water resources field have been examined: 1) allocate total project cost in proportion to a physical or nonphysical criterion; or 2) allocate assignable costs directly and prorate the remaining costs by a physical or nonphysical criterion. In general, these two approaches are simple to apply and easy to understand. In fact, these two approaches are currently accepted cost allocation methods used in accounting (Kaplan, 1982). However, these two approaches are unable to consistently provide an equitable cost allocation when an equitable cost allocation exists, i.e., sometimes these methods work and sometimes they fail. Furthermore, methods attempting to assign costs directly may be influenced by the configuration of the facilities and may discourage cooperation when more than one configuration is being considered. This is particularly evident for our regional water network problem. For a theoretically sound method that is able to find an equitable cost allocation if one exists and is not influenced by the configuration of the facilities, concepts from cooperative n-person game theory are necessary. 64 Cooperative Game Theory Game theory has been with us since 1944 when the first edition of The Theory of Games and Economic Behavior by John Von Neumann and Oskar Morgenstern appeared. In particular we are interested in games wherein all of the players voluntarily agree to cooperate because it is mutually bene ficial. Furthermore, games are studied in three forms or levels of abstraction. The extensive form requires a com plete description of the rules of a game and is generally characterized by a game tree to describe every player's move. A game in normal form condenses the description of a game into sets of strategies for each player and is represented by a game matrix. However, most efforts in cooperative game theory have been with games in charac teristic function form whereby the description of a game is in terms of payoffs rather than rules or strategies. The characteristic function form appears to be the most appropriate for studying coalition formation which is an essential feature in cooperative games. Also, cooperative games can be of three types depending on whether the game is defined in terms of costs, savings, or values. To keep the notation as simple as possible, only cost games will be discussed. Introductory and intermediate material on coop erative game theory can be found in Schotter and Schwodiauer (1980), Jones (1980), Luce and Raiffa (1957), Lucas (1981), Rapoport (1970), Shubik (1982), and Owen (1982). 65 Concepts of Cooperative Game Theory Let N = {1,2,...,n} be the set of players in the game. Associated with each subset of S players in N is a charac teristic function c, which assigns a real number c(S) to each nonempty subset of S players. For cost games, the characteristic function, c(S), can be defined as the least cost or optimal solution for the S-member coalition if the N-S member or complementary coalition is not present. However, depending on how the problem is defined, alterna tive definitions for c(S) may be required. For example, Sorenson (1972) presents the following four alternative definitions for the characteristic cost function: c^S) value to coalition if S is given preference over N-S. C2(S) = value of coalition to S if N-S is not present, c^iS) = value of coalition in a strictly competitive game between coalition S and N-S, and c^(S) = value of coalition to S if N-S is given preference. If c(S) can be defined as the least cost solution for coalition S if N-S is not present, then the cost game is naturally subadditive; i.e., c(S) + c(T) > c(SUT) ShT = 0, S,TcN (4-6) 66 where 0 is the empty set; and S and T are any two disjoint subsets of N. Subadditivity is a natural consequence of c(S) because the worst S and T can do as a coalition is the cost of independent action; i.e., c(S) + c(T) = c(SUT) SOT = 0, S,TCN. (4-7) A coalition in which the players realize no savings from cooperation is said to be inessential. General reasons why subadditivity exists are discussed by Sharkey (1982a). The primary reason why subadditivity exists for our regional water network problem is because of the economies of scale in pipeline construction cost. For a single output cost function, C(q), economies of scale is defined by C(Aq) < Ac(q) (4-8) where q = output level, and for all A such that 1 < A Â£ 1 + e, e is a small positive number. This definition means that the average costs are declining in the neighborhood of the output q. From Sharkey (1982a), economies of scale is sufficient but not a necessary condi tion for subadditivity. Subadditivity is a more general 67 condition which allows for both increasing marginal cost and increasing average cost over some range of outputs. Solution concepts for cooperative cost games involve the following three general axioms of fairness (Heaney and Dickinson, 1982; Young et al., 1982); 1) Individual Rationality: Player i should not pay more than his go-it-alone cost, i.e., x(i) < c(i), Â¥ ieN, (4-9) where x(i) is the allocated cost or the charge to player i. 2) Group Rationality: The total cost of the grand coalition, c(N), must be apportioned among the N players; i.e., Ex(i)=c(N). (4-10) ieN 3) Subgroup Rationality: This final axiom extends the notion of individual rationality to include subgroups, i.e., no subgroup or subcoalition S should be apportioned a cost greater than its go-it-alone cost, or 1 x(i) < c(S), V SCN. (4-11) is S The set of solutions or charges satisfying the first two axioms is called the set of imputations, while the 68 additional restriction of the third axiom defines what is known as the core of the game. For subadditive cost games the set of imputations is not empty, but the core may be empty. Shapley (1971) has shown that the core always exists for convex games. A cost game is convex if c(S) + c(T) > c(SUT) + c(ShT) SOT i 0, V S,TcN (4-12) or equivalently, convexity can be written as c(SUi) c(S) > c(TUi) c(T) ScTcN {i}, ieN. (4-13) Convexity simply means the incremental cost for player i to join coalition T is less than or equal to the incremental cost for player i to join a subset of T. This notion of convexity is analogous to economies of scale and implies the game has a particular form of increasing returns to scale in coalition size. As will be shown, the more attractive the game, i.e., larger savings in project costs, the greater the chance that the game is convex; whereas, if the game is less attractive, i.e., lower savings in project costs, the poten tial for a nonconvex game or an empty core game is greater. To illustrate the concept of the core, assume a three- person cost game with the following characteristic function values: 69 c(l) = 35 c (2 ) = 45 c ( 3 ) = 50 c(12 ) = 66 c (13 ) = 75 c(23 ) = 87 c(12 3 ) = 100 This game is subadditive so each player has an incentive to cooperate; i.e., c(l) + c(2) + c ( 3 ) > c(12 3 ) c(l) + c (23 ) > c(123 ) c (2 ) + c (13 ) >_ c (123) c ( 3 ) + c (12 ) > c(123) c (1) + c (2 ) > c (12 ) c(l) + c (3 ) > c (13 ) c (2 ) + c (3 ) > c ( 23 ) . Furthermore, this game is convex; i.e., c (12 ) + c (13 ) > c(12 3 ) + c (1) c (12 ) + c (23 ) > c(12 3 ) + c (2 ) c (13 ) + c ( 23 ) > c(12 3 ) + c ( 3 ) . Using the three general axioms of fairness, the core conditions are as follows: x(l) < 35 x ( 2) < 45 x ( 3 ) < 50 x(l) + x(2) < 66 x(l) + x(3) < 75 x(2) + x(3) < 87 x(l) + x(2) + x(3) = 100. 70 The first three conditions determine the upper bounds on x(i), i = 1,2,3, while the last four conditions determine bounds on x(i), i = 1, r 2 , r 3 . e. t c(123 ) - c(23) = 13 < x(l) < 35 = c(l) c(123 ) - c(13 ) = 25 < x (2 ) < 45 = c ( 2 ) c(12 3 ) - c(12 ) = 34 < x (3) < 50 = c ( 3 ) For a three-person game, graphical examination of the core conditions and the nature of the charge vectors is possible using isometric graph paper (Heaney and Dickinson, 1982). As shown on Figure 4-2, each player is assigned a charge axis. The plane of triangle ABC, with vertices (100,0,0), (0,100,0), and (0,0,100), represents points satisfying group rationality (axiom 2); whereas, the smaller triangle abc represents the set of imputations satisfying both individual rationality (axiom 1) and group rationality (axiom 2). The vertices a, b, and c represent the charge vectors: [35, 15, 50], [5, 45, 50], and [35, 45, 20], respectively. Line ab represents the upper bound for player 3, i.e., x(3) = c(3), where c(123) c(3) is allocated between players 1 and 2. As we move along line ab from point a to point b, the allocation to player 1 decreases from c(l) to c(123) c(2) c(3), i.e., from 35 to 5, while the allocation to player 2 increases from c(123) c(l) - c(3) to c(2), i.e., from 15 to 45. Similar explanations can be given for lines be and ac. A more restrictive set of solutions satisfying subgroup rationality (axiom 3), 71 x( 3) Figure 4-2. Geometry of Core Conditions for Three- Person Cost Game Example. 72 the shaded area on triangle abc, is the core for this game. The geometry of the core for this convex game is a hexagon. Line de represents the lower bound for player 2 or the set of charges where c(13) is allocated between player 1 and player 3 with the remainder, c(123) c(13), going to player 2. Similar explanations can be given for lines fg and hi which are the lower bounds for players 1 and 3, respectively; and for lines id, gh, and ef which are the upper bounds for players 1, 2, and 3, respectively. If an allocation lies outside the core, an inequitable situation prevails. For instance, point Z in Figure 4-2 allocates player 2 a cost less than its lower bound, c(123)-c(13), which means c(13) increases or the cost allocated to players 1 and 3 increases. Clearly, player 1 and player 3 can do better by forming their own two-person coalition rather than subsidizing player 2. As mentioned earlier, the convexity of a game and its attractiveness are related. This relationship is illustrated in Table 4-10. When the costs for the two-person coalitions progressively decrease, there is less incentive for forming the grand coalition so the core becomes progressively smaller and the game becomes progressively more nonconvex. As a consequence of the core conditions for a three-person sub additive cost game, a condition can be derived to determine if a core exists. From subgroup rationality and group rationality, we have the following conditions: 73 Table 4-10. Core Geometry for Three-Person Cost Game Example. Characteristic Function c(1) = 35, c(2) = 45, c(3) = 50, c(123) = 100 Geometry Zc(ij) of Core c (12) c(13 ) c(23 ) 66 75 87 228 < } Hexagon 61 73 86 220 A> Pentagon 59 71 85 215 ZA Trapezoid 58 70 80 208 A Triangle 56 68 76 200 Point 55 65 72 192 x (2 ) x(1) // -\rx<3) //y Empty Source: Modified from Fischer and Gately, 1975, p. 27a. 7 4 x(l) + x(2) < c(12) x(l) + x(3) < c(13) x ( 2 ) + x ( 3 ) < c ( 2 3 ) x(l) + x(2) + x(3) = c(12 3) (4-14) Summing the subgroup rationality conditions gives 2[x(1) + x(2) + x ( 3 ) ] < c(12) + c(13) + c(23). (4-15) If the group rationality conditions are substituted into the above equation, then we have the following condition to determine if a core exists: 2 c(12 3) < c(12 ) + c(13 ) + c(2 3 ) . (4-16) Therefore, in Table 4-10, the core exists as long as the sum of the two-person coalitions is greater than 200 or twice the value of the grand coalition. When the sum of the two-person coalitions equals 200, the core reduces to a unique vector, i.e., X = [24, 32, 44]. Finally, when the sum of the two-person coalition is less than 200, then the core is empty. Unfortunately, for larger games there is no simple condition for checking the existence of a core; however, as we will see later, a check can be made using linear programming. 75 Unique Solution Concepts The three axioms of fairness defining the core of the game significantly reduce the set of admissible solutions. Unless the core is empty or is a unique vector, an infinite number of possible equitable charge vectors remain to be considered, so additional criteria are needed to select a unique charge vector. Numerous methods are available for selecting a unique charge vector; but the two most popular methods discussed in the literature are the Shapley value (Shapley, 1953; Heaney, 1983b; Shubik, 1962; Heaney et al., 1975; Littlechild, 1970) and the nucleolus (Schmeidler, 1969; Kohlberg, 1971; Suzuki and Nakayama, 1976). Shapley value. The Shapley value for player i is defined as the expected incremental cost for the coalition of adding player i. Thus, each player pays a cost equal to the incremental cost incurred by the coalition when that player enters. Since the coalition formation sequence is unknown, the Shapley value assumes an equal probability for all sequences of coalition formation, i.e., the probability of each player being the first to join is equal, as are the probabilities of joining second, third, etc. For an n person game there are n! orderings. The six sequences of coalition formation for a three-person game are as follows: (123) (213) (231) (132) (312) (321) 7 6 Therefore, the Shapley value or the cost to player 1 for a three-person game is $(1) = 1/3 c(1) + 1/6 [c(12) c(2)] + 1/6 [c(13 c(3)] + 1/3 [c(12 3) c(23)]. (4-17) Player 1 has 1/3 probability of entering the coalition as the first player and 1/3 probability of entering the coalition as the last player. In addition, player 1 has 1/6 probability of entering the coalition after player 2 and 1/6 probability of entering the coalition after player 3. Notice that [c(S+i) c(S)] is the incremental cost of adding player i to the S coalition. The general formula for the Shapley value for player i is 4> ( i ) = I a. (S) [ c {S) c ( S { i } ) ] Scn 1 (4-18) where (s-i)! (n s) rf! s is the number of players in coalition S, n! is the total number of possible sequences of coalition formation, (s-1)! is the number of arrangements for those players before S, and (n-1)! is the number of arrangements for those players after S. 77 For example, for i = 1, n = 3: a1(l) 0121/3! = 1/3 ax(12) = 1111/31 = 1/6 a1(13) = 1111/3! = 1/6 a1(123) = 2101/31 = 1/3 Total 1.0 Note that 2 i(i) = c(N). (4-19) i eN Furthermore, if the game is convex, the Shapley value lies in the center of the core (Shapley, 1971). The Shapley value is criticized for several reasons. It may fall outside the core for nonconvex games, and it may be computed even when the core does not exist (Hamlen, 1980). Furthermore, the Shapley value is computationally burdensome for large games. For an n-person game, the Shapley value for each player requires the computation of n 1 2 coefficients and incremental costs. For example, an eight player game requires 128 coefficients and incremental costs to calculate the charge for each player. 78 Loehman and Whinston (1976) attempted to reduce the computational burden of the Shapley value by relaxing the assumption that all sequences of coalition formation are equally likely. This generalized Shapley value allows using a priori information to eliminate impossible sequences of coalition formation. Unfortunately, when Loehman et al. (1979) applied the generalized Shapley value to an eight-player regional wastewater management problem, they got a solution outside the core (Heaney, 1983a). Littlechild and Owen (1973) developed the simplified Shapley value for games wherein the characteristic function is a cost function with the property that the cost of any subcoalition is equal to the cost of the largest player in the subcoalition. Although Littlechild and Thompson (1977) demonstrated the computational ease of the simplified Shapley value in their case study of airport landing fees consisting of 13,572 landings by 11 different types of aircraft, the use of the simplified Shapley value is restricted to games with these special properties. Before calculating the Shapley value for our regional water network problem, the total enumeration procedure described in the preceding chapter is used to find the following characteristic cost function values (see Appendix A) : c(1) = 646,000 c(2 ) = 2,420,095 c(3 ) = 1, 990,992 c(12) = 2,928,511 c(13) 2,586,638 c ( 2 3 ) 3,984,177 79 and 1 c (12 3) 4, 556, 409 2 c (12 3) 4, 556, 826 3 c (1, 23) = 4, 630, 177 4 c (12 ,3) = 4, 919, 503 5 c (13 ,2) = 5, 006, 734 w c dr 2,3) = 5, 057, 087 k t h where c (hi,j) is the cost of the k best regional water network consisting of pipelines from the well field to county h, from county h to county i, and from the well field . w to county j. Also, c (1,2,3) is the cost for each county to t h go-it-alone. The cost allocation associated with the k t h best regional water network, i.e., the kc network game, is simply found by setting c(N) equal to c (N). The Shapley values for all options available to the three counties are calculated in Appendix A and summarized in Table 4-11. Appendix A also checks whether each Shapley value satisfies core conditions. All of the network games in this example are nonconvex. Table 4-11 shows that the cost allocations for the optimal and the second best networks, i.e., the first two options, satisfy all core conditions; therefore, these cost allocations are in the core and are considered equitable. Furthermore, unlike the cost allocations using the direct costing methods discussed earlier, the cost allocations for these two comparable cost 80 Table 4-11. Cost Allocation for Three-County Example Using the Shapley Value. Cost Allocation to County i Is Cost ($) Allocation Zx(i) In Core? Option County 1 County 2 County 3 ($) (From Ap- (Rank) x(l) x(2) x(3) pendix A) 1 590,087 2,175,905 1,790,417 4,556,409 Yes 2 590,226 2,176,044 1,790,556 4,556,826 Yes 3 614,677 2,200,494 1,815,006 4,630,177 No 4 711,119 2,296,936 1,911,448 4,919,503 No 5 740,196 2,326,013 1,940,525 5,006,734 No 6 646,000 2,420,095 1,990,992 5,057,087 81 networks are nearly identical. The Shapley value divided the additional $417 for the second best network equally among the counties. Option 3 illustrates the failure of the Shapley value to consistently give a core solution for nonconvex games. As shown in Appendix A, the cost alloca tion for option 3 fails to satisfy subgroup rationality for the coalition consisting of county 2 and county 3; i.e., x(2) + x(3) > c(23 ) (4-20) Moreover, options 4 and 5 illustrate Shapley values for games with an empty core. The nonexistence of the core for network games with options 4 and 5 can be determined by using other game theory methods, e.g., nucleolus. A close examination of the core conditions for network games with options 4 and 5 reveals these games are no longer subaddi- tive. By defining c(N) as c (N), c(N) is no longer the least cost or optimal solution for the grand coalition. t h Consequently, the k best network game is not naturally subadditive even though c (N) may be less expensive than w c (N). In any event, because network games with options 4 and 5 are not subadditive, there is no incentive to cooperate. Therefore, options 4 and 5 no longer need to be considered by the counties. Nucleolus. The other popular method to obtain a unique charge vector is to find the nucleolus. For a cost game, 82 the fairness criterion used by the nucleolus is based on finding the charge vector which maximizes the minimum savings of any coalition. For each imputation in the core of a cost game, a 2n vector in R is defined. The components of this vector are arranged in increasing order of magnitude and are defined by e(S) = c(S) z x(i) V ScN. (4-21) ieS 2 n The imputation whose vector in R is lexicographically the largest is called the nucleolus of the cost game. Given two vectors, X = (x^,...,x ) and Y = (y^,...,y ), X is lexi cographically larger than Y if there exists some integer k, 1 < k < n, such that x^ = y^ for 1 _< j < k and x^. > y^ (Owen, 1982). Basically, e(S) represents the minimum savings of coalition S with respect to charge vector X. Obviously, the coalition with the least savings objects to charge vector X most strongly, and the nucleolus maximizes this minimum savings over all coalitions. The nucleolus can be found by solving at most n-1 linear programs (Kohlberg, 1972; Owen, 1974, 1982), where the first linear programming problem is maximize e(l) subject to e(l) + x(i) < c(i) Â¥ ieN (4-22) 83 e (1) + Â£ x ( i ) < c (S) Â¥ S N ie S Â£ x ( i ) = c (N) ie N x ( i ), e (1) > 0 The nucleolus is calculated by sequentially solving for e(l), then e(2), e(3), etc., where e(i) is the ifc^ smallest savings to any coalition. Unlike the Shapley value, the nucleolus always is in the core for games with nonempty core. In fact, the nucleolus is always unique. However, the nucleolus is criticized because it cannot be written down in explicit form (Spinetto, 1975), and that it is difficult to compute and use in practice (Gugenheim, 1983). Probably the most difficult problem with using the nucleolus is the acceptance of its notion of fairness as opposed to other prevailing notions of fairness without generating unending controversies and debates. The nucleolus is generally considered to be analogous to Rawls' (1971) welfare criteria: the utility function of the least well off individual is maximized. Other notable notions of fairness include (1) Nozick's (1974) procedural approach to justice, and (2) Varian's (1975) or Baumol's (1982) definition of 84 equitable distribution whereby no one prefers the consumption bundle of anyone else. Calculating the nucleolus for our regional water network problem using the linear programming problem (4-22) gives the results summarized in Table 4-12. Equitable cost allocations are given for the first three options, and the cost allocations for the optimal and the second best networks are essentially the same. The additional $417 for the second best network is apportioned as follows: County 1 $209 County 2 104 County 3 104 Total $417 No cost allocations are given for options 4 and 5 because these network games have empty cores. That is, the linear programming problem (4-22) is infeasible. Finally, Table 4-12 reveals that each of the three counties has an incentive to cooperate in order to implement the cheapest regional water network. Propensity to disrupt. Another unique solution concept worth mentioning because of its intuitive appeal is the concept of an individual player's "propensity to disrupt." Gately (1974) defined an individual player i's propensity to disrupt as a ratio of what the other players would lose if player i refused to cooperate over how much player i would lose by not cooperating. Mathematically, player i's 85 Table 4-12. Cost Allocation for Three-County Example Using the Nucleolus. Cost Allocation to (?) County i lx (i ) Is Cost Allocation Option (Rank) County 1 x(l) County 2 x (2 ) County 3 x (3 ) ($) In Core? 1 609,116 2,144,583 1,802,710 4,556,409 Yes 2 609,325 2,144,687 1,802,814 4,556,826 Yes 3 4 646,000 2,163,025 1,821,152 4,630,177 Yes 5 6 646,000 2,420,095 1,990,992 5,057,087 -- 86 propensity to disrupt, d(i), a charge vector, X = [x(1),...,x(n)], which is in the core is c(N-i) Â£ x(j ) d ( i ) = 21 (4-23) c ( i ) x(i ) The higher the propensity to disrupt, the greater a player's threat to the coalition; e.g., d(i) = 10 implies player i could impose a loss of savings to the other players 10 times as great as the loss of savings to player i. As an illustration, the propensity to disrupt is calculated for each of the counties using the nucleolus for the optimal network of our regional water network problem: X = [609,116; 2,144,583; 1,802,710]. d (1) c(23 x(2) x(3) c (1) x (1) = 1.0 d (2 ) _ c(13 ) x (1) x(3 ) = .63 c ( 2 ) x ( 2 ) d (3 ) _ c(12 ) x(1) x ( 2 ) = .93 c ( 3 ) x ( 3 ) The calculations show that none of the counties have a strong threat against the other two counties with the nucleolus charge vector. County 1 could impose a loss to the other two counties which equals the loss imposed on itself, while, county 2's or county 3's departure would hurt the departing county more than it would hurt the remaining two counties. 87 Gately suggested equalizing each player's propensity to disrupt as a final cost allocation solution. Subsequently, Littlechild and Vaidya (1976) have generalized Gately's concept of an individual player's propensity to disrupt to include a coalition S's propensity to disrupt. That is, a coalition S's propensity to disrupt is defined as the ratio of what the complementary coalition, N-S, stands to lose over what the coalition S itself stands to lose for a given charge vector. More recently, Chames et al. (1978 ) and Chames and Golany (1983) refined these propensity to disrupt concepts into a unique solution concept which appears to have some empirical support. Finally, Straffin and Heaney (1981) have shown that Gately's propensity to disrupt is exactly the alternative cost avoided method first proposed during the TVA project in 1935. The alternative cost avoided method is discussed in the section on the separable costs, remaining benefits method. Empty Core Solution Concepts Examining games with an empty core is an active area of research. An empty core implies that no equitable cost allocation exists, and results from games wherein the addi tional savings from forming the grand coalition is rela tively small. That is, the savings resulting from forming smaller coalitions are almost as much as the savings from forming the grand coalition. Therefore, proposed solution 88 concepts generally seek to relax the bounds on subgroup rationality until a "quasi" or "anti" core is created. Table 4-13 lists four methods for finding a charge vector for games with an empty core. In any case, given the modest amount of economic gain for games with an empty core, it may be more advantageous to forego the grand coalition in favor of smaller coalition formations as suggested by Heaney (1983a). Furthermore, engineering projects tend to have a large proportion of the costs common to all participants; consequently, one would expect these games to be very attractive and games with an empty core to be fairly rare. Nevertheless, the game theory approach does alert us that a problem exists in allocating costs for such cases. Cost Allocation in the Water Resources Field Straffin and Heaney (1981) showed that the criteria of fairness as expressed by equations (4-1) and (4-2) associated with cost allocation proposed by the TVA experts in the 1930's paralleled the development of the concepts of individual and subgroup rationality found in cooperative game theory. Given that full costs have to be recovered, the core conditions are equivalent to the fairness criteria for allocating cost originally proposed by the TVA experts. Therefore, current practice for allocating costs in the water resources field should require the solution be in the core of a game. |

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EFFICIENCY/EQUITY ANALYSIS OF WATER RESOURCES PROBLEMSâ€”A GAME THEORETIC APPROACH Bv ELLIOT KIN NG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1985 To my parents and my wife, Eileen, and children, Matthew, Michelle, Michael ACKNOWLEDGMENTS I would like to thank my chairman, Dr. James P. Heaney, for the many hours spent guiding this research. His encouragement, support, and friendship during my three years at the University of Florida have been invaluable. I would also like to thank the other members of my supervisory committee, Dr. Sanford V. Berg, Dr. Donald J. Elzinga, Dr. Wayne C. Huber, and Dr. Warren Viessman, for their time and support. In addition, I wish to thank the U.S. Air Force for giving me the opportunity to pursue the Ph.D. degree. Thanks are also due to several fellow students who have made my program enjoyable and memorable. In particuÂ¬ lar, I wish to thank Mr. N. Devadoss, Mr. Mun-Fong Lee, and Mr. Robert Ryczak. I would also like to give special thanks to Mr. Robert Dickinson for keeping an extra copy of the LP-80 and Mrs. Barbara Smerage for doing such an excellent job typing this manuscript. I am extremely grateful to my parents for instilling in me a desire to seek further eduction. Furthermore, I am especially thankful to my wife, Eileen, for typing initial drafts of this manuscript and for her love, encouragement, and sacrifices throughout my program. We will miss the iii croissants, pizzas, and hoagies that supplemented my late night studies. Finally, I wish to thank my children, Matthew, Michelle, and Michael, for their love and underÂ¬ standing during the countless times I have chased them out of my study. IV TABLE OF CONTENTS Page ACKNOWLEDGMENTS iii LIST OF TABLES vii LIST OF FIGURES ix ABSTRACT x CHAPTER 1 INTRODUCTION 1 2 LITERATURE REVIEW 4 Efficiency Analysis 4 Equity Analysis 5 Conclusions 8 3 EFFICIENCY ANALYSIS 10 Introduction 10 Partial Enumeration Techniques 12 Total Enumeration Techniques 15 Modeling Network Problems as Digraphs 16 The Total Enumeration Procedure 21 Computational Considerations 30 Summary 3 8 4 EQUITY ANALYSIS 39 Introduction 39 Cost Allocation for Regional Water Networks 4 0 Criteria for Selecting a Cost Allocation Method 45 Ad Hoc Methods 4 8 Defining Identifiable Costs as Zero... 49 Defining Identifiable Costs as Direct Costs 54 v Cooperative Game Theory 64 Concepts of Cooperative Game Theory... 65 Unique Solution Concepts 7 5 Empty Core Solution Concepts 87 Cost Allocation in the Water Resources Field 88 Separable Costs, Remaining Benefits Method 9 0 Minimum Costs, Remaining Savings Method 9 5 Allocating Cost Using Game Theory Concepts. 99 The k Best System 99 The Dummy Player 108 Comparing Methods 115 Summary 119 5 EFFICIENCY/EQUITY ANALYSIS 120 Introduction 120 Maximum Cost 122 Minimum Cost 129 Fairness Criteria 132 Summary 13 3 6 CONCLUSIONS AND RECOMMENDATIONS 135 APPENDIX EFFICIENCY/EQUITY ANALYSIS OF A THREE- COUNTY REGIONAL WATER NETWORK WITH NONLINEAR COST FUNCTION 142 REFERENCES 15 0 BIOGRAPHICAL SKETCH 159 vi LIST OF TABLES Table Page 3-1 Example of Total Enumeration Procedure for 3-Node Digraph 27 3-2 The Number of Independent Calculations to Find the Costs of Spanning Directed Trees for All Possible Subdigraphs 31 3-3 Summary of Computational Effort for Digraphs Shown in Figure 3-4 34 3-4 Efficiency Analysis of a Three-User Water Supply Network with Nonlinear Cost Function Using Lotus 1-2-3 37 4-1 Projected Population Growth and Projected Average Per Capita Demand 41 4-2 The Costs and Percent Savings for All Options 44 4-3 Cost Allocation Matrix 50 4-4 Cost Allocation of Optimal Network Based on Population 52 4-5 Cost Allocation of Optimal Network Based on Demand 5 3 4-6 Cost Allocation of Optimal Network with Use of Facilities Method 56 4-7 Cost Allocation for the Use of Facilities Method 57 4-8 Cost Allocation of Optimal Network with Direct Costing/Equal Apportionment of Remaining Costs Method 60 4-9 Cost Allocation for Direct Costing/Equal Apportionment of Remaining Costs Method 61 vii 4-10 Core Geometry for Three-Person Cost Game Example 7 3 4-11 Cost Allocation for Three-County Example Using the Shapley Value 80 4-12 Cost Allocation for Three-County Example Using the Nucleolus 85 4-13 Empty Core Solution Methods 89 4-14 Cost Allocation for Three-County Example Using the SCRB Method 94 4-15 Cost Allocation for Three-County Example Using the MCRS Method 98 4-16 Nominal Versus Actual Core Bounds for Optimal Network Game 100 4-17 Cost Allocations for the Optimal Network and the Second Best Network ($) 103 4-18 Cost Allocation for Option 3 as a Two-Person Game Using the SCRB Method 10 9 4-19 Comparing Cost Allocations for Option 3 as Two-Person Game and Three-Person Game Using the SCRB and MCRS Methods Ill 4-20 Core Bounds for Option 3 as a Three-Person Game 112 4-21 Core Bounds for Option 3 as a Three-Person Game with County 1 as a Dummy Player 114 4-22 Core Bounds for Option 3 as a Two-Person Game 116 4-23 Comparison of Methods Discussed for Allocating Costs of Water Resources Projects. 117 5-1 Using Independent Calculations from the Total Enumeration Procedure to Find c(i), c(S), and c(N) for the Three-County Regional Water Network Problem 121 5-2 Efficiency/Equity Analysis of the Optimal Network 12 6 viii LIST OF FIGURES Figure Page 3-1 Types of Cost Functions 13 3-2 Example Digraph Representing a Regional Water Network Problem for Three Users 18 3-3 Flow Diagram of Total Enumeration Procedure for n-Node Digraph 23 3-4 Examples of 3,4,5-Node Digraphs 33 4-1 Lengths of Interconnecting Pipelines 43 4-2 Geometry of Core Conditions for Three-Person Cost Game Example 71 4-3 Core for the Optimal Network Game (C (N ) = $4,556,409 ) 101 4-4 Core for the Second Best Network Game ( C (N) = $4,556,826 ) 102 4-5 Reduction in. Core as c(N) Increases from c X(N) to c (N) 107 IX Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy EFFICIENCY/EQUITY ANALYSIS OF WATER RESOURCES PROBLEMSâ€”A GAME THEORETIC APPROACH By Elliot Kin Ng August, 1985 Chairman: James P. Heaney Major Department: Environmental Engineering Sciences Successful regional water resources planning involves an efficiency analysis to find the optimal system that maxiÂ¬ mizes benefits minus costs, and an equity analysis to apporÂ¬ tion project costs. Traditionally, these two problems have been treated separately. This dissertation incorporates efficiency analysis and equity analysis into a single regional water resources planning model. A reliable total enumeration procedure is used to find the optimal system for regional water network problems. This procedure is easy to understand and can be implemented using readily available computer software. Furthermore, the engineer can use realistic cost functions or perform detailed cost analysis and, also, examine good suboptimal systems. In addition, this procedure finds the optimal system for each individual and each subgroup of individuals; hence, an equity x analysis can be accomplished using the theory of the core from cooperative n-person game theory. Game theory concepts are used to perform an equity analysis on the optimal system as well as good suboptimal systems. For any system, an equitable cost allocation exists if a core exists. However, if a game is not properly defined, even a cost allocation in the core may be inequitable. A rigorous procedure using core conditions and linear programming is described to determine the core bounds. An individual's lower core bound and upper core bound unambiguÂ¬ ously measure the individual's minimum cost and maximum cost, respectively. Traditional approaches for quantifying minimum cost and maximum cost assume that either a regional system involving the grand coalition is built or all the individuals will go-it-alone. However, this rigorous procedure accounts for the possibility that a relatively attractive system involving subgroups may form. Furthermore, this rigorous procedure gives a general quantitative definition of marginal cost and opportunity cost. Once the minimum cost and maximum cost for each individual are determined, a basis for equitable cost allocation is available. Finally, efficiency analysis and equity analysis are not separable problems but are related by the economics of all the opportunities available to all individuals in a project. xi CHAPTER 1 INTRODUCTION In situations where multiple purposes and groups can take advantage of economies of scale in production and/or distribution costs, a regional water resources system is an attractive alternative to separate systems for each purpose and each group. However, a regional system imposes complex economic, financial, legal, socio-political, and organizaÂ¬ tional problems for the water resources professionals. This dissertation examines two problems associated with regional water resources planning that are typically treated separately, yet are closely related. The first problem involves performing an efficiency analysis to determine the economically efficient or optimal regional system that maximizes benefits minus costs. Once the optimal regional system is determined, a major task still remains to allocate project costs; therefore, an equity analysis must be performed to apportion project costs in an equitable manner. This second problem is viewed from the perspective of each purpose and each group because they must each be convinced that the optimal regional system is their best alternative; otherwise, voluntary participation will be difficult. No doubt, each purpose's and each 1 2 group's decision to participate in the optimal regional system depends on its allocated cost, and not necessarily on what is best for the region. The prevailing belief is that efficiency analysis and equity analysis are separate problems and, therefore, research has either focused entirely on efficiency analysis or equity analysis. Research on efficiency analysis has mainly been on the application of partial enumeration techÂ¬ niques to find optimal regional systems, while research on equity analysis has continued to explore the application of concepts from cooperative game theory to allocate project costs. The purpose of this dissertation is to integrate efficiency analysis and equity analysis into a single regional water resources planning model characterized by economies of scale. The model to be presented incorporates a total enumeration procedure along with concepts from cooperative game theory for efficiency/equity analysis. The specific application is to determine the least cost regional water supply network and to determine a "fair" allocation of costs among the multiple users. Chapter 2 reviews selected works on efficiency analysis and equity analysis of water resources problems. Chapter 3 presents a reliable total enumeration procedure for effiÂ¬ ciency analysis of regional water supply network problems. However, unlike traditional partial enumeration techniques used for efficiency analysis that give only the optimal 3 solution, this procedure also gives all the optimal soluÂ¬ tions for each user and each subgroup of users which are necessary information to perform an equity analysis using concepts from cooperative game theory. In addition, this procedure gives all the suboptimal solutions. Chapter 4 shows how the information from the total enumeration procedure is used to perform an equity analysis of not only the optimal solution, but also "good" suboptimal solutions. Chapter 5 reveals how efficiency analysis and equity analysis are related. Finally, Chapter 6 summarizes the results and conclusions. CHAPTER 2 LITERATURE REVIEW Efficiency Analysis During the past two decades, the problem of finding the economically efficient or optimal regional water system has been extensively modeled as a mathematical optimization problem. A review of selected works on efficiency analysis of regional water systems that includes Converse (1972), Graves et al. (1972), McConagha and Converse (1973), Yao (1973), Joeres et al. (1974), Bishop et al. (1975), Jarvis et al. (1978), Whitlatch and ReVelle (1976), Brill and Nakamura (1978), and Phillips et al. (1982) indicates a variety of partial enumeration techniques, e.g., nonlinear programming, for finding optimal regional systems. These optimal regional systems can be a least cost system or a system that maximizes benefits minus costs. Generally, regional water resources planning problems exhibit economies of scale in cost and, therefore, involve nonlinear concave cost functions. Consequently, to a great extent, the selection of the partial enumeration optimization technique to apply to a particular problem depends on the characteriÂ¬ zation of the nonlinear concave cost functions. For instance, linear programming can be applied if the nonlinear 4 concave cost functions are represented by linear approximations. 5 Equity Analysis Unfortunately, successful regional planning is not merely knowing the optimal regional system but must also include an equity analysis to find an acceptable allocation of costs among the participants. Otherwise, the optimal system will be difficult to implement. Of the publications cited in the preceding paragraph, only McConagha and Converse (1973) dealt with both efficiency and equity in regional water planning. In addition to presenting a heuristic procedure for finding the least cost regional wastewater treatment facility for seven cities, they evaluÂ¬ ated the equity of several cost allocation procedures. Although they recognized that an equitable cost allocation should not charge any city or subgroup of cities more than the cost of an individual treatment facility, they did not include the possibility of subgroup formation in their analysis. Giglio and Wrightington (1972) introduced concepts from cooperative game theory as a way to consider the possibility of subgroup formation in allocating costs of water projects. However, their treatment of cooperative game theory was incomplete. Therefore, they concluded that the game theory approach rarely yields a unique cost allocation 6 and proceeded to recommend the separable costs, remaining benefits (SCRB) method or methods based on measure of polluÂ¬ tion. Shortly thereafter, several researchers applied popular unique solution concepts from game theory like the Shapley value and the nucleolus to allocate the costs of regional water systems. Heaney et al. (1975) applied the Shapley value to find an equitable cost allocation of common storage units for storm drainage for pollution control among competing users. Suzuki and Nakayama (1976) applied the nucleolus to assign costs for a water resources development along Japan's Sakawa and Sagami Rivers. Loehman et al. (1979) used a generalization of the Shapley value to allocate the costs of a regional wastewater system involving eight dischargers along the lower Meramec River near St. Louis, Missouri. Subsequently, Heaney (1979) established that the fairÂ¬ ness criteria used for allocating costs in the water resources field and the concepts used in cooperative game theory are equivalent. Moreover, Straffin and Heaney (1981) showed that a conventional method for allocating costs used by water resources engineers is identical to a unique soluÂ¬ tion concept used by game theorists. More recently, Young et al. (1982) compared proportionality methods, game theoretic methods, and the SCRB method for allocating cost and concluded that the game theoretic methods may be too complicated while the SCRB method may give inequitable cost 7 allocations. Meanwhile, Heaney and Dickinson (1982) revealed why the SCRB method may fail to give equitable cost allocations and proposed a modification of the SCRB method that uses game theory concepts along with linear programming to insure an equitable cost allocation can be found if one exists. The possibilities of using concepts from cooperative game theory as a basis for allocating costs of water projects continue to develop. In fact, concepts from coopÂ¬ erative game theory are gaining acceptance in other fields as well. Researchers in accounting are looking toward coopÂ¬ erative game theory as a possible solution to the arguments by Thomas (1969, 1974) that any cost allocation scheme in accounting is arbitrary and hence not fully defensible. Recent works by Jensen (1977), Hamlen et al. (1977, 1980), Callen (1978), and Balachandran and Ramakrishnan (1981) applied concepts from cooperative game theory to evaluate the equity of existing and proposed cost allocation schemes in accounting. Meanwhile, in economics, concepts from cooperative game theory are frequently used as a basis for evaluating subsidy-free and sustainable pricing policies for decreasing cost industries, e.g., the work of Loehman and Whinston (1971, 1974), Faulhaber (1975), Sorenson et al. (1976, 1978), Zajac (1978), Panzar and Willig (1977), Faulhaber and Levinson (1981), and Sharkey (1982b). 8 Conclusions Three conclusions can be made from reviewing the literature on efficiency analysis and equity analysis of regional water resources planning. First, there is a gap in the research to jointly examine efficiency and equity in regional water resources planning. In spite of a continual effort to find economically efficient regional water systems and equitable cost allocation procedures, no published work incorporates both efficiency analysis and equity analysis in a single regional water resources planning model using realistic cost functions. Heaney et al. (1975) and Suzuki and Nakayama (1976) used linear cost models while Loehman et al. (1979) used conventional cost curves. Secondly, the cost allocation literature in the water resources field has consistently allocated the costs of treatment and piping together even though federal guidelines suggest that piping cost be allocated separately from treatment cost to the responsible users (Loehman et al., 1979; U.S. Environmental Protection Agency, 1976). Finally, the cost allocation literature has dealt with allocating the cost of the optimal system. However, situations in practice may require that "good" suboptimal systems be considered; therefore, an acceptable cost allocation procedure should be able to allocate the costs of several systems under consideration in an equitable manner. These three conclusions formed the basis for the research undertaken in this dissertation. Chapter 3 begins integrating efficiency analysis and equity analysis by searching for a computational procedure 9 to simultaneously perform an efficiency analysis and calculate all the necessary information to perform an equity analysis using concepts from cooperative game theory. CHAPTER 3 EFFICIENCY ANALYSIS Introduction The importance of both efficiency analysis and equity analysis in planning regional water resources systems is well recognized. Over the years, researchers have applied methods ranging from simple cost-benefit analysis to sophisÂ¬ ticated mathematical programming techniques to search for economically efficient or optimal regional water resources systems. Yet, the implementation of regional systems is difficult unless an equitable financial arrangement is found to allocate project costs among individuals (or particiÂ¬ pants) in a project. Until recently, a theoretically sound basis for allocating costs has eluded the water resources professional. However, there is increasing interest in using the theory of the core from cooperative n-person game theory as a basis for allocating costs, e.g., see Suzuki and Nakayama (1976), Bogardi and Szidarovsky (1976), Loehman et al. (1979), Heaney and Dickinson (1982), and Young et al. (1982). The theory of the core is based on principles of individual, subgroup, and group rationality. This means that no individual or subgroup of individuals should be allocated a cost in excess of the cost of nonparticipation, 10 11 while total cost must be apportioned among all individuals. The cost of nonparticipation is simply the cost that each individual and each subgroup of individuals must pay to independently acquire the same level of service by the most economically efficient means. As a result, to evaluate efficiency/equity for a regional system with n individuals, it is necessary to determine 2n-l optimal solutions. Although the close association between efficiency analysis and equity analysis is recognized, there have been few attempts to incorporate these two analyses in regional water resources planning. A typical efficiency analysis usually ends with determining the optimal solution for a problem without addressing cost allocation, and a typical equity analysis begins by assuming the 2n-l optimal soluÂ¬ tions are available to accomplish the cost allocation. This disjointed approach to efficiency/equity analysis is fostered by a belief that these two problems are independent (James and Lee, 1971; Loughlin, 1977). Furthermore, reliable techniques for finding the 2n-l optimal solutions to accomplish an efficiency/equity analysis of most problems encountered in actual practice are unavailable. This chapter begins by evaluating the applicability of partial and total enumeration techniques for finding the 2n-l optimal solutions for problems with different types of cost functions. Subsequently, a computational procedure is described to examine a regional water supply network problem 12 wherein we need to find the economic optimum and a "fair" allocation of costs among the individuals in the project. In order to do the cost allocation we need to find the costs of the optimal systems for each individual and each subgroup of individuals since these costs are going to be the basis for cost allocation. Partial Enumeration Techniques The difficulty of finding the optimal solution for a particular problem depends on the nature of the cost funcÂ¬ tions. Generally, a cost function can be classified as either linear, convex, concave, S-shape, or irregular (see Figure 3-1). To find the optimal solution for problems with either linear or convex cost functions is straightforward using readily available and reliable linear programming codes. Accordingly, a vast body of overlapping theoretical results is available from classical economics and operations research, e.g., convex programming, for finding the optimal solution to problems with convex cost functions. However, problems with linear and convex cost functions are unable to characterize the economies of scale in cost typically encountered in regional water resources planning. The concave cost function is generally used to represent economies of scale, and several partial enumeraÂ¬ tion techniques are available for dealing with this cost 13 Figure 3-1. Types of Cost Functions. 14 function. One approach surveyed by Mandl (1981) is separable programming which takes advantage of readily available linear programming codes by using a piecewise linear approximation of the concave cost function. Unfortunately, this approach is rather tedious to use and guarantees only a local optimal solution. A second approach is to retain the natural concave cost function and apply a general nonlinear programming code. However, according to surveys by Waren and Lasdon (1979) and Hock and Schittkowski (1983), general nonlinear programming codes may converge to local optima and may be subject to other failures, e.g., termination of code. A final approach used by Joeres et al. (1974) and Jarvis et al. (1978) is to approximate the concave cost function with several fixed-charge cost functions and apply a mixed-integer programming code. This approach guarantees a globally optimal solution, but standard mixed-integer programming codes are expensive to use. More importantly, unresolved problems remain as to how to properly define a fixed charge problem. If the fixed charge formulation is used because it is computationally expedient, then the resulting cost estimates may distort the cost allocation procedure. Given the current status of partial enumeration techniques for finding the optimal solutions to perform efficiency/equity analysis for problems with concave cost functions, one can conclude that other methods must be used. Obviously, this conclusion applies 15 to problems with S-shape and irregular cost functions as well. Total Enumeration Techniques Total enumeration techniques can be used to find the optimal solution for a problem regardless of the types of cost functions involved. The ability to handle irregular cost functions is especially important because this type of cost function is frequently used by state-of-the-art cost estimating models like CAPDET, i.e., Computer Assisted Procedure for Design and Evaluation of Wastewater Treatment Systems (U. S. Army Corps of Engineers, 1978) and MAPS, i.e., Methodology for Areawide Planning Studies (U. S. Army Corps of Engineers, 1980). For example, in MAPS, the cost function for constructing a force main is composed of separate cost functions for pipes, excavation, appurtenÂ¬ ances, and terrain. Furthermore, each of these cost funcÂ¬ tions is based on site-specific conditions. For instance, the cost function for pipe includes the cost of purchasing, hauling, and laying the pipe and depends on the material, diameter, length, and maximum pressure. No doubt, the composite site-specific cost function for a force main may be nonlinear, nonconvex, multimodel, and discontinuous. Another advantage with a total enumeration technique is that it presents and ranks all of the alternative soluÂ¬ tions. Unlike partial enumeration techniques which only 16 present the optimal solution for consideration, total enumeration techniques allow examination of suboptimal solutions which may be preferable when factors other than cost are considered. For example, proven engineering design or socio-political values are difficult to incorporate into an optimization model even if the problem is well defined, so the optimal solution may be so unrealistic that another solution must be selected. Depending on the size of the problem, a possible drawback with total enumeration techniques may be the compuÂ¬ tational effort to enumerate all possible solutions. However, for some problems, total enumeration may be the only meaningful approach. For these problems, the challenge with using a total enumeration approach is to find ways to reduce the computational effort by applying mathematical techniques or engineering considerations. After a discusÂ¬ sion on modeling network problems as digraphs, a total enumeration procedure that does not require extensive compuÂ¬ tational effort to find the least cost network for each individual and each group of individuals is presented. Modeling Network Problems as Digraphs Consider a situation wherein an existing water supply source, S, is going to serve n users with demands of Q^, . . . , Q , respectively. Assume that the water source is able to supply the total demand by the n users without 17 facility expansion except for a new regional water network. Furthermore, consider a particular system with three users that can be served directly by the source, and engineering considerations, e.g., gravity flow, have determined that it is feasible to send water from user 1 to both user 2 and user 3, and from user 2 to user 3. For this particular system, assume the total cost function for constructing a pipeline is rather simple. From Sample (1983), the total cost function for constructing a pipeline is characterized by economies of scale and can be expressed as a linear function of distance and a nonlinear function of flow; or C = aQbL (3-1) where C = total cost of pipeline, dollars Q = quantity of flow, mgd L = length of pipeline, feet, and a, b = parameters, 0
Given this situation, the objective of the regional waterauthority is to determine the least cost water network for each user and each group of users in order to perform efficiency/equity analysis. This problem can be modeled as a digraph or directed graph (see Figure 3-2) consisting of nodes to represent the source and users, and directed arcs to represent all 18 Figure 3-2. Example Digraph Representing a Regional Water Network Problem for Three Users. 19 possible interconnecting pipelines. If water can be sent in either direction between two users, then the pipeline is represented by two oppositely directed arcs. Consequently, any regional water network problem can be modeled by a digraph. Before continuing, a few brief definitions and concepts are necessary since the nomenclature used in the network and graph theory literature is not standardized. A digraph or directed graph, D(X,A), consists of a finite set of nodes, X, and a finite set of directed arcs, A. A directed arc is denoted by (i,j) where the direction of the arc (shown by an arrow) is from node i to node j; node i is called the initial node and node j is called the terminal node. A subdigraph of D(X,A) has a set of nodes that is a subset of X but contains all the arcs whose initial and terminal nodes are both within this subset. A path from node i to node j is simply a sequence of directed arcs from node i to node j. An elementary path is a path that does not use the same node more than once. A circuit is an elementary path with the same initial and terminal node. A directed tree or an arborescence is a digraph without a circuit for which every node, except the node called the root, has one arc directed into it while the root node has no arc directed into it. A spanning directed tree of a digraph is a directed tree that includes every node in the digraph. If a cost, C(i,j) is associated with every arc 20 (i,j) of a digraph, then the cost of a directed tree is defined as the sum of the costs of the arcs in the directed tree. Finally, a minimum spanning directed tree of a digraph is the spanning directed tree of the digraph with the least cost. For the reader desiring more information regarding networks and graphs, numerous texts are available, e.g., Christofides (1975), Minieka (1978), and Robinson and Foulds (1980 ) . The problem of finding the least cost water network for each user and each group of users is the same as finding the minimum spanning directed tree rooted at node S for all possible subdigraphs as well as the digraph shown in Figure 3-2. In general, not every digraph has a spanning directed tree; however, for a realistic problem one can assume a pipeline is available to serve all individuals participating in a regional system. Thus, a spanning directed tree exists for digraphs representing realistic regional water network problems. Although algorithms are found in Gabow (1977) and Camerini et al. (1980a, 1980b) for finding the minimum spanning directed tree or the K best spanning directed trees, these algorithms assume a linear cost model in which the cost on each arc is given prior to initiating the algorithm. As a result, these algorithms are not applicable to problems with nonlinear costs on each arc. That is, the cost along each arc cannot be determined in advance 21 because the cost is a function of the quantity of flow along the arc; yet, the quantity of flow along the arc is a function of the path in which the arc belongs. The Total Enumeration Procedure The procedure for enumerating and calculating the costs of all the spanning directed trees for all possible subÂ¬ digraphs as well as the digraph is based on recognizing that a large number of spanning directed trees of a digraph can be constructed from specific spanning directed trees of subdigraphs. These specific spanning directed trees are characterized by one arc emanating from the root node and are referred to as "essential spanning directed trees." In contrast, "inessential spanning directed trees" are characÂ¬ terized by more than one arc emanating from the root node. The procedure sequentially calculates the costs of essential spanning directed trees for subdigraphs with increasing number of nodes, until the costs of essential spanning directed trees are calculated for all possible subdigraphs and for the digraph. Meanwhile, the cost of each inessenÂ¬ tial spanning directed tree for all possible subdigraphs as well as the digraph is calculated simply by summing the costs of essential spanning directed trees of subdigraphs that are associated with each arc emanating from the root node of the inessential spanning directed tree. That is, each arc emanating from the root node belongs to an 22 essential spanning directed tree of a subdigraph. By applyÂ¬ ing this procedure the costs of all the spanning directed trees can be systematically enumerated for all possible subdigraphs as well as the costs of all the spanning directed trees for the digraph. As a result, the least cost network for each user and each group of users is found. In the following discussion, "n-node" means the number of nodes, not including the root node, is n; e.g., an i-node digraph or subdigraph consists of i+1 nodes if the root node is counted. The total enumeration procedure for the n-node digraph is summarized by the flow diagram shown in Figure 3-3. Step 1 begins the procedure for evaluating all subdiÂ¬ graphs consisting of the root node and one other node, i.e., the 1-node subdigraphs. Step 2 initializes a count of the number of combinaÂ¬ tions of i-node subdigraphs evaluated. Step 3 generates all possible combinations of i-node subdigraphs from the n-node digraph. The number of possible combinations is (^). For example, the 3-node digraph shown 3 in Figure 3-2 has (or three possible 2-node subdigraphs, i.e., subdigraphs consisting of the following sets of nodes {S,1,2 }, ÃS,1,3), and (S,2,3>. Step 4 selects one i-node subdigraph not previously selected and enumerates all of its spanning directed trees. A spanning directed tree may not exist in a case where a path does not exist from the root node to every node in the 23 Figure 3-3. Flow Diagram of Total Enumeration Procedure for n-Node Digraph not every node in the i-node i-node subdigraph, i.e., subdigraph has an arc directed into it. Actually, only the essential spanning directed trees need to be enumerated. The enumeration of inessential spanning directed trees is simply done by finding all possible combinations of i-node digraphs from the entire set of essential spanning directed trees enumerated previously, i.e., all essential spanning directed trees for all possible subdigraphs of the i-node subdigraph. This process substantially reduces the effort involved in enumerating all the spanning directed trees for an i-node subdigraph because a large number of spanning directed trees are inessential. If the i-node subdigraph is unusually large and dense, algorithms are available in Chen and Li (1973), Christofides (1975), and Minieka (1978) for generating spanning directed trees. If necessary, a procedure in Chen (1976) can be used to compute the number of spanning directed trees of an i-node subdigraph or an n-node digraph. A directed tree matrix, M, is defined for a digraph, where equals the number of arcs directed into node i and iru ^ is equal to the negative of the number of arcs in parallel from node i to node j. The number of spanning directed trees rooted at node S for the digraph defined by M is given by the determinant of the minor submatrix resulting from deleting the Sth row and 25 column of M. Applying this procedure to the 3-node digraph in Figure 3-2 gives the following directed tree matrix. S 1 2 3 S 1 2 3 0 -1 -1 -1 0 1-1-1 0 0 2 -1 0 0 0 3 / The determinant of the minor submatrix resulting from deletÂ¬ ing the Sth row and column is six, so there are six spanning directed trees rooted at node S for this digraph. Step 5 calculates the cost of each spanning directed tree enumerated in Step 4. The cost for each essential spanning directed tree is calculated independently. HowÂ¬ ever, the cost for each inessential spanning directed tree is simply calculated by summing the costs of essential spanning directed trees of subdigraphs calculated previously that are associated with the arcs emanating from the root node. For inessential spanning directed trees the costs can be calculated along with the enumeration process described in Step 4. Step 6 ranks all the spanning directed trees for the i-node subdigraph according to cost. The minimum spanning directed tree is the least cost network for the users associated with the set of nodes for the i-node subdigraph. 26 Step 7 checks the counter to see if all possible combinations of i-node subdigraphs have been evaluated. If not, Step 8 advances the counter by one before returning to Step 4 to evaluate another i-node subdigraph. If all of the possible combinations of i-node subdigraphs have been evaluated, the procedure goes to Step 9 and begins the evaluation of subdigraphs with i+1 nodes. Step 10 checks if the n-node digraph has been evaluÂ¬ ated. If not, the procedure returns to Step 2 and proceeds to evaluate the subdigraphs with i+1 nodes; otherwise, the procedure terminates. The total enumeration procedure is illustrated in Table 3-1 using the regional water network problem modeled by the 3-node digraph shown in Figure 3-2. During the first iteration all combinations of 1-node subdigraphs are evaluated. For this simple case 3 three combinations, i.e., (^) = 3, are evaluated. FurtherÂ¬ more, each combination has only one spanning directed tree, and the one spanning directed tree is essential. As a result, the cost of the spanning directed tree for each combination must be calculated. Obviously each spanning directed tree is the least cost network for the associated user. During the second iteration, three combinations, 3 i.e., (2) =3, of 2-node subdigraphs are evaluated. In this case, each combination has two spanning directed trees, but the cost of only one spanning directed tree needs 27 Table 3-1. Example of Total Enumeration Procedure for 3-Node Digraph Iteration i-Node i Subdigraphs Spanning Directed Trees for i-Node Subdigraph Are Spanning Directed Trees Essential? i = l ÃS, 1} Yes (S, 2} Â®- â€” (S, 3} Â®â€” â€” i=2 {S,l,2} (D-^Â®â€” {S, 1,3 } No Yes {S , 2,3 } No Yes No 28 Table 3.1. Continued. Iteration i-Node i Subdigraphs Spanning Directed Are Spanning Trees for i-Node Directed Trees Subdigraph Essential? 29 to be calculated. The cost of the inessential spanning directed tree is simply found by summing the costs of the corresponding essential spanning directed trees calculated during the first iteration. The minimum spanning directed tree for each combination is the least cost network for the associated group of users. Finally, for the third iteraÂ¬ tion, i.e., i=n, the 3-node digraph is being evaluated. This 3-node digraph has six spanning directed trees, and these six spanning directed trees can be enumerated by inspection. The four inessential spanning directed trees can be enumerated by simply finding all possible combinations of 3-node digraphs from the essential spanning directed trees generated during the first and second iterations. Thus, only two independent calculations are necessary to find the costs of the essential spanning directed trees. MeanÂ¬ while, the cost of the four inessential spanning directed trees is calculated simply by summing the costs of essential spanning directed trees for subdigraphs previously calcuÂ¬ lated during the first two iterations. For example, in Table 3-1, the cost for the inessential spanning directed tree consisting of the set of arcs {(S,3), (S,l), (1,2)} is determined by summing the costs of the two essential spanning directed trees consisting of the sets of arcs {(S,3) } and {(S,l), (1,2)} associated with the two subÂ¬ digraphs consisting of the sets of nodes ÃS,3} and {S,l,2}, respectively. Therefore, eight independent calculations are 30 necessary to find the costs of the six spanning directed trees for the digraph, and only two of the six spanning directed trees are essential. In fact, the eight indepenÂ¬ dent calculations enable us to find all 2n-l or seven optimal solutions necessary to perform efficiency/ equity analysis. Table 3-2 shows that the number of independent calculations necessary to find the cost of all the spanning directed trees for all possible subdigraphs is simply equal to the number of independent calculations to find the cost of all the spanning directed trees for the digraph less the number of essential spanning directed trees for the digraph. Consequently, for our 3-node digraph, six independent calculations are necessary to find the optimal solution for each user and each subgroup of users. For the balance of this chapter, the optimal solution for each user and each subgroup of users will be referred to as the 2n-2 optimal solutions. Finally, all suboptimal solutions are enumerated for all possible subdigraphs as well as for the digraph. Computational Considerations Although the number of independent calculations necesÂ¬ sary to find the costs of all the spanning directed trees for all possible subdigraphs as well as the digraph is uniquely determined by the configuration of the digraph, we can get a sense of the computational effort by examining the 31 Table 3-2. The Number of Independent Calculations to Find the Costs of Spanning Directed Trees for All Possible Subdigraphs. Independent Calculation Is Independent CalcuÂ¬ lation Used to Find the Costs of Spanning Directed Trees for the Digraph? Is Independent CalcuÂ¬ lation Used to Find the Costs of Spanning Directed Trees for All Possible Subdigraphs? Â©-*Â© Yes Yes 0â€”H0) Yes Yes (D-KD Yes Yes Yes Yes Yes Yes Â©-Â»Â©-Â»Â© Yes Yes dX Â© Yes No Ql Â©v Â© ) Yes No >(?/ Total Number of Yes 8 6 32 three digraphs shown in Figure 3-4. For the 3-node digraph, six independent calculations are necessary to find the costs of the four spanning directed trees for the digraph, and only one of the four spanning directed trees is essential. More importantly, 12 calculations are necessary to find the seven optimal solutions, but only 6 of the 12 calculations (50%) are independent. Furthermore, only five independent calculations are necessary to find the 2n-2 optimal soluÂ¬ tions. For the 4-node digraph, 10 independent calculations are necessary to find the cost of the eight spanning directed trees for the digraph, and only one of the eight spanning directed trees is essential. For this digraph, 33 calculations are necessary to find the 15 optimal solutions, but only 10 of the 33 calculations (30%) are independent. Moreover, only nine independent calculations are necessary to find the 2n-2 optimal solutions. Finally, for the 5-node digraph, 19 independent calculations are necessary to find the costs of the 24 spanning directed trees for the digraph, but only 2 of the 24 spanning directed trees are essential. In this case, 109 calculations are necessary to find the 31 optimal solutions, but only 19 of the 109 calculations (17%) are independent. From these 19 independent calculations, only 17 are necessary to find the 2n-2 optimal solutions. As we can see, summarized in Table 3-3, a large number of the spanning directed trees of a digraph are inessential. 33 3 gure ^~4 â€¢ Ex *mPles Â°f 3 ,4,5~Nocl e oÂ¿ 9rÂ¿iPhi Table 3- 3. Summary of Computational Effort for Digraphs Shown in Figure 3-4. Digraph 2n-l Optimal Solutions Number of Spanning Directed Trees Number of Inessential Spanning Directed Trees Number of Calculations to Find 2n-l Optimal Solutions Number of Independent Calculations to Find 2n-l Optimal Solutions (%) Number of Independent Calculations to Find 2n-2 Optimal Solutions 3-node 7 4 3 12 6 (50%) 5 4-node 15 8 7 33 10 (30%) 9 5-node 31 24 22 109 19 (17%) 17 u> 4^ 35 Also, the percentage of independent calculations decreases as the number of nodes for a digraph increases. The 5-node digraph in Figure 3-4 shows that the actual number of independent calculations necessary to determine the 31 optimal solutions to perform efficiency/equity analyÂ¬ sis of a regional water network problem involving five users is rather small. In fact, a regional water network serving five users may be considered a fairly large network. As larger systems form, increases in transactions costs because of multiple political jurisdictions, growing administrative complexity, etc., may eventually offset the gains from a regional system. In any case, real regional water network problems probably involve fairly small and sparse networks. That is, large networks can usually be broken down into smaller networks for analysis based on natural geographical and hydrological features, political boundaries, etc. Also, in actual problems there may not be that many choices for routing pipelines. Thus, the number of independent calculations necessary to calculate the 2n-l optimal solutions for a realistic regional water network should not be unreasonable. One of the advantages of using this total enumeration procedure is that it can be accomplished on a personal computer using readily available software. Thus, decision makers involved with planning and negotiating a regional water network can have easy access to information to aid the 36 decision-making process. For instance, the procedure can be implemented using the extremely "user friendly" Lotus 1-2-3 spreadsheet software package. Lotus 1-2-3 has the mathematical functions to handle calculations involving nonlinear cost functions or involving detailed cost analysis. A sample Lotus 1-2-3 printout is shown in Table 3-4 for a hypothetical water network problem modeled by the 3-node digraph shown in Figure 3-2. This printout should be self-explanatory. The top portion of the printout contains the data for the problem, and the bottom portion is the calculations associated with the total enumeration proÂ¬ cedure. The sorting capabilities of Lotus 1-2-3 allow automatic ranking of all the feasible solutions according to cost. Moreover, the Lotus 1-2-3 electronic spreadsheet automatically recalculates all values associated with a formula whenever a new value is entered or an existing value is changed. This automatically gives the total enumeration procedure the capability for sensitivity analysis. For example, the set of all feasible solutions ranked according to cost can be evaluated as the economies of scale, as represented by the value of b in equation (3-1), is varied over a specific range of values. Thus, for a regional network problem of realistic size, all the feasible solutions can be enumerated using a spreadsheet software package. 37 Table 3-4. Efficiency Analysis of a Three-User Water Supply Network with Nonlinear Cost Function Using Lotus 1-2-3. Sara Distance : L(i,j) is the distance in feet from i to j L (S, 1) = L(S,2) = 17320 L(S,3)= 26000 L(1,2)= 3325k L(1,3)= 13130 L(2,3)= 19673 15500 Demand : Q(i) is the demand in mgd for user i Q(1)= 1 Q(2)= Cost Function: a(Q~b)L a= 6 Q (3) = 38 b= 7 3.51 Calculations With C(i..j)[x]= Cost of network [x] for Total Enumeration i..j ; C(i..j)= L Procedure east cost C(1)[S1]= 646000 C(2)[S2]=2463343. C(3) [S3]=2012935. C(12)[SI,12]= 2984140. C (12)[SI;S2]= 3109348. C(12)= 2934140. C(13)[SI,13]= 2618975. C(13)[S1;S3]= 2658986. C(13)= 2518975. C(23) [S2,23]= 4061294. C(23)[32;S3]= 4476835. C(23)= 4061294. C(123)[SI,12,23]= 4648439. C (123)[SI,12;S3]= 4997126. C(123) [SI,12,13]= 4640756. C (123)[SI,13;S2]= 5032324. C (123) [SI;S2,23]= 4737294. C(123)[SI;S2;S3]= 5122335. C(123)= 4640756. Sort C(123) in ascending order Paths Cost C(123)[SI,12,13]= 4643755. C (123) [SI,12,23]= 4548439. C(123)[SI;S2,23]= 4707294. C (123)[SI,12;S3]= 4997126. C(123)[S1,13;S2]= 5082824. C(123)[S1;S2;S3]= 5122835. BEST C(123)= 4540756 38 Summary A total enumeration procedure for finding the optimal solutions necessary for efficiency/equity analysis of realistic regional water network problems is presented. The procedure can be easily understood and applied by engineers with little knowledge or experience in operations research techniques. Furthermore, the procedure allows the engineers to handle all problems regardless of the types of cost function involved or to perform detailed cost analysis. Finally, if the optimal solution is impractical for implementation, all suboptimal solutions ranked according to cost are readily available for consideration. CHAPTER 4 EQUITY ANALYSIS Introduction Proposed regional water resources systems involve multiple purposes and groups who must somehow share the cost of the entire project. The project may focus on construcÂ¬ tion of a large dam which serves numerous purposes such as water supply, flood control, and recreation. Also, canals from the dam direct the water to nearby users. A signifiÂ¬ cant portion of the total cost of this project may involve elements which serve more than one purpose and/or group. These costs are referred to as joint or common costs. In such cases, it is possible to find the optimal or the most economically efficient regional system, i.e., the one that maximizes benefits minus costs. However, a major effort remains to somehow apportion the project cost in an equitable manner. In fact, the importance of the financial analysis to apportion project cost is not limited to the optimal system but includes any other integrated systems being considered for implementation as well. This chapter examines principles of cost allocation using concepts from cooperative n-person game theory. An 39 40 example regional water network is used to illustrate these principles. Cost Allocation for Regional Water Networks A hypothetical situation similar to options contained in the West Coast Regional Water Supply Authority's master plan for Hillsborough, Pasco, and Pinellas counties in Florida (Ross et al., 1978) is now considered. Phase I (1980-1985) of the plan recommends the use of groundwater from existing and newly developed well fields to satisfy water demands in the tri-county area. For this hypothetical problem, assume that an existing well field is the most high quality and cost effective water supply source (S) available for three counties (1, 2, and 3) with projected demands of 1, 6, and 3 million gallons per day (mgd), respectively. The demand for each county is based on projected population growth and average per capita demand over a period of 5 years (see Table 4-1). Assume that the existing well field is currently operating below its capacity of 20 mgd and can satisfy the additional 10 mgd demanded by the three counties. In addition, assume that no facility expansion is required except for a new regional water network. FurtherÂ¬ more, each county can be served directly by the well field, and engineering considerations, e.g., gravity flow, have determined that water can be sent from county 1 to both county 2 and county 3, and from county 2 to county 3. The 41 Table 4-1. Projected Population Growth and Projected Average Per Capita Demand. County Projected Population Growth Projected Average Per Capita Demand (gal/cap-day) Projected Additional Demand (mgd) 1 8,000 125 1 2 40,000 150 6 3 18,750 160 3 Total 66,750 â€” 10 Weighted Average ... 150 .. 42 lengths of all possible interconnecting pipelines are shown in Figure 4-1. For our hypothetical problem, assume that the total cost of constructing a pipeline has strong economies of scale and is C = 38Q'^L, where C is total cost of pipeline in dollars, Q is quantity of flow in mgd, and L is the length of pipeline in feet. Given the problem just described, the cost of a pipeÂ¬ line serving county 1 alone is $646,000; the cost of a pipeline serving county 2 alone is $2,420,095; and the cost of a pipeline serving county 3 alone is $1,990,992. The total cost for three individual pipelines is $5,057,087. However, when the costs for all the options available to these three counties are enumerated using the procedure outlined in the preceding chapter, we see that the counties can do better by cooperating (see calculations in Appendix A using Lotus 1-2-3). There may be a slight difference between the numbers used in the text and the numbers in Appendix A because of rounding off. Also, cost data are only significant to the nearest thousand dollars. If the three counties cooperate, they can construct the least cost or optimal network consisting of pipelines from the well field to county 1, from county 1 to county 2, and from county 2 to county 3 (see Table 4-2). This optimal network costs $4,556,409 and represents a savings of 9.9% or $500,678 when compared with the cost for three individual pipelines. Obviously, constructing the optimal network is 3 tt<3d -1 â– 1>e .rvQ at- cÂ° ec .*<3 ?VPâ‚¬ f i-Q Xii( 44 Table 4-2. The Costs and Percent Savings for All Options. Option (Rank) Cost ($) Savings (%) 1 4,556,409 9.90 2 4,556,826 9.89 3 4,630,177 8.44 4 4,919,503 2.72 5 5,006,734 1. 00 6 5,057,087 0 45 in the best interest of the three counties, but to implement this least cost network, an equitable way to allocate the cost among the three counties must be found. This financial problem is known as a cost allocation problem. The complexÂ¬ ity is introduced because the counties share common pipes. Criteria for Selecting a Cost Allocation Method Several sets of criteria for selecting a cost allocaÂ¬ tion method are found in the literature. For the water resources field, criteria for allocating costs date back to the Tennessee Valley Authority (TVA) project in 1935 when prominent authorities were brought together to address the cost allocation problem. They developed the following set of criteria for allocating costs (Ransmeier, 1942, pp. 220-221) : 1. The method should have a reasonable logical basis. It should not result in charging any objective with a greater investment than the fair capitalized value of the annual benefit of this objective to the consumer. It should not result in charging any objective with a greater investÂ¬ ment than would suffice for its development at an alternate single purpose site. Finally, it should not charge any two or more objectives with a greater investment than would suffice for alternate dual purpose or multiple purpose improvement. 2. The method should not be unduly complex. 3. The method should be workable. 4. The method should be flexible. 5. The method should apportion to all purposes present at a multiple purpose enterprise a share in the overall economy of the operation. 46 This set of criteria developed for the water resources field is similar to the following set of criteria proposed by Claus and Kleitman (1973) for allocating the cost of a network: 1. The method must be easy to use and underÂ¬ standable to users. They must be able to predict the effects of changes in their service demands. 2. The method must have stability against system breakup. It should not be an advantage to one or more users to secede from the system. Thus, there are limits to which a method can subsidize one user or class of user at the expense of others. 3. It is desirable, though not necessary, that the costing be stable under evolutionary changes in the system or under mergers of users. 4. It is again desirable that the method should preserve the substance and appearance of nonÂ¬ discrimination among users. 5. If the method represents a change from present usage it is desirable that transition to the new method be easy. From these two sets of criteria, the most important criterion for selecting a method to allocate the cost of a regional water network is the method's ability to ensure stability or prevent breakup of the network. That is, the method should not allocate cost in a manner whereby an individual or a subgroup of individuals can acquire the same level of service by a less expensive alternative. OtherÂ¬ wise, the individual or subgroup of individuals will conÂ¬ sider their allocated cost inequitable or unfair and secede from the regional network for a less expensive alternative. Heaney (1979) has expressed these fairness criteria for an equitable cost allocation mathematically as follows: 1) x(i) < minimum [b(i), c(i)] VieN (4-1) where x (i) cost allocated to individual i b (i ) benefit of individual i c(i) = the alternative cost to individual i of independent action, and N set of all individuals; i.e., N = {1,2 , . . . ,n }. r â€¢ â€¢ â€¢ r This criterion simply means that individual i should not be charged a cost greater than the minimum of individual i's benefit and alternative cost for independent action. 2) Z x(i) _< minimum [b(S), c(S)] V Scn i eS (4-2) where c(S) = alternative cost to subgroup S of independent action, and b(S) = benefit of subgroup S. This second criterion extends the first criterion to include subgroup of individuals as well. These two fairness criteria are now used to evaluate some simple and seemingly fair cost allocation schemes for our regional water network problem. Throughout this chapter, we will assume for our regional water network problem that each county's and each 48 subgroup of counties1 alternative cost of independent action is less than or equal to each county's and each subgroup of counties' benefits, respectively; i.e., c(i) = minimum [b(i), c(i)] V ieN, and (4-3) c(S) = minimum [b(S), c(S)] V ScN. Ad Hoc Methods Over the years, many ad hoc methods have been proposed or used to apportion the costs of water resources projects (Goodman, 1984). In general, ad hoc methods used in the water resources field for allocating costs can be described as follows: allocate certain costs that are considered identifiable to an individual directly and prorate the remaining costs, i.e., total project cost less the sum of all identifiable costs, among all the individuals in the project by some physical or nonphysical criterion. MatheÂ¬ matically, this can be expressed as follows: x ( i ) = x(i)id + iMi)*rc (4-4) where x (i) x (i) id ip (i) rc cost allocated to individual i, costs identifiable to individual i, prorating factor for individual i, and remaining costs, i.e., c(N) 49 Furthermore, the requirement that Z (i) = 1.0 should be ieN obvious. James and Lee (1971) summarize 18 ways for allocating the costs of water projects depending on the definition of identifiable costs and the basis for prorating the remaining costs (see Table 4-3). Basically, the differences among these 18 methods are the following three ways of defining identifiable costs: 1) zero, 2) direct or assignable costs, or 3) separable costs; and the following six ways of prorating remaining costs: 1) equal, 2) unit of use, 3) priority of use, 4) net benefit, 5) alternative cost, or 6) the smaller of net benefit or alternative cost. The next two sections analyze the effects of defining identifiable costs as either zero or direct costs. A detailed treatment of separable costs, i.e., the difference between total project costs with and without an individual, is given in the section on the separable costs, remaining benefits method. Defining Identifiable Costs as Zero The simplest way to allocate costs is to define identiÂ¬ fiable costs as equal to zero and prorate total project cost by some physical or nonphysical criterion. For example, population and demand are two ways to prorate total project 50 Table 4-3. Cost Allocation Matrix. Definition of Identifiable Cost Basis for Prorating Remaining Costs A. Zero B. Direct Cost C. Separable Cost a. Equal Aa Ba Ca b. Unit of Use Ab Bb Cb c. Priority of Use Ac Be Cc d. Net Benefit Ad Bd Cd e. Alternative Cost Ae Be Ce f. Smaller of d. or e. Af Bf Cf Source: Modified from James and Lee, 1971, p. 533. 51 cost (Young et al., 1982). Using these two ways to prorate the cost of the optimal network for our regional water network problem gives the following cost allocations (see calculations in Table 4-4 and Table 4-5): Proportional to Population County 1 $ 546,769 County 2 2,733,845 County 3 1,275,795 $4,556,409 Proportional to Demand County 1 $ 455,641 County 2 2,733,845 County 3 1,366,923 $4,556,409 Although these cost allocations are simple to calculate and easy to understand, they fail to implement the optimal network because county 2 considers these cost allocations unfair. In contrast to counties 1 and 3, county 2 loses money by being allocated a cost in excess of its go-it-alone costs using either of these two methods. Consequently, county 2 would rather acquire a pipeline by itself than cooperate with counties 1 and 3 to construct the optimal network. The principal failure with these proportionality Table 4-4. Cost Allocation of Optimal Network Based on Population. County i Population Percent of Total Population Allocated Cost ($) x (i) Go-It-Alone Cost ($) c (i ) Is x ( i ) < c ( i ) ? 1 8,000 12 546,769 646,000 Yes 2 40,000 60 2,733,845 2,420,095 No 3 18,750 28 1,275,795 1,990,992 Yes Total 66,750 100 4,556,409 5,057,087 â€” Ul ro Table 4-5. Cost Allocation of Optimal Network Based on Demand County i Demand (mgd) Percent of Total Demand Allocated Cost ($) x (i) Go-It-Alone Cost ($) c (i ) Is x(i) < c(i)? 1 1 10 455,641 646,000 Yes 2 6 60 2,733,845 2,420,095 No 3 3 30 1,366,923 1,990,992 Yes Total 10 100 4,556,409 5,057,087 â€” U1 u> 54 methods is that they do not recognize explicitly each individual's contribution to total project cost. Defining Identifiable Costs as Direct Costs A way to recognize each individual's contribution to total project cost is by defining identifiable costs as those costs that can be directly assigned, and prorating the remaining costs by some physical or nonphysical criterion such as use or number of individuals; i.e., x (i) = x(i)direct + 4Â»(i)-re (4-5) where x(i),. = direct cost or assignable cost ireC to individual i. Although this direct costing approach intuitively seems fair, inequitable and unpredictable cost allocations can result. To illustrate, two direct costing methods are applied to our regional water network problem. A common approach to allocating remaining costs is by some physical measure of each individual's use of the common facilities; this method is generally referred to as the use of facilities method (Loughlin, 1977; Goodman, 1984). This traditional method is easy to understand and apply because quantitative information on a physical measure of use is generally available. In the water resources field, use can be measured in terms of the storage capacity and/or the 55 quantity of water flow provided by the common facilities. For our regional water network problem, the flow to each county is the obvious measure of use to apportion the costs of common pipelines since the assumed cost function depends on the flow. In the case of the optimal network, the only direct cost is the cost of the pipeline from county 2 to county 3 serving county 3, and the use of facilities method gives the following cost allocation (see calculations in Table 4-6). $ 204,283 2,221,299 2,130,827 $4,556,409 County 1 County 2 County 3 Total Unfortunately, this cost allocation does not implement the optimal network because county 3 can do substantially better by going alone, i.e., $1,990,992 versus paying $2,130,827. In addition to giving an inequitable cost allocation for the optimal network, the use of facilities method can promote noncooperation if other networks are also being considered. Table 4-7 shows the cost allocations for all possible options available to the three counties using the use of facilities method. Suppose the "second best" network or option 2 is also being considered by the counties. The second best network consists of the pipelines from the well field to county 1, from county 1 to county 2, and from county 1 to county 3. This second best network costs Table 4-6. Cost Allocation of Optimal Network with Use of Facilities Method Pipeline S-l 1-2 2-3 Total Cost ($) Go-It-Alone Cost ($) c (i ) Length (f t) 17,000 13,100 15,500 â€” â€” Q (mgd) 10 9 3 â€” â€” Pipeline Cost ($) 2,042,832 1,493,400 1,020,177 4,556,409 â€” Cost for County 1 ($) Q=1 mgd 204,283 0 0 204,283 646,000 Cost for County 2 ($) Q=6 mgd 1,225,699 995,600 0 2,221,299 2,420,095 Cost for County 3 ($) Q=3 mgd 612,850 497,800 1,020,177 2,130,827 1,990,992 Ln 57 Table 4-7. Cost Allocation for the Use of Facilities Method. Cost Allocation to ($) County i Zx (i) Is Cost Allocation Option (Rank) County 1 x(l) County 2 x (2 ) County 3 x (3 ) (?) Equitable? 1 204,283 2,221,299 2,130,827 4,556,409 No x(3)>c(3 ) 2 204,283 2,445,055 1,907,488 4,556,826 No x(2)>c(2 ) 3 646,000 1,976,000 2,008,177 4,630,177 No x(3)>c(3 ) 4 244,165 2,684,346 1,990,992 4,919,503 No x(2)>c(2) 5 323,000 2,420,095 2,263,639 5,006,734 No x(3)>c(3) 6 646,000 2,420,095 1,990,992 5,057,087 â€” (4) (5) (6) 58 $4,556,826 or $417 more than the optimal network; so, both networks are essentially comparable in cost, and either network might be considered the least cost network. In fact, the second best network becomes the optimal network if the economies of scale or the value of b in the cost function is .51 instead of .50 (see Table 3-4). NeverÂ¬ theless, applying the use of facilities method to this second best network gives the following cost allocation. $ 204,283 2,445,055 1,907,4 88 $4,556,826 County 1 County 2 County 3 In this case, the cost allocation fails to implement the second best network because county 2 is better off going alone, i.e, paying $2,420,095 rather than $2,445,055. Furthermore, if we examine the cost allocation for the optimal network and the second best network, another problem is evident. Although the costs for the two networks are $417 apart, the difference in costs between the two networks for county 2 and county 3 is enormous. Consequently, this cost allocation method imposes another obstacle for the counties to cooperate and implement either one of the two networks. County 2 strongly opposes the second best network because of its substantially higher cost while county 3 strongly opposes the optimal network for the same reason. This problem is even more serious when more options are considered by the counties. Table 4-7 indicates tremendous 59 differences in allocated cost for each county depending on the network, thereby making cooperation very difficult. This situation shows the danger for individuals to simply accept the least cost network without carefully examining all of their options if the use of facilities method for allocating costs is chosen. Another simple way to prorate the remaining costs is to divide it equally among the individuals associated with the common facilities (see calculations for optimal network in Table 4-8). Table 4-9 shows the cost allocations using this egalitarian approach and indicates that none of the cost allocations for options with savings are equitable. At first glance, the cost allocation for option 5 appears equitable because each county is charged a cost less than or equal to its go-it-alone cost. However, closer examinaÂ¬ tion reveals that counties 1 and 2 can do better as a coalition. They can construct a pipeline from the well field to county 1 and from county 1 to county 2, i.e., option 4, for $2,928,511 rather than pay the sum of their costs for option 5, i.e., $3,066,095. Unfortunately, a transition from option 5 to option 4 causes county 1 to lose money, i.e., $854,577 for option 4 versus $646,000 for option 5. To further complicate matters, option 5 only gives a 1% savings and requires county 1 to cooperate with county 3 to build a pipeline without getting any savings. Table 4-8. Cost Allocation of Optimal Network with Direct Costing/Equal Apportionment of Remaining Costs Method. Pipeline S-l 1-2 2-3 Total Cost ($) Go-It-Alone Cost($) c (i ) Length (ft) 17,000 13,100 15,500 â€” â€” Q (mgd) 10 9 3 â€” â€” Pipeline Cost ($) 2,042,832 1,493,400 1,020,177 4,556,409 â€” Cost for County 1 ($) Q=1 mgd 680,944 0 0 680,944 646,000 Cost for County 2 ($) Q=6 mgd 680,944 746,700 0 1,427,644 2,420,095 Cost for County 3 ($) Q=3 mgd 680,944 746,700 1,020,177 2,447,821 1,990,992 o Table 4-9. Cost Allocation for Direct Costing/Equal Apportionment of Remaining Costs Method 61 Cost Allocation to County i Option (Rank) County 1 x (1) ($) County 2 x (2 ) County 3 x (3) f x ( i ) ($) Is Cost Allocation Equitable? 1 680,944 1,427,644 2,447,821 4,556,409 No X(1)>c(1) x(3)>c(3) 2 680,944 1,900,300 1,975,582 4,556,826 No x(l)>c(l) 3 646,000 1,482,000 2,502,177 4,630,177 No x(3)>c(3 ) 4 854,577 2,073,934 1,990,992 4,919,503 No X(1)>c(1) 5 646,000 2,420,095 1,940,639 5,006,734 No x(1)+x(2)> c (12 ) 6 646,000 2,420,095 1,990,992 5,057,087 (4) (5) (6) 62 Given these observations, the stability of option 5 as a regional water network is at best questionable. Again, if the allocated costs for counties 2 and 3 for the optimal network are compared to the second best network, a similar situation like the one discussed for the use of facilities method exists. That is, counties 2 and 3 face substantially different costs for these two networks with comparable costs. Thus, assigning direct costs does not help eliminate inequitable cost allocations. In fact, direct costing methods can impose additional obstacles to cooperation. This occurs because the assignment of direct costs depends on the configuration of the facilities. For instance, the cost of the pipeline from county 2 to county 3 for our regional water network problem can be a direct cost or a joint cost depending on the network. The cost of the pipeÂ¬ line is a direct cost for county 3 if the second best network, i.e., option 2, is being considered; yet, the cost of the pipeline is a joint cost for counties 2 and 3 if the optimal network, i.e., option 1, is being considered. These changes in the cost classification for the pipeline from county 2 to county 3 contribute to the tremendous difference in the cost allocations for counties 2 and 3 for the two comparable cost networks. This situation indicates an additional criterion not addressed by Claus and Kleitman (1973) for selecting a procedure to allocate network cost. 63 The cost allocation procedure should be independent of network configuration; otherwise, the cost allocation proÂ¬ cedure can promote noncooperation if more than one network is being considered. In summary, two approaches for allocating costs in the water resources field have been examined: 1) allocate total project cost in proportion to a physical or nonphysical criterion; or 2) allocate assignable costs directly and prorate the remaining costs by a physical or nonphysical criterion. In general, these two approaches are simple to apply and easy to understand. In fact, these two approaches are currently accepted cost allocation methods used in accounting (Kaplan, 1982). However, these two approaches are unable to consistently provide an equitable cost allocation when an equitable cost allocation exists, i.e., sometimes these methods work and sometimes they fail. Furthermore, methods attempting to assign costs directly may be influenced by the configuration of the facilities and may discourage cooperation when more than one configuration is being considered. This is particularly evident for our regional water network problem. For a theoretically sound method that is able to find an equitable cost allocation if one exists and is not influenced by the configuration of the facilities, concepts from cooperative n-person game theory are necessary. 64 Cooperative Game Theory Game theory has been with us since 1944 when the first edition of The Theory of Games and Economic Behavior by John Von Neumann and Oskar Morgenstern appeared. In particular we are interested in games wherein all of the players voluntarily agree to cooperate because it is mutually beneÂ¬ ficial. Furthermore, games are studied in three forms or levels of abstraction. The extensive form requires a comÂ¬ plete description of the rules of a game and is generally characterized by a game tree to describe every player's move. A game in normal form condenses the description of a game into sets of strategies for each player and is represented by a game matrix. However, most efforts in cooperative game theory have been with games in characÂ¬ teristic function form whereby the description of a game is in terms of payoffs rather than rules or strategies. The characteristic function form appears to be the most appropriate for studying coalition formation which is an essential feature in cooperative games. Also, cooperative games can be of three types depending on whether the game is defined in terms of costs, savings, or values. To keep the notation as simple as possible, only cost games will be discussed. Introductory and intermediate material on coopÂ¬ erative game theory can be found in Schotter and Schwodiauer (1980), Jones (1980), Luce and Raiffa (1957), Lucas (1981), Rapoport (1970), Shubik (1982), and Owen (1982). 65 Concepts of Cooperative Game Theory Let N = {1,2,...,n} be the set of players in the game. Associated with each subset of S players in N is a characÂ¬ teristic function c, which assigns a real number c(S) to each nonempty subset of S players. For cost games, the characteristic function, c(S), can be defined as the least cost or optimal solution for the S-member coalition if the N-S member or complementary coalition is not present. However, depending on how the problem is defined, alternaÂ¬ tive definitions for c(S) may be required. For example, Sorenson (1972) presents the following four alternative definitions for the characteristic cost function: c.(S) = value to coalition if S is given preference over N-S. C2(S) = value of coalition to S if N-S is not present, c^iS) = value of coalition in a strictly competitive game between coalition S and N-S, and c^(S) = value of coalition to S if N-S is given preference. If c(S) can be defined as the least cost solution for coalition S if N-S is not present, then the cost game is naturally subadditive; i.e., c(S) + c(T) > c(SUT) ShT = 0, S,TcN (4-6) 66 where 0 is the empty set; and S and T are any two disjoint subsets of N. Subadditivity is a natural consequence of c(S) because the worst S and T can do as a coalition is the cost of independent action; i.e., c(S) + c(T) = c(SUT) SOT = 0, S,TCN. (4-7) A coalition in which the players realize no savings from cooperation is said to be inessential. General reasons why subadditivity exists are discussed by Sharkey (1982a). The primary reason why subadditivity exists for our regional water network problem is because of the economies of scale in pipeline construction cost. For a single output cost function, C(q), economies of scale is defined by C(Aq) < Ac(q) (4-8) where q = output level, and for all A such that 1 < A Â£ 1 + e, e is a small positive number. This definition means that the average costs are declining in the neighborhood of the output q. From Sharkey (1982a), economies of scale is sufficient but not a necessary condiÂ¬ tion for subadditivity. Subadditivity is a more general 67 condition which allows for both increasing marginal cost and increasing average cost over some range of outputs. Solution concepts for cooperative cost games involve the following three general axioms of fairness (Heaney and Dickinson, 1982; Young et al., 1982); 1) Individual Rationality: Player i should not pay more than his go-it-alone cost, i.e., x(i) _< c(i), Â¥ ieN, (4-9) where x(i) is the allocated cost or the charge to player i. 2) Group Rationality: The total cost of the grand coalition, c(N), must be apportioned among the N players; i.e., Ex(i)=c(N). (4-10) ieN 3) Subgroup Rationality: This final axiom extends the notion of individual rationality to include subgroups, i.e., no subgroup or subcoalition S should be apportioned a cost greater than its go-it-alone cost, or 1 x(i) < c(S), V SCN. (4-11) ie S The set of solutions or charges satisfying the first two axioms is called the set of imputations, while the 68 additional restriction of the third axiom defines what is known as the core of the game. For subadditive cost games the set of imputations is not empty, but the core may be empty. Shapley (1971) has shown that the core always exists for convex games. A cost game is convex if c(S) + c(T) > c(SUT) + c(ShT) SOT i 0, V S,TcN (4-12) or equivalently, convexity can be written as c(SUi) - c(S) > c(TUi) - c(T) ScTcN - {i}, ieN. (4-13) Convexity simply means the incremental cost for player i to join coalition T is less than or equal to the incremental cost for player i to join a subset of T. This notion of convexity is analogous to economies of scale and implies the game has a particular form of increasing returns to scale in coalition size. As will be shown, the more attractive the game, i.e., larger savings in project costs, the greater the chance that the game is convex; whereas, if the game is less attractive, i.e., lower savings in project costs, the potenÂ¬ tial for a nonconvex game or an empty core game is greater. To illustrate the concept of the core, assume a three- person cost game with the following characteristic function values: 69 c(l) = 35 c (2 ) = 45 c ( 3 ) = 50 c(12 ) = 66 c (13 ) = 75 c(23 ) = 87 c(12 3 ) = 100 This game is subadditive so each player has an incentive to cooperate; i.e., c(l) + c(2) + c ( 3 ) > c(12 3 ) c(l) + c (23 ) > c(123 ) c (2 ) + c (13 ) > c (123) c ( 3 ) + c (12 ) > c(12 3 ) c (1) + c (2 ) > c (12 ) c(l) + c (3 ) > c (13 ) c (2 ) + c (3 ) > c ( 23 ) . Furthermore, this game is convex; i.e., c (12 ) + c (13 ) > c(12 3 ) + c (1) c (12 ) + c (23 ) > c(12 3 ) + c (2 ) c (13 ) + c ( 23 ) > c(12 3 ) + c ( 3 ) . Using the three general axioms of fairness, the core conditions are as follows: x(l) < 35 x ( 2) < 45 x ( 3 ) < 50 x(l) + x(2) < 66 x(l) + x(3) < 75 x(2) + x(3) < 87 x(l) + x(2) + x(3) = 100. 70 The first three conditions determine the upper bounds on x(i), i = 1,2,3, while the last four conditions determine bounds on x(i), i = 1 i -2, r 3 , Ã . e. â–º t c (123 ) - c(23) = 13 < x(l) < 35 = c(l) c(123) - c(13 ) = 25 < x (2 ) < 45 = c ( 2 ) c(12 3 ) - c(12 ) = 34 < x ( 3 ) < 50 = c ( 3 ) For a three-person game, graphical examination of the core conditions and the nature of the charge vectors is possible using isometric graph paper (Heaney and Dickinson, 1982). As shown on Figure 4-2, each player is assigned a charge axis. The plane of triangle ABC, with vertices (100,0,0), (0,100,0), and (0,0,100), represents points satisfying group rationality (axiom 2); whereas, the smaller triangle abc represents the set of imputations satisfying both individual rationality (axiom 1) and group rationality (axiom 2). The vertices a, b, and c represent the charge vectors: [35, 15, 50], [5, 45, 50], and [35, 45, 20], respectively. Line ab represents the upper bound for player 3, i.e., x(3) = c(3), where c(123) - c(3) is allocated between players 1 and 2. As we move along line ab from point a to point b, the allocation to player 1 decreases from c(l) to c(123) - c(2) - c(3), i.e., from 35 to 5, while the allocation to player 2 increases from c(123) - c(l) - c(3) to c(2), i.e., from 15 to 45. Similar explanations can be given for lines be and ac. A more restrictive set of solutions satisfying subgroup rationality (axiom 3), 71 x( 3) Figure 4-2. Geometry of Core Conditions for Three- Person Cost Game Example. 72 the shaded area on triangle abc, is the core for this game. The geometry of the core for this convex game is a hexagon. Line de represents the lower bound for player 2 or the set of charges where c(13) is allocated between player 1 and player 3 with the remainder, c(123) - c(13), going to player 2. Similar explanations can be given for lines fg and hi which are the lower bounds for players 1 and 3, respectively; and for lines id, gh, and ef which are the upper bounds for players 1, 2, and 3, respectively. If an allocation lies outside the core, an inequitable situation prevails. For instance, point Z in Figure 4-2 allocates player 2 a cost less than its lower bound, c(123)-c(13), which means c(13) increases or the cost allocated to players 1 and 3 increases. Clearly, player 1 and player 3 can do better by forming their own two-person coalition rather than subsidizing player 2. As mentioned earlier, the convexity of a game and its attractiveness are related. This relationship is illustrated in Table 4-10. When the costs for the two-person coalitions progressively decrease, there is less incentive for forming the grand coalition so the core becomes progressively smaller and the game becomes progressively more nonconvex. As a consequence of the core conditions for a three-person subÂ¬ additive cost game, a condition can be derived to determine if a core exists. From subgroup rationality and group rationality, we have the following conditions: 73 Table 4-10. Core Geometry for Three-Person Cost Game Example. Characteristic Function c(1) = 35, c(2) = 45, c(3) = 50, c(123) = 100 Geometry Zc(ij) of Core c (12) c(13 ) c(23 ) 66 75 87 228 < } Hexagon 61 73 86 220 A> Pentagon 59 71 85 215 ZA Trapezoid 58 70 80 208 A Triangle 56 68 76 200 â€¢ Point 55 65 72 192 x (2 ) x(1) // -\rx<3) //y Empty Source: Modified from Fischer and Gately, 1975, p. 27a. 7 4 x(l) + x(2) < c(12) x(l) + x(3) < c(13) x ( 2 ) + x ( 3 ) < c ( 2 3 ) x(l) + x(2) + x(3) = c(12 3) (4-14) Summing the subgroup rationality conditions gives 2â€¢[x(1) + x(2) + x ( 3 ) ] < c(12) + c(13) + c(23). (4-15) If the group rationality conditions are substituted into the above equation, then we have the following condition to determine if a core exists: 2 * c(12 3) < c(12 ) + c(13 ) + c(2 3 ) . (4-16) Therefore, in Table 4-10, the core exists as long as the sum of the two-person coalitions is greater than 200 or twice the value of the grand coalition. When the sum of the two-person coalitions equals 200, the core reduces to a unique vector, i.e., X = [24, 32, 44]. Finally, when the sum of the two-person coalition is less than 200, then the core is empty. Unfortunately, for larger games there is no simple condition for checking the existence of a core; however, as we will see later, a check can be made using linear programming. 75 Unique Solution Concepts The three axioms of fairness defining the core of the game significantly reduce the set of admissible solutions. Unless the core is empty or is a unique vector, an infinite number of possible equitable charge vectors remain to be considered, so additional criteria are needed to select a unique charge vector. Numerous methods are available for selecting a unique charge vector; but the two most popular methods discussed in the literature are the Shapley value (Shapley, 1953; Heaney, 1983b; Shubik, 1962; Heaney et al., 1975; Littlechild, 1970) and the nucleolus (Schmeidler, 1969; Kohlberg, 1971; Suzuki and Nakayama, 1976). Shapley value. The Shapley value for player i is defined as the expected incremental cost for the coalition of adding player i. Thus, each player pays a cost equal to the incremental cost incurred by the coalition when that player enters. Since the coalition formation sequence is unknown, the Shapley value assumes an equal probability for all sequences of coalition formation, i.e., the probability of each player being the first to join is equal, as are the probabilities of joining second, third, etc. For an n person game there are n! orderings. The six sequences of coalition formation for a three-person game are as follows: (123) (213) (231) (132) (312) (321) 7 6 Therefore, the Shapley value or the cost to player 1 for a three-person game is $(1) = 1/3 c(1) + 1/6 [c(12) - c(2)] + 1/6 [c(13 - c(3)] + 1/3 [c(12 3) - c(23)]. (4-17) Player 1 has 1/3 probability of entering the coalition as the first player and 1/3 probability of entering the coalition as the last player. In addition, player 1 has 1/6 probability of entering the coalition after player 2 and 1/6 probability of entering the coalition after player 3. Notice that [c(S+i) - c(S)] is the incremental cost of adding player i to the S coalition. The general formula for the Shapley value for player i is 4> ( i ) = I a. (S) [ c {S) - c ( S - { i } ) ] Scn 1 (4-18) where (s-i)! (n - s)Ã rf! s is the number of players in coalition S, n! is the total number of possible sequences of coalition formation, (s-1)! is the number of arrangements for those players before S, and (n-1)! is the number of arrangements for those players after S. 77 For example, for i = 1, n = 3: a1(l) 0121/3! = 1/3 ax(12) = 1111/31 = 1/6 a1(13) = 1111/3! = 1/6 a1(123) = 2101/31 = 1/3 Total 1.0 Note that 2 i(i) = c(N). (4-19) i eN Furthermore, if the game is convex, the Shapley value lies in the center of the core (Shapley, 1971). The Shapley value is criticized for several reasons. It may fall outside the core for nonconvex games, and it may be computed even when the core does not exist (Hamlen, 1980). Furthermore, the Shapley value is computationally burdensome for large games. For an n-person game, the Shapley value for each player requires the computation of n 1 2 coefficients and incremental costs. For example, an eight player game requires 128 coefficients and incremental costs to calculate the charge for each player. 78 Loehman and Whinston (1976) attempted to reduce the computational burden of the Shapley value by relaxing the assumption that all sequences of coalition formation are equally likely. This generalized Shapley value allows using a priori information to eliminate impossible sequences of coalition formation. Unfortunately, when Loehman et al. (1979) applied the generalized Shapley value to an eight-player regional wastewater management problem, they got a solution outside the core (Heaney, 1983a). Littlechild and Owen (1973) developed the simplified Shapley value for games wherein the characteristic function is a cost function with the property that the cost of any subcoalition is equal to the cost of the largest player in the subcoalition. Although Littlechild and Thompson (1977) demonstrated the computational ease of the simplified Shapley value in their case study of airport landing fees consisting of 13,572 landings by 11 different types of aircraft, the use of the simplified Shapley value is restricted to games with these special properties. Before calculating the Shapley value for our regional water network problem, the total enumeration procedure described in the preceding chapter is used to find the following characteristic cost function values (see Appendix A) : c(1) = 646,000 c(2 ) = 2,420,095 c(3 ) = 1, 990,992 c(12) = 2,928,511 c(13) 2,586,638 c ( 2 3 ) 3,984,177 79 and 1 c (12 3) 4, 556, 409 2 c (12 3) 4, 556, 826 3 c (1, 23) = 4, 630, 177 4 c (12 ,3) = 4, 919, 503 5 c (13 ,2) = 5, 006, 734 w c dr 2,3) = 5, 057, 087 k t h where c (hi,j) is the cost of the k best regional water network consisting of pipelines from the well field to county h, from county h to county i, and from the well field . w to county j. Also, c (1,2,3) is the cost for each county to t h go-it-alone. The cost allocation associated with the k t h best regional water network, i.e., the kc network game, is simply found by setting c(N) equal to c (N). The Shapley values for all options available to the three counties are calculated in Appendix A and summarized in Table 4-11. Appendix A also checks whether each Shapley value satisfies core conditions. All of the network games in this example are nonconvex. Table 4-11 shows that the cost allocations for the optimal and the second best networks, i.e., the first two options, satisfy all core conditions; therefore, these cost allocations are in the core and are considered equitable. Furthermore, unlike the cost allocations using the direct costing methods discussed earlier, the cost allocations for these two comparable cost 80 Table 4-11. Cost Allocation for Three-County Example Using the Shapley Value. Cost Allocation to County i Is Cost ($) Allocation Zx(i) In Core? Option County 1 County 2 County 3 ($) (From Ap- (Rank) x(l) x(2) x(3) pendix A) 1 590,087 2,175,905 1,790,417 4,556,409 Yes 2 590,226 2,176,044 1,790,556 4,556,826 Yes 3 614,677 2,200,494 1,815,006 4,630,177 No 4 711,119 2,296,936 1,911,448 4,919,503 No 5 740,196 2,326,013 1,940,525 5,006,734 No 6 646,000 2,420,095 1,990,992 5,057,087 â€” 81 networks are nearly identical. The Shapley value divided the additional $417 for the second best network equally among the counties. Option 3 illustrates the failure of the Shapley value to consistently give a core solution for nonconvex games. As shown in Appendix A, the cost allocaÂ¬ tion for option 3 fails to satisfy subgroup rationality for the coalition consisting of county 2 and county 3; i.e., x(2) + x(3) > c(23 ) . (4-20) Moreover, options 4 and 5 illustrate Shapley values for games with an empty core. The nonexistence of the core for network games with options 4 and 5 can be determined by using other game theory methods, e.g., nucleolus. A close examination of the core conditions for network games with options 4 and 5 reveals these games are no longer subaddi- tive. By defining c(N) as c (N), c(N) is no longer the least cost or optimal solution for the grand coalition. t h Consequently, the k best network game is not naturally subadditive even though c (N) may be less expensive than w c (N). In any event, because network games with options 4 and 5 are not subadditive, there is no incentive to cooperate. Therefore, options 4 and 5 no longer need to be considered by the counties. Nucleolus. The other popular method to obtain a unique charge vector is to find the nucleolus. For a cost game, 82 the fairness criterion used by the nucleolus is based on finding the charge vector which maximizes the minimum savings of any coalition. For each imputation in the core of a cost game, a 2n vector in R is defined. The components of this vector are arranged in increasing order of magnitude and are defined by e(S) = c(S) - z x(i) V ScN. (4-21) ieS 2 n The imputation whose vector in R is lexicographically the largest is called the nucleolus of the cost game. Given two vectors, X = (x^,...,x ) and Y = (y^,...fy ), X is lexiÂ¬ cographically larger than Y if there exists some integer k, 1 < k < n, such that x^ = y^ for 1 _< j < k and x^. > y^ (Owen, 1982). Basically, e(S) represents the minimum savings of coalition S with respect to charge vector X. Obviously, the coalition with the least savings objects to charge vector X most strongly, and the nucleolus maximizes this minimum savings over all coalitions. The nucleolus can be found by solving at most n-1 linear programs (Kohlberg, 1972; Owen, 1974, 1982), where the first linear programming problem is maximize e(l) subject to e(l) + x(i) < c(i) Â¥ ieN (4-22) 83 e (1) + Â£ x ( i ) < c (S) Â¥ S N ie S Â£ x ( i ) = c (N) ie N x ( i ), e (1) > 0 The nucleolus is calculated by sequentially solving for e(l), then e(2), e(3), etc., where e(i) is the ifc^ smallest savings to any coalition. Unlike the Shapley value, the nucleolus always is in the core for games with nonempty core. In fact, the nucleolus is always unique. However, the nucleolus is criticized because it cannot be written down in explicit form (Spinetto, 1975), and that it is difficult to compute and use in practice (Gugenheim, 1983). Probably the most difficult problem with using the nucleolus is the acceptance of its notion of fairness as opposed to other prevailing notions of fairness without generating unending controversies and debates. The nucleolus is generally considered to be analogous to Rawls' (1971) welfare criteria: the utility function of the least well off individual is maximized. Other notable notions of fairness include (1) Nozick's (1974) procedural approach to justice, and (2) Varian's (1975) or Baumol's (1982) definition of 84 equitable distribution whereby no one prefers the consumption bundle of anyone else. Calculating the nucleolus for our regional water network problem using the linear programming problem (4-22) gives the results summarized in Table 4-12. Equitable cost allocations are given for the first three options, and the cost allocations for the optimal and the second best networks are essentially the same. The additional $417 for the second best network is apportioned as follows: County 1 $209 County 2 104 County 3 104 Total $417 No cost allocations are given for options 4 and 5 because these network games have empty cores. That is, the linear programming problem (4-22) is infeasible. Finally, Table 4-12 reveals that each of the three counties has an incentive to cooperate in order to implement the cheapest regional water network. Propensity to disrupt. Another unique solution concept worth mentioning because of its intuitive appeal is the concept of an individual player's "propensity to disrupt." Gately (1974) defined an individual player i's propensity to disrupt as a ratio of what the other players would lose if player i refused to cooperate over how much player i would lose by not cooperating. Mathematically, player i's 85 Table 4-12. Cost Allocation for Three-County Example Using the Nucleolus. Cost Allocation to ($) County i lx (i ) Is Cost Allocation Option (Rank) County 1 x(l) County 2 x (2 ) County 3 x (3 ) ($) In Core? 1 609,116 2,144,583 1,802,710 4,556,409 Yes 2 609,325 2,144,687 1,802,814 4,556,826 Yes 3 4 646,000 2,163,025 1,821,152 4,630,177 Yes 5 6 646,000 2,420,095 1,990,992 5,057,087 â€” 86 propensity to disrupt, d(i), a charge vector, X = [x(1),...,x(n)], which is in the core is c(N-i) - Â£ x(j ) d ( i ) = 3ZÃÃ (4-23) c ( i ) - x(i ) The higher the propensity to disrupt, the greater a player's threat to the coalition; e.g., d(i) = 10 implies player i could impose a loss of savings to the other players 10 times as great as the loss of savings to player i. As an illustration, the propensity to disrupt is calculated for each of the counties using the nucleolus for the optimal network of our regional water network problem: X = [609,116; 2,144,583; 1,802,710]. d (1) c(23 - x(2) - x(3 ) c (1) - x (1) = 1.0 d (2 ) _ c(13 ) - x (1) - x(3 ) = .63 c ( 2 ) - x ( 2 ) d (3 ) _ c(12 ) - x(1) - x ( 2 ) = .93 c ( 3 ) - x ( 3 ) The calculations show that none of the counties have a strong threat against the other two counties with the nucleolus charge vector. County 1 could impose a loss to the other two counties which equals the loss imposed on itself, while, county 2's or county 3's departure would hurt the departing county more than it would hurt the remaining two counties. 87 Gately suggested equalizing each player's propensity to disrupt as a final cost allocation solution. Subsequently, Littlechild and Vaidya (1976) have generalized Gately's concept of an individual player's propensity to disrupt to include a coalition S's propensity to disrupt. That is, a coalition S's propensity to disrupt is defined as the ratio of what the complementary coalition, N-S, stands to lose over what the coalition S itself stands to lose for a given charge vector. More recently, Chames et al. (1978 ) and Chames and Golany (1983) refined these propensity to disrupt concepts into a unique solution concept which appears to have some empirical support. Finally, Straffin and Heaney (1981) have shown that Gately's propensity to disrupt is exactly the alternative cost avoided method first proposed during the TVA project in 1935. The alternative cost avoided method is discussed in the section on the separable costs, remaining benefits method. Empty Core Solution Concepts Examining games with an empty core is an active area of research. An empty core implies that no equitable cost allocation exists, and results from games wherein the addiÂ¬ tional savings from forming the grand coalition is relaÂ¬ tively small. That is, the savings resulting from forming smaller coalitions are almost as much as the savings from forming the grand coalition. Therefore, proposed solution 88 concepts generally seek to relax the bounds on subgroup rationality until a "quasi" or "anti" core is created. Table 4-13 lists four methods for finding a charge vector for games with an empty core. In any case, given the modest amount of economic gain for games with an empty core, it may be more advantageous to forego the grand coalition in favor of smaller coalition formations as suggested by Heaney (1983a). Furthermore, engineering projects tend to have a large proportion of the costs common to all participants; consequently, one would expect these games to be very attractive and games with an empty core to be fairly rare. Nevertheless, the game theory approach does alert us that a problem exists in allocating costs for such cases. Cost Allocation in the Water Resources Field Straffin and Heaney (1981) showed that the criteria of fairness as expressed by equations (4-1) and (4-2) associated with cost allocation proposed by the TVA experts in the 1930's paralleled the development of the concepts of individual and subgroup rationality found in cooperative game theory. Given that full costs have to be recovered, the core conditions are equivalent to the fairness criteria for allocating cost originally proposed by the TVA experts. Therefore, current practice for allocating costs in the water resources field should require the solution be in the core of a game. 89 Table 4-13. Empty Core Solution Methods. Method Approach Source 1. Least Core or 8-Core Relax c(S) Shapley and Shubik, 1973; Young et al., 1982; Williams, 1982, 1983 2. Weak Least Core Relax c(S) or a-Core Shapley and Shubik, 1973; Young et al., 1982; Williams, 1982, 1983 3. Minimum Cost, Relax c(S) Remaining Savings Heaney and Dickinson, 1982 4 . Relax c(N) Chames and Golany, 1983 Homocore 90 Separable Costs, Remaining Benefits Method As discussed previously, ad hoc methods generally used in the water resources field allocate certain costs that are considered identifiable costs directly, and prorate the remaining costs by some criterion. The primary difference among these ad hoc methods is how identifiable costs are defined. We have already shown that defining identifiable costs as either zero or direct costs does not insure an equitable or core solution. We will now discuss several methods whereby identifiable costs are defined as separable costs. The recommended cost allocation method in the water resources field in the United States is the separable costs, remaining benefits (SCRB) method (Federal Inter-Agency River Basin Committee, 1950; Loughlin, 1977, 1978; Rossman, 1978; Heggen, 1980; Goodman, 1984). This method assigns each individual (or purpose) in a joint venture its separable costs and a share of the remaining costs in proportion to the remaining benefit, i.e., the minimum of the benefit or alternative cost less separable costs. Separable costs for individual i are defined as the difference between the cost of the joint venture with and without individual i. Separable costs include both the direct cost attributable to the entering individual and the incremental costs associated with a larger project because of the inclusion of another individual. Mathematically, separable costs are 91 sc (i) c(N) - c(N-i) Â¥ isN where sc (i) separable costs to individual i, C (N ) least cost system associated with group N, and c(N-i) = least cost system associated with subgroup N-i. After the separable costs for each individual have been allocated, the remaining costs to be assigned are called nonseparable costs (ncs), or nsc = c(n) - Â£ sc(i). (4-25) iÂ£N For the SCRB method the nonseparable costs are prorated on the basis of the remaining benefits; therefore, this prorated share is [min [b(i), c(i)] - sc(i)] ^iÂ£N ^min c(i)] - sc(i)}} where b(i) benefit of individual i c(i) = alternative cost of individual i if i acts independently, and 6 (i) prorating factor for the SCRB method. 92 The total charge to the individual is x (i ) sc(i) + 8 (i) â€¢ nsc. (4-27) The total charge, c(N), is c(N) lx (i) Isc(i) + Z 6 (i) â€¢ nsc (4-28) where IB(i) 1.0. The alternative justifiable expenditure method and the alternative cost avoided method are two variants of the SCRB method frequently mentioned in the water resources literaÂ¬ ture. The alternative justifiable expenditure method is recommended when data are not available for separable costs. Then, separable costs are defined as direct costs, and the nonseparable costs equal total project cost less the sum of all direct costs and are distributed in proportion to the remaining benefits. The alternative justifiable expenditure method is equivalent to the Louderback-Moriarity method recently proposed by Balachandran and Ramakrishnan (1981) in the accounting literature. The alternative cost avoided method is equivalent to Gately's propensity to disrupt with separable cost defined as before, but the nonseparable costs are distributed in proportion to the 93 alternative cost avoided, i.e., the difference between the alternatiave cost for the single purpose project and the separable costs. The SCRB method is now applied to our regional water network problem. The SCRB calculations are in Appendix A, and the results are summarized in Table 4-14. Like the nucleolus, the SCRB method gives equitable cost allocations for the first three options, and the three counties have an incentive to cooperate to implement the cheapest regional water network. Furthermore, Appendix A shows the SCRB method gives cost allocations for games without a core, i.e., solutions for network games for options 4 and 5 just like the Shapley value. Again, the cost allocations for the optimal network and the second best network are essentially the same. The additional $417 for the second best network follows: County 1 $211 County 2 89 County 3 117 Total $417 For our regional water network problem, each county's and each subgroup of counties' alternative cost of indeÂ¬ pendent action is less than or equal to each county's and each subgroup of counties' benefits; therefore, the SCRB method is identical to the alternative cost avoided method and Gately's propensity to disrupt method. For example, if 94 Table 4-14. Cost Allocation for Three-County Example Using the SCRB Method. Cost Allocation to ($) County i Â£ x ( i ) Is Cost Allocation In Core? Option (Rank) County 1 x(l) County 2 x (2 ) County 3 x (3 ) ($) (From ApÂ¬ pendix A) 1 604,369 2,165,958 1,786,082 4,556,409 Yes 2 604,580 2,166,047 1,786,199 4,556,826 Yes 3 4 646,000 2,178,677 1,805,500 4,630,177 Yes 5 6 646,000 2,420,095 1,990,992 5,057,087 â€” (3) (4) (6) 95 the SCRB charge vector for the optimal network is used: X = [604,369; 21,165,958; 1,786,082], the propensity to disrupt is d(i) = .77, i = 1,2,3. Since d(i) < 1, i = 1,2,3, each county is only a weak threat to the other two counties. Minimum Costs, Remaining Savings Method Although the SCRB method is recommended practice, it ignores subgroup rationality for projects with more than three individuals (Giglio and Wrightington, 1972; Young et al., 1982). The SCRB method only considers information on coalitions of size 1, (N-l), and N. As a result, the SCRB solution may be outside the core. Recently, Heaney and Dickinson (1982) showed the SCRB method may use infeasible upper bounds for apportioning the nonseparable costs in addition to ignoring information on subgroup rationality. As an improvement they proposed the minimum costs, remaining savings (MCRS) method as a generalized SCRB method. With the inclusion of all available information on subgroup rationality, the MCRS method uses linear programming to determine the minimum and maximum feasible costs for each individual. The feasible costs are then used as bounds to prorate the nonseparable costs just like the SCRB method; however, the feasible bounds now ensure the core conditions are met if the core is not empty. If the game is convex, the MCRS and the SCRB methods are identical. 96 Mathematically, the MCRS method can be stated as x (i) x(i) + 8 ( i ) â€¢ nsc min (4-29) where nsc = c (N ) - 1 x (i ) . . min i eN 6 ( i ) = [x(i) -x(i) . ] max min t x(i) -x(i) . ]} . â€ž max min J ieN and x(i) . and x(i) are found by solving the following min max 1 v 2n linear programs: max or min x(i) subject to x(i) _< c(i) Â¥ isN (4-30) 1 x(i) < c(S) Â¥ ScN ie S 1 x ( i ) = c (N) ieN x(i ) >0 Â¥ ieN In addition, the MCRS method can be used to determine if a game has a core by checking whether the linear programming problem (4-30) is feasible. If the linear programming problem (4-30) is infeasible, the game has an empty core. To find a unique charge vector to games with an empty core 97 using the MCRS method, the bounds or characteristic cost function values for the S-member coalitions are relaxed until a core appears. This procedure can be formulated by the following linear programming problem: minimize 0 subject to x(i) ieS Â£ x ( i ) = c (N ) ieN x ( i ) > 0 The MCRS method is now applied to our regional water network problem. The calculations are contained in Appendix A and the results are summarized in Table 4-15. Table 4-15 shows equitable cost allocations for the first three options. Again, the cost allocations for the optimal network and the second best network are essentially the same. The additional $417 for the second best network is allocated as follows: County 1 $217 County 2 1 County 3 199 Total $417 98 Table 4 -15. Cost MCRS Allocation for Three-County Example Using the Method. Cost Allocation to County i Is Cost ($) Allocation I x ( i ) In Core? Option (Rank) County 1 x(l) County 2 County 3 (?) (From Ap- x(2) x(3) pendix A) 1 606,861 2,151,206 1,798,342 4,556,409 Yes 2 607,078 2,151,207 1,798,541 4,556,826 Yes 3 4 646,000 2,163,025 1,821,152 4,630,177 Yes 5 6 646,000 2,420,095 1,990,992 5,057,087 â€” 99 Furthermore, the MCRS cost allocations in Table 4-15 encourage the counties to cooperate to construct the cheapÂ¬ est regional water network possible because the cost for each county progressively decreases with decreasing total project cost. Since none of the network games are convex, the MCRS solutions are different from the SCRB solutions because the MCRS method uses actual core bounds rather than nominal core bounds to apportion the nonseparable cost. For instance, the actual core bounds and the nominal core bounds for the optimal network game, shown in Table 4-16, illustrate the major difference between the MCRS method and the SCRB method. That is, the SCRB method distorts the allocation of the nonseparable cost by using infeasible bounds. Allocating Cost Using Game Theory Concepts The k Best System The cores for the optimal network game and the second best network game of our regional water network problem is shown in Figures 4-3 and 4-4 along with charge vectors for some of the cost allocation methods we discussed. By comÂ¬ paring Figures 4-3 and 4-4, we can see summarized in Table 4-17 that the SCRB and the game theory methods not only give equitable cost allocations for the two comparable cost networks, but each of these methods also gives almost 100 Table 4-16. Nominal Versus Actual Core Optimal Network Game. Bounds for Nominal Core Bounds Player i Lower Bound = c(N) - c(N-l) Upper Bound = c (i ) 1 572,232 646,000 2 1,969,771 2,420,095 3 1,627,898 1,990,992 Actual Core Bounds Player i Lower Bound From LP (4-30) with Objective Function Min x(i) Upper Bound From LP (4-30) with Objective Function Max x(i) 1 572,232 646,000 2 1,969,771 2,356,279 3 1,627,898 1,990,992 101 A Proportional to Population ^ Proportional to Demand â€¢ Use of Facilities o Direct Costing/Equal Apportionment of Remaining Costs x Shapley Value, Nucleolus, SCRB (Gately's Propensity to Disrupt), MCRS Figure 4-3. Core for the Optimal Network Game (C(N) = $4,556,409). 102 rA \/ v A ' " v A V w v -A v A \/ y Y A y A i 'X ( * i \ A ' \/ \ t \/ A /\ A A A \/A /\ A /\ /\ /? y W y Y ; X A \IK / ^ \/v \/ ^ a vA \ A S w/V v/\ \ / \ i X V A /V A /\ y y Y A /\ X | < i A \/ \t \/ \/ \/ \/ \/ A \/ V \z \/ \/ \/ \/ A/ \/ \/ ,/\ /N 1 A â€¢A 1 /\ \/ y v i y ^ /\ /I\ 7MA \ A / Y X ! X r: \/ / \V ^ \/ v / ,A J \,A \,A \ / v /'^ /1 \) / \ t\ <\ A A. A /V /\ ,A V I A | A i i a \( \/ \/ \/ \/ / \/ \/ \/ \/ \/ 1 \l/ i a A ,A A A /N \ /\ /\ /s ÃA 1 \/ \/ V V/ V y V Y Y Y \ V â€˜ f\, A , A, A f\, A, A, A, A., /\ Aw \ / \ w A \) z w W w w Y i X X i f V,/ yA A \,/x / \,/ X \ / / \,A Nl/N \/N \/ X rvÂ«. n/ v s . /r W \ A i'V ('V A a A, .Â¡u V V â€¢ / V Az v -Az A ÃÂ¿ C9 YA A fc: \j/ .L A , A /\ A , A A , A A A ' A\ A V w v V v/ v/ \/ w V V 1 \t A/ Ay A A/ A,/ A/ A., A, A,/ Aâ€ž Z\y Ay " \ A/' / / V / s V A A \/ J\ A / v V / V - V . . A / A, A / V, . \/ A y X ! :l tl \/ \f / \/ / \/ \ / \/ \/ ^ A \/ \/ \/ V Ay \ (\ A A, /\ A, \ Y. / Y A Y A . . A A , A V A ! A y V V 1 \ f 'X V A/ A,/ A,/ A, A , A A A A A / A A , Ay A zYr A > 1 r A A A /\ ' T~ A A w / V âœ“ A v A > . - J v ' r J\ y A, â€œA A A â– A â– ^ A T A A \ _ / \A \/ v A Y A A A A A, A A A A , /v A (1) V V V V V V \t V xU I A 1 A. | A Jk A 1 A o Use of Facilities o Direct Costing/Equal Apportionment of Remaining Costs x Shapley Value, Nucleolus, SCRB (Gately's Propensity to Disrupt), MCRS Figure 4-4. Core for the Second Best Network Game (C(N) = $4,556,826). Table 4-17. Cost Allocations for the Optimal Network and the Second Best Network ($). Method Network County 1 County 2 County 3 1. Optimal 590,087 2,175,905 1,790,417 Shapley 2. Second Best 590,226 2,176,044 1,790,556 Difference (2 - 1) 139 139 139 1. Optimal 609,116 2,144,583 1,802,710 Nucleolus 2. Second Best 609,325 2,144,687 1,802,814 Difference (2 - 1) 209 104 104 1. Optimal 604,369 2,165,958 1,786,082 SCRB (Gately's 2. Second Best 604,580 2,166,047 1,786,199 Propensity to Disrupt) Difference (2 - 1) 211 89 117 1. Optimal 606,861 2,151,206 1,798,342 MCRS 2. Second Best 607,078 2,151,207 1,798,541 Difference (2 - 1) 217 1 199 103 104 identical cost allocations for the two comparable cost networks. The game theory approach is able to give equitable solutions for the two comparable cost networks because the cost of independent action for each county and each subgroup of counties is recognized by the core conditions, i.e., the values of the characteristic cost function. Furthermore, the game theory approach is able to give nearly identical cost allocations for the two comparable cost networks because the core conditions for the two comparable cost networks are essentially the same except for the group rationality condition or the value of c(N), i.e., Core Conditions for the Optimal Network Game x(l) < 646,000 X (2 ) < 2,420,095 x (3 ) Â£ 1,990,992 x(l) + x (2 ) < 2,928,511 x(l) + x (3 ) < 2,586,638 X (2 ) + x (3 ) < 3,984,177 x (1) + x ( 2 ) + x (3 ) = 4,556,409 Core Conditions f or â€¢ the Second Best Network Game x(1) < 646,000 x(2 ) < 2,420,095 x ( 3 ) < 1,990,992 105 x (1) + x ( 2 ) x (1 ) + x ( 3 ) x ( 2 ) + x ( 3 ) x(l) + x(2) + x(3) < 2,928,511 < 2,586,638 < 3,984,177 = 4,556,826 The second best network naturally has a higher value for 1 2 c(N), i.e., c (N) < c (N); consequently, the second best network game is less attractive than the optimal network game in terms of cost. Therefore, the core of the second best network game is naturally smaller and more nonconvex than the core of the optimal network game, but this "reduced core" is a subset of the core for the optimal network game. This is shown by examining the following actual core bounds for the two comparable cost network games: Actual Core bounds for Optimal Network 572,232 < x(1) < 646,000 1,969,771 < x(2) < 2,356,279 1,627,898 < x(3) < 1,990,992 Actual Core Bounds for Second Best Network 572,649 < x(l) < 646,000 1,970,188 < x(2) < 2,355,862 1,628,315 < x(3) < 1,990,992 The core for the second best network has slightly higher lower bounds and slightly lower upper bounds compared with the core for the optimal network. 106 This natural reduction in the core as c(N) increases k-1 k from c (N) to c (N) is also shown in Figure 4-5 by examinÂ¬ ing nominal core bounds. Since the individual rationality t h core conditions are identical for the (k-1)1" best network t h game and the k best network game, the nominal upper core bounds (NUB) for both of these network games are identical, i.e., NUB^ ^ (i ) = NUB^ (i) = c(i), V ieN. Furthermore, since the subgroup rationality core conditions are identical t h t h for the (k-1) best network game and the kc best network game, the values of c(N-i), Â¥ iÂ£N, for both of these network games are identical. Consequently, as c(N) increases from k-1 k c (N) to c (N), the nominal lower core bounds (NLB), i.e., c(N) - c(N-i), are increasing by the same value for all individual i. Thus, Figure 4-5 shows that the increasing values of the nominal lower core bounds as c(N) increases k â€ 1 k from c (N) to C (N) is responsible for the reduction in the core. th The financial viability of the k best system can now be determined. As the value of c(N) increases progressively t h for the k best system, the core progressively reduces t h until the core possibly becomes empty. Therefore, all k best systems associated with games with a core can be conÂ¬ sidered financially viable since an equitable cost alloca- t h tion can be found. However, for all k best systems associated with games with an empty core, either an empty core cost allocation procedure is necessary, or these 107 Upper Bound NLB: Nominal Lower Bound Figure 4-5. Reduction in. Core as c(N) Increases from ck'x(N) to cK(N). 108 systems should not be considered because of the minimal economic gain or the loss of subadditivity. The Dummy Player In allocating the cost for option 3 (see Table 4-2) of our regional water network problem, there may be a temptaÂ¬ tion to simply treat this network as a two-person game involving counties 2 and 3 rather than a three-person game that also includes county 1. The cost allocation for two- person games is very simple. As shown by Heaney and Dickinson (1982), the core for a two-person game is a line and always exists. Furthermore, the two-person game is always convex, so the SCRB solution and the MCRS solution are identical and are located at the center of the core. If option 3 is treated as a two-person game, the cost allocation is trivial. County 1 simply pays its go-it-alone cost while counties 2 and 3 share the saving equally between themselves (see calculations in Table 4-18), i.e., County 1 $ 646,000 County 2 2,206,640 County 3 1,777,537 $4,630,177 Although this cost allocation satisfies core conditions regardless of whether option 3 is treated as a two-person game or a three-person game, there is a considerable differÂ¬ ence between this solution and the solutions by the SCRB and Table 4-18. Cost Allocation for Method. Option 3 as a Two-Person Game Using the SCRB County i SC(i)=C(N)-c(N-i) 6( i ) nsc =c(N) - Isc(i) Allocated Cost ($) x(i)=sc(i)+8(i) * nsc 2 1,993,185 1/2 426,910 2,206,640 3 1,564,082 1/2 426,910 1,777,537 c(2) = 2,420,095 c(3 ) = 1,990,992 c(23 ) = 3,984,177 109 110 the MCRS methods when option 3 is treated as a three-person game. This is shown in Table 4-19. Whether option 3 can be properly treated as a two-person game is important to know since county 2 and county 3 face substantially different costs. To determine whether county 1 can be excluded in the cost allocation for option 3, the core for option 3 as a three-person game is examined. The core conditions for option 3 as a three-person game are as follows: LI: x(l) < 646,000 L2 : x ( 2 ) < 2,420,095 L3 : x (3 ) < 1,990,992 L4 : x(l) + x (2 ) < 2,928,511 L5 : x (1) + x (3 ) < 2,586,638 L6 : x (2 ) + x (3 ) < 3,984,177 L7 : x (1) + x ( 2 ) + x (3 ) = 4,630,177 The core bounds for this three-person game are found by using linear programming problem (4-30). As expected, Table 4-20 shows county 1 simply pays its go-it-alone cost, i.e., $646,000. However, Table 4-20 indicates core constraint L4 is binding for x(2) and core constraint L5 is binding for 3 max x(3) . Therefore, constraints L4 and L5 involving max subcoalitions with county 1 are essential for determining the core for option 3 as a three-person game. ConseÂ¬ quently, county l's participation cannot be ignored when Table 4-19. Comparing Cost Allocations for Option 3 as Two-Person Game and Three-Person Game Using the SCRB and MCRS Methods. Approach (Method) Cost Allocation to County i ($) County 1 County 2 County 3 x(l) x(2) x(3) Is Cost Allocation in Core of Two- Person Game? Is Cost Allocation in Core of Three Person Game? Two-Person (SCRB or MCRS) 646,000 2,206,640 1,777,537 Yes Yes Three-Person (SCRB) 646,000 2,178,677 1,805,500 Yes Yes Three-Person (MCRS) 646,000 2,163,025 1,821,152 Yes Yes 112 Table 4-20. Core Bounds for Option 3 as a Three-Person Game. Shadow Price for Core Constraint from LP (4-30) Core Bound LI L2 L3 L4 L5 L6 L7 x (1) max 646,000 +1 x ( 2 ) = max 2,282,511 + 1 + 1 -1 x (3 ) max 1,940,638 + 1 +1 -1 x(l) . = min 646,000 -1 + 1 x (2 ) min 2,043,539 -1 + 1 x ( 3 ) . = mm 1,701,666 -1 + 1 113 determining the cost allocation for counties 2 and 3 for option 3. Suppose county 1 is a dummy player who contributes no savings to any coalition. This means subcoalitions (12) and (13) are inessential; i.e., c(12) = c(1) + c(2) = 3,066,095, and c(13) = c(l) + c(3) = 2,636,922. If county 1 is a dummy player, the core conditions for option 3 as a three-person game can be rewritten as follows: LI: x(l) < 646,000 L2 : x (2 ) < 2,420,095 L3 : x ( 3 ) < 1,990,992 L4 : x(l) + x (2 ) < 3,066,095 L5 : x(l) + x (3 ) < 2,636,992 L6 : x ( 2 ) + x (3 ) < 3,984,177 L7: x (1) + x ( 2 ) + x ( 3 ) - 4,630,177 Table 4-21 shows the core bounds for this game by solving linear programming problem (4-30). Again, county 1 simply pays its go-it-alone cost. However, Table 4-21 indicates core constraints L4 and L5 involving subcoalitions with county 1 are no longer binding for x(2) and x(3) , respectively. The binding core constraint for x(2)max is L2 and for x(3) is L3. Although Table 4-21 indicates core 114 Table 4-21. Core Bounds for Option 3 as a Three-Person Game with County 1 as a Dummy Player. Shadow Price for Core Constraint from LP (4-30) Core Bound LI L2 L3 L4 L5 L6 L7 x(l) max = 646,000 + 1 x (2 ) max = 2,420,095 + 1 x (3 ) max = 1,990,992 + 1 x(l) . mm = 646,000 -1 + 1 x ( 2 ) . mm = 1,993,185 -1 + 1 x ( 3 ) . mm = 1,564,082 -1 + 1 115 constraints L4 and L5 are binding for x(2) . and x(3) . , ^ min mm respectively, L4 and L5 are identical to L2 and L3, respecÂ¬ tively, because x(l) = 646,000. Consequently, when county 1 is a dummy player, the three-person game can be reduced to a two-person game with the following core conditions: L2 : x (2 ) _< 2,420,095 L3 : x (3 ) < 1,990,992 L6 : x ( 2 ) + x ( 3 ) = 3,984,177 Comparing Table 4-21 and Table 4-22 reinforces this result because the core bounds for the two-person game are identical to the core bounds for the three-person game with county 1 as a dummy player. In summary, an n-person game can be reduced to an (n-1)-person game only if the player removed from the game is a dummy player, i.e., a player who contributes no savings to any coalition. Otherwise, the cost allocation may be distorted even if it satisfies the core conditions for both the n-person game and the (n-1)-person game. Comparing Methods The methods we discussed are compared in Table 4-23. For any particular problem, any of these methods may sucÂ¬ cessfully find equitable cost allocation. However, the most suitable method for allocating cost in the water resources field appears to be the MCRS method. Although the game theory and the SCRB methods all give equitable cost 116 Table 4-22. Core Bounds for Option 3 as a Two-Person Game. Shadow Price for Core Constraint from LP (4-30) Core Bound L2 L3 L6 x (2 ) max 2,420,095 + 1 x (3 ) max 1,990,992 + 1 x ( 2 ) . = min 1,993,185 -1 + 1 x ( 3 ) . = mim 1,564,082 -1 + 1 Table 4-23. Comparison of Methods Discussed for Allocating Costs of Water Resources Projects. Method Feature ProporÂ¬ tionality Direct Costing Shapley Value Nucleolus SCRB MCRS 1. Will always get core solution for convex game No No Yes Yes Yes Yes 2. Will always get core soluÂ¬ tion for noncovex game No No No Yes No Yes 3. Applicable to empty core game No No No No No Yes 4. Can determine if core exists No No No Yes No Yes 5. Calculations easy for one or more systems Yes Yes No No Yes Yes 6. Independent of system configuration Yes No Yes Yes Yes Yes 7. Similar to methods used in water resources field Yes Yes No No Yes Yes 8. Easy to understand Yes Yes Yes No Yes Yes 9. Currently accepted accounting method Yes Yes No No No No Total Number of Yes 5 4 3 4 5 8 117 118 allocation if the game is convex, this convexity check may be burdensome for large games. In any case, both the Shapley value and the SCRB method may break down when the game is nonconvex. This is especially undesirable when suboptimal systems are evaluated since these games will naturally increase in nonconvexity with increasing value of c(N). In contrast, both the nucleolus and the MCRS methods give core solutions regardless of the degree of nonconvexity of a game. However, the MCRS method has an advantage over the nucleolus method in the water resources field because the MCRS method extends the presently recommended SCRB method. This avoids controversies over the acceptance of a different fairness criterion with using the nucleolus method. Moreover, while both the MCRS and nucleolus methods require solving multiple linear programming problems, the MCRS method is much easier to solve. The constraint sets for each of the 2n linear programming problems for the MCRS method are identical, whereas the nucleolus requires changing the constraint set for each of the possible n-1 linear programming problems. This makes the calculation of the nucleolus much more complex, especially when several systems are involved. However, even if several systems are involved, the sets of constraints are essentially unchanged except for the value of c(N) when using the MCRS method. Finally, the MCRS method is easier to understand and explain to the eventual decision makers which is also a criterion for selecting a cost allocation method. 119 Summary Several intuitively appealing ad hoc methods for allocating cost fail to give an equitable solution when an equitable solution exists. Moreover, in situations where several facility configurations are being considered, some ad hoc methods encourage noncooperation because these methods are not independent of the configuration of the facility. In order to overcome these shortcomings with ad hoc methods, concepts from cooperative game theory are t h necessary. The basis for determining whether the kc best system is financially viable is the existence of the core. This is because an equitable cost allocation exists to implement the system. However, a core solution may be "inequitable" if caution is not taken to include all nondummy players. Several game theoretic methods for allocating cost are examined, but the most suitable method for allocating cost in the water resource field appears to be the MCRS method. This conclusion is based on 1) reliability of finding an equitable cost allocation; 2) simplicity of computing the cost allocation for one or more systems; 3) adaptability to recommended methodology; and 4) ease of understanding. CHAPTER 5 EFFICIENCY/EQUITY ANALYSIS Introduction The cost allocation literature is replete with terminologies like opportunity cost, alternative cost, and marginal cost to represent the maximum or minimum amount that an individual should be charged. However, the precise meaning of these terms is obscured because no procedure for measurement is usually given. This chapter outlines a rigorous procedure to unambiguously quantify an individual's maximum cost and minimum cost for equity analysis. FurtherÂ¬ more, efficiency analysis and equity analysis are shown to be related by the costs of all opportunities available to all individuals in a project. Before the results in this chapter are discussed, recall that the computational effort used by the total enumeration procedure described in Chapter 3 to find c(N) is concurrrently used to find c(i) and c(S) with little addiÂ¬ tional effort. That is, the independent calculations are not only used to find c(N) but are also used to find c(i) and c(S). This important aspect of the procedure is illustrated in Table 5-1 using our three-county regional water network problem. 120 121 Table 5-1. Using Independent Calculations From the Total Enumeration Procedure to Find c(i),c(S), and C(N) for the Three-County Regional Water Network Problem. Independent Calculation: (SI); (S 2); (S3); (SI,12); (SI,13); (S2,23); (SI,12,23); (SI,12,13) Efficiency Analysis for c(N): c(123) = minimum [(SI,12,23); (SI,12,13); (SI,12) + (S3); (SI,13) + (S2); (SI) + (S 2,2 3); (SI) + (S2) + (S3)] Efficiency Analysis for c(i) and c(S): c(l) = (SI) c(2) = (S2) c(3) = (S3) c(12) = minimum [(SI,12); (SI) + (S2)] c(13) = minimum [(SI,13); (SI) + (S3)] c(23) = minimum [(S2,23); (S2) + (S3)] Note: (Si,ij) represents the cost of the water network consisting of pipelines from the well field to county i and from county i to county j. 122 Maximum Cost Opportunity cost (or alternative cost) is a concept used in economics to define the true cost of any action and is measured by the cost of the next best alternative that must be forgone when an action is taken (Nicholson, 1983). As a result, there is an opportunity cost for each individual associated with a regional water project because each individual must forego the opportunity to acquire the same level of service by either going-it-alone or joining a subcoalition. Although the costs for an individual to go-it-alone and for each subcoalition an individual can join to acquire the same level of service can be measured, an individual's opportunity cost cannot be determined just from these costs. This is because there is no way of specifying an individual's next best alternative without also knowing the individual's cost of joining each subcoalition. This means the cost of each subcoalition an individual can join must also be allocated. To further complicate matters, an individual's opportunity for joining a subcoalition depends on the opportunities available to the other individuals in a regional water project as well. For example, in a three- person game not every individual can join a two-person subcoalition. Kaplan (1982) correctly states that opporÂ¬ tunity costs are extremely important for decision making, yet difficult to measure because opportunity costs arise from transactions not executed. However, since opportunity 123 cost is the cost of an individual's next best alternative, the principle of individual rationality requires that the opportunity cost be the limit on how much an individual can be charged for joining the regional system. Otherwise, the individual can obviously do better by paying for his next best alternative rather than joining the regional system. Consequently, the maximum cost for individual i is equivalent to individual i's opportunity cost, and the maximum charge for individual i can be stated mathematically as follows: x (i ) < x ( i ) maximum Â¥ iÂ£N (5-1 where x(i) x (i ) maximum N = cost allocated to individual i, = maximum cost for individual i given individual i's opportunity to go-it- alone or join a subcoalition, and = set of all individuals; i.e., N = (l, 2,...,n ). This condition simply means individual i should be charged a cost less than or equal to individual i's cost of going-it- alone or joining a subcoalition. The maximum cost for individual i is now unambiguously quantifiable as the upper core bound for individual i, and is found by using the procedure outlined for the MCRS method to find the upper core bound for individual i, i.e., 124 x(i) maximum maximize x(i) (5-2) subject to x(i) ie S z x(i) c(N) ieN x ( i ) > 0 V ieN The interpretation of the upper core bound is now clear. The upper core bound for individual i is found by considering all the opportunities available to all individuals participating in a regional system. These opportunities, represented by the core conditions, include the possibility of all the individuals going-it-alone or some combination of subcoalition formation. Without knowing what will eventually take place, the core of a game accounts for all possibilities by representing all possible feasible solutions. These feasible solutions are found by taking the convex combination of the feasible core bounds. The feasible upper core bound determined from linear programming problem (5-2) represents the maximum cost for individual i. Any charges exceeding x(i) for individual i represent maximum solutions outside the core, and, consequently, individual i can do better by going-it-alone or joining a subcoalition. The upper core bounds for the optimal network game of our regional water network problem are now examined. This 125 is a nonconvex game whereby at least one subcoalition formaÂ¬ tion is relatively attractive in comparison to the grand coalition. From Table 5-2, the upper core bound for county 2 indicates county 2's maximum cost is $2,356,279 which is not simply the cost for county 2 of independent action, i.e., $2,420,095. The difference between the value of county 2's maximum cost, x(2) . , and go-it-alone cost, c(2), is due to the consideration by the core of not only county 2's opportunities for acquiring the same level of service by going-it-alone or joining a subcoalition, but also similar opportunities for county 1 and county 3 as well. To find which opportunities are relevant in determinÂ¬ ing x(2) . , linear programming problem (5-2) can be solved. Table 5-2 indicates core constraints L4 and L6 representing subcoalitions (12) and (23), respectively, are binding when solving for x(2) . This result means the opportunities of forming subcoalitions (12) and (23) limit how much county 2 can be charged. However, when linear programming problem (5-2) is solved for x(l) and x(3) . , Table 5-2 indicates that the only binding core maximum J 3 constraints are Ll and L3, respectively. Therefore, county l's and county 3's maximum costs are their respective go-it-alone costs. Linear programming problem (5-2) is an unambiguous and rigorous procedure to refute any claims by county 1 and/or county 3 that their maximum cost is less than c(l) and c(3), respectively, because of opportunities Table 5-2. Efficiency/Equity Analysis of the Optimal Network Efficiency Analysis Equity Analysis Binding Constraint Network Savings Cost ($) Core Condition m max min x(l) x (2 ) x ( 3 ) x(l ) x ( 2 ) x (3 ) c(123 ) 9.9 c(123 ) = 4,556,409 LI: x(l) < c(1) X c (1,23 ) 8.4 c(1)+c(23) = 4,630,177 L2: x(2) < c(2) c (12,3 ) 2.7 c(3)+c(12) = 4,919,503 L3: x(3) < c(3) X c (13,2 ) 1.0 c{2)+c(13) = 5,006,734 L4: x(1)+x(2) < c(12) X X c ( 1,2,3 ) 0 c(1)+c(2)+c(3) = 5,057,087 L5: x(1)+x(3) < c(13) X L6: x(2)+x(3 ) < c(23 ) X X L7 : x(1)+x(2)+x(3 ) = c(123) X X X Note: x(l) = 646,000; x(2) = 2,356,279 max max c(l) = 646,000- c(2) X(3)max= *'990,992; x(1) =572,232; x(2 ) = 1,969,771; x (3 ) . = 1,627,098 max min min min = 2,420,095; c(3) = 1,990,992; sc(l) =572,232; sc<2) = 1,969,771; sc(3) 1,627,890 127 to join a subcoalition. For example, even though subcoaliÂ¬ tion (13) is an essential coalition, Table 5-2 indicates that subcoalition (13) is never a factor in determining the maximum cost for any individuals in the game. Another aspect of properly quantifying maximum cost is in calculating the amount of savings from a regional water project. Normally, the amount of savings is based on each individual's go-it-alone cost; i.e., Savings (%) = 100 - [ C[Nl x 100] V ieN. (5-3) 2 c ( i ) ieN However, equation (5-3) assumes that either the regional water project involving the grand coalition is built or all the individuals will go-it-alone and, unfortunately, does not consider the possibility that a relatively attractive regional water project involving subcoalitions may be formed. To account for the possibility of relatively attractive subcoalition formations, the amount of savings from a regional water project should be defined in terms of maximum cost as defined by linear programming problem (5-2); i.e., Savings (%) 100 [- i C (N) 2 x ( i ) EN x 100] maximum V ieN. (5-4) 128 For our regional water network problem, the amount of savÂ¬ ings from the optimal network is 9.9% if equation (5-3) is used and is 8.7% if equation (5-4) is used. This result indicates failure to consider the possibility of attractive subcoalition formations can lead to an overestimation of maximum cost and, consequently, an overestimation of the amount of savings from a regional water project. For convex games, an individual's maximum cost is simply the individual's go-it-alone cost. A game is convex if none of the subcoalitions are attractive relative to the grand coalition. As discussed in Chapter 4, the nominal core bounds and the actual core bounds are identical if a game is convex. As a result, the maximum cost for each individual in a convex game is simply the individual's go-it-alone cost; i.e., x(i) . = c(i). This means in a ^ maximum convex game none of the subcoalition core constraints for linear programming problem (5-2) are binding; therefore, none of the individuals in a convex game can claim a maximum cost less than their go-it-alone cost because of opportuniÂ¬ ties to join a subcoalition. In summary, individual i's upper core bound represents individual i's maximum cost. If a game is convex, x(i) is simply equal to individual i's go-it-alone maximum ^ 1 ^ ^ cost, i.e., c(i); but, if a game is nonconvex, x(i) must be determined from linear programming problem (5-2). Whether a game is convex or nonconvex can be determined either by a convexity check using equation (4-12) or (4-13), or by comparing the nominal core bounds with the actual core bounds determined from linear programming problem (4-30). If a game is large, convexity check using equation (4-12) or (4-13) can be burdensome, and using linear programming problem (4-30) is more practical. In any event, determining whether a game is convex or nonconvex requires knowing the least cost system or characteristic cost function for each subcoalition, i.e., c(S). Furthermore, if a game is nonconvex, the importance of each subcoalition for equity analysis cannot be determined until linear programming problem (5-2) is solved. As we can see, efficiency analysis and equity analysis are related because the maximum costs which determine the maximum charges for the individuals in a regional project are found by considering the economics of all opportunities available to all individuals in the project. Minimum Cost A corresponding interpretation of an individual's lower core bound can be given. The lower core bound represents the minimum cost assignable to individual i for joining the grand coalition based on considering all the opportunities for all individuals in a project. Therefore, the minimum charge for individual i can be expressed mathematically as follows: 130 x (i) > X (i ) minimum V ieN (5-5) where x(i) = cost allocated to individual i, x (i ) minimum = minimum cost to individual i to join the grand coalition, and N = set of all individuals; i.e., N = (1,2,...,n }. This condition simply means individual i should be charged a cost greater or equal to individual i's cost of joining the grand coalition. The minimum cost for individual i is found by using the procedure outlined for the MCRS method to find the lower core bound for individual i, i.e., x(i) . . = minimize x(i) (5-6) minimum subject to x(i) _< c(i) V ieN Â£ x(i) < c(S) V ScN ie S Â£ x ( i ) = c (N) iÂ£N x ( i ) > 0 Â¥ ieN Any charges less than x(i) . . for individual i mean 3 minimum individual i is not paying for its minimum cost to join the 131 I grand coalition and represent solutions outside the core whereby individual i is being subsidized. The accepted practice of using separable cost or marginal cost as the minimum cost is based on the margin- ality principle that every individual should be charged at least the additional cost of being served (Young et al., 1982) and assumes that the minimum cost to join a coalition is last. The practicability of this assumption is troubling because not every individual can join a coalition last. In fact, a coalition may form without any clear understanding of the sequence of formation. Furthermore, Heaney (1979) has shown that the assumption is only true if a game is convex; i.e., c(N) - c(N - {i}) < c(S) - c[(S) - {i}] V SCN. (5-7) Fortunately, linear programming problem (5-6) eliminates any ambiguities in defining or quantifying the minimum cost for each individual in a project. Table 5-2 indicates that the minimum cost for each county of the optimal network for our regional water network problem is equal to each county's separable cost, i.e., the binding core constraints for x(i) . . is c(N) and c(N - {i}). minimum In summary, individual i's lower core bound represents individual i's minimum cost. If a game is convex, 132 x(i) . . is simply equal to individual i's separable minimum cost or marginal cost, i.e., x(i) . . = sc(i) = c(N) - c(N - {i}) V ieN. (5-8) minimum For nonconvex games, x(i) . . may or may not be identical to the separable cost or marginal cost for individual i; therefore, linear programming problem (5-6) must be solved. Fairness Criteria The fairness criteria for an equitable cost allocation expressed by equations (4-1) and (4-2) can now be simply and explicitly stated in terms of the core bounds, i.e., x(i) . < x(i) < x(i) min â€” â€” max V ieN. (5-9) This condition embodies the fairness criteria expressed by equations (4-1) and (4-2), yet clearly defines the range of costs that each individual i can be charged without violatÂ¬ ing individual rationality and/or subgroup rationality. Moreover, the lower and upper core bounds for each individual can be unambiguously quantified and interpreted as each individual's minimum cost and maximum cost, respectively. Finally, equation (5-9) can simplify the cost allocation procedure because the minimum cost for each individual is already determined, and only the remaining costs need to be allocated. Equation (4-4) can now be rewritten as follows: 133 x ( i ) = x ( i ) . min + ip ( i ) â€¢ rc V ieN (5-10) where x(i) = cost allocated to individual i = minimum x(i) from linear programming problem (5-6) ^(i) = prorating factor for individual i, and rc remaining costs i. e. , c (N) - Â£ i eN Therefore, any cost allocation procedure that is agreeable to the individuals participating in a regional project can be used to apportion the remaining costs as long as inequalities (5-9) are satisfied. For example, the remainÂ¬ ing costs might be prorated in proportion to a measure of use. Summary The interpretation of the core bounds is now clear. The lower core bound is a measure of minimum cost, while the upper core bound is a measure of maximum cost. The procedure for unambiguously measuring these costs is the same procedure used in the MCRS method for finding the core bounds. However, to determine the core bounds, an efficiency analysis is necessary to find the costs of all 134 opportunities available to all individuals in a project. Once the core bounds are found, they can be used as simple guidelines for allocating costs. CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS The motivation for this dissertation is based on the following three conclusions from reviewing the literature on efficiency analysis and equity analysis of regional water resources planning. First, no published work has incorporated efficiency analysis and equity analysis into a single regional water resources planning model using realistic cost functions. Secondly, the allocation of piping cost has not been examined separately from treatment cost. Thirdly, the cost allocation literature has not dealt with situations whereby good suboptimal systems are considered along with the optimal system. The first conclusion establishes the primary purpose of this dissertation. That is, to integrate efficiency analyÂ¬ sis and equity analysis into a single water resources planÂ¬ ning model using realistic cost functions. The selection of a regional water network problem is obviously based on the second conclusion. Finally, the third conclusion helped initiate a search for a reliable computational procedure to find good suboptimal systems that ultimately led to developÂ¬ ing a total enumeration procedure for efficiency/equity analysis of regional water network problems. 135 136 A major task with integrating efficiency analysis and equity analysis is finding a computational procedure. The principles of individual, subgroup, and group rationality from cooperative game theory give a theoretically sound basis for equity analysis. Consequently, successful efficiency/equity analysis depends on having a reliable method for finding not only the optimal regional system, but also the optimal system for each individual and each subÂ¬ group of individuals. Basically, either a partial enumeraÂ¬ tion or a total enumeration approach can be used to find these optimal solutions. Reliable partial enumeration techniques can be used for problems with well defined cost functions; but, for the types of cost function generally encountered in actual practice a total enumeration technique must be used. A reliable total enumeration procedure for finding the least cost water supply network for each individual, each subgroup of individuals, and the region is described. This procedure is easy to understand and use, and allows the engineer to use realistic cost functions or to perform detailed cost analysis. More importantly, the computational effort used by this procedure to find the optimal regional system can be concurrently used to find the optimal system for each individual and each subgroup of individuals with little additional effort. Furthermore, this procedure naturally gives all the suboptimal systems; therefore, good 137 suboptimal regional systems can be examined when factors other than cost are considered. Once a reliable method is available to find the optimal solutions, equity analysis can be accomplished using concepts from cooperative game theory. The financial viability of any system is based on the existence of a core because an equitable cost allocation can be found if a core exists. As the costs of suboptimal systems increase, the core naturally reduces in size until the core possibly becomes empty. Any system with an empty core is considered not financially viable because of the minimal economic gain or the loss of subadditivity. In comparing several ad hoc and game theory methods for allocating costs, the MCRS method appears to be the most suitable method for the water resources field. More importantly, the MCRS method gives a procedure for finding the core bounds by simply using the core conditions along with linear programming. The lower core bound and the upper core bound for an individual provide unambiguous measures of the individual's minimum cost and maximum cost, respectively. These costs are found by considering all the opportunities, represented by the core conditions, available to all individuals in a project. If the core conditions do not account for all the opportunities for each individual in a project to form an essential subcoalition, then the core bounds may be 138 distorted. In such cases, even a cost allocation in the core may be inequitable. Whether a game is convex or nonconvex is essential to how minimum cost, maximum cost, and savings are determined. The traditional approach is to assign minimum cost as the cost to join a coalition last, to assign maximum cost as the go-it-alone cost, and to calculate savings with respect to go-it-alone cost. These traditional approaches are acceptable only if a game is convex. For nonconvex games, these traditional approaches overlook opportunities to form good subcoalitions and, therefore, may distort the analysis by overestimating minimum cost, maximum cost, and savings. If a game is nonconvex, the minimum cost should be found by using linear programming problem (5-6), and the maximum cost should be found by using linear programming problem (5-2). Moreover, savings should be calculated with respect to the maximum cost as defined by linear programming problem (5-2). By knowing each individual's minimum cost and maximum cost, a basis for finding an equitable cost allocation is available. Since the minimum cost for each individual is already determined, decision makers simply need to agree on a method to apportion remaining costs without exceeding any individual's maximum cost. In summary, efficiency analysis and equity analysis in regional water resource planning are not separable problems. An efficiency analysis is necessary to find an 139 optimal or a good suboptimal system, but to implement this desirable system an equity analysis must be accomplished. Yet, accomplishing an equity analysis depends on an effiÂ¬ ciency analysis to find the optimal system for each individual and each subgroup of individuals to account for all opportunities available to all individuals in the project. Otherwise, each individual's minimum cost and maximum cost cannot be properly determined. Therefore, an efficiency analysis is incomplete unless it also provides the necessary information to accomplish an equity analysis. Finally, some thoughts for further research generated during the course of this dissertation are listed. 1. If different cost functions are used for different individuals in a regional project, a consistent set of accounting procedures is necessary to insure the cost functions are based on a comparable set of cost data. What is a consistent set of accounting procedures? 2. In any practical application, total project costs are not known precisely until after the project has been completed; therefore, a legitimate concern is how cost overruns are to be allocated. 3. Determine mathematical procedures to compute the total number of calculations to enumerate all 2n-l optimal solutions for any given digraph and to compute how many of these calculations are independent calculations. 140 4. Set up a computer model for detailed cost analysis, e.g., MAPS, and apply the procedures described in this dissertation to perform an efficiency/equity analysis of a real regional water network problem. 5. After a regional network is in place, how should the cost of expanding the network to serve new users be allocated. 6. A method to quantify transactions cost is necessary to better evaluate the financial gains of a regional system. APPENDIX EFFICIENCY/EQUITY ANALYSIS OF A THREE-COUNTY REGIONAL WATER NETWORK WITH NONLINEAR COST FUNCTION Appendix A contains the efficiency/equity calculations for tn three-county cost game example using Lotus 1-2-3. The results a presented as follows: Table 1 Efficiency Calculations With Total Enumeration Procedure Table 2 Cost Allocation for Option 1 Table 3 Cost Allocation for Option 2 Table 4 Cost Allocation for Option 3 Table 5 Cost Allocation for Option 4 Table 6 Cost Allocation for Option 5 Table 7 Template Used for Calculations Data Distance : L(i,j) is the distance in feet from i to j L (S,1)= 17000 L(S,3)= 30250 L(l,3)= 19670 L (S, 2) = 26000 L(1,2)= 13100 L(2,3) = 15500 Demand : Q(i) is the demand in mgd for player i Q(1)= 1 Q(2)= â€˜ 5 Q(3)= 3 Cost Function: a(Q~b)L a= 38 b= 3.5 Table 1 Calculations With Total Enumeration Procedure C(i.-j)[x]= Cost of network [x] for i..j ; C (i..j)= Least cost C(1)[S1]= 646000 C(2) [S2]=2420095. C(3) [S3] = =1990992. C(12)[SI,12]= 2923511. C(12)= 2928511. C(12)[S1;S2]= 3066095. C (13) [SI,13] = 2586638. C(13)= 2586638. C (13) [S1;S3] = 2636992. C(23) [S2,23] = 3984177. C(23)= 3934177. C (23)[S2;S3] = 4411088. C (123) [SI,12,23] = 4556409. 3 C(123)[S1,12;S3]= 4919503. 0 C (123)[SI,12,13] = 4556826. Of C(123)= 4556409. C(123) [S1,13;S2] = 5006734. 0 C (123)[S1;S2,23] = 4630177. 0 C (123)[S1;S2;S3] = 5057083. o Sort C(123) in ascending order (Yes= =l,Uo=0): Paths Cost Convex C (123)[SI,12,23] = 4556409. 0 C(123) [SI,12,13] = 4556826. 0 BEST C(123)[SI;S2,23]= 4630177. 0 C(123)= 4556409 C(123)[SI,12;S3]= 4919503. 0 C (123) [SI,13;S2] = 5006734. 0 C(123)[S1;S2;S3]= 5057083. 0 142 flj n 143 Table 2 Cost Allocation For C(123) = 4556409 Calculate the Shapley value (SV) for tine best C(123): SV(1)= 593037.3' SV(2)= 2175904. SV(3)= 1790416. Sura of SV(1)+SV(2)+SV(3)= 4556409 Check core conditions: (core test valid for subadditive gane only) IS core empty (yes=l,no=0)? 0 Does the Shapley value satisfy the core conditions? SV(1) Nominal Core Bounds: 572231.0 Sura of X(l)+X (2)+X(3)= 4556409 IS core empty (yes=l,no=0)? 0 Does the SCR3 value satisfy the core conditions? X(1) Actual Core Bounds Determined From LP: 572232 Sum of M(l)+M(2)+M(3)= 4556409 646000 2356279 1990992 M(3) = 1793342. IS core empty (yes=l,no=0)? 0 Does the MCRS value satisfy the core conditions? M(1) Table 3 Cost Allocation For C(123) = 4556323 Calculate the Shapley value (SV) for the best C(123): SV(1)= 590226.3 5V(2)= 2176043. SV(3)= 1790555. Sun of SV{1)+SV(2)+SV(3) = 4555826 Check core conditions: (core test valid for subadditive gane only) IS core empty (yes=l,no=0)? 0 Does the Shapley value satisfy the core conditions? SV(1) Nominal Core Bounds: 572648.0 Sun of X(1)+X(2)+X(3)= 4556826 IS core empty (yes=l,no=0)? 0 Does the SCRB value satisfy the core conditions? X(1) Actual Core Bounds Determined From LP: 572649 Sum of M(1)+M(2)+M(3)= 4556826 IS core empty (yes=l,no=0)? 0 Does the MCRS value satisfy the core conditions? M(1) 1 1 145 Table 4 Cost Allocation For C(123) = 4630177 Calculate the Shapley value (SV) for SV(1)= 614676.6â€™ SV(2)= 2200494. Sum of SV(1)+SV(2)+S V(3)= 4630177 the best C(123): SV(3)= 1315006. Check core conditions: (core test valid for subadditive game IS core empty (yes=l,no=3)? 3 Does the Shapley value satisfy tine core conditions? SV(1) SV(2) SV(3) SV(1)+SV(2) SV(1)+SV(3) SV(2) +SV (3) Calculate the SCRB value (X) for the Nominal Core Bounds: best C(123): 645999.0 2043538. 1731665. X(l)= 645999.4 X(2)= 2178677. Sum of X(1)+X(2)+X(3)= 4630177 X (3)= 1835499. IS core empty (yes=l,no=0)? 0 Does the SCRB value satisfy the core conditions? X(1) X (2) X (3) X (1) +X (2) X (1) +X (3) X (2) +X (3) Calculate the MCRS value (M) for the Actual Core Bounds Determined From LP 645000 Sum of M(1)+M(2)+M(3)= 4630177 best C(123): i â€¢ 646300 2282511 1940638 V(3)= 1821152 IS core empty (yes=l,no=0)? 0 Does the VCRS value satisfy the core conditions? M(1) M(2) M (3) M(l)+M(2) M(l) +M(3) M(2)+M(3) 146 Table 5 Cost Allocation For C(123) = 4919503 Calculate the Shapley value (SV) for the best C(123): SV(1)= 711118.6 SV(2)= 2296936. SV(3)= 1911448. Sum of SV(1)+SV(2)+SV(3)= 4919503 Check core conditions: (core test valid for subadditive game only) IS core empty (yes=l,no=0)? 1 Does the Shapley value satisfy the core conditions? SV(1) Nominal Core Bounds: 935325.0 Sum of X(1)+X(2)+X(3)= 4919503 IS core empty (yes=l,no=0)? 1 Does the SCRB value satisfy the core conditions? X(1) 0 0 1 1 Table 6 Cost Allocation For C(123) = 5006734 Calculate the Shapley value (SV) for the best C(123): SV(1)= 740195.6 SV(2)= 2326013. SV(3)= 1940525. Sum of SV(1)+SV(2)+SV(3)= 5006734 Check core conditions: (core test valid for subadditive game only) IS core empty (yes=l,no=0)? 1 Does the Shapley value satisfy the core conditions? SVdXC(l) 0 SV(2) 147 Calculate the SCRB value (X) for the best C(123): Nominal Core Bounds: 1022556. Sum of X(1)+X(2)+X(3)= 5006734 IS core empty (yes=l,no=0)? 1 Does the SCRB value satisfy the core conditions? X (1) X (2) X(3) X (1) +X (2) X (1) +X (3) X(2)+X(3) Table 7 Template Used for Calculations A 3 C D E F G H Data 203 204 Distance : L(i,j) is the distance in feet from i to j 205 L (S, 1) = 17000 L'S,3) = 30250 L(1,3)= 19670 205 L (S,2)= 26000 L(1,2)= 13100 L(2,3)= 15500 207 Demand : Q(i) is the demand in mgd for player i 2G3 Q(l) = 1 Q (2) = 6 Q (3) = 3 209 Cost Function: aCQ^bJL a= 33 b= 0.5 210 Calculations With Total Enumeration Procedure C(i..j)[x]= Cost of network [x] for i..j ; C(i..j)= Least cost for i..j C(1)[S1]= 646000 C(2)[S2]=2423095. C(3)[S3]=1990992. 215 C (12)[SI,12] = 2928511. C(12)= 2923511. 217 C (12)[S1;S2] = 3066095. 218 C (13)[SI,13] = 2586633. C(13)= 2585533. 220 C(13)[SI;S3]= 2636992. 221 C(23)[S2,23]= 3984177. C(23)= 3934177. 223 C(23)[S2;S3]= 4411038. 224 C(123)[SI,12,23] = 4556409. 0 226 C(123)[SI,12;S3]= 4919503. 0 227 C(123)[SI,12,13]= 4556326. 0 C(123)= 4556409. 228 C(123)[SI,13;S2]= 5006734. 0 229 C(123)[S1;S2,23]= 4630177. 0 230 C(123)[SI;S2;S2]= 5057088. 0 231 148 Sort C(123) in ascending order (Yes= 1,'-10=0) : 233 Paths Cost Convex 234 C(123)[SI,12,23]= 4556409. 0 235 C(123)[SI,12,13]= 4556826. 0 BEST 236 C(123)[SI;S2,23]= 4633177. 0 C(123)= 4556409 237 C(123)[SI,12;S3]= 4919503. 0 238 C(123)[SI,13;S2]= 5006734. 0 239 C(123)[S1;S2;S3]= 5057038. 0 240 241 Cost Allocation For C(123) = 4556439 243 Calculate the Shapley value (SV) for the best C(123): 245 SV(1)= 590087.3 SV(2)= 2175904. SV(3)= 1790416. 245 Son of SV(1)+SV(2)+SV(3)= 4556409 247 Check core conditions: (core test valid for subadditive game only) IS core empty (yes=l,no=3)? 0 250 Does the Shapley value satisfy the core conditions? 251 SV(1) 252 SV(2) 253 SV(3) 254 SV(1) +SV(2) 255 SV(1)+SV(3) 256 SV(2)+SV(3) 257 253 Calculate the SCRB value (X) for the best C(123): 260 Nominal Core Bounds: 261 572231.0 262 1969770. 263 1627897. 264 X (1)= 604369.fi X(2) = 2165957. X (3)= 1736032. 265 Sum of X(l)+X(2)+X(3)= 4556409 266 IS core empty (yes=l,no=0)7 â–º 0 268 Does the SCR3 value satisfy the core conditions? 269 X(1) 270 X (2) 271 X (3) 272 X(1)+X(2) 273 X (1) +X (3) 274 X(2)+X(3) 275 276 Calculate the MCRS value (N) for the Â¡ best C(123): 278 Actual Core Bounds Determined From LP: 279 572232 230 1969771 <.M (2 )< 2356279 231 1627893 232 M(l)= 606860.3 M(2)= 2151206. M(3)= 1793342. 233 Sum of M(1)+M(2)+M(3)= 4556409 234 149 IS core empty (yes=l,no=0)? 0 235 Does the MCRS value satisfy the core conditions? 287 M(1) 288 M(2) 239 M(3) 290 M (1) +M (2) 291 M(l) +M (3) 292 M(2)+M(3) 293 294 Partial Listing of Cell Formulas Total Enumeration Procedure: B215: +$E$210* ($C$209''$G$21C) *$B$206 D215: +$E$210*($E$209~$G$210)*$3$207 F215: +$E$210*($G$209~$G$210)*$D$206 C217: +$E$210*(($C$209+$E$209)~$G$210)*$S$206+$E$210*($E$209~$G$210)*$D $207 C217: @MIN ($C$217.. $C$218) C21S: +$B$215+$D$215 D226: +$E$210*(($C$209+$E$209+$G$209) ~$G$210)*$B$206+$E$210*(($E$209+$G $209) ~$G$210*$D$207+$E$210* ($G$209~$G$210) *$F$207 D227: +$C$217+$F$215 D223: +$E$210*(($C$209+$E$209+$G$209r$G$210)*$5$205+$E$210*($E$2S9~$G$ 210)*$D$207+$E$210*($G$209~$G$210)*$F$206 D229: +$C$220+$D$215 D230: +$B$215+$C$223 D231: +$3$215+$D$215+$F$215 E235: Â§IF(+$G$217+$G$220>=+$D$226+$S$215#ANDjf+$G$217+$G$223>=$D$226+$D$ 215#AND#+$G$220+$G$223>=$D$226+$F$215 Shapley value: B246: D247: E250: E252: E253: E254: E255: E256: E257: SCRB: C262: E262: B255: MCRS: B233: l/3*$3$215+l/6*($G$217-$D$215)+1/6*($G$220-$F$215)+1/3*($C$237-$G $223) +$B$246+$D$246+$F$246 9IF(+$G$217+$G$220+$G$223>=2*$G$237,0,1) 0IF(+$B$246<=+$B$215,1,0) @IF(+$D$246<=+$D$215,1,0) @IF(+$F$246<=+$F$215,1,0) 0IF(+$B$246+$D$246<=+$G$217,1,0) 0IF(+$B$246+$F$245<=+$G$220,1,0) 0IF(+$D$24o+$F$246<=+$G$223,1,0) +$G$237-$G$223 +$B$215 (($BS215-$C$262)/($B$215-SC$262+$D$215-$C$253+$F$215-$C$264))*($G $237-$C$262-$C$263-$C$264)+$C$262 (($E$280-$C$280)/($E$280-$C$280+$E$281-$C$2S1+$E$282-$C$282))*(SG $237-$C$280-$C$231-$C$232)+$C$280 REFERENCES Balachandran, B.V., and R.T.S. 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U.S. Environmental Protection Agency, Federal Guidelines, Industrial Cost Recovery Systems, MCD-45, General Service Administration, Denver, Colorado, February, 1976. Varian, H.R., Distributive justice, welfare economics, and the theory of justice, Journal of Philosophy and Public Affairs, 4, 223-247, 1975. Von Neumann, J., and 0. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, New Jersey, 1944. Waren, A.D., and L.S. Lasdon, The status of nonlinear programming software, Operations Research, 27(3), 431-456, 1979. Whitlatch, E.E., and C.S. ReVelle, Designing regionalized wastewater treatment systems, Water Resources Research, 12(4), 581-591, 1976. Williams, M.A., Empirical Applications of Cooperative Game Theory: The Distribution of Costs in Multiple Purpose River Developments, Ph.D. Dissertation, University of Chicago, Chicago, Illinois, 1982. Williams, M.A., Empirical Tests of Traditional and Game Theoretic Pricing Rules: The Distribution of Overhead Cost in Multiple Purpose River Developments, EPO #83-12, United States Dept, of Justice Antitrust Division, Washington, D.C., 1983. 158 Yao, K.M., Regionalization and water quality management, Journal of the Water Pollution Control Federation, 45(3), 407-411, 1973. Young, H.P., N. Okada, and T. Hashimoto, Cost allocation in water resources development, Water Resources Research, 18(3), 463-475, 1982. Zajac, E.E., Fairness or Efficiency: An Introduction to Public Utility Pricing, Ballinger Publishing Company, Cambridge, Massachusetts, 1978. BIOGRAPHICAL SKETCH Elliot Kin Ng was born August 9, 1950, in San Francisco, California, the son of Wah Hin Ng and Kit Har Yan. In 1968, he graduated from Lowell high school in San Francisco. He entered the University of California, Berkeley, in 1968 and received a Bachelor of Science in electrical engineering in 1972 and a Master of Science in electrical engineering in 1974. Subsequently, he spent two years working for Bechtel Incorporated, San Francisco, California, as an electrical engineer for the chemical and refinery division. He was commissioned in the U.S. Air Force as a captain in 1976. He held assignments at USAF School of Aerospace Medicine, Brooks AFB, Texas (EnvironÂ¬ mental, Safety and Facility Manager), USAF Occupational and Environmental Health Laboratory, Brooks AFB, Texas (ConÂ¬ sultant, Water Resources Engineer), and USAF Hospital, Wurtsmith AFB, Michigan (Bioenvironmental Engineer). During his assignments at Brooks AFB, Texas, he received a Master of Science degree in environmental management from the Univerity of Texas, San Antonio. He was selected by the Air Force Institute of Technology (AFIT) in 1981 for doctoral training in environmental engineering. In 1982, he entered the University of Florida to pursue the Doctor of Philosophy 159 160 degree. He was promoted to major in 1984. He is a registered professional engineer in the state of California. He and his wife, Eileen, have three children, Matthew Elliot (age 4), Michelle Eileen (age 2), and Michael Elliot (age 1). I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ames P. Heaney, Chairman Professor of Environmental Engineering Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Sanford V. Berg Associate Professor^of Economics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Donald J. Elzinga Professor of Industrial and Systems Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Wayne C. Huber Professor of Environmental Engineering Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Warren Viessman Professor of Environmental Engineering Sciences This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. 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