Resolving conflict on the trade-off curve generated by a multi-purpose sustainability model for the natural environment and farming under water bankruptcy

1. Introduction
The contradiction between increased demand for water and decreased water availability has led to excessive exploitation of water resources and, in some cases, encroachment on the share of water needed to preserve the natural environment. Today, managing water resources and finding optimum strategies for water allocation has attracted the attention of policy makers. Water scarcity, especially in arid and semi-arid areas, makes the situation dire and challenging for water allocation. Such physical restrictions cause competition for water among irrigation and drainage networks and the ecosystem. In order to achieve optimum sustainable strategies for allocation of water between multiple stakeholders, a set of conflicting (negatively or weakly correlated) and incommensurable objective functions should be addressed.
To reach optimum crop area and water allocation schemes, one can optimize the amount of water allocated to the plant by considering the crop response to water deficit. The incorporation of crop-water production function into the linear objective function for maximizing agricultural net benefit would transform it to a non-linear objective function. Therefore, the problem would become more complicated.
Regarding practical problems, the optimization of the ratios of variables and parameters would provide a deeper understanding for managers and decision makers than optimizing each of them in the form of a separate objective function. Thus, such formulation can be considered as a tool to assess the sustainability of agricultural ecosystems. Combining the objectives by taking advantage of ratios would facilitate the process of analysis of alternative solutions. One of the issues in dealing with multi-objective optimization problems is the choice of a solution from a usually large set of non-inferior alternative points that make up the optimal tradeoff (Pareto) curve. One can take advantage of social selection methods to resolve conflicts between the objectives of different stakeholders by identifying the best settlement solutions from among an optimal set of non-inferior alternative solutions.
The aim of the current study is to develop a non-linear multi-objective optimization model for water allocation which takes into account crop-water production function, reservoir continuity, and surface water balance. The developed multi-purpose framework is based on four sustainability indicators, two fractional functions and two simple functions.

2. Methodology
2.1. Water bankruptcy
The two main components of a bankruptcy situation are the amount of resources available and the claimed amounts of the stakeholders. In most water allocation cases, the first component can be easily equated with the amount of water available for distribution between users at a specific time and place.
The proportional cutback rule is one of the long-established principles for bankruptcy situations. It is widely employed to manage water resources under water scarcity in different parts of the world. According to the proportional cutback rule, allocated water to each water claimant is according to a proportion of its claimed volume of water. This proportion is defined as the ratio of total available water to total water demands of water claimants.

2.2. Deficit irrigation
Deficit irrigation is an optimization policy in which plants are subjected to different levels of water shortages. In each condition, the desirable level of water shortage and the resultant crop yield are determined through an optimization process on the basis of crop-water production function. This function describes the relationship between the gained crop yield and the total amount of water consumed by the plant through evapotranspiration.

2.3. Social choice approaches
Social decision-making methods are procedures that take into account all individual preferences in order to reach a social preference. In the current study, social choice rules such as the plurality, Borda count, median voting, majoritarian compromise, pairwise comparison, and Condorcet choice have been employed to choose a compromise point among a set of non-inferior alternative solutions.

2.4. Proposed sustainability model for farming and the ecosystem
In this research, we developed an optimization model for multi-purpose agricultural and environmental sustainability. The main goal of the proposed model was to address the optimal allocation of water between farming and the ecosystem in the Dorudzan reservoir-river system. The developed model consists of four objective functions, two fractional functions and two simple functions.
The first objective is the maximum ratio of net income –in terms of deficit irrigation– to blue water utilization. In order to optimize the virtual water content in agricultural products, one can divide the crop water requirement into two parts, green and blue water, and attempt to improve the water use efficiency of irrigation water. The second objective is the maximum ratio of total crop yields to total blue water utilization or, in other words, the minimum total blue virtual water content.
The third objective is the maximum fairness in water allocation or minimum total difference between the amount of allocated water to each irrigation district and the water share, which is determined by the bankruptcy, proportional cutback, rule.
The fourth objective is the minimum shortage in meeting the water demand of the Bakhtegan Lake ecosystem.
The main constraints of the proposed model are reservoir continuity, surface water balance at the downstream of the dam, crops’ acreages, and water consumption in each irrigation district.

3. Results and Discussion
We computed the average values of the 12-month standardized runoff index (SRI) for the 7-year periods (33 periods) from 1976-2015. According to the outcomes, the period of 2007 to 2014 water years, with an annual average SRI value of -1.25, was selected as a period of water bankruptcy for implementation of a proposed simulation-optimization model for multipurpose sustainability of the Droudzan reservoir-river system.
By solving the developed model using the Non-dominated Sorting Genetic Algorithm II (NSGA-II), we generated an optimal trade-off set that contained 637 management alternatives. Each generated alternative offered different values for the objective functions of the model.
There was no socially optimal management option based on the median voting and majoritarian compromise rules. This output revealed that none of the generated efficient management alternatives could obtain the approval of the majority of stakeholders, not only at first ranking level, but at all possible ranking levels (637 levels). Moreover, the compromise set derived by the Condorcet rule was an empty set. The latter output indicated that there was no management option in the generated optimal trade-off set, which agreed upon by all stakeholders through the pairwise comparison procedure.
The output of the plurality rule was a compromise set with four members given the fact that the voting took place between the four aspects of sustainability. Therefore, each of the four identified options by the plurality rule was the most desirable option based on one of the four aspects of sustainability in the system under consideration.
Among applied social choice techniques, only the Borda count and pairwise comparison approaches were able to identify a compromise set with a single member, Management Option No. 87, which could be recognized as the socially-optimal conflict-resolving alternative for the problem under investigation.
According to the conflict-resolving management alternative, the 7-year (the bankruptcy period) average of the cultivation areas allocated to wheat, barley and canola would be 16%, 24%, and 41%, respectively. This management option would supply 46% of the water demand of Bakhtegan Lake. In terms of relative satisfaction, this alternative has the second rank based on three sustainability indicators of the blue water productivity, allocating water for preserving the natural environment, and fair distribution of water among identified socially-optimal management options.

4. Conclusion
Given the nonlinear and multi-objective nature of the problem under investigation, the output of the developed model was an optimal set of non-inferior solutions that form the trade-off curve between four aspects of sustainability (objective functions) in the Dorudzan reservoir-river system. The current study also attempted to find a compromise solution among generated non-inferior alternative solutions by using social choice rules. Results of the proposed methodology provided valuable information on efficient water governance strategies and how to achieve multi-purpose agricultural and environmental sustainability in the study region.