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

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.     

Water Resources Allocation Using a Cooperative Game with Fuzzy Payoffs and Fuzzy Coalitions

In this paper, two fuzzy cooperative games are utilized for modeling equitable and efficient water allocation among water users in both inter-basin and intra-basin water allocation problems. The proposed all-inclusive water allocation approach consists of three main steps, following Sadegh et al. (Water Resour Manage 24(12):2991?C2310, 2010). In the first step, an initial water allocation is carried out using an optimization model taking into account an equity criterion. In the second step, the water users form crisp coalitions with fuzzy characteristic functions to increase the total net benefit of the system and also their own benefits. In the methodology used in this step, the water users do not need to have exact information about their payoffs and they can evaluate their payoffs as fuzzy numbers. In the second step, based on the Hukuhara difference of fuzzy numbers, optimum water allocation strategies are determined using a game with fuzzy characteristic function. In the third step, we applied another methodology which considers a class of fuzzy games with fuzzy characteristic functions and also fuzzy coalitions for water allocation. The methodology of this step is on the basis of the Hukuhara difference and the Choquet integral. The usefulness of the mentioned methodologies is studied by applying them to three defined real life scenarios in a case study of water allocation in Iran. The results showed that the proposed methodologies are professionally appropriate to real-world uncertain problems of equitable and economic inter-basin and intra-basin water resources allocations.     

A Fuzzy Variable Least Core Game for Inter-basin Water Resources Allocation Under Uncertainty

In this paper, a new solution concept, called Fuzzy Variable Least Core (FVLC), is developed for fuzzy cooperative games. The FVLC is able to incorporate fuzzy input variables and result in fuzzy benefit shares of players participating in a coalition. This solution concept is used for water and benefit allocation to water users in inter-basin water transfer systems considering the uncertainties associated with their benefit coefficients. In the proposed water allocation methodology, an Integrated Stochastic Dynamic Programming (ISDP) model is developed to obtain the water rights of players and economic water allocation policies. In the next step, the total net fuzzy benefit of the system is reallocated to water users in an equitable and rational way using a FVLC-based model. In this model, a new algorithm is proposed for converting a multilateral cooperative game with fuzzy variables to some fuzzy bilateral cooperative games, which are solved using the FLVC solution concept. The applicability and efficiency of the proposed methodology is examined by applying it to a large scale inter-basin water transfer project in Iran.     

Water and Pollution Discharge Permit Allocation to Agricultural Zones: Application of Game Theory and Min-Max Regret Analysis

This paper presents a new game theoretic methodology for water and discharge permit allocation to agricultural zones in shared rivers. The methodology consists of four main steps: (1) initial allocation of water rights and pollutant discharge permits, (2) forming possible coalitions and optimal water and discharge permit reallocation to water users participating in a coalition to increase their total net benefit, (3) equitable benefit reallocation by utilizing some solution concepts in cooperative game theory, and (4) identifying the best water and pollutant discharge permit allocation strategies by minimizing the maximum regret in the system. A new linear form for crop water production function is used in the objective function of the water allocation optimization models. To show the efficiency and applicability of the methodology, it is applied to the Karoon-Dez river system in Iran.     

A Nonlinear Interval Model for Water and Waste Load Allocation in River Basins

In this paper, a new methodology is proposed for simultaneous allocation of water and waste load in river basins. A nonlinear interval number optimization model is used to incorporate the uncertainties of model inputs and parameters. In this methodology, the bounds of the uncertain inputs are only required, not necessarily knowing their probability density or fuzzy membership functions. In the proposed model, the existing uncertainties in water demands and monthly available water are considered in the optimization model. Also the economic and environmental impacts of water allocation to the agricultural water users are taken into account. To have an equitable water and waste load allocation, benefits are reallocated to water users using some solution concepts of the cooperative game theory. Results of applying the methodology to the Dez river system in south-western part of Iran show its effectiveness and applicability for water and waste load allocation in an uncertain environment.     

利益相关者的水资源配置博弈

水资源的准公共产品属性及其在开发利用中存在的外部性使河流自然流态受到严重的扰动,本文基于非合作博弈博弈论,建立主从关系的用户博弈模型分析河流水资源分配,基于个体效益不能达到帕累托最优状态,采用合作博弈方法,建立水资源用户合作博弈模型。合作博弈增加了总效益,但使得上游用户的效益比非合作时减少,有必要对上游用户进行效益补偿:引入微观经济学的无差异曲线与合作联盟形成的条件构建合作效益分配的可行解集,并提出基于边际贡献的效益分配方法在可行解集中挑选唯一的效益分配解。研究结果表明,合作博弈总效益大于非合作博弈的总效益,达到帕累托最优状态;通过合作效益分配,处于先动优势的上游用户的分配效益大于非合作的效益,有利于行动占先者加入合作联盟。     

An Interval Parameter Model for Cooperative Inter-Basin Water Resources Allocation Considering the Water Quality Issues

In this paper, a new methodology based on interval optimization and game theory is developed for optimal operation of an inter-basin water transfer system considering efficiency, equity and sustainability criteria. A linear version of the agricultural water production function is proposed and used for incorporating deficit irrigation. The interval programming and cooperative game theory are utilized for equitable reallocation of benefits to water users in both water donor and receiving basins. To assess the sustainability of water allocation policies, water quality and environmental flow in the donor basin and groundwater table drawdown in the receiving basin are taken into consideration. The effectiveness of this methodology is examined by applying it to a large scale case study of water transfer from the Karoon river basin in south-west to the Rafsanjan plain in the central part of Iran.     

Equitable Waste Load Allocation in Rivers Using Fuzzy Bi-matrix Games

This paper presents a new game theoretic methodology for equitable waste load allocation in rivers utilizing fuzzy bi-matrix games, Non-dominated Sorting Genetic Algorithms II (NSGA-II), cooperative game theory, Bayesian Networks (BNs) and Probabilistic Support Vector Machines (PSVMs). In this methodology, at first, a trade-off curve between objectives, which are average treatment level of dischargers and fuzzy risk of low water quality, is obtained using NSGA-II. Then, the best non-dominated solution is selected using a non-zero-sum bi-matrix game with fuzzy goals. In the next step, to have an equitable waste load allocation, some possible coalitions among dischargers are formed and treatment costs are reallocated to discharges and side payments are calculated. To develop probabilistic rules for real-time waste load allocation, the proposed model is applied considering several scenarios of pollution loads and the results are used for training and testing BNs and PSVMs. The applicability and efficiency of the methodology are examined in a real-world case study of the Zarjub River in the northern part of Iran. The results show that the average relative errors of the proposed rules in estimating the treatment levels of dischargers are less than 5?%.     

Incorporating Economic and Political Considerations in Inter-Basin Water Allocations: A Case Study

When an inter-basin water transfer is expected among basins with some level of unfriendliness or hostility, ignoring political considerations, which are generally not integrated in economic investigations, can impede an integrated and efficient management. In this paper, a new economic-political methodology is proposed for the optimal and efficient allocation of water resources among water users in inter-basin water transfer systems. The proposed framework quantifies both the economic payoffs using an “n-person real fuzzy cooperative game”, and the political formation prospect of any coalition, using a Modified Political Accounting System (MPAS). The proposed economic-political methodology is applied to a large scale inter-basin water allocation problem including water donor and receiving basins struggling with water scarcity. The results show how including political considerations in the study may provide a more satisfactory solution compared to the just cost-effective water allocations.     

A multi-weight fuzzy Methodological Framework for Allocating Coalition Payoffs of Joint Water Environment Governance in Transboundary River Basins

To ensure transboundary river basins are protected as a system, forming water environment governance coalitions is important. However, accurate determination and allocation of the payoffs of joint environment governance under uncertain conditions is a challenging task. This study used a triangular fuzzy number to describe the payoff and solve the fuzzy allocation problem. The article also considered the importance of different coalitions and the effectiveness of the distribution. In addition, the authors constructed a multi-weight fuzzy allocation quadratic programming model and proposed an analytical solution to determine the triangular fuzzy allocation payoffs. These contributions were used to allocate the coalition payoff of basin-wide environment governance, and the Zhanghe River Basin is used as a case study. The results showed that the model produces outputs comparable with the Shapley value allocation method. Therefore, the proposed methodological framework can be used as an alternative allocation scheme that provides a reasonable solution during uncertainty.

     

Editorial Comments on the World Commission on Water for the 21stCentury

Abstract

In this study, in order to create a decision-making model on water resources projects, a hierarchy of criteria has been developed by public participation. The Value Management methodology has been used for extraction of the effective criteria and attributes in the scope of Integrated Water Resources Management (IWRM). The hierarchy is generic for water resources management in the Islamic Republic of Iran. Since the evaluations of alternatives with respect to some attributes are uncertain and vague, fuzzy set theory has been used. By merging fuzzy set theory and multi-attribute decision-making a new Decision Support System (DSS), namely FDM, has been developed to compare different alternatives. As an innovation, FDM accepts evaluations of alternatives with respect to the attributes as crisp variables, fuzzy variables, and linguistic variables. FDM embodies an expert system whose duty is to choose an appropriate method among the SAW, Fuzzy SAW, TOPSIS or Fuzzy TOPSIS based on the characteristics of the problem.

The central and Southeastern regions of Iran are considered arid regions, suffering from water shortages. In this paper, water transfers to the Zayanderud basin in Iran have been modeled by FDM. Successful application of this DSS in this study allows for its application by water authorities in other case studies.     

Water Resources Allocation in Transboundary River Basins Based on a Game Model Considering Inflow Forecasting Errors

Dynamic transboundary water resources allocation based on inflow prediction results is an important task for water resources management in river basins. This paper takes the watershed management agency as the leader and the associated area as the follower, and proposes a two-level asymmetric Nash-Harsanyi Leader-Follower game model considering inflow forecasting errors. In the proposed model, the Monte Carlo method is used to analyze the uncertainty of various stakeholder allocation results and the response regularity to the total water resource uncertainty. The quantitative relationship between the allocation results of stakeholders and the mean and standard deviation of total water resource uncertainty is subsequently established. The Huaihe River basin in China is selected as a case study. The results show the following: (1) the water allocated to the watershed management agency and three provinces has a normal distribution when the inflow forecasting error obeys the normal distribution; (2) the sum of the mean of the water allocated to stakeholders equals the mean of the forecast water resource and the sum of the standard deviations of the water allocated to stakeholders equals the standard deviation of the forecast water resource; (3) the mean and standard deviation of the allocation results have a good linear relationship with the mean and standard deviation of forecast water resource; (4) the distribution parameters of the stakeholder allocation results can be directly derived from the distribution parameters of the forecast information, thus aiding the stakeholders in making decisions and improving the practical value of the method.

     

A Game Theoretic Approach to Assess the Impacts of Major Investments on Transboundary Water Resources: The Case of the Euphrates and Tigris

There are varying and inconsistent claims among countries sharing the same river basins. While upstream countries are asserting that they regulate flows through built-up reservoirs (enabling flood prevention) and reduce seasonal flow variations (providing water during drought seasons), downstream countries criticize upstream countries for excessive withdrawals and declining annual flows. Behind the scene, almost every basin country unilaterally envisions of ambitious projects demanding extensive water withdrawals for their prospective development and growth. This study establishes a methodology to measure the rational economic and political impacts of extensive reservoir projects throughout a basin, using the case of the Euphrates and Tigris. This methodology uses both linear programming applications calculating country and coalition benefits, and game theory concepts (core and Shapley value) for evaluating the impacts of reservoirs. The Euphrates and Tigris River Basin Model (ETRBM) is extended by adding the time dimension. In conjunction with time, reservoir capacities are incorporated into the model as a new set of parameters enabling resources allocations over multiple periods. In a rational economic view, the existence of reservoirs, while providing certain advantages in allocating water overtime, provides varying geopolitical powers to upstream and downstream basin countries. As a result of model applications, it is observed that basin-wide coalitions may potentially eliminate construction of excessive reservoir capacities, and, therefore, unnecessary investment costs and evaporation losses. It is also observed that assuming the absence of downstream reservoirs, the upstream reservoirs provide significant, first, economic benefit to the all forms of coalitions with upstream country and associated parties and, second, power to the country having the reservoirs in any coalition scenario. However, again in a game theoretic perspective, the power of upstream countries looses some of its weight when downstream reservoirs are incorporated. During drought years, the availability of reservoirs ensures extensive core solutions and encourages a grand coalition among all parties.     

Incorporating Water Demand Management into a Cooperative Water Allocation Framework

The impact of a water demand management plan on a water system and its users is investigated within a comprehensive cooperative water allocation framework. In particular, a demand management plan is incorporated into a two-step multi-period fair water allocation model. A modified cooperative game is designed for the sharing of additional net benefits under the scenario having water demand management. The results indicate that cooperation among water users can yield more net benefits, and a water demand management plan is able to lead to a further increase of the aggregated net benefits by means of water transfers from less productive users to more productive ones. By utilizing the modified cooperative game, fair sharing of additional net benefits ensures that every water user can expect to receive more net benefits and thereby water users are motivated by incentives to implement a water demand management plan which in turn improves water use efficiency. The results demonstrate that the demand management plan can be of great assistance in some arid and semi-arid regions.     

Basin-wide Water Resources Planning by Integrating PSO Algorithm and MODSIM

Water resources planning and management at basin scale is such a large and complex problem that makes it essential to use effective modeling tools in order to obtain an optimum plan for river basins development. In this paper, a methodology is presented for optimized design and operation of the upstream Sirvan basin in Iran. The model proposed integrates MODSIM generalized river basin network flow model, with the capability of simulating various characteristics and features of water resources in a river basin, and Particle Swarm Optimization (PSO) algorithm. In the developed PSO-MODSIM model, the size of planned dams and water transfer systems, as design variables, and the relative priorities for meeting reservoir target storages, as operational variables, are varied and evolved using PSO algorithm. MODSIM is called to simulate the system performance and to evaluate the fitness of each set of those design and operational variables with respect to the model’s objective function. The PSO objective function is to maximize the total net benefit consisting of benefits due to supplying water to different types of water uses and construction costs of dams and water transfer and/or pumping systems. Varying the design and operational variables in MODSIM 8 is done using the MODSIM’s custom coding feature in VB.NET routine. The PSO-MODSIM model is used to size the planed water storage and transfer components of a river basin system and to allocate water resources optimally over time and space among competing demands, considering coordinated operation of the system components. The model results has been analyzed for different scenarios of water transfer from Sirvan to neighboring basins.     

Optimizing Multiple-Pollutant Waste Load Allocation in Rivers: An Interval Parameter Game Theoretic Model

In this paper, a new methodology is developed for optimal multiple-pollutant waste load allocation (MPWLA) in rivers considering the main existing uncertainties. An interval optimization method is used to solve the MPWLA problem. Different possible scenarios for treatment of pollution loads are defined and corresponding treatment costs are taken into account in an interval parameter optimization model. A QUAL2Kw-based water quality simulation model is developed and calibrated to estimate the concentration of the water quality variables along the river. Two non-cooperative and cooperative multiple-pollutant scenario-based models are proposed for determining waste load allocation policies in rivers. Finally, a new fuzzy interval solution concept for cooperative games, namely, Fuzzy Boundary Interval Variable Least Core (FIVLC), is developed for reallocating the total fuzzy benefit obtained from discharge permit trading among waste load dischargers. The results of applying the proposed methodology to the Zarjub River in Iran illustrate its effectiveness and applicability in multiple-pollutant waste load allocation in rivers.     

A Fuzzy Stochastic Dynamic Nash Game Analysis of Policies for Managing Water Allocation in a Reservoir System

In this paper a fuzzy dynamic Nash game model of interactions between water users in a reservoir system is presented. The model represents a fuzzy stochastic non-cooperative game in which water users are grouped into four players, where each player in game chooses its individual policies to maximize expected utility. The model is used to present empirical results about a real case water allocation from a reservoir, considering player (water user) non-cooperative behavior and also same level of information availability for individual players. According to the results an optimal allocation policy for each water user can be developed in addition to the optimal policy of the reservoir system. Also the proposed model is compared with two alternative dynamic models of reservoir optimization, namely Stochastic Dynamic Programming (SDP) and Fuzzy-State Stochastic Dynamic programming (FSDP). The proposed modeling procedures can be applied as an appropriate tool for reservoir operation, considering the interaction among the water users as well as the water users and reservoir operator.     

不确定性多目标模糊规划在水资源优化配置中的应用

在充分考虑水资源系统中供需水不确定性的基础上,基于多目标规划、模糊规划和区间规划原理,以经济效益、社会效益和环境效益最大为目标,以需水量、可供水量和不同子区用水部门间的用水公平性为主要约束,构建一种适用于多水源、多子区、多用户的考虑供需水不确定性的多目标模糊规划模型,以期通过平衡水资源配置系统中用户配置水量、缺水风险与系统收益三者的关系,实现系统综合效益最大。模型以衡水市历年供水、用水及社会经济等数据确定规划年的水资源参数及经济参数,采用区间参数反映系统中的不确定性,通过引入模糊隶属度函数,利用两步交互式算法,将多目标规划转化为单目标规划进行求解,以2025年为衡水市规划水平年,得到规划水平年下的11个子区、4种水源、4个用水行业的最优配水方案。结果表明：本研究制定的水资源优化配置方案可实现“外调水优先利用、地下水控制利用”,同时考虑用水的公平性约束后,在缺水条件下可有效控制水资源向每立方米水效益高的区域和部门流动,有利于多区域、多部门共担缺水风险;该水资源优化配置方案可有效缓解衡水市水资源供需矛盾,实现多水源和多目标之间的协同互补。研究成果可为河北省其他县域的水资源优化配置提供技术参...     

Bargaining Model of Synergistic Revenue Allocation for the Joint Operations of a Multi-Stakeholder Cascade Reservoir System

Given the institutional limitations of multi-stakeholders, exploring the synergistic revenue from the joint reservoir operations of a multi-stakeholder multi-reservoir system requires a synergistic revenue allocation mechanism to ensure a beneficial solution for multi stakeholders. This study established a synergistic revenue allocation model using bargaining game theory under the principles of equity, rationality, and efficiency. For the maximization the Nash product of gains in the utility of stakeholders and constraints on the feasibility of allocation plans considering all the possible formations of sub-coalitions, the analytic optimal solution of the bargaining model was derived using the first-order optimality condition. The optimal revenue allocation plan meets the conditions of the equal quasi-marginal utility function among stakeholders. The methodologies were applied to a hypothetical cascade reservoir system operated by multiple stakeholders. Compared with the revenue allocation plans obtained by a proportional rule method and the Shapley value method, the results corroborate that (1) the allocation plan of the bargaining model is jointly determined by the interval of the revenue range of each reservoir and the effectiveness of the sub-coalition constraints, indicating that the allocated synergistic revenue is positively correlated with the singleton contribution and team contribution on the total revenue of the grand coalition; (2) the difference in the plans obtained by the three methods is generally determined by the difference in equity definition; and (3) the synergistic revenue allocation plan obtained from the bargaining model is the highest homogenized among all reservoirs (stakeholders), which demonstrates that the revenue of those dominated stakeholders can be improved compared with other plans. The proposed methodologies provide new insights to guide benefit share decisions in multi-stakeholder reservoirs system.     

The Mekong Game: Achieving an All-win Situation

In recent years, there have been many quarrels among countries of the Mekong River Basin surrounding use of the water resources of that river. In particular, China’s behavior of constructing dams upstream has resulted in objections by many people in all countries downstream. Cooperative game theory has been applied to solving issues of building and utilizing dams on trans-border rivers. The central question is how to allocate the surplus benefits generated by the dams. This paper proposes to achieve an all-win situation in use of Mekong water resources through cooperation. A game named “The Mekong Game” was designed for those Ph.D. students and resource persons to play on the PROSPER.NET’s workshop in 2010. For a hypothetical case in Mekong Game, the Shapley Value Method’s results were given, which could be regarded as one of equitable results in theory, and the Mekong Game’s process and outcomes are introduced in this paper. Participants were divided into six groups representing the six involved “countries.” The process of meeting an agreement with an all-win idea was simulated through three round-table negotiations. For the hypothetical case, different set of results under different situations were compared and discussed, including no dam(s), dam(s) without cooperation, dam(s) with cooperation but without compensation and re-allocation, Shapley Value Method, and Mekong Game (bargaining and negotiation). The results reveal: although it is hard to get theoretical optimal aftermath considering other complicated factors than those can be calculated, the all-win situation is still possible through bargaining and negotiation processes, which should be much better than all-lose situation such as the current one.