Hamacher heronian mean operators for multi-critria decision-making under multi-valued picture fuzzy uncertain lingsuitic environment

Abstract

Picture fuzzy set (PFS) and linguistic term set (LTS) are two significant notions in multi-criteria decision-making (MCDM). In practice, decision-makers sometimes need utilize the multiple probable membership degrees for an uncertain linguistic term to express evaluation information. Motivated by these, to better convey the vagueness and uncertainty of cognitive information, multi-valued picture fuzzy uncertain linguistic set combining picture hesitant fuzzy set with uncertain linguistic term set is proposed. We firstly define the concepts of multi-valued picture fuzzy uncertain linguistic set and multi-valued picture fuzzy uncertain linguistic number. Hamacher operations are more general and flexible in information fusion, thus, Hamacher operations and comparison method are developed at the same time. Improved generalized Heronian Mean operator can simultaneously reflect correlations between values and prevent the redundant calculation. Then, two operators of improved generalized weighted Heronian mean and improved generalized geometric weighted Heronian mean in view of Hamacher operations are proposed. Meanwhile, some distinguished properties and instances of two operators are explored as well. Moreover, a novel MCDM approach applying the developed operators is constructed. Ultimately, an illustrative example on vendor selection is performed, and sensitivity analysis and comparison analysis are provided to verify the powerfulness of the proposed method.

Keywords

Hamacher improved generalized heronian operator multi-criteria decision-making multi-valued picture fuzzy uncertain linguistic set

1 Introduction

Multi-criteria decision-making (MCDM) exists in different domains, for instance culture, economy, politics, society, and military. Considering the uncertainty of actual information, utilizing fuzzy set (FS) to settle MCDM problem has become a hotspot in recent years. Zadeh [1] originally introduced FS, which utilizes a crisp value between 0 and 1 to present the membership. Later, Atanassov [2] presented intuitionistic fuzzy set (IFS), which adds non-membership to handle the weakness of FS. Their sum for membership and non-membership degrees lies in [0, 1]. Thus, IFS can simultaneously consider not only the membership and non-membership but also hesitancy degree. Nevertheless, IFS cannot directly express the neutral membership degree. Subsequently, picture fuzzy set (PFS) [3] was put forward in 2013, which can simultaneously take into account the degrees of positive membership, negative membership, and neutral membership, they take the value in [0,1]. PFS can reflect various answers of decision makers (DMs), which cannot be expressed by IFS accurately. Thus, the PFS is more practical on dealing with uncertain information than IFS. Then, some achievements on PFS [4 –6] and its extensions [7 –14] have been made, including the measures, aggregating operators, spherical fuzzy sets, and so on. In many cases, decision-makers may be hesitant when utilizing PFS to describe their assessment values, namely, they may employ a few probable crisp values lying in [0, 1] to depict the degree of positive, neutral, and negative, rather than a particular value. Therefore, Wang [15] presented the definition of picture hesitant fuzzy set (PHFS) to settle this issue, developed generalized weighted aggregating operators under picture hesitant fuzzy environment, and applied two examples to validate the superiorities of the novel approach.

However, in real MCDM problems, DMs may select linguistic term (LT) to convey fuzzy information rather than quantitative number. Thus, utilizing LT set (LTS) has attracted many attentions since it is proposed by Zadeh [16]. Generally, a membership of LT is 1, then, non-membership degree of LT is unable to be well-described. Then, to avoid this issue, intuitionistic linguistic set [17] and intuitionistic uncertain linguistic set [18], together with their extensions [19 –24], are defined and developed. As we all known, PFS as an extension of IFS is more suitable for expressing the vagueness of people. Thus, the combination of PFS and LTS has been concerned by scholars, many new linguistic sets are explored, for instance, picture fuzzy linguistic set (PFLS) [25], picture fuzzy uncertain linguistic set (PFULS) [26], hesitant picture 2-tuple linguistic set (HP2TLS) [27], linguistic picture fuzzy set (LPFS) [28], q-rung picture linguistic set (q-RPLS) [29], spherical linguistic fuzzy sets [30].

From the aforementioned analysis, we know some achievements on PFS and LTS have been made, however, with the increasing complication of cognitive information, the ambiguity and hesitancy of people commonly exist in real world, the existing researches cannot convey and cope with picture hesitant fuzzy uncertain linguistic cognitive situation. For example, in real MCDM problems, DMs believe degree of positive membership regarding uncertain LT [good, very good] is 0.5 or 0.6, degree of negative membership regarding uncertain LT [good, very good] is 0.2 or 0.3, and degree of neutral membership regarding uncertain LT [good, very good] is 0.1. Such decision information is beyond the scope of existing PFLS and its variations. To solve this issue, we present the definition of multi-valued picture fuzzy uncertain linguistic set (MVPFULS), consisting of two parts, the former part is an uncertain linguistic variable (ULV) to describe the qualitative assessment information, and the latter part is a multi-valued picture fuzzy element (MVPFE) to describe the quantitative assessment information on the former ULV, the MVPFE part is composed of several possible values on positive, neutral, and negative degrees. According to concept of MVPFULS, information mentioned above can be expressed as 〈 [good, very good] , ({ 0.5, 0.6 } { 0.1 } { 0.2, 0.3 }) 〉.

In coping with actual MCDM problems, aggregating operator fusing multiple input arguments plays a crucial role. Many agile aggregating operators have been introduced and applied, such as Bonferroni mean (BM) [31], Muirhead mean (MM) [32], Maclaurin symmetric mean (MSM) [33], weighted arithmetic averaging (WAA) [34], Induced ordered weighted averaging (IOWA) [35], weighted geometric average (WGA) [36], power average (PA) [37], Heronian mean (HM) [38]. Various operators have their own characteristics, so using various aggregating operator to fuse fuzzy information has attracted many attentions in MCDM problems. In recent years, a few aggregating operators have been employed to fuse picture fuzzy information. For instance, Wei [39] extended conventional arithmetic operator and geometric operator to picture fuzzy environment, some operators of PFWA, PFWG, PFOWA, PFOWG, PFHA, and PFHG were developed. Furthermore, Wei [40] also applied arithmetic and geometric operators based on Hamacher operations for fusing picture fuzzy information. Zhang [41] utilized Heronian mean operator based on Dombi operations to merge picture fuzzy numbers. Jana [42] used arithmetic and geometric operators based on Dombi operations to merge picture fuzzy numbers, PFDWA, PFDOWA, PFDHWA, PFDWG, PFDOWG, and PFDHWG operators were proposed to handle MADM problem. Wang [43] developed Muirhead mean operators to evaluate the risk of financial investment with picture fuzzy information. He [44] adopt Hamy mean operators based on Dombi operations to settle project evaluation under q-Rung picture fuzzy environment. Luo [45] expanded interaction partitioned Heronian operators to select hotel with picture fuzzy information. Ates [46] modified Bonferroni mean operators to fuse picture fuzzy information.

Commonly, there exist interactions among the criteria in actual MCDM environment. However, many operators mentioned above cannot consider these interaction relationships, and HM operator can effectively capture the correlations of input values. Meanwhile, HM operator has the advantage of avoiding the computational redundancy comparing with BM operator. Thus, many variations of HM operator have been developed and applied to settle MCDM problem with fuzzy information, for example, eneralized HM (GHM), improved GHM (IGHM), improved generalized weighting HM (IGWHM), and improved generalized geometry weighting HM (IGGWHM) [47 –49]. To date, HM operators have been successfully used for intuitionistic uncertain linguistic set [50], two-dimensional uncertain linguistic set [51], linguistic hesitant FS [52], single-valued neutrosophic set [53], neutrosophic hesitant FS [54], neutrosophic uncertain linguistic set [55], linguistic IFS [56], neutrosophic cubic set [57], interval-valued IFS [58], hesitant fuzzy linguistic set [59], q-rung orthopair FS [60], PHS [41], and multiple-valued picture fuzzy linguistic set [61]. Nevertheless, up to now, there is no research extending the IGHM operator to cope with MCDM problem under multi-valued picture fuzzy environment, especially for multi-valued picture fuzzy uncertain linguistic information, which can better express the uncertainty complex cognitive information.

At present, many aggregating operators commonly depend on algebraic, particular situation for Hamacher. Hamacher operations are a generalized form of algebraic operations and einstein operations [62 –64]. Therefore, some achievements employing aggregated operators based on Hamacher operational relations to settle MCDM problem with various fuzzy assessment information are more flexible and significant [65 –72]. However, until to now, the aggregated operators based on Hamacher operations have not been performed to fuse MVPFE and multi-valued picture fuzzy uncertain linguistic element (MVPFULN). Motivated by these, we will extend the IGHM operator on account of Hamacher operations to the MVPFULS.

Based on the above analysis, the main contributions and innovations of the work are:

Since the ambiguous and uncertainty of people in actual life, the MVPFULS combing PHFS with uncertain linguistic term set (ULTS) is defined to accurately express complex cognitive information.

Since the general and flexible of Hamacher operations, new operational rules based on Hamacher operations for MVPFULNs are developed, and the relevant comparative functions are provided as well.

Since the existence of correlations between criteria and the redundancy of calculation, the IGHM operators based on Hamacher operations are expanded to accommodate MVPFULS environment, two novel aggregating operators are proposed and some characteristics of them are also investigated respectively.

To show advantages of provided MCDM approach, an illustrative example with MVPFULS information is conducted, sensitivity and comparison analysis are performed.

The organization of the paper is arranged as below. Notions regarding ULTS, MVPFS, Hamacher operations, and HM operators are reviewed in Section 2. Definition of MVPFULS and MVPFULN are proposed, and the Hamacher operations and the comparative method for MVPFULNs are also developed in Section 3. Two novel operators of MVPFULHIGWHM and MVPFULHIGGWHM are constructed, and their characteristics and particular situation are exhibited as well in Section 4. Section 5 establishes an MCDM approach in terms of two operators. Section 6 shows an example utilizing the developed approach, along with the comparative and sensitive analysis. Section 7 gives summarizations.

2 Preliminaries

Here, we introduce a few notions containing uncertain linguistic term set (ULTS), multi-valued picture fuzzy set (MVPFS), Hamacher operations and generalized heronian mean (HM) operators, which are preliminaries of the remaindering analysis.

2.1 ULTS

A discrete linguistic term set (LTS) S = {s₁, s₂, … s_τ, } can easily describe the qualitative information, in which τ is an odd number and s_j (j = 1, 2, …, τ) denotes a probable linguistic term. For instance, if τ = 7, the LTS S are stated as: $\begin{matrix} S = {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}} \\ = {\begin{matrix} very poor, poor, slightly poor, medium \\ slightly good, good, very good \end{matrix}} . \end{matrix}$

To diminish information loss in the procedure of computation, a continuous LTS $\bar{S} = {s_{α} | α \in R}$ is proposed by expanding the discrete LTS S.

Definition 1. [18] Let $\tilde{s} = [s_{a}, s_{b}]$ be an uncertain linguistic variable (ULV), satisfying the below conditions $a \leq b, s_{a} \leq s_{b}, s_{a}, s_{b} \in \bar{S}$ . For arbitrary two ULVs ${\tilde{s}}_{j} = | s_{a_{j}}, s_{b_{j}} |$ and ${\tilde{s}}_{k} = [s_{a_{k}}, s_{b_{k}}], λ \geq 0$ , the relating operations are given as below.

${\tilde{s}}_{j} \oplus {\tilde{s}}_{k} = [s_{a_{j} + a_{k}}, s_{b_{j} + b_{k}}]$ ;

${\tilde{s}}_{j} \otimes {\tilde{s}}_{k} = [s_{a_{j} \times a_{k}}, s_{b_{j} \times b_{k}}]$ ;

$λ {\tilde{s}}_{j} = [s_{λ a_{j}}, s_{λ b_{j}}]$ ;

$({\tilde{s}}_{j})^{λ} = [s_{a_{j}} λ, s_{b_{j}} λ]$ .

2.2 Picture hesitant fuzzy set

To better depict the hesitant MCDM information, Wang [15] provided the definition of picture hesitant fuzzy set (PHFS). It is stated as multi-valued PFS (MVPFS).

Definition 2. [15] Let X be a fixed objects space, B is a picture hesitant FS (PHFS) in X, which is denoted by $B = {〈 x, {\tilde{μ}}_{B} (x), {\tilde{η}}_{B} (x), {\tilde{ν}}_{B} (x) 〉 | x \in X},$

Here x is an element on $X, {\tilde{μ}}_{B} (x) = {μ | μ \in {\tilde{μ}}_{B} (x)}$ , ${\tilde{η}}_{B} (x) = {η | η \in {\tilde{η}}_{B} (x)}$ , ${\tilde{ν}}_{B} (x) = {ν | ν \in {\tilde{ν}}_{B} (x)}$ are triple sets containing values in [0,1], indicating the probable positive, neutral, and negative membership degrees, respectively. Satisfying: μ ∈ [0, 1], η ∈ [0, 1], ν ∈ [0, 1] and 0 ≤ μ⁺ + η⁺ + ν⁺ ≤ 1. Where $μ^{+} = \max {μ | μ \in {\tilde{μ}}_{B} (x)}$ , $η^{+} = \max {η | η \in {\tilde{η}}_{B} (x)}$ , and $ν^{+} = \max {ν | ν \in {\tilde{ν}}_{B} (x)}$ .

For simplicity, a picture hesitant fuzzy element (PHFE) is expressed by $b = {\tilde{μ}, \tilde{η}, \tilde{ν}}$ , if there is only one element in X.

2.3 Hamacher operations

Hamacher operations containing Hamacher sum and product proposed by Hamacher [73], which is a generalizing form, Hamacher TN and Hamacher TCN is presented as below. $\begin{matrix} TN (x, y) = x \otimes y = \frac{xy}{γ + (1 - γ) (x + y - xy)}, \\ TCN (x, y) = x \oplus y = \frac{x + y - xy - (1 - γ) xy}{1 - (1 - γ) xy} . \end{matrix}$ Particularly, Hamacher operations will conclude to the algebraic operations if γ = 1. $TN (x, y) = xy, TCN (x, y) = x + y - xy .$ Hamacher operations will conclude to the Einstein operations if γ = 2. $\begin{matrix} TN (x, y) = \frac{xy}{1 + (1 - x) (1 - y)}, \\ TCN (x, y) = \frac{x + y}{1 + xy} . \end{matrix}$

2.4 HM operaters

HM considering correlations of input arguments was proposed and developed, such as generalized weighted HM (GWHM), generalized geometric weighted HM (GGWHM). To overcome the disadvantages of GWHM and GGWHM operators, the improved GWHM and GGWHM operators are provided.

Definition 3. [49] Let b_i (i = 1, 2, …, n) be a space of positive numbers, ω = (ω₁, ω₂, … ω_n) be a corresponding weight vector of b_i (i = 1, 2, …, n), and ω_i ≥ 0, $\sum_{i = 1}^{n} ω_{i} = 1, p, q \geq 0$ . Then, an IGWHM operator is shown as below. $\begin{matrix} {IGWHM}^{p, q} (b_{1}, b_{2}, \dots, b_{n}) \\ = {(\frac{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j} b_{i}^{p} b_{j}^{q}}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}})}^{\frac{1}{p + q}} \end{matrix}$ (1)

Definition 4. [49] Let b_i (i = 1, 2, …, n) be a space of positive numbers, ω = (ω₁, ω₂, … ω_n) be a corresponding weight vector for b_i (i = 1, 2, …, n), and ω_i ≥ 0, $\sum_{i = 1}^{n} ω_{i} = 1, p, q \geq 0$ . Then, an IGGWHM operator is shown as: $\begin{matrix} {IGGWHM}^{p, q} (b_{1}, b_{2}, \dots, b_{n}) \\ = \frac{1}{p + q} \prod_{i = 1}^{n} \prod_{j = i}^{n} {({pb}_{i} + {qb}_{j})}^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}} \end{matrix}$ (2)

3 Multi-valued picture fuzzy uncertain linguistic set

Multi-valued picture fuzzy uncertain linguistic set (MVPFULS) and multi-valued picture fuzzy uncertain linguistic number (MVPFULN) are firstly provided according to the aforementioned achievements, some relating operations for MVPFULNs in view of Hamacher are explored. Additionally, comparative approach is investigated to sort MVPFULNs.

3.1 MVPFULS and Hamacher operations

Definition 5. Assume X be a fixed objects space, an MVPFULS B in X is denoted by $B = {〈 x, [s_{θ (x)}, s_{σ (x)}], ({\tilde{μ}}_{B} (x), {\tilde{η}}_{B} (x), {\tilde{ν}}_{B} (x)) 〉 | x \in X}$ where $s_{θ (x)}, s_{σ (x)} \in \bar{S}$ , [ $s_{θ (x)}, s_{σ (x)}] \in \tilde{S}$ , $\bar{S}$ is a continuous LTS, ${\tilde{μ}}_{B} (x) = \cup_{μ \in \tilde{μ} B (x)} {μ}$ , ${\tilde{η}}_{B} (x) = \cup_{η \in \tilde{η} B (x)} {η}$ , ${\tilde{ν}}_{B} (x) = ⋃_{ν \in \tilde{ν} B (x)} {ν}$ are triple sets containing a few values in [0,1], separately representing possible degrees of positive, neutral, and negative of element x in X to B for the ULV [s_θ(x), s_σ(x)], meeting the below limits, 0 ≤ μ, η, ν ≤ 1, 0 ≤ μ⁺ + η⁺ + ν⁺ ≤ 1, where $μ^{+} = \cup_{μ \in \tilde{μ} B (x)} \max {μ}$ , $η^{+} = \cup_{η \in \tilde{η} B (x)} \max {η}$ , $ν^{+} = \cup_{ν \in \tilde{ν} B (x)} \max {ν}$ , where ${\tilde{π}}_{B} (x) = \cup_{π \in \tilde{π} B (x)} {π}$ indicate the refusal membership degree, where π = 1 - μ - η - ν. When X has only one element x, then $〈 [S_{θ (x)}, S_{σ (x)}], (\tilde{μ} (x), \tilde{η} (x), \tilde{ν} (x)) 〉$ is called an MVPFULN.

Definition 6. Let $b_{1} = 〈 [S_{θ (b_{1})}, S_{σ (b_{1})}], (\tilde{μ} (b_{1}), \tilde{η} (b_{1}), \tilde{ν} (b_{1})) 〉$ and $b_{2} = 〈 [S_{θ (b_{2})}, S_{σ (b_{2})}], (\tilde{μ} (b_{2}), \tilde{η} (b_{2}), \tilde{ν} (b_{2})) 〉$ be arbitrary two MVPFULNs, π = 1 - μ - η - ν. Hamacher operational relations for MVPFULNs are presented.

(1) $\begin{matrix} b_{1} \oplus b_{2} = 〈 [s_{θ (b_{1}) + θ (b_{2})}, s_{σ (b_{1}) + σ (b_{2})}], \\ (⋃ μ_{1} \in \tilde{μ} (b_{1}) \\ μ_{2} \in \tilde{μ} (b_{2}) {\frac{μ_{1} + μ_{2} - μ_{1} μ_{2} - (1 - γ) μ_{1} μ_{2}}{1 - (1 - γ) μ_{1} μ_{2}}}, \\ ⋃ η_{1} \in \tilde{η} (b_{1}) \\ η_{2} \in \tilde{η} (b_{2}) {\frac{η_{1} η_{2}}{γ + (1 - γ) (η_{1} + η_{2} - η_{1} η_{2})}}, \\ ⋃ ν_{1} \in \tilde{ν} (b_{1}) \\ ν_{2} \in \tilde{ν} (b_{2}) {\frac{ν_{1} ν_{2}}{γ + (1 - γ) (ν_{1} + ν_{2} - ν_{1} ν_{2})}}) 〉; \end{matrix}$ $(2) \begin{matrix} b_{1} \otimes b_{2} = 〈 [s_{θ (b_{1}) \times θ (b_{2})}, s_{σ (b_{1}) \times σ (b_{2})}], \\ (⋃ μ_{1} \in \tilde{μ} (b_{1}) \\ μ_{2} \in \tilde{μ} (b_{2}) {\frac{μ_{1} μ_{2}}{γ + (1 - γ) (μ_{1} + μ_{2} - μ_{1} μ_{2})}}, \\ ⋃ η_{1} \in \tilde{η} (b_{1}) \\ η_{2} \in \tilde{η} (b_{2}) {\frac{η_{1} + η_{2} - η_{1} η_{2} - (1 - γ) η_{1} η_{2}}{1 - (1 - γ) η_{1} η_{2}}}, \\ ⋃ ν_{1} \in \tilde{ν} (b_{1}) \\ ν_{2} \in \tilde{ν} (b_{2}) {\frac{ν_{1} + ν_{2} - ν_{1} ν_{2} - (1 - γ) ν_{1} ν_{2}}{1 - (1 - γ) ν_{1} ν_{2}}}) 〉; \end{matrix}$ ; $(3) \begin{matrix} λ b_{1} = 〈 [s_{λ θ (b_{1})}, s_{λ θ (b_{1})}], \\ (⋃ μ_{1} \in \tilde{μ} (b_{1}) \\ {\frac{(1 + (γ - 1) μ_{1})^{λ} - (1 - μ_{1})^{λ}}{(1 + (γ - 1) μ_{1})^{λ} + (γ - 1) (1 - μ_{1})^{λ}}}, \\ ⋃ η_{1} \in \tilde{η} (b_{1}) \\ {\frac{γ η_{1}^{λ}}{(1 + (γ - 1) (1 - η_{1}))^{λ} + (γ - 1) η_{1}^{λ}}}, \\ ⋃ ν_{1} \in \tilde{ν} (b_{1}) \\ {\frac{γ ν_{1}^{λ}}{(1 + (γ - 1) (1 - ν_{1}))^{λ} + (γ - 1) ν_{1}^{λ}}},) 〉; \end{matrix}$ $(4) \begin{matrix} b_{1}^{λ} = 〈 [s_{θ^{λ} (b_{1})}, s_{σ^{λ} (b_{1})}], \\ (⋃ μ_{1} \in \tilde{μ} (b_{1}) \\ {\frac{γ η_{1}^{λ}}{(1 + (γ - 1) (1 - μ_{1}))^{λ} + (γ - 1) μ_{1}^{λ}}}, \\ ⋃ η_{1} \in \tilde{η} (b_{1}) \\ {\frac{(1 + (γ - 1) η_{1})^{λ} - (1 - η_{1})^{λ}}{(1 + (γ - 1) η_{1})^{λ} + (γ - 1) (1 - η_{1})^{λ}}}, \\ ⋃ ν_{1} \in \tilde{ν} (b_{1}) \\ {\frac{(1 + (γ - 1) ν_{1})^{λ} - (1 - ν_{1})^{λ}}{(1 + (γ - 1) ν_{1})^{λ} + (γ - 1) (1 - ν_{1})^{λ}}}) 〉 . \end{matrix}$

When γ = 1, the above Hamacher operations are reduced to the algebraic operations for MVPFULNs. When γ = 2, the above Hamacher operations are reduced to the Einstein operations for MVPFULNs.

We know the computed values in Definition 6 are still MVPFULNs, satisfy operational relations as below.

Theorem 1. Let $b_{1} = 〈 [s_{θ (b_{1})}, s_{σ (b_{1})}], (\tilde{μ} (b_{1}), \tilde{η} (b_{1}), \tilde{ν} (b_{1})) 〉$ and $b_{2} = 〈 [s_{θ (b_{2})}, s_{σ (b_{2})}], (\tilde{μ} (b_{2}), \tilde{η} (b_{2}), \tilde{ν} (b_{2})) 〉$ be arbitrary two MVPFULNs, and λ₁, λ₂ ≥ 0, then

b₁ ⊕ b₂ = b₂ ⊕ b₁;

b₁ ⊗ b₂ = b₂ ⊗ b₁;

λ₁ (b₁ ⊕ b₂) = λ₁b₁ ⊕ λ₁b₂;

λ₁b₁ ⊕ λ₂b₁ = (λ₁ + λ₂) b₁;

$b_{1}^{λ_{1}} \otimes b_{1}^{λ_{2}} = b_{1}^{λ_{1} + λ_{2}}$ ;

$b_{1}^{λ_{1}} \otimes b_{2}^{λ_{1}} = (b_{1} \otimes b_{2})^{λ_{1}}$ ;

In view of Hamacher operations of MVPFULNs in Definition 6, all equations in Theorem 1 are true.

3.2 Comparative method

To rank MVPFULNs, the comparing function and the corresponding comparative method are provided as below.

Definition 7. Let $b = 〈 [s_{θ (b)}, s_{σ (b)}], (\tilde{μ} (b), \tilde{η} (b), \tilde{ν} (b)) 〉$ be a MVPFULN, we can calculate the function value of score E (b) and the function value of accuracy H (b) on account of the following formulas.

(1) $\begin{matrix} E (b) = \frac{\frac{s_{(θ (b) + σ (b))}}{2}}{3} \\ (\frac{1}{ℓ_{1}} \sum_{μ \in \tilde{μ} (b)} μ + \frac{1}{ℓ_{2}} \sum_{η \in \tilde{η} (b)} (1 - η) + \frac{1}{ℓ_{3}} \sum_{ν \in \tilde{ν} (b)} (1 - ν)) = \\ \frac{S_{(θ (b) + σ (b)) \cdot (\frac{1}{ℓ_{1}} \sum_{μ \in \tilde{μ} (b)} μ + \frac{1}{ℓ_{2}} \sum_{η \in \tilde{η} (b)} (1 - η) + \frac{1}{ℓ_{3}} \sum_{ν \in \tilde{ν} (b)} (1 - ν))}}{6} \end{matrix}$

(2) $\begin{matrix} H (a) = \frac{\frac{s_{(θ (b) + σ (b))}}{2}}{3} (\frac{1}{ℓ_{1}} \sum_{μ \in \tilde{μ} (b)} μ + \frac{1}{ℓ_{2}} \sum_{η \in \tilde{η} (b)} η + \frac{1}{ℓ_{3}} \sum_{ν \in \tilde{ν} (b)} ν) \\ = \frac{S_{(θ (b) + σ (b)) \cdot (\frac{1}{ℓ_{1}} \sum_{μ \in \tilde{μ} (b)} μ + \frac{1}{ℓ_{2}} \sum_{η \in \tilde{η} (b)} η + \frac{1}{ℓ_{3}} \sum_{ν \in \tilde{ν} (b)} ν)}}{6} \end{matrix}$

Where ℓ₁, ℓ₂, and ℓ₃ denotes the numbers of element in $\tilde{μ} (b)$ , $\tilde{η} (b)$ , and $\tilde{ν} (b)$ , respectively.

According to Definition 7, comparing approach for MVPFULNs as below can be gained.

Theorem 2. Let $b_{1} = 〈 [s_{θ (b_{1})}, s_{σ (b_{1})}], (\tilde{μ} (b_{1}), \tilde{η} (b_{1}), \tilde{ν} (b_{1})) 〉$ and $b_{2} = 〈 [s_{θ (b_{2})}, s_{σ (b_{2})}], (\tilde{μ} (b_{2}), \tilde{η} (b_{2}), \tilde{ν} (b_{2})) 〉$ be two MVPFULNs, then,

if E (b₁) > E (b₂), then b₁ ≻ b₂;

if E (b₁) = E (b₂), then,

if H (b₁) > H (b₂), then b₁ ≻ b₂;

if H (b₁) = H (b₂), then b₁ ∼ b₂;

4 Multi-valued picture fuzzy uncertain linguistic Hamacher improved generalized heronian mean operator

In this part, the conventional IGWHM and IGGWHM operators on account of Hamacher operations will be extended to fuse multi-valued picture fuzzy uncertain linguistic (MVPFUL) information. Then, two operators of the multi-valued picture fuzzy uncertain linguistic Hamacher improved generalized weighted HM (MVPFULHIGWHM) and the multi-valued picture fuzzy uncertain linguistic Hamacher improved generalized geometric weighted HM (MVPFULHIGGWHM) are introduced as below.

4.1 MVPFULHIGWHM operator

Definition 8. Let $b_{i} = 〈 [s_{θ (b_{i})}, s_{σ (b_{i})}], (\tilde{μ} (b_{i}), \tilde{η} (b_{i}), \tilde{ν} (b_{i})) 〉$ (i = 1, 2 … , n) be a space of MVPFULNs, ω = (ω₁, ω₂, …, ω_n) be a corresponding weight vector of b_i (i = 1, 2 … , n) and ω_i ≥ 0, $\sum_{i = 1}^{n} ω_{i} = 1, p, q \geq 0$ . The multi-valued picture fuzzy uncertain linguistic Hamacher improved generalized weighted HM (MVPFULHIGWHM) operator is calculated. $\begin{matrix} {MVPFULHIGWHM}^{p, q} (b_{1}, b_{2}, \dots, b_{n}) \\ = {(\frac{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j} b_{i}^{p} b_{j}^{q}}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}})}^{\frac{1}{p + q}} \end{matrix}$ (3)

Theorem 3. Let $b_{i} = 〈 [s_{θ (b_{i})}, s_{σ (b_{i})}], (\tilde{μ} (b_{i}), \tilde{η} (b_{i}), \tilde{ν} (b_{i})) 〉$ (i = 1, 2 … , n) be a space of MVPFULNs, and ω = (ω₁, ω₂, …, ω_n) be a corresponding weight ed vector of b_i (i = 1, 2 … , n) and ω_i ≥ 0, $\sum_{i = 1}^{n} ω_{i} = 1, p, q \geq 0$ . Based on Hamacher operations in Definition 6, the aggregating value employing MVPFULHIGWHM operator in Definition 8 is still a n MVPFULN and presented as Equation ( 4).

Here $ψ = \frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}$ ; $\begin{matrix} M_{ij} = & (1 + (γ - 1) (1 - μ_{i}))^{p} \\ (1 + (γ - 1) (1 - μ_{j}))^{q} + (γ^{2} - 1) μ_{i}^{p} μ_{j}^{q}; \end{matrix}$ $\begin{matrix} N_{ij} = & (1 + (γ - 1) (1 - μ_{i}))^{p} \\ (1 + (γ - 1) (1 - μ_{j}))^{q} - μ_{i}^{p} μ_{j}^{q}; \end{matrix}$ $\begin{matrix} X_{ij} = & (1 + (γ - 1) η_{i})^{p} (1 + (γ - 1) η_{j})^{q} \\ - (1 - η_{i}))^{p} (1 - η_{j})^{q}; \end{matrix}$ $\begin{matrix} Y_{ij} = & (1 + (γ - 1) η_{i})^{p} (1 + (γ - 1) η_{j})^{q} \\ + (γ^{2} - 1) (1 - η_{i})^{p} (1 - η_{j})^{q}; \end{matrix}$ $\begin{matrix} P_{ij} = (1 + (γ - 1) ν_{i})^{p} (1 + (γ - 1) ν_{j})^{q} \\ - (1 - ν_{i})^{p} (1 - ν_{j})^{q}; \end{matrix}$ $\begin{matrix} Q_{ij} = & (1 + (γ - 1) ν_{i})^{p} (1 + (γ - 1) ν_{j})^{q} \\ + (γ^{2} - 1) (1 - ν_{i})^{p} (1 - ν_{j})^{q}; \end{matrix}$ $\begin{matrix} {MVPFULHIGWHM}^{p, q} (b_{1}, b_{2}, \dots, b_{n}) = \\ 〈 [S_{{(\frac{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j} θ p (b_{i}) θ q (b_{j})}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}})}^{\frac{1}{p + q}},} S_{{(\frac{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j} σ p (b_{i}) σ q (b_{j})}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}})}^{\frac{1}{p + q}}}], \end{matrix}$ $\begin{matrix} (⋃ μ_{i} \in \tilde{μ} (b_{i}) \\ μ_{i} \in \tilde{μ} (b_{j}) \\ {\frac{γ {({(\prod_{i = 1}^{n} \prod_{j = i}^{n} M_{ij}^{ω_{i} ω_{j}})}^{ψ} - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} N_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}}}{{({(\prod_{i = 1}^{n} \prod_{j = i}^{n} M_{ij}^{ω_{i} ω_{j}})}^{ψ} + (γ^{2} - 1) {(\prod_{i = 1}^{n} \prod_{j = i}^{n} N_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}} + (γ - 1) {({(\prod_{i = 1}^{n} \prod_{j = i}^{n} M_{ij}^{ω_{i} ω_{j}})}^{ψ} - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} N_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}}}}, \end{matrix}$ $\begin{matrix} (⋃ η_{i} \in \tilde{η} (b_{i}) \\ η_{i} \in \tilde{η} (b_{j}) \\ {\frac{{({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Y_{ij}^{ω_{i} ω_{j}})}^{ψ} + (γ^{2} - 1) {(\prod_{i = 1}^{n} \prod_{j = i}^{n} X_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}} - {({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Y_{ij}^{ω_{i} ω_{j}})}^{ψ} - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} X_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}}}{{({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Y_{ij}^{ω_{i} ω_{j}})}^{ψ} + (γ^{2} - 1) {(\prod_{i = 1}^{n} \prod_{j = i}^{n} X_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}} + (γ - 1) {({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Y_{ij}^{ω_{i} ω_{j}})}^{ψ} - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} X_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}}}}, \end{matrix}$ $\begin{matrix} ⋃ ν_{i} \in \tilde{ν} (b_{i}) \\ ν_{j} \in \tilde{ν} (b_{j}) \\ {\frac{{({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Q_{ij}^{ω_{i} ω_{j}})}^{ψ} + (γ^{2} - 1) {(\prod_{i = 1}^{n} \prod_{j = i}^{n} P_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}} - {({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Q_{ij}^{ω_{i} ω_{j}})}^{ψ} - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} P_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}}}{{({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Q_{ij}^{ω_{i} ω_{j}})}^{ψ} + (γ^{2} - 1) {(\prod_{i = 1}^{n} \prod_{j = i}^{n} P_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}} + (γ - 1) {({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Q_{ij}^{ω_{i} ω_{j}})}^{ψ} - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} P_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}}}}) 〉, \end{matrix}$ (4)

Proof.

In terms of Definition 6, we get $\begin{matrix} b_{i}^{p} = 〈 [s_{θ} p_{(b_{i})}, s_{σ} p_{(b_{i})}], (⋃ μ_{i} \in \tilde{μ} (b_{i}) {\frac{γ μ_{i}^{p}}{(1 + (γ - 1) (1 - μ_{i}))^{p} + (γ - 1) μ_{i}^{p}}}, \\ ⋃ η_{i} \in \tilde{η} (b_{i}) {\frac{(1 + (γ - 1) η_{i})^{p} - (1 - η_{i})^{p}}{(1 + (γ - 1) η_{i})^{p} + (γ - 1) (1 - η_{i})^{p}}}, ⋃ ν_{i} \in \tilde{ν} (b_{i}) {\frac{(1 + (γ - 1) ν_{i})^{p} - (1 - ν_{i})^{p}}{(1 + (γ - 1) ν_{i})^{p} + (γ - 1) (1 - ν_{i})^{p}}}) 〉, \end{matrix}$ $\begin{matrix} b_{i}^{q} = 〈 [s_{θ} q_{(b_{j})}, s_{σ} q_{(b_{j})}], (⋃ μ_{j} \in \tilde{μ} (b_{j}) {\frac{γ μ_{j}^{q}}{(1 + (γ - 1) (1 - μ_{j}))^{q} + (γ - 1) μ_{j}^{q}}}, \\ ⋃ η_{j} \in \tilde{η} (b_{j}) {\frac{(1 + (γ - 1) η_{j})^{q} - (1 - η_{j})^{q}}{(1 + (γ - 1) η_{j})^{q} + (γ - 1) (1 - η_{j})^{q}}}, \\ ⋃ ν_{j} \in \tilde{ν} (b_{j}) {\frac{(1 + (γ - 1) ν_{j})^{q} - (1 - ν_{j})^{q}}{(1 + (γ - 1) ν_{j})^{q} + (γ - 1) (1 - ν_{j})^{q}}}) 〉, \end{matrix}$ And, $\begin{matrix} b_{i}^{p} b_{j}^{q} = 〈 [s_{θ} p_{(b_{i})} \cdot θ^{q} (b_{j}), s_{σ} p_{(b_{i})} \cdot σ^{q} (b_{j})], \\ (⋃ μ_{i} \in \tilde{μ} (b_{i}) \\ μ_{j} \in \tilde{μ} (b_{j}) {\frac{γ μ_{i}^{p} μ_{j}^{q}}{(1 + (γ - 1) (1 - μ_{i}))^{p} (1 + (γ - 1) (1 - μ_{j}))^{q} + (γ - 1) μ_{i}^{p} μ_{j}^{q}}}, \\ ⋃ η_{i} \in \tilde{η} (b_{i}) \\ η_{j} \in \tilde{η} (b_{j}) {\frac{(1 + (γ - 1) η_{i})^{p} (1 + (γ - 1) η_{j})^{q} - (1 - η_{i})^{p} (1 - η_{j})^{q}}{(1 + (γ - 1) η_{i})^{p} (1 + (γ - 1) η_{j})^{q} + (γ - 1) (1 - η_{i})^{p} (1 - η_{j})^{q}}}, \\ ⋃ ν_{i} \in \tilde{ν} (b_{i}) \\ ν_{j} \in \tilde{ν} (b_{j}) {\frac{(1 + (γ - 1) ν_{i})^{p} (1 + (γ - 1) ν_{j})^{q} - (1 - ν_{i})^{p} (1 - ν_{j})^{q}}{(1 + (γ - 1) ν_{i})^{p} (1 + (γ - 1) ν_{j})^{q} + (γ - 1) (1 - ν_{i})^{p} (1 - ν_{j})^{q}}}) 〉; \end{matrix}$ Then, $\begin{matrix} ω_{i} ω_{j} b_{i}^{p} b_{j}^{q} = 〈 [s_{ω_{i} ω_{j} θ^{p} (b_{i}) \cdot θ^{q} (b_{j})}, s_{ω_{i} ω_{j} θ^{p} (b_{i}) \cdot σ^{q} (b_{j})}], (⋃ μ_{i} \in \tilde{μ} (b_{i}) \\ μ_{j} \in \tilde{μ} (b_{j}) {\frac{M_{ij}^{ω_{i} ω_{j}} - N_{ij}^{ω_{i} ω_{j}}}{M_{ij}^{ω_{i} ω_{j}} + (γ - 1) N_{ij}^{ω_{i} ω_{j}}}}, \\ (⋃ η_{i} \in \tilde{η} (b_{i}) \\ η_{j} \in \tilde{η} (b_{j}) {\frac{γ X_{ij}^{ω_{i} ω_{j}}}{Y_{ij}^{ω_{i} ω_{j}} + (γ - 1) X_{ij}^{ω_{i} ω_{j}}}}, (⋃ ν_{i} \in \tilde{ν} (b_{i}) \\ ν_{j} \in \tilde{ν} (b_{j}) {\frac{γ P_{ij}^{ω_{i} ω_{j}}}{Q_{ij}^{ω_{i} ω_{j}} + (γ - 1) P_{ij}^{ω_{i} ω_{j}}}}) 〉 \end{matrix}$ Therefore, $\begin{matrix} \sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j} b_{i}^{p} b_{j}^{q} = 〈 [S_{\sum_{i = 1}^{n}} \sum_{j = i}^{n} ω_{i} ω_{j} θ^{p} (b_{i}) \cdot θ^{q} (b_{j}), S_{\sum_{i = 1}^{n}} \sum_{j = i}^{n} ω_{i} ω_{j} σ^{p} (b_{i}) \cdot σ^{q} (b_{j})], \\ (⋃ μ_{i} \in \tilde{μ} (b_{i}) \\ μ_{j} \in \tilde{μ} (b_{j}) {\frac{\prod_{i = 1}^{n} \prod_{j = i}^{n} M_{ij}^{ω_{i} ω_{j}} - \prod_{i = 1}^{n} \prod_{j = i}^{n} N_{ij}^{ω_{i} ω_{j}}}{\prod_{i = 1}^{n} \prod_{j = i}^{n} M_{ij}^{ω_{i} ω_{j}} + (γ - 1) \prod_{i = 1}^{n} \prod_{j = i}^{n} N_{ij}^{ω_{i} ω_{j}}}}, \\ ⋃ η_{i} \in \tilde{η} (b_{i}) \\ η_{j} \in \tilde{η} (b_{j}) {\frac{γ \prod_{i = 1}^{n} \prod_{j = i}^{n} X_{ij}^{ω_{i} ω_{j}}}{\prod_{i = 1}^{n} \prod_{j = i}^{n} Y_{ij}^{ω_{i} ω_{j}} + (γ - 1) \prod_{i = 1}^{n} \prod_{j = i}^{n} X_{ij}^{ω_{i} ω_{j}}}}, \\ ⋃ \\ {nu}_{i} \in \tilde{ν} (b_{i}) \\ ν_{j} \in \tilde{ν} (b_{j}) {\frac{γ \prod_{i = 1}^{n} \prod_{j = i}^{n} P_{ij}^{ω_{i} ω_{j}}}{\prod_{i = 1}^{n} \prod_{j = i}^{n} Q_{ij}^{ω_{i} ω_{j}} + (γ - 1) \prod_{i = 1}^{n} \prod_{j = i}^{n} P_{ij}^{ω_{i} ω_{j}}}}) 〉 \end{matrix}$ Furthermore, $\begin{matrix} \frac{1}{\sum_{i = 1}^{n} \sum_{j = 1}^{n} ω_{i} ω_{j}} \sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j} b_{i}^{p} b_{j}^{q} = \\ 〈 [S_{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}} \sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j} θ^{p} (b_{i}) \cdot θ^{q} (b_{j}), S_{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}} \sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j} σ^{p} (b_{i}) \cdot σ^{q} (b_{j})], \\ (⋃ μ_{1} \in \tilde{μ} (b_{1}) μ_{j} \in \tilde{μ} (b_{j}) {\frac{{(\prod_{i = 1}^{n} \prod_{j = i}^{n} M_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}} - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} N_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}}}{{(\prod_{i = 1}^{n} \prod_{j = i}^{n} M_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}} + (γ - 1) {(\prod_{i = 1}^{n} \prod_{j = i}^{n} N_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}}}}, ⋃ η_{1} \in \tilde{η} (b_{i}) η_{j} \in \tilde{η} (b_{j}) {\frac{γ {(\prod_{i = 1}^{n} \prod_{j = i}^{n} X_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}}}{{(\prod_{i = 1}^{n} \prod_{j = i}^{n} Y_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}} + (γ - 1) {(\prod_{i = 1}^{n} \prod_{j = i}^{n} X_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}}}}, ⋃ ν_{1} \in \tilde{ν} (b_{i}) ν_{j} \in \tilde{ν} (b_{j}) {\frac{γ {(\prod_{i = 1}^{n} \prod_{j = i}^{n} P_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}}}{{(\prod_{i = 1}^{n} \prod_{j = i}^{n} Q_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}} + (γ - 1) {(\prod_{i = 1}^{n} \prod_{j = i}^{n} P_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}}}}) 〉 \end{matrix}$ Thus, $\begin{matrix} {(\frac{1}{\sum_{i = 1}^{n} \sum_{j = 1}^{n} ω_{i} ω_{j}} \sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j} b_{i}^{p} b_{j}^{q})}^{\frac{1}{p + q}} = 〈 [S_{{(\frac{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j} θ^{p} (b_{i}) \cdot θ^{q} (b_{j})}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}})}^{\frac{1}{p + q}}}, S_{{(\frac{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j} σ^{p} (b_{i}) \cdot σ^{q} (b_{j})}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}})}^{\frac{1}{p + q}}}], \\ (⋃ μ_{i} \in \tilde{μ} (b_{i}) μ_{j} \in \tilde{μ} (b_{j}) {\frac{γ {({(\prod_{i = 1}^{n} \prod_{j = i}^{n} M_{ij}^{ω_{i} ω_{j}})}^{ψ} - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} N_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}}}{{({(\prod_{i = 1}^{n} \prod_{j = i}^{n} M_{ij}^{ω_{i} ω_{j}})}^{ψ} + (γ^{2} - 1) {(\prod_{i = 1}^{n} \prod_{j = i}^{n} N_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}} + (γ - 1) {({(\prod_{i = 1}^{n} \prod_{j = i}^{n} M_{ij}^{ω_{i} ω_{j}})}^{ψ} - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} N_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}}}}, ⋃ η_{i} \in \tilde{η} (b_{i}) η_{j} \in \tilde{η} (b_{j}) {\frac{{({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Y_{ij}^{ω_{i} ω_{j}})}^{ψ} + (γ^{2} - 1) {(\prod_{i = 1}^{n} \prod_{j = i}^{n} X_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}} - {({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Y_{ij}^{ω_{i} ω_{j}})}^{ψ} - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} X_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}}}{{({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Y_{ij}^{ω_{i} ω_{j}})}^{ψ} + (γ^{2} - 1) {(\prod_{i = 1}^{n} \prod_{j = i}^{n} X_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}} + (γ - 1) {({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Y_{ij}^{ω_{i} ω_{j}})}^{ψ} - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} X_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}}}}, ⋃ ν_{i} \in \tilde{ν} (b_{i}) ν_{j} \in \tilde{ν} (b_{j}) {\frac{{({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Q_{ij}^{ω_{i} ω_{j}})}^{ψ} + (γ^{2} - 1) {(\prod_{i = 1}^{n} \prod_{j = i}^{n} P_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}} - {({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Q_{ij}^{ω_{i} ω_{j}})}^{ψ} - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} P_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}}}{{({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Q_{ij}^{ω_{i} ω_{j}})}^{ψ} + (γ^{2} - 1) {(\prod_{i = 1}^{n} \prod_{j = i}^{n} P_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}} + (γ - 1) {({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Q_{ij}^{ω_{i} ω_{j}})}^{ψ} - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} P_{ij}^{ω_{i} ω_{j}})}^{ψ})}^{\frac{1}{p + q}}}}) 〉 \end{matrix}$

Therefore, the Equation (4) in Theorem 3 is right. The proposed MVPFULHIGWHM operator satisfies the desirable characteristics, for instance idempotency, monotonicity, boundedness.

Theorem 4. (Idempotency) Let b_i (i = 1, 2, …, n) be a space of MVPFULNs, and $b_{i} = b = 〈 [s_{θ (b)}, s_{σ (b)}], (\tilde{μ} (b), \tilde{η} (b), \tilde{ν} (b)) 〉$ (i = 1, 2, …, n), where $\tilde{μ} (b) = μ, \tilde{η} (b) = η, \tilde{ν} (b) = ν$ , then MVPFULHIGWHM^p,q (b₁, b₂, …, b_n) = b.

Theorem 5. (Monotonicity) Let $b_{i} = 〈 [s_{θ (b_{i})}, s_{σ (b_{i})}], (\tilde{μ} (b_{i}), \tilde{η} (b_{i}), \tilde{ν} (b_{i})) 〉$ and $b_{i} ’ = 〈 [s_{θ (b_{i}^{'})}, s_{σ (b_{i} ’)}], (\tilde{μ} (b_{i} ’), \tilde{η} (b_{i} ’), \tilde{ν} (b_{i} ’)) 〉$ be two collections of MVPFULNs (i = 1, 2, …, n), if s_{θ(b_i)} ≤ s_{θ(b_i’)}, s_{σ(b_i)} ≤ s_{σ(b_i’)}, and $d \tilde{μ} (b_{i}) \leq \tilde{μ} (b_{i} ’), \tilde{η} (b_{i}) \geq \tilde{η} (b_{i} ’), \tilde{ν} (b_{i}) \geq \tilde{ν} (b_{i} ’)$ for any i. Then, MVPFULHIGWHM^p,q (b₁, b₂, …, b_n) ≤ MVPFULHIGWHM^p,q (b₁’, b₂’, …, b₃’).

Theorem 6. (Boundedness) Let $b_{i} = 〈 [s_{θ (b_{i})}, s_{σ (b_{i})}], (\tilde{μ} (b_{i}), \tilde{η} (b_{i}), \tilde{ν} (b_{i})) 〉$ (i = 1, 2, …, n) be a set of MVPFULNs. Then, $min_{i} {b_{i}} \leq {MVPFULHIGWHM}^{p, q} (b_{1}, b_{2}, \dots, b_{n})$ ≤ $max_{i} {b_{i}}$ . Next, a few special instances of MVPFULHIGWHM operator relating to different parameter γ are investigated.

(1) When γ = 1, then MVPFULHIGWHM operator will reduce to multi-valued picture fuzzy uncertain linguistic Algebraic IGWHM (MVPFULAIGWHM) operator as presented in Equation (5). $\begin{matrix} {MVPFULHIGWHM}^{p, q} (b_{1}, b_{2}, \dots, b_{n}) \\ = 〈 [S_{{(\frac{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j} θ^{p} (b_{i}) \cdot θ^{q} (b_{j})}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}})}^{\frac{1}{p + q}}}, S_{{(\frac{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j} σ^{p} (b_{i}) \cdot σ^{q} (b_{j})}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}})}^{\frac{1}{p + q}}}], \end{matrix}$ $\begin{matrix} (⋃ μ_{i} \in \tilde{μ} (b_{i}) μ_{j} \in \tilde{μ} (b_{j}) {{(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} {(1 - μ_{i}^{p} μ_{j}^{q})}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}})}^{\frac{1}{p + q}}}, ⋃ η_{i} \in \tilde{η} (b_{i}) η_{j} \in \tilde{η} (b_{j}) {1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} {(1 - (1 - η_{i})^{p} (1 - η_{j})^{q})}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}})}^{\frac{1}{p + q}}}, ⋃ ν_{i} \in \tilde{ν} (b_{i}) ν_{j} \in \tilde{ν} (b_{j}) {1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} {(1 - (1 - ν_{i})^{p} (1 - ν_{j})^{q})}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}})}^{\frac{1}{p + q}}}) 〉 \end{matrix}$ (5)

(2) When γ = 2, MVPFULHIGWHM will change to multi-valued picture fuzzy uncertain linguistic Einstein IGWHM (MVPFULNIGWHM) operator. $\begin{matrix} {MVPFULHIGWHM}^{p, q} (b_{1}, b_{2}, \dots, b_{n}) = 〈 [S_{{(\frac{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j} θ^{p} (b_{i}) θ^{q} (b_{j})}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}})}^{\frac{1}{p + q}}}, S_{{(\frac{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j} σ^{p} (b_{i}) σ^{q} (b_{j})}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}})}^{\frac{1}{p + q}}}], \end{matrix}$ $\begin{matrix} (⋃ μ_{i} \in \tilde{μ} (b_{i}) μ_{j} \in \tilde{μ} (b_{j}) {\frac{2 {({(\prod_{i = 1}^{n} \prod_{j = i}^{n} M_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}} - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} N_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}})}^{\frac{1}{p + q}}}{{({(\prod_{i = 1}^{n} \prod_{j = i}^{n} M_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}} + 3 {(\prod_{i = 1}^{n} \prod_{j = i}^{n} N_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}})}^{\frac{1}{p + q}} + {({(\prod_{i = 1}^{n} \prod_{j = i}^{n} M_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}} - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} N_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}})}^{\frac{1}{p + q}}}}, ⋃ η_{i} \in \tilde{η} (b_{i}) η_{j} \in \tilde{η} (b_{j}) {\frac{{({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Y_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}} + 3 {(\prod_{i = 1}^{n} \prod_{j = i}^{n} X_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}})}^{\frac{1}{p + q}} - {({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Y_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}} - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} X_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}})}^{\frac{1}{p + q}}}{{({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Y_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}} + 3 {(\prod_{i = 1}^{n} \prod_{j = i}^{n} X_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}})}^{\frac{1}{p + q}} + {({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Y_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}} - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} X_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}})}^{\frac{1}{p + q}}}}, ⋃ ν_{i} \in \tilde{ν} (b_{i}) ν_{j} \in \tilde{ν} (b_{j}) {\frac{{({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Q_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}} + 3 {(\prod_{i = 1}^{n} \prod_{j = i}^{n} P_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}})}^{\frac{1}{p + q}} - {({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Q_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}} - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} P_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}})}^{\frac{1}{p + q}}}{{({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Q_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}} + 3 {(\prod_{i = 1}^{n} \prod_{j = i}^{n} P_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}})}^{\frac{1}{p + q}} + {({(\prod_{i = 1}^{n} \prod_{j = i}^{n} Q_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}} - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} P_{ij}^{ω_{i} ω_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} ω_{i} ω_{j}}})}^{\frac{1}{p + q}}}}) 〉 \end{matrix}$ (6)

Here, $M_{ij} = (2 - μ_{i})^{p} (2 - μ_{j})^{q} + 3 μ_{i}^{p} μ_{j}^{q}$ ; $N_{ij} = (2 - μ_{i})^{p} (2 - μ_{j})^{q} - μ_{i}^{p} μ_{j}^{q}$ ; X_ij = (1 + η_i) ^p (1 + η_j) ^q - (1 - η_i) ^p (1 - η_j) ^q;

Y_ij = (1 + η_i) ^p (1 + η_j) ^q + 3 (1 - η_i) ^p (1 - η_j) ^q;

P_ij = (1 + ν_i) ^p (1 + ν_j) ^q - (1 - ν_i) ^p (1 - ν_j) ^q;

Q_ij = (1 + ν_i) ^p (1 + ν_j) ^q + 3 (1 - ν_i) ^p (1 - ν_j) ^q;

4.2 MVPFULHIGGWHM operator

Definition 9. Let $b_{i} = 〈 [s_{θ (b_{i})}, s_{σ (b_{i})}], (\tilde{μ} (b_{i}), \tilde{η} (b_{i}), \tilde{ν} (b_{i})) 〉$ (i = 1, 2, …, n) be a space of MVPFULNs, ω = (ω₁, ω₂, …, ω_n) be a corresponding weighting vector of b_i (i = 1, 2, …, n) and ω_i ≥ 0, $\sum_{i = 1}^{n} ω_{i} = 1, p, q \geq 0$ . Then, the multi-valued picture fuzzy uncertain linguistic Hamacher improved generalized geometric weighted HM (MVPFULHIGGWHM) is provided. $\begin{matrix} {MVPFULHIGGWHM}^{p, q} (b_{1}, b_{2}, \dots, b_{n}) \\ = \frac{1}{p + q} \prod_{i = 1}^{n} \prod_{j = i}^{n} ({pb}_{i} + {qb}_{i})^{\frac{2 (n + 1 - i)}{n (n + 1)} \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}} \end{matrix}$ (7)

Theorem 7. Let $b_{i} = 〈 [s_{θ (b_{i})}, s_{σ (b_{i})}], (\tilde{μ} (b_{i}), \tilde{η} (b_{i}), \tilde{ν} (b_{i})) 〉$ (i = 1, 2, …, n) be a space of MVPFULNs, ω = (ω₁, ω₂, …, ω_n) be a corresponding weight vector of b_i (i = 1, 2, …, n), ω_i ≥ 0, $\sum_{i = 1}^{n} ω_{i} = 1, p, q \geq 0$ . Based on Hamacher operations in Definition 6, the aggregating value employing MVPFULHIGGWHM operator in Definition 9 is still an MVPFULN and presented as follows.

Here, $ξ_{ij} = \frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}$ $\begin{matrix} N_{ij} = (1 + (γ - 1) μ_{i})^{p} (1 + (γ - 1) μ_{j})^{q} \\ + (γ^{2} - 1) (1 - μ_{i})^{p} (1 - μ_{i})^{q}; \end{matrix}$ $\begin{matrix} M_{ij} = (1 + (γ - 1) μ_{i})^{p} (1 + (γ - 1) μ_{j})^{q} \\ - (1 - μ_{i})^{p} (1 - μ_{i})^{q}; \end{matrix}$ $\begin{matrix} X_{ij} = (1 + (γ - 1) (1 - η_{i}))^{p} (1 + (γ - 1) \\ (1 - η_{j}))^{q} + (γ^{2} - 1) η_{i}^{p} η_{j}^{q}; \end{matrix}$ $\begin{matrix} Y_{ij} = (1 + (γ - 1) (1 - η_{i}))^{p} (1 + (γ - 1) \\ (1 - η_{j}))^{q} - η_{i}^{p} η_{j}^{q}; \end{matrix}$ $\begin{matrix} P_{ij} = (1 + (γ - 1) (1 - ν_{i}))^{p} (1 + (γ - 1) \\ (1 - ν_{j}))^{q} + (γ^{2} - 1) ν_{i}^{p} ν_{j}^{q}; \end{matrix}$ $\begin{matrix} Q_{ij} = (1 + (γ - 1) (1 - ν_{i}))^{p} (1 + (γ - 1) \\ (1 - ν_{j}))^{q} - ν_{i}^{p} ν_{j}^{q}; \end{matrix}$ $\begin{matrix} {MVPFULHIGWHM}^{p, q} (b_{1}, b_{2}, \dots, b_{n}) = 〈 [S_{\frac{1}{p + q} \prod_{i = 1}^{n} \prod_{j = i}^{n} {(p θ (b_{i}) + q θ (b_{j}))}^{ξ_{ij}}}, S_{\frac{1}{p + q} \prod_{i = 1}^{n} \prod_{j = i}^{n} {(p σ (b_{i}) + q σ (b_{j}))}^{ξ_{ij}}}], \\ (⋃ μ_{i} \in \tilde{μ} (b_{i}) μ_{j} \in \tilde{μ} (b_{j}) {\frac{{(\prod_{i = 1}^{n} \prod_{j = i}^{n} N_{ij}^{ξ_{ij}} + (γ^{2} - 1) \prod_{i = 1}^{n} \prod_{j = i}^{n} M_{ij}^{ξ_{ij}})}^{\frac{1}{p + q}} - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} N_{ij}^{ξ_{ij}} - \prod_{i = 1}^{n} \prod_{j = i}^{n} M_{ij}^{ξ_{ij}})}^{\frac{1}{p + q}}}{{(\prod_{i = 1}^{n} \prod_{j = i}^{n} N_{ij}^{ξ_{ij}} + (γ^{2} - 1) \prod_{i = 1}^{n} \prod_{j = i}^{n} M_{ij}^{ξ_{ij}})}^{\frac{1}{p + q}} + (γ - 1) {(\prod_{i = 1}^{n} \prod_{j = i}^{n} N_{ij}^{ξ_{ij}} - \prod_{i = 1}^{n} \prod_{j = i}^{n} M_{ij}^{ξ_{ij}})}^{\frac{1}{p + q}}}}, ⋃ η_{i} \in \tilde{η} (b_{i}) η_{j} \in \tilde{η} (b_{j}) {\frac{γ {(\prod_{i = 1}^{n} \prod_{j = i}^{n} X_{ij}^{ξ_{ij}} - \prod_{i = 1}^{n} \prod_{j = i}^{n} Y_{ij}^{ξ_{ij}})}^{\frac{1}{p + q}}}{{(\prod_{i = 1}^{n} \prod_{j = i}^{n} X_{ij}^{ξ_{ij}} + (γ^{2} - 1) \prod_{i = 1}^{n} \prod_{j = i}^{n} Y_{ij}^{ξ_{ij}})}^{\frac{1}{p + q}} + (γ - 1) {(\prod_{i = 1}^{n} \prod_{j = i}^{n} X_{ij}^{ξ_{ij}} - \prod_{i = 1}^{n} \prod_{j = i}^{n} Y_{ij}^{ξ_{ij}})}^{\frac{1}{p + q}}}}, ⋃ ν_{i} \in \tilde{ν} (b_{i}) ν_{j} \in \tilde{ν} (b_{j}) {\frac{γ {(\prod_{i = 1}^{n} \prod_{j = i}^{n} P_{ij}^{ξ_{ij}} - \prod_{i = 1}^{n} \prod_{j = i}^{n} Q_{ij}^{ξ_{ij}})}^{\frac{1}{p + q}}}{{(\prod_{i = 1}^{n} \prod_{j = i}^{n} P_{ij}^{ξ_{ij}} + (γ^{2} - 1) \prod_{i = 1}^{n} \prod_{j = i}^{n} Q_{ij}^{ξ_{ij}})}^{\frac{1}{p + q}} + (γ - 1) {(\prod_{i = 1}^{n} \prod_{j = i}^{n} P_{ij}^{ξ_{ij}} - \prod_{i = 1}^{n} \prod_{j = i}^{n} Q_{ij}^{ξ_{ij}})}^{\frac{1}{p + q}}}}) 〉 \end{matrix}$ (8)

The proof is analogue to that of Theorem 3.

Furthermore, MVPFULHIGGWHM operator has some characteristics of idempotency, monotonicity, and boundedness.

Theorem 8. (Idempotency) Let b_i (i = 1, 2, …, n) be a space of MVPFULNs, and $b_{i} = b = 〈 [s_{θ (b)}, s_{σ (b)}], (\tilde{μ} (b), \tilde{η} (b), \tilde{ν} (b)) 〉$ (i = 1, 2, …, n), where $\tilde{μ} (b) = μ, \tilde{η} (b) = η, \tilde{ν} (b) = ν$ , then MVPFULHIGGWHM^p,q (b₁, b₂, …, b_n) = b.

Theorem 9. (Monotonicity) Let $b_{i} = 〈 [s_{θ (b_{i})}, s_{σ (b_{i})}], (\tilde{μ} (b_{i}), \tilde{η} (b_{i}), \tilde{ν} (b_{i})) 〉$ and $b_{i} ’ = 〈 [s_{θ (b_{i} ’)}, s_{σ (b_{i} ’)}], (\tilde{μ} (b_{i} ’), \tilde{η} (b_{i} ’), \tilde{ν} (b_{i} ’)) 〉$ (i = 1, 2, …, n) be two collections of MVPFULNs, if s_{θ(b_i)} ≤ s_{θ(b_i’)}, s_{σ(b_i)} ≤ s_{σ(b_i’)}, and $\tilde{μ} (b_{i}) \leq \tilde{μ} (b_{i}), \tilde{η} (b_{i}) \geq \tilde{η} (b_{i}), \tilde{ν} (b_{i}) \geq \tilde{ν} (b_{i})$ for any i. Then, MVPFULHIGGWHM^p,q (b₁, b₂, …, b_n) ≤ MVPFULHIGGWHM^p,q (b₁, b₂, …, b_n).

Theorem 10. (Boundedness) Let $b_{i} = 〈 [s_{θ (b_{i})}, s_{σ (b_{i})}], (\tilde{μ} (b_{i}), \tilde{η} (b_{i}), \tilde{ν} (b_{i})) 〉$ (i = 1, 2, …, n) be a set of MVPFULNs. Then, $min_{i} {b_{i}} \leq$ ${MVPFULHIGGWHM}^{p, q} (b_{1}, b_{2}, \dots, b_{n}) \leq max_{i} {b_{i}}$ .

The proof of Theorems 8–10 is analogue to Theorems 4–6. Thus, the process is deleted.

Next, a few special instances of MVPFULHIGGWHM operator relating to different parameter γ are investigated.

(1) When γ = 1, then MVPFULHIGGWHM operator will reduce to multi-valued picture fuzzy uncertain linguistic Algebraic improved generalized geometric weighted HM (MVPFULAIGGWHM) as presented in Equation (9). $\begin{matrix} {MVPFULHIGGWHM}^{p, q} (b_{1}, b_{2}, \dots, b_{n}) = \\ 〈 [S_{\frac{1}{p + q} \prod_{i = 1}^{n} \prod_{j = i}^{n} {(p θ (b_{i}) + q θ (b_{j}))}^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}}}, S_{\frac{1}{p + q} \prod_{i = 1}^{n} \prod_{j = i}^{n} {(p σ (b_{i}) + q σ (b_{j}))}^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}}}], \\ (⋃ μ_{i} \in \tilde{μ} (b_{i}) μ_{j} \in \tilde{μ} (b_{j}) {1 - {(1 - \prod_{i = 1}^{n} \prod_{j = i}^{n} (1 - (1 - μ_{i})^{p} (1 - μ_{j})^{q})^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}})}^{\frac{1}{p + q}}}, ⋃ η_{i} \in \tilde{η} (b_{i}) η_{j} \in \tilde{η} (b_{j}) {{(1 - \prod_{i = 1}^{n} \prod_{j = i}^{n} (1 - η_{i}^{p} η_{j}^{q})^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}})}^{\frac{1}{p + q}}}, ⋃ ν_{i} \in \tilde{ν} (b_{i}) ν_{j} \in \tilde{ν} (b_{j}) {{(1 - \prod_{i = 1}^{n} \prod_{j = i}^{n} (1 - ν_{i}^{p} ν_{j}^{q})^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}})}^{\frac{1}{p + q}}}) 〉 \end{matrix}$ (9)

(2) When γ = 2, MVPFULHIGGWHM operator will change to multi-valued picture fuzzy uncertain linguistic Einstein improved generalized geometric weighted HM (MVPFULAIGGWHM) as presented in Equation (10).

Here, N_ij = (1 + μ_i) ^p (1 + μ_j) ^q + 3 (1 - μ_i) ^p (1 - μ_j) ^q;

M_ij = (1 + μ_i) ^p (1 + μ_j) ^q - (1 - μ_i) ^p (1 - μ_j) ^q;

$X_{ij} = (2 - η_{i})^{p} (2 - η_{j})^{q} + 3 η_{i}^{p} η_{j}^{q}$ ;

$Y_{ij} = (2 - η_{i})^{p} (2 - η_{j})^{q} - η_{i}^{p} η_{j}^{q}$ ;

$P_{ij} = (2 - ν_{i})^{p} (2 - ν_{j})^{q} + 3 ν_{i}^{p} ν_{j}^{q}$ ;

$Q_{ij} = (2 - ν_{i})^{p} (2 - ν_{j})^{q} - ν_{i}^{p} ν_{j}^{q}$ ;

$\begin{matrix} {MVPFULHIGGWHM}^{p, q} (b_{1}, b_{2}, \dots, b_{n}) = \\ 〈 [S_{\frac{1}{p + q} \prod_{i = 1}^{n} \prod_{j = i}^{n} {(p θ (b_{i}) + q θ (b_{j}))}^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}}}, S_{\frac{1}{p + q} \prod_{i = 1}^{n} \prod_{j = i}^{n} {(p σ (b_{i}) + q σ (b_{j}))}^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}}}], \end{matrix}$ $(\begin{array}{l} \cup \begin{array}{l} μ_{i} \in \tilde{μ} (b_{i}) \\ μ_{j} \in \tilde{μ} (b_{j}) \end{array} {\frac{{(\prod_{i = 1}^{n} \prod_{j = i}^{n} N_{i j} {^{\frac{2 (n + 1 - i)}{n (n + 1)}}}^{\cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}} + 3 \prod_{i = 1}^{n} \prod_{j = i}^{n} M_{i j} {^{\frac{2 (n + 1 - i)}{n (n + 1)}}}^{\cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}})}^{\frac{1}{p + q}} - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} N_{i j}^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}} - \prod_{i = 1}^{n} \prod_{j = i}^{n} M_{i j}^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}})}^{\frac{1}{p + q}}}{{(\prod_{i = 1}^{n} \prod_{j = i}^{n} N_{i j}^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}} + 3 \prod_{i = 1}^{n} \prod_{j = i}^{n} M_{i j}^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}})}^{\frac{1}{p + q}} + {(\prod_{i = 1}^{n} \prod_{j = i}^{n} N_{i j} {^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot}}^{\frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}} - \prod_{i = 1}^{n} \prod_{j = i}^{n} M_{i j}^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}})}^{\frac{1}{p + q}}}}, \\ \cup \begin{array}{l} η_{i} \in \tilde{η} (b_{i}) \\ η_{j} \in \tilde{η} (b_{j}) \end{array} {\frac{2 {(\prod_{i = 1}^{n} \prod_{j = i}^{n} X_{i j}^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}} - \prod_{i = 1}^{n} \prod_{j = i}^{n} Y_{i j}^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}})}^{\frac{1}{p + q}}}{{(\prod_{i = 1}^{n} \prod_{j = i}^{n} X_{i j}^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}} + 3 \prod_{i = 1}^{n} \prod_{j = i}^{n} Y_{i j}^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}})}^{\frac{1}{p + q}} + {(\prod_{i = 1}^{n} \prod_{j = i}^{n} X_{i j} {^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot}}^{\frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}} - \prod_{i = 1}^{n} \prod_{j = i}^{n} Y_{i j}^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}})}^{\frac{1}{p + q}}}}, \\ \cup \begin{array}{l} ν_{i} \in \tilde{ν} (b_{i}) \\ ν_{j} \in \tilde{ν} (b_{j}) \end{array} {\frac{(2 \prod_{i = 1}^{n} \prod_{j = i}^{n} P_{i j}^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}} - \prod_{i = 1}^{n} \prod_{j = i}^{n} Q_{i j}^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}} \frac{1}{p + q})}{{(\prod_{i = 1}^{n} \prod_{j = i}^{n} P_{i j}^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}} + 3 \prod_{i = 1}^{n} \prod_{j = i}^{n} Q_{i j}^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}})}^{\frac{1}{p + q}} + {(\prod_{i = 1}^{n} \prod_{j = i}^{n} P_{i j}^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}} - \prod_{i = 1}^{n} \prod_{j = i}^{n} Q_{i j}^{\frac{2 (n + 1 - i)}{n (n + 1)} \cdot \frac{ω_{j}}{\sum_{k = i}^{n} ω_{k}}})}^{\frac{1}{p + q}}}} \end{array}) 〉$ (10)

5 MCDM approach with MVPFUL information

We provide a novel MCDM approach using the aforementioned developed operators with MVPFUL information in this section.

We can describe the MCDM problem as below. Let A = {A₁, A₂, …, A_m} be m alternatives, and C = {C₁, C₂, …, C_n} be n criteria, ω = (ω₁, ω₂, …, ω_n) is the corresponding weighted vector of n criteria, here ω_j ∈ [0, 1], and $\sum_{j = 1}^{n} ω_{j} = 1$ . Let R = [a_ij] _m×n be a multi-valued picture fuzzy uncertain linguistic decision matrix, $a_{ij} = 〈 [s_{θ (a_{ij})}, s_{σ (a_{ij})}], ({\tilde{μ}}_{a_{ij}}, {\tilde{η}}_{a_{ij}}, {\tilde{ν}}_{a_{ij}}) 〉$ takes the form of MVPFULN given by experts. Where ${\tilde{μ}}_{a_{ij}}$ indicates the probable degrees of positive membership for alternative A_i (i = 1, 2, …, m) regarding criteria C_j (j = 1, 2, …, n) relating to the uncertain LT value [s_{θ(a_ij)}, s_{σ(a_ij)}], ${\tilde{η}}_{a_{ij}}$ indicates the probable degrees of neutral membership for alternative A_i (i = 1, 2, …, m) regarding criteria C_j (j = 1, 2, …, n) relating to the uncertain linguistic term value [s_{θ(a_ij)}, s_{σ(a_ij)}], ${\tilde{ν}}_{a_{ij}}$ indicates the probable degrees of negative membership for alternative A_i (i = 1, 2, …, m) regarding criteria C = {C₁, C₂, …, C_n} relating to the uncertain linguistic term value [s_{θ(a_ij)}, s_{σ(a_ij)}].

Next, we can make decision according to the scheme of proposed algorithm shown in Fig. 1. The detail procedure for selecting best alternative is presented as follows.

Fig. 1

The scheme of proposed algorithm.

Step 1. Transform MVPFUL matrix.

In general, there exist two evaluating criteria in MCDM problem, that is, the benefit criteria and cost criteria. The higher the benefit criterion value is, the result is better, and the lower the cost criterion is, the effect is better. Thus, to keep the unity of all criteria, we can get benefit criteria by converting cost criteria. According to the following formulas, the original MVPFUL matrix R = [a_ij] _m×n should be converted into normalized matrix B = [b_ij] _m×n.

${\begin{array}{l} 〈 [s_{θ (a_{i j})}, s_{σ (a_{i j})}], ({\tilde{μ}}_{a_{i j}}, {\tilde{η}}_{a_{i j}}, {\tilde{ν}}_{a_{i j}}) 〉 & b e n e f i t c r i t e r i a \\ 〈 [s_{τ - σ (a_{i j})}, s_{τ - θ (a_{i j})}], ({\tilde{ν}}_{a_{i j}}, {\tilde{η}}_{a_{i j}}, {\tilde{μ}}_{a_{i j}}) 〉 & \cos t c r i t e r i a \end{array}$

If they are all benefit criteria, there is no need to transform the original matrix, namely B = R = [a_ij] _m×n.

Step 2. Compute information fusion value regarding each alternative.

We will obtain integrated assessment value b_i (i = 1, 2, …, m) for all alternatives A_i (i = 1, 2, …, m) utilizing the MVPFULHIGWHM operator in Definition 8 or MVPFULHIGGWHM operator in Definition 9. $\begin{matrix} b_{i} = {MVPFULHIGWHM}^{p, q} (b_{i 1}, b_{i 2}, \dots, b_{in}) \\ (i = 1, 2, \dots, m) \end{matrix}$ $\begin{matrix} b_{i} = {MVPFULHIGGWHM}^{p, q} (b_{i 1}, b_{i 2}, \dots, b_{in}) \\ (i = 1, 2, \dots, m) \end{matrix}$

Step 3. Compute the comparative function result.

According to formulas of Definition 7, we can compute the score function E (b_i) of b_i (i = 1, 2, …, m). If E (b_i) = E (b_j) for MVPFULNs b_i and b_j, then the accuracy function H (b_i) and H (b_j) need to be computed.

Step 4. Sort alternatives and derive the optimal one.

According to Theorem 2, we can rank all alternatives in terms of the comparative function values, and choose the optimal alternative.

6 An example and result analysis

To check the availability and superiority of the developed MCDM method utilizing the proposed operators under MVPFUL environment, a practical example [15] along with sensitivity and comparative analysis is conducted in this section.

6.1 Implementation

Suppose an enterprise want to purchase an appropriate enterprise resource planning (ERP) system from different vendors. Here A_i (i = 1, 2, 3, 4, 5) indicates five potential vendors, C_j (j = 1, 2, 3, 4) indicates four evaluation criteria, where C₁ is technology, C₂ is strategy implementation, C₃ is capability, C₄ is prestige. And ω = (0.2, 0.1, 0.3, 0.4) is the weighting vector of four criteria. Considering the hesitancy of experts and the description of linguistic information, the assessment value of each alternative A_i (i = 1, 2, 3, 4, 5) regarding criteria C_j (j = 1, 2, 3, 4) takes the expression format of MVPFULN. The corresponding original MVPFULN decision matrix R = [a_ij] _5×4 can be shown at the top of the next page.

Where LTS $\begin{matrix} S = {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7},} \\ = {\begin{matrix} pretty poor, slightly poor, poor, \\ medium, good, slightly good, pretty good \end{matrix}} . \end{matrix}$

Next, we will apply the developed MVPFULHIGWHM operator and MVPFULHIGGWHM operator to obtain the optimal ERP vendor. For simplicity, we assume γ = p = q = 1.

Step 1. Transform MVPFUL matrix.

Because they are all benefit types, it is unnecessary to transform R = [a_ij] _5×4.

Step 2. Compute information fusion value regarding each alternative.

We can derive the comprehensive value a_i for alternative A_i (i = 1, 2, 3, 4, 5) employing the MVPFULHIGWHM operator in Definition 8 as follows. $\begin{matrix} a_{1} = 〈 [s_{2.7956}, s_{3.7927}], \\ (\begin{matrix} {0.3324, 0.3556, 0.3546, 0.3773}, \\ {0.4427, 0.4839, 0.4546, 0.4920}, \\ {0.0481, 0.0566, 0.0524, 0.0615} \end{matrix}) 〉 \end{matrix}$ $\begin{matrix} a_{2} = 〈 \\ \begin{matrix} [s_{3.4818}, s_{4.4773}], \\ (\begin{matrix} {0.4046, 0.4960, 0.4316, 0.5203, 0.4527, 0.5394}, \\ {0.2827, 0.2892, 0.2885, 0.2950, 0.2897, 0.2963, 0.2955, 0.3022}, \\ {0.0889, 0.0968, 0.0943, 0.1025} \end{matrix}) \end{matrix} 〉 \end{matrix}$ $a_{3} = 〈 \begin{matrix} [s_{4.1528}, s_{5.1486}], \\ (\begin{matrix} {0.3833, 0.5042, 0.4349, 0.5482, 0.4620, 0.5705}, \\ {\begin{matrix} 0.2076, 0.2449, 0.2159, 0.2545, 0.2296, 0.2703, 0.2149, 0.2530, \\ 0.2235, 0.2629, 0.2377, 0.2792 \end{matrix}}, \\ {0.0471, 0.0595} \end{matrix}) \end{matrix} 〉$ $a_{4} = 〈 \begin{matrix} [s_{4.1787}, s_{5.1754}], \\ (\begin{matrix} {0.3088, 0.3215, 0.3468, 0.3593}, \\ {0.4621, 0.5166, 0.4819, 0.5369, 0.4738, 0.5298, 0.4941, 0.5507}, \\ {0.0578} \end{matrix}) \end{matrix} 〉$ $a_{5} = 〈 \begin{matrix} [s_{3.3489}, s_{4.3448}], \\ (\begin{matrix} {0.3069, 0.3497, 0.3991}, {0.4268, 0.4544, 0.4435, 0.4714}, \\ {0.0665, 0.0704} \end{matrix}) \end{matrix} 〉$ $\begin{matrix} R = [a_{ij}]_{5 \times 4} = \\ [\begin{matrix} 〈 [s_{3}, s_{4}], (\begin{matrix} {0.43, 0.53}, \\ {0.33}, \\ {0.06, 0.09} \end{matrix}) 〉 & 〈 [s_{4}, s_{5}], (\begin{matrix} {0.76, 0.89}, \\ {0.05, 0.08}, \\ {0.03} \end{matrix}) 〉 & 〈 [s_{2}, s_{3}], (\begin{matrix} {0.42}, \\ {0.35}, \\ {0.12, 0.18} \end{matrix}) 〉 & 〈 [s_{3}, s_{4}], (\begin{matrix} {0.08}, \\ {0.75, 0.89}, \\ {0.02} \end{matrix}) 〉 \\ 〈 [s_{2}, s_{3}], (\begin{matrix} {0.53, 0.65, 0.73}, \\ {0.10, 0.12}, \\ {0.08} \end{matrix}) 〉 & 〈 [s_{2}, s_{3}], (\begin{matrix} {0.13}, \\ {0.53, 0.64}, \\ {0.12, 0.21} \end{matrix}) 〉 & 〈 [s_{4}, s_{5}], (\begin{matrix} {0.33}, \\ {0.77, 0.82}, \\ {0.10, 0.13} \end{matrix}) 〉 & 〈 [s_{4}, s_{5}], (\begin{matrix} {0.58, 0.73}, \\ {0.15}, \\ {0.08} \end{matrix}) 〉 \\ 〈 [s_{4}, s_{5}], (\begin{matrix} {0.72, 0.86, 0.91}, \\ {0.33}, \\ {0.02} \end{matrix}) 〉 & 〈 [s_{4}, s_{5}], (\begin{matrix} {0.07}, \\ {0.05, 0.09}, \\ {0.05} \end{matrix}) 〉 & 〈 [s_{3}, s_{4}], (\begin{matrix} {0.04}, \\ {0.65, 0.72, 0.85}, \\ {0.05, 0.10} \end{matrix}) 〉 & 〈 [s_{5}, s_{6}], (\begin{matrix} {0.45, 0.68}, \\ {0.18, 0.26}, \\ {0.06} \end{matrix}) 〉 \\ 〈 [s_{3}, s_{4}], (\begin{matrix} {0.77, 0.85}, \\ {0.09}, \\ {0.05} \end{matrix}) 〉 & 〈 [s_{3}, s_{4}], (\begin{matrix} {0.65, 0.74}, \\ {0.10, 0.16}, \\ {0.1} \end{matrix}) 〉 & 〈 [s_{4}, s_{5}], (\begin{matrix} {0.02}, \\ {0.78, 0.89}, \\ {0.05} \end{matrix}) 〉 & 〈 [s_{5}, s_{6}], (\begin{matrix} {0.08}, \\ {0.65, 0.84}, \\ {0.06} \end{matrix}) 〉 \\ 〈 [s_{2}, s_{3}], (\begin{matrix} {0.70, 0.81, 0.90}, \\ {0.05}, \\ {0.02} \end{matrix}) 〉 & 〈 [s_{4}, s_{5}], (\begin{matrix} {0.68}, \\ {0.08}, \\ {0.13, 0.21} \end{matrix}) 〉 & 〈 [s_{3}, s_{4}], (\begin{matrix} {0.05}, \\ {0.77, 0.87}, \\ {0.06} \end{matrix}) 〉 & 〈 [s_{4}, s_{5}], (\begin{matrix} {0.13}, \\ {0.65, 0.75}, \\ {0.09} \end{matrix}) 〉 \end{matrix}] \end{matrix}$

We can derive the comprehensive value a_i for alternative A_i (i = 1, 2, 3, 4, 5) employing the MVPFULHIGGWHM operator in Definition 9 as follows. $a_{1} = 〈 [s_{2.8899}, s_{3.9005}], (\begin{matrix} {0.3406, 0.3565, 0.3634, 0.3804}, \\ {0.4320, 0.5092, 0.4370, 0.5135}, \\ 0.0592, 0.0765, 0.0665, 0.0836 \end{matrix}) 〉$ $\begin{matrix} [s_{3.0681}, s_{4.0875}], \\ a_{2} = 〈 (\begin{matrix} {0.2576, 0.2827, 0.2730, 0.2985, 0.2821, 0.3078}, \\ {0.4181, 0.4422, 0.4445, 0.4684, 0.4214, 0.4453, 0.4475, 0.4713}, \\ {0.0929, 0.1009, 0.1102, 0.1184} \end{matrix}) 〉 \end{matrix}$ $\begin{matrix} [s_{4.0125,} s_{5.0208}], \\ a_{3} = 〈 (\begin{matrix} {0.2414, 0.2831, 0.2546, 0.2968, 0.2589, 0.3013}, \\ {\begin{matrix} 0.2655, 0.2850, 0.2949, 0.3136, 0.3645, 0.3810, 0.2731, 0.2928, \\ 0.3022, 0.3210, 0.3709, 0.3876 \end{matrix}}, \\ {0.0461, 0.0593} \end{matrix}) 〉 \end{matrix}$ $\begin{matrix} [s_{3.8409}, s_{4.8497}], \\ a_{4} = 〈 (\begin{matrix} {0.2255, 0.2346, 0.2333, 0.2427}, \\ {0.4745, 0.5557, 0.5301, 0.6082, 0.4850, 0.5650, 0.5394, 0.6164}, \\ {0.0627} \end{matrix}) 〉 \end{matrix}$ $a_{5} = 〈 \begin{matrix} [s_{3.1962}, s_{4.2128}], \\ (\begin{matrix} {0.2785, 0.2929, 0.3036}, {0.4594, 0.4988, 0.5082, 0.5461}, \\ {0.0747, 0.0912} \end{matrix}) \end{matrix} 〉$

Step 3. Compute the comparative function value.

We can obtain the score function value E (a_i) based on the formula in Definition 7 as presented in Table 1.

Table 1
The score values using developed operators

Operator E (a₁) E (a₂) E (a₃) E (a₄) E (a₅)

MVPFULH 2.0104 2.7673 3.3940 2.7603 2.3523

IGWHM

MVPFULH 2.0550 2.0670 2.8590 2.3530 2.1061

IGGWHM

Operator	E (a₁)	E (a₂)	E (a₃)	E (a₄)	E (a₅)
MVPFULH	2.0104	2.7673	3.3940	2.7603	2.3523
IGWHM
MVPFULH	2.0550	2.0670	2.8590	2.3530	2.1061
IGGWHM

Step 4. Sort alternatives and derive the optimal one.

According to score value above and the comparison approach in Theorem 2, the ranking result of ERP vendors are gained as presented in Table 2.

Table 2

Ranking order using developed operators

Operator	Ranking order
MVPFULHIGWHM	A₃≻ A₂ ≻ A₄≻ A₅ ≻ A₁
MVPFULHIGGWHM	A₃≻ A₄ ≻ A₅≻ A₂ ≻ A₁

In Table 2, the orders utilizing MVPFULHIGWHM operator and MVPFULHIGGWHM operator are slightly different, the optimal vendor is always A₃, and A₁ is always the worst ERP vendor.

6.2 Sensitivity analysis

Considering the impact of diverse parameters γ, p, q and aggregated operators, we conduct sensitive analysis to investigate the variation trends of ranking results.

The variation trends of score values with different parameters γ, p, q using MVPFULHIGWHM operator are presented in Figs. 2 –6, and the variation trends of score values with different parameters γ, p, q using MVPFULHIGGWHM operator are presented in Figs. 7 –11.

Fig. 2

The score values using MVPFULHIGWHM operator when p = q = 1, γ∈ [1,10, 1,10].

Fig. 3

The score values using MVPFULHIGWHM operator when γ = 1, p = 1, q∈ [0,10].

Fig. 4

The score values using MVPFULHIGWHM operator when γ = 1, q = 1, p∈ [0,10].

Fig. 5

The score values using MVPFULHIGWHM operator when γ = 2, p = 1, q∈ [0,10].

Fig. 6

The score values using MVPFULHIGWHM operator when γ = 2, q = 1, p∈ [0,10].

Fig. 7

The score values using MVPFULHIGGWHM operator when p = q = 1, γ∈ [1,10].

Fig. 8

The score values using MVPFULHIGGWHM operator when γ = 1, p = 1, q∈ [0,10].

Fig. 9

The score values using MVPFULHIGGWHM operator when γ = 1, q = 1, p∈ [0,10].

Fig. 10

The score values using MVPFULHIGGWHM operator when γ = 2, p = 1, q∈ [0,10].

Fig. 11

The score values using MVPFULHIGGWHM operator when γ = 2, q = 1, p∈ [0,10].

From Figs. 2 –11, it can be clearly observed that different parameters γ, p, q and aggregating operators may cause different score function values and ranking orders. The following analysis results can be derived.

For the MVPFULHIGWHM operator in Figs. 2 –6, A₃ is always the best vendor, and A₁ is always the worst one. For the MVPFULHIGGWHM operator in Figs. 7 –11, A₃ is the best selection, and A₁ or A₂ is the worst selection.

In Figs. 2 and 7, p, q are assigned a specific value, namely p = q = 1, and γ is given the value belonging to [1,10, 1,10]. The score results of each alternative using MVPFULHIGWHM operator are larger when the values of parameter γ are bigger, and the order is always A₃ ≻ A₄ ≻ A₂ ≻ A₅ ≻ A₁, while the score values of each alternative using MVPFULHIGGWHM operator are smaller when the values of parameter γ are bigger, and the ranking is always A₃ ≻ A₄ ≻ A₅ ≻ A₂ ≻ A₁. A₃ is always the best vendor, and the worst vendor is always A₁ utilizing MVPFULHIGWHM operator and MVPFULHIGGWHM operator.

In Figs. 3 –6, when γ = 1 or γ = 2, we can observe the variation trends of score values with different values of parameter p or parameter q utilizing MVPFULHIGWHM operator. In Figs. 3 and 4, when the parameter γ is equal to 1, the MVPFULHIGWHM operator will reduce to MVPFULAIGWHM operator. In Figs. 5 and 6, when the parameter γ is equal to 2, the MVPFULHIGWHM operator will reduce to MVPFULNIGWHM operator. The sensitive analysis of parameter q in Fig. 5 is similar to that in Fig. 3 when p = 1, q ∈ [0, 10], and the sensitive analysis of parameter p in Fig. 6 is similar to that in Fig. 4 when q = 1, p ∈ [0, 10]. In Figs. 3 –6, the sorting results of all alternatives may change with the increase of parameter p or parameter q, nevertheless, A₃ is always the best one, and A₁ is always the worst selection.

In Figs. 8 –11, when γ = 1 or γ = 2, we can observe the variation trends of score values with different values of parameters p or q utilizing MVPFULHIGGWHM operator. In Figs. 8 and 9, when the parameter γ is equal to 1, the MVPFULHIGGWHM operator is reduced to the MVPFULAIGGWHM operator. In Figs. 10 and 11, when the parameter γ is equal to 2, the MVPFULHIGGWHM operator is reduced to the MVPFULNIGGWHM operator. The sensitive analysis of parameter p in Fig. 10 is similar to that in Fig. 8 when p = 1, q ∈ [0, 10], and the sensitive analysis of parameter q in Fig. 11 is similar to that in Fig. 9 when q = 1, p ∈ [0, 10]. The sorting results of all alternatives may change with the increase of parameter p or parameter q, nevertheless, A₃ is always the best one, and A₁ or A₂ is always the worst selection.

6.3 Comparison analysis

To investigate the superiorities of developed MCDM approach with the MVPFULHIGWHM operator and MVPFULHIGGWHM operator, we explore a comparative analysis with existing approaches [15, 26] in this subsection.

The comparative analysis consists of two aspects. One is our method is employed to cope with the instance developed by Wei [26] under picture uncertain linguistic environment, the other is our method is employed to settle the instance developed by Wang [15] under picture hesitant fuzzy environment. Considering the interrelationship and simplification, two parameters p and q are set to 1.

The comparison results with Wei’s method [26] are provided in Table 3.

Table 3
Comparative analysis with PFULS

Methods Ranking orders The best vendor The worst vendor

The method proposed by Wei [26] using PULWBM A₃≻ A₄ ≻ A₅≻ A₂ ≻ A₁ A ₃ A ₁

The method proposed by Wei [26] using PULWGBM A₃≻ A₄ ≻ A₅≻ A₂ ≻ A₁ A ₃ A ₁

The proposed method using MVPFULHIGWHM A₃≻ A₂ ≻ A₄≻ A₅ ≻ A₁ A ₃ A ₁

The proposed method using MVPFULHIGGWHM A₃≻ A₄ ≻ A₅≻ A₁ ≻ A₂ A ₃ A ₂

Methods	Ranking orders	The best vendor	The worst vendor
The method proposed by Wei [26] using PULWBM	A₃≻ A₄ ≻ A₅≻ A₂ ≻ A₁	A ₃	A ₁
The method proposed by Wei [26] using PULWGBM	A₃≻ A₄ ≻ A₅≻ A₂ ≻ A₁	A ₃	A ₁
The proposed method using MVPFULHIGWHM	A₃≻ A₂ ≻ A₄≻ A₅ ≻ A₁	A ₃	A ₁
The proposed method using MVPFULHIGGWHM	A₃≻ A₄ ≻ A₅≻ A₁ ≻ A₂	A ₃	A ₂

From Table 3, we can clearly observe that the orders employing PULWBM operator and PULWGBM operator are the same in Wei’s method, and the orders derived by our approach in term of Algebraic operations are different from those derived by Wei’s method. Nevertheless, no matter what, A₃ is always the best vendor, A₁ or A₂ is the worst selection.

The reasons leading to different ranking orders are listed as below. Firstly, the MCDM method developed by Wei [26] extended conventional BM operators to picture uncertain linguistic situation, and presented PULWBM and PULWGBM operators based on Algebraic operations. These two operators can reflect the correlations between input values a_i and a_j, where i ≠ j. However, the calculation between a_i and a_j is redundant, and which cannot consider the relationship between input value a_i and itself. The method developed in this manuscript can not only reflect the interrelationship between a_i and a_j avoiding redundant calculation, but also consider the correlation between a_j and itself. Thus, the proposed operators in this manuscript are superior to the operators in ref. [26]. Secondly, if there is only one value for positive, neutral, and negative of MVPFULS, then the MVPFULS in this manuscript is reduced to PFULS in ref. [26]. Additionally, when γ = 1, the Hamacher operations in this paper are reduced to the Algebraic operations in ref. [26]. Therefore, the MVPFULS and its operations are an extension of PFULS and its corresponding operations. Thirdly, the score function proposed by Wei doesn’t consider the neutral degree, but the comparative functions defined in this manuscript take into account three membership degrees, which can reflect more information.

The comparison results with Wang’s method [15] are exhibited in Table 4.

Table 4

Comparative results with MVPFS

Methods	Ranking orders	The best alternative	The worst alternative
The method proposed by Wang [15] using GPHFWA operator	A₃≻ A₂ ≻ A₅≻ A₁ ≻ A₄	A ₃	A ₄
The method proposed by Wang [15] using GPHFWG operator	A₃≻ A₂ ≻ A₁≻ A₅ ≻ A₄	A ₃	A ₄
The proposed method using MVPFULHIGWHM	A₃≻ A₂ ≻ A₅≻ A₁ ≻ A₄	A ₃	A ₄
The proposed method using MVPFULHIGGWHM	A₃≻ A₁ ≻ A₂≻ A₅ ≻ A₄	A ₃	A ₄

From Table 4, we can clearly observe orders employing IGWHM operator in this manuscript is the same as that utilizing GPHFWA operator in Wei’s method, and the ranking orders utilizing IGGWHM operator based on Algebraic operations in this manuscript are slightly different from those derived by Wei’s method. Nevertheless, no matter what, A₃ is always the best vendor, A₄ is always the worst one.

The reasons leading to different results are listed as below. The generalized picture hesitant fuzzy weighted aggregating operators [15] neglect the interrelationships among input arguments. The operational rules for PHFES are based on Algebraic operations, which is a particular case of Hamacher operations developed in this manuscript. Additionally, the PHFS defined in Ref. [15] can depict the hesitancy of experts, which employ some values to convey quantitative information, but cannot describe qualitative information. The MVPFULS developed in this manuscript can not only describe quantitative information, but also apply uncertain linguistic term set to better depict qualitative information. Thus, our approach is more agile and general in coping with actual MCDM issues comparing with Wang’s method [15].

7 Conclusion

Although FS is an effective method to handle MCDM in real life, it cannot represent all kinds of uncertainty information in different situation. To better express picture hesitant fuzzy uncertain linguistic cognitive information, we define the concept of MVPFULS, which is an extension of IFS, PFS, PHFS, ULS, PFLS, and PFULS, which is more general and practical for solving complicated cognitive information.

The major contributions of our work are listed below. Firstly, in terms of relative researches, we originally defined the concepts of MVPFULS and MVPFULN. Meanwhile, the operating laws based on Hamacher operations together with the comparison method for MVPFULNs are alos provided. Secondly, motivated by the virtues of IGHM operators, two aggregating operators, that is, the MVPFULHIGWHM operator and MVPFULHIGGWHM operator are investigated. Some promising characteristics and special instances of two operators are presented. Thirdly, MCDM approach with MVPFULNs based on the developed operators is established. Ultimately, an example with sensitivity and comparison analysis is performed to show the application and powerfulness of our method.

According to analysis conducted above, the main advantages of the approach under MVPFULS environment are summarized.

(1) The MVPFULS is more suitable for conveying hesitant quantitative and uncertain qualitative information in MCDM problem, which has superiorities of MVPFS and ULS.

(2) The developed operators in terms of Hamacher operations are more general with parameter γ. Since the Algebraic and Einstein operations are particular cases of γ = 1 and γ = 2, respectively.

(3) The proposed MCDM method is more flexible. Since it can reflect the correlations among input values, the parameters p and q in MVPFULHIGWHM operator and MVPFULHIGGWHM operator can also be given by experts according to their preferences.

In future, different aggregated operators for settling MVPFULN will be further explored and applied to handle different MCDM problem. Meanwhile, we will further discuss the distance measures and entropy of MVPFULS.

Conflicts of interest

The authors declare no conflict of interest.

Footnotes

Acknowledgments

The authors are very grateful to the anonymous reviewers for their valuable comments and suggestions to help improve the overall quality of this paper.

This work was supported by the Social Science Foundation of Hubei Province (No. 20ZD065).

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Hamacher heronian mean operators for multi-critria decision-making under multi-valued picture fuzzy uncertain lingsuitic environment

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 ULTS

2.2 Picture hesitant fuzzy set

2.3 Hamacher operations

2.4 HM operaters

3.1 MVPFULS and Hamacher operations

3.2 Comparative method

4 Multi-valued picture fuzzy uncertain linguistic Hamacher improved generalized heronian mean operator

4.1 MVPFULHIGWHM operator

6.1 Implementation

Table 1 The score values using developed operators Operator E (a1) E (a2) E (a3) E (a4) E (a5) MVPFULH 2.0104 2.7673 3.3940 2.7603 2.3523 IGWHM MVPFULH 2.0550 2.0670 2.8590 2.3530 2.1061 IGGWHM

Conflicts of interest

Footnotes

Acknowledgments

References

Table 1
The score values using developed operators

Operator E (a₁) E (a₂) E (a₃) E (a₄) E (a₅)

MVPFULH 2.0104 2.7673 3.3940 2.7603 2.3523

IGWHM

MVPFULH 2.0550 2.0670 2.8590 2.3530 2.1061

IGGWHM