An interval-valued fuzzy statistical group decision making approach with new evaluating indices for sustainable supplier selection problem

Abstract

This study presents a new soft computing approach based on failure mode and effects analysis (FMEA)’s concept for sustainable supplier selection problem (SSSP) in the light of multi-attributes decision analysis. The approach determines weights of experts by considering interval-valued fuzzy sets (IVFSs) and asymmetric uncertainty information simultaneously. The presented group decision approach assesses the sustainable suppliers as indicated by risks of economic, social and environmental measurements. Ideas of fuzzy possibilistic statistical modeling are brought into the soft computing approach. New definitions are enhanced by fuzzy possibilistic statistical positive and negative ideal solutions. In addition, new relations of fuzzy evaluating and prioritizing of the risks are presented. Then, an application example is tackled by the presented soft computing approach to exhibit its capacity by three measurements of the sustainability for the evaluation of sustainable suppliers.

Keywords

Sustainable supplier selection problem soft computing approach group decision making process weights of experts interval-valued fuzzy possibilistic modeling

1 Introduction

Organizations had endeavored to lessen costs and to enhance proficiency of the sustainable supply chain networks (SSCNs) by which they conveyed items to the customers at the correct cost and at the correct circumstances [1]. Vital sourcing is one of the fastest growing fields of managing the SSCNs. For instance, raw materials and parts are bought from the suppliers. Selecting suppliers can be critical task of sustainable supply chains (SSCs) since it accomplishes high-quality items, bring down expenses with more noteworthy consumer loyalty and helps expand benefit [1 –4]. Sustainable supplier management could regard how organizations can appraise and choose sustainable suppliers. Also, overseeing sustainable supply risks may be a basic segment of dealing with the SSCNs. Therefore, it is essential to their prosperity to comprehend the sources of risks and how to best oversee them [5]. As a result of sustainable suppliers as a fundamental part of the SSCM when the managers handle them, new models to involve risks can motivate researchers for applying diverse sustainable assessment factors and for concentrating on the risk evaluation to choose sustainable supplier with low risks. Indeed, overseeing risks and vulnerabilities in the appraisal of sustainable supplier alternatives regarding various assessment factors and dealing with the SSCM’ conditions are the central worries of the researchers.

Blackhurst et al. [6] developed a decision approach with attributes’ weights to compute risk priority for suppliers in automotive industry. Kull and Talluri [7] employed a decision method according to the AHP and GP methods for evaluating suppliers by their risks. Matook et al. [8] applied factor analysis and functional benchmarking to report supplier risk profiles and to appraise the low-risk alternatives. Dalalah et al. [9] provided a combined model with the GDM and DEMATEL methods for the supplier selection problem. Goebel et al. [10] provided elements regarding the purchasing and SSCNs behavior for the SSSP. Chai et al. [11] recognized trends for the supplier evaluation problem. Viswanadham and Samvedi [2] designed a model for the supplier selection with AHP and TOPSIS and FST methods. Paul [3] developed a methodology for a supply chain that regarded different selection attributes for dealing with supply risks along with a rule-based FIS. Giannakis and Papadopoulos [12] presented a combined model for data analysis with the FMEA to appraise risks. Govindan and Jepsen [13] proposed a decision model based on the ELECTRE TRI-C method for sorting suppliers with risks. Modak et al. [14] extended cooperative and non-cooperative model of a closed-loop supply chain. Panda et al. [15] appraised impacts of corporate social responsibility, and they investigated channel coordination in a socially responsible manufacturer-retailer closed-loop supply chain.

In this study, a new soft computing approach with fuzzy possibilistic statistical model, group decision making, asymmetric uncertainty information and FMEA’s concept is presented for the SSSP. In fact, a new group decision process is proposed under an interval-valued fuzzy condition in view of compromise solution programming and three possibilistic mean, standard deviation and the cube-root of skewness matrices. Furthermore, a new weighting method of experts based on fuzzy possibilistic statistical ideas is presented for processing the group decision making (GDM) with asymmetric uncertainty data. Also, new separation measures are introduced in view of interval valued fuzzy possibilistic statistical concepts and new ranking index by the FMEA’ idea for the SSSP. The presented soft computing approach could support the GDM in reporting a best sustainable supplier properly. Then, an application example is presented by the soft computing approach to exhibit its capacity by three measurements of the sustainability for the evaluation of sustainable supplier alternatives.

The study is organized as below: Section 2 presents the proposed soft computing approach. An application example is provided in Section 3. Also, discussions of results are given in this section. Section 4 involves conclusions.

2 Presented soft computing approach

In this section, a new interval-valued fuzzy group approach for the evaluation of sustainable suppliers is presented in SSCNs based on possibility theory and statistical concepts. The concept of FMEA is taken in the proposed soft computing approach. In fact, evaluation criteria are three conventional factors of the FMEA in the proposed evaluation and ranking process.

First, it is assumed that: TM = {TM_k| k = 1, …, p} as a set of team members or experts; SS = {SS_l| l = 1, …, u} as a finite set of sustainable suppliers; R = {R_i| i = 1, …, m} as a finite set of risks of three dimensions of the sustainability for each sustainable supplier; and C = {C_j| j = 1, …, n} as a finite set of risk assessment criteria for the SSSP.

Since the information of sustainable suppliers is uncertain with the GDM, the experts can consider an interval-valued fuzzy (IVF) ${\tilde{A}}_{ij}^{k}$ to estimate the judgment and opinion on sustainable risk (R_i) for each supplier (SS_i) with respect to risk assessment criterion C_i. The decision problem of SSSP according to risks of three dimensions of the sustainability with IVFSs and statistical concepts can be expressed as below: $\begin{matrix} {\tilde{A}}^{k} = [[({({\underline{a}}_{{ij}_{1}})}^{k}, {({\underline{a}}_{{ij}_{2}})}^{k}, {({\underline{a}}_{{ij}_{3}})}^{k}), \\ {({({\bar{a}}_{{ij}_{1}})}^{k}, {({\bar{a}}_{{ij}_{2}})}^{k}, {({\bar{a}}_{{ij}_{3}})}^{k})]]}_{m \times n} \end{matrix}$ (1)

For convenience, we denote:

$\begin{matrix} [({({\underline{a}}_{{ij}_{1}})}^{k}, {({\underline{a}}_{{ij}_{2}})}^{k}, {({\underline{a}}_{{ij}_{3}})}^{k}), ({({\bar{a}}_{{ij}_{1}})}^{k}, {({\bar{a}}_{{ij}_{2}})}^{k}, {({\bar{a}}_{{ij}_{3}})}^{k})] \\ = [({({\underline{a}}_{ij} - {\underline{ϑ}}_{ij})}^{k}, {({\underline{a}}_{ij})}^{k}, {({\underline{a}}_{ij} + {\underline{ξ}}_{ij})}^{k}), \\ ({({\bar{a}}_{ij} - {\bar{ϑ}}_{ij})}^{k}, {({\bar{a}}_{ij})}^{k}, {({\bar{a}}_{ij} + {\bar{ξ}}_{ij})}^{k})] \\ for kth TM ({TM}^{k}) . \end{matrix}$

According to the above-mentioned descriptions, steps of the proposed soft computing approach for the evaluation of sustainable suppliers are given as below:

Step 1. Proper risk assessment criteria are identified for the SSSP according to risks of three dimensions of the sustainability for each sustainable supplier (l = 1, …, u).

Step 2. Transform the aggregated IVF-matrix into the normalized matrix of sustainable supplier risks. To transform different criteria scales into a comparable scale, the linear scale transformation method is used and presented by:

$\begin{matrix} {({\tilde{y}}_{i j})}^{k} = {[({({\underline{y}}_{i j 1})}^{k}, {({\underline{y}}_{i j 2})}^{k}, {({\underline{y}}_{i j 3})}^{k}), ({({\bar{y}}_{i j 1})}^{k}, {({\bar{y}}_{i j 2})}^{k}, {({\bar{y}}_{i j 3})}^{k})]}_{m \times n} \\ = [(\frac{{({\underline{a}}_{i j} - {\underline{ϑ}}_{i j})}^{k}}{{({({\bar{a}}_{i j} + {\bar{ξ}}_{i j})}^{k})}^{+}}, \frac{{({\underline{a}}_{i j})}^{k}}{{({({\bar{a}}_{i j} + {\bar{ξ}}_{i j})}^{k})}^{+}}, \frac{{({\underline{a}}_{i j} + {\underline{ξ}}_{i j})}^{k}}{{({({\bar{a}}_{i j} + {\bar{ξ}}_{i j})}^{k})}^{+}}), \\ {(\frac{{({\bar{a}}_{i j} - {\bar{ϑ}}_{i j})}^{k}}{{({({\bar{a}}_{i j} + {\bar{ξ}}_{i j})}^{k})}^{+}}, \frac{{({\bar{a}}_{i j})}^{k}}{{({({\bar{a}}_{i j} + {\bar{ξ}}_{i j})}^{k})}^{+}}, \frac{{({\bar{a}}_{i j} + {\bar{ξ}}_{i j})}^{k}}{{({({\bar{a}}_{i j} + {\bar{ξ}}_{i j})}^{k})}^{+}})]}_{m \times n} \\ k = 1, ..., p a n d j \in Ω_{b} \end{matrix}$ (2) and $\begin{matrix} {({\tilde{y}}_{i j})}^{k} = {[({({\underline{y}}_{i j 1})}^{k}, {({\underline{y}}_{i j 2})}^{k}, {({\underline{y}}_{i j 3})}^{k}), ({({\bar{y}}_{i j 1})}^{k}, {({\bar{y}}_{i j 2})}^{k}, {({\bar{y}}_{i j 3})}^{k})]}_{m \times n} \\ = [(\frac{{({({\bar{a}}_{i j} - {\bar{ϑ}}_{i j})}^{k})}^{-}}{{({({\underline{a}}_{i j} + {\underline{ξ}}_{i j})}^{k})}^{+}}, \frac{{({({\bar{a}}_{i j} - {\bar{ϑ}}_{i j})}^{k})}^{-}}{{({\underline{a}}_{i j})}^{k}}, \frac{{({({\bar{a}}_{i j} - {\bar{ϑ}}_{i j})}^{k})}^{-}}{{({\underline{a}}_{i j} + {\underline{ϑ}}_{i j})}^{k}}), \\ {(\frac{{({({\bar{a}}_{i j} - {\bar{ϑ}}_{i j})}^{k})}^{-}}{{({\bar{a}}_{i j} + {\bar{ξ}}_{i j})}^{k}}, \frac{{({({\bar{a}}_{i j} - {\bar{ϑ}}_{i j})}^{k})}^{-}}{{({\bar{a}}_{i j})}_{k}}, \frac{{({({\bar{a}}_{i j} - {\bar{ϑ}}_{i j})}^{k})}^{-}}{{({\bar{a}}_{i j} + {\bar{ϑ}}_{i j})}^{k}})]}_{m \times n} \\ k = 1, ..., p a n d j \in Ω_{b} \end{matrix}$ (3) where Ω_b and Ω_c are the sets of benefit and cost criterion for the SSSP according to risks of three dimensions of the sustainability respectively, ${({({\bar{a}}_{ij} + {\bar{ξ}}_{ij})}^{k})}^{+} = max_{i} {({\bar{a}}_{ij} + {\bar{ξ}}_{ij})}^{k}$ , j = 1, …, n and k = 1, …, p, $({({\bar{a}}_{ij} - {\bar{ϑ}}_{ij})}^{k}) = min_{i} {({\bar{a}}_{ij} - {\bar{ϑ}}_{ij})}^{k}$ , j = 1, …, n and k = 1, …, p. The maximum rating of each sustainable supplier risk against each criterion and the minimum rating using the normalization process can be obtained.

Step 3. Provide ${\tilde{R}}^{*}$ as positive ideal decision (PID) based on all decisions with the GDM via the group decision concept of [16].

${\tilde{R}}^{*} = {[[{\underline{r}}_{i j 1}^{*}, {\underline{r}}_{i j 2}^{*}, {\underline{r}}_{i j 3}^{*}], ({\bar{r}}_{i j 1}^{*}), ({\bar{r}}_{i j 2}^{*}), ({\bar{r}}_{i j 3}^{*})]}_{m \times n}$ (4) where $\begin{array}{l} [({\underline{r}}_{i j 1}^{*}, {\underline{r}}_{i j 2}^{*}, {\underline{r}}_{i j 3}^{*}), ({\bar{r}}_{i j 1}^{*}, {\bar{r}}_{i j 2}^{*}, {\bar{r}}_{i j 3}^{*})] = [(\frac{1}{p} \sum_{k = 1}^{p} {({\underline{y}}_{i j} - {\underline{ϑ}}_{i j})}^{k}, \frac{1}{p} \sum_{k = 1}^{p} {({\underline{y}}_{i j})}^{k}, \frac{1}{p} \sum_{k = 1}^{p} {({\underline{y}}_{i j} + {\underline{ξ}}_{i j})}^{k},) (\frac{1}{p} \sum_{k = 1}^{p} {({\bar{y}}_{i j} - {\bar{ϑ}}_{i j})}^{k}, \frac{1}{p} \sum_{k = 1}^{p} {({\bar{y}}_{i j})}^{k}, \frac{1}{p} \sum_{k = 1}^{p} {({\bar{y}}_{i j} + {\bar{ξ}}_{i j})}^{k})] \\ i = 1, ..., m a n d j = 1, ..., n . \end{array}$

Step 4. Provide ${\tilde{R}}_{l}^{-}$ as the left negative ideal decision (LNID) based on all decisions with the GDM via the group decision concept of [16].

${\tilde{R}}_{l}^{-} = {[[{\underline{r}}_{i j 1}^{l}, {\underline{r}}_{i j 2}^{l}, {\underline{r}}_{i j 3}^{l}], ({\bar{r}}_{i j 1}^{l}, {\bar{r}}_{i j 2}^{l}, {\bar{r}}_{i j 3}^{l})]}_{m \times n}$ (5) where $\begin{matrix} [({\underline{r}}_{i j 1}^{l}, {\underline{r}}_{i j 2}^{l}, {\underline{r}}_{i j 3}^{l}), ({\underline{r}}_{i j 3}^{l}, {\underline{r}}_{i j 3}^{l}, {\underline{r}}_{i j 3}^{l})] = [(min_{1 \leq k \leq p} {({\underline{y}}_{ij} - {\underline{ϑ}}_{ij})}^{k}, min_{1 \leq k \leq p} {({\underline{y}}_{ij})}^{k}, min_{1 \leq k \leq p} {({\underline{y}}_{ij} + {\underline{ξ}}_{ij})}^{k}), \\ (min_{1 \leq k \leq p} {({\bar{y}}_{ij} - {\bar{ϑ}}_{ij})}^{k}, min_{1 \leq k \leq p} {({\bar{y}}_{ij})}^{k}, min_{1 \leq k \leq p} {({\bar{y}}_{ij} + {\bar{ξ}}_{ij})}^{k})] . i = 1, \dots, m and j = 1, \dots, n . \end{matrix}$

Step 5. Provide ${\tilde{R}}_{r}^{-}$ as right negative ideal decision (RNID) based on all decisions with the GDM via the group decision concept of [16]. ${\tilde{R}}_{r}^{-} = {[[{\underline{r}}_{i j 1}^{r}, {\underline{r}}_{i j 2}^{r}, {\underline{r}}_{i j 3}^{r}], ({\bar{r}}_{i j 1}^{r}, {\bar{r}}_{i j 2}^{r}, {\bar{r}}_{i j 3}^{r})]}_{m \times n}$ (6) Where ${[[{\underline{r}}_{i j 1}^{r}, {\underline{r}}_{i j 2}^{r}, {\underline{r}}_{i j 3}^{r}], ({\bar{r}}_{i j 1}^{r}, {\bar{r}}_{i j 2}^{r}, {\bar{r}}_{i j 3}^{r})]}_{m \times n} = \begin{array}{l} [(\min_{1 \leq k \leq p} {({\underline{y}}_{i j} - {\underline{ϑ}}_{i j})}^{k}, \min_{1 \leq k \leq p} {({\underline{y}}_{i j})}^{k}, \min_{1 \leq k \leq p} {({\underline{y}}_{i j} + {\underline{ξ}}_{i j})}^{k}), (\min_{1 \leq k \leq p} {({\bar{y}}_{i j} - {\bar{ϑ}}_{i j})}^{k}, \min_{1 \leq k \leq p} {({\bar{y}}_{i j})}^{k}, \min_{1 \leq k \leq p} {({\bar{y}}_{i j} + {\bar{ξ}}_{i j})}^{k})] . \\ i = 1, \dots, m and j = 1, \dots, n \end{array}$

Step 6. Provide separations of each expert decision from the PID, LNID and RNID, respectively.

where ${[[{\underline{r}}_{i j 1}^{r}, {\underline{r}}_{i j 2}^{r}, {\underline{r}}_{i j 3}^{r}], ({\bar{r}}_{i j 1}^{r}, {\bar{r}}_{i j 2}^{r}, {\bar{r}}_{i j 3}^{r})]}_{m \times n} = \begin{array}{l} [(\max_{1 \leq k \leq p} {({\underline{y}}_{i j} - {\underline{ϑ}}_{i j})}^{k}, \max_{1 \leq k \leq p} {({\underline{y}}_{i j})}^{k}, \max_{1 \leq k \leq p} {({\underline{y}}_{i j} + {\underline{ξ}}_{i j})}^{k}), (\max_{1 \leq k \leq p} {({\bar{y}}_{i j} - {\bar{ϑ}}_{i j})}^{k}, \max_{1 \leq k \leq p} {({\bar{y}}_{i j})}^{k}, \max_{1 \leq k \leq p} {({\bar{y}}_{i j} + {\bar{ξ}}_{i j})}^{k})] . \\ i = 1, \dots, m and j = 1, \dots, n \end{array}$ $\begin{array}{l} {\tilde{S}}_{k}^{*} = [({\underline{S}}_{k_{1}}^{*}, {\underline{S}}_{k_{2}}^{*}, {\underline{S}}_{k_{3}}^{*}), ({\bar{S}}_{k_{1}}^{*}, {\bar{S}}_{k_{2}}^{*}, {\bar{S}}_{k_{2}}^{*})] = \\ [(\sqrt{\sum_{i = 1}^{m} \sum_{j = 1}^{n} {({({\underline{y}}_{i j} - {\bar{ϑ}}_{i j})}^{k} - {\underline{r}}_{i j_{1}}^{*})}^{2}}, \sqrt{\sum_{i = 1}^{m} \sum_{j = 1}^{n} {({({\underline{y}}_{i j})}^{k} - {\underline{r}}_{i j_{1}}^{*})}^{2}}, \sqrt{\sum_{i = 1}^{m} \sum_{j = 1}^{n} {({({\underline{y}}_{i j} + {\underline{ξ}}_{i j})}^{k} - {\underline{r}}_{i j_{1}}^{*})}^{2}}), \\ (\sqrt{\sum_{i = 1}^{m} \sum_{j = 1}^{n} {({({\bar{y}}_{i j} - {\bar{ϑ}}_{i j})}^{k} - {\bar{r}}_{i j_{1}}^{*})}^{2}}, \sqrt{\sum_{i = 1}^{m} \sum_{j = 1}^{n} {({({\bar{y}}_{i j})}^{k} - {\bar{r}}_{i j_{1}}^{*})}^{2}}, \sqrt{\sum_{i = 1}^{m} \sum_{j = 1}^{n} {({({\bar{y}}_{i j} + {\bar{ξ}}_{i j})}^{k} - {\bar{r}}_{i j_{1}}^{*})}^{2}})], \end{array}$ (7)

$\begin{array}{l} {\tilde{S}}_{k}^{l} = [({\underline{S}}_{k}^{l}_{1}, {\underline{S}}_{k}^{l}_{2}, {\underline{S}}_{k}^{l}_{3}), ({\bar{S}}_{k}^{l}_{1}, {\bar{S}}_{k}^{l}_{2}, {\bar{S}}_{k}^{l}_{2})] = \\ [(\sqrt{\sum_{i = 1}^{m} \sum_{j = 1}^{n} {({({\underline{y}}_{i j} - {\underline{ϑ}}_{i j})}^{k} - {\underline{r}}_{i j}_{1}^{l})}^{2}}, \sqrt{\sum_{i = 1}^{m} \sum_{j = 1}^{n} {({({\underline{y}}_{i j})}^{k} - {\underline{r}}_{i j}_{2}^{l})}^{2}}, \sqrt{\sum_{i = 1}^{m} \sum_{j = 1}^{n} {({({\underline{y}}_{i j} + {\underline{ξ}}_{i j})}^{k} - {\underline{r}}_{i j}_{3}^{l})}^{2}}), \\ (\sqrt{\sum_{i = 1}^{m} \sum_{j = 1}^{n} {({({\bar{y}}_{i j} - {\bar{ϑ}}_{i j})}^{k} - {\bar{r}}_{i j}_{1}^{l})}^{2}}, \sqrt{\sum_{i = 1}^{m} \sum_{j = 1}^{n} {({({\bar{y}}_{i j})}^{k} - {\bar{r}}_{i j}_{2}^{l})}^{2}}, \sqrt{\sum_{i = 1}^{m} \sum_{j = 1}^{n} {({({\bar{y}}_{i j} + {\bar{ξ}}_{i j})}^{k} - {\bar{r}}_{i j}_{3}^{l})}^{2}})], \\ k = 1, \dots, p \end{array}$ (8) and $\begin{array}{l} {\tilde{S}}_{k}^{r} = [({\underline{S}}_{k}^{r}_{1}, {\underline{S}}_{k}^{r}_{2}, {\underline{S}}_{k}^{r}_{3}), ({\bar{S}}_{k}^{r}_{1}, {\bar{S}}_{k}^{r}_{2}, {\bar{S}}_{k}^{r}_{2})] = \\ [(\sqrt{\sum_{i = 1}^{m} \sum_{j = 1}^{n} {({({\underline{y}}_{i j} - {\underline{ϑ}}_{i j})}^{k} - {\underline{r}}_{i j}_{1}^{r})}^{2}}, \sqrt{\sum_{i = 1}^{m} \sum_{j = 1}^{n} {({({\underline{y}}_{i j})}^{k} - {\underline{r}}_{i j}_{2}^{r})}^{2}}, \\ \sqrt{\sum_{i = 1}^{m} \sum_{j = 1}^{n} {({({\underline{y}}_{i j} + {\underline{ξ}}_{i j})}^{k} - {\underline{r}}_{i j}_{3}^{r})}^{2}}), (\sqrt{\sum_{i = 1}^{m} \sum_{j = 1}^{n} {({({\bar{y}}_{i j} - {\bar{ϑ}}_{i j})}^{k} - {\bar{r}}_{i j}_{1}^{r})}^{2}}, \sqrt{\sum_{i = 1}^{m} \sum_{j = 1}^{n} {({({\bar{y}}_{i j})}^{k} - {\bar{r}}_{i j}_{2}^{r})}^{2}}, \\ \sqrt{\sum_{i = 1}^{m} \sum_{j = 1}^{n} {({({\bar{y}}_{i j} + {\bar{ξ}}_{i j})}^{k} - {\bar{r}}_{i j}_{3}^{r})}^{2}})], \\ k = 1, \dots, p \end{array}$ (9)

Step 7. Compute closeness coefficient of the expert decision versus ideal decisions with the GDM.

$\begin{array}{l} {\tilde{Q}}_{k} = [({\underline{Q}}_{k}_{1}, {\underline{Q}}_{k}_{2}, {\underline{Q}}_{k}_{3}), ({\bar{Q}}_{k}_{1}, {\bar{Q}}_{k}_{2}, {\bar{Q}}_{k}_{3})] = \\ [(𝒻_{k} \frac{{\underline{S}}_{k}^{l}_{1}}{\max_{k} {\underline{S}}_{k}^{l}_{1}} + 𝒻_{k} \frac{{\underline{S}}_{k}^{r}_{1}}{\max_{k} {\underline{S}}_{k}^{r}_{1}} - 𝒻_{k} \frac{{\underline{S}}_{k}^{*}_{1}}{\min_{k} {\underline{S}}_{k}^{*}_{1}}, 𝒻_{k} \frac{{\underline{S}}_{k}^{l}_{2}}{\max_{k} {\underline{S}}_{k}^{l}_{2}} + 𝒻_{k} \frac{{\underline{S}}_{k}^{r}_{2}}{\max_{k} {\underline{S}}_{k}^{r}_{2}} - 𝒻_{k} \frac{{\underline{S}}_{k}^{*}_{2}}{\min_{k} {\underline{S}}_{k}^{*}_{2}}, \\ 𝒻_{k} \frac{{\underline{S}}_{k}^{l}_{3}}{\max_{k} {\underline{S}}_{k}^{l}_{3}} + 𝒻_{k} \frac{{\underline{S}}_{k}^{r}_{3}}{\max_{k} {\underline{S}}_{k}^{r}_{3}} - 𝒻_{k} \frac{{\underline{S}}_{k}^{*}_{3}}{\min_{k} {\underline{S}}_{k}^{*}_{3}}), \\ (𝒻_{k} \frac{{\bar{S}}_{k}^{l}_{1}}{\max_{k} {\bar{S}}_{k}^{l}_{1}} + 𝒻_{k} \frac{{\bar{S}}_{k}^{r}_{1}}{\max_{k} {\bar{S}}_{k}^{r}_{1}} - 𝒻_{k} \frac{{\bar{S}}_{k}^{*}_{1}}{\min_{k} {\bar{S}}_{k}^{*}_{1}}, 𝒻_{k} \frac{{\bar{S}}_{k}^{l}_{2}}{\max_{k} {\bar{S}}_{k}^{l}_{2}} + 𝒻_{k} \frac{{\bar{S}}_{k}^{r}_{2}}{\max_{k} {\bar{S}}_{k}^{r}_{2}} - 𝒻_{k} \frac{{\bar{S}}_{k}^{*}_{2}}{\min_{k} {\bar{S}}_{k}^{*}_{2}}, \\ 𝒻_{k} \frac{{\bar{S}}_{k}^{l}_{3}}{\max_{k} {\bar{S}}_{k}^{l}_{3}} + 𝒻_{k} \frac{{\bar{S}}_{k}^{r}_{3}}{\max_{k} {\bar{S}}_{k}^{r}_{3}} - 𝒻_{k} \frac{{\bar{S}}_{k}^{*}_{3}}{\min_{k} {\bar{S}}_{k}^{*}_{3}})], \\ k = 1, \dots, p \end{array}$ (10) where $𝒻_{k} + ℊ_{k} + 𝒽_{k} = 1; k = 1, \dots, p .$

Step 8. Compute an integrated weight assigned to each expert. $\begin{matrix} {\tilde{H}}_{k} = {\tilde{γ}}_{k} \otimes {\tilde{Q}}_{k} = [({\underline{γ}}_{k_{1}}, {\underline{γ}}_{k_{2}}, {\underline{γ}}_{k_{3}}), ({\bar{γ}}_{k_{1}}, {\bar{γ}}_{k_{2}}, {\bar{γ}}_{k_{3}})] \\ \otimes [({\underline{Q}}_{k_{1}}, {\underline{Q}}_{k_{2}}, {\underline{Q}}_{k_{3}}), ({\bar{Q}}_{k_{1}}, {\bar{Q}}_{k_{2}}, {\bar{Q}}_{k_{3}})], \\ k = 1, \dots, p \end{matrix}$ (11) where ${\tilde{γ}}_{k}$ denote the relative importance of kth expert, ${\tilde{Q}}_{k}$ denote a relative importance which regards the measure of kth expert’s judgment to close to other experts.

Step 9. Establish the IVF-decision matrices of decision alternatives and aggregate them.

The aggregated IVF-decision matrices of the decision alternatives are presented with the arithmetic average with the following relation.

\begin{array}{l} \tilde{A} = \tilde{a}'_{i j} = {[({\underline{a}}_{i j}^{'} - {\underline{ϑ}}_{i j}^{'}, {\underline{a}}_{i j}^{'}, {\underline{a}}_{i j}^{'} + {\underline{ξ}}_{i j}^{'}), ({\bar{a}}_{i j}^{'} - {\bar{ϑ}}_{i j}^{'}, {\bar{a}}_{i j}^{'}, {\bar{a}}_{i j}^{'} + {\bar{ξ}}_{i j}^{'})]}_{m \times n} \\ = {[[(\frac{1}{p} \sum_{k = 1}^{p} {\underline{ℌ}}_{k}_{1} \otimes {\underline{y}}_{i j}_{1}^{k}, \frac{1}{p} \sum_{k = 1}^{p} {\underline{ℌ}}_{k}_{2} \otimes {\underline{y}}_{i j}_{2}^{k}, \frac{1}{p} \sum_{k = 1}^{p} {\underline{ℌ}}_{k}_{3} \otimes {\underline{a}}_{i j}_{3}^{k}), (\frac{1}{p} \sum_{k = 1}^{p} {\bar{ℌ}}_{k}_{1} \otimes {\bar{a}}_{i j}_{1}^{k}, \frac{1}{p} \sum_{k = 1}^{p} {\bar{ℌ}}_{k}_{2} \otimes {\bar{a}}_{i j}_{2}^{k}, \frac{1}{p} \sum_{k = 1}^{p} {\bar{ℌ}}_{k}_{3} \otimes {\bar{a}}_{i j}_{3}^{k})]]}_{m \times n} . \end{array}

(12)

Step 10. Construct the possibilistic interval mean matrix for the risk evaluation problem of the SSSP. The possibilistic interval mean (m_ij) of IVF ${\tilde{a}}_{ij}^{'} = [({\underline{a}}_{ij}^{'} - {\underline{ϑ}}_{ij}^{'}, {\underline{a}}_{ij}^{'}, {\underline{a}}_{ij}^{'} + {\underline{ξ}}_{ij}^{'}), ({\bar{a}}_{ij}^{'} - {\bar{ϑ}}_{ij}^{'}, {\bar{a}}_{ij}^{'}, {\bar{a}}_{ij}^{'} + {\bar{ξ}}_{ij}^{'})]$ are defined according to Equation (1a):

$\begin{array}{l} m_{i j} = [{\underline{m}}_{i j}, {\bar{m}}_{i j}] \\ = [\frac{(({\underline{a}}_{i j}^{'} - {\underline{ϑ}}_{i j}^{'}) + 2 ({\underline{a}}_{i j}^{'}) + ({\underline{a}}_{i j}^{'} + {\underline{ξ}}_{i j}^{'}))}{4}, \frac{(({\bar{a}}_{i j}^{'} - {\bar{ϑ}}_{i j}^{'}) + 2 ({\bar{a}}_{i j}^{'}) + ({\bar{a}}_{i j}^{'} + {\bar{ξ}}_{i j}^{'}))}{4}] \end{array}$ (13)

Then, the possibilistic mean interval matrix is constructed for the risk evaluation problem of the SSSP as follows:

ℳ = {[m_{i j}]}_{m \times n} = [\begin{matrix} \begin{matrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{matrix} & \begin{matrix} \dots & m_{1 n} \\ \dots & m_{2 n} \end{matrix} \\ \begin{matrix} ⋮ & ⋮ \\ m_{m 1} & m_{m 2} \end{matrix} & \begin{matrix} ⋱ & ⋮ \\ \dots & m_{m n} \end{matrix} \end{matrix}]

(14)

Step 11. Construct the possibilistic interval standard deviation matrix for the risk evaluation problem of the SSSP. The possibilistic interval standard deviation $(𝓈 𝒹_{i j})$ of IVF ${\tilde{a}}_{ij}^{'} = [({\underline{a}}_{ij}^{'} - {\underline{ϑ}}_{ij}^{'}, {\underline{a}}_{ij}^{'}, {\underline{a}}_{ij}^{'} + {\underline{ξ}}_{ij}^{'}), ({\bar{a}}_{ij}^{'} - {\bar{ϑ}}_{ij}^{'}, {\bar{a}}_{ij}^{'}, {\bar{a}}_{ij}^{'} + {\bar{ξ}}_{ij}^{'})]$ are determined according to Equation (2a): $𝓈 𝒹_{i j} = [{\underline{𝓈 𝒹}}_{i j}, {\bar{𝓈 𝒹}}_{i j}] = [\sqrt{\frac{33 {\underline{𝓏}}_{i j}^{3} + 21 {\underline{𝓏}}_{i j}^{2} {\underline{𝒷}}_{i j} + 11 {\underline{𝓏}}_{i j} {\underline{𝒷}}_{i j}^{2} - {\underline{𝒷}}_{i j}^{3}}{384 {\underline{𝓏}}_{i j}}}, \sqrt{\frac{33 {\bar{𝓏}}_{i j}^{3} + 21 {\bar{𝓏}}_{i j}^{2} {\bar{𝒷}}_{i j} + 11 {\bar{𝓏}}_{i j} {\bar{𝒷}}_{i j}^{2} - {\bar{𝒷}}_{i j}^{3}}{384 {\bar{𝓏}}_{i j}}}]$ (15) ${\underline{z}}_{i j} {\underline{𝒷}}_{i j}, {\bar{α}}_{i j}$ and ${\bar{𝒷}}_{i j}$ are determined according to the possibility theory in Appendix. Then, the possibilistic interval standard deviation matrix is constructed for the risk evaluation problem of the SSSP as follows:

S D = {[𝓈 𝒹_{i j}]}_{m \times n} = [\begin{matrix} \begin{matrix} 𝓈 𝒹_{11} & 𝓈 𝒹_{12} \\ 𝓈 𝒹_{21} & 𝓈 𝒹_{22} \end{matrix} & \begin{matrix} \dots & 𝓈 𝒹_{1 n} \\ \dots & 𝓈 𝒹_{2 n} \end{matrix} \\ \begin{matrix} ⋮ & ⋮ \\ 𝓈 𝒹_{m 1} & 𝓈 𝒹_{m 2} \end{matrix} & \begin{matrix} ⋱ & ⋮ \\ \dots & 𝓈 𝒹_{m n} \end{matrix} \end{matrix}]

(16)

Step 12. Construct the possibilistic interval cube-root of skewness matrix of the risk evaluation problem of the SSSP. The possibilistic cube-root of skewness $C 𝓇 𝓈_{i j}$ of IVF ${\tilde{a}}_{ij}^{'} = [({\underline{a}}_{ij}^{'} - {\underline{ϑ}}_{ij}^{'}, {\underline{a}}_{ij}^{'}, {\underline{a}}_{ij}^{'} + {\underline{ξ}}_{ij}^{'}), ({\bar{a}}_{ij}^{'} - {\bar{ϑ}}_{ij}^{'}, {\bar{a}}_{ij}^{'}, {\bar{a}}_{ij}^{'} + {\bar{ξ}}_{ij}^{'})]$ are determined according to Equation (3a).

C 𝓇 𝓈_{i j} = [{\underline{C 𝓇 𝓈}}_{i j}, {\bar{C 𝓇 𝓈}}_{i j}] = [\sqrt[3]{(\frac{{({\underline{σ}}_{i j}^{'} + {\underline{τ}}_{i j}^{'})}^{2}}{32}) ({\underline{σ}}_{i j}^{'} - {\underline{τ}}_{i j}^{'})}, \sqrt[3]{(\frac{{({\bar{σ}}_{i j}^{'} + {\bar{τ}}_{i j}^{'})}^{2}}{32}) ({\bar{σ}}_{i j}^{'} - {\bar{τ}}_{i j}^{'})}]

(17)

Then, the possibilistic interval cube-root of skewness matrix is constructed for the risk evaluation problem of the SSSP as follows: $C ℛ S = {[𝒸 𝓇 𝓈_{i j}]}_{m \times n} = [\begin{matrix} \begin{matrix} 𝒸 𝓇 𝓈_{11} & 𝒸 𝓇 𝓈_{12} \\ 𝒸 𝓇 𝓈_{21} & 𝒸 𝓇 𝓈_{22} \end{matrix} & \begin{matrix} \dots & 𝒸 𝓇 𝓈_{1 n} \\ \dots & 𝒸 𝓇 𝓈_{2 n} \end{matrix} \\ \begin{matrix} ⋮ & ⋮ \\ 𝒸 𝓇 𝓈_{m 1} & 𝒸 𝓇 𝓈_{m 2} \end{matrix} & \begin{matrix} ⋱ & ⋮ \\ \dots & 𝒸 𝓇 𝓈_{m n} \end{matrix} \end{matrix}]$ (18)

Step 13. Calculate possibilistic interval mean-entropy measure of each sustainable evaluation criterion. $ℰ 𝓃_{j} = [{\underline{ℯ}}_{j}, {\bar{ℯ}}_{j}] = [- \frac{1}{L n (𝓂)} \sum_{i = 1}^{m} {\bar{𝓂}}^{'}_{i j} L n ({\bar{𝓂}}^{'}_{i j}), - \frac{1}{L n (𝓂)} \sum_{i = 1}^{m} {\underline{𝓂}}^{'}_{i j} L n ({\underline{𝓂}}^{'}_{i j})]$ (19) $where {\bar{𝓂}}_{i j}^{'} = [{\underline{𝓂}}^{'}_{i j}, {\bar{𝓂}}^{'}_{i j}] = [\frac{{\underline{𝓂}}_{i j}}{m a x_{i} {\bar{𝓂}}_{i j}}, \frac{{\bar{𝓂}}_{i j}}{m a x_{i} {\bar{𝓂}}_{i j}}]$

Step 14. Calculate proposed entropy weight based on the possibilistic interval mean. $W_{j} = [{\underline{𝓌}}_{j}, {\bar{𝓌}}_{j}] = [1 - {\bar{ℯ}}_{j}, 1 - {\underline{ℯ}}_{j}]$ (20)Step 15. Define positive-ideal and negative-ideal vectors (PIV and NIV) of possibilistic interval mean for the risk evaluation problem of the SSSP. The PIV $(M^{*})$ and NIV $(M^{-})$ are calculated by: $ℳ^{*} = {ℳ_{1}^{*}, ℳ_{2}^{*}, \dots, ℳ_{n}^{*}} = {[{\underline{𝓂}}_{j}^{*}, {\bar{𝓂}}_{j}^{*}]} = {\max_{i} 𝓂_{i j} | i = 1, 2, \dots, m}$ (21) $ℳ^{−} = {ℳ_{1}^{−}, ℳ_{2}^{−}, \dots, ℳ_{n}^{−}} = {[{\underline{𝓂}}_{j}^{−}, {\bar{𝓂}}_{j}^{−}]} = {\max_{i} 𝓂_{i j} | i = 1, 2, \dots, m}$ (22)Step 16. Define PIV and NIV of possibilistic interval standard deviation. The PIV $S (D^{*})$ and NIV $(S D^{-})$ are determined by: $S D^{*} = {S D_{1}^{*}, S D_{2}^{*}, \dots, S D_{n}^{*}} = {[{\underline{𝓈 𝒹}}_{j}^{*}, {\bar{𝓈 𝒹}}_{j}^{*}]} = {\min_{i} 𝓈 𝒹_{i j} | i = 1, 2, \dots, m}$ (23) and $S D^{-} = {S D_{1}^{-}, S D_{2}^{-}, \dots, S D_{n}^{-}} = {[{\underline{𝓈 𝒹}}_{j}^{-}, {\bar{𝓈 𝒹}}_{j}^{-}]} = {\max_{i} 𝓈 𝒹_{i j} | i = 1, 2, \dots, m}$ (24)Step 17. Define PIV and NIV of possibilistic interval cube-root of skewness. The PIV $(C R S^{*})$ and NIV $(C R S^{-})$ are determined by: $C ℛ S^{*} = {C ℛ S_{1}^{*}, C ℛ S_{2}^{*}, \dots, C ℛ S_{n}^{*}} = {[{\underline{𝒸 𝓇 𝓈}}_{j}^{*}, {\bar{𝒸 𝓇 𝓈}}_{j}^{*}]} = {\min_{i} 𝒸 𝓇 𝓈_{i j} | i = 1, 2, \dots, m}$ (25) and $C ℛ S^{−} = {C ℛ S_{1}^{−}, C ℛ S_{2}^{−}, \dots, C ℛ S_{n}^{−}} = {[{\underline{𝒸 𝓇 𝓈}}_{j}^{−}, {\bar{𝒸 𝓇 𝓈}}_{j}^{−}]} = {\min_{i} 𝒸 𝓇 𝓈_{i j} | i = 1, 2, \dots, m}$ (26)Step 18. Calculate the separation measures matrix of each sustainable supplier risk according to possibilistic interval mean, standard deviation and cube-root of skewness from the PIV $(M^{*}$ , $S D^{*}$ , $C R S^{*})$ , respectively. $𝓂_{D_{i j}^{*}} (m_{i j}, ℳ_{j}^{*}) = \sqrt{{({\underline{𝓂}}_{j}^{*} - {\underline{𝓂}}_{i j})}^{2} + {({\bar{𝓂}}_{j}^{*} - {\bar{𝓂}}_{i j})}^{2}}$ (27) $s d_{D_{i j}^{*}} (s d_{i j}, S D_{j}^{*}) = \sqrt{{({\underline{𝓈 𝒹}}_{j}^{*} - {\underline{𝓈 𝒹}}_{i j})}^{2} + {({\bar{𝓈 𝒹}}_{j}^{*} - {\bar{𝓈 𝒹}}_{i j})}^{2}}$ (28) $𝒸 𝓇 𝓈_{D_{i j}^{*}} (𝒸 𝓇 𝓈_{i j}, C ℛ S_{j}^{*}) = \sqrt{{({\underline{𝒸 𝓇 𝓈}}_{j}^{*} - {\underline{𝒸 𝓇 𝓈}}_{i j})}^{2} + {({\bar{𝒸 𝓇 𝓈}}_{j}^{*} - {\bar{𝒸 𝓇 𝓈}}_{i j})}^{2}}$ (29)Step 19. Compute the separation measures matrix of each sustainable supplier risk according to possibilistic interval mean, standard deviation and cube-root of skewness number from the NIV $(M^{-}, S D^{-}, C R S^{-})$ , respectively. $𝓂_{D_{i j}^{-}} (𝓂_{i j}, ℳ_{j}^{-}) = \sqrt{{({\underline{𝓂}}_{j}^{-} - {\underline{𝓂}}_{i j})}^{2} + {({\bar{𝓂}}_{j}^{-} - {\bar{𝓂}}_{i j})}^{2}}$ (30) $𝓈 𝒹_{D_{i j}^{-}} (𝓈 𝒹_{i j}, S D_{j}^{-}) = \sqrt{{({\underline{𝓈 𝒹}}_{j}^{-} - {\underline{𝓈 𝒹}}_{i j})}^{2} + {({\bar{𝓈 𝒹}}_{j}^{-} - {\bar{𝓈 𝒹}}_{i j})}^{2}}$ (31) $𝒸 𝓇 𝓈_{D_{i j}^{-}} (𝒸 𝓇 𝓈_{i j}, C ℛ S_{j}^{-}) = \sqrt{{({\underline{𝒸 𝓇 𝓈}}_{j}^{-} - {\underline{𝒸 𝓇 𝓈}}_{i j})}^{2} + {({\bar{𝒸 𝓇 𝓈}}_{j}^{-} - {\bar{𝒸 𝓇 𝓈}}_{i j})}^{2}}$ (32)Step 20. Calculate proposed indices values of $η_{i}^{M +}$ , $η_{i}^{S D +}$ , $η_{i}^{C R S +}$ , $η_{i}^{M -}$ , $η_{i}^{S D -}$ and $η_{i}^{C R S -}$ for sustainable supplier risks according to mean, standard deviation, and cube-root of skewness (i = 1, 2, …, m). $η_{i}^{ℳ}^{+} = [\sum_{j = 1}^{n} {\underline{𝓌}}_{j} . 𝓂_{D_{i j}^{*}} (m_{i j}, ℳ_{j}^{*}), \sum_{j = 1}^{n} {\bar{𝓌}}_{j} . 𝓂_{D_{i j}^{*}} (m_{i j}, ℳ_{j}^{*})]$ (33) $η_{i}^{S D}^{+} = [\sum_{j = 1}^{n} {\underline{𝓌}}_{j} . 𝓈 𝒹_{D_{i j}^{*}} (𝓈 𝒹_{i j}, S D_{j}^{*}), \sum_{j = 1}^{n} {\bar{𝓌}}_{j} . 𝓈 𝒹_{D_{i j}^{*}} (𝓈 𝒹_{i j}, S D_{j}^{*})]$ (34) $η_{i}^{C ℛ S}^{+} = [\sum_{j = 1}^{n} {\underline{𝓌}}_{j} . 𝒸 𝓇 𝓈_{D_{i j}^{*}} (𝒸 𝓇 𝓈_{i j}, C ℛ S_{j}^{*}), \sum_{j = 1}^{n} {\bar{𝓌}}_{j} . 𝒸 𝓇 𝓈_{D_{i j}^{*}} (𝒸 𝓇 𝓈_{i j}, C ℛ S_{j}^{*})]$ (35) $η_{i}^{ℳ}^{-} = [\sum_{j = 1}^{n} {\underline{𝓌}}_{j} . 𝓂_{D_{i j}^{-}} (𝓂_{i j}, ℳ_{j}^{-}), \sum_{j = 1}^{n} {\bar{𝓌}}_{j} . 𝓂_{D_{i j}^{-}} (𝓂_{i j}, ℳ_{j}^{-})]$ (36) $η_{i}^{S D}^{-} = [\sum_{j = 1}^{n} {\underline{𝓂}}_{j} . 𝓈 𝒹_{D_{i j}^{-}} (𝓈 𝒹_{i j}, S D_{j}^{-}), \sum_{j = 1}^{n} {\bar{𝓂}}_{j} . 𝓈 𝒹_{D_{i j}^{-}} (𝓈 𝒹_{i j}, S D_{j}^{-})]$ (37) $η_{i}^{C ℛ S}^{-} = [\sum_{j = 1}^{n} {\underline{𝓂}}_{j} . 𝒸 𝓇 𝓈_{D_{i j}^{-}} (𝒸 𝓇 𝓈_{i j}, C ℛ S_{j}^{-}), \sum_{j = 1}^{n} {\bar{𝓂}}_{j} . 𝒸 𝓇 𝓈_{D_{i j}^{-}} (𝒸 𝓇 𝓈_{i j}, C ℛ S_{j}^{-})]$ (38)Step 21. Compute the proposed indices; α, β, and γ are within the range of 0 to 1. $ρ_{i} = α . η_{i}^{ℳ}^{+} + β . η_{i}^{S D}^{+} + γ . η_{i}^{C ℛ S}^{+}$ (39) $ϱ_{i} = (1 - α) . η_{i}^{M^{-}} + (1 - β) . η_{i}^{{SD}^{-}} + (1 - γ) . η_{i}^{{CRS}^{-}}$ (40)Step 22. Compute final proposed score (ℑ_i) of the risks for each sustainable supplier according to the preference order. The minimum value of the ℑ_i (risks score) demonstrates the better performance for the sustainable supplier (SS_l), θ within the range of 0 to 1. $ℑ_{i} = θ ρ_{i} + (1 - θ) ϱ_{i}$ (41)Step 23. Sustainable suppliers are ranked by increasing sorting of score (SS_l) values, l = 1, …, u, as below: $S c o r e (S S_{l}) = \sum_{i = 1}^{m} ℑ_{i}; \forall l$ (42)

3 Application of the proposed approach

To illuminate the proposed soft computing procedure, an application example is exhibited here for the SSSP. Selecting the suitable sustainable supplier is considered as the important choice issue. In the organization, six sustainable supplier candidates’ risks (R_i) are regarded for the long-term and mid-term planning, and a group of three managers from the organization (TM_k) are established for the SSSP with the least amount of risks according to three dimensions of the sustainability. TM₁ and TM₂ are optimistic; and TM₃ is pessimistic. In addition, the risk attitudes of the experts, including optimistic, neutral and pessimistic are regarded that lead to the asymmetric uncertainty information, and are described by linguistic variables. According to the literature, three principle assessment criteria of the FMEA, including O, S and D, are regarded for the GDM for the SSSP (C_j ; j = O, S, D). Risks of each sustainable supplier candidate as alternatives (R_i ; i = 1, 2, …, 12) in the decision matrixes are as follows:

Price performance value risk (R₁); On time delivery risk (R₂); Transportation cost risk (R₃); Organization commitment risk (R₄); Responsiveness risk (R₅); Rights of stakeholder’s risk (R₆); Work safety risk (R₇); Information disclosure risk (R₈); Environmental management system risk (R₉); Use of environment friendly technology risk (R₁₀); Pollution control initiatives risk (R₁₁); and Green design and recycling risk (R₁₂).

In this application example, the evaluation and selection problem of the sustainable supplier, linguistic variables are taken for the performance ratings of the sustainable suppliers’ risks according to three dimensions of the sustainability by Table 1. Then, the evaluation results about the weights and ratings that are transformed into IVF-numbers are given in Table 2 for the first sustainable supplier.

Table 1
Linguistic variables for values of sustainable suppliers’ risks

Linguistic variables Interval-valued fuzzy numbers

Very Poor (VP) [(0.00, 0.00, 1.00), (0.00, 0.00, 1.50)]

Poor (P) [(0.50, 1.00, 2.50), (0.00, 1.00, 3.50)]

Moderately Poor (MP) [(1.50, 3.00, 4.50), (0.00, 3.00, 5.50)]

Fair (F) [(3.50, 5.00, 6.50), (2.50, 5.00, 7.50)]

Moderately Good (MG) [(5.50, 7.00, 8.00), (4.50, 7.00, 9.50)]

Good (G) [(7.50, 9.00, 9.50), (5.50, 9.00, 10.00)]

Very Good (VG) [(9.50, 10.00, 10.00),(8.50, 10.00, 10.00)]

Linguistic variables	Interval-valued fuzzy numbers
Very Poor (VP)	[(0.00, 0.00, 1.00), (0.00, 0.00, 1.50)]
Poor (P)	[(0.50, 1.00, 2.50), (0.00, 1.00, 3.50)]
Moderately Poor (MP)	[(1.50, 3.00, 4.50), (0.00, 3.00, 5.50)]
Fair (F)	[(3.50, 5.00, 6.50), (2.50, 5.00, 7.50)]
Moderately Good (MG)	[(5.50, 7.00, 8.00), (4.50, 7.00, 9.50)]
Good (G)	[(7.50, 9.00, 9.50), (5.50, 9.00, 10.00)]
Very Good (VG)	[(9.50, 10.00, 10.00),(8.50, 10.00, 10.00)]

Table 2

Linguistic performance matrix of risks for sustainable supplier 1

Supplier A		Risk assessment
Sustainability dimensions	Sustainability-related risks	Occurrence			Severity			Ease of detection
Sustainability dimensions	Sustainability-related risks	TM ₁	TM ₂	TM ₃	TM ₁	TM ₂	TM ₃	TM ₁	TM ₂	TM ₃
Economics	(R₁)	G	MG	MG	G	MG	G	P	MP	P
	(R₂)	MG	MG	F	MG	F	F	P	MP	MP
	(R₃)	MG	MG	MG	MG	MG	F	MP	P	P
	(R₄)	MP	MP	MP	F	F	MG	MG	G	MG
	(R₅)	MG	G	MG	MG	MG	F	P	P	MP
Social	(R₆)	MG	MG	MG	G	MG	VG	P	MP	P
	(R₇)	MG	G	VG	MG	G	MG	MP	MP	P
	(R₈)	MG	G	F	MG	F	F	MP	P	MP
Environmental	(R₉)	MP	MP	P	MG	MG	MG	MG	G	G
	(R₁₀)	MG	MG	VG	MG	G	G	MP	P	P
	(R₁₁)	MG	F	G	MG	F	MG	MP	MP	MP
	(R₁₂)	MG	G	F	MG	MG	F	MP	MP	P

Table 3

Calculated TMs’ weights according to the soft computing approach (Steps 3 to 8)

Number of the supplier	Sustainable supplier (A)
Each TM’s weight	${\tilde{H}}_{1} = [(0.020, 0.032, 0.170), (0.018, 0.028, 0.199)]$
	${\tilde{H}}_{2} = [(0.020, 0.053, 0.158), (0.019, 0.089, 0.337)]$
	${\tilde{H}}_{3} = [(0.109, 0.126, 0.309), (0.071, 0.112, 0.336)]$
Number of the supplier	Sustainable supplier (B)
Each TM’s weight	${\tilde{H}}_{1} = [(0.018, 0.036, 0.180), (0.017, 0.100, 0.409)]$
	${\tilde{H}}_{2} = [(0.020, 0.058, 0.185), (0.020, 0.117, 0.418)]$
	${\tilde{H}}_{3} = [(0.115, 0.132, 0.286), (0.096, 0.146, 0.402)]$
Number of the supplier	Sustainable supplier (C)
Each TM’s weight	${\tilde{H}}_{1} = [(0.019, 0.036, 0.174), (0.019, 0.105, 0.376)]$
	${\tilde{H}}_{2} = [(0.020, 0.037, 0.159), (0.018, 0.065, 0.325)]$
	${\tilde{H}}_{3} = [(0.125, 0.107, 0.305), (0.101, 0.124, 0.385)]$

Calculated the weight of TMs from the IVF-matrixes can be transformed into normalized matrix of sustainable suppliers’ risks. The IVF-decision matrices of sustainable suppliers’ risks are provided by each expert and then are aggregated by Equation (12). Then, the possibilistic interval mean, standard deviation and cube-root of skewness matrixes for the SSSP are constructed. The possibilistic interval mean-entropy measure of each sustainable evaluation criterion are computed by Equations (19) and (20). Then, weights of these three evaluation criteria of risks are provided according to steps 3 to 8 for three sustainable suppliers as given in Table 3. PIV and NIV of possibilistic interval mean, and PIV and NIV of possibilistic interval standard deviation as well as the PIV and NIV of possibilistic interval cube-root of skewness are defined for the SSSP.

The separation measures of each sustainable supplier’s risks of possibilistic interval mean, standard deviation and cube-root of skewness from the PIV $(M^{*}$ , $S D^{*}$ , $C R S^{*})$ and from the NIV $(M^{-}, S D^{-}, C R S^{-})$ , are calculated, respectively. Finally, new separation measures are computed using Euclidean distance for the SSSP (α = β = γ = 0.7) . Computational results of the proposed indices and final values of ℑ_i for the sustainable suppliers are reported in Table 4.

Table 4

Proposed sustainable supplier ranking by the proposed model and IVF-SAW method

Indices	Sustainable supplier 1	Sustainable supplier 2	Sustainable supplier 3
Score (SS_l)	1.466	1.514	1.836
Sustainable supplier ranking	1	2	3
IVF-SAW method	18.20	11.25	9.86
Sustainable supplier ranking	1	2	3

In addition, results of the proposed soft computing approach with possibilistic interval mean, possibilistic interval standard deviation and possibilistic interval cube-root of skewness as well as the GDM process have been compared with IVF-SAW method and reported in Table 4. Comparing the proposed model with IVF-SAW method shows that the most suitable and the worst candidate sustainable suppliers are supplier 1 and supplier 3, respectively.

Finally, a sensitivity analysis on the TMs’ opinions has been reported in Table 5. Table 5 indicates the results of ranking according to ℑ_i values for each DM, individually. In fact, the application example is solved based on each DM’s decision matrix individually. Then, final reports of sustainable supplier ranking are presented in Table 5 for each DM (three conditions) and aggregated values of the DMs (i.e., main condition). The computational results indicate that opinions of the TMs or experts with asymmetric uncertainty information (i.e., optimistic and pessimistic attitudes) can effect on the final ranking of sustainable suppliers. Also, it illustrates the use of experts in the proposed new soft computing based on the GDM and interval-valued fuzzy sets with asymmetric uncertainty information.

Table 5

Sensitivity analysis on the TMs’ decision matrices

Conditions	Sustainable supplier 1	Sustainable supplier 2	Sustainable supplier 3	Sustainable supplier ranking
Main	1.466	1.514	1.836	1	2	3
TM₁	1.489	1.901	1.607	1	3	2
TM₂	1.662	1.477	1.902	2	1	3
TM₃	1.704	1.501	1.807	2	1	3

Some critical topics for the implementation of proposed soft computing approach with the GDM for the SSSP are reported as follows:

Firstly, the presented soft computing approach could be beneficial as combined in the assessment suitable alternative procedure in the SSCNs. Companies are profited with the group decision analysis model for reducing vulnerabilities of the SSSP. New measures as well as distinguish indices by considering the fuzzy possibilistic ideal positive and negative solutions could be properly employed for the SSSPs. Secondly, appraisement attributes of the SSSP could be generic, and minor adjustment could be employed satisfactorily for various SSCNs by determining the input parameters within the proposed group decision analysis. Thirdly, the decision structure provided in this study takes experts’ risk attitudes into account for the SSCNs. Introducing the risks in the computations could be satisfactorily performed by optimistic and pessimistic preferences of them.

4 Conclusions

Main objective of the sustainable supplier selection problem (SSSP) is to select the best alternative coordinating with sustainable capabilities of the company for sustainable supply chain networks (SSCNs). Suppliers can be taken account of the important sources of vulnerability. Proper supplier selection leads to build the sustainability and to reduce the risks in the SSCNs. This study introduced a new soft computing approach with failure mode and effects analysis (FMEA)’ idea for the SSSP with interval-valued fuzzy conditions. The soft computing approach was based on multi-attributes decision analysis, group decision making (GDM) and fuzzy possibilistic statistical modeling. The assessment, demonstrated by risks of economic, social and environmental measurements, depended on compromise programming ideas and matrices of possibilistic mean, standard deviation and cube-root of skewness. Moreover, a new weighting method of experts based on fuzzy possibilistic statistical ideas was presented in this study for processing the GDM with asymmetric uncertainty data. New definitions in the FMEA’ ideas were improved by fuzzy possibilistic statistical positive and negative ideal solutions. In addition, new relations of fuzzy evaluating and prioritizing of the risks were expanded. Finally, an application example was solved by the presented soft computing approach to exhibit its suitability by three measurements of the sustainability for the evaluation of sustainable suppliers. Moreover, a relative investigation was given with the IVF-SAW method from the literature for this SSSP, and the sensitivity analysis on the opinions of experts was reported. Indeed, computational results indicated that opinions of the experts with asymmetric uncertainty information affected on the final ranking of candidate sustainable suppliers.

Footnotes

Appendix

Fuzzy multi-attributes group decision making (FMAGDM): Fuzzy multi-attributes group decision making (FMAGDM) approach can take the most desirable candidate into account versus a set of evaluation factors; the information expressed by a group of experts is often uncertain and inexact (i.e., fuzzy) because of the high complexity of the conditions and the vagueness of inherent subjective nature of experts (e.g., [17 –24]). In recent years, some techniques were developed to handle FMAGDM problems by regarding the fuzzy sets theory under uncertainty with suitable outcomes (e.g., [25 –33]).

Possibility theory:Basic ideas and definitions about possibility theory are described (e.g., [34 –37]). Mean, variance and skewness of triangular fuzzy variable A = (a - τ, a, a + σ) are presented as [38]:

\renewcommand \theequation \arabic equationa

(1a) $E [A] = \frac{((a - τ) + 2 (a) + (a + σ))}{4}$ (2a) $V [A] = \frac{(33 α^{3} + 21 α^{2} β + 11 α β^{2} - β^{3})}{(384 α)}$

Where $α = max {((a) - (a - τ)), ((a + σ) - (a))}$ and $β = min {((a) - (a - τ)), ((a + σ) - (a))},$

and (3a) $\begin{matrix} S [A] = E [{(A - E [A])}^{3}] = \frac{{((a + σ) - (a - τ))}^{2}}{32} [((a + σ) - (a)) - ((a) - (a - τ))] . \end{matrix}$

Acknowledgments

The authors thank the anonymous referees for the valuable comments and recommendations that enhanced the primary version of this study.

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