WASPAS is the acronym of Weighted Aggregated Sum Product ASsessment method. It is a relatively new method, but it has been widely employed in the literature since its first introduction in 2012 by Zavadskas et al. WASPAS is a weighted combination of Weighted Sum Model (WSM) and Weighted Product Model (WPM) (Zavadskas et al., 2015a, 2015b). Extensions of WASPAS with fuzzy sets such as single-valued neutrosophic sets, interval valued intuitionistic fuzzy sets, and interval type-2 fuzzy sets have been also commonly studied in the literature. The common feature of all these extensions is the usage of linguistic terms including vague and imprecise assessments.

Fuzzy sets have been very popular in almost all branches of science since they have emerged in 1965 (Zadeh, 1965). Researchers (Zadeh, 1965; Smarandache, 1998; Grattan Guinness, 1976; Sambuc, 1975; Zadeh, 1975; Atanassov, 1986; Torra, 2010; Yager, 2013, 1986, 2017; Garibaldi and Ozen, 2007) have introduced many extensions of ordinary fuzzy sets in the literature. It starts from ordinary fuzzy sets and extends to recently developed types of fuzzy sets as shown in Fig. 1. In recent years, numerous researchers have utilized these extensions in the solution of multi-criteria decision-making problems. A classification of some recent publications after 2016 with respect to the type of extension 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. 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 (Cheng et al., 2016; Chiao, 2016).

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 (Chen and Chang, 2016; Yu and Xu, 2016; Xu et al., 2016).

Neutrosophic sets (NS): Smarandache (1998) 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 truthfulness, indeterminacy and falsity (Liu, 2016; Ma et al., 2016; Liu et al., 2016).

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 (Kutlu Gundogdu et al., 2018; Qin et al., 2016; He et al., 2016).

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) (Liu et al., 2017; Garg, 2016; Ren et al., 2016; Peng and Yang, 2016).

qRung orthopair fuzzy sets (qROFs): These sets have been introduced by Yager (2017) as an important way to express uncertain information, and they are an extension of the intuitionistic fuzzy sets and the Pythagorean fuzzy sets. Their eminent characteristic is that the sum of the qth power of the membership degree and the qth power of the degrees of non-membership is equal to or less than 1.

The spherical fuzzy sets (SFS) have been recently introduced by Kutlu Gundogdu and Kahraman (2018). SFS are based on the fundamentals of PFS and NS. The main differences between q-ROFs and SFS are the definition of hesitancy degree independently in SFS satisfying that the squared sum of membership, non-membership and hesitancy degrees is at most 1.

Pythagorean fuzzy sets (PFS) developed by Yager (2013), which had been called Intuitionistic type-2 fuzzy sets (IFS2) by Atanassov previously (Atanassov, 1999), are characterized by a membership degree and a nonmembership degree satisfying the condition that their squared sum is at most equal to one, which is a generalization of Intuitionistic Fuzzy Sets (IFS). Hesitancy degree in PFSs calculated by π p ˜ = 1 − μ p ˜ 2 ( u ) − ν p ˜ 2 ( u ) .

Similar to IFSs and PFSs, neutrosophic sets (NS) are represented by the three dimensions: a truthfulness degree, an indeterminacy degree, and a falsity degree (Smarandache, 1998). NS do not only deal with the hesitancy of the system but also decrease indecisiveness of inconsistent information. Thus, the truthfulness, falsity and indeterminacy values can be independently assigned (Smarandache, 1998).

Yang and Chiclana (2009) have proposed a new 3D spherical representation, which is called the spherical distance and allowed us to define a new distance function between intuitionistic fuzzy sets. On the surface of a sphere, the following condition is satisfied:

Let A ˜ = { ⟨ u , μ A ˜ ( u ) , ν A ˜ ( u ) ⟩ : u ∈ U } be an intuitionistic fuzzy set. They have (1) μ A ˜ + ν A ˜ + π A ˜ = 1 , which can be equivalently transformed to (2) x 2 + y 2 + z 2 = 1 , where x 2 = μ A ˜ ( u ), y 2 = ν A ˜ ( u ), z 2 = π A ˜ ( u ).

On a spherical surface, hesitancy can be calculated based on the given membership and non-membership values since the sum of these three parameters is exactly equal to 1 (Yang and Chiclana, 2009). Besides, they measure the spherical arc distance between two IFSs. Furthermore, Gong et al. (2016) introduced an approach generalizing Yang and Chiclana’s work. They applied the spherical distance measure to obtain the difference between two IFSs. They first introduced an ideal intuitionistic fuzzy estimation, and then by minimizing the spherical distance between the ideal opinion and each individual opinion in group decisions, they constructed a nonlinear optimization model.

The spherical fuzzy sets are based on the fact that the hesitancy of a decision maker can be assigned independently from membership and non-membership degrees, satisfying the following condition: (3) A ˜ s = { ⟨ u , ( μ A ˜ ( u ) , ν A ˜ ( u ) , π A ˜ ( u ) ⟩ | u ∈ U } , where μ A ˜ s : U → [ 0 , 1 ], ν A ˜ s : U → [ 0 , 1 ], π A ˜ s : U → [ 0 , 1 ] and 0 ⩽ μ A ˜ s 2 ( u ) + ν A ˜ s 2 ( u ) + π A ˜ s 2 ( u ) ⩽ 1, ∀ u ∈ U.

On the surface of the blue coloured sphere in Fig. 2, Eq. (3) becomes (4) μ A ˜ 2 ( u ) + ν A ˜ 2 ( u ) + π A ˜ 2 ( u ) = 1 .

Since Yang and Chiclana (2009) and Gong et al. (2016) only measure the arc distances on the surface of the sphere, Euclidean distance is not measured in these works. In our spherical fuzzy sets approach, the sphere is not solid but a spherical volume. Based on this fact, Euclidean distance measurement is meaningful. This also means that any two points within the spherical volume are also on the surface of another sphere; however, the sum given by Eq. (4) becomes less than one in this case (red coloured sphere in Fig. 3). Euclidean distance gives the shortest distance between two points in the space as in Fig. 2.

In this paper, we extend one of the most used multi-criteria decision making methods, WASPAS, to its spherical fuzzy version. We illustrate its application through an industrial robot selection problem.

The rest of this paper is organized as follows. Section 2 includes the literature review on WASPAS. Section 3 gives introductory definitions on 3D fuzzy sets. In Section 4, the preliminaries on SFS are given. Section 5 includes our novel proposed MCDM method called Spherical Fuzzy WASPAS method (SF-WASPAS) and Section 6 applies SF-WASPAS method to industrial robot selection problem and also includes a comparative analysis of SF-WASPAS and IF-TOPSIS. Finally, the last section presents the conclusions and suggestions for further research.

The publications on WASPAS method are summarized in Table 1.

In Fig. 3, the subject areas of the papers on WASPAS method are illustrated. Engineering applications is the top area with 32.4% while computer science and business-management place at the second and third ranks, respectively.

Since spherical fuzzy sets are the extension of IFS, PFS and NS, we briefly summarize these sets in the following.

Intuitionistic and Pythagorean fuzzy membership functions are composed of membership, non-membership and hesitancy parameters, which can be calculated by π I ˜ = 1 − μ − ν or π P ˜ = 1 − μ 2 − ν 2 , respectively. Neutrosophic membership functions are also defined by three parameters truthfulness, 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 4 illustrates the differences among IFS, PFS, NS and SFS.

In this section, we give the definition of SFS and summarize spherical distance measurement, arithmetic operations, aggregation and defuzzification operations (Kutlu Gundogdu and Kahraman, 2018).

Some operations are defined over the Spherical Fuzzy Sets (SFS) as below.

On the basis of relationship between SFS and PFS, we further define some novel operations for SFS as below:

Proofs of the above equations can be found in Kutlu Gundogdu and Kahraman (2018).

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 = { x 1 , x 2 , … , x m } ( m ⩾ 2) be a discrete set of m feasible alternatives and C = { c 1 , c 2 , … , c n } be a finite set of criteria and W = { w 1 , w 2 , … , w n } be the weight vector of all criteria which satisfies 0 ⩽ w j ⩽ 1 and ∑ j = 1 n w j = 1.

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

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

Step 2: Aggregate the judgments of each decision maker (DM) using Spherical Weighted Arithmetic Mean (SWAM). (27) SWAM w ( A s 1 , A s 2 , … , A s n ) = w 1 A s 1 + w 2 A s 2 + ⋯ + w n A s n = ⟨ [ 1 − ∏ i = 1 n ( 1 − μ A s i 2 ) w i ] 1 / 2 , ∏ i = 1 n ν A s i w i , [ ∏ i = 1 n ( 1 − μ A s i 2 ) w i − ∏ i = 1 n ( 1 − μ A s i 2 − π A s i 2 ) w i ] 1 / 2 ⟩ .

Step 2.1: Aggregate the criteria weights. All criteria may not be assumed to be of equal importance. In order to obtain weights, all the individual decision maker opinions for the importance of each criterion need to be aggregated.

Step 2.2: Construct aggregated spherical fuzzy decision matrix based on the opinions of decision makers. Denote the evaluation values of alternative x i ( 1 , 2 , … , m ) with respect to criterion C j ( 1 , 2 , … , n ) by C j ( x ˜ i ) = ( μ i j , ν i j , π i j ) and x ˜ i j = ( C j ( x ˜ i ) ) m × n is a spherical fuzzy decision matrix. For a MCDM problem with SFS, decision matrix x ˜ i j = ( C j ( x ˜ i ) ) m × n should be constructed as in Eq. (28). (28) x ˜ i j = ( C j ( x ˜ i ) ) m × n = ( μ 11 , ν 11 , π 11 ) ( μ 12 , ν 12 , π 12 ) … ( μ 1 n , ν 1 n , π 1 n ) ( μ 21 , ν 21 , π 21 ) ( μ 22 , ν 22 , π 22 ) … ( μ 2 n , ν 2 n , π 2 n ) ⋮ ⋮ ⋱ ⋮ ( μ m 1 , ν m 1 , π m 1 ) ( μ m 2 , ν m 2 , π m 2 ) … ( μ m n , ν m n , π m n ) . Decision makers also evaluate the decision criteria as given in Table 3.

Decision makers assess the alternatives with respect to the criteria as if they were benefit criteria such that they assign a lower linguistic term if it is a cost criterion.

Step 3: Calculate the score function value of each criterion in Table 3 and then normalize these values.

Step 3.1: Defuzzify the aggregated criteria weights by using the score function given in Eq. (29). (29) w j s = ( μ j − π j ) 2 − ( ν j − π j ) 2 . Note that: If it is less than 0, a small number is added to all criteria weights to provide a slightly greater number than zero.

Step 3.2: Normalize the aggregated criteria weights by using Eq. (30). (30) w ‾ j s = w j s ∑ j = 1 n w j s . Step 4: Calculate the results of Weighted Sum Model (WSM) as presented in Eq. (31). (31) Q ˜ i ( 1 ) = ∑ j = 1 n x ˜ i j w = ∑ j = 1 n x ˜ i j w ‾ j s .

Eq. (31) can be divided into two parts for ease of operations. First, the multiplication operator, then the addition operator is performed.

Step 4.1: Calculate the multiplication part of Eq. (31) by using Eq. (32). (32) x ˜ i j w = x ˜ i j w ‾ j s = ⟨ ( 1 − ( 1 − μ x ˜ i j 2 ) w j s ) 1 / 2 , ν x ˜ i j w j s , ( ( 1 − μ x ˜ i j 2 ) w j s ) − ( 1 − μ x ˜ i j 2 − π x ˜ i j 2 ) w j s ) 1 / 2 ⟩ .

Step 4.2: Calculate each addition term in Eq. (31) by using Eq. (33). (33) x ˜ i 1 w ⊕ x ˜ i 2 w = ⟨ ( μ x ˜ i 1 w 2 + μ x ˜ i 2 w 2 − μ x ˜ i 1 w 2 μ x ˜ i 2 w 2 ) 1 / 2 , ν x ˜ i 1 w ν x ˜ i 2 w , ( ( 1 − μ x ˜ i 2 w 2 ) π x ˜ i 1 w 2 + ( 1 − μ x ˜ i 1 w 2 ) π x ˜ i 2 w 2 − π x ˜ i 1 w 2 π x ˜ i 2 w 2 ) 1 / 2 ⟩ .

Step 5: Calculate the results of Weighted Product Model (WPM) as presented in Eq. (34). (34) Q ˜ i ( 2 ) = ∏ j = 1 n x ˜ i j w ‾ j s . Eq. (34) can be also divided into two parts for ease of operations. First, the exponential operator and then the multiplication operator is performed.

Step 5.1: Calculate the exponential part of Eq. (34) by using Eq. (35). (35) x ˜ i j w j s = ⟨ μ x ˜ i j w ‾ j s , ( 1 − ( 1 − ν x ˜ i j 2 ) w ‾ j s ) 1 / 2 , ( ( 1 − ν x ˜ i j 2 ) w ‾ j s − ( 1 − ν x ˜ i j 2 − π x ˜ i j 2 ) w ‾ j s ) 1 / 2 ⟩ .

Step 5.2: Calculate each multiplication term in Eq. (34) based on Eq. (36). (36) x ˜ i 1 w 1 s ⊗ x ˜ i 2 w 2 s = ⟨ μ x ˜ i 1 w 1 s μ x ˜ i 2 w 2 s , ( ν x ˜ i 1 w 1 s 2 + ν x ˜ i 2 w 2 s 2 − ν x ˜ i 1 w 1 s 2 ν x ˜ i 2 w 2 s 2 ) 1 / 2 , ( ( 1 − ν x ˜ i 2 w 2 s 2 ) π x ˜ i 1 w 1 s 2 + ( 1 − ν x ˜ i 1 w 1 s 2 ) π x ˜ i 2 w 2 s 2 − π x ˜ i 1 w 1 s 2 π x ˜ i 2 w 2 s 2 ) 1 / 2 ⟩ .

Step 6: Determine the threshold number λ and calculate Eqs. (37) and (38). (37) λ Q ˜ i ( 1 ) = ⟨ ( 1 − ( 1 − μ Q ˜ i ( 1 ) 2 ) λ ) 1 / 2 , ν Q ˜ i ( 1 ) λ , ( ( 1 − μ Q ˜ i ( 1 ) 2 ) λ − ( 1 − μ Q ˜ i ( 1 ) 2 − π Q ˜ i ( 1 ) 2 ) λ ) 1 / 2 ⟩ . (38) ( 1 − λ ) Q ˜ i ( 2 ) = ⟨ ( 1 − ( 1 − μ Q ˜ i ( 2 ) 2 ) ( 1 − λ ) ) 1 / 2 , ν Q ˜ i ( 2 ) ( 1 − λ ) , ( ( 1 − μ Q ˜ i ( 2 ) 2 ) ( 1 − λ ) − ( 1 − μ Q ˜ i ( 2 ) 2 − π Q ˜ i ( 2 ) 2 ) ( 1 − λ ) ) 1 / 2 ⟩ .

Step 7: Sum Eq. (37) and Eq. (38) as given by Eq. (39). (39) Q ˜ i = λ Q ˜ i ( 1 ) + ( 1 − λ ) Q ˜ i ( 2 ) .

Step 8: Defuzzify by using the score function as given in Eq. (29). We put the alternatives into order with respect to the decreasing values of score values. If the score values of two alternatives are equal, their accuracy function values might be considered as in Eq. (26).

Our proposed methodology is applied to an industrial robot selection problem. For this goal, mostly used five robots (6-axis robots X1, Scara robots X2, Dual-arm robots X3, Redundant robots X4, Cartesian robots X5) are evaluated. After a comprehensive literature review, four criteria have been determined, which are efficiency (C1), suitability (C2), automation (C3), and ergonomics (C4). The weights of three decision makers (DM1, DM2, DM3) having different experience levels are 0.4, 0.3 and 0.3, respectively.

First of all, the assessments for the criteria are collected from decision makers with respect to the goal, using the linguistic terms given in Table 2. All assessments are given in Tables 4, 5, and 6.

These judgments are aggregated using SWAM operator by considering the importance levels of decision makers. Aggregated decision matrix is obtained as in Table 7.

The linguistic importance weights of the criteria assigned by DMs are shown in Table 8.

The weight of each criterion obtained by using SWAM operator is presented in Table 9.

After the weights of the criteria have been determined, the defuzified and normalized criteria weights are calculated by utilizing Eqs. (33) and (34) as given in Table 10.

Based on Table 9 and Eqs. (32) and (33), Q ˜ i ( 1 ) is obtained as in Table 11. Based on the first column of Table 11 and Eq. (37), λ Q ˜ i ( 1 ) is calculated as given in the second column of Table 11.

According to Table 10 and Eqs. (35) and (36), Q ˜ i ( 2 ) is obtained as in Table 12. Based on the first column of Table 12 and Eq. (38), ( 1 − λ ) Q ˜ i ( 2 ) is calculated as given in the second column of Table 12.

In the next step, based on Tables 11 and 12, we can calculate the final value of SF-WASPAS using Eq. (39). They are given in Table 13.

From Table 13, the score value of each alternative is calculated based on Eq. (29) and given in Table 14.

The score values indicate that the best alternative is X1 and overall ranking is X 1 > X 3 > X 5 > X 4 > X 2X3>X5>X4>X2$]]>.

We compare the proposed SF-WASPAS with intuitionistic fuzzy TOPSIS (IF-TOPSIS) in this section. Table 15 presents the IF linguistic scale, which we use for comparison purposes.

In this comparison, the same judgments as given in Tables 4, 5 and 6 were used and aggregated using IFWA (Intuitionistic Fuzzy Weighted Average) operator given in Eq. (40) (Xu, 2007). Aggregated decision matrix is given in Table 16. (40) IFWA ( r i j ) = λ 1 r i j ( 1 ) ⊕ λ 2 r i j ( 2 ) ⊕ ⋯ ⊕ λ l r i j ( l ) = [ 1 − ∏ k = 1 l ( 1 − μ i j k ) λ k , ∏ k = 1 l ( ν i j k ) λ k , ∏ k = 1 l ( 1 − μ i j k ) λ k + ∏ k = 1 l ( ν i j k ) λ k ] .

The same criteria judgments given in Table 8 are used for this comparison. Opinions of decision makers on criteria are aggregated using IFWA operator and the weight of each criterion is presented in Table 17.