A Linguistic Information Granulation Model and Its Penalty Function-Based Co-Evolutionary PSO Solution Approach for Supporting GDM with Distributed Linguistic Preference Relations
Introduction
Decision-making problems are usually conducted through group decision-making (GDM) processes [5,11,26,30,32,37,44]. When addressing GDM problems, preference relations show sound performance when representing the outcomes of pairwise comparisons coming from decision-makers (DMs) [36]. In most real-world GDM scenarios, DMs cannot accurately express their preferences through numerical values given the complexity and uncertainty of problems themselves and the ambiguity inherent to human thinking. In such scenarios, linguistic variables [55], whose values are words instead of numbers, provide a flexible tool for DMs to express their preferences. As such, various linguistic expressions have been proposed to support GDM over the past decades, such as the linguistic 2-tuples [33], the hesitant fuzzy linguistic term sets (HFLTSs) [5], the probabilistic linguistic term sets (PLTSs) [39], and the linguistic distributions (LDs) [50,56,57]. Compared with linguistic 2-tuples, the latter three linguistic expressions (HFLTSs, PLTSs, and LDs) improve the flexibility of expression by allowing DMs to use multiple linguistic terms instead of a single one to express their preferences. Specifically, PLTSs and LDs are two different names for a similar concept [39], and HFLTSs quantitatively characterize the hesitant preferences of DMs by using several consecutive linguistic terms. Different from HFLTSs, certain symbolic proportion information over consecutive linguistic terms is provided by LDs to describe distributed preferences of DMs as distributed assessments. In recent years, based on the concepts of HFLTSs and LDs, the proportional HFLTSs [9] and the proportional interval type-2 HFLTSs [10] were proposed.
When such linguistic expressions as the HFLTSs and PLTSs are applied to record the outcomes of pairwise comparisons, the linguistic term-based preference relations, namely, the hesitant fuzzy linguistic preference relations [60] and the probabilistic linguistic preference relations [59] have been introduced. Specifically, Zhang et al. [57] proposed the distributed linguistic preference relations (DLPRs) with the use of LDs. It is evident that DLPRs are suitable for modeling the uncertainty and complexity of preference information coming from DMs in complex linguistic decision-making problems [[45], [50]]. DLPRs not only allows DMs to express their preference information using multiple linguistic terms, but also reflects the importance degrees or different proportions of the used linguistic terms. For example, suppose that a teacher uses {h1 = very poor, h2 = poor, h3 = equal, h4 = good, h5 = very good} to evaluate the comprehensive academic ability of two students and . The teacher evaluates the two students in five aspects, and conducts one test for each aspect. When all the tests are completed, compared with the performance of , the performance of is judged as in two tests, it is judged as in two tests, and in one test. Then, the comparison results of the two students in the five aspects can be described as a LD, that is, . In this study, we will also continue to focus on the GDM with DLPRs to enrich linguistic decision information management through information granulation [41].
When addressing linguistic decision problems, since linguistic information itself is not operational, the issue of how to make linguistic information operational is usually encountered. For this reason, the notion of computing with words (CWW) [26,33] has attracted much attention from researchers, and various linguistic computational models [6,13,15,16,33,55] within the formal mathematical framework of CWW have been developed for supporting operations on linguistic information over the past few decades, say the linguistic hierarchy model [6], the 2-tuple linguistic representation model [13,33], a model based on membership functions [55], a model based on fuzzy numbers [16], and a model based on ordinal scales [15]. In these linguistic computation models, both the distribution and the semantics of the given linguistic terms are established a priori [3,4]. Recent studies [3,4,41,58] have achieved the operational realization of linguistic information through information granulation with the help of the paradigm of information processing in granular computing [40]. Information granulation is a process of data abstraction and the derivation of knowledge from information, in which the information granules (e.g., fuzzy sets, rough sets, shadowed sets, and intervals) arise [40]. The process of the operational realization of linguistic information is consistent with information granulation. Under the information granulation framework, the question of how to achieve the operational version of linguistic information is commonly formulated as certain optimization problems, where the criteria of consistency and consensus, or their weighted averaging, are usually regarded as suitable optimization performance indices [3,4,58]. In these studies, the given linguistic terms’ distribution and semantics are optimized instead of being established a priori. It is evident that a great number of studies have been reported to enrich the field of linguistic information granulation in the context of GDM with linguistic information [3,4,58]. To the best of our knowledge, no studies introduce the information processing paradigm of granular computing to address the operational realization of linguistic information in GDM scenarios where DMs’ preference information over alternatives is described by means of DLPRs. Hence, the first objective of this study is to fill this gap by proposing a linguistic information granulation model for supporting GDM with DLPRs.
The proposed linguistic information granulation model is considered to be a constrained optimization problem (COP) [2] with some adjustable parameters. A common way of solving COPs is to convert them into unconstrained optimization problems (UCOPs) by first setting certain penalty functions [38]. After that, the unconstrained ones can be solved by using common optimization methods, such as particle swarm optimization (PSO) [25,43], genetic algorithm [48], and so on. However, the use of penalty functions induces some penalty factors whose values are hard to be determined. The settings for the penalty factors are subjective and empirical in some way [34]. Thus, the feasible domain associated with the UCOPs cannot be effectively approached when using those common optimization methods. Subsequently, evolutionary computing technology [14] has been introduced to deal with COPs [1,19,51,52]. Specifically, He and Wang proposed an effective co-evolutionary particle swarm optimization (CPSO) method [19]. With the CPSO, penalty factors can be adjusted adaptively by employing the notion of a co-evolutionary mechanism, which overcomes the problem of subjectively determining penalty factors. It should be noted that the CPSO loses effectiveness when encountering COPs with adjustable parameters. Developing a novel approach for solving such COPs with adjustable parameters as the proposed linguistic information granulation model is the second objective of this study.
To achieve the two above-mentioned objectives, this study consists of the following two parts:
- a)
First, an information granulation model based on two optimization criteria, that is, consistency and consensus, is proposed to arrive at the operational realization of linguistic information in the context of GDM with DLPRs. The linguistic information granulation in this study is formulated as a certain optimization problem where a combination of the consistency degree of individual DLPRs and consensus degree among individuals is regarded as its performance index. Considering that intervals are a commonly used form of information granules in information granulation, the granulation formalism under consideration also concerns this formal scheme.
- b)
Then, inspired by the idea of the co-evolutionary mechanism adopted in the CPSO, a novel solution approach, called penalty function-based co-evolutionary particle swarm optimization (PFCPSO), is developed with the combined use of penalty functions and CPSO for effectively addressing such COPs with adjustable parameters as the proposed information granulation model. Within the framework of the proposed PFCPSO approach, the penalty functions are used to transform COPs into UCOPs; and the penalty factors of the penalty functions and the adjustable parameters as well as the decision variables of the optimization problems at hand are simultaneously optimized through the co-evolutionary mechanism of two populations in CPSO.
In summary, there are mainly two unique contributions of this study. First is the presentation of the consistency and consensus-driven information granulation model for achieving the operational realization of linguistic information in the context of GDM with DLPRs. Both the distribution and the semantics of the given linguistic terms can be obtained by solving the information granulation model. The second is the development of the PFCPSO approach for coping with such COPs with adjustable parameters as the proposed linguistic information granulation model.
The remainder of this paper is organized as follows: In Section 2, some prerequisites are covered. Section 3 presents the consistency and consensus-driven information granulation model in the context of GDM with DLPRs. In Section 4, the original PFCPSO approach is introduced to cope with the proposed information granulation model. Section 5 discusses the applications of the proposals discussed in Sections 3 and 4. Finally, Section 6 concludes the study with suggestions for future work in this area.
Section snippets
Basic knowledge
Some basic concepts, such as fuzzy preference relations (FPRs), DLPRs, 2-tuple linguistic model and numerical scale model are briefly introduced in this section.
Consistency and consensus-driven information granulation model
Compared with numerical preference relations, preference relations expressed by linguistic variables provide DMs with more convenient and accurate ways to depict their preference information over alternatives. However, linguistic terms themselves are not operational, and thus they need to be made operational to solve linguistic decision problems. The problem of making linguistic information operational is still an open issue when solving GDM with DLPRs. A granulation of linguistic information
Proposed PFCPSO approach for solving information granulation model
In this section, inspired by the idea of the co-evolutionary mechanism adopted in the CPSO [19], a novel solution approach, called PFCPSO, is developed with a combined use of penalty functions and CPSO for addressing the proposed COP. To prepare for the presentation of the PFCPSO, the basic model of CPSO and its co-evolutionary mechanism are briefly introduced first.
Illustrative example
In this section, a comprehensive evaluation problem is analyzed using the proposals to demonstrate their applicability and validity. Through this illustrative example, the implementation process of the proposals is introduced in detail. Moreover, two comparative studies are investigated to discuss the effectiveness of the proposals.
Conclusions and future work
With its promising performance of DLPRs in expressing DMs’ preference information, linguistic GDMs of this type of preference relations have attracted considerable attention over the past years. However, the problem of making linguistic information operational in such linguistic decision scenarios is still an open issue. To achieve the operational realization of the granules of linguistic information in GDM with DLPRs, this study developed an information granulation model based on the criteria
Author Contributions
Qiang Zhang: Conceptualization, Writing - Original Draft, Funding acquisition;
Ting Huang: Software, Investigation, Methodology;
Xiaoan Tang: Writing - Review & Editing, Methodology;
Kaijie Xu: Validation;
Witold Pedrycz: Language improvement, Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This research was supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (No. 71521001), the National Natural Science Foundation of China (Nos. 71501055, 71690230, 71690235, 72071063, and 72071060), and the Fundamental Research Funds for the Central Universities (Nos. JZ2020HGQA0168 and PA2021KCPY0030).
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