(512b) Fairness-Guided Design of Water Distribution Systems for Agricultural Lands

Allocating resources and associated wealth (utility) among multiple stakeholders is a fundamental problem in social planning. Allocation is often guided by maximizing the total utility (the sum of players utilities). This solution, also known as the social welfare approach, is intuitive but has significant defficiencies. In particular, this approach might identify optimal allocations that are nonunique (different allocations give the same maximum social welfare) and that improperly capture system scales [1]. A wide variety of alternative utility allocation schemes have been explored in the literature. Rawls [2] proposed an allocation scheme that seeks to address the scaling issue of the social welfare by maximizing the utility of the smallest stakeholder. Unfortunately, this approach ignores large stakeholders and can yield non-unique solutions. Nash [3, 4] proposed an allocation approach that maximizes the product of the stakeholder utilities. This approach is equivalent to maximizing the sum of the logarithms of the utilities and thus captures stakeholder scales more naturally. Moreover, this approach provides a unique solution. Recently, it has been established that the Nash allocation approach maximizes entropy (a measure of diversity) and that it can be generalized by using a measure called proportional fairness. An important application of fair allocation schemes is infrastructure systems, which are systems that must be shared with a large number of stakeholders that cover a large spectrum of scales. For instance, electricity markets allocate supply and generation by maximizing the social welfare [5]. Proportional fairness schemes are used to allicate bandwidth in telecommunication networks and air traffic flow management [6]. Similarly, the Rawlsian allocation scheme has been used in networking, telecommunication, routing, and load balancing problems [7].

Fairness measures are also widely used to quantify income inequality. Recently, it has been establishes that fairness can also be measured using descriptive statistics, such as entropy [8, 1]. This conceptually makes sense, since the distribution of resources among a set of stakeholders can be interpreted as a task that seeks to shape a distribution of outcomes (a fundamental problem in statistics and stochastic programming) [9, 10]. Allocation of resources in agricultural systems is essential for maintaining a sufficiently diversified infrastructure that enables long-term sustainability [11]. The optimal water allocation problem has been studied by Li and Guo [12]. Here, they proposed a conflict resolution (multi-objective) optimization model that factors in economic, social, and ecological functions. A social welfare scheme is used as economic objective. Similarly, a conflict resolution model including the minimization of fresh water consumption and the minimization of the total annual cost in agricultural systems was proposed in Arredondo-Ramırez et al. [13]. Here, the authors observed that, when irrigation water is limited, crops water requirements cannot be met and this creates an economic conflicts between the stakeholders. Allocation model for optimizing regional water resources among various crops was reported in [14]. The problem considers insufficiency of water supply and the objective function is to maximize the total benefit of all the crops productions (using a social welfare approach). A stochastic programming method has been reported in the literature to conduct crop planning and water resource allocation under uncertainty. In that work, the formulation includes the maximization of the agricultural system benefit given limited water resources [15]. Recent work has emphasized on the need for developing design tools meet and prioritize water demand needs under scarcity events [16]. Here, one can use optimization techniques to design sophisticated water distribution networks that make optimal decisions on use, reuse, and recycling of water by accounting for the water exchange among crops, storage tanks, and treatment units. Existing work has designed such infrastructures by maximizing the overall utility of all crops [17]. In summary, the vast majority of work on resource allocation in agricultural systems has focused on maximizing the social welfare [18].

This work presents an optimization formulation for the design of distribution networks that allocate water and associated utility to multiple stakeholders in a fair manner. The developed model accounts for the water requirements, yields and sales of each crop, as well as for the costs related to the water exchange, storage and distribution. We use our framework to highlight deficiencies of the social welfare approach as well as to illustrate the benefits of using alternative fairness measures in utility allocation. We demonstrate that such measures become particularly critical under extreme events (e.g., with scarse resources).

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