# Power Improved Generalized Heronian Mean Operators Utilizing Hamacher Operations with Picture Fuzzy Information

Authors: Baolin Li; Lihua Yang
Journal: Complexity
 (2021)

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## Abstract

In multiple attribute decision-making (MADM), to better denote complicated preference information of decision-makers (DMs), picture fuzzy set (PFS) as an expansion of intuitionistic fuzzy set (IFS) has become a powerful tool in the recent years. Meanwhile, to remove the impact of abnormal data and capture the correlations among attributes in MADM issue, we propose the power improved generalized Heronian mean (PIGHM) operators in this paper, which have the merits of both power average (PA) operator and improved generalized Heronian mean (IGHM) operator. Additionally, Hamacher operations as a generalization of Algebraic operations and Einstein operations demonstrate good smooth approximate. Motivated by these, the main purpose is to explore PIGHM operators utilizing Hamacher operations to cope with MADM issue with picture fuzzy information. First, we introduce the Hamacher operations, the normalized hamming distance, and similarity measure of picture fuzzy numbers (PHNs). Second, based on these, two new picture fuzzy aggregating operators (AOs), the picture fuzzy Hamacher weighted power improved generalized Heronian mean (PFHWPIGHM) operator and the picture fuzzy Hamacher weighted geometric power improved generalized Heronian mean (PFHWGPIGHM) operator, are put forward, and some properties and special instances of proposed AOs are also investigated. Third, a new MADM model in terms of the PIGHM AOs is developed. Eventually, a practical MADM example, together with sensitivity analysis and comparative analysis, is conducted to verify the credibility and superiority of the new MADM model.


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## Body

## 1. Introduction

Intuitionistic fuzzy set (IFS), which was firstly defined by Atanassov [1] and is an extended form of fuzzy set (FS) proposed by Zadeh [2], can better depict the truth, the falsity, and indeterminacy memberships, where the degree of indeterminacy membership is dependent on the membership degrees of truth and falsity. However, in actual environment, the indeterminacy degree should be given by decision-makers (DMs) according to real situation. For instance, in a voting, sixty percent are for it, ten percent remain neutral, twenty percent are against it, and ten percent refuse to vote. Clearly, this circumstance is beyond the capability of IFS. Thus, picture fuzzy set (PFS) defined by Cuong [3], including the degrees of positive, neutral, and negative, can convey multiple responses of DMs.Since PFS can express different attitudes of DMs in complex situations, some studies regarding PFS have been conducted [4–9], concentrating on the aggregating operators (AOs), distance measure, cross entropy, and correlation coefficient. AOs, as an interesting direction owing to the diversity correlations among attributes, are an efficient tool for information fusion in MADM problem. Power average (PA) operator, which can decrease the impact of irrational data provided by DMs, was firstly defined by Yager [10]. Then, the PA operator has been applied to cope with different fuzzy information. Rani and Garg [11] applied PA operator and its extension to complex intuitionistic fuzzy set. Jiang et al. [12] proposed PA operator based on entropy and applied it to IFS. Ju et al. [13] extended PA AOs to q-rung orthopair fuzzy set (qROFS). Yang and Li [14] explored PA AOs to multivalued neutrosophic linguistic set (MVNLS). Garg and Kumar [15] constructed a MADM model utilizing PA AOs for linguistic IFS. Liu and Luo [16] extended PA AOs to simplified neutrosophic set (SNS). Garg and Nancy [17] developed PA AOs to linguistic single-valued neutrosophic set (LSVNS). Liu et al. [7] applied PA AOs to complex PFS. Considering the interrelationships among aggregating values, Heronian mean (HM) operator was initially introduced by Beliakov et al. [18]. Later, Sykora [19] proposed the generalized HM (GHM) operator. HM operator and GHM operator mentioned above do not consider the weights of input argument. Then, Yu and Wu [20] extended the GHM to interval-valued IFS (IVIFS) and established generalized weighted HM (GWHM) operator. Moreover, Yu [21] introduced generalized geometric weighted HM (GGWHM) operator with IFS. Nevertheless, it seems to be counterintuitive that the GWHM and GGWHM operators do not have the characteristics of reducibility and idempotency. Therefore, Chu and Liu [22] proposed novel weighted HM AOs to uncertain linguistic information, namely, weighted generalized HM (WGHM) operator and weighted generalized geometric HM (WGGHM) operator, and proved the properties of reducibility, idempotency, monotonicity, and boundedness. Li et al. [23] further extended weighted HM AOs to SVNS and put forward improved generalized weighted HM (IGWHM) and improved generalized weighted geometric HM (IGWGHM) operators. Subsequently, IGWHM and IGWGHM operators are extended to neutrosophic hesitant fuzzy set (NHFS) [24], normal neutrosophic set (NNS) [25], neutrosophic uncertain linguistic set (NULS) [26], and bipolar neutrosophic set (BNS) [27].Since each operator has its own merit, the combination of different operators has become a hotspot in MADM problem. Motivated by the superiorities of both PA operator and HM operator, Liu [28] combined them and proposed power Heronian (PH) operator and power weighted Heronian (PWH) operator to manage IVIFS. Later, Shi et al. [29] presented power geometric Heronian mean (PGHM) operator and weighted PGHM (WPGHM) operator to IFS. Zhao et al. [30] extended PH operator, weighted PH (WPH) operator, geometric PH (GPH) operator, and weighted GPH (WGPH) operator to SVNS. Jiang et al. [31] applied operators of PH, WPH, GPH, and WGPH to interval-valued dual HFS (IVDHFS). Ju et al. [32] combined PA and generalized HM operator and applied to fuse hesitant fuzzy linguistic numbers (HFLNs). Wang et al. [33] extended power generalized HM operator and its weighted form to q-rung orthopair hesitant fuzzy set (qROHFS). Moreover, rough PH operator [34] and power partitioned Heronian AOs [35, 36] have also been explored. As mentioned above, the combination of PA operator and HM operator has received the attention of many scholars. However, up to now, there is no research regarding the combination of PA operator and IGHM operator, which has more desirable properties than HM operator.For aggregating PFS, different AOs have been explored in the recent years. Wang et al. [37] extended weighted Muirhead mean (WMM) and weighted dual MM (WDMM) to picture fuzzy set and proposed PFWMM and PFWDMM operator for financial investment risk appraisal. Jana et al. [38] established some Dombi AOs under picture fuzzy environment. Liu and Zhang [39] explored operator of A-PFLWAA to MADM under PFS. Zhang et al. [40] introduced Heronian mean (HM) operators using Dombi operations to PFS. Ates and Akay [41] expanded some Bonferroni mean (BM) to picture fuzzy set to handle MADM, including PFBM, PFNWBM, and PFOWBM operators. Luo and Xing [42] proposed picture fuzzy interaction partitioned Heronian (PFIPH) operator for selecting hotel. Qin et al. [43] introduced the Dempster-Shafer theory to PFS and proposed power Muirhead mean (PMM) operators utilizing the proposed operational rules. Yang and Li [9] defined multiple-valued picture fuzzy linguistic set (MVPFLS) and extended generalized HM operators to it. Up to date, AOs for PFS are almost based on Algebraic operations. Besides, there are a few researches on fusing picture fuzzy numbers (PFNs) utilizing Einstein [44], Dombi [38, 40], and Hamacher [45, 46]. Hamacher operations, as an extension of Algebraic and Einstein operations, are more agile with a general parameter. Wang et al. [47] explored PA operator based on Hamacher to assess the quality of express service. However, in terms of the existing works, there is no research regarding exploring the combination of PA and IGHM operators to fuse picture fuzzy numbers (PFNs), especially using Hamacher operations.In sum, the motivations of the paper are presented as follows:(1)
In complex situations, PFS is more suitable for expressing more answers of DMs, such as support, neutral, against, and refusal, which is beyond the capabilities of FS and IFS. Thus, PFS is selected to represent decision-making information in the paper.(2)
In more sophisticated cases, interrelationships among attributes are commonly existed in MADM issue, and unreasonable data may also be provided by DMs. Nevertheless, existing picture fuzzy AOs cannot reflect the correlations between attributes and alleviate the influence of abnormal data simultaneously. Thus, to solve this problem, the combination of both PA and IGHM operators is proposed in the paper.(3)
In MADM problem, to fuse information, aggregating operator is usually based on different operations, and Hamacher operations are more general and agile. Thus, it is an interesting topic to investigate the power improved generalized Heronian mean (PIGHM) AOs utilizing Hamacher operations.Inspired by these factors, our contributions include the following: (1) We firstly combine PA and IGHM operators and propose some novel power improved generalized Heronian mean (PIGHM) AOs. (2) Based on Hamacher operations, we define novel weighted power improved generalized Heronian mean (WPIGHM) operator and weighted geometric power improved generalized Heronian mean (WGPIGHM) operator. (3) Based on the proposed AOs, we build a new MADM model to fuse PFNs.The framework of the paper is constructed as follows. In Section2, the notion and Hamacher operations of PFNs as well as distance measure and cosine coefficient are introduced. In Section 3, PIGHM AOs utilizing Hamacher operations to picture fuzzy information are developed. Moreover, their predominant properties and particular instances are also analyzed. In Section 4, the MADM method based on PIGHM AOs with PFS is proposed. In Section 5, example and analysis are performed to verify the reliability of the model. In Section 6, some remarks are provided.


## 2. Preliminaries

Here, PFS and its Hamacher operations are introduced, and some measures and related operators are investigated as well.


### 2.1. PFS

Definition 1.
(see [3]). A PFS H in Z is denoted as H=z,μHz,ηHz,νHz,|z∈Z, where μHz∈0,1, ηHz∈0,1, and νHz∈0,1 represent the positive, neutral, and negative memberships, respectively, meeting 0≤μHz+ηHz+νHz≤1, ∀z∈Z, and πHz=1−μHz−ηHz−νHz is called the refusal membership degree.
Particularly, if there is only one element inZ, H is defined by H=μH,ηH,νH, which is called picture fuzzy number (PFN).


### 2.2. Hamacher

Hamacher operations contain Hamacher producte⊗f=ef/γ+1−γe+f−ef and Hamacher sum e⊕f=e+f−ef−1−γef/1−1−γef. Some operations of PFNs utilizing Hamacher are presented below.Definition 2.
(see [45]). Let H=μH,ηH,νH and G=μG,ηG,νG be two PFNs, and λ>0. The Hamacher operations of PFNs are defined as follows:(1)
H⊕G=μH+μG−μHμG−1−γμHμG/1−1−γμHμG,ηHηG/γ+1−γηH+ηG−ηHηG,νHνG/γ+1−γνH+νG−νHνG(2)
H⊗G=μHμG/γ+1−γμH+μG−μHμG,ηH+ηG−ηHηG−1−γηHηG/1−1−γηHηG,νH+νG−νHνG−1−γνHνG/1−1−γνHνG(3)
λH=1+γ−1μHλ−1−μHλ/1+γ−1μHλ+γ−11−μHλ,γηHλ/1+γ−11−ηHλ+γ−1ηHλ,γνHλ/1+γ−11−νHλ+γ−1νHλ(4)
Hλ=γμHλ/1+γ−11−μHλ+γ−1μHλ,1+γ−1ηHλ−1−ηHλ/1+γ−1ηHλ+γ−11−ηHλ,1+γ−1νHλ−1−νHλ/1+γ−1νHλ+γ−11−νHλ


### 2.3. Measure

Definition 3.
The normalized Hamming distance measure between two PFNsH=μH,ηH,νH and G=μG,ηG,νG is denoted by(1)dH,G=12μH−μG+ηH−ηG+νH−νG,where dH,G meets these conditions: 0≤dH,G≤1 and dH,G=dG,H.
The smaller the distance between two PFNs is, the bigger the similarity between them is.Definition 4.
The similarity measure between two PFNsH=μH,ηH,νH and the ideal value H∗=1,0,0 is defined by(2)sH,H∗=μHμH2+ηH2+νH2.


### 2.4. PA and IGHM

Definition 5.
Letβff=1,2,…,h be multiple aggregated values, and the power average (PA) operator [10] and power geometric (PG) operator [48] are, respectively, denoted as(3)PAβ1,β2,…,βh=∑f=1h1+φβfβf∑f=1h1+φβf,PGβ1,β2,…,βh=∏f=1hβf1+φβf/∑f=1h1+φβf,where φβf=∑g=1,g≠fhsupβf,βg and supβf,βg=1−dβf,βg is the support between βf and βg, which meets the following conditions:(1)
supβf,βg∈0,1(2)
supβf,βg=supβg,βf(3)
supβf,βg≥supβf′,βg′
Ifdβf,βg≤dβf′,βg′, here, dβf,βg is the Hamming distance presented in Definition 3.Definition 6.
Lets,t≥0 and βff=1,2,…,h be multiple aggregated values, and let ω=ω1,ω2,…,ωh be a corresponding weight vector of βff=1,2,…,h, where ωf≥0 and ∑f=1hωf=1. Then, the improved generalized weighted Heronian mean (IGWHM) operator [23] and improved generalized geometric weighted Heronian mean (IGGWHM) operator [23] are, respectively, denoted as(4)IGWHMs,tβ1,β2,…,βh=∑f=1h∑g=fhωfωgβfsβgt∑f=1h∑g=fhωfωg1/s+t,IGGWHMs,tβ1,β2,…,βh=1s+t∏f=1h∏g=fhsβf+tβg2h+1−f/hh+1⋅ωg/∑k=fhωk,where the IGWHM operator has idempotency, monotonicity, and boundedness, and IGGWHM operator has reducibility, idempotency, monotonicity, and boundedness.


## 2.1. PFS

Definition 1.
(see [3]). A PFS H in Z is denoted as H=z,μHz,ηHz,νHz,|z∈Z, where μHz∈0,1, ηHz∈0,1, and νHz∈0,1 represent the positive, neutral, and negative memberships, respectively, meeting 0≤μHz+ηHz+νHz≤1, ∀z∈Z, and πHz=1−μHz−ηHz−νHz is called the refusal membership degree.
Particularly, if there is only one element inZ, H is defined by H=μH,ηH,νH, which is called picture fuzzy number (PFN).


## 2.2. Hamacher

Hamacher operations contain Hamacher producte⊗f=ef/γ+1−γe+f−ef and Hamacher sum e⊕f=e+f−ef−1−γef/1−1−γef. Some operations of PFNs utilizing Hamacher are presented below.Definition 2.
(see [45]). Let H=μH,ηH,νH and G=μG,ηG,νG be two PFNs, and λ>0. The Hamacher operations of PFNs are defined as follows:(1)
H⊕G=μH+μG−μHμG−1−γμHμG/1−1−γμHμG,ηHηG/γ+1−γηH+ηG−ηHηG,νHνG/γ+1−γνH+νG−νHνG(2)
H⊗G=μHμG/γ+1−γμH+μG−μHμG,ηH+ηG−ηHηG−1−γηHηG/1−1−γηHηG,νH+νG−νHνG−1−γνHνG/1−1−γνHνG(3)
λH=1+γ−1μHλ−1−μHλ/1+γ−1μHλ+γ−11−μHλ,γηHλ/1+γ−11−ηHλ+γ−1ηHλ,γνHλ/1+γ−11−νHλ+γ−1νHλ(4)
Hλ=γμHλ/1+γ−11−μHλ+γ−1μHλ,1+γ−1ηHλ−1−ηHλ/1+γ−1ηHλ+γ−11−ηHλ,1+γ−1νHλ−1−νHλ/1+γ−1νHλ+γ−11−νHλ


## 2.3. Measure

Definition 3.
The normalized Hamming distance measure between two PFNsH=μH,ηH,νH and G=μG,ηG,νG is denoted by(1)dH,G=12μH−μG+ηH−ηG+νH−νG,where dH,G meets these conditions: 0≤dH,G≤1 and dH,G=dG,H.
The smaller the distance between two PFNs is, the bigger the similarity between them is.Definition 4.
The similarity measure between two PFNsH=μH,ηH,νH and the ideal value H∗=1,0,0 is defined by(2)sH,H∗=μHμH2+ηH2+νH2.


## 2.4. PA and IGHM

Definition 5.
Letβff=1,2,…,h be multiple aggregated values, and the power average (PA) operator [10] and power geometric (PG) operator [48] are, respectively, denoted as(3)PAβ1,β2,…,βh=∑f=1h1+φβfβf∑f=1h1+φβf,PGβ1,β2,…,βh=∏f=1hβf1+φβf/∑f=1h1+φβf,where φβf=∑g=1,g≠fhsupβf,βg and supβf,βg=1−dβf,βg is the support between βf and βg, which meets the following conditions:(1)
supβf,βg∈0,1(2)
supβf,βg=supβg,βf(3)
supβf,βg≥supβf′,βg′
Ifdβf,βg≤dβf′,βg′, here, dβf,βg is the Hamming distance presented in Definition 3.Definition 6.
Lets,t≥0 and βff=1,2,…,h be multiple aggregated values, and let ω=ω1,ω2,…,ωh be a corresponding weight vector of βff=1,2,…,h, where ωf≥0 and ∑f=1hωf=1. Then, the improved generalized weighted Heronian mean (IGWHM) operator [23] and improved generalized geometric weighted Heronian mean (IGGWHM) operator [23] are, respectively, denoted as(4)IGWHMs,tβ1,β2,…,βh=∑f=1h∑g=fhωfωgβfsβgt∑f=1h∑g=fhωfωg1/s+t,IGGWHMs,tβ1,β2,…,βh=1s+t∏f=1h∏g=fhsβf+tβg2h+1−f/hh+1⋅ωg/∑k=fhωk,where the IGWHM operator has idempotency, monotonicity, and boundedness, and IGGWHM operator has reducibility, idempotency, monotonicity, and boundedness.


## 3. Power Improved Generalized Heronian Mean Operators

Based on Hamacher operations, two novel AOs, namely, the picture fuzzy Hamacher weighted power improved generalized Heronian mean (PFHWPIGHM) operator and the picture fuzzy Hamacher weighted geometric power improved generalized Heronian mean (PFHWGPIGHM) operator, are established. Besides, their properties and particular cases are explored as well.


### 3.1. PFHWPIGHM Operator

Definition 7.
Lets,t≥0 and βf=μf,ηf,νff=1,2,…,h be multiple PFNs, and the importance of PFN βf is represented by ωf, meeting ωf≥0 and ∑f=1hωf=1. Then, the picture fuzzy Hamacher weighted power improved generalized Heronian mean (PFHWPIGHM) operator is denoted as(5)PFHWPIGHMs,tβ1,β2,…,βh=⊕f=1h⊕g=fhωfωghεfωf/∑e=1hεeωeβfs⊗hεgωg/∑e=1hεeωeβgt∑f=1h∑g=fhωfωg1/s+t,where εf=1+φβf/∑e=1h1+φβe, φβf=∑g=1,g≠fhsupβf,βg, and supβf,βg=1−dβf,βg.Theorem 1.
Lets,t≥0 and βf=μf,ηf,νff=1,2,…,h be multiple PFNs, and the importance of PFN βf is represented by ωf, meeting ωf≥0 and ∑f=1hωf=1. Based on Hamacher operations in Definition 2, the aggregating value of PFHWPIGHM operator in Definition 7 is still a PFN and is shown as follows:(6)PFHWPIGHMs,tβ1,β2,…,βh=γ∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg+γ2−1∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg1/s+t+γ−1∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg+γ2−1∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t−∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg+γ2−1∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t+γ−1∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg+γ2−1∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t−∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg+γ2−1∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t+γ−1∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t.
Here,(7)Mfg=1+γ−1μfhζf+γ2−11−μfhζfs1+γ−1μghζg+γ2−11−μghζgt+γ2−11+γ−1μfhζf−1−μfhζfs1+γ−1μghζg−1−μghζgt,Nfg=1+γ−1μfhζf+γ2−11−μfhζfs1+γ−1μghζg+γ2−11−μghζgt−1+γ−1μfhζf−1−μfhζfs1+γ−1μghζg−1−μghζgt,Ofg=1+γ−11−ηfhζf+γ2−1ηfhζfs1+γ−11−ηghζg+γ2−1ηghζgt−1+γ−11−ηfhζf−ηfhζfs1+γ−11−ηghζg−ηghζgt,Pfg=1+γ−11−ηfhζf+γ2−1ηfhζfs1+γ−11−ηghζg+γ2−1ηghζgt+γ2−11+γ−11−ηfhζf−ηfhζfs1+γ−11−ηghζg−ηghζgt,Qfg=1+γ−11−νfhζf+γ2−1νfhζfs1+γ−11−νghζg+γ2−1νghζgt−1+γ−11−νfhζf−νfhζfs1+γ−11−νghζg−νghζgt,Rfg=1+γ−11−νfhζf+γ2−1νfhζfs1+γ−11−νghζg+γ2−1νghζgt+γ2−11+γ−11−νfhζf−νfhζfs1+γ−11−νghζg−νghζgt,ζf=εfωf∑e=1hεeωe,ζg=εgωg∑e=1hεeωe.Proof.
Based on Hamacher operations of PFNs presented in Definition2, we get(8)hζfβf=1+γ−1μfhζf−1−μfhζf1+γ−1μfhζf+γ−11−μfhζf,γηfhζf1+γ−11−ηfhζf+γ−1ηfhζf,γνfhζf1+γ−11−νfhζf+γ−1νfhζf,hζgβg=1+γ−1μghζg−1−μghζg1+γ−1μghζg+γ−11−μghζg,γηghζg1+γ−11−ηghζg+γ−1ηghζg,γνghζg1+γ−11−νghζg+γ−1νghζg,hζfβfs=γ1+γ−1μfhζf−1−μfhζfs1+γ−1μfhζf+γ2−11−μfhζfs+γ−11+γ−1μfhζf−1−μfhζfs,1+γ−11−ηfhζf+γ2−1ηfhζfs−1+γ−11−ηfhζf−ηfhζfs1+γ−11−ηfhζf+γ2−1ηfhζfs+γ−11+γ−11−ηfhζf−ηfhζfs,1+γ−11−νfhζf+γ2−1νfhζfs−1+γ−11−νfhζf−νfhζfs1+γ−11−νfhζf+γ2−1νfhζfs+γ−11+γ−11−νfhζf−νfhζfs,hζgβgt=γ1+γ−1μghζg−1−μghζgt1+γ−1μghζg+γ2−11−μghζgt+γ−11+γ−1μghζg−1−μghζgt,1+γ−11−ηghζg+γ2−1ηghζgt−1+γ−11−ηghζg−ηghζgt1+γ−11−ηghζg+γ2−1ηghζgt+γ−11+γ−11−ηghζg−ηghζgt,1+γ−11−νghζg+γ2−1νghζgt−1+γ−11−νghζg−νghζgt1+γ−11−νghζg+γ2−1νghζgt+γ−11+γ−11−νghζg−νghζgt.
Let(9)Af=1+γ−1μfhζf−1−μfhζf,Bf=1+γ−1μfhζf+γ2−11−μfhζf,Cf=1+γ−11−ηfhζf+γ2−1ηfhζf,Df=1+γ−11−ηfhζf−ηfhζf,Ef=1+γ−11−νfhζf+γ2−1νfhζf,Ff=1+γ−11−νfhζf−νfhζf,Ag=1+γ−1μghζg−1−μghζg,Bg=1+γ−1μghζg+γ2−11−μghζg,Cg=1+γ−11−ηghζg+γ2−1ηghζg,Dg=1+γ−11−ηghζg−ηghζg,Eg=1+γ−11−νghζg+γ2−1νghζg,Fg=1+γ−11−νghζg−νghζg.
Then,(10)hζfβfs⊗hζgβgt=γAfsAgtBfsBgt+γ−1AfsAgt,CfsCgt−DfsDgtCfsCgt+γ−1DfsDgt,EfsEgt−FfsFgtEfsEgt+γ−1FfsFgt,ωfωghζfβfs⊗hζgβgt=BfsBgt+γ2−1AfsAgtωfωg−BfsBgt−AfsAgtωfωgBfsBgt+γ2−1AfsAgtωfωg+γ−1BfsBgt−AfsAgtωfωg,γCfsCgt−DfsDgtωfωgCfsCgt+γ2−1DfsDgtωfωg+γ−1CfsCgt−DfsDgtωfωg,γEfsEgt−FfsFgtωfωgEfsEgt+γ2−1FfsFgtωfωg+γ−1EfsEgt−FfsFgtωfωg,⊕f=1h⊕g=fhωfωghζfβfs⊗hζgβgt=∏f=1h∏g=fhBfsBgt+γ2−1AfsAgtωfωg−∏f=1h∏g=fhBfsBgt−AfsAgtωfωg∏f=1h∏g=fhBfsBgt+γ2−1AfsAgtωfωg+γ−1∏f=1h∏g=fhBfsBgt−AfsAgtωfωg,γ∏f=1h∏g=fhCfsCgt−DfsDgtωfωg∏f=1h∏g=fhCfsCgt+γ2−1DfsDgtωfωg+γ−1∏f=1h∏g=fhCfsCgt−DfsDgtωfωg,γ∏f=1h∏g=fhEfsEgt−FfsFgtωfωg∏f=1h∏g=fhEfsEgt+γ2−1FfsFgtωfωg+γ−1∏f=1h∏g=fhEfsEgt+γ2−1FfsFgtωfωg.
Furthermore,(11)⊕f=1h⊕g=fhωfωghζfβfs⊗hζgβgt∑f=1h∑g=fhωfωg=∏f=1h∏g=fhBfsBgt+γ2−1AfsAgtωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhBfsBgt−AfsAgtωfωg1/∑f=1h∑g=fhωfωg∏f=1h∏g=fhBfsBgt+γ2−1AfsAgtωfωg1/∑f=1h∑g=fhωfωg+γ−1∏f=1h∏g=fhBfsBgt−AfsAgtωfωg1/∑f=1h∑g=fhωfωg,γ∏f=1h∏g=fhCfsCgt−DfsDgtωfωg1/∑f=1h∑g=fhωfωg∏f=1h∏g=fhCfsCgt+γ2−1DfsDgtωfωg1/∑f=1h∑g=fhωfωg+γ−1∏f=1h∏g=fhCfsCgt−DfsDgtωfωg1/∑f=1h∑g=fhωfωg,γ∏f=1h∏g=fhEfsEgt−FfsFgtωfωg1/∑f=1h∑g=fhωfωg∏f=1h∏g=fhEfsEgt+γ2−1FfsFgtωfωg1/∑f=1h∑g=fhωfωg+γ−1∏f=1h∏g=fhEfsEgt+γ2−1FfsFgtωfωg1/∑f=1h∑g=fhωfωg.
Let(12)Mfg=BfsBgt+γ2−1AfsAgt,Nfg=BfsBgt−AfsAgt,Ofg=CfsCgt−DfsDgt,Pfg=CfsCgt+γ2−1DfsDgt,Qfg=EfsEgt−FfsFgt,Rfg=EfsEgt+γ2−1FfsFgt.
Then,(13)⊕f=1h⊕g=fhωfωghζfβfs⊗hζgβgt∑f=1h∑g=fhωfωg=∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg+γ−1∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg,γ∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg+γ−1∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg,γ∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg+γ−1∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg.
Thus,(14)PFHWPIGHMs,tβ1,β2,…,βh=⊕f=1h⊕g=fhωfωghζfβfs⊗hζgβgt∑f=1h∑g=fhωfωg1/s+t=γ∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg+γ2−1∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg1/s+t+γ−1∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg1/s+t,∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg+γ2−1∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t−∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg+γ2−1∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t+γ−1∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t,∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg+γ2−1∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t−∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg+γ2−1∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t+γ−1∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t.
Therefore, Theorem1 is true.Theorem 2.
(idempotency). Lets,t≥0 and βf=μf,ηf,νff=1,2,…,h be multiple PFNs, and ω=ω1,ω2,…,ωhT, satisfying ωf=1/hf=1,2,…,h and ∑g=1hωg=1. When βf=βf=1,2,…,h, the PFHWPIGHM operator has the property of idempotency.Proof.
Sinceβf=β for each f, supβf,βg=1 and φβf=h−1.
Then,εf=1+φβf/∑e=1h1+φβe=1/h, hεfωf/∑e=1hεeωe=hωf, and hεgωg/∑e=1hεeωe=hωg.
Thus,(15)PFHWPIGHMs,tβ1,β2,…,βh=⊕f=1h⊕g=fhωfωghεfωf/∑e=1hεeωeβfs⊗hεgωg/∑e=1hεeωeβgt∑f=1h∑g=fhωfωg1/s+t=⊕f=1h⊕g=fhωfωgβs⊗βt∑f=1h∑g=fhωfωg1/s+t=βs+t1/s+t=β.
Therefore, Theorem2 is true.
Next, we will discuss some special instances of the PFHWPIGHM operator.(1)
Whenγ=1, the PFHWPIGHM operator is degenerated to picture fuzzy Algebraic weighted power improved generalized Heronian mean (PFAWPIGHM) operator as follows:(16)PFHWPIGHMs,tβ1,β2,…,βh=1−∏f=1h∏g=fh1−1−1−μfhεfωf/∑e=1hεeωes1−1−μghεgωg/∑e=1hεeωetωfωg1/∑f=1h∑g=fhωfωg1/s+t,1−1−∏f=1h∏g=fh1−1−ηfhεfωf/∑e=1hεeωes1−ηghεgωg/∑e=1hεeωetωfωg1/∑f=1h∑g=fhωfωg1/s+t,1−1−∏f=1h∏g=fh1−1−νfhεfωf/∑e=1hεeωes1−νghεgωg/∑e=1hεeωetωfωg1/∑f=1h∑g=fhωfωg1/s+t.(2)
Whenγ=2, the PFHWPIGHM operator is degenerated to picture fuzzy Einstein weighted power improved generalized Heronian mean (PFEWPIGHM) operator as follows:(17)PFHWPIGHMs,tβ1,β2,…,βh=2∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg+3∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg1/s+t+∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg1/s+t,∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg+3∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t−∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg+3∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t+∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t,∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg+3∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t−∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg+3∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t+∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t.
Here,(18)Mfg=1+μfhεfωf/∑e=1hεeωe+31−μfhεfωf/∑e=1hεeωes1+μghεgωg/∑e=1hεeωe+31−μghεgωg/∑e=1hεeωet+31+μfhεfωf/∑e=1hεeωe−1−μfhεfωf/∑e=1hεeωes1+μghεgωg/∑e=1hεeωe−1−μghεgωg/∑e=1hεeωet,Nfg=1+μfhεfωf/∑e=1hεeωe+31−μfhεfωf/∑e=1hεeωes1+μghεgωg/∑e=1hεeωe+31−μghεgωg/∑e=1hεeωet−1+μfhεfωf/∑e=1hεeωe−1−μfhεfωf/∑e=1hεeωes1+μghεgωg/∑e=1hεeωe−1−μghεgωg/∑e=1hεeωet,Ofg=2−ηfhεfωf/∑e=1hεeωe+3ηfhεfωf/∑e=1hεeωes2−ηghεgωg/∑e=1hεeωe+3ηghεgωg/∑e=1hεeωet−2−ηfhεfωf/∑e=1hεeωe−ηfhεfωf/∑e=1hεeωes2−ηghεgωg/∑e=1hεeωe−ηghεgωg/∑e=1hεeωet,Pfg=2−ηfhεfωf/∑e=1hεeωe+3ηfhεfωf/∑e=1hεeωes2−ηghεgωg/∑e=1hεeωe+3ηghεgωg/∑e=1hεeωet+32−ηfhεfωf/∑e=1hεeωe−ηfhεfωf/∑e=1hεeωes2−ηghεgωg/∑e=1hεeωe−ηghεgωg/∑e=1hεeωet,Qfg=2−νfhεfωf/∑e=1hεeωe+3νfhεfωf/∑e=1hεeωes2−νghεgωg/∑e=1hεeωe+3νghεgωg/∑e=1hεeωet−2−νfhεfωf/∑e=1hεeωe−νfhεfωf/∑e=1hεeωes2−νghεgωg/∑e=1hεeωe−νghεgωg/∑e=1hεeωet,Rfg=2−νfhεfωf/∑e=1hεeωe+3νfhεfωf/∑e=1hεeωes2−νghεgωg/∑e=1hεeωe+3νghεgωg/∑e=1hεeωet+32−νfhεfωf/∑e=1hεeωe−νfhεfωf/∑e=1hεeωes2−νghεgωg/∑e=1hεeωe−νghεgωg/∑e=1hεeωet.


### 3.2. PFHWGPIGHM Operator

Definition 8.
Lets,t≥0 and βff=1,2,…,h be multiple PFNs, and the importance of PFN βf is represented by ωf, meeting ωf≥0 and ∑f=1hωf=1. Then, the picture fuzzy Hamacher weighted geometric power improved generalized Heronian mean (PFHWGPIGHM) operator is denoted as(19)PFHWGPIGHMs,tβ1,β2,…,βh=1s+t⊗f=1h⊗g=fhsβfhεfωf/∑e=1hεeωe⊕tβghεgωg/∑e=1hεeωe2h+1−f/hh+1⋅ωg/∑k=fhωk,where εf=1+φβf/∑e=1h1+φβe, φβf=∑g=1,g≠fhsupβf,βg, and supβf,βg=1−dβf,βg.Theorem 3.
Lets,t≥0 and βf=μf,ηf,νff=1,2,…,h be multiple PFNs, and the importance of PFN βf is represented by ωf, meeting ωf≥0 and ∑f=1hωf=1. Based on Hamacher operations in Definition 2, the aggregating value of PFHWGPIGHM operator in Definition 8 is still a PFN and is shown as follows:(20)PFHWGPIGHMs,tβ1,β2,…,βh=∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk+γ2−1∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t−∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk+γ2−1∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t+γ−1∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t,γ∏f=1h∏g=fhOfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhPfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t∏f=1h∏g=fhOfg2h+1−f/hh+1⋅ωg/∑k=fhωk+γ2−1∏f=1h∏g=fhPfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t+γ−1∏f=1h∏g=fhOfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhPfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t,γ∏f=1h∏g=fhQfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhRfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t∏f=1h∏g=fhQfg2h+1−f/hh+1⋅ωg/∑k=fhωk+γ2−1∏f=1h∏g=fhRfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t+γ−1∏f=1h∏g=fhQfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhRfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t.
Here,(21)Mfg=1+γ−11−μfhζf+γ2−1μfhζfs1+γ−11−μghζg+γ2−1μghζgt−1+γ−11−μfhζf−μfhζfs1+γ−11−μghζg−μghζgt,Nfg=1+γ−11−μfhζf+γ2−1μfhζfs1+γ−11−μghζg+γ2−1μghζgt+γ2−11+γ−11−μfhζf−μfhζfs1+γ−11−μghζg−μghζgt,Ofg=1+γ−1ηfhζf+γ2−11−ηfhζfs1+γ−1ηghζg+γ2−11−ηghζgt+γ2−11+γ−1ηfhζf−1−ηfhζfs1+γ−1ηghζg−1−ηghζgt,Pfg=1+γ−1ηfhζf+γ2−11−ηfhζfs1+γ−1ηghζg+γ2−11−ηghζgt−1+γ−1ηfhζf−1−ηfhζfs1+γ−1ηghζg−1−ηghζgt,Qfg=1+γ−1νfhζf+γ2−11−νfhζfs1+γ−1νghζg+γ2−11−νghζgt+γ2−11+γ−1νfhζf−1−νfhζfs1+γ−1νghζg−1−νghζgt,Rfg=1+γ−1νfhζf+γ2−11−νfhζfs1+γ−1νghζg+γ2−11−νghζgt−1+γ−1νfhζf−1−νfhζfs1+γ−1νghζg−1−νghζgt,ζf=εfωf∑e=1hεeωe,ζg=εgωg∑e=1hεeωe.Proof.
Based on Hamacher operations of PFNs presented in Definition2, we get(22)βfhζf=γμfhζf1+γ−11−μfhζf+γ−1μfhζf,1+γ−1ηfhζf−1−ηfhζf1+γ−1ηfhζf+γ−11−ηfhζf,1+γ−1νfhζf−1−νfhζf1+γ−1νfhζf+γ−11−νfhζf,βghζg=γμghζg1+γ−11−μghζg+γ−1μghζg,1+γ−1ηghζg−1−ηfhζg1+γ−1ηghζg+γ−11−ηghζg,1+γ−1νghζg−1−νghζg1+γ−1νghζg+γ−11−νghζg,sβfhζf=1+γ−11−μfhζf+γ2−1μfhζfs−1+γ−11−μfhζf−μfhζfs1+γ−11−μfhζf+γ2−1μfhζfs+γ−11+γ−11−μfhζf−μfhζfs,γ1+γ−1ηfhζf−1−ηfhζfs1+γ−1ηfhζf+γ2−11−ηfhζfs+γ−11+γ−1ηfhζf−1−ηfhζfs,γ1+γ−1νfhζf−1−νfhζfs1+γ−1νfhζf+γ2−11−νfhζfs+γ−11+γ−1νfhζf−1−νfhζfs,tβghζg=1+γ−11−μghζg+γ2−1μghζgt−1+γ−11−μghζg−μghζgt1+γ−11−μghζg+γ2−1μghζgt+γ−11+γ−11−μghζg−μghζgt,γ1+γ−1ηghζg−1−ηghζgt1+γ−1ηghζg+γ2−11−ηghζgt+γ−11+γ−1ηghζg−1−ηghζgt,γ1+γ−1νghζg−1−νfhζgt1+γ−1νghζg+γ2−11−νghζgt+γ−11+γ−1νghζg−1−νghζgt.
Let(23)Af=1+γ−11−μfhζf+γ2−1μfhζf,Bf=1+γ−11−μfhζf−μfhζf,Cf=1+γ−1ηfhζf−1−ηfhζf,Df=1+γ−1ηfhζf+γ2−11−ηfhζf,Ef=1+γ−1νfhζf−1−νfhζf,Ff=1+γ−1νfhζf+γ2−11−νfhζf,Ag=1+γ−11−μghζg+γ2−1μghζg,Bg=1+γ−11−μghζg−μghζg,Cg=1+γ−1ηghζg−1−ηghζg,Dg=1+γ−1ηghζg+γ2−11−ηghζg,Eg=1+γ−1νghζg−1−νghζg,Fg=1+γ−1νghζg+γ2−11−νghζg.
Then,(24)sβfhζf⊕tβghζg=AfsAgt−BfsBgtAfsAgt+γ−1BfsBgt,γCfsCgtDfsDgt+γ−1CfsCgt,γEfsEgtFfsFgt+γ−1EfsEgt,sβfhζf⊕tβghζg2h+1−f/hh+1⋅ωg/∑k=fhωk=γAfsAgt−BfsBgt2h+1−f/hh+1⋅ωg/∑k=fhωkAfsAgt+γ−1BfsBgt2h+1−f/hh+1⋅ωg/∑k=fhωk+γ−1AfsAgt−BfsBgt2h+1−f/hh+1⋅ωg/∑k=fhωk,DfsDgt+γ−1CfsCgt2h+1−f/hh+1⋅ωg/∑k=fhωk−DfsDgt−CfsCgt2h+1−f/hh+1⋅ωg/∑k=fhωkDfsDgt+γ−1CfsCgt2h+1−f/hh+1⋅ωg/∑k=fhωk+γ−1DfsDgt−CfsCgt2h+1−f/hh+1⋅ωg/∑k=fhωk,FfsFgt+γ−1EfsEgt2h+1−f/hh+1⋅ωg/∑k=fhωk−FfsFgt−EfsEgt2h+1−f/hh+1⋅ωg/∑k=fhωkFfsFgt+γ−1EfsEgt2h+1−f/hh+1⋅ωg/∑k=fhωk+γ−1FfsFgt−EfsEgt2h+1−f/hh+1⋅ωg/∑k=fhωk,⊗f=1h⊗g=fhsβfhζf⊕tβghζg2h+1−f/hh+1⋅ωg/∑k=fhωk=γ∏f=1h∏g=fhAfsAgt−BfsBgt2h+1−f/hh+1⋅ωg/∑k=fhωk∏f=1h∏g=fhAfsAgt+γ−1BfsBgt2h+1−f/hh+1⋅ωg/∑k=fhωk+γ−1∏f=1h∏g=fhAfsAgt−BfsBgt2h+1−f/hh+1⋅ωg/∑k=fhωk,∏f=1h∏g=fhDfsDgt+γ−1CfsCgt2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhDfsDgt−CfsCgt2h+1−f/hh+1⋅ωg/∑k=fhωk∏f=1h∏g=fhDfsDgt+γ−1CfsCgt2h+1−f/hh+1⋅ωg/∑k=fhωk+γ−1∏f=1h∏g=fhDfsDgt−CfsCgt2h+1−f/hh+1⋅ωg/∑k=fhωk,∏f=1h∏g=fhFfsFgt+γ−1EfsEgt2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhFfsFgt−EfsEgt2h+1−f/hh+1⋅ωg/∑k=fhωk∏f=1h∏g=fhFfsFgt+γ−1EfsEgt2h+1−f/hh+1⋅ωg/∑k=fhωk+γ−1∏f=1h∏g=fhFfsFgt−EfsEgt2h+1−f/hh+1⋅ωg/∑k=fhωk.
Let(25)Mfg=AfsAgt−BfsBgt,Nfg=AfsAgt+γ−1BfsBgt,Ofg=DfsDgt+γ−1CfsCgt,Pfg=DfsDgt−CfsCgt,Qfg=FfsFgt+γ−1EfsEgt,Rfg=FfsFgt−EfsEgt.
Thus,(26)PFHWGPIGHMs,tβ1,β2,…,βh=1s+t⊗f=1h⊗g=fhsβfhεfωf/∑e=1hεeωe⊕tβghεgωg/∑e=1hεeωe2h+1−f/hh+1⋅ωg/∑k=fhωk=∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk+γ2−1∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t−∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk+γ2−1∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t+γ−1∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t,γ∏f=1h∏g=fhOfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhPfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t∏f=1h∏g=fhOfg2h+1−f/hh+1⋅ωg/∑k=fhωk+γ2−1∏f=1h∏g=fhPfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t+γ−1∏f=1h∏g=fhOfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhPfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t,γ∏f=1h∏g=fhQfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhRfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t∏f=1h∏g=fhQfg2h+1−f/hh+1⋅ωg/∑k=fhωk+γ2−1∏f=1h∏g=fhRfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t+γ−1∏f=1h∏g=fhQfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhRfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t.
Therefore, Theorem3 is true.Theorem 4.
(idempotency). Lets,t≥0 and βf=μf,ηf,νff=1,2,…,h be multiple PFNs, and ω=ω1,ω2,…,ωhT, satisfying ωf=1/nf=1,2,…,h and ∑g=1hωg=1. When βf=βf=1,2,…,h, the PFHWGPIGHM operator has the property of idempotency.Proof.
Sinceβf=β for each f, supβf,βg=1, and φβf=h−1, Then, εf=1+φβf/∑e=1h1+φβe=1/h, and hεfωf/∑e=1hεeωe=hωf=1. hεgωg/∑e=1hεeωe=hωg=1.
Thus,(27)PFHWGPIGHMs,tβ1,β2,…,βh=1s+t⊗f=1h⊗g=fhsβfhεfωf/∑e=1hεeωe⊕tβghεgωg/∑e=1hεeωe2h+1−f/hh+1⋅ωg/∑k=fhωk=1s+t⊗f=1h⊗g=fhsβ⊕tβ2h+1−f/hh+1⋅1/h+1−f=1s+tsβ⊕tβhh+1/22/hh+1=β.
Therefore, Theorem4 is true.
Next, we will discuss some special instances of the PFHWGPIGHM operator.(1)
Whenγ=1, the PFHWGPIGHM operator is degenerated to picture fuzzy Algebraic weighted geometric power improved generalized Heronian mean (PFAWGPIGHM) operator as follows:(28)PFHWGPIGHMs,tβ1,β2,…,βh=1−1−∏f=1h∏g=fh1−1−μfhεfωf/∑e=1hεeωes1−μghεgωg/∑e=1hεeωet2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t,1−∏f=1h∏g=fh1−1−1−ηfhεfωf/∑e=1hεeωes1−1−ηghεgωg/∑e=1hεeωet2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t,1−∏f=1h∏g=fh1−1−1−νfhεfωf/∑e=1hεeωes1−1−νghεgωg/∑e=1hεeωet2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t.(2)
Whenγ=2, the PFHWGPIGHM operator is degenerated to picture fuzzy Einstein weighted geometric power improved generalized Heronian mean (PFEWGPIGHM) operator as follows:(29)PFEWGPIGHMs,tβ1,β2,…,βh=∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk+3∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t−∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk+3∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t+∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t,2∏f=1h∏g=fhOfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhPfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t∏f=1h∏g=fhOfg2h+1−f/hh+1⋅ωg/∑k=fhωk+3∏f=1h∏g=fhPfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t+∏f=1h∏g=fhOfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhPfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t,2∏f=1h∏g=fhQfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhRfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t∏f=1h∏g=fhQfg2h+1−f/hh+1⋅ωg/∑k=fhωk+3∏f=1h∏g=fhRfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t+∏f=1h∏g=fhQfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhRfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t.
Here,(30)Mfg=2−μfhεfωf/∑e=1hεeωe+3μfhεfωf/∑e=1hεeωes2−μghεgωg/∑e=1hεeωe+3μghεgωg/∑e=1hεeωet−2−μfhεfωf/∑e=1hεeωe−μfhεfωf/∑e=1hεeωes2−μghεgωg/∑e=1hεeωe−μghεgωg/∑e=1hεeωet,Nfg=2−μfhεfωf/∑e=1hεeωe+3μfhεfωf/∑e=1hεeωes2−μghεgωg/∑e=1hεeωe+3μghεgωg/∑e=1hεeωet+32−μfhεfωf/∑e=1hεeωe−μfhεfωf/∑e=1hεeωes2−μghεgωg/∑e=1hεeωe−μghεgωg/∑e=1hεeωet,Ofg=1+ηfhεfωf/∑e=1hεeωe+31−ηfhεfωf/∑e=1hεeωes1+ηghεgωg/∑e=1hεeωe+31−ηghεgωg/∑e=1hεeωet+31+ηfhεfωf/∑e=1hεeωe−1−ηfhεfωf/∑e=1hεeωes1+ηghεgωg/∑e=1hεeωe−1−ηghεgωg/∑e=1hεeωet,Pfg=1+ηfhεfωf/∑e=1hεeωe+31−ηfhεfωf/∑e=1hεeωes1+ηghεgωg/∑e=1hεeωe+31−ηghεgωg/∑e=1hεeωet−1+ηfhεfωf/∑e=1hεeωe−1−ηfhεfωf/∑e=1hεeωes1+ηghεgωg/∑e=1hεeωe−1−ηghεgωg/∑e=1hεeωet,Qfg=1+νfhεfωf/∑e=1hεeωe+31−νfhεfωf/∑e=1hεeωes1+νghεgωg/∑e=1hεeωe+31−νghεgωg/∑e=1hεeωet+31+νfhεfωf/∑e=1hεeωe−1−νfhεfωf/∑e=1hεeωes1+νghεgωg/∑e=1hεeωe−1−νghεgωg/∑e=1hεeωet,Rfg=1+νfhεfωf/∑e=1hεeωe+31−νfhεfωf/∑e=1hεeωes1+νghεgωg/∑e=1hεeωe+31−νghεgωg/∑e=1hεeωet−1+νfhεfωf/∑e=1hεeωe−1−νfhεfωf/∑e=1hεeωes1+νghεgωg/∑e=1hεeωe−1−νghεgωg/∑e=1hεeωet.


## 3.1. PFHWPIGHM Operator

Definition 7.
Lets,t≥0 and βf=μf,ηf,νff=1,2,…,h be multiple PFNs, and the importance of PFN βf is represented by ωf, meeting ωf≥0 and ∑f=1hωf=1. Then, the picture fuzzy Hamacher weighted power improved generalized Heronian mean (PFHWPIGHM) operator is denoted as(5)PFHWPIGHMs,tβ1,β2,…,βh=⊕f=1h⊕g=fhωfωghεfωf/∑e=1hεeωeβfs⊗hεgωg/∑e=1hεeωeβgt∑f=1h∑g=fhωfωg1/s+t,where εf=1+φβf/∑e=1h1+φβe, φβf=∑g=1,g≠fhsupβf,βg, and supβf,βg=1−dβf,βg.Theorem 1.
Lets,t≥0 and βf=μf,ηf,νff=1,2,…,h be multiple PFNs, and the importance of PFN βf is represented by ωf, meeting ωf≥0 and ∑f=1hωf=1. Based on Hamacher operations in Definition 2, the aggregating value of PFHWPIGHM operator in Definition 7 is still a PFN and is shown as follows:(6)PFHWPIGHMs,tβ1,β2,…,βh=γ∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg+γ2−1∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg1/s+t+γ−1∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg+γ2−1∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t−∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg+γ2−1∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t+γ−1∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg+γ2−1∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t−∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg+γ2−1∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t+γ−1∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t.
Here,(7)Mfg=1+γ−1μfhζf+γ2−11−μfhζfs1+γ−1μghζg+γ2−11−μghζgt+γ2−11+γ−1μfhζf−1−μfhζfs1+γ−1μghζg−1−μghζgt,Nfg=1+γ−1μfhζf+γ2−11−μfhζfs1+γ−1μghζg+γ2−11−μghζgt−1+γ−1μfhζf−1−μfhζfs1+γ−1μghζg−1−μghζgt,Ofg=1+γ−11−ηfhζf+γ2−1ηfhζfs1+γ−11−ηghζg+γ2−1ηghζgt−1+γ−11−ηfhζf−ηfhζfs1+γ−11−ηghζg−ηghζgt,Pfg=1+γ−11−ηfhζf+γ2−1ηfhζfs1+γ−11−ηghζg+γ2−1ηghζgt+γ2−11+γ−11−ηfhζf−ηfhζfs1+γ−11−ηghζg−ηghζgt,Qfg=1+γ−11−νfhζf+γ2−1νfhζfs1+γ−11−νghζg+γ2−1νghζgt−1+γ−11−νfhζf−νfhζfs1+γ−11−νghζg−νghζgt,Rfg=1+γ−11−νfhζf+γ2−1νfhζfs1+γ−11−νghζg+γ2−1νghζgt+γ2−11+γ−11−νfhζf−νfhζfs1+γ−11−νghζg−νghζgt,ζf=εfωf∑e=1hεeωe,ζg=εgωg∑e=1hεeωe.Proof.
Based on Hamacher operations of PFNs presented in Definition2, we get(8)hζfβf=1+γ−1μfhζf−1−μfhζf1+γ−1μfhζf+γ−11−μfhζf,γηfhζf1+γ−11−ηfhζf+γ−1ηfhζf,γνfhζf1+γ−11−νfhζf+γ−1νfhζf,hζgβg=1+γ−1μghζg−1−μghζg1+γ−1μghζg+γ−11−μghζg,γηghζg1+γ−11−ηghζg+γ−1ηghζg,γνghζg1+γ−11−νghζg+γ−1νghζg,hζfβfs=γ1+γ−1μfhζf−1−μfhζfs1+γ−1μfhζf+γ2−11−μfhζfs+γ−11+γ−1μfhζf−1−μfhζfs,1+γ−11−ηfhζf+γ2−1ηfhζfs−1+γ−11−ηfhζf−ηfhζfs1+γ−11−ηfhζf+γ2−1ηfhζfs+γ−11+γ−11−ηfhζf−ηfhζfs,1+γ−11−νfhζf+γ2−1νfhζfs−1+γ−11−νfhζf−νfhζfs1+γ−11−νfhζf+γ2−1νfhζfs+γ−11+γ−11−νfhζf−νfhζfs,hζgβgt=γ1+γ−1μghζg−1−μghζgt1+γ−1μghζg+γ2−11−μghζgt+γ−11+γ−1μghζg−1−μghζgt,1+γ−11−ηghζg+γ2−1ηghζgt−1+γ−11−ηghζg−ηghζgt1+γ−11−ηghζg+γ2−1ηghζgt+γ−11+γ−11−ηghζg−ηghζgt,1+γ−11−νghζg+γ2−1νghζgt−1+γ−11−νghζg−νghζgt1+γ−11−νghζg+γ2−1νghζgt+γ−11+γ−11−νghζg−νghζgt.
Let(9)Af=1+γ−1μfhζf−1−μfhζf,Bf=1+γ−1μfhζf+γ2−11−μfhζf,Cf=1+γ−11−ηfhζf+γ2−1ηfhζf,Df=1+γ−11−ηfhζf−ηfhζf,Ef=1+γ−11−νfhζf+γ2−1νfhζf,Ff=1+γ−11−νfhζf−νfhζf,Ag=1+γ−1μghζg−1−μghζg,Bg=1+γ−1μghζg+γ2−11−μghζg,Cg=1+γ−11−ηghζg+γ2−1ηghζg,Dg=1+γ−11−ηghζg−ηghζg,Eg=1+γ−11−νghζg+γ2−1νghζg,Fg=1+γ−11−νghζg−νghζg.
Then,(10)hζfβfs⊗hζgβgt=γAfsAgtBfsBgt+γ−1AfsAgt,CfsCgt−DfsDgtCfsCgt+γ−1DfsDgt,EfsEgt−FfsFgtEfsEgt+γ−1FfsFgt,ωfωghζfβfs⊗hζgβgt=BfsBgt+γ2−1AfsAgtωfωg−BfsBgt−AfsAgtωfωgBfsBgt+γ2−1AfsAgtωfωg+γ−1BfsBgt−AfsAgtωfωg,γCfsCgt−DfsDgtωfωgCfsCgt+γ2−1DfsDgtωfωg+γ−1CfsCgt−DfsDgtωfωg,γEfsEgt−FfsFgtωfωgEfsEgt+γ2−1FfsFgtωfωg+γ−1EfsEgt−FfsFgtωfωg,⊕f=1h⊕g=fhωfωghζfβfs⊗hζgβgt=∏f=1h∏g=fhBfsBgt+γ2−1AfsAgtωfωg−∏f=1h∏g=fhBfsBgt−AfsAgtωfωg∏f=1h∏g=fhBfsBgt+γ2−1AfsAgtωfωg+γ−1∏f=1h∏g=fhBfsBgt−AfsAgtωfωg,γ∏f=1h∏g=fhCfsCgt−DfsDgtωfωg∏f=1h∏g=fhCfsCgt+γ2−1DfsDgtωfωg+γ−1∏f=1h∏g=fhCfsCgt−DfsDgtωfωg,γ∏f=1h∏g=fhEfsEgt−FfsFgtωfωg∏f=1h∏g=fhEfsEgt+γ2−1FfsFgtωfωg+γ−1∏f=1h∏g=fhEfsEgt+γ2−1FfsFgtωfωg.
Furthermore,(11)⊕f=1h⊕g=fhωfωghζfβfs⊗hζgβgt∑f=1h∑g=fhωfωg=∏f=1h∏g=fhBfsBgt+γ2−1AfsAgtωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhBfsBgt−AfsAgtωfωg1/∑f=1h∑g=fhωfωg∏f=1h∏g=fhBfsBgt+γ2−1AfsAgtωfωg1/∑f=1h∑g=fhωfωg+γ−1∏f=1h∏g=fhBfsBgt−AfsAgtωfωg1/∑f=1h∑g=fhωfωg,γ∏f=1h∏g=fhCfsCgt−DfsDgtωfωg1/∑f=1h∑g=fhωfωg∏f=1h∏g=fhCfsCgt+γ2−1DfsDgtωfωg1/∑f=1h∑g=fhωfωg+γ−1∏f=1h∏g=fhCfsCgt−DfsDgtωfωg1/∑f=1h∑g=fhωfωg,γ∏f=1h∏g=fhEfsEgt−FfsFgtωfωg1/∑f=1h∑g=fhωfωg∏f=1h∏g=fhEfsEgt+γ2−1FfsFgtωfωg1/∑f=1h∑g=fhωfωg+γ−1∏f=1h∏g=fhEfsEgt+γ2−1FfsFgtωfωg1/∑f=1h∑g=fhωfωg.
Let(12)Mfg=BfsBgt+γ2−1AfsAgt,Nfg=BfsBgt−AfsAgt,Ofg=CfsCgt−DfsDgt,Pfg=CfsCgt+γ2−1DfsDgt,Qfg=EfsEgt−FfsFgt,Rfg=EfsEgt+γ2−1FfsFgt.
Then,(13)⊕f=1h⊕g=fhωfωghζfβfs⊗hζgβgt∑f=1h∑g=fhωfωg=∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg+γ−1∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg,γ∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg+γ−1∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg,γ∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg+γ−1∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg.
Thus,(14)PFHWPIGHMs,tβ1,β2,…,βh=⊕f=1h⊕g=fhωfωghζfβfs⊗hζgβgt∑f=1h∑g=fhωfωg1/s+t=γ∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg+γ2−1∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg1/s+t+γ−1∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg1/s+t,∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg+γ2−1∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t−∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg+γ2−1∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t+γ−1∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t,∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg+γ2−1∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t−∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg+γ2−1∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t+γ−1∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t.
Therefore, Theorem1 is true.Theorem 2.
(idempotency). Lets,t≥0 and βf=μf,ηf,νff=1,2,…,h be multiple PFNs, and ω=ω1,ω2,…,ωhT, satisfying ωf=1/hf=1,2,…,h and ∑g=1hωg=1. When βf=βf=1,2,…,h, the PFHWPIGHM operator has the property of idempotency.Proof.
Sinceβf=β for each f, supβf,βg=1 and φβf=h−1.
Then,εf=1+φβf/∑e=1h1+φβe=1/h, hεfωf/∑e=1hεeωe=hωf, and hεgωg/∑e=1hεeωe=hωg.
Thus,(15)PFHWPIGHMs,tβ1,β2,…,βh=⊕f=1h⊕g=fhωfωghεfωf/∑e=1hεeωeβfs⊗hεgωg/∑e=1hεeωeβgt∑f=1h∑g=fhωfωg1/s+t=⊕f=1h⊕g=fhωfωgβs⊗βt∑f=1h∑g=fhωfωg1/s+t=βs+t1/s+t=β.
Therefore, Theorem2 is true.
Next, we will discuss some special instances of the PFHWPIGHM operator.(1)
Whenγ=1, the PFHWPIGHM operator is degenerated to picture fuzzy Algebraic weighted power improved generalized Heronian mean (PFAWPIGHM) operator as follows:(16)PFHWPIGHMs,tβ1,β2,…,βh=1−∏f=1h∏g=fh1−1−1−μfhεfωf/∑e=1hεeωes1−1−μghεgωg/∑e=1hεeωetωfωg1/∑f=1h∑g=fhωfωg1/s+t,1−1−∏f=1h∏g=fh1−1−ηfhεfωf/∑e=1hεeωes1−ηghεgωg/∑e=1hεeωetωfωg1/∑f=1h∑g=fhωfωg1/s+t,1−1−∏f=1h∏g=fh1−1−νfhεfωf/∑e=1hεeωes1−νghεgωg/∑e=1hεeωetωfωg1/∑f=1h∑g=fhωfωg1/s+t.(2)
Whenγ=2, the PFHWPIGHM operator is degenerated to picture fuzzy Einstein weighted power improved generalized Heronian mean (PFEWPIGHM) operator as follows:(17)PFHWPIGHMs,tβ1,β2,…,βh=2∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg+3∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg1/s+t+∏f=1h∏g=fhMfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhNfgωfωg1/∑f=1h∑g=fhωfωg1/s+t,∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg+3∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t−∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg+3∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t+∏f=1h∏g=fhPfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhOfgωfωg1/∑f=1h∑g=fhωfωg1/s+t,∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg+3∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t−∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg+3∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t+∏f=1h∏g=fhRfgωfωg1/∑f=1h∑g=fhωfωg−∏f=1h∏g=fhQfgωfωg1/∑f=1h∑g=fhωfωg1/s+t.
Here,(18)Mfg=1+μfhεfωf/∑e=1hεeωe+31−μfhεfωf/∑e=1hεeωes1+μghεgωg/∑e=1hεeωe+31−μghεgωg/∑e=1hεeωet+31+μfhεfωf/∑e=1hεeωe−1−μfhεfωf/∑e=1hεeωes1+μghεgωg/∑e=1hεeωe−1−μghεgωg/∑e=1hεeωet,Nfg=1+μfhεfωf/∑e=1hεeωe+31−μfhεfωf/∑e=1hεeωes1+μghεgωg/∑e=1hεeωe+31−μghεgωg/∑e=1hεeωet−1+μfhεfωf/∑e=1hεeωe−1−μfhεfωf/∑e=1hεeωes1+μghεgωg/∑e=1hεeωe−1−μghεgωg/∑e=1hεeωet,Ofg=2−ηfhεfωf/∑e=1hεeωe+3ηfhεfωf/∑e=1hεeωes2−ηghεgωg/∑e=1hεeωe+3ηghεgωg/∑e=1hεeωet−2−ηfhεfωf/∑e=1hεeωe−ηfhεfωf/∑e=1hεeωes2−ηghεgωg/∑e=1hεeωe−ηghεgωg/∑e=1hεeωet,Pfg=2−ηfhεfωf/∑e=1hεeωe+3ηfhεfωf/∑e=1hεeωes2−ηghεgωg/∑e=1hεeωe+3ηghεgωg/∑e=1hεeωet+32−ηfhεfωf/∑e=1hεeωe−ηfhεfωf/∑e=1hεeωes2−ηghεgωg/∑e=1hεeωe−ηghεgωg/∑e=1hεeωet,Qfg=2−νfhεfωf/∑e=1hεeωe+3νfhεfωf/∑e=1hεeωes2−νghεgωg/∑e=1hεeωe+3νghεgωg/∑e=1hεeωet−2−νfhεfωf/∑e=1hεeωe−νfhεfωf/∑e=1hεeωes2−νghεgωg/∑e=1hεeωe−νghεgωg/∑e=1hεeωet,Rfg=2−νfhεfωf/∑e=1hεeωe+3νfhεfωf/∑e=1hεeωes2−νghεgωg/∑e=1hεeωe+3νghεgωg/∑e=1hεeωet+32−νfhεfωf/∑e=1hεeωe−νfhεfωf/∑e=1hεeωes2−νghεgωg/∑e=1hεeωe−νghεgωg/∑e=1hεeωet.


## 3.2. PFHWGPIGHM Operator

Definition 8.
Lets,t≥0 and βff=1,2,…,h be multiple PFNs, and the importance of PFN βf is represented by ωf, meeting ωf≥0 and ∑f=1hωf=1. Then, the picture fuzzy Hamacher weighted geometric power improved generalized Heronian mean (PFHWGPIGHM) operator is denoted as(19)PFHWGPIGHMs,tβ1,β2,…,βh=1s+t⊗f=1h⊗g=fhsβfhεfωf/∑e=1hεeωe⊕tβghεgωg/∑e=1hεeωe2h+1−f/hh+1⋅ωg/∑k=fhωk,where εf=1+φβf/∑e=1h1+φβe, φβf=∑g=1,g≠fhsupβf,βg, and supβf,βg=1−dβf,βg.Theorem 3.
Lets,t≥0 and βf=μf,ηf,νff=1,2,…,h be multiple PFNs, and the importance of PFN βf is represented by ωf, meeting ωf≥0 and ∑f=1hωf=1. Based on Hamacher operations in Definition 2, the aggregating value of PFHWGPIGHM operator in Definition 8 is still a PFN and is shown as follows:(20)PFHWGPIGHMs,tβ1,β2,…,βh=∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk+γ2−1∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t−∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk+γ2−1∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t+γ−1∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t,γ∏f=1h∏g=fhOfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhPfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t∏f=1h∏g=fhOfg2h+1−f/hh+1⋅ωg/∑k=fhωk+γ2−1∏f=1h∏g=fhPfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t+γ−1∏f=1h∏g=fhOfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhPfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t,γ∏f=1h∏g=fhQfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhRfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t∏f=1h∏g=fhQfg2h+1−f/hh+1⋅ωg/∑k=fhωk+γ2−1∏f=1h∏g=fhRfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t+γ−1∏f=1h∏g=fhQfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhRfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t.
Here,(21)Mfg=1+γ−11−μfhζf+γ2−1μfhζfs1+γ−11−μghζg+γ2−1μghζgt−1+γ−11−μfhζf−μfhζfs1+γ−11−μghζg−μghζgt,Nfg=1+γ−11−μfhζf+γ2−1μfhζfs1+γ−11−μghζg+γ2−1μghζgt+γ2−11+γ−11−μfhζf−μfhζfs1+γ−11−μghζg−μghζgt,Ofg=1+γ−1ηfhζf+γ2−11−ηfhζfs1+γ−1ηghζg+γ2−11−ηghζgt+γ2−11+γ−1ηfhζf−1−ηfhζfs1+γ−1ηghζg−1−ηghζgt,Pfg=1+γ−1ηfhζf+γ2−11−ηfhζfs1+γ−1ηghζg+γ2−11−ηghζgt−1+γ−1ηfhζf−1−ηfhζfs1+γ−1ηghζg−1−ηghζgt,Qfg=1+γ−1νfhζf+γ2−11−νfhζfs1+γ−1νghζg+γ2−11−νghζgt+γ2−11+γ−1νfhζf−1−νfhζfs1+γ−1νghζg−1−νghζgt,Rfg=1+γ−1νfhζf+γ2−11−νfhζfs1+γ−1νghζg+γ2−11−νghζgt−1+γ−1νfhζf−1−νfhζfs1+γ−1νghζg−1−νghζgt,ζf=εfωf∑e=1hεeωe,ζg=εgωg∑e=1hεeωe.Proof.
Based on Hamacher operations of PFNs presented in Definition2, we get(22)βfhζf=γμfhζf1+γ−11−μfhζf+γ−1μfhζf,1+γ−1ηfhζf−1−ηfhζf1+γ−1ηfhζf+γ−11−ηfhζf,1+γ−1νfhζf−1−νfhζf1+γ−1νfhζf+γ−11−νfhζf,βghζg=γμghζg1+γ−11−μghζg+γ−1μghζg,1+γ−1ηghζg−1−ηfhζg1+γ−1ηghζg+γ−11−ηghζg,1+γ−1νghζg−1−νghζg1+γ−1νghζg+γ−11−νghζg,sβfhζf=1+γ−11−μfhζf+γ2−1μfhζfs−1+γ−11−μfhζf−μfhζfs1+γ−11−μfhζf+γ2−1μfhζfs+γ−11+γ−11−μfhζf−μfhζfs,γ1+γ−1ηfhζf−1−ηfhζfs1+γ−1ηfhζf+γ2−11−ηfhζfs+γ−11+γ−1ηfhζf−1−ηfhζfs,γ1+γ−1νfhζf−1−νfhζfs1+γ−1νfhζf+γ2−11−νfhζfs+γ−11+γ−1νfhζf−1−νfhζfs,tβghζg=1+γ−11−μghζg+γ2−1μghζgt−1+γ−11−μghζg−μghζgt1+γ−11−μghζg+γ2−1μghζgt+γ−11+γ−11−μghζg−μghζgt,γ1+γ−1ηghζg−1−ηghζgt1+γ−1ηghζg+γ2−11−ηghζgt+γ−11+γ−1ηghζg−1−ηghζgt,γ1+γ−1νghζg−1−νfhζgt1+γ−1νghζg+γ2−11−νghζgt+γ−11+γ−1νghζg−1−νghζgt.
Let(23)Af=1+γ−11−μfhζf+γ2−1μfhζf,Bf=1+γ−11−μfhζf−μfhζf,Cf=1+γ−1ηfhζf−1−ηfhζf,Df=1+γ−1ηfhζf+γ2−11−ηfhζf,Ef=1+γ−1νfhζf−1−νfhζf,Ff=1+γ−1νfhζf+γ2−11−νfhζf,Ag=1+γ−11−μghζg+γ2−1μghζg,Bg=1+γ−11−μghζg−μghζg,Cg=1+γ−1ηghζg−1−ηghζg,Dg=1+γ−1ηghζg+γ2−11−ηghζg,Eg=1+γ−1νghζg−1−νghζg,Fg=1+γ−1νghζg+γ2−11−νghζg.
Then,(24)sβfhζf⊕tβghζg=AfsAgt−BfsBgtAfsAgt+γ−1BfsBgt,γCfsCgtDfsDgt+γ−1CfsCgt,γEfsEgtFfsFgt+γ−1EfsEgt,sβfhζf⊕tβghζg2h+1−f/hh+1⋅ωg/∑k=fhωk=γAfsAgt−BfsBgt2h+1−f/hh+1⋅ωg/∑k=fhωkAfsAgt+γ−1BfsBgt2h+1−f/hh+1⋅ωg/∑k=fhωk+γ−1AfsAgt−BfsBgt2h+1−f/hh+1⋅ωg/∑k=fhωk,DfsDgt+γ−1CfsCgt2h+1−f/hh+1⋅ωg/∑k=fhωk−DfsDgt−CfsCgt2h+1−f/hh+1⋅ωg/∑k=fhωkDfsDgt+γ−1CfsCgt2h+1−f/hh+1⋅ωg/∑k=fhωk+γ−1DfsDgt−CfsCgt2h+1−f/hh+1⋅ωg/∑k=fhωk,FfsFgt+γ−1EfsEgt2h+1−f/hh+1⋅ωg/∑k=fhωk−FfsFgt−EfsEgt2h+1−f/hh+1⋅ωg/∑k=fhωkFfsFgt+γ−1EfsEgt2h+1−f/hh+1⋅ωg/∑k=fhωk+γ−1FfsFgt−EfsEgt2h+1−f/hh+1⋅ωg/∑k=fhωk,⊗f=1h⊗g=fhsβfhζf⊕tβghζg2h+1−f/hh+1⋅ωg/∑k=fhωk=γ∏f=1h∏g=fhAfsAgt−BfsBgt2h+1−f/hh+1⋅ωg/∑k=fhωk∏f=1h∏g=fhAfsAgt+γ−1BfsBgt2h+1−f/hh+1⋅ωg/∑k=fhωk+γ−1∏f=1h∏g=fhAfsAgt−BfsBgt2h+1−f/hh+1⋅ωg/∑k=fhωk,∏f=1h∏g=fhDfsDgt+γ−1CfsCgt2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhDfsDgt−CfsCgt2h+1−f/hh+1⋅ωg/∑k=fhωk∏f=1h∏g=fhDfsDgt+γ−1CfsCgt2h+1−f/hh+1⋅ωg/∑k=fhωk+γ−1∏f=1h∏g=fhDfsDgt−CfsCgt2h+1−f/hh+1⋅ωg/∑k=fhωk,∏f=1h∏g=fhFfsFgt+γ−1EfsEgt2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhFfsFgt−EfsEgt2h+1−f/hh+1⋅ωg/∑k=fhωk∏f=1h∏g=fhFfsFgt+γ−1EfsEgt2h+1−f/hh+1⋅ωg/∑k=fhωk+γ−1∏f=1h∏g=fhFfsFgt−EfsEgt2h+1−f/hh+1⋅ωg/∑k=fhωk.
Let(25)Mfg=AfsAgt−BfsBgt,Nfg=AfsAgt+γ−1BfsBgt,Ofg=DfsDgt+γ−1CfsCgt,Pfg=DfsDgt−CfsCgt,Qfg=FfsFgt+γ−1EfsEgt,Rfg=FfsFgt−EfsEgt.
Thus,(26)PFHWGPIGHMs,tβ1,β2,…,βh=1s+t⊗f=1h⊗g=fhsβfhεfωf/∑e=1hεeωe⊕tβghεgωg/∑e=1hεeωe2h+1−f/hh+1⋅ωg/∑k=fhωk=∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk+γ2−1∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t−∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk+γ2−1∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t+γ−1∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t,γ∏f=1h∏g=fhOfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhPfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t∏f=1h∏g=fhOfg2h+1−f/hh+1⋅ωg/∑k=fhωk+γ2−1∏f=1h∏g=fhPfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t+γ−1∏f=1h∏g=fhOfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhPfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t,γ∏f=1h∏g=fhQfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhRfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t∏f=1h∏g=fhQfg2h+1−f/hh+1⋅ωg/∑k=fhωk+γ2−1∏f=1h∏g=fhRfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t+γ−1∏f=1h∏g=fhQfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhRfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t.
Therefore, Theorem3 is true.Theorem 4.
(idempotency). Lets,t≥0 and βf=μf,ηf,νff=1,2,…,h be multiple PFNs, and ω=ω1,ω2,…,ωhT, satisfying ωf=1/nf=1,2,…,h and ∑g=1hωg=1. When βf=βf=1,2,…,h, the PFHWGPIGHM operator has the property of idempotency.Proof.
Sinceβf=β for each f, supβf,βg=1, and φβf=h−1, Then, εf=1+φβf/∑e=1h1+φβe=1/h, and hεfωf/∑e=1hεeωe=hωf=1. hεgωg/∑e=1hεeωe=hωg=1.
Thus,(27)PFHWGPIGHMs,tβ1,β2,…,βh=1s+t⊗f=1h⊗g=fhsβfhεfωf/∑e=1hεeωe⊕tβghεgωg/∑e=1hεeωe2h+1−f/hh+1⋅ωg/∑k=fhωk=1s+t⊗f=1h⊗g=fhsβ⊕tβ2h+1−f/hh+1⋅1/h+1−f=1s+tsβ⊕tβhh+1/22/hh+1=β.
Therefore, Theorem4 is true.
Next, we will discuss some special instances of the PFHWGPIGHM operator.(1)
Whenγ=1, the PFHWGPIGHM operator is degenerated to picture fuzzy Algebraic weighted geometric power improved generalized Heronian mean (PFAWGPIGHM) operator as follows:(28)PFHWGPIGHMs,tβ1,β2,…,βh=1−1−∏f=1h∏g=fh1−1−μfhεfωf/∑e=1hεeωes1−μghεgωg/∑e=1hεeωet2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t,1−∏f=1h∏g=fh1−1−1−ηfhεfωf/∑e=1hεeωes1−1−ηghεgωg/∑e=1hεeωet2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t,1−∏f=1h∏g=fh1−1−1−νfhεfωf/∑e=1hεeωes1−1−νghεgωg/∑e=1hεeωet2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t.(2)
Whenγ=2, the PFHWGPIGHM operator is degenerated to picture fuzzy Einstein weighted geometric power improved generalized Heronian mean (PFEWGPIGHM) operator as follows:(29)PFEWGPIGHMs,tβ1,β2,…,βh=∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk+3∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t−∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk+3∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t+∏f=1h∏g=fhNfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhMfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t,2∏f=1h∏g=fhOfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhPfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t∏f=1h∏g=fhOfg2h+1−f/hh+1⋅ωg/∑k=fhωk+3∏f=1h∏g=fhPfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t+∏f=1h∏g=fhOfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhPfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t,2∏f=1h∏g=fhQfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhRfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t∏f=1h∏g=fhQfg2h+1−f/hh+1⋅ωg/∑k=fhωk+3∏f=1h∏g=fhRfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t+∏f=1h∏g=fhQfg2h+1−f/hh+1⋅ωg/∑k=fhωk−∏f=1h∏g=fhRfg2h+1−f/hh+1⋅ωg/∑k=fhωk1/s+t.
Here,(30)Mfg=2−μfhεfωf/∑e=1hεeωe+3μfhεfωf/∑e=1hεeωes2−μghεgωg/∑e=1hεeωe+3μghεgωg/∑e=1hεeωet−2−μfhεfωf/∑e=1hεeωe−μfhεfωf/∑e=1hεeωes2−μghεgωg/∑e=1hεeωe−μghεgωg/∑e=1hεeωet,Nfg=2−μfhεfωf/∑e=1hεeωe+3μfhεfωf/∑e=1hεeωes2−μghεgωg/∑e=1hεeωe+3μghεgωg/∑e=1hεeωet+32−μfhεfωf/∑e=1hεeωe−μfhεfωf/∑e=1hεeωes2−μghεgωg/∑e=1hεeωe−μghεgωg/∑e=1hεeωet,Ofg=1+ηfhεfωf/∑e=1hεeωe+31−ηfhεfωf/∑e=1hεeωes1+ηghεgωg/∑e=1hεeωe+31−ηghεgωg/∑e=1hεeωet+31+ηfhεfωf/∑e=1hεeωe−1−ηfhεfωf/∑e=1hεeωes1+ηghεgωg/∑e=1hεeωe−1−ηghεgωg/∑e=1hεeωet,Pfg=1+ηfhεfωf/∑e=1hεeωe+31−ηfhεfωf/∑e=1hεeωes1+ηghεgωg/∑e=1hεeωe+31−ηghεgωg/∑e=1hεeωet−1+ηfhεfωf/∑e=1hεeωe−1−ηfhεfωf/∑e=1hεeωes1+ηghεgωg/∑e=1hεeωe−1−ηghεgωg/∑e=1hεeωet,Qfg=1+νfhεfωf/∑e=1hεeωe+31−νfhεfωf/∑e=1hεeωes1+νghεgωg/∑e=1hεeωe+31−νghεgωg/∑e=1hεeωet+31+νfhεfωf/∑e=1hεeωe−1−νfhεfωf/∑e=1hεeωes1+νghεgωg/∑e=1hεeωe−1−νghεgωg/∑e=1hεeωet,Rfg=1+νfhεfωf/∑e=1hεeωe+31−νfhεfωf/∑e=1hεeωes1+νghεgωg/∑e=1hεeωe+31−νghεgωg/∑e=1hεeωet−1+νfhεfωf/∑e=1hεeωe−1−νfhεfωf/∑e=1hεeωes1+νghεgωg/∑e=1hεeωe−1−νghεgωg/∑e=1hεeωet.


## 4. MADM Model

A novel MADM model using the proposed operators to manage PFNs is illustrated as follows. Suppose that there arem alternatives represented by L=L1,L2,…,Lm and h attributes represented by A=A1,A2,…,Ah, and the importance of each attribute is denoted by ωff=1,2,…,h, satisfying ∑f=1hωf=1 and ωf≥0. The assessment value of alternative Li relating to attribute Af takes the form of PFN: αif=μαif,ηαif,ναifi=1,2,…,m;f=1,2,…,h, where 0≤μαif,ηαif,ναif≤1 and 0≤μαif+ηαif+ναif≤1. The initial matrix D=αifm×hi=1,2,…,m;f=1,2,…,h is provided by experts.The MADM model with PFN is shown in Figure1 and the detailed steps are presented as follows.Step 1. Obtain the normalized matrix D˜=βifm×h.The assessment valueαif with the cost attribute Af needs to be transformed into βif based on the following formula:(31)βif=μαif,ηαif,ναif,for benefit attribute,ναif,ηαif,μαif,for cost attribute.Here,i=1,2,…,m;f=1,2,…,h.Step 2. Compute the supports supβif,βig.The supports are calculated based on the following formula:(32)supβif,βig=1−dβif,βig.Here,i=1,2,…,m;f,g=1,2,…,h, and dβif,βig is the normalized Hamming distance between PFNs βif and βig provided in Definition 3.Step 3. Compute the power weights εif.(33)εif=1+φβif∑e=1h1+φβie,whereφβif=∑g=1,g≠fhsupβif,βig, and i=1,2,…,m;f,g=1,2,…,h.Step 4. Compute the weights ζif of PFN βif.(34)ζif=εifωf∑e=1hεieωe,whereζif≥0 and ∑f=1hζif=1, and i=1,2,…,m;f,g=1,2,…,h.Step 5. Compute the global value βi of each alternative Li.The global valueβi can be obtained utilizing the PFHWPIGHM operator or the PFHWGPIGHM operator as defined in Theorems 1 and 3.(35)βi=PFHWPIGHMs,tβi1,βi2,…,βih,or(36)βi=PFHWGPIGHMs,tβi1,βi2,…,βih.Herei=1,2,…,m.Step 6. Compute the similarity of alternative Li and the ideal alternative L∗.According to Definition4, the similarity can be calculated based on the following formula:(37)sL,L∗=μβiμ2βi+η2βi+ν2βi.Step 7. Rank and select alternative(s).Figure 1
The proposed model.Based on the similarity measure calculated above, all alternatives are ranked, and the best vendor is obtained.


## 5. Example and Analysis

An example based on the MADM approach proposed above is applied to verify its practicability, and corresponding analysis is conducted as well.


### 5.1. Example

Assume that a company wants to purchase an ERP system from five vendors represented byLii=1,2,3,4,5, and there are four attributes represented by Aff=1,2,3,4 to evaluate different vendors, where A1 denotes function, A2 denotes fitness, A3 denotes capability, and A4 denotes reputation, whose weight vector is ω=0.2,0.1,0.3,0.4. The assessment value of each vendor Li with attribute Af is denoted by PFNαif, and the matrix D=αif5×4 below is provided:(38)D=αif5×4=0.53,0.33,0.090.89,0.08,0.030.42,0.35,0.180.08,0.89,0.020.73,0.12,0.080.13,0.64,0.21,0.03,0.82,0.130.73,0.15,0.080.91,0.03,0.020.07,0.09,0.050.04,0.85,0.100.68,0.26,0.060.85,0.09,0.050.74,0.16,0.100.02,0.89,0.050.08,0.84,0.060.90,0.05,0.020.68,0.08,0.210.05,0.87,0.060.13,0.75,0.09.To choose the best vendor, the detailed steps are shown as follows:Step 1. Obtain the normalized matrix D˜=βif5×4.Since all attributes are benefit type, the normalized matrixD˜=βif5×4 is equal to the original matrix D=αif5×4.Step 2. Compute the supports supβif,βig.Based on the normalized Hamming distance between PFNsβif and βig in Definition 3, we can get(39)dβ11,β12=0.335,dβ11,β13=0.110,dβ11,β14=0.540,dβ12,β13=0.445,dβ12,β14=0.815,dβ13,β14=0.520,dβ21,β22=0.625,dβ21,β23=0.725,dβ21,β24=0.015,dβ22,β23=0.180,dβ22,β24=0.610,dβ23,β24=0.710,dβ31,β32=0.465,dβ31,β33=0.885,dβ31,β34=0.250,dβ32,β33=0.420,dβ32,β34=0.395,dβ33,β34=0.635,dβ41,β42=0.115,dβ41,β43=0.815,dβ41,β44=0.765,dβ42,β43=0.750,dβ42,β44=0.690,dβ43,β44=0.060,dβ51,β52=0.220,dβ51,β53=0.855,dβ51,β54=0.770,dβ52,β53=0.785,dβ52,β54=0.670,dβ53,β54=0.115.Then,(40)supβ11,β12=supβ12,β11=0.665,supβ11,β13=supβ13,β11=0.890,supβ11,β14=supβ14,β11=0.460,supβ12,β13=supβ13,β12=0.555,supβ12,β14=supβ14,β12=0.185,supβ13,β14=supβ14,β13=0.480,supβ21,β22=supβ22,β21=0.375,supβ21,β23=supβ23,β21=0.275,supβ21,β24=supβ24,β21=0.985,supβ22,β23=supβ23,β22=0.820,supβ22,β24=supβ24,β22=0.390,supβ23,β24=supβ24,β23=0.290,supβ31,β32=supβ32,β31=0.535,supβ31,β33=supβ33,β31=0.115,supβ31,β34=supβ34,β31=0.750,supβ32,β33=supβ33,β32=0.580,supβ32,β34=supβ34,β32=0.605,supβ33,β34=supβ34,β33=0.365,supβ41,β42=supβ42,β41=0.885,supβ41,β43=supβ43,β41=0.185,supβ41,β44=supβ44,β41=0.235,supβ42,β43=supβ43,β42=0.250,supβ42,β44=supβ44,β42=0.310,supβ43,β44=supβ44,β43=0.940,supβ51,β52=supβ52,β51=0.780,supβ51,β53=supβ53,β51=0.145,supβ51,β54=supβ54,β51=0.230,supβ52,β53=supβ53,β52=0.215,supβ52,β54=supβ54,β52=0.330,supβ53,β54=supβ54,β53=0.885.Step 3. Compute the power weights εif.As an example,φβ11 can be computed as follows based on the formula φβif=∑g=1,g≠fhsupβif,βig:(41)φβ11=supβ11,β12+supβ11,β13+supβ11,β14=0.665+0.89+0.46=2.015.Then, we can get(42)φβif5×4=2.0151.4051.9251.1251.6351.5851.3851.6651.41.721.061.721.3051.4451.3751.4851.1551.3251.2451.445.Then, based on formula (33), we can get(43)εif5×4=0.28800.22970.27940.20300.25660.25170.23220.25950.24240.27470.20810.27470.23990.25440.24710.25860.23500.25350.24480.2666.Step 4. Compute the weights ζif of PFNβif.Based on formula (34), we can get(44)ζif5×4=0.23450.09350.34130.33060.20530.10070.27870.41530.19530.11070.25140.44260.19110.10140.29540.41210.18620.10040.29090.4225.Step 5. Compute the global value βi of each alternative Li.Letγ=2,s=t=1. The global value βi can be obtained utilizing the PFHWPIGHM operator as defined in Theorem 1.(45)β1=0.3779,0.4754,0.0420,β2=0.6544,0.2185,0.0637,β3=0.6804,0.2195,0.0374,β4=0.2992,0.6013,0.0335,β5=0.3504,0.5146,0.0388.Similarly, the global valueβi can be obtained utilizing the PFHWGPIGHM operator as defined in Theorem 3.(46)β1=0.3214,0.5766,0.1029,β2=0.3678,0.4453,0.1110,β3=0.3665,0.4298,0.0665,β4=0.1872,0.7498,0.0649,β5=0.2426,0.6444,0.0870.Step 6. Compute the similarity of alternative Li and the ideal alternative L∗.The similarity can be calculated based on formula (37).For the PFHWPIGHM operator,(47)sL1,L∗=0.6208,sL2,L∗=0.9445,sL3,L∗=0.9504,sL4,L∗=0.4449,sL5,L∗=0.5617.For the PFHWGPIGHM operator,(48)sL1,L∗=0.4811,sL2,L∗=0.6254,sL3,L∗=0.6444,sL4,L∗=0.2414,sL5,L∗=0.3496.Step 7. Rank and select alternative(s).The final ranking using the PFHWPIGHM operator or the PFHWGPIGHM operator is alwaysL3≻L2≻L1≻L5≻L4, and the best vendor is L3, while L4 is the worst one.


### 5.2. Sensitivity Analysis

In this part, we will discuss the impact of parametersγ,s, and t to the final ranking result.When parameterss=t=1, γ is assigned to different values, and the rankings utilizing the PFHWPIGHM operator and the PFHWGPIGHM operator are presented in Tables 1 and 2, respectively.Table 1
Rankings based on the PFHWPIGHM operator with different parameterγ.
γRankings1L3≻L2≻L1≻L5≻L42L3≻L2≻L1≻L5≻L43L3≻L2≻L1≻L5≻L45L3≻L2≻L1≻L5≻L410L3≻L2≻L1≻L5≻L415L3≻L2≻L1≻L5≻L420L3≻L2≻L1≻L5≻L450L3≻L2≻L1≻L5≻L4Table 2
Rankings based on the PFHWGPIGHM operator with different parameterγ.
γRankings1L3≻L2≻L1≻L5≻L42L3≻L2≻L1≻L5≻L43L3≻L2≻L1≻L5≻L45L3≻L2≻L1≻L5≻L410L3≻L2≻L1≻L5≻L415L2≻L3≻L1≻L5≻L420L2≻L3≻L1≻L5≻L450L2≻L3≻L1≻L5≻L4From Tables1 and 2, it can be clearly observed that the worst vendor is always L4, while the best vendor is always L3 utilizing the PFHWPIGHM operator, and the best vendor is L3 or L2 utilizing the PFHWGPIGHM operator.When parameterγ=1, s and t are assigned to different values, and the rankings utilizing the PFHWPIGHM operator and the PFHWGPIGHM operator are presented in Tables 3 and 4, respectively.Table 3
Rankings based on the PFHWPIGHM operator withγ=1.
s, tRankingss=t=0.5L3≻L2≻L1≻L5≻L4s=0.5,t=0L3≻L2≻L5≻L1≻L4s=1,t=0L3≻L2≻L5≻L1≻L4s=0,t=0.5L2≻L3≻L1≻L5≻L4s=0,t=1L2≻L3≻L1≻L5≻L4s=t=1L3≻L2≻L1≻L5≻L4s=2,t=1L3≻L2≻L5≻L1≻L4s=5,t=1L3≻L2≻L5≻L4≻L1s=1,t=2L2≻L3≻L1≻L5≻L4s=1,t=5L2≻L3≻L5≻L1≻L4s=t=2L3≻L2≻L5≻L1≻L4s=t=5L2≻L3≻L5≻L4≻L1s=t=10L2≻L3≻L5≻L4≻L1Table 4
Rankings based on the PFHWGPIGHM operator withγ=1.
s, tRankingss=t=0.5L3≻L2≻L1≻L5≻L4s=0.5,t=0L3≻L1≻L5≻L2≻L4s=1,t=0L1≻L3≻L2≻L5≻L4s=0,t=0.5L3≻L2≻L1≻L5≻L4s=0,t=1L3≻L2≻L1≻L5≻L4s=t=1L3≻L2≻L1≻L5≻L4s=2,t=1L2≻L3≻L1≻L5≻L4s=5,t=1L3≻L2≻L1≻L5≻L4s=1,t=2L3≻L2≻L1≻L5≻L4s=1,t=5L3≻L2≻L1≻L5≻L4s=t=2L3≻L2≻L1≻L5≻L4s=t=5L3≻L2≻L1≻L5≻L4s=t=10L3≻L2≻L1≻L5≻L4From Tables3 and 4, it can be clearly observed that the worst vendor is always L4 utilizing the PFHWGPIGHM operator, while the best vendor is L3 or L2 utilizing the PFHWPIGHM operator with different parameters s and t.When parameterγ=2, s and t are assigned to different values, and the rankings utilizing the PFHWPIGHM operator and the PFHWGPIGHM operator are presented in Tables 5 and 6, respectively.Table 5
Rankings based on the PFHWPIGHM operator withγ=2.
s, tRankingss=t=0.5L3≻L2≻L1≻L5≻L4s=0.5,t=0L3≻L2≻L1≻L5≻L4s=1,t=0L3≻L2≻L1≻L5≻L4s=0,t=0.5L2≻L3≻L1≻L5≻L4s=0,t=1L2≻L3≻L1≻L5≻L4s=t=1L3≻L2≻L1≻L5≻L4s=2,t=1L3≻L2≻L5≻L1≻L4s=5,t=1L3≻L2≻L5≻L4≻L1s=1,t=2L2≻L3≻L1≻L5≻L4s=1,t=5L2≻L3≻L5≻L1≻L4s=t=2L3≻L2≻L5≻L1≻L4s=t=5L2≻L3≻L5≻L4≻L1s=t=10L2≻L3≻L5≻L4≻L1Table 6
Rankings based on the PFHWGPIGHM operator withγ=2.
s, tRankingss=t=0.5L3≻L2≻L1≻L5≻L4s=0.5,t=0L3≻L1≻L5≻L2≻L4s=1,t=0L3≻L1≻L2≻L5≻L4s=0,t=0.5L3≻L2≻L1≻L5≻L4s=0,t=1L3≻L2≻L1≻L5≻L4s=t=1L3≻L2≻L1≻L5≻L4s=2,t=1L2≻L3≻L1≻L5≻L4s=5,t=1L3≻L2≻L1≻L5≻L4s=1,t=2L3≻L2≻L1≻L5≻L4s=1,t=5L3≻L2≻L1≻L5≻L4s=t=2L3≻L2≻L1≻L5≻L4s=t=5L3≻L2≻L1≻L5≻L4s=t=10L3≻L2≻L1≻L5≻L4From Tables5 and 6, it can be clearly observed that the best vendor is L3 or L2, while the worst vendor is always L4 utilizing the PFHWGPIGHM operator, and the worst vendor is L4 or L1 utilizing the PFHWPIGHM operator with different parameters s and t.From the sensitivity analysis above, we know that different parameters in the PFHWPIGHM operator and the PFHWGPIGHM operator may cause different ranking orders; thus, parametersγ,s, and t play an essential part in the MADM problem. When γ=1 or γ=2, picture fuzzy Hamacher operations can be degenerated to picture fuzzy Algebraic or Einstein operations, respectively. Therefore, the picture fuzzy Hamacher operators proposed in this paper are more general. Furthermore, parameter s or t can illustrate the risk preference of DMs who can decide the value of parameter s or t in terms of actual situation. Generally, the risk seeker will assign big value to parameter s or t. Therefore, the PFHWPIGHM operator and the PFHWGPIGHM operator are more flexible.


### 5.3. Comparison Analysis

To further demonstrate the superiorities of the proposed MADM method using the PFHWPIGHM operator and the PFHWGPIGHM operator, we compare the method proposed in this paper with the existing method proposed by Wei [45], based on the PFHWA operator and the PFHWG operator, and the method proposed by Ates and Akay [41], based on the PFNWBM operator. The rankings of different AOs are listed in Table 7.Table 7
Rankings of different operators.
OperatorRankingPFHWA [45], γ=2L3≻L2≻L1≻L5≻L4PFHWG [45], γ=2L3≻L1≻L2≻L5≻L4PFNWBM [41], s=t=1L3≻L2≻L1≻L5≻L4PFHWPIGHM,γ=2,s=t=1L3≻L2≻L1≻L5≻L4PFHWGPIGHM,γ=2,s=t=1L3≻L2≻L1≻L5≻L4From Table7, it is clearly observed that the ranking orders of different AOs are the same except the PFHWG operator. Besides, the best vendor is always L3, and the worst one is always L4 for all AOs presented above.Furthermore, a comparison between our method and these methods mentioned above as regards some characteristics is presented in Table8.Table 8
Comparison of characteristics between different methods.
CharacteristicWei [45]Ates and Akay [41]Proposed methodConsider correlations of attributesNoYesYesReduce the influence of biased dataNoNoYesOperational rulesHamacherAlgebraicHamacherNumber of parametersγs,tγ,s,tThe method of Wei [45] does not consider the correlations of attributes and the influence of biased data, and only one parameter is considered. The method of Ates and Akay [41] considers the correlations of attributes but cannot reduce the influence of biased data, and it is based on Algebraic operations, which is only a special case of Hamacher operations, and only two parameters are considered. In contrast, our method not only considers the correlations of attributes but also reduces the influence of biased data. Moreover, it is based on Hamacher operational rules and three parameters γ,s, and t. These characteristics make the proposed method more general and flexible.


## 5.1. Example

Assume that a company wants to purchase an ERP system from five vendors represented byLii=1,2,3,4,5, and there are four attributes represented by Aff=1,2,3,4 to evaluate different vendors, where A1 denotes function, A2 denotes fitness, A3 denotes capability, and A4 denotes reputation, whose weight vector is ω=0.2,0.1,0.3,0.4. The assessment value of each vendor Li with attribute Af is denoted by PFNαif, and the matrix D=αif5×4 below is provided:(38)D=αif5×4=0.53,0.33,0.090.89,0.08,0.030.42,0.35,0.180.08,0.89,0.020.73,0.12,0.080.13,0.64,0.21,0.03,0.82,0.130.73,0.15,0.080.91,0.03,0.020.07,0.09,0.050.04,0.85,0.100.68,0.26,0.060.85,0.09,0.050.74,0.16,0.100.02,0.89,0.050.08,0.84,0.060.90,0.05,0.020.68,0.08,0.210.05,0.87,0.060.13,0.75,0.09.To choose the best vendor, the detailed steps are shown as follows:Step 1. Obtain the normalized matrix D˜=βif5×4.Since all attributes are benefit type, the normalized matrixD˜=βif5×4 is equal to the original matrix D=αif5×4.Step 2. Compute the supports supβif,βig.Based on the normalized Hamming distance between PFNsβif and βig in Definition 3, we can get(39)dβ11,β12=0.335,dβ11,β13=0.110,dβ11,β14=0.540,dβ12,β13=0.445,dβ12,β14=0.815,dβ13,β14=0.520,dβ21,β22=0.625,dβ21,β23=0.725,dβ21,β24=0.015,dβ22,β23=0.180,dβ22,β24=0.610,dβ23,β24=0.710,dβ31,β32=0.465,dβ31,β33=0.885,dβ31,β34=0.250,dβ32,β33=0.420,dβ32,β34=0.395,dβ33,β34=0.635,dβ41,β42=0.115,dβ41,β43=0.815,dβ41,β44=0.765,dβ42,β43=0.750,dβ42,β44=0.690,dβ43,β44=0.060,dβ51,β52=0.220,dβ51,β53=0.855,dβ51,β54=0.770,dβ52,β53=0.785,dβ52,β54=0.670,dβ53,β54=0.115.Then,(40)supβ11,β12=supβ12,β11=0.665,supβ11,β13=supβ13,β11=0.890,supβ11,β14=supβ14,β11=0.460,supβ12,β13=supβ13,β12=0.555,supβ12,β14=supβ14,β12=0.185,supβ13,β14=supβ14,β13=0.480,supβ21,β22=supβ22,β21=0.375,supβ21,β23=supβ23,β21=0.275,supβ21,β24=supβ24,β21=0.985,supβ22,β23=supβ23,β22=0.820,supβ22,β24=supβ24,β22=0.390,supβ23,β24=supβ24,β23=0.290,supβ31,β32=supβ32,β31=0.535,supβ31,β33=supβ33,β31=0.115,supβ31,β34=supβ34,β31=0.750,supβ32,β33=supβ33,β32=0.580,supβ32,β34=supβ34,β32=0.605,supβ33,β34=supβ34,β33=0.365,supβ41,β42=supβ42,β41=0.885,supβ41,β43=supβ43,β41=0.185,supβ41,β44=supβ44,β41=0.235,supβ42,β43=supβ43,β42=0.250,supβ42,β44=supβ44,β42=0.310,supβ43,β44=supβ44,β43=0.940,supβ51,β52=supβ52,β51=0.780,supβ51,β53=supβ53,β51=0.145,supβ51,β54=supβ54,β51=0.230,supβ52,β53=supβ53,β52=0.215,supβ52,β54=supβ54,β52=0.330,supβ53,β54=supβ54,β53=0.885.Step 3. Compute the power weights εif.As an example,φβ11 can be computed as follows based on the formula φβif=∑g=1,g≠fhsupβif,βig:(41)φβ11=supβ11,β12+supβ11,β13+supβ11,β14=0.665+0.89+0.46=2.015.Then, we can get(42)φβif5×4=2.0151.4051.9251.1251.6351.5851.3851.6651.41.721.061.721.3051.4451.3751.4851.1551.3251.2451.445.Then, based on formula (33), we can get(43)εif5×4=0.28800.22970.27940.20300.25660.25170.23220.25950.24240.27470.20810.27470.23990.25440.24710.25860.23500.25350.24480.2666.Step 4. Compute the weights ζif of PFNβif.Based on formula (34), we can get(44)ζif5×4=0.23450.09350.34130.33060.20530.10070.27870.41530.19530.11070.25140.44260.19110.10140.29540.41210.18620.10040.29090.4225.Step 5. Compute the global value βi of each alternative Li.Letγ=2,s=t=1. The global value βi can be obtained utilizing the PFHWPIGHM operator as defined in Theorem 1.(45)β1=0.3779,0.4754,0.0420,β2=0.6544,0.2185,0.0637,β3=0.6804,0.2195,0.0374,β4=0.2992,0.6013,0.0335,β5=0.3504,0.5146,0.0388.Similarly, the global valueβi can be obtained utilizing the PFHWGPIGHM operator as defined in Theorem 3.(46)β1=0.3214,0.5766,0.1029,β2=0.3678,0.4453,0.1110,β3=0.3665,0.4298,0.0665,β4=0.1872,0.7498,0.0649,β5=0.2426,0.6444,0.0870.Step 6. Compute the similarity of alternative Li and the ideal alternative L∗.The similarity can be calculated based on formula (37).For the PFHWPIGHM operator,(47)sL1,L∗=0.6208,sL2,L∗=0.9445,sL3,L∗=0.9504,sL4,L∗=0.4449,sL5,L∗=0.5617.For the PFHWGPIGHM operator,(48)sL1,L∗=0.4811,sL2,L∗=0.6254,sL3,L∗=0.6444,sL4,L∗=0.2414,sL5,L∗=0.3496.Step 7. Rank and select alternative(s).The final ranking using the PFHWPIGHM operator or the PFHWGPIGHM operator is alwaysL3≻L2≻L1≻L5≻L4, and the best vendor is L3, while L4 is the worst one.


## 5.2. Sensitivity Analysis

In this part, we will discuss the impact of parametersγ,s, and t to the final ranking result.When parameterss=t=1, γ is assigned to different values, and the rankings utilizing the PFHWPIGHM operator and the PFHWGPIGHM operator are presented in Tables 1 and 2, respectively.Table 1
Rankings based on the PFHWPIGHM operator with different parameterγ.
γRankings1L3≻L2≻L1≻L5≻L42L3≻L2≻L1≻L5≻L43L3≻L2≻L1≻L5≻L45L3≻L2≻L1≻L5≻L410L3≻L2≻L1≻L5≻L415L3≻L2≻L1≻L5≻L420L3≻L2≻L1≻L5≻L450L3≻L2≻L1≻L5≻L4Table 2
Rankings based on the PFHWGPIGHM operator with different parameterγ.
γRankings1L3≻L2≻L1≻L5≻L42L3≻L2≻L1≻L5≻L43L3≻L2≻L1≻L5≻L45L3≻L2≻L1≻L5≻L410L3≻L2≻L1≻L5≻L415L2≻L3≻L1≻L5≻L420L2≻L3≻L1≻L5≻L450L2≻L3≻L1≻L5≻L4From Tables1 and 2, it can be clearly observed that the worst vendor is always L4, while the best vendor is always L3 utilizing the PFHWPIGHM operator, and the best vendor is L3 or L2 utilizing the PFHWGPIGHM operator.When parameterγ=1, s and t are assigned to different values, and the rankings utilizing the PFHWPIGHM operator and the PFHWGPIGHM operator are presented in Tables 3 and 4, respectively.Table 3
Rankings based on the PFHWPIGHM operator withγ=1.
s, tRankingss=t=0.5L3≻L2≻L1≻L5≻L4s=0.5,t=0L3≻L2≻L5≻L1≻L4s=1,t=0L3≻L2≻L5≻L1≻L4s=0,t=0.5L2≻L3≻L1≻L5≻L4s=0,t=1L2≻L3≻L1≻L5≻L4s=t=1L3≻L2≻L1≻L5≻L4s=2,t=1L3≻L2≻L5≻L1≻L4s=5,t=1L3≻L2≻L5≻L4≻L1s=1,t=2L2≻L3≻L1≻L5≻L4s=1,t=5L2≻L3≻L5≻L1≻L4s=t=2L3≻L2≻L5≻L1≻L4s=t=5L2≻L3≻L5≻L4≻L1s=t=10L2≻L3≻L5≻L4≻L1Table 4
Rankings based on the PFHWGPIGHM operator withγ=1.
s, tRankingss=t=0.5L3≻L2≻L1≻L5≻L4s=0.5,t=0L3≻L1≻L5≻L2≻L4s=1,t=0L1≻L3≻L2≻L5≻L4s=0,t=0.5L3≻L2≻L1≻L5≻L4s=0,t=1L3≻L2≻L1≻L5≻L4s=t=1L3≻L2≻L1≻L5≻L4s=2,t=1L2≻L3≻L1≻L5≻L4s=5,t=1L3≻L2≻L1≻L5≻L4s=1,t=2L3≻L2≻L1≻L5≻L4s=1,t=5L3≻L2≻L1≻L5≻L4s=t=2L3≻L2≻L1≻L5≻L4s=t=5L3≻L2≻L1≻L5≻L4s=t=10L3≻L2≻L1≻L5≻L4From Tables3 and 4, it can be clearly observed that the worst vendor is always L4 utilizing the PFHWGPIGHM operator, while the best vendor is L3 or L2 utilizing the PFHWPIGHM operator with different parameters s and t.When parameterγ=2, s and t are assigned to different values, and the rankings utilizing the PFHWPIGHM operator and the PFHWGPIGHM operator are presented in Tables 5 and 6, respectively.Table 5
Rankings based on the PFHWPIGHM operator withγ=2.
s, tRankingss=t=0.5L3≻L2≻L1≻L5≻L4s=0.5,t=0L3≻L2≻L1≻L5≻L4s=1,t=0L3≻L2≻L1≻L5≻L4s=0,t=0.5L2≻L3≻L1≻L5≻L4s=0,t=1L2≻L3≻L1≻L5≻L4s=t=1L3≻L2≻L1≻L5≻L4s=2,t=1L3≻L2≻L5≻L1≻L4s=5,t=1L3≻L2≻L5≻L4≻L1s=1,t=2L2≻L3≻L1≻L5≻L4s=1,t=5L2≻L3≻L5≻L1≻L4s=t=2L3≻L2≻L5≻L1≻L4s=t=5L2≻L3≻L5≻L4≻L1s=t=10L2≻L3≻L5≻L4≻L1Table 6
Rankings based on the PFHWGPIGHM operator withγ=2.
s, tRankingss=t=0.5L3≻L2≻L1≻L5≻L4s=0.5,t=0L3≻L1≻L5≻L2≻L4s=1,t=0L3≻L1≻L2≻L5≻L4s=0,t=0.5L3≻L2≻L1≻L5≻L4s=0,t=1L3≻L2≻L1≻L5≻L4s=t=1L3≻L2≻L1≻L5≻L4s=2,t=1L2≻L3≻L1≻L5≻L4s=5,t=1L3≻L2≻L1≻L5≻L4s=1,t=2L3≻L2≻L1≻L5≻L4s=1,t=5L3≻L2≻L1≻L5≻L4s=t=2L3≻L2≻L1≻L5≻L4s=t=5L3≻L2≻L1≻L5≻L4s=t=10L3≻L2≻L1≻L5≻L4From Tables5 and 6, it can be clearly observed that the best vendor is L3 or L2, while the worst vendor is always L4 utilizing the PFHWGPIGHM operator, and the worst vendor is L4 or L1 utilizing the PFHWPIGHM operator with different parameters s and t.From the sensitivity analysis above, we know that different parameters in the PFHWPIGHM operator and the PFHWGPIGHM operator may cause different ranking orders; thus, parametersγ,s, and t play an essential part in the MADM problem. When γ=1 or γ=2, picture fuzzy Hamacher operations can be degenerated to picture fuzzy Algebraic or Einstein operations, respectively. Therefore, the picture fuzzy Hamacher operators proposed in this paper are more general. Furthermore, parameter s or t can illustrate the risk preference of DMs who can decide the value of parameter s or t in terms of actual situation. Generally, the risk seeker will assign big value to parameter s or t. Therefore, the PFHWPIGHM operator and the PFHWGPIGHM operator are more flexible.


## 5.3. Comparison Analysis

To further demonstrate the superiorities of the proposed MADM method using the PFHWPIGHM operator and the PFHWGPIGHM operator, we compare the method proposed in this paper with the existing method proposed by Wei [45], based on the PFHWA operator and the PFHWG operator, and the method proposed by Ates and Akay [41], based on the PFNWBM operator. The rankings of different AOs are listed in Table 7.Table 7
Rankings of different operators.
OperatorRankingPFHWA [45], γ=2L3≻L2≻L1≻L5≻L4PFHWG [45], γ=2L3≻L1≻L2≻L5≻L4PFNWBM [41], s=t=1L3≻L2≻L1≻L5≻L4PFHWPIGHM,γ=2,s=t=1L3≻L2≻L1≻L5≻L4PFHWGPIGHM,γ=2,s=t=1L3≻L2≻L1≻L5≻L4From Table7, it is clearly observed that the ranking orders of different AOs are the same except the PFHWG operator. Besides, the best vendor is always L3, and the worst one is always L4 for all AOs presented above.Furthermore, a comparison between our method and these methods mentioned above as regards some characteristics is presented in Table8.Table 8
Comparison of characteristics between different methods.
CharacteristicWei [45]Ates and Akay [41]Proposed methodConsider correlations of attributesNoYesYesReduce the influence of biased dataNoNoYesOperational rulesHamacherAlgebraicHamacherNumber of parametersγs,tγ,s,tThe method of Wei [45] does not consider the correlations of attributes and the influence of biased data, and only one parameter is considered. The method of Ates and Akay [41] considers the correlations of attributes but cannot reduce the influence of biased data, and it is based on Algebraic operations, which is only a special case of Hamacher operations, and only two parameters are considered. In contrast, our method not only considers the correlations of attributes but also reduces the influence of biased data. Moreover, it is based on Hamacher operational rules and three parameters γ,s, and t. These characteristics make the proposed method more general and flexible.


## 6. Conclusion and Future Directions

On the basis of the related studies of AOs, the PIGHM operators, which combine the PA operator and the IGHM operator, are firstly explored in this paper. Considering the superiorities of PFS and Hamacher operations, two novel AOs, the PFHWPIGHM operator and the PFHWGPIGHM operator, are proposed, and some particular cases are also discussed. Further, based on these proposed AOs, a MADM model is constructed under PFS environment, and the detailed steps are also investigated. Eventually, an example on ERP vendor selection is conducted to demonstrate the practicality and credibility of the proposed model. Besides, we discuss the impact of different parametersγ,s, and t on the final ranking, and the comparable analysis from different aspects is also concluded.In the recent years, to better convey the vagueness and uncertainty of cognitive information, some different fuzzy sets and their applications have been proposed and developed, such as Pythagorean fuzzy set [47], intuitionistic fuzzy set [48], Pythagorean 2-tuple linguistic set [49], probabilistic linguistic set [50], picture fuzzy labelling of graphs [51], picture fuzzy soft set [52], bipolar picture fuzzy set [53], intuitionistic multiplicative set [54], and t-spherical fuzzy set [55]. In future research, it is meaningful to explore the application of the proposed AOs and MADM model to these fuzzy sets and apply them to different fields.



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*Source: 6261229-2021-07-16.xml*