A spherical fuzzy methodology integrating maximizing deviation and TOPSIS methods

https://doi.org/10.1016/j.engappai.2021.104212Get rights and content

Abstract

Due to the uncertainty and vagueness, ambiguity and subjectivity of the information in an intricate decision-making environment, the assessment data specified by experts are mostly fuzzy and uncertain. As an extension of Pythagorean fuzzy sets (PyFSs) and picture fuzzy sets (PFSs), spherical fuzzy sets (SFSs) are used frequently for presenting fuzzy and indeterminate information. In multi-criteria decision-making (MCDM) problems, the weights of criteria are not known generally. The maximizing deviation technique is a useful tool to handle such problems that we have partially or incomplete information about the criteria’ weights. This research expands the classical maximizing deviation technique to the spherical fuzzy maximizing deviation technique using single-valued (SV) and interval-valued (IV) spherical fuzzy sets to determine criteria weights. To rank the alternatives and specify the preeminent preference, we proposed the Interval Valued Spherical Fuzzy TOPSIS method based on the similarity measure instead of distance measure. For this purpose, we proposed an IVSF cosine similarity measure. To present its effectiveness and practicability, we apply the proposed methodology to an advertisement strategy selection problem, where IVSF sets are used to represent the evaluations about alternatives and criteria. A sensitivity analysis with different similarity measurements is performed to show the reliability of the proposed methodology.

Introduction

MCDM has been extensively used in several science fields, such as engineering, economics, and management in recent years. It has an essential role in modern decision science. The process of the MCDM is ranking all alternatives according to a list of criteria and selecting the optimal one among all options (Xing et al., 2019). Imprecision and uncertainty are two critical issues in MCDM problems. The fuzzy MCDM is a decision-making technique that uses fuzzy numbers to handle and measure impreciseness and vagueness (Sun et al., 2018; Farrokhizadeh et al., 2021). Due to the instinctive nature of human being thinking, evaluations over alternatives are always inexact and obscure in real decision-making problems. The necessity of dealing with uncertainty in real-world problems has originated different methodologies and theories (Rodríguez et al., 2016). The fuzzy set logic presented by Zadeh (1965) has a major impact on the solution of decision-making problems in an uncertain environment (Donyatalab et al., 2019). Since 1965, several extensions of general fuzzy sets extended by researchers. Type-two fuzzy sets developed by Zadeh (1975), intuitionistic fuzzy sets developed by Atanassov (1986), neutrosophic sets developed by Smarandache (2000), hesitant fuzzy sets developed by Torra (2010), Pythagorean fuzzy sets developed by Yager (2013), picture fuzzy sets developed by Cu’ò’ng (2015) and spherical fuzzy sets developed by Gündoǧdu and Kahraman (2019) are one of the popular extensions of ordinary fuzzy sets. In the Spherical fuzzy Sets (SFSs) that is the latest extension of fuzzy sets, an expert’s indeterminacy, membership and non-membership degrees can be expressed. Experts can assign all three parameters as long as they stay inside the unit sphere. This remarkable feature of the spherical fuzzy set distinguishes it from other fuzzy set models (Akram et al., 2020). Spherical fuzzy sets theory is useful and advantageous for handling uncertainty and imprecision in multiple attribute decision-making problems (Donyatalab et al., 2020). In real-world problems, it is not entirely rational to quantify the membership and non-membership degrees in the form of a single value number. To this purpose, Kutlu Gündoğdu and Kahraman (2019) developed the notion of IVSFSs. The researchers have proposed some decision-making processes by using spherical fuzzy sets in the literature (Ashraf et al., 2019b, Jin et al., 2019, Lathamaheswari et al., 2021)

Generally, in the MCDM problems, the weights of criteria are considered completely known. However, in the real world and practical MCDM problems, the weights of criteria are partially recognized or completely unrecognized due to various reasons such as lack of information or knowledge or time limitation. Thus, the classical methods are not suitable for this type of problems. Therefore, Yingming (1997) proposed the maximizing deviation method for such MCDM problems to specify the criteria’ weights. In the present research, we extend the maximizing deviation method to the spherical fuzzy sets with single and interval-valued numbers when the weights of criteria are partially recognized or completely unrecognized. After specifying the criteria weights using the proposed methods, the alternatives have to be ranked based on the experts’ evaluations, where the TOPSIS technique can be used. The logic of the classical TOPSIS is based on the distance measure so that it chooses the best option according to the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution. This study replaces this distance measure with the cosine similarity measure (CSM) and computes the similarity degree between each alternative and negative and positive reference index (RI). We developed the CSM for both single and interval-valued spherical fuzzy sets.

Cosine similarity, measures the similarity between two vectors of an inner product space. It is measured by the cosine of the angle between two vectors and determines whether two vectors are pointing in roughly the same direction. The Euclidean distance corresponds to the L2-norm of a difference between vectors whereas the cosine similarity is proportional to the dot product of two vectors and inversely proportional to the product of their magnitudes. If the similarity is more, that means you are close to ideal solutions. On the other hand, if the distance is less, that means you are close to ideal solutions. In the literature, both of these measures are often used in order to compare the alternatives with ideal solutions. In general, it is not expected these two approaches to yield a different result. However, cosine similarity has some specific application areas such as text similarity comparisons. Since there is no research in the literature on the extension of the spherical fuzzy sets to the similarity-based TOPSIS method, we prefer cosine similarity in this paper.

Thus, the uniqueness and contribution of this study is proposing the single and interval-valued spherical fuzzy maximizing deviation technique and integrated them with similarity measure based TOPSIS technique to solve the MCDM problems in such a way that the weight of the criteria are not predetermined.

The structure of the paper is formed as follows. Section 2 presents the preliminaries and a literature review on fuzzy TOPSIS method and maximizing deviation technique. Section 3 extends the maximizing deviation technique to SFSs and IVSFSs. In Section 4, we present the proposed method with the integration of maximizing deviation and TOPSIS. In Section 5, to validate the suggested method’s practicability and usefulness, we apply it to an advertisement strategy selection problem. Section 6 presents sensitivity and comparison analysis, and finally, in Section 7, the conclusion and discussion are given.

Section snippets

Spherical Fuzzy Sets (SFSs)

Spherical fuzzy sets are the newest addition of PFS and PyFS. Spherical fuzzy sets (SFSs) give a more extensive domain for the evaluator to explain and each decision-maker information.

Definition 1

A Single-valued spherical fuzzy set (SVSFSs) of the universe X which given by  (Gündoǧdu and Kahraman, 2019): Ãs={x,μÃsx,ϑÃsx,πÃsx|xX}where μÃsu,ϑÃsu,πÃsu:U0,1 are the degrees of membership, non-membership, and indeterminacy of x to ÃS, respectively, and 0μÃs2x+ϑÃs2(x)+πÃs2x1Then, 1μÃs2x+ϑÃs2(x)+π

Extension of maximizing deviation method using spherical fuzzy sets

In this section, we extend the maximizing deviation technique to single-valued and interval-valued spherical fuzzy extensions and establish a non-linear programming model to compute criteria’s weights.

Integration of maximizing deviation and TOPSIS methods based on the similarity measure

The TOPSIS method is one of the well-known MCDM techniques that chooses the best option so that it should have the shortest geometric distance from the positive ideal solution and the longest geometric distance from the negative ideal solution (Assari et al., 2012). In this section, we are going to use the similarity measure instead of distance measure in such a way that the alternative, which has the most similarity to positive ideal solution and the least similarity to negative ideal solution

An application to advertising strategy selection

Here, we apply our proposed hybrid method to advertising strategy selection in the product life period. The case company is one of the oldest and most famous food industry companies in Iran. The company wants to focus on product advertising to choose the best advertising strategy because sales of products are growing in the growth cycle of the life cycle. Experts introduced five alternatives to advertise products such as Television Ad. (A1), Radio Ad. (A2), newspaper Ad. (A3), Internet and SMS

Comparative and sensitivity analysis

To investigate the influence of the similarity measure on the results, we will perform a comparison and sensitivity analysis using different similarity measures in this section. To do this, two novel similarity measures for spherical fuzzy sets, Jaccard and square root similarity measures developed by Seyfi-Shishavan et al. (2020), will be utilized. The formula of these two similarity measures are given in Eqs. (52), (53), respectively. It is noteworthy that this analysis is just done for

Conclusion

This paper introduced a new hybrid method for the spherical fuzzy environment, which is the integration of maximizing deviation method and TOPSIS based on similarity measure. Also, we developed the cosine similarity measure for interval-valued spherical fuzzy sets. Decision-makers have a vital role in providing the information about options in decision-making problems. Due to time constraints, absence of information, knowledge or data, and experts’ incomplete expertise about the problem, the

CRediT authorship contribution statement

Elmira Farrokhizadeh: Conceptualization, Methodology, Writing - original draft. Seyed Amin Seyfi-Shishavan: Investigation, Writing, Validation. Fatma Kutlu Gündoğdu: Reviewing, Visualization, Investigation, Writing - review & editing. Yaser Donyatalab: Methodology, Validation. Cengiz Kahraman: Writing - review & editing. Seyyed Hadi Seifi: Visualization, Editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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