Probabilistic Hybrid Linguistic Approaches for Multiple Attribute Group Decision Making with Decision Hesitancy and the Prioritization of Attribute Relationships

Focusing on ill-structured multiple attribute decision-making (MADM) problems, including decision hesitancy and attribute prioritization relationships, this paper investigates appropriate approaches for decision making. Firstly, we introduce the probabilistic hybrid linguistic term set (P-HLTS) for capturing probabilistic preferences about possible linguistic labels belonging to a wide range of hesitant linguistic term sets. Entropy and distance measurements for P-HLTS are developed without arbitrary complementing operations. To facilitate decision making with attribute prioritization relationships, we present a probabilistic uncertain balanced linguistic-prioritized weighted average (PUBL-PWA) operator and the probabilistic uncertain balanced linguistic-induced prioritized ordered weighted average (PUBL-IPOWA) operator. In terms of the strength of the above tools, we further construct two multiple attribute group decision-making (MAGDM) approaches under P-HLTS environments, namely, an approach for decision-making situations where attribute prioritization relationships are known in advance and the relative importance of decision makers (DMs) or decision-making units (DMUs) is not required for consideration, and second approach for decision-making situations where both attribute prioritization relationships and the weighted vectors of DMs or DMUs are explicitly unknown. In general, our proposed approaches are more flexible and practical when considering heterogeneous opinions, avoiding information distortion brought about by complementing operation-based distance measures. Furthermore, illustrative application studies are conducted to verify our developed approaches.


Introduction
Multiple attribute decision-making (MADM) theory is an indispensable part of decision analysis theory, [1] and has been successfully applied to practical problems in many fields [2][3][4][5][6][7][8]. The main purpose of MADM is to find desirable solutions from finite alternatives according to assessments under a set of attributes. Due to increasing complexity and uncertainty in socioeconomic decision-making environments, single decision makers usually become less competent in evaluating all aspects of complicated decision-making problems [9,10]. Multiple attribute group decision-making (MAGDM) approaches thus have been developed by extending MADM to group settings, where a number of decision makers (DMs) or decision-making units (DMUs) are invited to present their assessments [5,11,12]. To deal with the uncertainty in decision maker preferences [13][14][15][16][17], fuzzy set theory [18] and its extensions have been introduced to enhance conventional MAGDM models, such as intuitionistic fuzzy sets [19,20], hesitant fuzzy sets [21], dual hesitant fuzzy sets [22], interval-valued dual hesitant fuzzy sets [23], and probabilistic dual hesitant fuzzy sets [24], among others.
However, when it comes to more complicated MAGDM problems, where problem structures are ill-defined for fuzzy quantification, Zadeh [25] has advocated the usage of linguistic variables for decision makers to qualitatively express their uncertain preferences. To this end, because of the merit in eliciting decision maker assessments more directly and precisely, linguistic variables have been continuously extended to accommodate various scenarios, such as unbalanced linguistic variables [26][27][28][29], uncertain linguistic variables [30][31][32][33], intuitionistic uncertain variables [34,35], hesitant fuzzy uncertain linguistic variables [15], and hesitant fuzzy unbalanced linguistic variables [36], among others. Although the above extensions of linguistic variables are very effective in situations where decision makers can approximate the most precise linguistic label in accordance with their assessment, Rodríguez et al. [37] revealed that decision makers are often cognitively irresolute among several possible linguistic labels. Taking this a step further, Pang et al. [11] and Liao et al. [38] point out the phenomena that DMs or DMUs usually will have different importance degrees in terms of those possible linguistic labels. Pang et al. [11] thus introduced the probabilistic linguistic term set (PLTS), which is capable of dealing with situations where probabilistic distributions cannot be acquired completely in reality, and is capable of carrying out probability information aggregation without a loss of information [24]. Since its introduction, PLTS has been successfully applied to solve practical MADM problems in various settings [38][39][40][41][42][43][44][45][46]. As can be seen, these studies all only allow decision makers to depict their cognitive models through balanced linguistic terms sets (BLTS) [25] which are uniformly and symmetrically distributed. However, field investigations [26,28] have revealed that decision makers usually tend to utilize nonuniform or asymmetrical linguistic term sets, referred to as unbalanced linguistic term sets (UBLTSs) [27], for expressing their assessments appropriately and flexibly. Additionally, when trying to obtain linguistic expressions that closely follow decision maker cognitive models, Rodríguez et al. [47] and Wang [48] have advocated for the adoption of consecutive and nonconsecutive comparative linguistic expressions. Therefore, in order to address different cognitive models where DMs or DMUs hold real problems, in this paper, we propose an effective expression tool for probabilistic hybrid linguistic term sets (P-HLTSs). P-HLTSs manage to enhance classical PLTS by accommodating a wide range of linguistic expression forms when considering diverse cognitive models, including balanced linguistic sets, unbalanced linguistic term sets, uncertain unbalanced linguistic term sets, comparative balanced linguistic term sets, and comparative unbalanced linguistic term sets. Comparatively, P-HLTS thus behaves in a more flexible and practical manner.
With respect to complicated decision-making situations which require decision information in the form of PLTS, pioneering efforts have been accumulatively paid to various effective MADM methodologies, such as that of Pang et al. [11], who introduced an extended TOPSIS-based MAGDM approach by use of proposed fundamental probabilistic linguistic averaging operators. Liu and Fei [49] developed two multiple attribute decision-making methods based on their probabilistic linguistic Archimedean Muirhead mean aggregation operators [49]. Liu and Li [50] designed an effective MAGDM approach to consider interrelationships among input arguments by use of generalized Maclaurin symmetric mean operators. Lin et al. [51] extended PLTS to define the probabilistic uncertain linguistic term set (PULTS) and devised a TOPSIS-based MAGDM method with the support of aggregation operators for PULTS. Liu et al. [40] developed an approach where attributes' weights were derived by their defined entropy measures for PLTS. Wu and Liao [41] studied a novel group decision-making approach based on ORESTE methods and PLTS under a QRD framework for design selection problems. Xiang et al. [42] studied an interactive venture capital group decision-making approach under the PLTS scenario, in which interactions among venture capitalists and entrepreneurs were considered to dynamically deduce weight information.
To deal with emergency decision-making problems, Gao et al. [43] developed an approach using probabilistic linguistic preference relations (PLPRs), where they equally synthesized subjective possibilities given by decision makers and objective possibilities which were obtained by case-based reasoning. Liao et al. [38] proposed an LINMAP-based group decision-making method where the developed linear programming models were used to calculate the weights of evaluative attributes. Some other classical decision-making methodologies have also been extended to PLTS environments, such as the probabilistic linguistic TODIM method [45] and the probabilistic linguistic VIKOR method [46]. However, to the best of our knowledge, the approaches in the literature are incapable of addressing the special type of decision-making problems which exist where decision information about attributes' prioritization relationships is unavailable. In fact, the phenomena of prioritization relationships is quite common when making decisions, due to limited knowledge or problematic complexity, presenting difficulties or unwillingness to provide complicated preference relationships, as required in AHP-like analytical models, while they are quite sure about the existence of prioritization relationships among evaluative attributes [52][53][54][55][56][57][58]. Therefore, to tackle ill-structured, complicated decision-making problems where attribute prioritization relationships exists, it is indispensable to study effective approaches that are simultaneously capable of taking the strength of PLTS and exploiting decision information denoted by attributes' prioritization relationships.
To do so, we firstly employ the newly defined probabilistic hybrid linguistic term set (P-HLTS) to empower DMs or DMUs in choosing the most appropriate linguistic term set for depicting their preferences. Secondly, in the light of Yager's [52,53] prioritized average (PA) operator theory, which manages to concurrently consider assessments under evaluative attributes and considering prioritization relationships among the attributes, we develop two prioritized average operators for MADM under P-HLTS environments, namely, the probabilistic uncertain balanced linguistic-prioritized weighted average (PUBL-PWA) operator and the probabilistic uncertain balanced linguistic-induced prioritized ordered weighted average (PUBL-IPOWA) operator. Further, based on the defined operational rules and the entropy measure of P-HLTS and the PUBL-PWA operator, we construct a MAGDM approach to solve decision-making problems where attributes' prioritization relationships have been determined in advance, and the relative importance of DMs or DMUs does not require consideration. Furthermore, to tackle more complicated decision-making problems, where attributes' prioritization relationships and weighting vectors for DMs or DMUs are both explicitly unknown, we utilize the PUBL-IPOWA operator to construct another MAGDM approach, in which we introduce a distance measure for P-HLTS to develop a divergence measure-based method for obtaining the unknown attributes' prioritization relationships and a similarity degree-based method for deriving weighting vectors for DMs or DMUs.
The content of this paper is organized as follows: In Section 2, we briefly introduce some preliminary information about probabilistic linguistic term sets (PLTSs) [11]. In Section 3, we firstly propose the concept of probabilistic hybrid linguistic term sets (P-HLTSs) and define some fundamental operational rules for P-HLTSs, then, a distance measure and entropy measure are both developed for P-HLTSs. In Section 4, we define the PUBL-PWA and PUBL-IPOWA operators. Their desirable properties are also studied. Subsequently, in Section 5, the details of the first and second approaches are presented. Next, we conduct a case study in order to verify the two approaches, considering an evaluation of the usability of a governmental website in Section 6. Finally, our conclusions are presented in Section 7.

Probabilistic Hesitant Fuzzy Set (P-HFS)
Definition 1. [59] If we let X be a finite set, then a probabilistic hesitant fuzzy set (P-HFS) on X is defined as follows: where ( ) l l p g represents the membership degree l g associated with the probability l p , ( ) h p is the probabilistic hesitant fuzzy element (P-HFE), and L is the number of all different membership degrees in ( ) h p .

Probabilistic Linguistic Term Set (PLTS)
In practice, decision makers may always have hesitancy among several possible linguistic terms when expressing their preferences, and the complete probabilistic distribution of these linguistic terms is usually not easily obtained accurately [24]. Pang et al. [11] thus have proposed the concept of a probabilistic linguistic term set (PLTS), which extends the hesitant fuzzy linguistic term set (HFLTS) [37] by adding probability information, without loss of any original compound decision information.

The Concept of Probabilistic Hybrid Linguistic Term Set (P-HLTS)
Recently, in their pioneering study, Lin et al. [51] introduced the concept of probabilistic uncertain linguistic sets (PULTSs), which allow decision makers to only express their preferences using balanced linguistic variables. Obviously, when adapting to various scenarios, decision makers are intrinsically inclined to use various linguistic variables when depicting their preferences, representatively including uncertain balanced linguistic term sets (UBLTSs) [51], uncertain unbalanced linguistic term set (UUBLTSs) [60,61], consecutive and nonconsecutive comparative balanced linguistic term sets (CBLTSs), or comparative unbalanced linguistic term sets (CUBLTSs) [48]. Therefore, in this section, we extend the definition of probabilistic hybrid linguistic term sets (P-HLTSs) in order to empower decision makers. For the purpose of acknowledging and differentiating our findings from the work of Lin et al. [51], in terms of differentiating PULTS from P-HLTS more clearly, we here denote PULTS as a probabilistic uncertain balanced linguistic term set (P-UBLTS), as shown in Definition 3.
where ( )  For practical usage, we can convert the various linguistic information forms shown above into uncertain balanced linguistic term sets (UBLTSs) by using corresponding transformation rules, shown as follows: Situation 1. l s  is selected from predefined unbalanced linguistic label sets (UUBLTSs) .
In this situation, P-HLTS actually changes into the new probabilistic uncertain unbalanced linguistic term set (P-UUBLTS) form, referred to as ˆ( ) L p  : By the transformation function y , whose procedures [27] have been detailed in Appendix A for further reference, the unbalanced linguistic term set can be transformed to a balanced linguistic term set. Thus, P-UUBLTS changes into P-UBLTS: Here, P-CBLTS is characterized by a set of generalized (either consecutive or non-consecutive) linguistic terms, therefore, we adopt Wang's [48]  By these rules above, we get the following: According to transformation functions between the comparative linguistic term set, unbalanced linguistic term set, and uncertain linguistic term set, various relationships between different probabilistic linguistic forms are shown in Figure 1.
In summary, our newly-defined probabilistic hybrid linguistic term set (P-HLTS) basically contains three new forms of probabilistic linguistic term sets, i.e., P-UUBLTS, P-CBLTS, and P-CUBLTS, which can be transformed into the same form of P-UBLTS [51]. Furthermore, we define the normalized version of P-HLTS in Definition 4. Since our newly defined P-HLTS can be transformed and unified into P-UBLTS [51], in the following section, we firstly detail the operational and comparison rules for P-UBLTS, then we propose novel distance and entropy measures for P-UBLTS in Section 3.3.

Basic Operational Rules and Comparison Rules for P-UBLTS
Pang et al. [11] defined the score and deviation degree for probabilistic balanced linguistic term sets (P-BLTSs). Lin et al. [51] extended these definitions and comparison rules into the environment of P-UBLTS, as shown in Definitions 6 and 7.

Definition 7. [51] If we let
a P-UBLTS number and ( ) k g be the subscript of P-UBLTS ( ) k L , then the score of ( ) L p  is given as follows: The deviation degree of ( ) L p is given as follows: Definition 8. [51] Considering two P-UBLTS numbers of 1 ( ) L p  and 2 ( ) L p  , then:

Proposed Distance Measure and Entropy Measure for P-UBLTS
When calculating the distance between any two P-UBLTS numbers, traditional distance measures [63] need to artificially complement, pessimistically or optimistically, certain numbers of elements to the unmatched P-UBLTS number. Either pessimistic complementing or optimistic complementing inevitably brings about information distortion to some extent. Therefore, in the following, we define another distance measure for P-UBLTS numbers without requiring any complementing operation.
Then, based on the normalized Hamming distance, we define a distance measure as follows: , then: Theorem 1. The distance measure 1 2 ( ( ), ( )) d L p L p   for P-UBLTSs satisfies the following properties: (1) (3) The entropy of fuzzy sets, first mentioned by Zadeh [64] and later widely employed in aggregation operator theories [40,52], is a useful tool to measure the fuzziness of information. Inspired by the entropy measures developed by Liu et al. [40] for probabilistic linguistic term sets (PLTS), we introduce a novel entropy measure for P-UBLTS here.
Theorem 2. The entropy defined in the Definition 10 satisfies the following requirements:

Prioritized Aggregation Operators for P-UBLTS
Considering that elementary WA and OWA operators cannot accommodate MADM problems with prioritized relationships among evaluative attributes, Yager [52,53]developed another two operators, namely, the fundamental prioritized average (PA) operator [52] and the prioritized ordered weighted aggregation (POWA) operator. Conventional PA and POWA operators generally assume that prioritization relationships among attributes can be determined by DMs or DMUs. When confronting ill-structured, complicated decision situations, the required attributes' prioritization relationships usually are explicitly unknown and can be objectively derived as order-inducing variables based on acquired decision information [65][66][67][68][69]. Therefore, to facilitate MADM under P-HLTS environments, where decision hesitancy and prioritization relationships between attributes exist, we present the probabilistic uncertain balanced linguistic prioritized weighted average (PUBL-PWA) operator. Furthermore, based on the conventional prioritized aggregation operator [52,53] and induced aggregation operator [65,66], we also develop a probabilistic uncertain balanced linguistic induced prioritized ordered weighted average (PUBL-IPOWA) . Desirable properties of the PUBL-PWA and PUBL-IPOWA are also investigated here.

Definition 11. Given a collection of P-UBLTS numbers
denotes the j k th uncertain balanced linguistic term and its probability respectively. The PUBL-PWA operator can be defined as follows: is the entropy of ( ) j L p  as calculated according to Definition 11.
Based on the operations of P-UBLTSs, the PUBL-PWA operator can be rewritten as in Theorem 3.

Theorem 3. If we let
Proof.
(3) Suppose n t = , then Theorem 3 is right, and we have the following: Then, when So, when 1 n k = + , Theorem 3 is also correct.
According to steps (1), (2) and (3), we conclude that Theorem 3 is right for all values of n . □ Theorem 4. The PUBL-PWA operator holds the following properties: (2) Boundedness: The PUBL-PWA operator lies between the max and min operators:

Proof.
(1) Assume that ( ) is any permutation of ( ) , there exists one and only one Thus, based on Theorem 3, we have the following: Based on the operations of P-UBLTSs, the PUBL-IPOWA operator can be rewritten as in Theorem 6.

Theorem 6. If we let
be a collection of P-UBLTS numbers, and Proof. Similar to the proof of Theorem 4, Theorem 6 also can be proved by the mathematical induction method, thus, detailed proof steps are omitted here. □ Additionally, in resemblance to the proof of Theorem 5, the PUBL-IPOWA operator also holds the properties of commutativity and boundedness.

Approaches for MAGDM under Probabilistic Hybrid Linguistic Environments with Decision Hesitancy and Attribute Prioritization Relationships
Aiming to address practical, complicated multiple attribute decision-making problems where decision hesitancy and prioritization relationships exist among the evaluated attributes, in this section, we employ the aforementioned expression tool of P-HLTS and its prioritized aggregation operators to develop two effective MAGDM approaches.
Here, we let Focusing on a specific scenario in which attribute prioritization relationships can be determined in advance and the relative importance of DMs or DMUs are not considered, we firstly construct the following approach. Step I-3. Calculate the prioritized weights

Approach I. MAGDM under P-HLTS environments with given prioritization relationships among the evaluated attributes
Step I-4. Obtain the aggregate results Step I -5. Calculate ( ) j E r  and ( ) j r s  according to Equations (14) and (17).
Step I-6. Based on the rules described in Definition 8, rank all the alternatives ( 1,2,..., ) j A j m = and select the most desirable one(s).
Next, regarding decision situations where the relative importance of DMs or DMUs is required but cannot be obtained according to the extant knowledge, or a prioritization relationship does exist among attributes yet cannot be determined based on the extant knowledge of DMs or DMUs, we constructed another approach. During processing in in the second approach, by use of the proposed distance measure for P-HLTS in Definition 9 we develop a method, as shown in Equation (30), that is based on similarity degrees [9] between decision matrices to derive the unknown weight vectors for DMs or DMUs. In essence, the similarity degree-based method allocates higher relative importance to the decision maker or decision-making unit whose decision matrix holds a shorter overall distance from others. To objectively determine the unknown group opinion on prioritization relationships among evaluative attributes, we introduce a divergence measure-based method, as shown in Equations (31) and (32) The divergence measure-based method is grounded in the fact that an attribute under which alternative assessments hold higher divergence is more effective in distinguishing alternatives than certain attributes that provide similar assessments for all alternatives. Therefore, in accordance with the descending order of all divergence measures, we can derive the prioritization relationships among the attributes. The proposed approach is detailed as follows:

Approach II. MAGDM under P-HLTS environments with unknown attribute prioritization relationships and unknown weights for DMs or DMUs
Step II-1. Transform each individual probabilistic hybrid linguistic decision matrix Step where d is the distance measure that we defined in Definition 9.
Step II-3. Derive the order inducing vector i e according to descending order of divergence measures of the assessments under each attribute, where we denote the divergence measure by i d : Step II-4. Calculate prioritized levels , such that the following is true: Step II-5. Calculate the prioritized weights Step II-6. Obtain the overall group aggregation results ( 1, 2, ..., ) of each alternative in the group matrix by applying the PUBL-IPOWA operator, where the following is true: Step II-7. See Step I-5.
For more clarity, Figure 2 shows flowcharts of the proposed approaches.

Output rank order of alternatives
Step

I-1 & Step II-1. Transform individual probabilistic linguistic decision matrices of various information forms into decision matrices in the unified form of probabilistic uncertain balanced linguistic term sets
Step I-5 & Step II-7. Calculate and Step I-6 & Step II-8. Based on the rules described in Definition 3.6, rank all the alternatives

Multiple attributes group decision making problems with preference information in the form of P-HLTS
Step II-4. Calculate prioritized levels Step II-5. Calculate group prioritized weights associated with PUBL-IPOWA operator Step II-6. Obtain overall group decision result of alternatives by use of PUBL-IPOWA operator

Derive order-inducing vector according to descending order of divergence measures of assessments under each attribute
Step II-2. Aggregate all individual decision matirx into the group decision matrix by use of PUBL-WA operator

Similarity degrees (SDs) between different individual decision matrices
Step II-3. Transform group decision matrix to the reordered group matrix Step I-2. Construct a synthesized group decision matrix in the form of P-UBLTS Step I-4. Obtain aggregate results of each alternatives in synthesized group decision matrix by applying PUBL-PWA operator Step I-3. Calculate prioritized weights associated with PUBL-PWA operator

Case Study on Governmental Website Usability Evaluation
Along with increasing complexity in their socioeconomic environments, the functionalities of modern governments have diversified to a great extent, especially in developing countries, which are full of socioeconomic dynamics. In order to deliver quality services to stakeholders and increase organizational effectiveness and efficiency, governments at all levels in different areas are trying to establish e-government strategies with the power of information and communication technologies [70]. For example, to guide urbanization rationally and effectively, the Chinese government has put forward national strategies to advocate characteristic towns that combine the functionalities of characteristic industry clusters, cities, communities, and culture and tourism. According to the strategic urbanization system, authorities at the provincial level have been organized and assigned with responsibilities to monitor and guide the development of local characteristic towns. Naturally, fostering and advancing the development of e-governments becomes an imperative part of management tasks for these provincial authorities. According to Baker [71], Clemmensen, and Katre [72], the existing literature has widely argued that e-government efforts will be stifled if e-government websites are not optimally designed to have good usability. Therefore, governments at different levels, such as the aforementioned Chinese provincial authorities, should adopt appropriate decision-making approaches to provide benchmark websites and share excellent experiences among their supervised institutions, so as to continuously improve overall excellence levels in building e-governments. Basically, six attributes of accessibility, information architecture, legitimacy, navigation, online services, and user-help and feedback are commonly utilized to comprehensively evaluate the usability of websites [71]. Essentially, a group of stakeholders should be involved during the process of evaluation, because a single decision maker generally cannot comprehensively consider these six attributes. As seen, MAGDM approaches exhibit intrinsic suitability in solving the complicated problems of comprehensive website usability evaluation.
Suppose that the administrative department in one of Chinese provincial authorities is rallying three decision-making units, namely, users, website developers, and academia experts, in order to determine the benchmarking alternative(s) from four websites     Case I: Suppose that all three decision-making units have reached a consensus on the prioritization relationships among the evaluative attributes, that is 3 , we here firstly apply the first proposed approach to determine the most desirable alternative website(s). The steps for this are organized as follows: Step I-1. Transform individual decision matrix 1 R in the form of P-CUBLTS, 2 R in the form of P-CBLTS, and 3 R in the form of P-UUBLTS into three decision matrices of 1 R  , 2 R  and 3 R  in the form of probabilistic uncertain balanced linguistic term sets, as shown in Tables 4-6.
Step I-2. Based on the three individual decision matrices of 1 R  , 2 R  and 3 R  , we then construct a synthesized group decision matrix 6 4 ( ) ij R r=   , as shown in Table 7.
Step I-5. Calculate ( ) j E r  according to Equations (14)- (16), where we have the following: Step I-6. Rank the alternatives according to descending order of ( ) j E r  , where we then have the following: Therefore, the best alternative website should be 3 A . Table 4. Transformed decision matrix   Table 6. Transformed decision matrix  Table 7. Group synthesized decision matrix R  . Case II: Suppose that the decision-making units have reached a consensus that there is a prioritization relationship among the evaluative attributes, but they cannot explicitly determine the prioritization relationships. In this case, the administrative department advocates that the relative importance of the decision-making units should be differentiated objectively according to their given assessments. Then, we may apply the second approach to solve the above complicated decision-making problem.
Step II-2. Determine the weighting vector for the three decision-making units according to Equation ( Table 8. Step II-3. By calculating the divergence measure i d according to Equations (31) and (32), we derive an order-inducing vector e in accordance with the descending order of i d , as listed in Table   9. Then, we can transform to the reordered group decision matrix according to the order-inducing vector e .  Table 9. Distance values and the corresponding order-inducing vector. Step II-4. Calculate prioritized levels by Equations (33) and ( Step II-5. Suppose ( ) f x x = , then, according to Equation (35), we obtain the attributes' weighting vectors as follows: Step II-6. Obtain overall group aggregation results ( 1,2,3,4) j r j =  by Equation (36). Please note that details about ( 1,2, 3, 4) j r j =  are omitted here for brevity.

Attributes
Step II-7. Calculate ( ) j E r  according to Equations (14)- (16), and we have: Step II-8. Rank the alternatives according to descending order of ( ) j E r  : Therefore, the best alternative is 2 A .

Comparative Experiments with Various Configurations of h and e
To further verify our approaches, in this section, we conduct more experiments on the two approaches with different configurations of the weighting vector, h , for the DMUs and the attributes' prioritization relationships, e . The experimental data and the obtained ranking results are presented in Table 10. prioritization relations e , the alternative 2 A was recognized as the best website. As shown in , derived in experiment II-3, has tiny differences from {1/3, 1/3, 1/3} h = , adopted in experiment II-2, experiment II-3 outputs the same ranking order as in experiment II-2. As can be seen from the above observation and analysis, decision makers' opinions on attributes' prioritization relationships substantially influences the decision results. Both approaches manage to accommodate the important opinions in their decision-making processes and reflect influences in ranking results accordingly. Generally, our proposed approaches present effective methodologies for MAGDM problems in P-HLTS environments where weighting vectors for DMs or DMUs and attribute prioritization relationships are known or explicitly unknown.

Comparative Experiments with Different Approaches
Consequently, because the various expression forms included in our proposed probabilistic hybrid linguistic term set (P-HLTS) can be transformed into the form of a probabilistic uncertain linguistic set [51] and both references [51], and since our research has been conducted to study MADM in group decision-making settings, we used the original synthesized group decision-making matrix in [51] for comparative experiments. The weight h for decision makers is thus not considered. Then, we configured our first proposed approach as shown in the experiment I-3 in Table 10, that is, we objectively calculate the order-inducing vector e based on the synthesized group decision matrix adopted in [51], then compare the corresponding ranking results as collected in Table 11.
In light of the comparative method in [51], which was constructed based on classic TOPSIS method, we here denote their method as extended TOPSIS in Table 11. Please note that our adapted first approach takes the synthesized group decision matrix, thereby using objectively derived e to guide the decision-making process, thus, the adapted first approach can be seen as a special case of our second proposed approach. Furthermore, to further verify the ranking results of our approach, we also constructed another approach to solve the same problem, named prioritized TOPSIS in Table 11, by integrating the TOPSIS framework and the e -guided attribute weights.
As seen from Table 11, after introducing the e -guided priority relationships among the evaluative attributes, both the adapted approach I and prioritized TOPSIS approach output different ranking results from the results generated by the extended TOPSIS [51]. Both the extended TOPSIS and the prioritized TOPSIS approaches inherit the robust decision-making process of the classic TOPSIS method, but the difference lies in that the prioritized TOPSIS approach integrates e -guided priority relationships among attributes and generates the ranking result of 1 Additionally, as can be seen via comparison of the result obtained by our proposed approach, both adapted approach I and prioritized TOPSIS indicate the best alternative as 4 A and the worst alternative as 1 A . Therefore, generally speaking, for situations where attributes' prioritization relationships are explicitly unknown, our proposed approaches provide a way of objectively determining e -guided priority relationships and effectively integrate priority relationships in MAGDM under P-HLTS environments in order to derive rational decision results.

Sensitivity Analysis
When tackling complicated multiple attribute decision-making problems where prioritization relationships exist among evaluative attributes, Yager [52] and Yager [53] have presented effective ways of deriving decision-making results by utilizing prioritized aggregation operators. During information aggregation, priority relationships among attributes and scope relationships among attributes generally should be taken into consideration [52,53]. Normalized priority importance weights for attributes can be calculated according to specified priority relationships. For decision-making scenarios of high complexity, where no concrete vectors exist for describing the scope of relationships among attributes, Yager [52,53,73] Table 12.
As shown in Table 12, we have carried out experiments on approach I (on the group synthesized decision matrix in Table 7) and approach II (on the aggregated group decision matrix in Table 8 { , , , , , } C C C C C C C = ). Due to complexity and uncertainty in practical scenarios, the values of some parts of parameters X and attributes C usually have to be expressed in uncertain forms, including linguistic variables and probabilistic hybrid linguistic term sets, etc. Especially, similar to the description in the preceding case study, priority relationships also will exist among the attributes as guided by specific operational management arrangements. Additionally, the task of this type of website usability evaluation presents problems regarding the priority attributes which are used to make the best decision (optimal X ) with the collected data and given guiding priority relationships among attributes.
More specifically, a technical system's (e-governmental website) usability function is defined by the vector of parameters 1 2 { , ,..., } n X X X X = . Operation of the technical system is determined by six characteristics (six attributes  Table 13. Table 13. Collected values of the parameters and characteristics of the technical system.

Parameters Characteristics
Step III-3. Solve the above vector problem of mathematical programming with equivalent attributes [79].
Step III-4. According to specific operational management arrangements, decision makers decide to choose priority attributes and determine the numerical value of the corresponding priority attributes.
Step III-5. With the given attribute priority, Mashunin and Mashunin's [78] methods are used to obtain the optimal parameter vector oo X within the assigned error range.
As pointed by Mashunin and Mashunin [78], the above approach of optimal decision-making with an assigned attribute priority is based on the axioms with the use of the normalization of attributes and the max-min principle, and the accuracy of choosing such an optimal parameter vector depends on a predetermined error range. Regarding the specific implementation of the vector optimization methods, of which the third approach presented here depends on, one can refer to references [78,79] for great insight in this regard. After the acceptable optimal vector oo X is obtained and the optimal parameters are further analyzed according to concrete scenarios, managerial suggestions for operational improvements thus can be deduced more reasonably.

Conclusions
Aiming to deal with ill-structured, complicated problems under linguistic MADM scenarios, we have introduced a more comprehensive expression tool for P-HLTSs to depict decision hesitancy concerning possible linguistic labels and the probabilistic preferences on those linguistic labels. P-HLTSs enhance classical probabilistic linguistic term sets through encompassing a wide range of linguistic expression forms when answering diverse cognitive models, including balanced linguistic sets, unbalanced linguistic term sets, uncertain unbalanced linguistic term sets, comparative balanced linguistic term sets, and comparative unbalanced linguistic term set. For MADM under complicated situations, where decision makers cannot determine concrete weighting vectors for attributes but have opinions on attributes' prioritization relationships, we have developed two important information aggregation operators (PUBL-PWA and PUBL-IPOWA), used to accommodate the specific decision opinions. Then, based on the above tools, with respect to practical MADM problems where weighting vector for DMs or DMUs and attributes' prioritization relationships are known or explicitly unknown, we have constructed two effective MAGDM approaches, respectively. Additionally, through application study and comparative experiments we have validated the two approaches.
Since the newly-introduced expression tool of P-HLTS behaves more flexibly and comprehensively when eliciting probabilistic linguistic decision information, future research efforts should be directed to studying effective aggregation operators for various practical contexts, thereby developing appropriate MADM methodologies for real applications. Another promising research direction should be well-explored and exploited by integrating MADM methodologies and vector optimization theories.  ) ( )