Linear uncertain extensions of the minimum cost consensus model based on uncertain distance and consensus utility

Highlights

• An equivalent analytical formula of linear uncertain distance measure is proposed.

• A consensus utility function describing the degree of opinion aggregation is defined.

• In consensus, opinions are generalized from real numbers to uncertain distributions.

• Three new models based on uncertain distance and consensus utility are proposed.

Abstract

Uncertainty theory adopts the belief degree and uncertainty distribution to ensure good alignment with a decision-maker’s uncertain preferences, making the final decisions obtained from the consensus-reaching process closer to the actual decision-making scenarios. Under the constraints of the uncertain distance measure and consensus utility, this article explores the minimum-cost consensus model under various linear uncertainty distribution-based preferences. First, the uncertain distance is used to measure the deviation between individual opinions and the consensus through uncertainty distributions. A nonlinear analytical formula is derived to avoid the computational complexity of integral and piecewise function operations, thus reducing the calculation cost of the uncertain distance measure. The consensus utility function defined in this article characterizes the adjustment value and degree of aggregation of individual opinions. Three new consensus models are constructed based on the consensus utility and linear uncertainty distribution. The results show that, in complex group decision-making contexts, the uncertain consensus models are more flexible than traditional minimum-cost consensus models: compared with the high volatility of the adjusted opinions in traditional deterministic consensus models with crisp number-based preferences, the variation trends of both individual adjusted opinions and the collective opinion with a linear uncertainty distribution are much smoother and the fitting range is closer to reality. The introduction of the consensus utility not only reflects the relative changes of individual opinions, but also accounts for individual psychological changes during the opinion-adjustment process. Most importantly, it reduces the cost per unit of consensus utility, facilitates the determination of the optimal threshold for the consensus utility, and improves the efficiency of resource allocation.

Introduction

The consumption of various resources is an important foundation for mankind’s socio-economic development. Because stakeholders with ample resources generally have absolute power in their decision making, the rational allocation of limited resources becomes an important proposition on fairness and development for human society. For example, with the recent coronavirus (COVID-19) pandemic, selective treatment caused by shortages of medical resources led to serious issues concerning social equity and social values. Research on resource allocation optimization is a hot topic, and there is no exception in the realm of group decision-making (GDM) involving experts, with their own preferences or attitudes, who are aiming to identify a common agreement from among several alternatives or feasible solutions [1], [2], [3], [4], [5]. Such common solutions can be regarded as a consensus, which is generally reached through a consensus-reaching process (CRP) [6], [7], [8] among different decision-makers (DMs). In fact, the notion of consensus can range from a full and unanimous agreement (i.e., hard consensus) to some more flexible circumstances with different degrees of partial agreement (i.e., soft consensus) [7], [9], [10], [11]. In real-world GDM, reaching a hard consensus is uneconomical and rather unlikely [10]. To the best of our knowledge, soft consensus is the mainstream target in research on consensus decision-making.

Practically, shifting DMs’ opinions towards a point of mutual consent often necessitates laborious, dynamic, and interactive negotiation and compromise, which takes time, requires effort and large amounts of resources, and is usually coordinated and supervised by a moderator [2], [10], [12], [13]. Thus, the minimum-cost consensus model (MCCM) has been proposed by Ben-Arieh and Easton [1] and Zhang et al. [14] to deal with single- and multi-criteria decision-making problems with a linear cost, where the aim is to obtain the optimal values of experts’ adjusted opinions and the collective opinion. Later, Ben-Arieh et al. [2] adopted a quadratic cost function to reach a consensus with a minimum cost and determine the maximum number of experts reaching a consensus under a limited budget. Dong et al. [15] initiated a minimum adjustment consensus model (MACM), and Zhang et al. [14] presented a generalized model with aggregation operators, which is a connection between MACM and MCCM. Subsequently, researchers have explored MCCM from diverse perspectives, such as the minimum cost–maximum return (i.e., primal–dual modeling) perspective [12], [13], [16], [17], unit cost optimization perspective [18], [19], [20], [21], preference relations and group consensus issues [22], [23], [24], [25], [26], and under the complex GDM context [27], [28], [29], [30], [31], [32], [33]; for a systematic literature review, see Zhang et al. [34]. Basically, the DMs’ preferences reported in the literature on MCCM are mostly represented by crisp numbers or intervals, neglecting the implicit complex distribution features of their opinions. This results in some information loss and the problem of distortion.

Besides the consensus cost, assessing the efficiency of CRPs is an important issue in GDM. Zhang et al. [7] propose the following criteria to compare the efficiency of CRPs: number of adjusted DMs, number of adjusted alternatives, number of adjusted preference values, the distance between the original and the adjusted preference information, and the number of consensus rounds. In this article, we mainly focus on the distance between the original and the adjusted preference information. In general, there are two commonly used methods of measuring the distance: the first considers the distance towards the consensus using the Manhattan distance [1], [12], [16], [20] or Euclidean distance [2], [20], [21], while the second is based on measures of the similarity or dissimilarity among DMs [4], [35], [36], [37]. These measures are defined from a mathematical perspective, and correspond to an absolute measure of decision efficiency. In reality, however, individual psychological changes (e.g., characterized as utility) in CRPs should also be considered. Without loss of generality, utility refers to a specific value assigned by a DM to a particular result according to his/her own preferences [38]. Because there are significant differences in both DMs (e.g., individual knowledge, experience, skills, abilities) and decision-making environments [39], [40], [41], individuals are likely to exhibit distinct attitudes and satisfaction levels in different GDM problems. Moreover, the monotonicity and concavity/convexity of utility functions are of great importance in reflecting the DMs’ preference structures [42]. In practice, utility can be represented by the membership degree proposed in Zadeh’s fuzzy theory [43], which ranges from zero to one. Much of the extant research [44], [45], [46] shows that depicting utility in the form of the membership degree can accurately reflect DMs’ subjective judgments. Therefore, referring to the connotation of membership degree, the consensus utility function is proposed in this article to assess the efficiency of CRPs. This function is computed based on the final collective opinion and the amount of adjustment to individual opinions. Compared with previous decision efficiency measures, our proposed method reflects the relative change in individual opinions, and also takes the DMs’ psychological changes into account during the opinion-adjustment process.

Originating from probability theory, the foundations of uncertainty theory were developed by Liu [47] to solve problems associated with events where the frequency does not necessarily correspond to the probability. For instance, when considering events with a low frequency of occurrence and high degree of uncertainty (e.g., disaster risk analysis, financial risk analysis), the reliability of the event happening is mainly evaluated by domain experts, and so the results are strongly subjective. More specifically, the advantages and characteristics of uncertainty theory compared with other theories dealing with uncertain events (e.g., probability theory, possibility theory, and Dempster–Shafer evidence theory) were identified by Gong et al. [48]. To date, uncertainty theory has achieved considerable success in practical applications [47], [49], [50], [51], [52], [53]. Moreover, Gong et al. [54] explored minimum cost consensus modeling under various linear uncertain-constrained scenarios. By introducing Liu’s theory into social network modeling, Gong et al. [55] quantified trust chains and whole networks by developing a multi-trust transitive aggregation model based on criteria associated with path length and trust quality. To cope with the information loss and distortion problems in interval operations, Guo et al. [56] explored ranking problems with incomplete linear uncertain preference relations by adopting a belief degree and an inverse uncertainty distribution. Although uncertainty theory can successfully simulate the DM’s preference, there has been little consensus research incorporating Liu’s uncertainty theory. Because real-life GDM problems are uncertain and changeable, and DMs’ judgments typically include complex implicit information, adopting the belief degree and uncertainty distribution to simulate DMs’ preferences is more in line with the DMs’ real preference expressions during consensus modeling. For example, when estimating whether rain will fall in a certain area, due to the influence of terrain, area and data, the probability, size and range of rainfall can only be judged by experts in the field, who may have different experiences and styles (e.g. optimistic and pessimistic), and can only give the belief degree of rainfall and the distribution of precipitation. The specific form is interval-type latitude and longitude data, which corresponds to linear uncertainty distribution.

The approach proposed in this paper aims to further promote the traditional MCC modeling theory [1], [14]. Specifically, the goal of this article is to introduce uncertainty theory into research on the MCCM. To focus our discussion on the essentials, the basic consensus mechanism of previous MCCMs is retained: (1) the rectilinear distance is used to quantify the change in individual opinions (i.e., opinion deviation between a DM’s adjusted opinion and their initial opinion) or to judge whether the collective opinion is within an acceptable range (i.e., the opinion deviation between the DM’s adjusted opinion and the collective opinion is within a predetermined threshold); (2) weighted averaging (WA) operators [57] are employed to obtain the collective opinion by aggregating the DMs’ adjusted opinions; (3) consensus should be realized through CRPs with the minimum total consensus cost. In traditional MCCMs [1], [12], [13], [14], [16], DMs’ preferences are expressed primarily by crisp numbers, disregarding the situation that DMs’ judgments are interval values or with specific distribution characteristics, while people can only give the approximate range of the evaluation opinions in an uncertain environment, so it is more reasonable to fit their opinions by adopting the data in interval type or the distribution; in addition, the efficiency of GDM is assessed by opinion deviations, neglecting the variation trends of individual satisfaction or utility during the opinion-adjustment process, while the use of consensus utility can specifically reflect the degree of individual acceptance of the collective opinion, which is of great significance to reach soft consensus. To overcome these shortcomings and achieve these goals, this article explores the MCC modeling and its extended problems under the constraints of an uncertain distance measure and consensus utility, where the DMs’ preferences obey a linear uncertainty distribution. The main contributions and originality of this proposal can be summarized as follows:

• On account of the characteristics of uncertainty distributions, this article adopts a linear uncertainty distribution to fit individual preferences, and proposes a new kind of distance measure to compute the deviation between individual adjusted opinions and initial opinions (or the collective opinion derived from CRPs). Our new measure reduces the computational cost caused by implicit integral or stochastic operations in the uncertain distance measure used in Liu’s uncertainty theory.

• Previous decision efficiency measures are calculated based on distances to the collective opinion or the similarity/dissimilarity among experts’ preferences. This article takes individual psychological utility (satisfaction level of adjusting initial opinions) into account, and develops a new definition of the consensus utility in the form of a quadratic function of the individual initial opinion, adjusted opinion, and collective opinion. The proposed efficiency measure reflects the adjustment value and the degree of aggregation of DMs’ opinions in GDM, ensuring that the final decisions simultaneously satisfy the realistic requirements of low cost and high consensus utility.

• Following the research idea of the basic MCCM with crisp number preferences, this article develops three new consensus models, namely a linear uncertain consensus model (LUCM), consensus utility-based MCCM ($\gamma$-MCCM), and consensus utility-based LUCM ($\gamma$-LUCM). These four consensus models are compared and analyzed through a detailed discussion of their relationships and transformation methods, as well as some numerical examples.

The remainder of this article is structured as follows. Section 2 reviews some preliminaries on uncertainty theory, presents a general construction framework of traditional MCCMs, and analyzes the linkage between MCCM and MACM. Considering that DMs’ opinions are fitted using a linear uncertainty distribution, Section 3 describes an uncertain distance that measures the deviation between DMs’ linear uncertain opinions, and builds an initial MCCM with linear uncertain variables (i.e., LUCM). By incorporating the DM’s psychological utility, a new definition of consensus utility in the form of a quadratic function of the DM’s initial opinion, adjusted opinion, and the collective opinion is derived in Section 4. Consensus utility-based MCCM/LUCM (i.e., $\gamma$-MCCM and $\gamma$-LUCM) are also proposed in this section. Section 5 presents the results of numerical examples used to analyze the four consensus models considered in this paper. Then these four models are discussed in Section 6. Finally, the conclusions to this study and future research directions are elaborated in Section 7.