A method for optimizing the layout of medical and health service facilities based on queuing networks

Abstract

Preventive medical services are crucial as they can intervene in a timely manner at the early stage of diseases, saving a large amount of medical expenditure and enhancing social welfare. Recognizing the importance of preventive medical services, the present invention proposes a software for the design of a preventive medical facility network considering the congestion effect to maximize the total social utility. Due to the two-level decision-making structure between the system manager and facility users, the present invention constructs a two-level non-linear integer programming model. The upper level is the problem of medical facility location and service capacity planning under budget constraints, while the lower level is the user choice equilibrium problem considering the congestion effect. To solve this two-level programming model, the present invention uses the genetic algorithm (GA) to solve the upper-level problem and the method of successive averages (MSA) to solve the lower-level problem. The software is developed using the free and open-source R language. Experiments show that the software can be used for the design of a preventive medical facility network under budget constraints and can also provide decision-making references for budget preparation.

Description

Technical Field:

[0001] The present invention proposes an optimization method for the layout of medical and health service facilities based on queuing networks, belonging to the technical field of public services. Background Art:

[0002] Preventive medical and health services are crucial for the government because they can reduce the likelihood and severity of potentially life-threatening diseases through early detection. Preventive medicine includes many services, such as influenza prevention, vaccination, cancer screening, hepatitis screening, and smoking cessation programs, etc. As is well known, prevention is always better than cure. It can save a large amount of money for the government and improve the well-being of the whole society. However, the preventive medical and health services in many countries and regions of the world are not satisfactory at present.

[0003] The present invention focuses on the location selection of preventive medical and health service facilities and the planning and design of related service capabilities. Different from traditional facility location studies, users do not choose the nearest facility but choose according to the attractiveness of medical facilities. Therefore, it is crucial to understand how users make choices. From the perspective of user behavior selection, previous studies can be divided into two types: (1) the system optimal model, that is, the system decision-maker guides users where to go; (2) the user choice model, that is, users can freely choose facilities. Traditional facility location studies are usually the system optimal model with distance as the main determinant. However, if the number of people demanding a certain facility is too large, the queuing waiting time will be too long, and it is easy to cause the problem of facility congestion. Vidyarthi and Kuzgunkaya considered the queuing waiting time in the facility and proposed a system optimization model for the planning of preventive medical and health facilities. Davari et al. not only took the queuing time as a constraint condition but also incorporated demand fairness and fuzzy attractiveness using multi-objective optimization. Recently, Risanger et al. proposed a system optimal model to select pharmacies for COVID-19 testing, where the demand is an exponential decay function of distance. In short, in the system optimal model, each user is assigned to a facility by the system decision-maker. However, in the medical and health service industry, since users can usually freely choose facilities, it is more appropriate to adopt the user choice model.

[0004] User selection models address the problem of how users choose facilities. They can be further classified into two categories: (1) non-equilibrium allocation, which does not consider the competitive behavior among users; (2) equilibrium allocation, which takes into account the competitive behavior among users. More specifically, there are three ways of non-equilibrium allocation. The most popular one is the all-or-nothing allocation, i.e., the winner-takes-all allocation, where the time (or distance) for users to reach the facility is regarded as the main determinant, and users are assumed to seek services from the nearest medical facility. Additionally, the congestion effect of the facility is also considered in the nearest all-or-nothing allocation. For example, Zhang et al. tried to incorporate the waiting time into the total time, while Davari et al. and Dogan et al. took the waiting time as a constraint. However, it is unrealistic to assume that all users from the same demand node choose services from the same nearest facility. In fact, users can have more flexibility in choosing facilities. The second is the Huff-type allocation, which allocates part of the demand to the facility according to the attractiveness of the facility and the travel time of the user. The famous gravity model is a special case of the Huff-type allocation. For a given specific parameter, the Huff-type allocation will degenerate into the gravity model. The third is the multinomial logit allocation, where the characteristics of users and unobserved attributes can be included in the utility function, but the waiting time of the facility is not considered and is only taken as a constraint, which is not in line with reality. In summary, non-equilibrium allocation still dominates in facility selection because it avoids the complexity of equilibrium problems.

[0005] In contrast, recent research has started to adopt equilibrium allocation to consider the congestion effect of facilities. Empirical studies have found that waiting time is important for users in services and medical settings. However, waiting time is not an exogenous variable like travel time, but an endogenous variable. More specifically, a shorter waiting time attracts users, but this in turn prolongs the waiting time for facility services. This means that the equilibrium problem between waiting time and the number of users must be considered. There are two ways of equilibrium allocation: (1) deterministic user equilibrium allocation, where the waiting time of the facility is an essential part of the deterministic utility; (2) stochastic user equilibrium allocation, which further includes a stochastic component to reflect the unobserved utility. It is assumed that users patronize the facility with the maximum (stochastic) utility. The game among users will reach a Nash equilibrium state, namely the so-called user equilibrium. In the equilibrium state, each user is satisfied with the facility they patronize, that is, people from the same demand node can obtain the same utility even if they go to different facilities. The description of recent user selection behavior has been further improved. Kucukyazici et al. adopted latent class analysis to incorporate user preferences into the design of the cancer screening facility network. Krohn et al. further introduced medical quality into the utility function of user selection. It can be expected that research will continue to develop in the direction of more realistic user selection behavior. Summary of the Invention:

[0006] Technical problem: The technical problem to be solved by the present invention is how to determine the location and service capacity planning and design of preventive medical facilities among candidate locations under the constraint of a limited budget, so as to maximize the total social utility of the system.

[0007] Technical solution: The present invention aims to propose an optimization method for the layout of medical and health service facilities based on a queuing network. This problem is formulated as a bilevel non-linear integer programming model. The upper level is the problem of medical facility location and service capacity planning with a budget constraint. The lower level is the user selection equilibrium problem, which determines the specific facilities to which users are assigned. In this study, a genetic algorithm (GA) is used to solve the upper-level problem, and the method of successive averages (MSA) is used to solve the lower-level problem. The technical solution of the present invention includes the following steps:

[0008] 1. Modeling method

[0009] Let the road network G=(N, L) consist of a set of nodes N and a set of road segments L. Nodes represent urban communities or road intersections, and road segments are the main traffic arteries. We assume that the number of users requiring preventive medical services at node i (i∈N) per unit time is h i . The set of candidate locations for medical facilities is M, and the set of selected locations The shortest travel time from node i to location j is denoted as t ij . The government has an available budget control B, through which one or more service desks can be established at the selected facility locations. We assume that the service desks are homogeneous, the service time follows an exponential distribution, and on average, μ users are served per unit time. We also assume that users are homogeneous, they arrive at each facility following a Poisson distribution, and the queuing rule is first-come, first-served (FCFS). These assumptions are reasonable for facilities that do not require appointments, which applies to most routine services. Therefore, each facility here is an M / M / s queuing system, where M represents that users arrive or leave following a Markov (or Poisson) distribution, or equivalently, following an exponential inter-arrival or service time distribution, and s represents the number of service desks in the medical facility.

[0010] The purpose of this problem is to determine the location and service capacity of each medical facility, that is, the number of service desks, under the constraint of budget B, so as to maximize the total utility of the system. For this purpose, we define three sets of decision variables:

[0011]

[0012] s j = the number of service points at location j,

[0013] x ij = the number of users from population node i to location j j belongs to M.

[0014] Suppose we get the solution S = {j: j ∈ M, y j = 1}, then we have:

[0015]

[0016] Use λ j to represent the arrival rate of users at facility j We can get:

[0017]

[0018] 1.1 Utility Evaluation Function

[0019] Users choose facilities based on the attractiveness of the facilities. Therefore, it is crucial to understand how users make choices. The user choice model is essentially based on the utility function defined by the attractiveness of the facilities. Use U ij to represent the observable utility when a user from demand node i receives service at location j. It mainly consists of three parts. (1) u j , the inherent attractiveness of location j. This may include internal factors such as parking convenience, the appearance of the facility, and the reputation of the practitioners. (2) t ij , the shortest travel time from origin node i to destination facility j. (3) The expected waiting time of users at location j, including queuing time and service time, is a function of the arrival rate λ j and the number of service desks s j . Since it is an M / M / s j queuing system at node j, for any S j ≥ 1, it can be obtained according to classical queuing theory It can be represented by a set of equations

[21] :

[0020]

[0021]

[0022]

[0023]

[0024] where L j is the expected queue length expressed in terms of the number of users, p0 is the probability of no users, and ρ j is the service intensity. Note that it is assumed that the stability condition of the queue is satisfied.

[0025] To combine these three costs, we assume U ijis in the form of a traditional linear additive function. A similarly reasonable assumption is that U ij and u j are positively correlated, and with t ij and negatively correlated. Therefore, U ij can be expressed as:

[0026]

[0027] where β1 and β2 represent the coefficients of travel time and waiting time respectively, and can be estimated using measured data. Note that in addition to these specific costs, the utility function can be extended based on the available data to incorporate other observed attributes.

[0028] Note that there is a dependency between the arrival rate λ j and the expected waiting time . According to our model, λ j is the sum of x ij , and x ij in turn depends on U ij , which in turn depends on which in turn depends on λ j . That is, λ j indirectly depends on itself. Since we are considering a network of competing facilities, this means that we need to solve a user equilibrium problem to determine the demand allocation x ij .

[0029] 1.2 User Equilibrium Model

[0030] Assume that users adopt a utility maximization decision rule, that is, users choose the facility with the highest observable utility. Let represent the highest utility generated by users at demand node i, that is:

[0031]

[0032] Assume that the facility planning scheme S and service capacity s j are known. At user equilibrium, no user wants to change their choice. Therefore, the equilibrium condition can be expressed as:

[0033]

[0034] where and represent the utility of users at node i accessing medical facility j and the highest utility of users at node i at user equilibrium respectively. In addition, it should be noted that

[0035]

[0036] where is the user arrival rate of facility j at user equilibrium, is the number of users from demand node i to facility location j at user equilibrium.

[0037] The equilibrium condition (9) states that if there is a user flow from node i to facility j, then the user utility of node i for facility j must be equal to the maximum utility Otherwise, it is not higher than the maximum value. This model implies that each user selects the service facility with the highest observed attractiveness.

[0038] To find in formula (9),

[0039]

[0040] The constraints are:

[0041]

[0042]

[0043] where

[0044]

[0045] Theorem 1 For a given facility planning scheme S and S j , the mathematical programming (10)-(13) is equivalent to (9).

[0046] Proof To prove that this mathematical programming is equivalent to (9), we transform it into a Lagrangian function with only non-negative constraints, that is,

[0047]

[0048] where w in the objective function i is the Lagrange multiplier of constraint (11).

[0049] According to the Karush-Kuhn-Tucker (KKT) conditions, the optimal conditions for this Lagrangian function are:

[0050]

[0051]

[0052]

[0053]

[0054] Obviously, (17) is equivalent to (11). Equations (15) and (16) imply that

[0055]

[0056] Note that

[0057]

[0058] Therefore, formula (19) can be further rewritten by formula (20) as:

[0059]

[0060] It can also be restated in its complementary form as:

[0061]

[0062]

[0063]

[0064] Formula (21) means that if there is a demand flow x ij > 0, the utility U ij is equal to w i , if there is no demand flow, i.e., x ij = 0, then the utility U ij is not greater than w i . Therefore, the Lagrange multiplier w i can be interpreted as the maximum utility generated by users at node i Therefore, formula (21) is equivalent to formula (9). Therefore, we can obtain the equilibrium flow by solving this mathematical programming problem.

[0065] 1.3 Bilevel programming model

[0066] The whole problem considered here is a bilevel decision-making structure, where the upper-level problem is to determine the facility location and the related service capacity, and the lower-level problem is to determine the equilibrium flow of users from the demand nodes to the facility locations given the upper-level decisions.

[0067] In practice, there is usually only a limited budget to support the establishment and operation of preventive medical facilities. This budget constraint can be used to consider the cost differences between establishing and operating medical service facilities in different areas of the city. Let be the fixed construction cost of facility j (j ∈ M), and let c v be the unit cost of adding service desks in the facility. In addition, for cost-effectiveness considerations, only when the user demand exceeds the minimum workload requirement R minFacilities can be established only when... In addition, under site constraint conditions, the number of service desks in facility j cannot exceed a limited size.

[0068] We consider the goal of maximizing the total social utility of the system, that is, the overall observable utility of users. The upper-level model of the medical facility network design can be established as follows:

[0069]

[0070] The constraints are as follows:

[0071]

[0072]

[0073]

[0074]

[0075]

[0076]

[0077]

[0078]

[0079] Among them, x ij is determined by the following lower-level model:

[0080]

[0081] The constraints are as follows:

[0082]

[0083]

[0084] The objective function (25) is to maximize the utility of the entire system. Constraint (26) ensures that at least one service desk is allocated to each proposed facility, while ensuring the non-negativity of the decision variable s j . Constraint (27) limits the number of service desks to a limited size. Constraint (28) defines the arrival rate λ j . Constraint (29) represents the stability condition of the queue. Constraint (30) stipulates that the arrival rate of the proposed facility must meet the minimum workload requirement. In constraint (31), T represents the penalty cost, ensuring that users can only obtain services from the established facilities. Constraint (32) is budget control. Constraint (33) is the feasible region of the decision variables.

[0085] 2. Solution Method

[0086] Since the bilevel programming model is highly nonlinear and contains integer decision variables, it is difficult to solve exactly. Therefore, we focus on efficient heuristic algorithms that have been successfully applied in preventive healthcare network design [4, 13]. Our solution algorithm follows a bilevel framework. For the lower-level problem, the method of successive averages (MSA) is used to solve the user equilibrium model. This allocation algorithm determines the equilibrium flow of users to facilities. For the upper-level problem, a metaheuristic algorithm, namely a genetic algorithm with an elitist strategy, is proposed to find the optimal locations and service capacities.

[0087] 2.1 Allocation Algorithm for the Lower Level

[0088] For the given upper-level decisions S and s j , the lower-level problem is to solve for the equilibrium flow. The algorithm adopted is an iterative method called the method of successive averages. Let k be the iteration counter and K be the maximum number of iterations. Let ε be a pre-determined tolerance parameter. θ k , k = 1,..., K, is the step size parameter during iteration, which takes values between 0 and 1. The specific calculation steps are given here:

[0089] Step 0 (Initialization): Determine appropriate values for ε and K, set k = 0; set

[0090]

[0091] Step 1 (Calculate Utility): Set k := k + 1; according to formula (2), calculate λ j , calculate the shortest path travel time t using Dijkstra's algorithm ij , i ∈ N, j ∈ S. Calculate the waiting time according to formulas (3)-(6) Calculate U according to formula (7) ij , i ∈ N, j ∈ S; according to formula (8), obtain i ∈ N.

[0092] Step 2 (All-or-Nothing Allocation): Calculate the flow x′ according to the all-or-nothing rule ij , that is, allocate all the demands of users to the facility he is most interested in:

[0093]

[0094] Step 3 (Generate Search Direction): Define j ∈ S, as the search direction.

[0095] Step 4 (Flow Update): Update the user flow j∈S, step size parameter θ k Defined as:

[0096]

[0097] Step 5 (stop condition): If continuous and To achieve relative error, or k>K, set and stop; otherwise, proceed to step 1. The relative error is defined as:

[0098]

[0099] In each iteration, the algorithm finds x in step 3 ij A new search direction, and then in step 4 update x by the step size ij The whole process is repeated until any of the stopping conditions in step 5 is met. The step size of each iteration is θ k is preset. Set θ k There are many ways to do this. Generally speaking, we should reduce θ as k increases. k To ensure convergence, we set θ k is the inverse of the number of iterations k+1. Note that in step 4 The update result makes it possible for the arrival rate of the facility to be greater than the maximum value, which violates the stability condition (29). There are usually two ways to solve this problem: one is to reduce the step size, and the other is to set a larger penalty time.

[0100] 2.2 Upper-level location selection algorithm

[0101] We build a genetic algorithm based on an elitist selection strategy to solve the upper-level problem, as it is one of the most successful metaheuristic algorithms for solving combinatorial optimization problems, with the ability to explore other regions of the feasible space and avoid local optima.

[0102] In the genetic algorithm, each chromosome represents a solution to the problem, and the quality of the solution is represented by fitness. In this study, integer encoding is used to represent chromosomes. Each chromosome is composed of several integers like genes. Each gene corresponds to a potential position in M, and its value represents the number of assigned service stations. If there are no assigned service stations, the medical facility will not be located at that location. We implement the genetic algorithm in the following steps:

[0103] Step 0 (initialization): Set the parameters used, including the population size N pop , maximum number of generations Gen, crossover probability p c , mutation probability p m, generation label gen = 1, elite part p e .

[0104] Step 1 (Generation of Initial Population): Randomly generate feasible solutions as the initial population N of chromosomes pop , scattered throughout the range of possible solutions. If it is judged to be infeasible according to the constraint conditions, generate another one until it is feasible.

[0105] Step 2 (Calculation of Fitness): For each chromosome in the population, generate a fitness value, that is, the objective function value. Use it to evaluate the performance of each chromosome in the population.

[0106] Step 3 (Generation of New Population)

[0107] Step 3.1 (Selection): According to the fitness values evaluated in Step 2, mark the best-performing p e part as the elite and discard the worst-performing p e part.

[0108] Step 3.2 (Crossover): The remaining (1 - p e )N pop chromosomes are used for the crossover operation. These chromosomes are randomly paired. The probability of performing crossover is p c . If two chromosomes are selected for crossover, randomly determine a gene position to cross to generate two offspring as new chromosomes. If the newly generated chromosomes are infeasible according to the constraints in the upper-level model, try another gene position until they are feasible.

[0109] Step 3.3 (Mutation): Determine the mutation of a chromosome with probability p m . Randomly select two genes at least one of which is positive and exchange their values. If the new chromosome is infeasible, try another two gene positions until it is a feasible offspring.

[0110] Step 3.4 (Elite): Generate a new population. After genetic operations, there are still (1 - p e )N pop feasible chromosomes. Add the p e N pop elites marked in Step 3.1 to ensure the population size N pop . This allows the best chromosomes in the current generation to continue unchanged to the next generation. It ensures that the quality of the solution does not degrade from one generation to the next. Let the generation label be gen := gen + 1.

[0111] Step 4 (Stop Iteration): If the maximum number of generations is reached, that is, gen ≥ Gen, terminate the iteration process and output the result. Otherwise, go back to Step 2.

[0112] 3. Programming Method

[0113]

[0114]

[0115]

[0116]

[0117]

[0118]

[0119]

[0120]

[0121]

[0122] Beneficial Effects: A painful lesson learned from the COVID-19 pandemic is that preventive healthcare services are crucial for governments as they can reduce the likelihood and severity of potentially life-threatening diseases through early detection. Prevention is always better than cure. It can save governments a significant amount of money and enhance the well-being of society as a whole. The present invention proposes a method for designing a preventive healthcare facility network considering the crowding effect under a limited budget constraint to maximize the total social utility of the system, which can provide a scientific basis and an operable method for the location selection and service capacity decision-making of preventive healthcare facilities and has important application value. Description of the Drawings:

[0123] Figure 1 is a model framework diagram;

[0124] Figure 2 is the Sioux Falls test network;

[0125] Figure 3 is the evolutionary process of the genetic algorithm;

[0126] Figure 4 is the sensitivity analysis under variable budget control. Detailed Implementation Manner:

[0127] We conducted a computational experiment to evaluate the effectiveness of the proposed model and algorithm. The experiment used the Sioux Falls network widely adopted in network design. This is a Figure 2The medium-scale network shown. The network consists of 24 nodes and 76 links. For the computational experiment, it is assumed that there are 8 demand nodes and 8 candidate locations. Thus, there are a total of 64 origin-destination pairs. The travel time and length of link a (a ∈ L) are denoted as t a and l a , and the numerical values are shown in Table 1. Assuming that the driving speed of each link is 30 miles per hour (mile / h), the link length can be converted into the link travel time. The preventive medical demand data calculated by the number of users per hour (users / hour) is shown in Table 2.

[0128] Table 1 Characteristics of the Sioux Falls Network

[0129]

[0130]

[0131] Table 2 Medical Demand Data of the Sioux Falls Network

[0132]

[0133] Based on the proposed model and solution algorithm, the following parameter values were used in the experimental study.

[0134] Problem parameters:

[0135] · The service rate μ of a single service desk = 6 users / hour;

[0136] · The fixed facility attractiveness u j = 0;

[0137] · The sensitivity coefficient β1 to travel time = 1 and the sensitivity coefficient β2 to waiting time = 1;

[0138] · The maximum number of service desks

[0139] · The fixed facility construction cost

[0140] · The unit service desk cost c v = 1;

[0141] · The budget control B = 50;

[0142] · The minimum workload R min = 10 users / hour;

[0143] Continuous average parameters:

[0144] · The maximum number of iterations K = 100;

[0145] · Error tolerance ε = 0.01;

[0146] Genetic algorithm parameters:

[0147] · Population size N pop = 200;

[0148] · Maximum number of generations Gen = 20;

[0149] · Crossover probability p c = 0.5;

[0150] · Mutation probability p m = 0.2:

[0151] · Probability of elitism p e = 0.1.

[0152] These algorithms are programmed using the free and open-source language R 3.6.3. All runs are performed on a personal computer equipped with a 3.6 GHz Intel i7-4790 CPU and 16 GB of memory. In this experiment, the genetic algorithm stops after running for 1.61 hours. As Figure 3 shown, after 11 generations, the evolutionary process begins to stabilize. Therefore, it can be concluded that the final result is an approximate optimal solution. Table 3 reports the optimal solution, where four possible locations are selected to establish preventive medical facilities, namely nodes 3, 7, 21, and 23, with the corresponding number of service desks being 20, 5, 13, and 12 respectively. Users of demand nodes can be assigned to more than one facility, such as nodes 13 and 20. However, other demand nodes indicate that users from the same node usually also visit the same facility. Table 4 shows that users select the facility with the highest utility, and users from the same demand node obtain approximately the same utility even when going to different facilities.

[0153] Table 3 Optimal planning solution and equilibrium state of preventive medical facilities

[0154]

[0155] Table 4 Utility matrix between demand nodes and facility locations

[0156]

[0157] Sensitivity analysis is always beneficial as it can provide valuable management insights. Here, sensitivity analysis with different budget controls is conducted, which is also a cost-benefit analysis. The budget is increased from 45 to 75 in steps of 5. The results are as Figure 4As shown, the horizontal axis represents budget control and the vertical axis represents the total system utility. Since only travel time and waiting time are used to define the utility function, the individual utility is negative, and the total system utility is also negative. Obviously, the marginal benefit is decreasing. The decision maker cannot obtain the same return with the same additional investment. There is an optimal budget control where the marginal cost equals the marginal benefit.

[0158] As Figure 4 shown, the relationship between utility (benefit) and budget (cost) can be modeled by polynomial regression. Let f represent the total system utility and B represent budget control. The polynomial regression can be expressed as:

[0159] f(B) = α o + α1B + α2B 2 , (37)

[0160] where α0 is the intercept, α1 is the coefficient of B, and α2 is the coefficient of B 2 . The values of these coefficients can be estimated using the results of sensitivity analysis. When the marginal benefit equals the marginal cost, the optimal budget B * can be obtained. That is:

[0161]

[0162] Taking this sensitivity analysis as an example. The parameter estimates of formula (37) are shown in Table 5. The hypothesis test shows that these parameters are all significant at the 0.05 level. Therefore, we can reject the null hypothesis. The adjusted R 2 is 0.884, indicating that the polynomial regression fits the data well. From formula (38), the optimal budget is 57.9. It is worthwhile to increase investment before the optimal budget. However, it is unwise to continue increasing investment after the optimal budget because the output will be less than the input.

[0163] Table 5 Estimated Parameters of Polynomial Regression

[0164]

Claims

1. An optimization method for the layout of medical and health service facilities based on queuing networks, the method comprising the following technical features: (1) Represent it as a two-layer programming model, with the system manager at the upper layer and the facility users at the lower layer; (2) The upper-layer model is a non-linear integer programming model constructed with the maximization of the total social utility of the system as the objective function, the limited budget input as the constraint condition, and the location and service capacity of preventive medical facilities as the decision variables, where the user demand allocation of each facility is determined by the lower-layer model, and the upper-layer model is expressed as: where M is the set of candidate nodes for medical facilities, N is the set of nodes in the road network, S is the set of selected locations, B is the budget control available to the government, x ij is the number of users from population node i to location j, y j is a 0-1 variable for facility selection, y j = 1 indicates that the facility is located at location j, y i = 0 indicates that the facility is not located at location j, u j is the inherent attractiveness of location j, β1 and β2 represent the coefficients of travel time and waiting time respectively, μ represents the average number of users served per unit time, t ij is the shortest travel time from node i to location j, t′ ij is the travel time from node i to location j, λ j represents the arrival rate of users at facility j, s j represents the number of service points at location j, Z + is the set of positive integers, is the expected waiting time of users at location j, including queuing time and service time, which is a function of the arrival rate λ j and the number of service desks s j of, R min is the minimum workload requirement, is the maximum number of service desks in facility j, T represents the penalty cost to ensure that users can only obtain services from the constructed facilities, is the fixed construction cost of facility j, c v is the unit cost of adding service desks in the facility; (3) The lower-layer model is a user choice equilibrium problem considering the congestion effect. It is assumed that users choose the medical facility with the maximum utility, but too many users will lead to an extended queuing time, an increase in user costs, and some users will switch to other facilities. The result of the game among users will reach a user equilibrium state, where queuing theory is used to calculate the queuing time; (4) A heuristic algorithm is designed for solving, where the continuous average algorithm is used for the lower-layer model and the genetic algorithm with an elite strategy is used for the upper-layer model; (5) The above models and algorithms are implemented using the R language.

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