Circular supplier selection using interval-valued intuitionistic fuzzy sets

Abstract

In attempts to foster sustainability, where the effective use of natural resources and renewable energy is important, the problem of circular supplier selection (CSS) has become a central issue. To address this problem, for the first time, we propose a group decision model based on an integrated Analytic hierarchy process (AHP) and Complex proportional assessment (COPRAS) methodology in an Interval-valued intuitionistic fuzzy-sets (IVIFS) environment for dealing with uncertainty that shapes decision makers’ judgements in solving the CSS problem. Based on experts’ opinions and a review of relevant literature in the field, three main categories of criteria are developed: economic, social, and circular. The main contributions of the study are, first, to develop specific CSS criteria, and second, to propose a novel integrated method for the CSS process using IVIFS with AHP and COPRAS. After using the AHP method to calculate the weights of individual criteria, we employ the COPRAS method, in an IVIFS environment, to rank potential circular suppliers, testing the validity of the proposed method via a case study involving a multinational cement company from Turkey. We also conduct a one at-a-time sensitivity analysis to demonstrate the effect of criteria weights on the results of the proposed approach, and compare the results of the approach with those obtained using other multi-criteria decision-making methods.

Appendix

This section presents some basic definitions of IFS and IVIFS.

Definition 1

Atanassov (1986) introduced an IFS, which is the generalisation of ordinary fuzzy sets (Zadeh, 1965). An IFS $A$ in X is denoted by Eq. (15).

$$\,A = \left\{ {\,\langle x,\,\mu_{A} (x),v_{A} (x)\rangle \,\left| {\,x \in X} \right.} \right\}$$

(15)

where $\mu_{A}(x):X\, \to [0,1]$ and $v_{A}(x):X\, \to {[0,1]}$ with the condition $0 \le \mu_{A} (x) + v_{A} (x) \le 1$ for $\forall x \in X$. The numbers $\mu_{A} (x)$ and $v_{A} (x)$ symbolise the membership function and non-membership function of the element x to set $A$, respectively. For each IFS $A$ in X, the hesitancy degree of $x \in X$ to set $A$ is defined as $\pi_{A} (x) = 1 - \mu_{A} (x) - v_{A} (x)$, and it is obvious that $0 \le \pi_{A} (x) \le 1,\,$ for $\forall x \in X$.

Definition 2

Let D[0,1] is defined as the set of all closed subintervals of the interval [0,1] and X (X≠0) is a universe of discourse. IVIFS $\tilde{A}$ in X is defined by Atanassov and Gargov, (1989) as in Eq. (16).

$$\,\tilde{A} = \left\{ {\langle x,\,\mu_{{\tilde{A}}} (x),v_{{\tilde{A}}} (x)\rangle \,\left| {\,x \in X} \right.} \right\}$$

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where $\mu_{{\tilde{A}}} (x):X\, \to D\,{[0,1]}$ and $v_{{\tilde{A}}} (x):X\, \to D\,{[0,1]}$ satisfy the condition $0 \le \sup \mu_{{\tilde{A}}}^{{}} (x) + \sup v_{{\tilde{A}}}^{{}} (x) \le 1$, for $\forall x \in X$. The intervals $\mu_{{\tilde{A}}} (x)$ and $v_{{\tilde{A}}} (x)$ are the membership function and non-membership function of the element x to set $\tilde{A}$, respectively. Thus, for each $x \in X,$$\mu_{{\tilde{A}}} (x)$ and $v_{{\tilde{A}}} (x)$ are defined as closed intervals and their lower and upper numbers can be displayed by $\mu_{{\tilde{A}}}^{ - } (x),\,\mu_{{\tilde{A}}}^{ + } (x),\,v_{{\tilde{A}}}^{ - } (x)$, and $v_{{\tilde{A}}}^{ + } (x)$, respectively. Thus, IVIFS $\tilde{A}$ is presented as in Eq. (17).

$$\,\tilde{A} = \left\{ {\,\langle x,\,{[}\mu_{{\tilde{A}}}^{ - } (x),\,\mu_{{\tilde{A}}}^{ + } (x){]},\,{[}v_{{\tilde{A}}}^{ - } (x),\,v_{{\tilde{A}}}^{ + } (x){]}\,\rangle \,\left| {\,x \in X} \right.} \right\}$$

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where $0 \le \mu_{{\tilde{A}}}^{ + } (x) + \,v_{{\tilde{A}}}^{ + } (x) \le 1,\,\mu_{{\tilde{A}}}^{ - } (x) \ge 0,\,v_{{\tilde{A}}}^{ - } (x) \ge 0$. For each element x, the hesitancy degree of an IVIFS of $x \in X$ in $\tilde{A}$ is defined as (Ye, 2012):

$$\pi_{{\tilde{A}}} (x) = \, \left( {\,{[}1 - \mu_{{\tilde{A}}} (x) - v_{{\tilde{A}}} (x){]}\,} \right) = \left( {\,{[}1 - \mu_{{\tilde{A}}}^{ + } (x) - v_{{\tilde{A}}}^{ + } (x){]}\,{,}\,{ [}1 - \mu_{{\tilde{A}}}^{ - } (x) - v_{{\tilde{A}}}^{ - } (x){]}\,} \right)$$

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In addition, let $\mu_{{\tilde{A}}} (x) = [\mu_{{\tilde{A}}}^{ - } , \, \mu_{{\tilde{A}}}^{ + } ]$ and $v_{{\tilde{A}}} (x) = [v_{{\tilde{A}}}^{ - } , \, v_{{\tilde{A}}}^{ + } ]$, and thus IVIFS $\tilde{A}$ is then denoted by

$$\tilde{A} = \left( {\,{[}\mu_{{\tilde{A}}}^{ - } ,\mu_{{\tilde{A}}}^{ + } {]}\,,\,\,{[}v_{{\tilde{A}}}^{ - } ,v_{{\tilde{A}}}^{ + } {]}\,} \right)$$

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Definition 3

The basic arithmetic expressions used for IVIFNs are as follows (Atanassov, 1994):

Let $\tilde{A} = \left( {\,{[}\mu_{{\tilde{A}}}^{ - } ,\mu_{{\tilde{A}}}^{ + } {]},\,\,{[}v_{{\tilde{A}}}^{ - } ,v_{{\tilde{A}}}^{ + } {]}\,} \right)$ and $\tilde{B} = \left( {\,{[}\mu_{{\tilde{B}}}^{ - } ,\mu_{{\tilde{B}}}^{ + } {]},\,\,{[}v_{{\tilde{B}}}^{ - } ,v_{{\tilde{B}}}^{ + } {]}\,} \right)$ be two IVIFS and $\lambda \ge 0$. Then,

$$\tilde{A} \le \tilde{B} \Leftrightarrow \mu_{{\tilde{A}}}^{ - } \le \mu_{{\tilde{B}}}^{ - } ,\,\mu_{{\tilde{A}}}^{ + } \le \mu_{{\tilde{B}}}^{ + } ,\,v_{\dddot B}^{ - } \le v_{{\tilde{A}}}^{ - } ,\,v_{\dddot B}^{ + } \le v_{{\tilde{A}}}^{ + }$$

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$$\tilde{A} + \tilde{B} = \left( {\,\left[ {\,\mu_{{\tilde{A}}}^{ - } + \mu_{{\tilde{B}}}^{ - } - \,\mu_{{\tilde{A}}}^{ - } \mu_{{\tilde{B}}}^{ - } ,\,\mu_{{\tilde{A}}}^{ + } + \mu_{{\tilde{B}}}^{ + } - \,\mu_{{\tilde{A}}}^{ + } \mu_{{\tilde{B}}}^{ + } ]\,,\,[v_{{\tilde{A}}}^{ - } v_{{\tilde{B}}}^{ - } ,\,v_{{\tilde{A}}}^{ + } v_{{\tilde{B}}}^{ + } } \right]\,} \right)$$

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$$\tilde{A} - \tilde{B} = \left( {\left[ {\frac{{\mu_{{\tilde{A}}}^{ - } + \mu_{{\tilde{B}}}^{ - } }}{{1 - \mu_{{\tilde{B}}}^{ - } }},\frac{{\mu_{{\tilde{A}}}^{ + } + \mu_{{\tilde{B}}}^{ + } }}{{1 - \mu_{{\tilde{B}}}^{ + } }}} \right]\,,\,\left[ {\frac{{v_{{\tilde{A}}}^{ - } }}{{v_{{\tilde{B}}}^{ - } }},\frac{{v_{{\tilde{A}}}^{ + } }}{{v_{{\tilde{B}}}^{ + } }}} \right]} \right)$$

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$$\tilde{A} \times \tilde{B} = \left( {\left[ {\mu_{{\tilde{A}}}^{ - } \mu_{{\tilde{B}}}^{ - } ,\mu_{{\tilde{A}}}^{ + } \mu_{{\tilde{B}}}^{ + } } \right],\left[ {v_{{\tilde{A}}}^{ - } + v_{{\tilde{B}}}^{ - } - v_{{\tilde{A}}}^{ - } v_{{\tilde{B}}}^{ - } ,v_{{\tilde{A}}}^{ + } + v_{{\tilde{B}}}^{ + } - v_{{\tilde{A}}}^{ + } v_{{\tilde{B}}}^{ + } } \right] } \right)$$

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$$\tilde{A}\ / \;\tilde{B} = \left( {\left[ {\frac{{\mu_{{\tilde{A}}}^{ - } }}{{\mu_{{\tilde{B}}}^{ - } }},\frac{{\mu_{{\tilde{A}}}^{ + } }}{{\mu_{{\tilde{B}}}^{ + } }}} \right]\,,\,\left[ {\frac{{v_{{\tilde{A}}}^{ - } - v_{{\tilde{B}}}^{ - } }}{{1 - v_{{\tilde{B}}}^{ - } }},\frac{{v_{{\tilde{A}}}^{ + } - v_{{\tilde{B}}}^{ + } }}{{1 - v_{{\tilde{B}}}^{ + } }}} \right]} \right)$$

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$$\lambda\tilde{A}=\left[1-\left(1-\mu_{\tilde{A}}^{-}\right)^{\lambda},1-\left(1-\mu_{\tilde{A}}^{+}\right)^{\lambda}\right],\left[v_{{\tilde{A}}}^{-\lambda},v_{{\tilde{A}}}^{+\lambda}\right]$$

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$$\tilde{A}^{\lambda } = \left[ {(\mu_{{\tilde{A}}}^{ - } )^{\lambda } \,,\,(\mu_{{\tilde{A}}}^{ + } )^{\lambda } } \right]\,,\,\left[ {1 - (1 - v_{{\tilde{A}}}^{ - } )^{\lambda } \,,\,1 - (1 - v_{{\tilde{A}}}^{ + } )^{\lambda } } \right]$$

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Definition 4

Let $\tilde{A} = \left( {\,{[}\mu_{{\tilde{A}}}^{ - } ,\mu_{{\tilde{A}}}^{ + } {]},\,\,{[}v_{{\tilde{A}}}^{ - } ,v_{{\tilde{A}}}^{ + } {]}\,} \right)$ be an IVIFN. $\tilde{A}$ is defuzzified using Eq. (27) (Oztaysi et al., 2017).

$$A = 1 - \frac{{\mu_{{\tilde{A}}}^{ - } + \mu_{{\tilde{A}}}^{ + } + (1 - v_{{\tilde{A}}}^{ - } ) + (1 - v_{{\tilde{A}}}^{ + } ) + \mu_{{\tilde{A}}}^{ - } * \mu_{{\tilde{A}}}^{ + } - \sqrt {(1 - v_{{\tilde{A}}}^{ - } ) * (1 - v_{{\tilde{A}}}^{ + } )} }}{4}$$

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Definition 5

The score function $S(\tilde{A})$ and accuracy function $H(\tilde{A})$ of an IVIFN can be defined as follows (Wu & Chiclana, 2012):

$$S(\tilde{A}) = \frac{{\mu_{{\tilde{A}}}^{ - } + \mu_{{\tilde{A}}}^{ + } - v_{{\tilde{A}}}^{ - } - v_{{\tilde{A}}}^{ + } }}{2}$$

(28)

$$H(\tilde{A}) = \frac{{\mu_{{\tilde{A}}}^{ - } + \mu_{{\tilde{A}}}^{ + } + v_{{\tilde{A}}}^{ - } + v_{{\tilde{A}}}^{ + } }}{2}$$

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Definition 6

Interval-valued intuitionistic fuzzy weighted averaging (IVIFWA) operator is used to aggregate DMs’ opinions on criteria and alternatives. For the set of all IVIFNs, $\tilde{\alpha }_{i} = \left( {\,{[}\mu_{i}^{ - } ,\mu_{i}^{ + } {]}\,,\,\,{[}v_{i}^{ - } ,v_{i}^{ + } {]}\,} \right),\;i = (1,2, \ldots ,n)$ is a collection of IVIFNs and DMs’ weights $w{}_{i} = (w_{1} ,w_{2} , \ldots ,w_{n} )^{T} ,w_{i} \in {[0,1]}\,,\,\sum\nolimits_{i = 1}^{n} {w{}_{i}} = 1$. Thus, IVIFWA operator can be extended to include w, and given as (Xu & Cai, 2009):

$$IVIFWA_{w} (\tilde{\alpha }_{1} ,\tilde{\alpha }_{2} , \ldots ,\tilde{\alpha }_{n} ) = \left[ {1 - \prod\limits_{i = 1}^{n} {(1 - \mu_{i}^{ - } )^{{w_{i} }} } ,\,\,1 - \prod\limits_{i = 1}^{n} {(1 - \mu_{i}^{ + } )^{{w_{i} }} } } \right]\,,\,\left[ {\prod\limits_{i = 1}^{n} {(v_{i}^{ - } )^{{w_{i} }} } ,\,\,\prod\limits_{i = 1}^{n} {(v_{i}^{ + } )^{{w_{i} }} } } \right]$$

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