Conflicts and resolutions in managing water allocation at the watershed scale

The network river basin computer model ACQUANET (LabSid, 2018) is a simpler version of MODSIM (Labadie, 2015), a well-known model/software for solving water allocation problems for multiyear periods. For known hydrological conditions, ACQUANET simulates the operation of reservoirs located at most upstream locations of a watershed and allocates water to downstream users according to initial storages in reservoirs, specified operating rule curves at reservoirs, and the given priority scheme in overall water distribution. A physical model, usually tree-structured, is automatically modified into a close capacitated network model with a set of no-storage nodes and numerous physical and (added) artificial links. As such, a new model is a closed network, which must be in balance during the given time step (which is defined as one calendar month). After input data are submitted to the model, the multiyear allocation problem is solved as a chain of consecutive monthly optimizations, transferring the necessary information from month-to-month to preserve continuous time flow. This way ACQUANET behaves as a network simulation–optimization model.

The model is strictly deterministic, with the base assumption that hydrological conditions are known for 1 month ahead. At the beginning of any month within the multi-year period, the model “knows” what inflows will happen during this month, the required storage levels to be met in reservoirs at the end of that month, the demands at all demand points, capacities at all links within a system (rivers, canals, pipelines, transfer waterways, etc.), and what are the priorities of all stated targets (reservoirs’ storage levels and downstream demands).

The 1-month allocation problem is stated as network linear programming problem (1)min F=ΣijCijXij,for all i and j \min \,F = \Sigma _{ij} C_{ij} X_{ij} ,{\rm{for}}\,{\rm{all}}\,i\,{\rm{and}}\,j subject to balance requirements at all nodes (2)ΣjXji-ΣjXij=0,for all i\Sigma _j X_{ji} - \Sigma _j X_{ij} = 0,{\rm{for}}\,{\rm{all}}\,i and satisfaction of flow conditions at all links (3)Lij≤Xij≤Uij,for all i and jL_{ij} \le X_{ij} \le U_{ij} ,{\rm{for}}\,{\rm{all}}\,i\,{\rm{and}}\,j where i and j are used to identify nodes in a network; X_ij is the flow in the link [i,j]; C_ij is the unit cost of flow through the link [i,j]; and L_ij and U_ij are lower and upper limits on the flow through the link [i,j].

The allocation problems (1)–(3) are solved for each month using given network parameters (inflows, demands, rule curves at reservoirs, capacities on links, demands’ priorities scheme, etc.). The solver in ACQUANET is based on a Lagrangian overrelaxation algorithm, the network solver is written in Fortran, and its interface is created with Visual Basic. The ACQUANET model is selected for its simplicity, user friendliness, and possibilities to include into analysis irrigation and power generation—which suits to both case studies selected in Serbia and Portugal.

Among the methods for solving discrete optimization problems with more than one criterion and a number of alternatives, the AHP (Saaty, 1980) is one of the most used methods in both individual and group environments. This method efficiently handles one of the key issues in decision-making: eliciting judgments from the DM about the importance of a given set of decision elements regarding the overall goal, with criteria set introduced to locally prioritize (numerically) exposed judgments. If a problem can be structured hierarchically, then a ratio scale can serve as an effective tool to enable this hierarchy by performing pair-wise comparisons.

The core of the AHP lies in presenting the problem as a hierarchy (illustrated in Figure 3) and pairwise comparing the hierarchical elements using Saaty’s 9-point scale (Saaty, 1980). In this way, the importance of one element over another is expressed with regards to the element in the higher level. The AHP creates the so-called local comparison matrices at all levels of a hierarchy and performs logical syntheses of their (local) priority vectors. The major feature of the AHP is that it can thus include in the same framework a variety of tangible and intangible goals, attributes, and other decision elements. In addition, it reduces complex decisions to a series of pairwise comparisons, implementing a structured, repeatable, and justifiable decision-making approach.

Figure 3

In the standard AHP, an eigenvector method is used to derive the weights from local matrices. After the local weights are calculated at all levels of the hierarchy, the synthesis consists of multiplying the criterion-specific weight of the alternative by the corresponding criterion weight and summing up the results to obtain composite weights for the alternative with respect to the goal. This procedure is unique for all alternatives and all criteria.

The AHP is designed to support decision-making processes in both individual and group contexts. Its recent applications in water management that can be found, for example, in Blagojevic et al. (2016a, b), Amineh et al. (2017), Pluchinotta et al. (2018), Karlsson-Vinkhuyzen et al. (2018), and Srdjevic et al. (2018).

In a group decision-making process, both consensus and consistency need to be pursued and sought after. A solution with a high level of consensus is desirable. Many researchers focus on how to define acceptable level of consensus and, in turn, how to achieve it (Moreno-Jimenez et al., 2008; Alonso et al., 2010; Bezerra et al., 2014; Blagojevic et al., 2016b; Brandt et al., 2017; Dong et al., 2017). Beside methods proposed in given literature, there are also other formal mathematical methods for reaching the consensus, such as consensus convergence modeling or central tendency methods (middle “value” is measured using the mean, median, or mode). One of the best-known formal models is the consensus convergence model (Lehrer and Wagner, 1981), where by repeatable mathematical procedure and through mutual respect, the DMs do not only achieve consensus on the issue under consideration but also agree on the overall relative weight of each member of the group.

This model is considered a suitable conflict resolution method in water management problems because its mathematical structure captures the typical situation of disagreement. Refusing to change one’s opinion is equivalent, in mathematical terms, to assigning a null weight to other members and full weight to oneself (Hartmann et al., 2009). This situation is pure dogmatism, which is widely seen as unacceptable in modern decision-making.

The central idea of the consensus convergence model is assigning the stakeholders’ beliefs about the expertise of other epistemic stakeholders on the issue at hand (Hartmann et al., 2009). The weight of respect, w_ij, describes the regard stakeholder, i, has for the opinion or expertise of stakeholder, j, and ∑j=1nωij=1\sum\limits_{j = 1}^n {\omega _{ij} = 1} for the group of n stakeholders.

Here we use an adapted version of the consensus convergence model presented in Regan et al. (2006). The procedure is based on the original model introduced by Lehrer and Wagner (1981), which uses the weights of respect assigned by each stakeholder, and modified model defined by Regan et al. (2006). The later model proposes using the weights of respect based on the strength of differences in criteria weights assigned by individuals in the group.

In this model, we can assume that initial criteria weights of n stakeholders are p10p_1^0 , p20p_2^0 , ..., pn0p_n^0 and a metric that calculates weights of respect is (4)ωij=1-|pi0-pj0|∑j=1n1-|pi0-pj0|\omega _{ij} = {{1 - \left| {p_i^0 - p_j^0 } \right|} \over {\sum\limits_{j = 1}^n {1 - \left| {p_i^0 - p_j^0 } \right|} }} where i refers to the individual who is assigning the weights, j refers to the individual being assigned a weight, and n is the number of group members.

The weights of respect are used to create n × n size matrix W(5)W=[w1w2⋯w1nw2w2⋯w1n⋯⋯⋯⋯wn1wn2⋯wn]W = \left[ {\matrix{{w_1 } & {w_2 } & \cdots & {w_{1n} } \cr {w_2 } & {w_2 } & \cdots & {w_{1n} } \cr \cdots & \cdots & \cdots & \cdots \cr {w_{n1} } & {w_{n2} } & \cdots & {w_{{\textbf {\emph{n}}}} } } } \right]

If P is a vector of initial criteria weights, consensual vector of criteria weights can be obtained by the iterative equation (6)Pc=WPc-1P_c = WP_{c - 1}

The procedure is repeated until the values of criteria weights in vectors P_c and P_c-1 are equal within the tolerance error limit. Convergence is guaranteed if weights of respect are constant throughout the iteration process for each DM.

Participants in the decision-making process (stakeholders) are identified by responsible public water management authority (PWMA). A group of 23 invited individuals are briefed on main problems related to long-term planning and management of the hydro system, and particularly on a methodology applicable to resolving existing conflicts among water users. The final intent of a meeting was to gather parties around the same table and try to solve the common problem.

The participants are explained how to act in the decision-making session aimed to reach a consensus about a strategy aiming to achieve well-balanced system use and satisfaction of prescribed system purposes and users’ expectations and also respecting the defined system capacity requirements and societal wider interests. A discussion helped to elaborate the most important decision-making issues and to define a global goal to identify the most desired long-term management strategy. Three criteria (identified as economic, social and ecological) and five purposes of the system (irrigation [IR], drainage [DR], used waters [UW]/effluent from industry, industrial supply [IS], and other purposes [OP]) are adopted as evaluation filters that will apply to management strategies.