Large-scale group decision making with multiple stakeholders based on probabilistic linguistic preference relation
Introduction
In real-life decision-making processes, because some decision-making problems benefit from multiple stakeholders, a large number of multiple stakeholders should take part in the decision-making process [1], [2], [3], [4]. To gain successful strategic supplier selection, a soft decision model involving multiple stakeholders was proposed by means of interval and hesitant fuzzy methodology [1]. In order to choose a location for a waste treatment facility, the large amount of data from standard stakeholders was analysed using the SMAA-O method [2]. With the development of e-democracy and social networks, a larger number of experts take part in the decision process [3]. A structured framework was established to identify the key service processes based on customer perspectives [4]. Stakeholders are defined as any individual or group that can affect or be affected by an organisation [5]. Each of these groups plays an important role in the development of society and enterprise growth. For example, a large number of stakeholders from different bodies of knowledge are needed to select sustainable suppliers, and a large number of citizens from different professional backgrounds are invited to provide their opinions about waste disposal sites.
Large-scale group decision making (LGDM) is different from the traditional group decision making, which uses a small-group with 3–5 decision makers (DMs). In recent years, a larger number of DMs are taken into account in decision-making processes because of the strong support of a networked environment and societal demands. As a result, LGDM problems have received increasing attention in recent years [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. LGDM methods can be roughly classified into four types, namely, clustering-based LGDM, consensus-reaching LGDM, LGDM methods and LGDM support systems. Labella et al. [4] performed a comparative analysis for different classical consensus-reaching processes applied to LGDM problems. Palomares et al. [6] proposed a graphical monitoring tool to manage LGDM. Palomares et al. [7] further presented a consensus model based on detecting and managing non-cooperative behaviours. Pérez et al. [8] put forward a consensus model in which experts’ importance levels are considered. Quesada et al. [9] utilised a uninorm-based weighting scheme to deal with non-cooperative behaviours. Xu et al. [10] proposed a dynamical consensus method based on an exit-delegation mechanism. Xu et al. [11] further studied a consensus model to manage minority opinions and non-cooperative behaviour. Liu et al. [12] proposed a weight value determination method for multiple groups based on subjective and objective information. Further, Liu et al. [13] proposed a two-layer weight determination model based on linguistic information. Zhang and Guo [14] proposed a LGDM model based on multigranular linguistic distribution information. Moreover, Song and Li [15] put forward a LGDM model based on incomplete multi-granular probabilistic linguistic term sets. Wu and Xu [16] presented a consensus model in LGDM in which the clusters are allowed to change. Zhang et al. [17] put forward a consensus model for heterogeneous LGDM. Wu and Liu [18] used an interval type-2 fuzzy equivalence clustering method to solve LGDM problems.
Existing related studies have made significant contributions to LGDM analysis. However, some stakeholders may provide incomplete preference relations because of the limits of their professional knowledge and experience or time pressure. On the one hand, a stakeholder may be unfamiliar with a certain object and thus unable to provide preferences associated with it, or a stakeholder may be unwilling to express his/her opinions regarding some pairs of objects because of emotional factors. In such cases, it would be sensible not to force an expert to express ‘false’ preferences for these objects, and thus, an incomplete preference relation would be constructed in which some elements are missing. Actually, situations in which all stakeholders are able to efficiently express their preferences for all available options might be considered the exception rather than the rule. Indeed, the above scenario requires all stakeholders to possess a precise or sufficient level of knowledge of the whole problem, including the ability to discriminate the degree to which some options are better than others, to address it. These assumptions can be seen as unrealistic in many decision-making situations, especially those involving a large number of alternatives with conflicting and/or dynamic sources of information [19]. Thus, incomplete preference information may be given by multiple stakeholders. Various incomplete preference relations have been continuously developed in recent years [20], [21], [22], [23], [24]. Therefore, it is necessary to establish an LGDM model based on incomplete preference information. Furthermore, for heterogeneous multiple stakeholders, it is more suitable to use linguistic term sets [25], [26], [27] or fuzzy numbers [28], [29], [30] to express preference information. A variety of linguistic models, including the fuzzy number-based, 2-tuple linguistic and virtual linguistic models, have been introduced to deal with practical problems [31], [32], [33], [34]. Hence, in this study, each stakeholder utilises an incomplete linguistic preference relation (LPR) to represent his/her preference relation for alternatives and then the probabilistic LPR (PLPR) proposed by Zhang et al. [35] with incomplete probabilities is constructed for multiple stakeholders to represent the group’s preference.
In this paper, our aim is to establish an LGDM model based on PLPR with incomplete probabilities using mathematical programming. First, we define the expected consistency of the PLPR and establish the probability computation model for the PLPR to obtain the missing occurrence probabilities and priority weights of alternatives. Then, to improve the consistency level of PLPR, the expected consistency index (ECI) of the PLPR is developed and then calculated. For a PLPR with an unsatisfactory level of consistency, an algorithm is proposed to obtain the PLPR with an acceptable expected level of consistency. In the following, the contributions of the proposed LGDM model with respect to previous studies are summarised.
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An LGDM model with multiple stakeholders is proposed based on PLPR with incomplete probability values. This model is more flexible than previous methods.
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A probability computation model is established based on the expected multiplicative consistency of PLPR to obtain its missing probability values.
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A consistency-improving algorithm is proposed to obtain a PLPR with a satisfactory level of consistency.
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The proposed model is thus feasible for practical purposes as it can permit a flexible number of stakeholders.
The rest of this paper is organised as follows. Section 2 reviews some basic knowledge of linguistic information, LPR and PLPR, and provides a justification of the probabilistic linguistic term set (PLTS). In Section 3, we propose a probability computation model for PLPR based on its expected level of consistency. In Section 4, a consistency-improving algorithm is proposed and a LGDM model is established. In Section 5, an investment selection problem is solved using proposed method and we then compare our suggested method with previous methods. Some concluding remarks are given in Section 6.
Section snippets
Preliminaries
In this section, we review basic knowledge about linguistic representation models and two preference relations: the LPR and PLPR.
Probability computation model of a PLPR with incomplete probabilities based on the expected multiplicative consistency
To obtain the complete corresponding probabilities of a PLPR, a probability computation model is established in this section based on the expected consistency of the PLPR. To do this, we first present the multiplicative consistency of an LPR as follows:
Theorem 1 Let be an LPR, if it satisfies the following conditions Then, is known as an LPR with multiplicative consistency. where is the given linguistic scale set and denotes the priority
Consistency improvement and the LGDM model
It is well-known that consistency is a significant issue widely considered in decision-making problems based on preference relations [44], [45], [46], [47], [48]. Dong et al. [44] presented a consistency index for linguistic preference relation and developed a consistency measure method for linguistic preference relation. Kou and Lin [45] proposed a consistency index for a pair-wise comparison matrix based on cosine maximisation method. Kou et al. [46] adapted Hadamard model to mitigate
Investment selection problem and comparative analysis
In this section, a real investment choice problem is solved by our method and we then analyse and summarise the differences of the proposed method with previous methods.
Conclusions
In recent years, as information technology has developed, it has become possible for multiple stakeholders take part in decision-making meetings. Because of the limited knowledge of DMs and environmental complexity, certain stakeholders may provide incomplete LPRs. Then, a PLPR with incomplete probabilities is constructed to represent group preferences. Therefore, one goal of this paper is to compute the probability for a PLPR. Based on the defined expected consistency, we proposed a
Acknowledgements
The authors would like to thank the editors and anonymous reviewers for their insightful and constructive commendations that have led to this improved version of the paper. This research was supported in part by grants from the National Natural Science Foundation of China (#71601032).
Declaration of competing interest
No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to //doi.org/10.1016/j.asoc.2019.04.036
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