Research on supplier performance evaluation system based on data mining with triangular fuzzy information

Abstract

From the beginning of 21th, enterprise’s competition predominance depends on capability of green supply chain. Suppliers are fountains of green supply chain, and the performance will directly influence the whole supply chain performance. The green supplier performance management is a key and difficult question in green supply chain manage system, which contains two cores, supplier performance evaluation and performance improvement. With the practice of two items, enterprise can control supplier performance continuously, and take improve actions in time, make the supplier performance to achieve requested level. Finally, enterprise can optimize the competition capability. In this paper, we investigate the multiple attribute decision making (MADM) problems for supplier performance evaluation with triangular fuzzy information. Motivated by the idea of Bonferroni mean and Einstein operations, we develop the aggregation techniques called the triangular fuzzy Einstein Bonferroni mean (TFEBM) operator for aggregating the triangular fuzzy information. For the situations where the input arguments have different importance, we then define the triangular fuzzy weighted Einstein Bonferroni mean (TFWEBM) operator, based on which we develop the procedure for multiple attribute decision making under the triangular fuzzy environments. Finally, a practical example for supplier performance evaluation is given to verify the developed approach.

Keywords

Multiple attribute decision making (MADM)triangular fuzzy numbers Bonferroni mean triangular fuzzy Bonferroni mean (TFBM) operator triangular fuzzy weighted Bonferroni mean (TFWBM) operator Chinese traditional medicine industry modernization development

1 Introduction

Sustainability is becoming to play an important role in supply chain management. Companies are increasingly expected to extend their sustainability efforts beyond their own operations to include those of their suppliers and to meet their customers’ sustainability expectations [1]. Traditionally, organizations consider criteria such as price, quality, flexibility, and delivery when evaluating supplier’s performance. In this way companies need efficient ways to select their suppliers with regard to their sustainability policies [1]. Now, many organizations based on the triple bottom line (TBL) approach have considered environmental, social, and economic concerns and have measured their suppliers’ sustainability performance [2]. Given the rise in environmental and resource conservation importance, green supply chain management has seen growth within industry. In addition, balancing economic development and environmental development is one of the critical issues faced by managers to help organizations maintain a strategically competitive position. Green supplier management as one of important part of green supply chain is also critical issue for effective green supply chain management. Therefore, research on making the green supplier selection decision, evaluating green supplier development programs and investigating how organizations manage performance of their suppliers in green supplier management is timely and necessary. There are extended models in the literature that examine supporting facts for major dimension on TBL. Carter supposes economic, environmental, and social as major aspects and organizational culture, transparency, risk management, and strategy as supporting aspects for major dimensions in his sustainable supply chain management framework [3]. There are several evaluation models for supplier selection and evaluation in the literature. Methodologies typically found in reviews of supplier selection approaches include weighted linear model approaches, mixed integer programming, analytical hierarchy process, linear and goal programming models, matrix methods, clustering methods, human judgment models, statistical analysis, and neural networks/case-based reasoning approaches[4 –14].

In this paper, we investigate the multiple attribute decision making (MADM) problems for supplier performance evaluation with triangular fuzzy information. Motivated by the idea of Bonferroni mean [15 –22] and Einstein operations [23 –29], we develop the aggregation techniques called the triangular fuzzy Einstein Bonferroni mean (TFEBM) operator for aggregating the triangular fuzzy information. For the situations where the input arguments have different importance, we then define the triangular fuzzy weighted Einstein Bonferroni mean (TFWEBM) operator, based on which we develop the procedure for multiple attribute decision making under the triangular fuzzy environments. Finally, a practical example for supplier performance evaluation is given to verify the developed approach.

2 Preliminaries

2.1 Triangular fuzzy numbers

In this section, we briefly describe some basic concepts and basic operational laws related to triangular fuzzy numbers.

Definition 1. [30] A triangular fuzzy numbers $\tilde{a}$ can be defined by a triplet (a^L, a^M, a^U). The membership function $μ_{\tilde{a}} (x)$ is defined as: $μ_{\tilde{a}} (x) = {\begin{matrix} 0, & x < a^{L}, \\ \frac{x - a^{L}}{a^{M} - a^{L}}, & a^{L} \leq x \leq a^{M}, \\ \frac{x - a^{U}}{a^{M} - a^{U}}, & a^{M} \leq x \leq a^{U}, \\ 0, & x \geq a^{U} . \end{matrix}$ (1) where 0 < a^L ≤ a^M ≤ a^U, a^L and a^U stand for the lower and upper values of the support of $\tilde{a}$ , respectively, and a^M for the modal value.

Definition 2. [30] Basic operational laws related to triangular fuzzy numbers: $\begin{matrix} \tilde{a} \oplus \tilde{b} & = & [a^{L}, a^{M}, a^{U}] \oplus [b^{L}, b^{M}, b^{U}] \\ = & [a^{L} + b^{L}, a^{M} + b^{M}, a^{U} + b^{U}] \end{matrix}$ $\begin{matrix} \tilde{a} \otimes \tilde{b} & = & [a^{L}, a^{M}, a^{U}] \otimes [b^{L}, b^{M}, b^{U}] \\ \approx [a^{L} b^{L}, a^{M} b^{M}, a^{U} b^{U}] \end{matrix}$ $\begin{matrix} λ \otimes \tilde{a} & = & λ \otimes [a^{L}, a^{M}, a^{U}] \\ = & [λ a^{L}, λ a^{M}, λ a^{U}], λ > 0 . \end{matrix}$

Definition 3. [31] Let $\tilde{b} = [b^{L}, b^{M}, b^{U}]$ and $\tilde{a} = [a^{L}, a^{M}, a^{U}]$ be two triangular fuzzy numbers, then the degree of possibility of a ≥ b is defined as $\begin{matrix} p (a \geq b) \\ = λ max {1 - max [\frac{b^{M} - a^{L}}{a^{M} - a^{L} + b^{M} - b^{L}}, 0], 0} \\ + (1 - λ) max {1 - max [\frac{b^{U} - a^{M}}{a^{U} - a^{M} + b^{U} - b^{M}}, 0], 0} \end{matrix}$ (2)

where the value λ is an index of rating attitude. It reflects the decision maker’s risk-bearing attitude. If λ > 0.5, the decision maker is risk lover. If λ = 0.5, the decision maker is neutral to risk. If λ < 0.5, the decision maker is risk avertor.

Xu [32] and Fan and Wang [33] developed the fuzzy ordered weighted averaging (FOWA) operator. Xu [34] introduced the fuzzy ordered weighted geometric (FOWG) operator. Xu and Wu [35] proposed the fuzzy induced ordered weighted averaging (FIOWA) operator. Xu and Da [36] developed the fuzzy induced ordered weighted geometric (FIOWG) operator. Xu [37] developed some fuzzy harmonic mean operators, such as fuzzy weighted harmonic mean (FWHM) operator, fuzzy ordered weighted harmonic mean (FOWHM) operator, fuzzy hybrid harmonic mean (FHHM) operator. Wei [38] proposed the fuzzy ordered weighted harmonic averaging(FOWHA) operator. Wei [39] developed the fuzzy induced ordered weighted harmonic mean (FIOWHM) operator and applied it to the group decision making. Wei [40] proposed the generalized triangular fuzzy correlated averaging operator and applied these operators to multiple attribute decision making.

2.2 Bonferroni mean

Bonferroni [15] originally introduced a mean type aggregation operator, called Bonferroni mean, which can provide for aggregation lying between the max, min operators and the logical “or” and “and” operators, which was defined as follows:

Definition 4. [15] Let p, q ≥ 0 and a_i (i = 1, 2, ⋯ , n) be a collection of non-negative real numbers. Then the aggregation functions:

$\begin{matrix} {BM}^{p, q} (a_{1}, a_{2}, \dots, a_{n}) \\ = {(\frac{1}{n (n - 1)} \sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} a_{i}^{p} a_{j}^{q})}^{\frac{1}{p + q}} \end{matrix}$ (3) is called the Bonferroni mean (BM) operator.

Some special cases of the BM^p,q are shown as follows:

If q = 0, then the BM^p,q is reduced to the following generalized mean operator:

${BM}^{p, 0} (a_{1}, a_{2}, \dots, a_{n}) = {(\frac{1}{n} \sum_{i = 1}^{n} a_{i}^{p})}^{\frac{1}{p}}$

If p = q = 1, then Equation (4) reduces to the following mean: ${BM}^{1, 1} (a_{1}, a_{2}, \dots, a_{n}) = {(\frac{1}{n (n - 1)} \sum_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} a_{i} a_{j})}^{\frac{1}{2}}$

2.3 Einstein operations

The set theoretical operators have had an important role since in the beginning of fuzzy set theory. All types of the particular operators were included in the general concepts of the t-norms and t-conorms, which satisfy the requirements of the conjunction and disjunction operators, respectively. There are various t-norms and t-conorms families can be used to perform the corresponding intersections and unions of IFSs. Einstein operations includes the Einstein product and Einstein sum, which are examples of t-norms and tconorms, respectively. They are defined as follows [23]:

Einstein product ⊗_ɛ is a t-norm and Einstein sum ⊕_ɛ is a t-conorm, where $\begin{matrix} a \oplus_{ɛ} b & = & \frac{a + b}{1 + a \cdot b}, \\ a \otimes_{ɛ} b & = & \frac{a \cdot b}{1 + (1 - a) \cdot (1 - b)}, \\ \forall (a, b) \in {[0, 1]}^{2} \end{matrix}$

3 Triangular fuzzy Einstein Bonferroni mean operators

The Bonferroni mean (BM) operator [15] and Einstein operations [23], however, have usually been used in situations where the input arguments are the non-negative real numbers. We shall extend the BM operators and Einstein operations to accommodate the situations where the input arguments are triangular fuzzy numbers. In this section, we shall investigate the BM operators and Einstein operations under triangular fuzzy environments. Based on Definition 4 and Section 2.3, we give the definition of the triangular fuzzy Einstein Bonferroni mean (TFEBM) operator as follows:

Definition 5. Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers, and let p, q > 0. If $\begin{matrix} {TFEBM}^{p, q} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({\tilde{a}}_{i}^{p} \otimes {\tilde{a}}_{j}^{q}))}^{\frac{1}{p + q}} \\ = [{(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({(a_{i}^{L})}^{p} \otimes {(a_{j}^{L})}^{q}))}^{\frac{1}{p + q}}, {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({(a_{i}^{M})}^{p} \otimes {(a_{j}^{M})}^{q}))}^{\frac{1}{p + q}}, \\ {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({(a_{i}^{U})}^{p} \otimes {(a_{j}^{U})}^{q}))}^{\frac{1}{p + q}}] = [{(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (\frac{{(a_{i}^{L})}^{p} {(a_{j}^{L})}^{q}}{1 + (1 - {(a_{i}^{L})}^{p}) (1 - {(a_{j}^{L})}^{q})}))}^{\frac{1}{p + q}}, \\ {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (\frac{{(a_{i}^{M})}^{p} {(a_{j}^{M})}^{q}}{1 + (1 - {(a_{i}^{M})}^{p}) (1 - {(a_{j}^{M})}^{q})}))}^{\frac{1}{p + q}}, {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (\frac{{(a_{i}^{U})}^{p} {(a_{j}^{U})}^{q}}{1 + (1 - {(a_{i}^{U})}^{p}) (1 - {(a_{j}^{U})}^{q})}))}^{\frac{1}{p + q}}] \end{matrix}$ (4)

then TFEBM^p,q is called the triangular fuzzy Einstein Bonferroni mean (TFEBM) operator.

It can be easily proved that the TFEBM operator has the following properties.

Theorem 1.(Idempotency) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers. If all ${\tilde{a}}_{j} ({\tilde{a}}_{j} = [a_{j}^{L}, a_{j}^{M}, a_{j}^{U}])$ are equal,i.e. ${\tilde{a}}_{j} ({\tilde{a}}_{j} = [a_{j}^{L}, a_{j}^{M}, a_{j}^{U}]) = \tilde{a} (\tilde{a} = [a^{L}, a^{M}, a^{U}])$ for allj, then ${TFEBM}^{p, q} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a}$ (5)

Theorem 2. (Boundedness) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers, and let ${\tilde{a}}^{-} \underset{j}{= min} {\tilde{a}}_{j}, {\tilde{a}}^{+} \underset{j}{= min} {\tilde{a}}_{j}$

Then ${\tilde{a}}^{-} \leq {TFEBM}^{p, q} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \leq {\tilde{a}}^{+}$ (6)

Theorem 3.(Monotonicity) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ and ${\tilde{a}}_{i}^{'} = [a_{i}^{' L}, a_{i}^{' M}, a_{i}^{' U}] (i = 1, 2, \dots, n)$ be two sets of triangular fuzzy numbers, if ${\tilde{a}}_{j} \leq {\tilde{a}}_{j}^{'}$ , for allj, then

$\begin{matrix} {TFEBM}^{p, q} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ \leq {TFEBM}^{p, q} ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (7)

Theorem 4.(Commutativity) Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ and ${\tilde{a}}_{i}^{'} = [a_{i}^{' L}, a_{i}^{' M}, a_{i}^{' U}] (i = 1, 2, \dots, n)$ be two sets of triangular fuzzy numbers, where ${\tilde{a}}_{i}^{'} = [a_{i}^{' L}, a_{i}^{' M}, a_{i}^{' U}] (i = 1, 2, \dots, n)$ is any permutation of ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ , then

$\begin{matrix} {TFEBM}^{p, q} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = {TFEBM}^{p, q} ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (8)

Considering that the input arguments may have different importance, here we define the triangular fuzzy weighted Einstein Bonferroni mean (TFWEBM) operator.

Definition 6. ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers and p, q > 0, w = (w₁, w₂, ⋯ , w_n) ^T is the weight vector of ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ , where w_i indicates the importance degree of ${\tilde{a}}_{i}$ , satisfying w_i > 0 (i = 1, 2, ⋯ , n), and $\sum_{i = 1}^{n} w_{i} = 1$ . If

$\begin{matrix} {TFWEBM}_{w}^{p, q} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({(w_{i} {\tilde{a}}_{i})}^{p} \otimes {(w_{j} {\tilde{a}}_{j})}^{q}))}^{\frac{1}{p + q}} \\ = [{(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({(w_{i} a_{i}^{L})}^{p} \otimes {(w_{j} a_{j}^{L})}^{q}))}^{\frac{1}{p + q}}, {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({(w_{i} a_{i}^{M})}^{p} \otimes {(w_{j} a_{j}^{M})}^{q}))}^{\frac{1}{p + q}}, \\ {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({(w_{i} a_{i}^{U})}^{p} \otimes {(w_{j} a_{j}^{U})}^{q}))}^{\frac{1}{p + q}}] = [{(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (\frac{{(w_{i} a_{i}^{L})}^{p} {(w_{j} a_{j}^{L})}^{q}}{1 + (1 - {(w_{i} a_{i}^{L})}^{p}) (1 - {(w_{j} a_{j}^{L})}^{q})}))}^{\frac{1}{p + q}}, \\ {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (\frac{{(w_{i} a_{i}^{M})}^{p} {(w_{j} a_{j}^{M})}^{q}}{1 + (1 - {(w_{i} a_{i}^{M})}^{p}) (1 - {(w_{j} a_{j}^{M})}^{q})}))}^{\frac{1}{p + q}}, {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (\frac{{(w_{i} a_{i}^{U})}^{p} {(w_{j} a_{j}^{U})}^{q}}{1 + (1 - {(w_{i} a_{i}^{U})}^{p}) (1 - {(w_{j} a_{j}^{U})}^{q})}))}^{\frac{1}{p + q}}] \end{matrix}$ (9)

then ${TFWEBM}_{w}^{p, q}$ is called the triangular fuzzy weighted Einstein Bonferroni mean (TFWEBM) operator.

4 An approach to multiple attribute decision making with triangular fuzzy information

In this section, we shall utilize the triangular fuzzy weighted Einstein Bonferroni mean (TFWEBM) operator to multiple attribute decision making for supplier performance evaluation with triangular fuzzy information. Let A = {A₁, A₂, ⋯ , A_m} be a discrete set of alternatives, G ={ G₁, G₂, ⋯ , G_n } be the set of attributes, whose weight vector is ω = (ω₁, ω₂, ⋯ , ω_n), with ω_j ≥ 0, j = 1, 2, ⋯ , n, $\sum_{j = 1}^{n} ω_{j} = 1$ . Suppose that $R = {({\tilde{a}}_{ij})}_{m \times n} = {[r_{ij}^{L}, r_{ij}^{M}, r_{ij}^{U}]}_{m \times n}$ is the decision making matrix, where ${\tilde{a}}_{ij}$ is the preference value, which take the form of triangular fuzzy numbers, given by the decision maker, for the alternative A_i ∈ A with respect to the attribute G_j ∈ G.

Then, we utilize the triangular fuzzy weighted Einstein Bonferroni mean (TFWEBM) operator to develop an approach to multiple attribute decision making problems with triangular fuzzy information, which can be described as following:

Step 1. Utilize the decision information given in matrix $\tilde{R}$ , and the TFWEBM operator (in general, we can take p = q = 1) $\begin{matrix} {\tilde{r}}_{i} = (r_{i}^{L}, r_{i}^{M}, r_{i}^{U}) = {TFWEBM}_{w}^{p, q} ({\tilde{r}}_{i 1}, {\tilde{r}}_{i 2}, \dots, {\tilde{r}}_{in}) \\ = {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} k, l = 1 \\ k \neq l \end{matrix}}^{n} ({(w_{k} {\tilde{r}}_{ik})}^{p} \otimes {(w_{l} {\tilde{r}}_{il})}^{q}))}^{\frac{1}{p + q}} \\ = [{(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} k, l = 1 \\ k \neq l \end{matrix}}^{n} ({(w_{k} r_{ij}^{L})}^{p} \otimes {(w_{l} r_{il}^{L})}^{q}))}^{\frac{1}{p + q}}, {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} k, l = 1 \\ k \neq l \end{matrix}}^{n} ({(w_{k} r_{ij}^{M})}^{p} \otimes {(w_{l} r_{il}^{M})}^{q}))}^{\frac{1}{p + q}}, \\ {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} k, l = 1 \\ k \neq l \end{matrix}}^{n} ({(w_{k} r_{ij}^{U})}^{p} \otimes {(w_{l} r_{il}^{U})}^{q}))}^{\frac{1}{p + q}}] = [{(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} k, l = 1 \\ k \neq l \end{matrix}}^{n} (\frac{{(w_{k} r_{ij}^{L})}^{p} {(w_{l} r_{il}^{L})}^{q}}{1 + (1 - {(w_{k} r_{ij}^{L})}^{p}) (1 - {(w_{l} r_{il}^{L})}^{q})}))}^{\frac{1}{p + q}}, \\ {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} k, l = 1 \\ k \neq l \end{matrix}}^{n} (\frac{{(w_{k} r_{ij}^{M})}^{p} {(w_{l} r_{il}^{M})}^{q}}{1 + (1 - {(w_{k} r_{ij}^{M})}^{p}) (1 - {(w_{l} r_{il}^{M})}^{q})}))}^{\frac{1}{p + q}}, {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} k, l = 1 \\ k \neq l \end{matrix}}^{n} (\frac{{(w_{k} r_{ij}^{U})}^{p} {(w_{l} r_{il}^{U})}^{q}}{1 + (1 - {(w_{k} r_{ij}^{U})}^{p}) (1 - {(w_{l} r_{il}^{U})}^{q})}))}^{\frac{1}{p + q}}] \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{matrix}$ (10)

Step 2. To rank these collective overall preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ , we first compare each ${\tilde{r}}_{i}$ with all the ${\tilde{r}}_{j} (j = 1, 2, \dots, m)$ by using Equation (2). For simplicity, we let $p_{ij} = p ({\tilde{r}}_{i} \geq {\tilde{r}}_{j})$ , then we develop a complementary matrix as P = (p_ij) _m×m, where p_ij ≥ 0, p_ij + p_ji = 1, p_ii = 0.5, i, j = 1, 2, ⋯ , n.

Summing all the elements in each line of matrix P, we have $p_{i} = \sum_{j = 1}^{m} p_{ij}, i = 1, 2, \dots, m .$ (11)

Step 3. Rank all the alternatives X_i (i = 1, 2, ⋯ , m) and select the best one(s) in accordance with the collective overall preference values p_i (i = 1, 2, ⋯ , m).

5 Numerical example

As marketing economic operations increase, the demand of enterprise insides control for domestic and overseas manager becomes more accurate. The management of supplier, which is the forward position of controlling business costs, has the characteristics of extensive management scope, dispersed control node, numerous evaluation index, complex internal relations and so on. The result of performance appraisal of management over supplier has a significant impact on the success of the enterprise’s management over the supplier. Therefore, lots of enterprises are devoted to look for right method of performance appraisal based on scientific and reasonable theoretical basis. The performance appraisal of the supplier of enterprise was completely and deeply analyzed based on large numbers of literature and practical investigation. In this section, we utilize a practical multiple attribute decision making problems for supplier performance evaluation to illustrate the application of the developed approaches. The company employs some external professional organizations (or experts) to aid this decision-making. The expert team selects four attributes to evaluate the modernization development of Chinese traditional medicine industry of five cities: (1) G₁: debt paying ability, (2) G₂: operation capability, (3) G₃: earning capacity; (4) G₄: the development capability. The five possible supplier companies A_i (i = 1, 2, ⋯ , 5) are to be evaluated using the triangular fuzzy numbers by the decision makers under the above four attributes (whose weighting vector is ω = (0.2, 0.1, 0.3, 0.4)), and construct the following matrix $A = {({\tilde{a}}_{ij})}_{5 \times 4}$ is shown as follows:

$\tilde{A} = [\begin{matrix} [0 . 76, 0 . 78, 0 . 81] [0 . 47, 0 . 50, 0 . 53] \\ [0 . 45, 0 . 52, 0 . 50] [0 . 54, 0 . 57, 0 . 60] \\ [0 . 62, 0 . 64, 0 . 67] [0 . 78, 0 . 80, 0 . 81] \\ [0 . 83, 0 . 84, 0.85] [0 . 82, 0 . 84, 0.85] \\ [0 . 68, 0 . 70, 0 . 76] [0 . 79, 0 . 82, 0 . 80] \end{matrix} \begin{matrix} [0 . 65, 0 . 70, 0 . 75] [0 . 57, 0 . 61, 0 . 65] \\ [0 . 73, 0 . 75, 0 . 78] [0 . 52, 0 . 62, 0 . 68] \\ [0 . 80, 0 . 82, 0 . 83] [0 . 75, 0 . 78, 0 . 80] \\ [0 . 67, 0 . 70, 0 . 73] [0 . 82, 0 . 83, 0.85] \\ [0 . 63, 0 . 64, 0 . 66] [0 . 63, 0 . 64, 0 . 60] \end{matrix}]$

In the following, in order to select the most desirable supplier companies, we utilize the TFWEBM operator to develop an approach to multiple attribute decision making problems for supplier performance evaluation with triangular fuzzy information, which can be described as following:

Step 1. Aggregate all triangular fuzzy preference value ${\tilde{r}}_{ij} (j = 1, 2, \dots, n)$ by using the TFWEBM to derive the overall triangular fuzzy preference values ${\tilde{r}}_{i} (i = 1, 2, 3, 4, 5)$ of the supplier companies A_i. $\begin{matrix} {\tilde{r}}_{1} = [0 . 1852, 0.2003, 0 . 2133] \\ {\tilde{r}}_{2} = [0 . 1813, 0 . 1921, 0 . 2049] \\ {\tilde{r}}_{3} = [0 . 2033, 0 . 2314, 0 . 2548] \\ {\tilde{r}}_{4} = [0 . 1833, 0 . 2011, 0 . 2210] \\ {\tilde{r}}_{5} = [0 . 1827, 0 . 1975, 0 . 2140] \end{matrix}$

Step 2. Rank all the supplier companies A_i (i = 1, 2, ⋯ , 5) in accordance with the preference degree p_i (i = 1, 2, ⋯ , 5): A₃ ≻ A₄ ≻ A₂ ≻ A₁ ≻ A₅, and thus the most desirable supplier company is A₃.

6 Conclusion

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