Water Resources Allocation Using Solution Concepts of Fuzzy Cooperative Games: Fuzzy Least Core and Fuzzy Weak Least Core

Water Resour Manage (2011) 25:2543–2573DOI 10.1007/s11269-011-9826-x

Water Resources Allocation Using Solution Conceptsof Fuzzy Cooperative Games: Fuzzy Least Coreand 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 aimfor optimal allocation of available water resources and associated benefits to waterusers in a river basin. The results of these solution concepts are compared withthe results of some traditional fuzzy and crisp games, namely Fuzzy Shapley Value,Crisp Shapley Value, Least Core, Weak Least Core and Normalized Nucleolus. It isshown 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 thegroup rationality criterion. This paper consists of two steps. In the first step, anoptimization model is used for initial water allocation to stakeholders. In the secondstep, fuzzy coalitions are defined and participation rates of water users (players) inthe fuzzy coalitions are optimized in order to reach a maximum net benefit. Then, thetotal net benefit is allocated to the players in a rational and equitable way using FuzzyLeast Core, Fuzzy Weak Least Core and some traditional fuzzy and crisp games.The effectiveness and applicability of the proposed methodology is examined usinga 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. SadeghDepartment of Civil and Environmental Engineering,The Henry Samuely School of Engineering,University of California, Irvine, CA, USAe-mail: [email protected]

R. Kerachian (B)School of Civil Engineering and Center of Excellence for Engineeringand Management of Civil Infrastructures, College of Engineering,University of Tehran, Tehran, Irane-mail: [email protected]

2544 M. Sadegh, R. Kerachian

1 Introduction

According to UNESCAP (2000), a water allocation scheme should consider threeprinciples of equity, efficiency and sustainability. The principle of equity means fairdistribution of water resources among stakeholders in a river basin. Efficiency meansthat the best possible economic solution to the problem of water allocation shouldbe found by minimizing the costs and maximizing the benefits. Lastly, the principleof sustainability means that water resources should be utilized economically and inan environmentally friendly manner both today and in the future.

As discussed by Madani (2010), when analyzing, operating or designing a complexwater project, a decision maker must ensure that the project is not only physically,environmentally, financially and economically feasible, but also socially and polit-ically feasible. This is challenging for engineers who conventionally measure theperformance of water projects in economic, financial, and physical terms. Optimiza-tion 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 behaviorsof stakeholders and policy decision makers to reach an optimal outcome and/orthe attainability of such outcome from the status quo. Game theory can provide aframework for studying the strategic actions of individual decision makers to developmore broadly acceptable solutions (Madani 2010).

There are different methods introduced in the literature for allocation of waterresources and corresponding costs and benefits. Some methods (e.g. the methodsproposed by Seyam et al. 2000; Van der Zaag et al. 2002) allocate water resources andassociated costs or benefits based on a predefined criterion such as water demandor 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 beenused for optimal allocation of water resources in river basins considering someeconomic and environmental criteria. However, most of them fail to address allprinciples 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 waterand environmental resources management. Theses approaches allocate resources inan economically optimal way considering the physical constraints and environmentalissues of the project. They also reallocate the total benefit to the stakeholders in afair and equitable way. Therefore, as mentioned by Wang et al. (2003, 2007), a gametheory-based water allocation approach can satisfy all principles of equity, efficiencyand sustainability. In the remaining part of this section, the main applications ofgame theory in the field of water and environmental resources management arediscussed.

Young et al. (1982) compared different methods for apportioning costs of watersupply development projects. They examined the advantages and disadvantages ofsome apportioning methods such as Separable Cost Remaining Benefits (SCRB),Shapley Value and some core-based games and applied these methods to a case studyin Sweden.

Tisdell and Harrison (1992) used different cooperative games to estimate thedistribution of income after trading water rights between six representative farmsand to allocate water to these farms in Queensland, Australia. They compared theresults of some allocation methods, namely Two-Part Allocation, Two-Part Alloca-tion with Consumptive Use, Volumetric Allocation and Volumetric Allocation with

Water Resources Allocation Using Solution Concepts... 2545

Consumptive Use, based on Rawlsian criterion of social justice. They discussed thattheir methodology can become an important policy evaluation tool.

Lejano and Davos (1995) applied a solution concept called “Normalized Nucleo-lus” to a water reuse project in southern California. They compared the results of thismethod with two traditional solution concepts, namely Shapley Value and Nucleolusand showed that the Normalized Nucleolus offers more assurance for multi-agencycoalitions to join the grand coalition.

Luss (1999) considered a number of resource allocation problems between com-peting activities and defined a performance function for each of them. He used aLexicographic Minimax approach to allocate resources to stakeholders optimally andequitably. A Lexicographic Minimax solution determines that performance functionvalues cannot be improved without violating a constraint. This approach determinesthe lexicographical smallest vector whose elements, the performance function values,are sorted in a non-increasing order. He discussed that this approach can be usefulto solve various resource allocation problems.

Wang et al. (2003) proposed a model for equitable, efficient and sustainablewater allocation among stakeholders in a river basin. Their model consisted of twosteps. Firstly, water is initially allocated to stakeholders based on their legal rightsor agreements, then water and net benefits are reallocated to promote equitablecooperation of stakeholders and to achieve an efficient water allocation. They usedcooperative 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 SeaBasin. In the first step, they used a priority-based maximal flow programming methodfor equitable allocation of initial water rights. In this step, it is believed that toensure the reasonable water uses, minimum water demands should be met in theallocation process as far as possible. The priority-based maximal flow programmingmethod assigns higher priorities to all minimum water demands and lower prioritiesto all maximum water demands. In the second step, they used a cooperative waterallocation model to achieve equity, efficiency and sustainability in their allocations.

Wang et al. (2008) proposed an algorithm for cooperative water allocation in riverbasins. In their work, water is initially allocated based on some legal and physicalconstraints. Then, to achieve an equitable condition, water is reallocated by usinga game theoretic model. The algorithm was applied to a real-life case study inCanada.

Yu and Zhang (2009) studied the fuzzy core for the games with fuzzy coalitionsin which, the fuzzy core coincides with the fuzzy imputation for each of the fuzzycoalitions.

Kucukmehmetogl (2009) studied the rational economic and political impacts ofextensive reservoir projects. He used both linear programming and traditional gametheory concepts of the Core and Shapley value to evaluate the impacts of reservoirsin the Euphrates and Tigris River Basin. He showed that as a result of applicationof the proposed model, coalitions may potentially eliminate the construction ofnew reservoirs and consequently can decrease the investment costs and evaporationlosses.

Mahjouri and Ardestani (2010) utilized some well-known crisp cooperative gamesnamely, Shapely Value, Separable Costs Remaining Benefits (SCRB) and MaximumCosts Remaining Savings (MCRS) for inter-basin water allocation considering thewater quantity and quality issues.


Getirana and Malta (2010) defined six different scenarios to analyze the existingconflict among three groups of irrigators in Rio de Janeiro State, Southeastern Brazil.Irrigators use a canal to supply their water demands. The hydraulic constraints andlimitations of the canal cause water unavailability and vulnerability of irrigating landsto floods. The Graph Model for Conflict Resolution, which solves non-cooperativegames, was then applied to evaluate the scenarios. They showed that using thismethodology and considering the current situations, the conflict can be resolved.

Sensarma and Okada (2010) introduced a new perspective on conflict and coop-eration analysis, where the game can change considering confronting players’ threatsand promises. They showed how confrontation can effectively change to cooperationin a case study in Japan.

Salazar et al. (2010) used a three person linear game to develop a water distrib-ution model in Mexican Valley, Mexico. They used non-symmetric Nash bargainingmethod to evaluate the different water distribution scenarios. The results of theirmodel showed that none of the scenarios would satisfy the domestic water demandso further investments along with higher water usage efficiency is needed to resolvethe problem.

Sadegh et al. (2010) proposed a game theory-based model for optimal operation ofan inter-basin water transfer project in Iran. They used the traditional Fuzzy ShapleyValue (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 fuzzycoalitions rather than crisp ones.

In this paper, two new solution concepts for fuzzy core-based games, namelyFuzzy Least Core (FLC) and Fuzzy Weak Least Core (FWLC) are developed andused for basin-wide water allocation. These two games aim at extending the coreof fuzzy coalitions when the core does not exist. The proposed water allocationmethodology consists of two steps: First, initial water allocation to players (waterusers) considering an equity criterion. Second, water and net benefit reallocationusing the proposed solution concepts. The main objective of the methodology is tomaximize the total net benefit of both the system and each player through formingfuzzy coalitions.

To evaluate the efficiency of the methodology, it is applied to a large scalewater allocation problem in southern Iran and the results of FLC and FWLCsolution concepts are compared with the results of some traditional fuzzy and crispgames, namely FSV, Least Core, Weak Least Core, Shapley Value and NormalizedNucleolus.

In the following section, an overview of the traditional fuzzy and crisp games anddetails 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 non-cooperative games. These two branches differ in how they formalize interdepen-dence among the players. In the non-cooperative game theory, a game is a detailedmodel of all the moves available to the players. In contrast, cooperative game theorydoes not specify a game by an exact description of the strategic environment, includ-ing the moves’ orders, the set of actions at each move and the payoff consequences


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 ofallocating payoff to each player. In cooperative games, predictions are based on thepayoff 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 areal-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 coalitionif they want to participate in it. In contrast, players can use only part of their resourcesto participate in a fuzzy coalition and the other portion to participate in other fuzzycoalitions. Crisp and fuzzy games can be used for allocating benefits, which arerespectively 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}, wheren is the total number of players. A crisp coalition S is a subset of N, and the class ofall 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 coalitionis non-empty and it is zero if the coalition is empty. In this paper, rational and supper-additive cooperative games are discussed. In crisp cooperative games, for each twodisjoint 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:∑

i∈N

φi = ν (N) (4)

φi ≥ ν ({i}) , ∀i ∈ N (5)

The vector φ = (φ1, φ2, ..., φn) is called an imputation for the game v and φi showsthe payoff to the player i by the game v.

There are several methods to obtain imputations for a game v, among them core-based games and Shapley Value are of more importance. The core concept impliesthat:

1. Payoff allocated to a player must not be less than what it can earn withoutparticipating in the coalition. This feature is called individual rationality.

2. Summation of payoffs allocated to each group of players as a result of partici-pating in the grand coalition must not be less than what they can earn withoutparticipating in it. This feature is called group rationality. Group rationalityincludes individual rationality.


3. Summation of payoffs allocated to all players must be equal to the amount oftotal profits produced in the game.

The following set represents the core of a game v, which contains all non-dominatedimputations for the game (Yu and Zhang 2009):

C (N) ={

φi ∈ R+|∑

i∈N

φi = ν (N) ,∑

i∈S

φi ≥ ν (S) ∀S ∈ P (N)

}(6)

Another crisp cooperative game is Shapley Value which is a way to assign aunique payoff to each player. The payoff allocated to a player by Shapley Value isproportional to its average marginal contribution to each coalition. It can be viewedas an index for measuring the power of each player in a game (Winter 2002). ShapleyValue is represented as (Young et al. 1982):

φi (ν) =∑

i∈S⊆N

(|S| − 1)! (|N| − |S|)!|N|!

[ν (S) − ν (S\{i})] (7)

where:

φi(ν) Payoff allocated to player i by Shapley Value game (Shapley Value ofplayer 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 itmay 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 asolution for the game is to relax the inequalities defining the core, which is calledextending the core. Two methods of extending the core are Least Core and WeakLeast Core.

Least Core is defined by imposing a uniform tax ε to all the coalitions other thanthe grand coalition. This tax encourages the whole group to stick together. The leastε for which an imputation φ exists and satisfies the following constraints, should becomputed:

∑

i∈S

φi ≥ ν (S) − ε ∀S ⊂ N

∑

i∈N

φi = ν (N) (8)

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, thetax ε will get a negative amount. A linear programming must be solved to computethe 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


associated coalition. So, the least tax ε that satisfies the following constraints must becomputed:

∑

i∈S

φi ≥ ν (S) − ε |S| ∀S ⊂ N

∑

i∈N

φi = ν (N) (9)

The Weak Least Core is the set of all imputations φ satisfying Eq. 9 for the computedtax ε.

Solution concept of Least Core (Nucleolus) is based on absolute savings and themajor concern about this concept is that most economic decisions are determinedby rates of savings rather than absolute savings. So, solution concept of NormalizedNucleolus, which uses the normalized excess function, was suggested by Lejano andDavos (1995). Excess function for a coalition S is

∑i∈S

φi − ν (S), which should be more

than zero. Normalized Nucleolus is defined by imposing a tax ε to the normalizedexcess function. It is needed to compute the least ε for which, an imputation φ

satisfying the following constraints, exists (Lejano and Davos 1995):∑i∈S

φi

ν (S)≥ 1 − ε ∀S ⊂ N

∑

i∈N

φi = ν (N) (10)

The Normalized Nucleolus is the set of all imputations φ satisfying Eq. 10 for thecomputed tax ε.

2.2 Fuzzy Games

Since Zadeh (1965) introduced the concept of fuzzy sets, it has been employed innumerous fields of research including water resources management. A fuzzy set isdefined by a membership function mapping the elements of a universe to the unitinterval [0, 1]. Aubin (1974) introduced a new type coalition in which players partiallytake part. He illustrated this type of coalition with an n-dimensional vector in whichthe ith component indicates the participation degree of player i in the coalition. Asthe general form of this type of coalition is partially similar to a fuzzy set, he namedit “fuzzy coalition”. This notion of so-called fuzziness has been widely used in theliterature, 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 theparticipation 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 bye� = (0, 0, ..., 0). The fuzzy coalition s is denoted by s = ∑

i∈ssi.ei. ei is a vector in which

the ith component is equal to one and other components are zero. The characteristicfunction of this game is defined as:

ν : L (N) → R+, ν(e�

) = 0, R+ = {r ∈ R|r ≥ 0} . (11)


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, thesolution allocates all the benefits produced in the game to the players. Secondly, theresults 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 themare the same. Lastly, if a player does not participate in a game, its payoff through thisgame 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.∑i∈N

ϕi (s) = ν (s) ,

3. ϕi (s) ≥ ν (is) ∀i ∈ Supp (s) ,

(16)

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 re-

sources brought to fuzzy coalition s.

The identification of characteristic functions for games with fuzzy coalitions isusually difficult, but they can be represented as fuzzy forms of the correspondingcrisp 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(

C̃ (s))

is defined as (Yu and Zhang 2009):

C̃ (s) =⎧⎨

⎩ϕ ∈ R+|∑

i∈N

ϕi (s) = ν (s) ,∑

i∈q

ϕi (s) ≥ ν (q) ∀q ⊆ s

⎫⎬

⎭ (17)

Li and Zhang (2009) proposed a general Fuzzy Shapley function, which does nothave the limitations of characteristic functions presented in previous works. If v is a


fuzzy characteristic function and s is a fuzzy coalition, ν(s) is the worth created by theN members participating in the fuzzy coalition s. The Fuzzy Shapley Value of playeri with participation rate si is presented as (Li and Zhang 2009):

ϕi (ν) =∑

i∈s⊆L(N)

(|s| − 1)! (|N| − |s|)!|N|!

⎡

⎣ν

⎛

⎝∑

j∈s

s j.e j

⎞

⎠ − ν

⎛

⎝∑

j∈s\i

s j.e j

⎞

⎠

⎤

⎦ (18)

where,

|s| The number of players which participate in the fuzzy coalition s witha participation rate greater than zero;

ν

(∑j∈s

s j.e j

)Characteristic function (worth) of fuzzy coalition s;

ν

( ∑j∈s\i

s j.e j

)Characteristic function (worth) of fuzzy coalition s without player i.

Total fuzzy payoff of each player is equal to the summation of its fuzzy payoffsobtained from different fuzzy coalitions:

ϕi =∑

s∈L(N)

ϕi (s) (19)

In games with fuzzy coalitions, as in crisp games, extension of the fuzzy core can beimplemented. Here, we introduce two such methods. They are called Fuzzy LeastCore (FLC) and Fuzzy Weak Least Core (FWLC) methods. Young et al. (1982)showed that the crisp versions of these two solution concepts are strong managementtools in the context of water resources management.

FLC is defined by imposing a uniform tax ε to all the fuzzy coalitions which aresubsets of fuzzy coalition s. This tax motivates the players to work together in eachfuzzy coalition s. To obtain the fuzzy imputation ϕ, the least tax ε which satisfies thefollowing constraints, should be computed:

∑

i∈s

ϕi (s) ≥ ν (q) − ε ∀q ⊆ s

∑

i∈s

ϕi (s) = ν (s) ∀s ∈ L (N) (20)

where, s and q are fuzzy coalitions. FLC is the set of all imputations ϕ satisfyingEq. 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 playersin the fuzzy coalition. So, the least ε that satisfies the following constraints should becomputed:

∑

i∈s

ϕi (s) ≥ ν (q) − ε ×∑

i∈q

pri (q) ∀q ⊆ s

∑

i∈s

ϕi (s) = ν (s) ∀s ∈ L (N) (21)

where, pri(q) is the participation rate of player i in fuzzy coalition q.


In FWLC, as in Weak Least Core, the tax ε needs a coefficient related tothe coalition’s cardinality. In fuzzy coalitions, players participate partially in thecoalitions, so this coefficient can be the average participation rate of the playerstimes the cardinality of the coalition. Average participation rate of fuzzy coalition

q is

∑i∈t

pri(q)

|q| , so:

∑

i∈s

ϕi (s) ≥ ν (q) − ε ×∑i∈t

pri (q)

|q| × |q| ∀q ⊆ s or

∑

i∈s

ϕi (s) ≥ ν (q) − ε ×∑

i∈q

pri (q) ∀q ⊆ s (22)

The FWLC is the set of all imputations ϕ satisfying Eq. 21. In FWLC and FLCsolution concepts, the total fuzzy payoff of each player is calculated using Eq. 19. Inthe following sections, the proposed fuzzy core-based solution concepts are appliedto 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 infuzzy coalitions in order to increase their benefits. Net benefits of fuzzy coalitions areallocated to players using FSV, FLC and FWLC. Player 1 is an agricultural districtwhich mainly produces potato, player 2 is an agro-industrial sector which cultivatestomato and has some facilities for processing agricultural products (for example itcan produce tomato paste and snacks), and player 3 is another agro-industrial sectorwhich has farms and crop processing facilities. Available water resources and netbenefit 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 toproduce snacks from its agricultural product. These two players share their resourcesand facilities to increase their total benefit in fuzzy coalition 1. For all possible fuzzycoalitions, 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 fuzzycoalition with has the highest net benefit coefficient. Therefore, they have to use theirresources to partially participate in different coalitions. The assumed participation

Table 1 Participation rates of players in fuzzy coalitions, available water resources for players andnet benefit coefficients of players in the numerical example

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)

Player 1 27

27

37 0 7 2

Player 2 27

37 0 2

7 14 1

Player 3 14 0 1

412 10 2

∗F.C.: Fuzzy coalition


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 2

7 of its water resources to participatein fuzzy coalition 1. This player also uses 2

7 of its water resources to participate infuzzy coalition 2 and the remaining portion is used for participation in fuzzy coalition3. This player does not participate in fuzzy coalition 4.

The worth (characteristic function) of each coalition is the product of its netbenefit coefficient and the available amount of water in it. For example, playersparticipate in fuzzy coalition 1 with the following characteristic functions:

ν(1, 1) = 2 × 27

× 7 = 4, ν (2, 1) = 1 × 27

× 14 = 4,

ν (3, 1) = 2 × 14

× 10 = 5,

ν ({1, 2} , 1) = 2 ×(

27

× 7 + 27

× 14)

= 12,

ν ({1, 3} , 1) = 2.22 ×(

27

× 7 + 14

× 10)

= 10,

ν ({2, 3} , 1) = 2 ×(

27

× 14 + 14

× 10)

= 13,

ν ({1, 2, 3} , 1) = 2.118 ×(

27

× 7 + 27

× 14 + 14

× 10)

= 18

where:

ν(i, 1) The amount of benefit (worth) which is individually produced byplayer i, with the amount of water it uses to participate in fuzzycoalition 1;

ν({i, j}, 1) The amount of benefit (worth) that the coalition {i, j}, which is a sub-coalition of fuzzy coalition 1, produces with the amount of water thissub-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 theamount of water it uses to participate in fuzzy coalition 1. The amount of water thisplayer uses to participate in fuzzy coalition 1 is equal to 2

7 × 7 = 2. Considering thenet benefit coefficient of this player, the worth of this portion of its water resourcesis equal to 2 × 2 = 4. In calculation of ν({1, 2}, 1), the net benefit coefficient of sub-coalition {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 fuzzycoalition 1. ν({1, 2, 3}, 1), which is the worth of fuzzy coalition 1, is calculated in thesame way.


It is noticeable that the coalition of players 1, 2 and 3 is not convex. According tothe definition of convexity, coalition s is convex if each two sub-coalitions q and ksatisfy 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 theconvexity 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 beused to allocate the total net benefit of fuzzy coalition 1 to the players participatingin it. Table 3 represents the allocated benefits to the players. The results show that allsolution concepts satisfy the individual rationality criteria but the group rationalityis only satisfied by FLC and FWLC. The results of FSV violate the group rationalitybecause:

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 fuzzycoalition 1. As suggested by Young et al. (1982), the existence of the group rationalityprovides economic incentives for players to participate in a coalition. Therefore, theproposed FLC and FWLC solution concepts can provide more reliable results.

Similarly, for the players participating in fuzzy coalitions 2, 3 and 4, the character-istic 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 withthe 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 alsopresented in Table 3. The total payoff allocated to each player is the summationof its payoffs from different fuzzy coalitions (Eq. 19). For example, the total payoffallocated 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

)


Tab

le3

Pay

offs

allo

cate

dto

play

erip

arti

cipa

ting

infu

zzy

coal

itio

nj(

ϕ(i,

j))

bydi

ffer

entf

uzzy

gam

esan

dth

eop

tim

alva

lue

ofth

eta

x(ε

)in

the

num

eric

alex

ampl

e

Fuz

zyco

alit

ion

1F

uzzy

coal

itio

n2

Fuz

zyco

alit

ion

3F

uzzy

coal

itio

n4

Tot

alpa

yoff

FSV

FL

CF

WL

CF

SVF

LC

FW

LC

FSV

FL

CF

WL

CF

SVF

LC

FW

LC

FSV

FL

CF

WL

C

Pla

yer

15.

174.

674.

677

76.

46.

616.

66.

760

00

17.8

318

.27

18.7

8P

laye

r2

6.33

7.67

7.67

99

9.6

00

05.

456

622

.72

22.6

721

.33

Pla

yer

36.

55.

665.

660

00

5.6

5.61

5.45

12.5

512

1223

.66

23.2

724

.1ε

–−0

.33

−0.6

1–

−3−8

.4–

−0.6

1−1

.78

−5.0

9−2

––

––


where, ϕ(1) is the total payoff allocated to player 1 through its participation indifferent fuzzy coalitions.

4 Case Study

In Iran, precipitation and available water resources are distributed unevenly. Incentral Iran, low rate of precipitation, increasing water demands and groundwaterover drafting have brought severe stresses on the groundwater resources which arethe only dependable water resources in this area. In recent years, the mentioned factshave raised undeniable need for transferring water from a nearby water basin. In thispaper, the proposed methodology is applied to an inter-basin water transfer projectfrom the Karoon river basin in south-western Iran to the Rafsanjan basin in centralIran. The main characteristics of the receiving and donor basins are explained in thissection.

The Karoon river basin includes two important rivers of Karoon and Dez. Thesetwo rivers join together and form the great Karoon river which ends in the PersianGulf. Yearly average discharge of Dez river is about 8.5 billion cubic meters whileyearly average discharge of Karoon river is 11.9 billion cubic meters. These riversare the main source of water for supplying the demands of 1.2 million hectaresof agricultural lands and several agro-industrial complexes in Khuzestan provincein Iran, since the groundwater quality in this area is not suitable for agriculturalpurposes (Mahab-Ghods Consulting Engineers 2004). The Karoon river, whichincorporates one fifth of Iran’s surface water resources, has a very large basin withan area of more than 100,000 km2. The average annual precipitation in this basinvaries from 150 mm in low level lands to 1,800 mm in the mountains. The length ofthe Karoon river is about 890 km and this river has four main tributaries which oneof them is the Solegan river. The average crop pattern of the agricultural lands in theKaroon basin is illustrated in Table 4. Suger cane is the only crop produced by theagro-industrial units in the Karoon river basin (Mahab-Ghods Consulting Engineers2004).

Iran Water Resources Management Company (IWRMC) has a plan to build adam on the Solegan river and transfer water to the Rafsanjan basin. According toMahab-Ghods Consulting Engineers (2004), total water demand of the Karoon riverbasin is 24.9 billion cubic meters while the total discharge of the great Karoon river is20.4 billion cubic meters. This fact shows that Karoon river basin has some problemssupplying 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 areaof about 20,000 km2 has an annual average precipitation rate of 170 mm. Groundwa-ter resources are the only dependable water resources in this basin, as rivers of thisbasin are seasonal and their annual discharge is insignificant (about one million cubicmeters). 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 (%) (Mahjouriand Ardestani 2010)

Crop type Wheat Potato Tomato Sugar beet Watermelon Corn Date Other crops

Percentage 40 4 8 8 7 8 5 20


supplied from groundwater resources. This fact along with low rate of precipitationand high rate of evaporation resulted in groundwater overexploitation. Total waterdischarge from this aquifer is about 737 million cubic meters per year which is 240million cubic meters more than the safe yield of it. Groundwater overexploitationresulted in major problems in this aquifer like groundwater table drawdown, landsubsidence and rise in groundwater salinity (Mahab-Ghods Consulting Engineers2004). According to Rahnama (2007), average land subsidence of 30 cm/year andgroundwater table drawdown of 80 cm per year are the results of groundwateroverexploitation in this basin. Therefore, IWRMC have decided to supply the waterdemands of this basin by transferring 250 million cubic meters of water from theSolegan river annually (Mahab-Gods Consulting Engineers 2004). Figure 1 depictsthe 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 ofthe Solegan reservoir, which are supposed to be supplied by this reservoir. Thesemonthly water demands are presented in Table 6. The environmental water demandin 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

KaroonBasin

Rafsanjan Plain

Persian Gulf

Fig. 1 The locations of the basin of origin and the water receiving basin in the study area


Table 5 The main characteristics of the inter-basin water transfer system (Mahjouri and Ardestani2010)

Solegan reservoir Water transfer system

Minimum Maximum Reservoir Spillway Tunnel Pipelinereservoir reservoir height capacity Length Design Inner Length Design Innervolume volume (m) (m3/s) (km) discharge diameter (km) discharge diameter(m3) (m3) (m3/s) (m) (m3/s) (m)

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 PersianGulf has also been considered (Dezab Consulting Engineers 2001). The agriculturalwater demands are divided into four sectors. These four sectors (players) are:Khuzestan modern agro-industrial sector, Khuzestan old agro-industrial sector andKhuzestan local agricultural sector in the Karoon basin and Rafsanjan agriculturalsector in the Rafsanjan basin. It is assumed that under any circumstances, theenvironmental and municipal sectors would receive their shares from the reservoir.

Since the Solegan river flow is much less than the total water demand, a modelbased on game theory is developed to optimally allocate water resources to thestakeholders (players). Game theory has been widely used by researchers in thecontext of water resources management where there are different stakeholders withconflicting 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 utilizedfor optimal water allocation to the agricultural and agro-industrial sectors in thestudy area.

In this case study, each coalition contains at least one agro-industry, which hassome facilities for processing the products of the agricultural sectors. In other words,a local agricultural sector can cooperate with an agro-industrial sector and thiscooperation 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 Environmental Rafsanjan Khuzestan Khuzestan Khuzestansector sector agricultural local old agro- modern agro-

sector agricultural industrial industrialsector sector sector

January 0.2 1.6 0 5.6 0.8 0.6February 0.4 2 0 6.6 0.9 0.7March 1 4 1.2 13.8 1.9 1.5April 0.8 5.6 17 18.8 2.5 2.1May 1.2 7.2 39.8 24.8 3.3 2.7June 1.2 8 49.6 27 3.6 3.0July 1.6 10 47 34.2 4.6 3.8August 1.4 9 45 30.4 4.1 3.3September 1.2 7.4 38 25 3.3 2.7October 0.8 4.6 28.2 15.4 2.1 1.7November 0.4 2.6 8.8 9 1.2 1.0December 0.2 1.8 0.2 6.2 0.9 0.7

The water demands have been adapted from Mahjouri and Ardestani (2010)


by the agro-industrial sector can be used for processing the crops produced by localagricultural sector. As an example, tomato which is produced by the local agriculturalsector in the Khuzestan basin can be stored and processed to produce tomato pasteby an agro-industrial sector in a coalition, while an individual local agricultural sectoris 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 beenobtained 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 achievethis objective, an optimization model is used to determine the optimum participationrates of players in fuzzy coalitions. Different solution concepts for fuzzy games arealso utilized to allocate the total net benefit to the players.

As one of the players (Rafsanjan agricultural sector) has a considerable net benefitcoefficient in comparison with other players, there is a tendency to fully supply thewater demand of this player. This fact makes other players reluctant to partiallyparticipate in coalitions that do not include Rafsanjan agricultural sector. Therefore,in this paper, Rafsanjan agricultural sector is not included in fuzzy coalitions. Theproposed methodology comprises two main steps. In the first step, initial watershares of players, which are proportional to their water demands, are determined.In the second step, different fuzzy coalitions are defined for players other thanRafsanjan agricultural sector and the participation rates of players in fuzzy coalitionare determined so that their net benefits get maximized. The proposed solutionconcepts are also utilized to reallocate the net benefit. In the following sections, moredetails 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 bythe Solegan reservoir. For initial water allocation, an equity criterion can be usedfor apportioning water resources among the competing users. Various methods havebeen proposed in the literature for initial water allocation based on different criteriasuch as users’ priorities, legal water rights or water management agreements.

Initial water allocation should consider the water rights of users. An economicobjective function can be used for initial water allocation while the water rights areconsidered 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. Inother words, the allocated water to each of the four competing users is assumed to beproportional to its demands without considering economic aspects. This assumptiondoes not affect the value of the total benefit of the system and the final waterallocation to the stakeholders, but it affects the final payoffs of the players. In theinitial water allocation model, the constraints include the water continuity equationsin Solegan reservoir, the physical constraints of the water supply system and theconstraints related to the environmental flow (in-stream flow) downstream of thisreservoir. The decision variables of the model are the monthly allocated water shares


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 andmain constraints of the initial water allocation model are considered as follows:

Minimize Z =28∑

y=1

12∑

m=1

(Rm,y − Dm,y

)2 (23)

Subject to:

X1,m,y

d1,m,y= X2,m,y

d2,m,y= ... = X4,m,y

d4,m,y, ∀m, y (24)

4∑

i=1

Xi,m,y + Insm,y ≤ Rm,y, ∀m, y (25)

Rm,y ≤ Rmax, ∀m, y (26)

Smin ≤ Sm,y ≤ Smax, ∀m, y (27)

⎧⎪⎪⎨

⎪⎪⎩

Sm+1,y = Sm,y + Im,y − Rm,y − Lm,y, m = 1, 2, ..., 11, ∀y

S1,y+1 = Sm,y + Im,y − Rm,y − Lm,y, m = 12, ∀y

S1,1 = S12,Y

(28)

Dm,y = d1,m,y + d2,m,y + d3,m,y + d4,m,y + Insm,y (29)

where, Z is the cumulative square deviation between allocated water to stakeholdersand their water demands in the planning horizon, i is the index of the competinguser (player), Xi,m,y is the allocated water to player i in month m of year y (millioncubic meters), Dm,y is the total water demand in month m of year y (million cubicmeters) and di,m,y is the water demand of player i in month m of year y (million cubicmeters). Lm,y is the total water loss during month m in year y due to evaporationand infiltration (million cubic meters) and Im,y is the inflow during month m in yeary (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 cubicmeters), 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 ofthe 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-linearfunction of the deviation of monthly allocated water from the water demand. Thispenalizes an extreme water shortage or flood event. Constraint 24 assures that wateris proportionally shared among the players considering their water demands. Basedon Eq. 28, the final reservoir storage is set to be equal to initial storage. Initial waterrights obtained in this section are used in the fuzzy game.


5.2 Reallocation of Water and Net Benefits

In this section, fuzzy cooperative games are used for reallocation of water and netbenefits to coalitions and players in a fair and equitable manner. Cooperative gametheory with rational coalitions can help players gain more profits. Firstly, watershould be reallocated to fuzzy coalitions so that the total net benefit of the systemgets maximized. This maximized net benefit will be shared among the players later.In fuzzy coalitions, players participate partially in the coalitions, so the amount ofresources they use to participate in a fuzzy coalition s is determined as:

x (s) =3∑

i=1

pri (s) × A (i) ∀s (30)

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.

s =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1: a coalition including Khuzestan modern and oldagro-industrial sectors

2: a coalition including Khuzestan modern agro-industrialand local agricultural sectors

3: a coalition including Khuzestan old agro-industrial andlocal agricultural sectors

4: a coalition including Khuzestan modern and oldagro-industrial and local agricultural sectors

Equation 30 describes that the amount of water brought to fuzzy coalition s equalsthe summation of the amount of water allocated to each player participating in thisfuzzy coalition times its participation rate. Equation 31 shows that the worth of acoalition 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 thehighest efficiency. In other words, if an amount of water is allocated to a coalitionmore than its capacity, the net benefit coefficient of the coalition can no longerremain as presented in Table 8.


Firstly, an optimization model is developed to maximize the total net benefit ofthe system:

Maximize T =4∑

s=1

ν (s) (32)

Subject to:

0 ≤ pri (s) ≤ 1 (33)

∑

s

pri (s) = 1, ∀i (34)

x (s) =3∑

i=1

pri (s) × A (i) , ∀s (35)

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) (37)

ϕi (s) = 12

[ν (i, s) − ν

(e�

)] + 12

[ν (s) − ν (s\i, s)

], ∀s, i (38)

ϕi (s) ≥ ν (i, s) , ∀s, i (39)

where:

T Total net benefits of the system;b(i) Net benefit coefficient of player i;ϕi(s) Fuzzy Shapley Value of player i through participation in fuzzy coalition s;

ν(i, s) The amount of benefit player i produces individually using the resources itbrings 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 coalitions is between zero and one. The second constraint denotes that the summation of par-ticipation rates of each player to all coalitions should be equal to one. The third andfourth constraints present the characteristic functions of fuzzy coalitions. The fifthand sixth constraints denote the amount of benefits a player can gain through non-participation and participation in fuzzy coalitions. The seventh constraint assuresthat a player will gain more benefits through participation in fuzzy coalitions ratherthan non-participation. The Fuzzy Shapley Value formula is used in this model as arepresentative of cooperative games to specify the allocated benefits to players as a


Table 7 Net benefit coefficient of player i (b(i))

i = 1 i = 2 i = 3 i = 4(Khuzestan modern (Khuzestan old (Khuzestan local (Rafsanjanagro-industrial agro-industrial agricultural agriculturalsector) sector) sector) sector)

b(i) (dollars/m3) 0.33 0.3 0.28 1.1277

result of participation in fuzzy coalitions. It is used to assure that players participatein rational coalitions. Other cooperative games can be used instead of the FuzzyShapely Value in this model.

Then, FSV, FLC and FWLC solution concepts are used to reallocate the netbenefits of fuzzy coalitions. Inputs of these solution concepts are characteristicfunctions 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’ capacitieswhich are presented in Tables 7 and 8. In the following section, the results of applyingthe 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 arepresented and they are compared with some traditional fuzzy and crisp cooperativegames, namely FSV, Least Core, Weak Least Core, Shapley Value and NormalizedNucleolus.

The first step in the proposed methodology provides optimal monthly reservoirreleases and distributes water resources among all the players with respect to theirwater demands. For example, the initial monthly allocated water to the Rafsanjanbasin is presented in Fig. 2. The results of this step are used as inputs for the nextstep, which provides the fuzzy payoffs of the players. The benefits are calculated foreach water year, so these monthly allocations should be converted to annual waterallocations to be used in the characteristic functions. The annual time series of initialwater 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 fuzzycoalitions convert to crisp ones. In this case, players use all of their initial waterrights to participate in the grand coalition (coalition 4 in our case study), and thetotal benefit of the grand coalition is allocated to the players using the equationsof 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 participatingin 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 usingwater (C(s))

s = 1 s = 2 s = 3 s = 4

B(s) (dollars/m3) 0.335 0.35 0.31 0.3C(s) (m3) 25 × 106 114.5 × 106 80.4 × 106 240 × 106


0

10

20

30

40

50

60

Jun-1978 Jun-1983 Jun-1988 Jun-1993 Jun-1998 Jun-2003

Allo

cate

d W

ater

an

d W

ater

Dem

and

(M

CM

) Water DemandAllocated Water

Fig. 2 Monthly allocated water to the Rafsanjan basin based on the results of the initial waterallocation model (million cubic meters)

time series of annual payoffs allocated to the players using different crisp games arepresented in Figs. 5, 6 and 7.

Then, fuzzy coalitions are formed and the solution concepts of FSV, FLC andFWLC 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 itswhole amount of available water resources for this participation.

Firstly, the optimization model defined by Eqs. 32–39 is used to compute theoptimal participation rates of players in different fuzzy coalitions as well as the worthof fuzzy coalitions. Then, Eqs. 18, 20 and 21 are used to calculate payoffs allocated toplayers using FSV, FLC and FWLC solution concepts, respectively. These solution

0

5

10

15

20

25

30

1978 1983 1988 1993 1998 2003year

Allo

cate

d W

ater

(m

illio

n m

) Player 1

Player 2

3

Fig. 3 Annual allocated water to the Khuzestan modern (player 1) and old (player 2) agro-industrialsectors, based on the results of the initial water allocation model


0

50

100

150

200

250

1978 1983 1988 1993 1998 2003

year

Allo

cate

d W

ater

(m

illio

n m

)

Player 3

Player 4

3

Fig. 4 Annual allocated water to the Khuzestan local agricultural (player 3) and Rafsanjan agricul-tural (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 fuzzycoalition. 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 fuzzycoalitions. 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 coalition4 is zero. Participation rate of Khuzestan local agricultural sector in the first fuzzycoalition, participation rate of Khuzestan old agro-industrial sector in the secondfuzzy coalition and participation rate of Khuzestan modern agro-industrial sectorin the third fuzzy coalition are also zero, as they do not participate in these fuzzycoalitions.

Figures 8, 9 and 10 show that players have a tendency to participate in a fuzzycoalition that has a higher net benefit coefficient. For example, as the second fuzzy

0

2

4

6

8

10

12

14

1978 1983 1988 1993 1998 2003

year

Allo

cate

d P

ayo

ff (

Millio

n D

ollars)

Crisp Least Core Crisp Weak Least Core

Crisp Shapley Value Crisp Normalized Nucleolus

Fig. 5 The annual payoffs paid to the Khuzestan modern agro-industrial sector based on the resultsof different crisp games


0

2

4

6

8

10

12

1978 1983 1988 1993 1998 2003year

Allo

cate

d P

ayof

f (M

illio

n D

olla

rs) Crisp Least Core Crisp Weak Least Core


Fig. 6 The annual payoffs paid to the Khuzestan old agro-industrial sector based on the results ofdifferent crisp games

coalition has the highest net benefit coefficient, players are willing to participate inthis coalition with their maximum participation rates.

Figure 11 demonstrates the values of reallocated water to fuzzy coalitions. Asshown in this figure, all amounts of water shares are reallocated to the second fuzzycoalition, which has the highest net benefit coefficient, until it reaches its capacity ofconsuming water. The water reallocation process is repeated for other coalitions inthe same way, until depletion of all available water resources. Reallocated water toeach fuzzy coalition should also be distributed among its players, considering thecrop pattern of the coalition and the water demands of players. For example, inthe second fuzzy coalition, which consists of a local agricultural sector and an agro-

0

10

20

30

40

50

60

70

80

1978 1983 1988 1993 1998 2003year

Allo

cate

d P

ayo

ff (

Mill

ion

Do

llars

) Crisp Least Core Crisp Weak Least Core


Fig. 7 The annual payoffs paid to the Khuzestan local agricultural sector based on the results ofdifferent crisp games


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1978 1983 1988 1993 1998 2003year

Par

tici

pat

ion

Rat

e

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 toproduce crops and the rest is allocated to the modern agro-industrial sector for bothproducing crops and processing the crops produced in the coalition. As an example,

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1978 1983 1988 1993 1998 2003

year

Par

tici

pat

ion

Rat

e

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))


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1978 1983 1988 1993 1998 2003

year

Par

tici

pat

ion

Rat

e

pr (3,2) pr (3,3)

Fig. 10 Participation rate of the Khuzestan local agricultural sector in fuzzy coalition 2 (pr(3, 2)) andfuzzy coalition 3 (pr(3, 3))

Fig. 12 represents the allocated water to players before and after participating infuzzy 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 Karoonriver basin by using different solution concepts of the fuzzy games. Also, these figuresshow 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

0

20

40

60

80

100

120

140

160

1978 1983 1988 1993 1998 2003year

Rea

lloca

ted

Wat

er (

mill

ion

m )3

Fuzzy Coalition 1

Fuzzy Colition 2

Fuzzy Coalition 3

Fig. 11 The reallocated water shares to fuzzy coalitions 1, 2 and 3


0

20

40

60

80

100

120

140

160

180

1978 1983 1988 1993 1998 2003year

Am

ou

nt

of

Wat

er (

MC

M)

share of player 3 before participating in fuzzy coalition 2share 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"

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 gamessatisfy the individual rationality criterion.

If the players participate in fuzzy coalitions, the total net benefit of the systemduring the planning horizon is 1,566.251 million dollars whereas this amount woulddecrease to 1,467.436 million dollars, if they participate in crisp coalitions. Table 9presents the total net benefits allocated to each player during the planning horizonbased on different fuzzy and crisp games. As shown in this table, each player willearn more benefits by participating in fuzzy games rather than crisp games. This factshows 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 arealmost the same. The maximum difference between payoffs allocated to playersusing FLC, FWLC and FSV is 12.3%. The FSV solution concept, which considersthe marginal contributions of players to fuzzy coalitions, can provide more reliableresults, if it does not violate the group rationality criterion. The FLC and FWLC donot have this limitation.

The Fuzzy Shapley Value provides a solution (imputation) which is located inthe center of the core when the core exists and fuzzy coalition is convex (Heaneyand Dickinson 1982). When fuzzy coalition is not convex, the solution of the FuzzyShapley 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 planninghorizon

Shapley value Least core Weak least core Normalizednuclelus

Fuzzy Crisp Fuzzy Crisp Fuzzy Crisp Crisp

Player 1 259.4265 212.5732 286.7089 212.487 266.0919 212.5856 151.5018Player 2 159.76 152.4621 159.9032 152.6345 164.1282 152.4621 149.2143Player 3 1,147.064 1,102.401 1,119.639 1,102.314 1,136.031 1,102.388 1,166.72


0

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1978 1983 1988 1993 1998 2003

year

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FSV gameFLC gameFWLC gamewithout participating in any coalition

Fig. 13 The annual payoffs paid to the Khuzestan modern agro-industrial sector based on the resultsof different solution concepts of the fuzzy games

As it was shown in the numerical example, Fuzzy Shapley Value does not satisfythe group rationality criterion in the first fuzzy coalition, which is not convex. Thissituation makes the non-convex coalitions unstable. However, the proposed fuzzycore-based solution concepts always satisfy the group rationality criterion, when thecore exists, and they make more reliable coalitions when coalitions are not convex.

In this case study, all fuzzy games satisfy the group rationality criterion, whichmeans players that constitute a coalition cannot gain more profits by participatingin a sub-coalition. Satisfaction of group rationality ensures the stability of our fuzzycoalitions.

0

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1978 1983 1988 1993 1998 2003year

Fin

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ff (

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Fig. 14 The annual payoffs paid to the Khuzestan old agro-industrial sector based on the results ofdifferent solution concepts of the fuzzy games


0

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1978 1983 1988 1993 1998 2003year

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Fig. 15 The annual payoffs paid to the Khuzestan local agricultural sector based on the results ofdifferent solution concepts of the fuzzy games

Another requirement of these games is that if the total benefit of the system in-creases, each participant should receive more benefits. Conversely, if the total benefitdecreases, no player should receive more benefits than what it received before. Thisproperty is called monotonicity (Megiddo 1974). Megiddo (1974) showed that thecrisp Least Core game is not monotone. As shown in Figures 13, 14 and 15, whenthe total benefit of the system increases, based on the results of different solutionconcepts of fuzzy games, all the players receive more benefits and variations ofannual allocated payoffs to different players are similar. It can prove that, in ourcase study, proposed solution concepts are monotone.

The proposed methodology efficiently allocates water resources and maximizesthe total net benefit of the system, while considering the environmental constraints.Moreover, this methodology reallocates the maximized net benefit to water usersin a fair and equitable way. Therefore, it can be concluded that this methodologycan easily be utilized for water allocation considering the efficiency, equity andsustainability criteria.

7 Summary and Conclusion

One of the main issues in the field of water resources management is optimalallocation of shared water resources to competing water users. In this paper, twonew solution concepts for fuzzy cooperative games, namely Fuzzy Least Core andFuzzy Weak Least Core, were developed. The results of these solution conceptswere compared with the results of some traditional fuzzy and crisp cooperativegames through a numerical example and a real world case study. In the proposedwater allocation methodology, an optimization model was used to obtain the initialallocated water shares to the players. Then, participation rates of players in differentfuzzy 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


using Fuzzy Least Core, Fuzzy Weak Least Core and Fuzzy Shapley Value. Theresults showed that players will obtain more benefits if they participate in fuzzycoalitions rather than either participating in crisp coalitions or not participating inany coalition. The proposed methodology can be easily utilized for water allocationconsidering the efficiency, equity and sustainability criteria. It was also shown thatthe proposed solution concepts of FLC and FWLC do not have the limitation of FSVin satisfying group rationality. The main limitation of the proposed methodology isthat it is deterministic. In future works, it can be extended to consider the existinguncertainties of the monthly reservoir inflows and water demands as well as theuncertainties of the net benefit coefficients of players and coalitions.

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Water Resources Allocation Using Solution Concepts of Fuzzy Cooperative Games: Fuzzy Least Core and Fuzzy Weak Least Core