#### Abstract

This paper proposes an extended technique for order preference by similarity to ideal solution (TOPSIS) for partner selection in a virtual enterprise (VE). The imprecise and fuzzy information of the partner candidate and the risk preferences of decision makers are both considered in the group multiattribute decision-making model. The weighted possibilistic mean values are used to handle triangular fuzzy numbers in the fuzzy environment. A ranking procedure for partner candidates is developed to help decision makers with varying risk preferences select the most suitable partners. Numerical examples are presented to reflect the feasibility and efficiency of the proposed TOPSIS. Results show that the varying risk preferences of decision makers play a significant role in the partner selection process in VE under a fuzzy environment.

#### 1. Introduction

A virtual enterprise (VE) is a type of temporary alliance of independent, geographically dispersed organizations that aim to share skills and resources to exploit low-cost fast-changing market opportunities and achieve high-quality customer satisfaction [1–6]. VEs have become prevalent because of the increasing customer demands in the present global economy and the increasing complexity and diminishing life cycle of products. A VE is formed and dissolved by the appearance and disappearance of market opportunities, respectively.

A VE faces many important issues throughout its life cycle. Given the role of partners in the success of a VE, the proper selection of partners has received considerable research attention. Literature on partner selection can be divided into two categories.

The first literature category involves the information environment, wherein the verdicts of decision makers on candidate partners can be expressed in precise values. Various approaches have been proposed for VE partner selection. Talluri and Baker [7] proposed a two-phase mathematics programming method. Wu et al. [8] developed an integer-programming method to minimize the transportation cost by geographic position and transportation approach. Ip et al. [1] described a risk-based partner selection problem and developed a mathematical programming model. Jarimo and Pulkkinen [9] presented a mixed-integer linear programming model to configure the virtual organization. Zhao et al. [10] developed a nonlinear integer-programming model to solve VE partner selection problems with precedence and due date constraints. Ng [11] proposed a weighted linear program to solve the multicriteria supplier selection problem. Other researchers simultaneously considered several factors such as cost, quality, credit, time, and risk for VE partner selection, thus making the partner selection problem for VE a type of multiattribute decision-making (MADM) problem. MADM methods, such as analytic hierarchy process (AHP), data envelopment analysis (DEA), and neural networks (NNs), have been developed to solve problems in VE partner selection. Sha and Che [12] developed a partner selection model based on AHP, multiattribute utility theory, and integer programming (IP) for multicriteria virtual integration; Sari et al. [13] applied the NN method to assess the performance of a particular partner in a VE. Wu [14] combined DEA, decision trees, and NNs to assess supplier performance. Liou [15] proposed a hybrid model to help airline companies select suitable partners for strategic alliances.

In reality, the decision makers are generally unsure of their judgement on candidates because the information about these candidates is uncertain and vague [4]. Thus, the second literature category considers the real-life vagueness and uncertainty of the partner selection process. Such problems can only be resolved by fuzzy set theory. Mikhailov [16] developed a fuzzy preference programming method for VE partner selection. Kahraman et al. [17] applied fuzzy AHP to select the supplier that highly satisfies the determined criteria. Wang and Lin [18] developed a fuzzy hybrid decision-aid model to select the best partner. Golec and Kahya [19] proposed a fuzzy model for competency-based employee evaluation and selection. Guneri et al. [20] presented an integrated fuzzy and linear programming approach for selecting suppliers and presented a set of linguistic values that are expressed in trapezoidal fuzzy numbers to assess the weights and ratings of the supplier selection criteria. Moreover, Crispim and Soua [3, 21] applied a fuzzy TOPSIS algorithm to rank alternative VE configurations. Ye and Li [4] proposed two MADM models with interval values to solve partner selection problems with partial information. Ye [5] proposed an extended TOPSIS method with interval-valued intuitionistic fuzzy numbers. Tseng [22] used linguistic preferences to describe the weights of green supply chain criteria and applied grey and fuzzy set theories to rank alternatives. Shaw et al. [23] used fuzzy AHP and fuzzy multiobjective linear programming to select the suitable supplier for the development of a low carbon supply chain. Liao and Kao [24] proposed an integrated fuzzy TOPSIS and multichoice goal programming approach to select suppliers in a supply chain.

However, these previous studies did not consider the risk preferences of decision makers in their proposed methods. The existence of contradicting methods in practice reflects the varying risk preference and aversion to risk neutrality of managers, particularly when they are placed under an uncertain environment. Given that risk-averse and risk-prone decision makers behave differently under similar situations, the risk preferences of decision makers must be considered when setting applicable and tailored decisions.

This paper uses fuzzy set theory to solve the vague and uncertain problems in the partner selection process. In contrast to the studies in the second literature category, we consider the risk preferences of decision makers to develop a partner selection approach that is consistent with actual practices. An extended TOPSIS is proposed in this study. The proposed extended TOPSIS considers the imprecise and uncertain information of partner candidates and the risk preferences of decision makers in the group MADM model. Weighted possibilistic mean values are used to manage triangular fuzzy numbers and establish the proposed TOPSIS model.

The rest of the paper is organized as follows. Section 2 introduces the basic concepts on the possibilistic mean values of fuzzy numbers. Section 3 presents the group MADM problem with triangular fuzzy numbers. Section 4 outlines the development of the extended TOPSIS. Section 5 presents an illustrative example. Section 6 summarizes the paper.

#### 2. Possibilistic Mean Values of Fuzzy Numbers

In this section, we introduce some basic concepts related to possibilistic mean values of fuzzy numbers. First, we assume that a fuzzy number is a fuzzy set of the real line with a normal, fuzzy convex, and continuous membership function of bounded support. The family of fuzzy numbers can be denoted by . A -level set of a fuzzy number is defined by as and(the closure of the support of ), as [25, 26].

*Definition 1 (see [25, 26]). *Let be a fuzzy number with . The lower and upper possibilistic mean values of fuzzy number with -level set can be defined as and .

*Definition 2 (see [25]). *The interval-valued possibilistic mean of can be defined as . However, the crisp possibilistic mean value of is defined as .

According to Definition 2, Zhang et al. [26] defined the weighted possibilistic mean of fuzzy number as

If , then ; if , then ; if , then . Obviously, for different , we can give different importance to the lower and upper possibilistic mean values.

Researchers have focused on the computing of different approximations of fuzzy numbers, within which triangular fuzzy numbers and trapezoidal fuzzy numbers are the most popular fuzzy analysis methods. In this paper, we use triangular fuzzy numbers to express the judgments of decision makers on the information of the candidates. Let be a triangular fuzzy number with center , left width , and right width ; . According to Definition 2, a -level set of the fuzzy number can easily be computed as

According to Definition 1, we can get

Therefore, the interval-valued possibilistic mean value and crisp possibilistic mean value of can be written as

Finally, the weighted possibilistic mean value of fuzzy number is computed as

#### 3. Partner Selection Problem Description and Notations

Let us consider a core enterprise getting a bid for a large project consisting of several subprojects. The core enterprise cannot complete the whole project only by its own ability. Therefore, the core enterprise has to select partners and form a VE to complete the project. The partner selection problem of a VE is described as follows: represents a finite set of possible candidates, represents a finite set of attributes according to the desirability judgment on a candidate.

Assuming the candidates’ information is imprecise and fuzzy, the decision maker utilizes a triangular fuzzy number with center , left-width , and right-width ; to evaluate candidate with respect to attribute . Let be the decision matrix in the form of triangular fuzzy numbers. Then the group MADM problem with triangular fuzzy numbers can be expressed in matrix format as follows:

Each decision maker will elicit weights for attribute as , where . belongs to and sums to one [18]; that is,

The values of different attributes have different dimensions. Thus, the fuzzy decision matrix should be standardized with matrix in order to reduce disturbance in the final results. Let be the standardized decision matrix in the form of triangular fuzzy numbers, where .

In general, there are two attribute categories for candidates: benefit type and cost type. The higher the benefit type value is, the better it will be. It is opposite for the cost type. We describe the normalized formula for triangular fuzzy numbers of the benefit type as follows:

Similarly, the formula for the triangular fuzzy number of the cost type is described as follows:

Now the triangular fuzzy number is the normalized form of triangular fuzzy number .

According to formula (5), we can get the weighted possibilistic mean value of triangular fuzzy number as follows: where reflects the decision maker ’s risk preference coefficient; a larger implies more risk-prone decision maker, whereas a smaller indicates a more risk-averse decision maker. Specially, , , and represent that the decision maker is extremely risk averse, risk neutral, and extremely risk prone, respectively.

Let , be the decision matrix with the weighted possibilistic mean value of triangular fuzzy number, where .

#### 4. An Extended TOPSIS Approach with Weighted Possibilistic Mean Values for Partner Selection

Yoon [27] developed a TOPSIS for multiattribute decision making. Now, TOPSIS is widely applied across different areas among numerous MADM methods and has received interest from both researchers and practitioners [28]. Hwang and Yoon [29] originally proposed TOPSIS to select the best candidate with a finite number of criteria. Kuo et al. [30] applied fuzzy SAW and fuzzy TOPSIS to select the location of an international distribution center in Asia Pacific. Liao and Kao [24] used fuzzy TOPSIS to select supplier in supply chain management. Boran et al. [31] employed an intuitionistic fuzzy TOPSIS approach to select a sales manager. Behzadian et al. [28] conducted a state-of-the-art literature review on TOPSIS research. Tian et al. [32] developed fuzzy TOPSIS model via chi-square test for information source selection. In this paper, we proposed a new extended TOPSIS method with weighted possibilistic mean values to solve partner selection problem of a VE.

Based on the TOPSIS concept, the chosen candidate should have the shortest distance from the positive ideal partner solution (PIPS) and the farthest distance from the negative ideal partner solution (NIPS). Hence, we identify PIPS and NIPS for each decision maker as follows: where .

By using the -dimensional Euclidean distance, the separation of each candidate from the PIPS for the decision maker is given as

Similarly, the separation of each candidate from the NIPS for the decision maker is given as

A closeness coefficient is defined to determine the ranking of all candidates once and of each candidate for decision maker is calculated.

The relative closeness of the candidate for decision maker is defined as

Obviously, for decision maker , the candidate is closer to the PIPS and farther from the NIPS as approaches to 1. Hence, according to the closeness coefficient, the ranking of all candidates can be determined, and decision maker selects the best one among a set of feasible candidates.

In addition, the group separation measure of each candidate is combined through an operation for all decision makers (). The two group separation measures of the PIPS and NIPS, and , respectively, can be computed by the following formulas [4, 33]:

Here, we take the geometric mean operation to combine all individual separation measures by the following formulas:

Hence, the relative closeness of the candidate is defined as

The procedure to find out the best partner with the extended TOPSIS method for group MADM with weighted possibilistic mean values is developed as follows.

*Step 1. *Make up a group decision-making team .

*Step 2. *Generate a set of possible candidates for subproject .

*Step 3. *Design a set of attributes .

*Step 4. *Construct the decision matrix with triangular fuzzy numbers.

*Step 5. *Each decision maker elicits weights for the attribute as , where .

*Step 6. *Compute normalized ratings. This step tries to transform various attribute dimensions into the nondimensional attributes, which allows comparison between the attributes. Formulas (8) and (9) are used for computing the normalized triangular fuzzy numbers .

*Step 7. *Calculate weighted possibilistic mean value of triangular fuzzy number by using formula (10).

*Step 8. *Identify PIPS and NIPS by using formula (11).

*Step 9. *Compute separation measure between candidate and PIPS for each decision maker by using formula (12).

*Step 10. *Calculate separation measure between candidate and NIPS for each decision maker by using formula (13).

*Step 11. *Calculate the closeness coefficient of candidate for the decision maker by using formula (14).

*Step 12. *Rank the preference order of all candidates according to the closeness coefficient and select the best partner for decision maker .

*Step 13. *Aggregate the separation measure for the group by using formula (16).

*Step 14. *Calculate the closeness coefficient of candidate for the group by using formula (17).

*Step 15. *Rank the preference order of all candidates according to the closeness coefficient and select the best one.

#### 5. Numerical Example

We present a numerical example to illustrate our proposed approach. Four decision makers, namely, , , , and , from the core enterprise are selected to form the decision-making team. These decision makers are then tasked to select the best partner out of four candidates, namely, , , , and , in forming a new VE. The six attributes of each candidate, including cost (), past performance (), relationship closeness (), completion probability (), time (), and quality (), are considered in the selection process. and are included in the cost-type attributes, whereas , , , and are included in the benefit-type attributes. Figure 1 shows the hierarchical structure of the partner selection problem, and Table 1 shows the decision matrix that contains the triangular fuzzy numbers and weights of the six attributes of each decision maker.

The selection process is conducted by the procedures outlined in Section 4. We calculate the normalized decision matrix of each decision maker with triangular fuzzy numbers, as well as the normalized decision matrix for each decision maker with weighted possibilistic mean values; the results are shown in Tables 2 and 3.

We identify the PIPS and NIPS of each decision maker by using (11); the results are shown in Table 4. The positive and negative separation measures between each candidate for each decision maker are calculated by using (12) and (13), respectively; the results are shown in Table 5.

The closeness coefficient is used to determine the preference order of each candidate for each decision maker; the results are shown in Table 6 and Figures 2, 3, 4, and 5.

For , the candidates are ranked as when ( = superior to) but are ranked as when. For , the candidates are ranked as when or 0.3 but are ranked as when .Therefore, is identified as the best candidate for , and or may be chosen as the partner for .

Decision makers may experience problems in reaching a consensus during a group decision-making process. Therefore, these decision makers must combine their assessments to form a highly reasonable evaluation. We obtain the aggregated group separation measures of all candidates by using (16); the results are shown in Table 7. The aggregated closed closeness coefficients and ranking of all candidates are calculated by using (17); the results are shown in Table 7 and Figure 6. is identified as the best candidate in all four cases.

#### 6. Discussions and Conclusions

We have proposed an extended TOPSIS for partner selection in a VE. Different from traditional methods, our proposed method considers the varying risk preferences of decision makers. We also use weighted possibilistic mean values to manage candidate information formed by triangular fuzzy numbers.

We perform sensitivity analyses by using a wide range of risk preference coefficient of decision makers to determine the effects of risk preferences on candidate rank. The value of may range between zero and one, which covers three categories, namely, extremely risk averse, risk neutral, and extremely risk prone. Our numerical examples show that the ranking of candidates changes with the shifting risk preferences of decision makers. For example, for the first decision maker (), the fourth candidate () is selected as the best partner when is extremely risk averse () and the first candidate () is selected as the best partner when is less risk averse, risk neutral, or extremely risk prone (). Other decision makers exhibit the same behavior in their selections. These observations confirm that the risk preferences of decision makers must be considered to produce results that closely reflect real-life practices. Our proposed approach can guide the core enterprise of a VE in selecting a suitable partner, particularly during situations when decision makers exhibit varying risk preferences.

Our extended TOPSIS method may also be applied for other purposes, such as for performance evaluation and for the selection of green suppliers, information systems, facility locations, construction technique alternatives, and human resource arrangements. Future research may investigate the group MADM model with the weighted possibilistic mean values of trapezoidal fuzzy numbers and include the weighted possibilistic variance of fuzzy numbers in the MADM model.

#### Acknowledgments

This research is supported by Natural Science Foundation of China (71001041, 71172075, 71371006, and 71090403), Program for New Century Excellent Talents in University (NCET-13-0219), Research Fund for the Doctoral Program of Higher Education of China (20130172110029), and the Fundamental Research Funds for the Central Universities, SCUT, (2013ZZ0093, x2gsD2133310).