A Two-sided Matching Decision-making Approach based on PROMETHEE Under the Probabilistic Linguistic Environment

Matching problems in daily life can be effectively solved by two-sided matching decision-making (TSMDM) approaches. The involved matching intermediary is to match two sides of subjects. This paper proposes a TSMDM approach based on preference ranking organization method (PROMETHEE) under the probabilistic linguistic environment. The probabilistic linguistic evaluations are firstly normalized and transformed to the benefit types. Then, the preference degrees of a subject over other subjects from the same side are obtained by using six types of preference function. Afterwards, groups of preference degrees of a subject are aggregated to the preference indexes by considering the weights of criteria. Hereafter, the preference degrees of a subject over other subjects from the same side are aggregated to the outgoing flow, while the preference degrees of other subjects from the same side over this subject are aggregated to the incoming flow. Furthermore, the net-flows, which is recognized as the satisfaction degrees are calculated by using outgoing flows to minus incoming flows. On the basis of this, the multi-objectives TSMDM model is built by considering the matching aspirations. A model with respect to the matching aspirations is built and solved by using the Lagrange function. The multi-objectives TSMDM model is further transformed to the single-objective model, the solution of which is the matching scheme. A matching problem related to the intelligent technology intermediary is solved to verify the effectiveness and the feasibility of the proposed approach.


Introduction
Two-sided matching decision-making (TSMDM) problems involve three elements: two sides of subjects and an intermediary. The task of the intermediary is to match two sides of subjects one by one, where the subjects are discrete and finite. Gale and Shapley (1962) introduced the TSMDM problems by investigating the college admissions and stability of marriage. On the basis of this, we recount the TSMDM problems by the example of marriage. Suppose there are three females and four males are to be matched by a marriage intermediary, then three females construct a side while four males construct the other side. The evaluations of females given by males and that of males given by females are previously obtained by the marriage intermediary, the task of which is to match males and females according to these evaluations. Since a male can only match a female, all females are matched, three males are matched and a male is not matched obviously. To deal with the matching problems, the approach the intermediary utilized is called the TSMDM approach. Since Gale and Shapley (1962) proposed the deferred-acceptance algorithm, lots of TSMDM approaches are extracted and further employed to the practical problems, such as the matching for technological knowledge suppliers and demanders (Han et al. 2018;Liu and Li 2017), the matching for two-way referral system , the matching in smart environmental protection (Yue et al. 2019).
Evaluations are important information for subjects to measure the subjects from the other side. Fan et al. (2017) considered the condition where the subjects can only give their uncertain preference ordinal and further proposed a TSMDM method considering elation and disappointment of subjects. Liang et al. (2020) considered the incomplete preference ordinal environment and extracted a TSMDM approach, where the missing information is expressed by using the similarity measures.
Considering the fuzziness of the subjects when giving their evaluations, Wang et al. (2016), Yu and Xu (2019), Yin and Li (2018) proposed several TSMDM approaches, where the evaluations are expressed as intuitionistic fuzzy sets. Yue et al. (2016) extended the TSMDM approach to accommodate the interval-valued intuitionistic fuzzy environment. Yue and Zhang (2020)   considered the hesitancy of the subjects and developed TSMDM approaches, where the evaluations are hesitant fuzzy sets. Although the evaluations of preference ordinal are qualitative, the expression of which is not simple enough for subjects. Except the preference ordinal, the above mentioned evaluations are all quantitative, which is complicated for subjects to express. Zadeh (1975) pointed out the linguistic information can be described as the linguistic variables, which are quantitative mathematical symbols. On the basis of this, Xu (2012) introduces the linguistic terms (LTs). The LTs are simple enough for subjects to express their qualitative evaluations. In other words, the qualitative information is quantitatively expressed by using the LTs. To take into account the linguistic information of subjects, Lin et al. (2018) introduced an enhanced TSMDM approach, where the evaluations are 2-tuple linguistic preferences. Rodriguez et al. (2012) defined hesitant fuzzy linguistic term sets (HFLTSs) based on hesitant fuzzy sets and the LTs. The HFLTSs can effectively express the qualitative information in a hesitant condition, where the subjects prefer to give several LTs. Pang et al. (2016) pointed out the drawback of the HFLTSs that the probabilities of the involved LTs are identical, they further defined probabilistic linguistic term sets (PLTSs), in which the probabilities can be varied by subjects according to their preference and degree of belief. PLTSs are mainly implemented to describe the evaluations in decision-making problems. Wang et al. (2019) introduced two distances of the PLTSs and further proposed a distance-based multi-criteria group decision-making approach.  presented several entropy measures for the PLTSs, including the fuzzy entropy, the hesitant entropy and the total entropy. Liao et al. (2017) established a linear programming method for multicriteria decision-making, where the evaluations are PLTSs. Gou and Xu (2016) defined several novel basic operational laws for the PLTSs based on equivalent transformation function. By considering the disparity degrees of the LTs, Liao et al. (2019) also defined several novel operational laws for the PLTSs. Wu et al. (2018) proposed a multi-criteria decision-making method based on the expectation function of the PLTSs and the improved Borda rule. Kobina et al. (2017) defined the probabilistic linguistic power aggregation operators, on the basis of this, they further extracted a multi-criteria group decision-making method. The application of the PLTSs in the decision-making methods is proved to be effective. Since the subjects are mainly human who prefer to conveniently give their oral evaluations, it is necessary to introduce the PLTSs to the TSMDM approaches.
Comparison is usually used for decision-making. Among existing decision-making approaches, the preference ranking organization method (PROMETHEE) is a typical one for comparison. Brans and Vinckes (1985) introduced the PROMETHEE to ensure it is easily accepted and understood by the decision-maker. Later, several modified decision-making methods were developed to extend the PROMETHEE (Chen et al. 2011;Dağdeviren 2008;Vinodh and Girubha 2012;Vujosevic and Popovic 2016). The objects are compared in PROMETHEE under six different criteria, which is the most outstanding feature. However, the above mentioned PROMETHEE are developed under the real number environment and did not consider the fuzzy condition. It is worthy to note that Liu and Li (2018) and  introduced the PROMETHEE, where the evaluations are the PLTSs.
The case studies in these two decision-making methods combining the PROMETHEE and the PLTSs exhibit that the application of the PLTSs is more precise and convenient when comparing several objects.
On the basis of these analyses, we propose a TSMDM approach based on PROMETHEE, where the evaluations are the PLTSs. The contributions of this paper are summarized as follows: (1) The preference functions for the PLTSs are defined based on the PROMETHEE.
(2) A novel matching aspiration model is built.
(3) A TSMDM approach under the probabilistic linguistic environment is proposed.
The rest of this paper is organized as follows: Section 2 reviews several basic concepts of the PLTSs and the two-sided matching. Section 3 proposes a TSMDM approach based on PROMETHEE in details. Section 4 exhibits an illustrative example by using the proposed approach.
Section 5 conducts sensitivity analysis and comparative analysis. Conclusion is pointed out in Section 6.

Preliminaries
In this section, several basic concepts with respect to the PLTSs and the two-sided matching are reviewed. Xu (2012) defined the linguistic term set (LTS) as   , , 1,0,1, ,

PLTSs
where the mid linguistic term 0 s represents an assessment of "indifference",  is an integer and 1   , and with the rest of the linguistic terms being placed symmetrically around it. LTs 12 , s s S   and satisfy two conditions: (1) If 12   , then 1 2 ss   ; (2) The negation operator is defined:   Lp. We call a PLTS that has been ordered, assigned the ignored probability, added the smallest LT the normalized PLTSs.
Definition 4 (Pang et al. 2016). Let Based on the mean-variance rule, Pang et al. (2016) defined the comparison method for the PLTSs.
Definition 5 (Pang et al. 2016 The comparison method proposed by Pang et al. (2016) can only reflect the qualitative relationship between two PLTSs, the quantitative difference between them cannot be obtained. Lin and Xu (2017) defined the normalized Hamming distance for the PLTSs.

Two-sided Matching
subject sets. Without loss of generality, we let nm  .
Definition 7 (Lin et al. 2018). A matching is a mapping In some conditions such as the matching of personnel and jobs, the company not only conducts the matching process, but also participates into the matching as the side of jobs. The intermediary is considered to be the third part in this research. The main task of the TSMDM approaches is to match two sides of subjects one by one and make the matching scheme satisfy all subjects. (1) and (2) in the Definition 7 imply that the subjects may not be totally matched, i.e., the number of the matching pairs is less than m . In such cases, there remain some subjects from side A and B that are not matched, which violate the task of the TSMDM approaches. Since the number of the subjects from side A is less than that of side B , i.e. nm  , we redefine the condition (1)

TSMDM Approach
The TSMDM problem is firstly described in this section, then we propose a TSMDM approach.

Description of TSMDM Problem
The subject sets A and B are identical to the subject sets in Definition 7. The criteria set To solve this problem is to find an optimal matching scheme for sides A and B by considering the above information.

Matching Process
The criteria in decision-making problems can be sorted into two kinds: benefit types and cost types. A higher evaluation is better for the benefit criterion, while a lower one is better for the cost criterion. For a group of criteria including benefit ones and cost ones, the identical operations for the evaluations under these criteria can cause confusion. Hence, transforming the cost type of evaluations to the benefit type is necessary. For the evaluation

if it is under the benefit criteria, then holds its value, if
it is under the cost criteria, then the involved LTs should be transformed by means of their negation operator. We further define a transforming function as: Brans and Vincke (1985) classified the criteria into six types and defined distinct preference function for them. However, the condition they considered is that the evaluations are real numbers and their difference can be easily measured by the subtraction operator. The operation is not that simple in the fuzzy environment. The subtraction operator is not defined for the PLTSs. The contribution of the subtraction operator is to calculate the difference between two real numbers.
Distances are usually used for measuring the difference between two PLTSs. We extend the preference functions to accommodate the probabilistic linguistic environment based on distance. On the aspect of evaluations of j B to   12   12   1  2  1  2  1  2   1  2  1  2   11  ,,   11  ,,  13  1  1  1  1  1  1  1  , , ,      1  11  12  13  14  15  16   ,  , , , , For the distance-based preference function 2  21  22  23  24  25  26   ,  , , , , Similarly, the incoming flows can be defined as: Subsequently, the net-flows are defined as: Since the higher the 1 ij a is, the higher the satisfaction of j B to 1 i A will be, we redefine ( 1) . . 1, 1, 2, , ,  (23) and (24) ( 2) . . 1, 0 1,  (29) we find two closed mutual satisfaction degrees make the matching aspiration larger, in other words, the matching aspirations are obtained in the light of the difference between two satisfaction degrees rather than the values of them. If we only consider the matching aspirations and make them to reach their maximum, then two subjects may be matched whose mutual satisfaction degrees are closed but lower than others, which is irrational. If we only consider the satisfaction degrees, although we can reach the maximum of the comprehensive satisfaction degrees, the satisfaction degrees of two matched subjects may be far apart. Although the summation of two mutual satisfaction degrees of two matched subjects is larger, the matching aspiration between this two subjects may be lower, which indicates the lower success rate for matching them.   2 M simultaneously considers the matching aspirations and the satisfaction degrees and effectively overcomes the above mentioned issues. Equation (30) integrates the matching aspirations and the satisfaction degrees by the multiplicative operator. The multiplicative operator can be seen as a restriction, it makes the comprehensive satisfaction degrees reach their maximum and ensures the satisfaction degrees of two matched subjects close enough.

Decision Steps
The decision process can be summarized as the following steps.
Step 1: Obtain the probabilistic linguistic evaluations and build evaluation matrices Step 2: Normalize the evaluations and transform the evaluations under the cost criteria to the benefit types by utilizing equation (2).
Step 3: Classify the criterion 1 C  and calculate the preference degree of j B to i A compared with other subjects of side A under 1 C  by utilizing equations (3)-(8). Classify the criterion 2 C  and calculate the preference degree of i A to j B compared with other subjects of side B under 2 C  by utilizing equations (9)-(14).
Step 4: Aggregate the preference degrees to the preference index for sides B and A by utilizing equations (15) and (16) respectively.
Step 5: Calculate the outgoing flows for sides B and A by utilizing equations (17) and (18) respectively. Calculate the incoming flows for sides B and A by utilizing equations (19) and (20) respectively.
Step 8: Build model   Step 9: Transform the multi-objectives TSMDM model   Step 10: Solve   3 M to obtain the matching scheme.

An Illustrative Example
Intelligent technology intermediary is a matching intermediary, the main business of which is to polymerize the intelligent technology suppliers and the demanders according to their requirements and mutual evaluations. The intelligent technology intermediary can help the intelligent technology suppliers and the demanders to reduce their cost for searching each other and market survey. The intelligent technology suppliers are individuals or companies who own the knowledge or property for the technology, while the demanders are enterprises who are seeking the technology. There are two distinguished advantages of the intelligent technology intermediary. One is that it is third part and do not participate into the matching process, the recommendation of which is objective. The other is that it gathers the information with respect to the intelligent technology and assesses the suppliers and the demanders previously, the suggestion of which is valuable. In a period, an intelligent technology firm in Fuzhou has received the requests from three demanders and four suppliers, the demanders look for face recognition technology while the suppliers tend to find partners. Owing to the financial pressure, a demander can only purchase the technology from one supplier. It is necessary to avoid the technology monopoly, hence, a supplier can only collaborate with one demander. The demanders are 1 Step 1: Obtain the probabilistic linguistic evaluations. The evaluation matrix   Step 2: The evaluations in Table 1-Tbale 4 are normalized to the form in Table 5-Tbale 8 respectively. The evaluations are transformed by using equation (2). Since the criteria are benefit, the evaluations are not changed.   Step 4: Aggregate the preference degrees to the preference indexes by using equations (15) and (16) respectively.
Step 5: The outgoing flows for sides B and A are calculated by using equations (17) and (18) respectively. The incoming flows for side B and A are calculated by using equations (19) and (20) respectively.
Step 6: The satisfaction degrees of i A given by j B and j B given by i A are calculated by using equations (21) and (22) respectively. The satisfaction degrees of i A given by j B are shown as Table 9, while the satisfaction degrees of j B given by i A are shown as Table 10. Step 7: Build multi-objectives TSMDM model   1 M .
Step 8: Build model   2 M . By solving the   2 M , we obtain the values of ij  as shown as Table 11. Step 9: Transform the   1 M to   3 M . Since there is no declaration of weights of two sides, we let 0.5 AB  .

Analyses
In this section, we conduct sensitivity analysis and comparative analysis for the proposed TSMDM approach.

Sensitivity Analysis
Weights of sides A and B are A  and B  respectively, which are used to measure the importance of two sides. The weights are not always 0.5 because there are several reasons to make the change of the value of weights, such as cost, remuneration, and reputation. The matching scheme may change according to the value of weights. By varying the value of A  and B  from zero to one consistently, we obtain a group of matching schemes associated with the values of goal function shown as Table 12. On the basis of the two-sided matching introduced in Definition 7, three conditions are satisfied, the criteria, the weights of criteria, the probabilistic linguistic evaluations, and the matching aspirations are all considered and integrated into the proposed TSMDM approach. The stability of the proposed approach is verified in the light of the sensitivity analysis.

Comparative Analysis
We compare the proposed approach with the TSMDM method proposed by Yue et al. (2016) in this subsection.   The comparative analysis is separated into four parts as follows: (1) Although this matching scheme is the same as that obtained from the proposed approach, the matching aspirations are different. In Yue et al. (2016)s' method, although 2 A matches 1 B , the matching aspiration between them is the lowest no matter in the first column or in the second row. The lowest matching aspiration for both 2 A and 1 B means the lowest success rate between them, which is contradict with the matching scheme. In the proposed TSMDM approach, the matching aspiration between 2 A and 1 B is 0.1466, which is the largest in the first column and second row. Hence, 2 A matches 1 B in the matching scheme obtained from the proposed approach is reasonable.
(2) Yue et al. (2016) use the IVIFSs to express the evaluations. However, the IVIFSs can only describe the quantitative evaluations rather than the qualitative ones. The PLTSs we introduced can describe the qualitative evaluations given by the subjects, which are more convenient.
(3) Yue et al. (2016) straightforwardly transform the fuzzy evaluations to their scores, which may cause the loss of information. We compare the probabilistic linguistic evaluations and derive the mutual preference degrees for two subjects. We consider six conditions and further define different preference functions for six types of criteria.
(4) Yue et al. (2016) don't consider the condition of multi-criteria, which is not useful for reallife problems. We furnish multiple criteria for two sides of subjects to help them evaluate each other comprehensively, which is useful.

Conclusion
TSMDM approaches are widely used in real life. Considering the comparison between the subjects, we propose a TSMDM approach based on the PROMETHEE, where the evaluations are PLTSs. Six preference functions are firstly extended to accommodate the probabilistic linguistic environment. The subtraction in preference functions are substituted by the distance between the PLTSs. The condition of an evaluation is larger than another but their difference is lower than a threshold is separated to two conditions, that is, a probabilistic linguistic evaluation is larger than another but their distance is lower than a threshold. By using the PROMETHEE, the subjects are compared to other subjects from the same side under each criterion. The net-flows, which are redefined as the satisfaction degrees are also obtained. The matching aspirations are proper tools to measure the success rates for matching two subjects, thus they are combined in the multi-objectives TSMDM model. A novel model aiming to calculate the values of matching aspirations is built, and is further solved by using the Lagrange function. We introduce the weights of two sides and further transform the multi-objectives TSMDM model to the single-objective TSMDM model with the derived value of matching aspirations. The proposed TSMDM approach is employed to resolve a matching problem for an intelligent technology intermediary in Fuzhou to verify the feasibility and the effectiveness. The sensitivity analysis is conducted to test the stability of the proposed approach in eleven conditions where the weights of two sides are varied consistently. The matching scheme is unchanged no matter how the weights vary, hence, the proposed TSMDM approach is stable. The comparative analysis between the proposed TSMDM approach and the existing TSMDM method considering the matching aspirations shows the progressiveness in four parts: the value of matching aspirations, the expression of evaluations, the transformation of evaluations, and the condition of multi-criteria.
In the future, we will consider several more complicated conditions for defining the preference functions under the probabilistic linguistic environment. Also, we will investigate the weight determining method for the TSMDM approaches.