Soft Comput

DOI 10.1007/s00500-014-1584-2

METHODOLOGIES AND APPLICATION

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

Xiuzhi Sang · Xinwang Liu © Springer-Verlag Berlin Heidelberg 2015

Abstract 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).

Communicated by V. Loia.

X. Sang (B)

College of Finance, Nanjing Agricultural University,

Nanjing 210095, Jiangsu, China e-mail: hellen1280@hotmail.com

X. Liu

School of Economics and Management, Southeast University,

Nanjing 210096, Jiangsu, China e-mail: xwliu@seu.edu.cn

Keywords Fuzzy TOPSIS · Interval type-2 fuzzy sets ·

Karnik–Mendel (KM) algorithm · Analytical solution 1 Introduction

The fuzzy set theory introduced by Zadeh (1965) has achieved a great success in various fields. Later, Zadeh (1975) introduced the type-2 fuzzy sets (T2 FSs), which was an extension of the fuzzy set, and the membership values are type-1 fuzzy sets on interval [0, 1]. Mendel (2001) further generalized the interval fuzzy set and defined the notion of

IT2 FSs, which has been found useful to deal with vagueness and uncertainty in decision problems, such as perceptual computing (Mendel and Wu 2010; Mendel et al. 2010), control system (Wu and Tan 2006; Wagner and Hagras 2010;

Wu 2012), time-series forecasting (Khosravi et al. 2012;

Chakravarty and Dash 2012; Miller et al. 2012), information aggregation (Zhou et al. 2010, 2011; Huang et al. 2014) and decision-making (Chen and Lee 2010; Wang et al. 2012;

Chen and Wang 2013).

MADM is a widespread method, which is applied to find the most desirable alternatives according to the information about attributes and weights provided by decision makers (Damghani et al. 2013; Xu 2010). TOPSIS, introduced by

Yoon and Hwang (1981), uses the similarity to ideal solution to solve MADM problems, where the performance ratings and weights are given as crisp values. Later, Triantaphyllou and Lin (1996) introduced fuzzy TOPSIS method based on fuzzy arithmetic operations. Chen (2000) extended TOPSIS method to fuzzy group decision-making situations. Wang and Elhag (2006) proposed a fuzzy TOPSIS method based on alpha level sets. Wang and Lee (2007) generalized TOPSIS method in fuzzy MADM environment. Chen and Tsao (2008) and Ashtiani et al. (2009) extended the TOPSIS method 123

X. Sang, X. Liu to interval-valued fuzzy numbers environment. Boran et al. (2009) proposed an intuitionistic fuzzy TOPSIS method for supplier selection problem. Li (2010) proposed TOPSISbased NLP methodology with interval-valued intuitionistic fuzzy sets. Tan (2011) introduced a multi-criteria intervalvalued intuitionistic fuzzy group decision-making method using Choquet integral-based TOPSIS method. Robinson and AmirtharajE (2011) developed TOPSIS method under triangular intuitionistic fuzzy sets. Behzadian et al. (2012) summarized the research on TOPSIS applications and methodologies. In addition to the developments of fuzzy TOPSIS in traditional type-1 fuzzy formats, a notable progress was the appearance of interval type-2 fuzzy TOPSIS method proposed by Chen and Lee (2010). They first computed the ranking values of the IT2 FSs elements in weighted decision matrix, then counted the crisp relative distance through traditional TOPSIS computing process. However, both the defuzzification from the very beginning and the crisp distance computation are approximate, which do not realize the IT2 FSs formats crossing the whole computing process, and lead to decision information loss.

In this paper, we provide an analytical solution to IT2

FSs-based TOPSIS model with KM algorithm. KM algorithm (Karnik and Mendel 2001) is a kind of the standard way to compute the centroid and perform type reduction for type-2 fuzzy sets and systems (Hagras 2007; Mendel 2007a, 2013). It transforms the fractional nonlinear programming problems into identifying the switch points of α levels, which is monotonically and superexponentially convergent to the optimal solution (Mendel and Liu 2007). Some applications of KM algorithm in decision-making have also been proposed. Wu and Mendel (2007) used the KM algorithm to compute the linguistic weighted average (LWA) of type-2 fuzzy sets. Liu and Mendel (2008) proposed a new α-cut algorithm for solving the fuzzy weighted averaging (FWA) problem with the KM algorithm. Liu et al. (2012) proposed the analytical solution to FWA with KM algorithm. Liu and

Wang (2013) introduced the analytical solution to generalized FWA with KM algorithm as well.

Based on KM algorithm (Karnik and Mendel 2001; Liu et al. 2012), we propose an analytical solution to the TOPSIS model with IT2 FSs variables. First, similar to the case of type-1 fuzzy TOPSIS method (Kao and Liu 2001;