Research Article  Open Access
Feng Wang, Junjun Mao, "Approach to Multicriteria Group Decision Making with ZNumbers Based on TOPSIS and Power Aggregation Operators", Mathematical Problems in Engineering, vol. 2019, Article ID 3014387, 18 pages, 2019. https://doi.org/10.1155/2019/3014387
Approach to Multicriteria Group Decision Making with ZNumbers Based on TOPSIS and Power Aggregation Operators
Abstract
Decisions strongly rely on information, so the information must be reliable, yet most of the realworld information is imprecise and uncertain. The reliability of the information about decision analysis should be measured. Znumber, which incorporates a restraint of evaluation on investigated objects and the corresponding degree of confidence, is considered as a powerful tool to characterize this information. In this paper, we develop a novel approach based on TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method and the power aggregation operators for solving the multiple criteria group decision making (MCGDM) problem where the weight information for decision makers (DMs) and criteria is incomplete. In the MCGDM, the evaluation information made by DMs is represented in the form of linguistic terms and the following calculation is performed using Znumbers. First, we establish an optimization model based on similarity measure to determine the weights of DMs and a linear programming model with partial weight information provided by DMs based on distance measure to determine the weights of criteria. Subsequently, decision matrices from all the DMs are aggregated into a comprehensive evaluation matrix utilizing the proposed ZWAPA operator or ZWGPA operator. Then, those considered alternatives are ranked in accordance with TOPSIS idea and the feature of Zevaluation. Finally, a practical example about supplier selection is given to demonstrate the detailed implementation process of the proposed approach, and the feasibility and validity of the approach are verified by comparisons with some existing approaches.
1. Introduction
Since the fuzzy set theory was coined by Zadeh [1], it has been widely used in many application fields as a tool that is capable of capturing the certainty of information and depicting peopleâ€™s subjective thinking, especially decisionmaking problems with uncertain, imprecise even incomplete information. After that, the theory has been developing and type2 fuzzy set [2], fuzzy multiset [3], intuitionistic fuzzy set [4â€“6], hesitant fuzzy set [7, 8] and their interval forms [9â€“11], various fuzzy numbers like triangular fuzzy number [12], trapezoidal fuzzy number [13], etc. appear successively. The classic fuzzy set and its generalizations and extensions successfully address lots of challenges from the complexity of practical problems but they do not take well into consideration the reliability of the information that is provided. The reliability of the evaluation is an important aspect in solving the decision making problem where it will affect the final outcome of the evaluation. Znumber, a novel fuzzy number with confidence degree, is pioneered by Zadeh [14] to overcome this limitation. Znumber is a 2tuple fuzzy numbers that includes the restriction of the evaluation and the reliability of the judgment. We use the first fuzzy number to represent the uncertainty of evaluation and the second fuzzy number for a measure of certainty or other related concepts such as sureness, confidence, reliability, strength of truth, or probability for the first fuzzy number. Znumber has more ability to describe the knowledge of human since it considers the level of confidence of evaluators. Owing to the inclusion of the reliability factor, Znumbers involve more uncertainties than fuzzy numbers. They can provide us with additional degree of freedom to represent the uncertainty and fuzziness of the real situation. It provides a new perspective for us in the decision making area. Therefore, using Znumbers to model and process uncertainty of the reality is more reasonable and applicable than using fuzzy numbers.
Traditionally the given evaluations in multiple criteria decision making (MCDM) are real numbers. As the complexity of the problem we investigate, DMsâ€™ opinions are usually expressed only in a fairly vague manner where it cannot be measured numerically. Instead, they normally use terms like â€śpoor,â€ť â€śfair,â€ť and â€śgoodâ€ť to express the judgment on an alternative, which is closer to peopleâ€™s perception. Then, the problem of how to compute with terms or words arises. It has gained enough attention by some researchers and they use typically fuzzy number to represent and compute linguistic terms which is able to seize the impreciseness of this information and handle subjective judgment of human well. In recent years, linguistic variables whose values are words or short sentences from natural or artificial languages have been researched extensively and various related fruits [14â€“21] are achieved. Because of the superiority of Znumber in expressing vague information, utilizing Znumber to characterize the issue with words is a new perspective. It is noted that the two components of Znumbers are mostly described in natural language [14]. For DMs, the scale of the provided linguistic words is limited, so it is possible that two alternatives get the same semantic assessment under some criterion but small difference may exist between them. Znumber can distinguish this difference because expertsâ€™ confidence degrees are different when giving the criterion values. The second component of Znumber is precisely capable of playing this role. At present the research involving Znumbers mainly focuses on the decision making issue with linguistic variables. In most illdefined decision environment, it is preferable for a DM to employ linguistic variables rather than real numbers when making assessments.
In decision making issue, the core is to determine the ranking order of alternatives. Hence choosing an effective method to rank Znumbers gets very important when conducting decision making problem in Znumber information. Kang et al. [22] devised a method to transform a Znumber into a fuzzy number based on the fuzzy expectation of the second component of Znumber with Centroid Method, which highlights the importance of the first part in a Znumber compared to the second and reduces successfully a Znumber to a classical fuzzy number. Due to that, many researchers usually first converted Znumbers to fuzzy numbers on the basis of the method provided in [22] and then implemented the related calculation on the fuzzy numbers that has been converted in the course of decision making. For example, authors in [23] first integrated the preference values given by DMs to a single value, then converted the integrated result (Znumber) to generalized fuzzy number, and finally calculated the spread and the fuzzy score value of standardized fuzzy number to attain ranking order of Znumbers. Authors in [24] employed the same way to transform Znumbers to fuzzy numbers before ordering Znumbers, the whole course that rank Znumbers was analogous to [23], and the research made lots of analysis and discussion on different Znumber combinations to investigate the effectiveness of the proposed ranking approach. Kang and his team focused on solving supplier selection problem by utilizing the methodology presented in that paper including two parts: one changed a Znumber to a traditional fuzzy number according to the fuzzy expectation; the other worked out the optimal priority weight for supplier selection with the improved genetic algorithm and the extended fuzzy AHP under Znumber environment in [25]. In the next few years, they developed a new uncertainty measure of fuzzy set considering the influence of fuzziness measure and the range (or cardinality) of the fuzzy set and proposed a method of generating Znumber based on the OWA weights using maximum entropy considering the attitude (preference) of the decision maker in [26, 27]. In 2008, they proposed a stable strategies analysis based on the utility of Znumber and apply it to the evolutionary games in [28]. The paper took full advantage of the structure of Znumbers to simulate the procedure of humans competition and cooperation. The methodology extends the classical evolutionary games into linguisticbased ones, which is quite applicable in the real situations. Authors in [29] integrated a Jaccard similarity measure of Znumbers to solve MCGDM problem. After linguistic values of DMs are converted to Znumbers, the method used the idea from Kang et al. [22] to transform Znumbers into trapezoidal fuzzy numbers (TpFNs) and then applied the Jaccard similarity measure between the expected intervals of TpFNs and the ideal alternative to attain the final decision. Authors in [30] put forward a multilayer methodology for ranking Znumbers, which consists of two layers: converting Znumbers to generalised TpFNs as the first layer and using CPS method based on centroid point and spread to rank the generalised fuzzy numbers that have been converted, as the second layer. The methodology is applicable to both positive and negative data values for alternatives and is considered as a generic decision making procedure. Different from those just mentioned literatures, Xiao [31] first converted Znumber to an intervalvalued fuzzy set with footprint of uncertainty and then defuzzified the interval fuzzy sets as crisp numbers with the wellknown KM algorithm [32]. Authors in [33] converted directly Znumbers to crisp numbers by performing the multiplication operation on the two parts of Znumbers (represented by TFNs) where both TFNs were defuzzified to crisp numbers via the graded mean integration and then multiplied into a crisp number. There is no doubt that the approach seems a bit rough and the information of Znumbers originally included is markedly reduced.
The above references always convert Znumbers to classical fuzzy numbers or generalized fuzzy numbers in handling the linguistic data before performing the real decision. This way can result in the loss of some valuable Zinformation and may give rise to the unreasonable phenomenon that two different Znumbers may be converted to the identical fuzzy number. Some researchers have been aware of the drawback and made some improvements. Zamri et al. [34] applied the fuzzy TOPSIS to Znumbers to handle uncertainty in the construction problem. In that paper, the distances of two parts in Zevaluations from the PIS and the NIS were calculated, respectively, without any interaction. The structure of Znumbers was not heavily damaged in the whole decision process such that less useful information of Znumbers was lost. Peng and Wang [35] developed an innovative method to address MCGDM problems where the weight information for DMs and criteria is incompletely known by introducing hesitant uncertain linguistic Znumbers (HULZNs). First it defined operations and distance of HULZNs by means of linguistic scale functions, then developed two power aggregation operators for HULZNs, and finally incorporated the proposed operators and the VIKOR model to solve the ERP system selection problem. The method system combined the advantages of Znumbers and linguistic models with deep theory and rigorous logic. Jiang et al. [36] gave an improved method for ranking generalized fuzzy numbers, which took into consideration the weight of centroid points, degrees of fuzziness, and the spreads of fuzzy numbers. Subsequently, this method was extended for ranking Znumbers and the final step of the procedures revised the score function of Znumbers in view of different status of the two parts of Znumbers. The methodology retains the fuzzy information of the reliability instead of converting the reliability part of a Znumber to a crisp number as many existing methods did and takes the full advantage that Znumber can express more vague information compared to other fuzzy numbers.
In general, there are not too many researches on Znumbers so far and relevant achievements are limited. This paper tries to gain some breakthrough in this aspect on the basis of the aforementioned references. Suppose that there is a collection of Znumbers . is the main part of and is the subsidiary factor. can influence the ranking result but cannot decide the result, while can. Therefore, the weight of should be larger than when processing the decision making problem with Znumbers. Besides, the characteristics of Znumber should attempt to be kept and the information of Znumbers should be retained as much as possible. In view of these considerations, we will do some improved work and develop another MCGDM approach by applying the simple and practical TOPSIS method to the decision information characterized by Znumbers where the weight information of DMs is entirely unknown and the weight for criteria is partly known. The decision procedures presented in consideration of various data weights manage to hold the previous amount of information. With these points, the remainder of this work is organized as follows: Section 2 briefly reviews some necessary concepts such as Znumber and triangular fuzzy number (TFN) and defines the operations of Znumbers whose both parts are expressed by TFNs. Section 3 defines the distance of Znumbers from the perspective of TFNs and shows the properties the distance measure possesses. Section 4 develops two power aggregation operators by extending the PA operator to Znumbers and discusses comprehensively their properties. In Section 5, a MCGDM methodology in Zinformation is proposed, which contains two phases: the first phase is to determine successively the weight vectors for DMs and criteria by establishing optimization models under the two circumstances that the prior information is completely unknown and incompletely unknown, respectively; the second makes the concrete decision procedures which combines the power aggregation operators towards Znumbers and the TOPSIS method. Section 6 shows the decisionmaking course through a numerical example about supplier selection and conducts a careful comparison with the existing approaches to illustrate the validity and superiority of the suggested approach. Finally, the conclusions for the main points of this paper are drawn in Section 7.
2. Preliminaries
This section mainly introduces some basic concepts relating to Znumber, which will be needed in subsequent sections.
Definition 1 (see [14]). A Znumbers is an ordered pair of fuzzy numbers, denoted by , which is associated with a realvalued uncertain variable . The first component of Znumber, , is a restriction on the values that can take. The second component is a measure of reliability of the first component such as confidence, sureness, strength of belief, and probability or possibility.
A Znumber gives information about not only uncertain variable, but also the reliability of the information. The first component is the principal part of a Znumber while the second one is just an explanation for . Typically, and are perceptionbased values which are described in natural language. For example, â€śtravel time by car from Berkeley to San Francisco, about 30 min, usually,â€ť â€śdegree of Robertâ€™s honesty, high, not sure,â€ť and â€śtomorrowâ€™s weather is warm, likely.â€ť For simplicity, these natural languages are presented by linguistic terms, that is, â€ś30 min, usuallyâ€ť, â€śhigh, not sure,â€ť and â€śwarm, likely.â€ť
Due to the complexity of real problems, many domain experts give their opinions via fuzzy numbers. For instance, in a new product launch price forecast, one expert may give his opinion like this: the lowest price is 2 dollars, the most possible price may be 3 dollars, and the highest price will not exceed 4 dollars. Thus we can use a triangular fuzzy number (2, 3, 4) to express the expertâ€™s idea. Throughout this paper, the linguistic variables from the two components of Znumbers, and , are represented by triangular fuzzy numbers. Accordingly, the triangular fuzzy number is defined below.
Definition 2 (see [12]). A triangular fuzzy number can be denoted by a triplet , whose membership function is determined as follows:
For convenience, we signify the triangular fuzzy number with TFN for short in what follows. The TFN expressed by (1) can be shown in Figure 1.
Lacking of typical properties for Znumber forces us to simplify its representation. In order to take advantage of Znumberâ€™s trait and the relation of the two components in Znumberâ€™s structure, we introduce Kang et al.â€™s conversion thinking [22].
Let a Znumber and convert the reliability of into a crisp value using Centroid Method where denotes an algebraic integration.
Since , (6) becomes the following equation owing to the membership function of a TFN:
Subsequently, take the centroid value of the reliability as the weight of the restriction and add the weight from the second part to the first part . Thereby the weighted Znumber can be written as .
Because both parts of Znumber are characterized by TFNs in the whole article, the partial operations of Znumbers will follow those of TFNs. In light of the structure and meaning of Znumbers, we define some operational laws concerning Znumbers combining with the operations of TFNs.
Definition 3. Assume that are any two Znumbers; then normalize their second components as
,
,
where
(1) , ;
(2) , , ;
(3) , ;
(4) , , .
According to (2),
In Definition 3, the reliability measure of Znumbers is unified in a scale of 0 to 1 in order to be compared easily by normalizing them. In the operations of the first components of Znumbers, they are given their respective weights from the degradation value of the corresponding second components by (2) in order that the supplementary role of the confidence component to the restriction component in a Znumber is highlighted. In Section 4, two power aggregation operators on Znumbers are developed and construction of the operators will involve the operations of Definition 3. In the last application part of this paper, we will conduct a large amount of calculations where the four operation rules above will be used frequently.
Next, we give the ranking rule for Znumbers used for decisionmaking.
Definition 4. Assume that are arbitrary two Znumbers and are all normal fuzzy numbers, .
If , then ;
If , thus we have the following: (i), then ;(ii), then .
For the preference orders of the fuzzy numbers , there have been varieties of existing methods to determine them.
3. Distance Measure of ZNumbers
There have been numerous distance formulas for previous fuzzy numbers such as Hamming distance, Euclidean distance, and Hausdorff distance. With the aid of the concept of crossentropy in information theory, we construct a new distance formula to measure the discrimination uncertain information of TFNs before defining the distance measure of Znumbers.
Definition 5. Let and be TFNs, used for describing the uncertain information of two objects; then their crossentropy is defined by
It is noticeable that (4) is meaningless if or ; thereupon we make the following extension.
Remark 6. For the cases and , we reverse the position of and of (4); that is, turn into , that is, ; if and , naturally the distance between them should be 0; hence we take the cross entropy at this point.
The cross entropy measure in fuzzy sets is able to discriminate effectively different fuzzy information, so it can be considered as an information distance. Despite its asymmetry of (4), it is necessary to symmetrize the crossentropy so as to become a real distance measure as follows:
Clearly, the distance measure of (5) satisfies the three basic properties that ordinary distance measure in fuzzy environment has (see Theorem 7)
Theorem 7. Let be two arbitrary TFNs and ; then
(1) ;
(2) ;
(3) .
Proof. Items (1) and (2) apparently hold, so we primarily prove item (3).
and , . Let ( is a constant); then is a monotonic function. Thus if or , the solution is unique. Apparently the only solution is ; that is, .
In this paper, both components of Znumber are characterized by TFNs, so the distance between Znumbers can be broken down into the distance for the two TFNs. Thus, we can give the distance formula for Znumbers as follows.
Definition 8. For two arbitrary Znumbers , where , then where .
From (5) and (6), we easily know that satisfies the following properties.
If are two Znumbers, then
(1) ;
(2) ;
(3) .
Through the distance measure between Znumbers in Definition 8, we notice that the influence factor of the first component of Znumber is higher than the second.
4. Power Aggregation Operator for ZNumbers
Power average (PA) operator was pioneered by Yager [37] in 2001. As an effective data aggregation tool, it allows input values to support each other and provide more versatility. This section manages to apply PA operator in the situation where the input arguments are Znumbers and derive two aggregation operators with desirable properties based on the PA operator.
Definition 9. PA operator is a mapping: that is given by the following variate function: where and is called the support for from , which satisfies the following three properties:
(1) ;
(2) ;
(3) , if .
Motivated by the PA operator and the geometric average operator (GA), Xu and Yager [38] developed a power geometric (PG) operator as follows.
Definition 10. PG operator is a mapping: that is given by the following variate function: where is the same as that of Definition 9.
Combining the PA operator and the weighted arithmetic average (WAA) operator, we can deduce a weighted arithmetic power average operator with Znumbers ZWAPA.
Definition 11. Let be a collection of Znumbers, let be the family of all Znumbers, and corresponds to the weight of , with and ; then the ZWAPA operator is this mapping: , which is defined as where and is the support for from with the following conditions:
(1) ;
(2) ;
(3) , if , where is the distance measure between Znumbers.
From Definition 11, it can be deduced that the support measure can be used to measure the proximity of a preference value in the form of Znumber provided by a DM to another. The closer the two preferences and , the smaller the distance of both, and the more they support each other.
It is evident that the aggregated result using ZWAPA operator is still a Znumber. Moreover, it can be easily proved that ZWAPA operator has the following properties.
Theorem 12. Let be a collection of Znumbers; if for all , then where is a constant.
Theorem 12 implies that when all the supports between Znumbers are equal, the ZWAPA operator reduces to the WAA operator.
Theorem 13 (commutativity). Let be a collection of Znumbers and let be any permutation of . If the weights are irrelevant to the position of arguments, then
Theorem 14 (idempotency). Let be a set of Znumbers; if for all , then
Theorem 15 (boundedness). Let be a set of Znumbers; then where and .
Combining PA operator with the weighted geometric average (WGA) operator, we can obtain a weighted geometric power average operator in Znumbers case ZWGPA.
Definition 16. Let be a collection of Znumbers, let be the set of all Znumbers, and is the weight of , with and ; then the ZWGPA operator is a mapping: , which is expressed as where is the same as the counterpart of Definition 11.
In compliance with the operations of Znumbers and Definition 16, the resulting value determined by ZWGPA operator is still a Znumber.
Being similar to ZWAPA operator, ZWGPA operator also possesses the following properties.
Theorem 17. Let be a collection of Znumbers; if for all , then where is a constant.
Theorem 17 indicates that when all the support degrees are equal, the ZWGPA operator reduces to the WGA operator. Furthermore, ZWGPA operator has also the three properties: commutativity, idempotency, and boundedness.
Obviously, the ZWAPA and ZWGPA operators are two nonlinear weighted aggregation tools, where the weight (not the weight of , ) depends upon the input arguments and allows these values being aggregated to support and reinforce each other, which are able to retain the input information well and take fully into account the interrelation among the inputs.
5. A MCGDM Method with ZNumbers
This section will build a framework for group decision making under Znumber environment; that is, DMsâ€™ evaluation values are expressed with Znumbers. This process contains the determination of weights of DMs and criteria by establishing optimization models and the ranking of all the alternatives by means of the thought of TOPSIS.
Before presenting elaborate operation, we need to draw the outline of MCGDM problem. Suppose alternatives set , criteria set , whose weight vector is with and , and decisionmakers set , whose weight vector is with and . Generally the set is divided into two types, and , where denotes the set of benefit criteria and represents the set of cost criteria; besides, and . Furthermore, the evaluation information of with respect to is denoted by . It is a Znumber and transformed by the semantic information of the th DM , with . Eventually, the decision matrix for the DM is produced.
5.1. WeightDetermining Method for DMs and Criteria in MCGDM
The determination of DMsâ€™ weights and criteria is an important research topic in MCGDM because they will have vital impacts on the final decision. Different DMs possess different knowledge backgrounds and professional degrees, so they cannot be assigned optional weights or given equal weight. Moreover, an alternative or object is evaluated via multiple criterion indexes and these criteria play different roles in the final decision. Therefore, they should also be given different weights. When weights information is completely unknown or partly unknown, we need to excavate fully clues from the decision matrices provided by DMs.
In previous studies, DMsâ€™ weights are subjectively given on the basis of experience with some subjective randomness. In this research, an objective approach for working out DMsâ€™ weights will be adopted. Similarity measure, a means that distinguishes diverse specialization degrees of DMs for decisionmaking problems, will be utilized under Znumber information. If the overall similarity of evaluation values in the decision matrix given by the th DM is greater than the overall similarity of the decision matrix from the th DM , indicating that provides less inconsistent and conflicting decision information than among DMs and plays a relatively important role in decisionmaking process, then should be endowed bigger weight than . In contrast, decision values a DM provides are figured out as smaller overall similarity; then this DM will be endowed smaller weight.
This article does not put forward similarity measure for Znumbers but distance measure has been suggested. In compliance with the relation between distance and similarity, we can transform distance measure into similarity measure by using According to the analysis above, the assignment principle of DMsâ€™ weight vector is to make the total similarity that DMsâ€™ evaluations form reaches the maximum. Motivated by Xuâ€™s work [39], we establish the following programming model to obtain the weights of DMs:To solve the above model, we construct Lagrange function where is a Lagrange multiplier. Let all the partial derivatives of the function be 0; we have By solving the above equations, we can get
Then normalize as to ensure that satisfies and . Now (20) is standardized as
It is (21) that is weight coefficient of DMs. By inspecting the formula structure, we can find out that the bigger the overall similarity degree that a DM corresponds to, the bigger the weighting value he or she can be endowed. Hence, the result decided by (21) accords with the aforementioned requirement about the weights assignment.
When it comes to criterion weights, after aggregating preference information of all the DMs by utilizing aggregation operators, we apply the wellknown maximizing deviation method [40] to Znumbers and still establish an optimization model to acquire weights of criteria from the aggregated matrix . If there are marked differences between preference values of alternatives under a criterion , then has a strong power that distinguishes distinct alternatives and the criterion is considered relatively important in choosing the best alternative. It will account for a relatively high weight. On the contrary, if there are more similar performance values under one criterion, this criterion will be assigned a small weight. In this paper, the deviation between any two alternativesâ€™ preference values is measured by the distance measure of Znumbers. In this way, the optimal criterion weight distribution should make the total deviation of all alternatives with respect to all criteria maximized. However, the information about criterion weights is not entirely unknown in many actual circumstances. Evaluators usually have a subjective judgment or limitation on weight coefficients of criteria but the given weighting information is often inadequate and imprecise, and partial weights only fall within the range set by them out of the complexity of practical situation. In this case, we are supposed to combine the subjective weighting method and the objective weighting method.
Assume that the set of known weighting information is denoted by . We construct another programming model with some constraints utilizing the maximizing deviation idea of the reference [39]:where is a set of constraint conditions that criterion weights must satisfy in terms of the real situation.
Model (M2) is a linear programming model. We can use the specialized LINGO software to work out the optimal solution of the model (M2).
5.2. Approach to MCGDM with ZNumber Information
In this research, the evaluations made by DMs take the form of linguistic variables and are represented by TFNs. The experts are asked to specify ratings for alternatives over evaluation factors using seven linguistic values varying from â€śVery Lowâ€ť to â€śVery Highâ€ť as restriction part and the linguistic scale ranging from â€śNot Sureâ€ť to â€śVery Sureâ€ť as reliability part. The transformations between linguistic values and TFNs about two parts of each evaluation are shown in Tables 1 and 2, respectively. The corresponding two scales of TFNs are presented graphically in Figures 2 and 3, respectively. By referring to linguistic variables of Tables 1 and 2, DMs give their preference ratings on alternatives under all criteria. Each preference involves two components: the restriction rating from Table 1 and the matching reliability measure using the short sentences of Table 2.


Based on the abovementioned knowledge preparation and the evaluation information given by DMs for decision problem, we integrate the suggested power aggregation operators and the popular TOPSIS method to develop a MCGDM approach in Znumber context. TOPSIS, proposed by Hwang and Yoon [41], is a kind of method to solve MADM problems, which aims at choosing the alternative with the shortest distance from the positive ideal solution (PIS) and the farthest distance from the negative ideal solution (NIS). TOPSIS is simple in the operation and swift in the calculation. It is able to make full use of the information in the original data, reduce effectively the information loss, and improve the precision in the final results. In addition, it is suitable for both small sample data and large sample. Therefore, the method is widely used for tackling the ranking problems in real situations. In this paper, the main procedures are demonstrated in Figure 4 including 3 stages, and the detailed process is presented below.
Step 1 (construct decision matrixes). According to the transformation between words and TFNs in Tables 1 and 2, all the evaluations provided by DMs are converted into Znumbers and a set of decision matrixes are obtained, where the preference value denotes the numerical evaluation (including the restriction and reliability ) of the alternative with respect to the criterion , provided by the th DM .
Step 2 (normalize the evaluation information). In order to eliminate the influence of different dimensions and the difference in priority that different types of criteria may bring, we normalize all preference values to the same magnitude grade and make all the research objects comparable. Concretely, let be normalized as , where , and is unchanged.
Step 3 (calculate the supports). Set where can be computed by (6). Obviously, the support degree completely satisfies support conditions (1)(3) listed in Definition 11.
Step 4 (calculate the comprehensive support with DMsâ€™ weights and weighting coefficient for each performance value). Utilize (21) to acquire DMsâ€™ weights and calculate the comprehensive weighted support degree of by the other performance value and compute the weights associated with the performance value where .
Step 5 (aggregate the evaluation values for all DMs). Utilize the ZWAPA operator (9) or ZWGPA operator (14) to aggregate all the individual decision matrix as the collective decision matrix . That is, let be weight of in Step 4 and utilize the WAA or WGA operator to aggregate decision evaluations from all DMs.
Step 6 (compute the criterion weights). Based on the integrated decision matrix , construct the model (M2) in terms of a practical problem and obtain the weight vector of criteria by solving the (M2).
Step 7 (determine the PIS and the NIS for alternatives on the integrated matrix ). The PIS , numerically expressed by , here , where The NIS , expressed by , here , where
Step 8 (compute the relative closeness degree). According to the distance formula Eq. (5), compute the weighted distance between the restraint components of the alternative and the corresponding PIS of and the weighted distance associated with the alternative Analogously we can get the weighted distance and about the confidence component of the alternative from the PIS and the NIS , respectively. and then calculate the relative closeness of the alternative to the ideal alternative , in form of , which is a pair of numbers derived from the two components of Znumbers.
Step 9 (determine the ranking order of all the alternatives). First we define such a priority relation between any two pairs and .
If , then .
If , then we have the following:(i) if , then ; (ii) if , then .
Rank all alternatives through the above ordering rule on . Notice that the first part of each pair of closeness coefficients plays a leading role when compared with each other. A desirable alternative should be as close to the PIS and as far from the NIS, as possible. As a result, its two closeness coefficients in the pair will be as large as possible. Therefore, we can use the closeness coefficients to prioritize alternatives and choose the best one. The larger the two closeness degrees, the better the corresponding alternative. The best alternative is the one with the greatest relative closeness to the ideal solution.
6. Application in Supplier Selection
The selection of suppliers is an important issue as companyâ€™s face in the business management where raw materials and components represent a significant percentage of the total product cost. The importance of supplier selection has increased also from outsourcing initiatives in which companies rely more on suppliers to improve the quality of their products, to reduce their costs, or to focus on a specific part of their operations. Thus, supplier selection constitutes a strategic decision [42]. Numerous companies take this work as the one of key tasks in outside cooperative trade. Therefore, how to select a satisfying supplier is crucial. Recently researches involving this issue have triggered enough attention and have obtained fruitful achievements. In this section, we apply the suggested algorithm in this background and make some simple comparisons and discussion against several existing decisionmaking approaches after deriving consequences of this problem.
This case study is about the supplier selection problem concerning an automobile manufacturing company. Due to its activity, the company needs a certain amount of raw materials such as iron, wire, and tire and has to coordinate with a number of suppliers. Through collecting plenty of information on material producing companies, there are six qualified suppliers being able to supply the required raw materials, denoted by . To promote stable and healthy development of the manufacturing company, it is very necessary to use a reasonable algorithm for selecting trustworthy supplier(s). For this purpose, the company invites some experts and organizes a professional team to evaluate performances of the potential suppliers by the consideration of various factors. The team consists of three experts (DMs), denoted by . They define five criteria to evaluate these alternatives including quality, price/cost, technological capability, partnership, and ontime delivery. After a heated discussion, partial information regarding the weights of criteria is provided as like the format S.H. Kim, B.S. Ahn, and Z.S. Xu previously mentioned in [43, 44]. Further, each DM gives their respective evaluative ratings of suppliers with respect to those setting criteria by employing linguistic terms of Table 1 (for restriction) and Table 2 (for reliability). It is noted that each criterion value given by experts is actually the combination of two linguistic ratings. The entire evaluations made by the three DMs are listed in Tables 3â€“5, and the hierarchical structure of this problem is displayed in Figure 5.



6.1. Illustration of the Proposed Approach
Now we take advantage of the proposed approach and implement its steps step by step to make the final choice of suppliers.
Step 1. By the conversion relation of words and TFNs in Tables 1 and 2, the evaluation information provided by is turned into three numerical decision matrices .
Step 2. Evidently, only is a costtype index among all the criteria and the rest are benefit index. In the light of normalization formula Eq. (23), let all the evaluation values of become , where , .
Step 3. By (24), compute the support degrees between any two normalized matrices and obtain three support matrices between diverse pairwise decision matrices as follows:
Step 4. Utilize (21) to figure out DMsâ€™ weights and by (25) calculate all the comprehensive support with the weights. Finally figure out the weight in relation to by (26) and get the following three weighting matrixes :