Multi-criteria outranking method based on probability distribution with probabilistic linguistic information

Highlights

• The probability distribution is introduced for modelling PLTSs.

• The concordance and discordance indices of PLTSs are defined.

• Four kinds of novel binary relations for PLTSs are proposed.

• An innovative multi-criteria outranking method is developed.

Abstract

The probabilistic linguistic term set (PLTS) is a powerful tool for describing qualitative evaluations derived from teams of experts, and it has adequate description capability in identifying preferences among different evaluations. The structure of PLTSs is complex, however, and many existing studies do not deal with probabilistic linguistic information appropriately. Hence, this study explores the simple and effective processing of PLTSs and develops an applicable multi-criteria decision-making (MCDM) method to address real-world problems. First, PLTSs are characterised as probability distributions, and the corresponding cumulative distribution functions are presented. In this manner, the concordance and discordance indices of PLTSs are defined by the systematic comparison between different cumulative distribution functions. Subsequently, four kinds of novel binary relations for PLTSs are proposed. Then, an innovative multi-criteria outranking method is developed by modelling pseudo-criteria and implementing outranking aggregation and exploitation. Finally, an illustrative example concerning new energy selection is provided to elucidate the application of the developed method. The strengths of this method are verified further by some analyses and discussions.

Introduction

Decision-making activities prevail in human life. Over the past half century, various decision-making methods, such as aggregation operator-based methods (Xu, 2004), information measure-based methods (Chen, 2000) and outranking relation-based methods (Aouam, Chang, & Lee, 2003), have been developed to address real-world problems. In practical decision processes, decision-makers (DMs) prefer to use natural languages to express their opinions because of the inherent preference of human nature. To characterise qualitative decision-making information, Zadeh (1975) introduced the concept of linguistic variables. With motivation stemming from this idea, many linguistic concepts and linguistic decision-making methods have been studied extensively.

Herrera, Herrera-Viedma, and Verdegay (1996) proposed the concept of linguistic term sets (LTSs) to describe linguistic variables and defined the linguistic distance and linguistic consensus degree. Furthermore, Herrera, Herrera-Viedma, and Martínez (2000) proposed multi-granularity LTSs and developed a fusion method of multi-granularity linguistic information. The multi-granularity linguistic model can deal with multiple linguistic information derived from experts in different fields and is a useful technique for group decision making (GDM) (Zhu et al., 2015, Tian et al., 2018). To deal with non-uniform and asymmetrical linguistic evaluation information, Herrera, Herrera-Viedma, and Martínez (2008) proposed the concept of unbalanced LTSs. This linguistic representation model has been applied to solve various practical problems owing to its capability of explicitly distinguishing the semantics of linguistic evaluations (Pei and Zheng, 2017, Fu and Liao, 2019). To accommodate intuitionistic fuzzy numbers (IFNs) in qualitative environments, intuitionistic linguistic fuzzy numbers (Liu, 2013) and linguistic intuitionistic fuzzy numbers (Chen, Liu, & Pei, 2015) were developed by combining IFNs and linguistic variables.

In some situations, DMs prefer to provide their evaluations using several linguistic values due to uncertain environments and limited knowledge. For solving such problems, hesitant fuzzy LTSs (HFLTSs) (Rodriguez, Martinez, & Herrera, 2012) were proposed on the basis of the elicitation of hesitant fuzzy sets. Many studies on HFLTSs, such as preference relations (Wu & Xu, 2016), aggregation operators (Lee & Chen, 2015), projection models (Wu, Xu, Ren, & Liao, 2018) and outranking relations (Liao, Yang, & Xu, 2018), have been conducted to handle decision-making problems. The HFLTS is a useful tool for depicting the hesitant linguistic evaluations of DMs; however, a limitation is that the importance (weight) of linguistic values in HFLTS is ignored. In fact, DMs may prefer a linguistic value to others; likewise, a linguistic value may be provided by several DMs and others may be derived from one DM. Naturally, the weights or preferences of linguistic values should be distinguished in such cases.

To overcome the existing defects in HFLTSs, Pang, Wang, and Xu (2016) presented the novel concept of probabilistic LTSs (PLTSs). In this concept, the evaluation information can be characterised using linguistic values with the corresponding probabilities. The linguistic values can describe the evaluations of DMs qualitatively, and the probabilities can reflect the preferences of DMs with respect to different evaluations. Since the introduction of PLTSs, they have received considerable attention, and numerous studies have been conducted. On the one hand, the basic theories and extended concepts of PLTSs have been investigated. Gou and Xu (2016) defined new operations for PLTSs so that the probability information in PLTSs can be retained completely after computation. After discussing the drawbacks of the comparison method of PLTSs (Pang et al., 2016), Bai, Zhang, Qian, and Wu (2017) proposed the possibility degree of PLTSs based on diagram analysis to compare different PLTSs. Wang, Wang, and Zhang (2019) introduced a generalised distance and a Hausdorff distance for PLTSs. Liu and Li (2019) proposed a series of probabilistic linguistic Maclaurin symmetric mean aggregation operators. Moreover, some new concepts, such as probabilistic linguistic vector-term sets (Zhai, Xu, & Liao, 2016) and probabilistic uncertain LTSs (Lin, Xu, Zhai, & Yao, 2018), were introduced motivated by PLTSs. On the other hand, some studies focused on models and methods with PLTSs. Liao, Jiang, Xu, Xu, and Herrera (2017) established a probabilistic linguistic linear programming model to address multi-criteria decision-making (MCDM) problems. Liu and You (2017) developed a TODIM method to address MCDM problems with PLTSs. Liang, Kobina, and Quan (2018) proposed a series of probabilistic linguistic Bonferroni mean (PLBM) aggregation operators and developed a multi-criteria group decision-making (MCGDM) method by combining PLBM with grey relational analysis (GRA). Wu and Liao (2018) proposed a multi-expert MCDM method by incorporating PLTSs into quality function deployment. In addition, many scholars applied PLTSs to address real-world problems, including venture capital (Cheng, Gu, & Xu, 2018), hotel selection (Peng, Zhang, & Wang, 2018), enterprise resource planning system selection (Chen, Wang, & Wang, 2019) and financial technology selection (Mao, Wu, Dong, Wan, & Jin, 2019).

Nevertheless, these studies have some shortcomings in the handling of probabilistic linguistic information. First, the linguistic terms in PLTSs are handled using the subscript or transformation function. Such scheme converts linguistic terms to crisp numbers and simplifies calculation. However, the fuzziness and randomness involved in linguistic variables are ignored completely. In this case, original evaluation information is lost and distorted inevitably. Second, the probability information in PLTSs is not processed appropriately. For example, the probability part is discarded after operation (Pang et al., 2016), and the probability is not calculated in the operational rules (Gou & Xu, 2016). Third, the measures of PLTSs (Liao et al., 2017, Liu and You, 2017, Liang et al., 2018) are defined simply depending on the products of linguistic terms and their probabilities. This strategy ignores the measure of probabilities and weakens their function. In conclusion, the processing of PLTSs in some existing studies is questionable.

According to the analyses above, the primary motivations of this study are summarised as follows:

• Although the PLTS is a useful tool for describing decision-making information, it contains more elements and has a more complicated structure than other linguistic presentation models. The linguistic term and probability in PLTSs are not handled appropriately in some existing studies. The key points of dealing with PLTSs are that the uncertainties of linguistic variables need to be maintained and the condition ${\sum }_{i=1}^{\#x_i}{p}_i⩽1$ (as described in Definition 1) needs to be satisfied. Considering these points, this study introduces probability distribution to analyse the structure of PLTSs, and the corresponding cumulative distribution function is then presented. In this manner, the concordance and discordance indices of PLTSs are defined by systematically comparing different cumulative distribution functions, and some novel binary relations for PLTSs are proposed.

• To reflect the qualitative characteristics of human preference and handle complex decision-making problems, the PLTS theory needs to be incorporated into decision-making models. Many existing decision-making methods with PLTSs have some shortcomings because of invalid processing of probabilistic linguistic information. Moreover, aggregation operator-based methods have high computational complexity because many elements in PLTSs need to be integrated. Therefore, effective and feasible PLTS methods that can be easily applied to address a wide range of real-world problems in organisation management, project evaluation and risk analysis need to be developed. Relation models have lower computational complexity than function models, but probabilistic linguistic outranking relations have not been studied in depth. Therefore, this study incorporates the proposed outranking concepts of PLTSs into classic relation models to develop an innovative MCDM method.

The rest of this paper is presented as follows. In Section 2, some basic concepts are reviewed briefly. In Section 3, the concordance and discordance indices and some novel binary relations for PLTSs are proposed. In Section 4, an innovative multi-criteria outranking method under probabilistic linguistic environments is developed. In Section 5, a practical new energy selection problem is handled to verify the developed method, and some analyses and discussions are then conducted. Finally, concluding remarks are drawn in Section 6.