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Multi-Criteria Decision Analysis for the Purposes of Groundwater Control System Design

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Abstract

The best way for an engineer or scientist to express their knowledge, experience and opinions is day-to-day verbal communication. When a decision needs to be made about an optimal groundwater control system, the decision-making criteria need not always be numerical values. If fuzzy logic is used in multi-criteria decision-making, the criteria are described by linguistic variables that can be represented through fuzzy membership and expert judgment is used to describe such a system. Prior hydrodynamic modeling of the aquifer regime defines the management scenarios for groundwater control and provides an indication of their effectiveness. In this paper, the fuzzy analytic hierarchy process is applied to deal with a trending decision problem such as the selection of the optimal groundwater management system. Linguistic variables are used to evaluate all the criteria and sub-criteria that influence the final decision and the numerical weights of each alternative are determined by mathematical calculations. The paper presents a part of the algorithm – fuzzy optimization in hydrodynamic analysis, which leads to the selection of the optimal groundwater control system. The proposed method is applied in a real case study of an open-cast mine.

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Acknowledgements

Our gratitude goes to the Ministry of Education, Science and Technological Development of the Republic of Serbia for financing projects “OI176022“, „TR33039” and „III43004“.

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Authors and Affiliations

  1. University of Belgrade, Faculty of Mining and Geology, Department of Hydrogeology, Belgrade, Serbia

    D. Bajić, D. Polomčić & J. Ratković

Authors
  1. D. Bajić
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  2. D. Polomčić
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  3. J. Ratković
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Corresponding author

Correspondence to D. Bajić.

Appendix

Appendix

The present appendix details the calculations of the priority weight vectors (Step 3), the application of the aggregation principle (Step 4) and the calculations of the fuzzy decision matrix and fuzzy performance matrix (Step 5).

Step 3: determination of priority weight vectors. The “weights” in a matrix were determined using the so-called fuzzy extent analysis and the models shown for the criteria, sub-criteria and alternatives. According to the criteria matrix A, the priority weight vectors of the technical criteria (ω1), economic criteria (ω2) and ia (ω3) are:

$$ \begin{array}{l}{\omega}_1=\frac{\left[1,1,1\right]+\left[\frac{2}{3},1,2\right]+\left[\frac{1}{2},1,\frac{3}{2}\right]}{\left[1,1,1\right]+\left[\frac{2}{3},1,2\right]+\left[\frac{1}{2},1,\frac{3}{2}\right]+\left[\frac{1}{2},1,\frac{3}{2}\right]+\left[1,1,1\right]+\left[1,\frac{3}{2},2\right]+\left[\frac{2}{3},1,2\right]+\left[\frac{1}{2},\frac{2}{3},1\right]+\left[1,1,1\right]}\hfill \\ {}{\omega}_1=\left(0.166,0.328,0.660\right)\hfill \\ {}{\omega}_2=\frac{\left[\frac{1}{2},1,\frac{3}{2}\right]+\left[1,1,1\right]+\left[1,\frac{3}{2},2\right]}{\left[1,1,1\right]+\left[\frac{2}{3},1,2\right]+\left[\frac{1}{2},1,\frac{3}{2}\right]+\left[\frac{1}{2},1,\frac{3}{2}\right]+\left[1,1,1\right]+\left[1,\frac{3}{2},2\right]+\left[\frac{2}{3},1,2\right]+\left[\frac{1}{2},\frac{2}{3},1\right]+\left[1,1,1\right]}\hfill \\ {}{\omega}_2=\left(0.192,0.382,0.660\right)\hfill \\ {}{\omega}_3=\frac{\left[\frac{2}{3},1,2\right]+\left[\frac{1}{2},\frac{2}{3},1\right]+\left[1,1,1\right]}{\left[1,1,1\right]+\left[\frac{2}{3},1,2\right]+\left[\frac{1}{2},1,\frac{3}{2}\right]+\left[\frac{1}{2},1,\frac{3}{2}\right]+\left[1,1,1\right]+\left[1,\frac{3}{2},2\right]+\left[\frac{2}{3},1,2\right]+\left[\frac{1}{2},\frac{2}{3},1\right]+\left[1,1,1\right]}\hfill \\ {}{\omega}_3=\left(0.166,0.290,0.587\right)\hfill \end{array} $$

The ultimate criteria weights are:

$$ \omega =\left[\begin{array}{c}\hfill {\omega}_1\hfill \\ {}\hfill {\omega}_2\hfill \\ {}\hfill {\omega}_3\hfill \end{array}\right]=\left[\begin{array}{c}\hfill \left(0.166,0.328,0.660\right)\hfill \\ {}\hfill \left(0.192,0.382,0.660\right)\hfill \\ {}\hfill \left(0.166,0.290,0.587\right)\hfill \end{array}\right] $$

Then the priority weight vectors (ω) of the sub-criteria were determined relative to the considered criterion. The sub-criteria weights, in the order given, are:

  • T1: time

$$ \begin{array}{l}{\omega}_{T_1}=\frac{\left[1,1,1\right]+\left[\frac{1}{5},\frac{1}{3},1\right]+\left[3,5,7\right]+\left[5,7,9\right]+\left[1,3,5\right]}{\left[1,1,1\right]+\dots +\left[1,3,5\right]+\dots +\left[\frac{1}{7},\frac{1}{5},\frac{1}{3}\right]+\dots +\left[\frac{1}{9},\frac{1}{7},\frac{1}{5}\right]+\dots +\left[\frac{1}{5},\frac{1}{3},1\right]+\dots}\hfill \\ {}{\omega}_{T_1}=\left(0.152,0.331,0.684\right)\hfill \end{array} $$

  • T2: hydrogeologic conformity

$$ \begin{array}{l}{\omega}_{T_2}=\frac{\left[1,3,5\right]+\left[1,1,1\right]+\left[3,4,5\right]+\left[5,7,9\right]+\left[5,6,7\right]}{\left[1,1,1\right]+\dots +\left[1,3,5\right]+\dots +\left[\frac{1}{7},\frac{1}{5},\frac{1}{3}\right]+\dots +\left[\frac{1}{9},\frac{1}{7},\frac{1}{5}\right]+\dots +\left[\frac{1}{5},\frac{1}{3},1\right]+\dots}\hfill \\ {}{\omega}_{T_2}=\left(0.223,0.425,0.803\right)\hfill \end{array} $$

  • T3: efficiency

$$ \begin{array}{l}{\omega}_{T_3}=\frac{\left[\frac{1}{7},\frac{1}{5},\frac{1}{3}\right]+\left[\frac{1}{5},\frac{1}{4},\frac{1}{3}\right]+\left[1,1,1\right]+\left[1,2,3\right]+\left[1,3,5\right]}{\left[1,1,1\right]+\dots +\left[1,3,5\right]+\dots +\left[\frac{1}{7},\frac{1}{5},\frac{1}{3}\right]+\dots +\left[\frac{1}{9},\frac{1}{7},\frac{1}{5}\right]+\dots +\left[\frac{1}{5},\frac{1}{3},1\right]+\dots}\hfill \\ {}{\omega}_{T_3}=\left(0.050,0.131,0.287\right)\hfill \end{array} $$

  • T4: flexibility

$$ \begin{array}{l}{\omega}_{T_4}=\frac{\left[\frac{1}{9},\frac{1}{7},\frac{1}{5}\right]+\left[\frac{1}{9},\frac{1}{7},\frac{1}{5}\right]+\left[\frac{1}{3},\frac{1}{2},1\right]+\left[1,1,1\right]+\left[1,1,1\right]}{\left[1,1,1\right]+\dots +\left[1,3,5\right]+\dots +\left[\frac{1}{7},\frac{1}{5},\frac{1}{3}\right]+\dots +\left[\frac{1}{9},\frac{1}{7},\frac{1}{5}\right]+\dots +\left[\frac{1}{5},\frac{1}{3},1\right]+\dots}\hfill \\ {}{\omega}_{T_4}=\left(0.038,0.056,0.101\right)\hfill \end{array} $$

  • T5: reliability

$$ \begin{array}{l}{\omega}_{T_5}=\frac{\left[\frac{1}{5},\frac{1}{3},1\right]+\left[\frac{1}{7},\frac{1}{6},\frac{1}{5}\right]+\left[\frac{1}{5},\frac{1}{3},1\right]+\left[1,1,1\right]+\left[1,1,1\right]}{\left[1,1,1\right]+\dots +\left[1,3,5\right]+\dots +\left[\frac{1}{7},\frac{1}{5},\frac{1}{3}\right]+\dots +\left[\frac{1}{9},\frac{1}{7},\frac{1}{5}\right]+\dots +\left[\frac{1}{5},\frac{1}{3},1\right]+\dots}\hfill \\ {}{\omega}_{T_5}=\left(0.038,0.057,0.125\right)\hfill \end{array} $$

  • E1: capital expenditure

$$ \begin{array}{l}{\omega}_{E_1}=\frac{\left[1,1,1\right]+\left[\frac{1}{5},\frac{1}{3},1\right]+\left[\frac{1}{5},\frac{1}{4},\frac{1}{3}\right]}{\left[1,1,1\right]+\left[\frac{1}{5},\frac{1}{3},1\right]+\left[\frac{1}{5},\frac{1}{4},\frac{1}{3}\right]+\left[1,3,5\right]+\left[1,1,1\right]+\left[\frac{1}{3},\frac{1}{2},1\right]+\left[3,4,5\right]+\left[1,2,3\right]+\left[1,1,1\right]}\hfill \\ {}{\omega}_{E_1}=\left(0.076,0.121,0.267\right)\hfill \end{array} $$

  • E2: operating expenses

$$ \begin{array}{l}{\omega}_{E_2}=\frac{\left[1,3,5\right]+\left[1,1,1\right]+\left[\frac{1}{3},\frac{1}{2},1\right]}{\left[1,1,1\right]+\left[\frac{1}{5},\frac{1}{3},1\right]+\left[\frac{1}{5},\frac{1}{4},\frac{1}{3}\right]+\left[1,3,5\right]+\left[1,1,1\right]+\left[\frac{1}{3},\frac{1}{2},1\right]+\left[3,4,5\right]+\left[1,2,3\right]+\left[1,1,1\right]}\hfill \\ {}{\omega}_{E_2}=\left(0.127,0.344,0.802\right)\hfill \end{array} $$

  • E3: maintenance costs

$$ \begin{array}{l}{\omega}_{E_3}=\frac{\left[3,4,5\right]+\left[1,2,3\right]+\left[1,1,1\right]}{\left[1,1,1\right]+\left[\frac{1}{5},\frac{1}{3},1\right]+\left[\frac{1}{5},\frac{1}{4},\frac{1}{3}\right]+\left[1,3,5\right]+\left[1,1,1\right]+\left[\frac{1}{3},\frac{1}{2},1\right]+\left[3,4,5\right]+\left[1,2,3\right]+\left[1,1,1\right]}\hfill \\ {}{\omega}_{E_3}=\left(0.273,0.535,1.031\right)\hfill \end{array} $$

  • Ž1: drawdown

$$ \begin{array}{l}{\omega}_{{\overset{\check{}}{Z}}_1}=\frac{\left[1,1,1\right]+\left[\frac{3}{2},2,\frac{5}{2}\right]+\left[\frac{5}{2},3,\frac{7}{2}\right]}{\left[1,1,1\right]+\left[\frac{3}{2},2,\frac{5}{2}\right]+\left[\frac{5}{2},3,\frac{7}{2}\right]+\left[\frac{2}{5},\frac{1}{2},\frac{2}{3}\right]+\left[1,1,1\right]+\left[\frac{2}{3},1,\frac{3}{2}\right]+\left[\frac{2}{7},\frac{1}{3},\frac{2}{5}\right]+\left[\frac{2}{3},1,\frac{3}{2}\right]+\left[1,1,1\right]}\hfill \\ {}{\omega}_{{\overset{\check{}}{Z}}_1}=\left(0.383,0.554,0.776\right)\hfill \end{array} $$

  • Ž2: quality of quantity of pumped groundwater

$$ \begin{array}{l}{\omega}_{{\overset{\check{}}{Z}}_2}=\frac{\left[\frac{2}{5},\frac{1}{2},\frac{2}{3}\right]+\left[1,1,1\right]+\left[\frac{2}{3},1,\frac{3}{2}\right]}{\left[1,1,1\right]+\left[\frac{3}{2},2,\frac{5}{2}\right]+\left[\frac{5}{2},3,\frac{7}{2}\right]+\left[\frac{2}{5},\frac{1}{2},\frac{2}{3}\right]+\left[1,1,1\right]+\left[\frac{2}{3},1,\frac{3}{2}\right]+\left[\frac{2}{7},\frac{1}{3},\frac{2}{5}\right]+\left[\frac{2}{3},1,\frac{3}{2}\right]+\left[1,1,1\right]}\hfill \\ {}{\omega}_{{\overset{\check{}}{Z}}_2}=\left(0.158,0.231,0.351\right)\hfill \end{array} $$

  • Ž3: energy efficiency

$$ \begin{array}{l}{\omega}_{{\overset{\check{}}{Z}}_3}=\frac{\left[\frac{2}{7},\frac{1}{3},\frac{2}{5}\right]+\left[\frac{2}{3},1,\frac{3}{2}\right]+\left[1,1,1\right]}{\left[1,1,1\right]+\left[\frac{3}{2},2,\frac{5}{2}\right]+\left[\frac{5}{2},3,\frac{7}{2}\right]+\left[\frac{2}{5},\frac{1}{2},\frac{2}{3}\right]+\left[1,1,1\right]+\left[\frac{2}{3},1,\frac{3}{2}\right]+\left[\frac{2}{7},\frac{1}{3},\frac{2}{5}\right]+\left[\frac{2}{3},1,\frac{3}{2}\right]+\left[1,1,1\right]}\hfill \\ {}{\omega}_{{\overset{\check{}}{Z}}_3}=\left(0.149,0.215,0.322\right)\hfill \end{array} $$

The ultimate weights of the technical (ωT), economic (ωE) and environmental (ωŽ) sub-criteria were:

$$ \begin{array}{cc}\hfill {\omega}_T=\left[\begin{array}{c}\hfill {\omega}_{T_1}\hfill \\ {}\hfill {\omega}_{T_2}\hfill \\ {}\hfill {\omega}_{T_3}\hfill \\ {}\hfill {\omega}_{T_4}\hfill \\ {}\hfill {\omega}_{T_5}\hfill \end{array}\right]=\left[\begin{array}{l}\left(0.152,0.331,0.684\right)\\ {}\left(0.223,0.425,0.803\right)\\ {}\left(0.050,0.131,0.287\right)\\ {}\left(0.038,0.056,0.101\right)\\ {}\left(0.038,0.057,0.125\right)\end{array}\right]\hfill & \hfill {\omega}_E=\left[\begin{array}{c}\hfill {\omega}_{E_1}\hfill \\ {}\hfill {\omega}_{E_2}\hfill \\ {}\hfill {\omega}_{E_3}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill \left(0.166,0.328,0.660\right)\hfill \\ {}\hfill \left(0.192,0.382,0.660\right)\hfill \\ {}\hfill \left(0.166,0.290,0.587\right)\hfill \end{array}\right]\hfill \\ {}\hfill {\omega}_{\overset{\check{}}{Z}}=\left[\begin{array}{c}\hfill {\omega}_{{\overset{\check{}}{Z}}_1}\hfill \\ {}\hfill {\omega}_{{\overset{\check{}}{Z}}_2}\hfill \\ {}\hfill {\omega}_{{\overset{\check{}}{Z}}_3}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill \left(0.383,0.554,0.776\right)\hfill \\ {}\hfill \left(0.158,0.231,0.351\right)\hfill \\ {}\hfill \left(0.149,0.215,0.322\right)\hfill \end{array}\right]\hfill & \hfill \hfill \end{array} $$

Further, the weights (x) of the alternatives in all Y matrices (where the alternatives were evaluated by pairwise comparison against each sub-criterion) were calculated applying fuzzy extent analysis, taking into account all the sub-criteria, in the following order:

$$ \begin{array}{ll}{x}_{T_1}=\left[\begin{array}{l}{x}_{A_1}\\ {}{x}_{A_2}\\ {}{x}_{A_3}\end{array}\right]=\left[\begin{array}{l}\left(0.356,0.669,1.218\right)\\ {}\left(0.112,0.217,0.406\right)\\ {}\left(0.075,0.114,0.218\right)\end{array}\right]\hfill & {x}_{T_2}=\left[\begin{array}{l}{x}_{A_1}\\ {}{x}_{A_2}\\ {}{x}_{A_3}\end{array}\right]=\left[\begin{array}{l}\left(0.076,0.121,0.267\right)\\ {}\left(0.236,0.420,0.802\right)\\ {}\left(0.164,0.459,1.031\right)\end{array}\right]\hfill \\ {}{x}_{T_3}=\left[\begin{array}{l}{x}_{A_1}\\ {}{x}_{A_2}\\ {}{x}_{A_3}\end{array}\right]=\left[\begin{array}{l}\left(0.061,0.092,0.175\right)\\ {}\left(0.261,0.460,0.821\right)\\ {}\left(0.212,0.448,0.874\right)\end{array}\right]\hfill & {x}_{T_4}=\left[\begin{array}{l}{x}_{A_1}\\ {}{x}_{A_2}\\ {}{x}_{A_3}\end{array}\right]=\left[\begin{array}{l}\left(0.283,0.390,0.556\right)\\ {}\left(0.214,0.349,0.563\right)\\ {}\left(0.143,0.261,0.443\right)\end{array}\right]\hfill \\ {}{x}_{T_5}=\left[\begin{array}{l}{x}_{A_1}\\ {}{x}_{A_2}\\ {}{x}_{A_3}\end{array}\right]=\left[\begin{array}{l}\left(0.083,0.134,0.267\right)\\ {}\left(0.229,0.407,0.802\right)\\ {}\left(0.164,0.459,1.031\right)\end{array}\right]\hfill & {x}_{E_1}=\left[\begin{array}{l}{x}_{A_1}\\ {}{x}_{A_2}\\ {}{x}_{A_3}\end{array}\right]=\left[\begin{array}{l}\left(0.045,0.055,0.082\right)\\ {}\left(0.300,0.457,0.670\right)\\ {}\left(0.256,0.487,0.914\right)\end{array}\right]\hfill \\ {}{x}_{E_2}=\left[\begin{array}{l}{x}_{A_1}\\ {}{x}_{A_2}\\ {}{x}_{A_3}\end{array}\right]=\left[\begin{array}{l}\left(0.049,0.067,0.102\right)\\ {}\left(0.322,0.475,0.761\right)\\ {}\left(0.196,0.458,0.900\right)\end{array}\right]\hfill & {x}_{E_3}=\left[\begin{array}{l}{x}_{A_1}\\ {}{x}_{A_2}\\ {}{x}_{A_3}\end{array}\right]=\left[\begin{array}{l}\left(0.056,0.078,0.122\right)\\ {}\left(0.263,0.470,0.879\right)\\ {}\left(0.212,0.451,0.879\right)\end{array}\right]\hfill \\ {}{x}_{{\overset{\check{}}{Z}}_1}=\left[\begin{array}{l}{x}_{A_1}\\ {}{x}_{A_2}\\ {}{x}_{A_3}\end{array}\right]=\left[\begin{array}{l}\left(0.444,0.649,0.912\right)\\ {}\left(0.169,0.290,0.506\right)\\ {}\left(0.050,0.061,0.089\right)\end{array}\right]\hfill & {x}_{{\overset{\check{}}{Z}}_2}=\left[\begin{array}{l}{x}_{A_1}\\ {}{x}_{A_2}\\ {}{x}_{A_3}\end{array}\right]=\left[\begin{array}{l}\left(0.355,0.552,0.785\right)\\ {}\left(0.250,0.391,0.665\right)\\ {}\left(0.048,0.057,0.081\right)\end{array}\right]\hfill \\ {}{x}_{{\overset{\check{}}{Z}}_3}=\left[\begin{array}{l}{x}_{A_1}\\ {}{x}_{A_2}\\ {}{x}_{A_3}\end{array}\right]=\left[\begin{array}{l}\left(0.355,0.602,0.912\right)\\ {}\left(0.245,0.339,0.547\right)\\ {}\left(0.049,0.059,0.081\right)\end{array}\right]\hfill & \hfill \end{array} $$

Step 4 - application of the aggregation principle: The triangular fuzzy numbers were multiplied in this step – the criterion weights obtained in the previous steps were multiplied by the weights of their sub-criteria. This operation resulted in the ultimate sub-criteria “weights” (ω’). All the criteria and sub-criteria were thus aggregated into a single level, as the aggregation principle “removed” one level and simplified the system. The calculations are shown below:

$$ \begin{array}{c}\hfill {\omega^{\hbox{'}}}_T={\omega}_1\otimes {\omega}_T=\left[\begin{array}{c}\hfill {\omega}_1\otimes {\omega}_{T_1}\hfill \\ {}\hfill {\omega}_1\otimes {\omega}_{T_2}\hfill \\ {}\hfill {\omega}_1\otimes {\omega}_{T_3}\hfill \\ {}\hfill {\omega}_1\otimes {\omega}_{T_4}\hfill \\ {}\hfill {\omega}_1\otimes {\omega}_{T_5}\hfill \end{array}\right]=\left[\begin{array}{l}\left(0.166,0.328,0.660\right)\otimes \left(0.152,0.331,0.684\right)\\ {}\left(0.166,0.328,0.660\right)\otimes \left(0.223,0.425,0.803\right)\\ {}\left(0.166,0.328,0.660\right)\otimes \left(0.050,0.131,0.287\right)\\ {}\left(0.166,0.328,0.660\right)\otimes \left(0.038,0.056,0.101\right)\\ {}\left(0.166,0.328,0.660\right)\otimes \left(0.038,0.057,0.125\right)\end{array}\right]\hfill \\ {}\hfill {\omega^{\hbox{'}}}_E={\omega}_2\otimes {\omega}_E=\left[\begin{array}{c}\hfill {\omega}_2\otimes {\omega}_{E_1}\hfill \\ {}\hfill {\omega}_2\otimes {\omega}_{E_2}\hfill \\ {}\hfill {\omega}_2\otimes {\omega}_{E_3}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill \left(0.192,0.382,0.660\right)\otimes \left(0.166,0.328,0.660\right)\hfill \\ {}\hfill \left(0.192,0.382,0.660\right)\otimes \left(0.192,0.382,0.660\right)\hfill \\ {}\hfill \left(0.192,0.382,0.660\right)\otimes \left(0.166,0.290,0.587\right)\hfill \end{array}\right]\hfill \\ {}\hfill {\omega^{\hbox{'}}}_{\overset{\check{}}{Z}}={\omega}_3\otimes {\omega}_{\overset{\check{}}{Z}}=\left[\begin{array}{c}\hfill {\omega}_3\otimes {\omega}_{{\overset{\check{}}{Z}}_1}\hfill \\ {}\hfill {\omega}_3\otimes {\omega}_{{\overset{\check{}}{Z}}_2}\hfill \\ {}\hfill {\omega}_3\otimes {\omega}_{{\overset{\check{}}{Z}}_3}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill \left(0.166,0.290,0.587\right)\otimes \left(0.383,0.554,0.776\right)\hfill \\ {}\hfill \left(0.166,0.290,0.587\right)\otimes \left(0.158,0.231,0.351\right)\hfill \\ {}\hfill \left(0.166,0.290,0.587\right)\otimes \left(0.149,0.215,0.322\right)\hfill \end{array}\right]\hfill \end{array} $$

The ultimate sub-criteria weights were:

$$ \begin{array}{ll}{\omega^{\hbox{'}}}_{T_1}=\left(0.025,0.108,0.451\right)={W}_1\hfill & {\omega^{\hbox{'}}}_{T_2}=\left(0.037,0,139,0.530\right)={W}_2\hfill \\ {}{\omega^{\hbox{'}}}_{T_3}=\left(0.008,0.043,0.190\right)={W}_3\hfill & {\omega^{\hbox{'}}}_{T_4}=\left(0.006,0.018,0.067\right)={W}_4\hfill \\ {}{\omega^{\hbox{'}}}_{T_5}=\left(0.006,0.019,0.082\right)={W}_5\hfill & {\omega^{\hbox{'}}}_{E_1}=\left(0.015,0.046,0.176\right)={W}_6\hfill \\ {}{\omega^{\hbox{'}}}_{E_2}=\left(0.024,0.131,0.529\right)={W}_7\hfill & {\omega^{\hbox{'}}}_{E_3}=\left(0.052,0.204,0.680\right)={W}_8\hfill \\ {}{\omega^{\hbox{'}}}_{{\overset{\check{}}{Z}}_1}=\left(0.064,0.161,0.455\right)={W}_9\hfill & {\omega^{\hbox{'}}}_{{\overset{\check{}}{Z}}_2}=\left(0.026,0.067,0.206\right)={W}_{10}\hfill \\ {}{\omega^{\hbox{'}}}_{{\overset{\check{}}{Z}}_3}=\left(0.025,0.063,0.189\right)={W}_{11}\hfill & \hfill \end{array} $$

Step 5: The fuzzy decision matrix and fuzzy performance matrix were solved. The fuzzy decision matrix was obtained on the basis of fuzzy extent analysis of the alternatives:

$$ \begin{array}{l}X=\left[\begin{array}{l}\left(0.356,0.669,1.218\right)\left(0.076,0.121,0.267\right)\left(0.061,0.092,0.175\right)\left(0.283,0.390,0.556\right)\\ {}\left(0.112,0.217,0.406\right)\left(0.236,0.420,0.802\right)\left(0.261,0.460,0.821\right)\left(0.214,0.349,0.563\right)\\ {}\left(0.075,0.114,0.218\right)\left(0.164,0.459,1.031\right)\left(0.212,0.448,0.874\right)\left(0.143,0.261,0.443\right)\end{array}\right.\hfill \\ {}\begin{array}{l}\kern3em \left(0.083,0.134,0.267\right)\left(0.045,0.055,0.082\right)\left(0.049,0.067,0.102\right)\left(0.056,0.078,0.122\right)\\ {}\kern3em \left(0.229,0.407,0.802\right)\left(0.300,0.457,0.670\right)\left(0.322,0.475,0.761\right)\left(0.263,0.470,0.879\right)\\ {}\kern3em \left(0.164,0.459,1.031\right)\left(0.256,0.487,0.914\right)\left(0.196,0.458,0.900\right)\left(0.212,0.451,0.879\right)\end{array}\hfill \\ {}\left.\begin{array}{l}\kern3em \left(0.444,0.649,0.912\right)\left(0.355,0.552,0.785\right)\left(0.355,0.602,0.912\right)\\ {}\kern3em \left(0.169,0.290,0.506\right)\left(0.250,0.391,0.665\right)\left(0.245,0.339,0.547\right)\\ {}\kern3em \left(0.050,0.061,0.089\right)\left(0.048,0.057,0.081\right)\left(0.049,0.059,0.081\right)\end{array}\right]\hfill \end{array} $$

The fuzzy performance matrix represented the overall effectiveness of each of the alternatives across the sub-criteria:

$$ \begin{array}{l}Z=\left[\begin{array}{l}\left(0.009,0.072,0550\right)\left(0.003,0.017,0.141\right)\left(0.001,0.004,0.033\right)\left(0.002,0.007,0.037\right)\\ {}\left(0.003,0.024,0.183\right)\left(0.009,0.059,0.425\right)\left(0.002,0.020,0.156\right)\left(0.001,0.006,0.038\right)\\ {}\left(0.002,0.012,0.099\right)\left(0.006,0.064,0.546\right)\left(0.002,0.019,0.166\right)\left(0.001,0.005,0.030\right)\end{array}\right.\hfill \\ {}\begin{array}{l}\kern3em \left(0.001,0.003,0.022\right)\left(0.001,0.003,0.014\right)\left(0.001,0.009,0.054\right)\left(0.003,0.016,0.083\right)\\ {}\kern3em \left(0.001,0.008,0.066\right)\left(0.004,0.021,0.118\right)\left(0.008,0.062,0.403\right)\left(0.014,0.096,0.598\right)\\ {}\kern3em \left(0.001,0.009,0.085\right)\left(0.004,0.022,0.161\right)\left(0.005,0.060,0.476\right)\left(0.011,0.092,0.598\right)\end{array}\hfill \\ {}\left.\begin{array}{l}\kern3em \left(0.028,0.104,0.415\right)\left(0.009,0.037,0.162\right)\left(0.009,0.038,0.172\right)\\ {}\kern3em \left(0.011,0.047,0.230\right)\left(0.007,0.026,0.137\right)\left(0.006,0.021,0.103\right)\\ {}\kern3em \left(0.003,0.010,0.041\right)\left(0.001,0.004,0.017\right)\left(0.001,0.004,0.015\right)\end{array}\right]\hfill \end{array} $$

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Bajić, D., Polomčić, D. & Ratković, J. Multi-Criteria Decision Analysis for the Purposes of Groundwater Control System Design. Water Resour Manage 31, 4759–4784 (2017). https://doi.org/10.1007/s11269-017-1777-4

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