A Novel Group Decision-Making Method with Adjustment Willingness in a Distributed Hesitant Fuzzy Linguistic Environment

Abstract

This research aims to construct a group decision-making (GDM) method that considers decision makers’ (DMs’) willingness to adjust in a distributed hesitant fuzzy linguistic (DHFL) environment. First, to address the practical scenario where DMs may express preferences using multiple linguistic values with explicit preference strengths, this paper extends the distributed hesitant fuzzy linguistic preference relation (DHFLPR) and supplements missing probabilities. Second, we integrate multiplicative consistency and consensus within a DHFL environment to construct two preference optimization models, whose objective functions are to minimize the overall adjustment based on DMs’ willingness to adjust, thus making the decision more consistent with actual environments. Finally, the viability and effectiveness of the new method are validated by numerical examples. The results show that our new method allows individual preferences to quickly meet the consistency requirement while maximally preserving their original preferences. Additionally, the DHFLPRs maintain the fuzziness and hesitancy in the new preferences, and effectively address the issue of unequal importance among distinct linguistic preference values.

1. Introduction

Decision making is the process of making an intelligent choice or decision based on specific goals, values, information, and resources when faced with multiple alternatives. Decision making usually involves evaluating different alternatives, comparing them, and then choosing one or more of them in order to achieve a specific goal or attain a desired outcome. As one of most common expressions of decision information, preference relations arise from pairwise comparisons of alternatives or criteria and are mainly of linguistic [1,2,3], fuzzy [4,5,6], and multiplicative [7,8] types. Since preference relationships correspond more closely to people’s daily assessment habits and are easily provided by decision makers (DMs), they have therefore attracted significant attention since appearing [9,10,11].

With the exponential growth of knowledge and information, an increasing number of decision problems necessitate group decision making (GDM) rather than a reliance on a single DM. As human beings’ knowledge and attention are limited, and their social status, life background, and work experience are not identical, multiple DMs with different theoretical knowledge and practical experience therefore need to be brought together for GDM. As an effective decision-making method, GDM is able to fully utilize collective superiority and intelligence, which attracts sufficient attention in operations research and management [12,13,14,15].

As a result of the ambiguity of DMs’ perception, individuals often rely on linguistic variables to articulate their preferences in real-world scenarios [16,17,18]. Zadeh [19] introduced the fuzzy linguistic method, which utilizes natural language instead of numerical values to convey qualitative decision information. Subsequently, scholars [20,21] have defined linguistic preference relations (LPRs), providing a more intuitive representation of DMs’ preference information. However, in complex decision environments, DMs frequently find themselves wavering between several linguistic preference values and struggle to assign a definitive value. To solve this problem, Rodriguez et al. [22,23] have proposed a notion for a hesitant fuzzy linguistic term set (HFLTS), an extension of the linguistic term set (LTS) that accommodates multiple linguistic values. Building on this foundation, Zhu and Xu [24] further gave hesitant fuzzy LPRs (HFLPRs). HFLPR-based GDM methods can capture DMs’ uncertainty levels from both fuzzy and hesitant perspectives during the collection and processing of decision information. However, these assume that equal importance among different linguistic preference values, without expressing different degrees of favoritism towards different linguistic preference values or differences in the number of experts involved in decision making.

The consistency issue in some preference-relation-based GDM problems has received significant attention, which has been classified into two parts: individual consistency and the consistency among DMs, with the latter also known as consensus. Individual consistency [25,26,27,28] aims to ensure that preference relations provided by DMs are logical and coherent. It serves as a crucial indicator of the scientific rigor of the decision-making process and its potential to yield accurate outcomes, and is essential for making the right decisions. Consensus [29,30,31,32], on the other hand, aims to derive a collective opinion that is shared by a high majority of DMs who are engaged in a GDM process. A variety of methods are given in the existing literature to address consistency problems in GDM. For instance, Zhou and Xu [33,34] constructed models using multiplicative and additive consistency formulas, respectively, to fill in missing probabilistic information, and tested and improved consistency. Li and Wang [35,36] analyzed the expected preference matrix by using additive and multiplicative consistency, respectively, and gave an improved model of consistency based on parametric iteration, and also portrayed a consensus index of the group through defining a distance from individual preferences to collective preferences. Wu et al. [37] investigated and rectified additive consistency issues in preference relations and employed localized feedback strategies to enhance consensus. Due to limitations in DMs’ cognitive processes, achieving complete consistency in preference relations is difficult. Therefore, Zhu et al. [38] introduced the concept of acceptable consistency in their study. Research has shown that preference relations which depend on individual consistency and consensus are effective in ensuring the rationality and feasibility in GDM outcomes.

In general, the traditional iterative preference matrix adjustment strategy [39,40] is currently predominantly used to gradually achieve the expected consistency requirements. However, such traditional method has two shortcomings. First, the iterative adjustment process is complicated, the computational workload is large, and the number of iterations easily reaches its upper limit, while the preference relation is still not enough to satisfy the consistency requirement. Second, such methods have a single goal of mathematical consistency and completely ignore the willingness of DMs to adjust. The willingness to adjust reflects the DM’s acceptance of modifications to the original preference information. For example, some key preferences may need to be restricted in the scope of adjustment, and traditional methods cannot embed such constraints, resulting in adjusted preference relations that deviate from the actual decision-making scenarios.

Aiming at the above problems, this paper proposes a GDM method in a DHFL environment that integrates the DMs’ willingness to adjust. Specifically, the willingness to adjust is manifested in two respects. First, at the individual level, by setting constraints on the adjustment amount of preference strength, allowing DMs to flexibly control the adjustment range based on risk preferences. Second, at the group level, an optimization model is constructed with the objective function of minimizing the overall adjustment strength to maximize the retention of original preference information while satisfying the consistency requirement. This design not only overcomes the shortcomings of traditional iterative methods that ignore the DM’s willingness to adjust, but also provides a more relevant solution for GDM in complex environments.

In a distributed hesitant fuzzy linguistic (DHFL) environment, DMs express preferences through multiple linguistic values and their preference strength distributions. While this mechanism effectively captures the ambiguity and hesitancy of decision information, it complicates adjustment process. Traditional methods rely on iterative algorithms to enforce the adjustment of inconsistent preference relations, ignoring two key real-world requirements. The first is the DM’s acceptance of the scope of the adjustment. In complex decision-making scenarios, DMs may have different degree of willingness to accept preference modifications due to information asymmetry or divergent interests. Ignoring this difference may result in adjusted preferences that deviate from the actual context and reduce the credibility of the decision outcome. The second is the need for preservation of original preference information. The advantage of distributed HFLPRs (DHFLPRs) is the preservation of the DM’s hesitant characteristics through the distribution of preference strengths. If the adjustment process unconstrainedly modifies the preference strength distribution, it may destroy the fuzzy semantics of the original data and lead to distorted information. Therefore, considering the DM’s willingness to adjust in a DHFL environment is not only the key to enhance the practical acceptability of the decision results, but also a necessary condition to maintain the integrity of fuzzy information.

This paper extends the DHFLPR, accommodating both complete and incomplete hesitation fuzzy linguistic elements (CHFLEs and ICHFLEs) to indicate the level of preference for pairwise comparisons between alternatives. Aiming at the problems in the consistency adjustment strategy, this paper combines the multiplicative consistency analysis and consensus analysis in a DHFL environment, and constructs two preference optimization models aiming to minimize the overall adjustment strength, so as to make the decision-making results more reasonable and more in line with the actual decision-making environment.

The rest of this manuscript is arranged as follows. Section 2 contains a few basic definitions and notions that provide the logical foundation for what follows. Section 3 is a description of the DHFLPRs and presents the consistency analysis. In Section 4, this paper proposes two preference optimization model and a GDM method that consider DMs’ willingness to adjust in a DHFL environment. Section 5 shows a numerical case in order to validate the effectiveness of our new method. Finally, Section 6 concludes the paper.

2. Preliminaries

In practical decision-making problems, DMs often prefer to express their preferences in language over numerical scales, as they may not be able to accurately choose the right values to express their preferences, and in some cases, evaluating alternatives using qualitative variables is better suited to deal with complex problems.

2.1. Linguistic Preference Relations

In practical decision making, DMs often express preferences through natural language rather than numerical values. In order to process such information in a structured way, an LTS needs to be defined, which consists of an ordered set of fuzzy linguistic labels.

Definition 1.

([41]) Assume $S=\{s_{\alpha }|\alpha =-t,\dots ,0,\dots ,t\}$ is a subscript-symmetric LTS in which $t$ is a positive number, while $s_{-t}$ and $s_t$ are, respectively, the minimum and maximum values of $S$. $S$ satisfies the following properties:

$$\left\{\begin{array}{ll}{s}_{\alpha }>s_{\beta },\hfill & \alpha >\beta ,\hfill \\ neg(s_{\alpha })=s_{-\alpha }.\hfill & \hfill \end{array}\right.$$

(1)

Subsequently, Xu [21] further extends this to a successive LTS $\overline{S}=\{s_{\alpha }|-t\le \alpha \le t\}$. If there exists $s_{\alpha }\in S$, then $s_{\alpha }$ is said to be a primitive linguistic term, and otherwise $s_{\alpha }$ is said to be an extended linguistic term. As a general rule, DMs will use original linguistic terms when evaluating alternatives, while extended linguistic terms will only be present in operations.

With respect to two linguistic terms of any kind $s_{\alpha },s_{\beta }\in \overline{S}$ and $\lambda ,{\lambda }_1,{\lambda }_2\in [0,1]$, Xu [42] introduces several calculation rules that are shown below:

$$\left\{\begin{array}{l}{s}_{\alpha +\beta }=s_{\alpha }+s_{\beta },\\ \lambda s_{\alpha }=s_{\lambda \alpha },\\ ({\lambda }_1+{\lambda }_2)s_{\alpha }={\lambda }_1{s}_{\alpha }\oplus {\lambda }_2{s}_{\alpha },\\ \lambda (s_{\alpha }\oplus s_{\beta })=\lambda s_{\alpha }\oplus \lambda s_{\alpha }.\end{array}\right.$$

(2)

In an effort to minimize the impact of human expertise and capacity limitations, and to obtain the best alternative sorting results, people tend to compare alternatives pairwise and construct preference relations. Therefore, preference relations have become the common form of DMs to express preference information.

Definition 2.

([20]) Assume that $L$ is an LPR for the given alternative set $X=\{x_1,\dots ,x_n\}$, which is expanded as follows:

$$L={(l_{ij})}_{n\times n}=\left[\begin{array}{cccc}{l}_{11}& l_{12}& \cdots & l_{1n}\\ l_{21}& l_{22}& \cdots & l_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ l_{n1}& l_{n2}& \cdots & l_{nn}\end{array}\right]$$

(3)

where $l_{ij}$ denotes the value of linguistic preference of $x_i$ with respect to $x_j$, and $l_{ij}\in S,l_{ij}\oplus l_{ji}=s_0,l_{ii}=s_0$.

2.2. Hesitant Fuzzy Preference Relations

When DMs want to assess the preference between an alternative compared to another, the preference information offered by a DM may contain a group of evaluated values instead of one, due to the complexity of the decision content and a great deal of uncertainty.

Definition 3.

([43]) Assume $U$ is a finite non-empty field, and then a hesitant fuzzy set (HFS) $B$ over $U$ has the following definition:

$$B=\{\langle x,h_B\rangle |x\in U\}$$

(4)

where $h_B=({\mu }_B^1,\dots ,{\mu }_B^{\#h_B})$ is known as a hesitant fuzzy element (HFE), which is a collection of several values belonging to a closed interval $[0,1]$ that represents the degree of affiliation of $x$ to $B$, and $\#h_B$ stands for the quantity of affiliation degrees in $h_B$.

Definition 4.

([44]) Given a finite set $X=\{x_1,\dots ,x_n\}$, assume a hesitant fuzzy preference relation (HFPR) with respect to $X$ being in the form of $H={(h_{ij})}_{n\times n}$, where $h_{ij}=(h_{ij}^1,\dots ,h_{ij}^{\#h_{ij}})$ is an HFE that denotes the possible preference of $x_i$ with respect to $x_j$, and $\#h_{ij}$ denotes the quantity of preference degrees in $h_{ij}$. An element $h_{ij}$ in $H$ satisfies the following:

$$\left\{\begin{array}{l}{h}_{ij}^l\oplus h_{ji}^{\#h_{ij}-l+1}=s_0,\hfill \\ h_{ij}^l<h_{ij}^{l+1},\hfill \\ \#h_{ij}=\#h_{ji},\hfill \\ h_{ii}=s_0.\hfill \end{array}\right.$$

(5)

Definition 5.

([10]) Assume that $H={(h_{ij})}_{n\times n}$ is an HFPR against the set of alternatives $X$. And with any $i,j,k\in N$, we have

$$\begin{array}{cc}{h}_{ij}^l\cdot h_{jk}^l\cdot h_{ki}^l=h_{ji}^l\cdot h_{kj}^l\cdot h_{ik}^l,& \hspace{1em}l=1,\dots ,\#h_{ij}\end{array}.$$

(6)

and then it is said that the HFPR $H$ is satisfied with multiplicative consistency, where $h_{ij}^l$ stands for the $l$th smallest ranked preference degree in $h_{ij}$.

2.3. Hesitant Fuzzy Linguistic Preference Relations

Solving complex problems featured by high uncertainty, DMs may find themselves hesitating between several linguistic terms when expressing preferences. For that, Rodriguez et al. gave the HFLTSs by combing the characteristics of HFSs and linguistic variables.

Definition 6.

([23]) Given an LTS $S$, then the HFLTS $h_S$ is one of finite and ordered subsets of $S$.

For example, $S=\{s_{\alpha }|\alpha =-4,\dots ,0,\dots ,4\}$ is an LTS and $h_S=\{s_{-3},s_{-1},s_4\}$ is an HFLTS.

Definition 7.

([24]) Let there be a given finite set $X=\{x_1,\dots ,x_n\}$ and a discrete LTS $S=\{s_{\alpha }|\alpha =-t,\dots ,0,\dots ,t\}$. An HFLPR on $X$ can be denoted by $H={(h_{ij})}_{n\times n}$, and $h_{ij}=(h_{ij}^1,\dots ,h_{ij}^{\#h_{ij}})$ denotes the value of the linguistic preference of $x_i$ with respect to $x_j$. An element $h_{ij}$ in $H$ satisfies the following:

$$\left\{\begin{array}{l}{h}_{ij}^l\oplus h_{ji}^{\#h_{ij}-l+1}=s_0,\hfill \\ h_{ij}^l<h_{ij}^{l+1},\hfill \\ \#h_{ij}=\#h_{ji},\hfill \\ h_{ii}=s_0.\hfill \end{array}\right.$$

(7)

where $\#h_{ij}$ denotes the quantity of preference degrees in $h_{ij}$, and note that the HFLPR degenerates into an LPR when $\#h_{ij}=1$.

3. Distributed Hesitant Fuzzy Linguistic Preference Relations and Consistency Analysis

3.1. Distributed Hesitant Fuzzy Linguistic Preference Relations

An LPR comprises individual linguistic terms that reflect a DM’s preference information concerning a pairwise comparison between alternatives. However, DMs may hesitate to express a preference for an alternative. In such cases, the preference information may contain multiple preference values rather than one.

Example 1.

Suppose a company is planning to purchase a commercial vehicle in the near future for business expansion. To select a suitable commercial vehicle, the DM must choose the brand that best aligns with their preferences from a set of six brands, as shown in

Table 1. The alternative set.

${\mathbf{x}}_1$	${\mathbf{x}}_2$	${\mathbf{x}}_3$	${\mathbf{x}}_4$	${\mathbf{x}}_5$	${\mathbf{x}}_6$
Cadillac	Lexus	Volvo	Toyota	Tesla	Audi

.

The DM expresses a preference evaluation as follows: degree of preference for $x_1$ with respect to $x_5$ is either $s_2$ or $s_3$; degree of preference for $x_2$ with respect to $x_3$ is either $s_{-1}$ or $s_0$; and degree of preference for $x_6$ with respect to $x_4$ is either $s_{-4}$ or $s_{-3}$. In such cases, accurately reflecting the DM’s true preference opinion using only $l_{15}=s_2$ or $l_{15}=s_3$ ($l_{23}=s_{-1}$ or $l_{23}=s_0$; $l_{64}=s_{-4}$ or $l_{64}=s_{-3}$) to denote the preference degree regarding $x_1$ relative to $x_5$ ($x_2$ relative to $x_3$; $x_6$ relative to $x_4$) becomes difficult.

Given the limitations of people’s perceptions and thinking, DMs may hold multiple degrees of preference for an alternative, and the importance of these degrees of preference varies among them. In such cases, the validity of HFLPR may be drastically reduced.

Example 2.

Assume that the DM in Example 1 (1) believes that the preference degree of alternative $x_1$ regarding $x_5$ is $s_2$ or $s_3$, but also believes that the probability of $s_2$ is 60% and the probability of $s_3$ is 40%; (2) believes that the preference degree of alter-native $x_2$ regarding $x_3$ is $s_{-1}$ or $s_0$, but also believes that the probability of $s_{-1}$ is 50% and the probability of $s_0$ is 50%; (3) believes that the preference degree of alternative $x_6$ regarding $x_4$ is $s_{-4}$ or $s_{-3}$, but also believes that the probability of $s_{-4}$ is 30% and the probability of $s_{-3}$ is 70%. Accurately representing a DM’s true preferences using HFLPRs is challenging, particularly when there are variations in preference strengths across multiple degrees of preference.

The traditional HFLPR assumes that each linguistic value has equal preference strength, but in practical decision making, DMs may assign different importance to different linguistic values. To solve this problem, this paper proposes DHFLPRs to quantify the importance difference by introducing a preference strength distribution. Specifically, DHFLPRs allow DMs to assign explicit probabilistic weights to each linguistic value and thus express their preference tendencies more precisely.

Definition 8.

Assume that $X=\{x_1,\dots ,x_n\}$ is a given fixed set, and $S=\{s_{\alpha }|\alpha =-t,\dots ,0,\dots ,t\}$ is a discrete LTS. An DHFLPR defined on $X$ may be represented by $H={(h_{ij})}_{n\times n}$. An element $h_{ij}$ in $H$ satisfies the following properties

$$\left\{\begin{array}{l}{h}_{ij}^l+h_{ji}^{\#h_{ij}-l+1}=s_0,\hfill \\ p_{ij}^l=p_{ji}^{\#h_{ij}-l+1},\hfill \\ h_{ij}^l<h_{ij}^{l+1},\hfill \\ 0\le {\displaystyle \sum _{l=1}^{\#h_{ij}}{p}_{ij}^l}\le 1,\hfill \\ \#h_{ij}=\#h_{ji},\hfill \\ h_{ii}=(s_0,1).\hfill \end{array}\right.$$

(8)

where $h_{ij}=\{(h_{ij}^l,p_{ij}^l)|l=1,\dots ,\#h_{ij}\}$ is a probability distribution over linguistic terms $h_{ij}^l$, which denotes the DHFL preference of $x_i$ over $x_j$, $\#h_{ij}$ denotes the quantity of preference degrees in $h_{ij}$, and $p_{ij}^l$ is the strength of preference for preference degree $h_{ij}^l$.

If ${\displaystyle \sum _{l=1}^{\#h_{ij}}{p}_{ij}^l}=1$, $H$ would be called a complete HFLPR (CHFLPR); similarly, if $0\le {\displaystyle \sum _{l=1}^{\#h_{ij}}{p}_{ij}^l}<1$, $H$ would be called an incomplete HFLPR (ICHFLPR).

When using the DHFLPR to record the preference evaluation of the DM in Example 2, the preference information for $x_1$ with respect to $x_5$ can be denoted as $h_{15}=\{(s_2,0.6),(s_3,0.4)\}$, the preference information for $x_2$ with respect to $x_3$ can be denoted as $h_{23}=\{(s_{-1},0.5),(s_0,0.5)\}$, and the preference information for $x_6$ with respect to $x_4$ can be denoted as $h_{64}=\{(s_{-4},0.3),(s_{-3},0.7)\}$.

The DHFLPR provides a structured basis for quantifying the DM’s willingness to adjust by introducing a preference strength distribution. Specifically, the relationship is reflected in the following three aspects:

(1)

Preference strength as a proxy for adjustment willingness: the distribution of preference strength for each linguistic term in the DHFLPR directly reflects the DM’s inclination towards different preference values. When adjustments are needed to satisfy consistency, the bounded range of preference strength adjustments indicates the DM’s tolerance of modifications to the original preference information.

(2)

Compatibility of ambiguity with adjustment flexibility: the DHFLPR allows DMs to express uncertainty in terms of multiple linguistic terms and their preference strength distributions. This ambiguity naturally supports the dynamics of the willingness to adjust, and by adjusting the probability distribution instead of a single value, the DM can achieve consistent optimization while preserving the fuzzy characteristics of the original preferences.

(3)

Mapping of optimization objective to willingness to adjust: the optimization model proposed in this paper takes minimizing the overall adjustment intensity as the objective function, which essentially translates willingness to adjust into mathematical constraints under the DHFLPR framework. For example, if a DM refuses to significantly modify the original preferences, their corresponding preference strength adjustments will be small and the optimization process will prioritize retaining their original input.

To facilitate the manipulation of linguistic terms, Gou et al. developed an equality function between HFLTSs and HFSs.

Definition 9.

([45]) Assuming that $S=\{s_{\alpha }|\alpha =-t,\dots ,0,\dots ,t\}$ is a given LTS, $h_S=\{s_{\alpha }^l|l=1,\dots ,\#h_S\}$ is an HFLTS, $s_{\alpha }^l\in S$, $h_B=\{{\mu }^l|l=1,\dots ,\#h_B\}$ is an HFS, and ${\mu }^l\in [0,1],\#h_B=\#h_S$, then an equality conversion function between linguistic term $s_{\alpha }^l$ and affiliation degree ${\mu }^l$ is

$$N(s_{\alpha }^l)={\displaystyle \frac{\alpha +t}{2t}}={\mu }^l$$

(9)

Definition 10.

Given that $H={(h_{ij})}_{n\times n}$ is a CHFLPR, the corresponding expected fuzzy matrix is denoted by $E={(e_{ij})}_{n\times n}$, and with any $i,j\in N,i\ne j$

$$e_{ij}={\displaystyle \sum _{l=1}^{\#h_{ij}}(N(h_{ij}^l)\cdot p_{ij}^l)}$$

(10)

Definition 11.

Given that $H={(h_{ij})}_{n\times n}$ is an ICHFLPR, the corresponding expected fuzzy matrix is denoted by $E={(e_{ij})}_{n\times n}$, where

$$e_{ij}={\displaystyle \sum _{l=1}^{\#h_{ij}}(N(h_{ij}^l)\cdot p_{ij}^l)}+(1-\lambda )\cdot N(h_{ij}^1)\cdot (1-{\displaystyle \sum _{l=1}^{\#h_{ij}}{p}_{ij}^l})+\lambda \cdot N(h_{ij}^{\#h_{ij}})\cdot (1-{\displaystyle \sum _{l=1}^{\#h_{ij}}{p}_{ij}^l})$$

(11)

which holds to any $i,j\in N,i\ne j$, and $e_{ii}=0.5,\lambda \in [0,1]$.

When $\lambda =0$, the minimum expected fuzzy preference value $e_{ij}^{-}$ can be obtained by allocating all unknown preference strengths $(1-{\displaystyle \sum _{l=1}^{\#h_{ij}}{p}_{ij}^l})$ in $h_{ij}$ to the linguistic preference value $h_{ij}^1$. When $\lambda =1$, the maximum expected fuzzy preference value $e_{ij}^{+}$ can be obtained by allocating all unknown preference strengths $(1-{\displaystyle \sum _{l=1}^{\#h_{ij}}{p}_{ij}^l})$ in $h_{ij}$ to the linguistic preference value $h_{ij}^{\#h_{ij}}$. Except for the above two cases, the expected fuzzy preference value in all other cases lies between the minimum and maximum expected fuzzy preference values, i.e., $e_{ij}\in (e_{ij}^{-},e_{ij}^{+})$. In practical decision problems, the allocation parameter for the unknown preference strength, i.e., the value of $\lambda$, is usually linked with the attitude of DMs in the face of uncertain risk. In general, DMs can be broadly categorized into three types, namely conservative, moderate, and radical, and based on the principle of equal distribution, the values of $\lambda$ corresponding to these three types of DMs are $[0,{\displaystyle \frac{1}{3}})$, $[{\displaystyle \frac{1}{3}},{\displaystyle \frac{2}{3}})$, and $[{\displaystyle \frac{2}{3}},1]$. In addition, when solving specific problems, the value of $\lambda$ needs to be taken into account not only for the classification of the type of DMs, but also the specific analysis of the content of the decision.

3.2. Consistency Analysis

Multiplicative consistency is a critical criterion in practical decision analysis for detecting logical inconsistencies in a DM’s evaluations. When multiplicative consistency is not met, it may be necessary to adjust the DM’s preference or assumptions to ensure decision outcomes are reasonable and reliable.

Definition 12.

([20]) Let $H$ be a DHFLPR on a finite set $X$, with an expected fuzzy matrix $E$. If it satisfies

$$\begin{array}{cc}{e}_{ij}\cdot e_{jk}\cdot e_{ki}=e_{ji}\cdot e_{kj}\cdot e_{ik},& i,j,k\in N.\end{array}$$

(12)

then it is concluded that the DHFLPR $H$ is satisfied with multiplicative consistency.

Example 3.

Let an LTS be $S=\{s_{\alpha }|\alpha =-4,\dots ,0,\dots ,4\}$, a finite set be $X=\{x_1,x_2,x_3\}$, and $H$ be a CHFLPR on $X$ with the following expression:

$$H=\left[\begin{array}{lll}\begin{array}{l}\{(s_0,1)\}\\ \\ \end{array}\hfill & \begin{array}{l}\{(s_{-2},0.5),\\ (s_{-1},0.5)\}\\ \end{array}\hfill & \begin{array}{l}\{(s_1,0.3),\\ (s_2,0.4),\\ (s_3,0.3)\}\end{array}\hfill \\ \begin{array}{l}\{(s_1,0.5),\\ (s_2,0.5)\}\\ \end{array}\hfill & \begin{array}{l}\{(s_0,1)\}\\ \\ \end{array}\hfill & \begin{array}{l}\{(s_{-1},0.25),\\ (s_0,0.25),\\ (s_1,0.5)\}\end{array}\hfill \\ \begin{array}{l}\{(s_{-3},0.3),\\ (s_{-2},0.4),\\ (s_{-1},0.3)\}\end{array}\hfill & \begin{array}{l}\{(s_{-1},0.5),\\ (s_0,0.25),\\ (s_1,0.25)\}\end{array}\hfill & \begin{array}{c}\{(s_0,1)\}\\ \\ \end{array}\hfill \end{array}\right]$$

From Equations (9) and (10), the expected fuzzy matrix can be obtained as follows:

$$E=\left[\begin{array}{ccc}0.5& 0.313& 0.75\\ 0.687& 0.5& 0.531\\ 0.25& 0.469& 0.5\end{array}\right]$$

It has been verified that there exists $e_{ij}\cdot e_{jk}\cdot e_{ki}\ne e_{ji}\cdot e_{kj}\cdot e_{ik}$, so the CHFLPR $H$ does not satisfy multiplicative consistency.

Theorem 1.

Assume $H={(h_{ij})}_{n\times n}$ to be a DHFLPR against a given set  $X$ , and  $E={(e_{ij})}_{n\times n}$ is the expected fuzzy matrix. If  $\overline{E}={({\overline{e}}_{ij})}_{n\times n}$ is an expected fuzzy matrix satisfying multiplicative consistency, then for any  $i,j\in N$.

$${\overline{e}}_{ij}={\displaystyle \frac{{\displaystyle \sum _{r=1}^n{e}_{ir}}}{{\displaystyle \sum _{r=1}^n{e}_{ir}}+{\displaystyle \sum _{r=1}^n{e}_{jr}}}}$$

(13)

Proof. 

The proof for any $i,j,k\in N$, such that we have

$$\begin{gathered} {\overline{e}}_{ij}\cdot {\overline{e}}_{jk}\cdot {\overline{e}}_{ki}={\displaystyle \frac{{\displaystyle \sum _{r=1}^n{e}_{ir}}}{{\displaystyle \sum _{r=1}^n{e}_{ir}}+{\displaystyle \sum _{r=1}^n{e}_{jr}}}}\cdot {\displaystyle \frac{{\displaystyle \sum _{r=1}^n{e}_{jr}}}{{\displaystyle \sum _{r=1}^n{e}_{jr}}+{\displaystyle \sum _{r=1}^n{e}_{kr}}}}\cdot {\displaystyle \frac{{\displaystyle \sum _{r=1}^n{e}_{kr}}}{{\displaystyle \sum _{r=1}^n{e}_{kr}}+{\displaystyle \sum _{r=1}^n{e}_{ir}}}} \\ {\overline{e}}_{ji}\cdot {\overline{e}}_{kj}\cdot {\overline{e}}_{ik}={\displaystyle \frac{{\displaystyle \sum _{r=1}^n{e}_{jr}}}{{\displaystyle \sum _{r=1}^n{e}_{jr}}+{\displaystyle \sum _{r=1}^n{e}_{ir}}}}\cdot {\displaystyle \frac{{\displaystyle \sum _{r=1}^n{e}_{kr}}}{{\displaystyle \sum _{r=1}^n{e}_{kr}}+{\displaystyle \sum _{r=1}^n{e}_{jr}}}}\cdot {\displaystyle \frac{{\displaystyle \sum _{r=1}^n{e}_{ir}}}{{\displaystyle \sum _{r=1}^n{e}_{ir}}+{\displaystyle \sum _{r=1}^n{e}_{kr}}}} \end{gathered}$$

So ${\overline{e}}_{ij}\cdot {\overline{e}}_{jk}\cdot {\overline{e}}_{ki}={\overline{e}}_{ji}\cdot {\overline{e}}_{kj}\cdot {\overline{e}}_{ik}$ holds. □

Example 4.

Based on Example 3, Equation (13) yields the expected fuzzy matrix satisfying multiplicative consistency as follows:

$$\overline{E}=\left[\begin{array}{ccc}0.5& 0.476& 0.562\\ 0.524& 0.5& 0.585\\ 0.438& 0.415& 0.5\end{array}\right]$$

Inference 1.

Let $H={(h_{ij})}_{n\times n}$ be a DHFLPR over the given set $X$, where $E={(e_{ij})}_{n\times n}$ is the expected fuzzy matrix, and $\overline{E}={({\overline{e}}_{ij})}_{n\times n}$ is the expected fuzzy matrix satisfying multiplicative consistency. Then, the DHFLPR $H$ satisfies multiplicative consistency if and only if the expected fuzzy matrix $E$ satisfies multiplicative consistency; i.e., $E=\overline{E}$.

It is evident that a sufficient and necessary condition for judging whether the DHFLPR $H$ satisfies multiplicative consistency is $E=\overline{E}$. Thus, the level of multiplicative consistency of the DHFLPR $H$ can be measured through a distance function between $E$ and $\overline{E}$.

As a result of the objective environment in which decision making takes place, obtaining preference relations that fully satisfy multiplicative consistency is often impractical. Consequently, an acceptable multiplicative consistency threshold is established, which can be calibrated according to the decision context and DMs’ preferences. The level of multiplicative consistency in the preference relation is then quantified. If this level falls within the acceptable threshold, no adjustment is required. Otherwise, adjustments are required to derive a new preference relation that meets the threshold.

If the DHFLPR $H$ does not completely satisfy multiplicative consistency, as is obvious from Inference 1 that $E\ne \overline{E}$, then there exists $e_{ij}\ne {\overline{e}}_{ij},i,j\in N$. Therefore, the extent of multiplicative consistency for $H$ can be measured by ${\displaystyle \frac{2}{n(n-1)}}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1,j>i}^n|e_{ij}-{\overline{e}}_{ij}|}}$. When ${\displaystyle \frac{2}{n(n-1)}}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1,j>i}^n|e_{ij}-{\overline{e}}_{ij}|}}=0$, $H$ is completely satisfies multiplicative consistency.

Definition 13.

([46]) Let ${\xi }_0$ be the threshold for acceptable multiplicative consistency. The index of multiplicative consistency for $H$ is

$$MCI(H)={\displaystyle \frac{2}{n(n-1)}}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1,j>i}^n|e_{ij}-{\overline{e}}_{ij}|}}$$

(14)

If $MCI(H)\le {\xi }_0$, then the DHFLPR $H$ is considered to satisfy acceptable multiplicative consistency.

When confronted with complex decision problems, individuals frequently encounter limitations in expertise and capacity, leading to decision outcomes that may lack comprehensiveness and accuracy. Given that different DMs harbor distinct theoretical knowledge and practical experience, employing a GDM method becomes more reasonable. It fully harnesses diverse perspectives and experiences, thereby augmenting the quality and credibility of decision making.

The goal of GDM is to achieve consistency among DMs, aiming for consensus. However, attaining complete consistency among DMs is often challenging. To effectively address this situation, it is common to establish a threshold for acceptable consensus and quantitatively assess the extent of consensus within the group. If the extent falls outside the acceptable threshold range, then the DMs’ preference information must be adjusted accordingly.

Suppose that $D=\{d_1,\dots ,d_m\}$ is a group of DMs. Each DM $d_k(k=1,\dots ,m)$ establishes a DHFLPR $H^k={(h_{ij}^k)}_{n\times n}$ by comparing the alternatives in set $X$ pairwise, resulting in $E^k={(e_{ij}^k)}_{n\times n}$, the expected fuzzy matrix. From Equation (14), an index of consistency between DMs $d_k$ and $d_g$ can be obtained:

$$MCI(d_k,d_g)={\displaystyle \frac{2}{n(n-1)}}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1,j>i}^n|e_{ij}^k-e_{ij}^g|}}$$

(15)

Definition 14.

([47]) Let ${\zeta }_0$ be the threshold for acceptable consensus. Aggregate indexes of consistency of the DMs $d_k$ and $d_g(g\ne k)$ to obtain the consensus index for $D$:

$$GCI(D)={\displaystyle \frac{1}{m}}{\displaystyle \sum _{k=1}^{m}MCI(d_k)}={\displaystyle \frac{1}{m(m-1)}}{\displaystyle \sum _{k=1}^m{\displaystyle \sum _{g=1,g\ne k}^{m}MCI(d_k,d_g)}}$$

(16)

If $GCI(D)\le {\zeta }_0$ is met, then the group $D$ is considered to have achieved an acceptable consensus.

4. Preference Optimization Models in Distributed Hesitant Fuzzy Linguistic Environments

Due to the cognitive and analytical limitations of DMs, preference relations established by DMs often exhibit inadequate multiplicative consistency in real-world GDM scenarios. To address this challenge, the systematic adjustment of DMs’ preference information is required to obtain a new set of preference relations that satisfy acceptable multiplicative consistency, while simultaneously ensuring preference relations of the group achieve acceptable consensus.

4.1. Preference Optimization Models in a Complete Hesitant Fuzzy Linguistic Environment

Assume there exists a GDM problem where $S=\{s_{\alpha }|\alpha =-t,\dots ,0,\dots ,t\}$ is a finite LTS, $X=\{x_1,\dots ,x_n\}$ is a set of given and finite alternatives, and $D=\{d_1,\dots ,d_m\}$ is a given group of DMs. Let $H^k={(h_{ij}^k)}_{n\times n}$ be a CHFLPR established by DM $d_k(k=1,\dots ,m)$, satisfying ${\displaystyle \sum _{l=1}^{\#h_{ij}^k}{p}_{ij}^{kl}}=1$, where $h_{ij}^k=\{(h_{ij}^{kl},p_{ij}^{kl})|l=1,\dots ,\#h_{ij}^k\}$ is a CHFLE, $E^k={(e_{ij}^k)}_{n\times n}$ is the expected fuzzy matrix, ${\overline{E}}^k={({\overline{e}}_{ij}^k)}_{n\times n}$ is the expected fuzzy matrix satisfying multiplicative consistency, ${\xi }_0$ is the threshold for acceptable multiplicative consistency, and ${\zeta }_0$ is the acceptable consensus threshold.

If all $MCI(H^k)\le {\xi }_0$ and $GCI(D)\le {\zeta }_0$, no adjustment is required; conversely, an adjustment to the preference strength $p_{ij}^{kl}$ is required. Let the adjusted preference relation be ${\tilde{H}}^k={({\tilde{h}}_{ij}^k)}_{n\times n}$, where ${\tilde{h}}_{ij}^k=\{(h_{ij}^{kl},p_{ij}^{kl}+q_{ij}^{kl})|l=1,\dots ,\#h_{ij}^k\}$, $q_{ij}^{kl}$ are the adjustment amounts of the preference strength $p_{ij}^{kl}$, and

$$\left\{\begin{array}{l}{q}_{ij}^{kl}\in [-p_{ij}^{kl},1-p_{ij}^{kl}],\hfill \\ {\displaystyle \sum _{l=1}^{\#h_{ij}^k}{q}_{ij}^{kl}}=0.\hfill \end{array}\right.$$

(17)

The adjustment strength, denoted as ${\theta }_{ij}^k$, of the DM $d_k$ for the linguistic preference $h_{ij}^k$, can be calculated by

$${\theta }_{ij}^k={\displaystyle \sum _{l=1}^{\#h_{ij}^k}(N(h_{ij}^{kl})\cdot q_{ij}^{kl})}$$

(18)

From Equations (10) and (18), the expected preference value of linguistic preference ${\tilde{h}}_{ij}^k$ can be obtained:

$$E({\tilde{h}}_{ij}^k)=e_{ij}^k+{\theta }_{ij}^k$$

(19)

Taking into account the adjustment strength ${\theta }_{ij}^k$ of all DMs on the linguistic preference $h_{ij}^k$, the overall adjustment strength can be obtained. Under the condition of satisfying a specific consistency, if the overall adjustment strength is smaller—i.e., the adjusted preference information is closer to DMs’ original preference information—it means that DMs’ willingness to adjust is higher. Therefore, taking an objective function to minimize the overall adjustment strength, a model was constructed as follows:

$$\begin{array}{l}\min {\displaystyle \sum _{k=1}^m{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^n|{\theta }_{ij}^k|}}}\\ \mathrm{s}.\mathrm{t}.\left\{\begin{array}{l}{\displaystyle \frac{2}{n(n-1)}}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1,j>i}^n|E({\tilde{h}}_{ij}^k)-{\overline{e}}_{ij}^k|}}\le {\xi }_0,\hfill \\ {\displaystyle \frac{1}{m(m-1)}}{\displaystyle \sum _{k=1}^m{\displaystyle \sum _{g=1,g\ne k}^m[\frac{2}{n(n-1)}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1,j>i}^n|E({\tilde{h}}_{ij}^k)-E({\tilde{h}}_{ij}^g)|}}]\le {\zeta }_0,}}\hfill \\ q_{ij}^{kl}\ge -p_{ij}^{kl},\hfill \\ q_{ij}^{kl}\le 1-p_{ij}^{kl},\hfill \\ {\displaystyle \sum _{l=1}^{\#h_{ij}^k}{q}_{ij}^{kl}}=0.\hfill \end{array}\right.\end{array}$$

(20)

From Equations (10), (18), and (19), Model (20) could be translated into the following:

$$\begin{array}{l}\min {\displaystyle \sum _{k=1}^m{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1,j>i}^n|{\displaystyle \sum _{l=1}^{\#h_{ij}^k}(N(h_{ij}^{kl})\cdot q_{ij}^{kl})}|}}}\\ \mathrm{s}.\mathrm{t}.\left\{\begin{array}{l}{\displaystyle \frac{2}{n(n-1)}}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1,j>i}^n|{\displaystyle \sum _{l=1}^{\#h_{ij}^k}(N(h_{ij}^{kl})\cdot p_{ij}^{kl})}}}+{\displaystyle \sum _{l=1}^{\#h_{ij}^k}(N(h_{ij}^{kl})\cdot q_{ij}^{kl})}-{\overline{e}}_{ij}^k|\le {\xi }_0,\hfill \\ {\displaystyle \frac{1}{m(m-1)}}{\displaystyle \sum _{k=1}^m{\displaystyle \sum _{q=1,q\ne k}^m[\frac{2}{n(n-1)}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1,j>i}^n|{\displaystyle \sum _{l=1}^{\#h_{ij}^k}(N(h_{ij}^{kl})\cdot p_{ij}^{kl})}+{\displaystyle \sum _{l=1}^{\#h_{ij}^k}(N(h_{ij}^{kl})\cdot q_{ij}^{kl})-}}}}}\hfill \\ {\displaystyle \sum _{l=1}^{\#h_{ij}^g}(N(h_{ij}^{gl})\cdot p_{ij}^{gl})}-{\displaystyle \sum _{l=1}^{\#h_{ij}^g}(N(h_{ij}^{gl})\cdot q_{ij}^{gl})|}]\le {\zeta }_0,\hfill \\ q_{ij}^{kl}\ge -p_{ij}^{kl},\hfill \\ q_{ij}^{kl}\le 1-p_{ij}^{kl},\hfill \\ {\displaystyle \sum _{l=1}^{\#h_{ij}^k}{q}_{ij}^{kl}}=0.\hfill \end{array}\right.\end{array}$$

(21)

The first constraint of model (21) indicates that each adjusted preference relation must satisfy acceptable multiplicative consistency, avoiding logical contradictions and improving decision science. The second constraint requires that group preferences need to reach a consensus threshold, which reduces group disagreement and enhances the acceptability of the results. The third and fourth constraints of model (21) denote the upper and lower bounds on the amount of preference strength adjustment, respectively, to respect the DM’s willingness to adjust and prevent overcorrection. Additionally, the last constraint of model (21) ensures that the sum of the adjustments of the preferences within the same HFLE is zero; i.e., the redistribution of weights is “balanced”.

Solving Model (21), the minimum overall adjustment strength and the adjustment amounts $q_{ij}^{kl}$ for the preference strengths $p_{ij}^{kl}$ can be obtained, which in turn leads to a new group of preference relations that satisfy acceptable consistency. If the value of the objective function is zero, and for all $q_{ij}^{kl}=0$, the preference relations $H^k$ satisfy acceptable consistency without any adjustment. If the overall adjustment strength is zero, and there exists $q_{ij}^{kl}\ne 0$, then it means that the preference relations $H^k$ satisfy acceptable consistency, but there exist more than one group of preference relations that satisfy the consistency requirement. When the overall adjustment strength is not zero in value, it means that the preference relations $H^k$ do not satisfy acceptable consistency and need to be adjusted accordingly. And $q_{ij}^{kl}$ represents the adjustment amount of the preference strength $p_{ij}^{kl}$.

4.2. Preference Optimization Models in an Incomplete Hesitant Fuzzy Linguistic Environment

The previous subsection explored the construction of a preference optimization model aimed at minimizing the overall adjustment strength as the objective function, considering DMs’ willingness to adjust in a complete hesitant fuzzy linguistic environment. By maximizing a retention of DMs’ original preference information, the model facilitates preference relations to quickly meet acceptable consistency requirements. Similarly, this subsection delves into preference optimization models in an incomplete hesitant fuzzy linguistic environment

Let $H^k$ be an ICHFLPR established for DM $d_k$ on an alternative set $X=\{x_1,\dots ,x_n\}$, satisfying $0\le {\displaystyle \sum _{l=1}^{\#h_{ij}^k}{p}_{ij}^{kl}}<1$, where $h_{ij}^k=\{(h_{ij}^{kl},p_{ij}^{kl})|l=1,\dots ,\#h_{ij}^k\}$ is an ICHFLE, and other conditions are similar to those in Section 4.1.

When $MCI(H^k)>{\xi }_0$ or $GCI(D)>{\zeta }_0$ exists, an adjustment to the preference strength $p_{ij}^{kl}$ is necessary; otherwise, no adjustment is needed. Let ${\tilde{H}}^k={({\tilde{h}}_{ij}^k)}_{n\times n}$ represent the adjusted preference relations, where ${\tilde{h}}_{ij}^k=\{(h_{ij}^{kl},p_{ij}^{kl}+q_{ij}^{kl})|l=1,\dots ,\#h_{ij}^k\}$, $q_{ij}^{kl}$ de-notes the amounts of adjustment for the preference strengths $p_{ij}^{kl}$, and

$$\left\{\begin{array}{l}{q}_{ij}^{k1}\in [-p_{ij}^{k1}-(1-\lambda )\cdot (1-{\displaystyle \sum _{l=1}^{\#h_{ij}^k}{p}_{ij}^{kl}}),1-p_{ij}^{k1}-(1-\lambda )\cdot (1-{\displaystyle \sum _{l=1}^{\#h_{ij}^k}{p}_{ij}^{kl}})],\hfill \\ q_{ij}^{kl}\in [-p_{ij}^{kl},1-p_{ij}^{kl}],l=2,\dots ,\#h_{ij}^k-1,\hfill \\ q_{ij}^{k\#h_{ij}^k}\in [-p_{ij}^{k\#h_{ij}^k}-\lambda \cdot (1-{\displaystyle \sum _{l=1}^{\#h_{ij}^k}{p}_{ij}^{kl}}),1-p_{ij}^{k\#h_{ij}^k}-\lambda \cdot (1-{\displaystyle \sum _{l=1}^{\#h_i^k}{p}_{ij}^{kl}})],\hfill \\ {\displaystyle \sum _{l=1}^{\#h_{ij}^k}{q}_{ij}^{kl}}=0.\hfill \end{array}\right.$$

(22)

The adjustment strength ${\theta }_{ij}^k$ of DM $d_k$ in linguistic preference $h_{ij}^k$ can be calculated by

$${\theta }_{ij}^k={\displaystyle \sum _{l=1}^{\#h_{ij}^k}(N(h_{ij}^{kl})\cdot q_{ij}^{kl})}$$

(23)

With the help of Equations (11) and (23), the expected preference value of linguistic preference ${\tilde{h}}_{ij}^k$ is obtained:

$$E({\tilde{h}}_{ij}^k)=e_{ij}^k+{\theta }_{ij}^k$$

(24)

Similarly, making the adjustment strength $h_{ij}^k$ for all DMs in linguistic preference accounted for obtaining the overall adjustment strength. Therefore, there is a model that could to be constructed by minimizing the overall adjustment strength as follows:

$$\begin{array}{l}\min {\displaystyle \sum _{k=1}^m{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^n|{\theta }_{ij}^k|}}}\\ \mathrm{s}.\mathrm{t}.\left\{\begin{array}{l}{\displaystyle \frac{2}{n(n-1)}}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1,j>i}^n|E({\tilde{h}}_{ij}^k)-{\overline{e}}_{ij}^k|}}\le {\xi }_0,\hfill \\ {\displaystyle \frac{1}{m(m-1)}}{\displaystyle \sum _{k=1}^m{\displaystyle \sum _{g=1,g\ne k}^m[\frac{2}{n(n-1)}}}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1,j>i}^n|E({\tilde{h}}_{ij}^k)-E({\tilde{h}}_{ij}^g)|}}]\le {\zeta }_0,\hfill \\ q_{ij}^{k1}\in [-p_{ij}^{k1}-(1-\lambda )\cdot (1-{\displaystyle \sum _{l=1}^{\#h_{ij}^k}{p}_{ij}^{kl}}),1-p_{ij}^{k1}-(1-\lambda )\cdot (1-{\displaystyle \sum _{l=1}^{\#h_{ij}^k}{p}_{ij}^{kl}})],\hfill \\ q_{ij}^{kl}\in [-p_{ij}^{kl},1-p_{ij}^{kl}],l=2,\dots ,\#h_{ij}^k-1,\hfill \\ q_{ij}^{k\#h_{ij}^k}\in [-p_{ij}^{k\#h_{ij}^k}-\lambda \cdot (1-{\displaystyle \sum _{l=1}^{\#h_{ij}^k}{p}_{ij}^{kl}}),1-p_{ij}^{k\#h_{ij}^k}-\lambda \cdot (1-{\displaystyle \sum _{l=1}^{\#h_i^k}{p}_{ij}^{kl}})],\hfill \\ {\displaystyle \sum _{l=1}^{\#h_{ij}^k}{q}_{ij}^{kl}}=0.\hfill \end{array}\right.\end{array}$$

(25)

Model (25) could be translated as follows by using Equations (11), (23), and (24):

$$\begin{array}{l}\min {\displaystyle \sum _{k=1}^m{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1,j>i}^n|{\displaystyle \sum _{l=1}^{\#h_{ij}^k}(N(h_{ij}^{kl})\cdot q_{ij}^{kl})}|}}}\\ \mathrm{s}.\mathrm{t}.\left\{\begin{array}{l}{\displaystyle \frac{2}{n(n-1)}}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1,j>i}^n|{\displaystyle \sum _{l=1}^{\#h_{ij}^k}(N(h_{ij}^{kl})\cdot p_{ij}^{kl})}+{\displaystyle \sum _{l=1}^{\#h_{ij}^k}(N(h_{ij}^{kl})\cdot q_{ij}^{kl})}-{\overline{e}}_{ij}^k|\le {\xi }_0,}}\hfill \\ {\displaystyle \frac{1}{m(m-1)}}{\displaystyle \sum _{k=1}^m{\displaystyle \sum _{g=1,g\ne k}^m[\frac{2}{n(n-1)}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1,j>i}^n|{\displaystyle \sum _{l=1}^{\#h_{ij}^k}(N(h_{ij}^{kl})\cdot p_{ij}^{kl})}+{\displaystyle \sum _{l=1}^{\#h_{ij}^k}(N(h_{ij}^{kl})\cdot q_{ij}^{kl})}-}}}}\hfill \\ {\displaystyle \sum _{l=1}^{\#h_{ij}^g}(N(h_{ij}^{gl})\cdot p_{ij}^{ql})}-{\displaystyle \sum _{l=1}^{\#h_{ij}^g}(N(h_{ij}^{gl})\cdot q_{ij}^{gl})}]\le {\zeta }_0,\hfill \\ q_{ij}^{k1}\in [-p_{ij}^{k1}-(1-\lambda )\cdot (1-{\displaystyle \sum _{l=1}^{\#h_{ij}^k}{p}_{ij}^{kl}}),1-p_{ij}^{k1}-(1-\lambda )\cdot (1-{\displaystyle \sum _{l=1}^{\#h_{ij}^k}{p}_{ij}^{kl}})],\hfill \\ q_{ij}^{kl}\in [-p_{ij}^{kl},1-p_{ij}^{kl}],l=2,\dots ,\#h_{ij}^k-1,\hfill \\ q_{ij}^{k\#h_{ij}^k}\in [-p_{ij}^{k\#h_{ij}^k}-\lambda \cdot (1-{\displaystyle \sum _{l=1}^{\#h_{ij}^k}{p}_{ij}^{kl}}),1-p_{ij}^{k\#h_{ij}^k}-\lambda \cdot (1-{\displaystyle \sum _{l=1}^{\#h_i^k}{p}_{ij}^{kl}})],\hfill \\ {\displaystyle \sum _{l=1}^{\#h_{ij}^k}{q}_{ij}^{kl}}=0.\hfill \end{array}\right.\end{array}$$

(26)

The first and second constraints of model (26) have the same meanings as expressed by the first two constraints of model (21), respectively. The third constraint of model (26) denotes a lower and upper limits on adjustment amount $q_{ij}^{k1}$ for the minimum preference strength $p_{ij}^{k1}$; meanwhile, the fourth constraint denotes a lower and upper limits on adjustment amount $q_{ij}^{kl}$ for the intermediate preference strength $p_{ij}^{kl}$; and, the fifth constraint denotes a lower and upper limits on adjustment amount $q_{ij}^{k\#h_{ij}^k}$ for the maximum preference strength $p_{ij}^{k\#h_{ij}^k}$. The last constraint of model (26) has the same meaning as that of model (21). Furthermore, Model (26) and Model (21) are both single-objective nonlinear programming models with constraints, and the solving process and corresponding result analysis are similar to Model (21).

4.3. Solving Algorithm

Combining the above aspects of the analysis, the following Algorithm 1 considers DMs’ willingness to adjust within a DHFL environment.

Algorithm 1: The new GDM process
Step 1: For a specific GDM problem, determine an LTS $S$, a given alternative set $X=\{x_1,\dots ,x_n\}$, and a DM set $D=\{d_1,\dots$$,d_m\}$. Step 2: The DM establishes a DHFLPR by comparing the alternatives pairwise. Step 3: Set the allocative parameter $\lambda$ and use Equations (9)–(11) to obtain the expected fuzzy matrix $E^k$. Then, use Equation (13) to derive the expected fuzzy matrix ${\overline{E}}^k$ that satisfies multiplicative consistency. Step 4: Set a threshold ${\xi }_0$ for acceptable multiplicative consistency and a threshold ${\zeta }_0$ for acceptable consensus, and then calculate $MCI(H^k)$ and $GCI$ $(D)$ using Equations (14) and (16), respectively. Step 5: If all $MCI(H^k)\le {\xi }_0$ and $GCI(D)\le {\zeta }_0$, go to step 7; else, continue to step 6. Step 6: If $MCI(H^k)>{\xi }_0$ or $GCI(D)>{\zeta }_0$ exists, optimization is conducted using either Model (21) or Model (26) to obtain a new set of preference relations ${\tilde{H}}^k$ that meet the consistency requirement. Step 7: Calculate the expected fuzzy matrix ${\tilde{E}}^k$ using Equations (9) and (10). Step 8: Calculate the priority weight ${\omega }_i^k$ of DM $d_k$ with respect to alternative $x_i$
${\omega }_i^k={\displaystyle \frac{1}{n}}{\displaystyle \sum _{j=1}^n{\tilde{e}}_{ij}^k}$	(27)
Step 9: Calculate the overall priority weight $\omega$ for all alternatives:
$\omega =({\omega }^1,\dots ,{\omega }^m)\times \left[\begin{array}{cccc}{\omega }_1^1& {\omega }_2^1& \cdots & {\omega }_n^1\\ {\omega }_1^2& {\omega }_2^2& \cdots & {\omega }_n^2\\ ⋮& ⋮& \ddots & ⋮\\ {\omega }_1^m& {\omega }_2^m& \cdots & {\omega }_n^m\end{array}\right]$	(28)
where ${\omega }^k$ is a weight for DM $d_k$. Step 10: The result of sorting of the alternatives is obtained according to the overall priority weight $\omega =({\omega }_1,\dots ,{\omega }_n)$.

5. Case Study

5.1. Case Background

A leading manufacturing conglomerate, facing strategic challenges in developing a new market entry plan. Hence, it has conducted a comprehensive market analysis through questionnaires and in-depth interviews to determine the best market strategy. The questionnaire covered a wide range of topics, including the trends in the international market, competitive dynamics, interests of potential partners, and cross-cultural consumer behavior in the market. For the in-depth interviews, the company invited industry experts, in-house employees, and potential partners in international markets to obtain a deeper grasp of their perspectives, experiences, and suggestions. Four alternatives were discussed, including international market expansion, strategic cooperation and alliance, digital transformation, and AI-driven innovation. The company’s senior leadership team has four members, namely, chief executive officer (CEO), chief operating officer (COO), chief marketing officer (CMO), and chief financial officer (CFO), which means that four key executives with different backgrounds and decision-making preferences will work together to develop the ultimate new market strategy.

5.2. Decision-Making Analysis

Step 1: Based on the case context, the alternative set is shown in

Table 2. The alternative set.

${\mathbf{x}}_1$	${\mathbf{x}}_2$	${\mathbf{x}}_3$	${\mathbf{x}}_4$
International market expansion	Strategic cooperation and alliance	Digital transformation	AI-driven innovation

, the LTS is shown in

Table 3. The linguistic term set.

Notation	Meaning
s−5	mostly poor
s−4	extremely poor
s−3	very poor
s−2	poor
s−1	slightly poor
s0	fair
s1	slightly good
s2	good
s3	very good
s4	extremely good
s5	mostly good

, and the DM set $D=\{d_1:CEO,d_2:COO,d_3:CMO,d_4:CFO\}$.

Step 2: The DM $d_k$ provides a DHFLPR $H^k$ based on the specialized knowledge and experience skills acquired.

$$\begin{gathered} H^1=\left[\begin{array}{cccc}\{(s_0,1)\}& \begin{array}{l}\{(s_{-1},0.6),\\ (s_0,0.3),\\ (s_2,0.1)\}\end{array}& \begin{array}{l}\{(s_2,0.3),\\ (s_3,0.7)\}\end{array}& \begin{array}{l}\{(s_{-2},0.8),\\ (s_{-1},0.2)\}\end{array}\\ \begin{array}{l}\{(s_{-2},0.1),\\ (s_0,0.3),\\ (s_1,0.6)\}\end{array}& \{(s_0,1)\}& \begin{array}{l}\{(s_{-3},0.5),\\ (s_1,0.2),\\ (s_4,0.3)\}\end{array}& \begin{array}{l}\{(s_0,0.2),\\ (s_1,0.2),\\ (s_2,0.6)\}\end{array}\\ \begin{array}{l}\{(s_{-3},0.7),\\ (s_{-2},0.3)\}\end{array}& \begin{array}{l}\{(s_{-4},0.3),\\ (s_{-1},0.2),\\ (s_3,0.5)\}\end{array}& \{(s_0,1)\}& \begin{array}{l}\{(s_{-1},0.9),\\ (s_1,0.1)\}\end{array}\\ \begin{array}{l}\{(s_1,0.2),\\ (s_2,0.8)\}\end{array}& \begin{array}{l}\{(s_{-2},0.6),\\ (s_{-1},0.2),\\ (s_0,0.2)\}\end{array}& \begin{array}{l}\{(s_{-1},0.1),\\ (s_1,0.9)\}\end{array}& \{(s_0,1)\}\end{array}\right]H^2=\left[\begin{array}{cccc}\{(s_0,1)\}& \begin{array}{l}\{(s_{-3},0.8),\\ (s_{-2},0.2)\}\end{array}& \begin{array}{l}\{(s_1,0.4),\\ (s_2,0.3),\\ (s_3,0.3)\}\end{array}& \begin{array}{l}\{(s_{-4},0.25),\\ (s_{-2},0.5),\\ (s_0,0.25)\}\end{array}\\ \begin{array}{l}\{(s_2,0.2),\\ (s_3,0.8)\}\end{array}& \{(s_0,1)\}& \begin{array}{l}\{(s_{-1},0.2),\\ (s_0,0.5),\\ (s_1,0.3)\}\end{array}& \begin{array}{l}\{(s_3,0.4),\\ (s_4,0.6)\}\end{array}\\ \begin{array}{l}\{(s_{-3},0.3),\\ (s_{-2},0.3),\\ (s_{-1},0.4)\}\end{array}& \begin{array}{l}\{(s_{-1},0.3),\\ (s_0,0.5),\\ (s_1,0.2)\}\end{array}& \{(s_0,1)\}& \begin{array}{l}\{(s_{-2},0.5),\\ (s_2,0.5)\}\end{array}\\ \begin{array}{l}\{(s_0,0.25),\\ (s_2,0.5),\\ (s_4,0.25)\}\end{array}& \begin{array}{l}\{(s_{-4},0.6),\\ (s_{-3},0.4)\}\end{array}& \begin{array}{l}\{(s_{-2},0.5),\\ (s_2,0.5)\}\end{array}& \{(s_0,1)\}\end{array}\right] \\ {H}^3=\left[\begin{array}{cccc}\{(s_0,1)\}& \begin{array}{l}\{(s_{-2},0.3),\\ (s_2,0.2),\\ (s_4,0.1)\}\end{array}& \begin{array}{l}\{(s_{-4},0.3),\\ (s_{-1},0.6),\\ (s_0,0.1)\}\end{array}& \begin{array}{l}\{(s_{-3},0.2),\\ (s_{-1},0.4)\}\end{array}\\ \begin{array}{l}\{(s_{-4},0.1),\\ (s_{-2},0.2),\\ (s_2,0.3)\}\end{array}& \{(s_0,1)\}& \begin{array}{l}\{(s_{-3},0.7),\\ (s_1,0.3)\}\end{array}& \begin{array}{l}\{(s_{-3},0.1),\\ (s_{-1},0.4),\\ (s_0,0.5)\}\end{array}\\ \begin{array}{l}\{(s_0,0.1),\\ (s_1,0.6),\\ (s_4,0.3)\}\end{array}& \begin{array}{l}\{(s_{-1},0.3),\\ (s_3,0.7)\}\end{array}& \{(s_0,1)\}& \begin{array}{l}\{(s_0,0.4),\\ (s_1,0.6)\}\end{array}\\ \begin{array}{l}\{(s_1,0.4),\\ (s_3,0.2)\}\end{array}& \begin{array}{l}\{(s_0,0.5),\\ (s_1,0.4),\\ (s_3,0.1)\}\end{array}& \begin{array}{l}\{(s_{-1},0.6),\\ (s_0,0.4)\}\end{array}& \{(s_0,1)\}\end{array}\right]H^4=\left[\begin{array}{cccc}\{(s_0,1)\}& \begin{array}{l}\{(s_{-3},0.4),\\ (s_1,0.6)\}\end{array}& \begin{array}{l}\{(s_{-2},0.3),\\ (s_0,0.2),\\ (s_2,0.3)\}\end{array}& \begin{array}{l}\{(s_1,0.6),\\ (s_2,0.3),\\ (s_4,0.1)\}\end{array}\\ \begin{array}{l}\{(s_{-1},0.6),\\ (s_3,0.4)\}\end{array}& \{(s_0,1)\}& \begin{array}{l}\{(s_2,0.5),\\ (s_4,0.5)\}\end{array}& \begin{array}{l}\{(s_{-4},0.8),\\ (s_0,0.2)\}\end{array}\\ \begin{array}{l}\{(s_{-2},0.3),\\ (s_0,0.2),\\ (s_2,0.3)\}\end{array}& \begin{array}{l}\{(s_{-4},0.5),\\ (s_{-2},0.5)\}\end{array}& \{(s_0,1)\}& \begin{array}{l}\{(s_3,0.2),\\ (s_4,0.2)\}\end{array}\\ \begin{array}{l}\{(s_{-4},0.1),\\ (s_{-2},0.3),\\ (s_{-1},0.6)\}\end{array}& \begin{array}{l}\{(s_0,0.2),\\ (s_4,0.8)\}\end{array}& \begin{array}{l}\{(s_{-4},0.2),\\ (s_{-3},0.2)\}\end{array}& \{(s_0,1)\}\end{array}\right] \end{gathered}$$

Step 3: Set the allocative parameter $\lambda =0.5$ and calculate the expected fuzzy matrix $E^k$ using Equations (9)–(11) as shown below.

$$\begin{gathered} E^1=\left[\begin{array}{cccc}0.5& 0.46& 0.77& 0.32\\ 0.54& 0.5& 0.49& 0.62\\ 0.23& 0.51& 0.5& 0.42\\ 0.68& 0.38& 0.58& 0.5\end{array}\right]E^2=\left[\begin{array}{cccc}0.5& 0.22& 0.69& 0.3\\ 0.78& 0.5& 0.51& 0.86\\ 0.31& 0.49& 0.5& 0.5\\ 0.7& 0.14& 0.5& 0.5\end{array}\right] \\ {E}^3=\left[\begin{array}{cccc}0.5& 0.56& 0.32& 0.32\\ 0.44& 0.5& 0.32& 0.43\\ 0.68& 0.68& 0.5& 0.56\\ 0.68& 0.57& 0.44& 0.5\end{array}\right]E^4=\left[\begin{array}{cccc}0.5& 0.44& 0.5& 0.66\\ 0.56& 0.5& 0.8& 0.18\\ 0.5& 0.2& 0.5& 0.85\\ 0.34& 0.82& 0.15& 0.5\end{array}\right] \end{gathered}$$

Continue to calculate the expected fuzzy matrix ${\overline{E}}^k$ by Equation (13) that satisfies multiplicative consistency, and the results are shown below.

$$\begin{gathered} {\overline{E}}^1=\left[\begin{array}{cccc}0.5& 0.488& 0.553& 0.489\\ 0.512& 0.5& 0.564& 0.501\\ 0.447& 0.436& 0.5& 0.437\\ 0.511& 0.499& 0.563& 0.5\end{array}\right]{\overline{E}}^2=\left[\begin{array}{cccc}0.5& 0.392& 0.487& 0.483\\ 0.608& 0.5& 0.596& 0.59\\ 0.513& 0.404& 0.5& 0.495\\ 0.517& 0.41& 0.505& 0.5\end{array}\right] \\ {\overline{E}}^3=\left[\begin{array}{cccc}0.5& 0.501& 0.413& 0.437\\ 0.499& 0.5& 0.411& 0.436\\ 0.587& 0.589& 0.5& 0.525\\ 0.563& 0.564& 0.475& 0.5\end{array}\right]{\overline{E}}^3=\left[\begin{array}{cccc}0.5& 0.507& 0.506& 0.537\\ 0.493& 0.5& 0.499& 0.53\\ 0.494& 0.501& 0.5& 0.531\\ 0.463& 0.47& 0.469& 0.5\end{array}\right] \end{gathered}$$

Step 4: Set ${\xi }_0=0.1$ and ${\zeta }_0=0.15$, and calculate $MCI(H^k)$ and $GCI(D)$ using Equations (14) and (16), respectively. After the calculation, we have $MCI(H^1)=0.107,MCI(H^2)=$$0.153,MCI(H^3)=0.067,MCI(H^4)=0.194$, so there exist preference relations that do not satisfy acceptable multiplicative consistency, and $GCI(D)=0.134$, so the decision group satisfies acceptable consensus.

Step 5: Model (26) is then employed to derive a new set of preference relations ${\tilde{H}}^k$ that satisfy the consistency requirement.

$$\begin{gathered} {\tilde{H}}^1=\left[\begin{array}{cccc}\{(s_0,1)\}& \begin{array}{l}\{(s_{-1},0.26),\\ (s_0,0.44),\\ (s_2,0.3)\}\end{array}& \begin{array}{l}\{(s_2,0.55),\\ (s_3,0.45)\}\end{array}& \begin{array}{l}\{(s_{-2},0.64),\\ (s_{-1},0.36)\}\end{array}\\ \begin{array}{l}\{(s_{-2},0.3),\\ (s_0,0.44),\\ (s_1,0.26)\}\end{array}& \{(s_0,1)\}& \begin{array}{l}\{(s_{-3},0.4),\\ (s_1,0.38),\\ (s_4,0.22)\}\end{array}& \begin{array}{l}\{(s_0,0.41),\\ (s_1,0.33),\\ (s_2,0.26)\}\end{array}\\ \begin{array}{l}\{(s_{-3},0.45),\\ (s_{-2},0.55)\}\end{array}& \begin{array}{l}\{(s_{-4},0.22),\\ (s_{-1},0.38),\\ (s_3,0.4)\}\end{array}& \{(s_0,1)\}& \begin{array}{l}\{(s_{-1},0.77),\\ (s_1,0.23)\}\end{array}\\ \begin{array}{l}\{(s_1,0.36),\\ (s_2,0.64)\}\end{array}& \begin{array}{l}\{(s_{-2},0.26),\\ (s_{-1},0.33),\\ (s_0,0.41)\}\end{array}& \begin{array}{l}\{(s_{-1},0.23),\\ (s_1,0.77)\}\end{array}& \{(s_0,1)\}\end{array}\right]{\tilde{H}}^2=\left[\begin{array}{cccc}\{(s_0,1)\}& \begin{array}{l}\{(s_{-3},0.71),\\ (s_{-2},0.29)\}\end{array}& \begin{array}{l}\{(s_1,0.54),\\ (s_3,0.46)\}\end{array}& \begin{array}{l}\{(s_{-4},0.28),\\ (s_{-2},0.41),\\ (s_0,0.31)\}\end{array}\\ \begin{array}{l}\{(s_2,0.29),\\ (s_3,0.71)\}\end{array}& \{(s_0,1)\}& \begin{array}{l}\{(s_{-1},0.27),\\ (s_0,0.39),\\ (s_1,0.34)\}\end{array}& \begin{array}{l}\{(s_3,0.58),\\ (s_4,0.42)\}\end{array}\\ \begin{array}{l}\{(s_{-3},0.46),\\ (s_{-1},0.54)\}\end{array}& \begin{array}{l}\{(s_{-1},0.34),\\ (s_0,0.39),\\ (s_1,0.34)\}\end{array}& \{(s_0,1)\}& \begin{array}{l}\{(s_{-2},0.66),\\ (s_2,0.34)\}\end{array}\\ \begin{array}{l}\{(s_0,0.31),\\ (s_2,0.41),\\ (s_4,0.28)\}\end{array}& \begin{array}{l}\{(s_{-4},0.42),\\ (s_{-3},0.58)\}\end{array}& \begin{array}{l}\{(s_{-2},0.34),\\ (s_2,0.66)\}\end{array}& \{(s_0,1)\}\end{array}\right] \\ {\tilde{H}}^3=\left[\begin{array}{cccc}\{(s_0,1)\}& \begin{array}{l}\{(s_{-2},0.33),\\ (s_2,0.25),\\ (s_4,0.42)\}\end{array}& \begin{array}{l}\{(s_{-4},0.47),\\ (s_{-1},0.4),\\ (s_0,0.13)\}\end{array}& \begin{array}{l}\{(s_{-3},0.52),\\ (s_{-1},0.48)\}\end{array}\\ \begin{array}{l}\{(s_{-4},0.42),\\ (s_{-2},0.25),\\ (s_2,0.33)\}\end{array}& \{(s_0,1)\}& \begin{array}{l}\{(s_{-3},0.74),\\ (s_1,0.26)\}\end{array}& \begin{array}{l}\{(s_{-3},0.21),\\ (s_{-1},0.33),\\ (s_0,0.46)\}\end{array}\\ \begin{array}{l}\{(s_0,0.13),\\ (s_1,0.4),\\ (s_4,0.47)\}\end{array}& \begin{array}{l}\{(s_{-1},0.26),\\ (s_3,0.74)\}\end{array}& \{(s_0,1)\}& \begin{array}{l}\{(s_0,0.32),\\ (s_1,0.68)\}\end{array}\\ \begin{array}{l}\{(s_1,0.48),\\ (s_3,0.52)\}\end{array}& \begin{array}{l}\{(s_0,0.46),\\ (s_1,0.33),\\ (s_3,0.21)\}\end{array}& \begin{array}{l}\{(s_{-1},0.68),\\ (s_0,0.32)\}\end{array}& \{(s_0,1)\}\end{array}\right]{\tilde{H}}^4=\left[\begin{array}{cccc}\{(s_0,1)\}& \begin{array}{l}\{(s_{-3},0.33),\\ (s_1,0.67)\}\end{array}& \begin{array}{l}\{(s_{-2},0.27),\\ (s_0,0.28),\\ (s_2,0.45)\}\end{array}& \begin{array}{l}\{(s_2,0.39),\\ (s_4,0.61)\}\end{array}\\ \begin{array}{l}\{(s_{-1},0.67),\\ (s_3,0.33)\}\end{array}& \{(s_0,1)\}& \begin{array}{l}\{(s_2,0.88),\\ (s_4,0.12)\}\end{array}& \begin{array}{l}\{(s_{-4},0.44),\\ (s_0,0.56)\}\end{array}\\ \begin{array}{l}\{(s_{-2},0.45),\\ (s_0,0.28),\\ (s_2,0.45)\}\end{array}& \begin{array}{l}\{(s_{-4},0.12),\\ (s_{-2},0.88)\}\end{array}& \{(s_0,1)\}& \begin{array}{l}\{(s_3,0.72),\\ (s_4,0.28)\}\end{array}\\ \begin{array}{l}\{(s_{-4},0.61),\\ (s_{-2},0.39)\}\end{array}& \begin{array}{l}\{(s_0,0.56),\\ (s_4,0.44)\}\end{array}& \begin{array}{l}\{(s_{-4},0.28),\\ (s_{-3},0.72)\}\end{array}& \{(s_0,1)\}\end{array}\right] \end{gathered}$$

Step 6: To derive the expected fuzzy matrix ${\tilde{E}}^k$ from Equations (9) and (10), and the results of the calculation are as follows.

$$\begin{gathered} {\tilde{E}}^1=\left[\begin{array}{cccc}0.5& 0.534& 0.745& 0.336\\ 0.466& 0.5& 0.506& 0.585\\ 0.255& 0.494& 0.5& 0.446\\ 0.664& 0.415& 0.554& 0.5\end{array}\right]{\tilde{E}}^2=\left[\begin{array}{cccc}0.5& 0.229& 0.692& 0.306\\ 0.771& 0.5& 0.507& 0.842\\ 0.308& 0.493& 0.5& 0.436\\ 0.694& 0.158& 0.564& 0.5\end{array}\right] \\ {\tilde{E}}^3=\left[\begin{array}{cccc}0.5& 0.625& 0.272& 0.348\\ 0.375& 0.5& 0.378& 0.404\\ 0.728& 0.622& 0.5& 0.568\\ 0.652& 0.596& 0.432& 0.5\end{array}\right]{\tilde{E}}^4=\left[\begin{array}{cccc}0.5& 0.468& 0.536& 0.822\\ 0.532& 0.5& 0.724& 0.324\\ 0.464& 0.276& 0.5& 0.828\\ 0.178& 0.676& 0.172& 0.5\end{array}\right] \end{gathered}$$

Step 7: Calculate the priority weight ${\omega }_i^k$ of alternative $x_i$ for DM $d_k$ on the basis of Equation (27), and the findings are presented in

Table 4. Priority weights.

${\mathbf{\omega }}_{\mathbf{i}}^{\mathbf{k}}$	${\mathbf{x}}_1$	${\mathbf{x}}_2$	${\mathbf{x}}_3$	${\mathbf{x}}_4$
$d_1$	0.529	0.509	0.424	0.533
$d_2$	0.432	0.655	0.434	0.479
$d_3$	0.436	0.414	0.605	0.545
$d_4$	0.582	0.52	0.517	0.382

.

Step 8: Set the weights of the DMs as 0.3, 0.2, 0.3, and 0.2, respectively, then calculate overall priority weight $\omega$ for all the alternatives using Equation (28).

$$\begin{array}{c}\omega =\left(0.3,0.2,0.3,0.2\right)\times \left(\begin{array}{cccc}0.529& 0.509& 0.424& 0.533\\ 0.432& 0.655& 0.434& 0.479\\ 0.436& 0.414& 0.605& 0.545\\ 0.582& 0.52& 0.517& 0.382\end{array}\right)\\ =\left(0.492,0.512,0.499,0.496\right)\end{array}$$

Step 9: After sorting, it can be observed that alternative $x_2$ corresponds to the highest priority weight. Therefore, strategic cooperation and alliance is deemed the optimal alternative.

5.3. Comparative Analysis

(1)

Comparison of alternative ranking with different consensus thresholds

To validate the rationality of the consensus threshold (${\zeta }_0=0.15$) chosen in this paper, we tested the effect of multiple consensus thresholds on the ranking of alternatives. Based on the same case (market strategy choice) as above with the initial preference data, other parameters (e.g., multiplicative consistency thresholds) are fixed, and only the consensus thresholds are adjusted to observe the stability of the ranking of alternatives.

Table 5. Ranking of alternatives with different consensus thresholds.

Consensus Threshold	Ranking of Alternatives
0.1	$x_3\succ x_4\succ x_1\succ x_2$
0.15	$x_2\succ x_3\succ x_4\succ x_1$
0.2	$x_2\succ x_3\succ x_4\succ x_1$
0.25	$x_2\succ x_3\succ x_4\succ x_1$
0.3	$x_2\succ x_1\succ x_3\succ x_4$

gives the results of ranking the alternatives under different consensus thresholds.

When the consensus threshold is set to a strict threshold (${\zeta }_0=0.1$), the ranking of alternatives is significantly different from the results of other thresholds ($x_3$ is the optimal alternative), resulting in distortion of the original information due to the high consensus requirement forcing a significant adjustment of preferences. When the consensus threshold is set to the medium threshold (${\zeta }_0=0.15,0.2,0.25$), the alternative ranking results are consistent, indicating that the threshold change in this interval does not trigger ranking fluctuations and the system is robust. When the consensus threshold is set to a relaxed threshold (${\zeta }_0=0.3$), the ranking of alternatives changes ($x_1$ priority rises), as the low consensus requirement allows for more original preferences to be retained, but may mask key conflicts (e.g., the potential risk of $x_1$).

In summary, there are three advantages of choosing ${\zeta }_0=0.15$ as the consensus threshold in this paper. First, the consensus threshold in this paper is located at the critical point of the sorting stability interval, which preserves the original preference information of DMs and ensures the effective integration of group opinions. Second, compared to strict and loose thresholds, the threshold in this paper achieves an optimal balance between the strength of the adjustment and the reliability of the results, which is suitable for most scenarios. Third, the ranked consistency of the medium threshold intervals suggests that the thresholds in this paper are not isolated optima, but represent a class of feasible thresholds, further supporting their generalizability.

(2)

Comparison of the ranking of alternatives using different methods

In order to verify the effectiveness of our new method, two other extant methods are further analyzed, that is, the traditional iterative adjustment method [39] and the probabilistic hesitant fuzzy optimization method [15], for ranking the afore-mentioned alternatives (i.e., the selection of the best market strategy), and the results are shown in

Table 6. Ranking of alternatives with different methods.

Method	Ranking of Alternatives	Optimal Alternative	Worst Alternative
This paper	$x_2\succ x_3\succ x_4\succ x_1$	$x_2$	$x_1$
Ref. [39]	$x_4\succ x_3\succ x_2\succ x_1$	$x_4$	$x_1$
Ref. [15]	$x_2\succ x_4\succ x_3\succ x_1$	$x_2$	$x_1$

.

The methods either in this paper or introduced in Ref. [15] both chose $x_2$ as the optimal alternative, indicating consistency in the ability to express complex preferences in a distributed language environment with a probabilistic optimization framework. However, Ref. [39] adopted $x_4$ as the optimal alternative due to unconstrained iteration, and its ranking results deviate significantly from other methods, reflecting its tendency to overcorrect. The method in this paper prioritizes $x_3$ over $x_4$, which stems from the limitations of the adjustment willingness constraint on the range of preference strength adjustment to avoid over-modification of the original fuzzy semantics. All methods ranked $x_1$ as the worst alternative, validating the internal consistency of the input data and further supporting the reliability of the comparison experiment.

Through comparative experiments, the method in this paper demonstrates the following three advantages in ranking alternatives. First, the preference distortion mitigation of traditional iterative approaches is suppressed by constraining the adjustment range to ensure the consistency of the ranking logic. Second, distributed linguistic modeling addresses the limitations of the rigid uniform distribution assumption and precisely captures the heterogeneous importance of linguistic values. Third, the lightweight optimization model avoids complex iteration or planning process, and the computational efficiency is improved for real-time decision-making needs.

6. Conclusions

To overcome the limitations of HFLPRs in portraying DMs’ preferences in complex decision problems, this paper extends the DHFLPRs. The preference relation maintains the fuzziness and hesitancy of the HFLPR for voicing the DM’s preference, but also addresses the issue of varying degrees of importance of different linguistic preference values within the same HFLE. Additionally, in order to address the problems arising from traditional iterative methods ignoring the DM’s willingness to adjust when adjusting preference information that does not meet consistency requirements, as well as the complexity and computational burden of the adjustment process. This paper constructs a preference optimization model and a GDM method by minimizing the overall adjustment strength as the objective function in a DHFL environment, while considering DMs’ willingness to adjust. This approach aims to maximize the conservation of DMs’ original preference information while allowing the preference relations to expedite predetermined consistency requirements.

Despite these advancements, the following directions can be further explored in the future. Combining dynamic preference modeling to analyze time-series decision-making scenarios, using distributed computing technology to scale up to large-scale groups, and integrating multimodal data (e.g., text, images) with non-cooperative behavioral game mechanisms to enhance the applicability of the method in high-complexity scenarios such as medical diagnosis and disaster emergency response. At the same time, lightweight decision support tools are developed to promote the transformation of theoretical results into practical applications.