Multiple attribute decision making based on probabilistic interval-valued intuitionistic hesitant fuzzy set and extended TOPSIS method

Abstract

Interval-valued intuitionistic hesitant fuzzy set (IVIHFS) has a key role in multiple attribute decision making (MADM) problems due to its ability to represent the decision maker’s hesitant opinions using preferred and non-preferred intervals. In this paper, we develop an interactive decision-making approach to solve multi-attribute group decision making (MAGDM) problems with incomplete weight information using probabilistic interval-valued intuitionistic hesitant fuzzy set (P-IVIHFS), which is an extension of IVIHFS. The assessments provided by the decision makers for individual alternatives regarding different attributes are expressed using probabilistic interval-valued intuitionistic hesitant fuzzy elements (P-IVIHFEs). Linear programming (LP) is used to obtain the optimal weights of attributes from the partially known weight information. Moreover, we extend the technique for order preference by similarity to ideal solution (TOPSIS) method in the framework of P-IVIHFS for the ranking purpose. Finally, we have solved a numerical example for the supplier selection problem using the proposed method to illustrate the applicability of the proposed approach. The comparative study demonstrates the suitability of the proposed approach over the existing methods.

Keywords

Probabilistic interval-valued intuitionistic hesitant fuzzy set multi-attribute group decision making similarity measure linear programming TOPSIS

1 Introduction

Nowadays researchers have paid more attention to multi-attribute decision-making (MADM) problems under imprecise and uncertain environments. As a result, many MADM methods have been developed, such as the weighted sum model based on some aggregation operators [2, 3], TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) [4, 5], VIKOR (Vlsekriterijumska Optimizacija I Kompromisno Resenje) [6, 7], ELECTRE (ELimination Et Choix Traduisant la REalite) method [8, 9], PROMETHEE (Preference Ranking Organization METHOD for Enrichment Evaluation) [10], etc. Among these methods, the TOPSIS method has the following distinctive advantages [11, 12]: (a) it is easy to understand, and a simple computation process can solve it; (b) the reasonable human choice with a clear logic; and (c) it can well reflect the gap between any two alternatives. In the last few years, the TOPSIS method has been extended to deal with different types of attribute values, such as interval values [13], linguistic variables [14], and intuitionistic fuzzy values [15]. Moreover, some interactive methods have been proposed for solving the MADM problems in which the attribute values are expressed in real numbers [16], triangular fuzzy numbers [17], linguistic labels or intuitionistic fuzzy values [18].

Fuzzy set theory was first introduced by Zadeh [1] in 1965. As an extension of the fuzzy set [1], hesitant fuzzy set (HFS) was introduced by Torra and Narukawa [19]. During the last few years, many contributions are found, where researchers have applied HFSs to MAGDM problems under various types of situations [20 , 59– 62]. According to the attribute weight information, the considered decision making situations can be mainly classified as the following three types: (a) the decision making situation where the attribute weights are completely known; (b) the decision making situation where the attribute weights are completely unknown; (c) the decision making situation where the attribute weights are partially known. Some MAGDM methods [22– 26 , 56– 58] have been presented based on interval-valued intuitionistic fuzzy sets (IVIFSs), where IVIFSs is the generalization of intuitionistic fuzzy sets (IFSs) [27]. The membership value and the non-membership value of an element belonging to an IVIFS are represented by interval values in [0, 1] instead of real values in [0, 1]. In [28], Chen presented an interval-valued intuitionistic fuzzy (IVIF) permutation method with likelihood-based preference functions for dealing with multiple criteria decision analysis problems. In [29], Chen and Chiou presented a multi-attribute decision making (MADM) method using IVIFSs, particle swarm optimization techniques and the evidential reasoning methodology. In [30], Chen et al. presented a MADM method based on interval-valued intuitionistic fuzzy weighted average operators, the Karnik-Mendel algorithms and a fuzzy ranking method for intuitionistic fuzzy values based on likelihood-based comparison relations of intervals. In [31], Chen and Tsai presented a MADM method based on the interval-valued intuitionistic fuzzy ordered weighted geometric averaging (IVIFOWGA) operator and the interval-valued intuitionistic fuzzy hybrid geometric averaging (IVIFHGA) operator to deal the company investment problems. In [32], Chen et al. presented a multi-criteria fuzzy decision-making method based on IVIFSs. In [33], Hui and Xu presented a MADM method with interval-valued intuitionistic fuzzy information based on the technique for order preference by similarity to ideal solution (TOPSIS) method [34]. Li [35] presented a nonlinear-programming method for MADM using IVIFSs and the TOPSIS method, where the TOPSIS method considers two reference points, i.e., the selected alternative has the shortest distance from the positive ideal solution (PIS) and the farthest distance from the negative ideal solution (NIS), which has successfully been applied to deal with decision-making problems [35, 36] in IVIF environments. Zhu [37] proposed the basic concept of probabilistic hesitant fuzzy set (P-HFS) by using probability to describe preference information. Yue et al. [38] developed some probabilistic hesitant fuzzy aggregation operators and applied them to decision making. Zeng et al. [39] presented the uncertain probabilistic ordered weighted averaging distance (UPOWAD) operator, which uses distance measures in a unified framework between the probability and the OWA operator. Moreover, Liu and Teng [66] defined some probabilistic linguistic muirhead mean aggregation operators and studied a MADM approach based on the defined operators. Liu and Li [67] extended the PROMETHEE II method in the framework of probabilistic linguistic information. An extended probabilistic linguistic TODIM (PL-TODIM) method is given in [68]. Liu and Li [69] proposed a novel decision-making method based on probabilistic linguistic information. Application of possibilistic hesitant fuzzy linguistic information to analyze failure modes is given in [63] and linguistic distribution-based optimization approach is studied in [65]. A comprehensive review of various consensus reaching process to achieve common agreement among decision makers is found in [64].

In many decision-making problems, when the exact information is not available, the decision makers (DMs) prefer to express their opinions using interval values consisting both of the membership and non-membership values. Again when the DMs are hesitant about their opinions, then they present their opinions based on interval-valued intuitionistic hesitant fuzzy elements (IVIHFEs). IVIHFEs can handle many complex situations due to having interval-valued and hesitant opinions. When the complexity is much, necessarily uncertainty becomes more. In these highly uncertain environments, IVIHFEs might be no longer useful when the DMs consider the frequency of occurrences of those IVIHFEs. P-IVIHFS [41] was recently introduced to overcome such deep uncertainties and a few articles are found where researchers used it in decision-making problems. As decision-making method, TOPSIS and VIKOR are well known to the multi-criteria decision making (MCDM) researchers due to their intrinsic ability to decision making arena by respectively implementing “closeness to ideal solution” and “maximum group utility for the majority” concept. In order to include the various kinds of uncertainties in decision making, TOPSIS and VIKOR methods were extended by fuzzy set and its various extensions. Consequently, in the last few years, many real-life uncertain decision-making problems were solved by these compromise programming methods. However, to the best of our knowledge, no such study is found in the literature, where TOPSIS and VIKOR methods are extended using P-IVIHFS to gain combined benefit from them. Therefore, to fill up this research gap and explore more applicability of P-IVIHFS in real life decision-making problem, we continue to study on P-IVIHFS and its application in TOPSIS and VIKOR based decision-making paradigm.

This paper aims to propose a decision-making approach which can address the situations to deal with both the subjective and objective nature of uncertainties, where the attributes’ weights are incompletely known. Subjective nature of uncertainty is managed by membership, non-membership and hesitancy degrees that describe that vagueness of an element in a set, whereas objective nature of uncertainty is managed by the probability that describes the frequency of occurrence of an element in a set. To achieve this, we propose a MCDM approach based on the probabilistic interval-valued intuitionistic hesitant fuzzy set (P-IVIHFS), linear programming (LP), and extended TOPSIS method. P-IVIHFS is used to represent the decision makers’ opinions and LP is used to find the optimal weights from the incompletely known weights. This study extends the TOPSIS and VIKOR method using P-IVIHFS. Two illustrative examples are provided to demonstrate the proposed approach. Firstly, the numerical example shows the improvement of our approach over the existing approaches when the P-IVIHFS is considered. We have also shown a sensitivity analysis to observe the impact of attribute weight changes on the ranking of alternatives. Secondly, the supplier selection problem is used to explore the applicability of our approach in practical real-life problems.

The rest of this paper is organized as follows. Section 2 briefly reviews the basic relevant ideas. In Section 3, we present our proposed method. A numerical example is given in Section 4. The corresponding result discussion and sensitivity analysis is depicted in Section 5. We applied the proposed approach in supplier selection problem in Section 6 followed by comparative analysis given in Section 7. Finally, we conclude in Section 8.

2 Preliminaries

In this section, we briefly recall IVIFS, IVIHFSs, and P-IVIHFSs.

Definition 1. [46] An IVIFS A in the universe of discourse X ={ x₁, x₂, …, x_n } can be represented by $A = {〈 x_{j}, {\tilde{μ}}_{A} (x_{j}), {\tilde{ν}}_{A} (x_{j}) 〉 | x_{j} \in X}$ where ${\tilde{μ}}_{A} (x_{j})$ and ${\tilde{ν}}_{A} (x_{j})$ are respectively the interval-valued membership function and the interval-valued non-membership function of the IVIFS A, ${\tilde{μ}}_{A} (x_{j}) = [μ_{A}^{-} (x_{j}), μ_{A}^{+} (x_{j})]$ , ${\tilde{ν}}_{A} (x_{j}) = [ν_{A}^{-} (x_{j}), ν_{A}^{+} (x_{j})]$ , $0 \leq μ_{A}^{-} (x_{j}) \leq μ_{A}^{+} (x_{j}) \leq 1$ , $0 \leq ν_{A}^{-} (x_{j}) \leq ν_{A}^{+} (x_{j}) \leq 1$ , $0 \leq μ_{A}^{+} (x_{j}) + ν_{A}^{+} (x_{j}) \leq 1$ and j = 1, 2, …, n. The degree of indeterminacy ${\tilde{π}}_{A} (x_{j})$ of the element x_j belonging to the IVIFS A is represented by ${\tilde{π}}_{A} (x_{j}) = [π_{A}^{-} (x_{j}), π_{A}^{+} (x_{j})]$ where $π_{A}^{-} (x_{j}) = 1 - μ_{A}^{+} (x_{j}) - ν_{A}^{+} (x_{j})$ and $π_{A}^{+} (x_{j}) = 1 - μ_{A}^{-} (x_{j}) - ν_{A}^{-} (x_{j})$ , j = 1, 2, …, n.

Definition 2. An IVIHFS E on the universe of discourse X is defined in terms of a function that when applied to X returns a subset of a set of all IVIF numbers. Mathematically it is represented by an object of the following form E = {x, h_E (x)} , ∀ x ∈ X. Here h_E (x) is a set of some IVIF numbers denoting the possible membership degree intervals and non-membership degree intervals of the element x ∈ X to the set E. Interval-valued intuitionistic hesitant fuzzy element (IVIHFE) is denoted by $\hat{h} = h_{E} (x)$ , where $α \in \hat{h}$ be an interval-valued intuitionistic fuzzy number (IVIFN) denoted by $α = (μ_{α}, ν_{α}) = ([μ_{α}^{-}, μ_{α}^{+}], [ν_{α}^{-}, ν_{α}^{+}])$ .

Some operations on IVIHFEs $\tilde{h}$ , ${\tilde{h}}_{1}$ and ${\tilde{h}}_{2}$ are given below. $\begin{matrix} {\tilde{h}}^{c} = {α^{c} | α \in \tilde{h}} = {[ν_{α}^{-}, ν_{α}^{+}], [μ_{α}^{-}, μ_{α}^{+}] | α \in \tilde{h}} \\ {\tilde{h}}_{1} \cup {\tilde{h}}_{2} = {max (α_{1}, α_{2}) | α_{1} \in {\tilde{h}}_{1} and α_{2} \in {\tilde{h}}_{2}} \\ = {[max (μ_{α_{1}}^{-}, μ_{α_{2}}^{-}), max (μ_{α_{1}}^{+}, μ_{α_{2}}^{+})], \\ [min (ν_{α_{1}}^{-}, ν_{α_{2}}^{-}), min (ν_{α_{1}}^{+}, ν_{α_{2}}^{+})] | α_{1} \in {\tilde{h}}_{1} and α_{2} \in {\tilde{h}}_{2}} \end{matrix}$

$\begin{matrix} {\tilde{h}}_{1} \cap {\tilde{h}}_{2} = {min (α_{1}, α_{2}) | α_{1} \in {\tilde{h}}_{1} and α_{2} \in {\tilde{h}}_{2}} \\ = {[min (μ_{α_{1}}^{-}, μ_{α_{2}}^{-}), min (μ_{α_{1}}^{+}, μ_{α_{2}}^{+})], \\ [max (ν_{α_{1}}^{-}, ν_{α_{2}}^{-}), max (ν_{α_{1}}^{+}, ν_{α_{2}}^{+})] | α_{1} \in {\tilde{h}}_{1} and α_{2} \in {\tilde{h}}_{2}} \end{matrix}$

$\begin{matrix} {\tilde{h}}_{1} \oplus {\tilde{h}}_{2} = {[μ_{α_{1}}^{-} + μ_{α_{2}}^{-} - μ_{α_{1}}^{-} μ_{α_{2}}^{-}, μ_{α_{1}}^{+} + μ_{α_{2}}^{+} - μ_{α_{1}}^{+} μ_{α_{2}}^{+}], \\ [ν_{α_{1}}^{-} ν_{α_{2}}^{-}, ν_{α_{1}}^{+} ν_{α_{2}}^{+}] | α_{1} \in {\tilde{h}}_{1} and α_{2} \in {\tilde{h}}_{2}} \end{matrix}$

Example 1. Suppose that a consumer wants to buy a car. He/she mainly consider the comfort of the car. When the customer is not sure about the comfort, s(he) may express his/her opinion in the way that comfort could be 50 to 60 but not less than 20 to 30 out of 100-point scale. Now this opinion can be expressed as ([0.5, 0.6], [0.2, 0.3]). Here the interval for membership degree ${\tilde{μ}}_{A} (x) = [μ_{A}^{-} (x), μ_{A}^{+} (x)] = [0.5, 0.6]$ and interval for non-membership degree ${\tilde{ν}}_{A} (x) =$ $[ν_{A}^{-} (x), ν_{A}^{+} (x)] = [0.2, 0.3]$ , where ${\tilde{π}}_{A} (x) = [π_{A}^{-} (x), π_{A}^{+} (x)] = [0.1, 0.3]$ be the interval for hesitation margin.

Definition 3. [41] Let X be a fixed set, then a probabilistic interval-valued intuitionistic hesitant fuzzy set (P-IVIHFS) on X is defined as an object denoted by $\tilde{E} = {x, h_{\tilde{E}} (x)}, \forall x \in X,$ where $h_{\tilde{E}} (x) = {〈 x_{j}, ({\tilde{μ}}_{A} (x_{j}), {\tilde{ν}}_{A} (x_{j})), p_{A} (x_{j}) 〉 | x_{j} \in X}$ and p_A (x_j) ∈ [0, 1] be the probability associated with each element which is used to express the possibility degree of its corresponding IVIHFN. For convenience, a probabilistic interval-valued intuitionistic hesitant fuzzy element (P-IVIHFE) is denoted by $\tilde{h} = {\tilde{h}}_{E} (x)$ and P is the set of all P-IVIHFEs on X.

Example 2. Let us consider Example 1. Here we consider that the customer is 60% sure about the comfort could be from 50 to 60 but not less than 20 to 30. Then this kind of situation can be expressed as ([0.5, 0.6], [0.2, 0.3], 0.6). Here ${\tilde{μ}}_{A} (x) = [μ_{A}^{-} (x), μ_{A}^{+} (x)] = [0.5, 0.6]$ and ${\tilde{ν}}_{A} (x) = [ν_{A}^{-} (x), ν_{A}^{+} (x)] = [0.2, 0.3]$ where ${\tilde{π}}_{A} (x) = [π_{A}^{-} (x), π_{A}^{+} (x)] = [0.1, 0.3]$ with 60% of surety i.e., probability of surety of the customer is p_A (x) = 0.6.

Definition 4. [45] Let the universe of discourse X = {x₁, x₂, …, x_n} and $A = {〈 x_{j}, {\tilde{μ}}_{A} (x_{j})$ , ${\tilde{ν}}_{A} (x_{j}) 〉 | x_{j} \in X}$ , $B = {〈 x_{j}, {\tilde{μ}}_{B} (x_{j}),$ ${\tilde{ν}}_{B} (x_{j}) 〉 |$ x_j ∈ X} be two IVIFSs where j = 1, 2, …, n. Let w_j be the weight vector of the elements x_j, 0 ≤ w_j ≤ 1and $\sum_{j = 1}^{n} w_{j} = 1$ . The similarity degree of A and B is denoted as S (A, B) and defined as $S (A, B) = 1 - \sum_{j = 1}^{n} w_{j} [\begin{matrix} \frac{1}{8} (| μ_{A}^{-} - μ_{B}^{-} | + | μ_{A}^{+} - μ_{B}^{+} | + | ν_{A}^{-} - ν_{B}^{-} | + | ν_{A}^{+} - ν_{B}^{+} |) \\ + \frac{1}{2} \max {| μ_{A}^{-} - μ_{B}^{-} |, | μ_{A}^{+} - μ_{B}^{+} |, | ν_{A}^{-} - ν_{B}^{-} |, | ν_{A}^{+} - ν_{B}^{+} |} \end{matrix}]$

Here the similarity measure S (A, B) between the IVIFSs A and B is based on the weighted Hamming distance and the Hausdorff metric, which satisfies the following properties:

S (A, B) ∈ [0, 1],

S (A, B) =1 if and only if A = B,

S (A, B) = S (B, A),

If C is an IVIFS and A ⊆ B ⊆ C, then S (A, C)≤S (A, B) and S (A, C)≤S (B, C).

3 Proposed method

In this section, we propose a new MADM method based on P-IVIHFSs, the LP methodology and the TOPSIS method. We also extend the proposed approach using VIKOR method.

Let A ={ A₁, A₂, …, A_m } be the finite set of alternatives and S ={ s₁, s₂, …, s_n }be the finite set of attributes. The weight vector w = (w₁, w₂, …, w_n) ^T of the attributes s_j (j = 1, 2, …, n) is determined by DMs, where 0 ≤ w_j ≤ 1, j = 1, 2, …, n, and $\sum_{j = 1}^{n} w_{j} = 1$ .

The proposed MADM method is described using the following steps.

Step 1. Firstly we determine the probabilistic interval-valued intuitionistic hesitant fuzzy matrix (P-IVIHFM) to represent the opinions of decision-makers regarding the attributes S ={ s₁, s₂, …, s_n } with respect to the alternatives A ={ A₁, A₂, …, A_m } using P-IVIHFEs. P-IVIHFM is formed using the Hadamard product between ${({\tilde{d}}_{ij})}_{m \times n}$ and p_i j which is defined as ${\tilde{h}}_{ij} = {\tilde{d}}_{ij} \circ p_{ij}$ , where the symbol “∘” indicates the multiplication of the corresponding elements in the sets of ${\tilde{d}}_{ij}$ and p_ij. Here ${({\tilde{d}}_{ij})}_{m \times n} = {([μ_{ij}^{-}, μ_{ij}^{+}], [ν_{ij}^{-}, ν_{ij}^{+}])}_{m \times n}$ be the decision matrix formed by using the opinion from the DMs and p_i jindicates the possible membership degrees of the alternative A_i under the attribute s_j.

Hence P-IVIHFM $\tilde{H}$ is defined as $\tilde{H} =$ $({\tilde{h}}_{ij})_{m \times n} = ({\tilde{d}}_{ij} \circ p_{ij})_{m \times n} = (([a_{ij}^{-}, a_{ij}^{+}], [b_{ij}^{-}, b_{ij}^{+}]$ )) _m×n where $([a_{ij}^{-}, a_{ij}^{+}], [b_{ij}^{-}, b_{ij}^{+}])$ be the element corresponding to an alternative A_i (i = 1, 2, …, m) with respect to the attributes s_j (j = 1, 2, …, n) with probability degree p_ij.

Step 2. The probabilistic interval-valued intuitionistic hesitant fuzzy positive ideal solution (P-IVIHFPIS) T⁺ and probabilistic interval-valued

intuitionistic hesitant fuzzy negative ideal solution (P-IVIHFNIS) T^- are determined as given below.

Consider the collection of benefit attributes as J₁ and collection of cost attributes as J₂ and J₁ ∩ J₂ =

φ. Assume that the weight of attributes s_j given by the decision maker is ${\tilde{w}}_{j} = ([q_{j}, y_{j}], [z_{j}, g_{j}])$ where 0 ≤ q_j ≤ y_j ≤ 1, 0 ≤ z_j ≤ g_j ≤ 1 and 0 ≤ y_j + g_j ≤ 1.

A positive ideal solution T⁺ and negative ideal solution T^- of P-IVIHFEs are defined below. $\begin{matrix} T^{+} & = {(max_{i} {h_{ij}} : s_{j} \in J_{1}), (min_{i} {h_{ij}} : \\ s_{j} \in J_{2}) | 1 \leq i \leq m and 1 \leq j \leq n} \\ = {t_{1}^{+}, t_{2}^{+}, \dots, t_{n}^{+}} \end{matrix}$

Here if s_j ∈ J₁, then $t_{j}^{+} = ([max_{i} a_{ij}^{-}, max_{i} a_{ij}^{+}],$ $[min_{i} b_{ij}^{-}, min_{i} b_{ij}^{+}]) = ([a_{j}^{+}, b_{j}^{+}], [c_{j}^{+}, d_{j}^{+}])$ . If s_j ∈ J₂, then $t_{j}^{+} = ([min_{i} a_{ij}^{-}, min_{i} a_{ij}^{+}], [max_{i} b_{ij}^{-}, max_{i} b_{ij}^{+}]) = ([a_{j}^{+}, b_{j}^{+}], [c_{j}^{+}, d_{j}^{+}])$ . $\begin{matrix} T^{-} & = {(min_{i} {h_{ij}} : C_{j} \in J_{1}), (max_{i} {h_{ij}} : \\ C_{j} \in J_{2}) | 1 \leq i \leq m and 1 \leq j \leq n} \\ = {t_{1}^{-}, t_{2}^{-}, \dots, t_{n}^{-}} \end{matrix}$

Here if S_j ∈ J₁, then $t_{j}^{-} = ([min_{i} a_{ij}^{-}, min_{i} a_{ij}^{+}]$ , $[max_{i} b_{ij}^{-}, max_{i} b_{ij}^{+}]) = ([a_{j}^{-}, b_{j}^{-}], [c_{j}^{-}, d_{j}^{-}])$ . If s_j ∈ J₂, then $t_{j}^{-} = ([max_{i} a_{ij}^{-}, max_{i} a_{ij}^{+}], [min_{i} b_{ij}^{-}, min_{i} b_{ij}^{+}]) = ([a_{j}^{-}, b_{j}^{-}], [c_{j}^{-}, d_{j}^{-}])$ , where 1 ≤ j ≤ n.

Step 3. The degree of similarity $N_{i}^{+}$ between the elements at the i^throw of the P-IVIHFM $\tilde{H} = ({\tilde{h}}_{ij})_{m \times n} = ([a_{ij}^{-}, a_{ij}^{+}], [b_{ij}^{-}, b_{ij}^{+}])_{m \times n}$ and the elements in the obtained P-IVIHFPIS T⁺is computed below.

$\begin{matrix} N_{i}^{+} = 1 - \sum_{j = 1}^{n} w_{j}^{*} [\begin{matrix} \frac{1}{8} (| a_{ij}^{-} - a_{j}^{+} | + | a_{ij}^{+} - b_{j}^{+} | + | b_{ij}^{-} - c_{j}^{+} | + | b_{ij}^{+} - d_{j}^{+} |) \\ + \frac{1}{2} \max {| a_{ij}^{-} - a_{j}^{+} |, | a_{ij}^{+} - b_{j}^{+} |, | b_{ij}^{-} - c_{j}^{+} |, | b_{ij}^{+} - d_{j}^{+} |} \end{matrix}] \end{matrix}$ where $N_{i}^{+} \in [0, 1]$ .

Similarly, the degree of similarity $N_{i}^{-}$ between the elements at the i^th row of the P-IVIHFM $\tilde{H} = {({\tilde{h}}_{ij})}_{m \times n} = {([a_{ij}^{-}, a_{ij}^{+}], [b_{ij}^{-}, b_{ij}^{+}])}_{m \times n}$ and the elements in the obtained P-IVIHFNIS T^-is computed as $\begin{matrix} N_{i}^{-} = 1 - \sum_{j = 1}^{n} w_{j}^{*} [\begin{matrix} \frac{1}{8} (| a_{ij}^{-} - a_{j}^{-} | + | a_{ij}^{+} - b_{j}^{-} | + | b_{ij}^{-} - c_{j}^{-} | + | b_{ij}^{+} - d_{j}^{-} |) \\ + \frac{1}{2} \max {| a_{ij}^{-} - a_{j}^{-} |, | a_{ij}^{+} - b_{j}^{-} |, | b_{ij}^{-} - c_{j}^{-} |, | b_{ij}^{+} - d_{j}^{-} |} \end{matrix}] \end{matrix}$ where $N_{i}^{-} \in [0, 1]$ .

Here $w_{j}^{*}$ is the optimal weight of the attributes which can be obtained from solving the LP methodology as defined in the next step.

Step 4. The LP model is constructed using the following expression. $\begin{matrix} \begin{matrix} max L = \sum_{i = 1}^{m} (N_{i}^{+} - N_{i}^{-}) \\ subject to q_{j} \leq w_{j}^{*} \leq 1 - z_{j} and \sum_{j = 1}^{n} w_{j}^{*} = 1 \end{matrix} \end{matrix}$

Step 5. Solving the above LP, we can obtain the optimal weights $w_{1}^{*}, w_{2}^{*}, \dots, w_{n}^{*}$ of the attributes.

Step 6. Based on optimal weights, calculate $N_{i}^{+}$ and $N_{i}^{-}$ as defined in Step 3.

Step 7. Calculate the relative closeness coefficient δ_i of each alternative A_i as $\begin{matrix} δ_{i} = \frac{N_{i}^{+}}{N_{i}^{+} + N_{i}^{-}} \end{matrix}$

Now the alternatives are ranked depending upon the value of δ_i.

The stepwise approaches to extend the proposed method using the VIKOR method is given below.

Step 1. Initially, the P-IVIHFPIS $f_{j}^{+}$ and P-IVIHFNIS $f_{j}^{-}$ for all the attributes are computed.

If the attribute j represents a benefit, then $f_{j}^{+} = max_{i} h_{ij}$ , $f_{j}^{-} = min_{i} h_{ij}$ .

If the criterion j represents a cost, then $f_{j}^{+} = min_{i} h_{ij}$ , $f_{j}^{-} = max_{i} h_{ij}$ . h_ij is defined above (Step 1) in the TOPSIS based approach.

Step 2. Next, the π_i and σ_i values are computed for i = 1, 2, …, m, which represent the average and the worst group scores for the alternative A _i, respectively, with the relation $π_{i} = \sum_{j = 1}^{n} w_{j}^{*} (f_{j}^{+} - h_{ij}) / (f_{j}^{+} - f_{j}^{-})$ and $σ_{i} = max_{j} [w_{j}^{*} (f_{j}^{+} - h_{ij}) / (f_{j}^{+} - f_{j}^{-})]$

Here, $w_{j}^{*}$ are the optimal weights of the criteria obtained using the LP methodology defined in Step 4 in the earlier approach. The smaller values of π_i and σ_i correspond to the better average and the worse group scores for the alternative A _i, respectively. $Here (f_{j}^{+} - h_{ij}) = [(| a_{ij}^{-} - a_{j}^{+} | +$ $| a_{ij}^{+} - b_{j}^{+} | + | b_{ij}^{-} - c_{j}^{+} | + | b_{ij}^{+} - d_{j}^{+} |)]$ and $(f_{j}^{+} - f_{j}^{-}) = [(| a_{j}^{+} - a_{j}^{-} | +$ $| b_{j}^{+} - b_{j}^{-} | + | c_{j}^{+} - c_{j}^{-} | + | d_{j}^{+} - d_{j}^{-} |)]$

Step 3. Compute the Q_i values for i = 1, 2, …, m with the relation $Q_{i} = v (π_{i} - π^{+}) / (π^{-} - π^{+}) + (1 - v)$ $(σ_{i} - σ^{+}) / (σ^{-} - σ^{+})$ where $π^{+} = min_{i} π_{i}$ , $π^{-} = max_{i} π_{i}$ and $σ^{+} = min_{i} σ_{i}$ , $σ^{-} = max_{i} σ_{i}$

v is the weight of the decision-making strategy.

Step 4. Rank the alternatives by sorting each π, σ and Q value in increasing order.

4 Numerical example

This section illustrates the proposed approach using a numerical example. For the purpose of comparison, we have used some information described in [40]. Suppose that A₁, A₂, A₃ and A₄ are four alternatives and S₁, S₂ and S₃ are three attributes which are basically benefit attributes. We consider the weights as IVIF numbers, where w₁ = ([0.10, 0.40] , [0.20, 0.55]), w₂ = ([0.20, 0.50] , [0.15, 0.45]), and w₃ = ([0.25, 0.60] , [0.15, 0.38]), where h₁ = 0.10, y₁= 0.40, z₁= 0.20, g₁= 0.55, h₂= 0.20, y₂= 0.50, z₂= 0.15, g₂= 0.45, h₃= 0.25, y₃= 0.60, z₃= 0.15 and g₃= 0.38.

[Step 1] Assume that the opinion of decision makers are represented by P-IVIHFMs as given below. $\begin{matrix} \overset{S_{1} S_{2} S_{3}}{\begin{matrix} \begin{matrix} A_{1} \\ A_{2} \end{matrix} \\ A_{3} \\ A_{4} \end{matrix} [\begin{matrix} [\begin{matrix} (([0.4, 0.5], [0.3, 0.4]), 0.5), \\ (([0.5, 0.6], [0.2, 0.3]), 0.5) \end{matrix}] [\begin{matrix} (([0.4, 0.6], [0.2, 0.4]), 0.4), \\ (([0.5, 0.7], [0.1, 0.3]), 0.5) \end{matrix}] [\begin{matrix} (([0.1, 0.3], [0.5, 0.6]), 0.3), \\ (([0.3, 0.4], [0.4, 0.5]), 0.5) \end{matrix}] \\ [\begin{matrix} (([0.6, 0.7], [0.2, 0.3]), 0.6), \\ (([0.5, 0.6], [0.3, 0.4]), 0.6) \end{matrix}] [\begin{matrix} (([0.6, 0.7], [0.2, 0.3]), 0.7), \\ (([0.5, 0.6], [0.3, 0.4]), 0.6) \end{matrix}] [\begin{matrix} (([0.4, 0.7], [0.1, 0.2]), 0.5), \\ (([0.1, 0.4], [0.4, 0.6]), 0.6) \end{matrix}] \\ [\begin{matrix} (([0.3, 0.6], [0.3, 0.4]), 0.4), \\ (([0.1, 0.3], [0.4, 0.7]), 0.5) \end{matrix}] [\begin{matrix} (([0.5, 0.6], [0.3, 0.4]), 0.5), \\ (([0.6, 0.7], [0.2, 0.3]), 0.5) \end{matrix}] [\begin{matrix} (([0.5, 0.6], [0.1, 0.3]), 0.6), \\ (([0.6, 0.7], [0.2, 0.3]), 0.6) \end{matrix}] \\ [\begin{matrix} (([0.7, 0.8], [0.1, 0.2]), 0.7), \\ (([0.5, 0.6], [0.2, 0.3]), 0.6) \end{matrix}] [\begin{matrix} (([0.6, 0.7], [0.1, 0.3]), 0.7), \\ (([0.5, 0.6], [0.3, 0.4]), 0.6) \end{matrix}] [\begin{matrix} (([0.3, 0.4], [0.1, 0.2]), 0.4), \\ (([0.4, 0.5], [0.2, 0.3]), 0.5) \end{matrix}] \end{matrix}]} \end{matrix}$

$\begin{matrix} \overset{S_{1} S_{2}, S_{3}}{\begin{matrix} \begin{matrix} A_{1} \end{matrix} \\ \begin{matrix} A_{2} \end{matrix} \\ \begin{matrix} A_{3} \end{matrix} \\ A_{4} \end{matrix} [\begin{matrix} [\begin{matrix} ([0.20, 0.25], [0.15, 0.20]), \\ ([0.25, 0.30], [0.10, 0.15]) \end{matrix}] [\begin{matrix} ([0.16, 0.24], [0.08, 0.16]), \\ ([0.25, 0.35], [0.05, 0.15]) \end{matrix}] [\begin{matrix} ([0.03, 0.09], [0.15, 0.18]), \\ ([0.15, 0.20], [0.20, 0.25]) \end{matrix}] \\ [\begin{matrix} ([0.36, 0.42], [0.12, 0.18]), \\ ([0.30, 0.36], [0.18, 0.24]) \end{matrix}] [\begin{matrix} ([0.42, 0.49], [0.14, 0.21]), \\ ([0.30, 0.36], [0.18, 0.24]) \end{matrix}] [\begin{matrix} ([0.20, 0.35], [0.05, 0.10]), \\ ([0.06, 0.24], [0.24, 0.36]) \end{matrix}] \\ [\begin{matrix} ([0.12, 0.24], [0.12, 0.16]), \\ ([0.05, 0.15], [0.20, 0.35]) \end{matrix}] [\begin{matrix} ([0.25, 0.30], [0.15, 0.20]), \\ ([0.30, 0.35], [0.10, 0.15]) \end{matrix}] [\begin{matrix} ([0.30, 0.36], [0.06, 0.18]), \\ ([0.36, 0.42], [0.12, 0.18]) \end{matrix}] \\ [\begin{matrix} ([0.49, 0.56], [0.07, 0.14]), \\ ([0.30, 0.36], [0.12, 0.18]) \end{matrix}] [\begin{matrix} ([0.42, 0.49], [0.07, 0.21]), \\ ([0.30, 0.36], [0.18, 0.24]) \end{matrix}] [\begin{matrix} ([0.12, 0.16], [0.04, 0.08]), \\ ([0.20, 0.25], [0.10, 0.15]) \end{matrix}] \end{matrix}]} \end{matrix}$

[Step 2] Since the attributes S₁, S₂ and S₃ are benefit attributes, i.e., J₁ ={ S₁, S₂, S₃ }, we can get the P-IVIHFPIS as $\begin{matrix} T^{+} = {(max_{i} {{\tilde{h}}_{ij}} : S_{j} \in J_{1}), (min_{i} {{\tilde{h}}_{ij}} : \\ S_{j} \in J_{2}) | 1 \leq i \leq 4 and 1 \leq j \leq 3} \\ = {t_{1}^{+}, t_{2}^{+}, t_{3}^{+}} \end{matrix}$ where $\begin{matrix} \begin{matrix} t_{1}^{+} = {\begin{matrix} [\begin{matrix} [max (0.20, 0.36, 0.12, 0.49), max (0.25, 0.42, 0.24, 0.56)], \\ [min (0.15, 0.12, 0.12, 0.07), min (0.20, 0.18, 0.16, 0.14)] \end{matrix}], \\ [\begin{matrix} [max (0.25, 0.30, 0.05, 0.30), max (0.30, 0.36, 0.15, 0.36)], \\ [min (0.10, 0.18, 0.20, 0.12), min (0.15, 0.24, 0.35, 0.18)] \end{matrix}] \end{matrix}} \\ = {[[0.49, 0.56], [0.07, 0.14]], [0.30, 0.36], [0.10, 0.15]} \end{matrix} \end{matrix}$

Similarly $\begin{matrix} t_{2}^{+} = {[[0.42, 0.49], [0.07, 0.16]], [[0.30, 0.36], \\ [0.05, 0.15]]}, t_{3}^{+} = {[[0.30, 0.36], [0.04, 0.08]], \\ [[0.36, 0.42], [0.10, 0.15]]} \end{matrix}$

That is, $\begin{matrix} \begin{matrix} t_{1}^{+} = {[[a_{11}^{+}, b_{11}^{+}], [c_{11}^{+}, d_{11}^{+}]], [[a_{21}^{+}, b_{21}^{+}], \\ [c_{21}^{+}, d_{21}^{+}]]} = {[[0.49, 0.56], [0.07, 0.14]], \\ [0.30, 0.36], [0.10, 0.15]} \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} t_{2}^{+} = {[[a_{12}^{+}, b_{12}^{+}], [c_{12}^{+}, d_{12}^{+}]], [[a_{22}^{+}, b_{22}^{+}], \\ [c_{22}^{+}, d_{22}^{+}]]} = {[[0.42, 0.49], \\ [0.07, 0.16]], [[0.30, 0.36], [0.05, 0.15]]} \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} t_{3}^{+} = {[[a_{13}^{+}, b_{13}^{+}], [c_{13}^{+}, d_{13}^{+}]], [[a_{23}^{+}, b_{23}^{+}], \\ [c_{23}^{+}, d_{23}^{+}]]} = {[[0.30, 0.36], [0.04, 0.08]], \\ [[0.36, 0.42], [0.10, 0.15]]} \end{matrix} \end{matrix}$

In the same way, since S₁, S₂ and S₃ are benefit attributes, i.e., J₁ ={ S₁, S₂, S₃ }, we can get the P-IVIHFPIS as $\begin{matrix} T^{-} = {(min_{i} {h_{ij}} : S_{j} \in J_{1}), \\ (max_{i} {h_{ij}} : S_{j} \in J_{2}) | 1 \leq i \leq 4 and \\ 1 \leq j \leq 3} = {t_{1}^{-}, t_{2}^{-}, t_{3}^{-}} \end{matrix}$ where $\begin{matrix} \begin{matrix} t_{1}^{-} = {\begin{matrix} [\begin{matrix} [min (0.20, 0.36, 0.12, 0.49), min (0.25, 0.42, 0.24, 0.56)], \\ [max (0.15, 0.12, 0.12, 0.07), max (0.20, 0.18, 0.16, 0.14)] \end{matrix}], \\ [\begin{matrix} [min (0.25, 0.30, 0.05, 0.30), min (0.30, 0.36, 0.15, 0.36)], \\ [max (0.10, 0.18, 0.20, 0.12), max (0.15, 0.24, 0.35, 0.18)] \end{matrix}] \end{matrix}} \\ = {[[0.12, 0.24], [0.15, 0.20]], [0.05, 0.15], [0.20, 0.35]} \end{matrix} \end{matrix}$ $\begin{matrix} t_{2}^{-} = { & [[0.16, 0.24], [0.15, 0.21]], [[0.25, 0.35], \\ [0.18, 0.24]]} \end{matrix}$ $\begin{matrix} t_{3}^{-} = { & [[0.03, 0.09], [0.15, 0.18]], [[0.06, 0.20], \\ [0.24, 0.36]]} \end{matrix}$

That is, $\begin{matrix} t_{1}^{-} & = {[[a_{11}^{-}, b_{11}^{-}], [c_{11}^{-}, d_{11}^{-}]], [[a_{21}^{-}, b_{21}^{-}], [c_{21}^{-}, d_{21}^{-}]]} \\ = {[[0.12, 0.24], [0.15, 0.20]], [0.05, 0.15], \\ [0.20, 0.35]} \end{matrix}$ $\begin{matrix} t_{2}^{-} & = {[[a_{12}^{-}, b_{12}^{-}], [c_{12}^{-}, d_{12}^{-}]], [[a_{22}^{-}, b_{22}^{-}], [c_{22}^{-}, d_{22}^{-}]]} \\ = {[[0.16, 0.24], [0.15, 0.21]], [[0.25, 0.35], \\ [0.18, 0.24]]} \end{matrix}$ $\begin{matrix} t_{3}^{-} & = {[[a_{13}^{-}, b_{13}^{-}], [c_{13}^{-}, d_{13}^{-}]], [[a_{23}^{-}, b_{23}^{-}], [c_{23}^{-}, d_{23}^{-}]]} \\ = {[[0.03, 0.09], [0.15, 0.18]], [[0.06, 0.20], \\ [0.24, 0.36]]} \end{matrix}$

[Step 3] Degree of similarity $N_{1}^{+}$ between the 1^st row of the P-IVIHFM $\tilde{H} = {({\tilde{h}}_{ij})}_{m \times n} =$ ${([a_{ij}^{-}, a_{ij}^{+}], [b_{ij}^{-}, b_{ij}^{+}])}_{m \times n}$ and the elements in the obtained P-IVIHFPIS T⁺ is calculated as follows. $\begin{matrix} \begin{matrix} N_{1}^{+} = 1 - w_{1} [\begin{matrix} \frac{1}{8} (\begin{matrix} | 0.20 - 0.49 | + | 0.25 - 0.56 | + | 0.15 - 0.07 | + | 0.20 - 0.14 | \\ + | 0.20 - 0.30 | + | 0.30 - 0.36 | + | 0.10 - 0.10 | + | 0.15 - 0.15 | \end{matrix}) \\ + \frac{1}{2} \max {\begin{matrix} | 0.20 - 0.49 |, | 0.25 - 0.56 |, | 0.15 - 0.07 |, | 0.20 - 0.14 |, \\ | 0.20 - 0.30 |, | 0.30 - 0.36 |, | 0.10 - 0.10 |, | 0.15 - 0.15 | \end{matrix}} \end{matrix}] \\ - w_{2} [\begin{matrix} \frac{1}{8} (\begin{matrix} | 0.16 - 0.42 | + | 0.24 - 0.49 | + | 0.08 - 0.07 | + | 0.16 - 0.16 | \\ + | 0.25 - 0.30 | + | 0.35 - 0.36 | + | 0.05 - 0.05 | + | 0.15 - 0.15 | \end{matrix}) \\ + \frac{1}{2} \max {\begin{matrix} | 0.16 - 0.42 |, | 0.24 - 0.49 |, | 0.08 - 0.07 |, | 0.16 - 0.16 |, \\ | 0.25 - 0.30 |, | 0.35 - 0.36 |, | 0.05 - 0.05 |, | 0.15 - 0.15 | \end{matrix}} \end{matrix}] \\ - w_{3} [\begin{matrix} \frac{1}{8} (\begin{matrix} | 0.03 - 0.30 | + | 0.09 - 0.36 | + | 0.15 - 0.04 | + | 0.18 - 0.08 | \\ + | 0.15 - 0.36 | + | 0.20 - 0.42 | + | 0.20 - 0.10 | + | 0.25 - 0.15 | \end{matrix}) \\ + \frac{1}{2} \max {\begin{matrix} | 0.03 - 0.30 |, | 0.09 - 0.36 |, | 0.15 - 0.04 |, | 0.18 - 0.08 |, \\ | 0.15 - 0.36 |, | 0.20 - 0.42 |, | 0.20 - 0.10 |, | 0.25 - 0.15 | \end{matrix}} \end{matrix}] \end{matrix} \end{matrix}$ i.e., $N_{1}^{+} = 1 - 0.2675 w_{1} - 0.2025 w_{2} - 0.3075 w_{3}$ . Similarly, we calculate $N_{2}^{+} = 1 - 0.13625 w_{1} - 0.1075 w_{2} - 0.27125 w_{3}$ , $N_{3}^{+} = 1 - 0.375 w_{1} - 0.1625 w_{2} - 0.07125 w_{3}$ , $\begin{matrix} N_{4}^{+} = 1 - 0.02125 w_{1} - 0.09875 w_{2} - 0.18875 w_{3} \end{matrix}$

Similarly, the degree of similarity $N_{1}^{-}$ between the elements at the 1st row of the decision matrix $\tilde{H} = {({\tilde{h}}_{ij})}_{m \times n} = {([a_{ij}^{-}, a_{ij}^{+}], [b_{ij}^{-}, b_{ij}^{+}])}_{m \times n}$ and the elements in the obtained interval-valued intuitionistic NIS T^- is computed as $\begin{matrix} \begin{matrix} N_{1}^{-} = 1 - w_{1} [\begin{matrix} \frac{1}{8} (\begin{matrix} | 0.20 - 0.12 | + | 0.25 - 0.24 | + | 0.15 - 0.15 | + | 0.20 - 0.20 | \\ + | 0.25 - 0.05 | + | 0.30 - 0.15 | + | 0.10 - 0.20 | + | 0.15 - 0.35 | \end{matrix}) \\ + \frac{1}{2} \max {\begin{matrix} | 0.20 - 0.12 |, | 0.25 - 0.24 |, | 0.15 - 0.15 |, | 0.20 - 0.20 |, \\ | 0.25 - 0.05 |, | 0.30 - 0.15 |, | 0.10 - 0.20 |, | 0.15 - 0.35 | \end{matrix}} \end{matrix}] \\ - w_{2} [\begin{matrix} \frac{1}{8} (\begin{matrix} | 0.16 - 0.16 | + | 0.24 - 0.24 | + | 0.08 - 0.15 | + | 0.16 - 0.21 | \\ + | 0.25 - 0.25 | + | 0.35 - 0.35 | + | 0.05 - 0.18 | + | 0.15 - 0.24 | \end{matrix}) \\ + \frac{1}{2} \max {\begin{matrix} | 0.16 - 0.16 |, | 0.24 - 0.24 |, | 0.08 - 0.15 |, | 0.16 - 0.21 |, \\ | 0.25 - 0.25 |, | 0.35 - 0.35 |, | 0.05 - 0.18 |, | 0.15 - 0.24 | \end{matrix}} \end{matrix}] \\ - w_{3} [\begin{matrix} \frac{1}{8} (\begin{matrix} | 0.03 - 0.03 | + | 0.09 - 0.09 | + | 0.15 - 0.15 | + | 0.18 - 0.18 | \\ + | 0.15 - 0.06 | + | 0.20 - 0.20 | + | 0.20 - 0.24 | + | 0.25 - 0.36 | \end{matrix}) \\ + \frac{1}{2} \max {\begin{matrix} | 0.03 - 0.03 |, | 0.09 - 0.09 |, | 0.15 - 0.15 |, | 0.18 - 0.18 |, \\ | 0.15 - 0.06 |, | 0.20 - 0.20 |, | 0.20 - 0.24 |, | 0.25 - 0.36 | \end{matrix}} \end{matrix}] \end{matrix} \end{matrix}$ i.e., $N_{1}^{-} = 1 - 0.1925 w_{1} - 0.1075 w_{2} - 0.085 w_{3}$ . Similarly, we calculate $N_{2}^{-} = 1 - 0.22875 w_{1} - 0.2025 w_{2} - 0.21125 w_{3}$ , $N_{3}^{-} = 1 - 0.2025 w_{1} - 0.0925 w_{2} - 0.33125 w_{3}$ $\begin{matrix} N_{4}^{-} = 1 - 0.35375 w_{1} - 0.21125 w_{2} - 0.21875 w_{3} \end{matrix}$

[Step 4 & 5] Solving the LP model, the optimal weights $w_{1}^{*}$ , $w_{2}^{*}$ and $w_{3}^{*}$ of the attributes S₁, S₂ and S₃ are respectively found as $w_{1}^{*} = 0.55$ , $w_{2}^{*} = 0.20$ and $w_{3}^{*} = 0.25$ .

[Step 6] The degree of similarity $N_{i}^{+}$ between the elements at the i^th row of the P-IVIHFM and the elements of the P-IVIHFPIS T⁺is computed as $N_{1}^{+} = 0 . 7355$ , $N_{2}^{+} = 0 . 83575$ , $N_{3}^{+} = 0 . 743438$ and $N_{4}^{+} = 0 . 921375$ . Similarly, the degree of similarity $N_{i}^{-}$ between the elements at the i th row of the P-IVIHFM and the elements of the P-IVIHFN T^- is computed as $N_{1}^{-} = 0 . 851375$ , $N_{2}^{-} = 0 . 780875$ , $N_{3}^{-} = 0 . 787313$ and $N_{4}^{-} = 0 . 7085$ .

[Step 7] The relative closeness coefficient of each of the alternatives are computed as follows. $\begin{matrix} δ_{1}^{*} = \frac{N_{1}^{+}}{N_{1}^{+} + N_{1}^{-}} = \frac{0 . 7355}{0 . 7355 + 0 . 851375} = 0.46349 \\ δ_{2}^{*} = \frac{N_{2}^{+}}{N_{2}^{+} + N_{2}^{-}} = \frac{0 . 83575}{0 . 83575 + 0 . 780875} = 0.516972 \\ δ_{3}^{*} = \frac{N_{3}^{+}}{N_{3}^{+} + N_{3}^{-}} = \frac{0 . 743438}{0 . 743438 + 0 . 787313} = 0.485669 \\ δ_{4}^{*} = \frac{N_{4}^{+}}{N_{4}^{+} + N_{4}^{-}} = \frac{0 . 921375}{0 . 921375 + 0 . 7085} = 0.565304 \end{matrix}$

Since $δ_{4}^{*} > δ_{2}^{*} > δ_{3}^{*} > δ_{1}^{*}$ , so the preference order of the alternative is A₄ > A₂ > A₃ > A₁.

Next, we analyze the extended VIKOR method using the same example explained above.

[Step 1] Since the attributes S ₁ , S ₂ and S ₃ are benefit attributes, i.e., J₁ ={ S₁, S₂, S₃ } the P-IVIHFPISs are computed as follows. $\begin{matrix} \begin{matrix} f_{1}^{+} = {\begin{matrix} [\begin{matrix} [max (0.20, 0.36, 0.12, 0.49), max (0.25, 0.42, 0.24, 0.56)], \\ [min (0.15, 0.12, 0.12, 0.07), min (0.20, 0.18, 0.16, 0.14)] \end{matrix}], \\ [\begin{matrix} [max (0.25, 0.30, 0.05, 0.30), max (0.30, 0.36, 0.15, 0.36)], \\ [min (0.10, 0.18, 0.20, 0.12), min (0.15, 0.24, 0.35, 0.18)] \end{matrix}] \end{matrix}} \\ = {[[0.49, 0.56], [0.07, 0.14]], [0.30, 0.36], [0.10, 0.15]} \end{matrix} \end{matrix}$

Similarly, $\begin{matrix} f_{2}^{+} = { & [[0.42, 0.49], [0.07, 0.16]], [[0.30, 0.36], \\ [0.05, 0.15]]} \end{matrix}$ $\begin{matrix} f_{3}^{+} = { & [[0.30, 0.36], [0.04, 0.08]], [[0.36, 0.42], \\ [0.10, 0.15]]} \end{matrix}$

That is, $\begin{matrix} f_{1}^{+} & = {[[a_{11}^{+}, b_{11}^{+}], [c_{11}^{+}, d_{11}^{+}]], [[a_{21}^{+}, b_{21}^{+}], [c_{21}^{+}, d_{21}^{+}]]} \\ = {[[0.49, 0.56], [0.07, 0.14]], [0.30, 0.36], \\ [0.10, 0.15]} \end{matrix}$ $\begin{matrix} f_{2}^{+} & = {[[a_{12}^{+}, b_{12}^{+}], [c_{12}^{+}, d_{12}^{+}]], [[a_{22}^{+}, b_{22}^{+}], [c_{22}^{+}, d_{22}^{+}]]} \\ = {[[0.42, 0.49], [0.07, 0.16]], [[0.30, 0.36], \\ [0.05, 0.15]]} \end{matrix}$ $\begin{matrix} f_{3}^{+} & = {[[a_{13}^{+}, b_{13}^{+}], [c_{13}^{+}, d_{13}^{+}]], [[a_{23}^{+}, b_{23}^{+}], [c_{23}^{+}, d_{23}^{+}]]} \\ = {[[0.30, 0.36], [0.04, 0.08]], [[0.36, 0.42], \\ [0.10, 0.15]]} \end{matrix}$

Similarly, the P-IVIHFNISs are computed as $\begin{matrix} \begin{matrix} f_{1}^{-} = {\begin{matrix} [\begin{matrix} [min (0.20, 0.36, 0.12, 0.49), min (0.25, 0.42, 0.24, 0.56)], \\ [max (0.15, 0.12, 0.12, 0.07), max (0.20, 0.18, 0.16, 0.14)] \end{matrix}], \\ [\begin{matrix} [min (0.25, 0.30, 0.05, 0.30), min (0.30, 0.36, 0.15, 0.36)], \\ [max (0.10, 0.18, 0.20, 0.12), max (0.15, 0.24, 0.35, 0.18)] \end{matrix}] \end{matrix}} \\ = {[[0.12, 0.24], [0.15, 0.20]], [0.05, 0.15], [0.20, 0.35]} \end{matrix} \end{matrix}$ $\begin{matrix} f_{2}^{-} = { & [[0.16, 0.24], [0.15, 0.21]], [[0.25, 0.35], \\ [0.18, 0.24]]} \end{matrix}$ $\begin{matrix} f_{3}^{-} = { & [[0.03, 0.09], [0.15, 0.18]], [[0.06, 0.20], \\ [0.24, 0.36]]} \end{matrix}$

That is, $\begin{matrix} f_{1}^{-} & = {[[a_{11}^{-}, b_{11}^{-}], [c_{11}^{-}, d_{11}^{-}]], [[a_{21}^{-}, b_{21}^{-}], [c_{21}^{-}, d_{21}^{-}]]} \\ = {[[0.12, 0.24], [0.15, 0.20]], [0.05, 0.15], \\ [0.20, 0.35]} \end{matrix}$

$\begin{matrix} f_{2}^{-} & = {[[a_{12}^{-}, b_{12}^{-}], [c_{12}^{-}, d_{12}^{-}]], [[a_{22}^{-}, b_{22}^{-}], [c_{22}^{-}, d_{22}^{-}]]} \\ = {[[0.16, 0.24], [0.15, 0.21]], [[0.25, 0.35], \\ [0.18, 0.24]]} \end{matrix}$

$\begin{matrix} f_{3}^{-} & = {[[a_{13}^{-}, b_{13}^{-}], [c_{13}^{-}, d_{13}^{-}]], [[a_{23}^{-}, b_{23}^{-}], [c_{23}^{-}, d_{23}^{-}]]} \\ = {[[0.03, 0.09], [0.15, 0.18]], [[0.06, 0.20], \\ [0.24, 0.36]]} \end{matrix}$

[Step 2] Next the π_i and σ_i values for i = 1, ..., m are computed as given below. $\begin{matrix} \begin{matrix} π_{1} = w_{1} [(\begin{matrix} | 0.49 - 0.20 | + | 0.56 - 0.25 | + | 0.07 - 0.15 | + | 0.14 - 0.20 | + \\ | 0.20 - 0.30 | + | 0.30 - 0.36 | + | 0.10 - 0.10 | + | 0.15 - 0.15 | \end{matrix})] / \\ [(\begin{matrix} | 0.49 - 0.12 | + | 0.56 - 0.24 | + | 0.07 - 0.15 | + | 0.14 - 0.20 | + \\ | 0.30 - 0.05 | + | 0.36 - 0.15 | + | 0.10 - 0.20 | + | 0.15 - 0.35 | \end{matrix})] \\ + w_{2} [(\begin{matrix} | 0.42 - 0.16 | + | 0.49 - 0.24 | + | 0.07 - 0.08 | + | 0.16 - 0.16 | + \\ | 0.25 - 0.30 | + | 0.35 - 0.36 | + | 0.05 - 0.05 | + | 0.15 - 0.15 | \end{matrix})] / \\ [(\begin{matrix} | 0.42 - 0.16 | + | 0.49 - 0.24 | + | 0.07 - 0.15 | + | 0.16 - 0.21 | + \\ | 0.30 - 0.25 | + | 0.36 - 0.35 | + | 0.05 - 0.18 | + | 0.15 - 0.24 | \end{matrix})] \\ + w_{3} [(\begin{matrix} | 0.30 - 0.03 | + | 0.36 - 0.09 | + | 0.04 - 0.15 | + | 0.08 - 0.18 | + \\ | 0.15 - 0.36 | + | 0.20 - 0.42 | + | 0.20 - 0.10 | + | 0.25 - 0.15 | \end{matrix})] / \\ [(\begin{matrix} | 0.30 - 0.03 | + | 0.36 - 0.09 | + | 0.04 - 0.15 | + | 0.08 - 0.18 | + \\ | 0.36 - 0.06 | + | 0.42 - 0.20 | + | 0.10 - 0.24 | + | 0.15 - 0.36 | \end{matrix})] \end{matrix} \end{matrix}$ and

$\begin{matrix} σ_{1} = max {\begin{matrix} w_{1} [(\begin{matrix} | 0.49 - 0.20 | + | 0.56 - 0.25 | + | 0.07 - 0.15 | + | 0.14 - 0.20 | + \\ | 0.20 - 0.30 | + | 0.30 - 0.36 | + | 0.10 - 0.10 | + | 0.15 - 0.15 | \end{matrix})] / \\ [(\begin{matrix} | 0.49 - 0.12 | + | 0.56 - 0.24 | + | 0.07 - 0.15 | + | 0.14 - 0.20 | + \\ | 0.30 - 0.05 | + | 0.36 - 0.15 | + | 0.10 - 0.20 | + | 0.15 - 0.35 | \end{matrix})], \\ w_{2} [(\begin{matrix} | 0.42 - 0.16 | + | 0.49 - 0.24 | + | 0.07 - 0.08 | + | 0.16 - 0.16 | + \\ | 0.25 - 0.30 | + | 0.35 - 0.36 | + | 0.05 - 0.05 | + | 0.15 - 0.15 | \end{matrix})] / \\ [(\begin{matrix} | 0.42 - 0.16 | + | 0.49 - 0.24 | + | 0.07 - 0.15 | + | 0.16 - 0.21 | + \\ | 0.30 - 0.25 | + | 0.36 - 0.35 | + | 0.05 - 0.18 | + | 0.15 - 0.24 | \end{matrix})], \\ w_{3} [(\begin{matrix} | 0.30 - 0.03 | + | 0.36 - 0.09 | + | 0.04 - 0.15 | + | 0.08 - 0.18 | + \\ | 0.15 - 0.36 | + | 0.20 - 0.42 | + | 0.20 - 0.10 | + | 0.25 - 0.15 | \end{matrix})] / \\ [(\begin{matrix} | 0.30 - 0.03 | + | 0.36 - 0.09 | + | 0.04 - 0.15 | + | 0.08 - 0.18 | + \\ | 0.36 - 0.06 | + | 0.42 - 0.20 | + | 0.10 - 0.24 | + | 0.15 - 0.36 | \end{matrix})] \end{matrix}} \end{matrix}$

Table 1
Sensitivity analysis for extended TOPSIS method

[w₁, w₂, w₃] δ ₁ δ ₂ δ ₃ δ ₄ Alternatives rank

[0.20,0.20,0.60] 0.447998 0.500476 0.533763 0.533055 δ₃ ≥ δ₄ ≥ δ₂ ≥ δ₁

[0.30,0.20,0.50] 0.452377 0.505276 0.520403 0.542315 δ₄ ≥ δ₃ ≥ δ₂ ≥ δ₁

[0.40,0.20,0.40] 0.456794 0.510006 0.506744 0.551538 δ₄ ≥ δ₂ ≥ δ₃ ≥ δ₁

[0.50,0.20,0.30] 0.46128 0.514667 0.492774 0.560725 δ₄ ≥ δ₂ ≥ δ₃ ≥ δ₁

[0.60,0.20,0.20] 0.46574 0.51926 0.478482 0.569874 δ₄ ≥ δ₂ ≥ δ₃ ≥ δ₁

[0.70,0.20,0.10] 0.470271 0.523788 0.463858 0.578988 δ₄ ≥ δ₂ ≥ δ₁ ≥ δ₃

[0.10,0.30,0.60] 0.447862 0.500553 0.536247 0.526154 δ₃ ≥ δ₄ ≥ δ₂ ≥ δ₁

[0.10,0.40,0.50] 0.452028 0.505397 0.525854 0.528521 δ₄ ≥ δ₃ ≥ δ₂ ≥ δ₁

[0.10,0.50,0.40] 0.456151 0.510136 0.515646 0.53086 δ₄ ≥ δ₃ ≥ δ₂ ≥ δ₁

[0.10,0.60,0.30] 0.46233 0.514776 0.505619 0.533172 δ₄ ≥ δ₂ ≥ δ₃ ≥ δ₁

[0.10,0.70,0.20] 0.464275 0.519318 0.495767 0.535457 δ₄ ≥ δ₂ ≥ δ₃ ≥ δ₁

[0.80,0.10,0.10] 0.470626 0.523791 0.459586 0.586059 δ₄ ≥ δ₂ ≥ δ₁ ≥ δ₃

[0.60,0.30,0.10] 0.469923 0.523784 0.467952 0.571973 δ₄ ≥ δ₂ ≥ δ₁ ≥ δ₃

[0.50,0.40,0.10] 0.469581 0.523781 0.471877 0.565013 δ₄ ≥ δ₂ ≥ δ₃ ≥ δ₁

[0.40,0.50,0.10] 0.469246 0.523777 0.475644 0.558108 δ₄ ≥ δ₂ ≥ δ₃ ≥ δ₁

[0.30,0.60,0.10] 0.468917 0.523774 0.479262 0.551257 δ₄ ≥ δ₂ ≥ δ₃ ≥ δ₁

[0.20,0.70,0.10] 0.468594 0.52377 0.482741 0.544459 δ₄ ≥ δ₂ ≥ δ₃ ≥ δ₁

[0.10,0.80,0.10] 0.468277 0.523767 0.486087 0.537715 δ₄ ≥ δ₂ ≥ δ₃ ≥ δ₁

[0.10,0.10,0.80] 0.439403 0.490543 0.557611 0.521333 δ₃ ≥ δ₄ ≥ δ₂ ≥ δ₁

[w₁, w₂, w₃]	δ ₁	δ ₂	δ ₃	δ ₄	Alternatives rank
[0.20,0.20,0.60]	0.447998	0.500476	0.533763	0.533055	δ₃ ≥ δ₄ ≥ δ₂ ≥ δ₁
[0.30,0.20,0.50]	0.452377	0.505276	0.520403	0.542315	δ₄ ≥ δ₃ ≥ δ₂ ≥ δ₁
[0.40,0.20,0.40]	0.456794	0.510006	0.506744	0.551538	δ₄ ≥ δ₂ ≥ δ₃ ≥ δ₁
[0.50,0.20,0.30]	0.46128	0.514667	0.492774	0.560725	δ₄ ≥ δ₂ ≥ δ₃ ≥ δ₁
[0.60,0.20,0.20]	0.46574	0.51926	0.478482	0.569874	δ₄ ≥ δ₂ ≥ δ₃ ≥ δ₁
[0.70,0.20,0.10]	0.470271	0.523788	0.463858	0.578988	δ₄ ≥ δ₂ ≥ δ₁ ≥ δ₃
[0.10,0.30,0.60]	0.447862	0.500553	0.536247	0.526154	δ₃ ≥ δ₄ ≥ δ₂ ≥ δ₁
[0.10,0.40,0.50]	0.452028	0.505397	0.525854	0.528521	δ₄ ≥ δ₃ ≥ δ₂ ≥ δ₁
[0.10,0.50,0.40]	0.456151	0.510136	0.515646	0.53086	δ₄ ≥ δ₃ ≥ δ₂ ≥ δ₁
[0.10,0.60,0.30]	0.46233	0.514776	0.505619	0.533172	δ₄ ≥ δ₂ ≥ δ₃ ≥ δ₁
[0.10,0.70,0.20]	0.464275	0.519318	0.495767	0.535457	δ₄ ≥ δ₂ ≥ δ₃ ≥ δ₁
[0.80,0.10,0.10]	0.470626	0.523791	0.459586	0.586059	δ₄ ≥ δ₂ ≥ δ₁ ≥ δ₃
[0.60,0.30,0.10]	0.469923	0.523784	0.467952	0.571973	δ₄ ≥ δ₂ ≥ δ₁ ≥ δ₃
[0.50,0.40,0.10]	0.469581	0.523781	0.471877	0.565013	δ₄ ≥ δ₂ ≥ δ₃ ≥ δ₁
[0.40,0.50,0.10]	0.469246	0.523777	0.475644	0.558108	δ₄ ≥ δ₂ ≥ δ₃ ≥ δ₁
[0.30,0.60,0.10]	0.468917	0.523774	0.479262	0.551257	δ₄ ≥ δ₂ ≥ δ₃ ≥ δ₁
[0.20,0.70,0.10]	0.468594	0.52377	0.482741	0.544459	δ₄ ≥ δ₂ ≥ δ₃ ≥ δ₁
[0.10,0.80,0.10]	0.468277	0.523767	0.486087	0.537715	δ₄ ≥ δ₂ ≥ δ₃ ≥ δ₁
[0.10,0.10,0.80]	0.439403	0.490543	0.557611	0.521333	δ₃ ≥ δ₄ ≥ δ₂ ≥ δ₁

We calculate π₁ = 0.654936 and σ₁ = 0.311321. Similarly, we get the values as π₂ = 0.410147 and σ₂ = 0.18333, π₃ = 0.669974 and σ₃ = 0.525786, π₄ = 0.187908 and σ₄ = 0.111917.

[Step 3] We compute Q_i values for i = 1, 2, 3, 4and consider changes from 0.1to 0.9. From the above result we conclude that A₄ ≥ A₂ ≥ A₁ ≥ A₃ for all values of v from 0.1to 0.9.

5 Result discussion and sensitivity analysis

In this study, we have proposed a decision-making method by extending the TOPSIS method using probabilistic interval-valued intuitionistic hesitant fuzzy set (P-IVIHFS) which is an extension of IVIHFS. In the last few years, many MADM methods were developed based on IVIHFS which considered only the subjective uncertainty whereas our study considers subjective as well as objective uncertainty. In addition to the information provided by IVIHFS, P-IVIHFS reflect the importance of each decision information using probability. Moreover, in the existing methods, weights of various attributes were known in advance using the membership values. Rather in the proposed method, the weights of the various attributes are partially known which are expressed as interval-valued intuitionistic fuzzy numbers. From the partial known information, we have optimized the weights of various attributes using the method of LP. As per our investigation, although the preference order of alternatives in the proposed method is different from the methods presented in [5 , 40], but the best alternative is found to be similar.

To depict the sensitivity analysis for extended TOPSIS method, we have reviewed the rank of the alternatives due to small variations in the weights of the attributes which shows the impact of attribute weights in the ranking process as given below in Table 1. We have also studied the variation of alternatives ranking due to the small changes in attributes weights for the extended VIKOR method which is shown in Table 2.

Table 2
Sensitivity analysis for extended VIKOR method (considering v = 0.5)

[w₁, w₂, w₃] Q ₁ Q ₂ Q ₃ Q ₄ Alternatives rank

[0.10,0.20,0.70] 0.700425 0.48144 0.202706 0.344287 Q₃ ≥ Q₄ ≥ Q₂ ≥ Q₁

[0.20,0.20,0.60] 0.641715 0.437251 0.282047 0.301093 Q₃ ≥ Q₄ ≥ Q₂ ≥ Q₁

[0.30,0.20,0.50] 0.583006 0.392757 0.372285 0.257898 Q₄ ≥ Q₃ ≥ Q₂ ≥ Q₁

[0.40,0.20,0.40] 0.524296 0.348264 0.462523 0.214704 Q₄ ≥ Q₂ ≥ Q₃ ≥ Q₁

[0.50,0.20,0.30] 0.476579 0.30377 0.552761 0.17151 Q₄ ≥ Q₂ ≥ Q₁ ≥ Q₃

[0.60,0.20,0.20] 0.489677 0.298117 0.642999 0.128315 Q₄ ≥ Q₂ ≥ Q₁ ≥ Q₃

[0.70,0.20,0.10] 0.502776 0.30087 0.733237 0.092085 Q₄ ≥ Q₂ ≥ Q₁ ≥ Q₃

[0.10,0.30,0.60] 0.644935 0.439062 0.256042 0.314194 Q₃ ≥ Q₄ ≥ Q₂ ≥ Q₁

[0.10,0.40,0.50] 0.589446 0.39638 0.309378 0.284102 Q₄ ≥ Q₃ ≥ Q₂ ≥ Q₁

[0.10,0.50,0.40] 0.533956 0.353699 0.362715 0.254009 Q₄ ≥ Q₃ ≥ Q₂ ≥ Q₁

[0.10,0.60,0.30] 0.53708 0.330146 0.416051 0.244809 Q₄ ≥ Q₂ ≥ Q₃ ≥ Q₁

[0.10,0.70,0.20] 0.556618 0.336522 0.469387 0.251774 Q₄ ≥ Q₂ ≥ Q₃ ≥ Q₁

[0.80,0.10,0.10] 0.527858 0.315725 0.799487 0.072019 Q₄ ≥ Q₂ ≥ Q₁ ≥ Q₃

[0.60,0.30,0.10] 0.477694 0.286015 0.666988 0.119861 Q₄ ≥ Q₂ ≥ Q₁ ≥ Q₃

[0.50,0.40,0.10] 0.452611 0.27116 0.600738 0.147636 Q₄ ≥ Q₂ ≥ Q₁ ≥ Q₃

[0.40,0.50,0.10] 0.471931 0.282029 0.534488 0.175412 Q₄ ≥ Q₂ ≥ Q₁ ≥ Q₃

[0.30,0.60,0.10] 0.506672 0.302319 0.50093 0.203187 Q₄ ≥ Q₂ ≥ Q₃ ≥ Q₁

[0.20,0.70,0.10] 0.541414 0.322609 0.511826 0.230963 Q₄ ≥ Q₂ ≥ Q₃ ≥ Q₁

[0.10,0.80,0.10] 0.576155 0.342899 0.522723 0.258738 Q₄ ≥ Q₂ ≥ Q₃ ≥ Q₁

[0.10,0.10,0.80] 0.755914 0.524426 0.16782 0.37438 Q₃ ≥ Q₄ ≥ Q₂ ≥ Q₁

[w₁, w₂, w₃]	Q ₁	Q ₂	Q ₃	Q ₄	Alternatives rank
[0.10,0.20,0.70]	0.700425	0.48144	0.202706	0.344287	Q₃ ≥ Q₄ ≥ Q₂ ≥ Q₁
[0.20,0.20,0.60]	0.641715	0.437251	0.282047	0.301093	Q₃ ≥ Q₄ ≥ Q₂ ≥ Q₁
[0.30,0.20,0.50]	0.583006	0.392757	0.372285	0.257898	Q₄ ≥ Q₃ ≥ Q₂ ≥ Q₁
[0.40,0.20,0.40]	0.524296	0.348264	0.462523	0.214704	Q₄ ≥ Q₂ ≥ Q₃ ≥ Q₁
[0.50,0.20,0.30]	0.476579	0.30377	0.552761	0.17151	Q₄ ≥ Q₂ ≥ Q₁ ≥ Q₃
[0.60,0.20,0.20]	0.489677	0.298117	0.642999	0.128315	Q₄ ≥ Q₂ ≥ Q₁ ≥ Q₃
[0.70,0.20,0.10]	0.502776	0.30087	0.733237	0.092085	Q₄ ≥ Q₂ ≥ Q₁ ≥ Q₃
[0.10,0.30,0.60]	0.644935	0.439062	0.256042	0.314194	Q₃ ≥ Q₄ ≥ Q₂ ≥ Q₁
[0.10,0.40,0.50]	0.589446	0.39638	0.309378	0.284102	Q₄ ≥ Q₃ ≥ Q₂ ≥ Q₁
[0.10,0.50,0.40]	0.533956	0.353699	0.362715	0.254009	Q₄ ≥ Q₃ ≥ Q₂ ≥ Q₁
[0.10,0.60,0.30]	0.53708	0.330146	0.416051	0.244809	Q₄ ≥ Q₂ ≥ Q₃ ≥ Q₁
[0.10,0.70,0.20]	0.556618	0.336522	0.469387	0.251774	Q₄ ≥ Q₂ ≥ Q₃ ≥ Q₁
[0.80,0.10,0.10]	0.527858	0.315725	0.799487	0.072019	Q₄ ≥ Q₂ ≥ Q₁ ≥ Q₃
[0.60,0.30,0.10]	0.477694	0.286015	0.666988	0.119861	Q₄ ≥ Q₂ ≥ Q₁ ≥ Q₃
[0.50,0.40,0.10]	0.452611	0.27116	0.600738	0.147636	Q₄ ≥ Q₂ ≥ Q₁ ≥ Q₃
[0.40,0.50,0.10]	0.471931	0.282029	0.534488	0.175412	Q₄ ≥ Q₂ ≥ Q₁ ≥ Q₃
[0.30,0.60,0.10]	0.506672	0.302319	0.50093	0.203187	Q₄ ≥ Q₂ ≥ Q₃ ≥ Q₁
[0.20,0.70,0.10]	0.541414	0.322609	0.511826	0.230963	Q₄ ≥ Q₂ ≥ Q₃ ≥ Q₁
[0.10,0.80,0.10]	0.576155	0.342899	0.522723	0.258738	Q₄ ≥ Q₂ ≥ Q₃ ≥ Q₁
[0.10,0.10,0.80]	0.755914	0.524426	0.16782	0.37438	Q₃ ≥ Q₄ ≥ Q₂ ≥ Q₁

6 The proposed method to solve the supplier selection problem

Supplier selection plays the most important role in logistics, production operation management and supply-chain management. Selection of the right suppliers has a huge impact on the supply chain performance which is necessary to increase customer satisfaction and to improve product and service quality. Researchers are paying more attention to sustainable supplier selection due to growing social and environmental awareness. In recent time, the companies and investors are taking risks in their foreign investments. But they depend on the information of the foreign place from their networks and local governments. In such cases, the multi-criteria group decision making (MCGDM) methods helps the companies and investors in minimizing their risk. The selection process for the preferable sustainable suppliers for an organization is a critical decision making process. Among the many MCDM methods proposed by the researchers for the selection of the best suppliers, IVIFSs based approaches are considered to be preferable as IVIFSs are capable to deal with the uncertainties. This study has solved the supplier problem using P-IVIHFS and extended TOPSIS method.

Table 3
Description of supplier selection criteria

Category Criterion Description

quality risk of the product c₁ acceptation rate of the product

c₂ on-time delivery rate

c₃ product qualification ratio

c₄ the remedy for quality problems

service risk c5 response to changes c₅ response to changes

c₆ technological and R&D support

c₇ ease of communication

supplier’s profile risk c₈ financial status

c₉ customer base

c₁₀ performance history

c₁₁ production facility and capacity

long-term cooperation risk c₁₂supplier’s delivery ratio

c₁₃ management level

c₁₄ technological capability

Category	Criterion Description
quality risk of the product	c₁ acceptation rate of the product
	c₂ on-time delivery rate
	c₃ product qualification ratio
	c₄ the remedy for quality problems
service risk c5 response to changes	c₅ response to changes
	c₆ technological and R&D support
	c₇ ease of communication
supplier’s profile risk	c₈ financial status
	c₉ customer base
	c₁₀ performance history
	c₁₁ production facility and capacity
long-term cooperation risk	c₁₂supplier’s delivery ratio
	c₁₃ management level
	c₁₄ technological capability

In this case study, a manufacturing company wants to select a suitable material supplier for the key components of new products. For that purpose, a decision-making group was formed including experts from each strategic decision area. Three alternatives/suppliers A₁, A₂ and A₃remained in competition after preliminary screening. The committee of experts identified 14 evaluative criteria for supplier selection which is presented in [47] Table 3. Initially decision makers’ provide their opinions using linguistic terms which are then mapped into interval-valued intuitionistic hesitant fuzzy numbers (IVIHFN) (Table 4). Opinions of decision makers (D₁, D₂, D₃) for the suppliers corresponding to each criteria are given in Table 5, where the linguistic terms, i.e., the IVIHFNs are characterized by probabilities which are known as probabilistic interval-valued intuitionistic hesitant fuzzy elements (P-IVIHFEs).

Table 4

Linguistic terms and IVIHFNs

Linguistic term	IVIHFS
very strong (VS)	[([0.73,0.83], [0.00,0.13]), ([0.82,0.92], [0.02,0.08])]
strong (S)	[([0.63,0.73], [0.10,0.23]), ([0.72,0.82], [0.04,0.12])]
medium (M)	[([0.43,0.53], [0.30,0.43]), ([0.52,0.62], [0.24,0.34])]
weak (W)	[([0.15,0.29], [0.45,0.64]), ([0.25,0.35], [0.40,0.52])]
very weak (VW)	[([0.00,0.14], [0.60,0.79]), ([0.12,0.25], [0.60,0.70])]

Table 5

Decision maker’s opinions using linguistic P-IVIHFEs

Criteta Suppl.		D ₁	D ₂	D ₃
C ₁	A ₁	(VW,0.6,0.5)	(VW,0.6,0.6)	(VW,0.7,0.6)
	A ₂	(VS,0.6,0.7)	(VS,0.7,0.6)	(VS,0.7,0.6)
	A ₃	(W,0.5,0.6)	(M,0.6,0.7)	(S,0.7,0.7)
C ₂	A ₁	(S,0.7,0.7)	(S,0.6,0.7)	(M,0.5,0.6)
	A ₂	(W,0.8,0.7)	(VW,0.7,0.6)	(W,0.6,0.7)
	A ₃	(VS,0.7,0.6)	(S,0.7,0.6)	(S,0.6,0.6)
C ₃	A ₁	(M,0.5,0.6)	(M,0.6,0.7)	(W,0.6,0.7)
	A ₂	(S,0.6,0.7)	(S,0.7,0.6)	(VS,0.6,0.7)
	A ₃	(W,0.7,0.6)	(VW,0.8,0.7)	(VW,0.8,0.7)
C ₄	A ₁	(S,0.8,0.7)	(S,0.7,0.6)	(S,0.8,0.7)
	A ₂	(M,0.6,0.7)	(M,0.6,0.7)	(M,0.6,0.7)
	A ₃	(VW,0.7,0.6)	(VW,0.6,0.7)	(VW,0.6,0.7)
C ₅	A ₁	(W,0.8,0.6)	(VW,0.7,0.6)	(W,0.7,0.6)
	A ₂	(M,0.5,0.6)	(W,0.6,0.7)	(S,0.8,0.6)
	A ₃	(W,0.6,0.7)	(W,0.8,0.7)	(W,0.7,0.6)
C ₆	A ₁	(VS,0.6,0.7)	(VS,0.6,0.6)	(VS,0.7,0.6)
	A ₂	(S,0.8,0.7)	(S,0.7,0.6)	(M,0.8,0.6)
	A ₃	(VS,0.8,0.7)	(S,0.8,0.7)	(VS,0.8,0.7)
C ₇	A ₁	(M,0.7,0.6)	(M,0.7,0.6)	(W,0.8,0.7)
	A ₂	(S,0.8,0.7)	(S,0.7,0.6)	(VS,0.8,0.7)
	A ₃	(W,0.8,0.7)	(W,0.8,0.7)	(W,0.7,0.6)
C ₈	A ₁	(M,0.7,0.6)	(M,0.7,0.6)	(W,0.6,0.7)
	A ₂	(S,0.8,0.7)	(S,0.7,0.7)	(M,0.7,0.6)
	A ₃	(VW,0.7,0.6)	(W,0.7,0.6)	(VW0.7,0.7)
C ₉	A ₁	(VS,0.8,0.7)	(VS,0.8,0.7)	(VS,0.7,0.6)
	A ₂	(W,0.7,0.6)	(M,0.6,0.7)	(S,0.8,0.7)
	A ₃	(VS,0.7,0.6)	(VS,0.8,0.7)	(VS,0.7,0.7)
C ₁₀	A ₁	(S,0.7,0.6)	(S,0.8,07)	(M,0.6,0.7)
	A ₂	(S,0.7,0.6)	(S,0.7,0.8)	(VS,0.7,0.6)
	A ₃	(S,0.6,0.7)	(S,0.6,0.7)	(VS,0.7,0.6)
C ₁₁	A ₁	(M,0.8,0.7)	(M,0.6,0.7)	(S,0.7,0.6)
	A ₂	(VS,0.8,0.7)	(S,0.6,0.7)	(S,0.8,0.7)
	A ₃	(VS,0.7,0.6)	(VS,0.6,0.7)	(VS,0.7,0.6)
C ₁₂	A ₁	(S,0.8,0.7)	(VS,0.6,0.7)	(VS,0.7,0.6)
	A ₂	(M,0.7,0.6)	(M,0.5,0.6)	(VW,0.6,0.7)
	A ₃	(VS,0.7,0.6)	(VS,0.6,0.7)	(VS,0.6,0.7)
C ₁₃	A ₁	(S,0.7,0.6)	(VS,0.7,0.6)	(S,0.8,0.7)
	A ₂	(W,0.8,0.7)	(W,0.7,0.6)	(W,0.7,0.6)
	A ₃	(VS,0.6,0.7)	(S,0.8,0.7)	(S,0.7,0.6)
C ₁₄	A ₁	(M,0.6,0.7)	(W,0.7,0.6)	(VW,0.8,0.7)
	A ₂	(S,0.6,0.7)	(M,0.5,0.7)	(VW,0.8,0.6)
	A ₃	(S,0.7,0.6)	(VS,0.8,0.7)	(VS,0.7,0.6)

The global weight of each criterion is obtained from the paper [47] from where the problem has been considered. Next we find the normalized global weights as if w_j = ([q_j, y_j] , [z_j, g_j]). Then the normalized global weights are given by ${\bar{w}}_{j} = ([q_{j}, y_{j}] / max [\sum_{j = 1}^{14} q_{j}, \sum_{j = 1}^{14} y_{j}], [z_{j}, g_{j}] /$ $max [\sum_{j = 1}^{14} z_{j}, \sum_{j = 1}^{14} g_{j}])$ where

w₁ = ([0 . 004, 0 . 035] , [0 . 089, 0 . 127]) ,

w₂ = ([0 . 025, 0 . 066] , [0 . 069, 0 . 01]) ;

w₃ = ([0 . 106, 0 . 11] , [0, 0 . 01]) ,

w₄ = ([0 . 008, 0 . 041] , [0 . 08, 0 . 119]) ,

w₅ = ([0 . 038, 0 . 084] , [0, 0 . 046]) ,

w₆ = ([0 . 066, 0 . 069] , [0 . 057, 0 . 06]) ,

w₇ = ([0 . 08, 0 . 082] , [0 . 044, 0 . 046]) ,

w₈ = ([0 . 112, 0 . 112] , [0 . 00, 0 . 002]) ,

w₉ = ([0 . 025, 0 . 025] , [0 . 141, 0 . 141]) ,

w₁₀ = ([0 . 073, 0 . 073] , [0 . 052, 0 . 054]) ,

w₁₁ = ([0 . 091, 0 . 091] , [0 . 022, 0 . 022]) ,

w₁₂ = ([0 . 015, 0 . 016] , [0 . 135, 0 . 135]) ,

w₁₃ = [([0.08, 0.085] , [0.016, 0.161])] and

w₁₄ = ([0 . 106, 0 . 107] , [0 . 004, 0 . 004]) ,

After performing the Hadamard product, the opinions of decision makers are represented by P-IVIHFMs as given below in Tabular form (Table 6), which is then formed as average decision matrix as shown in Table 7.

Table 6

Opinion of decision makers as P-IVIHFMs

In the next steps, we find the P-IVIHFPIS and P-IVIHFNIS as $\begin{matrix} T^{+} = { & (max_{i} {{\tilde{h}}_{ij}} : S_{j} \in J_{1}), \\ (min_{i} {{\tilde{h}}_{ij}} : S_{j} \in J_{2}) | 1 \leq i \leq 3 \\ and 1 \leq j \leq 14} \end{matrix}$ $\begin{matrix} t_{1}^{+} = {\begin{matrix} [\begin{matrix} [max (0, 0.49, 0.27), max (0.09, 0.55, 0.37)], \\ [min (0.38, 0, 0.15), min (0.49, 0.09, 0.23)] \end{matrix}], \\ [\begin{matrix} [max (0.06, 0.52, 0.34), max (0.14, 0.58, 0.40)], \\ [min (0.34, 0.01, 0.14), min (0.40, 0.05, 0.21)] \end{matrix}] \end{matrix}} \\ = {([0.49, 0.55], [0, 0.09]), ([0.52, 0.58], [0.01, 0.05])} \end{matrix}$

$\begin{matrix} t_{2}^{+} & = {([0.44, 0.51], [0.04, 0.13]), ([0.45, 0.51], [0.02, 0.06])}, \\ t_{3}^{+} & = {([0.42, 0.48], [0.04, 0.13]), ([0.5, 0.57], [0.02, 0.07])}, \\ t_{4}^{+} & = {([0.48, 0.56], [0.08, 0.17]), ([0.48, 0.54], [0.02, 0.08])}, \\ t_{5}^{+} & = {([0.27, 0.34], [0.17, 0.26]), ([0.3, 0.36], [0.13, 0.19])}, \\ t_{6}^{+} & = {([0.55, 0.63], [0, 0.1]), ([0.55, 0.62], [0.01, 0.05])}, \\ t_{7}^{+} & = {([0.51, 0.58], [0.05, 0.15]), ([0.5, 0.57], [0.02, 0.07])}, \\ t_{8}^{+} & = {([0.4, 0.5], [0.12, 0.21]), ([0.44, 0.50], [0.06, 0.12])}, \\ t_{9}^{+} & = {([0.56, 0.63], [0, 0.10]), ([0.54, 0.61], [0.01, 0.06])}, \\ t_{10}^{+} & = {([0.46, 0.53], [0.05, 0.14]), ([0.5, 0.56], [0.02, 0.07])}, \\ t_{11}^{+} & = {([0.49, 0.55], [0, 0.09]), ([0.52, 0.58], [0.01, 0.05])} \\ t_{12}^{+} & = {([0.48, 0.55], [0.03, 0.12]), ([0.52, 0.59], [0.02, 0.06])}, \\ t_{13}^{+} & = {([0.51, 0.53], [0.05, 0.14]), ([0.5, 0.57], [0.02, 0.07])}, \\ t_{14}^{+} & = {([0.51, 0.58], [0.02, 0.12]), ([0.52, 0.56], [0.01, 0.06])} \end{matrix}$ $\begin{matrix} T^{-} = { & (min_{i} {h_{ij}} : S_{j} \in J_{1}), \\ (max_{i} {h_{ij}} : S_{j} \in J_{2}) | 1 \leq i \leq 4 \\ and 1 \leq j \leq 14} \end{matrix}$ $\begin{matrix} \begin{matrix} t_{1}^{-} = {\begin{matrix} [\begin{matrix} [min (0, 0.49, 0.27), min (0.09, 0.55, 0.37)], \\ [max (0.38, 0, 0.15), max (0.49, 0.09, 0.23)] \end{matrix}], \\ [\begin{matrix} [min (0.06, 0.52, 0.34), min (0.14, 0.58, 0.40)], \\ [max (0.34, 0.01, 0.14), max (0.40, 0.05, 0.21)] \end{matrix}] \end{matrix}} \\ = {([0, 0.09], [0.38, 0.49]), ([0.06, 0.14], [0.34, 0.40])} \end{matrix} \end{matrix}$ $\begin{matrix} t_{2}^{-} & = {([0.11, 0.20], [0.31, 0.43]), ([0.20, 0.25], [0.26, 0.34])}, \\ t_{3}^{-} & = {([0.03, 0.17], [0.43, 0.57]), ([0.10, 0.19], [0.36, 0.43])}, \\ t_{4}^{-} & = {([0, 0.09], [0.34, 0.52]), ([0.08, 0.18], [0.4, 0.47])}, \\ t_{5}^{-} & = {([0.07, 0.20], [0.37, 0.50]), ([0.12, 0.19], [0.28, 0.35])}, \\ t_{6}^{-} & = {([0.42, 0.50], [0.13, 0.23]), ([0.41, 0.48], [0.07, 0.13])}, \\ t_{7}^{-} & = {([0.11, 0.25], [0.35, 0.49]), ([0.16, 0.22], [0.25, 0.33])}, \\ t_{8}^{-} & = {([0.03, 0.16], [0.39, 0.52]), ([0.10, 0.18], [0.34, 0.41])}, \\ t_{9}^{-} & = {([0.29, 0.40], [0.18, 0.28]), ([0.34, 0.40], [0.14, 0.21])}, \\ t_{10}^{-} & = {([0.40, 0.47], [0.16, 0.25]), ([0.43, 0.5], [0.07, 0.13])}, \\ t_{11}^{-} & = {([0.35, 0.42], [0.16, 0.25]), ([0.38, 0.45], [0.12, 0.18])}, \\ t_{12}^{-} & = {([0.17, 0.24], [0.24, 0.33]), ([0.23, 0.31], [0.12, 0.3])}, \\ t_{13}^{-} & = {([0.11, 0.27], [0.33, 0.47]), ([0.16, 0.22], [0.25, 0.33])}, \\ t_{14}^{-} & = {([0.12, 0.24], [0.33, 0.45]), ([0.2, 0.27], [0.28, 0.35])} \end{matrix}$

Table 7

Average decision matrix

Crit. Suppl.	A ₁	A ₂	A ₃
C ₁	$[\begin{matrix} ([0, 0.09], [0.38, 0.49]), \\ ([0.06, 0.14], [0.34, 0.40]) \end{matrix}]$	$[\begin{matrix} ([0.49, 0.55], [0, 0.09]), \\ ([0.52, 0.58], [0.01, 0.05]) \end{matrix}]$	$[\begin{matrix} ([0.27, 0.37], [0.15, 0.23]), \\ ([0.34, 0.40], [0.14, 0.21]) \end{matrix}]$
C ₂	$[\begin{matrix} ([0.35, 0.40], [0.09, 0.15]), \\ ([0.44, 0.50], [0.07, 0.12]) \end{matrix}]$	$[\begin{matrix} ([0.11, 0.20], [0.31, 0.43]), \\ ([0.20, 0.25], [0.26, 0.34]) \end{matrix}]$	$[\begin{matrix} ([0.44, 0.51], [0.04, 0.13]), \\ ([0.45, 0.51], [0.02, 0.06]) \end{matrix}]$
C ₃	$[\begin{matrix} ([0.19, 0.25], [0.2, 0.29]), \\ ([0.28, 0.35], [0.16, 0.24]) \end{matrix}]$	$[\begin{matrix} ([0.42, 0.48], [0.04, 0.13]), \\ ([0.5, 0.57], [0.02, 0.07]) \end{matrix}]$	$[\begin{matrix} ([0.03, 0.17], [0.43, 0.57]), \\ ([0.10, 0.19], [0.36, 0.43]) \end{matrix}]$
C ₄	$[\begin{matrix} ([0.48, 0.56], [0.08, 0.17]), \\ ([0.48, 0.54], [0.02, 0.08]) \end{matrix}]$	$[\begin{matrix} ([0.25, 0.30], [0.17, 0.25]), \\ ([0.34, 0.41], [0.16, 0.23]) \end{matrix}]$	$[\begin{matrix} ([0, 0.09], [0.34, 0.52]), \\ ([0.08, 0.18], [0.4, 0.47]) \end{matrix}]$
C ₅	$[\begin{matrix} ([0.07, 0.20], [0.37, 0.50]), \\ ([0.12, 0.19], [0.28, 0.35]) \end{matrix}]$	$[\begin{matrix} ([0.27, 0.34], [0.17, 0.26]), \\ ([0.3, 0.36], [0.13, 0.19]) \end{matrix}]$	$[\begin{matrix} ([0.1, 0.23], [0.32, 0.45]), \\ ([0.16, 0.22], [0.25, 0.33]) \end{matrix}]$
C ₆	$[\begin{matrix} ([0.46, 0.53], [0, 0.1]), \\ ([0.52, 0.58], [0.01, 0.05]) \end{matrix}]$	$[\begin{matrix} ([0.42, 0.50], [0.13, 0.23]), \\ ([0.41, 0.48], [0.07, 0.13]) \end{matrix}]$	$[\begin{matrix} ([0.55, 0.63], [0.03, 0.13]), \\ ([0.55, 0.62], [0.02, 0.07]) \end{matrix}]$
C ₇	$[\begin{matrix} ([0.24, 0.32], [0.26, 0.37]), \\ ([0.26, 0.33], [0.19, 0.25]) \end{matrix}]$	$[\begin{matrix} ([0.51, 0.58], [0.05, 0.15]), \\ ([0.5, 0.57], [0.02, 0.07]) \end{matrix}]$	$[\begin{matrix} ([0.11, 0.25], [0.35, 0.49]), \\ ([0.16, 0.22], [0.25, 0.33]) \end{matrix}]$
C ₈	$[\begin{matrix} ([0.23, 0.30], [0.23, 0.33]), \\ ([0.26, 0.33], [0.19, 0.25]) \end{matrix}]$	$[\begin{matrix} ([0.40, 0.50], [0.12, 0.21]), \\ ([0.44, 0.50], [0.06, 0.12]) \end{matrix}]$	$[\begin{matrix} ([0.03, 0.16], [0.39, 0.52]), \\ ([0.10, 0.18], [0.34, 0.41]) \end{matrix}]$
C ₉	$[\begin{matrix} ([0.56, 0.63], [0, 0.10]), \\ ([0.54, 0.61], [0.01, 0.06]) \end{matrix}]$	$[\begin{matrix} ([0.29, 0.40], [0.18, 0.28]), \\ ([0.34, 0.40], [0.14, 0.21]) \end{matrix}]$	$[\begin{matrix} ([0.53, 0.61], [0, 0.09]), \\ ([0.54, 0.61], [0.01, 0.06]) \end{matrix}]$
C₁0	$[\begin{matrix} ([0.40, 0.47], [0.11, 0.20]), \\ ([0.43, 0.5], [0.07, 0.13]) \end{matrix}]$	$[\begin{matrix} ([0.46, 0.53], [0.05, 0.14]), \\ ([0.5, 0.56], [0.02, 0.07]) \end{matrix}]$	$[\begin{matrix} ([0.42, 0.49], [0.16, 0.25]), \\ ([0.5, 0.56], [0.02, 0.07]) \end{matrix}]$
C₁1	$[\begin{matrix} ([0.35, 0.42], [0.16, 0.25]), \\ ([0.38, 0.45], [0.12, 0.18]) \end{matrix}]$	$[\begin{matrix} ([0.48, 0.55], [0.05, 0.14]), \\ ([0.52, 0.59], [0.02, 0.07]) \end{matrix}]$	$[\begin{matrix} ([0.49, 0.55], [0, 0.09]), \\ ([0.52, 0.58], [0.01, 0.05]) \end{matrix}]$
C₁2	$[\begin{matrix} ([0.48, 0.55], [0.03, 0.12]), \\ ([0.52, 0.59], [0.02, 0.06]) \end{matrix}]$	$[\begin{matrix} ([0.17, 0.24], [0.24, 0.33]), \\ ([0.23, 0.31], [0.12, 0.3]) \end{matrix}]$	$[\begin{matrix} ([0.46, 0.53], [0, 0.1]), \\ ([0.54, 0.61], [0.01, 0.06]) \end{matrix}]$
C₁3	$[\begin{matrix} ([0.48, 0.56], [0.05, 0.14]), \\ ([0.47, 0.54], [0.02, 0.07]) \end{matrix}]$	$[\begin{matrix} ([0.11, 0.27], [0.33, 0.47]), \\ ([0.16, 0.22], [0.25, 0.33]) \end{matrix}]$	$[\begin{matrix} ([0.46, 0.53], [0.05, 0.15]), \\ ([0.5, 0.57], [0.02, 0.07]) \end{matrix}]$
C₁4	$[\begin{matrix} ([0.12, 0.24], [0.33, 0.45]), \\ ([0.2, 0.27], [0.28, 0.35]) \end{matrix}]$	$[\begin{matrix} ([0.2, 0.27], [0.23, 0.33]), \\ ([0.31, 0.38], [0.19, 0.25]) \end{matrix}]$	$[\begin{matrix} ([0.51, 0.58], [0.02, 0.12]), \\ ([0.5, 0.56], [0.01, 0.06]) \end{matrix}]$

The Degree of similarities

N_{i}^{+}

and

N_{i}^{-}

are calculated using above formula as follows:

\begin{matrix} \begin{matrix} N_{1}^{+} = 1 - 0.66 w_{1} - 0.1 w_{2} - 0.31 w_{3} - 0.3 w_{5} \\ - 0.08 w_{6} - 0.36 w_{7} - 0.25 w_{8} - 0.1 w_{10} \\ - 0.22 w_{11} - 0.02 w_{12} - 0.03 w_{13} - 0.51 w_{14} \end{matrix} \end{matrix}

\begin{matrix} \begin{matrix} N_{2}^{+} = 1 - 0.31 w_{1} - 0.44 w_{2} - 0.28 w_{4} - 0.17 w_{6} \\ - 0.25 w_{8} - 0.33 w_{9} - 0.04 w_{11} - 0.4 w_{12} \\ - 0.51 w_{13} - 0.38 w_{14} \end{matrix} \end{matrix}

\begin{matrix} \begin{matrix} N_{3}^{+} = 1 - 0.28 w_{1} - 0.6 w_{3} - 0.63 w_{4} - 0.24 w_{5} \\ - 0.04 w_{6} - 0.52 w_{7} - 0.5 w_{8} - 0.02 w_{9} - 0.09 w_{10} \\ - 0.006 w_{11} - 0.02 w_{12} - 0.04 w_{13} - 0.12 w_{14} \end{matrix} \end{matrix}

\begin{matrix} \begin{matrix} N_{1}^{-} = 1 - 0.37 w_{2} - 0.32 w_{3} - 0.15 w_{6} - 0.16 w_{7} \\ - 0.26 w_{8} - 0.33 w_{9} - 0.04 w_{10} - 0.37 w_{12} - 0.48 w_{13} \end{matrix} \end{matrix}

\begin{matrix} \begin{matrix} N_{2}^{-} = 1 - 0.58 w_{1} - 0.6 w_{3} - 0.37 w_{4} - 0.3 w_{5} \\ - 0.52 w_{7} - 0.26 w_{8} - 0.13 w_{10} - 0.19 w_{11} \\ - 0.15 w_{14} \end{matrix} \end{matrix}

\begin{matrix} \begin{matrix} N_{3}^{-} = 1 - 0.39 w_{1} - 0.44 w_{2} - 0.06 w_{5} - 0.13 w_{6} \\ - 0.31 w_{9} - 0.07 w_{10} - 0.22 w_{11} - 0.41 w_{12} \\ - 0.47 w_{13} \end{matrix} \end{matrix}

Solving the LP model, the optimal weights are obtained as follows:

Table 8

Comparison with the existing methods

Method	Decision matrix	Criteria weights	Aggregation	Selection	Ranking
[48]	Provided by DMs	Unknown, calculated using linear programming	IVIFWA operator	Score function	A₃ > A₁ > A₂
[49]	Normalized satisfactory and dissatisfactory intervals	Provided by DMs	Induced IVIFWA operator	IVIFWA operator as score function	A₃ > A₂ > A₁
[50]	Provided by DMs	Subjective and entropy weights	Outranking matrix	Global superiority index	A₂ > (A₃ = A₁)
[51]	Provided by DMs	Maximizing consensus model	IVIFWA operator	TOPSIS	A₂ > A₁ > A₃
[52]	Provided by DMs	Provided by DMs	IVIFWA operator	TOPSIS with a geometric method	A₃ > A₂ > A₁
[53]	Pairwise comparison provided by DMs	IVIF entropy	IVIFWA operator	Relative weight	A₂ > A₁ > A₃
[54]	Pairwise Comparison provided by DMs	IVIF-ANP	IVIFOWA operator	Relative significance	A₂ > (A₃ = A₁)
[55]	Steady state of IVIFCM	Provided by DMs	No aggregation	Euclidean distance from target values	A₂ > A₁ > A₃
[47]	Provided by DMs	Local weights and weights from IVIFCM	IVIFWA operator	TOPSIS	A₂ > A₃ > A₁
Proposed method based on P-IVIHFS and TOPSIS	Provided by DMs	Global weights and weights calculated using LP	-	TOPSIS	A₁ > A₃ > A₂
Proposed method based on P-IVIHFS and VIKOR	Provided by DMs	Weights are similar as TOPSIS method	-	VIKOR	A₃ > A₁ > A₂

\begin{matrix} w_{1}^{*} & = 0.04, w_{2}^{*} = 0.025, w_{3}^{*} = 0.106, \\ w_{4}^{*} & = 0.008, w_{5}^{*} = 0.038, w_{6}^{*} = 0.066, \\ w_{7}^{*} & = 0.08, w_{8}^{*} = 0.112, w_{9}^{*} = 0.025, \\ w_{10}^{*} & = 0.073, w_{11}^{*} = 0.091, w_{12}^{*} = 0.015, \\ w_{13}^{*} & = 0.215 and w_{14}^{*} = 0.106 \end{matrix}

Using the above optimal weights, we find the relative closeness coefficient of each of the alternatives are as follows. $\begin{matrix} δ_{1}^{*} & = \frac{N_{1}^{+}}{N_{1}^{+} + N_{1}^{-}} = 0.500274, δ_{2}^{*} = 0.494165, \\ δ_{3}^{*} & = 0.487252 . \end{matrix}$

Since $δ_{1}^{*} > δ_{2}^{*} > δ_{3}^{*}$ so the preference order of the alternative is A₁ > A₂ > A₃.

Next, we perform the VIKOR method using the same example explained above.

[Step 1] We find the P-IVIHFPISs for the attributes are defined above. Similarly, the P-IVIHFNISs are computed as above.

[Step 2] Next the π_i and σ_i values for i = 1,…,m are computed as given below. $\begin{matrix} π_{1} & = 0 . 5398, σ_{1} = 0 . 106, π_{2} = 0 . 495574, \\ σ_{2} & = 0 . 215 and π_{3} = 0 . 4234, σ_{3} = 0 . 112 \end{matrix}$

[Step 3] Now we compute the values of Q_i for i = 1,2,3 and considering v = 0.5. $Q_{1} = 0.643776, Q_{2} = 0.936045 and Q_{3} = 0.124564 .$

From the above result, we conclude that A₃ > A₁ > A₂.

7 Comparative analysis

This section shows a comparative analysis which is given below in tabular form (Table 8).

The decision matrix used in our study is an extended version of the same given in [47]. In [48], the authors used interval-valued intuitionistic fuzzy numbers (IVIFNs) and applied fuzzy union and fuzzy integration based interval-valued intuitionistic fuzzy weighted average (IVIFWA) operator to generate the fragmental performance of each alternative and then linear programming model was used to calculate the exact criteria weights. Yue and Jia [49] calculated our normalized satisfactory and unsatisfactory intervals and aggregated them into an induced IVIFN. Xu and Shen [50] computed the entropy weights from the IVIFS decision matrix. TOPSIS method used by Zhang and Xu [51], where weights are completely known. TOPSIS method was used to rank IVIFNS in [52]. An IVIF-AHP based method is presented in [53], where the pairwise comparison matrices were obtained from the DMs so that these were consistent with the local weights of the criteria. In addition, the consistency ratio of the aggregated IVIFS decision matrix was below 0.1. In the network structure for the IVIF-ANP method [54], the pairwise comparison matrices were constructed with respect to each criterion. The linguistic terms were used for the pairwise comparison. Compared to the above-mentioned approaches, the proposed approach has used P-IVIHFS for prescribing the opinions of the decision makers and LP to optimize the attribute/criteria weights which are given as IVIFNs. Due to including the probabilistic characteristics, P-IVIHFS not only represent all possible decision information but also reflect the importance of each decision information.

8 Conclusions

In this paper, we have proposed a decision-making approach based on P-IVIHFSs, LP methodology and the extended TOPSIS method which can address the complex situations to deal with both subjective and objective nature of uncertainties, where the attributes’ weights are incompletely known. P-IVIHFS is used to represent the evaluating values of the alternatives corresponding to the attributes/criteria, where the concept of probability is embedded to provide the corresponding degree of possibility for the decision making information for the purpose of enhancement of reliability and effectiveness of the decision results. Instead of considering the exact weights of the attributes, this paper considers partial weight information using interval-valued intuitionistic fuzzy numbers which are then used to determine the optimal weights of the attributes based on LP method. The extension of TOPSIS and VIKOR methods have been implemented by encompassing the probability of the attributes with its interval-valued intuitionistic hesitant fuzzy values. This study has also performed a sensitivity analysis which shows the variation of alternatives’ ranking due to small changes in the attributes’ weights. To investigate the validity of the proposed method of real-life problems, we have applied it in supplier selection problem, where decision makers prescribe their opinions using linguistic P-IVIHFEs which are then converted into numerical values. The comparison of results with the relevant existing approaches shows the difference in ranking due to the addition of probability in the proposed approach. As a part of our future research work, we will consider P-IVIHFSs in real life decision making paradigm like finding temporary relief shelter locations in disasters. Researchers might also include the probability notation for various extension of fuzzy sets to solve the complex real-life problems.

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Criteria	Suppl.	D1	D2	D3
C ₁	A ₁	$[\begin{matrix} ([0, 0.08], [0.36, 0.47]), \\ ([0.06, 0.12], [0.3, 0.35]) \end{matrix}]$	$[\begin{matrix} ([0, 0.08], [0.36, 0.47]), \\ ([0.07, 0.15], [0.36, 0.42]) \end{matrix}]$	$[\begin{matrix} ([0, 0.10], [0.42, 0.55]), \\ ([0.07, 0.15], [0.36, 0.42]) \end{matrix}]$
	A ₂	$[\begin{matrix} ([0.44, 0.50], [0, 0.10]), \\ ([0.57, 0.64], [0.01, 0.06]) \end{matrix}]$	$[\begin{matrix} ([0.51, 0.58], [0, 0.09]), \\ ([0.49, 0.55], [0.01, 0.05]) \end{matrix}]$	$[\begin{matrix} ([0.51, 0.58], [0, 0.09]), \\ ([0.50, 0.55], [0.01, 0.05]) \end{matrix}]$
	A ₃	$[\begin{matrix} ([0.12, 0.18], [0.2, 0.26]), \\ ([0.15, 0.21], [0.24, 0.31]) \end{matrix}]$	$[\begin{matrix} ([0.26, 0.32], [0.18, 0.26]), \\ ([0.36, 0.43], [0.17, 0.24]) \end{matrix}]$	$[\begin{matrix} ([0.44, 0.51], [0.07, 0.16]), \\ ([0.50, 0.57], [0.03, 0.08]) \end{matrix}]$
C ₂	A ₁	$[\begin{matrix} ([0.44, 0.51], [0.07, 0.16]), \\ ([0.50, 0.57], [0.03, 0.08]) \end{matrix}]$	$[\begin{matrix} ([0.38, 0.44], [0.06, 0.14]), \\ ([0.50, 0.57], [0.03, 0.08]) \end{matrix}]$	$[\begin{matrix} ([0.22, 0.26], [0.15, 0.22]), \\ ([0.31, 0.37], [0.14, 0.20]) \end{matrix}]$
	A ₂	$[\begin{matrix} ([0.12, 0.23], [0.36, 0.51]), \\ ([0.36, 0.43], [0.17, 0.24]) \end{matrix}]$	$[\begin{matrix} ([0, 0.10], [0.42, 0.55]), \\ ([0.07, 0.15], [0.36, 0.42]) \end{matrix}]$	$[\begin{matrix} ([0.22, 0.26], [0.15, 0.22]), \\ ([0.18, 0.24], [0.28, 0.36]) \end{matrix}]$
	A ₃	$[\begin{matrix} ([0.51, 0.58], [0, 0.09]), \\ ([0.49, 0.55], [0.01, 0.05]) \end{matrix}]$	$[\begin{matrix} ([0.44, 0.51], [0.07, 0.16]), \\ ([0.43, 0.49], [0.02, 0.07]) \end{matrix}]$	$[\begin{matrix} ([0.38, 0.44], [0.06, 0.14]), \\ ([0.43, 0.49], [0.02, 0.07]) \end{matrix}]$
C ₃	A ₁	$[\begin{matrix} ([0.22, 0.26], [0.15, 0.22]), \\ ([0.31, 0.37], [0.14, 0.20]) \end{matrix}]$	$[\begin{matrix} ([0.26, 0.32], [0.18, 0.26]), \\ ([0.36, 0.43], [0.17, 0.24]) \end{matrix}]$	$[\begin{matrix} ([0.09, 0.17], [0.27, 0.38]), \\ ([0.18, 0.24], [0.28, 0.36]) \end{matrix}]$
	A ₂	$[\begin{matrix} ([0.38, 0.44], [0.06, 0.14]), \\ ([0.50, 0.57], [0.03, 0.08]) \end{matrix}]$	$[\begin{matrix} ([0.44, 0.51], [0.07, 0.16]), \\ ([0.43, 0.49], [0.02, 0.07]) \end{matrix}]$	$[\begin{matrix} ([0.44, 0.50], [0, 0.10]), \\ ([0.57, 0.64], [0.01, 0.06]) \end{matrix}]$
	A ₃	$[\begin{matrix} ([0.10, 0.29], [0.32, 0.45]), \\ ([0.15, 0.21], [0.24, 0.31]) \end{matrix}]$	$[\begin{matrix} ([0, 0.11], [0.48, 0.63]), \\ ([0.08, 0.18], [0.42, 0.49]) \end{matrix}]$	$[\begin{matrix} ([0, 0.11], [0.48, 0.63]), \\ ([0.08, 0.18], [0.42, 0.49]) \end{matrix}]$
C ₄	A ₁	$[\begin{matrix} ([0.50,, 0.58], [0.08, 0.18]), \\ ([0.50, 0.57], [0.03, 0.08]) \end{matrix}]$	$[\begin{matrix} ([0.44, 0.51], [0.07, 0.16]), \\ ([0.43, 0.49], [0.02, 0.07]) \end{matrix}]$	$[\begin{matrix} ([0.50, 0.58], [0.08, 0.18]), \\ ([0.50, 0.57], [0.02, 0.08]) \end{matrix}]$
	A ₂	$[\begin{matrix} ([0.26, 0.32], [0.18, 0.26]), \\ ([0.36, 0.43], [0.17, 0.24]) \end{matrix}]$	$[\begin{matrix} ([0.22, 0.26], [0.15, 0.22]), \\ ([0.31, 0.37], [0.14, 0.20]) \end{matrix}]$	$[\begin{matrix} ([0.26, 0.32], [0.18, 0.26]), \\ ([0.36, 0.43], [0.17, 0.24]) \end{matrix}]$
	A ₃	$[\begin{matrix} ([0, 0.10], [0.42, 0.55]), \\ ([0.08, 0.18], [0.36, 0.42]) \end{matrix}]$	$[\begin{matrix} ([0, 0.10], [0.42, 0.55]), \\ ([0.08, 0.18], [0.42, 0.49]) \end{matrix}]$	$[\begin{matrix} ([0, 0.08], [0.36, 0.47]), \\ ([0.08, 0.18], [0.42, 0.49]) \end{matrix}]$
C ₅	A ₁	$[\begin{matrix} ([0.12, 0.23], [0.36, 0.51]), \\ ([0.15, 0.21], [0.24, 0.31]) \end{matrix}]$	$[\begin{matrix} ([0, 0.10], [0.42, 0.55]), \\ ([0.07, 0.15], [0.36, 0.42]) \end{matrix}]$	$[\begin{matrix} ([0.10, 0.29], [0.32, 0.45]), \\ ([0.15, 0.21], [0.24, 0.31]) \end{matrix}]$
	A ₂	$[\begin{matrix} ([0.22, 0.26], [0.15, 0.22]), \\ ([0.32, 0.37], [0.14, 0.20]) \end{matrix}]$	$[\begin{matrix} ([0.09, 0.17], [0.27, 0.38]), \\ ([0.15, 0.21], [0.24, 0.31]) \end{matrix}]$	$[\begin{matrix} ([0.50, 0.58], [0.08, 0.18]), \\ ([0.43, 0.49], [0.02, 0.07]) \end{matrix}]$
	A ₃	$[\begin{matrix} ([0.09, 0.17], [0.27, 0.38]), \\ ([0.18, 0.24], [0.28, 0.36]) \end{matrix}]$	$[\begin{matrix} ([0.12, 0.23], [0.36, 0.51]), \\ ([0.15, 0.21], [0.24, 0.31]) \end{matrix}]$	$[\begin{matrix} ([0.10, 0.29], [0.32, 0.45]), \\ ([0.15, 0.21], [0.24, 0.31]) \end{matrix}]$
C ₆	A ₁	$[\begin{matrix} ([0.44, 0.50], [0, 0.10]), \\ ([0.57, 0.64], [0.01, 0.06]) \end{matrix}]$	$[\begin{matrix} ([0.44, 0.5], [0, 0.10]), \\ ([0.49, 0.55], [0.01, 0.05]) \end{matrix}]$	$[\begin{matrix} ([0.51, 0.58], [0, 0.09]), \\ ([0.49, 0.55], [0.01, 0.05]) \end{matrix}]$
	A ₂	$[\begin{matrix} ([0.50, 0.58], [0.08, 0.18]), \\ ([0.50, 0.57], [0.03, 0.08]) \end{matrix}]$	$[\begin{matrix} ([0.44, 0.51], [0.07, 0.16]), \\ ([0.43, 0.49], [0.02, 0.07]) \end{matrix}]$	$[\begin{matrix} ([0.34, 0.42], [0.24, 0.34]), \\ ([0.31, 0.37], [0.17, 0.24]) \end{matrix}]$
	A ₃	$[\begin{matrix} ([0.58, 0.66], [0, 0.10]), \\ ([0.57, 0.64], [0.01, 0.06]) \end{matrix}]$	$[\begin{matrix} ([0.50, 0.58], [0.08, 0.18]), \\ ([0.50, 0.57], [0.03, 0.08]) \end{matrix}]$	$[\begin{matrix} ([0.58, 0.66], [0, 0.10]), \\ ([0.57, 0.64], [0.01, 0.06]) \end{matrix}]$
C ₇	A ₁	$[\begin{matrix} ([0.30, 0.37], [0.21, 0.30]), \\ ([0.31, 0.37], [0.14, 0.20]) \end{matrix}]$	$[\begin{matrix} ([0.30, 0.37], [0.21, 0.30]), \\ ([0.31, 0.37], [0.14, 0.20]) \end{matrix}]$	$[\begin{matrix} ([0.12, 0.23], [0.36, 0.51]), \\ ([0.17, 0.24], [0.28, 0.36]) \end{matrix}]$
	A ₂	$[\begin{matrix} ([0.50, 0.58], [0.08, 0.18]), \\ ([0.50, 0.57], [0.03, 0.08]) \end{matrix}]$	$[\begin{matrix} ([0.44, 0.51], [0.07, 0.16]), \\ ([0.43, 0.49], [0.02, 0.07]) \end{matrix}]$	$[\begin{matrix} ([0.58, 0.64], [0, 0.10]), \\ ([0.57, 0.64], [0.01, 0.06]) \end{matrix}]$
	A ₃	$[\begin{matrix} ([0.12, 0.23], [0.36, 0.51]), \\ ([0.17, 0.24], [0.28, 0.36]) \end{matrix}]$	$[\begin{matrix} ([0.12, 0.23], [0.36, 0.51]), \\ ([0.15, 0.21], [0.24, 0.31]) \end{matrix}]$	$[\begin{matrix} ([0.10, 0.29], [0.32, 0.45]), \\ ([0.15, 0.21], [0.24, 0.31]) \end{matrix}]$
C ₈	A ₁	$[\begin{matrix} ([0.30, 0.37], [0.21, 0.30]), \\ ([0.31, 0.37], [0.14, 0.20]) \end{matrix}]$	$[\begin{matrix} ([0.30, 0.37], [0.21, 0.30]), \\ ([0.31, 0.37], [0.14, 0.20]) \end{matrix}]$	$[\begin{matrix} ([0.09, 0.17], [0.27, 0.38]), \\ ([0.17, 0.24], [0.28, 0.36]) \end{matrix}]$
	A ₂	$[\begin{matrix} ([0.50,, 0.58], [0.08, 0.18]), \\ ([0.50, 0.57], [0.02, 0.08]) \end{matrix}]$	$[\begin{matrix} ([0.44, 0.51], [0.07, 0.16]), \\ ([0.50, 0.57], [0.03, 0.08]) \end{matrix}]$	$[\begin{matrix} ([0.30, 0.37], [0.21, 0.30]), \\ ([0.31, 0.37], [0.14, 0.20]) \end{matrix}]$
	A ₃	$[\begin{matrix} ([0, 0.10], [0.42, 0.55]), \\ ([0.07, 0.15], [0.36, 0.42]) \end{matrix}]$	$[\begin{matrix} ([0.10, 0.29], [0.32, 0.45]), \\ ([0.15, 0.21], [0.24, 0.31]) \end{matrix}]$	$[\begin{matrix} ([0, 0.10], [0.42, 0.55]), \\ ([0.08, 0.18], [0.42, 0.49]) \end{matrix}]$
C ₉	A ₁	$[\begin{matrix} ([0.58, 0.66], [0, 0.10]), \\ ([0.57, 0.64], [0.01, 0.06]) \end{matrix}]$	$[\begin{matrix} ([0.58, 0.66], [0, 0.10]), \\ ([0.57, 0.64], [0.01, 0.06]) \end{matrix}]$	$[\begin{matrix} ([0.51, 0.58], [0, 0.09]), \\ ([0.49, 0.55], [0.01, 0.05]) \end{matrix}]$
	A ₂	$[\begin{matrix} ([0.10, 0.29], [0.32, 0.45]), \\ ([0.15, 0.21], [0.24, 0.31]) \end{matrix}]$	$[\begin{matrix} ([0.26, 0.32], [0.15, 0.22]), \\ ([0.36, 0.43], [0.16, 0.24]) \end{matrix}]$	$[\begin{matrix} ([0.50,, 0.58], [0.08, 0.18]), \\ ([0.50, 0.57], [0.02, 0.08]) \end{matrix}]$
	A ₃	$[\begin{matrix} ([0.51, 0.58], [0, 0.09]), \\ ([0.49, 0.55], [0.01, 0.05]) \end{matrix}]$	$[\begin{matrix} ([0.58, 0.66], [0, 0.10]), \\ ([0.57, 0.64], [0.01, 0.06]) \end{matrix}]$	$[\begin{matrix} ([0.51, 0.58], [0, 0.09]), \\ ([0.57, 0.64], [0.01, 0.06]) \end{matrix}]$
C ₁₀	A ₁	$[\begin{matrix} ([0.44, 0.51], [0.07, 0.16]), \\ ([0.43, 0.49], [0.02, 0.07]) \end{matrix}]$	$[\begin{matrix} ([0.50,, 0.58], [0.08, 0.18]), \\ ([0.50, 0.57], [0.02, 0.08]) \end{matrix}]$	$[\begin{matrix} ([0.26, 0.32], [0.18, 0.26]), \\ ([0.36, 0.43], [0.17, 0.24]) \end{matrix}]$
	A ₂	$[\begin{matrix} ([0.44, 0.51], [0.07, 0.16]), \\ ([0.43, 0.49], [0.02, 0.07]) \end{matrix}]$	$[\begin{matrix} ([0.44, 0.51], [0.07, 0.16]), \\ ([0.57, 0.65], [0.03, 0.10]) \end{matrix}]$	$[\begin{matrix} ([0.51, 0.58], [0, 0.09]), \\ ([0.49, 0.55], [0.01, 0.05]) \end{matrix}]$
	A ₃	$[\begin{matrix} ([0.38, 0.44], [0.06, 0.14]), \\ ([0.50, 0.57], [0.03, 0.08]) \end{matrix}]$	$[\begin{matrix} ([0.38, 0.44], [0.06, 0.14]), \\ ([0.50, 0.57], [0.03, 0.08]) \end{matrix}]$	$[\begin{matrix} ([0.51, 0.58], [0, 0.09]), \\ ([0.49, 0.55], [0.01, 0.05]) \end{matrix}]$
C ₁₁	A ₁	$[\begin{matrix} ([0.34, 0.42], [0.24, 0.34]), \\ ([0.36, 0.43], [0.17, 0.24]) \end{matrix}]$	$[\begin{matrix} ([0.26, 0.32], [0.18, 0.26]), \\ ([0.36, 0.43], [0.17, 0.24]) \end{matrix}]$	$[\begin{matrix} ([0.44, 0.51], [0.07, 0.16]), \\ ([0.43, 0.49], [0.02, 0.07]) \end{matrix}]$
	A ₂	$[\begin{matrix} ([0.58, 0.64], [0, 0.10]), \\ ([0.57, 0.64], [0.01, 0.06]) \end{matrix}]$	$[\begin{matrix} ([0.38, 0.44], [0.06, 0.14]), \\ ([0.50, 0.57], [0.03, 0.08]) \end{matrix}]$	$[\begin{matrix} ([0.50, 0.58], [0.08, 0.18]), \\ ([0.50, 0.57], [0.03, 0.08]) \end{matrix}]$
	A ₃	$[\begin{matrix} ([0.51, 0.58], [0, 0.09]), \\ ([0.49, 0.55], [0.01, 0.05]) \end{matrix}]$	$[\begin{matrix} ([0.44, 0.50], [0, 0.10]), \\ ([0.57, 0.64], [0.01, 0.06]) \end{matrix}]$	$[\begin{matrix} ([0.51, 0.58], [0, 0.09]), \\ ([0.49, 0.55], [0.01, 0.05]) \end{matrix}]$
C ₁₂	A ₁	$[\begin{matrix} ([0.50, 0.58], [0.08, 0.18]), \\ ([0.50, 0.57], [0.03, 0.08]) \end{matrix}]$	$[\begin{matrix} ([0.44, 0.50], [0, 0.10]), \\ ([0.57, 0.64], [0.01, 0.06]) \end{matrix}]$	$[\begin{matrix} ([0.51, 0.58], [0, 0.09]), \\ ([0.49, 0.55], [0.01, 0.05]) \end{matrix}]$
	A ₂	$[\begin{matrix} ([0.30, 0.37], [0.21, 0.30]), \\ ([0.31, 0.37], [0.14, 0.20]) \end{matrix}]$	$[\begin{matrix} ([0.22, 0.26], [0.15, 0.22]), \\ ([0.31, 0.37], [0.14, 0.20]) \end{matrix}]$	$[\begin{matrix} ([0, 0.08], [0.36, 0.47]), \\ ([0.08, 0.18], [0.42, 0.49]) \end{matrix}]$
	A ₃	$[\begin{matrix} ([0.51, 0.58], [0, 0.09]), \\ ([0.49, 0.55], [0.01, 0.05]) \end{matrix}]$	$[\begin{matrix} ([0.44, 0.50], [0, 0.10]), \\ ([0.57, 0.64], [0.01, 0.06]) \end{matrix}]$	$[\begin{matrix} ([0.44, 0.50], [0, 0.10]), \\ ([0.57, 0.64], [0.01, 0.06]) \end{matrix}]$
C ₁₃	A ₁	$[\begin{matrix} ([0.44, 0.51], [0.07, 0.16]), \\ ([0.43, 0.49], [0.02, 0.07]) \end{matrix}]$	$[\begin{matrix} ([0.51, 0.58], [0, 0.09]), \\ ([0.49, 0.55], [0.01, 0.05]) \end{matrix}]$	$[\begin{matrix} ([0.50, 0.58], [0.08, 0.18]), \\ ([0.50, 0.57], [0.03, 0.08]) \end{matrix}]$
	A ₂	$[\begin{matrix} ([0.12, 0.23], [0.36, 0.51]), \\ ([0.17, 0.24], [0.28, 0.36]) \end{matrix}]$	$[\begin{matrix} ([0.10, 0.29], [0.32, 0.45]), \\ ([0.15, 0.21], [0.24, 0.31]) \end{matrix}]$	$[\begin{matrix} ([0.10, 0.29], [0.32, 0.45]), \\ ([0.15, 0.21], [0.24, 0.31]) \end{matrix}]$
	A ₃	$[\begin{matrix} ([0.44, 0.50], [0, 0.10]), \\ ([0.57, 0.64], [0.01, 0.06]) \end{matrix}]$	$[\begin{matrix} ([0.50,, 0.58], [0.08, 0.18]), \\ ([0.50, 0.57], [0.03, 0.08]) \end{matrix}]$	$[\begin{matrix} ([0.44, 0.51], [0.07, 0.16]), \\ ([0.43, 0.49], [0.02, 0.07]) \end{matrix}]$
C ₁₄	A ₁	$[\begin{matrix} ([0.26, 0.32], [0.18, 0.26]), \\ ([0.36, 0.43], [0.17, 0.24]) \end{matrix}]$	$[\begin{matrix} ([0.10, 0.29], [0.32, 0.45]), \\ ([0.15, 0.21], [0.24, 0.31]) \end{matrix}]$	$[\begin{matrix} ([0, 0.11], [0.48, 0.63]), \\ ([0.08, 0.18], [0.42, 0.49]) \end{matrix}]$
	A ₂	$[\begin{matrix} ([0.38, 0.44], [0.06, 0.14]), \\ ([0.50, 0.57], [0.03, 0.08]) \end{matrix}]$	$[\begin{matrix} ([0.22, 0.26], [0.15, 0.22]), \\ ([0.36, 0.43], [0.17, 0.24]) \end{matrix}]$	$[\begin{matrix} ([0, 0.11], [0.48, 0.63]), \\ ([0.07, 0.15], [0.36, 0.42]) \end{matrix}]$
	A ₃	$[\begin{matrix} ([0.44, 0.51], [0.07, 0.16]), \\ ([0.43, 0.49], [0.02, 0.07]) \end{matrix}]$	$[\begin{matrix} ([0.58, 0.64], [0, 0.10]), \\ ([0.57, 0.64], [0.01, 0.06]) \end{matrix}]$	$[\begin{matrix} ([0.51, 0.58], [0, 0.09]), \\ ([0.49, 0.55], [0.01, 0.05]) \end{matrix}]$