A Multi-objective Approach for Transboundary River Water Allocation

Abstract

The allocation of water to the stakeholders of a large basin involves conflicting objectives, since increasing the allocated water to one stakeholder leads to a reduction in water allocated to other stakeholders. The consideration of conflicting objectives is inevitable when the basin is a transboundary basin, where a river crosses at least one political border, either a border within a nation or an international boundary. This paper proposes a multi-objective optimization model for sharing water among stakeholders of a transboundary river, assuming that the stakeholders cooperate. Here, the cooperation implies a balanced water allocation to stakeholders since shortage in each stakeholder have negative impacts on others. Each objective function of the multi-objective model represents the water profit of a stakeholder; which has to be maximized. To reach a cooperative solution, a new method for transforming the multi-objective formulation to a three-step single objective formulation is proposed. The solution guarantees each stakeholder's profit which is larger than a percentage of its highest possible profit obtained in the case when the percentage of profit is equal for all stakeholders. The proposed model formulation was applied to the Sefidrud River where eight provinces are the stakeholders competing for water resources of this basin.

1 Introduction

Water supports all aspects of human society such as agriculture and industry and at the same time, it is an integral part of ecosystems. Therefore, the competition of basins' stakeholders for more water resources utilization is at present a major issue for water authorities, especially in transboundary watersheds, due to insufficient available water for water demands satisfaction. The lack of collaboration between stakeholders in a transboundary watershed causes unfair water availability in the watershed and harmful consequences to rivers biodiversity. Thus, one-sided actions for supplying water requirements by a stakeholder are often ineffective, inefficient, or impossible (Mostert 2005) and cannot be sustainable.

Water allocation is one of the important tasks, which has to be set to guarantee a cooperative and sustainable management of transboundary water resources. While stakeholders' water portions are clear and specified; determined by an effective, efficient, and equitable way, there is no reason for water disputes between the stakeholders and one-sided actions. Therefore, the allocation of water between stakeholders is a compelling action for the cooperative management of transboundary water resources.

Allocating water within jurisdictions in ways that reflect water's importance for ecosystems and human societies and addresses an effective, efficient, and equitable water sharing is very challenging. To achieve this sort of water allocation “water allocation objectives should include economic, environmental, and social factors” (UNESCAP 2000). Water is a valuable and limited resource, and thus it should be used in the highest profitable way. The satisfaction of environmental water requirements should not be ignored in any water allocation study; because the environment indirectly influences the human well-being. The allocation of water to each region/stakeholder influences the inhabitants of a watershed in positive and negative ways. Therefore, social factors have to be taken into consideration in water allocation studies.

1.1 Background

Operations Research has a long tradition in agricultural and related natural resource industries (Plà et al. 2013); hence, water allocation models were mainly based on optimization techniques. Economic objective functions are often used in these models where these objectives have the advantage of summarizing all interests in a single financial metric (Harou et al. 2009). There are numerous water allocation models with economic objective functions, however we only referenced to a few most recent ones. Ali and Klein (2014) used a single objective model to maximize the net benefit of allocated water to agricultural, industrial, and domestic sectors in the Bow river sub basin of Southern Alberta. Roozbahani et al. (2013) introduced a single objective water allocation model for the Sefidrud basin, Iran. The maximization of agricultural water profit was the objective function of their model. Huang et al. (2012) developed a two-stage interval-quadratic programming method for agricultural water resources planning and management in the Tarim river basin, China. A nonlinear economic objective was the model's objective function. Divakar et al. (2011) developed a single objective model for optimal allocations of water to competing sectors such as agriculture, domestic, industry, and hydropower by maximizing the net economic benefits of bulk allocations to different sectors.

Water allocation is a multidimensional task and often several objectives, beside the economic indicator have to be considered simultaneously for achieving reasonable and efficient solutions. Hence, multi-objective models are successfully utilized for water allocation modelling in some cases in which we only cited to a few most recent studies. Rezapour Tabari and Yazdi (2014) developed a multi-objective optimization model based on inter-basin water resources and restoration of outer-basin water resources. They considered three water allocation objectives in their model: supplying inter-basin water demand, reducing the amount of water output from a basin boundary, and increasing water transfer to adjacent basins. Anghileri et al. (2013) proposed a multi-objective model for sharing water between competing uses in Alpine watershed in Italy. The model's objective functions were to minimize the shortage of irrigation demand and maximize the hydropower production. A genetic algorithm–based multi-objective model was developed for the Aharchay watershed, Iran by Ahmadi et al. (2012) to provide desirable water quality and quantity for various sectors while maximizing agricultural production in the upstream region, mitigating the unemployment (social) impacts of land use changes, and providing reliable water supply to the downstream region were the study's objectives. Han et al. (2011) developed a multi-objective model to allocate water resources with different water quality to multiple users with different water quality requirements for the Dalian city, China. The model's objective functions were the maximization of the net benefit of allocated water to agriculture, the minimization of sewage drainage and the maximization of greenbelt irrigation area. Liu et al. (2010) presented a multi-objective optimization model for the optimal allocation of water resources in saltwater intrusion areas by considering three objectives: maximizing economic interest and social satisfaction and minimizing the amounts of polluted water. The model was examined in the Pearl River Delta in China.

Water allocation becomes more complicated in a transboundary watershed where the stakeholders are often states/provinces or countries that confront with rising water requirements and as a result, the stakeholders have less concerned for the environmental water satisfaction. Hence, decision makers have to consider the environmental water supply with high priority in water allocation models for the transboundary basins.

There are a few studies on the water allocation of transboundary watersheds that consider all three water allocation indicators noted before; economic, environmental, and social factors. The Aral Sea basin is a transboundary watershed that several mathematical models were developed to allocate water between its stakeholders. McKinney and Cai (1997) applied a multi-objective model to this basin where maximizing the flow to the Aral Sea, the satisfaction of the basin's demands, and equalizing the distribution of water deficits between stakeholders were the water allocation objectives. Cai et al. (2002) presented a multi-objective model for water allocation of the Syr Darya River Basin (one of the two major rivers feeding the Aral Sea) by considering the risk of water supply to stakeholders such as the environment, equity in water allocation, and economic efficiency in water infrastructure development as water allocation indicators. Schlüter et al. (2005) did the same study for reaching an optimal ecologically sound water allocation pattern for the Amudarya River (second major rivers feeding the Aral Sea).

Kucukmehmetoglu and Guldmann (2010) developed a water allocation model for the Euphrates and Tigris River basin based on multi-objective optimization. The model's objective functions represented the benefits of countries from using water in different sectors included hydropower. The authors considered two out of three water allocation objectives; revenue and equity in their study, however the water supply to environment was not taken into consideration.

1.2 Organisation of the Paper

The structure of the paper is organized as follows. In the next section, the model formulation and the proposed method for transforming this model to a single objective linear model are presented. The characteristics of the Sefidrud watershed are briefly discussed in Section 3, followed by the model parameterisation in Section 4. The results of model are presented in Section 5, followed by the conclusion in Section 6.

2 Problem Formulation

The proposed model is a multi-objective model designed to maximize the profits of each riparian stakeholder subject to resources, water balance, environmental demand, and usage constraints. The profits are the net benefits derived from the allocated water to agricultural, urban, and industrial sectors in the each stakeholder. The unbalanced investment and consequently unfair income generation among the stakeholders of a transboundary basin is the main reason for water disputes. Obviously, there are no water disputes in the basin while the water could not make any profit. Based on this fact, the maximization of achieved profit for each stakeholder from its water share is selected to be the objective functions of our water allocation model. This well-chosen indicator can also play the role of a social factor while the model's results convince stakeholders that they achieve the maximum profits from the shared water resources.

The formulation of the proposed multi-objective model is based on the network of a basin. In a network presentation of the basin, the nodes stand for the supply and demand points and arcs represent reaches. The model is presented as follows:

Maximize:

$$ {Z}_k={\displaystyle \sum_{i_k=1}^{n_k}{\displaystyle \sum_t\left({\mathrm{p}}_{{}_{i_k}}^a\times {x}_{i_kt}^a+{\mathrm{p}}_{{}_{i_k}}^u\times {x}_{i_kt}^u+{\mathrm{p}}_{{}_{i_k}}^d\times {x}_{i_kt}^d\right)}} $$

(1)

Subject to:

$$ {\mathrm{SW}}_{i_kt}+{\displaystyle \sum_{l\in UN}{y}_{\left(l\to {i}_k\right)t}}={y}_{\left({i}_k\to j\right)t}+{x}_{{}_{i_kt}}^a+{x}_{{}_{i_kt}}^u+{x}_{{}_{i_kt}}^d\kern2em \forall i,t $$

(2)

$$ {y}_{\left({i}_k\to j\right)t}-{\mathrm{ED}}_{i_kt}\times {z}_{i_kt}\ge 0\kern1.12em \forall i,t $$

(3)

$$ {\displaystyle \sum_t{z}_{i_kt}}\ge \mathrm{R}\kern1em \forall i,t $$

(4)

$$ {x}_{{}_{i_kt}}^a\le {\mathrm{AD}}_{i_kt}\kern1em \forall i,t $$

(5)

$$ {x}_{{}_{i_kt}}^u\le {\mathrm{UD}}_{i_kt}\kern1em \forall i,t $$

(6)

$$ {x}_{{}_{i_kt}}^d\le {\mathrm{ID}}_{i_kt}\kern1em \forall i,t $$

(7)

$$ {x}_{{}_{i_kt}}^a\ge 0\kern1em \forall i,t $$

(8)

$$ {x}_{{}_{i_kt}}^u\ge 0\kern1em \forall i,t $$

(9)

$$ {x}_{{}_{i_kt}}^d\ge 0\kern1em \forall i,t $$

(10)

$$ {y}_{\left({i}_k\to j\right)t}\ge 0\kern1em \forall i,t $$

(11)

$$ {z}_{i_kt}=0\kern0.62em or\kern0.62em 1\kern1em \forall i,t $$

(12)

where:

k :

index for stakeholders

a :

index for agricultural sector

u :

index for urban sector

d :

index for industrial sector

t :

index for time step

i :

index for node

j :

neighbouring node downstream for node i

n _k :

number of nodes in stakeholder k

i _k :

node i where is in stakeholder k

${\mathrm{p}}_{i_k}^a$ :

water profit of agricultural sector in node i, for stakeholder k

${\mathrm{p}}_{i_k}^u$ :

water profit of urban sector in node i, for stakeholder k

${\mathrm{p}}_{i_k}^d$ :

water profit of industrial sector in node i, for stakeholder k

UN _i :

the set of upstream neighbouring nodes of node i

${x}_{{}_{i_kt}}^a$ :

allocated surface water to agriculture in node i, for stakeholder k, in time step t

${x}_{{}_{i_kt}}^u$ :

allocated surface water to urban need in node i, for stakeholder k, in time step t

${x}_{{}_{i_kt}}^d$ :

allocated surface water to industry in node i, for stakeholder k, in time step t

${\mathrm{SW}}_{i_kt}$ :

produced surface water in the catchment between node i, for stakeholder k, and nodes where are upstream, in time step t

${y}_{\left({i}_k\to j\right)t}$ :

transferred surface water from node i, for stakeholder k, to node j in time step t

${\mathrm{ED}}_{i_kt}$ :

environmental water need in node i, for stakeholder k, in time step t

${z}_{i_kt}$ :

binary variable, 1 if the water supply to environment in node i, for stakeholder k, in time step t is satisfied, otherwise 0

R:

reliability level for water supply to environment

${\mathrm{AD}}_{i_kt}$ :

agricultural water demand in node i, for stakeholder k, in time step t

${\mathrm{UD}}_{i_kt}$ :

urban water demand in node i, for stakeholder k, in time step t

${\mathrm{ID}}_{i_kt}$ :

industrial water demand in node i, for stakeholder k, in time step t.

The total net economic benefit for stakeholder k (Z _k ) is the sum of the net benefits of water usage to agriculture (${\mathrm{p}}_{{}_{i_k}}^a\times {x}_{i_kt}^a$), the net benefits of water usage to urban (${\mathrm{p}}_{{}_{i_k}}^u\times {x}_{i_kt}^u$), and the net benefits of water usage to industry (${\mathrm{p}}_{{}_{i_k}}^d\times {x}_{i_kt}^d$). Note that the relationship between the water profit of a sector (${\mathrm{p}}_{i_k}$) and allocated water to it (${x}_{{}_{i_kt}}$) is not always linear. In other words, there is no fixed water profit for all range of water allocation to a sector. For example, in the study of water allocation to various crops in an irrigated network with limited planted crop area, the marginal value of allocated water would be different based on the stages of plants growth and their needed water. Water stress during critical growth periods reduces yield and quality of crops, and thus farmers are willing to pay more for water needed. Therefore, the water profit of allocated water to the crops in this stage would be higher than other stages of plants growth. However, in this study, we aim to resolve water conflicts in watersheds where water disputes are in result of limited water resources and high desire of their stakeholders for water utilization in various sectors. In this case, we optimize the allocation of bulk water to a sector of the stakeholders and do not detail on the way that clients utilize the water. It would be extremely difficult to determine the real dynamic of profit as a function of water allocated to the stakeholders in such a large scale as provinces or countries. Therefore, in such studies it is reasonable to consider a fixed water profit for all range of allocated water.

The second point that has to be noted here is that the municipal water supply is often given a higher priority than other sectors by water authorities. Therefore, it is supposed that the value of ${\mathrm{p}}_{{}_{i_k}}^u$ is higher than ${\mathrm{p}}_{{}_{i_k}}^d$ and ${\mathrm{p}}_{{}_{i_k}}^a$.

Equation (2) represents the water balance constraint at node i. The water inputs to supply/demand node i are the tributary inflow ${\mathrm{SW}}_{i_kt}$, and transferred water from upstream nodes l (l∈UN _i where UN _i is the set of node i's neighbouring nodes upstream) to i, ${\displaystyle \sum_{l\in UN}{y}_{\left(l\to {i}_k\right)t}}$. On the other hand, water leaving node i is water withdrawal for agricultural (${x}_{{}_{i_kt}}^a$), industrial (${x}_{{}_{i_kt}}^d$), urban uses (${x}_{{}_{i_kt}}^u$), and water release to downstream node ${y}_{\left({i}_k\to j\right)t}$ (j is neighbouring node for node i downstream).

The amount of transferred water from node i to node j in time step t (${y}_{\left({i}_k\to j\right)t}$) has to be greater or equal to the environmental water requirement in node i (constraint (3)). To control the reliability of the environmental water supply, we consider a binary variable (${z}_{i_kt}$) in constraint (3) which is 1 if the environmental water need in time step t is satisfied. The sum of this binary variable over the time steps has to be more than or equal R, the reliability level of the environmental water supply (constraint (4)). In this study, we use the definition of the reliability introduced by Kundzewicz (1995) as the ratio of the time, when the volume of water supplied meets the demand, to the total time considered (temporal reliability).

The amounts of allocated water to a node for satisfying different water needs should not exceed to their water requirements. This limitation is imposed in the model by constraints (5) to (7). Finally, the decision variables of the model have to be non-negative which represent the amounts of allocated surface water to node i, transferred surface water from node i to node j, and the integer variables ensuring the reliability of water supply to environment in node i (constraints (8)-(12)).

2.1 Proposed Method for Finding a Pareto-Admissible Solution

A Pareto-admissible solution can also be generated by maximizing λ for each time step, while this ratio is the same for all stakeholders. λ is the ratio of a stakeholder's profit to its highest possible water profit. The proposed method is established upon this idea that all stakeholders have to receive the highest possible benefits from the basin's water resources in accordance with the objective functions of the multi-objective water allocation model but in an equitable manner. In this method, equitability is defined as the same percentage of the highest water profits for all stakeholders. Put differently, the proposed model maximizes the minimum water profit from allocated water to a stakeholder. The model to be solved is then:

Maximize:

$$ Z={\displaystyle \sum_t{\lambda}_t} $$

(13)

Subject to:

$$ {\displaystyle \sum_{i_k=1}^n\left({\mathrm{p}}_{{}_{i_k}}^a\times {x}_{i_kt}^a+{\mathrm{p}}_{{}_{i_k}}^u\times {x}_{i_kt}^u+{\mathrm{p}}_{{}_{i_k}}^d\times {x}_{i_kt}^d\right)\ge \left({\lambda}_t\times {f}_{kt}^{*}\right)}\kern1em \forall k,t $$

(14)

and constraints (2) to (12).

where:

λ _t ratio of a stakeholder's water profit to its highest possible water profit in time step t;

f ^* _kt highest possible water profit of the stakeholder k in time step t.

The model objective function is maximizing the summation of λ over time steps. Inequality (14) points to this idea that the amount of each stakeholder's water profit in time step t (${\displaystyle \sum_{i_k=1}^n\left({\mathrm{p}}_{{}_{i_k}}^a\times {x}_{i_kt}^a+{\mathrm{p}}_{{}_{i_k}}^u\times {x}_{i_kt}^u+{\mathrm{p}}_{{}_{i_k}}^d\times {x}_{i_kt}^d\right)}$) has to be greater than a ratio (λ _t ) of its highest water profit (f ^* _kt ) in time step t, while λ _t is the same for all stakeholders. This proposed method easily adapts the multi-objective water allocation model to a single objective model by reducing the multi objectives to a single objective and adding constraint (14). The proposed method is explained through three steps:

Step 1 finds the highest possible profit for each stakeholder (f ^* _kt ). For this purpose, a single objective optimization model is solved for each stakeholder. The objective function of this single objective model is to maximize the water profit of that particular stakeholder, subject to constraints (2)-(12). This model is called HWP (Highest Water Profit) model. The outputs of the HWP model are the values of decision variables when the water profit of the stakeholder is the largest possible value at each time step. Step 1 is repeated for all stakeholders until the highest possible profits for all stakeholders are determined. Note that due to satisfaction environmental water needs, the maximum allocated water to a stakeholder is often less than its water demand. Therefore, we use “highest possible profit” term instead of “highest profit”.

Step 2 determines the largest percentage of the highest possible profit for each time step. It could be achieved by the model that we explained above (expression (13) and constraints (2)-(12) and (14)). This model is denoted as the Highest Ratio of Highest Water Profit (HRHWP) model.

Step 3 finds the final water share of each stakeholder. For this aim, a single objective optimization model is solved for the basin. The objective function of this single objective model is to maximize the water profit of the basin, subject to constraints (2)-(12) and (14). In this model, the values of λ for time steps achieved from Step 2 are fixed in constraint (14). This model is referenced as the Final Water Profit (FWP) model. The function of this model is to maximize the profit of the basin while it maximizes the profit of a stakeholder with the lowest profit between other stakeholders. In other words, this model allocates equitably water to the stakeholders based on the results of Step 2 (when the value of the parameter λ is obtained) and then apportions the leftover of water among other stakeholders with respect to the water profits of them.

3 The Sefidrud Basin

3.1 The Basin Description

The multi-objective water allocation model and proposed solution method are applied to optimize the water allocation of the Sefidrud Basin, Iran. The river basin has a total drainage area of 59,217 km² and covers eight riparian administrative provinces, namely, Kordestan (Province 1), Hamedan (Province 2), Zanjan (Province 3), East Azarbaijan (Province 4), Ardabil (Province 5), Tehran (Province 6), Qazvin (Province 7) and Gilan (Province 8). The main waterway in the basin is the Sefidrud River, whose head tributaries, the Ghezelozan and Shahrud rivers, flow across the Kordestan province (Province 1) and Tehran province (Province 6), respectively. The name of head tributaries changes to Sefidrud after joining in the Gilan province. The Sefidrud River goes through the Gilan province (Province 8) and finally flows into the Caspian Sea (Figure 1). The current population of the basin is 2.1 million with future estimates of 2.24 million in 2025 (MGC 2011).

Fig 1

The Sefidrud Basin and its stakeholders

The watershed's total water yield is 7615 million cubic meters (MCM) whose the share of surface and groundwater are 6214 MCM and 1337 MCM, respectively (MGC 2011). Agriculture is the main user of water in this Basin and the water needs of other sectors can be ignored due to their low amounts and groundwater resource satisfaction. The annual agricultural water demand for 2007 (assuming as the current water need) was 3,701 MCM, however, it is estimated to be 7,270 MCM in 2025 as the future water demand (MGC 2011). The current and future basin agricultural demands, as well as the agricultural water profit for each province, are shown in Table 1. Note that the water profit of agricultural sector in the Sefidrud Basin was calculated in another study (MGC 2011) and the Iranian water authority officially has accepted its results. Hence, we used it in this study. The reader is referred to Roozbahani et al. (2013) for more details about the Basin water resources and demand.

Table 1 The current and future basin agricultural demand (Unit: MCM)

3.2 Water Disputes in the Basin

The Sefidrud Basin is an underdeveloped watershed and all stakeholders confront with rising water requirements for developing agricultural, industrial, and domestic sectors. Thus, there are severe competitions between the stakeholders for utilizing more water in their territories, especially between upstream and downstream stakeholders. Similar to other basins, mountains are upstream and flat plains are located downstream in this basin, and thus water resources of downstream stakeholder depend on transferred water from upstream regions. In the current situation, the upstream regions have a higher desire for water developments and want to utilize more water in their boundaries, which otherwise flows and benefits the downstream regions. However, the irrigation channel networks in downstream regions depend on the traditional flows from the upstream regions and hence do not want to compromise. These competitions have now turned into a serious water conflict between the provinces. In this circumstance, the Iranian water authority is under high social and political pressures to allocate more water and budgets for water developments to these stakeholders while there is no enough water to satisfy all water requirements. Furthermore, the stakeholders have a low priority for the environmental water satisfaction and the environment faces with the huge water shortages in the whole basin.

3.3 Basin Network

Main streamflow gauging stations in the basin are considered as nodes in the basin network. There are five shared nodes in this network. Five shared nodes are replaced by twelve dummy nodes (nodes 6 and 10 are shared between three provinces and nodes 20, 24, and 27 are shared between two provinces). The produced runoff and demand of nodes are obtained from previous studies of the basin (MGC 2011). The reader is referred to Roozbahani et al. (2013) for more details about the nodes' runoff and demand.

Apart from agricultural demand, the environment water supply is taken into consideration in water allocation planning decisions. The environmental water demands in the nodes were estimated, utilizing the modified Montana method (Torabi Palatkaleh et al. 2010a). In this method, the environmental water requirement of each node is calculated based on a percentage of the average monthly streamflow in the node. In this study, these percentages are 10 % for October to March and 30 % for April to September. This method has been officially accepted by the Iranian water authorities (Torabi Palatkaleh et al. 2010b).

The stylized network representation of the Basin is shown in Figure 2. This network includes supply/demand nodes scattered among 8 provinces. These supply/demand nodes are associated with the stream flow gauges and they provide surface water for the agricultural uses in their vicinities. The directed vertices of this graph represent the river reaches. These directions indicate the only possible way of water transfer between nodes. Each agricultural water need is supplied from the surface water resources. As noted, the conflict in water allocation arises from surface water resource, which is a common resource for all stockholders and is preferred water resource to supply agricultural demand.

Fig 2

The Sefidrud Basin network

4 Parameterisation

The Basin's water demand in 2025, which represents the water need of the Basin stakeholders for maximum social-economic developments (Table 1), was used in this study. According to completed study in the Basin (MGC 2011), there is no statistically significant trend (positive or negative) in the Basin discharge during 1975 to 2007 (50 years). Therefore, the monthly-recorded data for the Basin nodes during this period were used as the input to the optimization models (600 time intervals).

Since eight provinces constitute the Sefidrud Basin, eight mixed-integers linear programming models (the HWP models) were developed. The 90 % threshold of demand satisfaction in 90 % of years is to be a major reliability criterion for water supply systems in the Basin (Torabi Palatkaleh et al. 2010b). Therefore, the amount of R in the HWP models and the HRHWP is equal to 540 (The number of 540 is the product of the number of time steps (600) times to the threshold of demand satisfaction (90 %)).

5 Results

In this section, the results of the HWP and the HRHWP models will be discussed in sections 5.1 and 5.2 respectively, followed by the results of the FWP model in section 5.3. All models were solved using CPLEX libraries (IBM).

5.1 HWP Models Results

In this section, indexed models refer to the HWP models for the corresponding provinces. For example, Model 1 points to the HWP model which maximizes Province 1's profit. Table 2 shows the average profits of provinces over time steps given by the HWP models. The minimum profit for each province is distinguished by bold and underlined styles. As shown in this table, Model 1 causes the minimum profits for Provinces 2, 3, and 6. Model 3 also brings about the minimum profit for Provinces 4, 5 and 8. These data show that the interest of Province 1 is mainly in conflict of Provinces 2, 3, and 6 and the interest of Province 3 is mainly in conflict of Province 4, 5, and 8. In addition, the interest of Province 7 with Province 6 is in conflict. In brief, the allocation of water to Provinces 1 and 3 has the largest negative effects on the profits of other provinces.

Table 2 The profits of provinces given by the HWP models (Unit: Billion Rials)

Table 3 illustrates the ratio of provinces' profits given by the HWP models to their highest possible profit. As shown in this table, the Province 8' profits have fluctuated around 20 % (mainly less than 10 %) of its highest possible profit (3,807) while fluctuation is at least more than 40 % for other provinces. For example, the ratio of Province 2' profits given by the HWP models to its highest possible profit is varied from 14 % to 19 % (Table 3). In short, the water share of Province 8 from the Basin's surface water resource, which is located the Basin downstream, does not significantly depend upon the allocated water to other provinces.

Table 3 The ratio of provinces' profit given by the HWP models to their highest possible profits (Unit: %)

In order to determine the sensitivity of stakeholders to the environmental water supply, the HWP models were also run without imposing the environmental water requirements. The highest profits of provinces with and without imposing the environmental water supply to the models are presented in Table 4. By comparison the increase of provinces' profits in Table 4, it can be concluded that Province 6 is very sensitive to the environmental water supply while its highest profits increase 28 % without enforcing the models to satisfy the environmental water needs.

Table 4 The provinces' highest profits with and without imposing environmental water satisfaction to the HWP models (Unit: Billion Rials)

5.2 HRHWP Model Results

Figure 3 shows the results of the HRHWP model in the terms of λ for 600 time steps which are classified in seasons and various ranges. λ is always larger than 0.3 which it implies that the profits of stakeholders are always more than 30 % of their highest possible profits. The values of λ for the summer, when competition between the stakeholders is heightened, show that the categories of 0.5–0.6 and 0.6–0.7 have the highest frequency in this season. It implies that for 76 % of times in summer the surface water resource is allocated to the stakeholders in a way that their profit are at least half of their highest possible profits.

Fig 3

The histogram of λ in various seasons

The profits of provinces given by the HRHWP model are shown in Figure 4. As shown in this figure, the general trends of the water allocation for the Sefidrud Basin are such that the provinces' profits are closer to the highest possible profit in comparison to the lowest profit. Note that the lowest profit for each province was determined using the corresponding HWP model, where its objective function was to maximize the profit of all provinces except the corresponding province.

Fig 4

The highest, lowest, and proposed (by the HRHWP model) profits to provinces

To further clarify the HRHWP model performance, the model's results for time step 300 are presented in Table 5. The obtained λ by the model for this time step is 0.724 that means the stakeholders' profits for this time step is at least 72.4 % of their highest possible profits. The ratio of stakeholders' profits to their highest possible profits for this time step are calculated and presented in column (3). It shows that all stakeholders benefited equitability from the Basin surface water resource. As shown in Table 5, the minimum λ belongs to Province 2 and λ is more than 0.724 for other provinces. It means that Province 2 had the lowest profit in this time step and the model maximized the profit of this province.

Table 5 The HRHWP model's results for time step 300 (Unit: Million Rials)

5.3 FWP Model Results

Here we report the final results of water allocation obtained from the FWP model while satisfy the results obtained from the HRHWP model. The average yearly allocated water to each province is shown in Table 6. As presented in this table, the total surface water allocated to stakeholders is 3,360 MCM. According to these results, Provinces 8 and 2 have the largest and the smallest portions of the Basin's surface water with 1,294 MCM and 55 MCM, respectively. In addition, Provinces 8 and 1 have the smallest and the largest water shortages between stakeholders with 39 % and 73 %, respectively.

Table 6 The yearly averages of allocated water to provinces by the FWP model

The average monthly allocated water to the provinces is also presented in Figure 5. As shown, the competition over water utilization concentrates on the spring and summer (months 7 to 12) while the main parts of the stakeholders' water requirements in the autumn (months 1 to 3) are satisfied.

Fig 5

The maximum, minimum, and proposed (by the FWP model) water supplied to the stakeholders (The values are the average of 50 years)

5.4 Water Development Opportunities in the Sefidrud Basin

The capacity of the Sefidrud Basin for water development through new dams was also evaluated using the FWP model's results. Total water released to the Caspian Sea (the FWP model's output) and the monthly water supplies to environment in the last node in the Basin network are presented in Figure 6 (a). The total difference of these values was considered as the water development capability for the Sefidrud Basin (Figure 6 (b)). The annual average of water released to the Caspian Sea is 1,796 MCM where the portions of months 1 to 6 (autumn and winter) and months 7 to 12 (spring and summer) are 81 % and 19 %, respectively. This result shows that the water authorities are able to plan for regulating 1,845 MCM through constructing new dams. By considering the volumes of allocated water to stakeholders (3,360 MCM) and the total capacity of the new dams (1,796 MCM), the potential of the Basin surface water resource for satisfying the water requirements is 5,156 MCM. In other words, the Basin's stakeholders are able to use 5,156 MCM out of 6,214 MCM of the Basin's surface water resources when the satisfaction of the environmental water demand is considered a sustainable water development criterion for the water allocation modelling.

Fig 6

The monthly averages of water released to the Caspian Sea, the water supply to the environment in last node in the Basin network, and water released to the Caspian Sea, surpluses to the environmental demand

6 Conclusions

This work proposed a multi-objective model for resolving the water disputes in a transboundary watershed where the satisfaction of the environmental water requirements in the watershed is necessary. We considered the maximization of stakeholders' profits from the watershed surface water resource as the objective functions of this multi-objective model. This objective function is an indicator of how fair water is allocated among stakeholders. A three-step method was introduced to transform the multi-objective water allocation model to single objective models. This method was based on a fact that the allocated water to a stakeholder has to make the closest profit for it to its maximum possible profit.

The proposed multi-objective model and introduced method were applied to the Sefidrud Basin where water dispute between its stakeholders is an issue. The HWP models were developed to determine the highest possible profits of stakeholders. The results of these models showed that the water share of Province 8 (Gilan) from the Basin's surface water resource does not significantly depend upon the allocated water to other provinces. In addition, the interest of provinces in this Basin was in conflict of the interest of Provinces 1 and 3. Furthermore, the satisfaction of the environmental water needs was mainly negative effect on the profit of Province 6 when its profit increased if the water supply to the environment would not be considered in the models.

The HRHWP model was developed to determine the highest λ for each time step. This model maximized the profit of a stakeholder with minimum profit in each time step. The outcome of the HRHWP model showed that the profits of the Basin stakeholders are always larger than 30 % of their highest possible profits. It is also concluded that 76 % of times in summer the surface water resource is allocated to the stakeholders in a way that their profit are at least 50 % of their highest possible profits.

The FWP model determined the final water share of each stakeholder. This model maximized the profit of the basin while it maximizes the profit of a stakeholder with the lowest profit between other stakeholders. The results of the FWP model demonstrated that the Basin surface water resource can satisfy only 3,360 MCM from the stakeholders' water requirements when the shares of stakeholders are 286, 55, 762, 405, 212, 114, 232, and 1,294 MCM for Provinces 1 (the Kordestan), 2 (the Hamedan), 3 (the Zanjan), 4 (the East Azarbaijan), 5 (the Ardabil), 6 (the Tehran), 7 (the Qazvin) and 8 (the Gilan), respectively. Moreover, the Basin water authority can plan for constructing new dams to regular about 1,845 MCM more water in the Basin.

The research reported in this paper has two contributions. First contribution is water allocation problem modelling in a transboundary basin with multi-objective mixed-integer programming. Second ones is introduction of a new solution method to find an effective Pareto optimal solution for the problem. This study could be extended into several directions. First, the HWP models, the HRHWP model, and the FWP model should be improved with considering water infrastructure such as dams. Second, long-term dynamics should be introduced into the model, accounting for the possible impacts of climate change on water supply and demand. Third, the water quality should be accounted for, including salt intrusion on the productivity of agriculture downstream.