An extension of fuzzy TOPSIS for a group decision making with an application to tehran stock exchange

Highlights

• We propose three versions of fuzzy TOPSIS for solving group MADM problems.

• We apply fuzzy set theory to handle the imprecise information in the real-world problems.

• We take advantage of fuzzy-valued distance and fuzzy ranking method to provide a more rational decision-making process.

• We apply the proposed methods in the Tehran stock exchange.

Abstract

In financial markets, investors attempt to maximize their profits within a constructed portfolio with the aim of optimizing the tradeoffs between risk and return across the many stocks. This requires proper handling of conflicting factors, which can benefit from the domain of multiple criteria decision making (MCDM). However, the indexes and factors representing the stock performance are often imprecise or vague and this should be represented by linguistic terms characterized by fuzzy numbers. The aim of this research is to first develop three group MCDM methods, then use them for selecting undervalued stocks by dint of financial ratios and subjective judgments of experts. This study proposes three versions of fuzzy TOPSIS (Technique for Order Preference by Similarity to Ideal Solution): conventional TOPSIS (C-TOPSIS), adjusted TOPSIS (A-TOPSIS) and modified TOPSIS (M-TOPSIS) where a new fuzzy distance measure, derived from the confidence level of the experts and fuzzy performance ratings have been included in the proposed methods. The practical aspects of the proposed methods are demonstrated through a case study in the Tehran stock exchange (TSE), which is timely given the need for investors to select undervalued stocks in untapped markets in the anticipation of easing economic sanctions from a change in recent government leadership.

Introduction

Following the growth of capital markets especially stock exchange markets, a significant proportion of investor’s assets are indicated as a form of shares of companies, which are listed in the stock market. The classical theory of finance considers maximizing return along with minimum risk as a key feature for every successful investment. However, most of financial markets contain some different and effective factors [9]. In order to select the superior stocks for investment, many conflicting factors require to be investigated precisely to achieve the desirable results. Accordingly, the financial performance evaluation of companies in stock exchange markets would be considered as a multi-criteria decision-making (MCDM) problem.

MCDM methods are common approaches to structure information and decision evaluation in various problems with multiple and conflicting goals. Since decision making requires many considerations and often multi-dimensional details in complicated real-world situations, this research area is still attractive and has been widely used in many fields [66]. The MCDM techniques are generally divided into multi-objective decision making (MODM) and multi attribute decision making (MADM) techniques. MODM has been generally studied using mathematical programming methods with well-formulated theoretical frameworks where we have either an infinitive or a large number of alternative choices, the best of which should meet the decision maker (DM) restrictions and preference priorities [38]. When several conflicting criteria are found, the MADM techniques can be utilized to determine the best option among a set of possible alternatives. MADM takes part in a variety of actual situations, such as economic analysis, strategic planning, forecasting, medical diagnosis, venture capital and supply chain management [27]. Also, in the presence of many different conflicting criteria, MADM methods have received much attention for evaluation and ranking of the companies in the stock exchange literature (e.g., see Martel et al. [50] in the Tunisian Stock Exchange; Albadvi et al. [3] in the Tehran Stock Exchange; Xidonas et al. [68], [69] in the Athens Stock Exchange; Jerry Ho et al. [40] in the Taiwan Stock Exchange; Shen and Tzeng [60] in the Taiwan Stock Exchange).

Many MADM techniques have been developed by researchers which the most popular ones are dominant, maximin, maximax, lexicographic, permutation, simple additive weighting (SAW), analytic hierarchy process (AHP), elimination and choice expressing reality (ELECTRE), technique for order preference by similarity to ideal solution (TOPSIS), and linear programming techniques for multidimensional analysis of preference (LINMAP) methods [58], [27] .

Among the aforementioned MADM techniques, TOPSIS, introduced by Hwang and Yoon [39], is as a best-developed method for MADM problems on account of simple computation process and high flexibility. The policy of this method is based on selecting the most desirable alternative with considering the shortest distance from the positive ideal solution (PIS) and the farthest distance from the negative ideal solution (NIS).

In recent years, TOPSIS has been applied to the various fields such as product design [46], human resources management [17], quality control [70], location planning [5], human spaceflight mission simulators [61], risk assessment [62], and sustainability evaluation of the government bond funds [7].

An important limitation of the TOPSIS method is the necessity for precise measurement of the performance ratings and criteria weights [39] . However, in many real-life decision-making problems, the weights of the attributes and the ratings of the alternatives cannot be measured accurately when some DMs may give their judgments by using linguistic terms such as small, medium and large. Further, it is not conventionally easy to analyze complicated situations and the use of linguistic variables whose values are words or sentences in a natural or artificial language are indispensable. In this regard, the fuzzy sets theory is perfectly formulated by Zadeh [74], [75] to deal with and quantify ambiguity and linguistic concepts in decision-making problems [11], [76]. Since the inception of fuzzy sets theory in MCDM domain by Bellman and Zadeh [6] numerous methods were developed for this particular decision-making structure (e.g., see the surveys by Luhandjula [48], Chen and Hwang [18] and Fodor and Roubens [28]; fuzzy outranking by Hatami-Marbini and Tavana [34]; fuzzy MCDM applications by Hatami-Marbini et al. [35], [36]).

Fuzzy TOPSIS method was initially proposed by Triantaphyllou and Lin [64] in the context of MADM with the aim of, firstly, obtaining the fuzzy closeness coefficient (CC) for each alternative using fuzzy arithmetic operations and, secondly, providing the preference order of the alternatives by means of a defuzzification method. Along the line of the theoretical development for fuzzy TOPSIS, Chen [15] proposed the fuzzy version of TOPSIS method in the frame of group decision-making to calculate the distance between two triangular fuzzy numbers using a vertex method. Since Chen’s contribution in 2000, much development in fuzzy TOPSIS methods has attracted attention from researchers in both theory and practice (e.g., [21], [16], [25], [45], [72], [42], [47]).

Building on the idea of Chen [15] , Chu [21] developed a fuzzy TOPSIS method for selecting plant location in which the normalized weighted ratings were defuzzified at the outset in order to avoid complicated calculations embedded in fuzzy sets theory and obtain a crisp CC for the ranking order of alternatives. Chen et al. [16] presented a fuzzy TOPSIS method for supplier selection in supply chain management (SCM) where the distance calculated was the same as Chen [15] to acquire the crisp CC for each alternative. Reinterpreting fuzzy TOPSIS as a technique that deals with the linguistic information, Yong [73] transformed the defined triangular fuzzy numbers into crisp numbers based on the graded mean integration. The loss of fuzzy information is often derived from defuzzification methods in the earlier steps of the computations. To cope with this problem, Wang and Elhag [67] proposed a fuzzy TOPSIS method based on α-cut sets and fuzzy extension principle to formulate a nonlinear programming (NLP) solution procedure where the defuzzification was performed at the very end of decision process. In strategic management, Hatami- Marbini and Saati [33] considered the strategy selection dilemma as a group decision making problem by dint of fuzzy TOPSIS where the distance value of each alternative from the fuzzy PIS and fuzzy NIS was measured using Yao and Wu [71] ’s ranking fuzzy numbers method. Sadi-Nezhad and Khalili Damghani [59] proposed an alternative fuzzy TOPSIS method by placing emphasis on preference ratio and fuzzy distance measurement to attain the fuzzy CC for each alternative.

While one of the challenging and complex problems in fuzzy TOPSIS is the calculation of the distance between the fuzzy normalized rating of each alternative and the fuzzy ideal solutions, there has been limited attention paid to this issue according to our literature review (see Table 1). In addition, most of research studies on fuzzy TOPSIS suffer from a defuzzification technique in the early or middle stage of the problem solving since it leads to missing a great deal of information. In this regard, the obtained crisp CC in the fuzzy TOPSIS methods may not be acceptable and rational thanks to human reasoning embedded in fuzzy decision making problems. Therefore, the necessity of considering fuzzy distance measure from each alternative to the fuzzy PIS and fuzzy NIS in fuzzy TOPSIS is apparent.

In this paper, over and above discussing the main affecting factors on investment in stock market, we extend three fuzzy TOPSIS methods; conventional TOPSIS (C-TOPSIS) [39], adjusted TOPSIS (A-TOPSIS) [23] and modified TOPSIS (M-TOPSIS) [55] to effectively evaluate the companies in Tehran stock exchange (TSE), in which a proper fuzzy distance measure and fuzzy ranking method are applied in the presence of a group of DMs.

Although there are a number of methods for calculating the fuzzy distance measure in the literature, these methods usually suffer from some drawbacks [63], [65], [12], [32]. In this study, we utilize Guha and Chakraborty [32]’s method to compute fuzzy distance measure from the fuzzy rating associated to each alternative to the fuzzy ideal solution thanks to the following advantages: (i) utilizing generalized fuzzy numbers, (ii) considering the degree of confidence in DMs’ judgments, and (iii) utilizing “fuzzy similarity measure”. It is worth mentioning that Sadi-Nezhad and Khalili Damghani [59] as a similar study to our paper used Chakraborty and Chakraborty [12] that entails the shortcomings in some cases that have been treated by Guha and Chakraborty [32].

We finally require applying an efficient and proper method to rank the obtained fuzzy CCs while most existing methods have been non-discriminating and obtained non-intuitive results (e.g., see [8], [71], [51], [1]). This paper exploits a fuzzy ranking method proposed by Abbasbandy and Hajjari [2] to rank fuzzy CCs and render a proper relative order of alternatives at the final step. Abbasbandy and Hajjari’s method does not require a priori knowledge of all the alternatives as well as the computational effort. Moreover, the efficacy of this fuzzy ranking method in comparison with the existing methods was meticulously investigated in Abbasbandy and Hajjari [2].

The TSE began its operations over five decades ago in which it is a high potential and unique stock market in the Middle East by the inclusion of 38 industries and 339 active companies for investing and business due to its strategic location in the Middle East, a vast domestic market, quick access to neighboring markets involving Europe, Southeast Asia and East Asia markets. Since 2007, the TSE is categorized into three segmentations; (i) main market, (ii) second market, and (iii) corporate participation certificates market, with the aim of ameliorating supervision and compliance of the listed companies. It is notable that nine automotive and twenty-three automotive parts manufacturing industries with around 50% of daily trading stock on average, i.e.11.7 million dollars in 2013, can be recognized as the influential sectors in the TSE. Though there are many approaches for portfolio selection in the different stock exchanges, the TSE has received relatively little attention in the MCDM context [3], [26], [56].

Albadvi et al. [3] used PROMETHEE method for evaluating and ranking the industries and companies in each industry in the TSE. Fasanghari and Montazer [26] extended a fuzzy expert system for recommending stock portfolio in the TSE where the experts’ opinions have been gathered using the fuzzy Delphi method. Rezaie et al. [56] exploited a fuzzy data envelopment analysis (DEA) to prioritize the top 50 companies listed by the TSE in 2010. Roughly speaking, no TOPSIS-based approach has been undertaken to assess the industries and companies in the TSE, particularly the automotive and automotive parts manufacturing industries in the fuzzy environment, that is because we evaluate the potential profitable companies in these industries using the proposed TOPSIS methods.

To the best of our knowledge, this study as a decision support tool is the first one providing four main methodological and practical contributions: first, we expand the aforesaid literature on fuzzy TOPSIS by considering the confidence level of the DMs and generalized fuzzy numbers, second, we utilize a proper and powerful fuzzy-valued distance to calculate the fuzzy CC for each alternative, third, we implement defuzzification by using a proper fuzzy ranking method at the very end of decision process to reasonably stop missing information, and finally, we apply the proposed fuzzy TOPSIS methods to a real-world case study in the TSE to manage an investment portfolio as well as comparing our results with Wang and Elhag [67].

The rest of the paper is structured as follows: Section 2 briefly describes the basic preliminaries and definitions for TOPSIS. Section 3 presents the details of the proposed fuzzy TOPSIS methods. In Section 4, a case study in the TSE is presented to demonstrate the applicability and efficacy of our proposed methods. In Section 5, the conclusions and future research directions are provided.