# Applications of Cubic Schweizer–Sklar Power Heronian Mean to Multiple Attribute Decision-Making

Authors: Qaisar Khan; Muhammad Shahzad; Saqib Sharif; Mohammed Elgarhy; Mahmoud El-Morshedy; Suleman Nasiru
Journal: Complexity
 (2022)

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## Abstract

The cubic set (CS) is a basic simplification of several fuzzy notions, including fuzzy set (FS), interval-valued FS (IVFS), and intuitionistic FS (IFS). By the degrees of IVFS and FS, CS exposes fuzzy judgement, and this is a much more potent mathematical approach for dealing with information that is unclear, ambiguous, or indistinguishable. The article provides many innovative operational laws for cubic numbers (CNs) drawn on the Schweizer–Sklar (SS) t-norm (SSTN), and the SS t-conorm (SSTCN), as well as several desired properties of these operational laws. We also plan to emphasise on the cubic Schweizer–Sklar power Heronian mean (CSSPHM) operator, as well as the cubic Schweizer–Sklar power geometric Heronian mean (CSSPGHM) operator, in order to maintain the supremacy of the power aggregation (PA) operators that seize the complications of the unsuitable information and Heronian mean (HM) operators that contemplate the interrelationship between the input data being aggregated. A novel multiple attribute decision-making (MADM) model is anticipated for these freshly launched aggregation operators (AOs). Finally, a numerical example of enterprise resource planning is used to validate the approach’s relevance and usefulness. There is also a comparison with existing decision-making models.


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## Body

## 1. Introduction

Zadeh [1] proposed the fuzzy set (FS) as a procedure for expressing and transmitting precariousness and ambiguity. Since its inception, FS has attracted significant attention from intellectuals all over the world, who have calculated its factual and theoretical characteristics. Economic and business [2–4], genetic algorithms [5, 6], and supply chain management [7, 8] etc., are some of the most recent academic attempts at the theory and implementations of FSs. Ensuring the insertion of the notion of FS, various modifications of FSs were predicted, namely interval-valued FS [9], which explained the membership degree (MD) as a subclass of [0, 1] and Atanassov’s intuitionistic fuzzy set (AIFS) [10], which clarified the MD and nonmembership degree (NMD) as a single number in the [0, 1], with the constraint that sum of the two degrees must be less or equal to 1. As a consequence, IFS goes farther into explaining uncertainty and unreliability than FS. The attractive scenario occurs when the MDs of such an object is expressed in the form of IVFS and FS. Under such settings, the conformist IFS is unable to manage such data. To handle the aforementioned situation, Jun et al. [11] initiated the perception of the cubic set (CS). The aforementioned sets are special cases of CS. Mahmood et al. [12] proposed the concept of CNs and initiate some weighted aggregation operators (AOs) and apply these AOs to resolve multiple attribute decision-making (MADM) problems under a cubic environment.One of the key elements in the MADM process is the AOs. The AOs can blend many real numbers into a single one. Various AOs have different properties, namely, the PA operator offered by Yager [13], have the ability to remove the negative effects of uncomfortable information from last ranking results, and have been enlarged by numerous researchers from all over the world to figure out how to deal with different situations. Xu [14] enlarged the ordinary PA operator and delivered the IF power aggregation operator and implemented it in multiple attribute group decision-making (MAGDM), which can minimize the effects of inaccurate information. Some AOs, such as the Bonferroni mean (BM) operators [15], Heronian Mean (HM) operators [16], and Muirhead mean (MM) operators [17], as well as the Maclaurin symmetric mean (MSM) operator [18], ended up taking the connection between input arguments into consideration. BM and HM can take into account the connection between two input arguments, but MSM and MM operators can take into account the connection between any number of input opinions. These AOs were later extended to deal with a wide range of ambiguous circumstances [19–23].The majority of AOs use algebraic T-norm (TN) and T-conorm (TCN) to aggregate CNs. Currently, Ayub et al. [24] have presented a set of cubic fuzzy Dombi AOs that have been implemented on Dombi [25] TN and TCN and utilized to resolve MADM issues in a cubic fuzzy context. Dombi TN and Dombi TCN, as well as other TN and TCN, such as algebraic, Einstein, Hamacher, and Frank, are simplified in Archimedean TN (ATN) and Archimedean TCN (ATCN). On a generic parameter, Dombi TN and TCN outperform generic TN and TCN, providing more flexibility in the input dataset. Fahmi et al. [26], anticipated Einstein AOs for CNs and apply these AOs to solve MADM problems unde cubic information. Wan [27] and Wan and Dong [28] developed some power average/geometric operators for trapezoidal intuitionistic fuzzy (trIF) numbers and apply them to solve MAGDM problems under a trIF environment. Wan and Yi [29] initiated PA operators for trIFNs using strict t-norm and strict t-conorm. CS was further extended by Ali et al. [30] who introduced the concept of neutrosophic cubic set and give its applications in pattern recognition.Schweizer–Sklar (SS), TN (SSTN), and Schweizer–Sklar TCN (SSTCN) [31] are thorough ATN and ATCN instances, similar to the TN and TCN mentioned above. Because they contain a parameter that may be changed, SSTN and SSTCN are more flexible and superior to the prior techniques. Despite this, the majority of SS research has been on identifying the underlying theory and forms of SSTN and SSTCN [32, 33]. Recently, SS operational laws (OLs) were anticipated for interval-valued IFS (IVIFS) and IFS by Liu et al. [34] and Zhang [35], respectively, and predicted various power aggregation operators for these fuzzy structures. On the basis of SS OLs, Wang and Liu [36] projected MSM operators for IFS and apply them to resolve MADM problems. Liu et al. [37] also proposed SS OLs for single-valued neutrosophic (SVN) elements, as well as a variety of SS prioritised AOs for dealing with MADM issues in an SVN context. Zhang et al. [38] predicted and used certain MM operators for SVNS identified on SS OLs to solve MADM issues. By capturing the variable parameter from 0,∞, Nagarajan et al. [39] developed a couple of SS OLs for interval neutrosophic set (INS). For IN numbers, they anticipated various WA/WG AOs implemented on these SS OLs. The COPRAS was enhanced by Rong et al. [40], who predicted a new MAGDM technique based on SS OLs.From the above literature, it has been observed that the existing aggregation operators for CNs have only the capacity of removing the effect of awkward data or have the capacity of taking interrelationships among input arguments and a generic parameter. Yet, there are no such aggregation operators for CNs, which have the capacity of removing the effect of awkward data, can consider the interrelationship among input CNs, and also consist of the generic parameter. It has been observed that studies on various implementations of fuzzy MADM AOs depending on SS OLs have been published rapidly. Yet, no one has attempted to define cubic SS OLs and merge them with a power HM operator to deal with cubic information. As a consequence, we propose the following:(1)
The SS operations are considerably more adaptable and superior than the prior methods in terms of a variable parameter.(2)
Fortunately, there are many MADM difficulties in which the characteristics are linked, and many existing AOs can only alleviate such scenarios when the attributes are in the shape of real integers or other fuzzy formations.(3)
In the current situation, no such AOs exist which are drawn on SS OLs. In response to this limitation, we combined PA and HM operators with SS OLs to address MADM problems utilising cubic information.The subsequent are the urgencies and contributions of this effort as a result of important impacts from earlier studies as follows:(1)
Developing innovative SS ALs for CNs, describing their basic features, and using them in SS ALs that anticipate CSS power HM operators, CSS power geometric HM operators, and their weighted form(2)
Examining the commencing AOs’ basic features and exceptional cases(3)
Expecting the deployment of a MADM model on these commencing AOs(4)
Assessing enterprise resource planning (ERP) applications using a MADM model(5)
Confirming the feasibility and appropriateness of the launched MADM modelThis paper is structured in the following way to achieve these goals. Section2 introduces a variety of key concepts such as CSS, score and accuracy functions, PA, and HM operators. In Section 3, we look at a few SS OLs for CNs with general parameters that take values from −∞,0. Section 4 introduces the CSSPHM and CSSPGHM operators, as well as their weighted variants, and examines limited properties and detailed instances of the proposed AOs. In Section 5, a novel MADM model is established on these new AOs. A numerical example of enterprise resource planning is provided in order to verify the unassailability and compensations of the initiated approach. Finally, in Section 6, a brief conclusion is provided.


## 2. Preliminaries

In this portion, various essential conceptions namely, cubic set (CS), the Heronian mean (HM) operator, and their basic characteristics are reviewed.


### 2.1. The Cubic Set and Its Operational Laws

Definition 1.
(see [11]). Let U be a universe of discourse set. A CS is classified and mathematically indicated as follows:(1)CS=u″,Iu″,fu″for allu″∈U,where I″=IL″,IU″andf″ are IVFS and FS, respectively. For computational affluence, we shall label a cubic number (CN) by the ordered pair cn=IL,IU,f, where IL,IUandf are IVFN and FN, respectively. If f∈IL,IU, then it is said to be an internal cubic number and if f∉IL,IU, then it is said to be an external cubic number.
The OLs for CS were classified by Jun [11] and are established below as follows:Definition 2.
(see [11]). Let CS1andCS2 be any two CSS. Then,(2)CS1⊆CS2iffI1L″≤I2L″,I1U″≤I2U″,f1″≥f2″,for all″∈U,CS1=CS2iffI1L″=I2L″,I1U″=I2U″,f1″=f2″,CS1∪CS2=″,maxI1L″,I2L″,maxI1L″,I2U″,minf1″,f2″for all″∈U,CS1∩CS2=″,minI1L″,I2L″,minI1L″,I2U″,maxf1″,f2″for all″∈U.
For the comparison of two CNscn1andcn2 the score, accuracy functions, and comparison rules are designated as follows:(3)SoFcn1=I1L″+I1L″−f1″3,(4)ArYcn1=I1L″+I1U″+f1″3.
For comparison of two CNs, the comparison rules are listed below.(i)
IfSoFcn1>SoFcn2, then cn1 is superior to cn2 and is labelled by cn1>cn2(ii)
IfSoFcn1=SoFcn2and ArYcn1>ArYcn2, then cn1 is superior to cn2 and is labelled by cn1>cn2(iii)
IfSoFcn1=SoFcn2and ArYcn1=ArYcn2, then cn1 is same to cn2 and is labelled by cn1=cn2Definition 3.
(see [12]). Let the two CNs be cn1=I1L,I1U,f1 and cn2=I2L,I2U,f2. Then, the OLs for CNs are identified as go after:(5)cn1⊕cn2=I1L+I2L−I1LI2L,I1U+I2U−I1UI2U,f1f2;cn1⊗cn2=I1LI2L,I1UI2U,f1+f2−f1f2;ℑcn1=1−1−I1Lℑ,1−1−I1Uℑ,f1ℑ;ℑ>0,cn1ℑ=I1Lℑ,I1Uℑ,1−1−f1ℑ;ℑ>0,cn1c=1−I1U,1−I1L,1−f1.Definition 4.
Assume that the two CNs becn1=I1L,I1U,f1 and cn2=I2L,i2U,f2. Then, the normalized Hamming distance among cn1 and cn2 is labelled as go after:(6)DNE¯¯cn1,cn2=13I1L−I2L+I1U−I2U+f1−f2.


### 2.2. The PA Operator

Yagar [13] originated the acuity of the PA operator which is the vital AOs. The PA operator concentrated a variety of ineffectual consequences of inanely high or awkwardly low sentiments stated by professionals. The anticipated PA operator can merge a set of crisp numbers where the weighting vector is simply on the input information and is classified as go after.Definition 5.
(see [13]). Let Üuu=1,2,…,Θ be a faction of nonnegative real numbers. The PA operator is a function delineated by(7)PAÜ1,Ü2,…,ÜΘ=∑u=1Θ1+TÜuÜu∑u=1Θ1+TÜu,where(8)TÜu=∑v=1v≠uΘSupÜu,Üv,and SupÜu,Üv is the support degree for Üu from Üv, which meets the following axioms. (1) SupÜu,Üv∈0,1, (2) SupÜu,Üv=SupÜv,Üu, and (3) SupÜu,Üv≥SupA,M, if Üu−Üv<A−M.


### 2.3. Heronian Mean (HM) Operator

HM [16] operator is one of the substantial tools for aggregation, which can exemplify the interrelations of the input elements, and is demarcated as go after.Definition 6.
(see [16]). Let ∇=0,1,p,q≥0,HMp,q:∇m⟶∇, if HMp,q satisfies(9)HMp,qbc1,bc2,…,bcm=2m2+m∑G=1m∑H=imbcGpbcHp1/p+q.
Then, the mappingHMp,q is suspected to be an HM operator with constraints. The HM operator should certify the qualities of idempotency, boundedness, and monotonicity.


## 2.1. The Cubic Set and Its Operational Laws

Definition 1.
(see [11]). Let U be a universe of discourse set. A CS is classified and mathematically indicated as follows:(1)CS=u″,Iu″,fu″for allu″∈U,where I″=IL″,IU″andf″ are IVFS and FS, respectively. For computational affluence, we shall label a cubic number (CN) by the ordered pair cn=IL,IU,f, where IL,IUandf are IVFN and FN, respectively. If f∈IL,IU, then it is said to be an internal cubic number and if f∉IL,IU, then it is said to be an external cubic number.
The OLs for CS were classified by Jun [11] and are established below as follows:Definition 2.
(see [11]). Let CS1andCS2 be any two CSS. Then,(2)CS1⊆CS2iffI1L″≤I2L″,I1U″≤I2U″,f1″≥f2″,for all″∈U,CS1=CS2iffI1L″=I2L″,I1U″=I2U″,f1″=f2″,CS1∪CS2=″,maxI1L″,I2L″,maxI1L″,I2U″,minf1″,f2″for all″∈U,CS1∩CS2=″,minI1L″,I2L″,minI1L″,I2U″,maxf1″,f2″for all″∈U.
For the comparison of two CNscn1andcn2 the score, accuracy functions, and comparison rules are designated as follows:(3)SoFcn1=I1L″+I1L″−f1″3,(4)ArYcn1=I1L″+I1U″+f1″3.
For comparison of two CNs, the comparison rules are listed below.(i)
IfSoFcn1>SoFcn2, then cn1 is superior to cn2 and is labelled by cn1>cn2(ii)
IfSoFcn1=SoFcn2and ArYcn1>ArYcn2, then cn1 is superior to cn2 and is labelled by cn1>cn2(iii)
IfSoFcn1=SoFcn2and ArYcn1=ArYcn2, then cn1 is same to cn2 and is labelled by cn1=cn2Definition 3.
(see [12]). Let the two CNs be cn1=I1L,I1U,f1 and cn2=I2L,I2U,f2. Then, the OLs for CNs are identified as go after:(5)cn1⊕cn2=I1L+I2L−I1LI2L,I1U+I2U−I1UI2U,f1f2;cn1⊗cn2=I1LI2L,I1UI2U,f1+f2−f1f2;ℑcn1=1−1−I1Lℑ,1−1−I1Uℑ,f1ℑ;ℑ>0,cn1ℑ=I1Lℑ,I1Uℑ,1−1−f1ℑ;ℑ>0,cn1c=1−I1U,1−I1L,1−f1.Definition 4.
Assume that the two CNs becn1=I1L,I1U,f1 and cn2=I2L,i2U,f2. Then, the normalized Hamming distance among cn1 and cn2 is labelled as go after:(6)DNE¯¯cn1,cn2=13I1L−I2L+I1U−I2U+f1−f2.


## 2.2. The PA Operator

Yagar [13] originated the acuity of the PA operator which is the vital AOs. The PA operator concentrated a variety of ineffectual consequences of inanely high or awkwardly low sentiments stated by professionals. The anticipated PA operator can merge a set of crisp numbers where the weighting vector is simply on the input information and is classified as go after.Definition 5.
(see [13]). Let Üuu=1,2,…,Θ be a faction of nonnegative real numbers. The PA operator is a function delineated by(7)PAÜ1,Ü2,…,ÜΘ=∑u=1Θ1+TÜuÜu∑u=1Θ1+TÜu,where(8)TÜu=∑v=1v≠uΘSupÜu,Üv,and SupÜu,Üv is the support degree for Üu from Üv, which meets the following axioms. (1) SupÜu,Üv∈0,1, (2) SupÜu,Üv=SupÜv,Üu, and (3) SupÜu,Üv≥SupA,M, if Üu−Üv<A−M.


## 2.3. Heronian Mean (HM) Operator

HM [16] operator is one of the substantial tools for aggregation, which can exemplify the interrelations of the input elements, and is demarcated as go after.Definition 6.
(see [16]). Let ∇=0,1,p,q≥0,HMp,q:∇m⟶∇, if HMp,q satisfies(9)HMp,qbc1,bc2,…,bcm=2m2+m∑G=1m∑H=imbcGpbcHp1/p+q.
Then, the mappingHMp,q is suspected to be an HM operator with constraints. The HM operator should certify the qualities of idempotency, boundedness, and monotonicity.


## 3. Schweizer–Sklar Operational Laws for Cubic Numbers

In this portion, the SS OLs are commenced for CNs based on SSTN and SSTCN, and numerous underlying characteristics of SS OLs for CNs are explored.The SSTN and SSTCN [28, 29] are recognized as go after:(10)TSrSrÜ,Õ=ÜЖ+ÕЖ−11/Ж,TSrSr∗Ü,Õ=1−1−ÜЖ+1−ÕЖ−11/Ж,where Ж<0,Ü,Õ∈0,1.Additionally, whenЖ=0, we have TЖÜ,Õ=ÜÕ and TЖ∗Ü,Õ=Ü+Õ−ÜÕ. That is, SS TN and TCN reduce to algebraic TN and TCN.Now, based on SSTNTЖÜ,Õ and TCN TЖ∗Ü,Õ, we can permit the following definition of SRSR ALS of CNs.Definition 7.
Let the three CNs becn=IL,IU,f,cn1=I1L,I1U,f1, and cn2=I2L,I2U,f2. Then, the SS ALS for CNs are classified as follows:(11)cn1⊕cn2=1−1−I1LЖ+1−I2LЖ−11/Ж,1−1−I1UЖ+1−I2UЖ−11/Ж,f1Ж+f2Ж−11/Ж,(12)cn1⊗cn2=〈I1LЖ+I2LЖ−11/Ж,I1UЖ+I2UЖ−11/Ж,1−1−f1Ж+1−f2Ж−11/Ж,(13)ςcn=1−ς1−ILЖ−ς−11/Ж,1−ς1−IUЖ−ς−11/Ж,ςfЖ−ς−11/Ж,(14)cnς=ςILЖ−ς−11/Ж,ςIUЖ−ς−11/Ж,1−ς1−fЖ−ς−11/Ж.
Moreover, some worthy properties of the operational laws can be easily achieved.Theorem 1.
Letcn=IL,IU,f,cn1=I1L,I1U,f1 and cn2=I2L,I2U,f2 be any three CNs. Then,(15)cn1⊕SScn2=cn2⊕SScn1;cn1⊗SScn2=cn2⊗SScn1;λcn1⊕SScn2=λcn1⊕SSλcn2,λ≥0;λ1cn⊕SSλ2cn=λ1+λ2cn;λ1,λ2≥0;cnλ1⊗SScnλ2=cnλ1+λ2,λ1,λ2≥0;cn1λ⊗SScn2λ=cn1⊗SScn2λ,λ≥0.Proof.
The proof of 1 and 2 are easy, so we can only prove the remaining formulas.(16)λcn1⊕SScn2=1−1−I1LЖ+1−I2LЖ−11/Ж,1−1−I1UЖ+1−I2UЖ−11/Ж,f1Ж+f2Ж−11/Ж·λ=1−λ1−I1LЖ+1−I2LЖ−1−λ−11/Ж,1−λ1−I1UЖ+1−I2UЖ−1−λ−11/Жλf1Ж+f2Ж−1−λ−11/Ж,,=1−λ1−I1LЖ+λ1−I2LЖ−λ−λ−11/Ж,1−λ1−I1UЖ+λ1−I2UЖ−λ−λ−11/Ж,λf1Ж+λf2Ж−λ−λ−11/Ж,=1−λ1−I1LЖ+λ1−I2LЖ−2λ+11/Ж,1−λ1−I1UЖ+λ1−I2UЖ−2λ+11/Ж,λf1Ж+λf2Ж−2λ+11/Ж.
In the meantime, we can acquire that(17)λcn1⊕SSλcn2=1−λ1−I1LЖ−λ−1+λ1−I2LЖ−λ−1−11/Ж,1−λ1−I1UЖ−λ−1+λ1−I2UЖ−λ−1−11/Ж,λf1Ж−λ−1+λf2Ж−λ−1−11/Ж,=1−λ1−I1LЖ+λ1−I2LЖ−2λ−1−11/Ж,1−λ1−I1UЖ+λ1−I2UЖ−2λ−1−11/Ж,λf1Ж+λf2Ж−2λ−1−11/Ж,=1−λ1−I1LЖ+λ1−I2LЖ−2λ+11/Ж,1−λ1−I1UЖ+λ1−I2UЖ−2λ+11/Ж,λf1Ж+λf2Ж−2λ+11/Ж,=λcn1⊕SScn2.
Therefore,λcn1⊕SScn2=λcn1⊕SSλcn2,λ≥0; holds.(18)λ1cn⊕SSλ2cn=1−λ11−ILЖ−λ1−1+λ21−ILЖ−λ2−1−11/Ж,1−λ11−IUЖ−λ1−1+λ21−IUЖ−λ2−1−11/Ж1/Ж,λ1fЖ−λ1−1+λ2fЖ−λ2−1−1=1−λ¯11−ILЖ+λ21−ILЖ−λ1+λ2+11/Ж⋅1−λ11−IUЖ+λ21−IUЖ−λ1+λ2+11/Жλ1fЖ+λ2fЖ−λ1+λ2+11/Ж,=1−λ1+λ21−ILЖ−λ1+λ2+11/Ж,1−λ1+λ21−IUЖ−λ1+λ2+11/Ж,λ1+λ2fЖ−λ1+λ2+11/Ж=λ1+λ2cn.
The proofs of the other two parts are the same as the above two parts. Therefore, the proofs are omitted here.


## 4. The Cubic Schweizer–Sklar Power HM Operators

### 4.1. The CSSPHM and CSSPGHM Operators

In this segment, numerous new AOs namely, CSSPHM and CSSPGHM operators are anticipated by combining HM and GHM operators with PA operators to anticipate.Definition 8.
Letcnì=IìL,IìU,fì,ì=1,2,…,a¯ be a faction of CNs, and then the cubic Schweizer–Sklar power HM (CSSPHM) operator is clarified as follows:(19)CSSPHMÜ,Õcn1,cn2,…,cna¯=2a¯a¯+1∑ì,j=1j=ìa¯a¯1+Tcnì∑z=1a¯1+TcnzcnìÜ⊗SSa¯1+Tcnj∑z=1a¯1+TcnzcnjÕ1/Ü+Õ,where Ü,Õ≥0,Tcni=∑j=1j≠ìa¯supcni,cnj,supcni,cnj=1−DNE¯¯cni,cnj, and DNE¯¯cni,cnj can be figured by (6).
LetŸi=1+Tcni/∑z=1a¯1+Tcnz, then the definition of CSSPHM operator is correspondent to the following type:(20)CSSPHMÜ,Õcn1,cn2,…,cn†=2a¯a¯+1∑i,j=1j=ia¯a¯ŸicniÕ⊗SrSra¯ŸjcnjÜ1/Ü+Õ.Theorem 2.
LetÜ≥0,Õ≥0 and G,H grab no more than one value of 0 at a time, cni=IiL,IiU,fi be a set of CNs. Then, exploiting the CSSPHM operator, their merged values are CNs, and(21)CSSPHMÜ,Õcn1,cn2,…,cn†=1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−IiLℵ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjLЖ−a¯Ÿj−11/ЖЖ−Ü−+Õ1/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjUЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1−1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ŸifiЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯ŸjfjЖ−n⋄j−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж.Proof.
Firstly, we need to prove the following equation:(22)∑i,j=1j=ia¯a¯ŸicniÜ⊗SSa¯ŸjcnjÕ=1−∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−IiLЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjLЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ−a¯a¯+12−11/Ж,1−∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjUЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ−a¯a¯+12−11/Ж,∑i,j=1j=ia¯1−Ü1−a¯ŸifiЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯ŸjfjЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ−a¯a¯+12−11/Ж.
By SS OLs defined in Definition7, we get(23)a¯Ÿicni=1−a¯Ÿi1−IiLЖ−a¯Ÿi−11/Ж,1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/Ж,a¯ŸifiЖ−a¯Ÿi−11/Ж,and(24)a¯ŸicniÜ=Ü1−a¯Ÿi1−IiLЖ−a¯Ÿi−11/ЖЖ−Ü−11/Ж,Õ1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/ЖЖ−Ü−11/Ж,1−Ü1−a¯ŸifiЖ−a¯Ÿi−11/ЖЖ−Ü−11/Ж.
Similarly,(25)a¯Ÿjcnj=1−a¯Ÿj1−IjLЖ−a¯Ÿj−11/Ж,1−a¯Ÿj1−IjUЖ−a¯Ÿj−11/Ж,a¯ŸjfjЖ−a¯Ÿj−11/Ж,a¯ŸjcnjÕ=Õ1−a¯Ÿj1−IjLЖ−a¯Ÿj−11/ЖЖ−Õ−11/Ж,Õ1−a¯Ÿj1−IjUЖ−a¯Ÿj−11/ЖЖ−Õ−11/Ж,1−Õ1−a¯ŸjfjЖ−a¯Ÿj−11/ЖЖ−Õ−11/Ж.
So,(26)a¯ŸicniÜ⊗SSa¯ŸjcnjÕ=Ü1−a¯Ÿi1−IiLЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjLЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/Ж,Ü1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjUЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/Ж,1−Ü1−a¯ŸifiЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯ŸjfjЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/Ж,(1)
Whena¯=2 by Equation (17) and Equation (25), we have(27)∑i,j=1j=i2a¯ŸicniÜ⊗SSa¯ŸjcnjÕ=2Ÿ1cn1Ü⊗SS2Ÿ1cn1Õ⊕SS2Ÿ1cn1Ü⊗SS2Ÿ2cn2Õ⊕SS2Ÿ2cn2Ü⊗SS2Ÿ2cn2Õ=Ü1−2Ÿ11−I1LЖ−2Ÿ1−11/ЖЖ+Õ1−2Ÿ11−I1LЖ−2Ÿ1−11/ЖЖ−Ü−Õ+11/Ж,Ü1−2Ÿ11−I1UЖ−2Ÿ1−11/ЖЖ+Õ1−2Ÿ11−I1UЖ−2Ÿ1−11/ЖЖ−Ü−Õ+11/Ж,1−Ü1−2Ÿ1f1Ж−2Ÿ1−11/ЖЖ+Õ1−2Ÿ1f1Ж−2Ÿ1−11/ЖЖ−Ü−Õ+11/Ж〉⊕SSÜ1−2Ÿ11−I1LЖ−2Ÿ1−11/ЖЖ+Õ1−2Ÿ21−I2LЖ−2Ÿ2−11/ЖЖ−Ü−Õ+11/ЖÜ1−2Ÿ11−I1UЖ−2Ÿ1−11/ЖЖ+Õ1−2Ÿ21−I2UЖ−2Ÿ2−11/ЖЖ−Ü−Õ+11/Ж,1−Ü1−2Ÿ1f1Ж−2Ÿ1−11/ЖЖ+Õ1−2Ÿ2f2Ж−2Ÿ2−11/ЖЖ−Ü−Õ+11/Ж〉⊕SSÜ1−2Ÿ21−I2LЖ−2Ÿ2−11/ЖЖ+Õ1−2Ÿ21−I2LЖ−2Ÿ2−11/ЖЖ−Ü−Õ+11/ЖÜ1−2Ÿ21−I2UЖ−2Ÿ2−11/ЖЖ+Õ1−2Ÿ21−I2UЖ−2Ÿ2−11/ЖЖ−Ü−Õ+11/Ж,1−Ü1−2Ÿ2f2Ж−2Ÿ2−11/ЖЖ+Õ1−2Ÿ2f2Ж−2Ÿ2−11/ЖЖ−Ü−Õ+11/Ж〉.By using Equation (17), we get(28)∑i,j=1j=i2a¯ŸicniÜ⊗SSa¯ŸjcnjÕ=1−∑i,j=1j=i21−Ü1−a¯Ÿi1−IiLЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjLЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11ЖЖ−a¯a¯+12−11/Ж,1−∑i,j=1j=i21−Ü1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjUЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ−a¯a¯+12−11/Ж,∑i,j=1j=i21−Ü1−a¯ŸifiЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯ŸjfjЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ−a¯a¯+12−11/Ж.That is (22) is true a¯=2.(2)
Let us assume that Equation (24) is true a¯=z.(29)∑i,j=1j=iza¯ŸicniÜ⊗SSa¯ŸjcnjÕ=1−∑i,j=1j=iz1−Ü1−zŸi1−IiLЖ−zŸi−11/ЖЖ+Õ1−zŸj1−IjLЖ−zŸj−11/ЖЖ−Ü−Õ+11/ЖЖ−zz+12−11/Ж,1−∑i,j=1j=iz1−Ü1−zŸi1−IiUЖ−zŸi−11/ЖЖ+Õ1−zŸj1−IjUЖ−zŸj−11/ЖЖ−Ü−Õ+11/ЖЖ−zz+12−11/Ж,∑i,j=1j=iz1−Ü1−zŸifiЖ−zŸi−11/ЖЖ+Õ1−zŸjfjЖ−zŸj−11/ЖЖ−Ü−Õ+11/ЖЖ−zz+12−11/Ж.
Furthermore, whena¯=z+1(30)∑i,j=1j=iz+1a¯ŸicniÜ⊗SSa¯ŸjcnjÕ=∑i,j=1j=iz+1z+1ŸicniÜ⊗SSz+1ŸjcnjÕ⊕SS∑i=1zz+1ŸicniÜ⊗SSz+1Ÿz+1cnz+1Õ⊕SSz+1Ÿz+1cnz+1Ü⊗SSz+1Ÿz+1cnz+1Õ.
Firstly, we will show that(31)∑i=1zz+1ŸicniÜ⊗SSz+1Ÿz+1cnz+1Õ=1−∑i,j=1j=iz1−Ü1−z+1Ÿi1−IiLЖ−z+1Ÿi−11/ЖЖ+Õ1−z+1Ÿz+11−Iz+1LЖ−z+1Ÿz+1−11/ЖЖ−Ü−Õ+11/ЖЖ−z+1z+22−11/Ж,1−∑i,j=1j=iz1−Ü1−z+1Ÿi1−IiUЖ−z+1Ÿi−11/ЖЖ+Õ1−z+1Ÿz+11−Iz+1UЖ−z+1Ÿz+1−11/ЖЖ−Ü−Õ+11/ЖЖ−z+1z+22−11/Ж,∑i,j=1j=iz1−Ü1−z+1ŸifiЖ−z+1Ÿi−11/ЖЖ+Õ1−z+1Ÿz+1fz+1Ж−z+1Ÿz+1−11/ЖЖ−Ü−Õ+11/ЖЖ−z+1z+22−11/Ж.
We shall prove (31), on mathematical induction on z.(i)
Forz=2, we have(32)∑i=12z+1ŸicniÜ⊗SSz+1Ÿz+1cnz+1Õ=3Ÿ1cn1Ü⊗SS3Ÿ3cn3Õ⊕SS3Ÿ2cn2Ü⊗SS3Ÿ3cn3Õ=Ü1−3Ÿ11−I1LЖ−3Ÿ1−11/ЖЖ+Õ1−3Ÿ31−I3LЖ−3Ÿ3−11/ЖЖ−Ü−Õ+11/Ж,Ü1−3Ÿ11−I1UЖ−2Ÿ1−11/ЖЖ+Õ1−3Ÿ31−I3UЖ−3Ÿ3−11/ЖЖ−Ü−Õ+11/Ж,1−Ü1−3Ÿ1f1Ж−2Ÿ1−11/ЖЖ+Õ1−3Ÿ3f3Ж−3Ÿ3−11/ЖЖ−Ü−Õ+11/Ж〉⊕SSÜ1−3Ÿ21−I2LЖ−3Ÿ2−11/ЖЖ+Õ1−3Ÿ31−I3LЖ−3Ÿ3−11/ЖЖ−Ü−Õ+11/Ж,Ü1−3Ÿ21−I2UЖ−3Ÿ2−11/ЖЖ+Õ1−3Ÿ31−I3UЖ−3Ÿ3−11ЖЖ−Ü−Õ+11/Ж,1−Ü1−3Ÿ2f2Ж−3Ÿ2−11/ЖЖ+Õ1−3Ÿ3f3Ж−3Ÿ3−11/ЖЖ−Ü−Õ+11/Ж〉.By using (11), we get(33)=1−∑i=121−Ü1−3Ÿi1−IiLЖ−3Ÿi−11/ЖЖ+Õ1−3Ÿ31−I3LЖ−3Ÿ3−11/ЖЖ−Ü−Õ+11/ЖЖ−33+12−11/Ж,1−∑i=121−Ü1−3Ÿi1−IiUЖ−3Ÿi−11/ЖЖ+Õ1−3Ÿ31−I3UЖ−3Ÿ3−11/ЖЖ−Ü−Õ+11/ЖЖ−33+12−11/Ж,∑i=121−Ü1−3ŸifiЖ−3Ÿi−11/ЖЖ+Õ1−3Ÿ3f3Ж−3Ÿ3−11/ЖЖ−Ü−Õ+11/ЖЖ−33+12−11/Ж.(ii)
Suppose that Equation (28) is true z=ő.(34)∑i=1őő+1ŸicniÜ⊗SSő+1Ÿő+1cnő+1Õ=1−∑i=1ő1−Ü1−ő+1Ÿi1−IiLЖ−ő+1Ÿi−11/ЖЖ+Õ1−ő+1Ÿő+11−Iő+1LЖ−ő+1Ÿő+1−11/ЖЖ−Ü−Õ+11/ЖЖ−ő+1ő+22−11/Ж,1−∑i=1ő1−Ü1−ő+1Ÿi1−IiUЖ−ő+1Ÿi−11/ЖЖ+Õ1−ő+1Ÿő+11−Iő+1UЖ−ő+1Ÿő+1−11/ЖЖ−Ü−Õ+11/ЖЖ−ő+1ő+22−11/Ж,∑i=1ő1−Ü1−ő+1ŸifiЖ−ő+1Ÿi−11/ЖЖ+Õ1−ő+1Ÿő+1fő+1Ж−ő+1Ÿő+1−11/ЖЖ−Ü−Õ+11/ЖЖ−ő+1ő+22−11/Ж.
Then, whenz=ő+1, we have(35)∑i=1ő+1ő+2ŸicniÜ⊗SSő+2Ÿő+2cnő+2Õ=∑I=1ő+1ő+2ŸicniÜ⊗SSő+2Ÿő+2cnő+2Õ⊕SSő+2Ÿő+1cnő+1Ü⊗SSő+2Ÿő+2cnő+2Õ=1−∑i=1ő1−Ü1−ő+2Ÿi1−IiLЖ−ő+2Ÿi−11/ЖЖ+Õ1−ő+2Ÿő+21−Iő+2LЖ−ő+2Ÿő+2−11/ЖЖ−Ü−Õ+11/ЖЖ−ő+2ő+32−11/Ж,1−∑i=1ő1−Ü1−ő+2Ÿi1−IiUЖ−ő+2Ÿi−11/ЖЖ+Õ1−ő+2Ÿő+21−Iő+2UЖ−ő+2Ÿő+2−11/ЖЖ−Ü−Õ+11/ЖЖ−ő+2ő+32−11/Ж,∑i=1ő1−Ü1−ő+2ŸifiЖ−ő+2Ÿi−11/ЖЖ+Õ1−ő+2Ÿő+1fő+1Ж−ő+2Ÿő+1−11/ЖЖ−Ü−Õ+11/ЖЖ−ő+2ő+32−11/Ж⊕SS〈Ü1−ő+2Ÿő+11−Iő+1LЖ−ő+2Ÿő+1−11/ЖЖ+Õ1−ő+2Ÿő+21−Iő+2LЖ−ő+2Ÿő+2−11/ЖЖ−Ü−Õ+11/Ж,Ü1−ő+2Ÿő+11−Iő+1UЖ−ő+2Ÿő+1−11/ЖЖ+Õ1−ő+2Ÿő+21−Iő+2UЖ−ő+2Ÿő+1−11/ЖЖ−Ü−Õ+11/Ж,1−Ü1−ő+2Ÿő+1fő+1Ж−ő+2Ÿő+1−11/ЖЖ+Õ1−ő+2Ÿő+2fő+2Ж−ő+2Ÿő+2−11/ЖЖ−Ü−Õ+11/Ж〉.=1−∑i=1ő+11−Ü1−ő+2Ÿi1−IiLЖ−ő+2Ÿi−11/ЖЖ+Õ1−ő+2Ÿő+21−Iő+2LЖ−ő+2Ÿő+2−11/ЖЖ−Ü−Õ+11/ЖЖ−ő+2ő+32−11/Ж,1−∑i=1ő+11−Ü1−ő+2Ÿi1−IiUЖ−ő+2Ÿi−11/ЖЖ+Õ1−ő+2Ÿő+21−Iő+2UЖ−ő+2Ÿő+2−11/ЖЖ−Ü−Õ+11/ЖЖ−ő+2ő+32−11/Ж,∑i=1ő+11−Ü1−ő+2ŸifiЖ−ő+2Ÿi−11/ЖЖ+Õ1−ő+2Ÿő+2fő+2Ж−ő+2Ÿő+2−11/ЖЖ−Ü−Õ+11/ЖЖ−ő+2ő+32−11/Ж.
Therefore, (31) is true z=ő+1. Hence, (31) is true for all z. Similarly, we can prove other parts of (30). So, (30) becomes(36)∑i,j=1j=iz+1z+1ŸicniÜ⊗SSz+1ŸjcnjÕ=1−∑i,j=1j=iz+11−Ü1−z+1Ÿi1−IiLЖ−z+1Ÿi−11/ЖЖ+Õ1−z+1Ÿj1−IjLЖ−z+1Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ−z+1z+22−11/Ж,1−∑i,j=1j=iz+11−Ü1−z+1Ÿi1−IiUЖ−z+1Ÿi−11/ЖЖ+Õ1−z+1Ÿj1−IjUЖ−z+1Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ−z+1z+22−11/Ж,∑i,j=1j=iz+11−Ü1−z+1ŸifiЖ−z+1Ÿi−11/ЖЖ+Õ1−z+1ŸjfjЖ−z+1Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ−z+1z+22−11/Ж.
Hence, (22) is true a¯=z+1. Therefore, (22) is true for all †.
By (22), we can say that (14), is right. From (22), and the SS operational laws for CNs, we have(37)2a¯a¯+1∑i,j=1j=ia¯a¯ŸicniÜ⊗SSa¯ŸjcnjÕ=1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−IiLЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjLЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/Ж,1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjUЖ−a¯Ÿj−11ЖЖ−Ü−Õ+11/ЖЖ1/Ж,2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ŸifiЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯ŸjfjЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/Ж.
Furthermore,(38)2a¯a¯+1∑i,j=1j=ia¯a¯ŸicniÜ⊗SSa¯ŸjcnjÕ1/Ü+Õ=1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−IiLЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjLЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjUЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1−1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ŸifiЖ−a¯Ÿi−11ЖЖ+Õ1−a¯ŸjfjЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/ℵ.
Hence,(39)CSSPHMÜ,Õcn1,cn2,…,cn†=1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−IiLЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjLЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjUЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1−1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ŸifiЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯ŸjfjЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж.
Now, we confer some advantageous possessions of the projected CSrSrPHOM operator.Theorem 3 (idempotency).
LetÜ≥0,Õ≥0 and Ü,Õ capture no more than one value of 0 at a time, cni=IiL,IiU,fii=1,2,…,a¯ be a faction of CNs, and cni=IiL,IiU,fsi=cn=IL,IU,fi=1,2,…,a¯. Then,(40)CSSPHMÜ,Õcn1,cn2,…,cn†=cn=IL,IU,f.Proof.
Sincecni=IiL,IiU,fi=cn=IL,IU,fi=1,2,...,a¯, we have supcng,cnz=1∀g,z=1,2,…,a¯ so ⋄g=1/a¯∀g=1,2,…,a¯ and a¯Ÿg=a¯1/a¯=1∀g=1,2,…,a¯.
Then,(41)CSSPHMÜ,Õcn1,cn2,…,cna¯=CSSPHMÜ,Õcn,cn,…,cn=1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−1−ILЖ1/ЖЖ+Õ1−1−ILЖ1/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−1−IUЖ1/ЖЖ+Õ1−1−IUЖ1/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1−1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−fЖ+Õ1−fЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,=1Ü+ÕÜ+ÕILЖ−Ü−Õ+1−1Ü+Õ+11/Ж,1Ü+ÕÜ+ÕIUЖ−Ü−Õ+1−1Ü+Õ+11/Ж,1−1Ü+ÕÜ+Õ1−fЖ−Ü−Õ+1−1Ü+Õ+11/Ж=IL,IU,f=cn.Theorem 4.
(commutativity). Letcn1′,cn2′,…,cna¯′ be any permutation of cn1,cn2,…,cna¯. Then,(42)CSSPHMÜ,Õcn1′,cn2′,…,cna¯′=CSSPHMÜ,Õcn1,cn2,…,cna¯.Proof.
Sincecn1′,cn2′,…,cna¯′ is any permutation of cn1,cn2,…,cna¯, (43)CSSPHMÜ,Õcn1,cn2,…,cna¯=2a¯a¯+1∑i,j=1j=ia¯a¯ŸicniÜ⊗SSa¯ŸjcnjÕ1/Ü+Õ=2a¯a¯+1∑i,j=1j=ia¯a¯Ÿicnj′Ü⊗SSa¯Ÿjcnj′Õ1/Ü+Õ=CSSPHMÜ,Õcn1′,cn2′,…,cna¯′.Theorem 5 (boundedness).
Letcni=IiL,IiU,fii=1,2,…,a¯ be a faction of CNs, cn−=miniIiL,miniIiU,maxifi and cn+=maxiIiL,maxiIiU,minifi. Then,(44)cn−≤CSSPHMÜ,Õcn1,cn2,…,cna¯≤cn+.Proof.
By the judgement technique in Definition2, we have cni≥cnj′, and then based on Theorems 3 and 4, we have(45)CSSPHMÜ,Õcn1,cn2,…,cna¯≥CSSPHMÜ,Õcn1−,cn2−,…,cna¯−=cn−.
Similarly, we can have(46)CSSPHMÜ,Õcn1,cn2,…,cna¯≤CSSPHMÜ,Õcn1+,cn2+,…,cna¯+=cn+.
So, we have(47)CSSPHEMÜ,Õcn1−,cn2−,…,cna¯−≤CSSPHMÜ,Õcn1,cn2,…,cna¯≤CSSPHMÜ,Õcn1+,cn2+,…,cna¯+.
Hence,(48)cn−≤CSrSrPHOMÜ,Õcn1,cn2,…,cn†≤cn+.
By apportioning different values to the constraintsЖ,Ü and Õ, numerous different circumstances of the CSSPHM operator can be attained, and are specified below as follows:Case 1. IfЖ=0, then the CSSPHM operator relapses to the CPHM operator, which can be stated as follows:(49)CSSPHMÜ,Õcn1,cn2,…,cna¯=2a¯a¯+1∑i,j=1j=ia¯a¯ŸicniÜ⊗a¯ŸjcnjÕ1/Ü+Õ=1−∏i,j=1j=ia¯1−1−1−IiLa¯ŸiÜ1−1−IjLa¯ŸjÕ2/a¯a¯+11/Ü+Õ,1−∏i,j=1j=ia¯1−1−1−IiUa¯ŸiÜ1−1−IjUa¯ŸjÕ2/a¯a¯+11/Ü+Õ,1−1−∏i,j=1j=ia¯1−1−fia¯ŸiÜ1−fja¯ŸjÕ2/a¯a¯+11/Ü+Õ.Case 2. IfÕ⟶0, then the CSSPHM operator reverts to the cubic descending PA operator and is quantified as follows:(50)CSSPHMÜ,Õcn1,cn2,…,cna¯=limÕ⟶02a¯a¯+1∑i,j=1j=ia¯a¯ŸicniÜ⊗SSa¯ŸjcnjÕ1/Ü+Õ=2a¯a¯+1∑i=1a¯a¯+1−ia¯ŸicniÜ1/Ü=1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−Ü1−a¯Ÿi1−IiLЖ−a¯Ÿi−11/ЖЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж,1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−Ü1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/ЖЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж,1−1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−Ü1−a¯ŸifiЖ−a¯Ÿi−11ЖЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж.Case 3. IfÜ⟶0, then the CSSPHM operator reverts to the cubic ascending PA operator and is quantified as follows:(51)CSSPHM0,Õcn1,cn2,…,cna¯=limÜ⟶02a¯a¯+1∑i,j=1j=ia¯a¯ŸicniÜ⊗SSa¯ŸjcnjÕ1/Ü+Õ=2a¯a¯+1∑i=1a¯ia¯ŸicniÕ1/Õ=1Õ1−2a¯a¯+1∑i=1a¯i1−Õ1−a¯Ÿi1−IiLЖ−a¯Ÿi−11/ЖЖ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж,1Õ1−2a¯a¯+1∑i=1a¯i1−Õ1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/ЖЖ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж,1−1Õ1−2a¯a¯+1∑i=1a¯i1−Õ1−a¯ŸifiЖ−a¯Ÿi−11/ЖЖ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж.Case 4. IfÕ⟶0 and supcni,cnj=hh∈0,1for alli≠j, then the CSSPHM operator reverts to the cubic linear descending WA operator and is stated as follows:(52)CSSPHMÜ,0cn1,cn2,…,cna¯=limÕ⟶02a¯a¯+1∑i,j=1j=1a¯a¯ŸicniÜ⊗SSa¯ŸjcnjÕ1/Ü+Õ=2a¯a¯+1∑i=1a¯a¯+1−icniÜ1/Ü=1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−ÜIiLЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж,1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−ÜIiUЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж,1−1Ü1−2a¯a¯+1∑i=1†a¯+1−i1−Ü1−fiЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж.Certainly, the significant degree of CNscniÜi=1,2,…,a¯ is a¯,a¯−1,…,1.Case 5. IfÜ⟶0 and supcni,cnj=hh∈0,1for alli≠j, then the CSSPHOM operator reverts to the cubic linear ascending WA operator and is stated as follows:(53)CSSPHM0,Õcn1,cn2,…,cna¯=limÜ⟶02a¯a¯+1∑i,j=1j=1a¯a¯ŸicniÜ⊗SSa¯ŸjcnjÕ1/Ü+Õ=2a¯a¯+1∑i=1a¯icniÕ1/Õ=1Õ1−2a¯a¯+1∑i=1a¯i1−ÕIiLℵ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж,1Õ1−2a¯a¯+1∑i=1a¯i1−ÕIiUЖ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж,1−1Õ1−2a¯a¯+1∑i=1a¯i1−Õ1−fiЖ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж.Case 6. IfÜ=Õ=1/2 and supcni,cnj=hh∈0,1for alli≠j, then the CSSPHM operator reverts to the cubic basic HM operator and is stated as follows:(54)CSSPHM1/2,1/2cn1,cn2,…,cna¯=2a¯a¯+1∑i,j=1j=1a¯cni1/2⊗cnj1/2=1−2a¯a¯+1∑i,j=1j=ia¯1−12IiLЖ+12IjLЖ1/ЖЖ1/Ж,1−2a¯a¯+1∑i,j=1j=ia¯1−12IiUЖ+12IjUЖ1/ЖЖ1/Ж,2a¯a¯+1∑i,j=1j=ia¯1−121−fiЖ+121−fsjЖ1/ЖЖ1/Ж.Case 7. IfÜ=Õ=1, and supcni,cnj=hh∈0,1for alli≠j, then the CSSPHM operator reverts to the cubic linear HM operator, and is stated as follows:(55)CSSPHM1,1cn1,cn2,…,cna¯=2a¯a¯+1∑i,j=1j=1†cni⊗cnj1/2=121−2a¯a¯+1∑i,j=1j=i†1−IiLЖ+IjLЖ−11/ЖЖ1/ЖЖ+121/Ж,121−2a¯a¯+1∑i,j=1j=i†1−IiUЖ+IjUЖ−11/ЖЖ1/ЖЖ+121/Ж1−1−2a¯a¯+1∑i,j=1j=i†1−1−fiЖ+1−fjЖ−11/ЖЖ1/ЖЖ+121/Ж.Definition 9.
Letcni=IiL,IiU,fii=1,2,…,a¯ be a faction of CNs, then the cubic SS power geometric HM (CSSPGHM) operator is explicated as follows:(56)CSSPGHMÜ,Õcn1,cn2,…,cna¯=1Ü+Õ∏i,j=1j=ia¯Ücnia¯1+Tcni/∑z=1a¯1+Tcnz⊕SSÕcnja¯1+Tcnj/∑z=1a¯1+Tcnz2/a¯a¯+1,where Ü,Õ≥0,Tcni=∑j=1j≠ia¯supcni,cnj,supcni,cnj=1−DNE¯¯cni,cnj, and DNE¯¯cni,cnj can be totalized by (6).
LetŸi=1+Tcni/∑z=1a¯1+Tcnz, then the definition of CSSPGHM operator is correspondent to the subsequent formula:(57)CSSPGHMÜ,Õcn1,cn2,…,cna¯=1Ü+Õ∏i,j=1j=ia¯Ücnia¯Ÿi⊕SSÕcnja¯Ÿj2/a¯a¯+1.Theorem 6.
LetÜ≥0,Õ≥0 and Ü,Õ take no more than one value of 0 at a time, cni=IiL,IiU,fi be a faction of CNs. Then, exploiting the CSSPGHM operator, their merged values are CNs, and(58)CSSPGHMÜ,Õcn1,cn2,…,cna¯=1−1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ŸiIiLЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯ŸjIjLЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1−1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ŸiIiUЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯ŸjIjUЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−fiЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−fjЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж.Proof.
Proof of Theorem6 is similar to that of Theorem 2. Therefore, omitted here.
Now, we deliberate some privileged properties of the projected CSSPGHM operator.Theorem 7 (idempotency).
LetÜ≥0,Õ≥0, and Ü,Õ grab no more than one value of 0 at a time, cni=IiL,IiU,fii=1,2,…,a¯ be a faction of CNs, and cni=IiL,IiU,fi=cn=IL,IU,fi=1,2,…,a¯. Then,(59)CSSPGHMÜ,Õcn1,cn2,…,cna¯=cn=IL,IU,f.Theorem 8 (commutativity).
Letcn1′,cn2′,…,cna¯′ be any permutation of cn1,cn2,…,cna¯. Then,(60)CSSPGHMÜ,Õcn1′,cn2′,…,cna¯′=CSSPGHMÜ,Õcn1,cn2,…,cna¯.Theorem 9 (boundedness).
Letcni=IiL,IiU,fii=1,2,…,a¯ be a faction of CNs, cn−=miniIiL,miniIiU,maxifi and Θ+=maxiIiL,maxiIiU,minifi. Then,(61)cn−≤CSSPGHMÜ,Õcn1,cn2,…,cna¯≤cn+.The proofs of these properties are the same as for CSSPHM operators. Hence, these are executed here.By specifying distinctive values of the parametersЖ,Üand Õ, numerous specific AOs are obtained from the CSSPGHM operator, and are stated as given below:Case 1. IfЖ=0, then the CSSPGHM operator reverts to the CPGHM operator and is stated as follows:(62)CSSPGHMÜ,Õcn1,cn2,…,cna¯=1Ü+Õ∏i,j=1j=ia¯Ücnia¯Ÿi⊕Õcnja¯Ÿj2/a¯a¯+1=1−1−∏i,j=1j=ia¯1−1−IiLa¯ŸiÜ1−IjLa¯ŸjÕ2/a¯a¯+11/Ü+Õ,1−1−∏i,j=1j=ia¯1−1−IiUa¯ŸiÜ1−IjUa¯ŸjÕ2/a¯a¯+11/Ü+Õ,1−∏i,j=1j=ia¯1−1−1−fia¯ŸiÜ1−1−fja¯ŸjÕ2/a¯a¯+11/Ü+Õ.Case 2. IfÕ⟶0, then the CSSPGHM operator reverts to the cubic descending PG operator and is stated as follows:(63)CSSPGHMÜ,0cn1,cn2,…,cna¯=limÕ⟶01Ü+Õ∏i,j=1j=ia¯Ücnia¯Ÿi⊕Õcnja¯Ÿj2/a¯a¯+1=1Ü∏i=1a¯Ücnia¯Ÿìa¯+1−i2/a¯a¯+1=1−1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−Ü1−a¯ŸiIiLЖ−a¯Ÿi−11/ЖЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж,1−1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−Ü1−a¯ŸiIiUЖ−a¯Ÿi−11/ЖЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж,1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−Ü1−a¯Ÿi1−fiЖ−a¯Ÿi−11/ЖЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж.Case 3. IfÜ⟶0, then the CSSPGHM operator reverts to the cubic ascending PG operator and is stated as follows:(64)CSSPGHM0,Õcn1,cn2,…,cna¯=limG⟶01Õ∏i,j=1j=ia¯Ücnia¯Ÿi⊕Õcnja¯Ÿj2/a¯a¯+1=1Õ∏i=1a¯Õcnia¯Ÿii2/a¯a¯+1=1−1Õ1−2a¯a¯+1∑i=1a¯i1−Õ1−a¯ŸiIiLЖ−a¯Ÿi−11/ЖЖ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж,1−1Õ1−2a¯a¯+1∑i=1a¯i1−Õ1−a¯ŸiIiUЖ−a¯Ÿi−11/ЖЖ−Õ−11ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж,1Õ1−2a¯a¯+1∑i=1a¯i1−Õ1−a¯Ÿi1−fiЖ−a¯Ÿi−11/ЖЖ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж.Case 4. IfÕ⟶0 and supcni,cnj=hh∈0,1∀i≠j, then the CSSPGHM operator reverts to the cubic linear descending weighted geometric operator and is stated as follows:(65)CSSPGHMÜ,0cn1,cn2,...,cna¯=limÕ⟶01Ü∏i,j=1j=ia¯Ücnia¯Ÿi⊕Õcnja¯Ÿj2/a¯a¯+1=1Ü∏i=1a¯Ücnia¯+1−i2/a¯a¯+1=1−1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−Ü1−IiLЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж,1−1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−Ü1−IiUЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж,1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−Üfì−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж.Certainly, the significant degree of CNscnii=1,2,…,a¯ is a¯,a¯−1,…,1.Case 5. IfÕ⟶0 and supcni,cnj=hh∈0,1∀i≠j, then the CSSPGHM operator reverts to the cubic linear ascending weighted geometric operator and is stated as follows:(66)CSSPGHM0,Õcn1,cn2,…,cna¯=limÕ⟶01Ü+Õ∏i,j=1j=ia¯Ücnia¯Ÿi⊕Õcnja¯Ÿj2/a¯a¯+1=1Õ∏i=1a¯Õcnii2/a¯a¯+1,=1−1Õ1−2a¯a¯+1∑i=1a¯i1−Õ1−IiLЖ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж,1−1Õ1−2a¯a¯+1∑i=1a¯i1−Õ1−IiUЖ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж1Õ1−2a¯a¯+1∑i=1†i1−ÕfiЖ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж.Case 6. IfÜ=Õ=1/2 and supcni,cnj=hh∈0,1∀i≠j, then the CSSPGHM operator reverts to the cubic GHM operator and is stated as follows:(67)CSSPGHM1/2,1/2cn1,cn2,…,cna¯=1Ü+Õ∏i,j=1j=ia¯Ücnia¯Ÿi⊕Õcnja¯Ÿj2/a¯a¯+1=12∏i,j=1j=ia¯cni⊕cnj2/a¯a¯+1=2a¯a¯+1∑i,j=1j=ia¯1−121−IiLЖ+121−IjLЖ1/ЖЖ1/Ж,2a¯a¯+1∑i,j=1j=ia¯1−121−IiUЖ+121−IjUЖ1/ЖЖ1/Ж,1−2††+1∑i,j=1j=i†1−12fiЖ+12fjЖ1/ЖЖ1/Ж.Case 7. IfÜ=Õ=1 and supcni,cnj=hh∈0,1∀i≠j, then the CSSPGHM operator reverts to the cubic basic GHM operator and is stated as follows:(68)CSSPGHM1,1cn1,cn2,…,cna¯=1Ü+Õ∏i,j=1j=ia¯Ücnia¯Ÿi⊕Õcnja¯Ÿj2/a¯a¯+1=12∏i,j=1j=ia¯cni⊕cnj2/††+1=1−1−2a¯a¯+1∑i,j=1j=ia¯1−1−IiLЖ+1−jLЖ−11/ЖЖ1/ЖЖ+121/Ж,1−1−2a¯a¯+1∑i,j=1j=i†1−1−IiUЖ+1−IjUЖ−11/ЖЖ1/ЖЖ+121/Ж,121−2a¯a¯+1∑i,j=1j=i†1−fiЖ+fjЖ−11/ЖЖ1/ЖЖ+121/Ж.


### 4.2. CSSWPHM and CSSWPGHM Operators

In this portion, we instigate the CSSWPHM operator and the CSSWPGHM operator by taking the significance of the attributes.Definition 10.
LetÜ,Õ≥0,cni=IiL,IiU,fii=1,2,…,a¯ be a faction of CNs and ς=ς1,ς2,…,ςa¯T be the importance degree of cnii=1,2,…,a¯, where ςi≥0 and ∑i=1a¯ςi=1. The CSSWPHM operator is rationalized as follows:(69)CSSWPHMÜ,Õcn1,cn2,…,cna¯=2a¯a¯+1∑i,j=1j=ia¯a¯ςi1+Tcni∑z=1a¯ςz1+TcnzcniÜ⊗SSa¯ςj1+Tcnj∑z=1a¯ςz1+TcnzcnjÕ1/Ü+Õ,where Ü,Õ≥0,Tcni=∑j=1j≠i†supcni,cnj,supcni,cnj=1−DNE¯¯cni,cnj, and DNE¯¯cni,cnj can be reckoned by (6).
LetŸi=1+Tcni/∑z=1a¯1+Tcnz, then (69) be correspondent to the following shape:(70)CSSWPHMÜ,Õcn1,cn2,…,cna¯=2a¯a¯+1∑i,j=1j=ia¯a¯ςiŸicniÜ⊗SSa¯ŸjςjcnjÕ1/Ü+Õ.Theorem 10.
LetÜ≥0,Õ≥0 and Ü,Õ take no more than one value of 0 at a time, cni=IiL,IiU,fi be a faction of CNs. Then, exploiting the CSSWPHM operator, their merged values are CNs, and(71)CSSWPHMÜ,Õcn1,cn2,…,cna¯=1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ςiŸi1−IiLЖ−a¯ςiŸi−11/ЖЖ+Õ1−a¯ςjŸj1−IjLЖ−a¯ςjŸj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ςiŸi1−IiUЖ−a¯ςiŸi−11/ЖЖ+Õ1−a¯ςjŸj1−IjUЖ−a¯ςjŸj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1−1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ςiŸifiЖ−a¯ςiŸi−11/ЖЖ+Õ1−a¯ςjŸjfjЖ−a¯ςjŸj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж.Definition 11.
LetÜ,Õ≥0,cni=IiL,IiU,fii=1,2,…,a¯ be a faction of CNs and ς=ς1,ς2,…,ςa¯T be the significant degree of cnii=1,2,…,a¯, where ςi≥0 and ∑i=1a¯ςi=1. The CSSWPGHM operator is rationalized as follows:(72)CSSWPGHMÜ,Õcn1,cn2,…,cna¯=1Ü+Õ∏i,j=1j=ia¯Ücnia¯ςi1+Tcni/∑z=1a¯ςz1+Tcnz⊕SrSrÕcnja¯ςj1+Tcnj/∑z=1a¯ςz1+Tcnz2/a¯a¯+1,where Ü,Õ≥0,Tcni=∑j=1j≠ia¯supcni,cnj,supcni,cnj=1−DNE¯¯cni,cnj, and DNE¯¯cni,cnj can be processed by (6).
LetŸi=1+Tcni/∑z=1a¯1+Tcnz, then (72) is correspondent to the following type:(73)CSSWPGHMÜ,Õcn1,cn2,…,cn†=1Ü+Õ∏i,j=1j=ia¯Ücnia¯ςiŸi⊕SSÕcnja¯ςjŸj2/a¯a¯+1.Theorem 11.
LetÜ≥0,Õ≥0 and Ü,Õ take no more than one value of 0 at a time, cni=IiL,IiU,fii=1,2,…,a¯ be a faction of CNs. Then, employing the CSSWPGHM operator, their fused values are CNs, and(74)CSSWPGHMÜ,Õcn1,cn2,…,cna¯=1−1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ςiŸiIiLЖ−a¯ςiŸi−11/ЖЖ+Õ1−a¯ςjŸjIjLЖ−a¯ςjŸj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1−1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ςiŸiIiUЖ−a¯ςiŸi−11/ЖЖ+Õ1−a¯ςjŸjIjUЖ−a¯ςjŸj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,,1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ςiŸi1−fiЖ−a¯ςiŸi−11/ЖЖ+Õ1−a¯ςjŸj1−fjЖ−a¯jŸj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж.
The proofs of Theorems10 and 11 are the same as Theorems 2 and 6. Therefore, here we omit their proofs.


## 4.1. The CSSPHM and CSSPGHM Operators

In this segment, numerous new AOs namely, CSSPHM and CSSPGHM operators are anticipated by combining HM and GHM operators with PA operators to anticipate.Definition 8.
Letcnì=IìL,IìU,fì,ì=1,2,…,a¯ be a faction of CNs, and then the cubic Schweizer–Sklar power HM (CSSPHM) operator is clarified as follows:(19)CSSPHMÜ,Õcn1,cn2,…,cna¯=2a¯a¯+1∑ì,j=1j=ìa¯a¯1+Tcnì∑z=1a¯1+TcnzcnìÜ⊗SSa¯1+Tcnj∑z=1a¯1+TcnzcnjÕ1/Ü+Õ,where Ü,Õ≥0,Tcni=∑j=1j≠ìa¯supcni,cnj,supcni,cnj=1−DNE¯¯cni,cnj, and DNE¯¯cni,cnj can be figured by (6).
LetŸi=1+Tcni/∑z=1a¯1+Tcnz, then the definition of CSSPHM operator is correspondent to the following type:(20)CSSPHMÜ,Õcn1,cn2,…,cn†=2a¯a¯+1∑i,j=1j=ia¯a¯ŸicniÕ⊗SrSra¯ŸjcnjÜ1/Ü+Õ.Theorem 2.
LetÜ≥0,Õ≥0 and G,H grab no more than one value of 0 at a time, cni=IiL,IiU,fi be a set of CNs. Then, exploiting the CSSPHM operator, their merged values are CNs, and(21)CSSPHMÜ,Õcn1,cn2,…,cn†=1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−IiLℵ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjLЖ−a¯Ÿj−11/ЖЖ−Ü−+Õ1/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjUЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1−1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ŸifiЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯ŸjfjЖ−n⋄j−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж.Proof.
Firstly, we need to prove the following equation:(22)∑i,j=1j=ia¯a¯ŸicniÜ⊗SSa¯ŸjcnjÕ=1−∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−IiLЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjLЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ−a¯a¯+12−11/Ж,1−∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjUЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ−a¯a¯+12−11/Ж,∑i,j=1j=ia¯1−Ü1−a¯ŸifiЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯ŸjfjЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ−a¯a¯+12−11/Ж.
By SS OLs defined in Definition7, we get(23)a¯Ÿicni=1−a¯Ÿi1−IiLЖ−a¯Ÿi−11/Ж,1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/Ж,a¯ŸifiЖ−a¯Ÿi−11/Ж,and(24)a¯ŸicniÜ=Ü1−a¯Ÿi1−IiLЖ−a¯Ÿi−11/ЖЖ−Ü−11/Ж,Õ1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/ЖЖ−Ü−11/Ж,1−Ü1−a¯ŸifiЖ−a¯Ÿi−11/ЖЖ−Ü−11/Ж.
Similarly,(25)a¯Ÿjcnj=1−a¯Ÿj1−IjLЖ−a¯Ÿj−11/Ж,1−a¯Ÿj1−IjUЖ−a¯Ÿj−11/Ж,a¯ŸjfjЖ−a¯Ÿj−11/Ж,a¯ŸjcnjÕ=Õ1−a¯Ÿj1−IjLЖ−a¯Ÿj−11/ЖЖ−Õ−11/Ж,Õ1−a¯Ÿj1−IjUЖ−a¯Ÿj−11/ЖЖ−Õ−11/Ж,1−Õ1−a¯ŸjfjЖ−a¯Ÿj−11/ЖЖ−Õ−11/Ж.
So,(26)a¯ŸicniÜ⊗SSa¯ŸjcnjÕ=Ü1−a¯Ÿi1−IiLЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjLЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/Ж,Ü1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjUЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/Ж,1−Ü1−a¯ŸifiЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯ŸjfjЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/Ж,(1)
Whena¯=2 by Equation (17) and Equation (25), we have(27)∑i,j=1j=i2a¯ŸicniÜ⊗SSa¯ŸjcnjÕ=2Ÿ1cn1Ü⊗SS2Ÿ1cn1Õ⊕SS2Ÿ1cn1Ü⊗SS2Ÿ2cn2Õ⊕SS2Ÿ2cn2Ü⊗SS2Ÿ2cn2Õ=Ü1−2Ÿ11−I1LЖ−2Ÿ1−11/ЖЖ+Õ1−2Ÿ11−I1LЖ−2Ÿ1−11/ЖЖ−Ü−Õ+11/Ж,Ü1−2Ÿ11−I1UЖ−2Ÿ1−11/ЖЖ+Õ1−2Ÿ11−I1UЖ−2Ÿ1−11/ЖЖ−Ü−Õ+11/Ж,1−Ü1−2Ÿ1f1Ж−2Ÿ1−11/ЖЖ+Õ1−2Ÿ1f1Ж−2Ÿ1−11/ЖЖ−Ü−Õ+11/Ж〉⊕SSÜ1−2Ÿ11−I1LЖ−2Ÿ1−11/ЖЖ+Õ1−2Ÿ21−I2LЖ−2Ÿ2−11/ЖЖ−Ü−Õ+11/ЖÜ1−2Ÿ11−I1UЖ−2Ÿ1−11/ЖЖ+Õ1−2Ÿ21−I2UЖ−2Ÿ2−11/ЖЖ−Ü−Õ+11/Ж,1−Ü1−2Ÿ1f1Ж−2Ÿ1−11/ЖЖ+Õ1−2Ÿ2f2Ж−2Ÿ2−11/ЖЖ−Ü−Õ+11/Ж〉⊕SSÜ1−2Ÿ21−I2LЖ−2Ÿ2−11/ЖЖ+Õ1−2Ÿ21−I2LЖ−2Ÿ2−11/ЖЖ−Ü−Õ+11/ЖÜ1−2Ÿ21−I2UЖ−2Ÿ2−11/ЖЖ+Õ1−2Ÿ21−I2UЖ−2Ÿ2−11/ЖЖ−Ü−Õ+11/Ж,1−Ü1−2Ÿ2f2Ж−2Ÿ2−11/ЖЖ+Õ1−2Ÿ2f2Ж−2Ÿ2−11/ЖЖ−Ü−Õ+11/Ж〉.By using Equation (17), we get(28)∑i,j=1j=i2a¯ŸicniÜ⊗SSa¯ŸjcnjÕ=1−∑i,j=1j=i21−Ü1−a¯Ÿi1−IiLЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjLЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11ЖЖ−a¯a¯+12−11/Ж,1−∑i,j=1j=i21−Ü1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjUЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ−a¯a¯+12−11/Ж,∑i,j=1j=i21−Ü1−a¯ŸifiЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯ŸjfjЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ−a¯a¯+12−11/Ж.That is (22) is true a¯=2.(2)
Let us assume that Equation (24) is true a¯=z.(29)∑i,j=1j=iza¯ŸicniÜ⊗SSa¯ŸjcnjÕ=1−∑i,j=1j=iz1−Ü1−zŸi1−IiLЖ−zŸi−11/ЖЖ+Õ1−zŸj1−IjLЖ−zŸj−11/ЖЖ−Ü−Õ+11/ЖЖ−zz+12−11/Ж,1−∑i,j=1j=iz1−Ü1−zŸi1−IiUЖ−zŸi−11/ЖЖ+Õ1−zŸj1−IjUЖ−zŸj−11/ЖЖ−Ü−Õ+11/ЖЖ−zz+12−11/Ж,∑i,j=1j=iz1−Ü1−zŸifiЖ−zŸi−11/ЖЖ+Õ1−zŸjfjЖ−zŸj−11/ЖЖ−Ü−Õ+11/ЖЖ−zz+12−11/Ж.
Furthermore, whena¯=z+1(30)∑i,j=1j=iz+1a¯ŸicniÜ⊗SSa¯ŸjcnjÕ=∑i,j=1j=iz+1z+1ŸicniÜ⊗SSz+1ŸjcnjÕ⊕SS∑i=1zz+1ŸicniÜ⊗SSz+1Ÿz+1cnz+1Õ⊕SSz+1Ÿz+1cnz+1Ü⊗SSz+1Ÿz+1cnz+1Õ.
Firstly, we will show that(31)∑i=1zz+1ŸicniÜ⊗SSz+1Ÿz+1cnz+1Õ=1−∑i,j=1j=iz1−Ü1−z+1Ÿi1−IiLЖ−z+1Ÿi−11/ЖЖ+Õ1−z+1Ÿz+11−Iz+1LЖ−z+1Ÿz+1−11/ЖЖ−Ü−Õ+11/ЖЖ−z+1z+22−11/Ж,1−∑i,j=1j=iz1−Ü1−z+1Ÿi1−IiUЖ−z+1Ÿi−11/ЖЖ+Õ1−z+1Ÿz+11−Iz+1UЖ−z+1Ÿz+1−11/ЖЖ−Ü−Õ+11/ЖЖ−z+1z+22−11/Ж,∑i,j=1j=iz1−Ü1−z+1ŸifiЖ−z+1Ÿi−11/ЖЖ+Õ1−z+1Ÿz+1fz+1Ж−z+1Ÿz+1−11/ЖЖ−Ü−Õ+11/ЖЖ−z+1z+22−11/Ж.
We shall prove (31), on mathematical induction on z.(i)
Forz=2, we have(32)∑i=12z+1ŸicniÜ⊗SSz+1Ÿz+1cnz+1Õ=3Ÿ1cn1Ü⊗SS3Ÿ3cn3Õ⊕SS3Ÿ2cn2Ü⊗SS3Ÿ3cn3Õ=Ü1−3Ÿ11−I1LЖ−3Ÿ1−11/ЖЖ+Õ1−3Ÿ31−I3LЖ−3Ÿ3−11/ЖЖ−Ü−Õ+11/Ж,Ü1−3Ÿ11−I1UЖ−2Ÿ1−11/ЖЖ+Õ1−3Ÿ31−I3UЖ−3Ÿ3−11/ЖЖ−Ü−Õ+11/Ж,1−Ü1−3Ÿ1f1Ж−2Ÿ1−11/ЖЖ+Õ1−3Ÿ3f3Ж−3Ÿ3−11/ЖЖ−Ü−Õ+11/Ж〉⊕SSÜ1−3Ÿ21−I2LЖ−3Ÿ2−11/ЖЖ+Õ1−3Ÿ31−I3LЖ−3Ÿ3−11/ЖЖ−Ü−Õ+11/Ж,Ü1−3Ÿ21−I2UЖ−3Ÿ2−11/ЖЖ+Õ1−3Ÿ31−I3UЖ−3Ÿ3−11ЖЖ−Ü−Õ+11/Ж,1−Ü1−3Ÿ2f2Ж−3Ÿ2−11/ЖЖ+Õ1−3Ÿ3f3Ж−3Ÿ3−11/ЖЖ−Ü−Õ+11/Ж〉.By using (11), we get(33)=1−∑i=121−Ü1−3Ÿi1−IiLЖ−3Ÿi−11/ЖЖ+Õ1−3Ÿ31−I3LЖ−3Ÿ3−11/ЖЖ−Ü−Õ+11/ЖЖ−33+12−11/Ж,1−∑i=121−Ü1−3Ÿi1−IiUЖ−3Ÿi−11/ЖЖ+Õ1−3Ÿ31−I3UЖ−3Ÿ3−11/ЖЖ−Ü−Õ+11/ЖЖ−33+12−11/Ж,∑i=121−Ü1−3ŸifiЖ−3Ÿi−11/ЖЖ+Õ1−3Ÿ3f3Ж−3Ÿ3−11/ЖЖ−Ü−Õ+11/ЖЖ−33+12−11/Ж.(ii)
Suppose that Equation (28) is true z=ő.(34)∑i=1őő+1ŸicniÜ⊗SSő+1Ÿő+1cnő+1Õ=1−∑i=1ő1−Ü1−ő+1Ÿi1−IiLЖ−ő+1Ÿi−11/ЖЖ+Õ1−ő+1Ÿő+11−Iő+1LЖ−ő+1Ÿő+1−11/ЖЖ−Ü−Õ+11/ЖЖ−ő+1ő+22−11/Ж,1−∑i=1ő1−Ü1−ő+1Ÿi1−IiUЖ−ő+1Ÿi−11/ЖЖ+Õ1−ő+1Ÿő+11−Iő+1UЖ−ő+1Ÿő+1−11/ЖЖ−Ü−Õ+11/ЖЖ−ő+1ő+22−11/Ж,∑i=1ő1−Ü1−ő+1ŸifiЖ−ő+1Ÿi−11/ЖЖ+Õ1−ő+1Ÿő+1fő+1Ж−ő+1Ÿő+1−11/ЖЖ−Ü−Õ+11/ЖЖ−ő+1ő+22−11/Ж.
Then, whenz=ő+1, we have(35)∑i=1ő+1ő+2ŸicniÜ⊗SSő+2Ÿő+2cnő+2Õ=∑I=1ő+1ő+2ŸicniÜ⊗SSő+2Ÿő+2cnő+2Õ⊕SSő+2Ÿő+1cnő+1Ü⊗SSő+2Ÿő+2cnő+2Õ=1−∑i=1ő1−Ü1−ő+2Ÿi1−IiLЖ−ő+2Ÿi−11/ЖЖ+Õ1−ő+2Ÿő+21−Iő+2LЖ−ő+2Ÿő+2−11/ЖЖ−Ü−Õ+11/ЖЖ−ő+2ő+32−11/Ж,1−∑i=1ő1−Ü1−ő+2Ÿi1−IiUЖ−ő+2Ÿi−11/ЖЖ+Õ1−ő+2Ÿő+21−Iő+2UЖ−ő+2Ÿő+2−11/ЖЖ−Ü−Õ+11/ЖЖ−ő+2ő+32−11/Ж,∑i=1ő1−Ü1−ő+2ŸifiЖ−ő+2Ÿi−11/ЖЖ+Õ1−ő+2Ÿő+1fő+1Ж−ő+2Ÿő+1−11/ЖЖ−Ü−Õ+11/ЖЖ−ő+2ő+32−11/Ж⊕SS〈Ü1−ő+2Ÿő+11−Iő+1LЖ−ő+2Ÿő+1−11/ЖЖ+Õ1−ő+2Ÿő+21−Iő+2LЖ−ő+2Ÿő+2−11/ЖЖ−Ü−Õ+11/Ж,Ü1−ő+2Ÿő+11−Iő+1UЖ−ő+2Ÿő+1−11/ЖЖ+Õ1−ő+2Ÿő+21−Iő+2UЖ−ő+2Ÿő+1−11/ЖЖ−Ü−Õ+11/Ж,1−Ü1−ő+2Ÿő+1fő+1Ж−ő+2Ÿő+1−11/ЖЖ+Õ1−ő+2Ÿő+2fő+2Ж−ő+2Ÿő+2−11/ЖЖ−Ü−Õ+11/Ж〉.=1−∑i=1ő+11−Ü1−ő+2Ÿi1−IiLЖ−ő+2Ÿi−11/ЖЖ+Õ1−ő+2Ÿő+21−Iő+2LЖ−ő+2Ÿő+2−11/ЖЖ−Ü−Õ+11/ЖЖ−ő+2ő+32−11/Ж,1−∑i=1ő+11−Ü1−ő+2Ÿi1−IiUЖ−ő+2Ÿi−11/ЖЖ+Õ1−ő+2Ÿő+21−Iő+2UЖ−ő+2Ÿő+2−11/ЖЖ−Ü−Õ+11/ЖЖ−ő+2ő+32−11/Ж,∑i=1ő+11−Ü1−ő+2ŸifiЖ−ő+2Ÿi−11/ЖЖ+Õ1−ő+2Ÿő+2fő+2Ж−ő+2Ÿő+2−11/ЖЖ−Ü−Õ+11/ЖЖ−ő+2ő+32−11/Ж.
Therefore, (31) is true z=ő+1. Hence, (31) is true for all z. Similarly, we can prove other parts of (30). So, (30) becomes(36)∑i,j=1j=iz+1z+1ŸicniÜ⊗SSz+1ŸjcnjÕ=1−∑i,j=1j=iz+11−Ü1−z+1Ÿi1−IiLЖ−z+1Ÿi−11/ЖЖ+Õ1−z+1Ÿj1−IjLЖ−z+1Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ−z+1z+22−11/Ж,1−∑i,j=1j=iz+11−Ü1−z+1Ÿi1−IiUЖ−z+1Ÿi−11/ЖЖ+Õ1−z+1Ÿj1−IjUЖ−z+1Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ−z+1z+22−11/Ж,∑i,j=1j=iz+11−Ü1−z+1ŸifiЖ−z+1Ÿi−11/ЖЖ+Õ1−z+1ŸjfjЖ−z+1Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ−z+1z+22−11/Ж.
Hence, (22) is true a¯=z+1. Therefore, (22) is true for all †.
By (22), we can say that (14), is right. From (22), and the SS operational laws for CNs, we have(37)2a¯a¯+1∑i,j=1j=ia¯a¯ŸicniÜ⊗SSa¯ŸjcnjÕ=1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−IiLЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjLЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/Ж,1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjUЖ−a¯Ÿj−11ЖЖ−Ü−Õ+11/ЖЖ1/Ж,2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ŸifiЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯ŸjfjЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/Ж.
Furthermore,(38)2a¯a¯+1∑i,j=1j=ia¯a¯ŸicniÜ⊗SSa¯ŸjcnjÕ1/Ü+Õ=1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−IiLЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjLЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjUЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1−1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ŸifiЖ−a¯Ÿi−11ЖЖ+Õ1−a¯ŸjfjЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/ℵ.
Hence,(39)CSSPHMÜ,Õcn1,cn2,…,cn†=1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−IiLЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjLЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−IjUЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1−1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ŸifiЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯ŸjfjЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж.
Now, we confer some advantageous possessions of the projected CSrSrPHOM operator.Theorem 3 (idempotency).
LetÜ≥0,Õ≥0 and Ü,Õ capture no more than one value of 0 at a time, cni=IiL,IiU,fii=1,2,…,a¯ be a faction of CNs, and cni=IiL,IiU,fsi=cn=IL,IU,fi=1,2,…,a¯. Then,(40)CSSPHMÜ,Õcn1,cn2,…,cn†=cn=IL,IU,f.Proof.
Sincecni=IiL,IiU,fi=cn=IL,IU,fi=1,2,...,a¯, we have supcng,cnz=1∀g,z=1,2,…,a¯ so ⋄g=1/a¯∀g=1,2,…,a¯ and a¯Ÿg=a¯1/a¯=1∀g=1,2,…,a¯.
Then,(41)CSSPHMÜ,Õcn1,cn2,…,cna¯=CSSPHMÜ,Õcn,cn,…,cn=1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−1−ILЖ1/ЖЖ+Õ1−1−ILЖ1/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−1−IUЖ1/ЖЖ+Õ1−1−IUЖ1/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1−1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−fЖ+Õ1−fЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,=1Ü+ÕÜ+ÕILЖ−Ü−Õ+1−1Ü+Õ+11/Ж,1Ü+ÕÜ+ÕIUЖ−Ü−Õ+1−1Ü+Õ+11/Ж,1−1Ü+ÕÜ+Õ1−fЖ−Ü−Õ+1−1Ü+Õ+11/Ж=IL,IU,f=cn.Theorem 4.
(commutativity). Letcn1′,cn2′,…,cna¯′ be any permutation of cn1,cn2,…,cna¯. Then,(42)CSSPHMÜ,Õcn1′,cn2′,…,cna¯′=CSSPHMÜ,Õcn1,cn2,…,cna¯.Proof.
Sincecn1′,cn2′,…,cna¯′ is any permutation of cn1,cn2,…,cna¯, (43)CSSPHMÜ,Õcn1,cn2,…,cna¯=2a¯a¯+1∑i,j=1j=ia¯a¯ŸicniÜ⊗SSa¯ŸjcnjÕ1/Ü+Õ=2a¯a¯+1∑i,j=1j=ia¯a¯Ÿicnj′Ü⊗SSa¯Ÿjcnj′Õ1/Ü+Õ=CSSPHMÜ,Õcn1′,cn2′,…,cna¯′.Theorem 5 (boundedness).
Letcni=IiL,IiU,fii=1,2,…,a¯ be a faction of CNs, cn−=miniIiL,miniIiU,maxifi and cn+=maxiIiL,maxiIiU,minifi. Then,(44)cn−≤CSSPHMÜ,Õcn1,cn2,…,cna¯≤cn+.Proof.
By the judgement technique in Definition2, we have cni≥cnj′, and then based on Theorems 3 and 4, we have(45)CSSPHMÜ,Õcn1,cn2,…,cna¯≥CSSPHMÜ,Õcn1−,cn2−,…,cna¯−=cn−.
Similarly, we can have(46)CSSPHMÜ,Õcn1,cn2,…,cna¯≤CSSPHMÜ,Õcn1+,cn2+,…,cna¯+=cn+.
So, we have(47)CSSPHEMÜ,Õcn1−,cn2−,…,cna¯−≤CSSPHMÜ,Õcn1,cn2,…,cna¯≤CSSPHMÜ,Õcn1+,cn2+,…,cna¯+.
Hence,(48)cn−≤CSrSrPHOMÜ,Õcn1,cn2,…,cn†≤cn+.
By apportioning different values to the constraintsЖ,Ü and Õ, numerous different circumstances of the CSSPHM operator can be attained, and are specified below as follows:Case 1. IfЖ=0, then the CSSPHM operator relapses to the CPHM operator, which can be stated as follows:(49)CSSPHMÜ,Õcn1,cn2,…,cna¯=2a¯a¯+1∑i,j=1j=ia¯a¯ŸicniÜ⊗a¯ŸjcnjÕ1/Ü+Õ=1−∏i,j=1j=ia¯1−1−1−IiLa¯ŸiÜ1−1−IjLa¯ŸjÕ2/a¯a¯+11/Ü+Õ,1−∏i,j=1j=ia¯1−1−1−IiUa¯ŸiÜ1−1−IjUa¯ŸjÕ2/a¯a¯+11/Ü+Õ,1−1−∏i,j=1j=ia¯1−1−fia¯ŸiÜ1−fja¯ŸjÕ2/a¯a¯+11/Ü+Õ.Case 2. IfÕ⟶0, then the CSSPHM operator reverts to the cubic descending PA operator and is quantified as follows:(50)CSSPHMÜ,Õcn1,cn2,…,cna¯=limÕ⟶02a¯a¯+1∑i,j=1j=ia¯a¯ŸicniÜ⊗SSa¯ŸjcnjÕ1/Ü+Õ=2a¯a¯+1∑i=1a¯a¯+1−ia¯ŸicniÜ1/Ü=1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−Ü1−a¯Ÿi1−IiLЖ−a¯Ÿi−11/ЖЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж,1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−Ü1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/ЖЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж,1−1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−Ü1−a¯ŸifiЖ−a¯Ÿi−11ЖЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж.Case 3. IfÜ⟶0, then the CSSPHM operator reverts to the cubic ascending PA operator and is quantified as follows:(51)CSSPHM0,Õcn1,cn2,…,cna¯=limÜ⟶02a¯a¯+1∑i,j=1j=ia¯a¯ŸicniÜ⊗SSa¯ŸjcnjÕ1/Ü+Õ=2a¯a¯+1∑i=1a¯ia¯ŸicniÕ1/Õ=1Õ1−2a¯a¯+1∑i=1a¯i1−Õ1−a¯Ÿi1−IiLЖ−a¯Ÿi−11/ЖЖ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж,1Õ1−2a¯a¯+1∑i=1a¯i1−Õ1−a¯Ÿi1−IiUЖ−a¯Ÿi−11/ЖЖ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж,1−1Õ1−2a¯a¯+1∑i=1a¯i1−Õ1−a¯ŸifiЖ−a¯Ÿi−11/ЖЖ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж.Case 4. IfÕ⟶0 and supcni,cnj=hh∈0,1for alli≠j, then the CSSPHM operator reverts to the cubic linear descending WA operator and is stated as follows:(52)CSSPHMÜ,0cn1,cn2,…,cna¯=limÕ⟶02a¯a¯+1∑i,j=1j=1a¯a¯ŸicniÜ⊗SSa¯ŸjcnjÕ1/Ü+Õ=2a¯a¯+1∑i=1a¯a¯+1−icniÜ1/Ü=1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−ÜIiLЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж,1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−ÜIiUЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж,1−1Ü1−2a¯a¯+1∑i=1†a¯+1−i1−Ü1−fiЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж.Certainly, the significant degree of CNscniÜi=1,2,…,a¯ is a¯,a¯−1,…,1.Case 5. IfÜ⟶0 and supcni,cnj=hh∈0,1for alli≠j, then the CSSPHOM operator reverts to the cubic linear ascending WA operator and is stated as follows:(53)CSSPHM0,Õcn1,cn2,…,cna¯=limÜ⟶02a¯a¯+1∑i,j=1j=1a¯a¯ŸicniÜ⊗SSa¯ŸjcnjÕ1/Ü+Õ=2a¯a¯+1∑i=1a¯icniÕ1/Õ=1Õ1−2a¯a¯+1∑i=1a¯i1−ÕIiLℵ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж,1Õ1−2a¯a¯+1∑i=1a¯i1−ÕIiUЖ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж,1−1Õ1−2a¯a¯+1∑i=1a¯i1−Õ1−fiЖ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж.Case 6. IfÜ=Õ=1/2 and supcni,cnj=hh∈0,1for alli≠j, then the CSSPHM operator reverts to the cubic basic HM operator and is stated as follows:(54)CSSPHM1/2,1/2cn1,cn2,…,cna¯=2a¯a¯+1∑i,j=1j=1a¯cni1/2⊗cnj1/2=1−2a¯a¯+1∑i,j=1j=ia¯1−12IiLЖ+12IjLЖ1/ЖЖ1/Ж,1−2a¯a¯+1∑i,j=1j=ia¯1−12IiUЖ+12IjUЖ1/ЖЖ1/Ж,2a¯a¯+1∑i,j=1j=ia¯1−121−fiЖ+121−fsjЖ1/ЖЖ1/Ж.Case 7. IfÜ=Õ=1, and supcni,cnj=hh∈0,1for alli≠j, then the CSSPHM operator reverts to the cubic linear HM operator, and is stated as follows:(55)CSSPHM1,1cn1,cn2,…,cna¯=2a¯a¯+1∑i,j=1j=1†cni⊗cnj1/2=121−2a¯a¯+1∑i,j=1j=i†1−IiLЖ+IjLЖ−11/ЖЖ1/ЖЖ+121/Ж,121−2a¯a¯+1∑i,j=1j=i†1−IiUЖ+IjUЖ−11/ЖЖ1/ЖЖ+121/Ж1−1−2a¯a¯+1∑i,j=1j=i†1−1−fiЖ+1−fjЖ−11/ЖЖ1/ЖЖ+121/Ж.Definition 9.
Letcni=IiL,IiU,fii=1,2,…,a¯ be a faction of CNs, then the cubic SS power geometric HM (CSSPGHM) operator is explicated as follows:(56)CSSPGHMÜ,Õcn1,cn2,…,cna¯=1Ü+Õ∏i,j=1j=ia¯Ücnia¯1+Tcni/∑z=1a¯1+Tcnz⊕SSÕcnja¯1+Tcnj/∑z=1a¯1+Tcnz2/a¯a¯+1,where Ü,Õ≥0,Tcni=∑j=1j≠ia¯supcni,cnj,supcni,cnj=1−DNE¯¯cni,cnj, and DNE¯¯cni,cnj can be totalized by (6).
LetŸi=1+Tcni/∑z=1a¯1+Tcnz, then the definition of CSSPGHM operator is correspondent to the subsequent formula:(57)CSSPGHMÜ,Õcn1,cn2,…,cna¯=1Ü+Õ∏i,j=1j=ia¯Ücnia¯Ÿi⊕SSÕcnja¯Ÿj2/a¯a¯+1.Theorem 6.
LetÜ≥0,Õ≥0 and Ü,Õ take no more than one value of 0 at a time, cni=IiL,IiU,fi be a faction of CNs. Then, exploiting the CSSPGHM operator, their merged values are CNs, and(58)CSSPGHMÜ,Õcn1,cn2,…,cna¯=1−1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ŸiIiLЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯ŸjIjLЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1−1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ŸiIiUЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯ŸjIjUЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯Ÿi1−fiЖ−a¯Ÿi−11/ЖЖ+Õ1−a¯Ÿj1−fjЖ−a¯Ÿj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж.Proof.
Proof of Theorem6 is similar to that of Theorem 2. Therefore, omitted here.
Now, we deliberate some privileged properties of the projected CSSPGHM operator.Theorem 7 (idempotency).
LetÜ≥0,Õ≥0, and Ü,Õ grab no more than one value of 0 at a time, cni=IiL,IiU,fii=1,2,…,a¯ be a faction of CNs, and cni=IiL,IiU,fi=cn=IL,IU,fi=1,2,…,a¯. Then,(59)CSSPGHMÜ,Õcn1,cn2,…,cna¯=cn=IL,IU,f.Theorem 8 (commutativity).
Letcn1′,cn2′,…,cna¯′ be any permutation of cn1,cn2,…,cna¯. Then,(60)CSSPGHMÜ,Õcn1′,cn2′,…,cna¯′=CSSPGHMÜ,Õcn1,cn2,…,cna¯.Theorem 9 (boundedness).
Letcni=IiL,IiU,fii=1,2,…,a¯ be a faction of CNs, cn−=miniIiL,miniIiU,maxifi and Θ+=maxiIiL,maxiIiU,minifi. Then,(61)cn−≤CSSPGHMÜ,Õcn1,cn2,…,cna¯≤cn+.The proofs of these properties are the same as for CSSPHM operators. Hence, these are executed here.By specifying distinctive values of the parametersЖ,Üand Õ, numerous specific AOs are obtained from the CSSPGHM operator, and are stated as given below:Case 1. IfЖ=0, then the CSSPGHM operator reverts to the CPGHM operator and is stated as follows:(62)CSSPGHMÜ,Õcn1,cn2,…,cna¯=1Ü+Õ∏i,j=1j=ia¯Ücnia¯Ÿi⊕Õcnja¯Ÿj2/a¯a¯+1=1−1−∏i,j=1j=ia¯1−1−IiLa¯ŸiÜ1−IjLa¯ŸjÕ2/a¯a¯+11/Ü+Õ,1−1−∏i,j=1j=ia¯1−1−IiUa¯ŸiÜ1−IjUa¯ŸjÕ2/a¯a¯+11/Ü+Õ,1−∏i,j=1j=ia¯1−1−1−fia¯ŸiÜ1−1−fja¯ŸjÕ2/a¯a¯+11/Ü+Õ.Case 2. IfÕ⟶0, then the CSSPGHM operator reverts to the cubic descending PG operator and is stated as follows:(63)CSSPGHMÜ,0cn1,cn2,…,cna¯=limÕ⟶01Ü+Õ∏i,j=1j=ia¯Ücnia¯Ÿi⊕Õcnja¯Ÿj2/a¯a¯+1=1Ü∏i=1a¯Ücnia¯Ÿìa¯+1−i2/a¯a¯+1=1−1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−Ü1−a¯ŸiIiLЖ−a¯Ÿi−11/ЖЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж,1−1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−Ü1−a¯ŸiIiUЖ−a¯Ÿi−11/ЖЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж,1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−Ü1−a¯Ÿi1−fiЖ−a¯Ÿi−11/ЖЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж.Case 3. IfÜ⟶0, then the CSSPGHM operator reverts to the cubic ascending PG operator and is stated as follows:(64)CSSPGHM0,Õcn1,cn2,…,cna¯=limG⟶01Õ∏i,j=1j=ia¯Ücnia¯Ÿi⊕Õcnja¯Ÿj2/a¯a¯+1=1Õ∏i=1a¯Õcnia¯Ÿii2/a¯a¯+1=1−1Õ1−2a¯a¯+1∑i=1a¯i1−Õ1−a¯ŸiIiLЖ−a¯Ÿi−11/ЖЖ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж,1−1Õ1−2a¯a¯+1∑i=1a¯i1−Õ1−a¯ŸiIiUЖ−a¯Ÿi−11/ЖЖ−Õ−11ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж,1Õ1−2a¯a¯+1∑i=1a¯i1−Õ1−a¯Ÿi1−fiЖ−a¯Ÿi−11/ЖЖ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж.Case 4. IfÕ⟶0 and supcni,cnj=hh∈0,1∀i≠j, then the CSSPGHM operator reverts to the cubic linear descending weighted geometric operator and is stated as follows:(65)CSSPGHMÜ,0cn1,cn2,...,cna¯=limÕ⟶01Ü∏i,j=1j=ia¯Ücnia¯Ÿi⊕Õcnja¯Ÿj2/a¯a¯+1=1Ü∏i=1a¯Ücnia¯+1−i2/a¯a¯+1=1−1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−Ü1−IiLЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж,1−1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−Ü1−IiUЖ−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж,1Ü1−2a¯a¯+1∑i=1a¯a¯+1−i1−Üfì−Ü−11/ЖЖ−2a¯−i2a¯a¯+1−11/ЖЖ−1Ü−11/Ж.Certainly, the significant degree of CNscnii=1,2,…,a¯ is a¯,a¯−1,…,1.Case 5. IfÕ⟶0 and supcni,cnj=hh∈0,1∀i≠j, then the CSSPGHM operator reverts to the cubic linear ascending weighted geometric operator and is stated as follows:(66)CSSPGHM0,Õcn1,cn2,…,cna¯=limÕ⟶01Ü+Õ∏i,j=1j=ia¯Ücnia¯Ÿi⊕Õcnja¯Ÿj2/a¯a¯+1=1Õ∏i=1a¯Õcnii2/a¯a¯+1,=1−1Õ1−2a¯a¯+1∑i=1a¯i1−Õ1−IiLЖ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж,1−1Õ1−2a¯a¯+1∑i=1a¯i1−Õ1−IiUЖ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж1Õ1−2a¯a¯+1∑i=1†i1−ÕfiЖ−Õ−11/ЖЖ−a¯+i−12a¯a¯+1−11/ЖЖ−1Õ−11/Ж.Case 6. IfÜ=Õ=1/2 and supcni,cnj=hh∈0,1∀i≠j, then the CSSPGHM operator reverts to the cubic GHM operator and is stated as follows:(67)CSSPGHM1/2,1/2cn1,cn2,…,cna¯=1Ü+Õ∏i,j=1j=ia¯Ücnia¯Ÿi⊕Õcnja¯Ÿj2/a¯a¯+1=12∏i,j=1j=ia¯cni⊕cnj2/a¯a¯+1=2a¯a¯+1∑i,j=1j=ia¯1−121−IiLЖ+121−IjLЖ1/ЖЖ1/Ж,2a¯a¯+1∑i,j=1j=ia¯1−121−IiUЖ+121−IjUЖ1/ЖЖ1/Ж,1−2††+1∑i,j=1j=i†1−12fiЖ+12fjЖ1/ЖЖ1/Ж.Case 7. IfÜ=Õ=1 and supcni,cnj=hh∈0,1∀i≠j, then the CSSPGHM operator reverts to the cubic basic GHM operator and is stated as follows:(68)CSSPGHM1,1cn1,cn2,…,cna¯=1Ü+Õ∏i,j=1j=ia¯Ücnia¯Ÿi⊕Õcnja¯Ÿj2/a¯a¯+1=12∏i,j=1j=ia¯cni⊕cnj2/††+1=1−1−2a¯a¯+1∑i,j=1j=ia¯1−1−IiLЖ+1−jLЖ−11/ЖЖ1/ЖЖ+121/Ж,1−1−2a¯a¯+1∑i,j=1j=i†1−1−IiUЖ+1−IjUЖ−11/ЖЖ1/ЖЖ+121/Ж,121−2a¯a¯+1∑i,j=1j=i†1−fiЖ+fjЖ−11/ЖЖ1/ЖЖ+121/Ж.


## 4.2. CSSWPHM and CSSWPGHM Operators

In this portion, we instigate the CSSWPHM operator and the CSSWPGHM operator by taking the significance of the attributes.Definition 10.
LetÜ,Õ≥0,cni=IiL,IiU,fii=1,2,…,a¯ be a faction of CNs and ς=ς1,ς2,…,ςa¯T be the importance degree of cnii=1,2,…,a¯, where ςi≥0 and ∑i=1a¯ςi=1. The CSSWPHM operator is rationalized as follows:(69)CSSWPHMÜ,Õcn1,cn2,…,cna¯=2a¯a¯+1∑i,j=1j=ia¯a¯ςi1+Tcni∑z=1a¯ςz1+TcnzcniÜ⊗SSa¯ςj1+Tcnj∑z=1a¯ςz1+TcnzcnjÕ1/Ü+Õ,where Ü,Õ≥0,Tcni=∑j=1j≠i†supcni,cnj,supcni,cnj=1−DNE¯¯cni,cnj, and DNE¯¯cni,cnj can be reckoned by (6).
LetŸi=1+Tcni/∑z=1a¯1+Tcnz, then (69) be correspondent to the following shape:(70)CSSWPHMÜ,Õcn1,cn2,…,cna¯=2a¯a¯+1∑i,j=1j=ia¯a¯ςiŸicniÜ⊗SSa¯ŸjςjcnjÕ1/Ü+Õ.Theorem 10.
LetÜ≥0,Õ≥0 and Ü,Õ take no more than one value of 0 at a time, cni=IiL,IiU,fi be a faction of CNs. Then, exploiting the CSSWPHM operator, their merged values are CNs, and(71)CSSWPHMÜ,Õcn1,cn2,…,cna¯=1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ςiŸi1−IiLЖ−a¯ςiŸi−11/ЖЖ+Õ1−a¯ςjŸj1−IjLЖ−a¯ςjŸj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ςiŸi1−IiUЖ−a¯ςiŸi−11/ЖЖ+Õ1−a¯ςjŸj1−IjUЖ−a¯ςjŸj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1−1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ςiŸifiЖ−a¯ςiŸi−11/ЖЖ+Õ1−a¯ςjŸjfjЖ−a¯ςjŸj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж.Definition 11.
LetÜ,Õ≥0,cni=IiL,IiU,fii=1,2,…,a¯ be a faction of CNs and ς=ς1,ς2,…,ςa¯T be the significant degree of cnii=1,2,…,a¯, where ςi≥0 and ∑i=1a¯ςi=1. The CSSWPGHM operator is rationalized as follows:(72)CSSWPGHMÜ,Õcn1,cn2,…,cna¯=1Ü+Õ∏i,j=1j=ia¯Ücnia¯ςi1+Tcni/∑z=1a¯ςz1+Tcnz⊕SrSrÕcnja¯ςj1+Tcnj/∑z=1a¯ςz1+Tcnz2/a¯a¯+1,where Ü,Õ≥0,Tcni=∑j=1j≠ia¯supcni,cnj,supcni,cnj=1−DNE¯¯cni,cnj, and DNE¯¯cni,cnj can be processed by (6).
LetŸi=1+Tcni/∑z=1a¯1+Tcnz, then (72) is correspondent to the following type:(73)CSSWPGHMÜ,Õcn1,cn2,…,cn†=1Ü+Õ∏i,j=1j=ia¯Ücnia¯ςiŸi⊕SSÕcnja¯ςjŸj2/a¯a¯+1.Theorem 11.
LetÜ≥0,Õ≥0 and Ü,Õ take no more than one value of 0 at a time, cni=IiL,IiU,fii=1,2,…,a¯ be a faction of CNs. Then, employing the CSSWPGHM operator, their fused values are CNs, and(74)CSSWPGHMÜ,Õcn1,cn2,…,cna¯=1−1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ςiŸiIiLЖ−a¯ςiŸi−11/ЖЖ+Õ1−a¯ςjŸjIjLЖ−a¯ςjŸj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,1−1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ςiŸiIiUЖ−a¯ςiŸi−11/ЖЖ+Õ1−a¯ςjŸjIjUЖ−a¯ςjŸj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж,,1Ü+Õ1−2a¯a¯+1∑i,j=1j=ia¯1−Ü1−a¯ςiŸi1−fiЖ−a¯ςiŸi−11/ЖЖ+Õ1−a¯ςjŸj1−fjЖ−a¯jŸj−11/ЖЖ−Ü−Õ+11/ЖЖ1/ЖЖ−1Ü+Õ−11/Ж.
The proofs of Theorems10 and 11 are the same as Theorems 2 and 6. Therefore, here we omit their proofs.


## 5. An Application of SSPHM Operator to MADM

In this portion, we pertain to the aforementioned CSSPHM AOs to discover creative approaches for MADM under cubic environments. LetTr¯¯=tr¯¯1,tr¯¯2,…,tr¯¯g be the group of distant alternatives, the group of attributes is verbalized by Cei¯¯=Cei¯¯1,Cei¯¯2,…,Cei¯¯h, the importance degree of the attributes is epitomised by Wit¯¯=wit¯¯1,wit¯¯2,…,wit¯¯hT such that wit¯¯e∈0,1,∑e=1hwit¯¯e=1. In the procedure of decision making, the evaluation information about the alternative tr¯¯uu=1,2,…,g concerning the attribute Cei¯¯ww=1,2,…,h is expressed by a cubic decision matrix CD¯¯=cndeg×h, where cnde=ΨL,ΨU,ϒ is a CN.Afterwards, take a chance on actual decision scenarios where the weight vector of attributes has already been determined. As a result, we launch MADM methods based on the suggested CSSPHA operators.


### 5.1. MADM with Known Weight Vectors of Attributes

In this section, to pact with real decision circumstances in which the importance degrees of attributes are known in advance, we apply the CSSWPHM operator and CSSWPGHM operator to launch the following approach to solve MADM problems under cubic environments. To do so, instantly follow the steps given:Step 1. Locate supportSupcnde,cndx by utilizing the following formula:(75)Supcnde,cnex=1−DNE¯¯cnde,cnex,d=1,2,…,g,e=1,2,…,h,e≠x,whereDNE¯¯cnde,cnex is the distance measure and is intended by utilizing (6).Step 2. Discover the weighted support degreeTcnde that CN cnde collects from other CNs cn¯¯dxx=1,2,…,h;e≠x,(76)Tcnde=∑x=1,x≠ehsupcnde,cndx.Step 3. Establish a weighting vectorΞded=1,2,…,g,e=1,2,…,h associated with cnde ,(77)Ξde=gwit¯¯e1+Tcnde∑e=1hwit¯¯e1+Tcnde.Step 4. Employ CSSWPHOM or CSSWPGHOM operators to aggregate the evaluation valuescnded=1,2,…,g,e=1,2,…,h into an overall evaluation value cndd=1,2,…,g matching the alternatives tr¯¯dd=1,2,…,g;(78)cnd=CSSWPHOMζ,wit¯¯d1,cnd2,…,cnds,or(79)cnd=CSSWPGHOMζ,We¯¯cnd1,cnd2,…,cnds.Step 5. Locate the scoresSoF¯¯cnd for the overall CNs of the alternatives tr¯¯dd=1,2,…,g by manipulating Definition 2.Step 6. Rank all alternativestr¯¯dd=1,2,…,g and select the best one (s) with the ranking order cndd=1,2,…,g.


### 5.2. Illustrative Example

In this subpart, a numerical example adapted from [26] about enterprise resource planning in order to verify the unassailability and compensations of the initiated approach.Let us say a corporation decides to use an ERP system (enterprise resource planning). The specialist’s panel chosetr¯¯dg=1,2,3,4,5 five prospective investors after gathering all relevant information on ERP dealers and systems. Some external decision-making specialists are among the organization’s members. The group decides on five attributes Cei¯¯ee=1,2,3,4,5. To assess the alternatives, (1) function and technology Cei¯¯1, (2) strategic fitness Cei¯¯2, (3) the ability of the vendor Cei¯¯3, (4) reputation of the vendor Cei¯¯4, and (5) growth analysis of the vendor Cei¯¯5, with weight vectors of the attributes are 0.2,0.15,0.15,0.25,0.25T. CFN’s will be used by the expert’s committee to create the initial decision matrix given in Table 1. To solve this decision making, the following steps to be followed:Step 1. Analyse the supportSupcnded,cndxd by the following formula (61), and we have(80)S121=S211=0.9000,S131=S311=0.6667,S141=S411=0.7333,S151=S511=0.7667,S231=S321=0.7000,S241=S421=0.6333,S251=S521=0.8000,S341=S431=0.8333,S351=S531=0.8667,S451=S541=0.8333,S122=S212=0.6000,S132=S312=0.7667,S142=S412=0.7000,S152=S512=0.7000,S232=S322=0.8333,S242=S422=0.8333,S252=S522=0.8333,S342=S432=0.8667,S352=S532=0.7333,S452=S542=0.8000,S123=S213=0.7333,S133=S313=0.7000,S143=S413=0.9333,S153=S513=0.7000,S233=S323=0.7000,S243=S423=0.8333,S253=S523=0.9667,S343=S433=0.7333,S353=S533=0.7333,S453=S543=0.8000,S124=S214=0.8000,S134=S314=0.8667,S144=S414=0.8667,S154=S514=0.8667,S234=S324=0.8000,S244=S424=0.8667,S254=S524=0.7333,S344=S434=0.8667,S354=S534=0.9333,S454=S544=0.8000,S125=S215=0.7333,S135=S315=0.4333,S145=S415=0.7667,S155=S515=0.5,S235=S325=0.7000,S245=S425=0.9000,S255=S525=0.7667,S345=S435=0.6667,S355=S535=0.8667,S455=S545=0.7333.Step 2. Exploiting Equation (62), to discover the weighted support Tcnde that CN cnde collects from other CNs cn¯¯dxd,x=1,2,…,5;e≠x.,. For simplicity, we indicate Tcnde by Tde, we have(81)T11=3.0667,T12=3.0333,T13=3.0000,T14=3.0000,T15=3.2333,T21=2.7667,T22=3.1000,T23=3.2000,T24=3.2000,T25=3.0667,T31=3.0667,T32=3.2000,T33=2.8333,T34=3.2000,T35=3.1667,T41=3.4000,T42=3.2000,T43=3.4667,T44=3.4000,T45=3.3333,T51=2.4333,T52=3.1000,T53=2.6667,T54=3.0667,T55=2.8667.Step 3. Manipulating Equation (63), to locate the weight Ξded=1,....,5,e=1,...,5, we have(82)Ξ11=0.9975,Ξ12=0.7420,Ξ13=0.7359,Ξ14=1.2265,Ξ15=0.0173,Ξ21=0.9266,Ξ22=0.7565,Ξ23=0.7749,Ξ24=1.2915,Ξ25=0.0167,Ξ31=0.9895,Ξ32=0.7664,Ξ33=0.6995,Ξ34=1.2774,Ξ35=1.2672,Ξ41=1.0084,Ξ42=0.7219,Ξ43=0.7678,Ξ44=1.2605,Ξ45=1.2414,Ξ51=0.8953,Ξ52=0.8018,Ξ53=0.7171,Ξ54=1.3255,Ξ55=1.2603.Step 4Manipulating Equation (64) or (65), to locate the overall assessment values of each alternative, we have Ü=2,Õ=2,Ж=−2(83)tr1¯¯=0.2063,0.3921,0.5230,tr2¯¯=0.1599,0.3776,0.4751,tr3¯¯=0.3693,0.6628,0.5562,tr4¯¯=0.2892,0.5329,0.3297,tr5¯¯=0.4041,0.6646,0.3415Ortr1¯¯=0.3866,0.5985,0.6196,tr2¯¯=0.2869,0.5297,0.6201,tr3¯¯=0.3697,0.6298,0.6091,tr4¯¯=0.2973,0.5170,0.3474,tr5¯¯=0.3292,0.5763,0.4192.Step 5. Operating Equation (6) given in Definition 2, to locate the score values of overall trd¯¯d=1,…,5 CNs, to rank the alternatives, we have(84)SoF¯¯tr1¯¯=0.0251,SoF¯¯tr2¯¯=0.0208,SoF¯¯tr3¯¯=0.1586,SoF¯¯tr4¯¯=0.1641,SoF¯¯tr5¯¯=0.2424.OrSoF¯¯tr1¯¯=0.1218,SoF¯¯tr2¯¯=0.0655,SoF¯¯tr3¯¯=0.1301,SoF¯¯tr4¯¯=0.1556,SoF¯¯tr5¯¯=0.1621.Step 6. According to the score values, ranking order of the alternativestrd¯¯d=1,2,…,5 is as follows:(85)tr5¯¯>tr4¯¯>tr3¯¯>tr1¯¯>tr2¯¯ortr5¯¯>tr4¯¯>tr3¯¯>tr1¯¯>tr2¯¯.Table 1
Cubic decision making matrix.
Alternative/attributesCei¯¯1Cei¯¯2Cei¯¯3Cei¯¯4Cei¯¯5tr¯¯10.3,0.4,0.20.4,0.6,0.20.3,0.7,0.90.2,0.3,0.80.4,0.5,0.7tr¯¯20.1,0.2,0.70.3,0.7,0.20.2,0.4,0.30.4,0.5,0.40.3,0.8,0.6tr¯¯30.5,0.8,0.30.4,0.6,0.80.2,0.3,0.40.5,0.8,0.50.3,0.6,0.8tr¯¯40.2,0.4,0.30.3,0.8,0.20.3,0.5,0.50.4,0.5,0.20.3,0.4,0.6tr¯¯50.8,0.9,0.20.4,0.6,0.30.2,0.4,0.80.3,0.7,0.20.2,0.3,0.5Therefore, according to the ranking order, the best alternative istr5¯¯, while the worst one is tr2¯¯.


### 5.3. Impact of the ParameterЖ on Final Ranking Orders Applying CSrSrWPHM and CSrSrWPGHM Operators

In this subportion, the impact of the parameterЖ on last ranking orders employing CSSWPHM and CSSWPGHM operators is explored, and the value of parameters Ü=Õ=2 are permanent. For distinct values of the parameter Ж, the score values and ranking orders while operating CSSWPHM and CSSWPGHM operators are given in Table 2. One can observe from Table 2 that for different values of the parameter Ж the ranking orders are different. That is, employing the CSSWPHM operator and CSSWPGHM operator the best alternative is either tr5¯¯ or tr4¯¯ for different values of Ж while the worst alternative is either tr2¯¯ or tr1¯¯. One can also see from Table 2 that when the values of the parameter Ж decrease, the score values of the alternative trg¯¯g=1,…,5 increases while exploiting the CSSWPHM operator. Similarly, from Table 2, when the values of the parameter Ж decrease, the score values of the alternatives trg¯¯g=1,…,5 decreases.Table 2
Effect of the parameterЖ on final ranking order.
ParameterCSSWPHM operatorCSSWPGHM operatorRanking orderЖ=−3SoF¯¯tr1¯¯=0.0558,SoF¯¯tr2¯¯=0.0461,SoF¯¯tr3¯¯=0.1662,SoF¯¯tr4¯¯=0.1694,SoF¯¯tr5¯¯=0.2681.SoF¯¯tr1¯¯=0.0847,SoF¯¯tr2¯¯=0.0226,SoF¯¯tr3¯¯=0.1203,SoF¯¯tr4¯¯=0.1526,SoF¯¯tr5¯¯=0.1453.tr5¯¯>tr4¯¯>tr3¯¯>tr1¯¯>tr2¯¯ortr4¯¯>tr5¯¯>tr3¯¯>tr1¯¯>tr2¯¯Ж=−4SoF¯¯tr1¯¯=0.0882,SoF¯¯tr2¯¯=0.0741,SoF¯¯tr3¯¯=0.1788,SoF¯¯tr4¯¯=0.1782,SoF¯¯tr5¯¯=0.3019.SoF¯¯tr1¯¯=0.0505,SoF¯¯tr2¯¯=−0.0161,SoF¯¯tr3¯¯=0.1052,SoF¯¯tr4¯¯=0.1454,SoF¯¯tr5¯¯=0.1196.tr5¯¯>tr3¯¯>tr4¯¯>tr1¯¯>tr2¯¯ortr4¯¯>tr5¯¯>tr3¯¯>tr1¯¯>tr2¯¯Ж=−7SoF¯¯tr1¯¯=0.1557,SoF¯¯tr2¯¯=0.1420,SoF¯¯tr3¯¯=0.2193,SoF¯¯tr4¯¯=0.2158,SoF¯¯tr5¯¯=0.3889.SoF¯¯tr1¯¯=−0.0154,SoF¯¯tr2¯¯=−0.0914,SoF¯¯tr3¯¯=0.0475,SoF¯¯tr4¯¯=0.1166,SoF¯¯tr5¯¯=0.0394.tr5¯¯>tr3¯¯>tr4¯¯>tr1¯¯>tr2¯¯ortr4¯¯>tr3¯¯>tr5¯¯>tr1¯¯>tr2¯¯Ж=−10SoF¯¯tr1¯¯=0.1922,SoF¯¯tr2¯¯=0.1885,SoF¯¯tr3¯¯=0.2490,SoF¯¯tr4¯¯=0.2460,SoF¯¯tr5¯¯=0.4301.SoF¯¯tr1¯¯=−0.0502,SoF¯¯tr2¯¯=−0.1291,SoF¯¯tr3¯¯=0.0007,SoF¯¯tr4¯¯=0.0909,SoF¯¯tr5¯¯=−0.0079.tr5¯¯>tr3¯¯>tr4¯¯>tr1¯¯>tr2¯¯ortr4¯¯>tr3¯¯>tr5¯¯>tr1¯¯>tr2¯¯Ж=−20SoF¯¯tr1¯¯=0.2426,SoF¯¯tr2¯¯=0.2612,SoF¯¯tr3¯¯=0.2939,SoF¯¯tr4¯¯=0.2865,SoF¯¯tr5¯¯=0.4717.SoF¯¯tr1¯¯=−0.0964,SoF¯¯tr2¯¯=−0.1709,SoF¯¯tr3¯¯=−0.0581,SoF¯¯tr4¯¯=0.0455,SoF¯¯tr5¯¯=−0.0615.tr5¯¯>tr3¯¯>tr4¯¯>tr1¯¯>tr2¯¯ortr4¯¯>tr3¯¯>tr5¯¯>tr1¯¯>tr2¯¯Ж=−35SoF¯¯tr1¯¯=0.2677,SoF¯¯tr2¯¯=0.2954,SoF¯¯tr3¯¯=0.3128,SoF¯¯tr4¯¯=0.3070,SoF¯¯tr5¯¯=0.4860.SoF¯¯tr1¯¯=−0.1148,SoF¯¯tr2¯¯=−0.1856,SoF¯¯tr3¯¯=−0.0788,SoF¯¯tr4¯¯=0.0238,SoF¯¯tr5¯¯=−0.0806.tr5¯¯>tr3¯¯>tr4¯¯>tr1¯¯>tr2¯¯ortr4¯¯>tr3¯¯>tr5¯¯>tr1¯¯>tr2¯¯Ж=−50SoF¯¯tr1¯¯=0.2780,SoF¯¯tr2¯¯=0.3079,SoF¯¯tr3¯¯=0.3196,SoF¯¯tr4¯¯=0.3155,SoF¯¯tr5¯¯=0.4908.SoF¯¯tr1¯¯=−0.1212,SoF¯¯tr2¯¯=−0.1907,SoF¯¯tr3¯¯=−0.0859,SoF¯¯tr4¯¯=0.0159,SoF¯¯tr5¯¯=−0.0872.tr5¯¯>tr3¯¯>tr4¯¯>tr2¯¯>tr1¯¯ortr4¯¯>tr3¯¯>tr5¯¯>tr1¯¯>tr2¯¯Ж=−100SoF¯¯tr1¯¯=0.2895,SoF¯¯tr2¯¯=0.3213,SoF¯¯tr3¯¯=0.3268,SoF¯¯tr4¯¯=0.3248,SoF¯¯tr5¯¯=0.4958.SoF¯¯tr1¯¯=−0.1277,SoF¯¯tr2¯¯=−0.1958,SoF¯¯tr3¯¯=−0.0934,SoF¯¯tr4¯¯=0.0075,SoF¯¯tr5¯¯=−0.0940.tr5¯¯>tr3¯¯>tr4¯¯>tr2¯¯>tr1¯¯ortr4¯¯>tr3¯¯>tr5¯¯>tr1¯¯>tr2¯¯


### 5.4. Effect of the ParametersÜ,Õ on Final Ranking Order

In this subsegment, the effect of the parametersÜ,Õ on last ranking orders operating CSSWPHM and CSSWPGHM operators are inspected, and the value of parameter Ж=−2 are permanent. For distinct values of the parameters A,B, the score values and ranking orders while operating CSSWPHM and CSSWPGHM operators are quantified in Table 3. From Table 3, one can observe that when the values of the parameters Ü,Õ are different, the ranking orders obtained are different. That is, to employ the CSSWPHM operator and CSSWPGHM operator, the best alternative is either tr5¯¯ or tr4¯¯ or tr1¯¯ for different values of the parameters Ü,Õ, while the worst alternative is either tr2¯¯ or tr1¯¯ or tr3¯¯. One can also see from Table 3 that when the values of the parameters Ü,Õ increase, the score values of the alternatives trg¯¯g=1,…,5 decrease, while exploiting the CSSWPHM operator. Similarly, from Table 3, when the values of the parameter Ü,Õ decrease, the score values of the alternatives trg¯¯g=1,…,5 increase, while exploiting the CSSWPHM operator. The basic reason for this is because the above AOs are more adjustable since they are constituted of generic parameters, limit the influence of inconvenient information, and take into account the relationship between input information. As a result, the MADM model developed on these aggregation operators is more adaptable. As a consequence, the decision-maker may modify the values of these parameters to the specific requirements of the scenario.Table 3
Effect of the parameterÜ,Õ on final ranking order.
ParameterCSSWPHM operatorCSSWPGHM operatorRanking orderÜ=2,Õ=3SoF¯¯tr1¯¯=0.0170,SoF¯¯tr2¯¯=0.0176,SoF¯¯tr3¯¯=0.1532,SoF¯¯tr4¯¯=0.1639,SoF¯¯tr5¯¯=0.2316.SoF¯¯tr1¯¯=0.1285,SoF¯¯tr2¯¯=0.0802,SoF¯¯tr3¯¯=0.1281,SoF¯¯tr4¯¯=0.1544,SoF¯¯tr5¯¯=0.1603.tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯ortr5¯¯>tr4¯¯>tr1¯¯>tr3¯¯>tr2¯¯Ü=3,Õ=5SoF¯¯tr1¯¯=0.0041,SoF¯¯tr2¯¯=0.0108,SoF¯¯tr3¯¯=0.1473,SoF¯¯tr4¯¯=0.1614,SoF¯¯tr5¯¯=0.2185.SoF¯¯tr1¯¯=0.1458,SoF¯¯tr2¯¯=0.1010,SoF¯¯tr3¯¯=0.1316,SoF¯¯tr4¯¯=0.1565,SoF¯¯tr5¯¯=0.1665.tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯ortr5¯¯>tr4¯¯>tr1¯¯>tr3¯¯>tr2¯¯Ü=5,Õ=3SoF¯¯tr1¯¯=0.0096,SoF¯¯tr2¯¯=0.0104,SoF¯¯tr3¯¯=0.1556,SoF¯¯tr4¯¯=0.1603,SoF¯¯tr5¯¯=0.2337.SoF¯¯tr1¯¯=0.1502,SoF¯¯tr2¯¯=0.0867,SoF¯¯tr3¯¯=0.1418,SoF¯¯tr4¯¯=0.1622,SoF¯¯tr5¯¯=0.1810.tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯ortr5¯¯>tr4¯¯>tr1¯¯>tr3¯¯>tr2¯¯Ü=7,Õ=15SoF¯¯tr1¯¯=−0.0166,SoF¯¯tr2¯¯=0.0011,SoF¯¯tr3¯¯=0.1379,SoF¯¯tr4¯¯=0.1580,SoF¯¯tr5¯¯=0.1972.SoF¯¯tr1¯¯=0.1743,SoF¯¯tr2¯¯=0.1366,SoF¯¯tr3¯¯=0.1350,SoF¯¯tr4¯¯=0.1586,SoF¯¯tr5¯¯=0.1737.tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯ortr4¯¯>tr5¯¯>tr4¯¯>tr2¯¯>tr3¯¯Ü=15,Õ=7SoF¯¯tr1¯¯=−0.0073,SoF¯¯tr2¯¯=−0.0009,SoF¯¯tr3¯¯=0.1500,SoF¯¯tr4¯¯=0.1570,SoF¯¯tr5¯¯=0.2216.SoF¯¯tr1¯¯=0.1813,SoF¯¯tr2¯¯=0.1169,SoF¯¯tr3¯¯=0.1514,SoF¯¯tr4¯¯=0.1672,SoF¯¯tr5¯¯=0.1973.tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯ortr4¯¯>tr5¯¯>tr4¯¯>tr2¯¯>tr3¯¯Ü=12,Õ=25SoF¯¯tr1¯¯=−0.0237,SoF¯¯tr2¯¯=−0.0033,SoF¯¯tr3¯¯=0.1346,SoF¯¯tr4¯¯=0.1559,SoF¯¯tr5¯¯=0.1904.SoF¯¯tr1¯¯=0.1863,SoF¯¯tr2¯¯=0.1493,SoF¯¯tr3¯¯=0.1384,SoF¯¯tr4¯¯=0.1606,SoF¯¯tr5¯¯=0.1789.tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯ortr4¯¯>tr5¯¯>tr4¯¯>tr2¯¯>tr3¯¯Ü=25,Õ=12SoF¯¯tr1¯¯=−0.0144,SoF¯¯tr2¯¯=−0.0056,SoF¯¯tr3¯¯=0.1462,SoF¯¯tr4¯¯=0.1551,SoF¯¯tr5¯¯=0.2142.SoF¯¯tr1¯¯=0.1931,SoF¯¯tr2¯¯=0.1308,SoF¯¯tr3¯¯=0.1545,SoF¯¯tr4¯¯=0.1689,SoF¯¯tr5¯¯=0.2023.tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯ortr5¯¯>tr1¯¯>tr4¯¯>tr3¯¯>tr2¯¯Ü=30,Õ=50SoF¯¯tr1¯¯=−0.0307,SoF¯¯tr2¯¯=−0.0097,SoF¯¯tr3¯¯=0.1311,SoF¯¯tr4¯¯=0.1526,SoF¯¯tr5¯¯=0.1846.SoF¯¯tr1¯¯=0.2015,SoF¯¯tr2¯¯=0.1623,SoF¯¯tr3¯¯=0.1455,SoF¯¯tr4¯¯=0.1648,SoF¯¯tr5¯¯=0.1893.tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯ortr1¯¯>tr5¯¯>tr4¯¯>tr2¯¯>tr3¯¯


### 5.5. Comparison with Existing Approaches

In this section, the developed MADM model, which is predicated on this newly established novel AGOs, to various current approaches, namely, Mahmood et al. [12], Ayub et al. [24], and Fahmi et al. [26] MADM models. The comparison between these approaches and the proposed approach is given in Table 4. From Table 4, we can observe that the best alternative obtained from the existing approaches and the proposed approach is the same except for utilizing the CWHMDA operator, while the worst alternative obtained from the existing approaches and the proposed approach is different. The MADM model that will be implemented is based on the recently launched aggregation operators. To put it another way, these aggregation operators are suggested for CNs using Schweizer–Sklar operational, which are generic parameters that make the decision-making procedure more adaptable. In the meanwhile, existing MADM models are drawn on aggregation operators that are launched using algebraic operational laws or Dombi operational laws. The aggregation operator proposed by Mahmood et al. [12] is a simple weighted aggregation operator, which does not have the capacity of taking interrelationships or removing the effect of awkward data from the final ranking results. While, the aggregation operators developed by Ayub et al. [24], are drawn on Dombi operational laws, which have the capacity of tinterrelationships among input arguments and also consist of a generic parameter. The aggregation operators developed by Fahmi et al. [26] are simple weighted averaging operators based on Einstein operational laws for CNs. These aggregation operators do not have the characteristic of taking interrelationship among input arguments or removing the effect of awkward data from the final ranking results. Up to now, the existing aggregation operators for CNs have only the capacity of considering interrelationships among input arguments and cannot remove the effect of awkward data from the final ranking results. Whereas, anticipated aggregation operators will be able to remove the effect of awkward data while also taking into account the interrelationship between the input data at the same time. The predicted AOs also have the benefit of including generic parameters, which sorts the decision-making procedure more supple. As a result, while solving MADM models using cubic information, the initiated AOs are more practical and comparable in their application.Table 4
Comparison with existing approaches.
ApproachesScore valuesRanking ordersCWA [12]SoF¯¯tr1¯¯=0.1098,SoF¯¯tr2¯¯=0.1447,SoF¯¯tr3¯¯=0.1850,SoF¯¯tr4¯¯=0.1692,SoF¯¯tr5¯¯=0.2549tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯CWG [12]SoF¯¯tr1¯¯=0.0277,SoF¯¯tr2¯¯=0.0747,SoF¯¯tr3¯¯=0.1225,SoF¯¯tr4¯¯=0.1309,SoF¯¯tr5¯¯=0.1418tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯CEWA [26]SoF¯¯tr1¯¯=0.0962,SoF¯¯tr2¯¯=0.1348,SoF¯¯tr3¯¯=0.1762,SoF¯¯tr4¯¯=0.1635,SoF¯¯tr5¯¯=0.2378tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯CFWHMDA [26]SoF¯¯tr1¯¯=0.0029,SoF¯¯tr2¯¯=0.0735,SoF¯¯tr3¯¯=0.0077,SoF¯¯tr4¯¯=0.574,SoF¯¯tr5¯¯=0.0718tr2¯¯>tr5¯¯>tr4¯¯>tr3¯¯>tr1¯¯CFWGHMDA [26]SoF¯¯tr1¯¯=0.1210,SoF¯¯tr2¯¯=0.1635,SoF¯¯tr3¯¯=0.2144,SoF¯¯tr4¯¯=0.1574,SoF¯¯tr5¯¯=0.2784tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯Proposed AOsSoF¯¯tr1¯¯=0.0251,SoF¯¯tr2¯¯=0.0208,SoF¯¯tr3¯¯=0.1586,SoF¯¯tr4¯¯=0.1641,SoF¯¯tr5¯¯=0.2424tr5¯¯>tr4¯¯>tr3¯¯>tr1¯¯>tr2¯¯Proposed AOsSoF¯¯tr1¯¯=0.1218,SoF¯¯tr2¯¯=0.0655,SoF¯¯tr3¯¯=0.1301,SoF¯¯tr4¯¯=0.1556,SoF¯¯tr5¯¯=0.1621tr5¯¯>tr4¯¯>tr3¯¯>tr1¯¯>tr2¯¯


## 5.1. MADM with Known Weight Vectors of Attributes

In this section, to pact with real decision circumstances in which the importance degrees of attributes are known in advance, we apply the CSSWPHM operator and CSSWPGHM operator to launch the following approach to solve MADM problems under cubic environments. To do so, instantly follow the steps given:Step 1. Locate supportSupcnde,cndx by utilizing the following formula:(75)Supcnde,cnex=1−DNE¯¯cnde,cnex,d=1,2,…,g,e=1,2,…,h,e≠x,whereDNE¯¯cnde,cnex is the distance measure and is intended by utilizing (6).Step 2. Discover the weighted support degreeTcnde that CN cnde collects from other CNs cn¯¯dxx=1,2,…,h;e≠x,(76)Tcnde=∑x=1,x≠ehsupcnde,cndx.Step 3. Establish a weighting vectorΞded=1,2,…,g,e=1,2,…,h associated with cnde ,(77)Ξde=gwit¯¯e1+Tcnde∑e=1hwit¯¯e1+Tcnde.Step 4. Employ CSSWPHOM or CSSWPGHOM operators to aggregate the evaluation valuescnded=1,2,…,g,e=1,2,…,h into an overall evaluation value cndd=1,2,…,g matching the alternatives tr¯¯dd=1,2,…,g;(78)cnd=CSSWPHOMζ,wit¯¯d1,cnd2,…,cnds,or(79)cnd=CSSWPGHOMζ,We¯¯cnd1,cnd2,…,cnds.Step 5. Locate the scoresSoF¯¯cnd for the overall CNs of the alternatives tr¯¯dd=1,2,…,g by manipulating Definition 2.Step 6. Rank all alternativestr¯¯dd=1,2,…,g and select the best one (s) with the ranking order cndd=1,2,…,g.


## 5.2. Illustrative Example

In this subpart, a numerical example adapted from [26] about enterprise resource planning in order to verify the unassailability and compensations of the initiated approach.Let us say a corporation decides to use an ERP system (enterprise resource planning). The specialist’s panel chosetr¯¯dg=1,2,3,4,5 five prospective investors after gathering all relevant information on ERP dealers and systems. Some external decision-making specialists are among the organization’s members. The group decides on five attributes Cei¯¯ee=1,2,3,4,5. To assess the alternatives, (1) function and technology Cei¯¯1, (2) strategic fitness Cei¯¯2, (3) the ability of the vendor Cei¯¯3, (4) reputation of the vendor Cei¯¯4, and (5) growth analysis of the vendor Cei¯¯5, with weight vectors of the attributes are 0.2,0.15,0.15,0.25,0.25T. CFN’s will be used by the expert’s committee to create the initial decision matrix given in Table 1. To solve this decision making, the following steps to be followed:Step 1. Analyse the supportSupcnded,cndxd by the following formula (61), and we have(80)S121=S211=0.9000,S131=S311=0.6667,S141=S411=0.7333,S151=S511=0.7667,S231=S321=0.7000,S241=S421=0.6333,S251=S521=0.8000,S341=S431=0.8333,S351=S531=0.8667,S451=S541=0.8333,S122=S212=0.6000,S132=S312=0.7667,S142=S412=0.7000,S152=S512=0.7000,S232=S322=0.8333,S242=S422=0.8333,S252=S522=0.8333,S342=S432=0.8667,S352=S532=0.7333,S452=S542=0.8000,S123=S213=0.7333,S133=S313=0.7000,S143=S413=0.9333,S153=S513=0.7000,S233=S323=0.7000,S243=S423=0.8333,S253=S523=0.9667,S343=S433=0.7333,S353=S533=0.7333,S453=S543=0.8000,S124=S214=0.8000,S134=S314=0.8667,S144=S414=0.8667,S154=S514=0.8667,S234=S324=0.8000,S244=S424=0.8667,S254=S524=0.7333,S344=S434=0.8667,S354=S534=0.9333,S454=S544=0.8000,S125=S215=0.7333,S135=S315=0.4333,S145=S415=0.7667,S155=S515=0.5,S235=S325=0.7000,S245=S425=0.9000,S255=S525=0.7667,S345=S435=0.6667,S355=S535=0.8667,S455=S545=0.7333.Step 2. Exploiting Equation (62), to discover the weighted support Tcnde that CN cnde collects from other CNs cn¯¯dxd,x=1,2,…,5;e≠x.,. For simplicity, we indicate Tcnde by Tde, we have(81)T11=3.0667,T12=3.0333,T13=3.0000,T14=3.0000,T15=3.2333,T21=2.7667,T22=3.1000,T23=3.2000,T24=3.2000,T25=3.0667,T31=3.0667,T32=3.2000,T33=2.8333,T34=3.2000,T35=3.1667,T41=3.4000,T42=3.2000,T43=3.4667,T44=3.4000,T45=3.3333,T51=2.4333,T52=3.1000,T53=2.6667,T54=3.0667,T55=2.8667.Step 3. Manipulating Equation (63), to locate the weight Ξded=1,....,5,e=1,...,5, we have(82)Ξ11=0.9975,Ξ12=0.7420,Ξ13=0.7359,Ξ14=1.2265,Ξ15=0.0173,Ξ21=0.9266,Ξ22=0.7565,Ξ23=0.7749,Ξ24=1.2915,Ξ25=0.0167,Ξ31=0.9895,Ξ32=0.7664,Ξ33=0.6995,Ξ34=1.2774,Ξ35=1.2672,Ξ41=1.0084,Ξ42=0.7219,Ξ43=0.7678,Ξ44=1.2605,Ξ45=1.2414,Ξ51=0.8953,Ξ52=0.8018,Ξ53=0.7171,Ξ54=1.3255,Ξ55=1.2603.Step 4Manipulating Equation (64) or (65), to locate the overall assessment values of each alternative, we have Ü=2,Õ=2,Ж=−2(83)tr1¯¯=0.2063,0.3921,0.5230,tr2¯¯=0.1599,0.3776,0.4751,tr3¯¯=0.3693,0.6628,0.5562,tr4¯¯=0.2892,0.5329,0.3297,tr5¯¯=0.4041,0.6646,0.3415Ortr1¯¯=0.3866,0.5985,0.6196,tr2¯¯=0.2869,0.5297,0.6201,tr3¯¯=0.3697,0.6298,0.6091,tr4¯¯=0.2973,0.5170,0.3474,tr5¯¯=0.3292,0.5763,0.4192.Step 5. Operating Equation (6) given in Definition 2, to locate the score values of overall trd¯¯d=1,…,5 CNs, to rank the alternatives, we have(84)SoF¯¯tr1¯¯=0.0251,SoF¯¯tr2¯¯=0.0208,SoF¯¯tr3¯¯=0.1586,SoF¯¯tr4¯¯=0.1641,SoF¯¯tr5¯¯=0.2424.OrSoF¯¯tr1¯¯=0.1218,SoF¯¯tr2¯¯=0.0655,SoF¯¯tr3¯¯=0.1301,SoF¯¯tr4¯¯=0.1556,SoF¯¯tr5¯¯=0.1621.Step 6. According to the score values, ranking order of the alternativestrd¯¯d=1,2,…,5 is as follows:(85)tr5¯¯>tr4¯¯>tr3¯¯>tr1¯¯>tr2¯¯ortr5¯¯>tr4¯¯>tr3¯¯>tr1¯¯>tr2¯¯.Table 1
Cubic decision making matrix.
Alternative/attributesCei¯¯1Cei¯¯2Cei¯¯3Cei¯¯4Cei¯¯5tr¯¯10.3,0.4,0.20.4,0.6,0.20.3,0.7,0.90.2,0.3,0.80.4,0.5,0.7tr¯¯20.1,0.2,0.70.3,0.7,0.20.2,0.4,0.30.4,0.5,0.40.3,0.8,0.6tr¯¯30.5,0.8,0.30.4,0.6,0.80.2,0.3,0.40.5,0.8,0.50.3,0.6,0.8tr¯¯40.2,0.4,0.30.3,0.8,0.20.3,0.5,0.50.4,0.5,0.20.3,0.4,0.6tr¯¯50.8,0.9,0.20.4,0.6,0.30.2,0.4,0.80.3,0.7,0.20.2,0.3,0.5Therefore, according to the ranking order, the best alternative istr5¯¯, while the worst one is tr2¯¯.


## 5.3. Impact of the ParameterЖ on Final Ranking Orders Applying CSrSrWPHM and CSrSrWPGHM Operators

In this subportion, the impact of the parameterЖ on last ranking orders employing CSSWPHM and CSSWPGHM operators is explored, and the value of parameters Ü=Õ=2 are permanent. For distinct values of the parameter Ж, the score values and ranking orders while operating CSSWPHM and CSSWPGHM operators are given in Table 2. One can observe from Table 2 that for different values of the parameter Ж the ranking orders are different. That is, employing the CSSWPHM operator and CSSWPGHM operator the best alternative is either tr5¯¯ or tr4¯¯ for different values of Ж while the worst alternative is either tr2¯¯ or tr1¯¯. One can also see from Table 2 that when the values of the parameter Ж decrease, the score values of the alternative trg¯¯g=1,…,5 increases while exploiting the CSSWPHM operator. Similarly, from Table 2, when the values of the parameter Ж decrease, the score values of the alternatives trg¯¯g=1,…,5 decreases.Table 2
Effect of the parameterЖ on final ranking order.
ParameterCSSWPHM operatorCSSWPGHM operatorRanking orderЖ=−3SoF¯¯tr1¯¯=0.0558,SoF¯¯tr2¯¯=0.0461,SoF¯¯tr3¯¯=0.1662,SoF¯¯tr4¯¯=0.1694,SoF¯¯tr5¯¯=0.2681.SoF¯¯tr1¯¯=0.0847,SoF¯¯tr2¯¯=0.0226,SoF¯¯tr3¯¯=0.1203,SoF¯¯tr4¯¯=0.1526,SoF¯¯tr5¯¯=0.1453.tr5¯¯>tr4¯¯>tr3¯¯>tr1¯¯>tr2¯¯ortr4¯¯>tr5¯¯>tr3¯¯>tr1¯¯>tr2¯¯Ж=−4SoF¯¯tr1¯¯=0.0882,SoF¯¯tr2¯¯=0.0741,SoF¯¯tr3¯¯=0.1788,SoF¯¯tr4¯¯=0.1782,SoF¯¯tr5¯¯=0.3019.SoF¯¯tr1¯¯=0.0505,SoF¯¯tr2¯¯=−0.0161,SoF¯¯tr3¯¯=0.1052,SoF¯¯tr4¯¯=0.1454,SoF¯¯tr5¯¯=0.1196.tr5¯¯>tr3¯¯>tr4¯¯>tr1¯¯>tr2¯¯ortr4¯¯>tr5¯¯>tr3¯¯>tr1¯¯>tr2¯¯Ж=−7SoF¯¯tr1¯¯=0.1557,SoF¯¯tr2¯¯=0.1420,SoF¯¯tr3¯¯=0.2193,SoF¯¯tr4¯¯=0.2158,SoF¯¯tr5¯¯=0.3889.SoF¯¯tr1¯¯=−0.0154,SoF¯¯tr2¯¯=−0.0914,SoF¯¯tr3¯¯=0.0475,SoF¯¯tr4¯¯=0.1166,SoF¯¯tr5¯¯=0.0394.tr5¯¯>tr3¯¯>tr4¯¯>tr1¯¯>tr2¯¯ortr4¯¯>tr3¯¯>tr5¯¯>tr1¯¯>tr2¯¯Ж=−10SoF¯¯tr1¯¯=0.1922,SoF¯¯tr2¯¯=0.1885,SoF¯¯tr3¯¯=0.2490,SoF¯¯tr4¯¯=0.2460,SoF¯¯tr5¯¯=0.4301.SoF¯¯tr1¯¯=−0.0502,SoF¯¯tr2¯¯=−0.1291,SoF¯¯tr3¯¯=0.0007,SoF¯¯tr4¯¯=0.0909,SoF¯¯tr5¯¯=−0.0079.tr5¯¯>tr3¯¯>tr4¯¯>tr1¯¯>tr2¯¯ortr4¯¯>tr3¯¯>tr5¯¯>tr1¯¯>tr2¯¯Ж=−20SoF¯¯tr1¯¯=0.2426,SoF¯¯tr2¯¯=0.2612,SoF¯¯tr3¯¯=0.2939,SoF¯¯tr4¯¯=0.2865,SoF¯¯tr5¯¯=0.4717.SoF¯¯tr1¯¯=−0.0964,SoF¯¯tr2¯¯=−0.1709,SoF¯¯tr3¯¯=−0.0581,SoF¯¯tr4¯¯=0.0455,SoF¯¯tr5¯¯=−0.0615.tr5¯¯>tr3¯¯>tr4¯¯>tr1¯¯>tr2¯¯ortr4¯¯>tr3¯¯>tr5¯¯>tr1¯¯>tr2¯¯Ж=−35SoF¯¯tr1¯¯=0.2677,SoF¯¯tr2¯¯=0.2954,SoF¯¯tr3¯¯=0.3128,SoF¯¯tr4¯¯=0.3070,SoF¯¯tr5¯¯=0.4860.SoF¯¯tr1¯¯=−0.1148,SoF¯¯tr2¯¯=−0.1856,SoF¯¯tr3¯¯=−0.0788,SoF¯¯tr4¯¯=0.0238,SoF¯¯tr5¯¯=−0.0806.tr5¯¯>tr3¯¯>tr4¯¯>tr1¯¯>tr2¯¯ortr4¯¯>tr3¯¯>tr5¯¯>tr1¯¯>tr2¯¯Ж=−50SoF¯¯tr1¯¯=0.2780,SoF¯¯tr2¯¯=0.3079,SoF¯¯tr3¯¯=0.3196,SoF¯¯tr4¯¯=0.3155,SoF¯¯tr5¯¯=0.4908.SoF¯¯tr1¯¯=−0.1212,SoF¯¯tr2¯¯=−0.1907,SoF¯¯tr3¯¯=−0.0859,SoF¯¯tr4¯¯=0.0159,SoF¯¯tr5¯¯=−0.0872.tr5¯¯>tr3¯¯>tr4¯¯>tr2¯¯>tr1¯¯ortr4¯¯>tr3¯¯>tr5¯¯>tr1¯¯>tr2¯¯Ж=−100SoF¯¯tr1¯¯=0.2895,SoF¯¯tr2¯¯=0.3213,SoF¯¯tr3¯¯=0.3268,SoF¯¯tr4¯¯=0.3248,SoF¯¯tr5¯¯=0.4958.SoF¯¯tr1¯¯=−0.1277,SoF¯¯tr2¯¯=−0.1958,SoF¯¯tr3¯¯=−0.0934,SoF¯¯tr4¯¯=0.0075,SoF¯¯tr5¯¯=−0.0940.tr5¯¯>tr3¯¯>tr4¯¯>tr2¯¯>tr1¯¯ortr4¯¯>tr3¯¯>tr5¯¯>tr1¯¯>tr2¯¯


## 5.4. Effect of the ParametersÜ,Õ on Final Ranking Order

In this subsegment, the effect of the parametersÜ,Õ on last ranking orders operating CSSWPHM and CSSWPGHM operators are inspected, and the value of parameter Ж=−2 are permanent. For distinct values of the parameters A,B, the score values and ranking orders while operating CSSWPHM and CSSWPGHM operators are quantified in Table 3. From Table 3, one can observe that when the values of the parameters Ü,Õ are different, the ranking orders obtained are different. That is, to employ the CSSWPHM operator and CSSWPGHM operator, the best alternative is either tr5¯¯ or tr4¯¯ or tr1¯¯ for different values of the parameters Ü,Õ, while the worst alternative is either tr2¯¯ or tr1¯¯ or tr3¯¯. One can also see from Table 3 that when the values of the parameters Ü,Õ increase, the score values of the alternatives trg¯¯g=1,…,5 decrease, while exploiting the CSSWPHM operator. Similarly, from Table 3, when the values of the parameter Ü,Õ decrease, the score values of the alternatives trg¯¯g=1,…,5 increase, while exploiting the CSSWPHM operator. The basic reason for this is because the above AOs are more adjustable since they are constituted of generic parameters, limit the influence of inconvenient information, and take into account the relationship between input information. As a result, the MADM model developed on these aggregation operators is more adaptable. As a consequence, the decision-maker may modify the values of these parameters to the specific requirements of the scenario.Table 3
Effect of the parameterÜ,Õ on final ranking order.
ParameterCSSWPHM operatorCSSWPGHM operatorRanking orderÜ=2,Õ=3SoF¯¯tr1¯¯=0.0170,SoF¯¯tr2¯¯=0.0176,SoF¯¯tr3¯¯=0.1532,SoF¯¯tr4¯¯=0.1639,SoF¯¯tr5¯¯=0.2316.SoF¯¯tr1¯¯=0.1285,SoF¯¯tr2¯¯=0.0802,SoF¯¯tr3¯¯=0.1281,SoF¯¯tr4¯¯=0.1544,SoF¯¯tr5¯¯=0.1603.tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯ortr5¯¯>tr4¯¯>tr1¯¯>tr3¯¯>tr2¯¯Ü=3,Õ=5SoF¯¯tr1¯¯=0.0041,SoF¯¯tr2¯¯=0.0108,SoF¯¯tr3¯¯=0.1473,SoF¯¯tr4¯¯=0.1614,SoF¯¯tr5¯¯=0.2185.SoF¯¯tr1¯¯=0.1458,SoF¯¯tr2¯¯=0.1010,SoF¯¯tr3¯¯=0.1316,SoF¯¯tr4¯¯=0.1565,SoF¯¯tr5¯¯=0.1665.tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯ortr5¯¯>tr4¯¯>tr1¯¯>tr3¯¯>tr2¯¯Ü=5,Õ=3SoF¯¯tr1¯¯=0.0096,SoF¯¯tr2¯¯=0.0104,SoF¯¯tr3¯¯=0.1556,SoF¯¯tr4¯¯=0.1603,SoF¯¯tr5¯¯=0.2337.SoF¯¯tr1¯¯=0.1502,SoF¯¯tr2¯¯=0.0867,SoF¯¯tr3¯¯=0.1418,SoF¯¯tr4¯¯=0.1622,SoF¯¯tr5¯¯=0.1810.tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯ortr5¯¯>tr4¯¯>tr1¯¯>tr3¯¯>tr2¯¯Ü=7,Õ=15SoF¯¯tr1¯¯=−0.0166,SoF¯¯tr2¯¯=0.0011,SoF¯¯tr3¯¯=0.1379,SoF¯¯tr4¯¯=0.1580,SoF¯¯tr5¯¯=0.1972.SoF¯¯tr1¯¯=0.1743,SoF¯¯tr2¯¯=0.1366,SoF¯¯tr3¯¯=0.1350,SoF¯¯tr4¯¯=0.1586,SoF¯¯tr5¯¯=0.1737.tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯ortr4¯¯>tr5¯¯>tr4¯¯>tr2¯¯>tr3¯¯Ü=15,Õ=7SoF¯¯tr1¯¯=−0.0073,SoF¯¯tr2¯¯=−0.0009,SoF¯¯tr3¯¯=0.1500,SoF¯¯tr4¯¯=0.1570,SoF¯¯tr5¯¯=0.2216.SoF¯¯tr1¯¯=0.1813,SoF¯¯tr2¯¯=0.1169,SoF¯¯tr3¯¯=0.1514,SoF¯¯tr4¯¯=0.1672,SoF¯¯tr5¯¯=0.1973.tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯ortr4¯¯>tr5¯¯>tr4¯¯>tr2¯¯>tr3¯¯Ü=12,Õ=25SoF¯¯tr1¯¯=−0.0237,SoF¯¯tr2¯¯=−0.0033,SoF¯¯tr3¯¯=0.1346,SoF¯¯tr4¯¯=0.1559,SoF¯¯tr5¯¯=0.1904.SoF¯¯tr1¯¯=0.1863,SoF¯¯tr2¯¯=0.1493,SoF¯¯tr3¯¯=0.1384,SoF¯¯tr4¯¯=0.1606,SoF¯¯tr5¯¯=0.1789.tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯ortr4¯¯>tr5¯¯>tr4¯¯>tr2¯¯>tr3¯¯Ü=25,Õ=12SoF¯¯tr1¯¯=−0.0144,SoF¯¯tr2¯¯=−0.0056,SoF¯¯tr3¯¯=0.1462,SoF¯¯tr4¯¯=0.1551,SoF¯¯tr5¯¯=0.2142.SoF¯¯tr1¯¯=0.1931,SoF¯¯tr2¯¯=0.1308,SoF¯¯tr3¯¯=0.1545,SoF¯¯tr4¯¯=0.1689,SoF¯¯tr5¯¯=0.2023.tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯ortr5¯¯>tr1¯¯>tr4¯¯>tr3¯¯>tr2¯¯Ü=30,Õ=50SoF¯¯tr1¯¯=−0.0307,SoF¯¯tr2¯¯=−0.0097,SoF¯¯tr3¯¯=0.1311,SoF¯¯tr4¯¯=0.1526,SoF¯¯tr5¯¯=0.1846.SoF¯¯tr1¯¯=0.2015,SoF¯¯tr2¯¯=0.1623,SoF¯¯tr3¯¯=0.1455,SoF¯¯tr4¯¯=0.1648,SoF¯¯tr5¯¯=0.1893.tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯ortr1¯¯>tr5¯¯>tr4¯¯>tr2¯¯>tr3¯¯


## 5.5. Comparison with Existing Approaches

In this section, the developed MADM model, which is predicated on this newly established novel AGOs, to various current approaches, namely, Mahmood et al. [12], Ayub et al. [24], and Fahmi et al. [26] MADM models. The comparison between these approaches and the proposed approach is given in Table 4. From Table 4, we can observe that the best alternative obtained from the existing approaches and the proposed approach is the same except for utilizing the CWHMDA operator, while the worst alternative obtained from the existing approaches and the proposed approach is different. The MADM model that will be implemented is based on the recently launched aggregation operators. To put it another way, these aggregation operators are suggested for CNs using Schweizer–Sklar operational, which are generic parameters that make the decision-making procedure more adaptable. In the meanwhile, existing MADM models are drawn on aggregation operators that are launched using algebraic operational laws or Dombi operational laws. The aggregation operator proposed by Mahmood et al. [12] is a simple weighted aggregation operator, which does not have the capacity of taking interrelationships or removing the effect of awkward data from the final ranking results. While, the aggregation operators developed by Ayub et al. [24], are drawn on Dombi operational laws, which have the capacity of tinterrelationships among input arguments and also consist of a generic parameter. The aggregation operators developed by Fahmi et al. [26] are simple weighted averaging operators based on Einstein operational laws for CNs. These aggregation operators do not have the characteristic of taking interrelationship among input arguments or removing the effect of awkward data from the final ranking results. Up to now, the existing aggregation operators for CNs have only the capacity of considering interrelationships among input arguments and cannot remove the effect of awkward data from the final ranking results. Whereas, anticipated aggregation operators will be able to remove the effect of awkward data while also taking into account the interrelationship between the input data at the same time. The predicted AOs also have the benefit of including generic parameters, which sorts the decision-making procedure more supple. As a result, while solving MADM models using cubic information, the initiated AOs are more practical and comparable in their application.Table 4
Comparison with existing approaches.
ApproachesScore valuesRanking ordersCWA [12]SoF¯¯tr1¯¯=0.1098,SoF¯¯tr2¯¯=0.1447,SoF¯¯tr3¯¯=0.1850,SoF¯¯tr4¯¯=0.1692,SoF¯¯tr5¯¯=0.2549tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯CWG [12]SoF¯¯tr1¯¯=0.0277,SoF¯¯tr2¯¯=0.0747,SoF¯¯tr3¯¯=0.1225,SoF¯¯tr4¯¯=0.1309,SoF¯¯tr5¯¯=0.1418tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯CEWA [26]SoF¯¯tr1¯¯=0.0962,SoF¯¯tr2¯¯=0.1348,SoF¯¯tr3¯¯=0.1762,SoF¯¯tr4¯¯=0.1635,SoF¯¯tr5¯¯=0.2378tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯CFWHMDA [26]SoF¯¯tr1¯¯=0.0029,SoF¯¯tr2¯¯=0.0735,SoF¯¯tr3¯¯=0.0077,SoF¯¯tr4¯¯=0.574,SoF¯¯tr5¯¯=0.0718tr2¯¯>tr5¯¯>tr4¯¯>tr3¯¯>tr1¯¯CFWGHMDA [26]SoF¯¯tr1¯¯=0.1210,SoF¯¯tr2¯¯=0.1635,SoF¯¯tr3¯¯=0.2144,SoF¯¯tr4¯¯=0.1574,SoF¯¯tr5¯¯=0.2784tr5¯¯>tr4¯¯>tr3¯¯>tr2¯¯>tr1¯¯Proposed AOsSoF¯¯tr1¯¯=0.0251,SoF¯¯tr2¯¯=0.0208,SoF¯¯tr3¯¯=0.1586,SoF¯¯tr4¯¯=0.1641,SoF¯¯tr5¯¯=0.2424tr5¯¯>tr4¯¯>tr3¯¯>tr1¯¯>tr2¯¯Proposed AOsSoF¯¯tr1¯¯=0.1218,SoF¯¯tr2¯¯=0.0655,SoF¯¯tr3¯¯=0.1301,SoF¯¯tr4¯¯=0.1556,SoF¯¯tr5¯¯=0.1621tr5¯¯>tr4¯¯>tr3¯¯>tr1¯¯>tr2¯¯


## 6. Conclusion

The evaluation of the ERP (enterprise resource planning) system is one of the numerous applications of MADM. The goal of this article is to introduce a cubic set-based decision support as a practical way to explain ambiguity, reluctance, and uncertainty. This article makes a four-fold contribution. Firstly, inimitable Schweizer–Sklar operational rules for CNs are developed, and some of their key characteristics are examined. Secondly, using these inimitable Schweizer–Sklar operational laws, some CSSPHM operators are discussed, including the cubic Schweizer–Sklar power Heronian mean operator, the cubic Schweizer–Sklar power geometric Heronian mean operator, the cubic Schweizer–Sklar power weighted Heronian mean operator, and the cubic Schweizer–Sklar power weighted geometric Heronian mean operator, as well as their vital properties. We can see that some of the existing AOs are special instances of these freshly launched AOs by supplying particular values to the generic parameters. These AOs offer benefits over current AOs. While the current aggregate operators for CSS can only consider interrelationships among input data, the initiated AOs can remove the effect of awkward data, examine the interrelationship among the input data, and also have a general parameter at the same time. Finally, a MADM model is expected based on these AOs. The suggested technique is supported by numerical examples from enterprise resource planning. We also investigate the impact of the decision’s outcome using the recently released cubic fuzzy Schweizer–Sklar power Heronian mean aggregation operators. Then, we compare our work to that of others and also discussed the advantages of the proposed aggregation operators.In future, we will define Schweizer–Sklar operational laws for trapezoidal intuitionistic fuzzy numbers, dual hesitant fuzzy soft sets [41], Neutrosophic cubic sets [30], complex intuitionistic fuzzy sets [42], and extend several aggregation operators such as power average [27–29], robust aggregation operators [43], Choquet Integral for Spherical Fuzzy Sets [44], initiated for these structures and initiate some MADM models and apply these models to solve MADM problem under the said structure. We will also apply the anticipated approach to some new applications, such as detecting hate speech in social media [42], public transportation, technologies selection [45], traffic control, the digital twin model, and so on, or extend the anticipated model to some more extended form of CSS.



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*Source: 5920189-2022-11-08.xml*