Pythagorean fuzzy VIKOR approaches based on TODIM for evaluating internet banking website quality of Ghanaian banking industry
Introduction
In recent time, the rapid growth of information technology and its application to the business environment has incredibly changed and is still changing the businesses model in our world [1]. Internet-based electronic systems and their connections to the main body of traditional businesses are rapidly growing. The banking industry is no exception to this change, and more evident than ever before in Ghana. With the rapid development of Internet banking technology in Ghana, the stakeholders such as government, customers, financial institutions and the general public, have been attracted the requirements for evaluating the performance of the new technology [2]. For example, to the government and financial regulation institutions, they hope that the Internet banking websites can meet the regulated standards and the national laws. With regards to the customers, they may consider whether the website is easy to use and protect their privacy [3]. Obviously, the evaluation of Internet banking website is a very necessary thing. This process of evaluating the websites of Ghanaian baking industry involves the consideration of several standards or criterion and hence can be classified as a multi-criteria decision making (MCDM) problem.
MCDM generally discusses the determination of the best feasible solution or alternative(s) for a problem involving numerous criteria. Meanwhile, the alternatives are carefully evaluated under each criterion in advance. Over the years, the researchers have developed many MCDM techniques to aid the decision maker (DM). Vlsekriterijumska Optimizacija I Kompromisno Resenje (VIKOR) is one of the classical MADM methods, which was proposed by Opricovic [4]. VIKOR basically focuses on ranking and selecting from a set of alternatives in the presence of conflicting criterion. It can support the decision makers to make a final outcome via a compromised solution. Nevertheless, the decision environments are deemed very critical to the outcome that it produces. For the Internet banking website evaluation, the researchers face some inherent setbacks by using the traditional MCDM techniques. In the real life, the website evaluation is difficult for decision makers to precisely express their preferences. From the existing literature of VIKOR, many scholars have broadened its scope to the various types of fuzzy environments, e.g., triangular fuzzy numbers, intuitionistic fuzzy sets (IFSs) [5], [6], [7], hesitant fuzzy sets [8], [9], [10], [11] and interval type-2 fuzzy environment [12].
As a generalization of fuzzy sets (FSs), Yager [13], [14] recently presented a novel evaluation format of Pythagorean fuzzy sets (PFSs). It satisfies the situation that the square sum of its membership degree and non-membership degree is less than or equal to 1 [14]. Clearly, the PFSs have more power than FSs and IFSs to model the uncertainty in real-life MCDM problems [15], [16], [17], [18]. The extant literature has given great attention to the decision making problems with Pythagorean fuzzy information. For example, Dick et al. [19] studied the lattice-theoretic properties of PFSs, and then extended them to the unit disc of the complex plane. In light of the division and subtraction operations of PFSs, Peng and Yang [16] minutely investigated their properties. Peng and Yang [20] also introduced the Choquet integral into Pythagorean fuzzy information aggregation. By adopting the TODIM approach, Ren et al. [21] proposed a new approach for the MCDM problems with PFSs. Zhang [22] developed a new method based on similarity measure to handle Pythagorean fuzzy multiple criteria group decision-making problems (PFMCGDM) and discussed two new aggregation operators. In allusion to hierarchical multi-criteria Pythagorean fuzzy decision-making problems, Zhang [23] also designed a new closeness index-based Pythagorean fuzzy QUALIFLEX method. In real life decision making activities, it is very difficult to quantify the degrees of the membership and non-membership of PFSs with respect to a single numeric value [24]. To solve this problem, Peng and Yang [24] extended the PFSs to the concept of interval-valued Pythagorean fuzzy sets (IVPFSs). Hence, the PFSs have strong ability to express the information vagueness and uncertainty of the Internet banking website evaluation.
To this regard, this paper seeks to extend the traditional VIKOR method to adapt the Pythagorean fuzzy MCDM environment for the Internet banking website evaluation. In allusion to the complexity of the Internet banking website evaluation, it also may be necessary to take the DM’s risk attitude into consideration [21]. TODIM (an acronym in Portuguese for Interactive Multi-Criteria Decision Making) is a valuable tool to cope with the behavior character and widely used in decision making under risk and uncertainty [25], [26], which was developed based on the prospect theory [27]. With respect to the hybrid multiple attribute decision making (MADM) problem, Fan et al. [28] proposed an extended TODIM method. Under the interval type-2 fuzzy environment, Qin et al. [29] designed an extended TODIM multi-criteria group decision making method for green supplier selection. Based on the hesitant fuzzy environment, Zhang and Xu [30] developed some novel measured functions for the TODIM approach. With the above-mentioned results, TODIM can effectively depict the DM’s risk attitude of the Internet banking website evaluation and few studies discuss the Pythagorean fuzzy MCDM environment. Therefore, we modify the VIKOR method to handle the DM’s psychological behavior by introducing TODIM in the compromised decision-making process. In this paper, we design two types of strategies for the combination between TODIM and VIKOR. One is that we directly transform the evaluations of the decision matrix with the aid of TODIM and then utilize VIKOR. The other is that we only introduce the psychological behavior described in TODIM to modify the distance of VIKOR. Hence, the corresponding methods have been developed, i.e., Approaches I and II. During the decision procedure, we study Pythagorean fuzzy entropy and cross-entropy measures for determining the weight information of MCDM. Based on the Ghanaian banking industry, we further illustrate the validity and applicability of our proposed methods. This paper extends the applied field of VIKOR to Pythagorean fuzzy situations. In addition, we improve TODIM to make it adapt a more complex environment.
The remainder of the paper is organized as follows: In Section 2, we briefly review some basic concepts of the PFSs, the traditional VIKOR and the TODIM method. In Section 3, we study the determination of the weights of the criteria with the Pythagorean fuzzy entropy and cross-entropy measures. Based on the two types of strategies of the combination between TODIM and VIKOR, Section 4 establishes the algorithmic details and develops the corresponding Pythagorean fuzzy VIKOR approaches. In Section 5, we give an illustrative example to elaborate and verify our proposed approaches. Section 6 considers the situation with more information and discusses the differences among VIKOR, TODIM, Approach I and Approach II with the aid of Wilcoxon signed-rank test [31]. Section 7 concludes the paper and elaborates future studies.
Section snippets
Preliminaries
In this section, we briefly introduce some basic concepts of PFSs, and also discuss a description of the basic principle of both the TODIM technique and the VIKOR method in ranking alternatives in its crisp or classical form.
The determination of the criteria weights with thePythagorean fuzzy entropy and cross-entropy measures
For the MCDM problem with PFNs, its elements indicate the evaluation values of all alternatives with respect to each criterion. Let be a discrete set of alternatives, be the set of criteria and be the weight vector of the criteria, which satisfies and . The evaluation value of the alternative with respect to the criterion is denoted as . In this case, the Pythagorean fuzzy decision matrix
Pythagorean fuzzy VIKOR approaches based on TODIM for MCDM
In this section, we introduce TODIM into VIKOR under the Pythagorean fuzzy environment. By using the determination of the criteria weights of Section 3 and the basic description, we further designed Pythagorean fuzzy VIKOR approaches based on TODIM for MCDM with PFNs, i.e., Approaches I and II. We consider the psychological behavior of the decision maker and design two types of strategies for the combination between TODIM and VIKOR. In light of the different stages of the combination, we design
A case study
In Ghana, the era of commercial banking started way back in 1874. Later, the technology is adopted by the banking. The forerunners of the electronic banking system, namely Barclays Bank Ghana Limited and Standard Chartered Bank Ghana Limited, greatly influence the adoption of Internet banking and have changed the setting in the Ghanaian banking industry. The bank of Ghana serves as the main governmental body for the payment and settlement systems. There are currently 33 licensed banks,
Simulation and discussion
In this section, we further consider some new situations with more information. By randomly generating PFNs, we establish a new decision matrix which is shown in Table 8.
Assume that each criterion is the benefit criterion and set . The ranks of different methods are summarized in Table 9.
From Table 9, these results of Table 9 imply more obvious differences among five approaches. The traditional VIKOR [4] still meet the problem of weak discrimination ability. Compared with other
Conclusions
The TODIM method is a useful behavior decision making technique to solve the MCDM problems. With the aid of TODIM, the Pythagorean fuzzy VIKOR approaches characterized by the DMs’ behaviors are developed for the Internet banking website evaluation. On the one hand, this paper extends the applied field of VIKOR to Pythagorean fuzzy situations. On the other hand, we improve TODIM to make it adapt the more complex environment. By defining Pythagorean fuzzy entropy and cross-entropy measures, we
Acknowledgments
This work is partially supported by the National Natural Science Foundation of China (Nos. 71401026, 71432003, 71771155), the Fundamental Research Funds for the Central Universities of China (No. ZYGX2014J100) and the Social Science Planning Project of the Sichuan Province, China (No. SC15C009).
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