Intelligent Automation & Soft Computing DOI:10.32604/iasc.2021.016822
Article

Solid Waste Collection System Selection Based on Sine Trigonometric Spherical Hesitant Fuzzy Aggregation Information

Muhammad Naeem1, Aziz Khan2, Saleem Abdullah2,*, Shahzaib Ashraf3 and Ahmad Ali Ahmad Khammash4

1Deanship of Combined First Year, Umm Al-Qura University, Makkah, Saudi Arabia
2Department of Mathematics, Abdul Wali Khan University, Mardan, 23200, Pakistan
3Department of Mathematics and Statistics, Bacha Khan University, Charsadda, 24420, Pakistan
4Department of Mathematics, Deanship of Applied Sciences, Umm Al-Qura University, Makkah, Saudi Arabia
*Corresponding Author: Saleem Abdullah. Email: saleemabdullah@awkum.edu.pk
Received: 10 January 2021; Accepted: 12 February 2021

Abstract: Spherical fuzzy set (SFS) as one of several non-standard fuzzy sets, it introduces a number triplet (a,b,c) that satisfies the requirement a2+b2+c2≤1 to express membership grades. Due to the expression, SFS has a more extensive description space when describing fuzzy information, which attracts more attention in scientific research and engineering practice. Just for this reason, how to describe the fuzzy information more reasonably and perfectly is the hot that scholars pay close attention to. In view of this hot, in this paper, the notion of spherical hesitant fuzzy set is introduced as a generalization of spherical fuzzy sets. Some basic operations using sine trigonometric function are presented for spherical hesitant fuzzy sets. We define spherical hesitant fuzzy weighted average and spherical hesitant fuzzy weighted geometric aggregation operators. Based on these new aggregation operators, we propose a method for multi-criteria decision making (MCDM) in the spherical hesitant fuzzy information. Besides, a numerical real-life application about solid waste collection system selection is provided to demonstrate the validity of the proposed approaches along with relevant discussions, the merits of proposed approaches are also analyzed by validity test.

Keywords: Spherical fuzzy set; Hesitant fuzzy set; spherical hesitant fuzzy set; Sine trigonometric aggregation information; decision making

1  Introduction

The smart city idea is focused on the incorporation of information and communication technologies (ICTs) into city services to accumulate information for the allocation of assets and services, along with enhancing value of life and susceptibility. Difficulties and inefficient solutions in today’s cities raise a need for the smart apps to solve existing challenges effectively. In this background, many academics, architects, urban planners, and even municipal representatives have been drawn to smart city applications. In addition, a wide variety of implementations are available for the smart city concept in areas such as town planning, waste disposal, resource management and municipal services [1–3].

In particular, the problem of solid waste collection system selection is a multi-criteria decision-making problem that should also take into account environmental, social, economic and technological aspects. Due to the labor strength of the job and the use of the large number of vehicles in these processes, sample and transport are commonly considered to be the most critical and expensive phases of the process [4]. As ever, certain issues such as quick disposal of wastes from half-full bins and thus excessive fuel use of collection and transport vehicles, higher pollution levels, inefficient usage of city assets and services are raised in the predefined scheduling [5]. In this respect, the brand-new visible light communication (VLC) technique makes it possible to communicate ultra-fastly among terminals through light bulbs and to become an essential competitor for conventional radio frequency (RF) communication like Wi-Fi [6]. Without providing some other [7] communication method, VLC can supply a room’s interior lighting and data exchange at the same time.

Multiple criteria group decision making (MCGDM) method [8–14] is a significant and arising subject to depict an approach for choosing the finest alternative with group of the decision makers (DMs) and conditions. Two serious tasks are there in this procedure. The first one is to depict the atmosphere where the values of various attributes can be scrutinized successfully, while the aggregation of the depicted data is the second task. Generally, the data which depict the substances are frequently taken in the form of the deterministic or crisp in nature. Though, with the rising complications of the frameworks step by step, hardly data can be accumulated, from the records, assets and specialists, in crisp form. Therefore, to present the data more openly, a notion of fuzzy sets (FSs) [15] and its extended types for instance, Intuitionistic fuzzy set (InFSs) [16], Pythagorean fuzzy sets (PytFSs) [17], hesitant fuzzy sets (HeFSs) [18] etc., is applied by the scholars. Every element of InFS is indicated by an ordered pair, every pair is defined by a positive- membership degree (PMD) and a negative-membership grade (NMD). The totality of PMD and NMD is less than or equal to one. So, in that positions IFSs has no capacity to improve any proper outcome. To grip such state Yager [17,19] submitted the notion of Pythagorean FS (PytFS), which is the broad form of InFS. For PytFS, the square sum of PMD and NMD is less than or equal to 1. Spherical fuzzy set (SFS) [20] established by Ashraf et al. [21,22] which is the wide-ranging arrangement of the all the current approaches of FS in the fiction. SFSs can handle the vagueness more fruitfully and skillfully in decision making (DM) problems. Several DM approaches [23–27] has been established by the scholars with SF material to improve the SFS theory. Ashraf et al. [28] established the theory of the SF Dombi aggregation operators under the SF material. Chen et al. [29] established the logarithmic based (AOs) for SFNs to discuss the imprecision in DMPs. Jin et al [30] predicted the linguistic SF AOs under SF material. Rafiq et al. [31] offered the DMP established for the cosine similarity dealings according to SF material. Ashraf et al. [32] settled the spherical distance measure based on DM method in accordance with SF atmospheres. Ashraf et al. [33] adapted the SFS depiction of SF t-norms and t-conorms and described the TOPSIS based DM method in accordance with SF material. Zeng et al. [34] offered the SF rough set on the basis of TOPSIS methodology for dealing the imprecision in the process of SFSs. Torra [18] presented the idea of HeFS to stimulate the method of FS, which has a set of values without having a single value in the form of membership. HeFS is a leading tool for holding the indistinctness in DMPs.

In this paper, we offered the pioneering view of T- spherical HeFS (T-SHFS) by using the concept of SFS and HeFS to check the incredibility and imprecise figures in DMPs to reform the supreme alternative in conferring to list of criteria. A DMPs AOs acts the supreme role to aggregate the data. Since, for every aggregation procedure the rules of the operation perform a key role. It is necessary to construct fresh laws for the operation and aggregation of T-SHFNs. Consequently, the aim of this paper is to suggest some new laws for the operation of T-SHFSs. Therefore, by using the above stated proofs, we offered the MCGDM algorithm to grip the assessment material for T-SHFSs.

2  Preliminaries

This unit contains some basic definitions of FS, InFS, PytFS, SFS, HeFS and SHFS.

Definition 1. [18] Consider the ground set G≠φ . A hesitant FS (HeFS) k can be described as;

k={⟨g,hk(g)⟩|g∈G},

where hk(g) be any set having some values in [0,1].

Definition 2. [13] Consider the ground set G≠φ . A spherical FS k can be described as.

k={⟨g,Mk(g),Ik(g),Rk(g)⟩|g∈G},

where Mk:g→[0,1] be positive, Lk:g→[0,1] be neutral and Kk:g→[0,1] be negative membership degrees with the constraint [Mk(g)]2+[Ik(g)]2+[Rk(g)]2≤1, for all g∈G.

Definition 3: Consider the ground set G≠φ . A spherical hesitant FS k can be described as.

k={⟨g,M?(g),I?(g),R?(g)⟩|g∈G},

Where Mk(g)={κ|κ∈[0,1]},Ik(g)={δ|δ∈[0,1]}andRk(g)={∂|∂∈[0,1]}, denoted the positive membership degree (PMD), neutral membership degree (NeMD) and negative membership degree (NMD) with the constraint 0≤(κ+)2+(δ+)2+(∂+)2≤1, for all g∈R, such that

κ+=∪κ∈Mk(g)max⁡{κ},δ+=∪δ∈Lk(g)⁡max⁡{δ},and∂+=∪∂∈Kk(g)⁡max⁡{∂}.

Definition 4. [13] A T-spherical fuzzy set (T-SFS) k on G can be described as

k={(g,Mk(g),Ik(g),Rk(g)|g∈G},

where Mk,Ik,Rk(g):G→[0,1] denoted the PMD, NeMD and NMD of g∈G , respectively, and for each g∈G , it holds that (Mk(g))n+(Ik(g))n+(Rk(g))n≤1 for n≥1 . Analogous to its membership degrees, the degree of indeterminacy is given as πk(g)=1−(Mk(g))n−(Ik(g))n−(Rk(g))nn . For convenience, we call (Mk,Ik,Rk) is a T-spherical fuzzy set.

Definition 5. Suppose kg={Mg,Ig,Rg} ∈T−SpHeFS(k) (g∈G). The fundamental operational laws can be defined as follows;

(1) (k1)c=∪(κ1,δ1,∂1)∈(M1,L1,K1)⁡{K1,L1,M1};

(2) k1∪k2=∪(κg,δg,∂g)∈(Mg,Lg,Kg)⁡{max(Mg),min(Ig),min(Rg)};

(3) k1∩k2=∪(κg,δg,∂g)∈(Mg,Lg,Kg)⁡{min(Mg),min(Ig),max(Rg)};

Definition 6. Let k={M,I,R} , k1={M1,I1,R1} , and k2={M2,I2,R2} be the three T-Spherical hesitant fuzzy numbers (T-SHFNs), λ>0, kc indicates the complement of k and the operations of T-SHFNs are given below:

(1) k1⊕k2=∪κ1∈M1,δ1∈I1,∂1∈R1κ2∈M2,δ2∈I2,∂2∈R2⁡{κ1n+κ2n−κ1nκ1nn,δ1δ1,∂1∂2};

(2) λ.k=∪κ∈M,δ∈I,∂∈R⁡{1−(1−κn)λ,(δ)λ,(∂)λ};

(3) k1⊗k2=∪κ1∈M1,δ1∈I1,∂1∈R1κ2∈M2,δ2∈I2,∂2∈R2{{κ1κ2,δ1δ1,∂1n+∂2n−∂1n∂2nn};

(4) kλ=∪κ∈M,δ∈I,∂∈R⁡{(κ)λ,(δ)λ,1−(1−∂n)λ}.

3  New Sine Trigonometric Operational Laws for T-SHFS

We express a number of new operational laws for T-SHFSs in this part.

Definition 7. Consider a non-empty set G and k={(g,Mk(g),Ik(g),Rk(g)|g∈G} be T-SHFS, then a sine trigonometric operational laws (STOLs) of T-SHFS k can be described as in the following:

sin⁡k=∪κ∈M,δ∈I,∂∈R⁡{sin⁡(π2κk),1−sinn(π21−δknn)n,1−sinn(π21−∂knn)n},

It is obviously understood that the sin⁡k is also T-SHFS. As it is clear from the definition of T-SHFS, ∀g∈G , the functions κk ,δk and ,∂k satisfy: κk:G→[0,1],δk:G→[0,1],∂k:G→[0,1] and 0≤(κk(g))n+(δk(g))n+(∂k(g))n≤1 . Further, the membership function:

sin⁡(π2κk):G→[0,1],∀g∈G→sin⁡(π2κk(g))∈[0,1],

the neutral function:

1−sinn(π21−δknn)n:G→[0,1],∀g∈G→1−sinn(π21−δkn(g)n)n∈[0,1],

and the non-membership function:

1−sinn(π21−∂knn)n:G→[0,1],∀g∈G→1−sinn(π21−∂kn(g)n)n∈[0,1],

Therefore

sin⁡k={sin⁡(π2κk),1−sinn(π21−δknn)n,1−sinn(π21−∂knn)n}

is a T-SHFS.

Definition 8. Let Ki=(M,I,R) be T-SHFN. If

sin⁡k={sin⁡(π2κk),1−sinn(π21−δknn)n,1−sinn(π21−∂knn)n}

then sin⁡k is called a sine trigonometric operator, and the value sin⁡k is called sine trigonometric T-SHFN (ST-T-SHFN).

4  Sine Trigonometric Aggregation Operators

On the basis of STOL of T-SHFNs, we describe the below weighted averaging and geometric aggregation operators. Let ψ be the family of T-SHFNs.

Definition 9. Let ki=(Mi,Ii,Ri) be a set of n T-SHFNs, and suppose k:ψn→ψ . If

ST−T−SpHeFWA(k1,k2,…,kn)=ω1sin⁡k1⊕ω2sin⁡k2⊕…⊕ωnsin⁡kn

hence the mapping T-SpHeFWA is known as sine trigonometric T-SpHeF weighted averaging operator, where ω=(ω1,ω2,…,ωn)T is the weight information of sin⁡ki i.e., ωi>0 and ∑i=1m⁡ωi=1 .

Theorem 1. Consider the set ki=(Mi,Ii,Ri) of m T-SHFNs. Then, the aggregated value by using ST-T-SpHeFWA operator is also T-SHFN, then

ST−T−SpHeFWA(k1,k2,…,km)=∪κi∈Mi,δi∈Ii,∂i∈Ri⁡(1−∏i=1m⁡(1−sinn(π2κi))ωin,∏i=1m⁡(1−sinn(π21−δinn)n)ωi,∏i=1m⁡(1−sinn(π21−∂inn)n)ωi)

Proof: We prove the theorem using induction method. Since for each i , ki=(Mi,Ii,Ri) is ST-T-SpHeFWA which implies that Mi,Ii,Ri∈[0,1] and Min+Iin+Rin≤1 .

(1) For m=2 , we have ST−T−SpHeFWA(k1,k2)=ω1sin⁡k1⊕ω2sin⁡k2

As from Definition 3.1, we can see that sin⁡k1 and sin⁡k2 are ST-T-SHFNs and hence ω1sin⁡k1⊕ω2sin⁡k2 is also ST-T-SHFN. Also, for k1 and k2 , we have

ST−T−SpHeFWA(k1,k2)=ω1sin⁡k1⊕ω2sin⁡k2=∪κ1∈M1,δ1∈I1,∂1∈R1⁡{1−(1−sinn(π2κ1))ω1n,(1−sinn(π21−δ1nn)n)ω1,(1−sinn(π21−∂1nn)n)ω1}⊕∪κ2∈M2,δ2∈I2,∂2∈R2⁡{1−(1−sinn(π2κ2))ω2n,(1−sinn(π21−δ2nn)n)ω2,(1−sinn(π21−∂2nn)n)ω2}

=∪κi∈Mi,δi∈Ii,∂i∈Ri⁡{1−∏i=1n⁡(1−sinn(π2κi))ωin,∏i=1n⁡(1−sinn(π21−δinn)n)ωi,∏i=1n⁡(1−sinn(π21−∂inn)n)ωi}

Assume Eq. (1) holds for m=p . Now, for m=p+1 , we have

T−SpHeFWA(k1,k2,…,kp)⊕ωp+1sin⁡kp+1=∪κi∈Mi,δi∈Ii,∂i∈Ri⁡{1−∏i=1p⁡(1−sinn(π2κi))ωin,∏i=1p⁡(1−sinn(π21−δinn)n)ωi,∏i=1p⁡(1−sinn(π21−∂inn)n)ωi}⊕∪κk+1∈Mk+1,δk+1∈Ik+1,∂k+1∈Rk+1{1−(1−sinn(π2κk+1))ωk+1n,(1−sinn(π21−δp+1nn)n)ωp+1,(1−sinn(π21−∂p+1nn)n)ωp+1}

=∪κi∈Mi,δi∈Ii,∂i∈Ri⁡{1−∏i=1p+1⁡(1−sinn(π2κi))ωin,∏i=1p+1⁡(1−sinn(π21−δinn)n)ωi,∏i=1p+1⁡(1−sinn(π21−∂inn)n)ωi}

Hence, Eq. (1) also valid for m=p+1 . So, the result is valid for all positive integer m .

Definition 10. A sine-trigonometric T-SHFN ordered weighted average (ST-T-SpHeFOWA) operator is a mapping ST-T-SpHeFOWA :ψm→ψ , such that ω=(ω1,ω2,…,ωm)T , with ωi>0 and ∑i=1m⁡ωi=1,

ST−T−SpHeFOWA(k1,k2,…,km)=ω1sin⁡kρ(1)⊕ω2sin⁡kρ(2)⊕…⊕ωnsin⁡kρ(m)

where ρ is the permutation of (1,2,…,m) such that kρ(i−1)≥kρ(i) for i=2,3,…,m .

Theorem 2. For a collection of m T-SHFNs ki=(Mi,Ii,Ri) , the aggregated value by using ST-T-SpHeFOWA operator is still T-SHFN and given by

ST−T−SpHeFOWA(k1,k2,…,km)=∪κi∈Mi,δi∈Ii,∂i∈Ri⁡(1−∏i=1m⁡(1−sinn(π2κρ(i)))ωin,∏i=1m⁡(1−sinn(π21−δρ(i)nn)n)ωi,∏i=1m⁡(1−sinn(π21−∂ρ(i)nn)n)ωi)

.

Definition 11. A sine-trigonometric T-SHFN hybrid average (ST-T-SpHeFHA) operator is a mapping ST-T-SpHeFHA :ψm→ψ , such that ω=(ω1,ω2,…,ωm)T , with ωi>0 and ∑i=1m⁡ωi=1 , and

ST−T−SpHeFHA(k1,k2,…,km)=ω1sin⁡kρ(1)⊕ω2sin⁡kρ(2)⊕…⊕ωmsin⁡kρ(m)

where ρ is the permutation of (1,2,…,m) such that kρ(i−1)≥kρ(i) for i=2,3,…,m and ki=mωiki .

Theorem 3. For a collection of m T-SHFNs ki=(Mi,Ii,Ri) , the aggregated value by using ST-TSpHeFHA operator is still T-SHFN and given by

ST−TSHFHA(k1,k2,…,km)=∪κi∈Mi,δi∈Ii,∂i∈Ri⁡(1−∏i=1m⁡(1−sinn(π2κρ(i)))ωin,∏i=1m⁡(1−sinn(π21−δρ(i)nn)n)ωi,∏i=1m⁡(1−sinn(π21−∂ρ(i)nn)n)ωi)

.

Definition 12. Let ki=(Mi,Ii,Ri) is a set of m T-SHFNs and let ST−T−SpHeFWG:ψm→ψ. If

ST−T−SpHeFWG(k1,k2,…,km)=(sin⁡k1)ω1⊗(sin⁡k2)ω2⊗…⊗(sin⁡km)ωm

then the function ST-T-SpHeFWG is called sine trigonometric T-SpHeF weighted geometric operator, where ω=(ω1,ω2,…,ωm)T is the weight vector of sin⁡ki with ωi>0 and ∑i=1m⁡ωi=1 .

Theorem 4. Let ki=(Mi,Ii,Ri)(i=1,2,…,m) is a set of m T-SHFNs. Then, the aggregated value by using ST-TSpHeFWG operator is also T-SHFN and is given by

ST−TSpHeFWG(k1,k2,…,km)=∪κi∈Mi,δi∈Ii,∂i∈Ri⁡(∏i=1m⁡(sin⁡(π2κi))ωi,1−∏i=1m⁡(sinn(π21−δinn))ωin,1−∏i=1m⁡(sinn(π21−∂inn))ωin)

Definition 13. A sine trigonometric TSpHeF ordered weighted geometric (ST-TSpHeFOWG) operator is a mapping ST−q−ROFOWG:ψm→ψ , such that ω=(ω1,ω2,…,ωn)T , with ωi>0 and ∑i=1m⁡ωi=1 , and

ST−TSpHeFOWG(k1,k2,…,km)=(sin⁡kρ(1))ω1⊗(sin⁡kρ(2))ω2⊗…⊗(sin⁡kρ(n))ωm

where ρ is the permutation of (1,2,…,m) such that kρ(i−1)≥kρ(i) for i=2,3,…,m.

Theorem 5. Let ki=(Mi,Ii,Ri)(i=1,2,…,m) is a set of m T-SHFNs. Then the aggregated value by using ST-TSpHeFOWG operator is still T-SHFN and given by

ST−TSpHeFOWG(k1,k2,…,km)=∪κi∈Mi,δi∈Ii,∂i∈Ri⁡{∏i=1m⁡(sin⁡(π2κρ(i)))ωi,1−∏i=1m⁡(sinn(π21−δρ(i)nn))ωin,1−∏i=1m⁡(sinn(π21−∂ρ(i)nn))ωin}

Definition 14. A sine trigonometric T-SpHeF hybrid weighted geometric (ST-TSpHeFHG) operator is a mapping ST-TSpHeFHG :ϕm→ϕ , such that ω=(ω1,ω2,…,ωm)T , with ωi>0 and ∑i=1m⁡ωi=1 , and

ST−TSpHeFHG(k1,k2,…,km)=(sin⁡kρ(1))ϕ1⊗(sin⁡kρ(2))ϕ2⊗…⊗(sin⁡kρ(m))ϕm

where ρ is the permutation of (1,2,…,m) such that kρ(i−1)≥kρ(i) for i=2,3,…,m and ki=kmωi .

Theorem 6. Let ki=(Mi,Ii,Ri)(i=1,2,…,m) is a set of m T-SHFNs. Then the aggregated value by using ST-TSpHeFHG operator is still T-SHFN and given by

ST−TSpHeFHG(k1,k2,…,km)=∪κi∈Mi,δi∈Ii,∂i∈Ri⁡{∏i=1m⁡(sin⁡(π2κρ(i)))ωi,1−∏i=1m⁡(sinn(π21−δρ(i)nn))ωin,1−∏i=1m⁡(sinn(π21−∂ρ(i)nn))ωin}

As similar to ST-TSHFWA operator, the ST-TSpHeFOWA, ST-TSpHeFHA, ST-TSpHeFWG, ST-TSpHeFOWG and ST-TSpHeFHG operators satisfy the properties such as boundedness, monotonicity.

Fundamental Properties of the Proposed AOs

In this subsection, we scrutinized some relations between the suggested AOs and study their various major properties as given below.

Theorem 7. For two T-SHFNs k1 and k2 we have, sin⁡k1⊕sin⁡k2≥sin⁡k1⊗sin⁡k2 .

Proof: Let k1=(M1,I1,R1) and k2=(M2,I2,R2) be two T-SHFNs. Then, by using Definitions (i) & (ii) , we get

sin⁡k1⊕sin⁡k2=(∪κi∈Mi,δi∈Ii,∂i∈Ri⁡{1−∏i=12⁡(1−sinn(π2κi))n,∏i=12⁡1−sinn(π21−δinn)n,∏i=12⁡1−sinn(π21−∂inn)n})

and

sin⁡k1⊗sin⁡k2=((∏i=12⁡sin⁡(π2κi),1−∏i=12⁡sinn(π21−δinn)n,1−∏i=12⁡sinn(π21−∂inn)n))

Since for any two non-negative real numbers c and d , their arithmetic mean is greater than or equal to their geometric mean therefore, c+d2≥cd which follows that c+d−cd≥cd . Thus by taking c=sinn(pi2M1) and d=sinn(pi2M2) we have

1−(1−sinn(π2M1))(1−sinn(π2M2))≥(1−sinn(π2M1))(1−sinn(π2M2))

which further gives that

1−∏l=12⁡(1−sinn(π2Ml))n≥∏j=12⁡sin⁡(π2Ml)

Similarly, we can obtain

∏l=12⁡1−sinn(π21−Ilnn)n≤1−∏l=12⁡sinn(π21−Ilnn)n

And

∏l=12⁡1−sinn(π21−Rlnn)n≤1−∏l=12⁡sinn(π21−Rlnn)n.

Hence, by using Definition (i), we get

sin⁡k1⊕sin⁡k2≥sin⁡k1⊗sin⁡k2

Theorem 8. Let ki , k are T-SHFNs, then

(1) ST−TSpHeFWA(k1⊕k,k2⊕k,…,kn⊕k)≥ST−TSpHeFWA(k1⊗k,k2⊗k,…,kn⊗k) ; (2) ST−TSpHeFWG(k1⊕k,k2⊕k,,…,k,n⊕k,)≥ST−TSpHeFWG(k1⊗k,k2⊗k,…,kn⊗k)

Proof: We will prove part (1) only. Part (2) can be obtained in a similar way. For this, let ki=(Mi,Ii,Ri) & k=(M,I,R) . Since ki and k are T-SHFNs.

ST−TSpHeFWA(k1⊕k,k2⊕k,…,km⊕k)=(∪κi∈Mi,δi∈Ii,∂i∈Ri⁡(1−∏i=1m⁡(1−sinn(π21−(1−κin)(1−κn)n))ωin,∏i=1m⁡(1−sinn(π21−δinδnn)n)ωi,∏i=1m⁡(1−sinn(π21−∂in∂nn)n)ωi))