SciresolSciresolhttps://indjst.org/author-guidelinesIndian Journal of Science and Technology0974-564510.17485/IJST/v14i6.1838q-Rung Orthopair Dual Hesitant Fuzzy Bonferron Mean OperatorsAyubNausheennausheensuhail7@gmail.com1MalikAslam1Department of Mathematics, University of the PunjabLahore, 54590Pakistan1462021Abstract
Objectives/Methods: Taking into account the impreciseness and subjectiveness of decision makers (DMs) in complex decision-making situations, the assessment datum over alternatives given by DMs is consistently vague and uncertain. In meantime, to evaluate human’s hesitance, the q-rung orthopair dual hesitant fuzzy sets (q-RODHFSs) are defined which are more accurate for manipulation real MADM matters. To merge the datum in q-RODHFSs more precisely, in this research script, some Bonferroni mean (BM) operators in light of q-RODHFSs datum, which includes arbitrary number of being merged arguments, are developed and examined. Findings: Obviously, the novel defined operators can produce much accurate results than already existing methods. Additionally, some important measures of said BM operators are talked about and all the peculiar cases of them are studied which expresses that the BM operator is more dominant than others. Eventually, the MADM algorithm is furnished and the operators are utilized to choose the best alternative under q-rung orthopair dual hesitant fuzzy numbers (q-RODHFNs). Taking advantage of the novel operators and constructed algorithm, the developed operators are utilized in the MADM problems.
Atanassov1 conferred the concept of intuitionistic fuzzy set (IFS), as an advance form of fuzzy set (FS)2 . Every element enclosed in IFS was interpreted with the degree of membership γ and non-membership η, and their sum is restricted to 1, in mathematical form can be labeled as γ+η≤1. The IFS and hesitant fuzzy sets (HFSs) 3 has appealed many scholars’s consideration since its evolution. Likewise, as an impressive MADM technique, Pythagorean fuzzy sets (PFSs) 4 has appeared to outline the uncertainty and fuzziness of the assessment datum. It is also observed that, all the intuitionistic fuzzy decision-making problems are the special case of Pythagorean fuzzy decision-making problems, which means that the PFS is more powerful to handle the MADM problems. Wu and Wei5 developed few Hamacher aggregation operators under PFSs environment to amass PFSs datum. Peng etal.6 constructed a few novel distance measures utilizing PFSs information for use in MADM problems. Wei and Wei7 introduced a variety of cosine similarity measures for PFSs datum. Yet, practically, there may arise some relationships between more than one arguments, it is clear that previously studied collective operators are not authentic for such purpose. For the solution of such type of problems, the Bonferroni mean (BM) operator8 as a reputed information collecting tool which have capability to acknowledge the interrelationship of the arguments, have been explored. Liang etal.9 proposed some BM operators with PFSs information. Most likely, q-ROFS 10 are continuously expansive for the IFS, PFS and these two are its specific cases. Many researchers 11, 12, 13, 14, 15, 16, 17, 18, 19, 2021 developed a varriety of operators to aggregate the information presented q-ROFSs and its application in MADM. Taking advantage of the classical q-ROFSs, Liu and Liu22 derived the definition of q-rung orthopair fuzzy linguistic sets (q-ROFLSs) and developed a few power BM aggregation operators for q-ROFLSs datum. Xu etal.23 illustrated the concept of the q-rung otrhopair dual hesitant fuzzy set (q-RODHFS) and developed a few q-rung dual hesitant fuzzy HM operators for MADM.
Tang etal. 24 developed few Pythagorean fuzzy power aggregation operators and illustrated the idea of dual hesitant Pythagorean fuzzy sets (DHPFSs), as a combination of the PFSs and dual hesitant fuzzy sets (DHFSs) 25, 2627 also developed some Hamacher aggregation operators utilizing DHPFSs. Jiaetal.28 developed a wide range of distance measures based on connection numbers of set pair analysis with dual hesitant fuzzy sets. Wang etal.29 developed MM operators under DHPFSs datum. Apparently, there is no exploration led in light of BM operator to fuse q-RODHF information.
In past few years, numerous investigators studied the BM aggregation operators and their applications. The BM operations have the advantage of considering the relationship between the values being fused, thus the fused results are more reasonable and accurate. Clearly, DHq-ROFN is a meaningful tool to express evaluation information. BM operations are good to fuse evaluation information, so it’s worth to develop some BM operators under dual hesitant q-rung orthopair fuzzy environments. The main novelty and contribution of our manuscript is developing some new BM operators to aggregate the dual hesitant q-rung orthopair fuzzy information. Evidently, these operators have the following advantages. (1) The DHq-ROFS can not only extend the scope of the assessment information to depict more fuzzy information, but also consider the human’s hesitance, thus it is more useful and reasonable to derive decision-making results. (2) The BM operations can consider the relationship between fused arguments, obviously, BM operations are more suitable for handling practical MADM problems. Thus, it is of great significance to propose some new operators based on the dual hesitant q-rung orthopair fuzzy information and BM operations.
In the following text, we have developed a few BM aggregation operators to intertwine the q-RODHF datum. Furthermore, a portion of their alluring properties have additionally been considered and the unique instances of every operator is researched. At last, in light of these effective operators, a decision-making algorithm have been produced and a computative model is delineated to approve the methodology over some similar investigation with the current methodologies. To do as such, the rest of text is composed as pursues. Some basic knowledge about q-ROFSs, q-RODHFSs and BM have been reviewed in Section 2. In Section 3, we have talked about the BM and dual BM operators utilizing q-RODHFSs condition and then developed the q-rung orthopair dual hesitant fuzzy BM(q-RODHFBM) operator, the q-rung orthopair dual hesitant fuzzy weighted BM (q-RODHFWBM) operator, the q-rung orthopair dual hesitant fuzzy dual BM (q-RODHFDBM) operator and the q-rung orthopair dual hesitant fuzzy weighted dual BM (q-RODHFWDBM) operator. In Section 4, we will manufacture the MADM algorithm with q-RODHFNs. In Section 5, we will solve a numerical model for provider choice with q-RODHFNs and gave some similar investigation. Segment 6, finishes up the discussion with certain comments.
Preliminaries2.1 The q-RUNG ORTHOPAIR FUZZZY SET
The essential concepts and basic knowledge of q-rung orthopair fuzzy sets (q-ROFSs)10 are quickly evaluated as pursues.
Definition 2.1.10 Let χ be a universal set. A q-ROFS is an item owns the structure
O=x,γo(x),ηo(x)∣x∈χ
where the mapping γo(x):χ→[0,1] characterizes the membership degree and the mapping ηo(x):χ→[0,1] characterizes the non-membership degree of the component x∈χ to O, respectively, and, for each x∈χ, it satisfies
(γo(x))q+(ηo(x))q≤1,q≥1
The level of indeterminacy is described as:
πo(x)=(γo(x))q+(ηo(x))q-(γo(x))q(ηo(x))qq.
Generally, written as o=(γ,η) a q-ROFN.
Definition 2.2.10 Let o=(γ,η) be a q-ROFN, the score and acuracy function has the form:
to analyze the level of accuracy of the q-ROFN o=(γ,η) The bigger the value of T(o), the more the level of accuracy of the q-ROFN o is.
Now we describe the comparison rule between two q-ROFNs as pursues:
Definition 2.4.10 Let o1=(γ1,η1) and o2=(γ2,η2) be two q-ROFNs, L(o1)=12(1+γ1q-η1q) and L(o2)=12(1+γ2q-η2q) be score values of o1 and o2, respectively, and let T(o1)=γ1q+η1q and T(o2)=γ2q+η2q be the accuracy degrees of o1 and o2, respectively, then if L(o1)<L(o2), then o1≺o2; if L(o1)=L(o2), then (1) if T(o1)=T(o2), then o1=o2;(2) if T(o1)<T(o2), then o1≺o2.
Definition 2.5.10 Let o1=(γ1,η1),o2=(γ2,η2) and o=(γ,η) be three q-ROFNs, and some basic operations on them are defined as follows:
o1⊕o2=((γ1)q+(γ2)q-(γ1)q(γ2)qq,η1η2);
o1⊗o2=(γ1γ2,(η1)q+(η2)q-(η1)q(η2)qq);
λo=(1-(1-γq)λq,ηλ),λ>0;
(o)λ=(γλ,1-(1-ηq)λq),λ>0
o=(η,γ).
2.2 The q-RUNG ORTHOPAIR DUAL HESITANT FUZZZY SET
In the light of q-ROFSs10 and dual hesitant fuzzy sets 25, 26 Xu etal.23 introduced the idea and primary operations of the q-rung orthopair dual hesitant fuzzy sets (q-RODHFSs).
Definition 2.6.23 For any universal set χ, a q-rung orthopair dual hesitant fuzzy set (q-RODHFS) on χ is given as:
D=(⟨x,hO(x),gO(x)⟩|x∈χ)
Where hO(x)=∪ρ∈hO{ρ} and gO(x)=∪κ∈gO{κ} are two objects, also 0≤hO(x),gO(x)≤1, telling the membership (favorable) degrees and non-membership (unfavorable) degrees of the element x∈χ to the set D respectively, with the criteria:
∪ρ∈h(max(ρ))q+∪κ∈g(max(κ))q≤1
Where ρ∈ho(x),κ∈go(x) for all x∈χ. Instantly, the pair d(x)=(ho(x),go(x)) is called a q-rung orthopair dual hesitant fuzzy number (q-RODHFN) simply written as d=(h,g), with the criteria: ρ∈h,κ∈g,0≤ρ,κ≤1 and ∪ρ∈h(max(ρ))q+∪κ∈g(max(κ))q≤1.
Moreover, the relationship among q-RODHFNs could be communicated as:
Definition 2.7.23 For a q-RODHFN d=(ho,go), the score and accuracy functions are given as S(d)=12(1+1#h∑ρ∈hρq-1#g∑κ∈gκq) and T(d)=(1♯h∑ρ∈hρq+1♯g∑κ∈gκq) , where ♯h and ♯g are the numbers of the elements in h and g respectively, then, Let di=(hi,gi)(i=1,2) be any two q-RODHFNs, we have these comparison rules: if S(d1)>S(d2), then d1≻d2; if S(d1)=S(d2), then: (1) if T(d1)=T(d2), then d1=d2;(2) if T(d1)>T(d2), then d1≻d2.
Definition 2.8.23 Let d1=(h1,g1),d2=(h2,g2) and d=(h,g) be three q-RODHFNs, then, the basic working rules on the q-RODHFNs are defined as:
Bonferroni8 proposed the Bonferroni mean (BM) operator.
Definition 2.9.8 Suppose s,t≥0, and bi(i=1,2,…,τ) be nonnegative real numbers. If
BMs,tb1,b2,…,bτ=1τ(τ-1)∑i,j=1i≠jτbisbjt1s+t
Then we called BMs,t the Bonferroni mean (BM) operator.
2.4 The q-RODHFBM OPERATOR
This segment stretches out BM and to fuse the q-RODHFNs, we will introduce the q-rung orthopair dual hesitant fuzzy Bonferroni mean (q-RODHFBM) operator, besides, some valuable properties of q-RODHFBM operator are talked about.
Definition 2.10. Let dj=(hj,gj)(j=1,2,…,τ) be an assortment of q-RODHFNs. The q-rung orthopair dual hesitant fuzzy Bonferroni mean (q-RODHFBM) can be composed as:
q-RODHFBMs,td1,d2,…,dτ=1τ(τ-1)⊕i≠jτdis⊗djt1s+t
Theorem 1. Let dj=(hj,gj)(j=1,2,…,τ) be a list of q-RODHFNs. We can intertwine all the q-RODHFNs datum by utilizing the q-RODHFBM operator, the intertwined outcomes can be communicated in Eq.8, as pursues.
Proof. The proof is simple and obvious from definition (2.8).
Example 2.1. Let d_1=\{\{0.3,0.4\},\{0.5\}\},d_2=\{\{0.7\},\{0.1,0.2,0.6\}\}, and d_3=\{\{0.6\},\{0.3\}\} be three q-RODHFNs, and let s=1,t=1 and q=3 then according to Eq.8, we have
The fused outcomes of the membership function \rho, are displayed as below.
Similarly, we can find \rho_2=q-RODHFBM(0.4, 0.7, 0.6)=0.8013, and \rho=\{0.5585,0.8013\}. For the unfavorable (non-membership) function \kappa, the fused outcomes are displayed as. Alike, the values of \kappa_2, and \kappa_3, are \kappa_2= q-RODHFBM (0.5, 0.2, 0.3)=0.3557, \kappa_3 = q-RODHFBM(0.5, 0.6, 0.3)=0.4911, so we can find \kappa=\{0.3559,0.3557,0.4911\}. Therefore,
Remark 5. If s=1 and t\rightarrow0, then the q-RODHFBM will turn into the q-rung orthopair dual hesitant fuzzy geometric mean (q-RODHFGM) operator as shown below:
Remark 6. When s=t=1, then the q-RODHFBM will turn into the q-rung orthopair dual hesitant fuzzy interrelated square mean (q-RODHFISM) operator as shown below
To get better results in MADM, it’s good to take weighted attributes. In this segment we will introduce the q-rung orthopair dual hesitant fuzzy weighted Bonferroni mean (q-RODHFWBM) operator by this way.
Definition 2.11. Let d_j=(h_j,g_j)(j=1,2,\dots,\tau) be an assortment of q-RODHFNs with the weight vector w=(w_1,w_2,\dots,w_\tau)^T, there by satisfying w_i\in\lbrack0,1\rbrack and {\sum_{i=1}^\tau}w_i=1. If
Then we say q-RODHFWBM_\tau^{s,t} the q-rung orthopair dual hesitant fuzzy weighted Bonferroni mean operator.
Theorem 2. Let d_j=(h_j,g_j)(j=1,2,\dots,\tau) be an assortment of q-RODHFNs. The outcome value by using q-RODHFWBM operators is again a q-RODHFN, as shown below.
Example 2.2. Let d_1=\{\{0.3,0.4\},\{0.5\}\},d_2=\{\{0.7\},\{0.1,0.2,0.6\}\}, and d_3=\{\{0.6\},\{0.3\}\} be three q-RODHFNs, and let s=1,t=1 and q=3 then using Eq.(16), we get for the membership (favorable) function \rho, the final outcomes are given as below.
Alike, we can find \rho_2=q-RODHFBM(0.4, 0.7, 0.6)=0.4173, and \rho=\{0.4032,0.4173\}. For the non-membership (unfavorable) function \kappa, the final results are shown here. Alike, the results of \kappa_2, and \kappa_3, are \kappa_2= q-RODHFBM (0.5, 0.2, 0.3)=0.4569, \kappa_3 = q-RODHFBM(0.5, 0.6, 0.3)=0.5970, so we have \kappa=\{0.6799,0.4569,0.5970\}. Therefore,
2) For parameter s and t, there exist some important cases.
Remark 9. When t\rightarrow0, the q-RODHFWBM will turn into the q-rung orthopair dual hesitant fuzzy weighted arithmetic mean (q-RODHFWAM) as shown below:
Remark 10. If s=2 and t\rightarrow0, the q-RODHFWBM will turn into the q-rung orthopair dual hesitant fuzzy weighted square mean (q-RODHFWSM) as shown below:
Remark 11. If s=1 and t\rightarrow0, the q-RODHFWBM will turn into the q-rung orthopair dual hesitant fuzzy weighted geometric mean (q-RODHFWGM) operator as shown below:
Remark 12. When s=1 and t=1, the q-RODHFWBM will turn into the q-rung orthopair dual hesitant fuzzy weighted interrelated square mean (q-RODHFWISM) operator as shown below:
Then the name q-RODHFDBM{\;}^{s,t} stands for the q-rung orthopair dual hesitant fuzzy dual Bonferroni mean operator.
Theorem 3. Let d_j=(h_j,g_j)(j=1,2,\dots,\tau) be an assortment of q-RODHFNs. The resulted value by using q-RODHFDBM operator is again a q-RODHFN where as Eq.26, as shown here.
Based on operations (1)-(4) of the q-RODHFNs stated in Section 2, we can drive the following result.
Proof. From definition (2.8), the proof follows easily.
Example 2.3. Let d_1=\{\{0.3,0.4\},\{0.5\}\},d_2=\{\{0.7\},\{0.1,0.2,0.6\}\}, and d_3=\{\{0.6\},\{0.3\}\} be three q-RODHFNs, and let s=1,t=1 and q=3 then using Eq.(26), we get for the membership (favorable) function \rho, the ultimate outcomes are as below.
On same lines, we have found \rho_2= q-RODHFDBM(0.4, 0.7, 0.6)=0.5743, and \rho=\{0.5763,0.5743\}. For the non-membership (unfavorable) function \kappa, the ultimate outcomes are shown as . Alike, the results of \kappa_2, and \kappa_3, are \kappa_2=q-RODHFBM (0.5, 0.2, 0.3)=0.3404, \kappa_3 = q-RODHFDBM(0.5, 0.6, 0.3)=0.5745, so we can list \kappa=\{0.3303,0.3404,0.5745\}.
(2) For parameter s and t, there exist these important cases.
Remark 15. If t\rightarrow0, then q-RODHFDBM will turn into the q-rung orthopair dual hesitant fuzzy dual arithmetic mean (q-RODHFDAM) operator as shown below:
Remark 16. If s=2 and t\rightarrow0, then the q-RODHFDBM will turn into the q-rung orthopair dual hesitant fuzzy dual square mean (q-RODHFDSM) as shown below:
Remark 17. If s=1 and t\rightarrow0, then the q-RODHFDBM will turn into the q-rung orthopair dual hesitant fuzzy dual geometric mean (q-RODHFDGM) operator as shown below:
Remark 18. If s=1 and t=1, then the q-RODHFDBM will turn into the q-rung orthopair dual hesitant fuzzy interrelated square mean (q-RODHFDISM) operator as shown below:
In actual MADM, it’s good to assign weights to each attribute. In this segment, we shall explore the q-rung orthopair dual hesitant fuzzy weighted dual Bonferroni mean (q-RODHFWDBM) operator as pursues.
Definition 2.14. Let s,t>0 and a_i(i=1,2,\dots,\tau) be a set of nonnegative real numbers. If
Then q-RODHFWDBM{\;}^{s,t} stands for the q-rung orthopair dual hesitant fuzzy weighted dual Bonferroni mean operator.
Theorem 4. Let s,t>0 and d_j=(h_j,g_j)(j=1,2,\dots,\tau) be an assortment of q-RODHFNs. The aggregated result after utilizing q-RODHFWDBM operators is again a q-RODHFN where as Eq.34, as shown here.
(2) For parameter s and t, there exist the following vital cases.
Remark 21. When t\rightarrow0, the q-RODHFWDBM will turn into the q-rung orthopair dual hesitant fuzzy weighted dual arithmetic mean (q-RODHFWDAM) operator as shown below:
Remark 22. When s=1\;and\;t\rightarrow0, the q-RODHFWDBM will turn into the q-rung orthopair dual hesitant fuzzy weighted dual geometric mean (q-RODHFWDGM) operator as shown below:
Remark 23. When s=2\;and\;t\rightarrow0, the q-RODHFWDBM will turn into the q-rung orthopair dual hesitant fuzzy weighted dual square mean (q-RODHFWDSM) operator as shown below:
Remark 24. When s=1 and t=1, the q-RODHFWDBM will turn into the q-rung orthopair dual hesitant fuzzy weighted dual interrelated square mean (q-RODHFWDISM) operator as shown below:
Example 2.4. Let d_1=\{\{0.3,0.4\},\{0.5\}\},d_2=\{\{0.7\},\{0.1,0.2,0.6\}\}, and d_3=\{\{0.6\},\{0.3\}\} be three q-RODHFNs, and let s=1,t=1 and q=3 then utilizing Eq.(35), we have for the membership (favorable) function \rho, the final values are as below.
Alike, we can find \rho_2=q-RODHFDWBM(0.4, 0.7, 0.6)=0.9999, and \rho=\{0.9287,0.9999\}. For the non-membership (unfavorable) function \kappa, the ultimate values are shown below. Alike, the results of \kappa_2, and \kappa_3, are \kappa_2=q-RODHFDWBM (0.5, 0.2, 0.3)=0.9980, \kappa_3 = q-RODHFDWBM(0.5, 0.6, 0.3)=0.9856, so we can list \kappa=\{0.9952,0.9980,0.9856\}. Therefore,
q-RODHFDWBM(d_1,d_2,d_3)=\{\{0.9287,0.9999\},\{0.9952,0.9980,0.9856\}\}MODELS FOR MADM WITH q-RODHFNs
In the light of the q-RODHFWBM and q-RODHFWDBM operators, we shall furnish the model for MADM with q-RODHFNs. Let O=\{O_1,O_2,\dots,O_m\} be a discrete set of alternatives, and K=\{K_1,K_2,\dots,K_\tau\} be collection of attributes, w=\{w_1,w_2,\dots,w_\tau\} is the weight vector of the attribute K_j(j=1,2,\dots,\tau) where 0\leq w_j\leq1,{\sum_{j=1}^\tau}w_j=1. Suppose that d=(d_{ij})_{m\times\tau}=(h_{ij},g_{ij})_{m\times\tau} is the q-rung orthopair fuzzy decision matrix, where h_{ij} set specify the level that the alternative O_i satisfy the attribute K_j given by the decision maker, g_{ij} set specify the level that the alternative O_i doesn’t satisfy the attribute K_j given by the decision maker, \rho_{ij}\in h_{ij}\subset\lbrack0,1\rbrack,\kappa_{ij}\in g_{ij}\subset\lbrack0,1\rbrack,(\rho_{ij})^2+(\kappa_{ij})^2\leq1,i=1,2,\dots,m,j=1,2,\dots,\tau. In the accompanying, we will utilize the q-RODHFWBM and q-RODHFWDBM operator to the MADM problems for q-RODHFNs.
Step\;1: We take advantage of q-RODHFNs of the matrix \widetilde U, and utilize q-RODHFWBM operator to acquire d_i(i=1,2,\dots,m) of the alternative O_i.
Step\;2: Determine the scores L(d_i)(i=1,2,\dots,m) of the whole collection of q-RODHFNs d_i(i=1,2,\dots,m) and finally rank all the alternatives O_i(i=1,2,\dots,m) and then choose the exclusively optimal one(s). If the score values of two L(d_i) and L(d_k) are same, then we shall utilize the accuracy values T(d_i) and T(d_k) of the whole collection of q-RODHFNs and d_i and d_k, respectively, and then arrange the alternatives O_i and O_k with respect to the accuracy degrees T(d_i) and T(d_k)
Step\;3: Arrange all the alternatives O_i(i=1,2,\dots,m)
in descending order and select the optimal one(s) likewise L(d_i)(i=1,2,\dots,m)
Step\;4:End.
Application and Comparative Analysis4.1 Numerical Example
In this segment, we shall furnish an application to choose green providers in green inventory network the board (GINB) with q-RODHFNs. There are five possible green providers in GINB O_i(i=1,2,3,4,5) to decide. The specialists evaluate the five potential green providers with respect to the following attributes: 1. K_1 is the item quality factor; 2. K_2 is natural factors; 3. K_3 is conveyance factor; 4. K_4 is value factor. Five green providers O_i(i=1,2,3,4,5) are to be classified under q-RODHFNs with respect to four attributes with weight vector w=(0.4,0.3,0.1,0.2) displayed in Table 1.
In the accompanying, we take the advantage of the operators developed for provider selection in provide network board with q-rung orthopair dual hesitant fuzzy numbers (q-RODHFNs) datum.
Step 1: We take advantage of the decision datum in matrix \widetilde U, and the q-RODHFWBM operator to collect the collective preference values d_i of the provider in green inventory network the board O_i(i=1,2,3,4,5). The collective preference values d_i of the provider in green inventory network the board O_i(i=1,2,3,4,5) are listed below
Step 3: Rank all the providers O_i(i=1,2,3,4,5) likewise the scores S(O_i)(i=1,2,3,4,5) of the collective q-rung orthopair dual hesitant fuzzy numbers: O_2\succ O_3\succ O_4\succ O_5\succ O_1, and thus the most desirable supplier is O_2.
Based on the q-RODHFWDBM operator, in order to select the most desirable supplier, we can develop an approach to multiple attribute decision making problems with q-rung orthopair dual hesitant fuzzy information, which can be described as following:
Step\;\widetilde1: Aggregate all q-rung orthopair dual hesitant fuzzy value d_{ij}(j=1,2,3,4) by using the dual hesitant q-rung orthopair fuzzy weighted DBM (q-RODHFWDBM) operator to derive the overall q-rung orthopair dual hesitant fuzzy values d_i(i=1,2,3,4,5) of the supplier A_i. The overall performance values of all the supplier A_1 (here, we take q = 3,s=1,t=1) are given below,
Step\;\widetilde2: Calculate the scores s(A_i)(i=1,2,3,4,5) of the overall q-rung orthopair dual hesitant fuzzy values A_i(i=1,2,3,4,5) of the supplier A_i:
Step\;\widetilde3:Rank all the suppliers in supply chain management A_i(i=1,2,3,4,5) in accordance with the scores s(A_i)(i=1,2,3,4,5) of the overall dual hesitant q-rung orthopair fuzzy values A_i(i=1,2,3,4,5) by using definition 2.15: A_2\succ A_3\succ A_5\succ A_1\succ A_4, and thus the most desirable supplier is A_2. From the above analysis, it is easily seen that although the overall rating values of the alternatives are same by using two operators respectively.
4.2 Comparative analysis compared with existing magdm methods
To demonstrate the superiorities of the proposed method, we have compared our method with that (1) developed by Wang et al.’s 26 based on the dual hesitant fuzzy weighted averaging (DHFWA) operator, (2) presented by Tu et al.’s 27, based on the dual hesitant fuzzy weighted Bonferroni mean (DHFWBM) operator, (3) putforwarded by Tang 24, based on the dual hesitant Pythagorean fuzzy Heronian weighted averaging (DHPFHWA) operator, (4) proposed by, Xu et al.’s23 based on the dual hesitant Pythagorean fuzzy Heronian weighted averaging (DHPFHWA) operator. We utilized these methods to solve the above example, and the score functions and ranking results can be found in Table 2.
First of all, Wang et al.’s 26 and Tu et al.’s27 methods are based on DHFSs. Tang et al.’s24 method is based on DHPFSs. As mentioned above, DHFS and DHPFS are two special cases of q-RDHFS. When q = 1, then q-RDHFS is reduced to DHFS, and when q = 2, q-RDHFS is reduced to DHPFS. Evidently, q-RDHFS is more general and can describe a greater information range and process more information in the process of MAGDM. For instance, if an attribute value provided by DMs is {{0.1, 0.2, 0.6, 0.7}, {0.1, 0.4, 0.5}}, then obviously, the pair {{0.1, 0.2, 0.6, 0.7}, {0.1, 0.4, 0.5}} is not valid for DHFSs and DHPFSs. Thus, our method is more general, powerful, and can process more information in MAGDM. Wang et al.’s26 method is based on the simple weighted averaging operator. The drawback of this methods is that it does not consider the interrelationship between arguments. In other words, they assume all attributes are independent, which is not correct to some extent. In the abovementioned example, when choosing the most appropriate supplier, we need to consider not only the attribute values of each supplier but also the correlation between these attributes. Thus,Wang et al.’s26 method is not suitable for dealing with this problem. As our method has the ability to capture variable correlations, it is more reasonable than Wang et al.’s method for addressing this problem. Xu et al.’s23 is based on HM. Tu et al.’s27 and our methods based on Bonferroni mean (BM). The prominent characteristic of BM and HM is that both can consider the interrelationship between arguments. Therefore, all the three can process the interrelationship among attribute values. However, Xu et al.’s23 method and ours are better than Tu et al.’s27 method. In addition, as Tu et al.’s27 is a special case of our method (when q = 1), our method is more general, scientific, and applicable than Tu et al.’s27 method.
Conclusion
In this article, we have examined the MADM problems under q-RODHFNs. we have utilized the BM operator and established some BM aggregation operators with q-RODHFNs. We have developed (q-RODHFBM) operator, (q-RODHFWBM) operator, DBM operator, (q-RODHFDBM) operator and (q-RODHFWDBM) operator. Also, the important merits of the examined operators are talked about. Furthermore, we have endorsed q-RODHFWBM and q-RODHFWDBM operators to construct decision-making steps to handle the q-rung orthopair dual hesitant fuzzy MADM problems. Finally, we take a solid example for examining the green provider selection to exhibit our established model and to assert its efficiency and objectiveness. We have compared our results with q-RODHFWHM and q-RODHFWGHM operators, despite the fact that the results are minimal extraordinary and the ideal option is not changed. However, the q-RODHFWHM and q-RODHFWGHM operators just include the interrelationship of two arbitrary numbers but our introduced operators can include the interrelationship of any number arbitrary arguments, that indicates our established method is more decisive to handle the MADM problems. In the forthcoming, we will maintain our study about the MADM issues with the application and expansion of the presented operators to other realm.
1 nausheensuhail7@gmail.com
2 aslam.math@pu.edu.pk
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