Application of interval valued intuitionistic fuzzy TOPSIS for flood management

Abstract

The technique for order preference by similarity to ideal solution (TOPSIS) has been applied to numerous multi-criteria decision making (MCDM) problems where crisp numbers are utilised in defining linguistic evaluation. The interval valued intuitionistic fuzzy TOPSIS (IVIF TOPSIS) can offer a new decision making method in solving MCDM problems where interval valued intuitionistic fuzzy sets are utilised in defining linguistic terms. Differently from the TOPSIS, which directly utilised crisp numbers, this method introduces upper and lower intervals of memberships to capture wide arrays of uncertain and fuzzy information. In this paper, criteria and alternatives in flood management is investigated where the best alternative in flood mitigation approach can be identified using the IVIF TOPSIS. Four decision makers in flood management were invited to provide linguistic evaluation of seven alternatives with respect to seven criteria. Computational results indicate that the alternative ‘pumping station’ is identified as the best alternative of flood mitigation project. The findings of this study would benefit authority in suggesting the effective approach in flood mitigation initiatives.

Keywords

TOPSIS interval valued intuitionistic fuzzy set linguistic evaluation decision making flood management

1 Introduction

The technique for order preference by similarity to ideal solution (TOPSIS) method is a major area of interest within the field of decision making. The method was developed by Hwang and Yoon [8] and since then, the method has been applied to various multi-criteria decision making (MCDM) problems. The ultimate aim of TOPSIS is to rank the different alternatives by measuring their relative closeness to positive ideal solution and negative ideal solution. In TOPSIS, the performance ratings and the weights of the criterion are given in exact values or crisp values. Triantaphyllou and Lin [19] made an effort to develop the fuzzy TOPSIS method where performance ratings and the weights are made using fuzzy numbers and fuzzy arithmetic operations. The fuzzy TOPSIS was developed further by substituting fuzzy sets with dual memberships of intuitionistic fuzzy sets and become intuitionistic fuzzy TOPSIS. The development of fuzzy TOPSIS does not stop at intuitionistic fuzzy TOPSIS, instead interval valued intuitionistic fuzzy TOPSIS (IVIF TOPSIS) was developed [14, 23]. The IVIF TOPSIS is an extended technique for TOPSIS method for group decision making where interval valued intuitionistic fuzzy set (IVIFS) are used to deal with selection problem under uncertain and incomplete information [4, 9].

The TOPSIS method combined with interval valued intuitionistic fuzzy numbers via its linguistic terms has made various success in dealing with MCDM problems, thanks to the uncertain perception of decision maker’s. The beauty of IVIF TOPSIS lies on the interval values of dual memberships IVIFS, in which these intervals are embedded into the whole computation process of TOPSIS. One of the critical steps in TOPSIS is the way of weights of decision makers is obtained. According to Izadikhah [9], the weight of decision makers of TOPSIS are obtained using the concept of similarity measures. However, previous literature suggests that similarity measures are used to find degree of similarity between two sets and cannot adequately be used to decide the relative importance among decision makers. In contrast to this works, the present paper introduces equal weights of decision makers. In addition to that the concept of arithmetic mean is used to calculate weights of criteria. Moreover, rather than the application of normalised binary Hamming distance, this paper proposes a straightforward Euclidian distance between two IVIFSs as an edge to reduce the computational burden in obtaining positive ideal solution and negative ideal solution of separation measures of TOPSIS. These contributions would ease the computational process without losing the general framework of TOPSIS and preserve the dual memberships of IVIFS.

In the perspective of applications, previous studies have reported the applications of IVIF TOPSIS in solving many decision making problems. For example, Izadikhah [9] has extended TOPSIS method with interval valued intuitionistic fuzzy sets to solve the supplier selection problem under incomplete and uncertain information environment. Gupta et al., [5] integrated the TOPSIS and linear programming methods under interval-valued intuitionistic fuzzy environment and implemented it in solving a real-world case study in Taiwan urban areas where the best fuel buses were selected. Tooranloo et al., [18] integrated interval-valued intuitionistic fuzzy AHP (IVIF-AHP) and interval-valued intuitionistic fuzzy TOPSIS (IVIF-TOPSIS) to solve supplier evaluation and selection problem. Very recently, Hajek and Froelich [6] proposed the integration of TOPSIS with IVIFS cognitive maps and applied it to the supplier selection task. In another related research, Alaoui, and Tkiouat [1] combined the IVIFS with TOPSIS method to deal with the lack of sufficient statistical data in risk management of financial institutions. In another research related to supplier change management, Shi et al., [16] introduced an integrated approach of IVIFS-grey relational analysis-TOPSIS method for selection of green suppliers. Yue and Yue [21] combined the TOPSIS with information from questionnaire that is written in IVIFS and applied it to a perception research of satisfaction and loyalty among smartphone users. Kumar and Garg [10] proposed an extended TOPSIS combining the IVIFS and relative degree of closeness co-efficient. The proposed method based on set pair analysis was applied to investment selection.

It can be seen that most studies about the appli-cations of IVIF TOPSIS have focused on business, finance and management. So far, however, there has been little discussion about the applications of IVIF TOPSIS in solving flood management problem. Flood management is indeed a MCDM problem where several factors (criteria) could be the contributors to the flood occurrence [12 , 22]. Natural factors such as heavy rainfall, poor drainage, heavy widespread rain leads to land inundation, intense convection rain storms, and other local factors have become common factors in causing a significant number floods [13]. Human factors such as increase in impervious areas, disposal of solid wastes into rivers, obstruction and constriction in the rivers and sediments from land clearance and construction areas are also the rea-sons to cause flood problems. These are among the factors or criteria that become parts of the MCDM problems. Besides, there are also many alternatives that have been proposed to manage floods. Among the popular approaches or alternatives are constructing dam or reservoir, dikes, pumping station, flood barrier or flood wall, watershed, reten-tion pond and catchment areas. It is evidenced that the flood management is a multi-criteria problem, therefore, a fuzzy MCDM method that deal with interval of uncertainty is germane to the problem.

The IVIF TOPSIS can be used to deal with un-certainty in many other areas of management deci-sion problems and multiple criteria decision making problems. Hence, this paper aims to apply IVIF TOPSIS in suggesting the best alternative in miti-gating floods. The main contribution of this paper is the use of linguistic terms, which allows incor-porating IVIFS combines with TOPSIS in solving flood mitigation problem. Detailed explanations of the method used and its application to flood management are presented in the next sections. This paper is structured as follows. In Section 2, the basic notations, definitions and concepts of the IVIF are presented. Section 3 describes the methodology of this research. Implementation to a case of flood management is presented in Section 4. Section 5 concludes.

2 Preliminary

According to Hiroshi et al., [7] IVIFS emerges as an extension of both interval valued fuzzy sets and intuitionistic fuzzy sets. Atanassov and Gargov [2] generalized the concept of intuitionistic fuzzy sets to IVIFS, and explained some basic arithmetic operations of IVIFS. Xu and Cai [20] explained the concept of IVIFS, and provided some basic operational laws of IVIFS. All these concepts are defined as follows.

Definition 1. [2] An interval valued intuitionistic fuzzy set on a set x ≠ φ is an expression given by A = { 〈 x, μ_A (x) , v_A (x) 〉 |x ∈ X } . where μ_A : X → D [0, 1] , v_A : X → D [0, 1] , denote the degree of membership and non-membership with the condition 0 ≤ sup μ_A (x) + sup v_A ≤ 1.

Definitions 2. [2]. The intervals μ_A (x) and v_A (x) represent the degree of membership and non-membership of the element x to the set A respectively. Thus, for each x ∈ X, μ_A (x) and v_A (x) are closed intervals whose lower and upper end points are denoted by $μ_{AL} (x), μ_{AU} (x), v_{AL} (x)$ and v_AU (x) respectively.

Definition 3. [20]. Let A = ([a₁, b₁] , [c₁, d₁]) and B = ([a₂, b₂] , [c₂, d₂]) be any two IVIFSs, then their arithmetic operations are shown as follows:

A + B = ([a₁ + a₂ - a₁a₂, b₁ + b₂ - b₁b₂] , [c₁c₂, d₁d₂])

A ⊗ B = ([a₁a₂, b₁b₂] , [c₁ + c₂ - c₁c₂, d₁ + d₂ - d₁d₂])

$λ A = ([1 - (1 - a_{1})^{λ}, 1 - (1 - b_{1})^{λ}], [c_{1}^{λ}, d_{1}^{λ}]), λ > 0$

The IVIFSs are widely applied in decision analysis and information systems. The following section describes how IVIFSs are efficiently used in flood management problem under uncertain environment.

3 Methodology

This section describes the way this research is conducted in order to achieve the research objectives. The descriptions of criteria and alternatives, and the algorithm used in implementing the computation are included in this section.

3.1 Criteria, alternatives and linguistic terms

Based on literature in flood management, there are seven criteria that normally associated with flood occurrence. The criteria are economic factors (C1), social factors (C2), and environment factors (C3), technical factors (C4), political factors, (C5), legislative factors (C6) and management factors (C7). There are also several alternatives suggested to mitigate floods. Among the alternatives are dam or reservoir (A1), dikes (A2), pumping station(A3), flood barrier or flood wall (A4), watershed (A5), retention pond (A6), and catchment areas (A7). Four decision makers were invited to make evaluation on the most preferred alternatives with respect to the criteria.

The seven-scale linguistic terms defined in IVIFS are used by decision makers to evaluate alternatives with respect to criteria. These linguistic terms are widely used and applied in decision analysis and information systems. IVIFSs are helpful and efficient to apply and deal in many decision making problem under undetermined and uncertain environment. Table 1 shows the linguistic terms and their corresponding IVIFS.

Table 1
Linguistic terms and IVIFS [11]

Linguistic Terms IVIFS

Very Low (VL) ([0.00, 0.10], [0.85, 0.90])

Low (L) ([0.15, 0.25], [0.55, 0.60])

Medium Low (ML) ([0.30, 0.40], [0.45, 0.50])

Medium (M) ([0.50, 0.60], [0.25, 0.30])

Medium High (MH) ([0.60, 0.70], [0.15, 0.20])

High (H) ([0.70, 0.80], [0.05, 0.10])

Very High (VH) ([1.00, 1.00]), [0.00, 0.00])

Linguistic Terms	IVIFS
Very Low (VL)	([0.00, 0.10], [0.85, 0.90])
Low (L)	([0.15, 0.25], [0.55, 0.60])
Medium Low (ML)	([0.30, 0.40], [0.45, 0.50])
Medium (M)	([0.50, 0.60], [0.25, 0.30])
Medium High (MH)	([0.60, 0.70], [0.15, 0.20])
High (H)	([0.70, 0.80], [0.05, 0.10])
Very High (VH)	([1.00, 1.00]), [0.00, 0.00])

These linguistic terms are used to construct the decision matrix of TOPSIS. The next subsection describes the algorithm of IVIF TOPSIS where the linguistic terms are embedded in one of the computational steps.

3.2 Algorithm of interval valued intuitionistic fuzzy TOPSIS

The IVIF TOPSIS method determines the compromise ranking list and the compromise solution according to the measure of closeness to the ideal solution. The shortest distance to the positive ideal solution and the farthest distance to the negative ideal solution is ranked as the best alternatives. It is proposed that DMs evaluate the alternatives with respect to criterion by using the linguistic terms. Differently from Rouyendegh et al.,[15] where intuitionistic fuzzy numbered linguistic terms were introduced to TOPSIS, this paper proposes the inclusion of the IVIFS into the TOPSIS method. As a result of this merger, several new other contributions are made. Linguistic terms that are written in IVIFS are utilised to obtain aggregated IVIFS decision matrix and IVIFS weighting matrix, which later be transformed into aggregate weighted interval valued intuitionistic fuzzy decision matrix. In addition, our proposed method uses the concept of maximum and minimum membership of IVIFS in separation of positive ideal solution and negative ideal solution respectively. Appearance of IVIFS into the general structure of TOPSIS is made to overcome the issue of imprecision in linguistic judgement where interval values are expected to capture wider range of vagueness. The proposed method is carried out in ten computational steps. In Step 1 to step 5, the aggregated weighted decision matrix is obtained as an output of a combination of linguistic terms in IVIFS and the TOPSIS. Step 6 to Step 8 contribute to the separation of positive and negative ideal solution where the concept of averaged distances between IVIFS is used. The final two step are meant to create closeness coefficients of alternatives. The flow of computational steps that based on IVIFS are presented in Fig. 1.

Fig.1

Flow of computational steps of IVIF TOPSIS.

The steps of IVIF TOPSIS are proposed and summarized in the following algorithm.

Step 1: Generate a set of possible alternatives and criterion. Suppose that there are m feasible alternatives, denoted by A ={ A₁, A₂, …, A_m } and n criterions be C ={ C₁, C₂, …, C_n }.

Step 2: Generate a set of decision maker. Suppose that there are k decision makers, denoted by D ={ D₁, D₂, …, D_k }.

Step 3: Construct aggregated IVIF decision matrix Y_p of the pth decision maker and the average decision matrix $\bar{Y}$ . $Y_{p} = {(f_{ij}^{p})}_{m \times n} = \begin{matrix} C_{1} \\ C_{2} \\ ⋮ \\ C_{m} \end{matrix} \begin{matrix} \begin{matrix} A_{1} & A_{2} & \dots & A_{n} \end{matrix} \\ [\begin{matrix} f_{11}^{p} & f_{12}^{p} & \dots & f_{1 n}^{p} \\ f_{21}^{p} & f_{22}^{p} & \dots & f_{2 n}^{p} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ f_{m 1}^{p} & f_{m 2}^{p} & \dots & f_{mn}^{p} \end{matrix}] \end{matrix}$ (1) $\bar{Y} = {(f_{ij})}_{m \times n}, where f_{ij} = (\frac{f_{ij}^{1} \oplus f_{ij}^{2} \oplus \dots \oplus f_{ij}^{k}}{k}) .$

Step 4: Construct weighting matrix W of k decision maker and the average weighting matrix $\bar{W}$ . $W_{p} = {(w_{i}^{p})}_{1 \times m} = \begin{matrix} \begin{matrix} C_{1} & C_{2} & \dots & C_{m} \end{matrix} \\ [\begin{matrix} w_{1}^{p} & w_{2}^{p} & \dots & w_{m}^{p} \end{matrix}] \end{matrix}$ $\bar{W} = \begin{matrix} \begin{matrix} C_{1} & C_{2} & C_{3} & C_{4} & C_{5} & C_{6} & C_{7} \end{matrix} \\ [\begin{matrix} w_{1} & w_{2} & w_{3} & w_{4} & w_{5} & w_{6} & w_{7} \end{matrix}] \end{matrix}$ (2) where $\bar{W} = {(w_{i})}_{1 \times m}$ , and $w_{i} = \frac{w_{i}^{1} \oplus w_{i}^{2} \oplus w_{i}^{k}}{k}$

Step 5: Construct the aggregated weighted interval valued intuitionistic fuzzy decision matrix, D′. $D^{'} = D \otimes W = {(r_{ij}^{'})}_{m \times n}, r_{ij}^{'} = ([a_{ij}^{'}, b_{ij}^{'}], [c_{ij}^{'}, d_{ij}^{i}]) .$ (3)

Step 6: Determine positive ideal solution and negative ideal solution by using Equation (4).

$\begin{array}{l} {D^{'}}^{k +} = ({D^{'}}_{1}^{k +}, {D^{'}}_{2}^{k +}, ..., {D^{'}}_{m}^{k +}) \\ = (\begin{array}{l} 〈 [a_{1}^{k +}, b_{1}^{k +}], [c_{1}^{k +}, d_{1}^{k +}] 〉, \\ 〈 [a_{2}^{k +}, b_{2}^{k +}], [c_{2}^{k +}, d_{2}^{k +}] 〉, ..., \\ 〈 [a_{m}^{k +}, b_{m}^{k +}], [c_{m}^{k +}, c_{m}^{k +}] 〉 \end{array}) \end{array}$ (4)

$\begin{array}{l} {D^{'}}^{k -} = ({D^{'}}_{1}^{k -}, {D^{'}}_{2}^{k -}, ..., {D^{'}}_{m}^{k -}) \\ = (\begin{array}{l} 〈 [a_{1}^{k -}, b_{1}^{k -}], [c_{1}^{k -}, d_{1}^{k -}] 〉, \\ 〈 [a_{2}^{k -}, b_{2}^{k -}], [c_{2}^{k -}, d_{2}^{k -}] 〉, ..., \\ 〈 [a_{m}^{k -}, b_{m}^{k -}], [c_{m}^{k -}, c_{m}^{k -}] 〉 \end{array}) \end{array}$ (5) where $\begin{array}{l} {D^{'}}_{j}^{k +} = 〈 [a_{j}^{k +}, b_{j}^{k +}], [c_{j}^{k +}, d_{j}^{k +}] 〉 = \\ 〈 [\max_{i} a_{i j}^{k}, \max_{i} b_{i j}^{k}], [\min_{i} c_{i j}^{k}, \min_{i} d_{i j}^{k}] 〉, \end{array}$ $i = 1, . . ., n, j = 1, . . ., m, k = 1, . . ., K$ $\begin{array}{l} {D^{'}}_{j}^{k -} = 〈 [a_{j}^{k -}, b_{j}^{k -}], [c_{j}^{k -}, d_{j}^{k -}] 〉 = \\ 〈 [\min_{i} a_{i j}^{k}, \min_{i} b_{i j}^{k}], [\max_{i} c_{i j}^{k}, \max_{i} d_{i j}^{k}] 〉, \end{array}$ $i = 1, . . ., n, j = 1, . . ., m, k = 1, . . ., K$

Step 7: Calculate separation measures between the candidates and positive ideal solution for each decision maker by using Equation (6). $S_{i}^{+} (D_{i}, A^{+}) = {\frac{1}{4} \sum_{j = 1}^{n} [\begin{matrix} {(a_{ij} - a_{j}^{+})}^{2} + \\ {(b_{ij} - b_{j}^{+})}^{2} \\ + {(c_{ij} - c_{j}^{+})}^{2} + \\ {(d_{ij} - d_{j}^{+})}^{2} \end{matrix}]}^{\frac{1}{2}}$ (6) $i = 1, . . ., n, j = 1, . . ., m, k = 1, . . ., K$

Step 8: Calculate separation measures between the candidates and negative ideal solution for each decision maker by using Equation (7). $S_{i}^{-} (D_{i}, A^{-}) = {\frac{1}{4} \sum_{j = 1}^{n} [\begin{matrix} {(a_{ij} - a_{j}^{-})}^{2} + \\ {(b_{ij} - b_{j}^{-})}^{2} + \\ {(c_{ij} - c_{j}^{-})}^{2} + \\ {(d_{ij} - d_{j}^{-})}^{2} \end{matrix}]}^{\frac{1}{2}},$ (7) $i = 1, . . ., n, j = 1, . . ., m, k = 1, . . ., K$

Step 9: Calculate the closeness coefficient (RC_i) of $A_{i}^{'}$ by using Equation (8). ${RC}_{i} = \frac{S_{i}^{-}}{S_{i}^{-} + S_{i}^{+}}$ (8) $i = 1, 2, . . ., n$

Step 10: Rank the preference of alternatives according to ascending order of closeness coefficients. The best alternative is selected based on the highest closeness coefficient.

4 Implementation

In this paper, the IVIF TOPSIS is used to calculate the weights of each criterion and then ranked the alternatives based on the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution. The computational procedures of the IVIF TOPSIS of the case are implemented as follow

Step 1: Generate all possible alternatives and criteria. A ={ A₁, A₂, …, A₇ } and n criteria be C ={ C₁, C₂, …, C₇ }.

Step 2: Generate a set of decision makers. $D = {D_{1}, D_{2}, \dots, D_{4}} .$

Step 3: Construct aggregated IVIF decision matrix and the average decision matrix $\bar{Y}$ .

All the individual decision opinions is fussed into a group of opinion. The aggregated matrices from each decision maker are then averaged to construct an aggregated group decision matrix. The average of the group decision matrix is denoted as $\bar{Y}$ $\bar{Y} = \begin{matrix} \begin{matrix} A_{1} & A_{2} & A_{3} & A_{4} & A_{5} & A_{6} & A_{7} \end{matrix} \\ \begin{matrix} \begin{matrix} C_{1} \\ C_{2} \\ C_{3} \\ C_{4} \\ C_{5} \\ C_{6} \\ C_{7} \end{matrix} & [\begin{matrix} f_{11} & f_{12} & f_{13} & f_{14} & f_{15} & f_{16} & f_{17} \\ f_{21} & f_{22} & f_{23} & f_{24} & f_{25} & f_{26} & f_{27} \\ f_{31} & f_{32} & f_{33} & f_{34} & f_{35} & f_{36} & f_{37} \\ f_{41} & f_{42} & f_{43} & f_{44} & f_{45} & f_{46} & f_{47} \\ f_{51} & f_{52} & f_{53} & f_{54} & f_{55} & f_{56} & f_{57} \\ f_{61} & f_{62} & f_{63} & f_{64} & f_{65} & f_{66} & f_{67} \\ f_{71} & f_{72} & f_{73} & f_{74} & f_{75} & f_{76} & f_{77} \end{matrix}] \end{matrix} \end{matrix}$ where $\begin{matrix} f_{11} = \frac{MH + M + M + MH}{4} \\ = ([0 . 55, 0 . 65], [0 . 2, 0 . 25]) \\ \dots \\ \dots \\ \dots \\ f_{75} = ([0 . 5, 0 . 575], [0 . 3375, 0 . 375]), \\ f_{76} = ([0 . 45, 0 . 55], [0 . 325, 0 . 375]) \\ f_{77} = ([0 . 575, 0 . 65], [0 . 2625, 0 . 3]) \end{matrix}$

Step 4: Construct weighting matrix W of four decision makers and the average weighting matrix $\bar{W}$ .

The weighting matrix W of the four decision makers is constructed and the average weighting matrix $\bar{W}$ is calculated as below. $W_{1} = \begin{matrix} \begin{matrix} C_{1} & C_{2} & C_{3} & C_{4} & C_{5} & C_{6} & C_{7} \end{matrix} \\ [\begin{matrix} VH & VH & H & MH & H & MH & M \end{matrix}] \end{matrix}$ $W_{2} = \begin{matrix} \begin{matrix} C_{1} & C_{2} & C_{3} & C_{4} & C_{5} & C_{6} & C_{7} \end{matrix} \\ [\begin{matrix} M & MH & VH & M & MH & M & M \end{matrix}] \end{matrix}$ $W_{3} = \begin{matrix} \begin{matrix} C_{1} & C_{2} & C_{3} & C_{4} & C_{5} & C_{6} & C_{7} \end{matrix} \\ [\begin{matrix} VH & VH & H & H & MH & MH & H \end{matrix}] \end{matrix}$ $W_{4} = \begin{matrix} \begin{matrix} C_{1} & C_{2} & C_{3} & C_{4} & C_{5} & C_{6} & C_{7} \end{matrix} \\ [\begin{matrix} M & H & VH & H & M & M & M \end{matrix}] \end{matrix}$ $\bar{W} = \begin{matrix} \begin{matrix} C_{1} & C_{2} & C_{3} & C_{4} & C_{5} & C_{6} & C_{7} \end{matrix} \\ [\begin{matrix} w_{1} & w_{2} & w_{3} & w_{4} & w_{5} & w_{6} & w_{7} \end{matrix}] \end{matrix}$ where $\begin{matrix} w_{1} = (\frac{VH + M + VH + M}{4}) \\ = (\begin{matrix} [\frac{1 . 00 + 0 . 50 + 1 . 00 + 0 . 50}{4}, \frac{1 . 00 + 0 . 60 + 1 . 00 + 0 . 60}{4}], \\ [\frac{0 . 00 + 0 . 25 + 0 . 00 + 0 . 25}{4}, \frac{0 . 00 + 0 . 30 + 0 . 00 + 0 . 30}{4}] \end{matrix}) \\ = ([\frac{3}{4}, \frac{3 . 2}{4}], [\frac{0 . 5}{4}, \frac{0 . 60}{4}]) \\ = ([0 . 75, 0 . 8], [0 . 125, 0 . 15]) \end{matrix}$ $w_{2} = ([0 . 825, 0 . 875], [0 . 05, 0 . 075]),$ $w_{3} = ([0 . 85, 0 . 9], [0 . 025, 0 . 05]),$ $w_{4} = ([0 . 05, 0 . 625], [0 . 125, 0 . 175]),$ $w_{5} = ([0 . 6, 0 . 7], [0 . 15, 0 . 2]),$ $w_{6} = ([0 . 55, 0 . 65], [0 . 2, 0 . 25]),$ $w_{7} = ([0 . 55, 0 . 65], [0 . 2, 0 . 25])$

Step 5: Construct the aggregated weighted interval valued intuitionistic fuzzy decision matrix, D^′.

The aggregated weighted interval valued intuitionistic fuzzy decision matrix, D is then computed. $\begin{matrix} D^{'} = D \times W \\ = [\begin{matrix} D_{11}^{'} & D_{12}^{'} & D_{13}^{'} & D_{14}^{'} & D_{15}^{'} & D_{16}^{'} & D_{17}^{'} \\ D_{21}^{'} & D_{22}^{'} & D_{23}^{'} & D_{24}^{'} & D_{25}^{'} & D_{26}^{'} & D_{27}^{'} \\ D_{31}^{'} & D_{32}^{'} & D_{33}^{'} & D_{34}^{'} & D_{35}^{'} & D_{36}^{'} & D_{37}^{'} \\ D_{41}^{'} & D_{42}^{'} & D_{43}^{'} & D_{44}^{'} & D_{45}^{'} & D_{46}^{'} & D_{47}^{'} \\ D_{51}^{'} & D_{52}^{'} & D_{53}^{'} & D_{54}^{'} & D_{55}^{'} & D_{56}^{'} & D_{57}^{'} \\ D_{61}^{'} & D_{62}^{'} & D_{63}^{'} & D_{64}^{'} & D_{65}^{'} & D_{66}^{'} & D_{67}^{'} \\ D_{71}^{'} & D_{72}^{'} & D_{73}^{'} & D_{74}^{'} & D_{75}^{'} & D_{76}^{'} & D_{77}^{'} \end{matrix}] \end{matrix}$ $\begin{matrix} D_{11}^{'} {= f}_{11} \times w_{1} \\ = ([0 . 55, 0 . 65], [0 . 2, 0 . 25]) \times \\ ([0 . 75, 0 . 8], [0 . 125, 0 . 15]) \\ = ([0 . 4125, 0 . 52], [0 . 025, 0 . 0375]) \end{matrix}$ $\begin{matrix} D_{12}^{'} {= f}_{12} \times w_{2} \\ = ([0 . 65, 0 . 725], [0 . 1625, 0 . 2]) \times \\ ([0 . 75, 0 . 8], [0 . 125, 0 . 15]) \\ = ([0 . 4875, 0 . 58], [0 . 0203, 0 . 03]) \end{matrix}$ $D_{13}^{'} = ([0 . 4875, 0 . 58], [0 . 0203, 0 . 03]),$ $D_{14}^{'} = ([0 . 5437, 0 . 64], [0 . 0109, 0 . 0187]),$ $D_{15}^{'} = ([0 . 45, 0 . 56], [0 . 0188, 0 . 03]),$ $D_{16}^{'} = ([0 . 5063, 0 . 6], [0 . 0172, 0 . 0262])$ $\begin{matrix} D_{17}^{'} = ([0 . 5063, 0 . 6], [0 . 0172, 0 . 0262]) \\ D_{21}^{'} {= f}_{21} \times w_{2} \\ = ([0 . 575, 0 . 65], [0 . 2375, 0 . 275]) \times \\ ([0 . 825, 0 . 875], [0 . 05, 0 . 075]) \\ = ([0 . 4744, 0 . 5687], [0 . 0119, 0 . 0206]) \\ \dots \\ \dots \\ \dots \end{matrix}$ $D_{77}^{'} = ([0.3162, 0.4225], [0.0525, 0.075])$

Step 6: Determine positive ideal partner solution and negative ideal partner solution.

The positive ideal solution and negative ideal solution is determined using Equations (4) and (5). $A^{+} = = (\begin{matrix} 〈 [0.5525, 0.675], [0.0025, 0.0075] 〉, \\ 〈 [0.595, 0.6975], [0.0028, 0.0075] 〉, \\ 〈 [0.6163, 0.72], [0.0022, 0.0063] 〉, \\ 〈 [0.5525, 0.675], [0.0025, 0.0075] 〉, \\ 〈 [0.6587, 0.7425], [0.0025, 0.0063] 〉, \\ 〈 [0.5737, 0.675], [0.0034, 0.0087] 〉, \\ 〈 [0.7012, 0.7875], [0.0013, 0.0038] 〉 \end{matrix})$ $\overline{A} = (\begin{matrix} 〈 [0.3025, 0.4225], [0.04, 0.0625] 〉, \\ 〈 [0.2063, 0.3087], [0.08, 0.1125] 〉, \\ 〈 [0.1925, 0.2925], [0.085, 0.1187] 〉, \\ 〈 [0.1925, 0.2925], [0.085, 0.1187] 〉, \\ 〈 [0.2338, 0.3412], [0.07, 0.1] 〉, \\ 〈 [0.2338, 0.3412], [0.07, 0.0087] 〉, \\ 〈 [0.27, 0.385], [0.0625, 0.0875] 〉 \end{matrix})$

Step 7: Calculate separation measure between the candidates and positive ideal solution for each decision maker

The separation measure between the alternatives for positive ideal solution for each decision maker is calculated according to the Euclidean distance measure. $\begin{matrix} S_{1}^{+} (D_{1}, A^{+}) = {\frac{1}{4} \sum_{j = 1}^{n} [\begin{matrix} {(0.4125 - 0.5525)}^{2} + \\ {(0.52 - 0.675)}^{2} + \\ {(0.025 - 0.0025)}^{2} + \\ {(0.0375 - 0.0075)}^{2} + \\ {(0.4875 - 0.595)}^{2} + \\ {(0 . 58 - 0 . 6975)}^{2} + \\ {(0.0203 - 0.0028)}^{2} + \\ {(0.03 - 0.0075)}^{2} + \\ {(0.4875 - 0.6163)}^{2} + \\ {(0.58 - 0.72)}^{2} + \\ {(0.0203 - 0.0022)}^{2} + \\ {(0.03 - 0.0063)}^{2} + \\ {(0.5437 - 0.5525)}^{2} + \\ {(0.64 - 0.675)}^{2} + \\ {(0.0109 - 0.0025)}^{2} + \\ {(0.0187 - 0.0075)}^{2} + \\ {(0.45 - 0.6587)}^{2} + \\ {(0.56 - 0.7425)}^{2} + \\ {(0.0188 - 0.0025)}^{2} + \\ {(0.03 - 0.0063)}^{2} + \\ {(0.5063 - 0.5737)}^{2} + \\ {(0.6 - 0.675)}^{2} + \\ {(0.0172 - 0.0034)}^{2} + \\ {(0.0262 - 0.0087)}^{2} + \\ {(0.5063 - 0.7012)}^{2} + \\ {(0.6 - 0.7875)}^{2} + \\ {(0.0172 - 0.0013)}^{2} + \\ {(0.0262 - 0.0038)}^{2} \end{matrix}]}^{\frac{1}{2}} \\ = {\frac{1}{4} (0.2721)}^{\frac{1}{2}} = 0.2608 \end{matrix}$

Similarly, $S_{2}^{+} (D_{2}, A^{+}) = 0.2348, S_{3}^{+} (D_{3}, A^{+}) = 0$ $S_{4}^{+} (D_{4}, A^{+}) = 0.4527, S_{5}^{+} (D_{5}, A^{+}) = 0.656$ $S_{6}^{+} (D_{6}, A^{+}) = 0.7134, S_{7}^{+} (D_{7}, A^{+}) = 0.6529$

Step 8: Calculate separation measure between the candidates and NIPS for each decision maker.

The separation measure between the alternatives for negative ideal solution for each decision maker is calculated. $S_{1}^{-} (D_{1}, A^{-}) = {\frac{1}{4} \sum_{j = 1}^{n} [\begin{matrix} {(0.4125 - 0.3025)}^{2} + \\ {(0.52 - 0.4225)}^{2} \\ + {(0.025 - 0.04)}^{2} + \\ {(0.0375 - 0.0625)}^{2} + \\ {(0.4875 - 0.2063)}^{2} + \\ {(0 . 58 - 0 . 3087)}^{2} + \\ {(0.0203 - 0.08)}^{2} + \\ {(0.03 - 0.1125)}^{2} + \\ {(0.4875 - 0.1925)}^{2} + \\ {(0.58 - 0.2925)}^{2} + \\ {(0.0203 - 0.085)}^{2} + \\ {(0.03 - 0.1187)}^{2} + \\ {(0.5437 - 0.1925)}^{2} + \\ {(0.64 - 0.2925)}^{2} + \\ {(0.0109 - 0.085)}^{2} + \\ {(0.0187 - 0.1187)}^{2} + \\ {(0.45 - 0.2338)}^{2} + \\ {(0.56 - 0.3412)}^{2} + \\ {(0.0188 - 0.07)}^{2} + \\ {(0.03 - 0.1)}^{2} + \\ {(0.5063 - 0.2338)}^{2} + \\ {(0.6 - 0.3412)}^{2} + \\ {(0.0172 - 0.07)}^{2} + \\ {(0.0262 - 0.0087)}^{2} + \\ {(0.5063 - 0.27)}^{2} + \\ {(0.6 - 0.385)}^{2} + \\ {(0.0172 - 0.0625)}^{2} + \\ {(0.0262 - 0.0875)}^{2} \end{matrix}]}^{\frac{1}{2}}$ $= {\frac{1}{4} (0.9863)}^{\frac{1}{2}} = 0.4966,$ Similarly, $S_{2}^{-} (D_{2}, A^{-}) = 0.5287, S_{3}^{-} (D_{3}, A^{-}) = 0.7212$ $S_{4}^{-} (D_{4}, A^{-}) = 0.3060, S_{5}^{-} (D_{5}, A^{-}) = 0.0807$ $S_{6}^{-} (D_{6}, A^{-}) = 0.0149, S_{7}^{-} (D_{7}, A^{-}) = 0.0898$

Step 9: Calculate the closeness coefficient.

The closeness coefficient of the decision makers is then computed accordingly. ${RC}_{1} = \frac{S_{1}^{-}}{S_{1}^{-} + S_{1}^{+}} = \frac{0.4966}{0.4966 + 0.2608} = 0.6556$

Similarly,

RC₂ = 0.6925, RC₃ = 1.000

RC₄ = 0.4034, RC₅ = 0.1096

RC₆ = 0.0205, RC₇ = 0.1209

Step 10: Rank the preference order of all alternatives according to the closeness coefficient of the alternatives.

The preference order of all alternatives is ranked according to the closeness coefficient of the alternatives. $\begin{matrix} Rank (3) ≻ Rank (2) ≻ Rank (1) ≻ Rank (4) \\ ≻ Rank (7) ≻ Rank (5) ≻ Rank (6) \end{matrix}$

Thus, A₃ pumping station is selected as the most preferred alternative to manage the flood problems.

The result from IVIF TOPSIS shows that A₃, pumping station is the best ranked in terms of alternatives and followed by A₂, A₁, A₄, A₇, A₅ and A₆.

5 Conclusion

MCDM method is a technique that would give effective and useful framework in evaluating alternatives with respect to multiple and conflicted criteria. The purpose of this research was to determine the best alternative of flood control project using the IVIF TOPSIS method. About similar with other fuzzy MCDM methods, the IVIF TOPSIS method is an integration of IVIFS and TOPSIS where seven-scale linguistic terms are defined in IVIFS. Seven alternatives in flood management have been considered in this paper. These alternatives were evaluated with respect to seven criteria. The IVIF TOPSIS was applied to the case and successfully selecting the best alternatives of flood management. The closeness coefficients of alter-natives that computed from linguistic evaluation given by four decision makers are the ultimate output of the IVIF TOPSIS. Despite the differences in decision makers’ evaluation, the IVIF TOPSIS method concludes that ‘pumping station’ is the best way to mitigate or control floods. This result implicates that the local authority should provide more pumping stations from which rainfall would flows into a river. The result may be useful to the government particularly in solving flood occurrence where floods are considered as one of the most frequent environmental disasters in tropical countries. However, the stability of the results has yet to be explored and subjected to further investigation. Further work needs to be carried out to establish the stability of the final ranking using sensitivity analysis. Future studies on the current topic are also suggested in validating the results using other MCDM methods.

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