Model for evaluating the enterprise performance with triangular fuzzy information

Abstract

In the new knowledge-based economy era when the information technology have the rapid development and play an important role, the enterprise management environment facing great changes and it has entered the era of strategic management theme. Traditional performance evaluation focuses on the interests of stockholder, takes financial data as base and pays attention to the reflection of material assets and the result, therefore it is not capable and enough in a new economy era. Focus on enterprise business strategies, reflecting the interests of shareholders, taking into account the needs of stakeholders are more and more important in the future furious competition. In this paper, we investigate the multiple attribute decision making (MADM) problems for enterprise performance evaluation with triangular fuzzy information. We utilize the triangular fuzzy weighted Einstein Bonferroni mean (TFWEBM) operator to develop the procedure for multiple attribute decision making under the triangular fuzzy environments. Finally, a practical example for enterprise performance evaluation is given to verify the developed approach.

Keywords

Multiple attribute decision making (MADM)triangular fuzzy numbers triangular fuzzy weighted Bonferroni mean (TFWBM) operator enterprise performance evaluation

1. Introduction

The focus of strategic management research is how enterprises utilize appropriate strategies to create and maintain competitive advantages. The research on competition has grown exponentially in recent years. However, Mintzberg et al. noted that research on strategy has been criticized for its overly analytical orientation, upper management bias, lack of attention to action and learning, and neglect of the elements that lead to the creation of strategies [1]. Shrivastava note that research on organizational learning focuses on processes; this has the potential to offer insights into these identified drawbacks [2]. Brockman and Morgan believe that organizational learning is the basis for gaining a sustainable competitive advantage and a key variable in the enhancement of firm performance [3]. Tippins and Sohi state that firms that are able to learn have a better chance of sensing events and trends in the marketplace [4]. Furthermore, some studies provide evidence of a positive relationship between organizational learning and firm performance. For instance, Baker and Sinkula find that learning orientation has a direct effect on firm performance [5]. Ussahawanitchakit uses a cultural measure of learning and obtains similar results [6].

Enterprise performance appraisal is not only an outcome of market economy development on certain stage, but also a scientific method and an effective tool to supervise the enterprises for nations with mature market economy. It is the direction for modern enterprise management to learn successful experience of enterprise management from foreign market economy and apply enterprise performance appraisal for supervising and controlling the enterprises. As one of the methods of modern enterprise management, enterprise performance appraisal is enduring the challenges of rapidly developing economy and constantly updating enterprise management mode. It has received wider attention and deeper discussion. As for the work of enterprise performance appraisal in our country, how to follow the change of our economic and social environment and the trend of internationalization and how to establish a system of performance appraisal that fits our economic development plays an especially important operation significance for enhancing the health of our enterprises, improving the management level, sharpening the enterprise competitiveness and further improving the economic growth quality. The problem of enterprise performance with triangular fuzzy information is the classical multiple attribute decision making problem [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27].

In this paper, we investigate the multiple attribute decision making problems for enterprise performance evaluation with triangular fuzzy information. We utilize the triangular fuzzy weighted Einstein Bonferroni mean (TFWEBM) operator to develop the procedure for multiple attribute decision making under the triangular fuzzy environments. Finally, a practical example for enterprise performance evaluation is given to verify the developed approach.

2. Preliminaries

In this section, we briefly describe some basic concepts and basic operational laws related to triangular fuzzy numbers [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43].

Definition 1 [28]. A triangular fuzzy numbers $\tilde{{a}}$ can be defined by a triplet $\left({a^{L},a^{M},a^{U}}\right)$ . The membership function $\mu_{\tilde{{a}}}\left(x\right)$ is defined as:

$\mu_{\tilde{{a}}}\left(x\right)=\begin{cases}0,&x<a^{L},\\ {\displaystyle\frac{x-a^{L}}{a^{M}-a^{L}}},&a^{L}\leqslant x\leqslant a^{M},\\ \\ {\displaystyle\frac{x-a^{U}}{a^{M}-a^{U}}},&a^{M}\leqslant x\leqslant a^{U},\\ 0,&x\geqslant a^{U}.\\ \end{cases}$ (1)

where $0<a^{L}\leqslant a^{M}\leqslant a^{U}$ , $a^{L}$ 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 [29]. Let $\tilde{{b}}=\left[{b^{L},b^{M},b^{U}}\right]$ and $\tilde{{a}}=\left[{a^{L},a^{M},a^{U}}\right]$ be two triangular fuzzy numbers, then the degree of possibility of $a\geqslant b$ is defined as

$\displaystyle p\left({a\geqslant b}\right)=$ $\displaystyle\lambda\max\left\{{1{-}\max\left[{\frac{b^{M}-a^{L}}{a^{M}-a^{L}+% b^{M}-b^{L}},0}\right],0}\right\}$ $\displaystyle\quad∼{}+\left({1-\lambda}\right)\max\bigg{\{}1-$ $\displaystyle\quad∼{}\left.\max\left[{\frac{b^{U}-a^{M}}{a^{U}-a^{M}+b^{U}-b^{% M}},0}\right],0\right\}$ (2)

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

The Bonferroni mean (BM) operator [44, 45, 46, 47] and Einstein operations [48, 49, 50, 51, 52, 53, 54, 55], however, have usually been used in situations where the input arguments are the non-negative real numbers. Zhou [56] extended the BM operators and Einstein operations to accommodate the situations where the input arguments are triangular fuzzy numbers and proposed the triangular fuzzy Einstein Bonferroni mean (TFEBM) operator as follows:

Definition 3 [56]. Let $\tilde{{a}}_{i}=\left[{a_{i}^{L},a_{i}^{M},a_{i}^{U}}\right](i=1,2,$ $\ldots,n)$ be a set of triangular fuzzy numbers, and let $p,q>0$ . If

$\displaystyle\textit{TFEBM}^{p,q}({\tilde{{a}}_{1},\tilde{{a}}_{2},\ldots,% \tilde{{a}}_{n}})$ $\displaystyle=\left({\frac{1}{n(n-1)}\mathop{\oplus}\limits_{\begin{subarray}{% l}\scriptscriptstyle i,j=1\\ \scriptscriptstyle i\neq j\\ \end{subarray}}^{n}\left(\tilde{{a}}_{i}^{p}\otimes\tilde{{a}}_{j}^{q}\right)}% \right)^{\frac{1}{p+q}}$ $\displaystyle=\left[{\left({\frac{1}{n\left({n-1}\right)}\mathop{\oplus}% \limits_{{\begin{subarray}{l}\scriptscriptstyle i,j=1\\ \scriptscriptstyle i\neq j\\ \end{subarray}}}^{n}\left({\left({a_{i}^{L}}\right)^{p}\otimes\left({a_{j}^{L}% }\right)^{q}}\right)}\right)^{\frac{1}{p+q}},}\right.$ $\displaystyle\quad∼{}\left({\frac{1}{n\left({n-1}\right)}\mathop{\oplus}% \limits_{{\begin{subarray}{l}\scriptscriptstyle i,j=1\\ \scriptscriptstyle i\neq j\\ \end{subarray}}}^{n}\left({\left({a_{i}^{M}}\right)^{p}\otimes\left({a_{j}^{M}% }\right)^{q}}\right)}\right)^{\frac{1}{p+q}},$ $\displaystyle\quad∼{}\left.{\left({\frac{1}{n\left({n-1}\right)}\mathop{\oplus% }\limits_{{\begin{subarray}{l}\scriptscriptstyle i,j=1\\ \scriptscriptstyle i\neq j\\ \end{subarray}}}^{n}\left({\left({a_{i}^{U}}\right)^{p}\otimes\left({a_{j}^{U}% }\right)^{q}}\right)}\right)^{\frac{1}{p+q}}}\right]$ (3)

then $\textit{TFEBM}^{p,q}$ is called the triangular fuzzy Einstein Bonferroni mean (TFEBM) operator.

Considering that the input arguments may have different importance, here Zhou [56] further proposed the triangular fuzzy weighted Einstein Bonferroni mean (TFWEBM) operator.

Definition 4 [56]. $\tilde{{a}}_{i}=\left[{a_{i}^{L},a_{i}^{M},a_{i}^{U}}\right]\left({i=1,2,% \ldots,n}\right)$ be a set of triangular fuzzy numbers and $p,q>0$ , $w=\left({w_{1},w_{2},\ldots,w_{n}}\right)^{T}$ is the weight vector of $\tilde{{a}}_{i}=\left[{a_{i}^{L},a_{i}^{M},a_{i}^{U}}\right]\left({i=1,2,% \ldots,n}\right)$ , where $w_{i}$ indicates the importance degree of $\tilde{{a}}_{i}$ , satisfying $w_{i}>0\left({i=1,2,\ldots,n}\right)$ , and $\sum\nolimits_{i=1}^{n}{w_{i}}=1$ . If

$\displaystyle\textit{TFWEBM}_{w}^{p,q}\left({\tilde{{a}}_{1},\tilde{{a}}_{2},% \ldots,\tilde{{a}}_{n}}\right)$ $\displaystyle=\left({\frac{1}{n\left({n-1}\right)}\mathop{\oplus}\limits_{{% \begin{subarray}{l}i,j=1\\ i\neq j\\ \end{subarray}}}^{n}\left({\left({w_{i}\tilde{{a}}_{i}}\right)^{p}\otimes\left% ({w_{j}\tilde{{a}}_{j}}\right)^{q}}\right)}\right)^{\frac{1}{p+q}}$ $\displaystyle=\scalebox{.975}{$\left[{\left({\frac{1}{n\left({n-1}\right)}% \mathop{\oplus}\limits_{{\begin{subarray}{l}i,j=1\\ i\neq j\\ \end{subarray}}}^{n}\left({\left({w_{i}a_{i}^{L}}\right)^{p}\otimes\left({w_{j% }a_{j}^{L}}\right)^{q}}\right)}\right)^{\frac{1}{p+q}},}\right.$}$ (4)

then $\textit{TFWEBM}_{w}^{p,q}$ is called the triangular fuzzy wei- ghted Einstein Bonferroni mean (TFWEBM) operator.

3. Model for 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 enterprise performance evaluation with triangular fuzzy information. Let $A=\left\{{A_{1},A_{2},\ldots,A_{m}}\right\}$ be a discrete set of alternatives, $G=\left\{{G_{1},G_{2},\ldots,G_{n}}\right\}$ be the set of attributes, whose weight vector is $\omega=\left({\omega_{1},\omega_{2},\ldots,\omega_{n}}\right)$ , with $\omega_{j}\geqslant 0$ , $j=1,2,\ldots,n$ , $\sum\nolimits_{j=1}^{n}{\omega_{j}}=1$ . Suppose that $R=\left({\tilde{{a}}_{ij}}\right)_{m\times n}=\left[{r_{ij}^{L},r_{ij}^{M},r_{% ij}^{U}}\right]_{m\times n}$ 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 $A_{i}\in A$ with respect to the attribute $G_{j}\in G$ .

Then, we utilize the TFWEBM operator to solve the multiple attribute decision making problems with triangular fuzzy information, which can be described as following:

Step 3. Step 1.
Utilize the decision information given in matrix $\tilde{{R}}$ , and the TFWEBM operator (in general, we can take $p=q=1$ )

$\displaystyle\mspace{-75.0mu }\tilde{{r}}_{i}=\left({r_{i}^{L},r_{i}^{M},r_{i}% ^{U}}\right)$ $\displaystyle\mspace{-75.0mu }=\textit{TFWEBM}_{w}^{p,q}\left({\tilde{{r}}_{i1% },\tilde{{r}}_{i2},\ldots,\tilde{{r}}_{in}}\right)$ $\displaystyle\mspace{-75.0mu }\scalebox{1.05}{$=\left({\frac{1}{n\left({n-1}% \right)}\mathop{\oplus}\limits_{{\begin{subarray}{l}k,l=1\\ k\neq l\\ \end{subarray}}}^{n}\left({\left({w_{k}\tilde{{r}}_{ik}}\right)^{p}\otimes% \left({w_{l}\tilde{{r}}_{il}}\right)^{q}}\right)}\right)^{\frac{1}{p+q}}$}$ $\displaystyle\mspace{-75.0mu }\scalebox{.98}{$=\left[{\left({\frac{1}{n\left({% n-1}\right)}\mathop{\oplus}\limits_{{\begin{subarray}{l}k,l=1\\ k\neq l\\ \end{subarray}}}^{n}\left({\left({w_{k}r_{ij}^{L}}\right)^{p}\otimes\left({w_{% l}r_{il}^{L}}\right)^{q}}\right)}\right)^{\frac{1}{p+q}},}\right.$}$ $\displaystyle\mspace{-75.0mu }\scalebox{1.05}{$\left({\frac{1}{n\left({n-1}% \right)}\mathop{\oplus}\limits_{{\begin{subarray}{l}k,l=1\\ k\neq l\\ \end{subarray}}}^{n}\left({\left({w_{k}r_{ij}^{M}}\right)^{p}\otimes\left({w_{% l}r_{il}^{M}}\right)^{q}}\right)}\right)^{\frac{1}{p+q}},$}$ $\displaystyle\mspace{-75.0mu }\scalebox{1.05}{$\left.{\left({\frac{1}{n\left({% n-1}\right)}\mathop{\oplus}\limits_{{\begin{subarray}{l}k,l=1\\ k\neq l\\ \end{subarray}}}^{n}\left({\left({w_{k}r_{ij}^{U}}\right)^{p}\otimes\left({w_{% l}r_{il}^{U}}\right)^{q}}\right)}\right)^{\frac{1}{p+q}}}\right]$}$ $\displaystyle\mspace{-75.0mu }\scalebox{.85}{$=\left[{\left({\frac{1}{n\left({% n-1}\right)}\mathop{\oplus}\limits_{{\begin{subarray}{l}k,l=1\\ k\neq l\\ \end{subarray}}}^{n}\left({\frac{\left({w_{k}r_{ij}^{L}}\right)^{p}\left({w_{l% }r_{il}^{L}}\right)^{q}}{1+\left({1-\left({w_{k}r_{ij}^{L}}\right)^{p}}\right)% \left({1-\left({w_{l}r_{il}^{L}}\right)^{q}}\right)}}\right)}\right)^{\frac{1}% {p+q}},}\right.$}$ $\displaystyle\mspace{-75.0mu }\scalebox{.9}{$\left({\frac{1}{n\left({n-1}% \right)}\mathop{\oplus}\limits_{{\begin{subarray}{l}k,l=1\\ k\neq l\\ \end{subarray}}}^{n}\left({\frac{\left({w_{k}r_{ij}^{M}}\right)^{p}\left({w_{l% }r_{il}^{M}}\right)^{q}}{1+\left({1-\left({w_{k}r_{ij}^{M}}\right)^{p}}\right)% \left({1-\left({w_{l}r_{il}^{M}}\right)^{q}}\right)}}\right)}\right)^{\frac{1}% {p+q}},$}$ $\displaystyle\mspace{-75.0mu }\scalebox{.9}{$\left.{\left({\frac{1}{n\left({n-% 1}\right)}\mathop{\oplus}\limits_{{\begin{subarray}{l}k,l=1\\ k\neq l\\ \end{subarray}}}^{n}\left({\frac{\left({w_{k}r_{ij}^{U}}\right)^{p}\left({w_{l% }r_{il}^{U}}\right)^{q}}{1+\left({1-\left({w_{k}r_{ij}^{U}}\right)^{p}}\right)% \left({1-\left({w_{l}r_{il}^{U}}\right)^{q}}\right)}}\right)}\right)^{\frac{1}% {p+q}}}\right]$}$ $\displaystyle\mspace{-75.0mu }i=1,2,\ldots,m,j=1,2,\ldots,n.$ (5)
Step 2.
To rank these collective overall preference values $\tilde{{r}}_{i}\left({i=1,2,\ldots,m}\right)$ , we first comp- are each $\tilde{{r}}_{i}$ with all the $\tilde{{r}}_{j}\left({j=1,2,\ldots,m}\right)$ by using Eq. (2). For simplicity, we let $p_{ij}=p\left({\tilde{{r}}_{i}\geqslant\tilde{{r}}_{j}}\right)$ , then we develop a complementary matrix as $P=\left({p_{ij}}\right)_{m\times m}$ , where $p_{ij}\geqslant 0$ , $p_{ij}+p_{ji}=1$ , $p_{ii}=0.5$ , $i,j=1,2,\ldots,$ $n$ .

Summing all the elements in each line of matrix $P$ , we have

$p_{i}=\sum\limits_{j=1}^{m}{p_{ij}},i=1,2,\ldots,m.$ (6)
Step 3.
Rank all the alternatives $X_{i}(i=1,2,\ldots,$ $m)$ and select the best one (s) in accordance with the collective overall preference values $p_{i}\left({i=1,2,\ldots,m}\right)$ .

4. Numerical example

With the development of productivity and market competition, environmental pollution caused by maximizing shareholder value, waste of resources, poor product quality, lack of effective protection of the interests of employees and other issues are more and more serious, which greatly hampered the long-term stable development of enterprises. Stakeholder theory which was developed in the west in 60 s of the 20th century considered that any business has many stakeholders, and they carried on special-purpose investment to the company and bear the investment risks. Therefore, the enterprise’s survival and development depends on its ability to effectively deal with the relationship among various stakeholders, and shareholders is just only one of them. Theorists and practitioners gradually realized that it is a short-sighted behavior for enterprises if they simply pursue the goal of shareholder value in the production process, regardless of the interests of other stakeholders’ requirements. The best way to the long-term survival and prosperity of enterprises is to consider and meet the interests and demands of all important stakeholders. Enterprise and stakeholder has a mutually beneficial relationship. From the perspective of stakeholders, as an supervision system and management system, the enterprise performance evaluation system is not only an enterprise’s important self-monitoring, self-restraint, self-evaluation means, but also a vital instrumentation to effectively communicating with stakeholders, coordinating each stakeholder’s interests and finally achieving the strategic management goal of enterprise. In this section, we utilize a practical multiple attribute decision making problems for enterprise performance evaluation to illustrate the developed approaches. The expert team selects four attributes to evaluate the enterprise performance of five enterprises: (1) G ${}_{1}$ : financial performance; (2) G ${}_{2}$ : customer performance; (3) G ${}_{3}$ : internal processes of performance; (4) G ${}_{4}$ : t staff performance. The five possible enterprises $A_{i}\left({i=1,2,\ldots,5}\right)$ are to be evaluated using the triangular fuzzy numbers under the above four attributes (whose weighting vector is $\omega=\left({0.3,0.2,0.4,0.1}\right)$ ), and construct the following matrix $A=\left({\tilde{{a}}_{ij}}\right)_{5\times 4}$ is shown as follows:

$\displaystyle\tilde{\bm{A}}=\left[{\begin{array}[]{l}\left[{{\rm 0.80},{\rm 0.% 85},{\rm 0.90}}\right]\left[{{\rm 0.72},{\rm 0.76},{\rm 0.80}}\right]\\ \left[{{\rm 0.88},{\rm 0.90},{\rm 0.93}}\right]\left[{{\rm 0.67},{\rm 0.77},{% \rm 0.83}}\right]\\ \left[{{\rm 0.95},{\rm 0.97},{\rm 0.98}}\right]\left[{{\rm 0.90},{\rm 0.93},{% \rm 0.95}}\right]\\ \left[{{\rm 0.82},{\rm 0.85},{\rm 0.88}}\right]\left[{{\rm 0.97},{\rm 0.98},{% \rm 1.00}}\right]\\ \left[{{\rm 0.78},{\rm 0.79},{\rm 0.81}}\right]\left[{{\rm 0.78},{\rm 0.79},{% \rm 0.81}}\right]\\ \end{array}}\right.$ $\displaystyle\quad∼{}∼{}∼{}\left.{\begin{array}[]{l}\left[{{\rm 0.91},{\rm 0.9% 3},{\rm 0.96}}\right]\left[{{\rm 0.62},{\rm 0.65},{\rm 0.68}}\right]\\ \left[{{\rm 0.60},{\rm 0.67},{\rm 0.70}}\right]\left[{{\rm 0.69},{\rm 0.72},{% \rm 0.75}}\right]\\ \left[{{\rm 0.77},{\rm 0.79},{\rm 0.82}}\right]\left[{{\rm 0.93},{\rm 0.95},{% \rm 0.96}}\right]\\ \left[{{\rm 0.98},{\rm 0.99},{\rm 1.00}}\right]\left[{{\rm 0.97},{\rm 0.99},{% \rm 1.00}}\right]\\ \left[{{\rm 0.83},{\rm 0.85},{\rm 0.88}}\right]\left[{{\rm 0.94},{\rm 0.97},{% \rm 0.99}}\right]\\ \end{array}}\right]$

In the following, in order to select the most desirable enterprises, we utilize the TFWEBM operator to solve the multiple attribute decision making problems for enterprise performance evaluation.

Step 2. Step 1.
Aggregate all triangular fuzzy value $\tilde{{r}}_{ij}(j=1,2,\ldots,n)$ by using the TFWEBM to derive the overall triangular fuzzy preference values $\tilde{{r}}_{i}\left({i=1,2,3,4,5}\right)$ of the enterprises $A_{i}$ .

$\displaystyle\begin{array}[]{l}\tilde{{r}}_{1}=\left[{{\rm 0.1683,0.1807},{\rm 0% .1954}}\right]\\ \tilde{{r}}_{2}=\left[{{\rm 0.1543},{\rm 0.1707},{\rm 0.1840}}\right]\\ \tilde{{r}}_{3}=\left[{{\rm 0.1910},{\rm 0.2008},{\rm 0.2123}}\right]\\ \tilde{{r}}_{4}=\left[{{\rm 0.2156},{\rm 0.2261},{\rm 0.2378}}\right]\\ \tilde{{r}}_{5}=\left[{{\rm 0.2014},{\rm 0.2126},{\rm 0.2252}}\right]\\ \end{array}$
Step 2.
Rank all the enterprise $A_{i}(i=1,2,\ldots,5)$ in accordance with the preference degree $p_{i}\left({i=1,2,\ldots,5}\right)$ : $A_{4}\succ A_{5}\succ A_{3}\succ A_{1}\succ A_{2}$ , and thus the most desirable enterprises is $A_{4}$ .

5. Conclusion

Enterprises are the organic integrity of stockholders, creditor, customer, employee, government and other stakeholders. The stakeholder’s interest is different from each other. When evaluates an enterprise’ performance, it means that takes the stockholders’ benefits as the target and the only criterion would be obviously unilateral and unscientific. And some disadvantages have appeared in this kind of performance evaluating system. And the enterprises of public owner have double character: economic and society, so when we evaluate their performance, all the stakeholders’ benefits should be considered. We should set up a Performance Evaluating System on the base of stakeholders’ benefits. It is fit to the need of construction of coordinate society. In this paper, we investigate the multiple attribute decision making (MADM) problems for enterprise performance evaluation with triangular fuzzy information. We utilize the triangular fuzzy weighted Einstein Bonferroni mean (TFWEBM) operator to develop the procedure for multiple attribute decision making under the triangular fuzzy environments. Finally, a practical example for enterprise performance evaluation is given to verify the developed approach. In the future, our results may be further generalized by using the other uncertain decision making theories and methods [57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76].

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