EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6354 ISSN 1307-5543 – ejpam.com Published by New York Business Global Spherical Picture Fuzzy Sets with Application to Multicriteria Decision-Making Sadique Ahmad1, Badshah-e-Rome2,∗, Anisa Begum2, Naved Ahmad1 1 EIAS Data Science and Block Chain Laboratory, College of Computer and Information Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia 2 Department of Mathematics, Government Degree College Kabal Swat , Khyber Pakhtunkhwa, Pakistan Abstract. Since the introduction of fuzzy sets by Zadeh in 1965 [1], a lot of new theories regarding imprecision and uncertainty have been introduced. Some of these theories are extensions of fuzzy set theory, other try to handle imprecision and uncertainty in different way. The extensions of ordinary fuzzy sets are classified into two broad categories: 1. Intuitionistic fuzzy sets [2] and their versions, 2. Neutrosophic sets [3] and their versions. The first group extensions can be defined by a membership degree and a non-membership degree, whereas the second class of extensions can be defined by a membership degree (truthiness), a non-membership degree (falsity), and a hesitancy degree (indeterminacy). Spherical and picture fuzzy sets fall into the same group because of the def- inition of membership functions. The squared sum of membership, nonmembership, and hesitancy degrees is equal to or less than 1.0 in spherical fuzzy sets whereas it is valid for the first degree sum in picture fuzzy sets. In this paper, we unify the concepts of picture fuzzy set and spherical fuzzy set into a broad class and name it as spherical picture fuzzy set (SPFS). In SPFSs, every element of the universe is represented by a sphere. This unique geometrical representation is more adaptable and adequate for handling ambiguity in multi criteria decision-making. A new distance measure of spherical picture fuzzy sets is illustrated, and it is shown that it satisfies conditions of the distance measure. Besides investigating the structural properties of SPFS, set-theoretical operations along with some basic algebraic operations and aggregation operators are discussed. One of the most popular multi-criteria decision-making techniques, TOPSIS, is expanded to its SPFS form. To demonstrate the effectiveness and feasibility of the proposed SPFS-TOPSIS methodol- ogy for managing inherent vagueness in the given data, a numerical case study is analyzed wherein the methodology is applied to the pandemic hospital site selection problem. 2020 Mathematics Subject Classifications: 94D05,03B52, 03E72, 28E10 Key Words and Phrases: Spherical picture fuzzy set, Hesitancy degree, Multi-criteria decision making ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6354 Email addresses: saahmad@psu.edu.sa (S. Ahmad), baadeshah1@gmail.com(B.E Rome) anisagovt@gmail.com (A.Begum) nahmad@psu.edu.sa (N.Ahmad) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6354 2 of 20 1. Introduction The notion of fuzzy sets or fuzzy logic was first presented by Zadeh [1] in 1965. In that work Zadeh was implicitly advancing the thesis that one of the reasons humans are better at control than currently existing machines is that they are able to make effective decisions on the basis of imprecise linguistic information. Hence it should be possible to improve the performance of electromechanical controllers by modeling the way in which humans reason with this type of information. Usual fuzzy sets are described by member- ship degree (MND) and non-membership degree (NMND) from [0,1], which respectively show how strongly or weakly an element of universe of discourse is associated with the set. Flexibility in the boundary of fuzzy set is useful in tackling vagueness and uncer- tainty. Many researchers introduced several new extensions of ordinary fuzzy sets by describing membership functions [1],[4],[2],[5],[6],[3],[7],[8],[9],[10],[11],[12],[13]. Atanassov [2] presented the notion of intuitionistic fuzzy sets (IFSs) which consist of MD and NMD whose sum cannot exceed 1. In IFSs hesitancy of experts is taken into account. Torra [7] introduced hesitant fuzzy sets (HFSs) to work with a set of potential membership values of an element in a fuzzy set. F.E Boran et al. [14] introduced Pythagorean fuzzy sets (PyFSs) by adding a rather large ground of MNDs and NMNDs. The concept of neutrosophic sets was introduced by Smarandache [3]. In these sets, degree of truthfulness, falsity, and indeterminacy are linked with every element of the universe of discourse such that sum of these degrees cannot exceed 3. Coung [8] introduced picture fuzzy sets (PFSs) extending IFSs. PFSs were further extended by Gündoğdu and Kahraman [10] by initiating the notion of spherical fuzzy sets (SFSs) where each element is associated MD and NMD and HD. Circular Intuitionistic fuzzy sets (C- IFSs) was developed by Atanassov [15]. In Fig 1, the new extensions of usual fuzzy sets are displayed historically. In this article, we present the idea of spherical picture fuzzy set (SPFS), which extends and unifies the notions of C-IFSs and PFS. In SPFSs, every element of the universe is surrounded by a sphere. This unique geometric form is more adaptable and adequate to handle fuzzy information consisting of hesitance or ignorance. SPFSs can reduce data loss and more accurately capture the inherent ambiguity of objective problems in mathematical terms. Furthermore, by incorporating a radius parameter, spherical picture fuzzy sets can effectively integrate multiple vague data elements into a single SPFS. This approach can help in streamlining complex multi-attribute group decision-making processes. Combining affiliation, neutral attitude, non-affiliation, and radius into a spherical picture fuzzy set helps represent information more fully and makes it easier to handle vague or uncertain situations. For further details about multi-attribute group decision-making [16] and [17] should be consulted. Ranking of alternatives is one of the main steps in multi-attribute group decision-making processes. Score functions and distance evaluations are valuable tools for ranking. Chen [18] and other scholars initially proposed the score function within the framework of IFSs. Çakir [19] extended score function to C-IFSs. As fuzzy multi-attribute decision-making has evolved, an increasing number of score functions have been presented and used in S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6354 3 of 20 Figure 1: Evolution of fuzzy set decision-making processes. On the other hand, distance measures play an important role in IFSs. One of the main objectives of this research is to define and explore distance measures within spherical picture fuzzy sets. Recently some researchers have worked on fuzzy analysis [20]. 2. Preliminaries In this section some terms and definitions are provided which will be used in the main work of this manuscript. Definition 1. [1] Fuzzy set A in a universe of discourse Y is set of order pairs of the form A = {(y, ξA(y)), y ∈ Y }. Value of the membership function ξA(y) at y ∈ Y represents grade of membership of y in A. Definition 2. [2] An IFS A on a universal set Y is an object of the form A = {(y, ξA(y), ρA(y)|y ∈ Y )}, where ξA(y), ρA(y) ∈ [0, 1] are called the MD and NMD of y in A. ξA and ρA satisfy the following condition: ξA(y) + ρA(y) ≤ 1, ∀y ∈ Y. S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6354 4 of 20 Definition 3. [8] A PFSs A on a universe Y is an object in the form of A = {(y, ξA(y), ψA(y), ρA(y))|y ∈ Y }, where ξA(y), ψA(y), ρA(y) ∈ [0, 1] are called positive membership degree (MD), neutral membership degree (NtMD) and negative membership degree (ND) of y in A respectively, and ξA(y) + ψA(y) + ρA(y) ≤ 1,∀y ∈ Y. Definition 4. [8] Let A = {(y, ξA(y), ψA(y), ρA(y))|y ∈ Y }, be a PFSs then score function is defined as ξ(y) + ψ(y)− ρ(y). Definition 5. [8] Distances between two PFSs A and B, in Y = {y1, y2, ..., yn} are: (i) The normalized Harming distance H(A,B) = 1 n ∑n i=1(|ξA(yi)− ξB(yi)|+ |ψA(yi)− ψB(yi)|+ |ρA(yi)− ρB(yi)|) (ii) The normalized Euclidean distance E(A,B) = ( 1n ∑n i=1((ξA(yi)− ξB(yi)) 2 + (ψA(yi)−ψB(yi)) 2 + (ρA(yi)− ρB(yi)) 2)) 1 2 . 3. Definition and Properties of Spherical Picture Fuzzy Sets In this section some basic concepts and concerning spherical picture fuzzy set are presented for the development of main contents. Definition 6. Let Y be universe of discourse. Spherical picture fuzzy set (SPFS) is an object of the form S = {⟨y, ξ(y), ψ(y), ρ(y); γ⟩|y ∈ Y }, with 0 ≤ ξ(y) + ψ(y) + ρ(y) ≤ 1, where ξ : Y → [0, 1], ψ : Y → [0, 1] and ρ : Y → [0, 1] represent the membership, neutral, and non-membership functions of S and γ ∈ [0, 1] is radius of the sphere surrounding y ∈ Y , Each element y is SPFS is represented by a sphere with center (ξ(y), ψ(y), ρ(y)) and radius γ, whereas in PFS it is represented just by a point. PFS is a special type of SPFS with γ = 0. On the other hand a spherical picture fuzzy set with γ > 0 cannot be expressed as ordinary picture fuzzy set. Thus spherical picture fuzzy set is proper extension of PFS. ϕ(y) = 1− ξ(y)−ψ(y)− ρ(y) is known as degree of hesitancy of y ∈ Y with respect to S. Geometrical representation of SPFS is given in Fig 2. S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6354 5 of 20 Figure 2: SPFS geometrical representation Definition 7. Let P ∗ = {⟨l,m, n⟩|l,m, n ∈ [0, 1] and l +m + n ≤ 1}, be a picture fuzzy set then the associated spherical picture fuzzy Sr, may be expressed as: Sr = {⟨y, Cr(ξS(y), ψS(y), ρS(y))⟩|y ∈ Y }, where Cr is a function depicts a sphere with a radius of γ and a center of (ξS(y), ψS(y), ρS(y)), Cr(ξS(y), ψS(y), ρS(y)) = { ⟨l,m, n⟩|l,m, n ∈ [0, 1] and√ (ξS(y)− l)2 + (ψS(y)−m)2 + (ρS(y)− n)2 ≤ γ }⋂ P ∗ = { ⟨l,m, n⟩| √ (ξS(y)− l)2 + (ψS(y)−m)2 + (ρS(y)− n)2 ≤ γ : l +m+ n ≤ 1 } . Definition 8. From a given collection of (PFSs) {⟨mi,1, ni,1, oi,1⟩, ⟨mi,2, ni,2, oi,2⟩, ⟨mi,3, ni,3, oi,3⟩, .....} (SPFS) is calculated as follows: ⟨ξ(Si), ψ(Si), ρ(Si)⟩ = 〈∑ki j=1mi,j ki , ∑ki j=1 ni,j ki , ∑ki j=1 oi,j ki 〉 , (1) S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6354 6 of 20 where ki denote the number of decision makers. γi = max 1≤j≤ki √ (ξ(Si)−mi,j)2 + (ψ(Si)− ni,j)2 + (ρ(Si)− oi,j)2. (2) Some fundamental algorithms and operations for spherical picture fuzzy sets are pro- vided here to facilitate their application in multi-attribute decision making. Definition 9. Let S1 = {⟨x, ξ1(y), ψ1(y), ρ1(y); γ1⟩|y ∈ Y } and S2 = {⟨x, ξ2(y), ψ2(y), ρ2(y); γ2⟩|y ∈ Y }, be two SPFSs, then operation between them can be defined as follows: S1 ⊕min S2 = {⟨y, ξ1(y) + ξ2(y)− ξ1(y)ξ2(y), ψ1(y)ψ2(y), ρ1(y)ρ2(y); min(γ1, γ2)⟩|y ∈ Y }. S1 ⊕max S2 = {⟨y, ξ1(y) + ξ2(y)− ξ1(y)ξ2(y), ψ1(y)ψ2(y), ρ1(y)ρ2(y); max(γ1, γ2)⟩|y ∈ Y }. S1 ⊗min S2 = {⟨y, ξ1(y)ξ2(y), ψ1(y) + ψ2(y)− ψ1(y)ψ2(y), ρ1(y) + ρ2(y) − ρ1(y)ρ2(y);min(γ1, γ2)⟩|y ∈ Y }. S1 ⊗max S2 = {⟨y, ξ1(y)ξ2(y), ψ1(y) + ψ2(y)− ψ1(y)ψ2(y), ρ1(y) + ρ2(y) − ρ1(y)ρ2(y);max(γ1, γ2)⟩|y ∈ Y }. S1 ⊙ S2 = {⟨y, ξ1(y)ξ2(y), ψ1(y) + ψ2(y)− ψ1(y)ψ2(y), ρ1(y) + ρ2(y) − ρ1(y)ρ2(y); γ1 + γ2 2 ⟩|y ∈ Y }. (3) Definition 10. Let S = {⟨y, ξs(y), ψs(y), ρ1(y); γ1⟩|y ∈ Y } be a spherical picture fuzzy set. Subsequently, the score function of the spherical picture fuzzy set is Sc(S) = 1 3 (ξs − ψs − ρs + √ 2γ(2λ− 1)), λ = 0 and λ = 1 respectively show complete pessimism and optimism, while λ = 0.5 is indicator for nonchalance on behalf of the decision maker. 4. Distance analysis of spherical picture fuzzy set This paper examines the incorporation of hesitation degree into the spherical picture fuzzy set distance by assigning it to partial affirmation, impartial affirmation, and partial negation. This approach indirectly integrates hesitation degree into the distance metric, aiming to improve the handling of imprecise information in practice. Below is the initial definition of the assignment of HD to MD, NtMD, and NMD. Definition 11. Let S1 = {⟨y, ξS1(y), ψS1(y), ρS1(y), γS1⟩|yi ∈ Y } be a SPFS on Y along with the distribution of MD, NMD and NtMD by HD ϕA(yi) is defined as: CDS1 ϕ→ξ(y) = 1 2 [ϕS1(y) + 2ξS1(y)], S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6354 7 of 20 CDS1 ϕ→ψ(y) = 1 2 [ϕS1(y) + 2ψS1(y)], CDS1 ϕ→ρ(y) = 1 2 [ϕS1(y) + 2ρS1(y)]. The degree of hesitation can be seen as the decision maker’s hesitancy to accept, reject or support an uncertain object, as it reflects the unknown degree of knowledge. Since there may be a degree of partial negation as well as a degree of partial affirmation, and impartial affirmation in the hesitation degree, so the hesitation degree is divided equally into MD, NMD, and NtMD. In SPFS radius is obtained from the calculation of MD, NMD, and NtMD, Hence, if changes occur in these degrees, the radius will also change. On the basis of this, the radius of the SPFS is defined according to the newly assigned hesitation degree. Definition 12. Let there is a set of PFSs pairs {⟨mi,1, ni,1, oi,1⟩, ⟨mi,2, ni,2, oi,2⟩, ⟨mi,3, ni,3, oi,3⟩, .....} then the radius of the SPFS ⟨ξ(Si), ψ(Si), ρ(Si)⟩ = 〈∑ki j=1mi,j ki , ∑ki j=1 ni,j ki , ∑ki j=1 oi,j ki 〉 is calculated as γij = max 1≤j≤ki ∣∣∣∣ √( Dξi −Dmi )2 + ( Dψi −Dni )2 + ( Dρi −Doi )2∣∣∣∣ (4) where Dξi = ( ξ(Si) + 1 2 ϕ(Si) ) , Dmi = ( mi + 1 2 li ) Dψi = ( ψ(Si) + 1 2 ϕ(Si) ) , Dni = ( ni + 1 2 li ) Dρi = ( ρ(Si) + 1 2 ϕ(Si) ) , Doi = ( oi + 1 2 li ) ϕ(Si) = 1− ξ(Si)− ψ(Si)− ρ(Si), lij = 1−mi − ni − oi, next we define distance metric of SPFS under hesitancy degree. The degree is assigned based on the motivation of the distance metric for C-IFS given in literature [21]. S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6354 8 of 20 Definition 13. Let S1 = {⟨y, ξS1(y), ψS1(y), ρS1(y); γS1⟩|y ∈ Y }, and S2 = {⟨y, ξS2(y), ψS2(y), ρS2(y); γS2⟩|y ∈ Y }, be two spherical picture fuzzy sets on the universe set Y then the distance metric is defined as: ∂c(S1, S2) = |γS1 − γS2 |√ 2 + 1 2 ( (∆S1S2 ξ )2 + (∆S1S2 ψ )2 + (∆S1S2 ρ )2 + (∆S1S2 ϕ→ξ ) 2 + (∆S1S2 ϕ→ψ) 2 + (∆S1S2 ϕ→ρ) 2 ) 1 2 (5) where ∆S1S2 ξ =|ξS1(y)− ξS2(y)|, ∆S1S2 ψ =|ψS1(y)− ψS2(y)|, ∆S1S2 ρ =|ρS1(y)− ρS2(y)|, ∆S1S2 ϕ→ξ =|SDS1 ϕ→ξ − SDS2 ϕ→ξ|, ∆S1S2 ϕ→ψ =|SDS1 ϕ→ψ − SDS2 ϕ→ψ|, ∆S1S2 ϕ→ρ =|SDS1 ϕ→ρ − SDS2 ϕ→ρ|. The distance metric ∂c(A,B) is constructed in two main steps: first, the radius dif- ference between the two spherical picture fuzzy sets (SPFS) is calculated; second, the concept of distance of Euclidean in real space is applied. The formula for distance in a six-dimensional real space is constructed by examining (ξS1(y), ψS1(y), ρS1(y), SD S1 ϕ→ξ, SDS1 ϕ→ψ, SD S1 ϕ→ρ), and (ξS2(y), ψS2(y), ρS2(y), SD S2 ϕ→ξ, SD S2 ϕ→ψ, SD S2 ϕ→ρ), the coordinates of two points. Therefore the following theorem can be obtained. Theorem 1. Let S1 = {⟨y, ξS1(y), ψS1(y), ρS1(y); γS1⟩| y ∈ Y }, S2 = {⟨y, ξS2(y), ψS2(y), ρS2(y); γS2⟩| y ∈ Y }, and S3 = {⟨y, ξS3(y), ψS3(y), ρS3(y); γS3⟩| y ∈ Y }, be three SPFSs on universe of discourse Y . Then a) ∂c(S1, S2) ≥ 0, ∂c(S1, S2) = 0 if and only if S1 = S2. b) ∂c(S1, S2) = ∂c(S2, S1). c) ∂c(S1, S2) + ∂c(S2, S3) ≥ ∂c(S1, S3). Proof. To show that ∂c(S1, S2) is a distance metric, it will suffice to prove that it satisfies the three conditions given in Definition 13. a) Obviously ∂c(S1, S2) ≥ 0. If S1 = S2, then ∆S1S2 ξ = ∆S1S2 ψ = ∆S1S2 ρ = 0 , ∆S1S2 ξ→ϕ = ∆S1S2 ψ→ϕ = ∆S1S2 ρ→ϕ = 0, S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6354 9 of 20 and γS1 = γS2 = 0. Thus ∂c(S1, S2) = 0. Conversely, ∂c(S1, S2) = 0 implies that ∆S1S2 ξ = ∆S1S2 ψ = ∆S1S2 ρ = 0 and ξS1(y) = ξS2(y), ψS1(y) = ψS2(y), ρS1(y) = ρS2(y). Therefore S1 = S2. b) ∂c(S1, S2) = ∂c(S2, S1) is obvious. c) Consider |γS3 − γS1 | = |γS3 − γS2 + γS2 − γS1 | ≤ |γS3 − γS2 |+ |γS2 − γS1 |, which implies |γS3 − γS1 |√ 2 ≤ |γS1 − γS2 |√ 2 + |γS2 − γS3 |√ 2 . (6) Also it can be noticed that Dc(S1, S2) = ( 1 4 [ (∆S1S2 ξ )2 + (∆S1S2 ψ )2 + (∆S1S2 ρ )2 + (∆S1S2 ϕ→ξ ) 2 + (∆S1S2 ϕ→ψ) 2 + (∆S1S2 ϕ→ρ) 2 ]) 1 2 is Euclidean distance between the two points (ξS1(y), ψS1(y), ρS1(y), SD S1 ϕ→ξ, SD S1 ϕ→ψ, SD S1 ϕ→ρ) and (ξS2(y), ψS2(y), ρS2(y), SD S2 ϕ→ξ, SD S2 ϕ→ψ, SD S2 ϕ→ρ) in six-dimensional space divided by 2, therefor Dc(S1, S3) ≤ Dc(S1, S2) +Dc(S2, S3) (7) Adding (6) and (7), we have |γS3 − γS1 |√ 2 + ( 1 4 [ (∆S1S3 ξ )2 + (∆S1S3 ψ )2 + (∆S1S3 ρ )2 + (∆S1S3 ϕ→ξ ) 2 + (∆S1S3 ϕ→ψ) 2 + (∆S1S3 ϕ→ρ) 2 ]) 1 2 ≤ |γS1 − γS2 |√ 2 + ( 1 4 [ (∆S1S2 ξ )2 + (∆S1S2 ψ )2 + (∆S1S2 ρ )2 + (∆S1S2 ϕ→ξ ) 2 + (∆S1S2 ϕ→ψ) 2 + (∆S1S2 ϕ→ρ) 2 ]) 1 2 + |γS2 − γS3 |√ 2 + ( 1 4 [ (∆S2S3 ξ )2 + (∆S2S3 ψ )2 + (∆S2S3 ρ )2 + (∆S2S3 ϕ→ξ ) 2 + (∆S2S3 ϕ→ψ) 2 + (∆S2S3 ϕ→ρ) 2 ]) 1 2 , thus ∂c(S1, S2) satisfies the triangular inequality. Definition 14. let S1, S2 be two random SPFSs on the universe Y = {x1, x2, · · · , xn} then the normalized SPFS distance between S1 and S2 is defined as N∂c(S1, S2) = 1 n n∑ i=1 ( |∆S1S2 γ (i)| √ 2 + [ 1 4 (∆S1S2 ξ (i))2 + (∆S1S2 ψ (i))2 + (∆S1S2 ρ (i))2+ S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6354 10 of 20 (∆S1S2 ϕ→ξ (i)) 2 + (∆S1S2 ϕ→ψ(i)) 2 + (∆S1S2 ϕ→ρ(i)) 2 ] 1 2 ) . (8) Where ∆S1S2 γ (i) = |γS1(xi)− γS2(xi)| ∆S1S2 ξ (i) = |ξS1(xi)− ξS2(xi)|, ∆S1S2 ψ (i) = |ψS1(xi)− ψS2(xi)|, ∆S1S2 ρ (i) = |ρS1(xi)− ρS2(xi)|, ∆S1S2 ϕ→ξ (i) = |SDS1 ϕ→ξ(i)− SDS2 ϕ→ξ(i)|, ∆S1S2 ϕ→ψ(i) = |SDS1 ϕ→ψ(i)− SDS2 ϕ→ψ(i)|, ∆S1S2 ϕ→ρ(i) = |SDS1 ϕ→ρ(i)− SDS2 ϕ→ρ(i)|. To evaluate the feasibility and effectiveness of the newly proposed spherical picture fuzzy set (SPFS) distance measure, we apply it to a real-world multi-criteria decision- making (MCDM) problem. 5. Multi-Criteria Decision Making via Spherical Picture Fuzzy Sets TOPSIS is a well-established and effective approach for solving multi-attribute decision- making problems. In this paper, we introduce a SPFS−based TOPSIS method that incorporates the newly developed distance measure. This method enhances traditional approaches and improves the decision-making process by addressing uncertainties more comprehensively through spherical picture fuzzy sets. Step 1. Let us define the solution set for the TOPSIS problem as follows: σi = {σ1, σ2, ...σm}, i = 1, 2, ...,m; where σi represents the alternatives in the decision problem, and m represents number of available alternatives. The decision criteria set is defined as Cj = {C1, C2, ..., Cn}, j = 1, 2, ..., n; where Cj represents the criteria by which the alternatives are evaluated, and n is the total number of criteria. Let W = {W1,W2, ...,Wn}, be the weight vector associated with the criteria group, where Wj ≥ 0, n∑ j=1 Wi = 1; S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6354 11 of 20 which ensures that the total importance of the criteria adds up to 1. The decision maker (DM), with the assistance of experts from various fields, evaluates the alternatives based on the criteria. The experts provide judgments regarding the solutions and decision crite- ria in order to facilitate the process of decision making. Step 2. Analyze the perspectives of decision-makers, create an expert linguistic de- cision matrix, and use Table 1 to immediately convert qualitative data into picture fuzzy numbers (PFNs). Table 1: Picture fuzzy numbers semantic quantization table. Linguistic Value PFNs Certainly high value (CHV) ⟨0.8, 0.1, 0.0⟩ Very high value (VHV) ⟨0.4, 0.2, 0.3⟩ High value (HV) ⟨0.5, 0.3, 0.0⟩ Above average value (AAV) ⟨0.3, 0.3, 0.2⟩ Average value (AV) ⟨0.7, 0.1, 0.1⟩ Under average value (UAV) ⟨0.4, 0.3, 0.2⟩ Low value (LV) ⟨0.3, 0.4, 0.1⟩ Very low value (VLV) ⟨0.6, 0.2, 0.1⟩ Certainly low value (CLV) ⟨0.4, 0.3, 0.1⟩ Step 3. Create aggregated picture fuzzy numbers from the picture fuzzy pairs repre- senting the perspectives of several decision-makers for the same choice and decision criteria in the decision matrix. ⟨ξ(Ci), ψ(Ci), ρ(Ci)⟩ using equation(1). Then use equation(4) to de- termine the matching radius length in order to build a fuzzy decision matrixM = (xij)nm, with a spherical picture, where xij = {⟨ξij , ψij , ρij ; γij⟩} denotes the spherical picture fuzzy number of the alternative with respect to the criteria. Step 4. To obtain the weighted sum table of picture fuzzy conditions, quantify the weight information from Table 3 and determine the weights of the various Conditions. Next, to create the spherical picture fuzzy set criteria weight matrix W = (ωj)1×n, where ωj = {⟨ξj , ψj , ρj ; γj⟩}, the maximum radius r is determined using equation(4). Step 5. Create the decision matrix G = (gij)m× n, which is weighted. Each element gij = ⟨ξij , ψij , ρij , γij⟩ is calculated using the weight matrix W acquired in step 4, the spherical picture fuzzy set decision matrix M , and equation(3). S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6354 12 of 20 Step 6. Determine the best negative solution, G−, and the positive ideal solution, G+ for the choice matrix. G+ = {⟨(max i gij |j ∈ ℑ1), (min i gij |j ∈ ℑ2), (min i gij |j ∈ ℑ3)⟩|j = 1, 2, 3, ...n} (9) G− = {⟨(min i gij |j ∈ ℑ1), (max i gij |j ∈ ℑ2), (max i gij |j ∈ ℑ3)⟩|j = 1, 2, 3, ...n} (10) where g+j = {⟨ξ+j , ψ + j , ρ + j ; γ + j ⟩},g − j = {⟨ξ−j , ψ − j , ρ − j ; γ − j ⟩} represents the SPFS with highest and lowest membership degrees among j criteria. ℑ1,ℑ2,ℑ3 represent the beneficial criteria and the cost criteria. Table 2: Quantization table of weight information. Linguistic Value PFNs Certainly high importance (CHI) ⟨0.8, 0.1, 0.0⟩ Very high importance (VHI) ⟨0.4, 0.2, 0.3⟩ High importance (HI) ⟨0.5, 0.3, 0.0⟩ Above average importance (AAI) ⟨0.3, 0.3, 0.2⟩ Average importance (AI) ⟨0.7, 0.1, 0.1⟩ Under average importance (UAI) ⟨0.4, 0.3, 0.2⟩ Low importance (LI) ⟨0.3, 0.4, 0.1⟩ Very low importance (VLI) ⟨0.6, 0.2, 0.1⟩ Certainly low importance (CLI) ⟨0.4, 0.3, 0.1⟩ Step 7. Determine the distance between each option and the ideal solutions. The positive ideal solution, N∂+c (σi), and the negative ideal solution, N∂−c (σi), using the new distance equation(5) proposed in this paper. Step 8. Calculate the relative closeness coefficient RCC(σi) using normalized SPFS distances (8), rank the alternatives based on the RCC(σi), and finally select the best option. RCC(σi) = N∂−c (σi) N∂−c (σi) +N∂+c (σi) (11) 6. An Application to Pandemic Hospital Site Selection Using a hospital site example from the literature [22], the distance metric presented in this work is tested for validity below. For the public health systems, mass infectious diseases have always been an extremely challenging issue. Not only does it present a risk to events related to the lives of individuals, national and international public health even has the ability to trigger social unrest and have a serious detrimental effect on economic S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6354 13 of 20 growth. Many scholars are focusing their research on how to address widespread public health events in an effective manner. The issue of allocating medical resources will be discussed in this paper starting with the locations of Istanbul’s hospitals. Step 1. Firstly, seven locations for hospitals were determined, σ1-Bakirköy, σ2- Sancaktepe, σ3-Eyüp, σ4-Esenyurt, σ5-Çatalca, σ6-Tuzla, σ7-Ataşehir, which are randomly placed in different location of Istanbul. There are seven characteristics that must be taken into account during the selection process: ς1(Cost), ς2(Demographics), ς3(Environmental Factors), ς4(Transportation opportunities), ς5(Healthcare and medical practices), ς6(Infrastructure), ς7(Spread of the virus). This leads to the event set σ = {σ1, σ2, ..., σ7} and the criteria set C = {ς1, ς2, ..., ς7}. Additionally, three fuzzy multi-criteria decision-making experts, designated as DM1, DM2, and DM3, were chosen to serve as decision makers. Step 2. Based on their knowledge and the actual scenario, the decision makers (DMs) evaluated each proposal, and the resulting expert decision matrix is displayed in Table 3. Next, by quantifying semantic information, the qualitative evaluation data is converted into fuzzy sets (Table 1) Table 3: Expert decision sheet. Criterion DMs σ1 σ2 σ3 σ4 σ5 σ6 σ7 DM1 HV LV AV VLV AV AV AAV ς1 DM2 HV UAV AV LV UAV AV HV DM3 AAV LV AAV LV UAV AV HV DM1 AAV VHV VHV VHV UAV AAV HV ς2 DM2 AAV HV HV VHV LV AV HV DM3 AV VHV HV CHV LV AAV VHV DM1 LV AV UAV VLV HV AV UAV ς3 DM2 LV AV UAV LV AAV UAV AV DM3 UAV AAV LV LV HV AV AV DM1 HV AV AAV UAV UAV UAV AAV ς4 DM2 VHV AV HV UAV UAV LV AAV DM3 CHV AV AAV AV LV UAV HV DM1 AAV AV AAV AV UAV LV UAV ς5 DM2 AAV AV AAV AV LV UAV UAV DM3 HV UAV HV AAV AV AV AV DM1 UAV AV AAV CLV VHV AV LV ς6 DM2 UAV AAV AAV CLV CHV AAV LV DM3 LV HV HV VLV CHV AAV VLV DM1 HV HV HV VHV VLV LV HV ς7 DM2 VHV AAV AAV CHV CLV LV HV DM3 VHV HV HV CHV VLV UAV VHV S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6354 14 of 20 Step 3. The spherical picture fuzzy decision matrix displayed in Table 4 is created by combining the picture fuzzy evaluation data provided by the DMs in accordance with equations (2) and (4). Step 4. Table 2 was used to measure the data using the weight assessment criteria from Table 6 in the literature [23]. The weight data was then merged using equations (2) and (4) to produce the criteria weight matrix (Table 6 ). Step 5. As shown in Table 7, determine the spherical picture fuzzy decision matrix for each criterion Crj after applying the weights in accordance with equation (3). Step 6 and 7. Equations (9) and (10) are used to calculate the positive and negative ideal solutions for various criteria once the weighted decision matrix has been obtained. G+ = {⟨0.284, 0.361, 0.278, 0.277⟩, ⟨0.144, 0.515, 0.306, 0.165⟩, ⟨0.159, 0.533, 0.222, 0.150⟩, ⟨0.171, 0.511, 0.250, 0.165⟩, ⟨0.184, 0.489, 0.306, 0.180⟩, ⟨0.245, 0.422, 0.334, 0.200⟩, ⟨0.133, 0.578, 0.220, 0.175⟩}, G− = {⟨0.245, 0.417, 0.191, 0.246⟩, ⟨0.202, 0.438, 0.280, 0.219⟩, ⟨0.188, 0.510, 0.130, 0.188⟩, ⟨0.327, 0.340, 0.190, 0.120⟩, ⟨0.208, 0.444, 0.278, 0.207⟩, ⟨0.184, 0.488, 0.250, 0.178⟩, ⟨0.173, 0.533, 0.160, 0.213⟩}. S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6354 15 of 20 Table 4: Spherical picture fuzzy set decision matrix. Criterion σ1 σ2 ς1 ⟨0.433, 0.3, 0.067, 0.189⟩ ⟨0.333, 0.367, 0.133, 0.058⟩ ς2 ⟨0.433, 0.233, 0.166, 0.304⟩ ⟨0.433, 0.233, 0.2, 0.219⟩ ς3 ⟨0.333, 0.367, 0.133, 0.111⟩ ⟨0.567, 0.167, 0.133, 0.304⟩ ς4 ⟨0.567, 0.2, 0.1, 0.272⟩ ⟨0.7, 0.1, 0.1, 0.0000⟩ ς5 ⟨0.367, 0.3, 0.133, 0.188⟩ ⟨0.6, 0.167, 0.133, 0.249⟩ ς6 ⟨0.367, 0.333, 0.167, 0.111⟩ ⟨0.5, 0.233, 0.1, 0.245⟩ ς7 ⟨0.433, 0.233, 0.2, 0.218⟩ ⟨0.433, 0.3, 0.067, 0.188⟩ σ3 σ4 ς1 ⟨0.567, 0.167, 0.133, 0.303⟩ ⟨0.4, 0.333, 0.1, 0.238⟩ ς2 ⟨0.467, 0.267, 0.1, 0.219⟩ ⟨0.333, 0.167, 0.2, 0.340⟩ ς3 ⟨0.367, 0.333, 0.167, 0.111⟩ ⟨0.4, 0.333, 0.1, 0.238⟩ ς4 ⟨0.367, 0.3, 0.133, 0.207⟩ ⟨0.5, 0.233, 0.167, 0.249⟩ ς5 ⟨0.367, 0.3, 0.133, 0.188⟩ ⟨0.567, 0.167, 0.133, 0.303⟩ ς6 ⟨0.367, 0.3, 0.133, 0.465⟩ ⟨0.467, 0.267, 0.1, 0.145⟩ ς7 ⟨0.433, 0.3, 0.067, 0.188⟩ ⟨0.667, 0.133, 0.1, 0.381⟩ σ5 σ6 ς1 ⟨0.5, 0.233, 0.167, 0.249⟩ ⟨0.7, 0.1, 0.1, 0.0000⟩ ς2 ⟨0.333, 0.367, 0.133, 0.111⟩ ⟨0.433, 0.233, 0.166, 0.304⟩ ς3 ⟨0.433, 0.3, 0.067, 0.188⟩ ⟨0.6, 0.167, 0.133, 0.335⟩ ς4 ⟨0.367, 0.333, 0.167, 0.111⟩ ⟨0.367, 0.333, 0.167, 0.111⟩ ς5 ⟨0.467, 0.267, 0.133, 0.288⟩ ⟨0.467, 0.267, 0.133, 0.288⟩ ς6 ⟨0.667, 0.133, 0.2, 0.289⟩ ⟨0.433, 0.233, 0.167, 0.304⟩ ς7 ⟨0.533, 0.233, 0.1, 0.145⟩ ⟨0.333, 0.367, 0.133, 0.111⟩ σ7 ς1 ⟨0.433, 0.3, 0.066, 0.189⟩ ς2 ⟨0.467, 0.267, 0.1, 0.218⟩ ς3 ⟨0.6, 0.167, 0.133, 0.335⟩ ς4 ⟨0.367, 0.3, 0.133, 0.188⟩ ς5 ⟨0.5, 0.233, 0.167, 0.249⟩ ς6 ⟨0.4, 0.333, 0.1, 0.259⟩ ς7 ⟨0.467, 0.267, 0.1, 0.218⟩ S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6354 16 of 20 Table 5: Criteria weighting evaluation table. Criterion DM1 DM2 DM3 Cost Benefit ς1 AI AI AAI ✓ ς2 VHI VHI HI ✓ ς3 AAI HI HI ✓ ς4 HI HI VHI ✓ ς5 LI UAI UAI ✓ ς6 LI UAI UAI ✓ ς7 VLI LI LI ✓ Table 6: Criteria weight sheet Criterion Criteria Weight ς1 ⟨0.567, 0.167, 0.133, 0.304⟩ ς2 ⟨0.433, 0.233, 0.200, 0.219⟩ ς3 ⟨0.433, 0.300, 0.067, 0.188⟩ ς4 ⟨0.467, 0.267, 0.100, 0.219⟩ ς5 ⟨0.367, 0.333, 0.167, 0.111⟩ ς6 ⟨0.367, 0.333, 0.167, 0.111⟩ ς7 ⟨0.400, 0.333, 0.100, 0.238⟩ S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6354 17 of 20 Table 7: Weighted decision matrix Criterion σ1 σ2 σ3 σ4 σ5 σ6 σ7 ς1 ⟨0.245, 0.417, 0.191, 0.246⟩ ⟨0.189, 0.473, 0.248, 0.058⟩ ⟨0.322, 0.306, 0.248, 0.304⟩ ⟨0.227, 0.444, 0.220, 0.271⟩ ⟨0.284, 0.361, 0.278, 0.277⟩ ⟨0.397, 0.250, 0.220, 0.152⟩ ⟨0.246, 0.417, 0.190, 0.247⟩ ς2 ⟨0.188, 0.412, 0.333, 0.262⟩ ⟨0.188, 0.412, 0.360, 0.219⟩ ⟨0.202, 0.438, 0.280, 0.219⟩ ⟨0.231, 0.361, 0.360, 0.280⟩ ⟨0.144, 0.515, 0.306, 0.165⟩ ⟨0.188, 0.412, 0.333, 0.262⟩ ⟨0.202, 0.438, 0.280, 0.219⟩ ς3 ⟨0.144, 0.557, 0.191, 0.150⟩ ⟨0.246, 0.417, 0.191, 0.246⟩ ⟨0.159, 0.533, 0.222, 0.150⟩ ⟨0.173, 0.533, 0.160, 0.213⟩ ⟨0.188, 0.510, 0.130, 0.188⟩ ⟨0.260, 0.417, 0.191, 0.262⟩ ⟨0.260, 0.417, 0.191, 0.262⟩ ς4 ⟨0.265, 0.414, 0.190, 0.246⟩ ⟨0.327, 0.340, 0.190, 0.120⟩ ⟨0.171, 0.487, 0.220, 0.213⟩ ⟨0.234, 0.438, 0.250, 0.234⟩ ⟨0.171, 0.511, 0.250, 0.165⟩ ⟨0.171, 0.511, 0.250, 0.165⟩ ⟨0.171, 0.487, 0.220, 0.203⟩ ς5 ⟨0.135, 0.533, 0.278, 0.450⟩ ⟨0.220, 0.444, 0.278, 0.180⟩ ⟨0.135, 0.533, 0.278, 0.207⟩ ⟨0.208, 0.444, 0.278, 0.207⟩ ⟨0.171, 0.511, 0.278, 0.200⟩ ⟨0.171, 0.511, 0.278, 0.200⟩ ⟨0.184, 0.489, 0.306, 0.180⟩ ς6 ⟨0.135, 0.555, 0.306, 0.111⟩ ⟨0.184, 0.488, 0.250, 0.178⟩ ⟨0.135, 0.533, 0.278, 0.288⟩ ⟨0.171, 0.511, 0.250, 0.128⟩ ⟨0.245, 0.422, 0.334, 0.200⟩ ⟨0.159, 0.488, 0.306, 0.208⟩ ⟨0.147, 0.555, 0.250, 0.185⟩ ς7 ⟨0.227, 0.444, 0.220, 0.271⟩ ⟨0.173, 0.533, 0.160, 0.213⟩ ⟨0.173, 0.533, 0.160, 0.213⟩ ⟨0.267, 0.422, 0.190, 0.310⟩ ⟨0.213, 0.488, 0.190, 0.192⟩ ⟨0.133, 0.578, 0.220, 0.175⟩ ⟨0.187, 0.511, 0.190, 0.228⟩ Figure 3: Representation of alternatives on the basis of membership degrees The distances between each alternative and the positive and negative ideal solutions are then computed using the spherical picture fuzzy set distance measure, equation(5), as shown in Table 8. Table 8: The distance of each alternative to the positive and negative ideal solutions. Distance σ1 σ2 σ3 σ4 σ5 σ6 σ7 N∂+c (σi) 0.125 0.134 0.077 0.122 0.042 0.081 0.097 N∂−c (σi) 0.093 0.057 0.088 0.079 0.097 0.119 0.068 S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6354 18 of 20 Step 8. From the distance data obtained for different alternatives in Table 8, the relative closeness coefficient of each alternative to the positive ideal solution is calculated using Eq (11) and ranked. The results of the relative closeness coefficient calculation are shown in Table 9. Table 9: Relative closeness coefficients of alternatives to positive and negative ideal solutions. Alternatives σ1 σ2 σ3 σ4 σ5 σ6 σ7 RCC(σi) 0.427 0.298 0.533 0.393 0.698 0.595 0.412 The ranking of the different candidates according to the principle of maximum prox- imity RCC(σi) is σ5 ≻ σ6 ≻ σ3 ≻ σ1 ≻ σ7 ≻ σ4 ≻ σ2. Consequently optimal hospital site is σ5. 7. Conclusion Spherical picture fuzzy set is a relatively broad class of fuzzy set which unifies the concepts of picture fuzzy set and spherical fuzzy set. Spherical picture fuzzy set has a stronger ability to express uncertain information and can better reflect the essential characteristics of the objective world. In this paper, a new distance measure on the basis of spherical picture fuzzy sets is presented. Structural properties of SPFS along with some basic set-theoretical operations and aggregation operators are discussed. A numerical case study of pandemic hospital site selection is used to illustrate the effectiveness and rationality of SPFS-TOPSIS method in this paper. This method not only considers the three factors of membership degree, non- membership degree and radius, but also considers the potential association between hesitation degree, membership degree and non- membership degree. The limitation of this manuscript is that the distribution ratio about the hesitation parameter in the distance metric can have more forms, so the distribution ratio of hesi- tation can be a direction for future research. The distance metric proposed in this paper can further be considered to apply it to different multi-criteria decision models, and then solve a wider range of multi-criteria decision problems. In addition, the distance metric proposed in this paper can be used to solve problems related to pattern recognition and medical diagnosis. Hopefully the newly introduced concept of spherical picture fuzzy set and related proposed tools may provide a gateway for researchers to further dive in it. Acknowledgements The authors would like to thank Prince Sultan University for APC and support. S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6354 19 of 20 References [1] L. A. Zadeh. Fuzzy sets. Inf. Ctrl., 8(3):338–353, 1965. [2] K. T. Atanassov. Intuitionistic fuzzy sets. 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