In this article, a methodology is presented for water allocation to demands in new inter-basin water transfer projects. In this methodology, by proposing some criteria such as universality, equity, enforceability, adaptivity, and exclusivity, a new scheme is developed in order to determine the share of water users. For taking the available water uncertainty into account, the concept of conditional value at risk (CVaR) is used, in which the goal is to minimize an indicator of loss in low flow conditions. Also, in order to estimate the relative weights of criteria, a novel method is introduced. This method is founded on two essential information aggregation operators, namely, hybrid weighted averaging (HWA) and ordered weighted averaging (OWA). The significant property of this method is that the weighting process is done by considering the relative weights of experts and the viewpoint of the analyzer jointly. The results of the CVaR-based model are compared with those obtained from a long-term water allocation optimization model with monthly time steps. The performance of the proposed methodology is evaluated using available data from the Solakan–Rafsanjan water transfer project.

  • Proposing some criteria for estimating water rights in inter-basin water allocation.

  • Proposing a novel methodology to assess the relative weights of water users.

  • Developing a CVaR-based framework for water rights allocation under uncertainties.

  • Analyzing the CVaR-based model's sensitivity for different confidence levels.

  • Developing long-term optimization model to draw comparison against CVaR-based model.

Historically, water has been an essential foundation for population growth and human well-being. Nevertheless, the water resources have not been spread equally in a given area, and it begs for an extensive solution to supply water demands. Water supply alternatives have arisen to address this issue; one of the practices is inter-basin water transfer (Abed-Elmdoust & Kerachian 2012; Jafarzadegan et al. 2013). In inter-basin water transfer systems, water is allocated from a basin where water is abundantly available to a basin facing severe water scarcity (Jafarzadegan et al. 2014). After implementing a water transfer system, there has to be a detailed plan for water allocation, as they involve different stakeholders with conflicting interests (Sadegh et al. 2010). Some research into the inter-basin water allocation systems include Ballestero (2004) who proposed a novel decision stochastic approach to study the commercial facets of inter-basin water transfer systems, namely, the quantity of transferable water and the price. He argued that there has to be an agreement between different agencies controling the water resources, and so a numerical method was implemented. He concluded that the solution to this problem has to be the cross point of demand and supply curves. He chose a water transfer system in Spain as his case study, where the donor basin is located in central Spain where water is abundantly available, and the receiver basin is located in a more arid area of south-eastern Spain.

Ghassemi & White (2007, p. 59) studied various case studies in countries including the USA and Canada. They gathered and studied different variables in a given country like Australia and classified them into a number of major classes, namely, population, climatic variables, etc. They studied the water demand of each groundwater province of the country on the basis of their water use type. They highlighted the importance of the National Water Initiative in which a visionary master plan for long-term conservation of land and water resources of Australia is proposed.

The ultimate objective of water transfer studies of this kind is to optimize the water resources allocation in a given basin. Numerous studies have arisen to present a robust and novel model to address this matter (Karamouz et al. 2010). Nikoo et al. (2012) put forward a novel methodology using the concept of interval optimization and game theory. The optimal operation of an inter-basin water transfer system was arrived at using the efficiency, equity, and sustainability criteria. Jafarzadegan et al. (2013) proposed a fuzzy variable least core model to find the optimal water allocation. In their proposed methodology, an integrated stochastic dynamic programming model was used in order to arrive at the water share of each player and economic allocation policy. Jafarzadegan et al. (2014) developed a stochastic model to arrive at the optimal operation policy using the concept of conjunctive use of surface water resources in the donor basin and groundwater resources in the receiver basin. Liu et al. (2019) and Oussama et al. (2019) used different methods of the optimization models, such as particle swarm optimization model and genetic algorithm (GA), to arrive at the optimal operation policies in multi-reservoir systems. Sinha et al. (2020) have drawn an experimental methodology to evaluate the inter-basin water transfer schemes better using a multidisciplinary and transparent methodology.

In water allocation systems, there is always a level of uncertainty present in the process, as most of the elements concerning this process do not have a deterministic nature and behavior. Subsequently, different levels of risks are associated with water resources management (Yamout et al. 2007). There have been many efforts to quantify these risks, starting from the mathematical expected value (MEV) concept to the more modern concepts like central moments, and finally the concept of value at risk (VaR) (Cheng et al. 2003; Sarykalin et al. 2008). To overcome the deficiencies of VaR to incorporate risk in decision-making plans, the concepts of VaR was modified (Gupton et al. 1997). Nevertheless, the VaR had too many shortcomings, like the fact that the reduction of the risk data set into a single number might cause misleading interpretations of the outcomes, and so a new series of efforts started to address these concerns (Yamout et al. 2007). The concept of conditional value at risk (CVaR) was developed and, in time, gradually replaced VaR, as it was a better representative of the risk and its corresponding probability distribution. Researches regarding the application of the CVaR concept in different fields include Rockafellar & Uryasev (2000), who defined a new approach in order to optimize a portfolio of financial instruments to mitigate risk. They argued that portfolios with low CVaR are expected to have low VaR. They proposed a model to optimize the CVaR risk by using linear programming. For the normal distribution, the results of the VaR, the CVaR, and the variance minimization were compared. They argued that the VaR, CVaR, and variance minimization methods might provide similar results.

Webby et al. (2006), who used the concept of CVaR in order to minimize the loss regarding the decrease in the water level of Burley Griffin Lake, Australia. The constraints of this model were primarily supplying environmental flow and releasing the additional flow. They defined the loss function based on recreational use and landscape restoration as a function of the water level in the lake. They also took the precipitation as a random variable and investigated the distribution function of the precipitation. In their case study, CVaR was implemented as a decision provision for strategic planning and operative problems.

Piantadosi et al. (2008) presented a novel approach to determine policy in order to manage urban storm-water. They investigated this approach using the concept of CVaR. They implemented this method in a two-reservoir dam and assumed that the inflow was random, and the demand was constant. Stochastic dynamic programming was used in order to find an optimal policy that minimizes CVaR and a pumping policy that maximizes expected monetary value. After comparing these two policies, they concluded that they are not identical.

Yum et al. (2009) investigated the capability of CVaR as a criterion to study risk in a general decision-making and risk evaluation framework. They argued that VaR could be implemented in order to formulate the risk-based objective functions. In their article, with the assumption that the inflow to Cudfegong Dam, Australia, was indeterministic, and the values of VaR and CVaR are a risk function of lacking water supply. The minimized result of VaR and CVaR in a target period of 12 years was compared with the results of minimizing the MEV of loss for lacking water supply. They found that the CVaR can be taken as a more conservative criterion for loss.

Shao et al. (2011) developed a risk-based model to investigate the inherent uncertainties (CvaR-based inexact two-stage stochastic programming, CITSP). Their model was formulated by integrating a CVaR constraint into an inexact two-stage stochastic programming framework (inexact two-stage stochastic programming, ITSP). They applied the former model, CITSP, to a water allocation scenario with one reservoir and three water users. Their results illustrated that the CITSP would outperform the ITSP model, as it was more useful in order to show the attitude of the decision-maker to risk-taking condition. Consequently, CITSP would assist in finding a cost-effective management strategy.

Hu et al. (2016) proposed a multi-objective model for water allocation equality in order to alleviate the problems from efficiency risk. They used the Gini coefficient to optimize water allocation. They also incorporated the CVaR concept into the model constraints to mitigate the loss risk of the fluctuations in water availability. In their case study, they reached the conclusion that the model seems to be practical and rational, as it allows the authority to regulate water allocation strategies for a given river basin.

Tayebikhorami et al. (2020) studied the treated wastewater as an alternative water resource for non-potable uses like irrigation and industrial cooling. A novel CVaR-based multi-objective model was developed in order to take stakeholders' conflicting objectives into account and study potential resource losses related to inherent uncertainty present in the project. Their results showed the provision of a reasonable sharing policy, which would ultimately resolve the stakeholders' conflicts. They also concluded that during the optimization process, the outputs are sensitive to the confidence level.

In this article, after defining some main criteria for water rights allocation in new inter-basin water transfer systems, a CVaR-based methodology is developed for estimating the water rights under available water uncertainty. A loss function is defined regarding each criterion. In order to combine the loss functions, the hybrid weighted averaging (HWA) and ordered weighted averaging (OWA) operators are used. The target is to optimize the water rights by minimizing the combined CVaR-based loss function. At last, the results are compared with those obtained from a long-term water right allocation optimization model with monthly time steps. This is done to draw a detailed comparison between the proposed CVaR-based model and a classic optimization model, which implicitly incorporates the uncertainty of available water resources in the water donor basin.

The Karoon River is among the longest rivers in Iran. Karoon and its neighbor Dez River makes one-fifth of the whole surface water resources in Iran. The area of the Karoon Basin is approximately equal to 67,000 square kilometers and span over four provinces of Iran (Mahjouri & Ardestani 2011). The average annual precipitation in this basin varies between 150 mm and 1,800 mm. The average annual temperature is 7.5 degrees Celsius in the mountainous regions and 25 degrees Celsius in the southern parts of the basin (Karamouz et al. 2010; Shojaei et al. 2015). The long-term average annual flow (AAF) is about 11.9 billion cubic meters per year, but the annual water demand in this basin is higher than the inflow, an important reason that shows the difficulties in water allocation in this region. Many studies have been conducted on water quantity and quality assessment of this river (e.g. Kerachian & Karamouz 2005; Mahjouri & Ardestani 2010, 2011; Karamouz et al. 2008). One of the many tributaries of the great Karoon is the Solakan River, with a long-term average flow equal to 9.7 cubic meters per second and the annual inflow ranging from 64.7 to 302.9 million cubic meters per year. By constructing a new reservoir on this river, the Iran Water Resources Management Company is planning to allocate water to the Rafsanjan Plain. The area of Rafsanjan Plain is about 16,096 square kilometers. Due to low precipitation, most of the rivers in this region are seasonal. Another problem of this region is that due to over-exploitation, the level of groundwater is low, and it has brought many problems like extreme settlements and an extreme increase in the salinity of groundwater (Sadegh et al. 2010). A vast area of the Rafsanjan Plain is used to cultivate pistachio, and due to other issues discussed earlier, the responsible organizations planned to transfer water from one of the Karoon's tributaries to the Rafsanjan Plain in order to address these concerns. The proposed project transfers 250 million cubic meters per year to the Rafsanjan Basin. In this article, in addition to the environmental water demand downstream of the dam, four other water users are taken into account. Water users in this case study are presented in Table 1.

Table 1

Water users in the water donor and receiving basins

No.Water users
Khuzestan modern agro-industrial sector 
Khuzestan old agro-industrial sector 
Khuzestan local agricultural sector 
Rafsanjan agricultural sector 
Environmental sector 
No.Water users
Khuzestan modern agro-industrial sector 
Khuzestan old agro-industrial sector 
Khuzestan local agricultural sector 
Rafsanjan agricultural sector 
Environmental sector 

This inter-basin water transfer project can be studied in different aspects. In this article, the developed framework would be applied to this water transfer project. The location and a schematic configuration of the inter-basin water transfer project are shown in Figures 1 and 2. More details on this case study and the proposed water transfer project can be found in Sadegh et al. (2010) and Mahjouri & Ardestani (2011).

Figure 1

The geographical location of the inter-basin water transfer project.

Figure 1

The geographical location of the inter-basin water transfer project.

Close modal
Figure 2

A schematic illustration of the demands in the inter-basin water transfer project.

Figure 2

A schematic illustration of the demands in the inter-basin water transfer project.

Close modal

In this section, different parts of the methodology will be discussed. The proposed methodology is illustrated schematically in Figure 3.

Figure 3

The flow chart of the proposed methodology.

Figure 3

The flow chart of the proposed methodology.

Close modal

Determining criteria for water allocation

The optimal water allocation has always been an essential issue in water resources development plans. In this paper, several main criteria, namely universality, equity, enforceability, adaptivity, exclusivity, detached from the land title and use restriction, and environmental water right, are proposed for water rights allocation. These criteria and their sub-criteria are explained in the following sections.

Universality

Water resources are physically connected. Due to the interactions between different water resources, water allocation of a particular source might have positive or negative effects on rest (The Productivity Commission of Australia 2003). If these sources are not considered as a whole, the ones left outside the water rights system might become exhausted by overuse (The Productivity Commission of Australia 2003). Universality is an indicator that considers these interactions between various water resources. A water right system can be deemed as universal when the rights of access are protected. Formulation of universality can differ based on the characteristics of the local water resources, and so each case study requires a specific formulation. For a more mathematical approach, the universality is defined as the ratio of the water allocated to a user to the total resources. The following is the general formulation of universality (Sahebzadeh et al. 2013):
formula
(1)
where Rt,i is the allocated water right to the water user i in month t, Dt,i is the water demand of water user i in month t, and n is the number of months in the optimization model.
In our case study, because of the undesirable quality of groundwater in the donor basin (Khuzestan), it is not used in the agriculture industry. However, in the receiver basin (Rafsanjan), the interaction of local groundwater and the transferred water has to be taken into account. Due to the history of over-exploitation of groundwater resources in the receiver basin, the water elevation is lower than the acceptable values. According to all these explanations, the proper formulation of the universality in this case study would be the following equation:
formula
(2)
where Rt,4 is the allocated water to the 4th water user, Dt,4 water demand of 4th water user in month t, and n is the number of months in the optimization model.

Equity

Equity can be defined as a fair allocation of resources among all members of a system. For instance, providing sufficient water to sustain human rights for safety and well-being can be an aspect of equity. Based on the fact that equity is correlated with survival and level of welfare, fair water allocation has been a central issue among researchers, and it has led to introducing various indicators demonstrating the degree of equity present in a system (Zheng et al. 2011). In this study, three sub-criteria are defined in order to evaluate the equity in water allocation systems: water allocation proportional to the relative power of water users, water allocation proportional to deprivement of area, and water allocation proportional to demand type of water users.

(1) Water allocation proportional to the relative power of water users

Social scientists have proposed the operational definition of power as the capability of one party to influence another party in the desired direction. This influence can be considered as the changes in the mental and practical states of a party (Abed-Elmdoust & Kerachian 2014). In the water allocation studies, because of the inherent characteristics of the water resources, it is difficult to quantify the different aspects of water users and their powers. In order to solve these problems, many scientists have tried to propose some indicators that can quantify the power of each water user in inter-basin water transfer. Abed-elmdoust & Kerachian (2014) investigated political and social aspects of power, and they presented seven sub-criteria for evaluating the power of water users in an inter-basin water transfer system. These criteria are presented in Figure 4.

Figure 4

Different sub-criteria for evaluating the relative power of water users in an inter-basin water transfer system (Abed-Elmdoust & Kerachian 2014).

Figure 4

Different sub-criteria for evaluating the relative power of water users in an inter-basin water transfer system (Abed-Elmdoust & Kerachian 2014).

Close modal

These sub-criteria mentioned are discussed below (Abed-Elmdoust & Kerachian 2014):

  • (a)

    Economic power

In order to assess the economic power criterion, the following sub-criteria should be considered:

Regional gross income (RGI): RGI can be regarded as the value of the total incomes of the residents of a region both from inside and outside the area. RGI is positively inter-connected with the prosperity of an area, and so a region with high RGI values are considered to be more powerful.

The productivity of waters: The productivity of water can be quantified as gained profit per unit of water (the rate of water efficiency).

Existence of alternative water resources: Regions that have alternative water resources to rely on and have less dependency on the disputed resources are more powerful in negotiations.

Self-sufficiency: The more various the industrial and agricultural products are in a given region, the more self-governing the region becomes, and consequently, the more powerful it will be.

  • (b)

    Having agricultural and industrial infrastructures

An essential indicator of estimating the power of a region can be the existence of infrastructures and having a high potential for agricultural production. The more there is structure and potential for agricultural and industrial growth, the more it is likely that the water user is powerful.

  • (c)

    Receiving political support of government and parliament

In water transfer projects, another important factor in evaluating the power of a water user is geopolitical factors, ethnic, and demographic profiles in the regions. A policy that has an economic justification might be passed or rejected by officials based on these essential factors.

  • (d)

    Preference-based power

Preference-based power can be defined in order to evaluate the role and the effect of a water user in cooperation with other water users. A water user can be deemed as vital when the presence of a water user in a coalition transforms it from a losing coalition to a winning one. By defining the winning and losing coalition and assessing the vitality of the presence of a given water user, this power criterion is studied.

Finally, the relative power of each water users is found, and the criteria of the water allocation proportional with the relative power of water users can be computed using the following equations:
formula
(3)
formula
(4)
where Si,t is the value of expected water share of water user i in month t with respect to the relative power of that water user, Di,t is the water demand of water use i in month t, Ri,t is the allocated water to water user i in month t, ubi,t and lbi,t is upper and lower bound of the water share for water user i in month t, respectively, and m and n are numbers of water users and number of months in the planning period, respectively.
(2) Water allocation proportional to the deprivement of area

In many cases, water right allocation is done without any consideration of other criteria and only in order to reduce the poverty of a region or water users. The degree to which this criterion is taken into account is dependent on the region. Since there is no deprived area in this case study, this sub-criterion was not taken into account in this work.

(3) Water allocation proportional to demand type of water users

The type of water use is essential in allocation problems. For instance, potable water user is always prioritized over the agricultural and industrial water users. The degree to which this criterion is taken into account is dependent on the region. Since all water users in the donor and receiver basins are agricultural type, this sub-criterion was not taken into account in this work.

Enforceability

This criterion is defined in order to assess the degree to which the water allocation policies are acceptable to each water user. For instance, the closer the allocated water to a water user and its water demand, the more likely it is for the allocation policy to be accepted by the water users. This criterion can be formulated as the equation below:
formula
(5)
where Ri,t is the allocated water to the water user i in month t, Di,t is the water demand of the water user i in month t, m is the number of water users, and n is the number of months in the optimization model. In this case study, the equation below is the proper formulation of this criterion:
formula
(6)

Adaptivity

In emergency situations, when an unexpected event arises, i.e. drought, the governmental body can activate a reserve right, and so it would be able to satisfy the newly arisen situation (The Productivity Commission of Australia 2003). Adaptivity is defined in order to investigate the capability for altering a water rights system once the surrounding factors change, and it depicts the flexibility in water allocation plans (Zheng et al. 2011). Once the conditions for allocating a new reserve right are met, the government, using adaptive management, can gradually implement significant changes to allocation, and so the costs of the adjustment are reduced substantially. In this case study, adaptively is defined as:
formula
(7)
where V is annual water volume considered for the dam operating organization, which has to be left at the end of the water year in the reservoir, I is the yearly inflow to the reservoir, S1 is the stored volume in the dam at the beginning of the water year, and y is the number of years in the optimization model.

Exclusivity

Water rights can be defined as exclusive when ensuring the accrual of the costs and benefits of water rights to the water users. If the holder of the water share bear just a fraction of the costs of its action, or doesn't have the chance of enjoying the benefit from its share, the water would not be maximally used. How this criterion is taken into account is dependent on the characteristics of the region under study. In this study, in the donor basin, the water shares are exclusive, and all of the operation cost is accrued to the water user. In the receiver basin, with the assumption that there is a water transfer system, operation cost is paid by the water user. Therefore, it can be concluded that water shares are exclusive. To sum up, in this case study, this criterion does not affect the allocation.

Detached from the land title and use restrictions

To enable water users to trade and enter the water markets more freely, it is necessary to allocate the water rights detached from land and use. Tying water rights to land and use diminishes the chance of mediators to enter the market as mentioned earlier and, above all, it reduces flexibility and efficiency drastically. If the water right is not allocated detached from land or use, water will only be used in a specific location and for a particular set of goals. In this case study, water is transferred for the sole purpose of agricultural growth in the receiver zone. Furthermore, the water right in the receiver zone is attached to use, and water users in the area are not allowed to sell and trade their water rights. In the donor zone, water users are not restricted and they can freely trade their shares. In conclusion, for this criterion, it is preferred to use water in the donor zone. The equation below is the proper formulation of this criterion:
formula
(8)
where Rt,4 is the allocated water to the water user 4 in month t, Dt,4 water demand of water user 4 in month t, DLUR is the detached from the land title and use restrictions subcriteria, and n is the number of months in the optimization model.

Environmental water right (EWR)

Considering water rights for the environment in water allocation is a crucial matter for preserving the environment and wildlife. Various hydrological, hydraulic, and ecological methods are available for estimating the environmental flow. For instance, in the Tennant hydrological method (Tennant 1976), the environmental flow is computed as a percentage of the annual natural inflow of the river in a given location. For ensuring that the flow is natural, the anthropogenic share of the flow should be removed. In this model, ten percent of annual average flow (AAF) should be required for short-term survival of fishes, thirty percent of AAF for relatively good survival of fishes, and sixty percent for favorable life. In this paper, the following objective function is used for supplying environmental water demands:
formula
(9)
formula
(10)
where AAF is annual average flow, AF is annual flow, TE1t and TE2t are the environmental water share in month t based on the Tennant method, R5,t are environmental water demand in month t and allocated water to the environment in month t respectively, and n is the number of months during the planning horizon.

Determining the relative weights of criteria

In this case study, in order to compute the relative weights of each criterion, four water resources experts were asked to complete a questionnaire. This questionnaire was aimed at asking about different viewpoints of experts using a pairwise comparison matrix. In the next step, it is required to integrate these viewpoints. In cases of group decision-making (GDM), experts have different levels of expertise, and so it is not rational to weigh all of them equally. Generally, a GDM consisted of four significant steps. The first step is to evaluate and poll decision-makers, the second step is to homogenize the viewpoints, the third step is to integrate viewpoints and choose the optimal option, and the fourth step is to investigate the group agreement. One of the essential steps in GDM is the integration of different viewpoints. This integration is based on the concept of information aggregation, which can be defined as a mapping, as in the equation below (Yager 1993):
formula
(11)

Different information aggregation methods are developed, such as weighted arithmetic averaging, OWA, and HWA. In the following, the two essential concepts of OWA and HWA are explained.

OWA operator

This operator is introduced by Yager (1993) as follows:
formula
(12)
formula
(13)
where bi is the ith highest value in the set x with descending order, and wi is the weight vector. The vital concern in calculating the OWA is finding the weight vector (wi). There are two primary methods in finding the wi. In the first method, the wi is computed using the sample data, and in the second method, the wi is computed using linguistic quantifiers.
The first method is known as neatOWA. In this method, the wi is a function of values that will be aggregated in the future process. For the sake of brevity, the formulation of neatOWA and its potential application can be found in Liu & Lou (2006); Szidarovszky & Zarghami (2009); Yager (1993); and Yager & Filev (1999). The formulation of the neatOWA is as follows:
formula
(14)
formula
(15)
where wi is ith element of the weight vector and x = (x1,…,xn) is the set objects which are aggregated. α is the parameter of aggregation. The value of α is between 0 and ∞.
In the second method, proposed by Yager (1993) and Yager & Filev (1999) to find wi, Fuzzy linguistic quantifiers are used. The equation below is the formulation of the (wi) calculation:
formula
(16)
where Q is a function defined based on the interval of fuzzy identifier, i is the rank in the descendingly sorted data set of x and n is the number of components of set x. For the computation of Q, the equation below is proposed.
formula
(17)
where (a,b) is the interval of fuzzy quantifier, and r is equal to or . Values of a, b are the representatives of the decision-maker's viewpoint toward the method of aggregation of set x. These Fuzzy linguistic quantifiers are defined as Table 2.
Table 2

The definition of Fuzzy values of linguistic quantifiers

DescriptionValues of (a,b)
Maximum (0.3,0.8) 
Minimum (0,0.5) 
As much (0.5,1) 
DescriptionValues of (a,b)
Maximum (0.3,0.8) 
Minimum (0,0.5) 
As much (0.5,1) 
The difference between these quantifiers is the extent to which the decision-maker is willing to take the risk. For instance, the ‘as much’ quantifier is appropriate for a non-risk-taker decision-maker, and the minimum quantifier is suitable for the risk-taker decision-makers. In addition, the strictly monotonic equation below is also very useful in calculating the membership function of a quantifier:
formula
(18)
where α is the extent to which the decision-maker is a pessimist or risk-taker. To select α, it is advised to use a combination of the decision-maker's experience and organizational level (Table 3).
Table 3

The level of decision-makers' risk-taking

Description of the decision-maker
 Non-risk taker decision-maker 
 Neutral decision-maker 
 Risk-taker decision-maker 
Description of the decision-maker
 Non-risk taker decision-maker 
 Neutral decision-maker 
 Risk-taker decision-maker 

In other words, the OWA operator takes the characteristic of the decision-makers into account to assess the cumulative values by determining the value of α.

HWA operator

In the weighting process, in order to consider the importance and value of each variable simultaneously, the HWA operator is introduced (Xu & Da 2003). In this operator, all of the factors mentioned above are taken into account. The HWA operator can be formulated as the equations below:
formula
(19)
formula
(20)
formula
(21)
formula
(22)
where vj is the vector of weights in OWA operator, bj is the jth highest value of ordered set wixi, and the W is the weight vector of variables xi (i=1,2,…,n).

The proposed method to calculate the relative weights of criteria

In the first step, four experts who are thought to be knowledgeable about this water transfer project are asked to present their views about the relative importance of each criterion. The experts used the pairwise comparison matrix to assess the relative importance in which the components are quantified as a number between 1 and 7. Also, when the component of column i and row j is k, it is assumed that the component in column j and row i is . It is essential to mention, for reason mentioned earlier, the exclusivity criterion is not taken into account. The definition of each number in the pairwise comparison matrix is presented in Table 4.

Table 4

The description and the value of the questionnaire

ValueDescription
Two criteria have equal importance 
The importance of the criterion in the column is relatively higher than the criterion in the row 
The importance of the criterion in the column is higher than the criterion in the row 
The importance of the criterion in the column is much higher than the criterion in the row 
ValueDescription
Two criteria have equal importance 
The importance of the criterion in the column is relatively higher than the criterion in the row 
The importance of the criterion in the column is higher than the criterion in the row 
The importance of the criterion in the column is much higher than the criterion in the row 
The second step in the process is to calculate the importance of expert i, based on the views of other experts. The following equation shows the proper formulation of this step:
formula
(23)
where, is the similarity index between the viewpoint of expert i and expert j is component of column m, and row n in the pairwise comparison matrix of the expert i is the maximum possible difference between the viewpoints of two experts is the number of criteria studied in the pairwise comparison matrix is the number of experts asked to participate in this questionnaire
In this case study, the formulation for calculating the similarity index is like the equation below:
formula
(24)
where , here, is equal to 6.857; the number of experts is equal to 4; and the number of criteria studied is 6.
The third step is to combine the similarity indices () with the OWA operator in order to obtain the final similarity index of expert i (). In earlier sections, the two possible methods in the calculation of weights in the OWA operator were discussed. In this case study, for computing the weights, the following equations are used:
formula
(25)
formula
(26)
where α is a measure of how willing the decision-maker is to take the risk. In this article, the risk-taking state of (α < 1) is selected, and subsequently, α = 0.5 is chosen, which means that the decision-maker is willing to take some measures of risk. Consequently, the weight vector is like the following:
formula
(27)

Using the OWA operator, and based on the weight vector calculated earlier, the similarity indices will be integrated. The ultimate similarity index for all experts is normalized and presented in Table 5.

Table 5

The similarity index for different decision-makers

Experts
DM1 0.23 
DM2 0.26 
DM3 0.26 
DM4 0.26 
Experts
DM1 0.23 
DM2 0.26 
DM3 0.26 
DM4 0.26 
The fourth step of the process is to find the weight of expert i, which is a combination of the viewpoint of analyzer toward each expert i, and the weight corresponding with the opinion of other experts toward expert i. In order to find the weight of each criterion, the equations below are used:
formula
(28)
formula
(29)
where the values of α and are 0.4 and 0.6, respectively. Table 6 shows the values corresponding with the importance of each expert.
Table 6

The relative importance of decision-makers by the analyzer

DM1DM2DM3DM4
The importance of each expert based on the viewpoint of the analyzer (Di
DM1DM2DM3DM4
The importance of each expert based on the viewpoint of the analyzer (Di

Now, by having the two essential elements of the equation, the final weights can be calculated. The results are presented in Table 7.

Table 7

The final weights

Experts
DM1 0.23 0.143 0.19 
DM2 0.26 0.357 0.30 
DM3 0.26 0.214 0.24 
DM4 0.26 0.268 0.27 
Experts
DM1 0.23 0.143 0.19 
DM2 0.26 0.357 0.30 
DM3 0.26 0.214 0.24 
DM4 0.26 0.268 0.27 
Using the HWA operator, and based on the matrix of weights calculated in the last step, the pairwise comparison matrix will be integrated. The formulation of this step is presented as the equation below:
formula
(30)
where, is the component of the ultimate pairwise comparison matrix at row m column n is the weight of expert i, is the component of the pairwise comparison matrix of the expert i at row m, and column n.

The final pairwise comparison matrix of the criteria is shown in Table 8.

Table 8

The final pairwise comparison matrix

Criterion 1Criterion 2Criterion 3Criterion 4Criterion 5Criterion 6
Criterion 1 0.3 0.23 2.4 0.45 3.04 
Criterion 2 3.48 4.4 1.88 5.4 
Criterion 3 4.09 5.48 2.58 5.88 
Criterion 4 0.53 0.24 0.19 0.3 1.39 
Criterion 5 3.69 1.41 1.2 3.75 4.52 
Criterion 6 0.8 0.32 0.3 1.57 0.23 
Criterion 1Criterion 2Criterion 3Criterion 4Criterion 5Criterion 6
Criterion 1 0.3 0.23 2.4 0.45 3.04 
Criterion 2 3.48 4.4 1.88 5.4 
Criterion 3 4.09 5.48 2.58 5.88 
Criterion 4 0.53 0.24 0.19 0.3 1.39 
Criterion 5 3.69 1.41 1.2 3.75 4.52 
Criterion 6 0.8 0.32 0.3 1.57 0.23 

Finally, to reach the ultimate weights of the criteria, the OWA operator is applied to the matrix. Table 9 shows the ultimate criteria weights.

Table 9

The criteria weights

CriteriaUniversalityEquityEnforceabilityAdaptivityEnvironmental water userDetached from the land title and use
The ultimate weight (%) 8.94 26.53 31.44 3.82 24.55 4.72 
CriteriaUniversalityEquityEnforceabilityAdaptivityEnvironmental water userDetached from the land title and use
The ultimate weight (%) 8.94 26.53 31.44 3.82 24.55 4.72 

CVaR

Uncertainties in water resources management can always affect the efficiency of the policies enacted. One of the methods that can take these uncertainties into account is the CVaR. In order to understand the concept of the CVaR, the VaR has to be discussed first, because CVaR is defined based on VaR. VaR has three components: time horizon (T), confidence level (α), and loss value. VaR is described as the maximum probable loss at a given confidence level, which is computed by the cumulative function of probability distribution in random variables (Soltani et al. 2016). In risk-based planning with the concept of VaR, the goal is to minimize the VaR in a given time horizon (T) with a confidence level (α). Based on the explanation above, CVaR is the weighted average of losses, greater than or equal to VaR (Soltani et al. 2016). The proper formulation of these concepts would therefore be as:
formula
(31)
formula
(32)
where is the decision vector, is the vector representative of values of a random variable affected by the loss, is the loss function corresponding with x and y, is the confidence level, is the cumulative density of loss corresponding with the decision variable x, and E is the expected value operator. For more on this, the reader is encouraged to review Piantadosi et al. (2008); Rockafellar & Uryasev (2002); and Shao et al. (2011). In this article, using the criteria and their corresponding weights introduced in the earlier section, the objective function is defined as the following:
formula
(33)
where is the loss value for criterion j, and is the relative weight of criterion j.
With the definition of loss function as above, the CVaRα, which is the MEV of the area under the probability density distribution function for losses greater than VaRα, can be discretized as the following:
formula
(34)
where is the probability of loss, and lossi is the value of loss in area i.

It is important to note that the probability of occurrence of the loss is equivalent to the occurrence of probability corresponding annual inflow. The loss greater than 60% is equivalent to the inflow less than 40%, as the precise probability distribution function of the loss cannot be investigated, and so cannot be computed directly. With these two essential notions, can be calculated using the probability distribution function of annual inflow. This technique will be useful in the following formulation of the objective function.

Risk-based optimization model

The formulation of the optimization model is as follows:
formula
(35)
formula
(36)
formula
(37)
formula
(38)
formula
(39)
formula
(40)
where, is the value of the loss, is the probability of occurrence in ith sub-area in the discretized probability density distribution function of the loss function, is the value of criterion j, in sub-area i, is the weight of criterion j, is the water share of water user k in sub-area i in month t, is the water share percent of water user k in month t, is the annual inflow in sub-area i, are upper and lower limit of water share of water user i in month t, is the stored volume in the reservoir in month t concerning monthly inflow by the annual inflow in sub-area i, is the summation of leakage and evaporation from the reservoir in month t, is the monthly inflow in month t due to annual inflow of sub-area i, is the capacity of the monthly release, is a coefficient showing the monthly distribution of .

In this model, are the decision variables.

As was illustrated in the earlier sections, uncertainties in water allocation systems, which are results of uncertainties in inflow, are defined using the concept of CVaR. Three main steps which should be considered in this optimization model are discussed below.

Step 1: Finding the appropriate probability density distribution function

As discussed earlier, the annual inflow data for the Solegan River are available for 28 years. First, these data are required to be tested to ensure that the data set is normally distributed, and in order to do so, skewness and kurtosis are calculated. The Shapiro–Wilk test of normality is used to address this concern. Table 10 shows the values of these characteristics.

Table 10

Skewness and Kurtosis values

Acceptance threshold for normalityCalculated value
Skewness 2 to −2 0.661 
Kurtosis 2 to −2 1.192 
Significance level Greater than 0.05 0.254 
Acceptance threshold for normalityCalculated value
Skewness 2 to −2 0.661 
Kurtosis 2 to −2 1.192 
Significance level Greater than 0.05 0.254 

It can be claimed that the data set is normally distributed.

Step 2: Selecting the threshold

As discussed earlier, α is a representative of the risk that the decision-maker is willing to take. In this case study, the calculation is first presented with an α equal to 60%. In order to compute CVaR60%, there is a need to discretize the probability density function to the smaller sub-area. In this article, this continuous function is divided into fifteen discretized sub-areas. Then the average annual inflow in each of these sub-areas has to be calculated with respect to the probability distribution function for a given data set. The average annual inflow and the probability of occurrence in the given sub-area for α = 0.6 are presented in Table 11.

Table 11

The average annual inflow and the probability of occurrence in the given sub-area for α = 0.6

Sub-areaAverage annual inflow in each sub-area million cubic meters (MCM)The probability of occurrence in sub-areaSub-areaAverage annual inflow in each sub-area (MCM)The probability of occurrence in sub--areaSub-areaAverage annual inflow in each sub-area (MCM)The probability of occurrence in sub-area
80.256 5.23 28.185 3.23 11 113.77 1.29 
50.242 4.95 98.170 2.84 12 99.467 1.01 
19.228 4.6 156.68 2.4 13 85.16 0.78 
94.213 4.2 142.37 1.99 14 70.86 0.58 
59.199 3.75 10 128.074 1.62 15 56.55 0.43 
Sub-areaAverage annual inflow in each sub-area million cubic meters (MCM)The probability of occurrence in sub-areaSub-areaAverage annual inflow in each sub-area (MCM)The probability of occurrence in sub--areaSub-areaAverage annual inflow in each sub-area (MCM)The probability of occurrence in sub-area
80.256 5.23 28.185 3.23 11 113.77 1.29 
50.242 4.95 98.170 2.84 12 99.467 1.01 
19.228 4.6 156.68 2.4 13 85.16 0.78 
94.213 4.2 142.37 1.99 14 70.86 0.58 
59.199 3.75 10 128.074 1.62 15 56.55 0.43 

Step 3: Specifying the optimization model

The objective function of the model is to minimize the CVaR with a given confidence level. In this water right allocation model, the decision variable is the monthly water share percentage of water users. In order to calculate the objective function, the monthly inflow in each sub-area should be estimated, and therefore, the percentage of monthly share of annual discharge inflow using available data can be calculated. The percentages for all months are presented in Table 12. By using the data provided at the beginning of the case study and the model formulation presented earlier, the water share for each water user can be calculated. In Table 13, the main assumptions are presented.

Table 12

Average percentage of the annual inflow

MonthThe average percentage of annual inflowMonthThe average percentage of annual inflowMonthThe average percentage of annual inflow
Sep. 2.91 Jan. 12.36 May 3.54 
Oct. 5.75 Feb. 18.45 Jun. 2.01 
Nov. 9.00 Mar. 22.51 Jul. 1.70 
Dec. 8.91 Apr. 11.39 Aug. 1.69 
MonthThe average percentage of annual inflowMonthThe average percentage of annual inflowMonthThe average percentage of annual inflow
Sep. 2.91 Jan. 12.36 May 3.54 
Oct. 5.75 Feb. 18.45 Jun. 2.01 
Nov. 9.00 Mar. 22.51 Jul. 1.70 
Dec. 8.91 Apr. 11.39 Aug. 1.69 
Table 13

Assumptions in the optimization model

Assumptions in the optimization model
  
  
Assumptions in the optimization model
  
  

Finally, in the MATLAB environment and using the GA, the optimization model is solved.

The long-term optimization model

In this model, in order to take the existing uncertainties of the inflow into account, the long-term time series of the monthly flow can be used. It is essential to mention that in this method, instead of using the explicit definition of uncertainty in the model formulation, an implicit approach is implemented. In order to determine the average water share for each water user at the end of each month, the optimal amount of release from the reservoir and the corresponding allocations are found. Subsequently, by averaging these monthly allocations for each water user, its share of the monthly release is determined.

In the following, the optimization model is used for water allocation from a reservoir to n water users during t months:
formula
(41)
formula
(42)
formula
(43)
formula
(44)
where, is the value of criterion j, is the weight of criterion j, is the water allocated to water user i in month t, are upper and lower limit of water share of water user i in month t, is the inflow value in month t, is the summation of leakage and evaporation from the reservoir in month t, is the stored volume in the reservoir in the month t, Smin, Smax are the minimum and maximum of the stored volume, T is the number of months in the planning period.

In this model, are decision variables and MCM are million cubic meters.

The following details and assumptions should be considered in this long-term optimization model:
formula
(45)
formula
(46)
formula
(47)
formula
(48)

Finally, in the MATLAB environment and using GA, the optimization model is solved.

Sensitivity analysis of the CVaR-based model

In order to evaluate the sensitivity analysis of the results to the value of α, the results are presented for three values of α: 70, 60, and 40% in Table 14 and Figures 57. For the sake of brevity, only three water users' analyses are depicted. According to the definition of the first model, in the drought condition (or the high values for α), it is assumed that more water is allocated to water users with low water demands, as they are prioritized. In this case study, the third water user has the highest demand among all water users, with 40% of all water demands, and so the more severe the drought years are, the less water is allocated to the third water user.

Table 14

The share of water users from the total allocated water for different values of α

Water user 1Water user 2Water user 3Water user 4Water user 5Value of CVaR
α = 70% 10.11 11.89 18.7 29.67 29.62 0.0884 
α = 60% 9.02 10.76 20.21 29.61 30.4 0.1099 
α = 40% 7.75 9.32 26.41 26.01 30.51 0.1511 
Relative water demand (%) 4.46 5.35 40.14 27.25 29.22 – 
Water user 1Water user 2Water user 3Water user 4Water user 5Value of CVaR
α = 70% 10.11 11.89 18.7 29.67 29.62 0.0884 
α = 60% 9.02 10.76 20.21 29.61 30.4 0.1099 
α = 40% 7.75 9.32 26.41 26.01 30.51 0.1511 
Relative water demand (%) 4.46 5.35 40.14 27.25 29.22 – 
Figure 5

Results of the first model for different values of α (water user 1).

Figure 5

Results of the first model for different values of α (water user 1).

Close modal
Figure 6

Results of the first model for different values of α (water user 2).

Figure 6

Results of the first model for different values of α (water user 2).

Close modal
Figure 7

Results of the first model for different values of α (Water user 3).

Figure 7

Results of the first model for different values of α (Water user 3).

Close modal

The value calculated for CVaRα seems to be coherent with the assumption of the analyzer because it decreases with an increase in the value of α. Selecting the value of α is based on the viewpoint of the analyzer and the sensitivity to the level of severity of drought conditions. In other words, the more it is sensitive toward the drought condition, the more the value of α.

Results of the second model

The results of the second model are presented and discussed next.

Figure 8 is shows the fluctuation of the demand (measured based on volume in million cubic meters) and the allocated water throughout the planning horizon.

Figure 8

Allocated water and demand of the water users (a) water user 1 (b) water user 2 (c) water user 3 (d) water user 4 (e) water user 5.

Figure 8

Allocated water and demand of the water users (a) water user 1 (b) water user 2 (c) water user 3 (d) water user 4 (e) water user 5.

Close modal

For a more straightforward investigation, Table 15 shows the values of three characteristics of the model (reliability, resiliency, and vulnerability) are presented for three different levels of confidence (70%, 80%, 90%).

Table 15

The values of three characteristics for the long-term model in the planning period

Criterion90% (of demands)80% (of demands)70% (of demands)
Reliability 0.25 0.52 0.7 
Resiliency 0.1 0.21 0.26 
Vulnerability (MCM) 6,408.7 
Criterion90% (of demands)80% (of demands)70% (of demands)
Reliability 0.25 0.52 0.7 
Resiliency 0.1 0.21 0.26 
Vulnerability (MCM) 6,408.7 

It is essential to mention that the total demand of the study region in the planning period is 15,129 MCM and, as shown in Table 15, the total water deficit in the entire span of the planning period is equal to 6,408 MCM. About 42% of the water demand for water users is therefore not supplied, and it can be concluded that the demand is much higher than the available resources.

Comparison of the two models

As the first model allocates water with more focus on the low demand water users, the comparison of this model with the second one seems unjustifiable. However, the variation in the results of models can be analyzed and evaluated. Using the results of the second model, the average percentage of water share of water users of the annual inflow is calculated and compared with the results of the first model. Based on the definition of the first model, it is rational to assume that the model first allocates to water users with low water demands (water users 1, 2, 4) and then continues to allocate to the other water users. This is because the percentage of allocation is an essential factor in the loss function, and in order to minimize the function, the model tries to allocate to the water users with lower water demands first.

In order to compare the results of two models, three plots are presented in each of the given figures. The first one investigates the percentage of water share of the annual inflow to the reservoir, which is the decision variable of the first model, and the second one is the average percentage of water share of the annual inflow to the reservoir, which is derived from the results of the second model. The third one depicts the percentage of demand of each water user in each month relative to the summation of demands of the water users. For the sake of brevity, only the figures for the first three water users are illustrated (Figures 911).

Figure 9

Comparing the results of the two models (water user 1).

Figure 9

Comparing the results of the two models (water user 1).

Close modal
Figure 10

Comparing the results of two models (water user 3).

Figure 10

Comparing the results of two models (water user 3).

Close modal
Figure 11

Comparing the results of two models (third water user).

Figure 11

Comparing the results of two models (third water user).

Close modal

The most significant difference between the results of the first and second models is when they are allocating to water users with relatively higher demands. As shown in Figures 911, the first model in the early months when the demand of third water user is low tries to fulfill the water demands, but in the later months as the demand of the third water user increases, the amount of water allocated to it is drastically decreased.

Also, in the first model (the CVaR-based model), the percentage of water allocated to the other water users, especially the first and the second water users, is relatively more than that in the second model.

Table 16 shows that a considerable share of annual water is allocated to the fifth water user (the environmental water user, EWU). This can be justified by the equation defined for the EWU, and was taken into consideration in the definition of TE1. In the definition of EWU, if the model allocates water less than TE1, more loss is expected to occur (with a nonlinear inclination). It therefore seems rational that the model tries to allocate relatively high water share to EWU in situations where water is scarce.

Table 16

Share of water users from the total allocated water based on the results of the two models

Water user 1Water user 2Water user 3Water user 4Water user 5
CVaR model (α = 60%) 9.02 10.76 20.21 29.61 30.4 
Long-term model 6.56 7.69 28.81 24.06 32.88 
Relative water demand (%) 4.46 5.35 40.14 27.25 22.79 
Water user 1Water user 2Water user 3Water user 4Water user 5
CVaR model (α = 60%) 9.02 10.76 20.21 29.61 30.4 
Long-term model 6.56 7.69 28.81 24.06 32.88 
Relative water demand (%) 4.46 5.35 40.14 27.25 22.79 

In this study, a novel framework was developed to address the water rights allocation problem in inter-basin water transfer systems. To incorporate the inherent uncertainty of available water, the concept of CVaR was used. The HWA and OWA methods were also used to estimate the relative weights and combine criteria that have been defined to evaluate hydrologic, economic, and social aspects of water right allocation.

The Solakan–Rafsanjan inter-basin water transfer system has been one of the most crucial projects in Iran. This project, which is considered as the case study of the current paper, was defined in order to address the water shortages in arid and semi-arid areas of central Iran and to boost the agriculture sector of the Rafsanjan region.

The results showed a proclivity toward fulfilling the lower water demands first, as the model prioritized the minimization of such water users. This was apparent when the results of the CVaR-based model were compared to a long-term water right allocation optimization model with monthly steps. The sensitivity analysis of the values of confidence level (α) also showed that with an increase in the value of (α), the value of the CVaR-based model would decrease.

The methodology presented in this paper can be easily applied to water rights allocation in other water resources systems. In future studies, the proposed methodology could be extended to incorporate other uncertainties. One of the assumptions of this study is that the annual inflow to the reservoir in the donor basin has a constant monthly distribution. This limitation should be improved in future works. Another crucial issue in the discourse of today is global climate change, and its possible impacts on management of water bodies. Due to an average surface air temperature rise, the likelihood of extreme changes in all components of the climate system is increasing. The standard management schemes have been founded on stationarity and past hydrological experience, as it was assumed in this study. For the continuation of this research, it is highly advised to study this ongoing trend and its implication on water allocation based on projected dataset, instead of the history of a given water body.

The technical contribution made by Dr Armaghan Abed-Elmdoust is hereby acknowledged.

All relevant data are included in the paper or its Supplementary Information.

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