Extend TOPSIS-Based Two-Sided Matching Decision in Incomplete Indifferent Order Relations Setting Considering Matching Aspirations

This paper develops a method for two-sided matching decision in the environment of incomplete indifferent order relations. The two-sided matching decision problem with incomplete indifferent order relations and matching aspirations is firstly described. In order to solve this problem, the incomplete indifferent order relations are converted into the generalized Borda number matrices. The matching aspiration matrix can be determined based on the model calculation on the reciprocal differences of generalized Borda numbers. On this basis, the weighted satisfaction degree matrices are set up. The extended relative closeness matrices are determined by using an extended TOPSIS technique. Moreover, a two-sided matching model is developed. The two-sided matching alternative can be obtained by solving the model. For the purpose of illustration, an example including sensitivity analysis is presented.


INTRODUCTION
The two-sided matching decision involves how to match the agents of one side with the agents of the other side based on the preferences of the agents of both sides.The problems of two-sided matching decision exist widely in reality, such as stable marriage assignment (Kümmel et al., 2016;Cseh and Manlove, 2016;Doğan and Yıldız, 2016), college admission (Braun et al., 2014;Chen and Kao, 2014;Liu and Peng, 2015), employee selection (Wang et al., 2011;Mendes et al., 2010;Chen et al., 2016), and personnel assignment (Gallego and Larrain, 2012;Taylor, 2013;Gharote et al., 2015).Therefore, the two-sided matching decision is a hot topic with extensive actual backgrounds.Gale and Shapley (1962) initially research the problems of college admissions and marriage.In their studies, the concept of stable matching is proposed; then the existence and optimality of stable matching are given; at last, the deferred acceptance algorithm is developed.From then on, various different concepts, theories, techniques and algorithms have been presented with respect to the two-sided matching decision with different formats of information.For example, Li and Fan (2014) propose a stable two-sided matching method considering psychological behavior of agents on both sides to solve the two-sided matching problem with ordinal numbers.Castillo and Dianat (2016) study the truncation strategies in a centralized matching clearinghouse based on the deferred acceptance algorithm.Xu et al. (2015) propose the matching algorithms for one-to-one two-sided dynamic service markets.Chen et al. (2016) point out that the generalized median stable matchings exist in many-to-many two-sided matching markets when contracts are strong replaceable and satisfy the law of aggregate demand.Liang et al. (2015) propose a novel decision analysis method to solve the multiple target of satisfied and stable two-sided matching decision problem considering the preference ordering, where the targets could be satisfied, weak satisfied and stable, -satisfied and stable, satisfied and stable.
The existing studies enrich the theories of two-sided matching decision, and develop the different algorithms for solving the problems of two-sided matching decision with various formats of information, and expand the actual application background.However, on the one hand, the preferences provided by agents of two sides may be in the format of incomplete indifferent order relations in some practical problems owing to imprecise source of information containing unquantifiable information and incomplete information.In this case, the classical two-sided matching decision cannot effectively deal with these kinds of problems.On the other hand, the matching aspirations of agents of two sides are seldom considered in the existing studies.Therefore, how to investigate the problem of two-sided matching decision with incomplete indifferent order relations considering matching aspirations is a valuable research topic.In view of this, this paper presents a two-sided matching decision method with incomplete indifferent order relations considering matching aspirations based on an extended TOPSIS (technique for order performance by similarity to an ideal solution) method.It is well known that, the TOPSIS method is first developed by Hwang and Yoon (1981), and is one of the classical multi-attribute decision methods.The basic idea of TOPSIS method is that the selected alternative should have the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution (Hwang and Yoon, 1981;Yue, 2014).This article intends to apply the idea of TOPSIS into two-sided matching decision with incomplete indifferent order relations information.
The structure of this paper is organized as follows: Section 2 formulates the considered two-sided matching problem.Section 3 presents an extend TOPSIS-based method for two-sided matching decision.Section 4 gives an example.Section 5 concludes this paper.

THE CONSIDERED TWO-SIDED MATCHING PROBLEM
This paper considers the two-sided matching problem, where the preferences provided by the agents of two sides are in the format of incomplete indifferent order relations.And the research angle employed in this paper is matching aspiration.The notation of the considered two-sided matching problem is given as follows.
∂ be the incomplete indifferent order relation given by agent ℘  , where   ∂ denotes the number of agents of side Here, symbol "  " or "  " denotes "superior to" or "be equivalent to"; Let   be the matching aspiration between ∂ i and ℘  , which usually satisfies the characteristic of non-negativity and normalization.
Remark 1.In above presentation, the matching aspiration   is unknown.The determination method will be given in section 3.2.
Remark 2. The concept of two-sided matching can be seen by reference (Yue, 2014).Then we know that a twosided matching (or a two-sided matching alternative)  can be expressed by the union of the set of matching pair   and the set of single pair   .
Based on the above analysis, the problem researched here is how to obtain the reasonable two-sided matching alternative  based on the incomplete indifferent order relations , and the matching aspirations   ( ∈ ,  ∈ ).

Construction of the Normalized Borda Matrices
In order to handle with the incomplete indifferent order relations, the definitions of the generalized Borda numbers are introduced.

Contribution of this paper to the literature
• Rather than making a contribution to the relevant theories of stable two-sided matching, the focus of this study is to obtain the two-sided matching alternative, which can reflect the matching aspirations of agents.
• This study tries to use the generalized Borda numbers for handling incomplete indifferent order relations.
• The findings of this study can enhance our understanding of an extend TOPSIS technology in two-sided matching decision.

Determination of the Matching Aspirations
In order to determine the matching aspiration   , the following analysis combined with the absolute difference Remark 4. In Eq. ( 5), the case of �̅  ∂→℘ − ̅  ℘→∂ � = 0 may sometimes occur.At this time, Eq. ( 5) is meaningless.
In order to handle with this case, the denominator �̅  ∂→℘ − ̅  ℘→∂ � can be replaced by According to Remark 3, Eq. ( 5) can be further expressed by Based on the above analysis, the selection of   should make  ↔℘ greatest.Therefore, the objective function is established as Moreover, the following linear programming model (M-1) can be constructed, i.e., = 1;   ∈ (0, 1),  ∈ ,  ∈  Theorem 1.The optimal solution (noted as   * ) of model (M-1) is expressed by Then the partial derivatives of function  with respect to variables   and  can be computed, i.e.,

Building of the Extended Relative Closeness Matrices
Obviously,   ∂→℘ ∈ [0,1] ⋃ , and the greater   ∂→℘ is, the higher the satisfaction degree of agent ∂ i over ℘  is.
Similarly, with respect to weighted satisfaction degree matrix Obviously,   ℘→∂ ∈ [0,1] ⋃ , and the greater   ℘→∂ is, the higher the satisfaction degree of agent ℘  over ∂ i is.

Development of the Two-Sided Matching Model
Firstly, the 0-1 variable   is introduced, where According to the meanings of the extended relative closeness, maximization of the extended relative closeness can be regarded as the objective function.Furthermore, considering the constraint condition of one-to-one twosided matching, the following two-sided matching model (M-2) can be developed, i.e.,

Determination of the Two-Sided Matching Algorithm
In sum, an algorithm for solving the two-sided matching problem under the conditions of incomplete indifferent order relations considering matching aspirations is given.The steps of the algorithm are provided as follows.
Step Step 11.Transform model (M-2) into model (M-3) by using the linear weighted method.
Step 12. Determine the two-sided matching alternative by solving model (M-3).

ILLUSTRATED EXAMPLE
In this section, an example is used to illustrate the application of the proposed extend TOPSIS-based two-sided matching decision.
Suppose an oversea venture-capital company plans to invest a cell-phone company in Nan Chang of China.In order to enable the new cell-phone company to run smoothly, the manager intends to assign experienced staffs to vacant positions in the new factory.Each position in the new cell-phone company is held by one staff, and each staff is assigned to only one position.There are five vacant positions, which consist of a purchaser ( 1 ), a material handler ( 2 ), a production planner ( 3 ), a technician ( 4 ) and a quality inspector ( 5 ).Now seven experienced staffs ℘ 1 , ℘ 2 , …, and ℘ 7 who have multiple skills apply to the five positions.The decision makers from five position departments evaluate the staffs from four perspectives: personality characteristics, technical skill, previous experience, and human relationship skill.Seven staffs evaluate the positions from three perspectives: salary and welfare, development space, and work environment.The incomplete indifferent order relations ) are given below.In order to enhance the level of operating efficiency, the intermediary who specializes in human resource allocation is employed to show the two-sided matching alternative.
To solve the above problem, the proposed two-sided matching decision is used and the procedure is given as follows.
Step 1.According to the incomplete indifferent order relations × can be built by Eqs. ( 1a)-( 1c) and ( 2), which is shown in Table 1.
According to the incomplete indifferent order relations × can be built by Eqs.(3a)-( 3c) and ( 4), which is shown in Table 2.
Step 7. Based on the weighted satisfaction degree matrix   25), which is shown in Table 11.
In the following, we discuss the influence of weights  ∂ and  ℘ towards the two-sided matching alternative.
Case III.If  ∂ = 0.9,  ℘ = 0.1, then model (M-2) is transformed into model (M-3), where   ℘↔∂ = 0.9  ∂→℘ + 0.1  ℘→∂ .Similarly, through solving the model, the two-sided matching alternative  * can be obtained, where     In conclusion, the comparative analysis on the influence of weights  ∂ and  ℘ towards the two-sided matching alternative is shown in Table 15.From Table 15, we know that the two-sided matching alternative may be changed when weights  ∂ and  ℘ are changed.Therefore, weights  ∂ and  ℘ play an important role in determining the two-sided matching alternative.

CONCLUSIONS
This paper presented a decision method for solving the two-sided matching problem with incomplete indifferent order relations considering matching aspirations.The incomplete indifferent order relations were transformed into the generalized Borda number matrices, and the matching aspirations were calculated based on model calculation.Based on this, the weighted satisfaction degree matrices were built.Then, the extended relative closeness matrices can be determined by using an extended TOPSIS method.Furthermore, a two-sided matching model can be constructed.By solving the proposed model, the two-sided matching alternative was determined.An example with sensitivity analysis is also given to illustrate the effectiveness of the presented method.
Comparing with the existing research, the main contribution of this paper is as follows: (1) the generalized Borda numbers were adopted to handle incomplete indifferent order relations, which is a new idea; (2) the research angle was matching aspirations, and hence the obtained two-sided matching alternative could reflect the matching aspirations of agents; (3) an idea of extend TOPSIS was firstly introduced into two-sided matching decision, which was a novel idea; (4) the presented method developed the theory and method for two-sided matching decision with incomplete indifferent order relations.
The main limitation of this paper is that it only discussed the two-sided matching problem with less incomplete indifferent order relations.And the related theory of stable matching under the condition of incomplete indifferent order relations was not studied.
Hence, the following two aspects could be further studied.First, if the complete two-sided matching alternative cannot obtained based on the information of incomplete indifferent order relations, then this kind of two-sided matching decision problem should be further investigated.Second, the theory and property of stable matching with incomplete indifferent order relations should be further probed.

Figure 1 .
Figure 1.The two-sided matching alternative