# Multi-Criteria Group Decision-Making for Selection of Green Suppliers under Bipolar Fuzzy PROMETHEE Process

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## Abstract

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## 1. Introduction

- The PROMETHEE method is extended to a bipolar fuzzy PROMETHEE method to deal with the double-sided information of human reasoning. More generally, trapezoidal bipolar fuzzy numbers are used to obtain more accurate results.
- The personal interest or influence of decision makers towards the criteria is minimized by using the Shannon entropy weighting technique to calculate the normalized weights of criteria.
- The partial and complete ranking of alternatives are determined by applying the PROMETHEE I and PROMETHEE II, respectively.
- A numerical example for the selection of green suppliers is presented that shows the validity and authenticity of the proposed method.

## 2. Methodology of the Bipolar Fuzzy PROMETHEE Method

**Definition**

**1**

**.**Consider a non-empty universe of discourse X. A bipolar fuzzy set $\tilde{\mathbb{B}}$ on X is defined as

**Definition**

**2**

**.**A bipolar fuzzy number $\tilde{\mathbb{B}}$ is defined on real line R having the form

**Definition**

**3**

**.**A bipolar fuzzy number $\tilde{\mathbb{B}}=\u2329Y,Z\u232a=\u2329[{\vartheta}_{1},{\vartheta}_{2},{\vartheta}_{3},{\vartheta}_{4}],[{\delta}_{1},{\delta}_{2},{\delta}_{3},{\delta}_{4}]\u232a$ is called the trapezoidal bipolar fuzzy number, denoted by $\u2329({\vartheta}_{1},{\vartheta}_{2},{\vartheta}_{3},{\vartheta}_{4}),({\delta}_{1},{\delta}_{2},{\delta}_{3},{\delta}_{4})\u232a$, if its satisfaction and dissatisfaction degrees are described as follows:

**Definition**

**4**

**.**Consider a trapezoidal bipolar fuzzy number $\tilde{\mathbb{B}}=\u2329({\vartheta}_{1},{\vartheta}_{2},{\vartheta}_{3},{\vartheta}_{4}),({\delta}_{1},{\delta}_{2},{\delta}_{3},{\delta}_{4})\u232a$, which can be converted into a real number by applying the ranking function as:

#### 2.1. Procedure of Bipolar Fuzzy PROMETHEE Method

**Step****1.**- Identify the linguistic variables.Linguistic variables are used by decision-makers to determine the ratings of an alternative with respect to different criteria. It is most important to identify the relevant and appropriate set of linguistic variables and define their respective values. In this method, a set of seven linguistic variables in the form of trapezoidal bipolar fuzzy numbers are considered and shown in Figure 1. The values of these trapezoidal bipolar fuzzy numbers are taken from the numerical domain [0,1].
**Step****2.**- Construct a decision matrix.Suppose that the alternatives ${\mathbb{S}}_{\alpha}$ are evaluated on the basis of ${\mathcal{Q}}_{\beta}$ conflicting criteria which are assessed by every decision maker ${\mathcal{D}}_{\phi};\phantom{\rule{3.33333pt}{0ex}}\phi =1,2,\cdots ,r$. Then, r decision matrices are constructed containing the rating values of linguistic variables given by r decision-makers in the following manner:$$\mathcal{L}={\left[{\ell}_{\alpha \beta}^{\phi}\right]}_{p\times q}=\left[\begin{array}{cccc}{\ell}_{11}^{\phi}& {\ell}_{12}^{\phi}& \cdots & {\ell}_{1q}^{\phi}\\ {\ell}_{21}^{\phi}& {\ell}_{22}^{\phi}& \cdots & {\ell}_{2q}^{\phi}\\ .& .& .& .\\ .& .& .& .\\ .& .& .& .\\ {\ell}_{p1}^{\phi}& {\ell}_{p2}^{\phi}& ...& {\ell}_{pq}^{\phi}\end{array}\right],$$$$\begin{array}{c}\begin{array}{c}{\vartheta}_{\alpha \beta 1}=\frac{1}{\phi}\underset{\phi =1}{\sum ^{r}}{\vartheta}_{\alpha \beta 1}^{\phi},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\vartheta}_{\alpha \beta 2}=\frac{1}{\phi}\underset{\phi =1}{\sum ^{r}}{\vartheta}_{\alpha \beta 2}^{\phi},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\vartheta}_{\alpha \beta 3}=\frac{1}{\phi}\underset{\phi =1}{\sum ^{r}}{\vartheta}_{\alpha \beta 3}^{\phi},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\vartheta}_{\alpha \beta 4}=\frac{1}{\phi}\underset{\phi =1}{\sum ^{r}}{\vartheta}_{\alpha \beta 4}^{\phi},\hfill \\ {\delta}_{\alpha \beta 1}=\frac{1}{\phi}\underset{\phi =1}{\sum ^{r}}{\delta}_{\alpha \beta 1}^{\phi},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\delta}_{\alpha \beta 2}=\frac{1}{\phi}\underset{\phi =1}{\sum ^{r}}{\delta}_{\alpha \beta 2}^{\phi},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\delta}_{\alpha \beta 3}=\frac{1}{\phi}\underset{\phi =1}{\sum ^{r}}{\delta}_{\alpha \beta 3}^{\phi},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\delta}_{\alpha \beta 4}=\frac{1}{\phi}\underset{\phi =1}{\sum ^{r}}{\delta}_{\alpha \beta 4}^{\phi}.\hfill \end{array}\hfill \end{array}$$These aggregated values are used to construct an aggregated decision matrix as follows:$$\mathcal{L}={\left[{\ell}_{\alpha \beta}\right]}_{p\times q}=\left[\begin{array}{cccc}{\ell}_{11}& {\ell}_{12}& \cdots & {\ell}_{1q}\\ {\ell}_{21}& {\ell}_{22}& \cdots & {\ell}_{2q}\\ \vdots & \vdots & \vdots & \vdots \\ {\ell}_{p1}& {\ell}_{p2}& \cdots & {\ell}_{pq}\end{array}\right].$$
**Step****3.**- Rank the bipolar fuzzy numbers.The bipolar fuzzy numbers of aggregated values are then converted into the crisp values of real numbers by using the ranking function of bipolar fuzzy numbers as follows:$$\begin{array}{c}\begin{array}{cc}{t}_{\alpha \beta}=\hfill & \left(\left[\frac{{\vartheta}_{\alpha \beta 1}+{\vartheta}_{\alpha \beta 2}+{\vartheta}_{\alpha \beta 3}+{\vartheta}_{\alpha \beta 4}}{4}\right]+\left[\frac{-{\vartheta}_{\alpha \beta 1}-{\vartheta}_{\alpha \beta 2}+{\vartheta}_{\alpha \beta 3}+{\vartheta}_{\alpha \beta 4}}{2}\right]\right)-\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & \left(\left[\frac{{\delta}_{\alpha \beta 1}+{\delta}_{\alpha \beta 2}+{\delta}_{\alpha \beta 3}+{\delta}_{\alpha \beta 4}}{4}\right]+\left[\frac{-{\delta}_{\alpha \beta 1}-{\delta}_{\alpha \beta 2}+{\delta}_{\alpha \beta 3}+{\delta}_{\alpha \beta 4}}{2}\right]\right),\hfill \end{array}\hfill \end{array}$$
**Step****4.**- Determine the deviation by pairwise comparison.The deviation of alternatives is computed by the pairwise comparison of alternatives on the basis of criteria ${\mathcal{Q}}_{\beta}$ by using the following expression:$${d}_{\beta}({\mathbb{S}}_{\alpha},{\mathbb{S}}_{\sigma})={t}_{\beta}\left({\mathbb{S}}_{\alpha}\right)-{t}_{\beta}\left({\mathbb{S}}_{\sigma}\right),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\alpha ,\sigma =1,2,\cdots ,p,$$
**Step****5.**- Define the preference function.A preference function ${P}_{\beta}({\mathbb{S}}_{\alpha},{\mathbb{S}}_{\sigma})={F}_{\beta}\left[{d}_{\beta}({\mathbb{S}}_{\alpha},{\mathbb{S}}_{\sigma})\right]$ is defined to evaluate the preference of alternative ${\mathbb{S}}_{\alpha}$ regarding alternative ${\mathbb{S}}_{\sigma}$ on the basis of each criterion and has a value ranging from 0 to 1. If the value of the preference function is zero or negative then there is the indifference of the decision maker between the alternatives with respect to that criterion. On the other hand, a value closer to 1 shows a greater preference. This preference function represents the intensity of preference of an alternative ${\mathbb{S}}_{\alpha}$ over another alternative ${\mathbb{S}}_{\sigma}$ and is categorized as follows:
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- ${P}_{\beta}({\mathbb{S}}_{\alpha},{\mathbb{S}}_{\sigma})=0$ shows an indifference between ${\mathbb{S}}_{\alpha}$ and ${\mathbb{S}}_{\sigma}$, or no preference of ${\mathbb{S}}_{\alpha}$ over ${\mathbb{S}}_{\sigma}$;
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- ${P}_{\beta}({\mathbb{S}}_{\alpha},{\mathbb{S}}_{\sigma})\sim 0$ represents a weak preference of ${\mathbb{S}}_{\alpha}$ over ${\mathbb{S}}_{\sigma}$;
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- ${P}_{\beta}({\mathbb{S}}_{\alpha},{\mathbb{S}}_{\sigma})\sim 1$ represents a strong preference of ${\mathbb{S}}_{\alpha}$ over ${\mathbb{S}}_{\sigma}$;
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- ${P}_{\beta}({\mathbb{S}}_{\alpha},{\mathbb{S}}_{\sigma})=1$ shows a strict preference of ${\mathbb{S}}_{\alpha}$ over ${\mathbb{S}}_{\sigma}$.

**Step****6.**- Calculate the normalized weights.The weight value of each criterion shows the relative importance of that criterion towards the other criteria of that problem. These weight values may be completely or partially unknown for decision makers, and can be calculated by using various techniques or methods. If all the criteria have the same importance for a decision maker, then all weights can be assigned equal value. In this methodology, the entropy weight measuring information is used to enumerate the normalized weights of conflicting criteria. In order to calculate the weights by entropy measure, first we should normalize the decision values of each criterion ${\mathcal{Q}}_{\beta}\phantom{\rule{3.33333pt}{0ex}}(\beta =1,2,\cdots ,q)$ and obtain the projection values $\mathcal{P}\left(\alpha \beta \right)$ of criteria as follows:$$\mathcal{P}\left(\alpha \beta \right)=\frac{{t}_{\alpha \beta}}{{\displaystyle \sum _{\alpha =1}^{p}}{t}_{\alpha \beta}}.$$These projection values are then used to calculate the entropy value $\mathcal{E}\left(\beta \right)$ for each criterion as follows:$$\mathcal{E}\left(\beta \right)=-c\underset{\alpha =1}{\sum ^{p}}\mathcal{P}\left(\alpha \beta \right)log\left(\mathcal{P}\left(\alpha \beta \right)\right),$$$$div\left(\beta \right)=1-\mathcal{E}\left(\beta \right),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\beta =1,2,\cdots ,q.$$The divergence value $div\left(\beta \right)$ denotes the inherent contrast intensity of criteria ${\mathcal{Q}}_{\beta}$. The higher value of $div\left(\beta \right)$ shows that the criterion ${\mathcal{Q}}_{\beta}$ is considered as more important for that problem. Then, the weights of criteria are calculated as:$$w\left(\beta \right)=\frac{div\left(\beta \right)}{{\displaystyle \sum _{\beta =1}^{q}}div\left(\beta \right)},$$
**Step****7.**- Determine the multi-criteria preference index.When a preference function and weight is assigned to each criterion by a decision maker for the considered problem, then the multi-criteria preference index of alternatives is determined. The multi-criteria preference index ∏ is calculated as the weighted average of the preference functions ${P}_{\beta}$:$$\prod ({\mathbb{S}}_{\alpha},{\mathbb{S}}_{\sigma})=\frac{{\displaystyle \sum _{\beta =1}^{q}}w\left(\beta \right){P}_{\beta}({\mathbb{S}}_{\alpha},{\mathbb{S}}_{\sigma})}{{\displaystyle \sum _{\beta =1}^{q}}w\left(\beta \right)};\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\alpha \ne \sigma ,\phantom{\rule{3.33333pt}{0ex}}\alpha ,\sigma =1,2,\cdots ,p.$$Since the normalized weights are used in this method, Equation (12) is reduced as follows:$$\prod ({\mathbb{S}}_{\alpha},{\mathbb{S}}_{\sigma})={\displaystyle \sum _{\beta =1}^{q}}w\left(\beta \right){P}_{\beta}({\mathbb{S}}_{\alpha},{\mathbb{S}}_{\sigma});\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\alpha \ne \sigma ,\phantom{\rule{3.33333pt}{0ex}}\alpha ,\sigma =1,2,\cdots ,p.$$The multi-criteria preference index ∏ has a value between 0 and 1, such that
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- $\prod ({\mathbb{S}}_{\alpha},{\mathbb{S}}_{\sigma})\approx 0$ represents the weak preference of alternative ${\mathbb{S}}_{\alpha}$ over ${\mathbb{S}}_{\sigma}$ with respect to all criteria;
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- $\prod ({\mathbb{S}}_{\alpha},{\mathbb{S}}_{\sigma})\approx 1$ represents the strong preference of alternative ${\mathbb{S}}_{\alpha}$ over ${\mathbb{S}}_{\sigma}$ with respect to all criteria.

This preference index induces an outranking relation on the set $\mathbb{S}$ of alternatives which is further represented by an outranking graph. The nodes of this outranking graph are the alternatives and, between any two nodes ${\mathbb{S}}_{\alpha}$ and ${\mathbb{S}}_{\sigma}$, there are two arcs with values $\prod ({\mathbb{S}}_{\alpha},{\mathbb{S}}_{\sigma})$, and $\prod ({\mathbb{S}}_{\sigma},{\mathbb{S}}_{\alpha}),$ which have no particular relation. **Step****8.**- Find the preference order.The outranking relation is then used to obtain the ranking of alternatives, which may be partial or complete. The alternatives are ranked partially by using the PROMETHEE I, whereas complete ranking can be obtain by proceeding one more step of PROMETHEE II.
- (i)
- Ordering the alternatives by partial ranking or PROMETHEE I.For each alternative ${\mathbb{S}}_{\alpha}$ in the outranking graph, the leaving or outgoing flow is defined as:$${\xi}^{+}\left({\mathbb{S}}_{\alpha}\right)=\frac{1}{p-1}\sum _{{\mathbb{S}}_{\sigma}\in \mathbb{S}}\prod ({\mathbb{S}}_{\alpha},{\mathbb{S}}_{\sigma});\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\alpha \ne \sigma ,\phantom{\rule{3.33333pt}{0ex}}\alpha ,\sigma =1,2,\cdots ,p,$$Similarly, for each alternative ${\mathbb{S}}_{\alpha}$ in outranking graph, the entering or incoming flow is defined as:$${\xi}^{-}\left({\mathbb{S}}_{\alpha}\right)=\frac{1}{p-1}\sum _{{\mathbb{S}}_{\sigma}\in \mathbb{S}}\prod ({\mathbb{S}}_{\sigma},{\mathbb{S}}_{\alpha});\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\alpha \ne \sigma ,\phantom{\rule{3.33333pt}{0ex}}\alpha ,\sigma =1,2,\cdots ,p,$$The alternative which has the greater value of ${\xi}^{+}\left({\mathbb{S}}_{\alpha}\right)$ and the lower value of ${\xi}^{-}\left({\mathbb{S}}_{\alpha}\right)$ is chosen as the most suitable alternative. The outgoing and incoming flows determine the preferences as given in Equations (12) and (13), respectively.$$\begin{array}{c}\left\{\begin{array}{cc}{\mathbb{S}}_{\alpha}{P}^{+}{\mathbb{S}}_{\sigma}\hfill & \iff {\xi}^{+}\left({\mathbb{S}}_{\alpha}\right)>{\xi}^{+}\left({\mathbb{S}}_{\sigma}\right);\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall {\mathbb{S}}_{\alpha},{\mathbb{S}}_{\sigma}\in \mathbb{S},\hfill \\ {\mathbb{S}}_{\alpha}{I}^{+}{\mathbb{S}}_{\sigma}\hfill & \iff {\xi}^{+}\left({\mathbb{S}}_{\alpha}\right)={\xi}^{+}\left({\mathbb{S}}_{\sigma}\right);\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall {\mathbb{S}}_{\alpha},{\mathbb{S}}_{\sigma}\in \mathbb{S},\hfill \end{array}\right.\hfill \end{array}$$$$\begin{array}{c}\left\{\begin{array}{cc}{\mathbb{S}}_{\alpha}{P}^{-}{\mathbb{S}}_{\sigma}\hfill & \iff {\xi}^{-}\left({\mathbb{S}}_{\alpha}\right)<{\xi}^{-}\left({\mathbb{S}}_{\sigma}\right);\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall {\mathbb{S}}_{\alpha},{\mathbb{S}}_{\sigma}\in \mathbb{S},\hfill \\ {\mathbb{S}}_{\alpha}{I}^{-}{\mathbb{S}}_{\sigma}\hfill & \iff {\xi}^{-}\left({\mathbb{S}}_{\alpha}\right)={\xi}^{-}\left({\mathbb{S}}_{\sigma}\right);\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall {\mathbb{S}}_{\alpha},{\mathbb{S}}_{\sigma}\in \mathbb{S}.\hfill \end{array}\right.\hfill \end{array}$$The PROMETHEE I partial ordering $({P}_{1},{I}_{1},{R}_{1})$ is then obtained by taking the intersection of the two previously mentioned preferences as:$$\begin{array}{c}\left\{\begin{array}{cc}{\mathbb{S}}_{\alpha}{P}_{1}{\mathbb{S}}_{\sigma}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\left({\mathbb{S}}_{\alpha}\phantom{\rule{3.33333pt}{0ex}}\mathrm{outranks}\phantom{\rule{3.33333pt}{0ex}}{\mathbb{S}}_{\sigma}\right)\phantom{\rule{3.33333pt}{0ex}}\hfill & \mathrm{if}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\mathbb{S}}_{\alpha}{P}^{+}{\mathbb{S}}_{\sigma}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}{\mathbb{S}}_{\alpha}{P}^{-}{\mathbb{S}}_{\sigma},\hfill \\ \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\hfill & \mathrm{or}\phantom{\rule{3.33333pt}{0ex}}{\mathbb{S}}_{\alpha}{P}^{+}{\mathbb{S}}_{\sigma}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}{\mathbb{S}}_{\alpha}{I}^{-}{\mathbb{S}}_{\sigma},\hfill \\ \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\hfill & \mathrm{or}\phantom{\rule{3.33333pt}{0ex}}{\mathbb{S}}_{\alpha}{I}^{+}{\mathbb{S}}_{\sigma}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}{\mathbb{S}}_{\alpha}{P}^{-}{\mathbb{S}}_{\sigma};\hfill \\ {\mathbb{S}}_{\alpha}{I}_{1}{\mathbb{S}}_{\sigma}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\left({\mathbb{S}}_{\alpha}\phantom{\rule{3.33333pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{indifferent}\phantom{\rule{4.pt}{0ex}}\mathrm{to}\phantom{\rule{3.33333pt}{0ex}}{\mathbb{S}}_{\sigma}\right)\hfill & \mathrm{iff}\phantom{\rule{3.33333pt}{0ex}}{\mathbb{S}}_{\alpha}{I}^{+}{\mathbb{S}}_{\sigma}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}{\mathbb{S}}_{\alpha}{I}^{-}{\mathbb{S}}_{\sigma};\hfill \\ {\mathbb{S}}_{\alpha}{R}_{1}{\mathbb{S}}_{\sigma}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\left({\mathbb{S}}_{\alpha}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}{\mathbb{S}}_{\sigma}\phantom{\rule{3.33333pt}{0ex}}\mathrm{are}\phantom{\rule{4.pt}{0ex}}\mathrm{incomparable}\right)\phantom{\rule{3.33333pt}{0ex}}\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\hfill \end{array}$$Since all the alternatives are not comparable in PROMETHEE I, the computation of the net outranking flow of alternatives is as follows.
- (ii)
- Ordering the alternatives by complete ranking or PROMETHEE II.The net flow of each alternative is the difference of outgoing and incoming flows, which is computed as:$$\xi \left({\mathbb{S}}_{\alpha}\right)={\xi}^{+}\left({\mathbb{S}}_{\alpha}\right)-{\xi}^{-}\left({\mathbb{S}}_{\alpha}\right).$$The net flow provides the complete ordering of alternatives by avoiding any incomparability, the PROMETHEE II complete ranking $({P}_{2},{I}_{2})$ is given in Equation (16).$$\begin{array}{c}\left\{\begin{array}{cc}{\mathbb{S}}_{\alpha}{P}_{2}{\mathbb{S}}_{\sigma}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\left({\mathbb{S}}_{\alpha}\phantom{\rule{3.33333pt}{0ex}}\mathrm{outranks}\phantom{\rule{3.33333pt}{0ex}}{\mathbb{S}}_{\sigma}\right)\phantom{\rule{3.33333pt}{0ex}}\hfill & \mathrm{iff}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\xi \left({\mathbb{S}}_{\alpha}\right)>\xi \left({\mathbb{S}}_{\sigma}\right),\hfill \\ {\mathbb{S}}_{\alpha}{I}_{2}{\mathbb{S}}_{\sigma}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\left({\mathbb{S}}_{\alpha}\phantom{\rule{3.33333pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{indifferent}\phantom{\rule{4.pt}{0ex}}\mathrm{to}\phantom{\rule{3.33333pt}{0ex}}{\mathbb{S}}_{\sigma}\right)\phantom{\rule{3.33333pt}{0ex}}\hfill & \mathrm{iff}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\xi \left({\mathbb{S}}_{\alpha}\right)=\xi \left({\mathbb{S}}_{\sigma}\right).\hfill \end{array}\right.\hfill \end{array}$$Thus, all the alternatives can be compared on the basis of net flows $\xi \left({\mathbb{S}}_{\alpha}\right)$. The alternative with maximum net flow is observed as the most suitable alternative.The framework of the procedure of the bipolar fuzzy PROMETHEE method is provided in Figure 4. This framework consists of the goal of the selection procedure, the environmental and economical criteria, the alternatives for evaluation, the preference functions, and the net outranking flows of PROMETHEE I and PROMETHEE II in the form of partial and complete ranking, respectively.

This multi-criteria outranking approach is based on a series of computations, in which all steps remain the same other than the definition of the preference function and the computation of normalized weights. The normalized weights can be calculated by using an appropriate method according to the choice of decision values or the preference of the decision makers. The preference function defined in Step 5 is an irrational choice of preference function which depends on the nature of the criteria or the desire of the decision-makers. The choice of types of preference function is very important as it may change the net outranking flow or the ranking of alternatives.

#### 2.2. Preference Function

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

## 3. Green Supplier Selection

- ${\mathbb{S}}_{1}=$ MVG Food Marketing Sdn Bhd;
- ${\mathbb{S}}_{2}=$ CF Org Noodle Sdn Bhd;
- ${\mathbb{S}}_{3}=$ Hexa Food Sdn Bhd;
- ${\mathbb{S}}_{4}=$ SCS Food Manufacturing Sdn Bhd.

- ${\mathcal{Q}}_{1}=$ Cost of products (consists of transportation, purchasing, inventory, maintenance, holding, security etc.);
- ${\mathcal{Q}}_{2}=$ Quality of products (indicated by the principles, techniques, and practices of companies);
- ${\mathcal{Q}}_{3}=$ Service provided (low costs, high productivity, quick response, minimum wastage, no damage, etc.);
- ${\mathcal{Q}}_{4}=$ Delivery (at the correct time, at the right place, and in good condition);
- ${\mathcal{Q}}_{5}=$ Pollution control (an important criterion, as pollution is obtained as a byproduct of energy use in the production procedures);
- ${\mathcal{Q}}_{6}=$ Environmental management system (the environmental dimension has been recently added in assessment procedures);
- ${\mathcal{Q}}_{7}=$ Green packaging (a type of packaging which aims protect the environment by using environmentally friendly material).

**Step****1.**- The group of decision-makers decided to use linguistic variables to rate the alternatives with respect to different criteria for evaluation. A set of seven linguistic variables $\{Very\phantom{\rule{3.33333pt}{0ex}}good,\phantom{\rule{3.33333pt}{0ex}}Good,\phantom{\rule{3.33333pt}{0ex}}Medium\phantom{\rule{3.33333pt}{0ex}}good,\phantom{\rule{3.33333pt}{0ex}}Fair,\phantom{\rule{3.33333pt}{0ex}}Medium\phantom{\rule{3.33333pt}{0ex}}poor,\phantom{\rule{3.33333pt}{0ex}}Poor,\phantom{\rule{3.33333pt}{0ex}}Very\phantom{\rule{3.33333pt}{0ex}}poor\}$ is presented in Figure 1. These linguistic variables and their corresponding bipolar fuzzy numbers are given in Table 1.
**Step****2.**- The preference ratings of alternatives with respect to conflicting criteria given by decisions makers in the form of linguistic terms are shown in Table 2.The rating values of these linguistic variables in the form of trapezoidal bipolar fuzzy numbers were used as defined in Table 1, and the results are given in Table 3. The aggregated decision values of these trapezoidal bipolar fuzzy numbers were computed by employing the Equation (1), and an aggregated decision matrix was constructed as shown in Table 4.For example, the aggregated decision value of supplier ${\mathbb{S}}_{1}$ with respect to criterion ${\mathcal{Q}}_{1}$ was computed by arithmetic mean as,$$\begin{array}{cccc}\frac{1}{3}[0.8,0.7,0.7]=0.73,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\hfill & \frac{1}{3}[0.9+0.8+0.8]=0.83,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\hfill & \frac{1}{3}[1.0+0.8+0.8]=0.87,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\hfill & \frac{1}{3}[1.0,0.9,0.9]=0.93,\hfill \\ \frac{1}{3}[0.0,0.1,0.1]=0.07,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\hfill & \frac{1}{3}[0.0+0.2+0.2]=0.13,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\hfill & \frac{1}{3}[0.1+0.3+0.3]=0.23,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\hfill & \frac{1}{3}[0.2,0.3,0.3]=0.27.\hfill \end{array}$$
**Step****3.**- In this step, a simple decision matrix consisting of real numbers as entries was constructed for further calculations by using the ranking function of bipolar fuzzy numbers. Equation (2) was applied to the entries of Table 1, and the bipolar fuzzy numbers were converted to crisp values, which are shown in matrix T.$$T=\begin{array}{ccccc}\begin{array}{c}\\ {\mathcal{Q}}_{1}\\ {\mathcal{Q}}_{2}\\ {\mathcal{Q}}_{3}\\ {\mathcal{Q}}_{4}\\ {\mathcal{Q}}_{5}\\ {\mathcal{Q}}_{6}\\ {\mathcal{Q}}_{7}\end{array}& \begin{array}{c}{\mathbb{S}}_{1}\\ \left[\begin{array}{c}0.635\\ 0.665\\ 0.555\\ 0.370\\ 0.000\\ 0.100\\ 0.100\end{array}\right.\end{array}& \begin{array}{c}{\mathbb{S}}_{2}\\ 0.555\\ 0.455\\ 0.455\\ 0.555\\ 0.277\\ 0.455\\ 0.455\end{array}& \begin{array}{c}{\mathbb{S}}_{3}\\ 0.455\\ 0.525\\ 0.740\\ 0.665\\ 0.177\\ 0.277\\ 0.177\end{array}& \begin{array}{c}{\mathbb{S}}_{4}\\ \left.\begin{array}{c}0.635\\ 0.635\\ 0.370\\ 0.635\\ 0.525\\ 0.177\\ 0.177\end{array}\right]\end{array}\end{array}$$For instance, ${t}_{11}$ is the performance value of supplier ${\mathbb{S}}_{1}$ on the basis of criterion ${\mathcal{Q}}_{1}$, which was calculated as follows:$$\begin{array}{c}\begin{array}{cc}{t}_{11}\hfill & =\left(\left[\frac{0.73+0.83+0.87+0.93}{4}\right]+\left[\frac{-0.73-0.83+0.87+0.93}{2}\right]\right)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & -\left(\left[\frac{0.07+0.13+0.23+0.27}{4}\right]+\left[\frac{-0.07-0.13+0.23+0.27}{2}\right]\right)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & =(0.84+0.12)-(0.175+0.15)=0.635.\hfill \end{array}\hfill \end{array}$$
**Step****4.****Step****5.**- A preference function is required for the implementation of the PROMETHEE method. The preference function was used to define the deviation of any pair of alternatives on the basis of each criterion. In this step, the usual criterion preference function was used as defined in Definition 5, and the results are summarized in Table 6.
**Step****6.**- The weights of criteria specify the importance of each criterion towards the alternatives of the problem. In this method, the entropy weight information technique is used to calculate the normalized weights of criteria. The first step of this technique is the computation of projection values of criteria in order to normalize the decision values of criteria. The projection values for all criteria were calculated using Equation (4), and the results shown in Table 7. For example, $\mathcal{P}\left(11\right)$ is the projection value of criterion ${\mathcal{Q}}_{1}$ regarding the supplier ${\mathbb{S}}_{1}$ and was calculated as follows:$$\begin{array}{c}\mathcal{P}\left(11\right)=\frac{0.635}{0.635+0.555+0.455+0.635}=0.279.\hfill \end{array}$$The entropy value and the degree of divergence for each criterion were calculated by using the projection values given in Table 7, and deploying Equations (5) and (6), respectively, which were further utilized to determine the normalized weights of criteria. The results of entropy values, degrees of divergence, and weights of criteria are respectively shown in Table 8.
**Step****7.**- In this step, the multi-criteria preference index of each alternative is calculated, taking into account the weight criteria. The preference index of each alternative shows the value of preference of a supplier over other suppliers. The values of multi-criteria preference index were computed using Equation (9), and the results are summarized in Table 9.
**Step****8.**- This step concludes the whole procedure and the partial as well as net flows of alternatives are computed.
- (i)
- Positive and negative flows of alternatives (PROMETHEE I).The outgoing and incoming flows of alternatives are obtained which are positive and negative outranking flows, respectively. The positive outranking flow of an alternative determines how an alternative dominates all other alternatives and the negative flow shows how an alternative is dominated by all other alternatives. Equations (10) and (11) were used to calculate the outgoing and incoming flows of each alternative, and the results are given in Table 10. The partial ordering of suppliers was then obtained by taking the intersection of preorders ${P}^{+}$ and ${P}^{-}$, which is given as follows:$$\begin{array}{c}{\mathbb{S}}_{2}{P}_{1}{\mathbb{S}}_{1},\phantom{\rule{3.33333pt}{0ex}}{\mathbb{S}}_{2}{P}_{1}{\mathbb{S}}_{3},\phantom{\rule{3.33333pt}{0ex}}{\mathbb{S}}_{3}{P}_{1}{\mathbb{S}}_{1},\phantom{\rule{3.33333pt}{0ex}}{\mathbb{S}}_{4}{P}_{1}{\mathbb{S}}_{1},\phantom{\rule{3.33333pt}{0ex}}{\mathbb{S}}_{4}{P}_{1}{\mathbb{S}}_{3},\hfill \end{array}$$The results of positive outranking flow show that the supplier ${\mathbb{S}}_{2}$ is most preferable with greatest outgoing flow rate and determine that the supplier ${\mathbb{S}}_{2}$ is more dominant over all other suppliers. On the other hand, the negative outranking flow shows that the supplier ${\mathbb{S}}_{4}$ is less dominated by all other suppliers with minimum incoming flow value. These results are not able to compare all the alternatives nor to determine the most suitable supplier. For this reason, we need to calculate the net outranking flow of suppliers to obtain the complete preference of suppliers in the next step.
- (ii)
- Net flow of alternatives (PROMETHEE II).The net flow of each alternative was calculated by using Equation (15), which is the combination of positive and negative flows. The net outranking flow provides the complete ranking of alternatives by avoiding the incomparability. The results of the net flow of alternatives are shown in Table 11. These results can be justified by sketching the PROMETHEE diamond, which is a chart used to elaborate the results of PROMETHEE I and PROMETHEE II simultaneously. The positions of suppliers or complete ranking (PROMETHEE II) as well as the outgoing and incoming flows are also displayed in Figure 6.The net flow determines the complete ranking of suppliers in descending order. According to the results of Table 11 and the positions of suppliers in Figure 6, supplier ${\mathbb{S}}_{2}$ is the most preferable alternative and the ordering of alternatives is ${\mathbb{S}}_{2}\succ {\mathbb{S}}_{4}\succ {\mathbb{S}}_{3}\succ {\mathbb{S}}_{1}$.

## 4. Comparative Study

**Step****5.**- In this step, the combination of level and linear preference functions were adopted for conflicting criteria. The list of all criteria and their corresponding preference functions is given in Table 12.The linear preference function has a preference threshold value and the level function evaluates the criteria on the basis of preference and indifference threshold values, which are given by decision makers. In this multi-criteria decision-making problem, the indifference and preference threshold values were considered as $0.05$ and $0.1$, respectively, for both linear and level preference functions. The deviations between every pair of alternatives were obtained using Equations (6) and (7), and the results are shown in Table 13.
**Step****6.****Step****7.****Step****8.**- The partial and net flows are determined in this step.
- (i)
- The positive and negative outranking flows of the suppliers were computed by deploying the Equations (10) and (11), respectively. The respective results of outgoing and incoming flows for PROMETHEE I are shown in Table 15.The partial ordering of suppliers was then obtained by taking the intersection of preorders ${P}^{+}$ and ${P}^{-}$, which is given as follows:$$\begin{array}{c}{\mathbb{S}}_{2}{P}_{1}{\mathbb{S}}_{1},\phantom{\rule{3.33333pt}{0ex}}{\mathbb{S}}_{2}{P}_{1}{\mathbb{S}}_{3},\phantom{\rule{3.33333pt}{0ex}}{\mathbb{S}}_{3}{P}_{1}{\mathbb{S}}_{1},\phantom{\rule{3.33333pt}{0ex}}{\mathbb{S}}_{4}{P}_{1}{\mathbb{S}}_{1},\phantom{\rule{3.33333pt}{0ex}}{\mathbb{S}}_{4}{P}_{1}{\mathbb{S}}_{3},\hfill \end{array}$$
- (ii)
- The net outranking flow of each alternative was calculated by deploying Equation (15), and the results of PROMETHEE II are given in Table 16.It can be clearly seen that supplier ${\mathbb{S}}_{2}$ is chosen as the most preferable alternative under the combination of linear and level preference functions and the suppliers can be ordered as ${\mathbb{S}}_{2}\succ {\mathbb{S}}_{4}\succ {\mathbb{S}}_{3}\succ {\mathbb{S}}_{1}.$

## 5. Discussion and Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Framework of the bipolar fuzzy preference ranking organization method for enrichment of evaluations (PROMETHEE).

Linguistic Variable | Abbreviation | Bipolar Fuzzy Number |
---|---|---|

$Very\phantom{\rule{3.33333pt}{0ex}}good$ | $\mathbb{VG}$ | $\langle (0.8,0.9,1.0,1.0),(0.0,0.0,0.1,0.2)\rangle $ |

$Good$ | $\mathbb{G}$ | $\langle (0.7,0.8,0.8,0.9),(0.1,0.2,0.3,0.3)\rangle $ |

$Medium\phantom{\rule{3.33333pt}{0ex}}good$ | $\mathbb{MG}$ | $\langle (0.5,0.6,0.7,0.8),(0.2,0.3,0.4,0.5)\rangle $ |

$Fair$ | $\mathbb{F}$ | $\langle (0.4,0.5,0.5,0.6),(0.4,0.5,0.5,0.6)\rangle $ |

$Medium\phantom{\rule{3.33333pt}{0ex}}poor$ | $\mathbb{MP}$ | $\langle (0.2,0.3,0.4,0.5),(0.5,0.6,0.7,0.8)\rangle $ |

$Poor$ | $\mathbb{P}$ | $\langle (0.1,0.2,0.2,0.3),(0.7,0.8,0.8,0.9)\rangle $ |

$Very\phantom{\rule{3.33333pt}{0ex}}poor$ | $\mathbb{VP}$ | $\langle (0.0,0.0,0.1,0.2),(0.8,0.9,1.0,1.0)\rangle $ |

${\mathcal{D}}_{1}$ | ${\mathcal{D}}_{2}$ | ${\mathcal{D}}_{3}$ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathbb{S}}_{\mathbf{1}}$ | ${\mathbb{S}}_{\mathbf{2}}$ | ${\mathbb{S}}_{\mathbf{3}}$ | ${\mathbb{S}}_{\mathbf{4}}$ | ${\mathbb{S}}_{\mathbf{1}}$ | ${\mathbb{S}}_{\mathbf{2}}$ | ${\mathbb{S}}_{\mathbf{3}}$ | ${\mathbb{S}}_{\mathbf{4}}$ | ${\mathbb{S}}_{\mathbf{1}}$ | ${\mathbb{S}}_{\mathbf{2}}$ | ${\mathbb{S}}_{\mathbf{3}}$ | ${\mathbb{S}}_{\mathbf{4}}$ | |

${\mathcal{Q}}_{1}$ | $\mathbb{VG}$ | $\mathbb{MG}$ | $\mathbb{G}$ | $\mathbb{G}$ | $\mathbb{G}$ | $\mathbb{VG}$ | $\mathbb{G}$ | $\mathbb{G}$ | $\mathbb{G}$ | $\mathbb{G}$ | $\mathbb{MG}$ | $\mathbb{VG}$ |

${\mathcal{Q}}_{2}$ | $\mathbb{MG}$ | $\mathbb{G}$ | $\mathbb{G}$ | $\mathbb{VG}$ | $\mathbb{VG}$ | $\mathbb{MG}$ | $\mathbb{G}$ | $\mathbb{G}$ | $\mathbb{VG}$ | $\mathbb{G}$ | $\mathbb{G}$ | $\mathbb{G}$ |

${\mathcal{Q}}_{3}$ | $\mathbb{MG}$ | $\mathbb{G}$ | $\mathbb{VG}$ | $\mathbb{MG}$ | $\mathbb{G}$ | $\mathbb{G}$ | $\mathbb{VG}$ | $\mathbb{MG}$ | $\mathbb{VG}$ | $\mathbb{MG}$ | $\mathbb{G}$ | $\mathbb{G}$ |

${\mathcal{Q}}_{4}$ | $\mathbb{MG}$ | $\mathbb{VG}$ | $\mathbb{VG}$ | $\mathbb{G}$ | $\mathbb{MG}$ | $\mathbb{G}$ | $\mathbb{VG}$ | $\mathbb{VG}$ | $\mathbb{G}$ | $\mathbb{MG}$ | $\mathbb{MG}$ | $\mathbb{G}$ |

${\mathcal{Q}}_{5}$ | $\mathbb{F}$ | $\mathbb{F}$ | $\mathbb{F}$ | $\mathbb{G}$ | $\mathbb{F}$ | $\mathbb{G}$ | $\mathbb{F}$ | $\mathbb{G}$ | $\mathbb{F}$ | $\mathbb{MG}$ | $\mathbb{G}$ | $\mathbb{G}$ |

${\mathcal{Q}}_{6}$ | $\mathbb{F}$ | $\mathbb{MG}$ | $\mathbb{F}$ | $\mathbb{G}$ | $\mathbb{MG}$ | $\mathbb{G}$ | $\mathbb{MG}$ | $\mathbb{F}$ | $\mathbb{F}$ | $\mathbb{G}$ | $\mathbb{G}$ | $\mathbb{F}$ |

${\mathcal{Q}}_{7}$ | $\mathbb{MG}$ | $\mathbb{MG}$ | $\mathbb{F}$ | $\mathbb{G}$ | $\mathbb{F}$ | $\mathbb{G}$ | $\mathbb{G}$ | $\mathbb{G}$ | $\mathbb{F}$ | $\mathbb{G}$ | $\mathbb{F}$ | $\mathbb{G}$ |

${\mathbb{S}}_{1}$ | ${\mathbb{S}}_{2}$ | ${\mathbb{S}}_{3}$ | ${\mathbb{S}}_{4}$ | ||
---|---|---|---|---|---|

${\mathcal{D}}_{1}$ | ${\mathcal{Q}}_{1}$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.7,0.8,0.8,0.9),$ |

$(0.0,0.0,0.1,0.2)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | ||

${\mathcal{Q}}_{2}$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.8,0.9,1.0,1.0),$ | |

$(0.2,0.3,0.4,0.5)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.0,0.0,0.1,0.2)\rangle $ | ||

${\mathcal{Q}}_{3}$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.5,0.6,0.7,0.8),$ | |

$(0.2,0.3,0.4,0.5)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.0,0.0,0.1,0.2)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | ||

${\mathcal{Q}}_{4}$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.7,0.8,0.8,0.9),$ | |

$(0.2,0.3,0.4,0.5)\rangle $ | $(0.0,0.0,0.1,0.2)\rangle $ | $(0.0,0.0,0.1,0.2)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | ||

${\mathcal{Q}}_{5}$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.7,0.8,0.8,0.9),$ | |

$(0.4,0.5,0.5,0.6)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | ||

${\mathcal{Q}}_{6}$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.7,0.8,0.8,0.9),$ | |

$(0.4,0.5,0.5,0.6)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | ||

${\mathcal{Q}}_{7}$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.7,0.8,0.8,0.9),$ | |

$(0.2,0.3,0.4,0.5)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | ||

${\mathcal{D}}_{2}$ | ${\mathcal{Q}}_{1}$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.7,0.8,0.8,0.9),$ |

$(0.1,0.2,0.3,0.3)\rangle $ | $(0.0,0.0,0.1,0.2)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | ||

${\mathcal{Q}}_{2}$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.7,0.8,0.8,0.9),$ | |

$(0.0,0.0,0.1,0.2)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | ||

${\mathcal{Q}}_{3}$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.5,0.6,0.7,0.8),$ | |

$(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.0,0.0,0.1,0.2)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | ||

${\mathcal{Q}}_{4}$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.8,0.9,1.0,1.0),$ | |

$(0.2,0.3,0.4,0.5)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.0,0.0,0.1,0.2)\rangle $ | $(0.0,0.0,0.1,0.2)\rangle $ | ||

${\mathcal{Q}}_{5}$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.7,0.8,0.8,0.9),$ | |

$(0.4,0.5,0.5,0.6)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | ||

${\mathcal{Q}}_{6}$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.4,0.5,0.5,0.6),$ | |

$(0.2,0.3,0.4,0.5)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | ||

${\mathcal{Q}}_{7}$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.7,0.8,0.8,0.9),$ | |

$(0.4,0.5,0.5,0.6)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | ||

${\mathcal{D}}_{3}$ | ${\mathcal{Q}}_{1}$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.8,0.9,1.0,1.0),$ |

$(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.0,0.0,0.1,0.2)\rangle $ | ||

${\mathcal{Q}}_{2}$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.7,0.8,0.8,0.9),$ | |

$(0.0,0.0,0.1,0.2)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | ||

${\mathcal{Q}}_{3}$ | $\langle (0.8,0.9,1.0,1.0),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.7,0.8,0.8,0.9),$ | |

$(0.0,0.0,0.1,0.2)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | ||

${\mathcal{Q}}_{4}$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.7,0.8,0.8,0.9),$ | |

$(0.1,0.2,0.3,0.3)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | ||

${\mathcal{Q}}_{5}$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.5,0.6,0.7,0.8),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.7,0.8,0.8,0.9),$ | |

$(0.4,0.5,0.5,0.6)\rangle $ | $(0.2,0.3,0.4,0.5)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | ||

${\mathcal{Q}}_{6}$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.4,0.5,0.5,0.6),$ | |

$(0.4,0.5,0.5,0.6)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | ||

${\mathcal{Q}}_{7}$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.7,0.8,0.8,0.9),$ | $\langle (0.4,0.5,0.5,0.6),$ | $\langle (0.7,0.8,0.8,0.9),$ | |

$(0.4,0.5,0.5,0.6)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ | $(0.4,0.5,0.5,0.6)\rangle $ | $(0.1,0.2,0.3,0.3)\rangle $ |

${\mathbb{S}}_{1}$ | ${\mathbb{S}}_{2}$ | ${\mathbb{S}}_{3}$ | ${\mathbb{S}}_{4}$ | |
---|---|---|---|---|

${\mathcal{Q}}_{1}$ | $\langle (0.73,0.83,0.87,0.93),$ | $\langle (0.67,0.77,0.83,0.90),$ | $\langle (0.63,0.73,0.77,0.87),$ | $\langle (0.73,0.83,0.87,0.93),$ |

$(0.07,0.13,0.23,0.27)\rangle $ | $(0.10,0.17,0.27,0.33)\rangle $ | $(0.13,0.23,0.33,0.37)\rangle $ | $(0.07,0.13,0.23,0.27)\rangle $ | |

${\mathcal{Q}}_{2}$ | $\langle (0.70,0.80,0.90,0.93),$ | $\langle (0.63,0.73,0.77,0.87),$ | $\langle (0.70,0.80,0.80,0.90),$ | $\langle (0.73,0.83,0.87,0.93),$ |

$(0.07,0.10,0.20,0.30)\rangle $ | $(0.13,0.23,0.33,0.37)\rangle $ | $(0.10,0.20,0.30,0.30)\rangle $ | $(0.07,0.13,0.23,0.27)\rangle $ | |

${\mathcal{Q}}_{3}$ | $\langle (0.67,0.77,0.83,0.90),$ | $\langle (0.63,0.73,0.77,0.87),$ | $\langle (0.77,0.87,0.93,0.97),$ | $\langle (0.57,0.67,0.73,0.83),$ |

$(0.10,0.17,0.27,0.33)\rangle $ | $(0.13,0.23,0.33,0.37)\rangle $ | $(0.03,0.07,0.17,0.23)\rangle $ | $(0.17,0.27,0.37,0.43)\rangle $ | |

${\mathcal{Q}}_{4}$ | $\langle (0.57,0.67,0.73,0.83),$ | $\langle (0.67,0.77,0.83,0.90),$ | $\langle (0.70,0.80,0.90,0.93),$ | $\langle (0.73,0.83,0.87,0.93),$ |

$(0.17,0.27,0.37,0.43)\rangle $ | $(0.10,0.17,0.27,0.33)\rangle $ | $(0.07,0.10,0.20,0.30)\rangle $ | $(0.07,0.13,0.23,0.27)\rangle $ | |

${\mathcal{Q}}_{5}$ | $\langle (0.40,0.50,0.50,0.60),$ | $\langle (0.53,0.63,0.67,0.77),$ | $\langle (0.50,0.60,0.60,0.70),$ | $\langle (0.70,0.80,0.80,0.90),$ |

$(0.40,0.50,0.50,0.60)\rangle $ | $(0.23,0.33,0.40,0.47)\rangle $ | $(0.30,0.40,0.43,0.50)\rangle $ | $(0.10,0.20,0.30,0.30)\rangle $ | |

${\mathcal{Q}}_{6}$ | $\langle (0.43,0.53,0.57,0.67),$ | $\langle (0.63,0.73,0.77,0.87),$ | $\langle (0.53,0.63,0.67,0.77),$ | $\langle (0.50,0.60,0.60,0.70),$ |

$(0.33,0.43,0.47,0.57)\rangle $ | $(0.13,0.23,0.33,0.37)\rangle $ | $(0.23,0.33,0.40,0.47)\rangle $ | $(0.30,0.40,0.43,0.50)\rangle $ | |

${\mathcal{Q}}_{7}$ | $\langle (0.43,0.53,0.57,0.67),$ | $\langle (0.63,0.73,0.77,0.87),$ | $\langle (0.50,0.60,0.60,0.70),$ | $\langle (0.70,0.80,0.80,0.90),$ |

$(0.33,0.43,0.47,0.57)\rangle $ | $(0.13,0.23,0.33,0.37)\rangle $ | $(0.30,0.40,0.43,0.50)\rangle $ | $(0.10,0.20,0.30,0.30)\rangle $ |

${\mathcal{Q}}_{1}$ | ${\mathcal{Q}}_{2}$ | ${\mathcal{Q}}_{3}$ | ${\mathcal{Q}}_{4}$ | ${\mathcal{Q}}_{5}$ | ${\mathcal{Q}}_{6}$ | ${\mathcal{Q}}_{7}$ | |
---|---|---|---|---|---|---|---|

${\mathbb{S}}_{1}{\mathbb{S}}_{2}$ | $0.08$ | $0.21$ | $0.1$ | $-0.185$ | −0.277 | −0.355 | −0.355 |

${\mathbb{S}}_{1}{\mathbb{S}}_{3}$ | $0.18$ | $0.14$ | $-0.185$ | $-0.295$ | −0.177 | −0.177 | −0.077 |

${\mathbb{S}}_{1}{\mathbb{S}}_{4}$ | $0.0$ | $0.03$ | $0.185$ | $-0.265$ | −0.525 | −0.077 | −0.077 |

${\mathbb{S}}_{2}{\mathbb{S}}_{1}$ | $-0.08$ | $-0.21$ | $-0.1$ | $0.185$ | 0.277 | 0.355 | 0.355 |

${\mathbb{S}}_{2}{\mathbb{S}}_{3}$ | $0.1$ | $-0.07$ | $-0.285$ | $-0.11$ | 0.1 | 0.178 | 0.278 |

${\mathbb{S}}_{2}{\mathbb{S}}_{4}$ | $-0.08$ | $-0.18$ | $0.085$ | $-0.08$ | −0.248 | 0.278 | 0.278 |

${\mathbb{S}}_{3}{\mathbb{S}}_{1}$ | $-0.18$ | $-0.14$ | $0.185$ | $0.295$ | 0.177 | 0.177 | 0.077 |

${\mathbb{S}}_{3}{\mathbb{S}}_{2}$ | $-0.1$ | $0.07$ | $0.285$ | $0.11$ | −0.1 | −0.178 | −0.278 |

${\mathbb{S}}_{3}{\mathbb{S}}_{4}$ | $-0.18$ | $-0.11$ | $0.37$ | $0.03$ | −0.348 | 0.1 | 0.0 |

${\mathbb{S}}_{4}{\mathbb{S}}_{1}$ | $0.0$ | $-0.03$ | $-0.185$ | $0.265$ | 0.525 | 0.077 | 0.077 |

${\mathbb{S}}_{4}{\mathbb{S}}_{2}$ | $0.08$ | $0.18$ | $-0.085$ | $0.08$ | 0.248 | −0.278 | −0.278 |

${\mathbb{S}}_{4}{\mathbb{S}}_{3}$ | $0.18$ | $0.11$ | $-0.37$ | $-0.03$ | 0.348 | −0.1 | 0.0 |

${\mathcal{Q}}_{1}$ | ${\mathcal{Q}}_{2}$ | ${\mathcal{Q}}_{3}$ | ${\mathcal{Q}}_{4}$ | ${\mathcal{Q}}_{5}$ | ${\mathcal{Q}}_{6}$ | ${\mathcal{Q}}_{7}$ | |
---|---|---|---|---|---|---|---|

${\mathbb{S}}_{1}{\mathbb{S}}_{2}$ | 1 | 1 | 1 | 0 | 0 | 0 | 0 |

${\mathbb{S}}_{1}{\mathbb{S}}_{3}$ | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

${\mathbb{S}}_{1}{\mathbb{S}}_{4}$ | 0 | 1 | 1 | 0 | 0 | 0 | 0 |

${\mathbb{S}}_{2}{\mathbb{S}}_{1}$ | 0 | 0 | 0 | 1 | 1 | 1 | 1 |

${\mathbb{S}}_{2}{\mathbb{S}}_{3}$ | 1 | 0 | 0 | 0 | 1 | 1 | 1 |

${\mathbb{S}}_{2}{\mathbb{S}}_{4}$ | 0 | 0 | 1 | 0 | 0 | 1 | 1 |

${\mathbb{S}}_{3}{\mathbb{S}}_{1}$ | 0 | 0 | 1 | 1 | 1 | 1 | 1 |

${\mathbb{S}}_{3}{\mathbb{S}}_{2}$ | 0 | 1 | 1 | 1 | 0 | 0 | 0 |

${\mathbb{S}}_{3}{\mathbb{S}}_{4}$ | 0 | 0 | 1 | 1 | 0 | 1 | 0 |

${\mathbb{S}}_{4}{\mathbb{S}}_{1}$ | 0 | 0 | 0 | 1 | 1 | 1 | 1 |

${\mathbb{S}}_{4}{\mathbb{S}}_{2}$ | 1 | 1 | 0 | 1 | 1 | 0 | 0 |

${\mathbb{S}}_{4}{\mathbb{S}}_{3}$ | 1 | 1 | 0 | 0 | 1 | 0 | 0 |

$\mathcal{P}\left(\mathit{\alpha}\mathit{\beta}\right)$ | ${\mathcal{Q}}_{1}$ | ${\mathcal{Q}}_{2}$ | ${\mathcal{Q}}_{3}$ | ${\mathcal{Q}}_{4}$ | ${\mathcal{Q}}_{5}$ | ${\mathcal{Q}}_{6}$ | ${\mathcal{Q}}_{7}$ |
---|---|---|---|---|---|---|---|

${\mathbb{S}}_{1}$ | $0.279$ | $0.292$ | $0.262$ | $0.166$ | $0.0$ | $0.099$ | $0.11$ |

${\mathbb{S}}_{2}$ | $0.243$ | $0.20$ | $0.215$ | $0.249$ | $0.283$ | $0.451$ | $0.501$ |

${\mathbb{S}}_{3}$ | $0.20$ | $0.230$ | $0.349$ | $0.299$ | $0.181$ | $0.275$ | $0.195$ |

${\mathbb{S}}_{4}$ | $0.279$ | $0.279$ | $0.175$ | $0.285$ | $0.536$ | $0.175$ | $0.195$ |

${\mathcal{Q}}_{1}$ | ${\mathcal{Q}}_{2}$ | ${\mathcal{Q}}_{3}$ | ${\mathcal{Q}}_{4}$ | ${\mathcal{Q}}_{5}$ | ${\mathcal{Q}}_{6}$ | ${\mathcal{Q}}_{7}$ | |
---|---|---|---|---|---|---|---|

$\mathcal{E}\left(\beta \right)$ | $0.99$ | $0.99$ | $0.98$ | $0.98$ | $0.72$ | $0.90$ | $0.88$ |

$div\left(\beta \right)$ | $0.01$ | $0.01$ | $0.02$ | $0.02$ | $0.28$ | $0.10$ | $0.12$ |

$w\left(\beta \right)$ | $0.02$ | $0.02$ | $0.04$ | $0.04$ | $0.50$ | $0.18$ | $0.21$ |

Suppliers | ${\mathbb{S}}_{1}$ | ${\mathbb{S}}_{2}$ | ${\mathbb{S}}_{3}$ | ${\mathbb{S}}_{4}$ |
---|---|---|---|---|

${\mathbb{S}}_{1}$ | − | $0.08$ | $0.04$ | $0.06$ |

${\mathbb{S}}_{2}$ | $0.93$ | − | $0.91$ | $0.43$ |

${\mathbb{S}}_{3}$ | $0.97$ | $0.10$ | − | $0.26$ |

${\mathbb{S}}_{4}$ | $0.93$ | $0.58$ | $0.54$ | − |

Suppliers | ${\mathit{\xi}}^{+}\left({\mathbb{S}}_{\mathit{\alpha}}\right)$ | ${\mathit{\xi}}^{-}\left({\mathbb{S}}_{\mathit{\alpha}}\right)$ |
---|---|---|

${\mathbb{S}}_{1}$ | $0.060$ | $0.943$ |

${\mathbb{S}}_{2}$ | $0.757$ | $0.253$ |

${\mathbb{S}}_{3}$ | $0.443$ | $0.497$ |

${\mathbb{S}}_{4}$ | $0.683$ | $0.25$ |

Suppliers | $\mathit{\xi}\left({\mathbb{S}}_{\mathit{\alpha}}\right)$ |
---|---|

${\mathbb{S}}_{1}$ | $-0.883$ |

${\mathbb{S}}_{2}$ | $0.504$ |

${\mathbb{S}}_{3}$ | $-0.054$ |

${\mathbb{S}}_{4}$ | $0.433$ |

Criteria | Preference Function |
---|---|

Cost of products $\left({\mathcal{Q}}_{1}\right)$ | Linear |

Quality of products $\left({\mathcal{Q}}_{2}\right)$ | Level |

Services $\left({\mathcal{Q}}_{3}\right)$ | Level |

Delivery $\left({\mathcal{Q}}_{4}\right)$ | Level |

Pollution control $\left({\mathcal{Q}}_{5}\right)$ | Level |

Environmental management system $\left({\mathcal{Q}}_{6}\right)$ | Level |

Green packaging $\left({\mathcal{Q}}_{7}\right)$ | Level |

${\mathcal{Q}}_{1}$ | ${\mathcal{Q}}_{2}$ | ${\mathcal{Q}}_{3}$ | ${\mathcal{Q}}_{4}$ | ${\mathcal{Q}}_{5}$ | ${\mathcal{Q}}_{6}$ | ${\mathcal{Q}}_{7}$ | |
---|---|---|---|---|---|---|---|

${\mathbb{S}}_{1}{\mathbb{S}}_{2}$ | 1 | 1 | $0.5$ | 0 | 0 | 0 | 0 |

${\mathbb{S}}_{1}{\mathbb{S}}_{3}$ | 1 | $0.5$ | 0 | 0 | 0 | 0 | 0 |

${\mathbb{S}}_{1}{\mathbb{S}}_{4}$ | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

${\mathbb{S}}_{2}{\mathbb{S}}_{1}$ | 0 | 0 | 0 | 1 | 1 | 1 | 1 |

${\mathbb{S}}_{2}{\mathbb{S}}_{3}$ | 1 | 0 | 0 | 0 | $0.5$ | 1 | 1 |

${\mathbb{S}}_{2}{\mathbb{S}}_{4}$ | 0 | 0 | $0.5$ | 0 | 0 | 1 | 1 |

${\mathbb{S}}_{3}{\mathbb{S}}_{1}$ | 0 | 0 | 1 | 1 | 1 | 1 | $0.5$ |

${\mathbb{S}}_{3}{\mathbb{S}}_{2}$ | 0 | $0.5$ | 1 | $0.5$ | 0 | 0 | 0 |

${\mathbb{S}}_{3}{\mathbb{S}}_{4}$ | 0 | 0 | 1 | 0 | 0 | $0.5$ | 0 |

${\mathbb{S}}_{4}{\mathbb{S}}_{1}$ | 0 | 0 | 0 | 1 | 1 | $0.5$ | $0.5$ |

${\mathbb{S}}_{4}{\mathbb{S}}_{2}$ | 1 | 1 | 0 | $0.5$ | 1 | 0 | 0 |

${\mathbb{S}}_{4}{\mathbb{S}}_{3}$ | 1 | $0.5$ | 0 | 0 | 1 | 0 | 0 |

Suppliers | ${\mathbb{S}}_{1}$ | ${\mathbb{S}}_{2}$ | ${\mathbb{S}}_{3}$ | ${\mathbb{S}}_{4}$ |
---|---|---|---|---|

${\mathbb{S}}_{1}$ | − | $0.06$ | $0.03$ | $0.04$ |

${\mathbb{S}}_{2}$ | $0.93$ | − | $0.66$ | $0.41$ |

${\mathbb{S}}_{3}$ | $0.865$ | $0.07$ | − | $0.13$ |

${\mathbb{S}}_{4}$ | $0.735$ | $0.56$ | $0.53$ | − |

Suppliers | ${\mathit{\xi}}^{+}\left({\mathbb{S}}_{\mathit{\alpha}}\right)$ | ${\mathit{\xi}}^{-}\left({\mathbb{S}}_{\mathit{\alpha}}\right)$ |
---|---|---|

${\mathbb{S}}_{1}$ | $0.0433$ | $0.8433$ |

${\mathbb{S}}_{2}$ | $0.667$ | $0.23$ |

${\mathbb{S}}_{3}$ | $0.355$ | $0.4067$ |

${\mathbb{S}}_{4}$ | $0.6083$ | $0.1933$ |

Suppliers | $\mathit{\xi}\left({\mathbb{S}}_{\mathit{\alpha}}\right)$ |
---|---|

${\mathbb{S}}_{1}$ | $-0.80$ |

${\mathbb{S}}_{2}$ | $0.437$ |

${\mathbb{S}}_{3}$ | $-0.0517$ |

${\mathbb{S}}_{4}$ | $0.415$ |

Suppliers | Usual Criterion | Linear and Level Criteria |
---|---|---|

Preference Function | Preference Functions | |

Supplier ${\mathbb{S}}_{1}$ | 4 | 4 |

Supplier ${\mathbb{S}}_{2}$ | 1 | 1 |

Supplier ${\mathbb{S}}_{3}$ | 3 | 3 |

Supplier ${\mathbb{S}}_{4}$ | 2 | 2 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Akram, M.; Shumaiza; Al-Kenani, A.N.
Multi-Criteria Group Decision-Making for Selection of Green Suppliers under Bipolar Fuzzy PROMETHEE Process. *Symmetry* **2020**, *12*, 77.
https://doi.org/10.3390/sym12010077

**AMA Style**

Akram M, Shumaiza, Al-Kenani AN.
Multi-Criteria Group Decision-Making for Selection of Green Suppliers under Bipolar Fuzzy PROMETHEE Process. *Symmetry*. 2020; 12(1):77.
https://doi.org/10.3390/sym12010077

**Chicago/Turabian Style**

Akram, Muhammad, Shumaiza, and Ahmad N. Al-Kenani.
2020. "Multi-Criteria Group Decision-Making for Selection of Green Suppliers under Bipolar Fuzzy PROMETHEE Process" *Symmetry* 12, no. 1: 77.
https://doi.org/10.3390/sym12010077