https://orcid.org/0000-0002-5436-1763
Figures
Abstract
The novel multivalued neutrosophic aggregation operators are proposed in this paper to handle the complicated decision-making situations with correlation between specific information and partitioned parameters at the same time, which are based on weighted power partitioned Hamy mean (WMNPPHAM) operators for multivalued neutrosophic sets (MNS) proposed by combining the Power Average and Hamy operators. Firstly, the power partitioned Hamy mean (PPHAM) is capable of capture the correlation between aggregation parameters and the relationship among attributes dividing several parts, where the attributes are dependent definitely within the interchangeable fragment, other attributes in divergent sections are irrelevant. Secondly, because MNS can effectively represent imprecise, insufficient, and uncertain information, we proposed the multivalued neutrosophic PMHAM (WMNPHAM) operator for MNS and its partitioned variant (WMNPPHAM) with the characteristics and examples. Finally, this multiple attribute group decision making (MAGDM) technique is proven to be feasible by comparing with the existing methods to confirm this method’s usefulness and validity.
Citation: Wang M, Li T, Tian Y, Zhang K (2023) Multivalued neutrosophic power partitioned Hamy mean operators and their application in MAGDM. PLoS ONE 18(2): e0281734. https://doi.org/10.1371/journal.pone.0281734
Editor: Ronnason Chinram, Prince of Songkla University, THAILAND
Received: August 24, 2022; Accepted: January 31, 2023; Published: February 15, 2023
Copyright: © 2023 Wang et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper.
Funding: Shandong Provincial Social Science Planning Project (No.20CGLJ24); National Social Science Foundation of China (No.21BJY027). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introduction
The world is full of partial, imprecise, inconsistent, and uncertain data that can’t be characterized with precise numbers [1–3]. In order to deal with these complex problems, the MAGDM method can sort and adopt the superlative alternative from a set of complicated options [4]. Zadeh suggested fuzzy set to solve MAGDM problems in order to decrease information loss and increase assessment accuracy [5]. However, main limitation of fuzzy set is that it can’t handle complex fuzzy information adequately because its membership limit is only one value. As a result, Atanassov expanded fuzzy set to the intuitionistic fuzzy set [6–8]. However, in the face of conflicting, partial, and uncertain data, the foregoing has some limitations. Smarandache proposed the notion of neutrosophic sets to address this issue, which incorporated an independent indeterminacy membership function [9]. Such as the generalization of fuzzy sets, the single-valued neutrosophic set and multivalued neutrosophic sets are some of the achievements in this subject [10–17].
Because the decision makers typically cannot generate precise assessment values for every membership due to limited knowledge and experience [18–20], the MNS can manage the unclear information that has more than one value of membership offered by certain decision makers. As a result, MNS can cope with convoluted fuzzy information better than other fuzzy sets, but little research in this subject has been conducted, thus it is critical to investigate the MADM or MAGDM approach using MNS.
Aggregation operators, as a critical tool for solving MADM or MAGDM issues, may combine information from all qualities and decision makers and rate the options [21–29]. However, there has been little progress in the research of MNS aggregation operators thus far. As previously stated, ranking systems such as PROMETHEE [30, 31], TODIM [32, 33], and others were unable to produce complete values for many options. Furthermore, approaches based on aggregation operators can produce both ranking results and complete values [34].
Existing MAGDM approaches are unable to capture complex relationship patterns between characteristics properly. However, actual MADM or MAGDM situations with many attribute correlations need the use of developing operators as Power average operator, Maclaurin symmetric mean operator [35–39]. When compared with Heronian mean which was proposed by Hara [40], the Hamy mean operator can manage the interrelationships between numerous qualities more flexibly. Furthermore, Liu [41], Wu [42] point out that Hamy mean operator is a more powerful extension of MSM. However, the classic Hamy mean operator has several limitations, such as the fact that it can only aggregate crisp integers and cannot directly establish relationships among multiple-input arguments with partition structure.
It is critical to expand the Hamy mean operator to handle complicated MAGDM issues with ambiguous information and interrelationships of attributes, the novel multi-valued neutrosophic power partitioned hamy mean (MNPPHM) operator is presented in this paper, which is based on the Hamy operator. As a result, our research concentrates on the nominated operators to build the MAGDM technique using MNS. The contributions are:
- The PPHAM operator is extended to improve the ability of existing operators to handle the specific information and partitioned parameters.
- The PHAM operator (WMNPHAM) is presented with MNS, as well as its partitioned operator (WMNPPHAM).
- The developed operators’ properties are demonstrated with numerous unique examples.
- A practical MAGDM approach is generalized with the WMNPPHAM operator.
- The applicability and efficacy of the provided MAGDM technique is demonstrated.
The following is the structure of this paper. We introduce the fundamental principles and operating norms of MNS and Hamy mean operator operators in Section 2. Section 3 proposes the WMNPHAM and WMNPPHAM. In Section 4, we construct MAGDM technique with these operators. Section 5 provides an example that compares the proposed method to existing ways to demonstrate its viability. Finally, the findings are described in Section 6.
Preliminaries
The neutrosophic fuzzy set
Definition 1. As a universe of objects K, where q is a generic element in K, the single valued neutrosophic set Q in K is defined as:
(1)
where UK(q), DK(q) and MK(q) denote the truth-membership, the indeterminacy-membership and the falsity-membership of the element x ∈ X to the set K respectively. For each point q in Q, we have UK(q), DK(q), MK(q) ∈ [0,1], 0 ≤ UK(q) + DK(q) + MK(q) ≤ 3.
For simplicity, we may utilize the simpler form Q = (Uq, Dq, Mq) to represent single valued neutrosophic set, and the element q can be termed as a single valued neutrosophic number.
Definition 2. K is a nonempty fixed set, the multivalued neutrosophic set (MNS) in Q could be characterized as:
(2)
where
,
and
are three sets with some values in interval [0,1], and satisfying the limits:
and
.
The is known as a multivalued neutrosophic number which is indicated by the simplified symbol
.
Definition 3. Let I ∈ MNS, the complement of an MNS may be characterized as Ic stated as follows:
(3)
Definition 4. Two MNS and
, and k > 0, the basic operations can be defined as:
Definition 5. Such any MNS ,
(8)
is regarded as the score function of
, where
,
and
reflects the numbers of the values in
,
,
.
Definition 6. For any a MNS ,
(9)
Assume and
are two MNS, the comparison algorithm of MNS is determined by the following equations:
- If
and
, then
- If
and
, then
- If
and
, then
.
Definition 7. Assume , then the Hamming distance between them can be expressed in terms:
(10)
Hamy mean operator
Definition 8. The form of Hamy mean operator is:
(11)
Where (i1, i2, …,ij) explores every j-tuple combination and
is the binomial coefficient, and
.
Obviously, the Hamy mean operator has several characteristics:
- HAM(j) (0,0,…,0) = 0, HAM(j)(w, w,…,w) = w;
- HAM(j) (w1, w2,…,wn) ≤ HAM(j) (y1, y2,…,yn), if wi ≤ yi for all i;
- min{wi} ≤ HAM(j) (w1, w2,…,wn) ≤ max{wi}.
Power aggregation operators
Definition 9. The Power aggregation operator is the mapping Rn → R as:
(12)
Where
, and Sup(αi, αj) is the support as αi from αj. And certain qualities are detailed as follows:
- Sup(αi, αj) ∈ [0,1];
- Sup(αi, αj) = Sup(αj, αi)
- Sup(αi, αj) ≥ Sup(αe, αf), if |αe − αf|<|αi − αj|.
Hamy mean operators based on multivalued neutrosophic sets
In summary, we will investigate the Hamy mean operator and Power aggregation operator to deal with MNS and build MNPPHAM operator and WMNPPHAM operator, as well as explain various attributes and specific circumstances of these new operators, with the operating regulations of MNS.
MNPHAM operator
Definition 11. are MNS, and x = 1,2,…,m The MNPHAM operator is described this way:
(14)
where
is support degree, which meets:
, then
, where
indicate distance among any two neutrosophic sets signed by the Definition 7.
The denominator represents the binomial coefficient
and n is the balancing coefficient in the preceding Eq (14), we could note
(15)
then power weight vector is identified by (σ1, σ2,…,σn). As a result, Eq (14) can be documented in the following simplified form:
(16)
The following theorems could be derived from the operational rules of the MNS:
Theorem 1. Let be a MNS, the result of aggregation is still MNS.
Proof. Since
we have
and
so
then
Therefore,
In addition, the NPHAM has some features as:
(1) Theorem 2 (Idempotency). , for (i = 1, 2,…,n), then
(18)
Proof. Since , for (i = 1, 2,…,n), then
So, according to Theorem 1, we have
(2) Theorem 3 (monotonicity). Let and
be two MNS, and suppose
and
for all i, then
(19)
Proof. (1)Since for all i, then
So
(3) Since , for all i, then
So
we can get
(3) Theorem 4 (Boundedness). The MNHPHM operator situates within:
(20)
WMNPHAM operator
Definition 12. is a MNS, and weighted neutrosophic set power Hamy mean operator (WMNPHAM) is expressed in the form.
(21)
where
,
which meet the specified criteria:
then
, where
represent the measure of distance in definition 7. ω = (ω1,ω2,…,ωn)T is the weight vector of
such that ωi ∈ [0,1] and
.
Theorem 5. Assume be MNS, So the result of aggregation is still MNS.
Proof. This theorem’s proof is analogous to that of Theorem 1.
(1)Theorem 6 (Idempotency)., for all i, we have
(23)
(2) Theorem 7 (monotonicity). If and
be two neutrosophic sets, and suppose
and
for all i, then
(24)
Proof. (1) Since for all i, then
(3) Theorem 8 (Boundedness). The WNPHAM operator situates within:
(25)
Proof. Let , since
, according to Theorem 3, we could obtain:
And
Then
WMNPWPPHAM operator
Definition 13. Let be a collection of MNS, and the elements could well be separated into l parts P = (P1, P2,⋯,Pl), where
, t = (1, 2,⋯,l), Ps ⋂ Pt = ∅,
, the WMNPPHAM operator is described as follows:
(26)
Furthermore, the foregoing operators satisfy the Theorems of Idempotency, Commutativity, and Boundedness. However, the proofs are much like the proofs of the Theorems for MNPPHAM and WMNPPHAM operators, therefore the proof procedure is omitted here.
MAGDM approach with WMNPPHAM operators
In this part, we will utilize the WMNPPHAM operators to solve the MAGDM issue. For example, a MAGDM issue concludes a set of m alternatives A = {A1, A2,…,Am}, the decision makers D = {D1, D2,…,Dz} with their weight vector θ = (θ1, θ2,…,θz)T and a collection of n attributes C = {C1, C2,…,Cn} with ω = (ω1, ω2,…,ωn)T matching θl ∈ [0,1],, ωn ∈ [0,1],
. Taking into account the relationship of the attribute set C, C can be divided into l parts P1, P2,…,Pl, where
, t = (1, 2,…,l), Ps ⋂ Pt = ∅,
. |Pt| represents the number in partition Pt. Assume that the DM Dh(h = 1,2,…,z) evaluates his/her assessment using MNS
, which is the assessment of the attribute Cj regarding the alternative Ai. The decision matrix might indeed be compiled as follows:
(27)
To overcome this problem, the following phases of the new MAGDM technique might be taken:
Step 1. The gathered choice matrices Rh must be normalized into standard matrices SRh by converting the cost-type to the benefit-type.
Step 2. Calculate the supports
As Definition 7, is the Hamming-Hausdorff distance.
Step 3. Estimate the weights associated with the MN
.
Since
then the weights
combined by the
could be collected:
Step 4. Aggregate each expert’s evaluation information.
To composite the MNS , use the WMNPHAM operator described in Eq (30)
(30)
Step 5. Estimate the supports
Step 6. Estimate the weights τij
Since
the weights τij with the MN
can show as:
(32)
Step 7. Evaluate each alternative inside each partition Pt, using the WMNPHAM operator described by (27), where , i = 1, 2,…, m as:
(33)
Step 8. Use the Eqs. (8) and (9) to calculate the score values and accuracy values
.
Step 9. As shown by Definition 6, order the alternatives Ai and pick the most accomplished one(s).
Illustrative example
The investment firm seeks to choose the best option from the four agricultural brands, which are A1, A2, A3 and A4. Three characteristics are used to evaluate the four alternatives: (1) C1(the risk index), (2) C2(the growth index), (3) C3(environmental impact index), where C1 and C2 belong to the benefit type, C3 is of the cost type with their weight is ω = (0.35, 0.25, 0.4)T. Furthermore, there are three decision makers that can make quick choices based on their knowledge and experience. Their decisions may be written down in the form MNS by (i = 1,2,3,4, j = 1,2,3,4, h = 1,2,3) with their weight is θ = (0.3,0.5.0.2)T. The qualities may be separated into two portions based on their characteristics P1 = {C1, C2} and P2 = {C3}. In this situation, the decision maker can provide numerous possible values under each characteristic for the evaluations while adhering to the primary level of “excellent.” Then there is the decision matrix among the decision makers
could be collect as follows
Decision-making procedure
Step 1. Normalize the decision matrix.
The normalized MN decision matrix can be calculated using Eq (28) as:
Step 2. Compute the supports degree For convenience,
can denoted by Supkt. According to Eq (29), the Supkt (k,t = 1,2,3; k ≠ t) can be calculated:
Step 3. Estimate the weights with
.
The can be calculated as Tk (k = 1,2,3):
The weights can be formed by MN
using Eq (31).
are formed by σk (k = 1,2,3) as follows:
Step 4. Combine the evaluation information of every expert.
The collective multivalued neutrosophic decision matrix can be computed as:
Step 5. Retrieve the supports
could be retrieved using Eq (31) as follows:
Step 6. Compute the weights τij with MNS
The weighted support of the MNS
using Eq (32), by the other MNS
can be computed.
So the weights τij (j = 1,2,…,m) can be computed.
Step 7. Evaluate the information within each partition of attributes.
This step can estimate the collective evaluation values of each alternate within Pt by (20), which show in Table 1.
The value of the alternative αi can be calculated by WMNPPHAM operator in Table 2:
Step 8. Use the Eqs. (8) and (9) to calculate the score values.
Si can be calculated by Definition 5 as:
Step 9. Order all the alternatives.
The results in Step 4, we can get S1 > S2 >S4 >S3. So, the final rank of all the alternatives could be shown as A1 ≻ A2 ≻ A4 ≻ A3.
Influence of the parameter on the final result
The changing value of parameter x in the MNWHAM operator can be taken to demonstrate the effects on the ranking results in Table 3.
Comparison analysis
The efficacy and practicality of the suggested MAGDM technique by WMNPPHAM operators must be compared and verified, thus we perform a comparative analysis using the same illustrative case. The analysis might be made from the following aspects: techniques utilizing MNS with other operators and ways using the same operators with different discrete forms of neutrosophic numbers. Then these different ranking results could be shown as α1 ≻ α4 ≻ α2 ≻ α3 [43]. Clearly, the ideal choice is α1, whereas the worst alternative is α3.
We summarize the reasons for variances in the final rankings of all the examined approaches and the suggested methodology in Table 4. Not only can our approach consider interrelationships between any two qualities, numerous arguments, and membership and nonmembership, but it also has beneficial flexibility to represent preference and capacity to describe uncertainty. Furthermore, our technique may partition the attributes into discrete portions that include both the interdependence and the independence of the attributes. So we may infer that WMNPPHAM is more practical and efficient.
Conclusion
In this study, we propose the WMNPHAM and WMNPPHAM operators, which extend the Hamy mean and Power aggregation operator to the MNS. In addition, we describe the desirable qualities, create the score function, and use it to rank the choices. After that, based on the WMNPPHAM operator, we provide detailed procedures for solving MAGDM issues using multi-valued neutrosophic information. Furthermore, we compare the efficacy and practicality of the created technique to current methods.
Therefore, for addressing complex decision-making situations, these proposed novel multivalued neutrosophic aggregation operators can aggregate fuzzy information and partitioned parameters meantime, which can be used as a practical tool to solve the MADM challenges more efficiently and effectively. In the future, further research can expand them to other aspects of MADM or MAGDM, such as clustering algorithms or consistency analysis. Meanwhile, other fuzzy sets can be combined to deal with the practical problems, such as q-rung orthopair fuzzy 2-tuple linguistic sets [48], interval-valued intuitionistic fuzzy hypersoft set [49], and so on.
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