A new ranking method for TOPSIS and VIKOR under interval valued intuitionistic fuzzy sets and possibility measures

Abstract

In this work we propose an approach of multi-criteria decision making (MCDM) using TOPSIS and VIKOR methods under interval-valued intuitionistic fuzzy (IVIF) sets and possibility theory. We are interested to study positive and negative ideal solutions to propose new formulas. Indeed, these solutions are presented with many formulas in literature [1, 2] which could cause ambiguity [3]. Due to the importance of possibility theory in resolution of many problems, we propose to use possibility measure for positive ideal solution and necessity measure for negative ideal solution under interval valued intuitionistic fuzzy sets. According to this, TOPSIS and VIKOR are modified to obtain new approaches. The latter are applied to an example from literature using IVIF data. This example permits to assess the investment projects problem for ranking different projects. The found results showed different solutions from that existing in literature which can give more choice to decision makers with additional information due to use of possibility measures.

Keywords

Interval-valued intuitionistic fuzzy sets possibility measure multi-criteria decision making necessity measure TOPSIS VIKOR

1 Introduction

Multi-criteria decision making focuses to solve the most desirable alternative selection according to multiple criteria for a problem [4, 5]. It exists many techniques to resolve this problem such as AHP [6], ELECTRE [7], etc. We are interested to the technique for order preference by similarity to ideal solution (TOPSIS) which has been widely applied in numerous domains. TOPSIS is presented in literature with many versions according to the type of used data. The first version deals with crisp values where the information are certain and complete. Other versions deal with fuzzy sets and their generalizations which resolve problems with incomplete, imprecise and uncertain information. For example, Atanassov et al. [8] proposed method of multi-measurement tools for multi-criteria decision making using a score for ranking alternatives using intuitionistic fuzzy sets (IFSs). Boran et al. [4] applied TOPSIS method with IFSs, to select appropriate supplier in group decision making environment. Liu and Wang [9] solved MCDM problems where data are described with IFSs. They defined a series of score functions and evaluation functions that measure the degrees of satisfaction of decision makers. Cihat et al. [10] applied TOPSIS using IFSs to rank vehicle technologies. Fuzzy sets and intuitionistic fuzzy sets provide one membership degree. It exists situations when one membership degree may not be suitable, due to lack of exactitude. It will be needed an interval. This is the case of interval valued intuitionistic fuzzy sets where membership and non-membership degrees are expressed by intervals. IVIF set is an extension of both the intuitionistic fuzzy set (IFS) and interval valued fuzzy set. It is used in data definition of many MCDM problems. However, researches did not pay attention in their application. Especially in defining IVIF positive ideal solutions (IVIFPIS) and IVIF negative ideal solution (IVIFNIS). So, some of these solutions are not real and some others are defined taking not necessary information such as hesitation degree which can be deduced from membership and non-membership intervals, for more information see [3, 11]. Hence, to provide standard ideal solutions taking into consideration the maximum of information from proposed alternatives, we use possibility measures to compute IVIFPIS and IVIFNIS. In this context, Wang et al. [12] used IVIF TOPSIS based on a series of optimization models to determine attribute weight vector of criteria. In the same way, Huimin and Liying [5] applied TOPSIS with IVIF sets based on cross-entropy method for criteria weights. Other weight methods are defined in [13 –16] and [17]. Intepe et al. [18] used TOPSIS method with IVIF sets for the solution of technological forecasting technique.

Possibility theory proposed by Zadeh in [19] grained attention of many researchers and it has been used successfully in many domains, especially in MCDM problem resolution. Ye and Li [20] extended TOPSIS method to use possibility theory and triangular fuzzy numbers. The authors proposed a possibilistic mean value to construct the mean value matrix, witch is used to determine positive ideal solution (PIS) and negative ideal solution (NIS) and applied a possibilistic standard deviation matrix to determine the separation measure between alternatives using relative closeness coefficients. Wang et al. [21] proposed an extended TOPSIS model based on possibility theory using possibilistic standard variance matrix and possibilistic skewness matrix to determine the positive and negative ideal solutions. Others extended TOPSIS method from different perspectives are given in [22, 23] and [24]. For example, the authors [25 –27] used maximum and minimum operators to determine PIS and NIS which can lack information. So, its more realistic to represent them using necessity and possibility measures. The latter could take maximum information from criteria and alternatives values. Our aims is to compute PIS and NIS under interval-valued intuitionistic fuzzy environment using possibility and necessity measures.

This paper is organized as follows: In section 2, some preliminaries about IVIF sets, possibility theory, IVIF MCDM problems and aggregation operators are presented. In section 3 and 4, TOPSIS and VIKOR methods under interval valued intuitionistic fuzzy sets are detailed. In section 5, possibilistic IVIF-TOPSIS and possibilistic IVIF-VIKOR are developed for multiple criteria decision making problem resolution using possibility and necessity measures for computation IVIFPIS and IVIFNIS. In section 6, the proposed approaches are applied to evaluate investment projects and in section 7, a conclusion is deduced.

2 Preliminaries

2.1 Definition of IVIF sets

An IVIF set $\tilde{A}$ is defined by Atanassov [28] as: $\tilde{A} = {〈 x_{i}, [μ_{\tilde{A}}^{L} (x_{i}), μ_{\tilde{A}}^{U} (x_{i})], [ν_{\tilde{A}}^{L} (x_{i}), ν_{\tilde{A}}^{U} (x_{i})] 〉 | x_{i} \in X}$ (1) where $[μ_{\tilde{A}}^{L} (x_{i}), μ_{\tilde{A}}^{U} (x_{i})]$ and $[ν_{\tilde{A}}^{L} (x_{i}), ν_{\tilde{A}}^{U} (x_{i})]$ denote the intervals of membership and non membership degrees of an element $x_{i} \in \tilde{A}$ satisfying: $0 \leq μ_{\tilde{A}}^{U} (x_{i}) + ν_{\tilde{A}}^{U} (x_{i}) \leq 1$ , $0 \leq ν_{\tilde{A}}^{L} (x_{i}) \leq ν_{\tilde{A}}^{U} (x_{i}) \leq 1$ and $μ_{\tilde{A}}^{L} (x_{i}) \leq μ_{\tilde{A}}^{U} (x_{i})$ . The hesitation degree of an interval-valued intuitionistic fuzzy number x_i to set $\tilde{A}$ is defined as: $π_{\tilde{A}} (x_{i}) = [1 - μ_{\tilde{A}}^{U} (x_{i}) - ν_{\tilde{A}}^{U} (x_{i}), 1 - μ_{\tilde{A}}^{L} (x_{i}) - ν_{\tilde{A}}^{L} (x_{i})]$ . If $μ_{\tilde{A}}^{L} (x_{i}) = μ_{\tilde{A}}^{U} (x_{i})$ and $ν_{\tilde{A}}^{L} (x_{i}) = ν_{\tilde{A}}^{U} (x_{i})$ then $\tilde{A}$ is reduced to an IFS.

2.2 IVIF MCDM problem presentation

For a multi-criteria decision making problem, let A ={ A₁, A₂, …, A_m } be the set of alternatives, C ={ C₁, C₂, …, C_n } be the set of criteria and W = (w₁, w₂, …, w_n) ^T be the set of weights of criteria, where w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ .

The values of criteria are represented by interval valued intuitionistic fuzzy numbers ${\tilde{α}}_{ij} = ([a_{ij}, b_{ij}], [c_{ij}, d_{ij}])$ , where [a_ij, b_ij] and [c_ij, d_ij] are respectively, fuzzy membership degrees and fuzzy non-membership degrees of an alternative A_i over a criterion C_j, with $A_{i} = {〈 C_{j}, [μ_{A_{i}}^{L} (C_{j}), μ_{A_{i}}^{U} (C_{j})], [ν_{A_{i}}^{L} (C_{j}), ν_{A_{i}}^{U} (C_{j})] 〉 | C_{j} \in C}$ . Thus, an MCDM problem under IVIF sets can be concisely expressed with matrix ${\tilde{H}}_{m \times n}$ as:

$\tilde{H} = (\begin{matrix} C_{1} & C_{2} & \dots & C_{n} \\ A_{1} & ([a_{11}, b_{11}], [c_{11}, d_{11}]) & ([a_{12}, b_{12}], [c_{12}, d_{12}]) & \dots & ([a_{1 n}, b_{1 n}], [c_{1 n}, d_{1 n}]) \\ A_{2} & ([a_{21}, b_{21}], [c_{21}, d_{21}]) & ([a_{22}, b_{22}], [c_{22}, d_{22}]) & \dots & ([a_{2 n}, b_{2 n}], [c_{2 n}, d_{2 n}]) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ A_{m} & ([a_{m 1}, b_{m 1}], [c_{m 1}, d_{m 1}]) & ([a_{m 2}, b_{m 2}], [c_{m 2}, d_{m 2}]) & \dots & ([a_{mn}, b_{mn}], [c_{mn}, d_{mn}]) \end{matrix})$ (2)

2.3 Aggregation operators for interval valued intuitionistic fuzzy numbers

Aggregation operators permit to simplify the evaluation of IVIF sets and the comparison of two interval valued intuitionistic fuzzy Numbers (IVIFNs) using an accuracy function or a possibility measure. Xu [29] proposed some aggregation operators, such as the intuitionistic fuzzy ordered weighted averaging (IFOWA), intuitionistic fuzzy hybrid aggregation (IFHA) and the intuitionistic fuzzy weighted averaging (IFWA) operators, which are used for comparison between two interval valued intuitionistic fuzzy values, for more information see [30]. The aggregation operators could be applied on MCDM matrix (2) to simplify it. In the following we explain some aggregation operators existing in literature.

Let ${\tilde{a}}_{j} = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ , j = (1, 2, …, n) denote an interval valued intuitionistic fuzzy number.

Xu [31] defined two aggregation operators.

Interval-valued intuitionistic fuzzy weighted arithmetic averaging (IVIFWA) operator: $\begin{matrix} IVIFWA = ([1 - \prod_{j = 1}^{n} (1 - a_{j})^{w_{j}}, \\ 1 - \prod_{j = 1}^{n} (1 - b_{j})^{w_{j}}], \\ [\prod_{j = 1}^{n} (c_{j})^{w_{j}}, \prod_{j = 1}^{n} (d_{j})^{w_{j}}]) \end{matrix}$ (3) where w = (w₁, w₂, …, w_n) ^T is a weight vector of ${\tilde{a}}_{j}$ with w_j ∈ [0, 1], $\sum_{j = 1}^{n} w_{j} = 1$ .

Interval-valued intuitionistic fuzzy weighted geometric (IVIFWG) operator:

$\begin{matrix} IVIFWG = \\ ([\prod_{j = 1}^{n} (a_{j})^{w_{j}}, \prod_{j = 1}^{n} (b_{j})^{w_{j}}], \\ [1 - \prod_{j = 1}^{n} (1 - c_{j})^{w_{j}}, 1 - \prod_{j = 1}^{n} (1 - d_{j})^{w_{j}}]) \end{matrix}$ (4) and ω = (ω₁, ω₂, …, ω_n) ^T is the weight vector of $\tilde{α_{j}} (j = 1, 2, \dots, n)$ , ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1$ .

Also Xu [31] defined the score function and accuracy function to sort the interval-valued intuitionistic fuzzy numbers presented respectively as follow: $S (\tilde{a}) = \frac{1}{2} (a_{j} - c_{j} + b_{j} - d_{j})$ $h (\tilde{a}) = \frac{1}{2} (a_{j} + b_{j} + c_{j} + d_{j})$ where $S (\tilde{a}) \in [- 1, 1]$ and $h (\tilde{a}) \in [0, 1]$ . $S (\tilde{a})$ represent the deviation of the membership and non membership degrees and $h (\tilde{a})$ represents the mean value of membership and non membership degrees.

2.4 Possibility theory presentation

The possibility theory, investigated by Zadeh and Dubois, defines a pair of measures: possibility and necessity [19] and [32, 33]. Let A be a subset of states (s) and ∏ be a possibility distribution, which maps the set of interpretations Ω to the binary set {0, 1}, with the following conventions [32].

∏ (s) =0: the state s is rejected as impossible;

∏ (s) =1: the state s is totally possible

If s and s’ are two states and ∏ (s) > ∏ (s′), then s is considered to be a more plausible value than s’. ∏ maps the set Ω into a totally ordered scale (sc), then the valued of possibility distribution ∏ induces two set functions over Ω namely:The possibility function π (A) quantifies to what extent the event A is plausible.

The necessity function N (A) quantifies the certainty of A, ∀ A ⊂ Ω

The possibility measure π of A is defined in [0, 1] as:

$π (A) = sup_{s \in A} \prod (s)$ (5)π (A) evaluates to what extent A is consistent with ∏ and reflects the most normal situation in which A is true.

The degree of necessity (certainty) $N (A) = 1 - π (A^{c}) = inf_{s \notin A} (1 - \prod (s))$ (6)

where A^c is the complement of A.

The Necessity measure N (A) evaluates to what extent A is certainly implied by ∏ and reflects the more normal situation where A is false.

Note that if N (A) =1 then A is certainly true, and if N (A) =0 then A is not certain (A still be possible).

2.5 Possibility measures of IVIF numbers from literature applied to MCDM methods

In the following we present some measures of possibility from literature [34].

Let $\tilde{α_{1}} = ([a_{1}, b_{1}], [c_{1}, d_{1}])$ and $\tilde{α_{2}} = ([a_{2}, b_{2}], [c_{2}, d_{2}])$ two interval-valued intuitionistic fuzzy numbers in Ω, where Ω is the set of all IVIFN and $p (\tilde{α_{1}} \geq \tilde{α_{2}})$ is the possibility measure of two interval-valued intuitionistic fuzzy numbers $\tilde{α_{1}}$ and $\tilde{α_{2}}$ .

Zhang et al. [35] defined two possibility measures of two interval-valued intuitionistic fuzzy numbers as follow:

First measure:

$\begin{matrix} p_{1} (\tilde{α_{1}} \geq \tilde{α_{2}}) = \\ min (max (A (\tilde{α_{1}}) - A (\tilde{α_{2}}) + 0.5, 0), 1) \end{matrix}$ (7) where $A (\tilde{α_{1}}) = λ \frac{a + b}{2} + (1 - λ) \frac{a - c + b - d}{2}$ , λ ∈ [0, 1] and it represents the performance on the mean valued of its membership degree.

The authors showed that p₁ satisfies the following properties:

$0 \leq p_{1} (\tilde{α_{1}} \geq \tilde{α_{2}}) \leq 1$ ;

$p_{1} (\tilde{α_{1}} \geq \tilde{α_{2}}) = 1 \Leftrightarrow A (\tilde{α_{1}}) - A (\tilde{α_{2}}) \geq 0.5$ ;

$p_{1} (\tilde{α_{1}} \geq \tilde{α_{2}}) = 0 \Leftrightarrow A (\tilde{α_{1}}) - A (\tilde{α_{2}}) \leq - 0.5$

$p_{1} (\tilde{α_{1}} \geq \tilde{α_{2}}) + p_{1} (\tilde{α_{2}} \geq \tilde{α_{1}}) = 1$

Second measure:

$\begin{matrix} p_{2} (\tilde{α_{1}} \geq \tilde{α_{2}}) = γ min \\ (max (\frac{b_{1} - a_{2}}{b_{1} - a_{1} + b_{2} - a_{2}}, 0), 1) + (1 - γ) \\ min (max (\frac{d_{2} - c_{1}}{d_{1} - c_{1} + d_{2} - c_{2}}, 0), 1) \end{matrix}$ (8) where γ ∈ [0, 1] gives the decision makers’ preference on membership degree or non-membership degree. When the decision maker is optimal, γ ≥ 0.5 and when the decision maker is pessimistic, γ < 0.5.

p₂ satisfies the following properties:

$0 \leq p_{2} (\tilde{α_{1}} \geq \tilde{α_{2}}) \leq 1$ ;

$p_{2} (\tilde{α_{1}} \geq \tilde{α_{2}}) = 1 \Leftrightarrow b_{2} \leq a_{1}$ and d₁ ≤ c₂

$p_{2} (\tilde{α_{1}} \geq \tilde{α_{2}}) = 0 \Leftrightarrow b_{1} \leq a_{2}$ and d₂ ≤ c₁

$p_{2} (\tilde{α_{1}} \geq \tilde{α_{2}}) + p_{2} (\tilde{α_{2}} \geq \tilde{α_{1}}) = 1$

Wan et al. [36] defined possibility measure by the following formula: $\begin{matrix} p_{3} (\tilde{α_{1}} \geq \tilde{α_{2}}) = \frac{1}{2} [p ([a_{1}, b_{1}] \geq [a_{2}, b_{2}]) \\ + p ([c_{2}, d_{2}] \geq [c_{1}, d_{1}])] \end{matrix}$ (9) where p ([a₁, b₁] ≥ [a₂, b₂]) and p ([c₂, d₂] ≥ [c₁, d₁]) can be calculated using this formula: $p (a \geq b) = max {1 - max (\frac{b^{+} - a^{-}}{l_{a} + l_{b}}, 0), 0}$ (10) where l_a = a⁺ - a^- and l_b = b⁺ - b^-

3 IVIF TOPSIS defined in literature

IVIF sets are successfully applied to multi-criteria decision making [18 , 37–39] where a problem could be defined as in Section 2.2. In the following we present TOPSIS steps using IVIF sets from literature. knowing that it exists a unique approach in literature of IVIF TOPSIS [40].

Step 1: Compute the interval valued intuitionistic fuzzy positive-ideal solution (IVIFPIS noted A⁺) and interval valued intuitionistic fuzzy negative-ideal solution (IVIFNIS noted A^-) as: $A^{+} = (r_{1}^{+}, r_{2}^{+}, r_{n}^{+})$ (11) $A^{-} = (r_{1}^{-}, r_{2}^{-}, r_{n}^{-})$ (12) where $r_{j}^{+} = ([a_{j}^{+}, b_{j}^{+}], [c_{j}^{+}, d_{j}^{+}])$ , $r_{j}^{-} = ([a_{j}^{-}, b_{j}^{-}], [c_{j}^{-}, d_{j}^{-}])$ The following formulas determine $r_{j}^{+}$ interval valued positive ideal solution and $r_{j}^{-}$ interval valued negative ideal solution. $r_{j}^{+} = {\begin{matrix} a_{j}^{+} & = & max_{i} a_{ij} \\ b_{j}^{+} & = & max_{i} b_{ij} \\ c_{j}^{+} & = & min_{i} c_{ij} \\ d_{j}^{+} & = & min_{i} d_{ij} \end{matrix}$ (13) $r_{j}^{-} = {\begin{matrix} a_{j}^{-} & = & min_{i} a_{ij} \\ b_{j}^{-} & = & min_{i} b_{ij} \\ c_{j}^{-} & = & max_{i} c_{ij} \\ d_{j}^{-} & = & max_{i} d_{ij} \end{matrix}$ (14) where i = 1, 2, …, m, j = 1, 2, …, n. The authors proved that $r_{j}^{+}$ , $r_{j}^{-}$ are intervals valued intuitionistic fuzzy numbers (IVIFNs) with $d_{j}^{+} + b_{j}^{+} \leq 1$ and $a_{j}^{-} + c_{j}^{-} \leq 1$

Step 2: Determine the distance of each alternative from IVIFPIS and IVIFNIS:

The distance dist⁺ (A_i, A⁺) between each alternative A_i and positive ideal solution A⁺ can be calculated as: dist⁺ (A_i, A⁺) =

$\sqrt{\frac{1}{2} ((d_{\inf}^{+} (A_{i}, A^{+}))^{2} + (d_{\sup}^{+} (A_{i}, A^{+}))^{2})}$ (15) where ${dist}_{\inf}^{+} (A_{i}, A^{+}) =$ $\sqrt{\frac{1}{3} \sum_{j = 1}^{n} ((a_{ij} - a_{j}^{+})^{2} + (c_{ij} - c_{j}^{+})^{2} + (π_{ij}^{L} - π_{j}^{+ L})^{2})}$ ${dist}_{\sup}^{+} (A_{i}, A^{+}) =$ $\sqrt{\frac{1}{3} \sum_{j = 1}^{n} ((b_{ij} - b_{j}^{+})^{2} + (d_{ij} - d_{j}^{+})^{2} + (π_{ij}^{U} - π_{j}^{+ U})^{2})}$ where $π_{ij}^{L} = 1 - c_{ij} - d_{ij}$ and $π_{ij}^{U} = 1 - a_{ij} - b_{ij}$

The distance dist^- (A_i, A^-) between each alternative A_i and negative ideal solution A^- can be calculated as: dist^- (A_i, A^-) = $\sqrt{\frac{1}{2} ({dist}_{\inf}^{-} (A_{i}, A^{-}))^{2} + ({dist}_{\sup}^{-} (A_{i}, A^{-}))^{2})}$ (16) where ${dist}_{\inf}^{-} (A_{i}, A^{-}) =$ $\sqrt{\frac{1}{3} \sum_{j = 1}^{n} ((a_{ij} - a_{j}^{-})^{2} + (c_{ij} - c_{j}^{-})^{2} + (π_{ij}^{L} - π_{j}^{- L})^{2})}$ ${dist}_{\sup}^{-} (A_{i}, A^{+}) =$ $\sqrt{\frac{1}{3} \sum_{j = 1}^{n} ((b_{ij} - b_{j}^{-})^{2} + (d_{ij} - d_{j}^{-})^{2} + (π_{ij}^{U} - π_{j}^{- U})^{2})}$

Step 3: Compute the closeness coefficient to determine the ranking order of alternatives: $Co = \frac{{dist}_{i}^{-}}{{dist}_{i}^{-} + {dist}_{i}^{+}}$ (17) Where 0 ≤ Co ≤ 1.

Step 4: Rank alternatives in the descending order according to their closeness coefficient (Co) values. The best alternative is that having the higher value of Co. This value indicates that an alternative is closer to IVIFPIS and farther from IVIFNIS simultaneously.

4 IVIF-VIKOR defined in literature

The VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje -a Serbian name) is a method developed by Opricovic and Tzeng [41]. This technique started with L_p- (18) metric concepts [42]. The VIKOR method was developed for multi-criteria optimization in complex systems. Each alternative is evaluated according to each criterion. The performance of ranking is obtained by comparing the closeness measure to the ideal alternatives. $L_{p, i} = {\sum_{j = 1}^{n} {[w_{i} (A_{j}^{+} - A_{ij}) / (A_{j}^{+} - A_{j}^{-})]}^{p}}^{1 / p}$ (18) Let J₁ presents a collection of benefit criteria (the larger the C_j, the greater the preference), J₂ presents a collection of cost criteria (the smaller the C_j, the greater the preference), the interval-valued intuitionistic fuzzy positive-ideal solution, denoted by ${\tilde{α}}^{+}$ , and the interval-valued intuitionistic fuzzy negative-ideal solution, denoted by ${\tilde{α}}^{-}$ . Therefore, IVIF-VIKOR is presented in [43] by these steps:

Step 1: The IVIFPIS and the IVIFNIS are defined as follows: $\begin{matrix} {\tilde{α}}^{+} = & {C_{j}, [(max_{i} a_{ij}, max_{i} b_{ij}) | j \in J_{1}, \\ (min_{i} a_{ij}, min_{i} b_{ij}) | j \in J_{2}], \\ [(min_{i} c_{ij}, min_{i} d_{ij}) | j \in J_{1}, \\ (max_{i} c_{ij}, max_{i} d_{ij}) | j \in J_{2}] | i = 1, \dots, m} \end{matrix}$ (19) where ${\tilde{α}}_{j}^{+} = ([a_{j}^{+}, b_{j}^{+}], [c_{j}^{+}, d_{j}^{+}])$ , with j = 1, 2, …, n $\begin{matrix} {\tilde{α}}^{-} = & {C_{j}, [(min_{i} a_{ij}, min_{i} b_{ij}) | j \in J_{1}, \\ (max_{i} a_{ij}, max_{i} b_{ij}) | j \in J_{2}], \\ [(max_{i} c_{ij}, max_{i} d_{ij}) | j \in J_{1}, \\ (min_{i} c_{ij}, min_{i} d_{ij}) | j \in J_{2}]} \end{matrix}$ (20) where ${\tilde{α}}_{j}^{-} = ([a_{j}^{-}, b_{j}^{-}], [c_{j}^{-}, d_{j}^{-}])$ , with j = 1, 2, …, n

Step 2: Compute S_i and R_iS_i defines a compromise solution giving a ’maximum group utility’ and R_i defines the minimum individual regret of the ’opponent’. S_i and R_i are computed as follows: $\begin{matrix} S_{i} = & \sum_{j = 1}^{n} W_{j} \times \frac{dist (A_{ij}, A^{+})}{dist (A^{+}, A^{-})} \end{matrix}$ (21) $\begin{matrix} R_{i} = & max_{j} (W_{j} \times \frac{dist (A_{ij}, A^{+})}{dist (A^{+}, A^{-})}) \end{matrix}$ (22) where $\frac{dist (A_{ij}, A^{+})}{dist (A^{+}, A^{-})} = \frac{(| a_{j}^{+} - a_{ij} | + | b_{j}^{+} - b_{ij} | + | c_{j}^{+} - c_{ij} | + | d_{j}^{+} - d_{ij} |)}{(| a_{j}^{+} - a_{j}^{-} | + | b_{j}^{+} - b_{j}^{-} | + | c_{j}^{+} - c_{j}^{-} | + | d_{j}^{+} - d_{j}^{-} |)}$ where i = 1, 2, …, m and j = 1, 2, …, n

Step 3: Find the index values Q_i, i = 1, 2, …, m:the relations between R_i and S_i expressed by Q_i are defined by the following formulas:

$Q_{i} = {\begin{matrix} \frac{R_{i} - R^{-}}{R^{+} - R^{-}} if S^{+} = S^{-}, \\ \frac{S_{i} - S^{-}}{S^{+} - S^{-}} if R^{+} = R^{-}, \\ γ \frac{S_{i} - S^{-}}{S^{+} - S^{-}} + (1 - γ) \frac{R_{i} - R^{-}}{R^{+} - R^{-}}, otherwise \end{matrix}$ (23) where $\begin{matrix} S^{+} & = & min_{i} S_{i}, S^{-} = max_{i} S_{i}, \\ R^{+} & = & min_{i} R_{i}, R^{-} = max_{i} R_{i} \end{matrix}$ (24) where γ is the weight of the strategy of maximum group utility and (1 - γ) is the weight of the individual regret.

Step 4: Classify the alternatives:To rank alternatives, the values of S, R and Q are classified in an ascending order.

Step 5: Propose a compromise solution:It means that an alternative A′ is considered the best if it has the minimum value by the measure Q. A solution is considered as compromise if the conditions below are verified:Condition 1: $Q (A^{″}) - Q (A^{'}) \geq \frac{1}{m - 1}$ (25) where A″ is the second best alternative (second minimum) given by Q; m is the number of alternatives.Condition 2: Defines a stable solution in decision-making. The most appropriate alternative A′ found by Q must be also the best found by S or / and by R. The non verification of one condition leads to other compromise solutions consisting of:

Alternatives A′ and A″ if condition 2 is not verified

Alternatives A′, A″, …, A^M if condition 1 is not satisfied. The alternative A^M is found using the formula: $Q (A^{M}) - Q (A^{'}) < \frac{1}{m - 1}$

5 Proposition of interval valued intuitionistic fuzzy MCDM methods using possibility theory

5.1 Possibilistic IVIF-TOPSIS

In literature, it does not exist TOPSIS method using IVIF sets and possibility theory. In this section we propose an approach of TOPSIS based on IVIF sets and possibility measures to determine the PIS ans NIS under Interval Valued Intuitionistic Fuzzy Sets.

Step 1: Constitute the interval valued intuitionistic fuzzy decision matrix: $\tilde{H} = ({\tilde{r}}_{ij})_{m \times n} = ([a_{ij}, b_{ij}], [c_{ij}, d_{ij}])$ (2).

Step 2: Use interval-valued intuitionistic fuzzy weighted geometric IVIFWG operator (3) to aggregate decision matrix $\tilde{H}$ and to obtain the interval valued intuitionistic fuzzy numbers (IVIFN) (26) of each alternative A_i, i = 1, …, m. $IVIFN (A_{i}) = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ (26)

Step 3: Apply formula (8) to compute the possibility degree of interval valued intuitionistic fuzzy numbers (IVIFN (A₁), IVIFN (A₂),...,IVIFN (A_m)). Thus, we transform the interval valued intuitionistic fuzzy decision matrix $\tilde{H}$ into the following form of possibility matrix p_ij. $(p_{ij})_{m \times m} = (\begin{matrix} 0.5 & a_{12} & \dots & a_{1 m} \\ a_{21} & 0.5 & \dots & a_{2 m} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & 0.5 \end{matrix})$ (27)

Step 4: Determine the IVIFPIS denoted as A⁺ using (28) on average value of possibility degrees of each alternative IVIFN (A₁) , IVIFN (A₂) , …, IVIFN (A_m), instead of (11).

Then, compute the IVIFNIS, denoted as A^- using necessity measure (29) instead of (12). Thus, using the necessity measure to obtain negative ideal solution, it estimated to what extent each value outside A. Also A_i is not included in A. $A^{+} = 1 / m {\sum_{i = 1}^{m} p_{2} (A_{i})}$ (28) $A^{-} = inf_{A_{i} \notin A} 1 - p 2 (A_{i})$ (29) where the possibility measure p₂ (8) is defined in Section 2.5.

Step 5: Compute the euclidean distance measures $D_{i}^{+}$ (30) between each alternative and the IVIFPIS and $D_{i}^{-}$ (31) between each alternative and IVIFNIS. $D_{i}^{+} = \sqrt{1 / 2 \sum_{j = 1}^{n} w_{j} dist (A_{ij}, A^{+})}$ (30) where dist (A_ij, A⁺) = ${(a_{ij} - a_{j}^{+})^{2} + (b_{ij} - b_{j}^{+})^{2} + (c_{ij} - c_{j}^{+})^{2} + (d_{ij} - d_{j}^{+})^{2}}$

Where a_ij ∈ (p_ij) _m×m, $A^{+} = ([a_{j}^{+}, b_{j}^{+}], [c_{j}^{+}, d_{j}^{+}])$ and j = 1, …, n, i = 1, …, m $D_{i}^{-} = \sqrt{1 / 2 \sum_{j = 1}^{n} w_{j} dist (A_{ij}, A^{-})}$ (31) where dist (A_ij, A^-) = ${(a_{ij} - a_{j}^{-})^{2} + (b_{ij} - b_{j}^{-})^{2} + (c_{ij} - c_{j}^{-})^{2} + (d_{ij} - d_{j}^{-})^{2}}$ $A^{-} = ([a_{j}^{-}, b_{j}^{-}], [c_{j}^{-}, d_{j}^{-}])$ j = 1, …, n, i = 1, …, m

Step 6: Compute the relative closeness coefficient CC (32) of alternatives A_i using the distance positive ideal solution $D_{i}^{+}$ and the distance of negative ideal solution $D_{i}^{-}$ , as follow: $CC (A_{i}) = \frac{D_{i}^{-}}{D_{i}^{+} + D_{i}^{-}}$ (32) where 0 ≤ CC (A_i) ≤1.

Step 7: The alternative would be ranked according to the descending order of CC (A_i). Thus, the alternative with the highest value of CC (A_i) will be the best.

5.2 Possibilistic IVIF-VIKOR method

Step 1,2,3: Are same steps as those in IVIF-TOPSIS in section 5.

Step 4: We use the proposed formulas (28) and (29) exposed in section 5 to compute the possibility IVIPIS (A⁺) and possibility IVINIS (A^-) instead of (11)31 and (11)32.

Step 5: Compute S_i and R_i using possibility ideal solution A⁺ and possibility negative ideal solution A^-. $S_{i} = \sum_{j = 1}^{n} W_{j} \times \frac{D (P_{ij}, A^{+})}{D (A^{+}, A^{-})}$ (33) $R_{i} = max_{j} (W_{j} \times \frac{D (p_{ij}, A^{+})}{D (A^{+}, A^{-})})$ (34) where P_ij: The possibility decision matrix using formula (8). D (p_ij, A⁺): The distance measure between each alternative and PIS using formula (31) exposed in step 5 section 5. D (p_ij, A^-): The distance measure between each alternative and NIS using formula (30).

Step 6: Find the index values Q_i: the relations between R_i (33) and S_i (34), where Q_i are defined by (23) in section 4.

Step 7: Classify the alternatives:To rank these alternatives, the values Q_i are classified in an ascending order.

6 Application of proposed possibilistic methods

6.1 Description of data sets

The data set is borrowed from [44]. A management team should evaluate each investment to select the best among the investment projects considering four criteria. The aggregated matrix of experts preferences is presented as follows.

$(\begin{matrix} [0.6452, 0.7382], [0.0974, 0.2129] & [0.6347, 0.732], [0, 0.2155] & [0.6724, 0.7662], [0, 0.1838] & [0.6718, 0.7703], [0, 0.1764] \\ [0.5590, 0.6560], [0.1687, 0.2927] & [0.6997, 0.7984] [0, 0.1498] & [0.5148, 0.6388], [0, 0.2934] & [0.6724, 0.7662], [0, 0.1838] \\ [0.2000, 0.3492], [0.3845, 0.5721] & [0.3891, 0.4880], [0.3233, 0.4581] & [0.2959, 0.4056], [0.3845, 0.5345] & [0.6347, 0.7320], [0, 0.2155] \\ [0.6974, 0.7912], [0, 0.1587] & [0.6880, 0.7796], [0, 0.1719] & [0.6880, 0.77960], [0, 0.1719] & [0.7632, 0.8184], [0, 0.1501] \end{matrix})$ (35)

6.2 Application of possibilistic IVIF-TOPSIS method

Let $\tilde{α} = ([a_{ij}, b_{ij}], [c_{ij}, d_{ij}])$ be an evaluation value of the investment projects

Step 1: Use decision matrix $\tilde{H}$ (35) to evaluate the investment project.

Step 2: Compute the comprehensive evaluation of each services (alternatives) using geometric weighted average operator (4) to aggregate the decision matrix $\tilde{H}$ (35). Thus, we transform the IVIF decision matrix into IVIF numbers for each alternative presented as follows: IVI (A₁) = [0.6566, 0.7526] , [0.0216, 0.1960]IVI (A₂) = [0.6152, 0.7192] , [0.0386, 0.2254]IVI (A₃) = [0.3693, 0.4940] , [0.2677, 0.4418]IVI (A₄) = [0.7124, 0.7939] , [0, 0.1624]

Step 3: Use possibility measure (8) to calculate the possibility degree of each investment project interval-valued intuitionistic fuzzy number evaluation and to build the following possibility degree matrix:

$p_{ij} = (\begin{matrix} 0.5000 & 0.6255 & 1.0000 & 0.3222 \\ 0.3745 & 0.5000 & 1.0000 & 0.1957 \\ 0 & 0 & 0.5000 & 0 \\ 1.0000 & 0.8043 & 0.6778 & 0.5000 \end{matrix})$ (36)

Step 4: Determine the mean values of possibility degree for each alternatives using (28) to determine the IVIFPIS denoted with (A⁺) and the IVIFNIS (A^-) using the necessity measure (29). The corresponding results are presented as follow: A⁺ = ([0.6493, 0.2617] [0, 0.2068])A^- = ([0.6778, 0.8043] [1.0000, 0.3222])

Step 5: Calculate the distance between alternatives and positive ideal solutions and between alternatives and NIS.

D⁺ = [0.0861, 0.0980, 0.1151, 0.0074]Also we found the following results distance vector between alternatives and NIS.

D^- = [0.1363, 0.1839, 0.2522, 0.1784]

Step 6: Compute the closeness coefficient (32) using D⁺ and D^-.

Step 7: Rank alternatives according to descending order. The corresponding results are presented in Table 1 showing that the best alternative is A₄.

Table 1

Relative Closeness Coefficients of IVIF-TOPSIS

Alternatives	A ₁	A ₂	A ₃	A ₄
DPIS	0.0861	0.0980	0.1151	0.0074
DNIS	0.1363	0.1839	0.2522	0.1784
RCloC	0.6129	0.6524	0.6867	0.9601
Ranking	4	3	2	1

Using investment projects data sets selection described by matrix $\tilde{H}$ in section 6.1 we find the following classification order of alternatives: A₄ ≻ A₃ ≻ A₂ ≻ A₁.

6.3 Application of possibilistic IVIF-VIKOR method

Results of step 1, step 2, step 3, step 4 are same of those presented in Section 6.2. The results of remaining steps are:

Step 5: Compute S_i and R_i using possibility ideal solution A⁺ and possibility negative ideal solution A^- obtained in step 4.

S_i = 0.2538, 0.4759, 0.7446, 1.7324R_i = 0.2438, 0.2238, 0.2538, 0.2581

Step 6: Compute the relation Q_i between S_i and R_iQ_i = 0.0, 0.0751, 0.1660, 1.0

Step 6: Rank the alternativesThe values Q_i are sorted in descending order: Q_i = 1.0, 0.1660, 0.0751, 0.0⇒A₄ > A₃ > A₂ > A₁ Table 2 shows that the best alternative ranked first is A₄ and the worst alternative ranked last is A₁ using possibilistic IVIF-TOPSIS and possibilistic IVIF-VIKOR.

Table 2
Alternative Ranking Order Using Possibilistic IVIF-TOPSIS and Possibilistic IVIF-VIKOR

Alternatives A ₁ A ₂ A ₃ A ₄

Possibilistic IVIF-TOPSIS 4 3 2 1

Possibilistic IVIF-VIKOR 4 3 2 1

Alternatives	A ₁	A ₂	A ₃	A ₄
Possibilistic IVIF-TOPSIS	4	3	2	1
Possibilistic IVIF-VIKOR	4	3	2	1

6.4 Comparison of obtained results with that of some methods from literature

Table 3 presents the ranks of investment projects using proposed methods and methods from literature. The best alternative is:

A₄, using possibilistic IVIF-TOPSIS and possibilistic IVIF-VIKOR

A₃, using possibilistic IVIF ELECTRE II

A₁, using IVIF ELECTRE III, TOPSIS and COPRAS methods.

Alternatives A₂ and A₃ are ranked second or third by all methods so its rank as first alternative by ELECTRE II could be discarded especially that is also ranked last by TOPSIS [47] and CORPAS. ELECTRE III and TOPSIS [48] methods have inverted completely the results of proposed methods (instead of the rank from A₄ to A₁, their rank are from A₁ to A₄). It remains difficult to choose between A₁ and A₄ as they are ranked first or last.

Table 3
Results of Possibilistic IVIF-TOPSIS and IVIF-VIKOR applied on (2)6490 and some Methods from Literature

Alternatives A ₁ A ₂ A ₃ A ₄

Possibilistic IVIF-TOPSIS 4 3 2 1

Possibilistic IVIF-VIKOR 4 3 2 1

Possibilistic IVIF ELECTRE II [45] 3 2 1 3

IVIF ELECTRE III [46] 1 2 3 4

TOPSIS [47] 1 2 4 3

TOPSIS [48] 1 2 3 4

COPRAS [49] 1 2 4 3

Alternatives	A ₁	A ₂	A ₃	A ₄
Possibilistic IVIF-TOPSIS	4	3	2	1
Possibilistic IVIF-VIKOR	4	3	2	1
Possibilistic IVIF ELECTRE II [45]	3	2	1	3
IVIF ELECTRE III [46]	1	2	3	4
TOPSIS [47]	1	2	4	3
TOPSIS [48]	1	2	3	4
COPRAS [49]	1	2	4	3

7 Conclusion

In this work, we presented several possibility measures of IVIF sets from literature. In addition, we exposed method of aggregation between IVIF numbers. Therefore, we detailed IVIF-TOPSIS and IVIF-VIKOR method from literature. Hence, we proposed two approaches possibilistic IVIF-TOPSIS and possibilistic IVIF-VIKOR for multiple criteria decision making problems resolution using possibility and necessity measures for computation IVIFPIS and IVIFNIS. The advantage of these methods according to the existing ones, that they take more information about alternatives to constitute PIS and NIS. Also, the methods are more realistic especially considering the ambiguity found to constitute PIS and NIS in literature. IVIF sets combined with possibility theory can efficiently help the decision-maker for making decisions. Proposed approaches are applied for selecting the best project of investment. As perspective, possibility theory will be applied to other MCDM methods [50, 51] or other research topics such as classification using ATOVIC algorithm [52].

Footnotes

Acknowledgment

The authors would like to acknowledge the anonymous referees for their interesting comments permitting us to improve this article.

The research leading to these results has received funding from the Ministry of Higher Education and Scientific Research of Tunisia under the grant agreement number LR11ES48.

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A new ranking method for TOPSIS and VIKOR under interval valued intuitionistic fuzzy sets and possibility measures

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Definition of IVIF sets

5.1 Possibilistic IVIF-TOPSIS

6.1 Description of data sets

Table 2 Alternative Ranking Order Using Possibilistic IVIF-TOPSIS and Possibilistic IVIF-VIKOR Alternatives A 1 A 2 A 3 A 4 Possibilistic IVIF-TOPSIS 4 3 2 1 Possibilistic IVIF-VIKOR 4 3 2 1

Footnotes

Acknowledgment

References

Table 2
Alternative Ranking Order Using Possibilistic IVIF-TOPSIS and Possibilistic IVIF-VIKOR

Alternatives A ₁ A ₂ A ₃ A ₄

Possibilistic IVIF-TOPSIS 4 3 2 1

Possibilistic IVIF-VIKOR 4 3 2 1