A novel approach integrating AHP and TOPSIS under spherical fuzzy sets for advanced manufacturing system selection

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Abstract

The hybrid of AHP and TOPSIS has led researchers to integrate the combination with different extensions of fuzzy sets. The recently developed three-dimensional spherical fuzzy set is an extension of the fuzzy set, which is effective in handling uncertainty and quantifying expert judgements. In this paper, a novel framework is elaborated which combines AHP and TOPSIS with a spherical fuzzy set. Spherical fuzzy AHP is used to calculate the spherical fuzzy weights of the criteria, while spherical fuzzy TOPSIS is used to find the final rank of the alternatives. A new spherical fuzzy geometric mean formula is proposed for calculating the spherical fuzzy criteria weights. A new eleven-point spherical fuzzy linguistic term scale is presented, which can be used by the experts to quantify the preference. The proposed framework is applied to an advanced manufacturing system selection problem with six evaluation criteria and four alternatives. It is found that spherical fuzzy AHP-TOPSIS is effective in handling uncertainty in decision making and leads to robust and competitive results compared with state-of-the-art multi-criteria decision-making (MCDM) approaches.

Introduction

It is very difficult to make a decision in an environment characterized by vagueness and uncertainty. The introduction of fuzzy set theory has been instrumental in aiding decision makers to tackle this vagueness and uncertainty. The preference of experts is commonly expressed in linguistic terms, quantified using the Likert scale (Emovon et al., 2018). However, the Likert scale is unable to handle the ambiguity or vagueness of decision making as it assigns a single value to each linguistic term. With the advent of fuzzy set theory, linguistic terms are quantified using fuzzy numbers, which are successful in capturing the vagueness in preference. Normal fuzzy sets (type-1 fuzzy sets) are successful in capturing the vagueness but are unable to handle uncertainty associated with it. So, researchers have developed different extensions of fuzzy set theory to successfully handling data uncertainties.

Real-life problems contain multiple criteria and trade-offs, which should be considered during a decision-making process. Consequently, such decision making is referred to as multi-criteria decision making (MCDM), which can be classified into different categories such as, comparative/relative measurement methods, reference point methods, outranking methods and other methods (i.e., dominance, maxmin and minmax). The Analytical Hierarchical Process (AHP) is a relative measurement method, which can be implemented for ranking multiple alternatives by considering both qualitative and quantitative criteria (Saaty, 2008). AHP is used by many researchers to find the relative importance (weights) of criteria and sub-criteria (Prakash and Barua, 2015). The use of fuzzy set theory in AHP (use of a fuzzy number instead of a simple number) helps in capturing the vagueness in preference. Fuzzy AHP has been applied in a wide range of applications, notably: selecting best transportation projects by policymakers in the United States of America (Arslan, 2009), prioritizing sustainability aspects and indicators for material transportation problems (Calabrese et al., 2016), analysing and prioritizing different risks in the implementation of green supply chain practices (Mangla et al., 2015), evaluating a manufacturing plant’s sustainability factors (Jayawickrama et al., 2017), itemizing the barriers of reverse logistics (Lamba et al., 2019), comparing global climate models (Panjwani et al., 2019) and evaluating alternatives for E-waste’s collection and processing (Khoshand et al., 2019). Fuzzy axiomatic approach in which the information axiom under fuzzy environment is coupled with AHP (Maldonado-Macías et al., 2013), is another approach which is used in handling complex problems with quantitative and qualitative criteria (Maldonado-Macías et al., 2015). The approach has a limitation in that it only considers information using a membership function and does not consider other two aspects such as the degree of non-membership and the indeterminacy/degree of hesitancy.

The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is another MCDM method, which also selects the best alternative among many. Alternatives which have the least distance from the best or positive ideal result and have the greatest distance from the worst or negative ideal result, is considered as the best alternative (Behzadian et al., 2012). Fuzzy set theory coupled with TOPSIS helps decision makers to compute more reliable results, which are free of errors due to vagueness. Some of the applications of Fuzzy TOPSIS are: evaluating the ergonomic compatibility of advanced manufacturing technology (Maldonado-Macías et al., 2014), reviewing the factors for reverse logistics implementation in the Indian electronics industry (Agrawal et al., 2016), itemizing test cases to detect faults in software testing (Tahvili et al., 2016), ranking the criteria of flexible manufacturing systems (Siddiquie et al., 2017), and investigating the factors affecting the delay in electrical installation projects (Hamdan et al., 2019).

Considering the effectiveness of fuzzy AHP and Fuzzy TOPSIS, there has been an increasing interest among researchers and practitioners to integrate those approaches. Such integration has been implemented on different extensions of fuzzy set theory such as, type-1 fuzzy set (Singh and Sarkar, 2019), interval type 2 fuzzy set (Cevik Onar et al., 2014), neutrosophic sets (Junaid et al., 2020), Pythagorean Fuzzy Sets (PFS) (Ak and Gul, 2019), and even mixed fuzzy sets such as the mix of hesitant and interval type 2 fuzzy sets (Cevik Onar et al., 2014). The advantage of AHP to derive the criteria weights using pairwise comparison matrix and the ability of TOPSIS method to find the best alternative has led researchers to use this combination in a number of applications, such as the selection of a human resource manager for telecommunication company (Kusumawardani and Agintiara, 2015), selecting the chief inspectors of a bank (Esmaili Dooki et al., 2017), selecting the best cloud solution for big data projects (Boutkhoum et al., 2017), and selection of the best maintenance for thermal power plant (Panchal and Kumar, 2017).

Spherical fuzzy set (SFS) introduced by Kutlu Gündoğdu and Kahraman, 2019a, Kutlu Gündoğdu and Kahraman, 2019b, is the three-dimensional fuzzy set that was developed as the extension of intuitionistic fuzzy set, PFS and the neutrosophic logics, particularly to handle uncertainty during the quantification of experts judgements. The historical mapping of different extensions of fuzzy set (major development related to spherical fuzzy), along with their limitations, are shown in Fig. 1. In intuitionistic fuzzy set, the sum of the degree of membership and the degree of non-membership should lie between 0 to 1 and the indeterminacy/degree of hesitancy is calculated by subtracting this sum from 1 (Atanassov, 1989), making it more precise than type-1 fuzzy sets and type-2 fuzzy sets. Whereas, in PFS (Yager, 2013), the sum of its degree of membership and degree of non-membership can be greater than 1, but the sum of the square of its degree of membership and the square of its degree of non-membership should be less than or equal to 1. As Yager (2013) states,“The membership functions of PFS allow for a higher number of non-standard membership grades (its membership function) than intuitionistic fuzzy membership functions”, but indeterminacy/degree of hesitancy is dependent on the degree of membership and the degree of non-membership.

Smarandache (1999) introduced the concept of neutrosophic logic with terms like degree of truthfulness (T) equivalent to degree of membership, degree of falsehood (F) equivalent to degree of non-membership, and level of indeterminacy (I) equivalent to the indeterminacy/degree of hesitancy. In neutrosophic logic all three T, F and I should lie between 0 to 1, while the sum of T, F and I can lie between 0 to 3. This sum makes the function of T, F and I linear in 3D spaces.

However, as per Yang and Chiclana (2009), “In some cases the linear distance is not appropriate and in such cases nonlinear distance are more suitable”. Thus, in SFS, the sum of the degree of membership, non-membership degree and indeterminacy/degree of hesitancy can be greater than 1 but the square sum of the degree of membership, non-membership degree and indeterminacy/degree of hesitancy should lie between 0 and 1, consequently making it nonlinear. Also, the degree of membership, the degree of non-membership and indeterminacy/degree of hesitancy can be defined independently, which allows decision makers to define the decision-making problem with more information about criteria (greater flexibility) compared with IFS and PFS. The representation of IFS, PFS, neutrosophic set and SFS is shown in 3D space in Fig. 2. SFS also makes the decision-making process more intelligent (equivalent to human judgement) so that using SFS can lead to higher accuracy of assessment of alternatives in the decision-making process. Due to this advantage, SFS has recently been applied in a range of applications such as, warehouse site selection (Kutlu Gündoğdu and Kahraman, 2019a), selection of waste disposal site (Kutlu Gündoğdu et al., 2019), personnel selection problem (Kutlu Gündoğdu, 2020) and evaluation of design and technology of linear delta robot (Kutlu Gündoğdu and Kahraman, 2020b).

TOPSIS was first extended with SFS by(Kutlu Gündoğdu and Kahraman, 2019b). Later, Kutlu Gündoğdu and Kahraman (2020a) integrated AHP with SFS. In SFS-TOPSIS, the weights of the criteria are directly assigned without any comparison between the criteria. Whereas, in SFS-AHP, the final scores to rank the alternatives are calculated without determining the best and worst solutions. So, there is a need to combine AHP and TOPSIS under SFS for more accurate results. To our knowledge, there is no study which has combined AHP and TOPSIS under SFS. Therefore, this paper proposes a hybrid of two methods i.e. AHP and TOPSIS, which uses SFS theory in handling the key issues of uncertainty in decision making. The pairwise comparison matrix in AHP is formulated using spherical fuzzy number. This spherical fuzzy pairwise comparison matrix is used to calculate the spherical fuzzy weights of the criteria; whereas, spherical fuzzy TOPSIS is used to find the final rank of the alternatives. To calculate the spherical fuzzy weights of the criteria, Buckley’s geometric mean method is utilized. Novel features of this proposed framework are:

  • A new spherical fuzzy geometric mean formula for calculating the spherical criteria weights is proposed.

  • New eleven-point spherical fuzzy linguistic term scale is presented.

  • This is the first integrated approach of AHP and TOPSIS with the newly introduced SFS.

The remainder of this paper is organized into five sections. The literature review section highlights the work done in AHP-TOPSIS under different fuzzy sets. Section 3 explains the preliminaries of extended fuzzy set and arithmetic operations of SFS. A new combination of AHP-TOPSIS using the SFS is proposed in Section 4, while one numerical illustration is presented in Section 5. The sixth section highlights the managerial implications of the proposed method, which is followed by the conclusion section.

Section snippets

Literature review

In this study, the literature review is carried out based on the works which combine AHP and TOPSIS with different fuzzy sets as a MCDM technique. Literature on both type-1 fuzzy sets and extended fuzzy sets are considered as an avenue for AHP-TOPSIS combinations.

(i) Type-1 Fuzzy Sets with AHP-TOPSIS

In the literature, there is an abundance of papers published MCDM as a combination of AHP and TOPSIS under simple type-1 fuzzy sets. We considered the application of AHP-TOPSIS with type-1 fuzzy

Spherical Fuzzy sets

In an SFS (Kutlu Gündoğdu and Kahraman, 2019b), S̃ of the universe of discourse X is given by S̃=x,μS̃x,υS̃x,πS̃x|xX, where μS̃:X0,1,υS̃:X[0,1],πS̃:X[0,1], μS̃(x) is the degree of membership, υS̃x is the non-membership degree and πS̃x is the indeterminacy/degree of hesitancy. The expression is subject to condition 0 μS̃(x)2+υS̃(x)2+πS̃(x)2 1. For convenience, we call S̃=μS̃,υS̃,πS̃. While performing computation on a decision-making problem, different arithmetic operations of SFS are

Methodology

The integration of AHP and TOPSIS has been implemented by a number of researchers on different extensions of fuzzy set theory, as explained in the literature review. The added advantage of AHP to derive the criteria weights using a pairwise comparison matrix and the ability of the TOPSIS method to find the best alternative has made this combination a success.

Integration of AHP and TOPSIS with Spherical Fuzzy Sets

The flowchart of the methodology for integrating AHP, TOPSIS and SFS is shown in

Application to advanced manufacturing system selection

In the era of Industry 4.0 and the advancement of the digital framework, demand for products can be rapidly changed in real time. Such dynamic behaviour in manufacturing systems has pushed manufacturing companies to reduce the product life cycle and produce goods in small batches. With the help of advanced technologies like flexible manufacturing system (FMS), industries are capable of manufacturing ranges of products in different batch sizes with less changeover time. In the past, many

Managerial implications

The managerial implication of the developed novel spherical fuzzy AHP-TOPSIS model is that it can provide support to managers and decision makers for making strategic decisions and bring out robust and reliable results. This framework can be used by managers of different industries, in different applications like supplier selection, maintenance strategy selection, evaluation of robots under industrial condition, material handling equipment selection and many other decision-making processes.

Conclusions and recommendations for further work

This article illustrates a novel spherical fuzzy AHP-TOPSIS which has been proved to be successful in handling imprecision in the decision-making process. The newly developed three-dimensional extension of fuzzy sets (i.e., spherical fuzzy sets) are incorporated in different MCDM techniques. In the paper, spherical fuzzy AHP is used to calculate the weights of criteria, while spherical fuzzy TOPSIS is used to find the final ranking of the alternatives. A new spherical fuzzy geometric mean

CRediT authorship contribution statement

Manoj Mathew: Problem conceptualization, Data curation, Formal analysis, Investigation, Methodology, Resources, Software, Supervision, Validation, Visualization, Writing - original draft. Ripon K. Chakrabortty: Problem conceptualization, Investigation, Methodology, Resources, Supervision, Validation, Visualization, Writing - review & editing. Michael J. Ryan: Lead project investigator, Project administration, Writing - review & 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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