A rough set approach for determining weights of decision makers in group decision making

This study aims to present a novel approach for determining the weights of decision makers (DMs) based on rough group decision in multiple attribute group decision-making (MAGDM) problems. First, we construct a rough group decision matrix from all DMs’ decision matrixes on the basis of rough set theory. After that, we derive a positive ideal solution (PIS) founded on the average matrix of rough group decision, and negative ideal solutions (NISs) founded on the lower and upper limit matrixes of rough group decision. Then, we obtain the weight of each group member and priority order of alternatives by using relative closeness method, which depends on the distances from each individual group member’ decision to the PIS and NISs. Through comparisons with existing methods and an on-line business manager selection example, the proposed method show that it can provide more insights into the subjectivity and vagueness of DMs’ evaluations and selections.


Introduction
The aim of a multiple attribute decision-making (MADM) problem is to obtain alternatives' rankings or an optimal alternative selection by the decision information from each DM with respect to amount of criterias. Nowadays, MADM problems have been involved in various aspects of politics, economies, science, technology, culture, education and other fields [1][2][3][4][5][6][7][8].
However, along with the constantly expansion of criterias, it is nearly impossible for a single decision maker to make an appropriate judgment independently for a project [9][10][11][12][13][14]. Therefore, many companies and groups prefer to make a final decision through a panel of experts [15][16][17][18][19][20]. Each expert has his/her preference to each attribute based on his/her knowledge level and cognitive capability. As the preference information of each expert is always different in group decision-making problems, current research focus on the aggregation of decision information and priority order of group members [21].
French [22] proposed three major postulates and a variety of theorems to deal with the effects of group members' opinions. Theil [23] proposed an approach to define the weights of the linear combination of individual preference functions in committee decision problem. Bodily [24] developed a delegation process to setting the members' weights, which is obtained a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 using the theory of Markov chains. Mirkin and Fishburn [25] make use of eigenvectors method to gain weights information of group members. Martel and Ben Khelifa [26] use individual outranking criterias to determine the group members' weights. Ramanathan [27] developed an AHP method to obtain group members' weights, and aggregated group decisions. Fu and Yang [28] used a group consensus to address multiple attributive group decision problems, which is from evidential reasoning approach. Xu and Wu [29] proposed a discrete model to support the group consensus reaching process, in which the weights of experts is predefined. Zhou et al. [30] proposed the generalized logarithm chi-square method to aggregate group members' information. Zhang [31] presented several generalized Atanassov's intuitionistic fuzzy power geometric operators to aggregate input arguments. Yue [32] presented an extended TOPSIS method for ranking the order of decision makers and the order of alternatives. Efe [33] proposed an integration of fuzzy AHP and fuzzy TOPSIS to present the weights of decision makers with linguistic terms.
These methods mentioned above have made significant contributions to the determination of experts' weights and aggregation of experts' judgments in MAGDM. However, how to deal with the subjective and heuristic decisions of a group of experts in a simple and efficient way is still a question [34][35][36][37][38]. In order to address this question, an easy operation method in this paper is developed for determining weights of experts based on rough group decision.
Rough set theory, first proposed by Pawlak [11], is an effective and efficient tool to handle imprecision and vagueness information from DMs. As rough group decision originates from rough set theory, it can enable DMs to express true and objective evaluation without any priori information. Additionally, it can deal with a group of vague and subjective information at the same time.
The remainder of this paper is structured as follows. The following section gives a brief introduction to rough group decision. Then, we present the detailed description of the proposed method in group decision setting. Then, we compare the developed method in this study with other existing methods. Next, an illustrative example is given. Finally, the conclusions are made for the whole study.

Determination of the rough group decision
Here, we shall introduce some concepts about the rough group decision. Definition 1 ([39]). Let U be a universe including all DMs' decisions, X is an arbitrary decision of U. Assume that there is a set of each DM's judgements on attributes over alternatives, where i is the number of alternatives, j is the number of attributes and t is the number of DMs, i 2 {1,2,. . .,m}, j 2 {1,2,. . .,n}, k 2 {1,2,. . .,t}, t > 0. Assume the elements of set J are in ascending order . Then, the lower approximation and the upper approximation of v k ij are defined as: Upper approximation : In order to obtain the rough decision, the crisp decision v k ij , which contains vague and subjective information of a DM, should be converted into rough number form. As the geometric mean preserves the reciprocal property of pair-wise comparison matrixes, it is utilized to synthesize individual decisions from DMs.

Definition 2 ([40]).
A rough number is selected to represent the judgment v k ij , defined by its lower limit Limðv k ij Þ and upper limit Limðv k ij Þ as follows: x and y are from the lower and upper approximation for v k ij . N L and N U are defined as the numbers of judgements from the lower approximation and upper approximation of v k ij . Definition 3 ([41]). The rough number form RNðv k ij Þ of v k ij is obtained by using Eq (1)-Eq (4), where v kL ij and v kU ij are from the lower limit and upper limit of rough number RNðv k ij Þ in the kth decision matrix. The interval of boundary region (i.e. v kU ij À v kL ij ) indicates the vagueness degree. That is, a smaller interval boundary to a rough number means more precise. Then, the crisp decision v k ij is represented by the rough decision RNðv k ij Þ. Definition 4. In sum, the average rough interval RNðJÞ is obtained by using Eq (1)-Eq (5), ð8Þ v L ij and v U ij are from the rough number ½v L ij ; v U ij . t is the number of experts. Then, a set of each DM's decision, J, is represented by the average rough interval RNðJÞ. Definition 5. The average value of RNðJÞ is obtained as follows: ðRNðJÞÞ Ã , which is the median of the average rough interval RNðJÞ, can reflect the common aspirations and consistent judgements of DMs with respect to the set J.

Proposed approach to group decision making
In the following, the MAGDM problems under consideration with rough group decision shall be described in detail. . .,m) to be evaluated against n selection criteria u j (j = 1,2,. . .,n) with n criteria's weight w j (j = 1,2,. . .,n), which satisfies 0 w j 1 and X n j¼1 w j ¼ 1. Assume D = {d 1 ,d 2 ,. . .,d t } is a finite set of DMs, λ = {λ 1 ,λ 2 ,. . .,λ t } is the weight vector of all DMs, which fulfils λ k ! 0 and X t k¼1 l k ¼ 1.

Standardization of the decision matrix
Invite DMs to give the relative importance of m feasible alternatives under n attributes by using the one-nine scale of AHP method. The decision matrix of the kth DM is as follows: In general, MAGDM problems have benefit attributes (the larger the value is, the better) and cost attributes (the smaller the value is, the better). To acquire a dimensionless form, it is necessary to normalize each attribute value x k ij in decision matrix X k into a corresponding element y k ij in normalized decision matrix Y k by using Eqs (12) and (13) [34]. where and Then, it is clear that u j 2 [0,1], j 2 N.
As the attributes' weight vector fw k 1 ; w k 2 ; . . . ; w k n g is given by the kth DM, the weighted normalized decision matrix is constructed as

Definition of DMs' weights
Inspired by the idea of the rough group decision, the group decision matrix is built as follows: where . . . ; v t ij g. As mentioned above, e v ij is a set of each DM's judgements. Then, the average rough interval of e v ij is obtained by applying Eq (1)-Eq (8). The rough group decision matrix RV is obtained as follows: As known to all, TOPSIS has become a widely used technique for MAGDM. The main idea of TOPSIS is that the ideal alternative has the best level for all attributes, whereas the negative ideal has the worst level for all attributes. According to the idea of TOPSIS, we define RV Ã as PIS for all individual decision matrixes with Definition 5 of previous section. Then, the average matrix of rough group decision matrix is obtained by using Eq (9). where . And then, from a TOPSIS method perspective, both the upper limit and the lower limit matrix of the rough group decision matrix are potential to have the farthest distance from the average matrix. Thus, we divided the NIS into two parts: L-NIS RV À L and U-NIS RV À U .
where v L ij and v U ij are from the rough number ½v L ij ; v U ij . The separation of each individual decision matrix V k from the PIS RV Ã is calculated as: It is clear that the smaller the value of S þ k is, the more important the weight of the kth DM.
Similarly, the separation of each individual decision matrix V k from the NISs RV À L and RV À U are calculated as: It is clear that the larger the value of S LÀ k and S UÀ k are, the more important the weight of the kth DM.
Assuming that S þ k , S LÀ k and S UÀ k are all under consideration, we define relative closeness to rank the weights of DMs. The relative closeness of the kth DM in relation to ideal solutions is defined as: Assume the decision matrix of the kth DM is the positive ideal solution; then, S þ k ¼ 0 and C k = 1. So if C k = 1, the corresponding decision is absolutely the best decision.
According to Eqs (20)-(23), it can be inferred that if the individual matrix V k is close to RV + , V k is far from RV À L and RV À U . Therefore, we can define the weight of the kth DM as follows: where λ k ! 0 and X t k¼1 l k ¼ 1. Then, we can rank the weights of DMs according to Eqs (23) and (24).

Priority order of alternatives
With the weight of the kth DM, a group decision matrix Y is obtained by using the following formula Then, use the aggregation formula to summarize the ith row's elements of Y. Then, the overall attribute value y i of the alternative A i is obtained. According to the value y i , the priority order of those feasible alternatives can be ranked, and the best alternative can be chosen.

The presented algorithm
As described above, a method for determining the DMs' weights, based on the rough group decision, is shown as follows.
Step 2. Calculate the weighted normalized decision matrix V k by multiplying fw k 1 ; w k 2 ; . . . ; w k n g and Y k in Eq (14).
Step 3. Calculate the group decision matrix e V in Eq (15).
Step 4. Calculate the rough group decision matrix RV in Eq (16) by using Eq (1) to Eq (8).
Step 6. Calculate the separation from each individual decision to the ideal decisions, S þ k , S LÀ k and S UÀ k , by applying Eq (20)-Eq (22). Step 7. Calculate the relative closeness to the ideal solutions by using Eq (23).
Step 10. Summarize each line's elements of the collective decision matrix in Eq (26) and obtain an overall assessment value for each alternative.
Step 11. Rank the preference order of all alternatives according to their total assessment values.
The hierarchical structure of the proposed approach is summarized in Fig 1.

Comparisons between the proposed approach and existing approaches
In the following, we compared three approaches, the proposed approach, the method of Ye and Li [42] and the extended TOPSIS method of Yue [31]. Table 1 presents the differences between the two methods, the proposed method and the extended TOPSIS method of Ye and Li. First, the PIS and NIS are derived from alternatives, which are vectors, while in this paper, they are derived from rough group decision matrix, which are matrixes. Second, each DM's weight is different and determined by the distances from his/her decision matrix to PIS and NISs in this paper. That is to say, the weight of each DM is defined by the given data, not pre-defined, and reflects the gap between his/her preference and group preference to the feasible alternatives on attributes objectively. In addition, the developed approach's procedure in Fig 1 is simple and clear for high-dimensional data analysis in group setting.
In the method of Yue, the three benchmark matrixes (PIS, L-NIS and U-NIS) are defined through aggregation of DMs' decision information by using TOPSIS, while in this paper these matrixes are defined by rough group decision, which are based on rough number and rough boundary interval. The average rough boundary interval in Eq (17) from rough boundary intervals can reflect the vagueness degree of all DMs to attributes of alternatives. From this point of view, the smaller the interval, the lower the vagueness degree. In addition, both of the two methods take a group effect with PIS and NISs into account. That is, if the decision matrix  is far away from the NISs and close to the PIS, the decision is better. Therefore, the better the decision is, the more the DM's weight. The comparisons mentioned above are shown in Table 2.

Illustrative example
In the following, the proposed method shall be applied to a human resources management [43]. A company wants to hire an on-line business manager. Therefore, the company proposes several relevant tests, which are regarded as the evaluated benefit criterias.  Table 3. The worst decision represented by the upper limit and lower limit matrix of rough group decision relative closeness Priority order of alternatives Priority order of alternatives In accordance with the suggested steps mentioned above, each decision matrix given by experts in Table 3 shall be normalized to achieve nondimensionalization. Because of the benefit attributes of Table 3, we first normalize Table 3 into four normalized decision matrixes of Table 4 according to Step 1. In the normalized decision matrixes of Table 4, X 1 , X 2 , X 3 , X 4 shall be marked by Y 1 , Y 2 , Y 3 , Y 4 , respectively.
Then, the weights of attributes are shown in Table 5, which are given by the four experts. By using Step 2, each column vector of the normalized decision matrix is multiplied by the associated attributes' weight vector given by each expert in Table 5. Therefore, the weighted normalized decision matrixes are obtained in Table 6. By using Step 3 and Step 4, we can calculate the rough group decision matrix from the weighted normalized decision matrixes. Next, these important matrixes (RV + , RV À L and RV À U ) are shown in Table 7 by using Step 5. By using Step 6, the distances from each weighted normalized decision matrix to the ideal solutions (RV + , RV À L and RV À U ) are calculated. The results are summarized in Table 8. Next, the relative closeness by using Step 7, the weight vector of experts by using Step 8, and experts' priority ranking are calculated, respectively. These results mentioned above are all list in Table 8. The final experts' priority ranking obtained by the rough group decision method is shown as  The 6th column of Table 8 shows the weights of four invited experts. By Step 10, the Eq (25) is used to combine each DM's decision to the collective decisions, which are shown in the column 2 and 3 of Table 9. Next, the overall evaluations of 17 candidates are shown in column 4 by summarizing all data in each line of columns 2 and 3 of Table 9. Finally, the ranking for these candidates are obtained in the last column of Table 9. It is clear that the 16th candidate ranks the first, and the 12th candidate ranks the last.

Conclusions
This paper designs a novel method to determine the weights of experts based on rough group decision. The proposed approach utilizes rough group decision to aggregate the subjective and heuristic information of experts. The validation of this method in a human resources selection indicates that it can be regarded as an objective and effective evaluation tool in group decisionmaking. By contrast, the rough group method can effectively manage the subjectivity of experts in decision process and reflect the vagueness of experts objectively. Due to the amount of information, it will be easier and faster to solve these problems with software MATLAB. Although the method in this paper provides a simple and effective mechanism for weights of experts in group decision setting, it is only useful for real number form of attributes. Therefore, we shall extend the proposed approach to support other forms information on attributes, such as linguistic variables or fuzzy numbers in future work.

S1 File. This file contains all Supporting Figures A and Tables A-I. Figure
A in S1 File. Figure A shows the hierarchical structure of the proposed approach. Table A in S1 File. Note: "*" and "#" mark the first and the last candidate, respectively.
doi:10.1371/journal.pone.0172679.t009 Table A presents the differences and similarities between the extended TOPSIS of Ye and Li and the proposed method. Table B in S1 File. Table B presents the differences and similarities between the extended TOPSIS of Yue and the proposed method. Table C in S1 File. Table C lists the original data from four experts.