Decision-making based on probabilistic linguistic term sets without loss of information

Decision-making is human-centered and has inborn uncertainty with vague information expressed by natural language. To date, there are many models developed to facilitate the representations of the ill-structured information with uncertainty and vagueness. The first model is the famous fuzzy linguistic model by Zadeh [59], which relies on the membership function to express the non-crisp word or sentences in human language and has been a milestone in qualitative analysis in decision-making. Since then, a formal methodology, known as computing with words (CWW), in reasoning, computing and making decisions with natural language based information has come into being and soon became an eye-catching formulation in various fields. Notably, many linguistic models are introduced to handle the pervasive linguistic qualitative problems and can be fallen into two categories: the single linguistic term-based models and the multiple linguistic term-based models. Among the single linguistic term-based models, there are the symbolic linguistic computing model [56],the virtual linguistic computing model [55], the 2-tuple linguistic model [21] besides the Zadeh’s classical fuzzy linguistic variables. These models rely only on a single linguistic term representation of a linguistic variable, which suggests that one has to present his/her opinions without any hesitations. However, it is not the case in practice. Due to the incomplete information or knowledge, people may be uncertain or hesitant on the linguistic expressions and one single linguistic term is not adequate to portray the scenario accurately. Then, the multiple linguistic term-based models are proposed to cover the uncertainties with complex linguistic expressions, such as the hesitant fuzzy linguistic term set (HFLTS) [41], which is mathematically reformulated as hesitant fuzzy linguistic element (HFLE) [32]. The HFLTSs use a set of potential linguistic terms to effectively express the vague and complex linguistic statements, such as “between medium and tall”, “at least slightly good” and the like. The series of fruitful theoretical and practical results [31] betray that HFLTS or HFLE provides an effective way to capture the uncertainty in natural languages.

To express the preferences of DMs in a more flexible way, HFLTSs are extended to incorporate additional information for differentiating the given linguistic terms. The typical ones are the PLTS [38], equivalent to the proportional linguistic distribution assessment [19, 52]), the proportional HFLTSs [7]. Particularly, PLTS proposed by Pang et al. [38] assigns probabilities to the linguistic terms of an HFLTS and is more accurate in describing the vagueness in linguistic statements than HFLTS. Consequently, since the arising of PLTSs, the emerging model attracts extensive attentions from scholars in various fields and substantially advances the progress in CWW [30, 37]. The pertaining investigations can be distinguished into three categories. The first is focus on the PLTSs operations and notations [17, 28, 37, 38, 54, 58, 61], which is the basis for CWW and information aggregation; the second is about some PLTS-based methods for decision-making. Many classical methods are tailored to facilitate the PLTSs, such as the PLTS-based versions of MULTIMOORA [51], TODIM [36], ORESTE [49], VIKOR [63], ELECTRE II [33], ELECTRE III [28], LINMAP [29], QUALIFLEX [14], gained and lost dominance score method [50], DEMATEL [9], which bring more convenience for various applications over the PLTSs. The third is about the extensions or variants of PLTSs, such as the probabilistic uncertain linguistic term set [34], probabilistic linguistic vector-term set [60], interval-valued PLTS [3], Uncertain PLTS [23], dual PLTS [53]. The investigations around PLTSs is on-going vigorously. For a more comprehensive and detailed summaries on PLTSs, we recommend the most recent survey contributions [30, 37].

As is known, PLTS has the advantage in considering the ignorance in probabilistic information due to the partial information (the summation of probabilities for the given linguistic terms is less than one), then the normalization of PLTSs is inevitably necessary for further investigation. Until now, there are fruitful normalization methods for PLTSs, such as Average assignment [38], full-set assignment [28, 62], power-set assignment [37], envelope assignment [37], attitudes assignment [45], in tackling the incomplete probabilistic information. However, these normalization methods lead to information loss about the ignorance of probability information since the ignorance of probability is eliminated after the normalization. Then the authors in [37] presented an open problem on whether the normalization is necessary. Indeed, the ignorance of probability information is practically inevitable, then it is of vital importance to provide a reasonable process framework without any information loss, which is the very point which drives us to reconsider the process frame for the underlying probability information of the PLTSs in a different perspective.

In general, a PLTS has a set of linguistic terms with a probability distribution with normality, which suggests people have an exact and complete knowledge of probabilistic information. If the normality does not hold, according to [38], then it suggests that people have partial ignorance as a result of incomplete information or knowledge. Indeed, the partial ignorance actually means the ‘probabilistic information’ for the linguistic terms is non-probabilistic and is referred to a non-additive measure or capacity [16, 43]. In other words, for this case a PLTS can be seen as the set of linguistic terms with a set of probability distributions. Hence, in this work, we try to reformulate the PLTS from the perspective of ambiguity and provides a new insight to handle the ignorance of probability information through the non-additive probabilities or capacities (alternatively the multiple probability distributions). Furthermore, the classical DEMATEL is extended by the reformulated PLTSs. In detail, the probability information by the PLTSs is incorporated to tackle the uncertainty in the linguistic evaluation of the pair-wise comparisons and consequently used to synthesize the prominence-relation analysis.

Throughout the paper, we focus on the reformulation the classical PLTSs, and introduce a DEMATEL-based multiple attributes decision-making (MADM). In summary, the following aspects in this work can be highlighted. Here, we focus on the reformulation of the classical PLTS by transforming the ignorance of probabilities into multiple probability distributions in an ambiguity perspective. As a matter of fact, the reformulation is the accurate expression in the probability information, thus takes a step forward in exploring the experts-preference and knowledge. Further, in comparison (see Table 1), the DEMATEL-based aggregation method using the reformulated PLTSs in this work does not necessarily normalize the PLTSs beforehand and can well incorporate the different attitudes of the DMs for decision-making, thus does not result in any information loss in the process. Therefore, the improvement of the information exploration and utilization are the intrinsic starting point and drive for the employment of the reformulated PLTSs in DEMATEL. Compared to DEMATEL method based on the classical PLTS [9], or the linguistic term, or numerical expression, the method with the reformulated PLTS in this work has the advantage in effectively manifesting the experts-preference and knowledge, thus could lead to substantial improvement on the precision of decision-making.

The paper is organized as follows: as a preliminary, the next section provides some justifications on the ambiguity and the ignorance of the probability information about PLTSs; the redefinition and some notations of the PLTS are presented in the subsequent section followed by which the t-(co)norm-based operations over the reformulated PLTSs are focussed upon; then, the DEMATEL technology is extended using the reformulated PLTSs and then a PLTS DEMATEL-based aggregation method for decision-making is proposed; in the penultimate section, the illustrative example for the method is presented. In the end, some concluding remarks are presented.

In the literature of economics and psychology, the theories of choice have dominantly been the expected utility (EU) theory and the subjective expected utility (SEU) theory. It is assumed that in EU the pertaining probabilities are known while in SEU probabilities of states are subjectively or personally known. Actually, it is unbelievable to think of decision-making in which the probabilities are objectively presented. Thus, SEU is much more widely applied than EU. Nevertheless, there are much empirical evidence against SEU concerning the sharp distinction between whether the probabilities are (un)known. The distinctions are notated by various names, such as the risk vs uncertainty by Knight Frank [25], unambiguous vs ambiguous probabilities [13]. Generally, the term ambiguity refers to the case that the probability is unknown or partially known. Moreover, how much the decision-maker know about the probabilities does substantially influence their decisions [5]. The ignorance of probability, being a vital feature of PLTSs, can be seen as a snapshot of ambiguity. Therefore, ambiguity could be a point of departure to provide some new insights on PLTSs.

In general, a PLTS has the form as For the case \(\sum _{i=1}^mp_i= 1\), i.e., the PLTS has a set of linguistic terms with a probability distribution, which suggests people have an exact and complete knowledge of probabilistic information; for the case \(\sum _{i=1}^mp_i< 1\), according to [38], it suggests that people have partial ignorance as a result of incomplete information or knowledge. More precisely, the partial ignorance is referred to a situation that the probabilities are unknown or partially known, which is referred to ambiguity argued by Ellsberg [13] in his experimental results known as the Ellsberg paradox. Ellsberg paradox tells us that under ambiguity the decision-makers may rely on non-probabilistic measures which are the so called non-additive probabilities [43], or capacities [16] and are defined to be a set function \(v:\Sigma \rightarrow [0,1]\) satisfying \(v(\emptyset )=0\) and \(v(S)=0\), and \(v(E)\le v(E')\) whenever \(E\subseteq E'\). Here \(\Sigma \) is the event algebra of a state space S. It is shown [42] that a capacity v can be interpreted as a lower envelope on a set \(\hbox {core}(v)\) of considered possible probabilities P, i.e., \(v(E)=\inf _{P\in \hbox {core}(v)}\{P(E)\}.\) Intuitively, the decision-makers with ambiguity or partial probability entertain multiple probabilities as potential beliefs. Therefore, \(\sum _{i=1}^mp_i< 1\) in PLTSs actually means the ‘probabilistic information’ for the linguistic terms is non-probabilistic and should be referred to a non-additive measure or capacity. In other words, for this case a PLTS can be seen as the set of linguistic terms with a family of probability distributions.

For illustration, we provide the following example which is adapted from [38].

By the justifications of the above, a PLTS by Eq. (1) can be redefined as follows.

By the definition, if \(k=1\), then \({\mathscr {L}}(p)\) is equivalently reduced to the case \(\sum _{i=1}^mp_i=1\) in Eq. (1). In general, people only have partial or incomplete relevant information about the alternatives, and can hardly obtain a precise probability distribution for the potential linguistic terms. Instead, they may choose a set of possible probability distributions for the linguistic terms in compromise. Thus, the redefinition is practically and behaviorally reasonable. Additionally, if \(m=1\), i.e., only one linguistic term is given, the probability is necessary 1 which suggests a complete belief in the specific linguistic term. For this case, we would abbreviate the notation \(s_\alpha (1)\) by omitting the probability 1, and use \(s_\alpha \) instead. Here, it is assumed the partial belief focuses only on the case with multiple linguistic terms.

With an ambiguity perspective, the reformulation of the PLTS transforms the ignorance of probabilities into multiple probability distributions. Alternatively, the PLTS is a set of linguistic terms with multiple probability distributions. Note that \(p_i=\min _jp_{ij}\) implies that \(\sum _{i=1}^mp_i=\sum _{i=1}^m\min _jp_{ij}\le 1\). Thus, the multiple probability distributions representation of PLTS do not lead to any information loss over the ignorance of probabilities, and provides a smart way to tackle the probability normalization problem. By the reformulation the probability normalization for PLTS is not necessary any more and can be directly skipped for further investigations. In [38], the authors provided a normalization method to handle the ignored probability information and suggested that there may be some alternative methods on probability normalization. However, by these normalization methods the ignored probabilities are assigned to various linguistic term entities, and consequently removed. In other words, the normalization methods on the PLTS bring about much information loss on the ignored probabilities. Thus, Liao et al. [30] adopted a wait-and-see attitude about the normalization and proposed the open problem on whether it is necessary to normalize the PLTS. Here, the reformulation of the PLTS provides an ambiguity perspective into the partial ignorance of probabilistic information. Thus the PLTS considers the partial probability as a set of multiple probability distributions. Therefore, the redefinition of PLTS provides a compromised solution to the open problem [30, 37] on PLTS normalization through the multiple probability distribution-based representation of PLTSs. In the rest of the paper, we focus on the PLTSs by (3).

For the normalization of PLTSs having different sets of linguistic terms, we just have to add the unshared linguistic terms and assign zero probabilities to them. We use the dotted symbols \(\dot{{\mathscr {L}}}(p)\) for the normalized PLTSs.

To provide a ranking scheme for comparison, we introduce the score values and deviation degrees for the reformulated PLTSs, in which the behavioral (optimism or pessimism) attitudes can be incorporated.

In “Ambiguity and the ignorance of the probability information about PLTSs”, the worse case representation of the Choquet integral w.r.t. the capacity is an indication of ambiguity aversion. Also there are some empirical evidences in some recent work showing that people may have behaviors with ambiguity loving or neutrality [40]. Thus, the optimism index for the score value is necessary and suggests the degree of the optimism of a DM. The greater the index, the more optimistic the DM. With different choice of the optimism index, the attitudes of the DM can be demonstrated.

It can be checked that the normalization of the PLTSs has no influences on their score values and deviation degrees as a result of the assigned zero probabilities, and thus do not change their ranking scheme. But, the optimism index \(\pi \), which is an indicator of DM’s attitudes, is substantial for the ranking of PLTSs. To highlight the point, we provide the following example.

Here, since we can assign zero probabilities to the unshared linguistic terms without influencing their ranking, the two PLTSs are assumed to have the same set of ordered linguistic terms. In addition, we have the following properties for the distance between PLTSs.

In every field, information aggregation relies extensively on the underlying operations. PLTSs are no exceptions, since their introduction, the basic operations have been one of the hot topics in the area. The most recent work by Mi et al. [37] provides an extensively comprehensive summary on the topic. The typical operations are based on the crisp values of the linguistic terms by appropriate transformation, usually the (scaled) subscripts of the linguistic terms, such as the operations in [17, 38, 50].¹ The pertaining transformation may lead to information loss of the intrinsic linguistic implications, but intuitive and simple for applications. Here, the operations for the reformulated PLTSs are defined over the subscripts and the underlying probabilities in an element-by-element way using one of the important aggregation operators—t-(co)norms [24, 35, 48], whose notations are provided in Appendix.

By these definitions, the operations are closed in PLTSs by (3). Take \({\mathscr {L}}_1(p) \oplus {\mathscr {L}}_2(p)\) as an example, for each of the obtained elements \(L_{j_1j_2}(p)\), it holds that \(p_{i_1j_1}p_{i_2j_2}\ge 0\), and \(\sum _{i_1=1}^{m_1}\sum _{i_2=1}^{m_2}p_{i_1j_1}p_{i_2j_2}=\sum _{i_1=1}^{m_1}p_{i_1j_1}\sum _{i_2=1}^{m_2}p_{i_2j_2}=1,\) which implies \({\mathscr {L}}_1(p) \oplus {\mathscr {L}}_2(p)\) is a reformulated PLTS. In fact, these operations proceed with the two separate operations over the linguistic terms and the probabilities. For the operations \(\odot \) and \(\oplus \), by the definition, it may lead to repeated linguistic terms, which can be combined by summing up the pertaining probabilities. For illustration, we will only use the Frank and Hamacher t-(co)norms in the further analysis. Moreover, the following properties of these operations, i.e., symmetry, distributivity and associativity, can be derived with the properties of the underlying t-(co)norms or their generators.

By these operations for PLTSs in Definition 6, we can define some average operators for information aggregations, such as the weighted average and geometric average operators for PLTSs.

In this part, to illustrate the feasibility of the PLTS based method in applications, we use the reformulated PLTSs to provide a PLTS DEMATEL-based frameworks for MADM.

As usual, we assume \(\mathbf{X }=\{x_1,x_2,\ldots ,x_m\}\) is a set of alternatives, \(\mathbf{A }=\{a_1,a_2,\ldots ,a_n\}\) is a set of attributes with the attribute weighting vector \(w=(w_1,w_2,\ldots ,w_n)^{\mathrm{T}}\). The DM provides the evaluation of the alternative \(x_i\) on attribute \(a_j\) using the PLTS \({\mathscr {L}}_{ij}(p)\), then the decision matrix can be presented as follows.where \({\mathscr {L}}_{ij}(p)=\{L_{ij}^{(j_1)}|j_1=1,2,\ldots ,k_1\}\) with \(L_{ij}^{(j_1)}=\{s_{\alpha _{i}}(p_{ij}^{(i_1j_1)})|s_{\alpha _{i}}\in S,i=1,2,\ldots ,m_{ij},\) \( p_{ij}^{(i_1j_1)}\ge 0,\sum _{i=1}^{m_{ij}}p_{ij}^{(i_1j_1)}= 1\}\). In the following, we present the DEMATEL-based aggregation method for decision-making.

In this part, we provide an illustrative example, which is adapted from [38, 39] on the feasibility and the validity of the reformulated PLTS-based applications.

Probabilistic linguistic term set (PLTS) provides a much more effective model to compute with words and to express the uncertainty in the pervasive natural language by probability information. In this paper, to avoid loss of information, we redefine the classical PLTSs by multiple probability distributions from an ambiguity perspective and present some basic operations using Archimedean t-(co)norms. Different from the classical PLTSs, the reformulated PLTSs are not necessarily normalized beforehand for further investigations. Moreover, the multiple probability distributions based PLTSs facilitate the incorporation of the different attitudes of the DMs in their score values and the deviation, and thus the comparisons. In particular, the reformulated PLTSs can be incorporated in DEMATEL. In fact, the reformulation is the accurate expression in the probability information, thus takes a step forward in exploring the experts-preference and knowledge. Further, the reformulated PLTS requires no pre-stage normalization, thus does not result in any information loss in the process. Therefore, comparing to the existing DEMATEL methods under the classical PLTSs or numerical expression, the DEMATEL based on the reformulated PLTSs takes a step forward to exploring the experts’ probabilistic knowledge in a more accurate way without information loss.

From an ambiguity perspective, the paper concentrates on the reformulation of the classical PLTSs, which contributes a new insight into the PLTS-based methods and applications. With the reformulated PLTSs, we can provide some new thought to eliminate the pre-stage normalization and proceed with further investigations without any information loss. Thus, some known methods (see Table 1) for PLTS decision-making and some smart extensions of PLTSs [3, 23, 34, 53, 60] could be also restructured by the reformulated PLTSs in this work. In the future work, we would attempt to investigate the extended or improved DEMATEL methods [1, 10‐12, 47] using the reformulated PLTSs.