A multi-objective optimization model for doctor-patient matching and scheduling

Abstract

The application discloses a kind of doctor-patient matching and scheduling multi-objective optimization model, it is related to doctor-patient allocation technical field, and its technical solution main point is: the present application considers the heterogeneity time window constraint of multiple doctors and the degree of doctor-patient professional matching, constructs the multi-objective optimization model of doctor-patient matching and scheduling.For the random nature of medical service time, introduce stochastic programming method to model scheduling problem in uncertain environment, use sample approximation average method (SAA) to process random service time, and design improved progressive hedging algorithm (PHA) for efficient solution.Finally, through multiple simulation experiments and comparative analysis, the superiority of the proposed method in matching scheduling efficiency and scheme robustness is verified, which provides technical support for the construction of platform intelligent triage and scheduling system, and ultimately realizes the optimal allocation of resources to improve the collaborative performance of demand side.

Description

Technical Field

[0001] This invention relates to the field of doctor-patient allocation technology, and more specifically, to a multi-objective optimization model for doctor-patient matching and scheduling. Background Technology

[0002] With the rapid popularization of internet healthcare, online medical platforms have become an important channel for patients to access convenient medical services. Their demand-side coordination efficiency directly affects patients' healthcare experience and the platform's sustainable development. However, the current demand coordination efficiency of online medical platforms is constrained by inefficient demand identification and scheduling mechanisms, and there are many pain points that urgently need to be addressed in the process of matching supply and demand between doctors and patients.

[0003] From the demand side, patients expect to obtain convenient and high-quality medical services with minimal time cost. The realization of this demand heavily relies on the platform's matching and scheduling capabilities. However, in reality, patients' needs are often expressed in unstructured natural language. Traditional matching methods based on department tags, professional titles, or keyword searches struggle to bridge the semantic gap, often falling into two dilemmas: either the match is accurate but the waiting time is too long, and patients' needs cannot be responded to in a timely manner; or the doctor has time but the doctor's needs are not aligned with the patient's, making it difficult to guarantee the quality of the match. Related data shows that non-targeted consultations account for as much as 93.52%, with most patients lacking clear medical selection goals and heavily relying on the platform's matching and recommendation capabilities. Research indicates that the compatibility of the matched doctor and the waiting time significantly affect a patient's willingness to continue using the platform; inappropriate matching directly leads to patient churn.

[0004] From the supply side, platform doctors remain scarce, and most are part-time with limited online consultation time. Existing research indicates that online doctors tend to provide consultations on platforms with lower patient diversity and higher patient concentration, further exacerbating the difficulty of matching supply and demand. Simultaneously, the feasibility of implementing matching results is constrained by doctors' time windows and service capabilities, while scheduling feasibility, in turn, limits the matching selection space; the two are mutually coupled. However, existing research often treats matching and scheduling optimization separately, lacking a systematic approach that integrates matching and scheduling, making it difficult to achieve efficient collaboration between supply and demand.

[0005] Specifically, existing technologies have the following drawbacks:

[0006] 1. Insufficient characterization of the heterogeneity of doctors' time windows. While existing studies have considered the time window constraints of doctors and patients, most treat these time windows as homogeneous, failing to fully characterize the heterogeneity of available time windows for each service provider in online healthcare platforms where multiple doctors provide services simultaneously. In reality, there are significant differences in the online service periods, durations, and frequencies among different doctors, and this heterogeneity directly affects the feasibility of scheduling schemes and service efficiency.

[0007] 2. Insufficient consideration of patient service mode preferences. Online medical platforms typically offer multiple service modes such as text-based consultations and telephone consultations. Different patients have varying preferences for these service modes, and the distribution of service durations differs significantly across modes. Existing research rarely incorporates patient preferences for service modes into the matching and scheduling decision-making framework, making it difficult to meet the diverse service needs of patients.

[0008] 3. Insufficient integrated research on matching and scheduling. Existing research often treats patient-doctor matching and appointment scheduling separately: it rarely considers the constraints of scheduling feasibility on matching selection; scheduling research focuses on time arrangement optimization, and rarely considers the impact of matching quality on scheduling effectiveness. This separate research paradigm makes it difficult to achieve synergistic optimization of matching quality and scheduling efficiency.

[0009] In summary, current online healthcare platforms face numerous challenges in matching and scheduling doctors and patients, and the limitations of existing research make it difficult to build an efficient demand coordination mechanism. Summary of the Invention

[0010] The purpose of this invention is to provide a multi-objective optimization model for doctor-patient matching and scheduling, providing technical support for the construction of intelligent triage and scheduling systems on platforms, and ultimately achieving optimal resource allocation to improve demand-side collaborative efficiency.

[0011] The above-mentioned technical objective of the present invention is achieved through the following technical solution: a multi-objective optimization model for doctor-patient matching and scheduling, wherein the specific implementation process of the multi-objective optimization model includes the following two stages:

[0012] Phase 1: Construction The scenario-based matching and scheduling integration module [MP] is as follows:

[0013] (1)

[0014] (2)

[0015] (3)

[0016] (4)

[0017] (5)

[0018] (6)

[0019] The second stage involves constructing a scheduling module that considers random service times, further introducing spatiotemporal constraints on medical resources, including but not limited to the time window limit for available doctors and differences in patient service type needs; and finally, constructing a multi-objective optimization model that minimizes doctor-patient matching costs and patient waiting times while optimizing the efficiency of doctor resource utilization.

[0020] The second-stage scheduling module [SP] that considers random service time is as follows:

[0021] (25)

[0022] st (26)

[0023] (27)

[0024] (28)

[0025] (29)

[0026] (30)

[0027] (31)

[0028] (32)

[0029] (33)

[0030] (34)

[0031] (35)

[0032] (36)

[0033] The present invention is further configured such that the first stage of constructing the matching module is specifically as follows:

[0034] Suppose that there are a total of within standard time T. Patients need to be assigned, and patients should be recorded as... Patients are divided into two categories: those seeking text-based consultations, using asynchronous communication and describing their condition in text and / or image format; and those seeking video or voice consultations, using synchronous communication requiring real-time interaction between the doctor and patient. The group of patients seeking text-based consultations is denoted as […]. Video consultations with patients are a collection ,satisfy and Significant differences in service duration under different consultation models;

[0035] Within this standard time period T, there are J heterogeneous available doctors, and the service cycle will be... Divided into A time period, denoted as The duration of each time period is To accurately depict the diverse characteristics of doctors' available time windows, each doctor... Doctors can freely choose their working hours according to their own preferences. A binary variable is introduced to represent the available time window for doctors. This indicates that doctor j can provide services during the k-th time period. This indicates that the time period is unavailable, therefore for each doctor Its available time window set for ;

[0036] For ease of subsequent calculations, please provide to the doctor. The available time windows are renumbered, and the new index is denoted as . ,in, Let j be the number of available time windows for doctor j, and introduce variables. Let t be the number of the t-th available time window of doctor j in the global time slot. , where k is the t-th available global time slot number for doctor j;

[0037] Based on doctor-patient matching costs ; Statistical estimation of patients based on historical medical data The actual service time at doctor j's office was... ;use Let represent the decision variable in the matching problem, such as if patient i is matched with doctor. but ,otherwise In addition, a binary decision variable is introduced to formulate scheduling decisions for each doctor. Each of the doctors Each available time window There is Available locations, indexed as Customers assigned to smaller index positions will be served earlier; additionally, let's assume... Indicates in The total cost of patient waiting time, doctor idle time, and overtime work in the given scenario;

[0038] Based on the above information, the goal of the first phase of building the matching and scheduling integration module is to minimize the doctor-patient matching cost and expected cost; therefore, the first phase [MP] can be described as follows:

[0039] (1)

[0040] (2)

[0041] (3)

[0042] (4)

[0043] (5)

[0044] (6)

[0045] The objective function of the [MP] phase is to minimize the cost of matching patients with doctors and the total cost of the scheduling phase. The total cost of the scheduling phase includes patient waiting time, doctor idle time, and overtime costs. These are weight control parameters; Formula (2) indicates that each patient can only be seen by one doctor; Formula (3) indicates that a patient can only be assigned to one position within one available time period of the doctor they are seeing; Formula (4) restricts each available time window for each doctor j. Each position It can be occupied by a maximum of one patient; Formula (5) represents the position of the doctor within each available time window. Only the previous position It can only be occupied when it is occupied.

[0046] The present invention is further configured such that the second stage of constructing the scheduling module is as follows:

[0047] The goal of the second phase is to minimize the sum of patient waiting costs, physician idle time costs, and overtime costs. Indicates the first The t-th available time window for the th doctor Location allocation for reserved service time for patients, if ,but This means that if a patient is not assigned to a doctor The Available time windows The location, then, will not give the patient this location. Reserve service time;

[0048] 1) Characterization of time costs at t=1

[0049] First, the doctor First available time window The first position The waiting time is:

[0050] (7)

[0051] The waiting time for other positions is determined by the previous patient's waiting time, the actual service time, and the reserved service time; doctors The first available time window Location The waiting time and idle time are respectively expressed in equations (8) and (9):

[0052] (8)

[0053] (9)

[0054] Among them, when hour, ,otherwise ;when hour, ,otherwise ;

[0055] In online medical platforms, doctors may have multiple consecutive or discontinuous available working time windows within a time period. Their overtime within the available time windows depends not only on whether there is another time window, but also on whether the available time windows are consecutive and whether the treatment mode of the last patient served in that window is text, video, or voice.

[0056] when At that time, regardless of the type of patient served last within the time window, if the service duration exceeds the window, all excess time will be considered overtime. If the current time window and the next time window are consecutive, then when serving the last patient, the portion exceeding the current time window can be carried over to the next time window and is not counted as overtime, meaning the service is completed within the doctor's overall continuous available time. However, if the available time windows are not consecutive, the determination of overtime depends on the service type of the last patient. For text-based consultations, since their service is highly non-immediate, it is assumed that the doctor will continue to provide services to the patient in subsequent non-consecutive time windows, and therefore, the overtime for the current window is not counted. For video or voice consultations, if the service exceeds the scope of the current time window, the portion exceeding the current time window is considered as overtime for that time window.

[0057] Therefore, this article introduces virtual overtime hours. Based on the recursive relationship, the doctor's virtual overtime and free time are:

[0058] (10)

[0059] (11)

[0060] Furthermore, in the context Below, set up a doctor The location of the last patient served within the available time window is ,in Introducing binary variables ,when Indicates doctor In the time window The last patient served was a telephone patient, when Indicates doctor time window The last patient served was the one with images or text. This indicates that the doctor's available time window... Therefore, their actual overtime hours at that window are:

[0061] (12)

[0062] Among them, variables For doctors Available time window Numbering of global time slots Equation (12) indicates that if the doctor In time period There are subsequent time windows, and these time windows are available. and available time windows They are connected in the global time slots, that is... and Then the doctor In the time window If there is no actual overtime work, the corresponding virtual overtime work time is directly carried over to the next time window and recorded as the first patient waiting time in the next time window. If the two time windows are not consecutive, i.e. and If the last patient is a text / image patient, the virtual overtime period will be postponed to the next time window; if the last patient is a telephone patient, the doctor needs to work overtime to complete the service, and this will be recorded as a doctor's overtime work. In the time window The actual overtime hours. Therefore, the waiting time for patients between the two time windows is:

[0063] (13)

[0064] If the doctor has only one available time window, that is When a doctor is assigned to provide overtime services to patients, their actual overtime hours and free time are as follows:

[0065] (14)

[0066] (15)

[0067] 2) Time cost description

[0068] when At that time, based on the condition of the last patient served by the doctor in the previous available time window, determine The waiting time for the first appointment; doctor Available time window The first position The waiting time is:

[0069] (16)

[0070] Similarly, the waiting time for other positions is determined by the waiting time of the previous patient, the actual service time, and the reserved service time; therefore, the doctor Available time windows Location The waiting time and idle time are:

[0071] (17)

[0072] (18)

[0073] doctor In the The available time windows for virtual overtime and free time are:

[0074] (19)

[0075] (20)

[0076] Similar to equation (12), if the doctor If there is a subsequent available time window, then it is within the available time window. The actual overtime hours are:

[0077] (twenty one)

[0078] in, For the last patient served, there is .variable For doctors The The number of available time windows in the global time slot. Similarly, the waiting time for the patient between the two available time windows is:

[0079] (twenty two)

[0080] 3) Depicting the last available time slot for overtime and free time

[0081] The doctor's overtime and free time during the last available time window to serve the last patient is as follows:

[0082] (twenty three)

[0083] (twenty four)

[0084] Based on the above analysis, the scheduling module [SP] for the second stage is:

[0085] (25)

[0086] st (26)

[0087] (27)

[0088] (28)

[0089] (29)

[0090] (30)

[0091] (31)

[0092] (32)

[0093] (33)

[0094] (34)

[0095] (35)

[0096] (36)

[0097] The objective function (25) for the [SP] phase is to minimize the expected patient waiting cost, physician idle cost, and overtime cost; constraint (26) indicates that if the physician... Within the available time window Inner If a position is not assigned a patient, the reservation time for that position is 0; constraint (27) ensures that a position is assigned to a doctor. The total reserved time for patients is equal to the length of their available time window; constraint (28) linearizes recursive relations (8)(9) and (17)(18); constraint (29) linearizes recursive relations (10)(11) and (19)(20); (30) linearizes recursive relations (14)-(15) and (23)-(24); constraint (32) integrates recursive relations (12)(21); constraint (33) integrates recursive relations (13)(22).

[0098] In summary, the present invention has the following beneficial effects:

[0099] This invention constructs a multi-objective optimization model for doctor-patient matching and scheduling. By constructing a matching cost evaluation module based on multi-granularity text similarity and a multi-objective optimization resource matching module that considers random service time, it simultaneously optimizes the effectiveness of doctor-patient matching and the scheduling efficiency of heterogeneous time windows of multiple service providers.

[0100] Through a series of theoretical analyses and numerical experiments, the following main conclusions were obtained:

[0101] First, multi-granularity text feature extraction methods can effectively address the unstructured challenges of online medical data and significantly improve the accuracy of doctor-patient matching. Single-granularity feature extraction methods struggle to fully capture the semantic relationships in doctor-patient matching. Coarse-grained LDA topic models excel at identifying the macro-level correlation between patient condition descriptions and doctor preferences, but their ability to distinguish details is limited; fine-grained SBERT models can accurately quantify sentence-level semantic similarity. The coarse-fine granularity fusion method proposed in this invention extracts richer matching information from doctor-patient data through feature complementarity. Experimental results show that the matching accuracy of the fusion method is significantly improved compared to the single method. This discovery reveals the multi-level semantic features of the doctor-patient matching problem and provides a methodological reference for the algorithm design of intelligent matching on online medical platforms.

[0102] Second, the integrated optimization framework for doctor-patient matching and scheduling can achieve global optimum, while the separate optimization mode suffers from significant efficiency loss. The multi-objective optimization model for doctor-patient matching and scheduling constructed in this invention incorporates matching cost and scheduling cost (i.e., multi-objective optimization resources) into a unified objective function. It also incorporates the heterogeneous time window constraints of doctors and the two service types of patients into the model, considering doctor service capacity limitations and time window constraints in the first stage. This more realistically portrays the operating environment of online consultation platforms, avoiding the dilemma of optimal matching but infeasible scheduling, and overcoming the shortcomings of existing research that separates matching and scheduling and ignores their coupling relationship. Comparative experiments show that the integrated optimization method significantly outperforms the separate optimization method in terms of total cost, verifying the necessity and effectiveness of collaborative optimization of matching and scheduling, and providing a theoretical basis for optimizing the platform's scheduling mechanism and improving doctors' time utilization efficiency.

[0103] Third, the improved PHA algorithm can effectively enhance the solution efficiency of large-scale stochastic optimization problems. The synergistic effect of the variable progressive fixing strategy and the penalty parameter adaptive update strategy is the key to the algorithm's performance improvement. The standard PHA faces challenges such as slow convergence speed and unstable solution quality when solving the large-scale mixed-integer stochastic programming problem constructed in this invention. This invention proposes an improvement scheme from two levels: algorithm structure and parameter mechanism. The variable progressive fixing strategy monitors the consistency of decision variables across scenarios and pre-fixes converged variables, effectively reducing the problem size of subsequent iterations. The penalty parameter adaptive update strategy dynamically adjusts the penalty intensity based on the convergence improvement rate, balancing the trade-off between convergence speed and solution quality. Numerical experiments show that the improved PHA algorithm can produce higher quality and more stable solution results in a shorter running time, demonstrating good computational efficiency in large-scale problems. Attached Figure Description

[0104] Figure 1 This describes the performance of the SAA solution in different scenarios in Embodiment 4 of the present invention;

[0105] Figure 2 This is the optimal total cost comparison in Embodiment 5 of the present invention;

[0106] Figure 3 This is a comparison of operating costs and matching costs in Embodiment 5 of the present invention;

[0107] Figure 4 This is a comparison chart of scheduling costs and matching costs under different scale scenarios in Embodiment 5 of the present invention;

[0108] Figure 5 This is a comparison chart of scheduling cost and matching cost under different scale scenarios in Embodiment 5 of the present invention, where (1) is a comparison of the standardized costs of the three optimization strategies, and (2) is the cost improvement effect of integrated optimization. Detailed Implementation

[0109] The following is in conjunction with the appendix Figure 1-5 The present invention will be described in further detail below.

[0110] The relevant symbols and parameter settings involved in this invention are shown in Table 1:

[0111] Table 1 Symbol and Parameter Settings

[0112]

[0113] Example 1: A multi-objective optimization model for doctor-patient matching and scheduling. The specific implementation process of the multi-objective optimization model includes the following two stages:

[0114] Phase 1: Constructing the matching and scheduling integration module, as detailed below:

[0115] Consider random scenarios The quantity, let's say there are a total of [number] within standard time T. Patients need to be assigned, and patients should be recorded as... Patients are divided into two categories: those seeking text-based consultations, using asynchronous communication and describing their condition in text and / or image format; and those seeking video or voice consultations, using synchronous communication requiring real-time interaction between the doctor and patient. The group of patients seeking text-based consultations is denoted as […]. Video consultations with patients are a collection ,satisfy and Significant differences in service duration under different consultation models;

[0116] Within this standard time period T, there are J heterogeneous available doctors, and the service cycle will be... Divided into A time period, denoted as The duration of each time period is To accurately depict the diverse characteristics of doctors' available time windows, each doctor... Doctors can freely choose their working hours according to their own preferences. A binary variable is introduced to represent the available time window for doctors. This indicates that doctor j can provide services during the k-th time period. This indicates that the time period is unavailable, therefore for each doctor Its available time window set for ;

[0117] For ease of subsequent calculations, please provide to the doctor. The available time windows are renumbered, and the new index is denoted as . ,in, Let j be the number of available time windows for doctor j, and introduce variables. Let t be the number of the t-th available time window of doctor j in the global time slot. , where k is the t-th available global time slot number for doctor j;

[0118] The matching cost between doctors and patients is obtained based on the semantic analysis results of the first stage. ; Statistical estimation of patients based on historical medical data The actual service time at doctor j's office was... ;use Let represent the decision variable in the matching problem, such as if patient i is matched with doctor. but ,otherwise In addition, a binary decision variable is introduced to formulate scheduling decisions for each doctor. Each of the doctors Each available time window There is Available locations, indexed as Customers assigned to smaller index positions will be served earlier; additionally, let's assume... Indicates in The total cost of patient waiting time, doctor idle time, and overtime work in the given scenario;

[0119] Based on the above information, the goal of the first phase of building the matching module is to minimize the doctor-patient matching cost and expected cost; therefore, the first phase [MP] can be described as follows:

[0120] (1)

[0121] (2)

[0122] (3)

[0123] (4)

[0124] (5)

[0125] (6)

[0126] The objective function of the [MP] phase is to minimize the cost of matching patients with doctors and the total cost of the scheduling phase. The total cost of the scheduling phase includes patient waiting time, doctor idle time, and overtime costs. These are weight control parameters; Formula (2) indicates that each patient can only be seen by one doctor; Formula (3) indicates that patient z can only be assigned to one position within one available time period of the doctor who sees him; Formula (4) restricts each available time window of each doctor j. Each position It can be occupied by a maximum of one patient; Formula (5) represents the position of the doctor within each available time window. Only the previous position It can only be occupied when it is occupied.

[0127] The second stage involves constructing a scheduling module that considers the random service time of services and introducing spatiotemporal constraints on medical resources. These constraints include, but are not limited to, the time window limit for available doctors and the differences in patient service type requirements. Finally, a multi-objective optimization model is constructed to minimize doctor-patient matching costs and patient waiting time, as well as to optimize the efficiency of doctor resource utilization.

[0128] The second phase of building the scheduling module is as follows:

[0129] The goal of the second phase is to minimize the sum of patient waiting costs, physician idle time costs, and overtime costs. Indicates in the scene Next The doctor's first The first available time window Location allocation for reserved service time for patients, if ,but This means that if a patient is not assigned to a doctor The Available time windows The location, then, will not give the patient this location. Reserve service time;

[0130] 1) Characterization of time costs at t=1

[0131] First, the doctor First available time window The first position The waiting time is:

[0132] (7)

[0133] The waiting time for other positions is determined by the previous patient's waiting time, the actual service time, and the reserved service time; doctors The first available time window Location The waiting time and idle time are respectively expressed in equations (8) and (9):

[0134] (8)

[0135] (9)

[0136] Among them, when hour, ,otherwise ;when hour, ,otherwise ;

[0137] In online medical platforms, doctors may have multiple consecutive or discontinuous available working time windows within a time period. Their overtime within the available time windows depends not only on whether there is another time window, but also on whether the available time windows are consecutive and whether the treatment mode of the last patient served in that window is text, video, or voice.

[0138] when At that time, regardless of the type of patient served last within the time window, if the service duration exceeds the window, all excess time will be considered overtime. If the current time window and the next time window are consecutive, then when serving the last patient, the portion exceeding the current time window can be carried over to the next time window and is not counted as overtime, meaning the service is completed within the doctor's overall continuous available time. However, if the available time windows are not consecutive, the determination of overtime depends on the service type of the last patient. For text-based consultations, since their service is highly non-immediate, it is assumed that the doctor will continue to provide services to the patient in subsequent non-consecutive time windows, and therefore, the overtime for the current window is not counted. For video or voice consultations, if the service exceeds the scope of the current time window, the portion exceeding the current time window is considered as overtime for that time window.

[0139] Therefore, virtual overtime hours are introduced. Based on the recursive relationship, the doctor's virtual overtime and free time are:

[0140] (10)

[0141] (11)

[0142] Furthermore, in the context Below, set up a doctor The location of the last patient served within the available time window is ,in Introducing binary variables ,when Indicates doctor In the time window The last patient served was a telephone patient, when Indicates doctor time window The last patient served was the one with images or text. This indicates that the doctor's available time window... Therefore, their actual overtime hours at that window are:

[0143] (12)

[0144] Among them, variables For doctors Available time window Numbering of global time slots Equation (12) indicates that if the doctor In time period There are subsequent time windows, and these time windows are available. and available time windows They are connected in the global time slots, that is... and Then the doctor In the time window If there is no actual overtime, the corresponding virtual overtime will be directly postponed to the next time window and recorded as the first patient waiting time in the next time window.

[0145] If the two time windows are not contiguous, that is and If the last patient is a text / image patient, the virtual overtime period will be postponed to the next time window; if the last patient is a telephone patient, the doctor needs to work overtime to complete the service, and this will be recorded as a doctor's overtime work. In the time window The actual overtime hours. Therefore, the waiting time for patients between the two time windows is:

[0146] (13)

[0147] If the doctor has only one available time window, that is When a doctor is assigned to provide overtime services to patients, their actual overtime hours and free time are as follows:

[0148] (14)

[0149] (15)

[0150] 2) Time cost description

[0151] when At that time, based on the condition of the last patient served by the doctor in the previous available time window, determine The waiting time for the first appointment; doctor Available time window The first position The waiting time is:

[0152] (16)

[0153] Similarly, the waiting time for other positions is determined by the waiting time of the previous patient, the actual service time, and the reserved service time; therefore, the doctor Available time windows Location The waiting time and idle time are:

[0154] (17)

[0155] (18)

[0156] doctor In the The available time windows for virtual overtime and free time are:

[0157] (19)

[0158] (20)

[0159] Similar to equation (12), if the doctor If there is a subsequent available time window, then it is within the available time window. The actual overtime hours are:

[0160] (twenty one)

[0161] in, For the last patient served, there is .variable For doctors The The number of available time windows in the global time slot. Similarly, the waiting time for the patient between the two available time windows is:

[0162] (twenty two)

[0163] 3) Depicting the last available time slot for overtime and free time

[0164] The doctor's overtime and free time during the last available time window to serve the last patient is as follows:

[0165] (twenty three)

[0166] (twenty four)

[0167] Based on the above analysis, the scheduling module [SP] for the second stage is:

[0168] (25)

[0169] st (26)

[0170] (27)

[0171] (28)

[0172] (29)

[0173] (30)

[0174] (31)

[0175] (32)

[0176] (33)

[0177] (34)

[0178] (35)

[0179] (36)

[0180] The objective function (25) for the [SP] phase is to minimize the expected patient waiting cost, physician idle cost, and overtime cost; constraint (26) indicates that if the physician... Within the available time window Inner If a position is not assigned a patient, the reservation time for that position is 0; constraint (27) ensures that a position is assigned to a doctor. The total reserved time for patients is equal to the length of their available time window; constraint (28) linearizes recursive relations (8)(9) and (17)(18); constraint (29) linearizes recursive relations (10)(11) and (19)(20); (30) linearizes recursive relations (14)-(15) and (23)-(24); constraint (32) integrates recursive relations (12)(21); constraint (33) integrates recursive relations (13)(22).

[0181] Example 2: Solving the model in Example 1 using the SAA method to handle random service time.

[0182] The SAA method is an efficient scenario-based approach for solving stochastic programming problems and has been widely used to solve decision problems. This paper derives a deterministic mixed-integer linear programming problem based on the SAA model. In practice, a set of... Based on the probability distribution of a sample of independent and identically distributed actual patient service time, each sample scenario is divided into... The probability is set as The SAA method transforms the original two-stage stochastic programming problem into an approximately deterministic single-stage problem. To obtain the deterministic procedure for the entire problem, it is still necessary to linearize the relevant constraints and the nonlinear terms in the objective function regarding the decision variables.

[0183] First, the nonlinear constraints (31), (32), and (33) of the linearization scheduling stage are defined. For constraint (31), a binary variable is first introduced. ,when hour, . , The nonlinear constraint (31) can be transformed into a set of linearized expressions:

[0184] (37)

[0185] Then, linearize the constraint (32). Introduce binary variables. ,when hour, ;when hour, At the same time, continuous auxiliary variables are introduced. express The product term. The linearized constraints are as follows:

[0186] (38)

[0187] Finally, for constraint (33), a binary variable is introduced. ,like ,but ;like ,but and introduce For smaller positive numbers, the linearized constraints are as follows:

[0188] (39)

[0189] Based on this, the problem under study is represented as a deterministic mixed-integer linear programming problem [DF], which is the benchmark method in computational experiments. The model is as follows:

[0190] (40)

[0191] St(2)-(6),(26)-(30),(34)-(39)

[0192] in, Indicates the situation The actual service time of patients under the following variables , , , and Each represents a scenario The corresponding waiting time, idle time, and overtime time are then calculated. The Big-M parameter is introduced into the model (…). Reconstructing the original problem into a mixed-integer linear programming problem significantly impacts the model's computational performance; even with large problem sizes, finding the optimal solution within a reasonable timeframe remains challenging. Based on preliminary testing, when... , , At that time, the optimal solution of the corresponding [DF] cannot be obtained within 5 hours.

[0193] Example 3: Further solving [DF] in Example 2 based on the improved PHA

[0194] The SAA method transforms the original stochastic programming problem into an approximately deterministic problem through Monte Carlo sampling, enabling the acquisition of high-quality approximate solutions with finite samples. However, the resulting deterministic equivalent model remains a large-scale two-stage mixed integer programming problem, and the scale of variables and constraints in the [DF] model expands dramatically with the increase in the number of samples, significantly increasing computational complexity. The model provided in this invention contains a large number of binary integer variables in the first stage and involves mixed decisions of integer and continuous variables in the second stage. This unique structure makes the aforementioned traditional decomposition methods difficult to apply directly, further increasing the difficulty of solving the model. Therefore, based on SAA, PHA is used to solve the [DF] model.

[0195] The PHA algorithm uses augmented Lagrangian techniques to decompose the original multi-scenario problem into subproblems based on single scenarios, iteratively solves these subproblems, constructs aggregate solutions from the subproblems, and adjusts the decomposition parameters to guide the method towards a good hedging solution for the original problem whenever the solutions to the subproblems are inconsistent (i.e., they do not produce the same solution). The core principle is to relax the unexpectedness constraint and solve the scenario subproblems independently. The algorithm converges when the solutions to all subproblems satisfy the unexpectedness constraint. However, the time-linked nature of the model and the nonlinear logic of waiting and overtime work lead to complex subproblem structures. Furthermore, the introduction of consistency constraints significantly increases coordination difficulty due to all scenarios sharing matching and scheduling decisions, making direct application of PHA challenging. Therefore, this embodiment combines SAA and an improved PHA method to solve the DF model. First, based on the given scenario set, the formulas are separated and subproblems are defined. Then, a variable fixation method and a penalty update mechanism are used to improve the performance of the basic SAA-PHA.

[0196] To construct a decomposable structure, for variables , Each scene Its corresponding secondary variable was introduced. , Thus, the following equivalent reconstruction form [SDF] is obtained.

[0197] (41)

[0198] in, and This represents the consensus solution, used to approximate consistency across all scenarios. To obtain a separable consensus, the nearest neighbor method is used to estimate the consensus solution, as shown in the following formula.

[0199] (42)

[0200] (43)

[0201] and For the corresponding Lagrange multipliers, and This is the penalty parameter. The constraint must be satisfied in all scenarios. Unintended constraints are applied only to the following variables:

[0202] (44)

[0203] (45)

[0204] Finally, based on the above planning, application scenarios are decomposed, and each scenario is... The following scenario subproblems, referred to as SSDF, are defined, and each scenario is solved independently in each iteration of PHA. The objective function SSDF is as follows:

[0205] (46)

[0206] To address the online medical patient-doctor allocation problem with uncertain service duration, complex matching, and scheduling logic, a scenario decomposition strategy based on PHA is proposed. This method treats patient-doctor matching decisions and scheduling arrangements (i.e., first-stage decision variables) as key decisions requiring consensus across all scenarios. By introducing unexpected consistency constraints, it ensures that the final solution maintains optimal consistency under all uncertain scenarios. Variables such as service duration, waiting time, idle time, and overtime are treated as scenario-specific second-stage variables, satisfying specific scheduling constraints only within their respective scenarios. The PHA method relaxes the consistency constraints through an augmented Lagrange mechanism, transforming them into penalty terms and Lagrange multipliers in the objective function. This allows each scenario to independently and concurrently optimize its subproblems. After each iteration, the penalty parameters and multipliers are dynamically adjusted based on the global average solution and consistency error, achieving global convergence. The basic general steps of the PHA for solving online medical patient-doctor matching and scheduling are shown in Table 2, Algorithm 1.

[0207]

[0208] Table 2 Basic PHA Pseudocode

[0209] The algorithm employs an iterative mechanism. Let z denote the iteration counter. In step 1, the iteration counter, Lagrange multipliers, and penalty parameters are initialized. It is important to note that... This represents a non-negative constant initial value. In step 2, for each scenario... Solve the SSDF subproblem for each scenario and determine a matching decision and scheduling scheme for each patient. However, in the first iteration, terms containing Lagrange multipliers and penalty parameters are ignored when solving the SSDF model to obtain an initial solution. In step 3, consensus parameter values ​​are calculated by averaging the decision variable values ​​across all scenarios, with each scenario assigned the same weight. In step 4, the Lagrange multiplier parameter values ​​are updated based on the deviation between the scenario decision variable values ​​and the consensus parameter values. Finally, in step 5, it is checked whether the unexpected constraints are met. If the maximum deviation between all scenario solutions and the consensus solution does not exceed a preset tolerance... If the maximum number of iterations is reached, the algorithm converges and terminates; if the maximum number of iterations is reached... If convergence has not yet occurred, the current optimal solution is output and the process terminates; otherwise, the iteration counter is updated and a new round of iteration is started from step 2. The basic PHA algorithm, through the above iterative mechanism, can handle the uncertainties in the matching and scheduling problem in online medical platforms, generating robust allocation schemes that perform well under various patient service time scenarios. However, preliminary experiments show that the basic PHA requires an excessively long convergence time to obtain a solution, and its quality is generally found to be low. Therefore, the basic PHA is improved to address this issue.

[0210] Analysis revealed that the standard PHA faces the following challenges in solving large-scale online healthcare allocation problems. First, this problem involves numerous binary decision variables, requiring hundreds of iterations for convergence, often resulting in computation times of several hours or even days. This slow convergence speed severely limits the algorithm's application in real-world healthcare institutions. Second, the penalty parameter ρ is a key factor affecting PHA convergence performance. A parameter that is too small... The value leads to insufficient coordination between solutions in the scenario, making it difficult to reach a consensus solution; an excessively large value... However, excessively high values ​​can cause Lagrange multiplier oscillations, compromising algorithm stability. To address these bottlenecks, improvements are proposed at both the algorithm structure and parameter mechanism levels. At the algorithm structure level, a progressive variable fixing strategy (see Table 3) is designed to monitor the consistency of decision variables across scenarios. When a variable reaches a preset threshold, it is fixed in advance, reducing the scale of decision variables in subsequent iterations and thus shortening the solution time for sub-problems. At the parameter mechanism level, an adaptive penalty parameter update strategy is proposed (see Table 4). This strategy dynamically adjusts the penalty parameter based on the convergence improvement rate. When convergence is slow, the penalty intensity is increased to accelerate consistency establishment; when convergence is too fast, the penalty intensity is decreased to avoid over-constraint, achieving a balance between convergence speed and solution quality. The effects of these two improvement strategies will be systematically evaluated in numerical experiments.

[0211] (1) Variable fixation strategy based on exponential decay dynamic threshold

[0212] The computational complexity of mixed-integer programming (PHA) lies in the integer variables. In PHA iterations, fixing binary variables is an effective heuristic for accelerating convergence. Therefore, to reduce the computation time for solving subproblems and speed up the convergence process, some variables are fixed in advance before the termination criterion is met. Preliminary observations of the behavior of the first-stage decision variables throughout the iteration process reveal that when a certain matching variable... or scheduling variable When a variable exhibits high consistency across solutions in multiple scenarios, it typically retains the same value in subsequent iterations. The strategy's applicability to patients... Matching variables and scheduling variables Consistency across different scenarios is monitored. When the scenario average of a variable reaches a preset threshold, the variable is fixed to the corresponding value. As the variable fixing process is applied in the iterations, the number of decision variables in the scenario subproblems decreases, and the feasible region of other unfixed variables also shrinks accordingly, thus significantly reducing the computational burden of subsequent iterations. Before implementing the variable fixing strategy, a threshold level needs to be set to determine when scenario consistency is achieved. This threshold level depends on the current iteration number. Specifically, for binary decision variables... Calculate its average value across all scenarios:

[0213] (47)

[0214] when When it is assumed that the variable has reached a consensus in all scenarios, it is fixed at 1 in all scenarios; when At that time, it is fixed at 0. (Schedule variable) The fixed judgment process is exactly the same. The fixed strategy uses a dynamic threshold with an exponential decay mechanism. This threshold starts at 100% in the first iteration and then gradually decreases exponentially until it stabilizes at a certain lower limit. For example, when hour, ;when hour, ,in The attenuation rate, This is the asymptotic lower bound of the threshold. The number of iterations starting from the stability threshold; when hour, In the first iteration, the threshold is set to 100%, requiring complete consistency before fixing to prevent premature fixing from leading to convergence to a poor solution. In the early iteration stages ( The threshold decreases exponentially. For example, in the 5th iteration... In the 10th iteration A high threshold level in early iterations can prevent the risk of converging to a low-quality solution. As the number of iterations increases, the differences in variable values ​​across different scenario solutions gradually decrease, allowing the algorithm to use a lower threshold to fix more variables. When the number of iterations reaches... Afterward, the threshold stabilized at a low level and stopped decreasing. Detailed steps are given in Algorithm 2, as shown in Table 3.

[0215]

[0216] Table 3. Variable fixing strategy for the exponential decay dynamic threshold

[0217] Improved Algorithm 2 is called after step 3 and before step 4 in the basic PHA algorithm, receiving the current iteration. Scene solution and a fixed set of variables, based on dynamic thresholds The algorithm performs variable fixing checks. First, it calculates a dynamic threshold based on the current iteration count. Then, it sequentially checks each unfixed matching and scheduling variable, calculates its scenario average, and determines whether the fixing condition is met. Finally, it returns the updated set of fixed variables and the number of newly fixed variables. Fixed variables retain their values ​​in subsequent iterations of the scenario subproblem solution and are no longer used as decision variables. This is achieved by adding equality constraints to the solver or removing the variable directly from the model. Because the variable fixing process further narrows the feasible domain of relevant variables, the solution time for the scenario subproblem is significantly shortened. This gradual fixing strategy improves computational efficiency while ensuring solution quality; the specific effects will be verified in numerical experiments.

[0218] (2) Adaptive penalty parameter update strategy

[0219] The choice of penalty parameter value is crucial to ensuring that the PHA algorithm converges to a high-quality solution within a reasonable time. Setting an excessively high penalty parameter forces decision variables in different scenarios to converge quickly to a consensus solution in early iterations. However, since the consensus solution is updated progressively through averaging scenario solutions, the estimated quality of the consensus solution in early iterations is low. An excessively high penalty parameter can cause scenario solutions to prematurely lock into the vicinity of a low-quality consensus solution, limiting the algorithm's ability to explore better solutions and potentially leading to a suboptimal solution. On the other hand, an excessively low penalty parameter delays the convergence of decision variables, thus increasing the number of iterations. While the quality of the consensus solution gradually improves with increasing iterations, it requires a greater computational cost. Therefore, the PHA algorithm needs to dynamically adjust the penalty parameter in each iteration based on the current convergence state to balance solution quality and computational efficiency. Existing research has shown that using a dynamic update method to adjust parameter values ​​has significant advantages in terms of solution quality and running time. Since the decision variables in this study include a doctor-patient matching layer (…),… ) and time and location scheduling layer ( Two different dimensions of binary decision-making, for Shen's research The penalty parameter update method in the original code is improved by using the original residual as an indicator to measure the degree of divergence in the scenario solution. Specifically, The formula for calculating the original residual in the z-th iteration is:

[0220] (48)

[0221] (49)

[0222] The current convergence state is determined by monitoring the decrease in the original residual between adjacent iterations. Specifically, at each iteration, the strategy first calculates the convergence improvement rate. ,like This indicates that the residuals decrease, the solutions to the scenarios tend to be consistent, and the consensus improves. This indicates that the residuals are increasing and the consensus is deteriorating; A value close to 0 indicates that the algorithm has stalled or has converged. Based on this, an adaptive penalty parameter update strategy was constructed, and the update rules are shown in Table 4.

[0223]

[0224] Table 4. Fixed Penalty Parameter Strategy

[0225] The penalty parameter adaptive strategy employs a phased dynamic adjustment mechanism to ensure that the algorithm maintains good performance at different convergence stages. During the initialization phase ( All penalty parameters are uniformly set to... This lower initial value avoids imposing overly strong consistency constraints on the scenario solution during the algorithm's startup phase, preserving ample exploration space for the scenario subproblems. Simultaneously, the historical residuals are initialized to... This ensures that the first iteration does not trigger incorrect adjustments due to a lack of historical data. In the early stages of the iteration ( (In the mid-term), the strategy actively responds to changes in the convergence state based on the real-time calculated improvement rate. The strategy sets a slow improvement threshold. and rapidly improve threshold As a criterion for adjusting decisions. When When this occurs, it indicates that the convergence progress of this type of variable has stalled, and its penalty parameter is increased to [value missing]. The increase factor Control the adjustment range, upper limit To prevent excessive punishment; when When this occurs, it indicates that the convergence speed is exceptionally fast, and the penalty parameter is reduced to... The reduction coefficient Achieve gentle adjustment, lower bound Ensure basic consistency constraint strength; when the improvement rate is at When the interval is reached, the convergence speed is moderate, maintaining... Unchanged. In the later stages of convergence (near the termination criterion), when the original residuals of a certain type of variable... When the strategy determines that the variable has converged sufficiently, it automatically stops adjusting and maintains the current state. This prevents unnecessary oscillations caused by parameter perturbations when approaching the optimal solution. Furthermore, the strategy... and The variables maintain completely independent penalty parameter trajectories, fully respecting the heterogeneous convergence characteristics of different decision variables and avoiding the "one-sided" problem that may occur with traditional unified parameter methods. For example, if the convergence improvement rate of the matching variables is considered in the 5th iteration... (less than) Greater than This indicates that the consistency of matching decisions is established slowly, and the strategy adjusts the penalty parameter to... This strengthens the constraint on consistency; if the improvement rate of the concurrent scheduling variables... (greater than) This indicates that the decision variables are rapidly converging, and the strategy will adjust the penalty parameter to... The constraints are appropriately relaxed to reduce the difficulty of solving subproblems. Finally, the algorithm returns the updated penalty parameters. These parameters will be used in the next iteration. The updated parameters take effect in the objective function of the subproblem of the scenario, and the strength of the scenario consistency constraint is dynamically adjusted by modifying the weight coefficients in the augmented Lagrange term. The updated parameters also affect the update formula of the dual variable in step 4 of the basic PHA algorithm. and This ensures the co-evolution of the penalty term and the dual information.

[0226] Example 4: Analysis of the model provided by this invention based on a real-world case.

[0227] First, based on 1884 telephone online medical consultations conducted in 2024 on the online medical platform of a hospital in Zhengzhou, this study empirically characterizes the telephone consultation service time. Statistical results show that the mean call duration was 7.05 minutes, the standard deviation was 0.66 minutes, the median was 6.58 minutes, and the minimum and maximum were 3.0 minutes and 10.3 minutes, respectively. To accurately reflect the distribution characteristics of service time, the method of moments was used to fit the data, resulting in a Gamma (114.11, 0.06) distribution for doctor telephone consultation service time (mean 7.05 minutes, standard deviation 0.66 minutes). This distribution effectively characterizes the right skewness and variation of actual service time. Since the platform data lacks direct records of text-based consultation service time, a reasonable estimate of the doctor's processing time for text-based consultations was made by systematically reviewing relevant domestic and international literature and practical experience. Based on domestic and international literature and practical experience, it is assumed that the clinical time required for a doctor to handle one text-based consultation (completing a full consultation) in online medical services typically falls within the range of 10-20 minutes. Considering the service characteristics and the median level in the references, in the baseline scenario, it is assumed that the doctor service time for text-based consultations follows a Gamma distribution with a mean of 15 minutes. Given that text-based consultations involve text input and possible multi-turn interactions, their time variability is generally higher than that of real-time telephone consultations. Assuming a standard deviation of 2.5 minutes (coefficient of variation of approximately 16.7%), the time estimation method yields a Gamma(36, 0.42) distribution for text-based consultations (mean 15 minutes, standard deviation 2.5 minutes).

[0228] Secondly, the patient's unit waiting time cost, the doctor's unit idle time cost, and overtime time cost are set as... , , .patient For doctors Matching cost From uniform distribution All the calculation examples were solved using PyCharm with GUROBI. The detailed configuration is as follows: GUROBI Optimiser version 11.0.3 build v11.0.3 RC0. CPU model: Intel(R) Core(TM) i5-1155G7 @ 2.50GHz.

[0229] In addition, due to the presence of random factors in the problem, the scenario value, i.e., the value of |H|, needs to be determined before formal analysis. Figure 1 The figure shows the target mean of the model after solving using the SAA method under different number of scenarios. As can be seen from the figure, the average total cost stabilizes when the sample size reaches 100, meaning that the randomness of the problem in this study can be well captured by a sample size of 100. Therefore, |H|=100 is chosen as the standard scenario size for subsequent experiments.

[0230] To verify the effectiveness of the model, three solution methods were compared: SAA, basic PHA, and improved PHA. Three experimental designs were implemented. The sizes are respectively Five randomized case instances are generated for each set of parameters. The SAA case solution time limit is 20,000 seconds. For the SAA method, report the average cost of the best feasible solution obtained within the time limit and the average optimality gap of unsolved cases. For the basic PHA and improved PHA methods, report their average relative deviation from SAA, calculated using the following formula:

[0231] (50)

[0232] in and The optimal feasible solution objective values ​​obtained by the PHA and SAA methods (within the runtime range) are shown respectively, with negative values ​​indicating that PHA outperforms SAA. The average computation time for each method is also reported statistically in the experiments. Table 5 shows the results at different sample sizes. The results show the performance comparison of the three solution methods on different problem sizes.

[0233]

[0234] Table 5 Performance Comparison of Three Solution Methods

[0235] As shown in Table 5, the improved PHA algorithm has a significant efficiency advantage over directly solving the SAA. This is particularly evident for medium-sized computational examples. The SAA method yields low-quality solutions within a 20,000-second time limit, while the basic PHA method, although improving the objective value by an average of 9.28%, has a computation time as high as 12,047.28 seconds. For large-scale examples... The SAA method and the basic PHA both failed to obtain a feasible solution within the 20,000-second time limit. However, the improved PHA solved the problem in 2111.9 seconds, achieving an average improvement of 20.00% in the target value compared to SAA. Furthermore, when the basic PHA reached full convergence (53548.96 seconds), its deviation rate was 21.42%, while the improved PHA's deviation was only 1.42%. Therefore, the proposed improved PHA significantly improves solution efficiency while maintaining solution quality. This is mainly because the basic PHA converges slowly when dealing with large-scale problems, making it difficult to obtain high-quality solutions within the same time limit. The improved PHA, by introducing a variable fixing strategy with an exponentially decaying dynamic threshold and an adaptive penalty parameter update mechanism, effectively accelerates the algorithm's convergence process, demonstrating good effectiveness and efficiency in solving large-scale stochastic programming problems. Especially in the case of large sample sizes, the improved PHA has significant advantages over the traditional SAA method and the basic PHA algorithm, and has good practical application value.

[0236] Example 5: Sensitivity analysis of the model

[0237] (1) Matching cost coefficient under different scenarios Analysis

[0238] Some experimental parameters were set to analyze the matching cost coefficient under different scenarios. The impact on the optimal solution and objective value. Specifically, assuming a fixed number of 7 doctors, the impact on patient size... Sensitivity analysis was performed below. The value range is set to (0, 200). For example... Figure 2 As shown, the total cost is... The changes exhibit a significant scale effect. Within a smaller interval, the four curves intersect significantly, while... When the patient population is large, the curve exhibits a stable linear increasing trend, and the slope of the curve increases with the size of the patient population. To gain a deeper understanding of the composition of total costs and... Regarding its response mechanism, this invention decomposes the total cost into two components: scheduling cost and matching cost for analysis. Figure 3 Figures (1) and (2) show the total scheduling cost and the total matching cost (unweighted, i.e., removing the weighted average cost). The coefficients show distinctly different trends across different patient sizes.

[0239] It can be observed that low The total cost of an interval is primarily determined by the scheduling structure, and the scheduling cost is significantly affected by system load. Low-load scenarios ( The surplus of doctors generates significant idle costs; high-load scenarios ( The shortage of doctors has led to a sharp increase in patient waiting times and doctors' overtime costs; in contrast, the medium workload ( , This achieves an optimal balance between idle time, waiting time, and overtime, thus minimizing scheduling costs. Therefore, the total cost is low. The interval exhibits a non-monotonic structure with the lowest scale in the middle and the highest scale at both ends, forming a curve intersection. With As the number of patients increases, the weight of matching cost in the objective function gradually rises, and the slope of the total cost curve also increases with the number of patients. This is because matching cost has a linear cumulative effect with the number of patients. In other words, the total cost increases with the number of patients. The sensitivity equals the sum of the system matching costs. The more patients there are, the greater the cumulative matching costs, causing the cost curve to curve at the same level. Incremental changes result in larger absolute changes, creating a stratified structure where the high-load curve is steepest and the low-load curve is flattest. This difference explains why the total cost curve is lower at different scales. Intersections occur, and at high... The intervals show a linear increase. Furthermore, from... Figure 3 From (1), it can be observed that the total scheduling cost increases with... The changes exhibit a three-stage pattern: a slow initial rise, followed by a rapid rise, and then a period of stabilization. Furthermore, a higher patient-to-doctor ratio correlates with better outcomes. The stronger the sensitivity. Belongs to the 0-60 stage. The scheduling cost of the scenario increased by 600%; while The number of scenarios only increased by about 50%. This is mainly because... When the system is relatively small, the model primarily focuses on scheduling efficiency, with the system's idle, waiting, and overtime costs in a relatively balanced state; however, as the system grows... As the patient-to-doctor ratio increases, the model begins to prioritize matching quality, altering the original scheduling structure to achieve better matching. This leads to some doctors having concentrated workloads while others are idle, or patient wait times being extended to match patients with doctors of specific specialties, thus rapidly increasing scheduling costs. In scenarios with a high patient-to-doctor ratio, where doctor resources are already strained, this restructuring of the scheduling structure will trigger even more significant cost increases. Subsequently, the four curves entered a relatively stable phase, and the growth rate of scheduling costs converged significantly. This contrasts sharply with the continued divergence in total scheduling costs. Figure 3 (2) shows the matching cost in When the size is small, it is in the rapid convergence phase, as... Increasing the matching cost leads to stable convergence. The fundamental reason lies in the upper limit constraint of physician resources. With a fixed number of physicians (7) and a given matching cost, the overall matching quality the system can provide has an upper limit, regardless of the number of patients served. Once the number reaches 40, most patients have been assigned to higher-quality doctors, and the matching quality is close to the theoretical upper limit of this resource allocation. Further increasing... The value cannot exceed this upper limit, so the total matching cost tends to stabilize and the gap between different scale scenarios narrows significantly.

[0240] Figure 4 The scheduling and matching costs were shown to vary with different patient sizes. The changing dual-axis evolution trend. The trend is analyzed using the marginal rate of substitution, which refers to the scheduling cost required to reduce matching cost by one unit. From... Figure 4 The characteristics of changes in the marginal rate of substitution can be directly observed in this context. In low... In this interval, matching costs decrease rapidly while scheduling costs increase gradually, indicating that significant improvements in matching quality can be achieved with relatively low scheduling costs, resulting in a low marginal substitution rate. As the cost of matching increases, it gradually converges and stabilizes, while the cost of scheduling continues to rise. This means that continuing to pursue higher matching quality requires increasingly higher scheduling costs, leading to a sharp increase in the marginal rate of substitution. Using a scenario as an example, we can more intuitively understand the changing pattern of the marginal rate of substitution. When When the cost increases from 0.2 to 10, the matching cost decreases from 76.7 to 64.9, while the scheduling cost only increases from 124.8 to 152.8. The marginal substitution rate is approximately 2.4, meaning that reducing the matching cost by 1 unit requires only 2.4 units of scheduling cost, which is within the efficient range. When the cost increases from 10 to 40, the matching cost decreases by only 4.2, while the scheduling cost increases by 142, and the marginal substitution rate rises sharply to 33.8. After age 40, the marginal rate of substitution exceeds 68 and continues to increase. The current value can no longer substantially improve the quality of matching. However, the marginal rate of substitution changes differently at different scales. Therefore, when optimizing doctor-patient matching and scheduling on online healthcare platforms, it is necessary to dynamically adjust the ratio based on the doctor-patient ratio. The goal is to minimize patient-doctor matching costs without significantly increasing operating costs. This differentiated configuration strategy enables comprehensive optimization of matching quality, scheduling efficiency, and total system cost across various operational scenarios.

[0241] To verify the effectiveness of the ensemble optimization framework, the Min-Max normalization method was used to normalize the scheduling cost and matching cost respectively, and the normalized total cost under the three strategies was calculated. When α approaches zero, the model degenerates into a pure scheduling optimization problem; when... When the value is 200, the model approximates a pure matching optimization problem. The comparison results are as follows... Figure 5As shown, the ensemble optimization achieves improvements in standardized costs across all patient-scale scenarios. Taking a scenario as an example, the standardized total cost of pure scheduling optimization and pure matching optimization is close to 1.00, while the integrated optimization ( The total cost can be reduced to 0.44, an improvement of 56.2%. This result confirms the necessity of incorporating the doctor-patient matching and scheduling problem into a unified objective function. The ensemble framework can identify the synergistic effect between the two objectives and avoid the systemic imbalance caused by over-optimization of a single objective. The ensemble optimization achieved a significant reduction in standardized total cost across all patient-scale scenarios, achieving a better overall balance between scheduling efficiency and matching quality.

[0242] (2) Sensitivity analysis of different distribution tables of random parameters

[0243] To further explore the model's performance under different random distributions, the gamma distribution parameters of the actual service time for text / image and telephone consultations were modified for analysis. Specific parameter adjustment schemes are detailed in Table 6. The results in Table 6 show that the model has solvable boundaries under different strategies, indicating that the model of this invention is not limited to the initially specified distribution, thus confirming the model's universality.

[0244]

[0245] Table 6. Model solution results under different actual service duration intervals.

[0246] This specific embodiment is merely an explanation of the present invention and is not intended to limit the invention. After reading this specification, those skilled in the art can make modifications to this embodiment without contributing any inventive step, but such modifications are protected by patent law as long as they are within the scope of the claims of the present invention.

Claims

1. A multi-objective optimization model for doctor-patient matching and scheduling, characterized by: The specific implementation process of the multi-objective optimization model includes the following two stages: Phase 1: The first phase of the matching and scheduling integration model aims to minimize the sum of the following costs: doctor-patient matching cost and the expected cost of the second phase; based on this, the first phase integration module [MP] can be described as follows: （1） （2） （3） （4） （5） （6） The second stage: Given a match, this stage considers the scheduling problem of random service time in order to minimize the sum of patient waiting costs, doctor idle time and overtime costs; This stage introduces spatiotemporal constraints on medical resources, including but not limited to the time window limit for available doctors and the differences in patient service type needs; ultimately, a multi-objective optimization model is constructed to minimize doctor-patient matching costs and patient waiting time, and to optimize the efficiency of doctor resource utilization. exist In this scenario, the scheduling module [SP] that considers random service time in the second stage is: （25） s.t. (26) （27） （28） （29） （30） （31） （32） （33） （34） （35） （36）。 2. The multi-objective optimization model for doctor-patient matching and scheduling according to claim 1, characterized in that: The first stage of building the matching module is as follows: Consider random scenarios The quantity, let's say there are a total of [number] within standard time T. Patients need to be assigned, and patients should be recorded as... Patients are divided into two categories: those seeking consultation via text and images, who use asynchronous communication to describe their condition in the form of text and / or images; Video and voice consultations with patients utilize synchronous communication, requiring real-time interaction between both doctors and patients; patients receiving text-based consultations are recorded as follows: Video consultation patients are collected ,satisfy and Significant differences in service duration under different consultation models; Within this standard time period T, there are J heterogeneous available doctors, and the service cycle will be... Divided into A time period, denoted as The duration of each time period is ; To accurately characterize the diverse features of a doctor's available time window, each doctor Doctors can freely choose their working hours according to their own preferences. A binary variable is introduced to represent the available time window for doctors. This indicates that doctor j can provide services during the k-th time period. This indicates that the time period is unavailable, therefore for each doctor Its available time window set for ; For ease of subsequent calculations, please provide to the doctor. The available time windows are renumbered, and the new index is denoted as . ,in, Let j be the number of available time windows for doctor j, and introduce variables. Let t be the number of the t-th available time window for doctor j in the global time slot. , where k is the t-th available global time slot number for doctor j; The matching cost between doctors and patients is obtained based on the semantic analysis results of the first stage. ; Statistical estimation of patients based on historical medical data The actual service time at doctor j's office was... ;use Let represent the decision variable in the matching problem, such as if patient i is matched with doctor. but ,otherwise In addition, a binary decision variable is introduced to formulate scheduling decisions for each doctor. Each of the doctors Each available time window There is Available locations, indexed as Customers assigned to smaller index positions will be served earlier; additionally, let's assume... Indicates in The total cost of patient waiting time, doctor idle time, and overtime work in the given scenario; Based on the above information, the goal of the first phase of building the matching module is to minimize the doctor-patient matching cost and expected cost; therefore, the first phase [MP] can be described as follows: （1） （2） （3） （4） （5） （6） The objective function of the [MP] phase is to minimize the cost of matching patients with doctors and the total cost of the scheduling phase. The total cost of the scheduling phase includes patient waiting time, doctor idle time, and overtime costs. These are weight control parameters; Formula (2) indicates that each patient can only be seen by one doctor; Formula (3) indicates that a patient can only be assigned to one position within one available time period of the doctor they are seeing; Formula (4) restricts each available time window for each doctor j. Each position It can be occupied by a maximum of one patient; Formula (5) represents the position of the doctor within each available time window. Only the previous position It can only be occupied when it is occupied.

3. The multi-objective optimization model for doctor-patient matching and scheduling according to claim 2, characterized in that: The second phase of building a scheduling module that considers random service times is as follows: set up Indicates in the scene Next The doctor's first The first available time window Location allocation for reserved service time for patients, if ,but This means that if a patient is not assigned to a doctor The Available time windows The location, then, will not give the patient this location. Reserve service time; 1) Characterization of time costs at t=1 First, the doctor First available time window The first position The waiting time is: （7） The waiting time for other positions is determined by the previous patient's waiting time, the actual service time, and the reserved service time; doctors The first available time window Location The waiting time and idle time are respectively expressed in equations (8) and (9): （8） （9） Among them, when hour, ,otherwise ;when hour, ,otherwise ; In online medical platforms, doctors may have multiple consecutive or discontinuous available working time windows within a time period. Their overtime within the available time windows depends not only on whether there is another time window, but also on whether the available time windows are consecutive and whether the treatment mode of the last patient served in that window is text, video, or voice. when At that time, regardless of the type of patient served last within the time window, if the service duration exceeds the window, all excess time will be considered overtime; when If the current time window and the next time window are consecutive, then when serving the last patient, the portion exceeding the current time window can be carried over to the next time window and is not counted as overtime, meaning the service is completed within the doctor's overall continuous available time. However, if the available time windows are not consecutive, the determination of overtime depends on the service type of the last patient. For text-based consultations, since their service is highly non-immediate, it is assumed that the doctor will continue to provide services to the patient in subsequent non-consecutive time windows, and therefore, the overtime for the current window is not counted. For video or voice consultations, if the service exceeds the scope of the current time window, the portion exceeding the current time window is considered as overtime for that time window. Therefore, virtual overtime hours are introduced. Based on the recursive relationship, the doctor's virtual overtime and free time are: （10） （11） Furthermore, in the context Below, set up a doctor The location of the last patient served within the available time window is ,in Introducing binary variables ,when Indicates doctor In the time window The last patient served was a telephone patient, when Indicates doctor time window The last patient served was the one with images; therefore, it can be inferred that the doctor's available time window... Therefore, their actual overtime hours at that window are: （12） Among them, variables For doctors Available time window Numbering of global time slots Equation (12) indicates that if the doctor In time period There are subsequent time windows, and these time windows are available. and available time windows They are connected in the global time slots, that is... and Then the doctor In the time window If there is no actual overtime, the corresponding virtual overtime will be directly postponed to the next time window and recorded as the first patient waiting time in the next time window. If the two time windows are not contiguous, that is and If the last patient is a text / image patient, the virtual overtime period will be postponed to the next time window; if the last patient is a telephone patient, the doctor needs to work overtime to complete the service, and this will be recorded as a doctor's overtime work. In the time window The actual overtime hours. Therefore, the waiting time for patients between the two time windows is: （13） If the doctor has only one available time window, that is When a doctor is assigned to provide overtime services to patients, their actual overtime hours and free time are as follows: （14） （15） 2) Time cost description when At that time, based on the condition of the last patient served by the doctor in the previous available time window, determine The waiting time for the first appointment; doctor Available time window The first position The waiting time is: （16） Similarly, the waiting time for other positions is determined by the waiting time of the previous patient, the actual service time, and the reserved service time; therefore, the doctor Available time windows Location The waiting time and idle time are: （17） （18） doctor In the The available time windows for virtual overtime and free time are: （19） （20） Similar to equation (12), if the doctor If there is a subsequent available time window, then it is within the available time window. The actual overtime hours are: （21） in, For the last patient served, there is ;variable For doctors The The number of available time windows in the global time slot. Similarly, the waiting time for the patient between the two available time windows is: （22） 3) Depicting the last available time slot for overtime and free time The doctor's overtime and free time during the last available time window to serve the last patient is as follows: （23） （24） Based on the above analysis, the scheduling module [SP] for the second stage is: （25） s.t. (26) （27） （28） （29） （30） （31） （32） （33） （34） （35） （36） The objective function (25) for the [SP] phase is to minimize the expected patient waiting cost, physician idle cost, and overtime cost; constraint (26) indicates that if the physician... Within the available time window Inner If a position is not assigned a patient, the reservation time for that position is 0; constraint (27) ensures that a position is assigned to a doctor. The total reserved time for patients is equal to the length of their available time window; constraint (28) linearizes recursive relations (8)(9) and (17)(18); constraint (29) linearizes recursive relations (10)(11) and (19)(20); (30) linearizes recursive relations (14)-(15) and (23)-(24); constraint (32) integrates recursive relations (12)(21); constraint (33) integrates recursive relations (13)(22).

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