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A decision-making environment is full of uncertainty and complexity. Existing tools include fuzzy sets, soft sets, intuitionistic fuzzy sets, Pythagorean fuzzy sets (PFSs) and so on. Compared with intuitionistic fuzzy sets (IFSs), PFSs proposed by Yager have advantages in handling vagueness in the real world and possess good symmetry. The entropy measure is the most widespread form of uncertainty measure. In this paper, we improve the technique for order preference by similarity to an ideal solution (TOPSIS) method to better deal with multiple-attribute group decision making (MAGDM) problems based on Pythagorean fuzzy soft sets (PFSSs). To better determine the weights of attributes, we firstly define a novel Pythagorean fuzzy soft entropy which is more reasonable and valid. Meanwhile the entropy has good symmetry. Entropy for PFSSs which is used to determine the subjective weights of attributes is also defined. Then we introduce a measure to calculate integrated weights by combining objective weights and subjective weights. Based on the integrated weights, the TOPSIS method is generalized and improved to solve the MAGDM problem. A distance measure taking into account the characteristics of Pythagorean fuzzy numbers (PFNs) is used to calculate distance between alternatives and ideal solutions. Finally, the proposed MAGDM method is applied in the case of selecting a missile position. Compared with other methods, it is shown that the proposed method can rank alternatives more reasonably and have higher distinguishability.

A New Method for MAGDM Based on Improved TOPSIS and a Novel Pythagorean Fuzzy Soft Entropy