Multicriterion decision making in groundwater planning

The groundwater planning problems are often multiobjective. Due to conflicting objectives and non-linearity of the variables involved, several feasible solutions may have to be evolved rather than single optimal solution. In this study, the simulation model built on an Analytic Element Method (AEM) and the optimization model built on a Non-dominated Sorting Genetic Algorithm (NSGA-II) were coupled and applied to study a part of the Dore river catchment, France. The maximization of discharge, the minimization of pumping cost and the minimization of piping cost are the three objectives considered. 2105 non-dominated groundwater planning strategies were generated. K-Means cluster analysis was employed to classify the strategies, and clustering was performed for 3 to 25 clusters. A cluster validation technique, namely Davies–Bouldin (DB) index, was employed to find the optimal number of clusters of groundwater strategies which were found to be 20. Multicriterion Decision-Making (MCDM) techniques, namely VIKOR and TOPSIS, were developed to rank the 20 representative strategies. Both these decision-making techniques preferred representative strategy A5 (piping cost, pumping cost and discharge respectively of 880,000 Euro, 679,000 Euro and 1,263.1 m/s). The sensitivity analysis of parameter v in VIKOR suggested that there were changes in ranking pattern for various values of v. However, the first position remained unchanged.


INTRODUCTION
Growing water demand necessitates the development of cost-effective wells and transport system. This involves the development of groundwater pumping and transport systems. A major part of the total energy consumption, in a water distribution network, is for extracting the groundwater. This consumption becomes more dominant due to unmanaged extraction and corresponding diminishing of groundwater resources (Ahlfeld & Laverty ). Therefore, optimal wells and transport system are expected to achieve sustainable economic benefits and utilization of groundwater resources, avoiding reconstruction of pumping wells in view of declining groundwater resources in a multiobjective framework. The use of a simulation-optimization model was found suitable for different groundwater management problems (Onwunalu &   Therefore, the present study is focused on the application of clustering to find the representative solutions from generated Pareto front that can be used in multicriterion analysis.
Keeping these developments, the present study focuses on the following objectives: • To explore NSGA-II as the multiobjective optimization algorithm for generating non-dominated groundwater strategies to a part of the Dore river catchment, France.
• To develop a methodology for clustering and ranking of non-dominated strategies that can be used as the basis for groundwater policy studies.
Objectives are proposed to be achieved by employing the following four-stage procedure: 1. NSGA-II-based multiobjective optimization to generate non-dominated groundwater strategies based on objective functions: maximize discharge, minimize cost of piping and minimize cost of pumping.
2. K-Means based cluster analysis for grouping the number of strategies generated in stage 1 into clusters.
3. The DB cluster validation index for finding the optimum cluster size.
4. Application of two MCDM methods, TOPSIS and VIKOR, to select the best groundwater strategy.
To our knowledge, no earlier study is reported in groundwater planning incorporating approaches similar to the ones suggested in the present work.
The next section presents the study area and model development; Section 'Methodologies employed' describes the employed methodologies. Section 'Stage 1: model application' presents results related to stage 1 (multiobjective modelling), Section 'Stages 2 and 3: cluster analysis and validation' is related to stages 2 and 3 (clustering aspects) and Section 'Stage 4: application of VIKOR and TOPSIS' is related to stage 4 (ranking aspects). The last section presents the conclusions of the study.

STUDY AREA AND MODEL DEVELOPMENT
The study area consists of part of the Dore river catchment, (AEM). The aquifer is considered to be homogenous as it is located between the two rivers with alluvium property. The AEM is particularly developed to deal with simulationoptimization problems and it was found efficient (Gaur et al. a, b). The AEM has some advantages such as fast convergence as the head can be computed directly at the desired location without getting the head for a whole grid system. Here also, the authors are trying to utilize the benefits in the application of simulation-optimization techniques in groundwater management problem.
The calibration of AEM was performed using 12 available piezometric measurements (Gaur et al. a, b), Therefore, the AEM is coupled with Particle Swarm Optimization (PSO) so that potentials from all the known elements are stored in a single matrix. So for each run, this model does not calculate the value for the whole equation again and again. This coupling of the models reduces the convergence time (Gaur et al. a, b). The development of wells and transport systems is multiobjective, and the description of objectives is as follows: The maximization of discharge of water that can be extracted from the aquifer through the given number of wells is: where Q, h, P(Q) and P(h), N w respectively are discharge (m 3 /s), head (m), penalty terms and the number of wells.
The total cost of pumping from each well is to be minimized and is expressed as follows (Gaur et al. a, b): where H is the head from the water table in the aquifer to the water level in the storage tank. The height of the storage tank is 5 m; η is the combined efficiency of pump and prime mover; R E is the cost of electricity per kilowatt-hour (euros/ kwh); r is the rate of interest (euros/euros/year); T is the life of project (year); k P is the cost of pumping per kwh (euros); when T → ∞, r T ¼ 1/r. γ is the specific weight of water (N/m 3 ).
The total cost of a new pipe system is to be minimized and is expressed as follows: where A 2 is the cost of piping includes the earthwork and other miscellaneous items and L is the total length of pipes. Decision variables are co-ordinates and discharge of wells.
The constraints incorporated into the model are as follows: where h i,min is the minimum allowable head of groundwater at ith well; Q i,min and Q i,max are minimum and maximum discharge limits for ith well; S w,min is the minimum distance between any pair of wells; x j and y j are co-ordinates of the well, i.e. i ≠ j. More details of modelling are available in Gaur et al. (a, b). Limits of the drawdown were defined as 258 m, whereas minimum and maximum discharge limits were 120 to 280 m 3 /h. Retention time was considered 20 h.

METHODOLOGIES EMPLOYED
Even though a number of advanced multiobjective optimization techniques were developed and are available, we propose to explore NSGA-II as it is highly efficient in preserving elitism, crowding-distance mechanism, fast nondominated sorting mechanism, robustness and convergence Separation measure of each strategy a from the antiideal Ranking basis Higher C a value is preferred  • We used the system without parallelization; hence, there is no migration of the sub-population and consequently no effect of migration fraction.
• The creation function generated the initial population which was set as 'feasible'. This creates uniformly distributed points and handles linear constraints, if any.
• Tournament selection was chosen as the selection function for choosing parents for the next generations. For our problem, changing the mutation rate from 0.1 to 0.8 did not affect the range of solutions obtained; so it was again set to the default value of 0.1.
• Decreasing the crossover fraction to 0.5 to 0.2 caused the penalty solutions being persistent, and increasing the value from 0.8 to 1, had no significant effect in diversity.
Hence, the default value of 0.8 was chosen.
• Pareto fraction defines the number of Pareto solutions to be obtained from the previous Pareto fronts. On increasing the Pareto front population fraction from 0.35 at intervals of 0.25, a decrease in penalty solutions was observed. Finally, the value of 0.85 was chosen.
• If the change of best fitness was very small over 100 generations (stall generations) it was considered that the solution was stable and the Pareto front had stopped moving. So, the function tolerance was set to 10 À5 . The are multiplied by any constant number, it will only act as a scaling factor and the solution will follow the same trend.
2105 non-dominated groundwater planning strategies were generated. These were difficult to handle directly for ranking; K-Means in conjunction with the DB index was used for finding the optimal size of clusters that could be further processed for ranking of representative strategies.

Stages 2 and 3: cluster analysis and validation
The normalization approach was used for making the criterion dimensionless (Raju & Nagesh Kumar ). of strategies among 2105 falling in each cluster is presented in Figure 5. For example, A1(1946): 6.22% meant that 6.22% of 2105 strategies are part of cluster 1; 1946 is the representative strategy in cluster 1 and A1 is the notation to present the corresponding representative strategy. It is observed that, in some of the clusters, the division of strategies is almost uniform. The representative strategy for each cluster needs to be determined. The squared error between the weighted normalized strategy in that group and group mean is computed.

Stage 4: application of VIKOR and TOPSIS
Matlab based VIKOR and TOPSIS codes were developed for ranking of A1 to A20. The methodologies of VIKOR and TOPSIS are presented in Tables 1 and 2. The code had provision for browsing input file, weights file, as well as the normalization approach. In the VIKOR method, user can choose any value of ν (between 0 and 1). Figure 7 presents values of S j , R j , Q j for ν ¼ 0. However, a significant effect is observed in other strategies. Spearman rank correlation (Gibbons ) between S1-S2, S1-S3 and S2-S3 are found to be 0.911, 0.844 and 0.977, respectively, indicating a reasonably strong correlation between different scenarios.  The methodology in our opinion is robust and can be replicated with suitable modifications. However, the outcome may vary for different periods and locations which we propose to study in future with the present methodology.
Since a similar methodology was not applied for the Dore catchment in the past, there is no mechanism to verify the effectiveness of the present approach. However, we propose to collect more micro level data and conduct field surveys to validate or test the efficacy of the present approach.
Water managers can perform sensitivity analysis to assess the impact of one unit of piping cost/pumping cost on discharge which will help analyse the impact of changing piping material cost or electricity cost due to inflation. With this hypothesis, water managers can use this information to optimally locate the pumping wells and their corresponding discharges to accomplish the water demand of nearby city 'Thiers' in a unified manner with future economical scenarios. This also unlocks the path to place the results into pragmatic use. However, a few more challenges still remain. Some amount of conviction is needed for the field experts and end users about the reasonableness and ability of methodologies that suit their thinking and requirements before implementing. The approach suggested here has an academic and practical flavour and we are confident that it would assist water resources planners and researchers.
We will take this opportunity to collect more field data, field visits and targeted to compare the present methodology with traditional methodology for more meaningful inferences. Efforts will also be made to minimize computational time to perform simulation and optimization process seamlessly.

CONCLUSIONS
The present study discussed the methodology consisting of a four-stage procedure comprising the application of NSGA-II, K-Means cluster analysis, DB index and ranking techniques for groundwater planning problem of the Dore    river catchment, France. The following specific observations are made from the present study: • It is observed that representative strategy A5 (strategy number 1274 falling in cluster 5) (piping cost, pumping cost and discharge respectively of 880,000 Euro, 679,000 Euro and 1,263.1 m 3 /s) is found to be the best by both VIKOR and TOPSIS.
• Spearman rank correlation suggests not so strong correlation between VIKOR and TOPSIS. The sensitivity analysis of changing values of ν did not show any impact on the top-ranking strategy (A5).
• Performing clustering of the NSGA-II generated nondominated strategies that minimized duplication of data and facilitated getting at the optimum number of clusters.
• AEM-based groundwater model is efficient to handle simulation-optimization model efficiently.
• Location of the storage tank and corresponding piping cost significantly impact the optimal location of pumping wells.