Elsevier

Information Sciences

Volume 582, January 2022, Pages 415-438
Information Sciences

Two prospect theory-based decision-making models using data envelopment analysis with hesitant fuzzy linguistic information

https://doi.org/10.1016/j.ins.2021.09.022Get rights and content

Abstract

The ultimate goal of the optimal alternative selection decision-making is to achieve higher returns with as few risks as possible. However, the existing multi-attribute decision-making (MADM) methods rarely measure their returns and risks simultaneously. In addition, these methods have no clear explanation for how to improve the non-optimal alternatives. Considering this, this paper constructs the returns and risks of alternatives in an identical measurement system by extending DEA model, thereby selecting the optimal alternative more comprehensively and improving the non-optimal alternatives. First, considering the bounded rationality of experts, the prospect theory is introduced into the information evaluation using hesitant fuzzy linguistic term sets to fully characterize the experts' psychological behavior under uncertainty. Then, by constructing the return-risk ratio as the efficiency evaluation index, two prospect theory-based hesitant fuzzy linguistic superefficiency models, namely the risk-oriented (RIHFLS) and return-oriented (REHFLS) models, are proposed to fully rank the alternatives. Furthermore, these two models are extended to RIHFLPS and REHFLPS models considering different importance of attributes. By adjusting the returns and risks to the target values, the non-optimal alternatives can be enhanced to efficient state. Finally, the feasibility and superiority of the proposed methods are verified by an application example.

Introduction

In multi-attribute decision-making (MADM) problems, due to the ambiguity and complexity of problems, it is impossible for decision makers (DMs) to carry out their own cognitive evaluation in a completely precise environment. In order to solve this problem, many scholars have introduced the fuzzy sets into MADM problems, and have achieved a lot of research results [34], [41]. At present, some extended forms of fuzzy sets, such as intuitionistic fuzzy sets, q-rung orthopair fuzzy sets, 2-type fuzzy sets and hesitant fuzzy sets are widely used to quantitatively describe the cognitive preference of DMs for alternatives, but they are not suitable for dealing with qualitative information. This is because the uncertainty and fuzziness of human thinking make DMs unable to express their complete cognitive preferences through only one or several specific numerical values. On the contrary, qualitative evaluation can intuitively describe the characteristics of alternatives. The hesitant fuzzy linguistic term sets (HFLTSs) proposed by Rodriguez et al. [30] allow DMs to qualitatively describe their uncertainty feelings about an alternative with ordered and continuous subset of linguistic term set (LTS), which is more in line with the hesitation of DMs' cognitive thinking.

Since HFLTSs were proposed, the theory has been widely developed and applied, and many HFL-MADM methods have been proposed. One is the HFL-MADM methods based on information aggregation operators. These methods first obtain comprehensive evaluation information of alternatives through special information fuse rules, and then take the score value of comprehensive evaluation information as the ranking standard to achieve the optimal alternative selection. For example, in the study of Zhang and Wu [43], the weighted averaging operator, weighted geometric operator and ordered weighted averaging operator were used to aggregate the HFLTSs. Later, operators reflecting the interrelationships between elements, such as the Bonferroni mean (BM), Heronian mean (HM), Maclaurin symmetric mean (MSM) and Hamy mean (HAM), were also extended to the HFLTS environment [10], [23], [42]. The other is the HFL-MADM methods based on decision-making models. These methods take the relative dominance of alternatives obtained by special decision rules as the ranking standard. First, the classical TOPSIS and VIKOR methods were applied to solve the HFL-MADM problems [16], [32], [39]. Later, more advanced ranking methods were adopted. For example, Gou et al. [9] developed a hesitant fuzzy linguistic alternative queuing method based on the hesitant fuzzy linguistic entropy and cross-entropy measures. Liao et al. [19] extended the MULTIMOORA method with strong robustness to handle the HFL-MADM problem in which attributes are inter-dependent. The MADM methods considering the priority relationship of attributes have also been extended and applied. For instance, Liao et al. [15] presented a HFL-Thermodynamic method by integrating the PROMETHEE method to select the optimal provider. Liao et al. [18] investigated the ELECTRE II method in the HFLTS environment and proposed two new approaches based on ELECTRE II to solve the HFL-MADM problems. Recently, Wang et al. [37] proposed a double hierarchy HFL-ORESTE method based on the new score function of double hierarchy HFLTSs. In an identical information environment, Liu et al. [22] applied the TODIM method that considers the psychological behavior of experts to evaluate university courses.

By the above review, we find that the ranking standard of the existing HFL-MADM methods essentially reflects the return level provided by alternatives. Note that any business cooperation for profit will be accompanied by risks, and these risks cannot be completely eliminated. The ultimate goal of the optimal alternative selection is to achieve higher returns on the premise of reducing risks as much as possible. However, the existing HFL-MADM methods rarely consider their potential risks when ranking the alternatives. As a result, the relationship between returns and risks has not been well modeled, and the obtained decision-making results are lack of comprehensive. In addition, the existing HFL-MADM methods effectively respond to how to select the optimal alternative, but there is no clear explanation on how to improve the non-optimal alternatives. Enterprise competition is the catalyst for the development of the industry. Only by improving the competitiveness of weak enterprises (the non-optimal enterprises) can we promote the high-quality development of the whole industrial chain. For example, in terms of internal competition in enterprise, a company wants to select one of several branches for pilot business to expand the business field. If the adopted decision-making method can provide the improvement guidance for the non-optimal branches simultaneously, it will help them better participate in this new business chain in the future and thereby, realize the maximum benefit of decision cost and promote the overall development of the company. From the perspective of external competition, if an enterprise that fails to win the bid wants to change its weak position and win cooperation in the next bidding, the enterprise should improve its weaknesses relative to the advantageous enterprises to enhance its competitiveness. Through the virtuous circle competition between enterprises to continuously promote the overall development of the industry. In fact, these two defects also exist in the MADM method with other fuzzy information. Based on the two defects of HFL-MADM methods, we propose two main purposes of this paper. One is to model the returns and risks of alternatives in an identical measurement system by a certain method and thereby, obtain more objective and comprehensive results. The other is to provide the improvement suggestions for the non-optimal alternatives to assist them in making progress.

Data envelopment analysis (DEA) is a nonparametric system analysis method for evaluating the relative efficiency of decision making units (DMUs) with multiple inputs and multiple outputs. For performance evaluation of a certain production activity, the goal of DMUs is to produce higher outputs with as few inputs as possible. Obviously, the optimal alternative selection and performance evaluation are similar in the way of goal achievement, that is, to obtain higher returns/outputs with as few risks/inputs as possible. In view of this, in this study, the returns and risks of alternatives can be modeled in an identical measurement system by using DEA model to seek more objective decision-making results.

The original DEA model was put forward by Charnes et al. [3], which was defined as the CCR model. The CCR model assumes the constant return to scale (CRS), which means that when inputs increase byktimes, outputs also increase by the same proportion. Later, Banker et al. [2] put forward the BCC model with variable returns to scale (VRS) to apply to the production situation where outputs and inputs do not need to change in the same proportion. The proposal of two models opened up a new field for efficiency evaluation systems. In the process of continuous improvement, many DEA models have been proposed to adapt to different evaluation scenarios, such as superefficiency model [1], [21], cross-efficiency model [4], [20], series network model [6], [25], parallel network model [12], [28], series–parallel network model [31], DEA model with imprecise inputs and outputs [25], [26] and so on. By comparing the deviation degree between the inputs and outputs of DMU and the efficient production frontier, we can identify the inputs and outputs with the input excess and output shortage. When there is neither input excess nor output shortage, the DMU is on the frontier and is efficient; On the contrary, the DMU is inefficient. Among them, the inefficient DMUs can be enhanced to strongly efficient state by proportionate improvement and slack improvement. In other words, by adjusting the inputs and outputs to the target values, the inefficient DMUs can be promoted to the input–output balance state that there is neither input excess nor output shortage. In an actual production activity, these improvement measures can provide production guidance for the inefficient DMUs. This function of DEA model is not possessed by any MADM method. Therefore, when DEA model is applied to MADM problems, the purpose of providing the improvement suggestions for the non-optimal alternatives can also be realized.

Based on this advantage, Zhou et al. [44] proposed the hesitant fuzzy envelopment analysis (HFEA) model and hesitant fuzzy preference envelopment analysis (HFPEA) model to improve the non-optimal alternatives, which is a preliminary exploration for the combination of DEA model and MADM problem. However, the HFEA and HFPEA models are both extended from the CCR model. The CCR model may identify multiple efficient DMUs with efficiency value of 1, and the efficiency of these DMUs cannot be further discriminated. As a result, the HFEA and HFPEA models may not be able to identify the only optimal alternative. In contrast, the superefficiency model proposed by Andersen and Petersen [1], as an improved DEA model, can achieve the fully ranking of all the evaluated DMUs. Obviously, when DEA method is applied to decision-making process, the superefficiency model is easier to achieve the optimal alternative selection. Therefore, this paper will apply the superefficiency model to carry out the optimal alternative selection decision-making under HFLTSs. In the process of identifying the optimal one, the two defects of the existing HFL-MADM methods are also well eliminated.

Note that DEA model based on the input–output ratio clearly divides the indicators into the input and output categories. However, generally speaking, the evaluation attributes in MADM problems do not need to be divided into the two categories, only need to select the appropriate attributes according to the decision objectives. Therefore, a key problem in this paper is how to apply the superefficiency model to the HFL-MADM problems. As mentioned above, the return-risk goal of the optimal alternative selection is close to the output-input goal of DMUs in DEA model. Based on this fact, we can construct the return-risk ratio to replace the output-input ratio as the efficiency evaluation index, and thereby the superefficiency model is successfully applied to MADM environment. The efficiency of the evaluated alternative can be measured from the perspective of risk-oriented (the degree of risk minimization without lowering the return level) or return-oriented (the degree of return maximization without improving the risk level). Take the risk-oriented as an example, compared with other alternatives, if the risks of the evaluated alternative can continue to be reduced under the current return level, then the alternative is relative inefficient; on the contrary, when any attempt to reduce the risks while keeping the returns unchanged is impossible, the alternative is relative efficient.

When DEA model is extended to MADM problem, such as the HFEA and HFPEA models proposed Zhou et al. [44], it is assumed that experts are completely rational, which fail to consider the important role of experts’ risk attitude in the decision-making process. Due to cognitive constraints, experts are not purely rational when making decisions in the decision environment with risk and uncertainty. The decision-making process is also affected by the complex psychological mechanism. Prospect theory, originally proposed by Kahneman and Tversky [11], was used to describe the actual decision behavior of an individual under risk. Later, Tversky and Kahneman [35] proposed the cumulative prospect theory to improve the prospect theory which fails to explain the stochastic dominance phenomenon, and gave the mathematical expressions of value function and weighting function. As one of the most influential behavioral decision theories, prospect theory has been widely used and expanded in evaluation and decision-making. For example, Peng and Yang [27] proposed an interval-valued fuzzy soft set method based on the value and weighting functions to evaluate the investment projects with three risk states. Similarly, Liu et al. [24] presented a risk decision-making model based on these two functions to select the optimal emergency evacuation action for the disaster event with three possible states. In contrast, when the possible states of event are not considered, only the value function is widely used to explain the impact of psychological behavior of experts under uncertainty. For instance, Liao et al. [15], Zhou et al. [46] and Ren et al. [29] adjusted the initial evaluations to the prospect evaluations reflecting the psychological behavior of experts by extending the value function to the fuzzy environment. Wan et al. [36] used the value function to measure the prospect value of each alternative with respect to each attribute under hybrid fuzzy information representations. As pointed out by Zhou et al. [46], due to the bounded rationality of experts, the evaluations of experts are influenced by the gain-loss utility in comparison to the reference point. Particularly, when experts are allowed to give the cognitive evaluations of alternatives in the form of HFLTSs, they may hesitate in several linguistic expressions, and this uncertain evaluation process is more vulnerable to the complex psychological mechanism. Therefore, this paper introduces the value function of prospect theory to adjust the evaluations to fully characterize the psychological behavior of experts under uncertainty. By doing so, experts are allowed to express their prospect evaluations, that is, they are risk-averse when perceived gains or risk-seeking when perceived losses.

Based on the above concerns, some prospect theory-based hesitant fuzzy linguistic superefficiency models and corresponding MADM methods are developed to select the optimal alternative and improve the non-optimal alternatives. To achieve this, the innovative works of this study are summarized as follows:

  • (1)

    First, we propose a new hesitant fuzzy linguistic prospect value function. By the function, the initial hesitant fuzzy linguistic evaluations are adjusted to the hesitant fuzzy linguistic prospect evaluations reflecting the psychological behavior of experts under uncertainty. Compared with the hesitant fuzzy linguistic prospect value function of Liao et al. [15], the new value function well handles the relationship between losses and gains, that is, the prospect values of losses should always be less than the gains.

  • (2)

    Second, the score function and deviation function of hesitant fuzzy linguistic prospect evaluations are proposed to depict the returns and risks of alternatives, respectively. By constructing the return-risk ratio, two prospect theory-based hesitant fuzzy linguistic superefficiency models, namely the risk-oriented (RIHFLS) model and return-oriented (REHFLS) model, are proposed to derive the superefficiency scores of all alternatives and their fully ranking. Here, considering there is no an identical proportional change relationship between the risks and returns of alternatives, the proposed models are established under the condition of VRS.

  • (3)

    Then, the proposed models can identify the inefficiency components of the non-optimal alternatives. By adjusting the current returns and risks to the target values, the non-optimal alternatives can be enhanced to strongly efficient state in which the returns and risks are relatively balanced, thereby helping them enhance the competitiveness.

  • (4)

    Furthermore, considering different importance of attributes, the risk-oriented hesitant fuzzy linguistic preference superefficiency (RIHFLPS) model and the return-oriented hesitant fuzzy linguistic preference superefficiency (REHFLPS) model with weight preference relationships are proposed by extending the RIHFLS and REHFLS models, respectively.

  • (5)

    Finally, based on the proposed risk-oriented and return-oriented models, two new MADM methods are developed to select the optimal alternative and provide the improvement suggestions for the non-optimal alternatives.

The structure of this paper is as follows: Some necessary concepts are explained in Section 2. Then, in Section 3, a new hesitant fuzzy linguistic prospect evaluation method is proposed. Then, by constructed the return-risk ratio function of hesitant fuzzy linguistic prospect evaluations, a series of risk-oriented and return-oriented models are developed. In Section 4, based on the proposed risk-oriented and return-oriented models, we develop two MADM methods to select the optimal alternative and improve the non-optimal alternatives, respectively. Section 5 discusses feasibility and superiority of the proposed methods. Finally, some conclusions are presented.

Section snippets

Preliminaries

In this Section, some important research works for the linguistic term set (LTS), hesitant fuzzy linguistic term set (HFLTS) and DEA models are briefly reviewed.

Prospect theory-based hesitant fuzzy linguistic superefficiency models

Note that the evaluation attributes in MADM problems cannot be clearly divided into the input and output categories. Therefore, we need to apply DEA model to the decision-making problem through other ways. For the optimal alternative selection, its goal that achieves higher returns with as few risks as possible is similar to the idea of producing higher outputs with as few inputs as possible in DEA model. In view of this, we construct the return-risk ratio to replace the output-input ratio as

The two MADM methods based on the risk-oriented and return-oriented models

In the following, we elaborate on the application steps of the proposed models in the MADM problem with HFLEs. Taking the risk-oriented model as an example, we elaborate the specific optimal alternative selection steps in detail. According to whether exist preference relationships between attributes, the RIHFLS model (15) or RIHFLPS model (21) is selected to calculate the superefficiency scores of all alternatives and select the optimal one. At the same time, Definition 8 is used to provide

Case description and data collection

In this Section, the presented risk-oriented models (15), (21) are applied to address a practical problem where fresh logistics service providers are evaluated.

The emergence of fresh e-commerce provides a lot of convenience for consumers. With the development of economy and the fast pace of life, it has gradually become an indispensable shopping channel. In recent years, food safety problems are serious, and people have more strict requirements for the quality of fresh products. Online shopping

Conclusion

The ultimate goal of the optimal alternative selection is to achieve higher returns on the premise of reducing risks as much as possible. By constructing the return-risk ratio as the efficiency evaluation index, this paper extends DEA model to solve the MADM problem with HFLEs, and proposes two prospect theory-based hesitant fuzzy linguistic superefficiency models, namely the risk-oriented (RIHFLS and RIHFLPS) models and return-oriented (REHFLS and REHFLPS) models, to select the optimal

Compliance with ethical standards

(1) Disclosure of potential conflicts of interest

We declare that we do have no commercial or associative interests that represent a conflict of interests in connection with this manuscript. There are no professional or other personal interests that can inappropriately influence our submitted work.

(2) Research involving human participants and/or animals

This article does not contain any studies with human participants or animals performed by any of the authors.

CRediT authorship contribution statement

Hongxue Xu: Conceptualization, Methodology, Investigation, Validation, Software, Writing – original draft. Peide Liu: Conceptualization, Methodology, Formal analysis, Writing - review & editing, Supervision, Funding acquisition. Fei Teng: Investigation, Validation, Supervision, Writing - review & editing.

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.

Acknowledgment

This paper is supported by Major bidding projects of National Social Science Fund of China (No. 19ZDA080).

References (46)

  • H. Liao et al.

    Two new approaches based on ELECTRE II to solve the multiple criteria decision making problems with hesitant fuzzy linguistic term sets

    Appl. Soft Comput.

    (2018)
  • H.H. Liu et al.

    Cross-efficiency evaluation in data envelopment analysis based on prospect theory

    Eur. J. Oper. Res.

    (2019)
  • P. Liu et al.

    Double hierarchy hesitant fuzzy linguistic entropy-based TODIM approach using evidential theory

    Inf. Sci.

    (2021)
  • P. Liu et al.

    Normal wiggly hesitant fuzzy linguistic power Hamy mean aggregation operators and their application to multi-attribute decision-making

    Comput. Ind. Eng.

    (2020)
  • Y. Liu et al.

    Risk decision analysis in emergency response: a method based on cumulative prospect theory

    Comput. Oper. Res.

    (2014)
  • M.R. Mozaffari et al.

    Towards greener petrochemical production: Two-stage network data envelopment analysis in a fully fuzzy environment in the presence of undesirable outputs

    Expert Syst. Appl.

    (2021)
  • X. Peng et al.

    Algorithms for interval-valued fuzzy soft sets in stochastic multi-criteria decision making based on regret theory and prospect theory with combined weight

    Appl. Soft Comput.

    (2017)
  • X. Shi et al.

    Overall efficiency of operational process with undesirable outputs containing both series and parallel processes: a SBM network DEA model

    Expert Syst. Appl.

    (2021)
  • M. Tang et al.

    Managing information measures for hesitant fuzzy linguistic term sets and their applications in designing clustering algorithms

    Information Fusion

    (2019)
  • R. Tao et al.

    A dynamic group MCDM model with intuitionistic fuzzy set: Perspective of alternative queuing method

    Inf. Sci.

    (2021)
  • S.P. Wan et al.

    Prospect theory based method for heterogeneous group decision making with hybrid truth degrees of alternative comparisons

    Comput. Ind. Eng.

    (2020)
  • X. Wang et al.

    Assessment of traffic congestion with ORESTE method under double hierarchy hesitant fuzzy linguistic environment

    Appl. Soft Comput.

    (2020)
  • Z. Wu et al.

    Two MAGDM models based on hesitant fuzzy linguistic term sets with possibility distributions: VIKOR and TOPSIS

    Inf. Sci.

    (2019)
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