A novel VIKOR method using spherical fuzzy sets and its application to warehouse site selection

Abstract

The extensions of ordinary fuzzy sets such as intuitionistic fuzzy sets (IFS), Pythagorean fuzzy sets (PFS), and neutrosophic sets (NS), whose membership functions are based on three dimensions, aim at collecting experts’ judgments more informatively and explicitly. In the literature, generalized three-dimensional spherical fuzzy sets have been developed by Kutlu Gündoğdu and Kahraman (2019), including their arithmetic operations, aggregation operators, and defuzzification operations. Spherical Fuzzy Sets (SFS) are a new extension of Intuitionistic, Pythagorean and Neutrosophic Fuzzy sets, a SFS is characterized by a membership degree, a nonmembership degree, and a hesitancy degree satisfying the condition that their squared sum is equal to or less than one. These sets provide a larger preference domain in 3D space for decision makers (DMs). In this paper, our aim is to extend classical VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method to spherical fuzzy VIKOR (SF-VIKOR) method and to show its applicability and validity through an illustrative example and to present a comparative analysis between spherical fuzzy TOPSIS (SF-TOPSIS) and SF-VIKOR. We handle a warehouse location selection problem with four alternatives and four criteria in order to demonstrate the performance of the proposed SF-VIKOR method.

Keywords

Spherical fuzzy sets multicriteria decision making VIKOR warehouse location selection

1. Introduction

After the presentation of ordinary fuzzy sets by Zadeh (1965), they have been very popular in almost all branches of science. Various researchers [6 , 57] have developed several extensions of ordinary fuzzy sets as given in Fig. 1 with a historical order. Yager (2013) has renamed Atanassov’s intuitionistic fuzzy sets of second type (IFS2) as Pythagorean fuzzy sets (PFS). Hence PFS and IFS2 mean the same fuzzy sets thereafter [44].

Fig.1

Extensions of fuzzy sets.

In recent years, several researchers have utilized these extensions in the solution of multi-criteria decision making problems. A classification of some recent publications with respect to the types of fuzzy extensions is as follows:

Type-2 fuzzy sets (T2FS): The concept of a type-2 fuzzy set was introduced by Zadeh (1975) as an extension of the concept of an ordinary fuzzy set called a type-1 fuzzy set [28]. Such sets are fuzzy sets whose membership grades themselves are type-1 fuzzy sets; they are very useful in circumstances where it is difficult to determine an exact membership function for a fuzzy set [20 , 45].

Intuitionistic fuzzy sets (IFS): Intuitionistic fuzzy sets introduced by Atanassov (1986) enable defining both the membership and non-membership degrees of an element in a fuzzy set [8 , 58].

Hesitant fuzzy sets (HFS): Hesitant fuzzy sets can be used as a functional tool allowing many potential degrees of membership of an element to a set. These fuzzy sets force the membership degree of an element to be possible values between zero and one [1 , 62].

Pythagorean fuzzy sets (PFS): Atanassov’s intuitionistic fuzzy sets of second type (IFS2) or Yager’s Pythagorean fuzzy sets are characterized by a membership degree and a nonmembership degree satisfying the condition that the square sum of its membership degree and nonmembership degree is equal to or less than one, which is a generalization of Intuitionistic Fuzzy Sets (IFS) [15–17 , 65].

Neutrosophic sets (NS): Smarandache (1999) developed neutrosophic logic and neutrosophic sets (NSs) as an extension of intuitionistic fuzzy sets. The neutrosophic set is defined as the set where each element of the universe has a degree of truthiness, indeterminacy and falsity [6 , 41].

The spherical fuzzy sets (SFS) have been recently introduced by Kutlu Gündoğdu and Kahraman (2019) [9]. These sets are based on the fact that the hesitancy of a decision maker can be defined independently from membership and nonmembership degrees, satisfying the following condition [9]:

$0 ⩽ μ_{- A}^{2} (u) + v_{- A}^{2} (u) + π_{- A}^{2} (u) ⩽ 1 \forall u \in U$ (1)

On the surface of the sphere, Equation (1) becomes $μ_{\tilde{A}}^{2} (u) + ν_{\tilde{A}}^{2} (u) + π_{\tilde{A}}^{2} (u) = 1 \forall u \in U$ (2)

The idea behind SFS is to let decision makers to generalize other extensions of fuzzy sets by defining a membership function on a spherical surface and independently assign the parameters of that membership function with a larger domain. SFS are a synthesis of PFS and NS.

The VIKOR method was developed by S. Opricovic (1998) as an MCDM method to solve a discrete multi-criteria problem with non-commensurable and conflicting criteria [49]. The method aims at determining a compromise solution for ranking the alternatives by considering those criteria. A compromise solution is a feasible solution nearest to the ideal solution [50]. The classical VIKOR method has been extended to its fuzzy versions by using various types of fuzzy sets such as type-2 fuzzy VIKOR [36], intuitionistic fuzzy VIKOR [22, 53], and hesitant fuzzy VIKOR [13], Pythagorean fuzzy VIKOR [56].

To the best of our knowledge, spherical fuzzy VIKOR method has not yet been developed in the literature. The motivation of this paper is to extend the VIKOR method under spherical fuzzy environment, assuming that membership, non-membership and hesitancy degrees are independently assigned and their squared sum is equal to at most 1. In this study, the proposed SF-VIKOR decision making model is applied to a warehouse site selection problem since the experts’ evaluations of location alternatives involve impreciseness and vagueness. An illustrative case study containing four criteriaand four alternatives is presented in the application section.

The originality of the paper comes from the presentation of a novel SF-VIKOR and the application of the proposed method to warehouse site selection problem. The SF-VIKOR enables decision makers to independently reflect their hesitancies in the decision process by using a linguistic evaluation scale based on spherical fuzzy sets.

The rest of this paper is organized as follows. Section 2 includes the introductory definitions and the preliminaries on SFS. Section 3 summarizes a literature review on VIKOR method. Section 4 includes our proposed MCDM method: Spherical Fuzzy VIKOR (SF-VIKOR). Section 5 applies SF-VIKOR method to a warehouse site selection problem. Section 6 includes a comparative analysis between SF-VIKOR and SF-TOPSIS and a sensitivity analysis for SF-VIKOR. Finally, the study is concluded in the last section.

2. Spherical Fuzzy Sets: preliminaries

Intuitionistic and Pythagorean fuzzy membership functions are composed of membership, non-membership and hesitancy parameters, which can be calculated by $π_{\tilde{I}} = 1 - μ - ϑ$ or $π_{\tilde{p}} = \sqrt{1 - μ_{\tilde{p}}^{2} - ν_{\tilde{p}}^{2}}$ , respectively. Neutrosophic membership functions are also defined by three parameters truthiness, falsity and indeterminacy, whose sum can be between 0 and 3, and the value of each is between 0 and 1 independently. In spherical fuzzy sets, while the squared sum of membership, non-membership and hesitancy parameters can be between 0 and 1, each of them can be defined between 0 and 1 independently to satisfy that their squared sum is at most equal to 1. Figure 2 illustrates the differences among IFS, PFS, NS and SFS [7 , 44].

Fig.2

Geometric representations of IFS, PFS, NS and SFS.

In this section, we give the definition of SFS and summarize spherical distance measurement, arithmetic operations, aggregation operators and defuzzification operations.

Definition 1. Spherical Fuzzy Sets (SFS) ${\tilde{A}}_{S}$

A spherical fuzzy set ${\tilde{A}}_{S}$ of the universe of discourse U is given by ${\tilde{A}}_{S} = {〈 u, (μ_{{\tilde{A}}_{S}} (u), ν_{{\tilde{A}}_{S}} (u), π_{{\tilde{A}}_{S}} (u)) | u \in U}$ (3) where $μ_{{\tilde{A}}_{S}} : U \to [0, 1], ν_{{\tilde{A}}_{S}} (u) : U \to [0, 1], π_{{\tilde{A}}_{S}} : U \to [0, 1]$ and $0 ⩽ μ_{{\tilde{A}}_{S}}^{2} (u) + ν_{{\tilde{A}}_{S}}^{2} (u) + π_{{\tilde{A}}_{S}}^{2} (u) ⩽ 1 \forall u \in U$ (4)

For each u, the numbers $μ_{{\tilde{A}}_{S}} (u), ν_{{\tilde{A}}_{S}} (u)$ and $π_{{\tilde{A}}_{S}} (u)$ are the degree of membership, non-membership and hesitancy of u to ${\tilde{A}}_{S}$ , respectively (Kutlu Gündoğdu & Kahraman, 2019).

On the basis of relationship between SFS and PFS, Kutlu Gündoğdu & Kahraman (2019) further define some novel operations for SFS as below [9]:

Definition 2. Basic Operators

Union; ${\tilde{A}}_{S} \cup {\tilde{B}}_{S} = {max {μ_{{\tilde{A}}_{S}}, μ_{{\tilde{B}}_{S}}}, min {ν_{{\tilde{A}}_{S}}, v_{{\tilde{B}}_{S}}}, min {{(1 - ({(max {μ_{{\tilde{A}}_{S}}, μ_{{\tilde{B}}_{S}}})}^{2} + {(min {ν_{{\tilde{A}}_{S}}, v_{{\tilde{B}}_{S}}})}^{2}))}^{1 / 2}, max {π_{{\tilde{A}}_{S}}, π_{{\tilde{B}}_{S}}}}}$ (5)

Intersection; ${\tilde{A}}_{S} \cap {\tilde{B}}_{S} = {min {μ_{{\tilde{A}}_{S}}, μ_{{\tilde{B}}_{S}}}, max {ν_{{\tilde{A}}_{S}}, v_{{\tilde{B}}_{S}}}, max {{(1 - ({(min {μ_{{\tilde{A}}_{S}}, μ_{{\tilde{B}}_{S}}})}^{2} + {(max {ν_{{\tilde{A}}_{S}}, v_{{\tilde{B}}_{S}}})}^{2}))}^{1 / 2}, min {π_{{\tilde{A}}_{S}}, π_{{\tilde{B}}_{S}}}}}$ (6)

Addition; ${\tilde{A}}_{S} \oplus {\tilde{B}}_{S} = {{(μ_{{\tilde{A}}_{S}}^{2} + μ_{{\tilde{B}}_{S}}^{2} - μ_{{\tilde{A}}_{S}}^{2} μ_{{\tilde{B}}_{S}}^{2})}^{1 / 2}, v_{{\tilde{A}}_{S}} v_{{\tilde{B}}_{S}}, {((1 - μ_{{\tilde{B}}_{S}}^{2}) π_{{\tilde{A}}_{S}}^{2} + (1 - μ_{{\tilde{A}}_{S}}^{2}) π_{{\tilde{B}}_{S}}^{2} - π_{{\tilde{A}}_{S}}^{2} π_{{\tilde{B}}_{S}}^{2})}^{1 / 2}}$ (7)

Multiplication; ${\tilde{A}}_{S} \otimes {\tilde{B}}_{S} = {μ_{{\tilde{A}}_{S}} μ_{{\tilde{B}}_{S}}, {(v_{{\tilde{A}}_{S}}^{2} + v_{{\tilde{B}}_{S}}^{2} - v_{{\tilde{A}}_{S}}^{2} v_{{\tilde{B}}_{S}}^{2})}^{1 / 2}, {((1 - v_{{\tilde{B}}_{S}}^{2}) π_{{\tilde{A}}_{S}}^{2} + (1 - v_{{\tilde{A}}_{S}}^{2}) π_{{\tilde{B}}_{S}}^{2} - π_{{\tilde{A}}_{S}}^{2} π_{{\tilde{B}}_{S}}^{2})}^{1 / 2}}$ (8)

Multiplication by a scalar; λ > 0 $λ \cdot {\tilde{A}}_{S} = {{(1 - {(1 - μ_{{\tilde{A}}_{S}}^{2})}^{λ})}^{1 / 2}, v_{{\tilde{A}}_{S}}^{λ}, {({(1 - μ_{{\tilde{A}}_{S}}^{2})}^{λ} - {(1 - μ_{{\tilde{A}}_{S}}^{2} - π_{{\tilde{A}}_{S}}^{2})}^{λ})}^{1 / 2}}$ (9)

λ . Power of ${\tilde{A}}_{S}$ ; λ > 0 ${\tilde{A}}_{S}^{λ} = {μ_{{\tilde{A}}_{S}}^{λ}, {(1 - {(1 - v_{{\tilde{A}}_{S}}^{2})}^{λ})}^{1 / 2}, {({(1 - v_{{\tilde{A}}_{S}}^{2})}^{λ} - {(1 - v_{{\tilde{A}}_{S}}^{2} - π_{{\tilde{A}}_{S}}^{2})}^{λ})}^{1 / 2}}$ (10)

Definition 3. For these SFS ${\tilde{A}}_{S} = (μ_{{\tilde{A}}_{S}}, v_{{\tilde{A}}_{S}}, π_{{\tilde{A}}_{S}})$ and ${\tilde{B}}_{S} = (μ_{{\tilde{B}}_{S}}, v_{{\tilde{B}}_{S}}, π_{{\tilde{B}}_{S}})$ , the followings are valid under the condition λ, λ₁, λ₂ > 0. $i . {\tilde{A}}_{S} \oplus {\tilde{B}}_{S} = {\tilde{B}}_{S} \oplus {\tilde{A}}_{S}$ (11) $ii . {\tilde{A}}_{S} \otimes {\tilde{B}}_{S} = {\tilde{B}}_{S} \otimes {\tilde{A}}_{S}$ (12) $iii . λ ({\tilde{A}}_{S} \oplus {\tilde{B}}_{S}) = λ {\tilde{A}}_{S} \oplus λ {\tilde{B}}_{S}$ (13) $iv . λ_{1} {\tilde{A}}_{S} \oplus λ_{2} {\tilde{A}}_{S} = (λ_{1} + λ_{2}) {\tilde{A}}_{S}$ (14) $v . ({\tilde{A}}_{S} \otimes {\tilde{B}}_{S})^{λ} = {\tilde{A}}_{S}^{λ} \otimes {\tilde{B}}_{S}^{λ}$ (15) $vi . {\tilde{A}}_{S}^{λ_{1}} \otimes {\tilde{A}}_{S}^{λ_{2}} = {\tilde{A}}_{S}^{λ_{1} + λ_{2}}$ (16)

Definition 4. Spherical Weighted Arithmetic Mean (SWAM) with respect to, $w = (w_{1}, w_{2} . . . . . . ., w_{n}); w_{i} \in [0, 1]; \sum_{i = 1}^{n} w_{i} = 1$ , SWAM is defined as; $\begin{matrix} {SWAM}_{w} ({\tilde{A}}_{S 1}, . . . . . . ., {\tilde{A}}_{Sn}) \\ = w_{1} {\tilde{A}}_{S 1} + w_{2} {\tilde{A}}_{S 2} + . . . . . . + w_{n} {\tilde{A}}_{Sn} \\ = {{[1 - \prod_{i = 1}^{n} (1 - μ_{{\tilde{A}}_{S}}^{2})^{w_{i}}]}^{1 / 2}, \prod_{i = 1}^{n} v_{{\tilde{A}}_{S}}^{w_{i}}, \\ {[\prod_{i = 1}^{n} (1 - μ_{{\tilde{A}}_{S}}^{2})^{w_{i}} - \prod_{i = 1}^{n} (1 - μ_{{\tilde{A}}_{S}}^{2} - π_{{\tilde{A}}_{Si}}^{2})^{w_{i}}]}^{1 / 2}} \end{matrix}$ (17)

Definition 5. Spherical Weighted Geometric Mean (SWGM) with respect to, $w = (w_{1}, w_{2} . . . . . . ., w_{n}); w_{i} \in [0, 1]; \sum_{i = 1}^{n} w_{i} = 1$ , SWGM is defined as; $\begin{matrix} {SWGM}_{w} ({\tilde{A}}_{1}, . . . . . . ., {\tilde{A}}_{n}) \\ = {\tilde{A}}_{S 1}^{w_{1}} + {\tilde{A}}_{S 2}^{w_{2}} + . . . . . . + {\tilde{A}}_{Sn}^{w_{n}} \\ = {\prod_{i = 1}^{n} μ_{{\tilde{A}}_{Si}}^{w_{i}}, {[1 - \prod_{i = 1}^{n} (1 - v_{{\tilde{A}}_{Si}}^{2})^{w_{i}}]}^{1 / 2}, \\ {[\prod_{i = 1}^{n} (1 - v_{{\tilde{A}}_{Si}}^{2})^{w_{i}} - \prod_{i = 1}^{n} (1 - v_{{\tilde{A}}_{Si}}^{2} - π_{{\tilde{A}}_{Si}}^{2})^{w_{i}}]}^{1 / 2}} \end{matrix}$ (18)

3. A literature review on fuzzy VIKOR

One of the most popular MCDM methods, VIKOR focuses on ranking and sorting a set of alternatives against various, or possibly conflicting and non-commensurable decision criteria assuming that compromising is acceptable to resolve conflicts. Similar to some other MCDM methods, VIKOR relies on an aggregating function that represents closeness to the ideal solution. It introduces the ranking index based on the particular measure of closeness to the ideal solution and uses linear normalization to eliminate units of criterion functions [50].

As seen in Table 1, VIKOR has been integrated with different fuzzy extensions in a few fields of application in recent years. The publications on VIKOR method are summarized in Table 1.

Table 1
A literature review on fuzzy VIKOR

Year Authors Extension of VIKOR Application area

2002 Opricovic and Tzeng [51] VIKOR Analysis the planning strategies for reducing the future social and economic costs in the area with potential natural hazard

2010 Vahdani et al. [2] Interval-valued fuzzy sets Illustrative example

2011 Devi [23] Intuitionistic fuzzy sets Robot selection

2011 Liu and Wang [39] Interval-valued trapezoidal fuzzy numbers Illustrative example

2013 Zhao et al. [61] Interval-valued Intuitionistic fuzzy sets Illustrative example

2013 Liao and Xu [13] Hesitant fuzzy sets Service quality among domestic airlines

2013 Wan et al. [53] Triangular intuitionistic fuzzy numbers Personnel selection problem

2014 Mousavi et al. [47] Fuzzy GRA VIKOR (trapezoidal fuzzy numbers) Selection of material handling equipment

2015 You et al. [60] Fuzzy VIKOR(Inteval 2-tuple linguistic) Supplier selection

2015 Ghorabaee et al. [36] Interval type-2 fuzzy sets Project selection

2016 Mousavi et al. [48] Intuitionistic fuzzy sets Practical portfolio selection and material handling selection problems

2016 Wu et al. [63] Fuzzy VIKOR(with linguistic information) Supplier selection in nuclear power industry

2017 Rao et al. [3] Fuzzy VIKOR Supplier selection

2017 Wang and Cai [59] Fuzzy VIKOR Emergency supplier selection

2017 Soner et al. [37] Interval type 2 fuzzy sets Selection of maritime transportation

2017 Zhao et al. [22] Intuitionistic fuzzy sets Supplier Selection

2018 Chen [56] Pythagorean fuzzy VIKOR Numerical examples

2018 Emeç Akkaya [54] Fuzzy VIKOR &warehouse location selection

2018 Zhou et al. [11] Fuzzy extended VIKOR Mobile robot selection model for hospital pharmacy

2018 Khan et al. [30] Pythagorean hesitant fuzzy setting Numerical examples

Year	Authors	Extension of VIKOR	Application area
2002	Opricovic and Tzeng [51]	VIKOR	Analysis the planning strategies for reducing the future social and economic costs in the area with potential natural hazard
2010	Vahdani et al. [2]	Interval-valued fuzzy sets	Illustrative example
2011	Devi [23]	Intuitionistic fuzzy sets	Robot selection
2011	Liu and Wang [39]	Interval-valued trapezoidal fuzzy numbers	Illustrative example
2013	Zhao et al. [61]	Interval-valued Intuitionistic fuzzy sets	Illustrative example
2013	Liao and Xu [13]	Hesitant fuzzy sets	Service quality among domestic airlines
2013	Wan et al. [53]	Triangular intuitionistic fuzzy numbers	Personnel selection problem
2014	Mousavi et al. [47]	Fuzzy GRA VIKOR (trapezoidal fuzzy numbers)	Selection of material handling equipment
2015	You et al. [60]	Fuzzy VIKOR(Inteval 2-tuple linguistic)	Supplier selection
2015	Ghorabaee et al. [36]	Interval type-2 fuzzy sets	Project selection
2016	Mousavi et al. [48]	Intuitionistic fuzzy sets	Practical portfolio selection and material handling selection problems
2016	Wu et al. [63]	Fuzzy VIKOR(with linguistic information)	Supplier selection in nuclear power industry
2017	Rao et al. [3]	Fuzzy VIKOR	Supplier selection
2017	Wang and Cai [59]	Fuzzy VIKOR	Emergency supplier selection
2017	Soner et al. [37]	Interval type 2 fuzzy sets	Selection of maritime transportation
2017	Zhao et al. [22]	Intuitionistic fuzzy sets	Supplier Selection
2018	Chen [56]	Pythagorean fuzzy VIKOR	Numerical examples
2018	Emeç	Akkaya [54]	Fuzzy VIKOR &warehouse location selection
2018	Zhou et al. [11]	Fuzzy extended VIKOR	Mobile robot selection model for hospital pharmacy
2018	Khan et al. [30]	Pythagorean hesitant fuzzy setting	Numerical examples

A literature review on fuzzy VIKOR method using SCOPUS database gives 4,595 published papers in all fields. Among these, 514 papers mention fuzzy VIKOR in their titles, abstracts, or keywords. Yearly distribution of papers on fuzzy VIKOR is given in Fig. 3. As it is clearly seen, about 150 papers more per year are added to the previous year VIKOR publications.

Fig.3

Fuzzy VIKOR studies based on years 2010–2018.

Figure 4 illustrates the article numbers on fuzzy VIKOR with respect to their authors. Authors E. K. Zavadskas (with 121 publications) from Vilnius Gediminas Technical University, G.H. Tzeng (with 89 publications) from Harbin Institute of Technology, and Z. Xu (with 54 publications) from Nanjing University of Information Science and Technology are the most productive three researchers in this field.

Fig.4

Researchers studying on fuzzy VIKOR.

Fuzzy VIKOR method has been used in many different areas. These areas can be categorized as follows: engineering, computer science, business management, mathematics, environmental science, decision sciences, social sciences, energy, earth and planetary sciences, material science and other areas as represented in Fig. 5. Especially, in the computer science and engineering areas, the method has been extensively used.

Fig.5

Fuzzy VIKOR studies with respect to their research areas.

4. Extension of VIKOR with spherical fuzzy sets

The proposed SF-VIKOR method integrates the superiorities of Pythagorean fuzzy sets and neutrosophic sets as follows. It presents a larger definition space for the parameters of a SFS as in PFS. Hesitancy degree in a SFS can be assigned independently as in NS. The proposed sets eliminate the argued condition that the sum of membership, non-membership and hesitancy degrees is at most 3 in NS. Besides, the proposed method involves a new spherical fuzzy distance measurement, a new SF aggregation operator and a new SF defuzzification formula.

The proposed spherical fuzzy VIKOR method is composed of several steps as given in the following. Before giving these steps, we present the flow chart of the SF-VIKOR method in Fig. 6 in order to make it easily understandable.

Fig.6

SF-VIKOR proposed methodology.

A MCDM problem can be expressed as a decision matrix whose elements indicate the evaluation values of all alternatives with respect to each criterion under Spherical fuzzy environment. Let X ={ x₁, x₂, . . . . . . x_m } (m ⩾ 2) be a discrete set of m feasible alternatives and C ={ C₁, C₂, . . . . . . C_n } be a finite set of criteria, and w ={ w₁, w₂, . . . . . . w_n } be the weight vector of all criteria which satisfies 0 ⩽ w_j ⩽ 1 and $\sum_{j = 1}^{n} w_{j} = 1$ .

Step 1. Let DMs fill in the decision and criteria evaluation matrices using the linguistic terms given in Table 2 [9].

Table 2

Linguistic terms and their corresponding spherical fuzzy numbers

Linguistic terms	(μ, v, π)
Absolutely more Importance (AMI)	(0.9, 0.1, 0.1)
Very High Importance (VHI)	(0.8, 0.2, 0.2)
High Importance (HI)	(0.7, 0.3, 0.3)
Slightly More Importance (SMI)	(0.6, 0.4, 0.4)
Equally Importance (EI)	(0.5, 0.5, 0.5)
Slightly Low Importance (SLI)	(0.4, 0.6, 0.4)
Low Importance (LI)	(0.3, 0.7, 0.3)
Very Low Importance (VLI)	(0.2, 0.8, 0.2)
Absolutely Low Importance (ALI)	(0.1, 0.9, 0.1)

Step 2. Aggregate the judgments of each decision maker (DM) using Spherical Weighted Arithmetic Mean (SWAM) or Spherical Weighted Geometric Mean (SWGM) operators that are given in Definitions (4) and (5).

Construct aggregated spherical fuzzy decision matrix based on the opinions of decision makers.

Denote the evaluation values of Alternative X_i (i = 1, 2 . . . . . m) with respect to criterion C_j (j = 1, 2 . . . . . n) by $C_{j} ({\tilde{X}}_{i}) = (μ_{ij}, v_{ij}, π_{ij})$ and $D = {(C_{j} ({\tilde{X}}_{i}))}_{mxn}$ is a spherical fuzzy decision matrix. For a MCDM problem with SFS, decision matrix $\tilde{D} = {(C_{j} ({\tilde{X}}_{i}))}_{mxn}$ should be constructed as in Equation (19). $\tilde{D} = {(C_{j} ({\tilde{X}}_{i}))}_{mxn} = (\begin{matrix} \begin{matrix} (μ_{11}, v_{11}, π_{11}) & (μ_{12}, v_{12}, π_{12}) \begin{matrix} . & . & . \end{matrix} & (μ_{1 n}, v_{1 n}, π_{1 n}) \end{matrix} \\ \begin{matrix} \begin{matrix} (μ_{21}, v_{21}, π_{21}) \\ . \\ . \end{matrix} & \begin{matrix} (μ_{22}, v_{22}, π_{22}) \begin{matrix} . & . & . \end{matrix} \\ . \\ . \end{matrix} & \begin{matrix} (μ_{2 n}, v_{2 n}, π_{2 n}) \\ . \\ . \end{matrix} \end{matrix} \\ \begin{matrix} (μ_{m 1}, v_{m 1}, π_{m 1}) & (μ_{m 2}, v_{m 2}, π_{m 2}) \begin{matrix} . & . & . \end{matrix} & (μ_{mn}, v_{mn}, π_{mn}) \end{matrix} \end{matrix})$ (19)

Step 3. Defuzzify the aggregated decision matrix by using Equation (20) $Score (C_{j} ({\tilde{X}}_{i})) = {(2 μ_{ij} - π_{ij})}^{2} - {(v_{ij} - π_{ij})}^{2}$ (20)

Step 4. Aggregate the spherical fuzzy linguistic evaluations of the decision criteria, assigned by decision makers.

At this point, there are two possible ways to follow. The first possible way is to follow a partially fuzzy approach as follows:

Defuzzify the aggregated criteria weights by using the score function given in Equation (21). Normalize the aggregated criteria weights by using Equation (22). $w_{j}^{s} = {(2 μ_{ij} - π_{ij})}^{2} - {(v_{ij} - π_{ij})}^{2}$ (21) ${\bar{w}}_{j}^{s} = \frac{w_{j}^{s}}{\sum_{J = 1}^{n} w_{j}^{s}}$ (22)

The second way is to continue without defuzzifying the criteria weights. This approach is called full fuzzy approach.

Step 5. Determine the Spherical Fuzzy Positive Ideal Solution (SF-PIS) and the Spherical Fuzzy Negative Ideal Solution (SF-NIS), based on the score values obtained in Step 3.

For the SF-PIS, Equation (23) is used to find the maximum scores in the decision matrix. Based on the crisp maximum scores, the corresponding SF numbers are determined as in Equation(24). $X^{*} = {C_{j}, max_{i} < Score (C_{j} (X_{i})) > | j = 1, 2 . . . n}$ (23)

$\begin{matrix} {\tilde{X}}^{*} & = & {〈 C_{1}, (μ_{1}^{*}, v_{1}^{*}, π_{1}^{*}) 〉, 〈 C_{2}, (μ_{2}^{*}, v_{2}^{*}, π_{2}^{*}) 〉 . . . . . \\ 〈 C_{n}, (μ_{n}^{*}, v_{n}^{*}, π_{n}^{*}) 〉} \end{matrix}$ (24)

For the SF-NIS, Equation (25) is used to find the minimum scores in the decision matrix. Based on the crisp minimum scores, the corresponding SF numbers are determined as in Equation (26). $X^{-} = {C_{j}, min_{i} < Score (C_{j} (X_{i})) > | j = 1, 2 . . . n}$ (25)

$\begin{matrix} {\tilde{X}}^{-} & = & {〈 C_{1}, (μ_{1}^{-}, v_{1}^{-}, π_{1}^{-}) 〉, 〈 C_{2}, (μ_{2}^{-}, v_{2}^{-}, π_{2}^{-}) 〉 \\ . . . . . 〈 C_{n}, (μ_{n}^{-}, v_{n}^{-}, π_{n}^{-}) 〉} \end{matrix}$ (26)

Step 6. Calculate the values S_i and R_i by the relations according to partial fuzzy and full fuzzy approaches as in Equations (27) and (28),respectively. $S_{i} = \sum_{j = 1}^{n} {\bar{w}}_{j}^{s} \cdot D = \sum_{j = 1}^{n} {\bar{w}}_{j}^{s} \cdot \frac{D ({\tilde{X}}_{ij}, {\tilde{X}}_{j}^{*})}{D ({\tilde{X}}_{j}^{-}, {\tilde{X}}_{j}^{*})}$ (27) $R_{i} = max_{j} ({\bar{w}}_{j}^{s} \cdot D) = max_{j} ({\bar{w}}_{j}^{s} \cdot \frac{D ({\tilde{X}}_{ij}, {\tilde{X}}_{j}^{*})}{D ({\tilde{X}}_{j}^{-}, {\tilde{X}}_{j}^{*})})$ (28)

Alternatively, spherical fuzzy weights can be used to continue with the full fuzzy approach as in Equation (29). $\begin{matrix} D \cdot {\tilde{w}}_{j}^{s} & = & {{(1 - {(1 - μ_{{\tilde{w}}_{j}^{s}}^{2})}^{D})}^{1 / 2}, v_{{\tilde{w}}_{j}^{s}}^{λ}, \\ {({(1 - μ_{{\tilde{w}}_{j}^{s}}^{2})}^{D} - {(1 - μ_{{\tilde{w}}_{j}^{s}}^{2} - π_{{\tilde{w}}_{j}^{s}}^{2})}^{D})}^{1 / 2}} \end{matrix}$ (29)

$Score (D \cdot {\tilde{w}}_{j}^{s})$ can be calculated by using Equation (21) and then S_i and R_i values can be found as in Equations (30) and (31), respectively. $S_{i} = \sum_{j = 1}^{n} Score (D \cdot {\tilde{w}}_{j}^{s})$ (30) $R_{i} = max_{j} (Score (D \cdot {\tilde{w}}_{j}^{s}))$ (31)

At this point, there are three possible distance formulas to follow. Euclidean distance (Equations (32) and (33)), Xu and Zhang’s distance (Equations (34) and (35)) and spherical distance (Equations (36) and (37)) can be used in this step[5 , 64].

$\begin{matrix} D_{E} ({\tilde{X}}_{ij}, {\tilde{X}}_{j}^{*}) \\ = \sqrt{({(μ_{{\tilde{X}}_{ij}} - μ_{{\tilde{X}}_{j}^{*}})}^{2} + {(v_{{\tilde{X}}_{ij}} - v_{{\tilde{X}}_{j}^{*}})}^{2} + {(π_{{\tilde{X}}_{ij}} - π_{{\tilde{X}}_{j}^{*}})}^{2})} \end{matrix}$ (32) $\begin{matrix} D_{E} ({\tilde{X}}_{j}^{-}, {\tilde{X}}_{j}^{*}) \\ = \sqrt{({(μ_{{\tilde{X}}_{j}^{-}} - μ_{{\tilde{X}}_{j}^{*}})}^{2} + {(v_{{\tilde{X}}_{j}^{-}} - v_{{\tilde{X}}_{j}^{*}})}^{2} + {(π_{{\tilde{X}}_{j}^{-}} - π_{{\tilde{X}}_{j}^{*}})}^{2})} \end{matrix}$ (33) $\begin{matrix} D_{XZ} ({\tilde{X}}_{ij}, {\tilde{X}}_{j}^{*}) \\ = \frac{1}{2} (| μ_{{\tilde{X}}_{ij}}^{2} - μ_{{\tilde{X}}_{j}^{*}}^{2} | + | v_{{\tilde{X}}_{ij}}^{2} - v_{{\tilde{X}}_{j}^{*}}^{2} | + | π_{{\tilde{X}}_{ij}}^{2} - π_{{\tilde{X}}_{j}^{*}}^{2} |) \end{matrix}$ (34) $\begin{matrix} D_{XZ} ({\tilde{X}}_{j}^{-}, {\tilde{X}}_{j}^{*}) \\ = \frac{1}{2} (| μ_{{\tilde{X}}_{j}^{-}}^{2} - μ_{{\tilde{X}}_{j}^{*}}^{2} | + | v_{{\tilde{X}}_{j}^{-}}^{2} - v_{{\tilde{X}}_{j}^{*}}^{2} | + | π_{{\tilde{X}}_{j}^{-}}^{2} - π_{{\tilde{X}}_{j}^{*}}^{2} |) \end{matrix}$ (35) $\begin{matrix} D_{S} ({\tilde{X}}_{ij}, {\tilde{X}}_{j}^{*}) \\ = \frac{2}{π} (arccos (μ_{{\tilde{X}}_{ij}} μ_{{\tilde{X}}_{j}^{*}} + v_{{\tilde{X}}_{ij}} v_{{\tilde{X}}_{j}^{*}} + π_{{\tilde{X}}_{ij}} π_{{\tilde{X}}_{j}^{*}})) \end{matrix}$ (36) $\begin{matrix} D_{S} ({\tilde{X}}_{j}^{-}, {\tilde{X}}_{j}^{*}) \\ = \frac{2}{π} (arccos (μ_{{\tilde{X}}_{j}^{-}} μ_{{\tilde{X}}_{j}^{*}} + v_{{\tilde{X}}_{j}^{-}} v_{{\tilde{X}}_{j}^{*}} + π_{{\tilde{X}}_{j}^{-}} π_{{\tilde{X}}_{j}^{*}})) \end{matrix}$ (37) where S_i represents the separation measure of alternative i from the best value, and R_i represents the separation measure of alternative I from the worst value.

Step 7. Calculate S^*, S^-, R^*, R^- and Q_i values based on Equations (38) and (39), respectively. $\begin{matrix} S^{*} = min_{i} S_{i} \\ S^{-} = max_{i} S_{i} \\ R^{*} = min_{i} R_{i} \\ R^{-} = max_{i} R_{i} \end{matrix}$ (38) $Q_{i} = v \frac{(S_{i} - S^{*})}{(S^{-} - S^{*})} + (1 - v) \frac{(R_{i} - R^{*})}{(R^{-} - R^{*})}$ (39)

The indices min S_i and min R_i are related to a maximum majority rule, and a minimum individual regret of an opponent strategy, respectively. As well, v is introduced as the weight of the strategy of the maximum group utility. v is usually assumed to be 0.5.

Step 8. Select the best alternative. Finally, the best alternative with the minimum of Q_i is determined.

Propose as a compromise solution the alternative (a′) which is ranked the bet by the measure Q_i (minimum) if the following two conditions are satisfied:

C1:“Acceptable advantage”: $Q (a^{″}) - Q (a^{'}) ⩾ \frac{1}{m - 1}$ (40) where a″ the alternative with second position in the ranking list by m is the number of alternatives.

C2: “Acceptable stability in decision making”: Alternative a′ must also be the best ranked by S or/and R.

5. An illustrative example

Facility layout and location has been a well-established research area within operations research and management science. However, the originality of the paper comes from the proposed novel SF-VIKOR method in the warehouse site selection. Our proposed methodology is applied to the selection of a facility warehouse location. For this goal, proposed four subprovinces (X: Cihanbeyli, A2: Aksehir, A3: Eregli, A4: Meram) in Konya as shown in Fig. 7 are evaluated.

Fig.7

The map of Konya.

After a comprehensive literature review, four criteria have been determined. Criteria are infrastructure (C1), markets (C2), costs (C3), and labor characteristics (C4). In this structure, while “costs” is a non-beneficial criterion, the rest of them are beneficial. First of all, the assessments for the criteria are collected from decision maker with respect to the goal, using the linguistic terms given in Table 2. In the evaluation process, three decision makers (DM1, DM2, and DM3) are included who are three experienced engineers in supply chain and logistics management. The weights of these decision makers who have different experience levels are 0.3, 0.2 and 0.5, respectively. All assessments are given in Tables 3–5.

Table 3

Assessments of DM1

Alternatives	C ₁	C ₂	C ₃	C ₄
X ₁	HI	VHI	HI	HI
X ₂	SLI	LI	SLI	VLI
X ₃	VHI	AMI	HI	VHI
X ₄	SMI	HI	SMI	HI

Table 4

Assessments of DM2

Alternatives	C ₁	C ₂	C ₃	C ₄
X ₁	VHI	HI	VHI	HI
X ₂	VLI	ALI	LI	EI
X ₃	AMI	VHI	AMI	HI
X ₄	HI	EI	HI	SMI

Table 5

Assessments of DM3

Alternatives	C ₁	C ₂	C ₃	C ₄
X ₁	HI	HI	VHI	HI
X ₂	VLI	ALI	EI	SLI
X ₃	HI	AMI	AMI	VHI
X ₄	SMI	EI	HI	VHI

These judgments are aggregated using SWAM operator by considering the importance levels of decision makers. Aggregated decision matrix is obtained as in Table 6.

Table 6

Aggregated decision matrix

Alternatives	C ₁	C ₂	C ₃	C ₄
X ₁	(0.72, 0.28, 0.28)	(0.74, 0.27, 0.27)	(0.77, 0.23, 0.23)	(0.70, 0.30, 0.30)
X ₂	(0.28, 0.73, 0.29)	(0.19, 0.83, 0.19)	(0.44, 0.56, 0.45)	(0.38, 0.63, 0.39)
X ₃	(0.79, 0.21, 0.22)	(0.89, 0.11, 0.12)	(0.86, 0.14, 0.15)	(0.78, 0.22, 0.22)
X ₄	(0.62, 0.38, 0.38)	(0.58, 0.43, 0.44)	(0.67, 0.33, 0.33)	(0.74, 0.26, 0.27)

The linguistic importance weights of the criteria assigned by DMs are shown in Table 7.

Table 7

Importance weights of the criteria

Criteria	DM1	DM2	DM3
C ₁	HI	EI	EI
C ₂	VHI	AMI	HI
C ₃	HI	VHI	VHI
C ₄	SLI	LI	EI

The weight of each criterion obtained by using SWAM operator is presented in Table 8.

Table 8

Aggregation criteria weights

Criteria	Weight of each criterion
C ₁	(0.58, 0.43, 0.44)
C ₂	(0.79, 0.21, 0.22)
C ₃	(0.77, 0.23, 0.23)
C ₄	(0.44, 0.56, 0.45)

After the weights of the criteria have been determined, the defuzified and normalized criteria weights are calculated by utilizing Equations (21) and (22) as given in Table 9.

Table 9

Defuzzified and normalized criteria weights

Criteria	Weight of each criterion
C ₁	0.11916
C ₂	0.43071
C ₃	0.40964
C ₄	0.04048

Based on Table 6 and Equation (21), score function values are obtained as in Table 10. The highest values represent positive ideal solution while the lowest values represent negative ideal solution.

Table 10

Score function values based on SWAM operator

Alternatives	C ₁	C ₂	C ₃	C ₄
X ₁	1.367	1.444	1.744	1.210
X ₂	–0.127	–0.382	0.172	0.080
X ₃	1.834	2.732	2.473	1.817
X ₄	0.751	0.507	1.036	1.482

According to the best and worst scores, the corresponding Spherical Fuzzy Positive Ideal Solution (SF-PIS) and Spherical Fuzzy Negative Ideal Solution (SF-NIS) are given in Table 11.

Table 11

SF-PIS and SF-NIS

	C ₁	C ₂	C ₃	C ₄
X^*(Best)	(0.79, 0.21, 0.22)	(0.89, 0.11, 0,12)	(0.86, 0.14, 0.15)	(0.78, 0.22, 0.22)
X^-(Worst)	(0.28, 0.73, 0.29)	(0.19, 0.83, 0.19)	(0.44, 0.56, 0.45)	(0.38, 0.63, 0.39)

In the next step, based on Equations (38–40), we can calculate the values of S_i, R_i, and Q_i for each alternative based on Euclidean distance. They are given in Tables 12 and 13.

Table 12

The values of S_i, R_i, and Q_i for each alternative based on Euclidean distance

S _i	X ₁	0.2274
	X ₂	1.0000
	X ₃	0.0000
	X ₄	0.4804
R _i	X ₁	0.1117
	X ₂	0.4307
	X ₃	0.0000
	X ₄	0.2337
Q _i	X ₁	0.2433
	X ₂	1.0000
	X ₃	0.0000
	X ₄	0.5114

Table 13

The ranking of alternatives in ascending order by S_i, R_i and Q_i based on Euclidean distance

Ranking	1	2	3	4
S _i	X ₃	X ₁	X ₄	X ₂
R _i	X ₃	X ₁	X ₄	X ₂
Q _i	X ₃	X ₁	X ₄	X ₂

In the next step, based on Equations (38–40), we can calculate the values of S_i, R_i, and Q_i for each alternative based on spherical distance. They are given in Tables 14 and 15.

Table 14

The values of S_i, R_i, and Q_i for each alternative based on spherical distance

S _i	X ₁	0.6797
	X ₂	1.0000
	X ₃	0.5907
	X ₄	0.7625
R _i	X ₁	0.1063
	X ₂	0.4307
	X ₃	0.0000
	X ₄	0.2375
Q _i	X ₁	0.2175
	X ₂	1.0000
	X ₃	0.0000
	X ₄	0.4199

Table 15

The ranking of alternatives in ascending order by S_i, R_i and Q_i based on spherical distance

Ranking	1	2	3	4
S _i	X ₃	X ₁	X ₄	X ₂
R _i	X ₃	X ₁	X ₄	X ₂
Q _i	X ₃	X ₁	X ₄	X ₂

In the next step, based on Equations (38–40), we can calculate the values of S_i,R_i, and Q_i for each alternative based on Xu and Zhang’s distance. They are given in Tables 16 and 17.

Table 16

The values of S_i,R_i, and Q_i for each alternative based on Xu and Zhang’s distance

S _i	X ₁	0.2146
	X ₂	1.0000
	X ₃	0.0000
	X ₄	0.4742
R _i	X ₁	0.1117
	X ₂	0.4307
	X ₃	0.0000
	X ₄	0.2337
Q _i	X ₁	0.2146
	X ₂	1.0000
	X ₃	0.0000
	X ₄	0.4742

Table 17

The ranking of alternatives in ascending order by S_i, R_i and Q_i based on Xu and Zhang’s distance

Ranking	1	2	3	4
S _i	X ₃	X ₁	X ₄	X ₂
R _i	X ₃	X ₁	X ₄	X ₂
Q _i	X ₃	X ₁	X ₄	X ₂

According to the full fuzzy spherical Fuzzy VIKOR method, the values of S_i, R_i, and Q_i for each alternative and the rankings based on spherical distance, Zhang & Xu’s distance and Euclidean distance are given in Tables 18–20.

Table 18

The values of S_i, R_i, and Q_i for each alternative based on spherical distance (full fuzzy approach)

Alternatives	S _i	R _i	Q _i	Ranking based on S_i	Ranking based on R_i	Ranking based onQ_i
X ₁	3.1075	1.4416	0.1284	1	2	2
X ₂	5.1827	1.8337	1.0000	4	4	4
X ₃	3.4204	1.3060	0.0754	2	1	1
X ₄	4.2999	1.5667	0.5343	3	3	3

Table 19

The values of S_i, R_i, and Q_i for each alternative based on Xu and Zhang’s distance (full fuzzy approach)

Alternatives	S _i	R _i	Q _i	Ranking based on S_i	Ranking based on R_i	Ranking based onQ_i
X ₁	–2.4045	–0.2744	0.3329	2	2	2
X ₂	1.8454	0.8472	1.0000	4	4	4
X ₃	–4.0000	–1.0000	0.0000	1	1	1
X ₄	0.0622	0.4236	0.7328	3	3	3

Table 20

The values of S_i, R_i, and Q_i for each alternative based on Euclidean distance (full fuzzy approach)

Alternatives	S _i	R _i	Q _i	Ranking based on S_i	Ranking based on R_i	Ranking based onQ_i
X ₁	–2.3389	–0.2479	0.3457	2	2	2
X ₂	1.8454	0.8472	1.0000	4	4	4
X ₃	–4.0000	–1.0000	0.0000	1	1	1
X ₄	0.0766	0.4154	0.7318	3	3	3

All the results show that the ranking of the alternatives from the best to the worst is X₃ > X₁ > X₄ > X₂.

6. Comparative and sensitivity analyses

Kutlu Gündoğdu and Kahraman (2019) extended TOPSIS method to Spherical fuzzy TOPSIS (SF-TOPSIS) using spherical fuzzy sets [9]. SF-TOPSIS has been proposed in their paper and applied to the performance comparison of air condition suppliers successfully. In this study, our proposed methodology is compared with SF-TOPSIS for the site selection of warehouse.

In this comparison, the same aggregated decision matrix is used as given in Table 7. Xu and Zhang’s distances to positive and negative ideal solutions to SF-PIS and SF-NIS can be used as given Table 21.

Table 21
Xu and Zhang’s distances to positive and negative ideal solutions

Alternatives SF-PIS SF-NIS

X ₁ 0.053 0.201

X ₂ 0.250 0.000

X ₃ 0.000 0.250

X ₄ 0.084 0.185

Alternatives	SF-PIS	SF-NIS
X ₁	0.053	0.201
X ₂	0.250	0.000
X ₃	0.000	0.250
X ₄	0.084	0.185

Based on the classical closeness ratio formula, the ratios are calculated and presented in Table 22.

Table 22

Closeness ratio of each alternative

Alternatives	Closeness Ratio	Ranking
X ₁	0.7902	2
X ₂	0	4
X ₃	1	1
X ₄	0.6884	3

The closeness ratios based on SF-TOPSIS method indicate that the best alternative is X₃ and overall ranking is X₃ > X₁ > X₄ > X₂ similar with SF-VIKOR method.

We applied a sensitivity analysis by changing v, which is introduced as the weight of the strategy of the maximum group utility and the individual regret, and observed the robustness of the given decisions. Sensitivity analysis showed that very robust decisions have been obtained from SF-VIKOR as given in Fig. 8. Although the compromise solutions (Q) changed, the ranking of alternatives remained the same.

Fig.8

Sensitivity analysis.

7. Conclusion and future work

VIKOR method is a MCDM method developed for the solution of complex systems. The method determines compromise solutions for the problems including conflicting criteria. We presented a spherical fuzzy extension of VIKOR to determine fuzzy compromise solution, where both criteria weights and alternative evaluations could be spherical fuzzy sets. Spherical fuzzy VIKOR is based on the spherical fuzzy aggregation functions and different distance measurement methods. We have also presented both a partially fuzzy approach and a full fuzzy approach to spherical fuzzy VIKOR. Comparative and sensitivity analyses showed that the ranking obtained from the SF-VIKOR method is quite robust and similar to SF-TOPSIS method. Spherical fuzzy sets let the decision makers assign membership, non-membership, and hesitancy degrees independently under the constraint that their squared sum is at most 1. Spherical fuzzy sets are a kind of combination of Pythagorean (IFS2) fuzzy sets and neutrosophic fuzzy sets.

For further research, we suggest interval-valued SF-VIKOR (IVSF-VIKOR) method to be developed and the obtained results to be compared with the SF-VIKOR in this paper. Aggregation operators and distance measurement equations for IVSF sets need to be developed prior to IVSF-VIKOR method.

References

Farhadinia , Multiple criteria decision-making methods with completely unknown weights in hesitant fuzzy linguistic term setting, Knowledge-Based Systems 93 (2016), 135–144.

Vahdani ,

S.M.

Mousavi ,

Tavakkoli-Moghaddam and

Hashemi , A new design of the elimination and choice translating reality method for multi-criteria group decision-making in an intuitionistic fuzzy environment, Applied Mathematical Modelling 37 (2013), 1781–1799.

Rao ,

Xiao ,

Goh ,

Zheng and

Wen , Compound mechanism design of supplier selection based on multiattribute auction and risk management of supply chain, Comput. Ind. Eng 105 (2017), 63–75.

Yu ,

Zhang and

Xu , Group decision making under hesitant fuzzy environment with application to personnel evaluation, Knowledge-Based Systems 52 (2013), 1–10.

Szmidt and

Kacprzyk , Distances between intuitionistic fuzzy sets, Fuzzy Sets and Systems (2000), 505–518.

Smarandache , A unifying field in logics neutroso-phy:neutrosophic probability, set and logic, Rehoboth: American Research Press, 1999.

Smarandache , Neutrosophic set-a generalization of the intuitionistic fuzzy set. International journal of pure and applied mathematics (2003), 287.

Jin ,

Ni ,

Chen and

Li , Approaches to group decision making with intuitionistic fuzzy preference relations based on multiplicative consistency, Knowledge-Based Systems 97 (2016), 48–59.

Kutlu Giindogdu and

Kahraman , Spherical fuzzy sets and spherical fuzzy TOPSIS method. Journal of Intelligent & Fuzzy Systems, 36(1) (2019), 337–352.

10.

Kutlu Gundogdu ,

Kahraman &

H. N.

Civan , A novel hesitant fuzzy EDAS method and its application to hospital selection, Journal of Intelligent & Fuzzy Systems, 35(6) (2018), 6353–6365.

11.

Zhou ,

Wang and

Goh , Fuzzy extended VIKOR-based mobile robot selection model for hospital pharmacy, International Journal of Advanced Robotic Systems 15(4) (2018), 1–11.

12.

G.L.

Xu ,

S.P.

Wan ,

Wang ,

J.Y.

Dong and

Y.F.

Zeng , Mathematical programming methods for consistency and consensus in group decision making with intuitionistic fuzzy preference relations, Knowledge-Based Systems 98 (2016), 30–43.

13.

H.C.

Liao and

Z.S.

Xu , A VIKOR-based method for hesitant fuzzy multi-criteria decision-making, Fuzzy Optim. Decision Making 12(4) (2013), 373–392.

14.

Ma ,

Hu ,

Li and

Zhang , Toward trustworthy cloud service selection: A time-aware approach using interval neutrosophic set, Journal of Parallel and Distributed Computing 96 (2016), 75–94.

15.

Garg , A novel correlation coefficients between Pythagorean fuzzy sets and its applications to decision-making processes, International Journal of Intelligent Systems 31(12) (2016), 1234–1252.

16.

Garg , Generalized Pythagorean fuzzy geometric aggregation operators using Einstein t-norm and tconorm for multicriteria decision-making process, International Journal of Intelligent Systems 32(6) (2017a), 597–630.

17.

Garg and A., Novel, Improved Accuracy Function for Interval Valued Pythagorean Fuzzy Sets and Its Applications in the Decision-Making Process, International Journal of Intelligent Systems 32(12) (2017b), 1247–1260.

18.

Grattan-Guinness , Fuzzy membership mapped onto interval and many-valued quantities, Zeitschrift fur mathematische Logik und Grundladen der Mathematik 22(1) (1976), 149–160.

19.

J.M.

Garibaldi ,

Jaroszewski and

Musikasuwan , Non-stationary fuzzy sets, IEEE Transactions on Fuzzy Systems (2008), 1072–1086.

20.

Qin and

Liu , Multi-attribute group decision making using combined ranking value under interval type-2 fuzzy environment, Information Sciences 297 (2015), 293–315.

21.

Qin ,

Liu and

Pedrycz , Frank aggregation operators and their application to hesitant fuzzy multiple attribute decision making, Applied Soft Computing 41 (2016), 428–452.

22.

Zhao ,

X.Y.

You ,

H.C.

Liu and

S.M.

Wu , An extended VIKOR method using intuitionistic fuzzy sets and combination weights for supplier selection, Symmetry 9(9) (2017), 169.

23.

Devi , Extension of VIKOR method in intuitionistic fuzzy environment for robot selection, Expert Syst Appl 38(11) (2011), 14163–14168.

24.

K.P.

Chiao , The multi-criteria group decision making methodology using type 2 fuzzy linguistic judgments, Applied Soft Computing 49 (2016), 189–211.

25.

K.T.

Atanassov , Intuitionistic fuzzy sets, Fuzzy sets and Systems 20(1) (1986), 87–96.

26.

Wang ,

Shen and

Zhu , Dual hesitant fuzzy power aggregation operators based on Archimedean t-conorm and t-norm and their application to multiple attribute group decision making, Applied Soft Computing 38 (2016), 23–50.

27.

L.A.

Zadeh , Fuzzy Sets, Information and Control (1965), 338–353.

28.

L.A.

Zadeh , The concept of a linguistic variable and its application to approximate reasoning, Information Sciences (1975), 199–249.

29.

M.S.A.

Khan and

Abdullah , Interval-valued Pythagorean fuzzy GRA method for multiple-attribute decision making with incomplete weight information, International Journal of Intelligent Systems 33(8) (2018), 1689–1716.

30.

M.S.A.

Khan ,

Abdullah ,

Ali and

Amin , An extension of VIKOR method for multi-attribute decision-making under Pythagorean hesitant fuzzy setting, Granular Computing (2018), 1–14.

31.

M.S.A.

Khan ,

Abdullah ,

Yousaf Ali ,

Hussain and

Farooq , Extension of TOPSIS method base on Choquet integral under interval-valued Pythagorean fuzzy environment, Journal of Intelligent & Fuzzy Systems 34(1) (2018), 267–282.

32.

M.S.A.

Khan ,

Abdullah and

Lui , Gray Method for Multiple Attribute Decision Making with Incomplete Weight Information under the Pythagorean Fuzzy Setting, Journal of Intelligent Systems (Preprint) (2018).

33.

M.S.A.

Khan ,

Ali ,

Abdullah ,

Amin and

Hussain , New extension of TOPSIS method based on Pythagorean hesitant fuzzy sets with incomplete weight information, Journal of Intelligent & Fuzzy Systems 35(5) (2018), 5435–5448.

34.

M.S.A.

Khan ,

Abdullah ,

Ali ,

Siddiqui and

Amin , Pythagorean hesitant fuzzy sets and their application to group decision making with incomplete weight information, Journal of Intelligent & Fuzzy Systems 33(6) (2017), 3971–3985.

35.

M.S.A.

Khan ,

Abdullah ,

Ali ,

Amin and

Hussain , Pythagorean hesitant fuzzy Choquet integral aggregation operators and their application to multi-attribute decision-making, Soft Computing (2018), 1–17.

36.

M.K.

Ghorabaee ,

Amiri ,

J.S.

Sadaghiani and

E.K.

Zavadskas , Multi-criteria project selection using an extended VIKOR method with interval type-2 fuzzy sets, Int. J. Inf. Technol. Decis. Mak 14 (2015), 993–1016.

37.

Soner ,

Celik and

Akyuz , Application of AHP and VIKOR methods under interval type 2 fuzzy environment in maritime transportation, Ocean Eng 129 (2017), 107–116.

38.

Gupta ,

M.K.

Mehlawat and

Grover , Intuitionistic fuzzy multi-attribute group decision-making with an application to plant location selection based on a new extended VIKOR method, Information Sciences 370-371 (2016), 184–203.

39.

Liu and

Wang , An extended VIKOR method for multiple attribute group decision making based on generalized interval-valued trapezoidal fuzzy numbers, Scientific Research and Essays 6(4) (2011), 765–776.

40.

Liu ,

Zhang ,

Liu and

Wang , Multi-valued neu-trosophic number Bonferroni mean operators with their applications in multiple attribute group decision making, International Journal of Information Technology & Decision Making 15(05) (2016), 1181–1210.

41.

Liu , The aggregation operators based on archimedean 2t-Conorm and t-Norm for single-valued neutrosophic numbers and their application to decision making, International Journal of Fuzzy Systems 18(5) (2016), 849–863.

42.

Sambuc , Function Φ-Flous, Application al'aide au Diagnostic en Pathologie Thyroidienne. University of Marseille (1975).

43.

R.R.

Yager , On the theory of bags, (Multi sets), Int. Jou. Of General System 13(1) (1986), 23–37.

44.

Yager and

Abbasov , Pythagorean Membership Grades, Complex Numbers, and Decision Making, Int. J. of Intelligent Systems (2013), 436–452.

45.

S.H.

Cheng ,

S.M.

Chen and

Z.C.

Huang , Autocratic decision making using group recommendations based on ranking interval type-2 fuzzy sets, Information Sciences 361 (2016), 135–161.

46.

S.M.

Chen and

C.H.

Chang , Fuzzy multi-attribute decision making based on transformation techniques of intuitionistic fuzzy values and intuitionistic fuzzy geometric averaging operators, Information Sciences 352-353 (2016), 133–149.

47.

S.M.

Mousavi ,

Vahdani ,

Tavakkoli-Moghaddam and

Tajik , Soft computing based on a fuzzy grey group compromise solution approach with an application to the selection problem of material handling equipment, Int J Comp Integ M 27(6) (2014), 547–569.

48.

S.M.

Mousavi ,

Vahdani , &

S.S.

Behzadi , Designing a model of intuitionistic fuzzy VIKOR in multi-attribute group decision-making problems, Iranian Journal of Fuzzy Systems 13(1) (2016), 45–65.

49.

Opricovic , Multicriteria optimization of civil engineering systems, Fac. Civ. Q8 Eng. Belgrade 2 (1998), 5–21.

50.

Opricovic and

G.H.

Tzeng , Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS, Eur. J. Oper. Res 156 (2004), 445–455.

51.

Opricovic and

G.H.

Tzeng , Multicriteria planning of post-earthquake sustainable reconstruction, Computer-Aided Civil and Infrastructure Engineering 17(3) (2002), 211–220.

52.

S.P.

Wan , G., l Xu,

Wang and

J.Y.

Dong , A new method for Atanassov's intervalvalued intuitionistic fuzzy MAGDM with incomplete attribute weight information, Information Sciences 316 (2015), 329–347.

53.

S.P.

Wan ,

Q.Y.

Wang and

J.Y.

Dong , The extended VIKOR method for multiattribute group decision making with triangular intuitionistic fuzzy numbers, Knowledge-Based Systems 52 (2013), 65–77.

54.

ş.

Emec and

Akkaya , Stochastic AHP and fuzzy VIKOR approach for warehouse location selection problem, Journal of Enterprise Information Management 31(6) (2018), 950–962.

55.

T.Y.

Chen ,

H.P.

Wang and

Y-Y

Lu , A multicriteria group decision-making approach based on interval-valued intuitionistic fuzzy sets: A comparative perspective, Expert Systems with Applications 38 (2011), 7647–7658.

56.

T.Y.

Chen , Remoteness index-based Pythagorean fuzzy VIKOR methods with a generalized distance measure for multiple criteria decision analysis, Information Fusion 41 (2018), 129–150.

57.

Torra , Hesitant fuzzy sets, International Journal ofIntel-ligent Systems 25(6) (2010), 529–539.

58.

Wang ,

Zhu ,

Song and

Lei , Combination of unreliable evidence sources in intuitionistic fuzzy MCDM framework, Knowledge-Based Systems 97 (2016), 24–39.

59.

Wang and

Cai , A group decision-making model based on distance-based VIKOR with incomplete heterogeneous information and its application to emergency supplier selection, Kybernetes 46 (2017), 501–529.

60.

X.Y.

You ,

J.X.

You ,

H.C.

Liu and

Zhen , Group multi-criteria supplier selection using an extended VIKOR method with interval 2-tuple linguistic information, Expert Syst. Appl 42 (2015), 1906–1916.

61.

Zhao ,

Tang ,

Yang and

Huang , Extended VIKOR method based on cross-entropy for interval-valued intuitionistic fuzzy multiple criteria group decision making, Journal of Intelligent & Fuzzy Systems 25(4) (2013), 1053–1066.

62.

He ,

Shi and

Meng , Multiple attribute group decision making based on IVHFPBMs and a new ranking method for interval-valued hesitant fuzzy information, Computers & Industrial Engineering 99 (2016), 63–77.

63.

Wu ,

Chen ,

Zeng ,

Xu and

Yang , Supplier selection in nuclear power industry with extended VIKOR method under linguistic information, Appl. Soft Computing 48 (2016), 444–457.

64.

Xu and

Zhang , Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information, Knowledge-Based Systems 52 (2013), 53–64.

65.

Liu ,

Liu and

Pang , Pythagorean uncertain linguistic partitioned Bonferroni mean operators and their application in multi-attribute decision making, Journal of Intelligent & Fuzzy Systems 32(3) (2017), 2779–2790.

A novel VIKOR method using spherical fuzzy sets and its application to warehouse site selection

Abstract

Keywords

1. Introduction

Table 21 Xu and Zhang’s distances to positive and negative ideal solutions Alternatives SF-PIS SF-NIS X 1 0.053 0.201 X 2 0.250 0.000 X 3 0.000 0.250 X 4 0.084 0.185

References

Table 21
Xu and Zhang’s distances to positive and negative ideal solutions

Alternatives SF-PIS SF-NIS

X ₁ 0.053 0.201

X ₂ 0.250 0.000

X ₃ 0.000 0.250

X ₄ 0.084 0.185