A Large Group Decision Making Approach Based on TOPSIS Framework with Unknown Weights Information

. Large group decision making considering multiple attributes is imperative in many decision areas. The weights of the decision makers (DMs) is difficult to obtain for the large number of DMs. To cope with this issue, an integrated multiple-attributes large group decision making framework is proposed in this article. The fuzziness and hesitation of the linguistic decision variables are described by interval-valued intuitionistic fuzzy sets. The weights of the DMs are optimized by constructing a non-linear programming model, in which the original decision matrices are aggregated by using the interval-valued intuitionistic fuzzy weighted average operator. By solving the non-linear programming model with MATLAB ® , the weights of the DMs and the fuzzy comprehensive decision matrix are determined. Then the weights of the criteria are calculated based on the information entropy theory. At last, the TOPSIS framework is employed to establish the decision process. The divergence between interval-valued intuitionistic fuzzy numbers is calculated by interval-valued intuitionistic fuzzy cross entropy. A real-world case study is constructed to elaborate the feasibility and effectiveness of the proposed methodology.


Definition 3[21]
For two interval-valued intuitionistic fuzzy numbers (IVIFNs) , the standard Euclidean distance between A and B is:

Definition 4[22] The entropy of A=〈ൣu
The information entropy of the j th criterion c j is defined as: Then the weight of cj is calculated by:(1)

Definition 5[17]
A symmetric form of IVIF cross-entropy between IVIFNs A and B is defined as: where:

Methodology
The mathematical description of the MALGDM can be defined as following. The alternative set is: X={x 1 , x 2 ,…, is the ratings given by e k with regard to c j on x i . Then the logic flow of the proposed methodology is demonstrated in Figure 1.
Step 1 Determining the weights of DMs Step 3 Ranking alternatives 1 Determining PIS and NIS 2 Calculating cluster effect and regret value with cross-entropy Step 2 Calculating the weights of criteria The problem can be solved through three phases in detail.
Step 1: Calculating Λ by using non-linear programming optimization model.
The optimization model of Λ is constructed as following: where d൫R ෩ k , R ෩ ൯ is the Euclidean distance (Equation (2)) between R ෩ k and the fuzzy comprehensive decision matrix R ෩ : By Equation (9), the objective function is transformed to: Solve the model through MATLAB ® to obtain the optimal solution of Λ. Then the fuzzy comprehensive decision matrix R ෩ is acquired through IIFWA as: Step 2: Calculate the weight of criteria.
Calculating Ω based on Definition 4.
(1) Determining the PIS and the NIS.
The comprehensive decision matrix R i of x i is denoted as: The PIS and the NIS are denoted in the form of IVIFS as, respectively: (2) Calculate the weighted cross-entropy and relative closeness coefficient between each alternative and the PIS or NIS. D * (R i , PIS) is the weighted cross-entropy between each alternative and the PIS. D * (R i , NIS) is the weighted cross-entropy between each alternative and the NIS in the following. Then, the relative closeness degree (RCD) of each alternative is calculated, where0≤θ≤1, and θ is the attention degree. Finally, the alternatives are ranked based on RCDi (i=1, 2, …, n). The bigger the RCDi is, the better the alternative xi is.
Due to the decline of traditional manufacturing industry, automobile manufacturing enterprises are confronted with high cost and low margin pressure. Some enterprises even are confronted with the danger of survival in the fierce competition environment. Under this circumstance, a growing number of enterprises are preparing customer-oriented 5S (Sale, Spare-part, Service, Survey, Sustainability) service. There are four alternatives providing 5S service, denoted as x 1 , x 2 , x 3 , x 4 according to their own industry and management characteristics. The criteria of four alternatives C={c 1 , c 2 , c 3 , c 4 } are: the ability of satisfying customer demand, the controlling ability of operating costs, the sustainable development ability, and the enhancing ability of production and sales. The initial interval-valued intuitionistic fuzzy decision matrices of fifteen DMs are listed in Table 1.
According to the proposed MALGDM methodology, the alternatives are ranked through three steps in the following: Step 1: The optimized model of the weights of DMs is constructed by using Equation (8)~ (10). The iteration number is set as 1000, the optimization target value is eventually obtained as 6.467 as shown in Figure 2. The weights of the DMs as shown in Table 2. The fuzzy comprehensive decision matrix R ෩ is obtained using Equation (1), and the PIS and the NIS are determined using Equation (13)~(14) as shown in Table 3.
Step 2: The weights of the criteria are obtained using Equation (4)~(5), the results are listed in Table 4.
Step 3: The cross-entropy and relative closeness degree of alternatives are calculated by using Equation (15)~ (17), which are listed in Table 5.
According to the RCDs of each alternative, the alternatives are ranked as: x3> x2> x1>x4.     The sensitivity of the proposed methodology is analyzed in this section by adjusting the original evaluation data [23]. For the highest importance degree, the hesitancy degree of c2 in Table 4 is reduced to zero. The cross-entropy and the RCDs of each alternative are calculated by the proposed approach, the results are listed in Table 6. For comparison, the weighted Euclidean distance and RCDs of each alternative are calculated by using Equation (9), the results are listed in Table 7. The Radar Map of the three kinds of ranking results is shown in Figure 3 and finally the ideal alternative is x3. The ranking order does not change compared to the original order which can indicate that the applicability of the proposed approach is acceptable. Yet, the ranking order obtained by the weighted Euclidean distance under TOPSIS framework changes, that is, the ranks of x1 and x2 are swapped, which indicates that the proposed methodology is effective.    This paper reviews approaches dealing with MALGDM problem which is difficult to obtain the weights of the DMs and proposes an integrated MALGDM framework. The main contributions are as follows.
(1) First, the weights of the DMs are optimized by constructing a non-linear programming model, in which the original decision matrices are aggregated by using IIFWA operator. By solving the non-linear programming model with MATLAB ® , the weights of the DMs and the comprehensive decision matrix are determined.
(2) Second, the IVIFS is introduced to describe the membership degree and non-membership degree of the DMs. The IVIF-cross entropy is employed to measure the divergence degree between IVIFNs and the information entropy is employed to determine weights of criteria under TOPSIS framework.
A selection problem of alternatives providing 5S service is implemented as the case study. There are still some issues need to be considered in the future. For example, the consensus analysis can be introduced to increase the consensus between R ෩ k and R ෩ .