An analytical solution to the TOPSIS model with interval type-2 fuzzy sets

TOPSIS is a popular used model for multiple attribute decision-making problems. Recently, Chen and Lee (Exp Syst Appl 37(4):2790–2798, 2010) extended TOPSIS method to interval type-2 fuzzy sets (IT2 FSs) environment. They first compute the ranking values of the elements in fuzzy-weighted decision matrix, and used the ranking values to compute the crisp relative closeness through traditional TOPSIS computing process. Such ranking computation leads to the information loss of the weighted decision matrix. In this paper, we introduce an analytical solution to IT2 FSs-based TOPSIS model. First, we propose the fractional nonlinear programming (NLP) problems for fuzzy relative closeness. Second, based on Karnik–Mendel (KM) algorithm, the switch points of the NLP models are identified, and the analytical solution to IT2 FSs-based TOPSIS model can be obtained. Compared with Chen and Lee’s method, the proposed method operates the IT2 FSs directly and keeps the IT2 FSs formats in the whole process, and the result of which is precise in analytical form. In addition, some properties of the proposed analytical method are discussed, and the computing process is summarized as well. To illustrate the analytical solution, an example is given and the result is compared with that of Chen and Lee’s method (Exp Syst Appl 37(4):2790–2798, 2010).