Water Resour Manage (2011) 25:2543–2573 DOI 10.1007/s11269-011-9826-x
Water Resources Allocation Using Solution Concepts of Fuzzy Cooperative Games: Fuzzy Least Core and Fuzzy Weak Least Core Mojtaba Sadegh · Reza Kerachian
Received: 1 May 2010 / Accepted: 8 April 2011 / Published online: 3 May 2011 © Springer Science+Business Media B.V. 2011
Abstract In this paper, two new solution concepts for fuzzy cooperative games, namely Fuzzy Least Core and Fuzzy Weak Least Core are developed. They aim for optimal allocation of available water resources and associated benefits to water users in a river basin. The results of these solution concepts are compared with the results of some traditional fuzzy and crisp games, namely Fuzzy Shapley Value, Crisp Shapley Value, Least Core, Weak Least Core and Normalized Nucleolus. It is shown that the proposed solution concepts are more efficient than the crisp games. Moreover, they do not have the limitation of Fuzzy Shapley Value in satisfying the group rationality criterion. This paper consists of two steps. In the first step, an optimization model is used for initial water allocation to stakeholders. In the second step, fuzzy coalitions are defined and participation rates of water users (players) in the fuzzy coalitions are optimized in order to reach a maximum net benefit. Then, the total net benefit is allocated to the players in a rational and equitable way using Fuzzy Least Core, Fuzzy Weak Least Core and some traditional fuzzy and crisp games. The effectiveness and applicability of the proposed methodology is examined using a numerical example and also applying it to the Karoon river basin in southern Iran. Keywords Water resources allocation · Fuzzy least core · Fuzzy weak least core · Fuzzy Shapley value · Cooperative game theory · Benefit allocation
M. Sadegh Department of Civil and Environmental Engineering, The Henry Samuely School of Engineering, University of California, Irvine, CA, USA e-mail:
[email protected] R. Kerachian (B) School of Civil Engineering and Center of Excellence for Engineering and Management of Civil Infrastructures, College of Engineering, University of Tehran, Tehran, Iran e-mail:
[email protected]
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1 Introduction According to UNESCAP (2000), a water allocation scheme should consider three principles of equity, efficiency and sustainability. The principle of equity means fair distribution of water resources among stakeholders in a river basin. Efficiency means that the best possible economic solution to the problem of water allocation should be found by minimizing the costs and maximizing the benefits. Lastly, the principle of sustainability means that water resources should be utilized economically and in an environmentally friendly manner both today and in the future. As discussed by Madani (2010), when analyzing, operating or designing a complex water project, a decision maker must ensure that the project is not only physically, environmentally, financially and economically feasible, but also socially and politically feasible. This is challenging for engineers who conventionally measure the performance of water projects in economic, financial, and physical terms. Optimization techniques can find the optimal values of the decision variables in such terms. However, they might not be able to provide insights into the strategic behaviors of stakeholders and policy decision makers to reach an optimal outcome and/or the attainability of such outcome from the status quo. Game theory can provide a framework for studying the strategic actions of individual decision makers to develop more broadly acceptable solutions (Madani 2010). There are different methods introduced in the literature for allocation of water resources and corresponding costs and benefits. Some methods (e.g. the methods proposed by Seyam et al. 2000; Van der Zaag et al. 2002) allocate water resources and associated costs or benefits based on a predefined criterion such as water demand or population of water users. In some other methods (e.g. McKinney et al. 1999; Fredericks et al. 1998; Wurbs 2001), optimization and simulation models have been used for optimal allocation of water resources in river basins considering some economic and environmental criteria. However, most of them fail to address all principles of equity, efficiency and sustainability at the same time (Wang et al. 2007). There are also several game theoretic approaches in the literature proposed for water and environmental resources management. Theses approaches allocate resources in an economically optimal way considering the physical constraints and environmental issues of the project. They also reallocate the total benefit to the stakeholders in a fair and equitable way. Therefore, as mentioned by Wang et al. (2003, 2007), a game theory-based water allocation approach can satisfy all principles of equity, efficiency and sustainability. In the remaining part of this section, the main applications of game theory in the field of water and environmental resources management are discussed. Young et al. (1982) compared different methods for apportioning costs of water supply development projects. They examined the advantages and disadvantages of some apportioning methods such as Separable Cost Remaining Benefits (SCRB), Shapley Value and some core-based games and applied these methods to a case study in Sweden. Tisdell and Harrison (1992) used different cooperative games to estimate the distribution of income after trading water rights between six representative farms and to allocate water to these farms in Queensland, Australia. They compared the results of some allocation methods, namely Two-Part Allocation, Two-Part Allocation with Consumptive Use, Volumetric Allocation and Volumetric Allocation with
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Consumptive Use, based on Rawlsian criterion of social justice. They discussed that their methodology can become an important policy evaluation tool. Lejano and Davos (1995) applied a solution concept called “Normalized Nucleolus” to a water reuse project in southern California. They compared the results of this method with two traditional solution concepts, namely Shapley Value and Nucleolus and showed that the Normalized Nucleolus offers more assurance for multi-agency coalitions to join the grand coalition. Luss (1999) considered a number of resource allocation problems between competing activities and defined a performance function for each of them. He used a Lexicographic Minimax approach to allocate resources to stakeholders optimally and equitably. A Lexicographic Minimax solution determines that performance function values cannot be improved without violating a constraint. This approach determines the lexicographical smallest vector whose elements, the performance function values, are sorted in a non-increasing order. He discussed that this approach can be useful to solve various resource allocation problems. Wang et al. (2003) proposed a model for equitable, efficient and sustainable water allocation among stakeholders in a river basin. Their model consisted of two steps. Firstly, water is initially allocated to stakeholders based on their legal rights or agreements, then water and net benefits are reallocated to promote equitable cooperation of stakeholders and to achieve an efficient water allocation. They used cooperative game theory with crisp coalitions to reallocate the net benefits. Fang et al. (2005) used a two-step model for water allocation in the Aral Sea Basin. In the first step, they used a priority-based maximal flow programming method for equitable allocation of initial water rights. In this step, it is believed that to ensure the reasonable water uses, minimum water demands should be met in the allocation process as far as possible. The priority-based maximal flow programming method assigns higher priorities to all minimum water demands and lower priorities to all maximum water demands. In the second step, they used a cooperative water allocation model to achieve equity, efficiency and sustainability in their allocations. Wang et al. (2008) proposed an algorithm for cooperative water allocation in river basins. In their work, water is initially allocated based on some legal and physical constraints. Then, to achieve an equitable condition, water is reallocated by using a game theoretic model. The algorithm was applied to a real-life case study in Canada. Yu and Zhang (2009) studied the fuzzy core for the games with fuzzy coalitions in which, the fuzzy core coincides with the fuzzy imputation for each of the fuzzy coalitions. Kucukmehmetogl (2009) studied the rational economic and political impacts of extensive reservoir projects. He used both linear programming and traditional game theory concepts of the Core and Shapley value to evaluate the impacts of reservoirs in the Euphrates and Tigris River Basin. He showed that as a result of application of the proposed model, coalitions may potentially eliminate the construction of new reservoirs and consequently can decrease the investment costs and evaporation losses. Mahjouri and Ardestani (2010) utilized some well-known crisp cooperative games namely, Shapely Value, Separable Costs Remaining Benefits (SCRB) and Maximum Costs Remaining Savings (MCRS) for inter-basin water allocation considering the water quantity and quality issues.
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Getirana and Malta (2010) defined six different scenarios to analyze the existing conflict among three groups of irrigators in Rio de Janeiro State, Southeastern Brazil. Irrigators use a canal to supply their water demands. The hydraulic constraints and limitations of the canal cause water unavailability and vulnerability of irrigating lands to floods. The Graph Model for Conflict Resolution, which solves non-cooperative games, was then applied to evaluate the scenarios. They showed that using this methodology and considering the current situations, the conflict can be resolved. Sensarma and Okada (2010) introduced a new perspective on conflict and cooperation analysis, where the game can change considering confronting players’ threats and promises. They showed how confrontation can effectively change to cooperation in a case study in Japan. Salazar et al. (2010) used a three person linear game to develop a water distribution model in Mexican Valley, Mexico. They used non-symmetric Nash bargaining method to evaluate the different water distribution scenarios. The results of their model showed that none of the scenarios would satisfy the domestic water demand so further investments along with higher water usage efficiency is needed to resolve the problem. Sadegh et al. (2010) proposed a game theory-based model for optimal operation of an inter-basin water transfer project in Iran. They used the traditional Fuzzy Shapley Value (FSV) for allocation of benefits to water users in donor and receiving basins. They showed that water users can gain much more benefits by participating in fuzzy coalitions rather than crisp ones. In this paper, two new solution concepts for fuzzy core-based games, namely Fuzzy Least Core (FLC) and Fuzzy Weak Least Core (FWLC) are developed and used for basin-wide water allocation. These two games aim at extending the core of fuzzy coalitions when the core does not exist. The proposed water allocation methodology consists of two steps: First, initial water allocation to players (water users) considering an equity criterion. Second, water and net benefit reallocation using the proposed solution concepts. The main objective of the methodology is to maximize the total net benefit of both the system and each player through forming fuzzy coalitions. To evaluate the efficiency of the methodology, it is applied to a large scale water allocation problem in southern Iran and the results of FLC and FWLC solution concepts are compared with the results of some traditional fuzzy and crisp games, namely FSV, Least Core, Weak Least Core, Shapley Value and Normalized Nucleolus. In the following section, an overview of the traditional fuzzy and crisp games and details of FLC and FWLC solution concepts will be provided.
2 Crisp and Fuzzy Cooperative Games Game theory is usually divided into two branches, namely cooperative and noncooperative games. These two branches differ in how they formalize interdependence among the players. In the non-cooperative game theory, a game is a detailed model of all the moves available to the players. In contrast, cooperative game theory does not specify a game by an exact description of the strategic environment, including the moves’ orders, the set of actions at each move and the payoff consequences
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relative to all possible plays; instead, it reduces the data to the coalitional form. Cooperative game theory studies the interaction among coalitions with the aim of allocating payoff to each player. In cooperative games, predictions are based on the payoff opportunities available to each coalition, which is conveyed by a real number. The main advantage of this approach is its convenience in practical use, because a real-life situation fits to a coalitional form more easily (Winter 2002). Coalitions are categorized into two divisions: crisp and so-called fuzzy coalitions. In crisp coalitions, players should bring their whole available resources to a coalition if they want to participate in it. In contrast, players can use only part of their resources to participate in a fuzzy coalition and the other portion to participate in other fuzzy coalitions. Crisp and fuzzy games can be used for allocating benefits, which are respectively produced in crisp and fuzzy coalitions. 2.1 Crisp Games Here, the assumption is that the sets of feasible payoffs for each coalition are given. We consider the set of all players of the cooperative game as N = {1, 2, . . ., n}, where n is the total number of players. A crisp coalition S is a subset of N, and the class of all coalitions of S is denoted by P(S). Then, a characteristic function ν is defined as: ν : P (N) → R+ satisfying ν () = 0,
R+ = {r ∈ R|r ≥ 0}
(1)
Equation 1 says that a characteristic function of a coalition is positive if the coalition is non-empty and it is zero if the coalition is empty. In this paper, rational and supperadditive cooperative games are discussed. In crisp cooperative games, for each two disjoint coalitions S and T, a game is supper-additive if: ν (S ∪ T) ≥ ν (S) + ν (T) , ∀ S , T ∈ P (N) , S ∩ T =
(2)
And a crisp coalition is said to be convex if: ν (S ∪ T) + ν (S ∩ T) ≥ ν (S) + ν (T) , ∀ S , T ∈ P (N)
(3)
A solution to a crisp cooperative game is a vector φ = (φ1 , φ2 , ..., φn ) satisfying: φi = ν (N) (4) i∈N
φi ≥ ν ({i}) , ∀i ∈ N
(5)
The vector φ = (φ1 , φ2 , ..., φn ) is called an imputation for the game v and φ i shows the payoff to the player i by the game v. There are several methods to obtain imputations for a game v, among them corebased games and Shapley Value are of more importance. The core concept implies that: 1. Payoff allocated to a player must not be less than what it can earn without participating in the coalition. This feature is called individual rationality. 2. Summation of payoffs allocated to each group of players as a result of participating in the grand coalition must not be less than what they can earn without participating in it. This feature is called group rationality. Group rationality includes individual rationality.
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3. Summation of payoffs allocated to all players must be equal to the amount of total profits produced in the game. The following set represents the core of a game v, which contains all non-dominated imputations for the game (Yu and Zhang 2009): C (N) = φi ∈ R+ | φi = ν (N) , φi ≥ ν (S) ∀S ∈ P (N) (6) i∈N
i∈S
Another crisp cooperative game is Shapley Value which is a way to assign a unique payoff to each player. The payoff allocated to a player by Shapley Value is proportional to its average marginal contribution to each coalition. It can be viewed as an index for measuring the power of each player in a game (Winter 2002). Shapley Value is represented as (Young et al. 1982): φi (ν) =
(|S| − 1)! (|N| − |S|)! ν (S) − ν (S\{i}) |N|!
(7)
i∈S⊆N
where: φi (ν) Payoff allocated to player i by Shapley Value game (Shapley Value of player i); |N| Total number of players participating in the game; |S| Cardinality of coalition S (number of members in coalition S); ν(S\{i}) Characteristic function (worth) of coalition S without player i. The Shapley Value will be in the center of the core if the game is convex, and it may fall out of the core if the game is non-convex (Shapley 1971). There are situations in which the core does not exist, here the way to obtain a solution for the game is to relax the inequalities defining the core, which is called extending the core. Two methods of extending the core are Least Core and Weak Least Core. Least Core is defined by imposing a uniform tax ε to all the coalitions other than the grand coalition. This tax encourages the whole group to stick together. The least ε for which an imputation φ exists and satisfies the following constraints, should be computed: φi ≥ ν (S) − ε ∀S ⊂ N i∈S
φi = ν (N)
(8)
i∈N
The Least Core is the set of all imputations φ satisfying Eq. 8 for the computed ε (Shapley and Shubik 1973). Suppose the core exists, but a unique answer is needed, one way of narrowing down the choices is imposing the uniform tax ε (Young et al. 1982). In this case, the tax ε will get a negative amount. A linear programming must be solved to compute the tax ε. The method of Weak Least Core is to some extent similar to the Least Core. Here, the uniform tax ε will get a coefficient proportionate to the cardinality of the
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associated coalition. So, the least tax ε that satisfies the following constraints must be computed: φi ≥ ν (S) − ε |S| ∀S ⊂ N i∈S
φi = ν (N)
(9)
i∈N
The Weak Least Core is the set of all imputations φ satisfying Eq. 9 for the computed tax ε. Solution concept of Least Core (Nucleolus) is based on absolute savings and the major concern about this concept is that most economic decisions are determined by rates of savings rather than absolute savings. So, solution concept of Normalized Nucleolus, which uses the normalized excess function, was suggested by Lejano and Davos (1995). Excess function for a coalition S is φi − ν (S), which should be more i∈S
than zero. Normalized Nucleolus is defined by imposing a tax ε to the normalized excess function. It is needed to compute the least ε for which, an imputation φ satisfying the following constraints, exists (Lejano and Davos 1995): φi i∈S ≥ 1 − ε ∀S ⊂ N ν (S) φi = ν (N) (10) i∈N
The Normalized Nucleolus is the set of all imputations φ satisfying Eq. 10 for the computed tax ε. 2.2 Fuzzy Games Since Zadeh (1965) introduced the concept of fuzzy sets, it has been employed in numerous fields of research including water resources management. A fuzzy set is defined by a membership function mapping the elements of a universe to the unit interval [0, 1]. Aubin (1974) introduced a new type coalition in which players partially take part. He illustrated this type of coalition with an n-dimensional vector in which the ith component indicates the participation degree of player i in the coalition. As the general form of this type of coalition is partially similar to a fuzzy set, he named it “fuzzy coalition”. This notion of so-called fuzziness has been widely used in the literature, for example Butnariu (1980), Tsurumi et al. (2001), Branzei et al. (2004), Li and Zhang (2009) and Yu and Zhang (2009). To develop a cooperative game with so-called fuzzy coalitions, let N = {1, 2, ..., n} be the set of all players. s = (s1 , s2 , ..., sn ) is called a fuzzy coalition, when si shows the participation rate of player i in the fuzzy coalition s. si is bounded in the interval [0, 1]. The set of fuzzy coalitions is denoted by L(N) and the empty coalition is denoted by e = (0, 0, ..., 0). The fuzzy coalition s is denoted by s = si .ei . ei is a vector in which i∈s
the ith component is equal to one and other components are zero. The characteristic function of this game is defined as: ν : L (N) → R+ , ν e = 0, R+ = {r ∈ R|r ≥ 0} . (11)
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For two fuzzy coalitions q and k, q ⊆ k means: qi ≤ ki
∀i ∈ N.
(12)
Support set of fuzzy coalition s is also denoted by: Supp (s) = { i ∈ N|si > 0} .
(13)
Supper-additive fuzzy games are defined as: ν (q ∪ k) ≥ ν (q) + ν (k)
∀q, k ∈ L (N) , q ∩ k = 0.
(14)
A fuzzy game ν is convex if: ν (q ∪ k) + ν (q ∩ k) ≥ ν (q) + ν (k)
∀q, k ∈ L (N) .
(15)
Solution concepts of fuzzy cooperative games have some characteristics. Firstly, the solution allocates all the benefits produced in the game to the players. Secondly, the results satisfy individual rationality. Thirdly, for any game the solution is unique. Fourthly, if two players have the same impact on a game, the payoffs to both of them are the same. Lastly, if a player does not participate in a game, its payoff through this game is zero (Winter 2002). A fuzzy imputation ϕ : L (N) → R+ for the fuzzy coalition s in the fuzzy game ν is defined as: 1. ϕi (s) = 0 ∀i ∈ / Supp (s) , 2.
ϕi (s) = ν (s) ,
(16)
i∈N
3. ϕi (s) ≥ ν (is )
∀i ∈ Supp (s) ,
where, ϕ (s) = (ϕ1 (s) , ϕ2 (s) , ..., ϕn (s)) and: ϕi (s) Allocated payoff to player i in fuzzy coalition s; ν(s) Characteristic function (worth) of fuzzy coalition s; ν(is ) Amount of benefits that is produced individually by player i, using its resources brought to fuzzy coalition s. The identification of characteristic functions for games with fuzzy coalitions is usually difficult, but they can be represented as fuzzy forms of the corresponding crisp characteristic functions. Yu and Zhang (2009) extended the core of crisp game as an imputation
for a game with fuzzy coalitions. The core of fuzzy coalition s in the fuzzy game ν C˜ (s) is defined as (Yu and Zhang 2009): ⎧ ⎫ ⎨ ⎬ C˜ (s) = ϕ ∈ R+ | ϕi (s) = ν (s) , ϕi (s) ≥ ν (q) ∀q ⊆ s ⎩ ⎭ i∈N
(17)
i∈q
Li and Zhang (2009) proposed a general Fuzzy Shapley function, which does not have the limitations of characteristic functions presented in previous works. If v is a
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fuzzy characteristic function and s is a fuzzy coalition, ν(s) is the worth created by the N members participating in the fuzzy coalition s. The Fuzzy Shapley Value of player i with participation rate si is presented as (Li and Zhang 2009): ⎞ ⎛ ⎞⎤ ⎡ ⎛ (|s| − 1)! (|N| − |s|)! ⎣ν ⎝ ϕi (ν) = s j.e j⎠ − ν ⎝ s j.e j⎠⎦ (18) |N|! j∈s
i∈s⊆L(N)
j∈s\i
where, |s| The number of players which participate in the fuzzy coalition s with a participation rate greater than zero;
s j.e j Characteristic function (worth) of fuzzy coalition s; ν j∈s s j.e j Characteristic function (worth) of fuzzy coalition s without player i. ν j∈s\i
Total fuzzy payoff of each player is equal to the summation of its fuzzy payoffs obtained from different fuzzy coalitions: ϕi = (19) ϕi (s) s∈L(N)
In games with fuzzy coalitions, as in crisp games, extension of the fuzzy core can be implemented. Here, we introduce two such methods. They are called Fuzzy Least Core (FLC) and Fuzzy Weak Least Core (FWLC) methods. Young et al. (1982) showed that the crisp versions of these two solution concepts are strong management tools in the context of water resources management. FLC is defined by imposing a uniform tax ε to all the fuzzy coalitions which are subsets of fuzzy coalition s. This tax motivates the players to work together in each fuzzy coalition s. To obtain the fuzzy imputation ϕ, the least tax ε which satisfies the following constraints, should be computed: ϕi (s) ≥ ν (q) − ε ∀q ⊆ s i∈s
ϕi (s) = ν (s) ∀s ∈ L (N)
(20)
i∈s
where, s and q are fuzzy coalitions. FLC is the set of all imputations ϕ satisfying Eq. 20. FWLC is quite similar to FLC. In FWLC solution concept, the uniform tax ε gets a coefficient proportionate to the summation of participation rates of players in the fuzzy coalition. So, the least ε that satisfies the following constraints should be computed: ϕi (s) ≥ ν (q) − ε × pri (q) ∀q ⊆ s i∈s
i∈q
ϕi (s) = ν (s) ∀s ∈ L (N)
i∈s
where, pri (q) is the participation rate of player i in fuzzy coalition q.
(21)
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In FWLC, as in Weak Least Core, the tax ε needs a coefficient related to the coalition’s cardinality. In fuzzy coalitions, players participate partially in the coalitions, so this coefficient can be the average participation rate of the players times the cardinality of the coalition. Average participation rate of fuzzy coalition pri (q)
q is
i∈t
|q|
, so:
ϕi (s) ≥ ν (q) − ε ×
i∈s
ϕi (s) ≥ ν (q) − ε ×
pri (q)
i∈t
× |q|
|q|
i∈s
∀q ⊆ s or
pri (q) ∀q ⊆ s
(22)
i∈q
The FWLC is the set of all imputations ϕ satisfying Eq. 21. In FWLC and FLC solution concepts, the total fuzzy payoff of each player is calculated using Eq. 19. In the following sections, the proposed fuzzy core-based solution concepts are applied to a numerical example and a real-world case study in Iran.
3 Numerical Example Assume there are three players (N = {1, 2, 3}) who have decided to participate in fuzzy coalitions in order to increase their benefits. Net benefits of fuzzy coalitions are allocated to players using FSV, FLC and FWLC. Player 1 is an agricultural district which mainly produces potato, player 2 is an agro-industrial sector which cultivates tomato and has some facilities for processing agricultural products (for example it can produce tomato paste and snacks), and player 3 is another agro-industrial sector which has farms and crop processing facilities. Available water resources and net benefit coefficients of players (Dollars per m3 of water) are presented in Table 1. If player 1 cooperates with player 2, it can use the facilities provided by player 2 to produce snacks from its agricultural product. These two players share their resources and facilities to increase their total benefit in fuzzy coalition 1. For all possible fuzzy coalitions, net benefit coefficients (Dollars per m3 of water) are shown in Table 2. Due to some physical constraints and the capacity of crop processing facilities, it is not possible that players bring all their available water resources to the fuzzy coalition with has the highest net benefit coefficient. Therefore, they have to use their resources to partially participate in different coalitions. The assumed participation
Table 1 Participation rates of players in fuzzy coalitions, available water resources for players and net benefit coefficients of players in the numerical example
Player 1 Player 2 Player 3 ∗ F.C.:
Participation rate in fuzzy coalition
Available water
Net benefit coefficient
F.C. 1∗
F.C. 2
F.C. 3
F.C. 4
resources (m3 )
(dollars/m3 )
2 7 2 7 1 4
2 7 3 7
3 7
0
7
2
0
14
1
0
1 4
2 7 1 2
10
2
Fuzzy coalition
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Table 2 Net benefit coefficients of fuzzy coalitions in the numerical example Fuzzy coalition
1
2
3
4
Net benefit coefficient (dollars/m3 )
2.118
2
2.22
2
rates of players in different fuzzy coalitions are presented in Table 1. For example, the second row shows that the first player uses 27 of its water resources to participate in fuzzy coalition 1. This player also uses 27 of its water resources to participate in fuzzy coalition 2 and the remaining portion is used for participation in fuzzy coalition 3. This player does not participate in fuzzy coalition 4. The worth (characteristic function) of each coalition is the product of its net benefit coefficient and the available amount of water in it. For example, players participate in fuzzy coalition 1 with the following characteristic functions: 2 2 × 7 = 4, ν (2, 1) = 1 × × 14 = 4, 7 7 1 ν (3, 1) = 2 × × 10 = 5, 4 2 2 ν ({1, 2} , 1) = 2 × × 7 + × 14 = 12, 7 7 1 2 × 7 + × 10 = 10, ν ({1, 3} , 1) = 2.22 × 7 4 1 2 × 14 + × 10 = 13, ν ({2, 3} , 1) = 2 × 7 4 2 1 2 × 7 + × 14 + × 10 = 18 ν ({1, 2, 3} , 1) = 2.118 × 7 7 4 ν(1, 1) = 2 ×
where: ν(i, 1) The amount of benefit (worth) which is individually produced by player i, with the amount of water it uses to participate in fuzzy coalition 1; ν({i, j}, 1) The amount of benefit (worth) that the coalition {i, j}, which is a subcoalition of fuzzy coalition 1, produces with the amount of water this sub-coalition uses to participate in fuzzy coalition 1; ν({1, 2, 3}, 1) Worth of fuzzy coalition 1. ν(1, 1) shows the amount of benefit that player 1 can produce, individually, with the amount of water it uses to participate in fuzzy coalition 1. The amount of water this player uses to participate in fuzzy coalition 1 is equal to 27 × 7 = 2. Considering the net benefit coefficient of this player, the worth of this portion of its water resources is equal to 2 × 2 = 4. In calculation of ν({1, 2}, 1), the net benefit coefficient of subcoalition {1, 2}, which is equal to the net benefit coefficient of fuzzy coalition 2, is multiplied by the amount of water these two players use to participate in fuzzy coalition 1. ν({1, 2, 3}, 1), which is the worth of fuzzy coalition 1, is calculated in the same way.
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It is noticeable that the coalition of players 1, 2 and 3 is not convex. According to the definition of convexity, coalition s is convex if each two sub-coalitions q and k satisfy the following property: ν (q ∪ k) + ν (q ∩ k) ≥ ν (q) + ν (k) . While in this fuzzy coalition, the sub-coalitions of {1, 2} and {2, 3} violate the convexity criteria: ν (({1, 2} , 1) ∪ ({2, 3} , 1)) = ν ({1, 2, 3} , 1) = 18, ν (({1, 2} , 1) ∩ ({2, 3} , 1)) = ν ({2} , 1) = 4, ν ({1, 2} , 1) = 12,
ν {{2, 3} , 1} = 13,
18 + 4 < 12 + 13 The solution concepts of FSV (Eq. 18), FLC (Eq. 20) and FWLC (Eq. 21) shall be used to allocate the total net benefit of fuzzy coalition 1 to the players participating in it. Table 3 represents the allocated benefits to the players. The results show that all solution concepts satisfy the individual rationality criteria but the group rationality is only satisfied by FLC and FWLC. The results of FSV violate the group rationality because: 5.17 + 6.33 = 11.5 ≤ 12 ⇒ ϕ (1, 1) + ϕ (2, 1) ≤ ν ({1, 2} , 1) where, ϕ(i, 1) is the payoff allocated to player i as a result of its participation in fuzzy coalition 1. As suggested by Young et al. (1982), the existence of the group rationality provides economic incentives for players to participate in a coalition. Therefore, the proposed FLC and FWLC solution concepts can provide more reliable results. Similarly, for the players participating in fuzzy coalitions 2, 3 and 4, the characteristic functions are as follows: ν (1, 2) = 4,
ν (2, 2) = 6,
ν ({1, 2} , 2) = 16
ν (1, 3) = 6,
ν (3, 3) = 5,
ν ({1, 3} , 3) = 12.21
ν (2, 4) = 4,
ν (3, 4) = 10,
ν ({2, 3} , 4) = 18
where: ν(i, j) The amount of benefit (worth) player i can individually produce with the amount of water resources it uses to participate in fuzzy coalition j, ν({1, 2}, 2) Worth of fuzzy coalition 2, ν({1, 3}, 3) Worth of fuzzy coalition 3, ν({2, 3}, 4) Worth of fuzzy coalition 4. The results of fuzzy solution concepts for fuzzy coalitions 2, 3 and 4 are also presented in Table 3. The total payoff allocated to each player is the summation of its payoffs from different fuzzy coalitions (Eq. 19). For example, the total payoff allocated to player 1 using different fuzzy solution concepts is calculated as follows: ϕ (1) = 5.17 + 7 + 6.61 + 0 = 18.78 by the FSV game ϕ (1) = 4.67 + 7 + 6.6 + 0 = 18.27 by the FLC game ϕ (1) = 4.67 + 6.4 + 6.76 + 0 = 17.84 by the FWLC game
Player 1 Player 2 Player 3 ε
5.17 6.33 6.5 –
4.67 7.67 5.66 −0.33
4.67 7.67 5.66 −0.61
7 9 0 –
7 9 0 −3
FLC
Fuzzy coalition 2
FSV
FWLC
FSV
FLC
Fuzzy coalition 1 6.4 9.6 0 −8.4
FWLC 6.61 0 5.6 –
FSV 6.6 0 5.61 −0.61
FLC
Fuzzy coalition 3 6.76 0 5.45 −1.78
FWLC 0 5.45 12.55 −5.09
FSV 0 6 12 −2
FLC
Fuzzy coalition 4 0 6 12 –
FWLC 17.83 22.72 23.66 –
FSV 18.27 22.67 23.27 –
FLC
Total payoff 18.78 21.33 24.1 –
FWLC
Table 3 Payoffs allocated to player i participating in fuzzy coalition j (ϕ(i, j)) by different fuzzy games and the optimal value of the tax (ε) in the numerical example
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M. Sadegh, R. Kerachian
where, ϕ(1) is the total payoff allocated to player 1 through its participation in different fuzzy coalitions.
4 Case Study In Iran, precipitation and available water resources are distributed unevenly. In central Iran, low rate of precipitation, increasing water demands and groundwater over drafting have brought severe stresses on the groundwater resources which are the only dependable water resources in this area. In recent years, the mentioned facts have raised undeniable need for transferring water from a nearby water basin. In this paper, the proposed methodology is applied to an inter-basin water transfer project from the Karoon river basin in south-western Iran to the Rafsanjan basin in central Iran. The main characteristics of the receiving and donor basins are explained in this section. The Karoon river basin includes two important rivers of Karoon and Dez. These two rivers join together and form the great Karoon river which ends in the Persian Gulf. Yearly average discharge of Dez river is about 8.5 billion cubic meters while yearly average discharge of Karoon river is 11.9 billion cubic meters. These rivers are the main source of water for supplying the demands of 1.2 million hectares of agricultural lands and several agro-industrial complexes in Khuzestan province in Iran, since the groundwater quality in this area is not suitable for agricultural purposes (Mahab-Ghods Consulting Engineers 2004). The Karoon river, which incorporates one fifth of Iran’s surface water resources, has a very large basin with an area of more than 100,000 km2 . The average annual precipitation in this basin varies from 150 mm in low level lands to 1,800 mm in the mountains. The length of the Karoon river is about 890 km and this river has four main tributaries which one of them is the Solegan river. The average crop pattern of the agricultural lands in the Karoon basin is illustrated in Table 4. Suger cane is the only crop produced by the agro-industrial units in the Karoon river basin (Mahab-Ghods Consulting Engineers 2004). Iran Water Resources Management Company (IWRMC) has a plan to build a dam on the Solegan river and transfer water to the Rafsanjan basin. According to Mahab-Ghods Consulting Engineers (2004), total water demand of the Karoon river basin is 24.9 billion cubic meters while the total discharge of the great Karoon river is 20.4 billion cubic meters. This fact shows that Karoon river basin has some problems supplying its own water demands in some months of the year especially in dry years. The Rafsanjan basin is located in central Iran. This basin with a catchment’s area of about 20,000 km2 has an annual average precipitation rate of 170 mm. Groundwater resources are the only dependable water resources in this basin, as rivers of this basin are seasonal and their annual discharge is insignificant (about one million cubic meters). So, water demands of about 110,000 ha of pistachio orchards in this basin are
Table 4 The crop pattern of the Khuzestan agricultural sector in the Karoon Basin (%) (Mahjouri and Ardestani 2010) Crop type
Wheat
Potato
Tomato
Sugar beet
Watermelon
Corn
Date
Other crops
Percentage
40
4
8
8
7
8
5
20
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2557
supplied from groundwater resources. This fact along with low rate of precipitation and high rate of evaporation resulted in groundwater overexploitation. Total water discharge from this aquifer is about 737 million cubic meters per year which is 240 million cubic meters more than the safe yield of it. Groundwater overexploitation resulted in major problems in this aquifer like groundwater table drawdown, land subsidence and rise in groundwater salinity (Mahab-Ghods Consulting Engineers 2004). According to Rahnama (2007), average land subsidence of 30 cm/year and groundwater table drawdown of 80 cm per year are the results of groundwater overexploitation in this basin. Therefore, IWRMC have decided to supply the water demands of this basin by transferring 250 million cubic meters of water from the Solegan river annually (Mahab-Gods Consulting Engineers 2004). Figure 1 depicts the locations of the basin of origin and the water receiving basin in the study area. Main characteristics of the water transfer system are presented in Table 5. There are environmental, municipal and agricultural demands, downstream of the Solegan reservoir, which are supposed to be supplied by this reservoir. These monthly water demands are presented in Table 6. The environmental water demand in the Karoon river basin has been estimated by Dezab Consulting Engineers (2001) using the Tennant method (Tennant 1976). In estimation of the in-stream flow in
Caspian Sea
IRAN Karoon Basin Rafsanjan Plain
Persian Gulf
Fig. 1 The locations of the basin of origin and the water receiving basin in the study area
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Table 5 The main characteristics of the inter-basin water transfer system (Mahjouri and Ardestani 2010) Solegan reservoir Minimum reservoir volume (m3 )
Water transfer system
Maximum Reservoir Spillway Tunnel Pipeline reservoir height capacity Length Design Inner Length Design Inner volume (m) (m3 /s) (km) discharge diameter (km) discharge diameter (m3 ) (m) (m3 /s) (m) (m3 /s)
86.9 × 106 645 × 106 106
1,112
53.8
13.5
3.7
384
9
2–2.6
downstream reaches of the Karoon river, controlling the backwater from the Persian Gulf has also been considered (Dezab Consulting Engineers 2001). The agricultural water demands are divided into four sectors. These four sectors (players) are: Khuzestan modern agro-industrial sector, Khuzestan old agro-industrial sector and Khuzestan local agricultural sector in the Karoon basin and Rafsanjan agricultural sector in the Rafsanjan basin. It is assumed that under any circumstances, the environmental and municipal sectors would receive their shares from the reservoir. Since the Solegan river flow is much less than the total water demand, a model based on game theory is developed to optimally allocate water resources to the stakeholders (players). Game theory has been widely used by researchers in the context of water resources management where there are different stakeholders with conflicting objectives and the decision of each stakeholder can affect other players. In this paper, the FSV, FLC and FWLC solution concepts for fuzzy games are utilized for optimal water allocation to the agricultural and agro-industrial sectors in the study area. In this case study, each coalition contains at least one agro-industry, which has some facilities for processing the products of the agricultural sectors. In other words, a local agricultural sector can cooperate with an agro-industrial sector and this cooperation can be beneficial for both of them because the infrastructure provided
Table 6 The demands of the main water users which should be supplied from the Solegan reservoir (million cubic meters) Month
Municipal sector
Environmental sector
Rafsanjan agricultural sector
Khuzestan local agricultural sector
Khuzestan old agroindustrial sector
Khuzestan modern agroindustrial sector
January February March April May June July August September October November December
0.2 0.4 1 0.8 1.2 1.2 1.6 1.4 1.2 0.8 0.4 0.2
1.6 2 4 5.6 7.2 8 10 9 7.4 4.6 2.6 1.8
0 0 1.2 17 39.8 49.6 47 45 38 28.2 8.8 0.2
5.6 6.6 13.8 18.8 24.8 27 34.2 30.4 25 15.4 9 6.2
0.8 0.9 1.9 2.5 3.3 3.6 4.6 4.1 3.3 2.1 1.2 0.9
0.6 0.7 1.5 2.1 2.7 3.0 3.8 3.3 2.7 1.7 1.0 0.7
The water demands have been adapted from Mahjouri and Ardestani (2010)
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by the agro-industrial sector can be used for processing the crops produced by local agricultural sector. As an example, tomato which is produced by the local agricultural sector in the Khuzestan basin can be stored and processed to produce tomato paste by an agro-industrial sector in a coalition, while an individual local agricultural sector is forced to sell its products at the time of harvesting, when the prices are low. Information about the net benefit coefficients of players and coalitions has been obtained from the annual statistical reports of the Iran’s Ministry of Agriculture.
5 Water Allocation Model Formulation The main objective of the system is to obtain the maximum total net benefit, considering different possible fuzzy coalitions among the players. In order to achieve this objective, an optimization model is used to determine the optimum participation rates of players in fuzzy coalitions. Different solution concepts for fuzzy games are also utilized to allocate the total net benefit to the players. As one of the players (Rafsanjan agricultural sector) has a considerable net benefit coefficient in comparison with other players, there is a tendency to fully supply the water demand of this player. This fact makes other players reluctant to partially participate in coalitions that do not include Rafsanjan agricultural sector. Therefore, in this paper, Rafsanjan agricultural sector is not included in fuzzy coalitions. The proposed methodology comprises two main steps. In the first step, initial water shares of players, which are proportional to their water demands, are determined. In the second step, different fuzzy coalitions are defined for players other than Rafsanjan agricultural sector and the participation rates of players in fuzzy coalition are determined so that their net benefits get maximized. The proposed solution concepts are also utilized to reallocate the net benefit. In the following sections, more details about the main steps of the methodology are presented. 5.1 Initial Water Allocation In the existing plan, the water in Solegan river (the donor river) is controlled by the Solegan reservoir. For initial water allocation, an equity criterion can be used for apportioning water resources among the competing users. Various methods have been proposed in the literature for initial water allocation based on different criteria such as users’ priorities, legal water rights or water management agreements. Initial water allocation should consider the water rights of users. An economic objective function can be used for initial water allocation while the water rights are considered as constraints. In this case study, the water rights data was not available; therefore, initial water allocation is done based on the water demands of users. In other words, the allocated water to each of the four competing users is assumed to be proportional to its demands without considering economic aspects. This assumption does not affect the value of the total benefit of the system and the final water allocation to the stakeholders, but it affects the final payoffs of the players. In the initial water allocation model, the constraints include the water continuity equations in Solegan reservoir, the physical constraints of the water supply system and the constraints related to the environmental flow (in-stream flow) downstream of this reservoir. The decision variables of the model are the monthly allocated water shares
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to the four water users. The planning horizon is considered to be 28 years (1978– 2005). As suggested by Mahjouri and Ardestani (2010, 2011), objective function and main constraints of the initial water allocation model are considered as follows: Minimize Z =
12 28
Rm,y − Dm,y
2
(23)
y=1 m=1
Subject to: X2,m,y X4,m,y X1,m,y = = ... = , d1,m,y d2,m,y d4,m,y 4
Xi,m,y + Insm,y ≤ Rm,y ,
∀m, y
∀m, y
(24)
(25)
i=1
Rm,y ≤ Rmax ,
∀m, y
Smin ≤ Sm,y ≤ Smax , ⎧ Sm+1,y = Sm,y + Im,y − Rm,y − Lm,y , ⎪ ⎪ ⎨ S1,y+1 = Sm,y + Im,y − Rm,y − Lm,y , ⎪ ⎪ ⎩ S1,1 = S12,Y
∀m, y
(26)
(27)
m = 1, 2, ..., 11, ∀y m = 12, ∀y
Dm,y = d1,m,y + d2,m,y + d3,m,y + d4,m,y + Insm,y
(28)
(29)
where, Z is the cumulative square deviation between allocated water to stakeholders and their water demands in the planning horizon, i is the index of the competing user (player), Xi,m,y is the allocated water to player i in month m of year y (million cubic meters), Dm,y is the total water demand in month m of year y (million cubic meters) and di,m,y is the water demand of player i in month m of year y (million cubic meters). Lm,y is the total water loss during month m in year y due to evaporation and infiltration (million cubic meters) and Im,y is the inflow during month m in year y (million cubic meters). Insm,y is the required environmental flow (in-stream flow) in the river downstream of the reservoir in month m of year y (million cubic meters), Rm,y is the outflow from reservoir during month m in year y (million cubic meters), Sm,y is the reservoir storage at the beginning of month m in year y (million cubic meters), Rmax is the maximum allowable monthly water release from reservoir, Smin is the minimum water storage of the reservoir, Smax is the maximum water storage of the reservoir and Y is the total number of years in the planning horizon. Power two in the objective function (Eq. 23) shows that total loss is a non-linear function of the deviation of monthly allocated water from the water demand. This penalizes an extreme water shortage or flood event. Constraint 24 assures that water is proportionally shared among the players considering their water demands. Based on Eq. 28, the final reservoir storage is set to be equal to initial storage. Initial water rights obtained in this section are used in the fuzzy game.
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5.2 Reallocation of Water and Net Benefits In this section, fuzzy cooperative games are used for reallocation of water and net benefits to coalitions and players in a fair and equitable manner. Cooperative game theory with rational coalitions can help players gain more profits. Firstly, water should be reallocated to fuzzy coalitions so that the total net benefit of the system gets maximized. This maximized net benefit will be shared among the players later. In fuzzy coalitions, players participate partially in the coalitions, so the amount of resources they use to participate in a fuzzy coalition s is determined as: x (s) =
3
pri (s) × A (i) ∀s
(30)
i=1
where: A(i) Amount of water resources allocated to player i in the first step; x(s) Total amount of water resources brought to fuzzy coalition s; pri (s) Participation rate of player i in fuzzy coalition s. The characteristic function of fuzzy coalition s is considered as follows: ν (s) =
B (s) × x (s) if x (s) ≤ C (s) B (s) × C (s) otherwise
(31)
where: C(s) Capacity (total demand) of fuzzy coalition s to consume water; B(s) Net benefit coefficient of fuzzy coalition s per unit of water; ν(s) Characteristic function (worth) of fuzzy coalition s. ⎧ 1: a coalition including Khuzestan modern and old ⎪ ⎪ ⎪ ⎪ agro-industrial sectors ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2: a coalition including Khuzestan modern agro-industrial ⎪ ⎪ ⎨ and local agricultural sectors s= ⎪ 3: a coalition including Khuzestan old agro-industrial and ⎪ ⎪ ⎪ ⎪ local agricultural sectors ⎪ ⎪ ⎪ ⎪ ⎪ 4: a coalition including Khuzestan modern and old ⎪ ⎪ ⎩ agro-industrial and local agricultural sectors Equation 30 describes that the amount of water brought to fuzzy coalition s equals the summation of the amount of water allocated to each player participating in this fuzzy coalition times its participation rate. Equation 31 shows that the worth of a coalition depends on the amount of water it receives as well as its capacity. A coalition’s capacity is the maximum amount of water it can consume with the highest efficiency. In other words, if an amount of water is allocated to a coalition more than its capacity, the net benefit coefficient of the coalition can no longer remain as presented in Table 8.
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Firstly, an optimization model is developed to maximize the total net benefit of the system: Maximize T =
4
ν (s)
(32)
s=1
Subject to: 0 ≤ pri (s) ≤ 1
pri (s) = 1,
(33)
∀i
(34)
s
x (s) =
3
pri (s) × A (i) ,
∀s
(35)
i=1
ν (s) =
B (s) × x (s) if x (s) ≤ C (s) B (s) × C (s) otherwise
(36)
ν (i, s) = b (i) × pri (s) × A (i)
ϕi (s) =
1 1 ν (i, s) − ν e + ν (s) − ν (s\i, s) , 2 2 ϕi (s) ≥ ν (i, s) ,
∀ s, i
(37)
∀ s, i
(38)
(39)
where: T b (i) ϕi (s) ν(i, s)
Total net benefits of the system; Net benefit coefficient of player i; Fuzzy Shapley Value of player i through participation in fuzzy coalition s; The amount of benefit player i produces individually using the resources it brings to coalition s; ν(e ) Worth of an empty coalition, which is equal to zero; ν(s\i, s) Net benefit that coalition {s\i} produces individually with the amount of water resources it uses to participate in coalition s. According to the first constraint, the participation rate of player i in fuzzy coalition s is between zero and one. The second constraint denotes that the summation of participation rates of each player to all coalitions should be equal to one. The third and fourth constraints present the characteristic functions of fuzzy coalitions. The fifth and sixth constraints denote the amount of benefits a player can gain through nonparticipation and participation in fuzzy coalitions. The seventh constraint assures that a player will gain more benefits through participation in fuzzy coalitions rather than non-participation. The Fuzzy Shapley Value formula is used in this model as a representative of cooperative games to specify the allocated benefits to players as a
Water Resources Allocation Using Solution Concepts...
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Table 7 Net benefit coefficient of player i (b (i))
b (i) (dollars/m3 )
i=1 (Khuzestan modern agro-industrial sector)
i=2 (Khuzestan old agro-industrial sector)
i=3 (Khuzestan local agricultural sector)
i=4 (Rafsanjan agricultural sector)
0.33
0.3
0.28
1.1277
result of participation in fuzzy coalitions. It is used to assure that players participate in rational coalitions. Other cooperative games can be used instead of the Fuzzy Shapely Value in this model. Then, FSV, FLC and FWLC solution concepts are used to reallocate the net benefits of fuzzy coalitions. Inputs of these solution concepts are characteristic functions of fuzzy coalitions and participation rates of players in the coalitions, which were obtained using the optimization model. Inputs also include the players’ net benefit coefficients, coalitions’ net benefit coefficients and coalitions’ capacities which are presented in Tables 7 and 8. In the following section, the results of applying the proposed methodology to the case study are presented.
6 Results and Discussion In this section, the results of the developed solution concepts of FLC and FWLC are presented and they are compared with some traditional fuzzy and crisp cooperative games, namely FSV, Least Core, Weak Least Core, Shapley Value and Normalized Nucleolus. The first step in the proposed methodology provides optimal monthly reservoir releases and distributes water resources among all the players with respect to their water demands. For example, the initial monthly allocated water to the Rafsanjan basin is presented in Fig. 2. The results of this step are used as inputs for the next step, which provides the fuzzy payoffs of the players. The benefits are calculated for each water year, so these monthly allocations should be converted to annual water allocations to be used in the characteristic functions. The annual time series of initial water allocations to the four major players are represented in Figs. 3 and 4. When the participation rates of players are limited to be zero or one, the fuzzy coalitions convert to crisp ones. In this case, players use all of their initial water rights to participate in the grand coalition (coalition 4 in our case study), and the total benefit of the grand coalition is allocated to the players using the equations of Shapley Value, Least Core, Weak Least Core and Normalized Nucleolus games (Eqs. 7, 8, 9 and 10). In these games, if a player earns more benefits by participating in the grand coalition, it will join the coalition, otherwise it works individually. The
Table 8 Net benefit coefficient of fuzzy coalition s (B(s)) and capacity of fuzzy coalition s for using water (C(s)) B(s) (dollars/m3 ) C(s) (m3 )
s=1
s=2
s=3
s=4
0.335 25 × 106
0.35 114.5 × 106
0.31 80.4 × 106
0.3 240 × 106
Allocated Water and Water Demand (MCM)
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M. Sadegh, R. Kerachian
60
Water Demand Allocated Water
50 40 30 20 10
0 Jun-1978
Jun-1983
Jun-1988
Jun-1993
Jun-1998
Jun-2003
Fig. 2 Monthly allocated water to the Rafsanjan basin based on the results of the initial water allocation model (million cubic meters)
time series of annual payoffs allocated to the players using different crisp games are presented in Figs. 5, 6 and 7. Then, fuzzy coalitions are formed and the solution concepts of FSV, FLC and FWLC are utilized to reallocate the net benefits of the fuzzy coalitions. In this case, there is no obligation for a player, who participates in a fuzzy coalition, to use its whole amount of available water resources for this participation. Firstly, the optimization model defined by Eqs. 32–39 is used to compute the optimal participation rates of players in different fuzzy coalitions as well as the worth of fuzzy coalitions. Then, Eqs. 18, 20 and 21 are used to calculate payoffs allocated to players using FSV, FLC and FWLC solution concepts, respectively. These solution
Allocated Water (million m3)
30 Player 1 25
Player 2
20 15 10 5 0 1978
1983
1988
1993
1998
2003
year
Fig. 3 Annual allocated water to the Khuzestan modern (player 1) and old (player 2) agro-industrial sectors, based on the results of the initial water allocation model
Water Resources Allocation Using Solution Concepts...
2565
Allocated Water (million m3)
250
Player 3 Player 4
200 150 100 50 0 1978
1983
1988
1993
1998
2003
year
Fig. 4 Annual allocated water to the Khuzestan local agricultural (player 3) and Rafsanjan agricultural (player 4) sectors, based on the results of the initial water allocation model
concepts provide the payoffs allocated to players due to participation in each fuzzy coalition. Then, Eq. 19 is used to obtain the final payoffs of the players. Figures 8, 9 and 10 show the participation rates of players in different fuzzy coalitions. These participation rates maximize the total net benefit of the system. As the net benefit coefficient of fuzzy coalition 4 is less than the other coalitions, players do not tend to participate in it, so their participation rates in fuzzy coalition 4 is zero. Participation rate of Khuzestan local agricultural sector in the first fuzzy coalition, participation rate of Khuzestan old agro-industrial sector in the second fuzzy coalition and participation rate of Khuzestan modern agro-industrial sector in the third fuzzy coalition are also zero, as they do not participate in these fuzzy coalitions. Figures 8, 9 and 10 show that players have a tendency to participate in a fuzzy coalition that has a higher net benefit coefficient. For example, as the second fuzzy
Allocated Payoff (Million Dollars)
14 12
Crisp Least Core
Crisp Weak Least Core
Crisp Shapley Value
Crisp Normalized Nucleolus
10 8 6 4 2 0 1978
1983
1988
1993
1998
2003
year
Fig. 5 The annual payoffs paid to the Khuzestan modern agro-industrial sector based on the results of different crisp games
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M. Sadegh, R. Kerachian
Allocated Payoff (Million Dollars)
12 10
Crisp Least Core
Crisp Weak Least Core
Crisp Shapley Value
Crisp Normalized Nucleolus
8 6 4 2 0 1978
1983
1988
1993
1998
2003
year
Fig. 6 The annual payoffs paid to the Khuzestan old agro-industrial sector based on the results of different crisp games
coalition has the highest net benefit coefficient, players are willing to participate in this coalition with their maximum participation rates. Figure 11 demonstrates the values of reallocated water to fuzzy coalitions. As shown in this figure, all amounts of water shares are reallocated to the second fuzzy coalition, which has the highest net benefit coefficient, until it reaches its capacity of consuming water. The water reallocation process is repeated for other coalitions in the same way, until depletion of all available water resources. Reallocated water to each fuzzy coalition should also be distributed among its players, considering the crop pattern of the coalition and the water demands of players. For example, in the second fuzzy coalition, which consists of a local agricultural sector and an agro-
Allocated Payoff (Million Dollars)
80 70
Crisp Least Core
Crisp Weak Least Core
Crisp Shapley Value
Crisp Normalized Nucleolus
60 50 40 30 20 10 0 1978
1983
1988
1993
1998
2003
year
Fig. 7 The annual payoffs paid to the Khuzestan local agricultural sector based on the results of different crisp games
Water Resources Allocation Using Solution Concepts...
2567
1 0.9
Participation Rate
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1978
1983
1988
1993
1998
2003
year pr (1,2)
pr (1,1)
Fig. 8 Participation rate of the Khuzestan modern agro-industrial sector in fuzzy coalition 1 (pr(1, 1)) and fuzzy coalition 2 (pr(1, 2))
industrial sector, 65% of reallocated water is given to the local agricultural sector to produce crops and the rest is allocated to the modern agro-industrial sector for both producing crops and processing the crops produced in the coalition. As an example,
1 0.9
Participation Rate
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1978
1983
1988
1993
1998
2003
year pr ( 2,1 )
pr ( 2,3 )
Fig. 9 Participation rate of the Khuzestan old agro-industrial sector in fuzzy coalition 1 (pr(2, 1)) and fuzzy coalition 3 (pr(2, 3))
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M. Sadegh, R. Kerachian 1
0.9
Participation Rate
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1978
1983
1988
1993
1998
2003
year pr ( 3,3)
pr ( 3,2)
Fig. 10 Participation rate of the Khuzestan local agricultural sector in fuzzy coalition 2 (pr(3, 2)) and fuzzy coalition 3 (pr(3, 3))
Fig. 12 represents the allocated water to players before and after participating in fuzzy coalition 2. Figures 13, 14 and 15 represent final payoffs to Khuzestan modern agro-industrial, Khuzestan old agro-industrial and Khuzestan local agricultural sectors in the Karoon river basin by using different solution concepts of the fuzzy games. Also, these figures show the players’ income without participating in any coalition. As it was expected, the results show that the final payoffs of the players are more than what they gained
Reallocated Water (million m3)
160 Fuzzy Coalition 1
140
Fuzzy Colition 2
120
Fuzzy Coalition 3
100 80 60 40 20 0 1978
1983
1988
1993 year
Fig. 11 The reallocated water shares to fuzzy coalitions 1, 2 and 3
1998
2003
Water Resources Allocation Using Solution Concepts...
share of player 3 before participating in fuzzy coalition 2 share of player 3 after participating in fuzzy coalition 2" share of player 1 before participating in fuzzy coalition 2" share of player 1 after participating in fuzzy coalition 2"
180 160 Amount of Water (MCM)
2569
140 120 100 80 60 40 20 0 1978
1983
1988
1993
1998
2003
year
Fig. 12 Water shares of players before and after participating in fuzzy coalition 2
alone without participating in any coalition. It shows that the proposed fuzzy games satisfy the individual rationality criterion. If the players participate in fuzzy coalitions, the total net benefit of the system during the planning horizon is 1,566.251 million dollars whereas this amount would decrease to 1,467.436 million dollars, if they participate in crisp coalitions. Table 9 presents the total net benefits allocated to each player during the planning horizon based on different fuzzy and crisp games. As shown in this table, each player will earn more benefits by participating in fuzzy games rather than crisp games. This fact shows that fuzzy games are more efficient than the crisp ones. As shown in the Figures 13, 14 and 15, the results of FSV, FLC and FWLC are almost the same. The maximum difference between payoffs allocated to players using FLC, FWLC and FSV is 12.3%. The FSV solution concept, which considers the marginal contributions of players to fuzzy coalitions, can provide more reliable results, if it does not violate the group rationality criterion. The FLC and FWLC do not have this limitation. The Fuzzy Shapley Value provides a solution (imputation) which is located in the center of the core when the core exists and fuzzy coalition is convex (Heaney and Dickinson 1982). When fuzzy coalition is not convex, the solution of the Fuzzy Shapley Value may fall outside of the core, which is a very undesirable situation.
Table 9 The summation of payoffs paid to the Khuzestan modern (player 1) and old (player 2) agro-industrial and local agricultural (player 3) sectors by crisp and fuzzy games during the planning horizon
Player 1 Player 2 Player 3
Shapley value
Least core
Fuzzy
Crisp
Fuzzy
259.4265 159.76 1,147.064
212.5732 152.4621 1,102.401
286.7089 159.9032 1,119.639
Weak least core
Normalized nuclelus
Crisp
Fuzzy
Crisp
Crisp
212.487 152.6345 1,102.314
266.0919 164.1282 1,136.031
212.5856 152.4621 1,102.388
151.5018 149.2143 1,166.72
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18
FSV game FLC game FWLC game without participating in any coalition
Final payoff (million Dollars)
16 14 12 10 8 6 4 2 0 1978
1983
1988
1993
1998
2003
year
Fig. 13 The annual payoffs paid to the Khuzestan modern agro-industrial sector based on the results of different solution concepts of the fuzzy games
As it was shown in the numerical example, Fuzzy Shapley Value does not satisfy the group rationality criterion in the first fuzzy coalition, which is not convex. This situation makes the non-convex coalitions unstable. However, the proposed fuzzy core-based solution concepts always satisfy the group rationality criterion, when the core exists, and they make more reliable coalitions when coalitions are not convex. In this case study, all fuzzy games satisfy the group rationality criterion, which means players that constitute a coalition cannot gain more profits by participating in a sub-coalition. Satisfaction of group rationality ensures the stability of our fuzzy coalitions.
Final payoff (million Dollars)
14 FSV game FLC game FWLC game without participating in any coalition
12 10 8 6 4 2 0 1978
1983
1988
1993
1998
2003
year
Fig. 14 The annual payoffs paid to the Khuzestan old agro-industrial sector based on the results of different solution concepts of the fuzzy games
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2571
90
FSV game FLC game FWLC game without participating in any coalition
Final payoff (million Dollars)
80 70 60 50 40 30 20 10 0 1978
1983
1988
1993
1998
2003
year
Fig. 15 The annual payoffs paid to the Khuzestan local agricultural sector based on the results of different solution concepts of the fuzzy games
Another requirement of these games is that if the total benefit of the system increases, each participant should receive more benefits. Conversely, if the total benefit decreases, no player should receive more benefits than what it received before. This property is called monotonicity (Megiddo 1974). Megiddo (1974) showed that the crisp Least Core game is not monotone. As shown in Figures 13, 14 and 15, when the total benefit of the system increases, based on the results of different solution concepts of fuzzy games, all the players receive more benefits and variations of annual allocated payoffs to different players are similar. It can prove that, in our case study, proposed solution concepts are monotone. The proposed methodology efficiently allocates water resources and maximizes the total net benefit of the system, while considering the environmental constraints. Moreover, this methodology reallocates the maximized net benefit to water users in a fair and equitable way. Therefore, it can be concluded that this methodology can easily be utilized for water allocation considering the efficiency, equity and sustainability criteria.
7 Summary and Conclusion One of the main issues in the field of water resources management is optimal allocation of shared water resources to competing water users. In this paper, two new solution concepts for fuzzy cooperative games, namely Fuzzy Least Core and Fuzzy Weak Least Core, were developed. The results of these solution concepts were compared with the results of some traditional fuzzy and crisp cooperative games through a numerical example and a real world case study. In the proposed water allocation methodology, an optimization model was used to obtain the initial allocated water shares to the players. Then, participation rates of players in different fuzzy coalitions were optimized in order to reach a maximum total net benefit. In the next step, the total net benefit was reallocated in a rational equitable way
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using Fuzzy Least Core, Fuzzy Weak Least Core and Fuzzy Shapley Value. The results showed that players will obtain more benefits if they participate in fuzzy coalitions rather than either participating in crisp coalitions or not participating in any coalition. The proposed methodology can be easily utilized for water allocation considering the efficiency, equity and sustainability criteria. It was also shown that the proposed solution concepts of FLC and FWLC do not have the limitation of FSV in satisfying group rationality. The main limitation of the proposed methodology is that it is deterministic. In future works, it can be extended to consider the existing uncertainties of the monthly reservoir inflows and water demands as well as the uncertainties of the net benefit coefficients of players and coalitions.
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