Two matching agent sets, A and B, are given in this subsection. Let A={A1,A2,⋯,Am} denote the set of m matching objects, where Ai denotes the ith agent in A. Let B={B1,B2,⋯,Bn} be the set of n matching objects, where Bj denotes the jth agent in B, and thus satisfies n≥m≥2, M={1,⋯,m},N={1,⋯,n}.

Definition 1 [35]. Suppose μ: A∪B→A∪B is a one-to-one map, ∀Ai∈A, ∀Bj∈B, satisfying: i) μ(Ai)∈B; ii) μ(Bj)∈A∪{Bj}; iii) μ(Ai)=Bj if and only if μ(Bj)=Ai.

Then, μ refers to the matching scheme between set A and set B, where μ(Ai)=Bj denotes that Ai is matched with Bj in μ and the (Ai,Bj) is defined as a matching pair, μ(Ai)≠Bk denotes that Ai is not matched with Bk in μ. Specially, when μ(Bj)=Bj, then Bj is unmatched in μ. As n≥m≥2, each matching object in A must be matched with one matching object in B and n−m matching objects in B are unmatched.

HFS was first proposed by Torra [26] and further expressed in mathematical symbol by Xia et al. [36]. Unlike other extensions of the fuzzy set, HFS allows the membership degree of an element to a set presented by several possible values, which can express the uncertain information more comprehensively. However, the probability of each membership degree is the same. To better represent the preference information, by incorporating probability to describe the preference information in HFS, Zhu [27] proposed the concept of PHFS. Compared with HFSs, the following PHFSs can reserve more original information.

Definition 2 [27,32]. Let X be a fixed set. Then, a PHFS on X is H={<x,h(γi(x)|pi(x))>|x∈X}, where, h(γi(x)|pi(x)) is a set of elements and γi(x)|pi(x) is a subset of [0,1] denoting the hesitant fuzzy information with probabilities to the set H. For convenience, h(γi(x)|pi(x)) is called a probabilistic hesitant fuzzy element (PHFE), and PHFE h(γi(x)|pi(x)) is denoted by h(p), then h(p)={γi|pi,i=1,2,⋯,#h}, where pi is the probability of the membership degree γi and ∑i=1#hpi=1,γi|pi is considered as a term of the PHFE, #h is the total number of different membership degrees. In this paper, we arrange the terms γi|pi in increasing order based on the value of γi.

Remark 1: In Definition 2, ∑i=1#hpi=1 states that the PHFE includes the complete probabilistic information. ∑i=1#hpi<1 states that the probability information is incomplete. Therefore, the following formula can be used to transform the PHFE with incomplete probability information into the PHFE with complete probability information: pi¯=pi∑i=1#hpi,i=1,⋯,#h. For decision makers with different culture and knowledge backgrounds, it is possible that ∑i=1#hpi<1. In this paper, we assume that ∑i=1#hpi≤1.

Remark 2: In Definition 2, the PHFE h(p)={γi|pi,i=1,2,⋯,#h} of PHFS may be revised according to the multiplier of the membership degree γi and the corresponding probability pi, and then re-order the terms γipi in increasing order. For instance (0.5|0.4,0.6|0.2,0.7|0.4) can be revised as (0.12,0.2,0.28).

Definition 3 [27,32]: Let h(p),h1(p1),h2(p2) be three PHFEs and α>0, then some operation rules among them are defined as: i)

(h(p))α=∪γ∈h(p){γα|p};

Definition 4 [27,32]: Let h(p)={γi|pi,i=1,2,⋯,#h} be a PHFE, a score function is defined by: (1)S(h(p))=∑i=1#hγipi

According to the score function, the deviation function is defined by: (2)V(h(p))=∑i=1#h(γi−S(h(p)))2pi

According to the score functions, we can compare PHFEs h1(p) and h2(p) based on the following rules:

if S(h1)>S(h2), then h1(p)≻h2(p);

if S(h1)<S(h2), then h1(p)≺h2(p);

if S(h1)=S(h2): if V(h1)>V(h2), then h1(p)≺h2(p); if V(h1)<V(h2), then h1(p)≻h2(p);if V(h1)=V(h2), then h1(p)∼h2(p).

Zhang et al. [32] proposed the probabilistic hesitant fuzzy weighted averaging (PHFWA) operator and the probabilistic hesitant fuzzy weighted geometric (PHFWG) operator, both of which are very useful in decision-making using probabilistic hesitant fuzzy information.

Definition 5 [32]. Let h1(p1),h2(p2)⋯hn(pn) be PHFEs, and wj(∑j=1nwj=1) be the weight of PHFEs hj(pj), then a probabilistic hesitant fuzzy weighted averaging (PHFWA) operator and a probabilistic hesitant fuzzy weighted geometric (PHFWG) operator are stated as follows: (3)PHFWA(h1(p1),h2(p2)⋯hn(pn))=⊕j=1nwjhj(pj)=∪γj∈hj(pj){1−∏j=1n(1−γj)wj|p1⋯pn} (4)PHFWG(h1(p1),h2(p2)⋯hn(pn))=⊗j=1nhj(pj)wj=∪γj∈hj(pj){∏j=1nγjwj|p1⋯pn}

These two operators can transform the fuzzy decision information of the decision makers into aggregated decision information to identify the best option out of all feasible choices.

The definitions of the Hamming distance and the Euclidean distance of PHFEs are given as follows:

Definition 6 [29]. Let h1(p1) and h2(p2) be two PHFEs. If #h=#h2=#h1, the number of elements in h1(p1) and h2(p2) are the same, then the normalized probabilistic hesitant fuzzy Hamming distance measure between h1(p1) and h2(p2) is defined as dH(h1,h2), and the normalized probabilistic hesitant fuzzy Euclidean distance measure between h1(p1) and h2(p2) is defined as dE(h1,h2), (5)dH(h1,h2)=∑i=1#h|h1σ(i)(p1σ(i))−h2σ(i)(p2σ(i))|=∑i=1#h|γ1σ(i)p1σ(i)−γ2σ(i)p2σ(i)| (6)dE(h1,h2)=1#h∑i=1#h|h1σ(i)−h2σ(i)|2=1#h∑i=1#h|γ1σ(i)p1σ(i)−γσ(i)2p2σ(i)|2where γ1σ(i) and γ2σ(i) are the ith biggest values in h1(p1) and h2(p2).

Remark3: In most cases, #h2≠#h1. For the sake of calculation convenience, we need to treat the small-length probabilistic hesitancy fuzzy elements with equal length. The short-length probability hesitancy fuzzy element is supplemented by a smaller or larger membership degree element depending on the decision-makers’ various risk preferences. If the decision maker is risk-averse, add the element with the lowest value; if the decision maker likes taking risk, add the element with the highest value; if the decision maker is risk-neutral, add the average of the maximum and minimum values in the set. Let the probability of adding elements be 0, so that the correlation calculation remains unchanged even after adding elements. In this paper, the decision makers are assumed to be pessimists who expect unfavorable outcomes and may add the minimum value, and we define that the probability of each added value is 0. For example, h1(p1) and h2(p2) are two PHFEs, let h1(p1)=(0.5|0.5,0.7|0.5), h2(p2)=(0.5|0.3,0.6|0.3,0.7|0.4). Apparently, #h2>#h1; then, we extend h1(p1) until it has the same length as h2(p2), that is h1(p1)=(0.5|0,0.5|0.5,0.7|0.5). According to the Hamming distance measure, dH(h1,h2) can be calculated as: dH(h1,h2)=|0.5×0−0.3×0.5|+|0.5×0.5−0.6×0.3|+|0.7×0.5−0.7×0.4|=0.15+0.07+0.07=0.29.

Definition 7 [29]. The probabilistic hesitant fuzzy Positive Ideal Solution (PIS) is given as: h+(p)={1|1#h,1|1#h,⋯,1|1#h}, and the probabilistic hesitant fuzzy Negative Ideal Solution (NIS) is given as: h−(p)={0|1#h,0|1#h,⋯,0|1#h}, respectively.

The TOPSIS method, which was first proposed by Hwang and Yoon, is essential for processing information from multi-attribute evaluations and is a topic of much scholarly discussion [37]. According to the PHFSs Hamming distance measure and the basic idea of traditional TOPSIS method, let D− be the distance measure from the NIS and D+ be the distance measure from the PIS. Obviously, the larger the distance D+ the better the alternative, while the smaller the distance D− the better the alternative [33]. The alternative is defined as follows in order to represent the alternative closeness coefficient: (7)C=D−D++D−

The closeness coefficients C for each alternative are calculated for both, the distance D+ and the distanceD− simultaneously. The optimal solution is the one that comes closest to the positive ideal solution and is farthest from the negative ideal solution. From the above analysis, we have come to know that the higher the closeness coefficient, the better the alternative.

Two agent sets are given in the two-sided matching problem which is addressed in this paper. Considering the fuzziness of the thinking of the matching subject and the disparity in knowledge organization, the evaluation information given by the two-sided agent is presented in the form of PHFS, and the attribute weight information given is also incomplete. The problem studied in this paper is to promote the formation of equitable bilateral matching results based on the information of probabilistic hesitancy fuzzy sets given by two-sided agents. The main notations used in this section are presented as follows:

A={A1,A2,⋯,Am}: The set of matching objects on the side A, Ai(i=1,⋯,m) denotes the ith matching object in A.

B={B1,B2,⋯,Bn}: The set of matching objects on the side B, Bj denotes the j th matching object in B.

CA={C1A,C2A,⋯,CsA}: The set of attributes used by matching objects A when providing PHFE assessments over matching objects B, where CkA(k=1,⋯,s) is the kth attribute that the matching objects A pay attention to.

CB={C1B,C2B,⋯,CtB}: The set of attributes used by matching objects B when providing PHFE assessments over matching objects A, where ClB(l=1,⋯,t) is the lth attribute that the matching objects pay attention to.

wAi={w1Ai,w2Ai,⋯,wsAi}: The weight vector of the attribute used by Ai, where wkAi is the weight of the kth attribute in CA, wkAi>0,k=1,2⋯s and ∑k=1swkAi=1, i∈M.

wBj={w1Bj,w2Bj,⋯,wtBj}: The weight vector of the attributes used is the weight of the lth attribute in CB, wlbj>0,l=1,2⋯t and ∑l=1twlBj=1,j∈N.

HAi=(hijkA)n×s: The matrix about matching objects B provided by Ai, where hijkA is Ai’s PHFE assessment over Bj with respect to CkA.

HBj=(hijlB)m×t: The evaluation matrix from objects to Bj, where hijlB is Bj’s PHFE assessment over Ai with respect to ClB.

For the matching object Ai, let hijkA denote his/her assessment for the matching object Bj in accordance with the attribute CkA. The kth attribute’s weight in CA is due to wkAi, then Ai’s overall assessments of Bj can be obtained by hijA=∑k=1shijkAwkAi, all these HPFEs are contained in the probabilistic hesitant fuzzy matrixes HAi=(hijkA)n×s(i=1,⋯m). For the matching object Bj, let hijlB denote his/her assessment for the matching object Ai with respect to the attribute ClB. Owing to wlBj is the weight of the lth attribute in CB, then Bj’s overall assessments of Ai can be obtained by hijB=∑l=1thijlBwlBj, all these PHFEs are contained in the probabilistic hesitant fuzzy matrixes HBj=(hijlB)m×t(j=1,⋯n).

Weight plays an essential role in reaching a reasonable decision result. The matching subject is unable to provide a comprehensive attribute weight vector due to the diversity and fuzziness of the information in the real decision-making environment. At the same time, Wang [38] puts forward a maximum deviation method to determine the attribute weight, and the core idea is to give the attribute weight based on the evaluation information given by the decision-maker. Some models will be proposed to determine the attribute weight vector for a two-sided agent based on the maximum deviation method.

In relation to the kth attribute CkA in CA, if matching object Ai provides assessment to the matching object Bj and Bf, denoted by hijkA and hifkA, then Definition 6 states that: (8)dH(hijkA,hifkA)=∑y=1#h|hjσ(y)(pjσ(y))−hfσ(y)(pfσ(y))|=∑y=1#h|γjσ(y)pjσ(y)−γfσ(y)pfσ(y)|

The total deviation among Ai’s PHFE assessments over B can be calculated by: (9)dH(Aik)=∑j=1n∑f=1ndH(hijkA,hifkA)

We can evaluate the total variance between Ai’s PHFE assessments over B with regard to all attributes by considering the weights of each attribute. (10)dH(Ai)=∑k=1swkAidH(Aik)=∑k=1s∑j=1n∑f=1nwkAidH(hijkA,hifkA)

Then, a constrained condition ∑k=1s(wkAi)2=1 for the weight vector wAi={w1Ai,w2Ai,⋯,wsAi} is introduced based on Wang’s analysis. On this basis, the optimal weight vector is determined.

Therefore, a model for computing the attribute weight for Ai can be presented as: (11)max∑k=1s∑j=1n∑f=1nwkAidH(hijkA,hifkA)s.t.∑k=1s(wkAi)2=1wkAi≥0,k=1,⋯,s

By using the Lagrange multiplier method and normalizing the weight vector, we have (12)wkAi=∑j=1n∑f=1n∑y=1#h|γjσ(y)pjσ(y)−γfσ(y)pfσ(y)|∑k=1s∑j=1n∑f=1n∑y=1#h|γjσ(y)pjσ(y)−γfσ(y)pfσ(y)|(k=1,⋯,s)

Similar to that, the attribute weight vector for Bj, i.e., wBj={w1Bj,w2Bj,⋯,wtBj} can be calculated as: (13)wlBj=∑i=1m∑u=1m∑z=1#h|γiσ(z)piσ(z)−γuσ(z)puσ(z)|∑l=1t∑i=1m∑u=1m∑z=1#h|γiσ(z)piσ(z)−γuσ(z)puσ(z)|(l=1,⋯,t)

Furthermore, the matrices HA=(hijA)m×n and HB=(hijB)m×n can be calculated by aggregating the evaluation values based on the PHFWA operator.

First, the Hamming distance measure between the comprehensive evaluation of each matching agent and their corresponding positive and negative ideal evaluation is calculated, that is: (14)daij+=dH(hijA,hijA+),daij−=dH(hijA,hijA−),dbij+=dH(hijB,hijB+),dbij−=dH(hijB,hijB−)

Then, the closeness coefficients of each matching agent could be computed by: (15)φaij=daij−daij++daij−,φbij=dbij−dbij++dbij−

The degree of divergence within the matching agent sets could clearly be observed in the closeness coefficients of each matching. For the matching agent Ai to the matching agent Bj, the small the distance daij+ and the larger the distance daij−, the higher the closeness coefficient φaij. For the matching agent Bj to the matching agent Ai, the small the distance dbij+ and the larger the distance dbij−, the higher the closeness coefficient φbij. The higher the closeness coefficient φaij meaning the higher the satisfaction of the matching agent Ai to the matching agent Bj; the higher the closeness coefficient φbij meaning the higher the satisfaction of the matching agent Bj to the matching agent Ai. From the above analysis, we can find that the closeness coefficient information can be utilized to measure the satisfaction degree. Therefore, we can establish the satisfaction degree matrices SA=(φaij)mn and SB=(φbij)mn to represent the satisfaction degree of the matching agent Ai to the matching agent Bj, and the satisfaction degree of the matching agent Bj to the matching agent Ai, respectively.

Additionally, a TSMDM model could be developed to obtain the best matching. Suppose xij represents a binary variable, when xij=0; it implies that Ai is not matched with Bj; xij=1 implies that Ai is matched with Bj and thus we have a matching pair (Ai,Bj). Consequently, a multi-objective programming model can be constructed as follows based on thorough consideration of agent satisfaction maximization: (16)maxZA=∑i=1m∑j=1nφaijxij(a)maxZB=∑i=1m∑j=1nφbijxij(b)s.t.∑i=1mxij≤1,j=1,2,⋯n(c)∑j=1nxij≤1,i=1,2,⋯m(d)xij={0,1},i=1,2,⋯m;j=1,2,⋯n(e)

In model (16), formula (a) and formula (b) aim to maximize the overall satisfaction. And formula (c) and formula (d) refer to the one-to-one matching constrains.

We apply the linear weighted method to solve the multi-objective optimization model, and thus model (17) could be constructed as follows: (17)maxZ=wA∑i=1m∑j=1nφaijxij+wB∑i=1m∑j=1nφbijxij=∑i=1m∑j=1nφijxijs.t.∑i=1mxij≤1,j=1,2,⋯n∑j=1nxij≤1,i=1,2,⋯mxij={0,1},i=1,2,⋯m;j=1,2,⋯nwhere φij=wAφaij+wBφbij; wA and wB refer to the different importance of two-sided agent sets, which satisfies wA,wB∈(0,1),wA+wB=1. For both sides in the actual TSMDM, the goal ZA and ZB have the same status, we can allow wA=wB, if the goal ZA is more important than ZB. We can also let wA>wB. Finally, we can use Matlab or Lingo 11 to solve model (17).

In conclusion, the steps of the proposed TSMDM approach can be summed up as follows (see Fig. 1 below):

Step 1: Construct evaluation matrices HAi=(hijkA)m×s(i=1,⋯m) and HBj=(hijlB)n×t(j=1,⋯n) based on the PHFEs information given by two-sided agents.

Step 2: Compute the attribute weight vector wAi={w1Ai,w2Ai,⋯,wsAi} and wBj={w1Bj,w2Bj,⋯,wtBj}according to Eqs. (12) and (13).

Step 3: According to the Eq. (3), transform evaluation matrices HAi=(hijkA)m×s (i=1,⋯m) and HBj=(hijlB)n×t(j=1,⋯n) into evaluation matrices HA=(hijA)m×n and HB=(hijB)m×n based on the attribute weight vector wAi={w1Ai,w2Ai,⋯,wsAi} and wBj={w1Bj,w2Bj,⋯,wtBj}.

Step 4: According to Eq. (13), determine the distance matrix DA+=(daij+)mn,DA−=(daij−)mn,DB+=(dbij+)mn,DB−=(dbij−)mn by calculating the distance between the comprehensive evaluation of the matching agent and the positive and negative ideal evaluation value daij+,daij−,dbij+,dbij−.

Step 5: Construct the satisfaction degree matrices SA=(φaij)mn,SB=(φbij)mn by using Eq. (14).

Step 6: Construct model (16) based on the satisfaction matrix SA=(φaij)mn and SB=(φbij)mn, and then transform it into model (17) using the linear weighted method.

Step 7: Determine the matching scheme bysolving model (17).

Because of operational constraints and limited space, primary schools are unable to provide lunch for both students and teachers. Therefore, their lunch supply needs to be outsourced and completed by external catering companies. The local education authority of a city intends to provide lunch to students in three primary schools in order to improve public service and reduce chances for students to leave school. The local education authority conducts a comprehensive survey of the situation and information summary of the actual activity, and invites five catering companies to bid for three primary schools. An enterprise can only bid effectively in one school for a prolonged period of time due to the financial strain and service quality. To establish a compatible match and maintain a stable, strong partnership that benefits both sides, the local education authority makes the three primary schools Ai(i=1,2,3) evaluate five catering companies Bj(j=1,2,3,4,5) at four aspects of the management experience, safety, health and the professional qualifications of the company, which are regarded as the four attributes CkA(k=1,2,3,4). Moreover, let the five companies Bj(j=1,2,3,4,5) evaluate the three primary schools Ai(i=1,2,3) at these four given attributes CkB(k=1,2,3,4), which are the return of investment, the distance between school and company, the student amount, and the credibility of the school. To be able to obtain a reasonable evaluation, 3 schools and 5 companies invite experts to give the evaluation value anonymously based on the given attributes. By summarizing the expert assessments, they provide the assessment information to the education bureau/commission as a probability hesitation fuzzy set. We can take an example of the assessment of A1 to B1 with respect to the attribute C1A. If three experts support 0.5, one expert suggests 0.4, and one expert insists on 0.3, then the final assessment is determined by PHFE {0.5|0.6,0.4|0.2,0.3|0.2}. Certainly, the PHFS assessments will be more reasonable and include more information. Similarly, all the PHFS evaluation matrices of candidates on the other side are obtained and shown in Tables 1–8.