Abstract

Population growth coupled with increased urban and agricultural water use have exacerbated water shortages worldwide. Conflicts among water users frequently arise over scarce water. The application of conflict resolution methods has the potential to resolve such conflicts. Bankruptcy games is a branch of game theory applicable to problems dealing with conflict resolution. This study addresses water allocation to urban-industrial, agricultural, and environmental water uses downstream of the Zarrineh-roud dam, Iran, which diverts water from the Zarrineh-roud River, an important tributary to Lake Urmia. Lake Urmia has been severely stressed by reduction of its water inputs. Water allocation is posed in this study as a bankruptcy game in which the allocation to stakeholders is optimized with proportional (P), adjusted proportional, constrained equal award (CEA), and constrained equal losses methods. The CEA was chosen as the best allocation method based on performance criteria and the Bankruptcy Allocation Sustainability Index. Monthly, real-time, water allocation rule curves were calculated with genetic programming.

INTRODUCTION

Lake Urmia, Iran, has been impacted by diversion of some of its tributary water to meet agricultural, urban, and municipal water demands. The allocation of water to Lake Urmia and other water stakeholders in its drainage areas is posed in this work as a bankruptcy game and tackled with game theory. Game theory is a mathematical technique applied to the analysis of problems involving dynamic and conflicting interactions among agents in a strategic environment. The problems solved, which are basically about decision pitting rival parties, are framed as a game involving players trying to outdo each other. Game theory has been applied to fundamental problems in mathematics and economics (Von Neumann & Morgenstern 1944; Nash 1950; Shapley 1953; Bondareva 1963; Packel 1981; O'Neill 1982; Rubinstein 1982; Aumann & Maschler 1985; Curiel et al. 1988; Mas-Colell 1989; Dagan & Volij 1993).

Sheikhmohammady & Madani (2008) applied bankruptcy methods to the allocation of water, petroleum, and gas resources of the Caspian Sea to Azerbaijan, Iran, Kazakhstan, Turkmenistan, and Russia. Ansink & Weikard (2009) applied bankruptcy methods to the sequential allocation of resources. Madani (2010) applied two-by-two games to water resources problems. Zarezadeh et al. (2012) employed bankruptcy rules to allocate Qiziluzan-Sefidroud River's water to eight provinces in Iran. Madani & Zarezadeh (2012) demonstrated the effectiveness of bankruptcy procedures in solving water disputes with a hypothetical groundwater bankruptcy problem. Madani et al. (2014) introduced a bankruptcy procedure to resolve transboundary river water allocation conflict in which the stakeholders’ demands exceeded the total available water. Mianabadi et al. (2014) developed a bankruptcy method for water resources allocation. Their method considered the contribution of each beneficiary of water resources in accordance with the United Nation's Watercourses Convention (1997). Mianabadi et al. (2015) presented a weighted bankruptcy solution to the problem of scarce resources allocation with an application to the allocation of the Tigris River's water between Turkey, Syria, and Iraq. Sechi & Zucca (2015) proposed a method based on bankruptcy games to allocation the resources in complex supply systems under water-shortage conditions.

Performance criteria introduced by Hashimoto et al. (1982) are herein applied to evaluate the performance of model solutions to the water allocation problem under scarcity being solved. Several applications of classical optimization methods to water resources systems can be found in Simonovic & Mariño (1980), Rosenthal (1981), Yeh (1985), Hiew (1987), Barros et al. (2001), Labadie (2004) and also metaheuristic optimization tools (Wardlaw & Sharif 1999; Momtahen & Borhani Darian 2005; Jalali et al. 2007; Bozorg-Haddad et al. 2008; Afshar et al. 2009). Application of meta-heuristic methods to water resources systems, of which genetic programming (GP) is an example, can be found in Afshar et al. (2011), Fallah-Mehdipour et al. (2012), Fallah-Mehdipour et al. (2013a, 2013b, 2013c), Bozorg-Haddad et al. (2014), Asgari et al. (2015), Akbari-Alashti et al. (2014), Fallah-Mehdipour et al. (2014), Orouji et al. (2014), and Ashofteh et al. (2015), among others. The Hashimoto et al. (1982) reliability, resiliency and vulnerability indices have been applied in several studies, among which are those by Moy et al. (1986), Vogel & Bolognese (1995), Zongxue et al. (1998), Minville et al. (2009), and Sandoval-Solis (2011).

This paper employs nonlinear programming (NLP) and GP as optimization algorithms to solve bankruptcy games dealing with water allocation in the Urmia lake basin, Iran. This appears to be the first study implementation of GP for water allocation under uncertainty among competing stakeholders.

Methodology

This section presents three subsections: (1) reservoir operation modeling with NLP; (2) game theory and conflict resolution with the bankruptcy approach; and (3) real-time water allocation using GP.

Reservoir operation modeling with NLP

Reservoir water releases are commonly considered as the decision variables in reservoir operation problems. Alternatively, the reservoir operation rule curve may be the decision variable, in which case various coefficients and rule curve formulas of the type written in Equation (1) become the unknowns to be solved for (Fallah-Mehdipour et al. 2012):  
formula
(1)
in which = release volume during the tth time period (106 m3); = storage volume at the beginning of the tth time period (106 m3); = inflow volume to the reservoir during the tth time period (106 m3); and F is a linear or nonlinear function obtained with GP.
There are several formulas used as a rule curve such as linear equations in Equation (2) that have been previously applied in Bozorg-Haddad et al. (2008):  
formula
(2)
in which a, b, and c are equation coefficients obtained by regression analysis of time series data.
Bolouri-Yazdeli et al. (2014) proposed quadratic and cubic equations as shown in Equations (3) and (4) as rule curves:  
formula
(3)
 
formula
(4)
in which a′, b′, c′, d′, e′, f′, g′ are coefficients and decision variables which are calculated by optimization.

Equations (2)–(4) have a predetermined structure. Better equations may exist based on other kind of mathematical relations without predetermined linear or nonlinear structure (Fallah-Mehdipour et al. 2012). In this study, the reservoir operation rule curve equation is determined by GP and the results will be compared with standard operating policy (SOP) of a reservoir. The SOP is the historical, non-optimized, rule curve of a reservoir.

The minimization of the sum of relative deviations between release and demand volumes is herein considered as the objective function shown in Equation (5):  
formula
(5)
in which Z = objective function to meet downstream water demand; T = number of operation periods; and = downstream water demand during the tth period (106 m3).
The continuity Equation (6) governs the evolution of storage in a reservoir:  
formula
(6)
in which = reservoir storage volume at the beginning of the ()st time period (106 m3); = spill from reservoir during the tth time period (106 m3); and = evaporative reservoir losses during the tth time period (106 m3). The is calculated during each period based on Equation (7):  
formula
(7)
in which = average of reservoir surface during period t () which is calculated by Equation (8); and = evaporation depth during the tth time period (m).  
formula
(8)
in which is a function of storage obtained by fitting a cubic equation to the area-volume data.
The constraints on releases and storage are as follows:  
formula
(9)
 
formula
(10)
where and are the minimum and maximum values of reservoir storage (106 m3), respectively.

Hashimoto et al. (1982) evaluated reservoir performance using a reliability index that reflects the number of times a system incurs failure; a resiliency index for the time it takes a system to return to normal following a failure; and a vulnerability index that measures the intensity of failures. The former three indexes are employed in this study to describe the performance of a reservoir system.

Reliability

The reliability index takes two forms: (1) time-based reliability and (2) volumetric reliability. According to the definition by Hashimoto et al. (1982), the time-based reliability is the probability of non-failure periods. While, the volumetric reliability equals to ratio of releases summation to sum of demands during the operation period. The time-based and volumetric reliabilities are defined by Equations (11) and (12):  
formula
(11)
 
formula
(12)
in which and = time-based and volumetric reliability indexes; respectively. f= number of failure periods and T= number of operation periods.

Resiliency

This index measures a system's capacity to recover after failure. Hashimoto et al. (1982) defined resiliency by Equation (13):  
formula
(13)
in which = resiliency rate; and = number of failure series, where a failure series is a sequence of consecutive failure periods that is preceded and followed by non-failure periods.

Vulnerability

This index measures the severity of system failure. Hashimoto et al. (1982) defined the vulnerability index as the average of the maximum shortages that occur in each sequence of failure periods:  
formula
(14)
in which = vulnerability index; and = maximum shortage during the kth failure series.

System operators attempt to maximize reliability and resiliency and minimize the vulnerability.

Game theory and conflict resolution with bankruptcy allocation rules

The terms of the bankruptcy change with changes in the conflict between stakeholders who are engaged in a zero-sum competition whereby each stakeholder attempts to maximize his utility. When a system fails to meet the demands of each stakeholder, the basic question is how to allocate the available resources between the stakeholders. Answering this question is difficult when the utility functions and payoffs of the stakeholders are not well defined.

The following are the best-known bankruptcy methods applied to water allocation of a finite amount of water available:

Proportional (P) rule for water allocation

The share of each stakeholder is calculated on the basis of Equation (15). In accordance with the P method, the amount of allocated water is proportional to the demands of the stakeholders:  
formula
(15)
in which = water share of the ith stakeholder by applying the P method; = the water demand of the ith stakeholder, and = satisfaction coefficient that is always positive and is calculated with Equation (16) as follows:  
formula
(16)
in which E = the total available water, and C= sum of all the stakeholders’ water demands =.

Adjusted proportional rule for water allocation

This method first calculates the value of according to the values of E and C= as written in Equation (17):  
formula
(17)
in which = the initial allocation, which is the minimum amount of water that can be allocated to the ith stakeholder, and j = counter for all stakeholders except the ith stakeholder. Next, the value of the water allocated to each stakeholder is improved by Equation (18):  
formula
(18)
in which = improved water allocation to the ith stakeholder, and N = the set of all of the stakeholders.
In the third stage of this method, the water allocation to stakeholders () is calculated according to the improved allocations and the remaining water available as follows:  
formula
(19)
 
formula
(20)
in which = secondary allocation to the ith stakeholder, and = satisfaction coefficient corresponding to the state of improved water allocations. Lastly, the water allocation to the ith stakeholder obtained with the adjusted proportional (AP) method () equals the sum of the initial () and secondary () allocations as expressed by Equation (21):  
formula
(21)

Constrained equal awards rule for water allocation

Based on this rule, the water share of each stakeholder (or player) equals the smaller of the player's water demand (or claim) or the value according to Equations (22) and (23):  
formula
(22)
 
formula
(23)
in which the water share of each stakeholder (or player) by using the constrained equal award (CEA) method and Equation (22) is calculated by trial and error or with optimization algorithms; is calculated by Equation (24), which has one solution provided that .  
formula
(24)

Constrained equal losses rule for water allocation

This method allocates water to each stakeholder using Equation (25) as follows:  
formula
(25)
in which Max. (x, y) = the larger of the two arguments x or y, and is defined by Equation (24). Water allocations to the stakeholders are made at specific time steps, i.e. the system is a time-dependent system.

The P rule for water allocation

The P rule prescribes water allocation by solving the following minimization problem (Equations (28)–(31)) in which the minimization is with respect to the allocations to the ith stakeholder at the tth time step () defined as follows:  
formula
(26)
where = the relative allocation coefficient (or the satisfaction coefficient) to the ith stakeholder at the tth time step (state variable); = claim of the ith stakeholder at the tth time step;  
formula
(27)
= the maximum relative allocation coefficient at the tth time step (state variable).  
formula
(28)
Subject to:  
formula
(29)
 
formula
(30)
 
formula
(31)
in which = total resources available at the tth time step. The multiplication term in the objective function ensures the existence of a single answer for the optimization model and causes the allocation coefficients () to be close to each other (Madani et al. 2014). The lowest value of the objective function is achieved when the relative allocation coefficients are equal to each other, = for i = 1, 2, …., m, t = 1, 2…, T.

The AP rule for water allocation

The AP method addresses the water allocation to the ith stakeholder when all the other stakeholders have been awarded their water claims. The initial allocation to the ith stakeholder in this case equals the remaining water once the water allocations to all other stakeholders have been made. If there are no remaining resources, then the initial allocation to the ith stakeholder equals zero. The initial water allocations to all other stakeholders (stakeholder j = 1, 2, …, m, ji) are modified in the next step of the AP method. The modified claim of each stakeholder equals his initial allocation minus the amount of remaining water. The P rule is subsequently applied to the remaining resources and the modified or improved claims. Optimization of water allocation with the AP rule in dynamic, time-dependent, systems such as reservoir systems is as follows:  
formula
(32)
Subject to:  
formula
(33)
 
formula
(34)
 
formula
(35)
 
formula
(36)
 
formula
(37)
 
formula
(38)
 
formula
(39)

in which = total claims of the stakeholders except that of the ith stakeholder; = the initial allocation to the ith stakeholder at the tth time step; = the relative allocation coefficient applied to the adjustment of the claims; = the maximum value of the relative allocation coefficient applied to the claims; = adjusted claim of the ith stakeholder at the tth time step, and = allocation to the ith stakeholder at the tth time step when adjusting the claims (these are the decision variable). Based on the aforementioned equations, the Bi,t parameter represents the sum of other stakeholders’ claims against the player i in period t, which is employed to calculate the initial allocation to each stakeholder. Moreover, is less than , the adjusted claim of stakeholder i in time step t.

The CEA rule for water allocation

The CEA method first allocates the water claimed by the stakeholder with the smallest water claim. If there is water available after supplying the first stakeholder (i.e., if ), then this first stakeholder is removed from the water allocation process and the remaining water is distributed among the other stakeholders in a similar fashion, that is, satisfying with the extant smallest claim first, and proceeding sequentially (Madani & Dinar 2013). Optimal water allocation with the CEA rule in dynamic, time-dependent systems such as reservoirs is given by:  
formula
(40)
Subject to:  
formula
(41)
 
formula
(42)
 
formula
(43)
in which t= time period; = allocation to the ith stakeholder at the tth time step (decision variable), and = the maximum amount of allocation at the tth time period. The value of may be larger than one. Therefore, to achieve converge in the objective function (40) its second (multiplicative) term was divided by .

The constrained equal losses rule for water allocation

The constrained equal losses (CEL) rule allocates water by prioritizing those stakeholders with the largest demands, starting with the stakeholder with the largest demand, followed by the stakeholder with the second largest demand, and proceeding likewise and sequentially. The solution process is: CEL specifies a loss for each of player and substracts this loss from the claims of each stakeholder (). If is less than , the value of allocation to the player is zero. The final allocation to each stakeholder equals . Optimization modeling of resources allocation by CEL rule in time-dependent systems is as follows:  
formula
(44)
Subject to:  
formula
(45)
 
formula
(46)
 
formula
(47)

in which = the water shortage applied in supplying the water claim of the ith stakeholder at the tth time step, and = the maximum shortage among all the stakeholders at the tth time period.

Stability evaluation

The water allocation rules presented above are based on diverse concepts of fairness and introduce a variety of strategies for water allocation among stakeholders. The rules are unequally acceptable to water stakeholders, depending on perceived differential benefits resulting from them (Madani & Lund 2011). A common method to choose among the various rules is balloting by stakeholders to select one among them (Sheikhmohammady & Madani 2008; Madani et al. 2014). The plurality index measures the acceptability of allocation rules by determining the number of stakeholders who prefer each possible allocation rule, so that the rule with most supporters has the highest acceptability (Dinar & Howitt 1997).

It is possible that the plurality index selects a rule that does not earn the support of the majority of the stakeholders, rendering it of questionable acceptability. Other methods to measure the acceptability of allocation rules (Read et al. 2014); Loehman et al. (1979) employed the power index (), which was developed by Shapley & Shubik (1954). This index is described by Equation (48) and it evaluates the power of players in game theory problems in which the players look for the best way to accrue incremental benefits resulting from cooperative allocation, that is, allocation that is acceptable to the majority of stakeholders:  
formula
(48)
in which = incremental water allocated to the ith player; = ith player's allocation in the current situation (non-cooperative state), and m = the number of stakeholders.
High power-index values are associated with stakeholders with currently low water allocations who would benefit most with the allocation governed by the power index. For this reason, an alternative, more sustainable solution may be achieved when the power index is distributed among players (Dinar & Howitt 1997). One proposal to accomplish this is by employing the coefficient of variation of the players’ power index to evaluate the sustainability of water allocation solutions as a sustainability index ():  
formula
(49)
in which = standard deviation of the players’ power index, and = the average of the players’ power index. Low values of the sustainability index defined by Equation (49) indicates greater sustainability of water allocation.
Madani et al. (2014) proposed the Bankruptcy Power Index (BPI) for bankruptcy allocation of resources as follows:  
formula
(50)
 
formula
(51)
 
formula
(52)
in which = BPI for the ith stakeholder; and m = of the number of stakeholders.

According to the aforementioned equations, the BPI for ith stakeholder is calculated for planning horizon using allocation to the ith stakeholder () and the initial allocation to the ith stakeholder ().

The bankruptcy allocation sustainability index (BASI) employs the coefficient of variation of the players’ BPI to evaluate the various bankruptcy rules:  
formula
(53)
in which = standard deviation of the players’ BPI and = the average of the players’ BPI. High values of the BASI indicates that the allocation method is unstable and likely unsustainable.

Real-time water allocation with GP

GP is a metaheuristic algorithm which explores the decision space of an optimization problem randomly and calculates the decision variables’ values near to optimal solutions based on evolutionary- and naturally-inspired phenomena. At the first step, a few random possible solutions are generated that are metaphoric of chromosomes in genetics. Chromosomes (i.e. possible solutions) are evaluated with an objective function and are ranked according to the value of the objective function that they attain. During the optimization process, the populations that are generated in each step are modified randomly by crossover and mutation operators. GP is an artificial intelligence algorithm that expresses complex problems with mathematical and logical relations. Its search process and convergence to an optimal solution is similar to those of the genetic algorithm (GA), with the exception that the decision variables are not always expressed in numerical form, but, rather, include operators, functions, and coefficients. This property gives enhanced capabilities to GP for calculating appropriate relations in an optimization problem.

Table 1 lists resemblances and differences between the GA and the GP operators. GP treats the operators, functions, and coefficients of relations as decision variables, which constitutes a key advantage. As a result, no preset relation is imposed on the GP algorithm, and the best relation is obtained with operators, functions, and coefficients. The GP has ample capability to generate optimal reservoir operating rules as a function of reservoir inflow and storage.

Table 1

Comparison of the GA and GP

 Algorithm features
Algorithm name Basis of the algorithm Algorithmic process Decision variables Decision variable sets Generation of new solutions with the crossover operator Generation of new solutions with the mutation operator 
GA Based on evolution theory Random search and iteration Gene (represents numbers) Chromosome Exchanging several genes between two chromosomes randomly Generating new random values for numerical decision variables with random genes 
GP Based on evolution theory Random search and iteration Node (represents operators, functions, and numbers) Tree Exchanging several branches between some trees Randomly generating new operators, functions and coefficients in nodes 
 Algorithm features
Algorithm name Basis of the algorithm Algorithmic process Decision variables Decision variable sets Generation of new solutions with the crossover operator Generation of new solutions with the mutation operator 
GA Based on evolution theory Random search and iteration Gene (represents numbers) Chromosome Exchanging several genes between two chromosomes randomly Generating new random values for numerical decision variables with random genes 
GP Based on evolution theory Random search and iteration Node (represents operators, functions, and numbers) Tree Exchanging several branches between some trees Randomly generating new operators, functions and coefficients in nodes 

A code in GP consists of mathematical operators (for example ) and various functions (for example ) that create random relations between independent variables (consisting of reservoir inflow and storage) and water releases (), which serve as the dependent variable in reservoir operating rules. The release value obtained by GP is evaluated with the objective function (Equation (5)) in each algorithmic iteration. The GP generates a rule curve relation in each iteration and the values and their associated values of the objective function are calculated and ranked. The releases are improved from one iteration to the next with mutation and crossover operators until reaching a stopping criterion. The GP outputs consist of the operating rule curve, the optimal reservoir releases, the values of the objective function at the optimal solution, reservoir storage, and spill volumes.

Case study

The Zarrineh-roud River, Iran, supplies a large part of Lake Urmia's water. Lake Urmia has experienced declining water levels and ecosystems degradation in the past years due to diversions of its native inflow by various regional water users or stakeholders. The Zarrineh-roud reservoir is the largest dam in the Urmia lake basin and a key regulator of inflow to Lake Zurmia. The Zarrineh-roud Sub-basin was herein selected as the case study with the purpose of determining optimal operating rules for the Zarrineh-roud reservoir employing bankruptcy rules to optimally allocate water to local stakeholders.

The Urmia lake basin is located between northern latitudes 37°4′ and 38°17′ and eastern longitude 45° to 46°. Figure 1 shows the Urmia lake basin with locations of reservoirs which have more than 10 (106 m3) capacity. Lake Urmia's average annual evaporation and lake precipitation equal 4,467.9 × 106 m3 and 1,381.2 × 106 m3, respectively (Abbaspour & Nazaridoust 2007). The average annual volume of groundwater feeding Lake Urmia at normal level equals 210.7 × 106 m3 (Abbaspour & Nazaridoust 2007). The total average annual volume of water (except for lake precipitation) that enters the lake equals 2,675.7 × 106 m3. This implies that the average annual water shortage to meet Lake Urmia's ecological requirements equals 411 × 106 m3 (= (4,467.9 − 1,381.2 − 2,675.7) × 106 m3). Lake inflows support its ecosystem, including the brine shrimp (Artemia species). A summary of data applied in the calculation of the ecological needs of Lake Urmia is listed in Table 2. There are 17 main rivers in the Lake Urmia basin that constitute its tributary water sources. The volumes of water to be provided by the 17 rivers to correct the Lake Urmia water deficit are listed in Table 3. This work investigates several options to reduce river water-use and to eliminate or reduce Lake Urmia's water deficit.

Table 2

Lake Urmia's ecological claims

Quality-quantity indicators for Lake Urmia associated with ecological requirementsValue (unit)
Tolerance threshold of salt of Lake Urmia 240 (mg/L) 
Lake water level at salt threshold 1,274.1 (m) 
Ecological water surface 4,652.2 (km2
The volume of annual evaporation from the ecological surface 4,467.9 (106 m3
The volume of annual rainfall on the ecological surface 1,381.2 (106 m3
The Lake Urmia ecological water storage 3,086 (106 m3
Quality-quantity indicators for Lake Urmia associated with ecological requirementsValue (unit)
Tolerance threshold of salt of Lake Urmia 240 (mg/L) 
Lake water level at salt threshold 1,274.1 (m) 
Ecological water surface 4,652.2 (km2
The volume of annual evaporation from the ecological surface 4,467.9 (106 m3
The volume of annual rainfall on the ecological surface 1,381.2 (106 m3
The Lake Urmia ecological water storage 3,086 (106 m3
Table 3

Share of water supply to Lake Urmia by main rivers needed to correct the lake's water deficit

River nameLong-term average discharge (106 m3)Share of the water for Lake Urmia (106 m3)
Aji-chay 428.7 35.14 
Azarshahr-chay 31.41 2.57 
Qala-chay 82.75 6.78 
Javan-chay 11.28 0.92 
Sufi-chay 125.41 10.28 
Marduq-chay 86.44 7.08 
Leylan-chay 62.01 5.08 
Zarrineh-roud 1,838.87 150.71 
Simiineh-roud 560.38 45.93 
Mahabad-chay 274.85 22.53 
Godar-chay 398.67 32.67 
Baranduz-chay 281.31 23.06 
Shahr-chay 169.57 13.9 
Roze-chay 44.37 3.64 
Nazlu-chay 417.89 34.35 
Zola-chay 164.38 13.47 
Sinikh-chay 27.82 2.28 
Sum 5,006.13 411 
River nameLong-term average discharge (106 m3)Share of the water for Lake Urmia (106 m3)
Aji-chay 428.7 35.14 
Azarshahr-chay 31.41 2.57 
Qala-chay 82.75 6.78 
Javan-chay 11.28 0.92 
Sufi-chay 125.41 10.28 
Marduq-chay 86.44 7.08 
Leylan-chay 62.01 5.08 
Zarrineh-roud 1,838.87 150.71 
Simiineh-roud 560.38 45.93 
Mahabad-chay 274.85 22.53 
Godar-chay 398.67 32.67 
Baranduz-chay 281.31 23.06 
Shahr-chay 169.57 13.9 
Roze-chay 44.37 3.64 
Nazlu-chay 417.89 34.35 
Zola-chay 164.38 13.47 
Sinikh-chay 27.82 2.28 
Sum 5,006.13 411 
Figure 1

Locations of small, medium, and large reservoirs in Urmia basin.

Figure 1

Locations of small, medium, and large reservoirs in Urmia basin.

The geometric characteristics of the Zarrineh-roud reservoir

The maximum and minimum storages of the Zarrineh-roud reservoir are equal to 762 (106 m3) and 107.6 (106 m3), respectively. The Zarrineh-roud reservoir was built to supply a variety of water uses including municipal, industrial, and agricultural uses. Table 4 lists the values of the monthly requirements for each of these stakeholders. Table 5 lists the values of average amounts of precipitation, evaporation and net precipitation at Zarrineh-roud reservoir.

Table 4

The amounts of different water claims downstream from the reservoir

Water demand (106 m3)Sep.Oct.Nov.Dec.Jan.Feb.Mar.Apr.MayJun.Jul.Aug.Sum
Agricultural 88 0.0 0.0 0.0 0.0 0.0 22 102 229 245 22 179 1,091 
Environmental 1.56 4.91 8.32 9.52 12.2 80.48 159.84 125.02 11.94 3.47 1.81 1.46 420.55 
Urban and industrial 14.2 12.2 12.7 13.27 13.01 13.41 11.1 13.01 13.4 14.3 14.5 13.2 158.03 
Lake Urmia 0.56 1.76 2.98 3.41 4.37 28.84 57.28 44.8 4.29 1.24 0.65 0.52 150.71 
Sum 104.32 18.87 24.00 26.20 29.59 122.73 250.22 284.83 258.62 263.74 242.97 194.19 1,820.29 
Water demand (106 m3)Sep.Oct.Nov.Dec.Jan.Feb.Mar.Apr.MayJun.Jul.Aug.Sum
Agricultural 88 0.0 0.0 0.0 0.0 0.0 22 102 229 245 22 179 1,091 
Environmental 1.56 4.91 8.32 9.52 12.2 80.48 159.84 125.02 11.94 3.47 1.81 1.46 420.55 
Urban and industrial 14.2 12.2 12.7 13.27 13.01 13.41 11.1 13.01 13.4 14.3 14.5 13.2 158.03 
Lake Urmia 0.56 1.76 2.98 3.41 4.37 28.84 57.28 44.8 4.29 1.24 0.65 0.52 150.71 
Sum 104.32 18.87 24.00 26.20 29.59 122.73 250.22 284.83 258.62 263.74 242.97 194.19 1,820.29 
Table 5

Monthly average values of precipitation, evaporation, and net evaporation at the Zarrineh-roud reservoir

 Sep.Oct.Nov.Dec.Jan.Feb.Mar.Apr.MayJun.Jul.Aug.Sum
Average precipitation (mm) 60.78 127.64 115.4 86.46 80.75 136.99 202.52 219.62 79.00 26.09 7.69 16.69 1,125.62 
Evaporation (mm) 175.60 84.20 59.0 42.90 39.90 55.10 77.00 132.60 172.60 207.80 217.80 195.10 1,431.8 
Net evaporation (mm) 86.82 −43.44 −56.40 −43.56 −40.85 −81.89 −125.52 −87.02 93.80 181.71 210.11 178.41 272.18 
 Sep.Oct.Nov.Dec.Jan.Feb.Mar.Apr.MayJun.Jul.Aug.Sum
Average precipitation (mm) 60.78 127.64 115.4 86.46 80.75 136.99 202.52 219.62 79.00 26.09 7.69 16.69 1,125.62 
Evaporation (mm) 175.60 84.20 59.0 42.90 39.90 55.10 77.00 132.60 172.60 207.80 217.80 195.10 1,431.8 
Net evaporation (mm) 86.82 −43.44 −56.40 −43.56 −40.85 −81.89 −125.52 −87.02 93.80 181.71 210.11 178.41 272.18 

RESULTS

The results of the long-term operation of Zarrineh-roud reservoir

The optimized long-term operation of Zarrineh-roud reservoir was calculated with NLP for a 57-year period. The most recent 5 years were selected to test the calculated rule curves. The values of monthly water demand equaled the sum of the four downstream stakeholders’ claims. The lowest value of total water demand occurs in October and equals 18.87 × 106m3. The maximum value of water demand occurs in July and equals 74.263 × 106m3. The objective function of the optimization model (see Equation (5)) was calculated to be 40.497.

Figure 2 shows the optimal monthly reservoir releases calculated with NLP and water demand for a 684-month long period. It is seen in Figure 2 that there are large deficits of water supply from period 564 through 600. The values of the performance criteria are listed in Table 6. Table 6 shows that NLP achieved low resiliency and time-based reliability and high value of volumetric reliability.

Table 6

Values of the performance criteria for reservoir operation optimized with NLP

Performance criteria
Sum of relative deficits (Equation (5) the objective function)ResiliencyVolumetric reliabilityTime-based reliability
40.497 0.134 0.842 0.467 
Performance criteria
Sum of relative deficits (Equation (5) the objective function)ResiliencyVolumetric reliabilityTime-based reliability
40.497 0.134 0.842 0.467 
Figure 2

Values of total water demand and release during the operational period, (a) period 1 to 342 and (b) period 343 to 684. Change period to 343–684 in the horizontal axis.

Figure 2

Values of total water demand and release during the operational period, (a) period 1 to 342 and (b) period 343 to 684. Change period to 343–684 in the horizontal axis.

Water allocation to downstream stakeholders by means of P, AP, CEA, and CEL

Water allocations are released to downstream users during each operational period following the calculation of optimal releases. Each water receiver or beneficiary is considered as a player (stakeholder) in the four bankruptcy methods herein considered (P, AP, CEA, and CEL). Figure 3 depicts the water allocations assigned to each of the stakeholders with the four implemented methods. It is observed in Figure 3(a) that the CEL and CEA methods assigned the lowest and highest water allocations to the urban sectors, respectively. The CEL method assigned zero water to the urban sector in some periods. The P method also assigned very small amounts of water in dry periods. This is so because in dry periods the water deficit is high and water allocation to the urban sector is drastically reduced given that the P method divides the water deficit equally among stakeholders. The AP method produced low water allocations in dry periods similar to those calculated with the CEA method. The provision of urban water is a high priority, and applying high water deficits to this sector has detrimental economic and social consequences. Therefore, the CEL, AP, and P methods are not well suited to apportion water to the urban sector, which exhibits relatively low water claims, when compared with the agricultural sector.

Figure 3

Water allocations by the four bankruptcy methods to the (a) urban sector, (b) environmental sector, (c) Lake Urmia, and (d) agricultural sector.

Figure 3

Water allocations by the four bankruptcy methods to the (a) urban sector, (b) environmental sector, (c) Lake Urmia, and (d) agricultural sector.

It is seen in Figure 3(b) that the CEA and AP approaches assigned relatively large allocations to the environmental sector. The CEL method allocated the lowest water volume to the environmental sector. The CEA, AP, P and CEL methods calculated the largest, second largest, and third largest water allocations to the environmental sector, respectively (see Figure 3(c)). The CEL assigned the largest water allocation to the agricultural sector as shown in Figure 3(d), followed by the CEA, P, and AP methods which provided the second, third, and fourth largest water allocations to the agricultural sector, respectively. The CEL method, therefore, performed well in supplying stakeholders with large water claims. But the water allocations by the CEL method could have severely detrimental effects on other stakeholders.

Table 7 lists the values of the calculated four performance criteria herein applied to water allocations to stakeholders with bankruptcy methods. Recall that the performance criteria are the volumetric reliability, time-based reliability, resiliency, and vulnerability. It is evident in Table 7 that, concerning the water allocation to the urban sector, the CEA had the best performance because its values of the volumetric reliability, time-based reliability, and resiliency are higher (better) than those of the other methods, and the amount of vulnerability is the lowest (better). The AP method had the second best performance in the urban sector after that of the CEA method. Also, the CEL method was not successful in supplying this sector's water claim. The AP method performed best in providing environmental water needs having the best values of volumetric reliability and vulnerability. The CEA method had the best values of time-based reliability and resiliency in supplying the environmental needs. The best values of the performance criteria in supplying the lake's claim belong to the CEA, AP and P methods, respectively, in decreasing order of performance quality. The CEL method has acceptable performance in supplying the agriculture sector. Yet, in some cases it incurred significant failures, showing a relatively high vulnerability index. Overall, the CEA method performed best with water supply to the agricultural sector. The aforementioned index named the ‘Bankruptcy Allocation Stability Index’ (BASI) is commonly employed in bankruptcy games for the purposes of comparing and choosing the best method of conflict resolution. A low value of BASI signifies that an implemented allocation method has acceptable performance in the allocation problem in question. Table 8 lists the values of BASI for the four bankruptcy methods implemented in this work. The results shown in Table 8 indicate that the P method has the best acceptability having the lowest (best) BASI among the bankruptcy methods. It is followed in decreasing order of acceptability by the CEA, AP and CEL methods. The P method allocates available resources by considering the merits of the water claims by stakeholders relative to the available water, and assigns water shortage (or deficit) ratios that are consequent with those merits. The resulting water allocation is likely to be acceptable to stakeholders for the same reason that a low value of BASI has a likely high acceptability among stakeholders. Applying equal ratios of water deficit to the allocations to stakeholders is sound in situations when the various claims have very similar magnitudes. However, an allocation method chosen based on the BASI would be questionable when the water claims of stakeholders are significantly different to each other, such as is the case with the claims of the stakeholders located downstream of the case-study reservoir. These considerations make evident that it is necessary to evaluate an allocation method's BASI and the four performance criteria to make a more suitable selection method for water allocation. The results listed in Tables 7 and 8 establish that the CEA method achieved the best water allocations judged by the performance criteria and the BASI.

Table 7

Comparison of the performance criteria of the bankruptcy methods with respect to the water allocations to various stakeholders

Performance criteria
SectorVolumetric reliability
Time-based reliability
Resiliency
Vulnerability
StakeholderPAPCEACELPAPCEACELPAPCEACELPAPCEACEL
Urban-industrial 0.82 0.94 0.96 0.45 0.47 0.36 0.67 0.19 0.19 0.25 0.38 0.052 14.16 2.26 1.49 63.02 
Environmental 0.81 0.92 0.9 0.52 0.53 0.53 0.76 0.19 0.16 0.25 0.28 0.052 14.57 4.63 8.13 53.68 
Lake Urmia 0.82 0.87 0.96 0.45 0.56 0.43 0.94 0.19 0.21 0.25 0.57 0.052 14.07 8.6 3.61 59.87 
Agricultural 0.77 0.73 0.77 0.77 0.48 0.46 0.49 0.46 0.16 0.15 0.16 0.15 16.4 21.93 16.59 18.35 
Performance criteria
SectorVolumetric reliability
Time-based reliability
Resiliency
Vulnerability
StakeholderPAPCEACELPAPCEACELPAPCEACELPAPCEACEL
Urban-industrial 0.82 0.94 0.96 0.45 0.47 0.36 0.67 0.19 0.19 0.25 0.38 0.052 14.16 2.26 1.49 63.02 
Environmental 0.81 0.92 0.9 0.52 0.53 0.53 0.76 0.19 0.16 0.25 0.28 0.052 14.57 4.63 8.13 53.68 
Lake Urmia 0.82 0.87 0.96 0.45 0.56 0.43 0.94 0.19 0.21 0.25 0.57 0.052 14.07 8.6 3.61 59.87 
Agricultural 0.77 0.73 0.77 0.77 0.48 0.46 0.49 0.46 0.16 0.15 0.16 0.15 16.4 21.93 16.59 18.35 
Table 8

The values of BASI for all water allocation method

PAPCEACEL
BASI 0.075 0.57 0.368 0.67 
PAPCEACEL
BASI 0.075 0.57 0.368 0.67 

Figures 47 depict the water allocations to the urban, environmental, lake, and agricultural stakeholders, respectively, calculated with the CEA method. These figures confirm the CEA method's suitable performance in supplying the stakeholders with low water claims. This does not mean that that 100% of their water claims are supplied at all times. Rather, relatively low water deficits were applied to these stakeholders in dry periods. The CEA might seem unfair in comparison with the P method at first sight. Yet, further analysis reveals that the high water use by the agricultural sector has a destabilizing effect on water claims. For this reason, the agricultural water claim is managed such that, in dry periods, its water allocation is allocated rationally. This is accomplished by applying low water deficits to stakeholders with low claims (such as the urban sector) during times of heightened scarcity.

Figure 4

Water allocations to the urban sector by the CEA method, (a) period 1 to 342 and (b) period 343 to 684.

Figure 4

Water allocations to the urban sector by the CEA method, (a) period 1 to 342 and (b) period 343 to 684.

Figure 5

Water allocations to the environmental sector by the CEA method, (a) period 1 to 342 and (b) period 343 to 684.

Figure 5

Water allocations to the environmental sector by the CEA method, (a) period 1 to 342 and (b) period 343 to 684.

Figure 6

Water allocations to Lake Urmia by the CEA method, (a) period 1 to 342 and (b) period 343 to 684.

Figure 6

Water allocations to Lake Urmia by the CEA method, (a) period 1 to 342 and (b) period 343 to 684.

Figure 7

Water allocations to the agricultural sector by the CEA method, (a) period 1 to 342 and (b) period 343 to 684.

Figure 7

Water allocations to the agricultural sector by the CEA method, (a) period 1 to 342 and (b) period 343 to 684.

Calculation of monthly allocation rule-curves with GP

Real-time rule curves describing the monthly water allocations to the four stakeholders (urban, environmental, Lake Urmia, and agricultural) were calculated with GP, for a total of 48 (= 4 stakeholders times 12 months annually) real-time operating rules of the form given by Equation (54):  
formula
(54)
in which i= counter for stakeholders (i = 1, 2, 3, 4); k= counter of the periods (months) (k = 1, 2, 3, 4)); = allocation volume to the ith stakeholder in the kth period and = the function of the real-time water allocation equation for the ith stakeholder in the kth period (i.e. the rule curve i, k).

The GP was calibrated with the first 40 years of the data for the Zarrineh-roud reservoir (70% of the data) and tested with the last 17 years of data (30% of the data). The allocations to the stakeholders were written as functions of reservoir inflow during any period and of the reservoir storage at the beginning of the same period. These functions constitute the reservoir operation rules. Functions and operators were applied in GP to calculate the monthly rule curves. The objective function was the minimization of the root mean square error (RMSE) between calculated real-time and the long-term allocation volumes. The number of trees and chromosomes implemented in GP were 3 and 30, respectively, and the number of iterations were 200,000 in each of the five runs of GP. Three-dimensional (3D) charts displaying the monthly rule curves and the objective function (RMSE) in March are portrayed in Figure 8 as an example of the results concerning the calculated rule curves.

Figure 8

3D charts depicting the calculated monthly rule-curves and the value of the objective function (RMSE) in March, (a) urban sector, (b) environmental sector, (c) the Lake Urmia sector, and (d) the agricultural sector.

Figure 8

3D charts depicting the calculated monthly rule-curves and the value of the objective function (RMSE) in March, (a) urban sector, (b) environmental sector, (c) the Lake Urmia sector, and (d) the agricultural sector.

Evaluation of real-time water allocations

The real-time rule curves calculated with GP were tested with 5 years of monthly time series of reservoir inflow by evaluating its allocation of the Zarrineh-roud reservoir's water resources to downstream stakeholders. The GP-calculated rule curves were compared with the optimal releases and water allocations calculated with NL and CEA (or NLP-CEA). The NLP-CEA method gives the global optimal solution. The purpose of the comparison to test how well the real-time operation rules calculated with GP approximate the long-term water allocations obtained with NLP-CEA. Figure 9 shows the allocation to stakeholders by GP and NLP-CEA. Figure 9 demonstrates that GP approximates the NLP-CEA water allocations to the stakeholders (urban, Lake Urmia, environmental, and agricultural) with negligible errors. Of special significance in these figures is the difference between the water allocation values by the two methods in the periods when the claims are largest. Claims are largest during dry periods when the water system faces water shortages. Evidently, GP calculates near optimal water allocation during dry periods, thus making it a useful optimization method for real-time reservoir operation. Table 9 lists several correlation indexes between NLP-CEA's and GP's results. Table 9's results prove that the performance of GP in estimating the water allocations is acceptable. Table 10 lists the performance criteria of water allocations to stakeholders by GP and NLP-CEA. It is evident from Table 10 that the values of time-based reliability and resiliency are lower than those volumetric reliabilities. This means that both methods minimized water shortages in each period to prevent large shortages overall. The GP's performance criteria are quite close to those of the NLP-CEA method, proving its capacity to determine sound real-time water allocation to stakeholders.

Table 9

Various correlation indexes between GP and NLP-CEA

Stakeholder
CriteriaUrban-industrialEnvironmentalLake UrmiaAgricultural
RMSE 0.3685 7.36 3.18 15.96 
MAE 0.287 3.88 1.54 9.63 
0.93 0.95 0.96 0.89 
R2 0.865 0.9 0.92 0.79 
Stakeholder
CriteriaUrban-industrialEnvironmentalLake UrmiaAgricultural
RMSE 0.3685 7.36 3.18 15.96 
MAE 0.287 3.88 1.54 9.63 
0.93 0.95 0.96 0.89 
R2 0.865 0.9 0.92 0.79 

MAE, mean absolute error; RMSE, root mean square error.

Table 10

Comparison of GP and NLP-CEA with the four performance criteria

Performance criteria
Time-based reliability
Volumetric reliability
Resiliency
Vulnerability
StakeholderGPNLP-CEAGPNLP-CEAGPNLP-CEAGPNLP-CEA
Urban-industrial 0.0 0.0 0.93 0.97 0.0 0.0 1.13 0.027 
Environmental 0.0 0.0 0.603 0.72 0.0 0.0 13.17 7.88 
Lake Urmia 0.0 0.0 0.704 0.78 0.0 0.0 8.41 4.82 
Agricultural 0.0 0.262 0.48 0.685 0.0 0.143 36.47 11.09 
Performance criteria
Time-based reliability
Volumetric reliability
Resiliency
Vulnerability
StakeholderGPNLP-CEAGPNLP-CEAGPNLP-CEAGPNLP-CEA
Urban-industrial 0.0 0.0 0.93 0.97 0.0 0.0 1.13 0.027 
Environmental 0.0 0.0 0.603 0.72 0.0 0.0 13.17 7.88 
Lake Urmia 0.0 0.0 0.704 0.78 0.0 0.0 8.41 4.82 
Agricultural 0.0 0.262 0.48 0.685 0.0 0.143 36.47 11.09 
Figure 9

Water allocations obtained by NLP-CEA and the calculated real-time allocation rules to the (a) urban sector, (b) environmental sector, (c) Lake Urmia, and (d) agricultural sector.

Figure 9

Water allocations obtained by NLP-CEA and the calculated real-time allocation rules to the (a) urban sector, (b) environmental sector, (c) Lake Urmia, and (d) agricultural sector.

CONCLUDING REMARKS

Long-term optimal Zarrineh-roud reservoir releases were calculated with NLP. Thereafter, the four bankruptcy methods, P, AP, CEA and CEL, were applied to optimize water allocations among the four stakeholders receiving water from the reservoir. The CEA method exhibited the best water allocation performance based on time-based reliability, volumetric reliability, resiliency, and vulnerability criteria. Moreover, the acceptability of these methods was analyzed with the BASI stability index. The P method has the best acceptability among the four bankruptcy methods, followed by the CEA method. Overall, the CEA method was chosen as the best method of water allocations to downstream-reservoir stakeholders (urban, environmental, Lake Urmia, and agricultural).

The CEA method was applied in real-time with GP. GP calculated optimal monthly water allocations to the four stakeholders, thus yielding 48 optimized real-time rule curves. The GP-calculated rule curves were compared with the optimal water allocations from the NLP-CEA method over a 5-year period. The comparison results demonstrated that the GP water allocations approximate very closely the optimal allocations.

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