4-1- Groundwater Dam Site Selection with MCDM Approach
Nine different MCDM methods were used for the site selection problem, whose results are presented in Table 6. The programming process for all of these methods was conducted by using the MATLAB package. As Fig. 6 shows, alternative A1 (Tameh) was ranked first in 6 out of 9 methods. Alternative A5 (Hoseinabad) was ranked second in most of these methods. There are some differences between the methods in the ranking of other alternatives, which is one of the challenges of multi-criteria decision-making approach.
4-1-1- Coupling the Results (Hybrid-MCDM)
The variety of methods, and consequently, the variety of the solutions using different approaches lead to difference in the MCDM results. This has created uncertainty in the use of multi-criteria decision-making methods. For this purpose, in the present study, three algorithms, including Average, Borda, and Copeland, have been used to combine the results of different MCDM methods. These combining methods are described in detail in the research of Mohebali et al. (2019). In order to implement prioritization algorithms, it is necessary to create a table like Table 7 that shows the ranking of the alternatives in different methods. The “Average method” is calculated by averaging the rows of this matrix. According to this method, alternatives A1 and A5 occupy the first and second ranks (Table 7).
The second method is the Borda strategy. This method is a pairwise comparison between the ranks of different options in different MCDM methods. Borda matrix is performed by differencing the number of priority of row alternative from the priority of the column alternative in different MCDM methods. The final score for each alternative is obtained from the summation of the numbers in the row line of that alternative in the Borda matrix (Klamler 2005). Table 8 shows the results of combining different MCDM methods with the Borda strategy.
Copeland method is another simple majority strategy. In the Copeland method, having priority or not having priority is the important point. In the Copeland matrix, if the row alternative has priority to the column alternative, the number 1 is placed in the related element of the matrix. If the row alternative has not the priority to the column alternative, the number − 1 putting in the related element. Also, if row alternative and column alternative being equal, number 0 put in the element. The final score for each alternative is obtained from the summation of the numbers in the row line of that alternative (Klamler 2005, Mohebali et al. 2019). Table 9 shows the final results of the Copeland method for combining the solutions of different MCDM methods.
Finally, according to Table 10, the results of these three “Noise Free” methods are compared, and a single solution is presented for Hybrid-MCDM.
4-2- Groundwater Dam Site Selection with DEA Approach
4-2-1- Simple DEA
In this step, the same decision matrix formed in Table 5 is used here as well. The ten alternatives that are the groundwater dam options constitute the decision-making units (DMUs), and the 13 criteria presented in Fig. 6 constitute the inputs and outputs of the DEA algorithm.
In the Simple DEA method, as shown in Fig. 7, the three criteria C1, C3, and C9, are considered as inputs. In these three criteria the lowering values is better and thy called negative criteria. The other 10 criteria are considered as the output of the DEA model, and the higher values of them are preferred.
In the present study, the DEA problem has been solved by four methods, including CCR and BCC approaches by Input- and Output-oriented. The results of simple DEA method are presented in Table 11. As can be seen in Table 11, the “CCR Min-Input” and “CCR Max-Output” solutions are equal. In such cases, it is said that the CCR method is the correct DEA approach, and the solutions of the BCC model are not very desirable (Adler 2002). As shown in Table 11, in BCC Min-Input model, many options have the same rating and efficiency score. Therefore, it can be concluded that in a simple DEA approach, the CCR method has the best solutions to the problem of locating groundwater dams.
4-2-2- Socio-Economic DEA
The main purpose of constructing these kinds of dams is to prevent migration and preserve the population of rural areas. Due to the small scale of these projects, the socio-economic dimension of them is critical. Therefore, the socio-economic DEA method has been used to solve the site selection problem. In this case, according to Fig. 8, three socio-economic criteria affecting the construction of the dam, namely C3, C4, and C5, which include “Distance from villages”, “Population of downstream”, and “Surface area of benefited agricultural lands”, are considered as model outputs. The other 10 components are included as model inputs.
Since in solving the DEA problem, the inputs must be minimized, and the outputs should be maximized, so the numbers in the decision matrix should be changed. In three components C1, C3, and C9 that were used as output in the simple DEA method, the lower values are preferred. Now that we want to solve the problem with the socio-economic approach, C3 must be reversed and transferred to the outputs. The other components that were in the output must be reversed and transferred to the input. Afterward, the whole matrix is normalized by dividing it by the sum of each column. Table 12 shows the normalized matrix for socio-economic DEA calculations.
All calculations in four main DEA methods are performed using the “Supper-Efficiency” algorithm. The calculations were performed using MATLAB and Frontiers Analyst software. The results obtained for the socio-economic approach in all four methods are shown in Table 13. The solutions of the two CCR methods are the same and with considering this fact that the dominant approach in this issue is socio-economic and three socio-economic components are in the output of the model, the CCR Max-Output method have been chosen as the best model for solving the groundwater dam site selection problem. Therefore, options A1 and A5 are the first and second alternatives for the construction of a groundwater dam, which are consistent with the solutions of the Hybrid-MCDM method, which is obtained by combining nine methods and was noise-free. On the other hand, the advantage of the DEA method is that it does not need to be weighted by the experts and can only be solved by applying the decision matrix. Also, its programming is much easier than the Hybrid-MCDM method, and there is no need to solve 9 different methods and combining them. Another advantage of DEA is that it can be implemented for millions of options, while except for AHP, other MCDM methods for more than 1000 alternatives have critical computer programing limitations.
As shown in Table 14, the DEA can distinguish efficient alternatives from inefficient ones. As shown in this table, out of 10 alternatives, only five options are efficient. Table 14 also shows the benefit of using the supper efficiency method in solving DEA-assisted locating problems. As shown in standard efficiency mode, the first five options are efficient, and number 100 is assigned to them. The supper efficiency approach caused a distinction between these five options and ranking between them.
4-2-3- Efficiency Analysis
The most important advantage of the DEA method is that it makes it possible to examine the efficiency of different alternatives. Two of the most important performance studies in the DEA method are Potential Improvement Analysis (PIA) and Input/output Contribution (IOC). In the following, both of these methods are examined in evaluating the efficiency of underground dam construction sites.
4-2-3-1- Potential Improvement
In potential improvement analysis the coefficients u and v for each option are determined and the alternative projected on the efficiency fronts (BCC and CCR). As a result, in each option, an ideal state will be obtained for each of the components. The distance of each component to that ideal state is calculated by using Eq. 4:
$$Potential{\text{ }}Improvement=\frac{{(Target - Value)*100}}{{Value}}$$
4
The potential improvement values for option A1, which is the first priority for the construction of an underground dam among the 10 alternatives, are presented in Table 15. Negative values of potential improvement indicate that the potential of that component can be decreased. It means that in which percent of the reduction in a component, the alternative is still efficient and is in the ideal state. For example, Table 15 shows that there is a potential improvement that the “rainfall” criteria decreased by 68%, and the alternative A1 will still be in the ideal state. In the case of “Stream Depth” criteria, there is no potential improvement for this option, and the criteria are not far from its ideal state.
Figure 9 shows the potential improvement values for the four options A1, A2, A3, and A9. An interesting and convenient point in the construction of groundwater dams is that, as shown in Fig. 9, for option A1 in the “Stream Width” criteria, a 49% reduction potential is obtained. It means that in option A1, the length of the dam can be reduced by 49% to reach the ideal and efficient state. Due to the low-cost nature of the groundwater dams, this analysis can play an effective role in the costs reducing and is a useful guide for engineers and project managers for further decisions. In other words, this analysis shows that if the length of the dam is reduced by 49%, option A1 is still the most desirable option among these 10 options.
Another interesting point about the C3, C4, and C5 criteria is that, according to the socio-economic approach, if the population of option A1 decreases by more than 86% after the operation, this option will no longer be efficient.
4-2-3-2- Input/output Contribution
Another efficiency analysis that the DEA method presents is input/output contribution (IOC) analysis. In this analysis, by obtaining the values of weights u and v for each of the components of each option, the contribution percentage of a certain criterion in the total performance of an alternative can be determine. Figure 10 shows the results of this analysis for the four options A1, A2, A3, and A9. The numbers are normalized as well as rounded. This analysis, which acts as a kind of sensitivity analysis, can determine which criteria played a greater role in the performance of each alternative. For example, for option A1, which is the most desirable option for the construction of a groundwater dam, the components of “stream depth”, “population” and “area of benefited agricultural lands” have the greatest impact on the efficiency of this option.
Figure 10 shows the same influential components for the other options, and we see that according to the socio-economic approach in the analysis, the components C3, C4, and C5 had a high contribution for all four options, and they play an effective role in the efficiency of these dams. Such kind of analyzes provides the possibility of a more in-depth study in selecting the optimal alternative in the construction of water structures, especially groundwater dams.