Decision method for two-sided matching with interval-valued intuitionistic fuzzy sets considering matching aspirations

Abstract

This paper studied the two-sided matching problem based on interval-valued intuitionistic fuzzy sets and matching aspirations, the related concepts of which were described. In order to solve this problem, the interval-valued intuitionistic fuzzy matrices were transformed into interval score matrices and score matrices. Based on the score matrices, matching aspiration matrix, and matching matrix, a two-sided matching model under the conditions of one-to-one two-sided matching was developed. Therein, matching aspirations were calculated using the maximum reciprocal-deviation principle. Considering the statuses of all agents on both sides were the same in most situations, the two-sided matching model was converted into a single-objective optimization model using the linear weighted method. “Optimal” two-sided matching was obtained by solving the model. At the conclusion of the study, the feasibility and effectiveness of the proposed method was illustrated with an example of matching executives with positions.

Keywords

Two-sided matching interval-valued intuitionistic fuzzy set matching aspiration maximum reciprocal-deviation two-sided matching model

1 Introduction

Two-sided matching is a common problem that exists in real life. Examples include symmetric stable marriage, matching in advertising, college admissions, matching service providers and customers, assignments for trainees to software project requirements, etc. [7 , 20]. Gale and Shapley initially studied the stable marriage and college admissions models, and proposed the well-known Gale-Shapley algorithm [1]. This was followed by the study of various two-sided matching problems by many other scholars [2 , 18]. Because the “optimal” two-sided matching alternative can enhance the satisfaction degree of agents, studying the theories and methods for two-sided matching has important theoretical significance and practical application value.

At present, the theories and methods for two-sided matching with ordinal numbers, order relations, and linguistic terms have been useable. However, considering the complexity and fuzzy uncertainty of the two-sided matching problem, the cognitive limitation of agents, the preferences of two-sided agents are sometimes in the format of interval-valued intuitionistic fuzzy sets, which was an idea first introduced by Atanassov and Gargov and is an extension of the intuitionistic fuzzy sets [8, 9]. The difference is that the degrees of membership and non-membership are interval-valued numbers, not real numbers. Hence, it can depict the fuzzy nature of objective things more skillfully [9]. The theory of interval-valued intuitionistic fuzzy set, which is provided in Table 1, has been widely used in the field of decision making [5 , 21]. However, the theory is seldom applied to the field of two-sided matching decisions. If the two-sided matching decision is investigated from the perspective of matching aspiration, then the success rate of two-sided matching is improved. Therefore, studying the two-sided matching problem with interval-valued intuitionistic fuzzy sets from the point of view of matching aspiration is significant.

In view of this, this paper applied the theory of interval-valued intuitionistic fuzzy set to the field of two-sided matching decisions, and studied the problem from the perspective of matching aspiration. The results of this paper determined that the main advantage of the proposed method is that it is able to solve two-sided matching problems with interval-valued intuitionistic fuzzy sets and can reflect the matching aspirations of agents.

2 Preliminary

2.1 Interval-valued intuitionistic fuzzy set

Definition 1. By letting E be a non-empty set, then $\bar{I} = {< x, t_{\bar{I}} (x), f_{\bar{I}} (x) > | x \in E}$ is called the interval-valued intuitionistic fuzzy set (IVIFS) [9] on E. Here, the functions $t_{\bar{I}} (x) = [t_{\bar{I}}^{L} (x), t_{\bar{I}}^{U} (x)]$ and $f_{\bar{I}} (x) = [f_{\bar{I}}^{L} (x), f_{\bar{I}}^{U} (x)]$ denote the degree of membership $t_{\bar{I}} (x) : E \to int ([0, 1])$ and the degree of non-membership $f_{\bar{I}} (x) : E \to int ([0, 1])$ of element x ∈ E to set $\bar{I}$ , and satisfy $0 \leq t_{\bar{I}}^{U} (x) + f_{\bar{I}}^{U} (x) \leq 1, \forall x \in E$ , $t_{\bar{I}}^{U} (x) \geq t_{\bar{I}}^{L} (x) \geq 0$ , $f_{\bar{I}}^{U} (x) \geq f_{\bar{I}}^{L} (x) \geq 0$ .

Definition 2. By letting $g_{\bar{I}} (x) = [g_{\bar{I}}^{L} (x), g_{\bar{I}}^{U} (x)] = 1 - t_{\bar{I}} (x) - f_{\bar{I}} (x) = [1 - t_{\bar{I}}^{U} (x) - f_{\bar{I}}^{U} (x), 1 - t_{\bar{I}}^{L} (x) - f_{\bar{I}}^{L} (x)]$ then $g_{\bar{I}} (x)$ is called the degree of indeterminacy of x ∈ E [9] to set $\bar{I}$ . Obviously, $g_{\bar{I}} (x) \subseteq [0, 1], \forall x \in E$ . Specifically, if $g_{\bar{I}} (x) = 0$ , then $\bar{I}$ is reduced to a fuzzy set. For convenience, IVIFS $\bar{I} = {< x, t_{\bar{I}} (x), f_{\bar{I}} (x) > | x \in E}$ is denoted as $\bar{I} = < t_{\bar{I}}, f_{\bar{I}} >$ or $\bar{I} \dot{=} < [t_{\bar{I}}^{L}, t_{\bar{I}}^{U}], [f_{\bar{I}}^{L}, f_{\bar{I}}^{U}] >$ .

Definition 3. For IVIFS $\bar{I} = < t_{\bar{I}}, f_{\bar{I}} >$ , the interval score of $\bar{I}$ is denoted by $s_{\bar{I}} = (t_{\bar{I}} + α_{\bar{I}} g_{\bar{I}}) - (f_{\bar{I}} + β_{\bar{I}} g_{\bar{I}})$ (1)

Here $α_{\bar{I}}$ , $β_{\bar{I}}$ and $γ_{\bar{I}}$ denote the proportion of support, opposition, and abstention, respectively, in $g_{\bar{I}}$ , and satisfy $α_{\bar{I}} + β_{\bar{I}} + γ_{\bar{I}} = 1, α_{\bar{I}}, β_{\bar{I}}, γ_{\bar{I}} \geq 0 .$

Remark 1. The determination of redistribution of $g_{\bar{I}}$ should be based on the decision result of two-sided matching. (The specific method for determining the proportion is introduced in Section 4.1.)

2.2 Two-sided matching

In the two-sided matching decision problem, let ∂ = {∂₁, ∂₂, …, ∂_p} and ℘ = {℘ ₁, ℘ ₂, …, ℘ _q} be the set of two-sided agents, where ∂_i and ℘_j denote the ith agent of side ∂, and the jth agent of side ℘, and 2 ≤ p ≤ q. Let P = {1, 2, …, p} , Q = {1, 2, …, q}.

Definition 4. Let ϒ :∂ ∪ ℘ → ∂ ∪ ℘ be a one-to-one mapping. If for all ∂_i ∈ ∂ and ℘_j∈ ℘, the following propositions hold (i) ϒ (∂_i) ∈ ℘, (ii) ϒ (℘ _j) ∈ ∂ ∪ {℘ _j}, and (iii) ϒ (∂_i) = ℘ _j iff ϒ (℘ _j) = ∂_i. Then, ϒ is called the two-sided matching [3], where ϒ (∂_i) = ℘ _j (or matching pair ϒ (∂_i, ℘ _j)) denotes ∂_i and ℘_j are matched in ϒ, ϒ (℘ _j) = ℘ _j (or single pair (℘ _j, ℘ _j)) denotes ℘_j is not matched (or single).

Definition 5. Let ϒ be the two-sided matching, and then ϒ = ϒ_Two ∪ ϒ_One, where ϒ_Two denotes the set of matching pair, and ϒ_One denotes the set of single pair.

3 The problem of two-sided matching with interval-valued intuitionistic fuzzy sets considering matching aspirations

In the two-sided matching problem, let $\overset{↔}{A} = [{\overset{↔}{a}}_{ij}]_{p \times q}$ be the interval-valued intuitionistic fuzzy matrix from side ∂ to ℘, where ${\overset{↔}{a}}_{ij} = < [t_{{\overset{↔}{a}}_{ij}}^{L}, t_{{\overset{↔}{a}}_{ij}}^{U}], [f_{{\overset{↔}{a}}_{ij}}^{L}, f_{{\overset{↔}{a}}_{ij}}^{U}] >$ . Here, $[t_{{\overset{↔}{a}}_{ij}}^{L}, t_{{\overset{↔}{a}}_{ij}}^{U}]$ denotes the degree of interval-valued satisfaction for agent ∂_i towards ℘_j, and $[f_{{\overset{↔}{a}}_{ij}}^{L}, f_{{\overset{↔}{a}}_{ij}}^{U}]$ denotes the degree of interval-valued dissatisfaction for agent ∂_i towards ℘_j. Let $\overset{↔}{B} = [{\overset{↔}{b}}_{ij}]_{p \times q}$ be the interval-valued intuitionistic fuzzy matrix from side ℘ to ∂, where ${\overset{↔}{b}}_{ij} = < [t_{{\overset{↔}{b}}_{ij}}^{L}, t_{{\overset{↔}{b}}_{ij}}^{U}], [f_{{\overset{↔}{b}}_{ij}}^{L}, f_{{\overset{↔}{b}}_{ij}}^{U}] >$ . Here, $[t_{{\overset{↔}{b}}_{ij}}^{L}, t_{{\overset{↔}{b}}_{ij}}^{U}]$ denotes the degree of interval-valued satisfaction for agent ℘_j towards ∂_i, and $[f_{{\overset{↔}{b}}_{ij}}^{L}, f_{{\overset{↔}{b}}_{ij}}^{U}]$ denotes the degree of interval-valued dissatisfaction for agent ℘_j towards ∂_i. Let Δ = [δ_ij] _p×q be the matching aspiration matrix, where δ_ij denotes the matching aspiration between agents ∂_i and ℘_j. Let ϒ* be the “optimal” two-sided matching.

Remark 2. The determination of δ_ij is based on the satisfaction degrees of agents ∂_i and ℘_j, which usually satisfy the characteristics of non-negativity and normalization. The specific method for determining this is given in Section 4.2.

Based on the above analysis, the problem described in this paper is how to obtain “optimal” two-sided matching ϒ* based on interval-valued intuitionistic fuzzy matrices $\overset{↔}{A} = [{\overset{↔}{a}}_{ij}]_{p \times q}$ and $\overset{↔}{B} = [{\overset{↔}{b}}_{ij}]_{p \times q}$ , and the matching aspiration matrix Δ = [δ_ij] _p×q. The research strategy for this study is provided in Fig. 1.

4 The decision of two-sided matching with interval-valued intuitionistic fuzzy sets considering matching aspirations

4.1 The model of two-sided matching considering scores and matching aspirations

Firstly, according to Equation (1), interval-valued intuitionistic fuzzy matrices $\overset{↔}{A} = [< [t_{{\overset{↔}{a}}_{ij}}^{L}, t_{{\overset{↔}{a}}_{ij}}^{U}], [f_{{\overset{↔}{a}}_{ij}}^{L}, f_{{\overset{↔}{a}}_{ij}}^{U}] >]_{p \times q}$ and $\overset{↔}{B} = [< [t_{{\overset{↔}{b}}_{ij}}^{L}, t_{{\overset{↔}{b}}_{ij}}^{U}], [f_{{\overset{↔}{b}}_{ij}}^{L}, f_{{\overset{↔}{b}}_{ij}}^{U}] >]_{p \times q}$ were transformed into interval score matrices ${\bar{S}}_{\overset{↔}{A}} = [[s_{{\overset{↔}{a}}_{ij}}^{L}, s_{{\overset{↔}{a}}_{ij}}^{U}]]_{p \times q}$ and ${\bar{S}}_{\overset{↔}{B}} = [[s_{{\overset{↔}{b}}_{ij}}^{L}, s_{{\overset{↔}{b}}_{ij}}^{U}]]_{p \times q}$ , where $\begin{matrix} [s_{{\overset{↔}{a}}_{ij}}^{L}, s_{{\overset{↔}{a}}_{ij}}^{U}] \\ = ([t_{{\overset{↔}{a}}_{ij}}^{L}, t_{{\overset{↔}{a}}_{ij}}^{U}] + α_{{\overset{↔}{a}}_{ij}} [g_{{\overset{↔}{a}}_{ij}}^{L}, g_{{\overset{↔}{a}}_{ij}}^{U}]) \\ - ([f_{{\overset{↔}{a}}_{ij}}^{L}, f_{{\overset{↔}{a}}_{ij}}^{U}] + β_{{\overset{↔}{a}}_{ij}} [g_{{\overset{↔}{a}}_{ij}}^{L}, g_{{\overset{↔}{a}}_{ij}}^{U}]) \end{matrix}$ (2) $\begin{matrix} [s_{{\overset{↔}{b}}_{ij}}^{L}, s_{{\overset{↔}{b}}_{ij}}^{U}] \\ = ([t_{{\overset{↔}{b}}_{ij}}^{L}, t_{{\overset{↔}{b}}_{ij}}^{U}] + α_{{\overset{↔}{b}}_{ij}} [g_{{\overset{↔}{b}}_{ij}}^{L}, g_{{\overset{↔}{b}}_{ij}}^{U}]) \\ - ([f_{{\overset{↔}{b}}_{ij}}^{L}, f_{{\overset{↔}{b}}_{ij}}^{U}] + β_{{\overset{↔}{b}}_{ij}} [g_{{\overset{↔}{b}}_{ij}}^{L}, g_{{\overset{↔}{b}}_{ij}}^{U}]) \end{matrix}$ (3)

Remark 3. According to Remark 1, $α_{{\overset{↔}{a}}_{ij}}$ and $β_{{\overset{↔}{a}}_{ij}}$ should reflect the result of matching ∂_i with ℘_j. Then, the redistribution proportion can be given by the following: $\begin{matrix} α_{{\overset{↔}{a}}_{ij}} & = & \frac{1}{2 n} \sum_{j = 1}^{n} (t_{{\overset{↔}{a}}_{ij}}^{L} + t_{{\overset{↔}{a}}_{ij}}^{U}), β_{{\overset{↔}{a}}_{ij}} \\ = & \frac{1}{2 n} \sum_{j = 1}^{n} (f_{{\overset{↔}{a}}_{ij}}^{L} + f_{{\overset{↔}{a}}_{ij}}^{U}) \end{matrix}$ (4) $\begin{matrix} α_{{\overset{↔}{b}}_{ij}} & = & \frac{1}{2 m} \sum_{i = 1}^{m} (t_{{\overset{↔}{b}}_{ij}}^{L} + t_{{\overset{↔}{b}}_{ij}}^{U}), β_{{\overset{↔}{b}}_{ij}} \\ = & \frac{1}{2 m} \sum_{i = 1}^{m} (f_{{\overset{↔}{b}}_{ij}}^{L} + f_{{\overset{↔}{b}}_{ij}}^{U}) \end{matrix}$ (5) where $α_{a_{i}^{↔}}$ (or $β_{a_{i}^{↔}}$ ) can be explained as the average support proportion (or the average opposition proportion) of the result of matching ∂_i with the agent of side ℘; $α_{b_{j}^{↔}}$ (or $β_{b_{j}^{↔}}$ ) can be explained as the average support proportion (or the average opposition proportion) of the result of matching ℘_j with the agent of side ∂.

If the interval scores $[s_{{\overset{↔}{a}}_{ij}}^{L}, s_{{\overset{↔}{a}}_{ij}}^{U}]$ and $[s_{{\overset{↔}{b}}_{ij}}^{L}, s_{{\overset{↔}{b}}_{ij}}^{U}]$ are considered to be uniformly distributed, then the interval score matrices ${\bar{S}}_{\overset{↔}{A}} = {[[s_{{\overset{↔}{a}}_{ij}}^{L}, s_{{\overset{↔}{a}}_{ij}}^{U}]]}_{p \times q}$ and ${\bar{S}}_{\overset{↔}{B}} = {[[s_{{\overset{↔}{b}}_{ij}}^{L}, s_{{\overset{↔}{b}}_{ij}}^{U}]]}_{p \times q}$ can be transformed into the score matrices $S_{\overset{↔}{A}} = [s_{{\overset{↔}{a}}_{ij}}]_{p \times q}$ and $S_{\overset{↔}{B}} = [s_{{\overset{↔}{b}}_{ij}}]_{p \times q}$ , where the following is true: $s_{{\overset{↔}{a}}_{ij}} = (s_{{\overset{↔}{a}}_{ij}}^{L} + s_{{\overset{↔}{a}}_{ij}}^{U}) / - 2$ (6) $s_{{\overset{↔}{b}}_{ij}} = (s_{{\overset{↔}{b}}_{ij}}^{L} + s_{{\overset{↔}{b}}_{ij}}^{U}) / - 2$ (7)

Then, the 0-1 variable y_ij can be introduced, where $y_{ij} = {\begin{matrix} 1, & ϒ (\partial_{i}) = ℘_{j} \\ 0, & ϒ (\partial_{i}) \neq ℘_{j} \end{matrix}$ . Hence, the matching matrix Y = [y_ij] _p×q can be built. Based on the score matrices $S_{\overset{↔}{A}} = [s_{{\overset{↔}{a}}_{ij}}]_{p \times q}$ and $S_{\overset{↔}{B}} = [s_{{\overset{↔}{b}}_{ij}}]_{p \times q}$ , matching aspiration matrix Δ = [δ_ij] _p×q and matching matrix Y = [y_ij] _p×q, the model of two-sided matching can be constructed. The greater the interval score, the higher the degree of satisfaction. Hence, the interval score is taken as the objective of maximization. Furthermore, Model (M-1) of two-sided matching under the condition of one-to-one two-sided matching constraint can be developed as follows: $(M - 1) {\begin{matrix} max D_{\partial_{i}} = \sum_{j = 1}^{q} δ_{ij} s_{{\overset{↔}{a}}_{ij}} y_{ij}, i \in P \\ \max D_{℘_{j}} = \sum_{i = 1}^{p} δ_{ij} s_{{\overset{↔}{b}}_{ij}} y_{ij}, j \in Q \\ s . t . \sum_{j = 1}^{q} y_{ij} = 1; \sum_{i = 1}^{p} y_{ij} \leq 1; y_{ij} \in {0, 1}, \\ i \in P, j \in Q \end{matrix}$

4.2 Calculation of matching aspirations

In Model (M-1), the value of δ_ij is unknown. In order to determine δ_ij, the analysis combined with the value of absolute difference $| s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} |$ is given below.

If $| s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} |$ becomes increasingly greater, then the difference of the degree of satisfaction between ∂_i and ℘_j also becomes increasingly greater. In this case, the success rate of matching ∂_i with ℘_j will decrease. Namely, δ_ij will become smaller, and vice versa. For convenience sake, let $1 / - | s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} |$ be the reciprocal difference of $| s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} |$ . In order to enhance the success rate of matching in reality, further instructions are given below. If the greater $1 / - | s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} |$ is, then the greater the success rate of matching ∂_i with ℘_j will be. Correspondingly, δ_ij should be greater. If the smaller $1 / - | s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} |$ is, then the smaller the success rate of matching ∂_i with ℘_j will be. Correspondingly, δ_ij should also be smaller. Based on this idea, the total reciprocal difference for ∂_i towards all the agents of side ℘ (noted as V_{∂_i→℘}) can be represented by the following: $V_{\partial_{i} \to ℘} = \sum_{j = 1}^{q} \frac{δ_{ij}}{| s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} |}$ (8)

Remark 4. In Equation (8), the case of $| s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} | = 0$ may sometimes occur. At this time, Equation (8) is meaningless. In order to handle this case, the denominator $| s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} |$ is transformed as $2^{| s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} |}$ . Therefore, Equation (8) can be further expressed as follows: $V_{\partial_{i} \to ℘} = \sum_{j = 1}^{q} \frac{δ_{ij}}{2^{| s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} |}}$ (9)

Therefore, the total reciprocal difference for all the agents of side ∂ towards that of side ℘ (noted as V_∂→℘) can be represented by the following: $V_{\partial \to ℘} = \sum_{i = 1}^{p} \sum_{j = 1}^{q} \frac{δ_{ij}}{2^{| s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} |}}$ (10)

According to the above analysis, the selection of δ_ij should make V_∂→℘ greatest. Therefore, the objective function is established as the following: $max V_{\partial \to ℘} = \sum_{i = 1}^{p} \sum_{j = 1}^{q} \frac{δ_{ij}}{2^{| s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} |}}$ (11)

Moreover, the single-objective optimization Model (M-2) can be constructed as follows: $(M - 2) {\begin{matrix} max V_{\partial \to ℘} = \sum_{i = 1}^{p} \sum_{j = 1}^{q} \frac{δ_{ij}}{2^{| s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} |}}, \\ s . t . \sum_{i = 1}^{p} \sum_{j = 1}^{q} δ_{ij}^{2} = 1; δ_{ij} \in {0, 1}, \\ i \in P, j \in Q \end{matrix}$

Theorem 1. The optimal solution (noted as $δ_{ij}^{*}$ ) of Model (M-2) is expressed by the following: $δ_{ij}^{*} = \frac{1}{\sqrt{\sum_{i = 1}^{p} \sum_{j = 1}^{q} \frac{1}{2^{2 | s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} |}}} 2^{| s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} |}}$ (12)

Proof. Let Lagrange function $L = \sum_{i = 1}^{p} \sum_{j = 1}^{q} \frac{δ_{ij}}{2^{| s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} |}} + \frac{λ}{2} (\sum_{i = 1}^{p} \sum_{j = 1}^{q} δ_{ij}^{2} - 1)$ , then $\frac{\partial L}{\partial δ_{ij}} = \frac{1}{2^{| s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} |}} + λ δ_{ij}$ and $\frac{\partial L}{\partial λ} = \sum_{i = 1}^{p} \sum_{j = 1}^{q} δ_{ij}^{2} - 1$ . Let $\frac{\partial L}{\partial δ_{ij}} = 0$ and $\frac{\partial L}{\partial λ} = 0$ , then the following is obtained: $δ_{ij} = - \frac{1}{λ 2^{| s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} |}}$ (13) $\sum_{i = 1}^{p} \sum_{j = 1}^{q} δ_{ij}^{2} = 1$ (14)

Substitute Equation (13) into Equation (14), and the following is obtained: $λ = - \sqrt{\sum_{i = 1}^{p} \sum_{j = 1}^{q} \frac{1}{2^{2 | s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} |}}}$ (15)

Substitute Equation (15) into Equation (13), Equation (12) holds.

Remark 5. Considering the weighted vectors generally meet the normalization condition, $δ_{ij}^{*}$ should be normalized. By letting $δ_{ij}^{'} = δ_{ij}^{*} / - \sum_{i = 1}^{p} \sum_{j = 1}^{q} δ_{ij}^{*}$ , then the following is obtained: $δ_{ij}^{'} = \frac{1}{\sum_{i = 1}^{p} \sum_{j = 1}^{q} \frac{1}{2^{| s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} |}} 2^{| s_{{\overset{↔}{a}}_{ij}} - s_{{\overset{↔}{b}}_{ij}} |}}$ (16)

With Equation (16), the matching aspiration matrix $Δ^{'} = [δ_{ij}^{'}]_{p \times q}$ is set up.

4.3 Solution of the model of two-sided matching

Considering that all the agents of both sides are the same position in most actual cases, then Model (M-1) can be transformed into the following single-objective optimization Model (M-3): $(M - 3) {\begin{matrix} max D = \sum_{j = 1}^{q} \sum_{i = 1}^{p} (s_{{\overset{↔}{a}}_{ij}} + s_{{\overset{↔}{b}}_{ij}}) δ_{ij} y_{ij}, \\ s . t . \sum_{j = 1}^{q} y_{ij} = 1; \sum_{i = 1}^{p} y_{ij} \leq 1; y_{ij} \in {0, 1}, \\ i \in P, j \in Q \end{matrix}$

By substituting $δ_{ij}^{'}$ into Model (M-3), the following optimization Model (M-4) is obtained: $(M - 4) {\begin{matrix} max D = \sum_{j = 1}^{q} \sum_{i = 1}^{p} (s_{{\overset{↔}{a}}_{ij}} + s_{{\overset{↔}{b}}_{ij}}) δ_{ij}^{'} y_{ij}, \\ s . t . \sum_{j = 1}^{q} y_{ij} = 1; \sum_{i = 1}^{p} y_{ij} \leq 1; y_{ij} \in {0, 1}, \\ i \in P, j \in Q \end{matrix}$

By solving Model (M-4), the “optimal” matching matrix $Y^{*} = [y_{ij}^{*}]_{p \times q}$ can be obtained. Since Model (M-4) is a single-objective linear programming model, it can be solved by the existing mathematical optimization software.

4.4 The procedure of the decision of two-sided matching

In summary, a procedure of the decision of two-sided matching is presented as follows:

Step 1: Transform interval-valued intuitionistic fuzzy matrix $\overset{↔}{A} = [{\overset{↔}{a}}_{ij}]_{p \times q}$ into interval score matrix ${\bar{S}}_{\overset{↔}{A}} = {[[s_{{\overset{↔}{a}}_{ij}}^{L}, s_{{\overset{↔}{a}}_{ij}}^{U}]]}_{p \times q}$ using Equations (2) and (4). Transform interval-valued intuitionistic fuzzy matrix $\overset{↔}{B} = [{\overset{↔}{b}}_{ij}]_{p \times q}$ into interval score matrix ${\bar{S}}_{\overset{↔}{B}} = {[[s_{{\overset{↔}{b}}_{ij}}^{L}, s_{{\overset{↔}{b}}_{ij}}^{U}]]}_{p \times q}$ using Equations (3) and (5).

Step 2: Transform interval score matrices ${\bar{S}}_{\overset{↔}{A}} = {[[s_{{\overset{↔}{a}}_{ij}}^{L}, s_{{\overset{↔}{a}}_{ij}}^{U}]]}_{p \times q}$ and ${\bar{S}}_{\overset{↔}{B}} = {[[s_{{\overset{↔}{b}}_{ij}}^{L}, s_{{\overset{↔}{b}}_{ij}}^{U}]]}_{p \times q}$ into score matrices $S_{\overset{↔}{A}} = [s_{{\overset{↔}{a}}_{ij}}]_{p \times q}$ and $S_{\overset{↔}{B}} = [s_{{\overset{↔}{b}}_{ij}}]_{p \times q}$ using Equations (6) and (7), respectively.

Step 3: Develop the multiple-objective optimization Model (M-1) based on score matrices $S_{\overset{↔}{A}} = [s_{{\overset{↔}{a}}_{ij}}]_{p \times q}$ and $S_{\overset{↔}{B}} = [s_{{\overset{↔}{b}}_{ij}}]_{p \times q}$ , matching aspiration matrix Δ = [δ_ij] _p×q and matching matrix Y = [y_ij] _p×q.

Step 4: Build the matching aspiration matrix $Δ^{'} = [δ_{ij}^{'}]_{p \times q}$ using Equation (16).

Step 5: Convert Model (M-1) into Model (M-4) based on the matching aspiration matrix $Δ^{'} = [δ_{ij}^{'}]_{p \times q}$ .

Step 6: Obtain the “optimal” two-sided matching ϒ* by solving Model (M-4).

5 Example of matching executives with positions

A matching example between executives and positions was considered, as described in this section. Suppose a group company wants to assign executives to three CEO positions of subsidiary (∂₁, ∂₂, ∂₃). Five executives (℘₁, ℘ ₂, …, ℘ ₅) enter into the final match phase after preliminary screening. The three position experts evaluate five executives from the perspectives of previous work experience, management ability, dynamic learning ability, and the ability to manage change. Then, the interval-valued intuitionistic fuzzy matrix $\overset{↔}{A} = [{\overset{↔}{a}}_{ij}]_{3 \times 5}$ , which is shown in Table 2, is given. The five executives evaluate three positions from the perspectives of strategy of subsidiary, position environment, and personal attitude toward change. Then, the interval-valued intuitionistic fuzzy matrix $\overset{↔}{B} = [{\overset{↔}{b}}_{ij}]_{3 \times 5}$ , which is shown in Table 3, is given. Finally, the board of the group company will determine the ϒ* combined with the above preference information.

To solve the above problem, a simple description of the matching process is given below.

Steps 1-2: Transform interval-valued intuitionistic fuzzy matrix $\overset{↔}{A} = [{\overset{↔}{a}}_{ij}]_{3 \times 5}$ into score matrix $S_{\overset{↔}{A}} = [s_{{\overset{↔}{a}}_{ij}}]_{3 \times 5}$ using Equations (2), (4), and (6). Transform interval-valued intuitionistic fuzzy matrix $\overset{↔}{B} = [{\overset{↔}{b}}_{ij}]_{3 \times 5}$ into score matrix $S_{\overset{↔}{B}} = [s_{{\overset{↔}{b}}_{ij}}]_{3 \times 5}$ using Equations (3), (5), and (7).

Step 3: Develop the multiple-objective optimization Model (M-1) based on score matrices $S_{\overset{↔}{A}} = [s_{{\overset{↔}{a}}_{ij}}]_{3 \times 5}$ and $S_{\overset{↔}{B}} = [s_{{\overset{↔}{b}}_{ij}}]_{3 \times 5}$ , and match aspiration matrix Δ = [δ_ij] _3×5 and matrix Y = [y_ij] _3×5.

Step 4: Build the matching aspiration matrix $Δ^{'} = [δ_{ij}^{'}]_{3 \times 5}$ using Equation (16).

Step 5: Convert Model (M-1) into Model (M-4) based on matching aspiration matrix $Δ^{* *} = [δ_{ij}^{* *}]_{3 \times 5}$ .

Step 6: By solving Model (M-4), the “optimal” matching matrix $Y^{*} = [y_{ij}^{*}]_{3 \times 5}$ can be obtained. Therefore, the “optimal” two-sided matching is $ϒ^{*} = ϒ_{Two}^{*} \cup ϒ_{One}^{*}$ , where $ϒ_{Two}^{*} = {(\partial_{1}, ℘_{2}), (\partial_{2}, ℘_{4}), (\partial_{3}, ℘_{3})}$ , $ϒ_{One}^{*} = {(℘_{1}, ℘_{1}), (℘_{5}, ℘_{5})}$ . In other words, position ∂₁ matches with executive ℘₂, position ∂₂ matches with executive ℘₄, position ∂₃ matches with executive ℘₃; executives ℘₁ and ℘₅ are not matched.

Remark 6. It is necessary to point out that the existing methods cannot solve the two-sided matching problem considered in this paper. Hence, it is not necessary to compare the obtained results with previous studies.

6 Conclusion

This paper proposed a decision method for solving two-sided matching problems based on interval-valued intuitionistic fuzzy sets considering matching aspirations. The interval-valued intuitionistic fuzzy matrices were transformed into score matrices and a two-sided matching model considering scores and matching aspirations was developed. “Optimal” two-sided matching was obtained by solving the proposed model.

Compared to pre-existing research, the main contribution of this paper lies in the following: (1) The theory of interval-valued intuitionistic fuzzy set was introduced into the field of two-sided matching decisions, which are usually ignored in previous research; (2) The novel interval score was introduced to work with the interval-valued intuitionistic fuzzy set, which can utilize the degree of indeterminacy; (3) This research perspective was from the aspect of matching aspirations, and hence the obtained two-sided matching alternative reflected the matching aspirations of agents; (4) The research of this paper developed a decision theory and method for two-sided matching with interval-valued intuitionistic fuzzy sets.

The limitation of this paper is that it only preliminarily discussed the two-sided matching problem with the preferences of interval-valued intuitionistic fuzzy sets, and the related theory of stable matching under the condition of interval-valued intuitionistic fuzzy sets was not involved.

Hence, the following two courses could be further investigated. First, if the preferences of agents are other intuitionistic fuzzy information, then this kind of two-sided matching decision problem should be further researched. Second, the theory and property of stable matching with interval-valued intuitionistic fuzzy sets should be further explored.

Footnotes

Acknowledgments

This work was partly supported by the NSFC (Project Nos. 71261007, 71261006, 71361021), NSSFC (Project No. 15BGL005), Social Science Fund of Jilin Province (2016JD49) and Natural Science Fund of Jiangxi Province (Project Nos. 20151BAB201026, 20161BAB201025).

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