On spherical fuzzy distance measure and TAOV method for decision-making problems with incomplete weight information

Abstract

A spherical fuzzy set is one of the reliable tools to handle the uncertainties in the data with the help of the three membership degrees: Positive, neutral, and negative. Under this environment, the paper aims to present a novel decision-making algorithm named as MCDM-TAOV with some novel distance measures using matrix norms. In the literature, several kinds of distance measures occur, but they fail to meet certain axioms of the measure. To overwhelm this, the proposed measure satisfies all axiomatic requirements and overcomes the hindrances in the existing distance measures. Several counter-intuitive examples are provided to justify the superiority of the proposed measures. Furthermore, we addressed the total area based on orthogonal vector (TAOV) methodology to solve the MCDM (“multi-criteria decision-making”) problems. A distance-based criteria weight determination method is presented to compute the unknown criteria weight. The approach has been demonstrated by assessing the third-party reverse logistics provider (3PRLP) problem. The practical applicability, comparative analysis and advantages of the study with other decision-making methods are furnished to depict the efficiency of the moulded process.

Introduction

MCDM (“Multi-criteria decision-making”) is one of the most emerging topics to address decision-making problems (DMPs), where the target is to select the finest alternative(s) from the available resources. In the competitive era, our everyday life decisions are part of the MCDM, where the target is to select the most effective alternative from a collection of available options based on specific significant considerations. However, uncertainties always play a dominant role in any decision-making process. To address this, a concept of fuzzy set (FS) was introduced by Zadeh (1965) in 1965. A degree of membership, limited to [0,1], has been associated with each element, depicting the partial belongingness of the component of the set. Later on,  Atanassov (1986) extended the FS to the idea of IFS (“intuitionistic fuzzy set”) by adding the additional degree, named as degree of non-membership grade $\varrho$ along with membership grade $\sigma$ such theta $\sigma +\varrho \le 1$ for $\sigma ,\varrho \in [0,1]$. To consider the cases under which $\sigma +\varrho >1$,  Yager, 2013a, Yager, 2013b presented the notion of the PyFS (“Pythagorean fuzzy set”), by extending the wider region of the uncertainties with the condition ${\sigma }^2+{\varrho }^2\le 1$ for $\sigma ,\varrho \in [0,1]$. The region of the IFS contains the region of the PyFS. Under these environments, many scholars have defined several MCDM algorithms for solving the DMPs. For instance, Alcantud et al. (2020) investigated temporal IFSs. Wang and Li (2020) presented an MCDM algorithm based on the interaction power Bonferroni mean operators with PyFS features.  Wang and Garg (2021) utilize the Archimedean t-norms operators to define some interactive operators to address the DMPs under PyFS features. Akram et al., 2019, Akram et al., 2020, and  Biswas and Sarkar (2019) focused on developing decision-making techniques for group DMPs under the features of the PyFSs. Apart from that, many notable scholars have contributed to the construction of decision-making techniques and their applications under the different environments (Hajiagha et al., 2018, Akram et al., 2021b, Sharma et al., 2021, Ashraf et al., 2022, Qasem et al., 2021, Ali, 2022).

Although those mentioned above approaches are widely applicable for DMPs in our day-to-day life situations, these approaches are limited to only the problems where the decision is made in terms of either yes or no or both at the same time. However, there are many cases where decision-makers cannot make decisions using such tools. For instance, in the case of voting, a voter may have the option of voting in favor, against, or not voting at all. Such situations are not handled with the helps of FS, IFS and PyFS features. For this,  Cuong and Kreinovich (2013) initiated the notion of the PFS (“picture fuzzy set”) with three degrees, namely membership ($\sigma$), non-membership ($\varrho$) and refusal ($\vartheta$) such that $\sigma +\vartheta +\varrho \le 1$ for $\sigma ,\varrho ,\vartheta \in [0,1]$. Under this environment, researchers have presented several algorithms for DMPs. For instance,  Ashraf et al. (2019d) presented the aggregation operators based MCDM method to solve DMPs. Zhang et al. (2018), Wang et al. (2020) presented the picture fuzzy TOPSIS (“technique for order of preference by similarity to ideal solution”) method and their applications to different sectors to address the DMPs. The Dijkstra algorithm for shortest pathway network with picture fuzzy features is addressed by Akram et al. (2021a). Khan et al. (2021) addressed the bi-parametric distance and similarity measures for the pairs of PFSs and their applications to medical diagnostics problems.

Although PFS is extensively researched and implemented in several sectors, its scope for conveying information is still restricted. For instance, when an expert pay decision in such that way that $\sigma +\vartheta +\varrho >1$ then such situation is not addressed with this set. This hindrance is highlighted by Ashraf et al., 2019c, Ashraf and Abdullah, 2019 and hence presented the notion of the SFS (“spherical fuzzy set”) with three degrees of grades $\sigma ,\vartheta ,\varrho \in [0,1]$ such that ${\sigma }^2+{\vartheta }^2+{\varrho }^2\le 1$. Clearly, the domain of SFS is more wider than IFS, PyFS, PFS and hence it is considered as a more generalized extension of the FS. Keeping the advantages of SFS, several authors (Cengiz and Gündoğdu, 2021, Yuan et al., 2021, Dogan, 2021, Zahid et al., 2022) have presented their applications to solve the MCDM problems. Zhang et al. (2021) presented the MABAC (“multi-attributive border approximation area comparison”) method by integrating the concept of “cumulative prospect theory” and the entropy to evaluate suppliers’ performance using SFS features.  Ali (2021) constructed a score function with the SFS features and presented a MARCOS (“measurement of alternatives and ranking according to the compromise solution”) method for MCDM problems. In  Karaşan et al. (2021), authors integrated the CODAS (“combinative distance-based assessment”) approach with SFS features to address the problems. Mathew et al. (2020) integrated AHP (“analytic hierarchy process”) and TOPSIS approach to solve the system selection under SFS environment.  Karande and Chakraborty (2013) presented a spherical fuzzy Delphi-TOPSIS approach for MCDM problems.

TAOV (“Total Area based on Orthogonal Vectors”), developed by Hajiagha et al. (2018), is one of the newly MCDM methods based on the orthogonality of decision criteria. This MCDM method is composed of three important phases, namely, “initialization”, “orthogonalization”, and “comparison”. The major component of this method is to convert the decision matrix into the orthogonal decision matrix using PCA (“principal component analysis”). The TAOV method is based on the idea of using the Pythagorean theorem to calculate the distance between two places whose vectors are basically orthogonal to each other. In recent years, TAOV method has been widely used by the various researchers (Mokhtarzadeh et al., 2018, Aydın and Ege Oruç, 2022, Peng et al., 2021). For instance,  Mokhtarzadeh et al. (2018) applied the TAOV approach to explore a case study from an Iranian high-tech company in order to rank the most desirable technologies for the research and development division.  Aydın and Ege Oruç (2022) introduced a novel MCDM method by combining MAUT (“multiattribute utility theory”) and TAOV methods and applied it to the evaluation of renewable energy source alternatives. Peng et al. (2021) explored the q-rung orthopair fuzzy TAOV method with an aim to apply it to mobile edge caching scheme selection.

In the framework of the MCDM procedure, several approaches have been stated by the researchers to address the DMPs. However, an information measure is one of the reliable tools for DMPs. Distance measures are information measures to compute the distance between the pairs of FSs. In literature, several MCDM algorithms based on distance measures have been defined by the scholars (Ali et al., 2022b, Bashir et al., 2021, Bashir et al., 2019, Malik et al., 2018, Ali et al., 2022a). In terms of SFS environment,  Ashraf et al., 2019b, Ashraf et al., 2019a defined distance measures for SFS features. Mahmood et al. (2019) solve the medical diagnosis problems using distance measures. Khan et al. (2020) presented some improved distance and similarity measures between SFSs. Although these existing measures are well utilized to solve the problems, but at some certain cases, it occurs with several deficiencies such as (1) it fails to classify the different objects by giving the same results for two different SFSs; (2) it fails to satisfy some axioms of the measures and hence produces some “indistinguishable” and “inconsistent” results; (3) it occurs with the problems of “division by zero” problem.

From the above-mentioned literature, we observe some challenges which are stated as below.

• In terms of the utilization of distance measures, several authors (Ashraf et al., 2019b, Ashraf et al., 2019a, Mahmood et al., 2019, Khan et al., 2020) have conducted their research on distance measures for solving MCDM problems. However, these existing measures fail to meet the axiomatic definition of distance measure. Further, we observe several cases where these distances give either “inaccurate results” or “the same measure” for different SFNs (as described in Section 4).

• In the context of the MCDM problem, the criteria weight is a key feature of the process. In real-life situations, the information about the criteria weights cannot be overstated in advance. In other words, due to time restrictions, lack of information, knowledge, or data, and experts’ insufficient understanding of the topic, the information about criteria weights is usually and mostly somewhat known or entirely unknown. Thus, there is a proper need to consider this issue and to define a model that considers optimistic and pessimistic values when determining the weights of criteria.

• In the SFS decision-making process, all the existing methods assumed that the decision criteria are independent to avoid any overestimation of the scores. Thus, there is a need to define some more generalized MCDM method which can overcome this challenge.

To overcome the mentioned challenges, we construct a novel spherical fuzzy MCDM approach based on the TAOV method in this paper. We have defined some new axiomatic distance measures in the approach and hence it is based weight determination mechanism to explore the study. The key contributions of this article are enlisted as follows:

• A novel spherical fuzzy norm-based distance measure is formulated in the paper. Several counter-examples are introduced to analyze the drawbacks of the previous distance measures of SFSs.

• The devised spherical fuzzy distance measure is comprehensively compared and analyzed with other researchers’ works via numerical examples. The results illustrate that the originated distance outperforms other metrics to measure uncertainty and avoid counterintuitive cases.

• To introduce a new criteria weight determination method, based on the proposed distance measure, to handle the unknown criteria weight problems accurately.

• A new MCDM procedure based on the TAOV method is proposed in the study with SFS features. In this method, a PCA tool has been used to construct the orthogonal vectors related to the ratings; hence, the total area is computed to rank the elements.

• A practical case study involving the assessment of the 3PRLP (“third-party reverse logistics provider”) problem is taken to illustrate the suggested approach’s procedure. Additionally, the effects of the criteria weights on the ranking outcomes for the alternatives and comparison analysis are investigated in detail. Finally, a validity test and Spearman’s rank correlation test analysis are conducted to explain the result’s performance.

The rest of the paper is organized as follows. In Section 2, we briefly reviewed the concepts of SFSs and the existing distance measures. In Section 3, we proposed a norm-based distance measure and its demonstration and interpretation. Section 4 delineates the merits of the devised distance measure by making a comparison with existing measures. In Section 5, we develop the proposed MCDM procedure with the TAOV method and distance-based criteria weight determination under the background of SFS. The application of the proposed method is given in Section 6 as a numerical example. Further, the sensitivity and comparison analysis of the established MCDM method is also conducted in this section. Lastly, Section 7 ends with some concluding remarks.