An airport ground staff intelligent scheduling method and system based on a Bron-Kerbosch maximal clique algorithm

Abstract

The application discloses an airport ground staff intelligent scheduling method and system based on a Bron-Kerbosch maximal group algorithm, and the method comprises the following steps: obtaining staff information and task information from an external system, and performing data preprocessing and virtual staff supplement; splitting tasks according to manpower demand and generating an executable staff set; constructing a time conflict matrix and executing the Bron-Kerbosch maximal group algorithm; defining a decision variable, constructing a constraint condition and setting a multi-objective hierarchical optimization function; calling a mixed integer programming solver to solve the model; and generating an allocation scheme and performance statistical data. The application decomposes a complex task into independent subtasks through intelligent task splitting, constructs a time conflict matrix, adopts the Bron-Kerbosch algorithm to identify a maximal group, establishes a multi-objective mixed integer programming model to balance a task allocation rate and resource balance, and designs a virtual staff mechanism to ensure system robustness.

Description

Technical Field

[0001] This invention belongs to the field of airport operation management and intelligent scheduling technology, specifically involving an intelligent scheduling method and system for airport ground staff based on the Bron-Kerbosch maximal clique algorithm, which is used to solve the problem of intelligent allocation and optimized scheduling of airport ground service personnel in various tasks such as boarding, picking up, and special services. Background Technology

[0002] Currently, most airports still use manual scheduling, where dispatchers assign tasks based on experience. This method has the following significant problems:

[0003] 1. Lack of Task Splitting Mechanism: The existing system cannot intelligently handle the splitting of complex tasks. For example, the standard configuration for boarding a flight on an international flight requires 3 ground staff, but the system can only allocate the entire task as a whole, and cannot break it down into 3 independent sub-tasks to be assigned to different employees. This leads to: overly coarse allocation granularity, making it impossible to manage human resources granularly; when only 2 employees are available, the system cannot determine whether it can be downgraded (minimum 2 people are required to complete the task); and it cannot handle situations where some employees have already been pre-assigned (e.g., the task requires 3 people, but 1 person has already been assigned, so 2 more people need to be assigned).

[0004] 2. Traditional methods can only perform simple time overlap detection and cannot handle complex multi-task time conflict scenarios. Specific problems include: only judging whether the time of two tasks overlaps without considering employee resource sharing constraints; failing to identify complex conflict networks formed by multiple tasks (such as a ternary conflict group where tasks A and B conflict, B and C conflict, and A and C also conflict); and lacking a systematic algorithm for identifying conflict task sets, resulting in incomplete constraint modeling.

[0005] 3. The existing system lacks an intelligent dynamic adjustment mechanism for task priorities, and the priority handling mechanism is imperfect: for tasks that have been partially assigned to personnel, the system cannot automatically increase their priority to ensure execution, resulting in improper handling of pre-assigned tasks; critical tasks are not adequately guaranteed, and high-priority VIP tasks or urgent tasks cannot receive priority resource guarantees; minimum manpower requirements cannot be guaranteed, and the system cannot ensure that each task is assigned to at least the minimum required number of personnel.

[0006] 4. Insufficient Multi-Objective Optimization Capability: Airport ground handling scheduling needs to consider multiple conflicting objectives simultaneously, which existing methods cannot effectively balance. Traditional methods cannot reconcile the competitive relationship between task completion rate and resource balance. For individual employees, there is a conflict between completing as many tasks as possible and the fair distribution of employee workload; there is also a conflict between points balance and efficiency optimization, with the average distribution of employee points conflicting with maximizing task execution efficiency; time balance and skill matching are also not considered, as there is a conflict between balancing employee working hours and precisely matching task skill requirements. Manual scheduling often overlooks these aspects and struggles to find a balance point among multiple objectives.

[0007] 5. The existing system has insufficient capacity to handle abnormal situations and poor system robustness: when the employee information specified in the pre-assigned task is incomplete or incorrect, the system cannot automatically repair it and cannot handle missing data; when the constraints are contradictory (such as all eligible employees are already saturated), the system cannot provide alternative solutions to handle constraint conflicts; when flight time changes or employees take temporary leave occur, the system cannot quickly re-optimize to meet the airport's real-time update requirements.

[0008] Currently, technological developments in airport dispatching, both domestically and internationally, are mainly focused on the following areas:

[0009] 1. Rule-based scheduling system: It mainly relies on preset rules for allocation, which has poor flexibility and cannot handle complex combination of constraints.

[0010] 2. Application of heuristic algorithms: Genetic algorithms, particle swarm optimization, etc. are used, but there is a lack of special design for the characteristics of airport operations, resulting in unstable solution quality.

[0011] 3. Mathematical optimization methods: Some systems use linear programming, but most of them are single-objective optimizations, and the time conflict modeling is not accurate enough.

[0012] Therefore, there is a lack of comprehensive technical solutions that can accurately model complex temporal conflict relationships, handle multi-objective optimization, and possess good robustness. There is an urgent need to develop a new technical solution that can: intelligently decompose composite tasks and perform fine-grained allocation; accurately identify and model complex temporal conflict relationships; effectively balance multiple contradictory optimization objectives; possess powerful anomaly handling capabilities; and support the computational performance requirements of large-scale real-time scheduling. Summary of the Invention

[0013] To address the technical problems in existing technologies, such as the lack of task decomposition mechanisms, inaccurate time conflict identification, insufficient multi-objective optimization capabilities, and poor system robustness, this invention provides an intelligent scheduling method and system for airport ground staff based on the Bron-Kerbosch maximal clique algorithm. The method decomposes composite tasks into independent sub-tasks through intelligent task decomposition, constructs a time conflict matrix, identifies maximal cliques using the Bron-Kerbosch algorithm, establishes a multi-objective mixed integer programming model to balance task allocation rate and resource balance, and designs a virtual employee mechanism to ensure system robustness.

[0014] To achieve the above objectives, the present invention provides the following solution:

[0015] A method for intelligent scheduling of airport ground staff based on the Bron-Kerbosch maximal clique algorithm, the method comprising:

[0016] S1: Obtain employee and task information from external systems, and perform data preprocessing and virtual employee supplementation; the external systems include: human resource management system and flight operation system;

[0017] S2: Divide the task into several sub-tasks based on the manpower requirements of the task; generate a set of executable employees for each sub-task based on employee ability constraints, time constraints and dynamic priority assignment algorithm;

[0018] S3: Construct a conflict matrix based on time overlap and competition for employee resources; based on the conflict matrix, use the Bron-Kerbosch algorithm to identify the maximal clique set of conflict tasks;

[0019] S4: Based on the maximal clique set, decision variables are set, constraints are constructed, and a multi-objective hierarchical optimization function is set to obtain a multi-objective mixed integer programming model;

[0020] S5: Use a mixed-integer programming solver to perform hierarchical optimization of the multi-objective mixed-integer programming model, extract and output the employee task allocation scheme.

[0021] Preferably, in S1, employee information includes employee identifier, working time window, qualification information, work area authorization, and performance data; task information includes task identifier, time window, manpower requirements, constraints, and task personnel allocation details.

[0022] Preferably, in step S1, the method for supplementing virtual employees includes:

[0023] When a lock is assigned but employee information is missing, the virtual employee automatic compensation mechanism is activated. By automatically creating placeholder employee records, each lock constraint can find a corresponding employee index.

[0024] Preferably, in step S2, the number of task splits is equal to the larger of the standard manpower requirement and the number of task personnel allocation details.

[0025] Preferably, in step S2, the dynamic priority assignment algorithm is as follows: based on the pre-allocation of tasks and the minimum manpower requirements, the priority of each sub-task is intelligently adjusted; specifically, it includes:

[0026] The system calculates the pre-allocation status, including the number of locked and semi-locked tasks; it also calculates the manpower gap, i.e., the number of sub-tasks that still need to be guaranteed to meet the minimum manpower requirements; it assigns execution priorities: the priority of pre-allocated sub-tasks is increased to 100 times their original priority; to meet the minimum manpower requirements, the priority of the corresponding number of remaining sub-tasks is increased to 100 times in sequence; other ordinary sub-tasks retain their original priorities.

[0027] Preferably, in S2, the conditions for generating the executable employee set include: the employee qualifications meet the task requirements, the employee's work area covers the task location, and the employee's time window includes the task's time window.

[0028] Preferably, in step S3, the method for constructing the conflict matrix includes:

[0029] For any two subtasks, a two-condition judgment is performed. First, it is determined whether the time intervals overlap, that is, whether the end time of subtask i is greater than the start time of subtask j and whether the end time of subtask j is greater than the start time of subtask i. Second, the employee sharing situation is checked, that is, whether there is an intersection of the sets of allocable employees. When the time overlaps and employee sharing is simultaneously true, it is marked as a conflict in the conflict matrix.

[0030] In S3, the method for identifying the maximal clique set of conflict tasks using the Bron-Kerbosch algorithm includes:

[0031] The recursive backtracking strategy of the Bron-Kerbosch algorithm is adopted to maintain three sets: the current clique R, the candidate set P, and the exclusion set X. The algorithm introduces a pivot optimization strategy to start from an empty clique, and gradually expands the current clique by selecting the node with the most neighbors. At each time, a candidate node is selected to be added to the current clique, and the candidate set is updated to the nodes adjacent to that node. When both the candidate set and the exclusion set are empty, a maximal clique is output.

[0032] Preferably, in step S4, the decision variable is set as follows: a binary variable x[i,j] is defined for each combination of employee and subtask. When x[i,j]=1, it means that employee i is assigned to subtask j; otherwise, it means that employee i is not assigned.

[0033] Constraint Construction: Receive a list of maximal cliques and generate a set of conflict constraints for all subtasks in each maximal clique. Specifically, for each maximal clique {T1, T2, ..., Tk}, generate the constraint x[i, T1] + x[i, T2] + ... + x[i, Tk] ≤ 1 for each employee i, ensuring that any employee can be assigned to at most one subtask in the maximal clique. In addition to conflict constraints, generate personnel quantity constraints to ensure that the personnel requirements of each subtask are met, generate task score constraints to ensure that the total score of the assigned employees reaches the task requirements, generate working time constraints to ensure that the daily working time of each employee does not exceed the upper limit, and generate locking and pre-assignment constraints to ensure that the determined assignment relationship is executed.

[0034] Objective function setting: A two-level hierarchical structure is adopted. The first level objective includes maximizing the number of tasks completed and maximizing the number of pre-assigned tasks executed. The second level objective includes minimizing the variance of employee scores, minimizing the variance of working hours, and minimizing the number of virtual employees used.

[0035] Preferably, in step S5, the method of using a mixed-integer programming solver to perform hierarchical optimization of a multi-objective mixed-integer programming model and extracting and outputting the employee task allocation scheme includes: employing a two-round iterative solution strategy;

[0036] In the first round of solving, the constructed multi-objective mixed integer programming model MIP is input into the mixed integer programming solver Gurobi. The solution parameters are configured, including a time limit of 300 seconds and a relative error tolerance of MIPGap=0.11. The solver uses a branch and bound algorithm to search for feasible solutions that satisfy all constraints and make the first-level objective function optimal. During the search process, the solver maintains a branch tree, with each node representing a partial solution. The upper bound of each node is calculated through linear relaxation, and a pruning strategy is used to eliminate branches that cannot produce the optimal solution, thereby obtaining the optimal value of the first-level objective.

[0037] In the second round of solving, new constraints are added to the model, requiring that the first-level objective value cannot be lower than the optimal value obtained in the first round of solving. Then, the second-level objective function is optimized. After the solution is completed, the values ​​of all decision variables are extracted from the solver, the satisfaction of all constraints is verified, and the variable combination of x[i,j]=1 is converted into the actual employee-task allocation record, generating an allocation scheme report and an unfinished task list.

[0038] This invention also provides an intelligent scheduling system for airport ground staff based on the Bron-Kerbosch maximal clique algorithm. The system is used to implement the aforementioned method and includes: a data acquisition module, a task splitting module, a conflict identification module, an optimization modeling module, and a solution module.

[0039] The data acquisition module is used to acquire employee and task information from external systems, and to perform data preprocessing and virtual employee supplementation; wherein, the external systems include: human resource management system and flight operation system;

[0040] The task splitting module is used to split a task into several sub-tasks according to the manpower requirements of the task; and to generate a set of executable employees for each sub-task based on employee ability constraints, time constraints and dynamic priority assignment algorithm.

[0041] The conflict identification module is used to construct a conflict matrix based on time overlap and employee resource competition as judgment conditions; and to identify the maximal clique set of conflict tasks based on the conflict matrix using the Bron-Kerbosch algorithm.

[0042] The optimization modeling module is used to set decision variables, construct constraints, and set multi-objective hierarchical optimization functions based on the maximal clique set to obtain a multi-objective mixed integer programming model.

[0043] The solution module is used to perform hierarchical optimization of a multi-objective mixed integer programming model using a mixed integer programming solver, and to extract and output the employee task allocation scheme.

[0044] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0045] The significance of this invention lies in its ability to efficiently and fairly generate task and personnel scheduling schemes under different airport load scenarios. It solves the problem of complex computational complexity caused by the intricate mapping of relationships between tasks, employee qualifications, and employee points; and addresses the issue of solution space fragmentation caused by various task constraints. Simultaneously, it achieves a task allocation success rate of 95%, keeps the difference in points and working hours within 30%, and controls the solution time within 15 minutes, meeting the real-time scheduling needs of airports. Attached Figure Description

[0046] To more clearly illustrate the technical solution of the present invention, the drawings used in the embodiments are briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0047] Figure 1 This is a schematic diagram of a method for intelligent scheduling of airport ground staff based on the Bron-Kerbosch maximal clique algorithm according to an embodiment of the present invention.

[0048] Figure 2 This is a schematic diagram of the task pre-assignment process according to an embodiment of the present invention;

[0049] Figure 3This is a schematic diagram illustrating the two-level target solution in an embodiment of the present invention. Detailed Implementation

[0050] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0051] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0052] Example 1

[0053] The core technical idea of ​​this invention is to transform the airport ground handling scheduling problem into a multi-objective mixed integer programming problem with satisfied constraints, and to solve it through the following innovative technical means.

[0054] ① Intelligent task splitting technology: It breaks down complex tasks into independent sub-task units according to standard manpower requirements, so as to achieve fine allocation.

[0055] ② Application of the Bron-Kerbosch maximal clique algorithm: This transforms the time conflict identification problem into a maximal clique search problem in graph theory, accurately identifying all conflict task sets. The Bron-Kerbosch algorithm is a method for finding all maximal cliques in an undirected graph. A maximal clique is the largest set of vertices in a graph where every two vertices are connected by an edge, and no other vertices can be added while maintaining this property. In graph theory, it refers to a complete subgraph (where every two vertices are connected by an edge), and no other vertices can be added while maintaining the completeness of the subgraph property.

[0056] ③ Virtual Employee Compensation Mechanism: This mechanism addresses data gaps by automatically creating virtual employee records, ensuring system robustness. Virtual employees are placeholder employee records automatically created by the system to handle situations where employee information is missing in pre-assigned tasks.

[0057] ④ Multi-objective hierarchical optimization: Design two-level objective functions to ensure both task completion rate and balanced resource allocation.

[0058] ⑤ Mixed Integer Programming (MIP) Solution: Employs a mature mathematical optimization solver to ensure solution quality and computational efficiency. MIP is a mathematical optimization method where decision variables are partly integers (such as 0 or 1) and partly continuous real numbers. It seeks the optimal solution through linear constraints and an objective function.

[0059] To achieve the above objectives, the present invention provides the following technical solutions: Figure 1 As shown, this invention provides an intelligent scheduling method for airport ground staff based on the Bron-Kerbosch maximal clique algorithm, comprising the following steps:

[0060] S1. Data Acquisition and Preprocessing: Acquire employee information from the human resources management system and task information from the flight operation system, and perform data preprocessing and automatically create virtual employee records for tasks with missing employee information.

[0061] S2. Task Splitting and Matching Constraint Generation: The task is split into several sub-tasks based on the manpower requirements; an executable employee set for each sub-task is generated based on employee ability constraints and time constraints; dynamic priority assignment is performed based on the pre-allocation status.

[0062] S3. Time conflict identification based on Bron-Kerbosch algorithm: Construct a conflict matrix based on time overlap and employee resource competition as the criteria; use Bron-Kerbosch algorithm to identify the maximal clique set of conflict tasks;

[0063] S4. Multi-objective mixed-integer programming modeling: Define binary decision variables to represent the allocation relationship between employees and sub-tasks, and establish a constraint system including manpower constraints, time conflict constraints, and workload balance constraints; design two-level objective functions, with high-priority objectives maximizing the task allocation success rate and low-priority objectives minimizing the resource allocation imbalance.

[0064] S5. Model Solving and Result Output: A mixed integer programming solver is used for hierarchical optimization; the employee task allocation scheme is extracted and output.

[0065] Step S1: Data Acquisition and Preprocessing

[0066] This step establishes a complete data acquisition and preprocessing workflow, ensuring that the scheduling algorithm receives standardized and complete input data. The core innovation of this step lies in the automatic compensation mechanism for virtual employees and the intelligent processing of pre-allocated information.

[0067] 1.1: Employee Information Acquisition and Processing

[0068] Obtain basic employee data from a Human Resources Management System (HRMS), specifically including:

[0069] ① Identification information: Employee number, such as "EMP001", serves as a unique identifier for the employee; Employee name, such as "Zhang San", is used for result display; Employee ID, such as "uuid-12345", is a unique identifier within the system.

[0070] ② Time Constraint Information Group: Records employees' working time windows and shift dates. The system automatically converts time information into a unified numerical format and calculates working hours as the basis for time matching. Standardized processing of time data is a prerequisite for all subsequent time-related constraint checks. The system parses date and time strings into time objects and then converts them into numerical values ​​in minutes. This unified time representation facilitates the determination of time window inclusion relationships and overlap detection.

[0071] ③ Capability Constraint Information Group: This group includes qualification information and work area authorization. Qualification information uses a multi-value representation, supporting employees holding multiple qualifications, such as general service qualifications and special passenger service qualifications. The system parses multi-value strings and converts them into a structured list of qualifications. This list structure allows subsequent qualification matching checks to be completed quickly through set inclusion relationships. Work areas also support multi-region representation, allowing employees to work in multiple terminal areas, providing flexibility for cross-regional coordination and scheduling. It also includes organizational affiliation information, such as Ground Services Department.

[0072] ④ Performance Appraisal Information: Current daily work points, e.g., 50 points, used for daily workload balancing; actual daily working hours, e.g., 5.5 hours, used for hourly work balancing calculations; current monthly work points, e.g., 450 points, used for monthly fairness considerations. This data is used in subsequent workload balancing calculations to ensure the fairness of scheduling results. The system converts this performance data into numerical types, which serve as input parameters for the balance constraints in the optimization model.

[0073] 1.2: Task Information Acquisition and Processing

[0074] The mission information obtained from the flight operation system undergoes structured processing to form a standardized mission data model. Mission identification information includes a mission number and name, facilitating system recognition and user understanding. Time constraint information accurately records the mission start and end times, and the system automatically calculates the mission duration, providing fundamental data for time conflict detection.

[0075] The manpower requirements information employs a two-tiered design: standard manpower requirements represent the ideal number of personnel, while minimum manpower requirements represent the minimum acceptable number. This dual-requirement design provides flexibility for task breakdown and prioritization adjustments. Standard manpower requirements determine the number of task sub-tasks, while minimum manpower requirements are used to calculate the number of support sub-tasks, ensuring that critical tasks receive at least the minimum manpower support. Constraint information clearly defines task execution requirements, including required qualifications and task location; these constraints are used for subsequent employee-task matching checks.

[0076] 1.3: Pre-allocation information processing mechanism

[0077] Each task contains a detailed array of task personnel assignments, which is the core data structure of the pre-assignment mechanism. The assignment details record the pre-assignment relationship between tasks and employees, including association identification information, pre-assigned employee information, and release status control parameters.

[0078] The release state control parameter is the core of the pre-allocation mechanism, defining three allocation states: unallocated (value 0) indicates that the task / position can be freely assigned to any eligible employee; pre-allocated (value 1) indicates that the system recommends assigning it to a specific employee, but adjustments can be made when resources are scarce; and locked (value 2) indicates that the task / position must be assigned to a specific employee and cannot be changed. This three-level release state design establishes a balance between rigid constraints and flexible optimization, enabling the system to respect existing personnel arrangements while allowing for flexible adjustments when necessary to improve overall resource utilization. Figure 2 As shown.

[0079] 1.4: Automatic Compensation Mechanism for Virtual Employees

[0080] When encountering a situation where employee information is missing due to a locked assignment, the system activates an automatic virtual employee compensation mechanism. This is a key innovation of the invention, effectively solving the problem of model infeasibility caused by incomplete data.

[0081] The theoretical basis of the compensation mechanism: In mixed-integer programming models, locking assignment constraints require decision variables to take the value 1, i.e., forcing a specific task to be assigned to a specific employee. When the information of the specified employee is missing, the employee index corresponding to the constraint cannot be found in the employee list, causing the constraint equation to fail to be established, making the entire optimization model infeasible. The virtual employee mechanism ensures that each locking constraint can find a corresponding employee index by automatically creating placeholder employee records, thereby guaranteeing the feasibility of the model.

[0082] When the system detects missing employee information, it automatically creates a virtual employee record. Key design features of virtual employees include:

[0083] Exclusivity Constraint: Virtual employees have an exclusive characteristic; they can only execute their corresponding locked tasks and cannot be assigned to other tasks. This design is achieved by setting special matching rules during the employee task matching phase: when an employee is marked as a virtual employee, the system only establishes a matching relationship between it and the locked task that created that virtual employee, returning a mismatch for all other tasks. This constraint isolation mechanism ensures that virtual employees do not occupy the optional resources of other tasks, avoiding interference with normal scheduling.

[0084] Perfect Qualification Configuration: The system automatically configures all necessary qualifications and regional permissions for virtual employees, ensuring they meet all task constraints. The virtual employee's work time window precisely matches the task's time requirements, eliminating time-related limitations. This design guarantees that virtual employees unconditionally meet all execution conditions for their corresponding tasks.

[0085] Balance Isolation: Virtual employees do not participate in the balance calculation of points and working hours, avoiding impact on the fairness of real employees. When constructing balance constraints, the system only counts the workload of real employees, filtering by employee identification attributes. This design ensures that virtual employees are only used as a technical means of data repair and will not negatively affect the fairness of the scheduling scheme.

[0086] Through this compensation mechanism, the system can automatically repair more than 85% of data missing issues, significantly improving the system's robustness and usability. The essence of the virtual employee mechanism is to transform the "data incompleteness problem" into a "constraint satisfiability problem," by introducing placeholder variables that satisfy specific constraints to ensure that the optimization model always has a feasible solution.

[0087] Mathematical model of virtual employee automatic compensation mechanism:

[0088] ① Rules for generating virtual employees

[0089] For a set of tasks with locked but missing employees:

[0090] ;

[0091] in Indicates task The locked state, Indicates that the specified employee is in the employee set. It does not exist in [the context].

[0092] The system automatically generates a set of virtual employees:

[0093] ;

[0094] The expanded complete set of employees:

[0095] ;

[0096] in, A collection of real employees, m Refers to the actual number of employees;

[0097] ② Specificity constraint (ensuring that virtual employees only perform corresponding tasks)

[0098] For each ,in, To lock a set of tasks for which employees are missing, enforce the following constraints:

[0099] ;

[0100] Simultaneously set exclusion constraints:

[0101] ;

[0102] in, For a set of tasks; For summation, For task indexing, Exclude all tasks , Should virtual employees be assigned tasks? , The sum is zero. This exclusion constraint ensures that virtual employees only perform their assigned tasks and are not assigned to other tasks.

[0103] ③ Qualifications and time are automatically matched.

[0104] For virtual employees Configure the aptitude vectors to meet the task requirements. All qualification requirements:

[0105] ;

[0106] in, For the task Required qualifications For all... to be valid, This refers to the qualification type.

[0107] Set up virtual employee time windows to precisely match task execution times:

[0108]

[0109] in, Set the start time for virtual employees. The end time for virtual employees' work.

[0110] Make the time window include constraints Heng was established.

[0111] ④ Fairness segregation (virtual employees do not participate in the leveling calculation)

[0112] Introducing Filtering Coefficient :

[0113]

[0114] The balance target calculation only includes actual employees:

[0115]

[0116] in and For employees The cumulative points and cumulative working hours.

[0117] Step S2: Task splitting and matching constraint generation

[0118] The core of this step is to break down the complex task into independent sub-task units, determine the set of employees that can perform the task for each sub-task, and make intelligent priority adjustments.

[0119] 2.1: Intelligent Task Splitting Mechanism Based on Allocation Details

[0120] Task splitting is a key technology for decomposing complex tasks into independent allocation units. Traditional scheduling methods allocate the entire task as a whole, making it difficult to handle complex situations with partial pre-allocation. This invention innovatively achieves precise task splitting based on detailed task and personnel allocation information.

[0121] The theoretical basis for the task splitting mechanism: Airport ground service tasks typically require collaboration from multiple employees; for example, boarding service for a flight might require three ground staff. In optimization modeling, treating the task of three people as a whole fails to express partial constraints such as "one person has been pre-assigned to Zhang San." Task splitting decomposes multi-person tasks into multiple single-person task units, allowing each pre-assignment relationship to be precisely expressed through independent constraint equations, thereby achieving fine-grained scheduling control.

[0122] The splitting rules are designed according to the following principles: the basic splitting quantity equals the larger of the standard manpower requirement and the number of allocated details; when the number of allocated details differs from the standard manpower requirement, the actual number of allocated details prevails; when the number of allocated details is less than the minimum manpower requirement, the minimum manpower requirement is adjusted to equal the number of allocated details to ensure the feasibility of the constraints. This dynamic adjustment mechanism can automatically handle data inconsistencies and avoid model infeasibility due to data quality issues.

[0123] Taking flight boarding as an example, if the original task requires 3 people by standard criteria and 2 people by minimum, and the allocation details include information on 3 job positions, then the system will split the task into 3 independent sub-task units. Each sub-task inherits the original task's constraints such as time, qualifications, and location, and is associated with corresponding pre-allocation information. The split sub-tasks are completely identical in terms of time and space attributes, but are independent in terms of allocation relationships and priorities, which provides a foundation for subsequent fine-grained scheduling.

[0124] 2.2: Dynamic Priority Intelligent Assignment Algorithm

[0125] To ensure the priority execution of important and pre-assigned tasks, the system employs a dynamic priority assignment algorithm. This algorithm intelligently adjusts the priority of each subtask based on the pre-assignment status and minimum manpower requirements.

[0126] The mathematical principle behind the priority amplification mechanism: In the objective function of mixed-integer programming, task priority serves as a coefficient of the decision variable, and its absolute value directly determines the importance of the task in the optimization process. By amplifying the priority of pre-assigned tasks by a factor of 100, the system mathematically ensures that the optimizer will prioritize finding feasible allocation schemes for these tasks. The choice of this amplification factor is based on the following considerations: it needs to be large enough to ensure the absolute priority of pre-assigned tasks, but it cannot be too large to cause numerical instability. Experiments have verified that a factor of 100 achieves a good balance between priority protection and numerical stability.

[0127] The algorithm first calculates the pre-allocation status, including the number of locked tasks and the number of semi-locked tasks. Then, it calculates the manpower gap, which is the number of sub-tasks that still need to be guaranteed to meet the minimum manpower requirement. The calculation logic for the manpower gap is: minimum manpower requirement minus the number of pre-allocated sub-tasks. If the result is positive, it means that additional sub-tasks still need to be guaranteed to meet the minimum manpower requirement; if the result is zero or negative, it means that the pre-allocation has already met or exceeded the minimum requirement, and no additional guarantee is needed.

[0128] Finally, priority assignment is performed: the priority of pre-assigned subtasks is increased to 100 times their original priority; to meet minimum manpower requirements, the priority of the remaining subtasks is increased to 100 times their original priority in sequence; other ordinary subtasks retain their original priority. This hierarchical assignment strategy forms a three-level priority system: the highest level is pre-assigned tasks (mandatory execution), the second highest level is guaranteed tasks (ensuring minimum manpower), and the lowest level is flexible tasks (executed when resources are sufficient).

[0129] For example, a task originally had a priority of 5, a minimum manpower requirement of 2 people, and a standard manpower requirement of 3 people. One subtask was locked. The system calculated a manpower shortage of 1, so it set the priority of the locked subtask to 500, the priority of another guaranteed subtask to 500, and the priority of the remaining ordinary subtask remained at 5. This design ensures that at least two subtasks can be executed, meeting the minimum manpower requirement.

[0130] 2.3: Establishing Employee-Task Matching Relationships

[0131] The system determines the set of executable employees for each subtask, which is the foundation of constraint modeling. Matching checks include three dimensions: qualification matching checks verify that the employee qualification list contains the qualifications required for the task; time window matching checks ensure that the employee's working hours fully cover the task execution time; and regional constraint matching checks verify that the employee's authorized work area covers the task execution location.

[0132] The significance of constraint modeling for matching relationships: The set of executable employees defines the valid range of values ​​for decision variables. For employees not in the set of executable employees, the system forces the corresponding decision variable to be 0 when establishing constraints, thus prohibiting the assignment relationship. This pre-screening mechanism significantly reduces the search space of the optimization model and improves solution efficiency. Simultaneously, the accuracy of the matching relationships directly affects the executability of the scheduling scheme. Only when the matching check is completely correct can it be guaranteed that the generated assignment scheme will not fail in actual execution due to insufficient employee capabilities or time conflicts.

[0133] The system employs targeted strategies for handling special cases. For virtual employees, a matching relationship is established only with their corresponding locked tasks. For locked-assigned tasks, even if the regular matching conditions are not fully met, the system still forces a matching relationship between the assigned employee and the task to ensure strict enforcement of the locking constraints. This forced matching mechanism reflects the principle that the rigid requirements of business rules take precedence over technical constraints: when management explicitly designates an employee to perform a task, the system should respect this decision even if the employee's technical qualifications are slightly inadequate. The person assigned the task is not subject to any constraints, including qualifications, employee working hours, and work location.

[0134] Through the above processing, each subtask obtained an executable employee index set, laying the foundation for subsequent conflict detection and optimization modeling.

[0135] Step S3: Time-conflict graph theory modeling based on the Bron-Kerbosch algorithm

[0136] This step is the core innovative technology of this invention, transforming the complex problem of temporal conflict identification into a maximal clique search problem in graph theory. This transformation not only improves the accuracy of conflict identification but also provides a complete theoretical foundation for subsequent constraint generation.

[0137] 3.1: Precise Modeling of Time Conflict Relationships

[0138] Traditional time conflict detection methods rely solely on simple time overlap judgments, failing to accurately handle complex conflict relationships under employee-shared constraints. This invention innovatively proposes a conflict judgment criterion based on employee sharing.

[0139] The dual-condition theory of conflict determination states that the necessary and sufficient conditions for scheduling conflicts between two tasks are: overlapping task times (i.e., the time windows of the two tasks intersect); and competition for employee resources (i.e., the sets of available employees for the two tasks intersect). Simple time overlap is insufficient to constitute a conflict, because if the two tasks do not share available employees, they can be assigned to different employees without conflict. This dual-condition determination accurately describes the essence of scheduling conflicts: conflicts stem from competition for limited resources (employees) within the same time period.

[0140] The system constructs an n×n conflict matrix, where n is the total number of subtasks. For each pair of tasks, the system first checks for time overlap, determining if the later start time is earlier than the earlier end time. Then, it checks for employee sharing, calculating the intersection of the executable employee sets for the two tasks. When both time overlap and employee sharing are true, a 1 is marked at the corresponding position in the conflict matrix, indicating a conflict. The conflict matrix is ​​a symmetric matrix with diagonal elements of 0 (tasks do not conflict with themselves), and a matrix element value of 1 indicating a conflict between the corresponding two tasks.

[0141] Mathematical model of time conflict relationship:

[0142] ① Task time window definition

[0143] Task The execution time is represented as a half-open interval:

[0144] ;

[0145] in For the start time, The end time is specified. Half-open intervals are used to strictly distinguish the boundaries between adjacent tasks.

[0146] ② Time overlap determination function

[0147] Define a time conflict indicator function:

[0148] ;

[0149] in For indicator functions, This is a time conflict indicator function. It is an indicator function (the condition is true and takes the value 1). For the task The start and end times, For the task The start and end times, This is a logical AND operation. The time conflict indicator function is used to determine if the time windows of two tasks overlap.

[0150] This function captures overlapping time windows:

[0151] (Time overlap);

[0152] (Time can be serialized);

[0153] in, , For the task , The executable set of employees, This is a time overlap indicator function;

[0154] ③ Executable employee set definition

[0155] Task The executable employee set is defined as employees who meet the triple constraints of qualification, region, and time:

[0156] ;

[0157] in: For employees, For employee qualification collection, For mission qualification requirements, Authorize areas for employees, The location for mission execution. Provide a working time window for employees;

[0158] ④ Employee resource competition determination function

[0159] Define employee-shared instruction functions:

[0160] ;

[0161] in, , For the task , A set of executable employees;

[0162] Determine whether two tasks are competing for the same employee resources:

[0163] There are common potential executors, and there is competition for resources;

[0164] The assigned employees are completely disjoint and can be executed in parallel;

[0165] ⑤ Comprehensive Conflict Adjacency Matrix

[0166] Define the conflict adjacency matrix The elements are:

[0167] ;

[0168] Matrix properties:

[0169] symmetry: ;

[0170] The diagonal is zero: ;

[0171] Sparsity: In practical problems, sparsity is 60%-80%;

[0172] ⑥ Weighted conflict intensity modeling

[0173] Combined with task priority Constructing weighted conflict edge weights:

[0174] ;

[0175] in, For weighted conflict edge weights, The maximum priority of the two tasks. Assign weights to task priorities and assign weights to conflicts by modeling the conflict intensity to reflect the importance of the conflict.

[0176] Border weights reflect the importance of conflict scheduling and provide a basis for ranking conflict intensity for subsequent maximal clique constraints.

[0177] 3.2: Application of the Bron-Kerbosch Maximum Clique Algorithm

[0178] The Bron-Kerbosch algorithm is a classic algorithm for finding all maximal cliques in a graph. This invention innovatively applies it to temporal conflict identification. The theoretical basis of graph theory modeling is as follows: each subtask is considered a vertex in the graph, and the conflict relationships between tasks are considered edges. The graph corresponding to the conflict matrix is ​​called a conflict graph. In a conflict graph, a clique is a subset of vertices where every two vertices are connected to each other, meaning that all tasks within the clique are pairwise conflicting. A maximal clique is a clique that cannot have any more vertices added while maintaining its clique property; it represents the largest set of mutually exclusive tasks.

[0179] Define a conflict diagram :

[0180] Vertex set: (Each vertex corresponds to one task);

[0181] Edge set: (Edges represent conflict relationships);

[0182] in, , For the task;

[0183] group Satisfy the condition that any two vertices are adjacent:

[0184] ;

[0185] in, For the set of unassigned tasks;

[0186] A maximal clique cannot have its clique property preserved by adding more vertices:

[0187] ;

[0188] The set of maximal cliques is denoted as ,in, For the first A very large cluster.

[0189] The Bron-Kerbosch algorithm maintains three sets:

[0190] The team members currently being built;

[0191] : Potential candidate nodes to join the current group;

[0192] Nodes that have completed the search;

[0193] Define recursive operators :

[0194] ;

[0195] Recursive logic: Choosing a pivot For each implement:

[0196] ;

[0197] Then from Move to .in For nodes The adjacency set.

[0198] The initial call begins the search from an empty clique, with all vertices as candidates and no processed nodes.

[0199] ;

[0200] in, It is an empty set (the initial clique is empty). For all vertices (initial candidate set).

[0201] For each maximal group And every employee Construct constraints:

[0202] ;

[0203] in, For binary decision variables, employees Should the task be executed? The allocation threshold is defined as:

[0204]

[0205] here For employees The set of locked tasks. This definition ensures that:

[0206] Normal situation: Each employee can select a maximum of one task within the group; special circumstances: if an employee has A locked task in In the middle, then Allow execution indivual;

[0207] The correspondence between maximal cliques and scheduling constraints: Each maximal clique represents a set of pairwise conflicting tasks. These tasks cannot be assigned to the same employee simultaneously due to time overlap and shared employee constraints. In mixed-integer programming models, each maximal clique corresponds to a constraint equation: for any employee, the number of tasks assigned to them within that maximal clique does not exceed 1. This transformation from graph theory structure to mathematical constraints precisely expresses complex time conflict relationships as linear constraints, enabling the optimizer to systematically avoid all possible time conflicts.

[0208] The algorithm's integrity guarantee: The Bron-Kerbosch algorithm, through systematic recursive search, can find all maximal cliques in the graph without omission. The algorithm maintains three sets: the current clique set represents the members currently being built; the candidate set represents nodes that may join the current clique; and the exclusion set represents nodes that have already been processed. The algorithm's recursive logic ensures the integrity of the search: for each node in the candidate set, the algorithm attempts to add it to the current clique, and then recursively continues the search in the subproblems formed by its adjacent nodes; by recording processed nodes in the exclusion set, the algorithm avoids repeatedly searching for the same clique structure.

[0209] The principle of the pivot optimization strategy: The standard Bron-Kerbosch algorithm has high time complexity. This invention adopts a pivot optimization strategy to improve efficiency. The mathematical basis for pivot selection: Select the node with the highest degree in the candidate set and exclusion set as the pivot point. Degree represents the number of connections between the node and other nodes; a node with a high degree means it conflicts with more tasks. The core idea of ​​optimization is: for candidate nodes adjacent to the pivot point (i.e., in conflict), if they are added to a clique, the pivot point will inevitably be unable to join that clique; and all maximal cliques containing the pivot point can be discovered during subsequent processing of the pivot point. Therefore, the algorithm prioritizes processing candidate nodes that are not adjacent to the pivot point, which can reduce recursive branches by about 50%, significantly improving search efficiency without sacrificing completeness.

[0210] The algorithm uses a backtracking mechanism to traverse all possible node combinations, ensuring that all maximal cliques are found. When both the candidate set and the exclusion set are empty, the current clique set constitutes a maximal clique, and the system records it. The correctness of the recursion termination conditions is as follows: when the candidate set is empty, it means there are no new nodes that can be added to the current clique while maintaining the clique property; when the exclusion set is also empty, it means the current clique is not a subset of previously discovered cliques. Both conditions being satisfied simultaneously guarantees that the discovered clique is maximal.

[0211] This invention makes two key optimizations to the standard algorithm: first, it sorts nodes by degree, prioritizing nodes with high connectivity to identify large-scale conflict clusters earlier; second, it adopts a pivot selection strategy, reducing recursive branches by approximately 50% and significantly improving algorithm efficiency. The heuristic significance of degree sorting is that tasks with high degree conflict with more other tasks, which are often the key constraints in scheduling. Prioritizing these tasks allows for the earlier discovery of large maximal clusters, rapidly shrinking the subsequent search space and thus improving overall search efficiency.

[0212] 3.3: Conflict Path Optimization and Constraint Simplification

[0213] Maximal clique identification can generate a large number of conflicting constraints, and directly applying all constraints would significantly increase model complexity. The system employs conflict path optimization techniques to reduce the number of redundant constraints while ensuring constraint integrity.

[0214] The optimization strategy includes: analyzing the inclusion relationships between different maximal cliques to eliminate redundant constraints covered by other constraints; evaluating the importance of constraints based on the priority weights of tasks in conflicting paths; and merging similar constraints to reduce the number of constraints. The principle of constraint redundancy analysis is as follows: if maximal clique A is a subset of maximal clique B, then the constraints corresponding to maximal clique B implicitly include the constraints corresponding to maximal clique A. In this case, the constraints of maximal clique A are redundant and can be safely removed without affecting the feasible region of the model. By eliminating such redundant constraints, the system can reduce the number of constraints by more than 30% while maintaining constraint integrity, thus improving the model's solution efficiency.

[0215] Step S4: Multi-objective hierarchical mixed-integer programming modeling

[0216] 4.1: Design of Decision Variables and Basic Constraints

[0217] The system uses a binary allocation matrix as the main decision variable to represent the allocation relationship between employees and tasks. A value of 1 indicates that the employee is assigned to the task, and a value of 0 indicates that the employee is not assigned. The dimensions of the decision variable are designed as follows: the row index of the binary matrix corresponds to the employee, and the column index corresponds to the subtask. The matrix size is m×n (m is the number of employees, and n is the number of subtasks). This matrix representation intuitively describes the essence of the allocation problem and facilitates the expression of various constraints.

[0218] The system also defines auxiliary variables for calculating employee workload, including cumulative points and cumulative working hours. To achieve balanced optimization, the system introduces balancing variables, representing the maximum and minimum points, maximum and minimum working hours among all employees. The modeling role of the auxiliary variables: These variables transform the nonlinear maximum and minimum value functions into linear constraints and optimization objectives, ensuring the entire model maintains a linear structure, making it suitable for solving using efficient linear programming algorithms.

[0219] The basic constraint system includes five types of constraints. The manpower constraint ensures that each subtask can be assigned to at most one employee. The mathematical expression of the constraint is: for each subtask j, the sum of the allocation variables of all employees on that task does not exceed 1. This guarantees the exclusivity of the subtask and avoids duplicate allocation.

[0220] The time conflict uniqueness constraint is constructed based on maximal cliques, ensuring that each employee selects at most one task from each conflicting task group. The graph theory basis for this constraint is: for each maximal clique (representing a set of mutually exclusive tasks), for each employee, the sum of their assignment variables across all tasks within that maximal clique does not exceed 1. This constraint mathematically prohibits the possibility of an employee executing conflicting tasks simultaneously. In special cases, employees are allowed to execute multiple locked tasks: when a maximal clique contains multiple locked tasks assigned to the same employee, the right-hand side of the constraint is adjusted to the number of locked tasks assigned to that employee. This design respects the rigidity of pre-assignment.

[0221] The cumulative integral constraint defines an employee's total integral as the sum of their monthly base integral and daily task integral. The linearized expression of this constraint is: the cumulative integral variable for employee i equals their monthly base integral (a constant term) plus the sum of the products of all task integrals and their corresponding allocation variables. This is a standard linear equality constraint. The cumulative working hours constraint defines an employee's total working hours as the sum of the durations of all assigned tasks. The modeling method is similar to the cumulative integral constraint.

[0222] Extreme value constraints establish the relationship between balance variables and actual workload, used for subsequent balance calculations. The modeling technique for extreme value constraints is: the maximum integral variable must be greater than or equal to the cumulative integral of each employee, and the minimum integral variable must be less than or equal to the cumulative integral of each employee. By minimizing the difference between the maximum and minimum values ​​in the objective function, the optimizer automatically drives these variables to converge to their true maximum and minimum values, thereby achieving balance optimization.

[0223] For locked assignment tasks, the system forces the assignment of the task to a designated employee by setting the lower bound of a variable to 1, ensuring that the locking constraint is strictly enforced. The forced constraint is implemented by setting both the lower and upper bounds of the corresponding decision variable to 1, so that the variable can only take the value 1. Mathematically, this is equivalent to an equality constraint, but setting variable limits makes it easier for the optimizer to process.

[0224] 4.2: Design of a two-level objective function, such as... Figure 3 As shown.

[0225] This invention employs a hierarchical optimization design, achieving hierarchical optimization through target priority weight settings. The first layer has high target priority, focusing on maximizing task allocation success rate and pre-allocation execution rate. The second layer has low target priority, focusing on optimizing resource allocation balance.

[0226] The mathematical principle of hierarchical optimization: In a two-tiered objective, there are first-priority and second-priority objectives. When searching for the optimal solution, the optimizer prioritizes satisfying the higher-priority objective. At the same priority level, improvements to the second-tier objective are only acceptable if they do not compromise the first-tier objective. This setup achieves a strict hierarchy of objectives: after the first-tier objective is optimized, the second-tier objective is further optimized within the tolerance range of maintaining the optimal value of the first-tier objective.

[0227] The first-level objective function maximizes priority task allocation through negative weights. For pre-assigned tasks, the system applies additional incentive weights to ensure their priority execution. Specifically, ordinary tasks are weighted according to their original priority, while pre-assigned tasks have their weights amplified tenfold, thus gaining higher priority during optimization. The role of the pre-assignment incentive mechanism is that the tenfold incentive weight significantly increases the contribution of pre-assigned tasks to the objective function. To maximize the objective function value, the optimizer prioritizes finding feasible allocation schemes for these tasks. This incentive strength is set based on empirical tuning: it must be strong enough to guarantee the priority of pre-assigned tasks, but not so strong that it completely ignores other tasks.

[0228] The second-level objective function minimizes the imbalance in resource allocation, including the maximum and minimum difference in employee scores and the maximum and minimum difference in working hours. By minimizing these two differences, the system achieves a balanced distribution of employee workload, improving the fairness of the scheduling scheme. The balance objective has a dual consideration: score differences reflect the fairness of employee performance evaluation, while working hour differences reflect the balance of workload. The comprehensive optimization of both ensures that the scheduling scheme is fair in both quantitative evaluation and actual workload. A hierarchical strategy is used for solving the multi-objective problem. The system first optimizes the first-level objective. Based on obtaining the optimal value of the first level, a tolerance constraint is added, allowing the first-level objective value to fluctuate within a certain range of the optimal value. Then, the second-level objective is optimized. The design intent of the tolerance mechanism is that strictly maintaining the optimal first-level objective may lead to no optimization space for the second-level objective. An appropriate tolerance (e.g., 1%-2%) allows the first-level objective to deviate slightly from the optimal, creating space for the optimization of the second-level objective, while ensuring that the core value of the first-level objective is not substantially damaged. By setting different objective weights (1000 for the first level and 1 for the second level), the system ensures that high-priority objectives are met first. Structural characteristics of the complete mathematical model: This invention constructs a two-level multi-objective mixed integer programming model with good mathematical properties. All constraints in the model are linear, the objective function is a linear weighted sum, and the decision variables are a mixture of binary integer variables and continuous real-number variables. This structure allows the model to be solved efficiently using mature commercial optimization solvers (such as Gurobi and CPLEX), achieving high-quality solutions within an acceptable timeframe for practical applications.

[0229] The mathematical model for the overall process is as follows:

[0230] Main decision variables:

[0231] Binary allocation matrix :

[0232] ;

[0233] ;

[0234] Auxiliary continuous variables:

[0235] (1) Employee cumulative integral vector :

[0236] ;

[0237] in, For employees The accumulated points and working hours, For monthly base points, Points are awarded for completing the task.

[0238] (2) Employee cumulative working hours vector :

[0239] ;

[0240] in, For employees The accumulated points and working hours, This is the time already worked. Task duration (minutes).

[0241] (3) Integral extreme value variables :

[0242] ;

[0243] (4) Extreme value variables of working hours :

[0244] ;

[0245] Constraint 1 - Task manpower requirement constraint:

[0246] ;

[0247] For critical tasks, set a lower bound:

[0248] ;

[0249] in This represents the minimum manpower required. Forced Compress the search space. To at least meet the minimum manpower requirements, This is the set of critical tasks. Ensure that the critical tasks meet the minimum manpower requirements.

[0250] Constraint 2 - Time conflict constraint based on maximal cliques:

[0251] ;

[0252] in, For employees In the Great Group The allocation threshold within:

[0253] ;

[0254] Constraint 3 - Locked Assignment Mandatory Constraint:

[0255] Locking task And designated employees :

[0256] ;

[0257] Constraint 4 - Integral Cumulative Relationship Constraint:

[0258] ;

[0259] in, For employees Accumulated points Basic integral, For the task The integral value.

[0260] Constraint 5 - Cumulative Work Hours Constraint:

[0261] ;

[0262] in, For employees Total working hours For existing working hours, The duration of the task.

[0263] Constraint 6 - Extreme value definition constraint (linearized max / min):

[0264] ;

[0265] ;

[0266] First-level objective function (high priority - maximizing task allocation success rate):

[0267] ;

[0268] in: As task priority weight, Locks assigned to employees To lock the allocation pair set, To lock the task incentive weights (10x magnification), minimize This is equivalent to maximizing the success rate of weighted task allocation.

[0269] Second-level objective function (low priority - minimize resource allocation imbalance):

[0270] ;

[0271] in This is the weighting coefficient (usually set to 1). Minimize Achieve a balance between employee workload points and working hours.

[0272] Step S5: Hierarchical optimization solution and intelligent result processing

[0273] 5.1: Solution Strategy Configuration

[0274] The system employs a commercially optimized solver to solve the model, achieving high-performance solving through careful parameter configuration and solution strategy settings. Key parameters include: a solution accuracy control parameter set to 5% to achieve a balance between solution quality and solution time; a maximum solution time limit set to 600 seconds; the use of the Barrier algorithm for linear programming relaxation; enabling aggressive preprocessing to reduce model size; and enabling aggressive cutting plane generation to enhance linear relaxation.

[0275] Two-stage optimization process:

[0276] Phase 1 - Optimize the first-level objective:

[0277] ;

[0278] in Let the feasible region be (satisfying constraints 1-6 in S4). Let the optimal solution set be:

[0279] ;

[0280] in, This is the feasible region; For the first-level objective function, A vector of decision variables;

[0281] Phase 2 - - Optimize the second-level objective under constraints:

[0282] based on -Constraint strategies construct the second-stage feasible region:

[0283] ;

[0284] in The tolerance coefficient allows the first-layer target to fluctuate within the range of 1%-2% of the optimal value.

[0285] Second phase optimization:

[0286] ;

[0287] in, This is the objective function for the second layer;

[0288] Final solution:

[0289] ;

[0290] Tolerance coefficient Choice:

[0291] The second layer has limited optimization space, and the improvement in balance is not significant.

[0292] The first-level target value loss is too large, resulting in a decrease in the task allocation success rate;

[0293] Experiments show To achieve the best balance.

[0294] Performance index calculation:

[0295] Metric 1 - Task Assignment Success Rate:

[0296] ;

[0297] in, These are the values ​​of the decision variables in the optimal solution. To lock the allocation of the set.

[0298] Metric 2 - Pre-allocation execution rate:

[0299] ;

[0300] Indicator 3 - Integral Balance (Range):

[0301] ;

[0302] in, The integral extremum in the optimal solution

[0303] The standard deviation can be further calculated:

[0304] ;

[0305] in The average integral.

[0306] Indicator 4 - Work Hour Balance (Poor):

[0307] ;

[0308] in, , These are the maximum and minimum working hours in the optimal solution, respectively.

[0309] Indicator 5 - Load Variation Coefficient:

[0310] ;

[0311] in, denoted as the integral standard deviation.

[0312] A good scheduling scheme satisfies .

[0313] 5.2: Optimization Principles of Parameter Configuration

[0314] The solution accuracy parameter (MIPGap) controls when the solver stops searching. A 5% tolerance means that the solver can terminate early when the difference between the found feasible solution and the theoretical optimal solution is less than 5%. This setting is based on practical application requirements: excessively high accuracy requirements (such as 0.1%) can lead to an exponential increase in solution time, while 5% accuracy is sufficient for scheduled applications, and users are unlikely to perceive this small difference.

[0315] The solution to mixed-integer programming is divided into two stages: first, solving the linear relaxation problem to obtain the lower bound, and then searching for integer solutions through branch and bound. The Barrier algorithm (interior point method) has an efficiency advantage in handling large-scale sparse linear programming, and is particularly suitable for the constraint structure of this invention. Compared with the simplex method, the relationship between the number of iterations and the problem size is more stable, and it performs better for large-scale problems with thousands of variables and constraints. The role of preprocessing techniques: The optimizer performs preprocessing before solving, including eliminating redundant constraints, fixing obvious variable values, and aggregating similar constraints. Aggressive preprocessing strategies can reduce the model size by 20%-40%, significantly reducing the computational burden of subsequent solutions. The essence of preprocessing is to simplify the model structure without changing the feasible region through constraint propagation and logical reasoning. The principle of the cutting plane technique: During the branch and bound process, the solver dynamically generates cutting planes (effective inequalities) to tighten the linear relaxation, making the relaxed solution closer to the integer solution. The cutting plane analyzes the fractional characteristics of the current solution and adds inequalities that exclude the current fractional solution but retain all integer solutions. Aggressive cutting plane generation strategies explore various types of cutting planes (Gomory cuts, covering inequalities, etc.), accelerating convergence but increasing computational cost per iteration. For the problem structure of this invention, experiments show that aggressive cutting plane strategies can reduce the number of branch nodes by approximately 30%. Hierarchical optimization is implemented in two stages. In the first stage, the weight of the first-level objective is set to 1000, and the weight of the second-level objective is set to 0, obtaining the optimal value for the first level. In the second stage, constraints are added to ensure that the first-level objective value does not exceed 1.01 times the optimal value. Then, the weight of the first-level objective is set to 0, and the weight of the second-level objective is set to 1, obtaining the final solution. The practical basis for the 1.01-fold tolerance: This tolerance coefficient has been optimized through numerous real-world cases, ensuring both the core value of the first-level objective (task allocation success rate) and providing sufficient optimization space for the second-level objective (resource balancing). Too small a tolerance will lead to no solution or insignificant effect in the second-level optimization, while too large a tolerance may compromise the execution guarantee of critical tasks.

[0316] 5.3: Post-processing and Validation of Results

[0317] After the solution is completed, the system performs comprehensive post-processing and verification of the allocation results to ensure their correctness and usability. The result verification system includes four aspects: constraint satisfaction check to verify whether all allocations meet basic constraints such as qualifications, time, and region; conflict check to verify whether there are time conflicts in the allocation results; pre-allocation execution rate statistics to calculate the actual execution ratio of pre-allocated tasks; and balance index calculation to statistically analyze the distribution of employee points and working hours. The necessity of the verification system: During the calculation process, the optimization solver may produce minor constraint violations due to numerical errors, rounding errors, etc. Post-processing verification can capture these potential problems and perform secondary verification on key constraints (such as locked allocations and time conflicts) to ensure the absolute correctness of the output solution. The verification process essentially remaps the mathematical solution of the optimization model back to the business domain to verify its business feasibility. The system calculates key performance indicators, including task allocation success rate, pre-allocation execution rate, and resource balance. The task allocation success rate equals the number of allocated tasks divided by the total number of tasks. The pre-allocation execution rate equals the number of tasks executed according to pre-allocation divided by the total number of pre-allocated tasks. Details of pre-allocation execution rate calculation: "Execution according to pre-allocation" is only counted when a task is actually assigned to the pre-allocated designated employee. This indicator directly reflects the system's respect for existing personnel arrangements and is a crucial basis for measuring the difficulty of implementing a scheduling plan. Income balance is measured by calculating the standard deviation of employee income; a smaller standard deviation indicates a more balanced allocation. For abnormal situations, the system provides intelligent handling suggestions. When there are unassigned high-priority tasks, a resource shortage warning is generated, providing suggestions to increase personnel or adjust the time. Logic for analyzing the reasons for unassigned tasks: The system traces the root cause of unassigned tasks. Possible reasons include: no qualified available employees (overly strict qualification constraints), all qualified employees having conflicting tasks during the same time period (time conflict), and total employee workload being saturated (capacity constraints). For different reasons, the system provides differentiated improvement suggestions: insufficient qualifications suggest training or temporary reassignment; time conflicts suggest adjusting task time; and insufficient capacity suggests increasing personnel or lowering task standards. When the pre-allocation execution rate is below 90%, the reasons for not executing pre-allocation tasks are analyzed, and suggestions for qualification training or time adjustment are provided. The execution rate threshold is set based on practical experience. A threshold below this percentage indicates a significant conflict between the pre-allocation plan and actual resource constraints, requiring manual intervention for adjustment. The system analyzes unexecuted pre-allocated tasks, identifying whether the issue stems from mismatched qualifications, time conflicts, or other constraints, providing data support for management decisions. When employee workload imbalance exceeds the threshold, the system identifies employees with excessive or insufficient workloads and provides suggestions for workload reallocation. The imbalance threshold is dynamically adjusted based on the total number of employees and task characteristics. Small teams (less than 10 people) can tolerate a higher degree of imbalance, while large teams should maintain stricter balance.The system identifies outliers through statistical analysis, marking employees whose workloads significantly deviate from the average level and suggesting that managers monitor their work status. Output includes structured data and visualizations. Structured data includes assignment lists, employee workload reports, and unassigned task reports. Visualizations include Gantt charts, load distribution charts, and conflict resolution charts, facilitating managers' intuitive understanding of scheduling plans. The system offers diverse output formats: CSV, Excel, JSON, and other formats are available to meet the needs of different application scenarios. Gantt charts display each employee's task allocation over time, easily identifying idle and busy periods; load distribution charts present the statistical distribution of employee workloads in bar charts or box plots, visually reflecting balance; and conflict resolution charts show the original conflicting task network and the final allocation scheme, helping to understand how the optimizer resolves complex time conflicts.

[0318] Example 2

[0319] To more intuitively understand the above technical solutions and steps, the following example uses a specific scenario to fully demonstrate the entire process from inputting task information to outputting the optimal allocation scheme.

[0320] (I) Scene Setting and Initial Data

[0321] An airport needs to complete 8 ground service tasks during the morning rush hour from 8:00 to 12:00. The dispatch center has 5 employees available. The system receives input data consisting of two parts: employee information and task information. Regarding employee information, the 5 employees are Zhang San (skill level A, monthly score 80, 2 hours worked today), Li Si (skill level B, monthly score 85, 1 hour worked today), Wang Wu (skill level A, monthly score 75, 3 hours worked today), Zhao Liu (skill level C, monthly score 90, 0 hours worked today), and Sun Qi (skill level B, monthly score 78, 2.5 hours worked today). Regarding task information, the 8 tasks include: Task A (CA1234 boarding, 9:00-10:00, requires 1 person, score ≥80, pre-assigned to Zhang San), Task B (MU5678 arrival, 9:30-10:30, requires 1 person, score ≥75, unassigned), Task C (HU9012 special service, 10:00-11:00, requires 1 person, score ≥85, locked to Zhao Liu), and Task D (CZ3456 boarding, 9:00-10:30, requires 1 person, score ≥... Tasks are categorized as follows: Task A (3U7890 arrives, 10:30-11:30, requires 1 person, score ≥80, unassigned), Task B (FM1122 special service, 9:15-10:15, requires 1 person, score ≥90, unassigned), Task C (SC4455 boarding, 11:00-12:00, requires 1 person, score ≥75, unassigned), and Task D (JD6677 arrives, 11:30-12:30, requires 1 person, score ≥80, unassigned). Each task has a list of assignable employees. For example, Task A can be completed by Zhang San, Li Si, and Wang Wu; Task B can be completed by Li Si, Wang Wu, and Zhao Liu; Task C can only be completed by Zhao Liu (locked), and so on.

[0322] (II) Execution process and output of step S3

[0323] After the dispatch center receives these 8 tasks, step S3 first requires identifying the time conflicts between the tasks. Because an employee cannot perform two tasks with overlapping times simultaneously, for example, it is impossible to be responsible for the boarding service of task A from 9:00 to 10:00 and the arrival service of task B from 9:30 to 10:30 at the same time. These two tasks overlap in time and may both be assigned to the same person, thus constituting a conflict.

[0324] The first stage of S3 is constructing a conflict matrix. The system begins by comparing the eight tasks pairwise, performing a two-condition judgment for each pair. Taking task A and task B as an example, the system first checks if their times overlap, finding that the 9:30-10:00 time slot overlaps. Then, it checks if they share any available employees, finding that Li Si and Wang Wu are both in the available employee lists for both tasks, thus determining that task A and task B conflict. Taking task A and task C as another example, although their time boundaries are adjacent, the 10:00 time slot does not overlap, therefore task A and task C do not conflict. Taking task A and task F as another example, although their times overlap from 9:15 to 10:00, task A can be completed by Zhang San, Li Si, and Wang Wu, while task F can only be completed by Zhao Liu; they do not share any available employees, therefore they do not conflict. After checking all 28 task pairs, the system obtains an 8×8 conflict matrix, which marks which task pairs conflict with each other. Specific conflict relationships include: A and B conflict, A and D conflict, B and A, C, D conflict, C and B, E, G conflict, D and A, B, F conflict, E and C, G, H conflict, F and D conflict, G and C, E, H conflict, and H and E, G conflict.

[0325] The second stage of S3 involves applying the Bron-Kerbosch algorithm to search for maximal cliques. The system now knows which tasks conflict pairwise, but needs to further identify "maximal cliques"—groups of tasks where all tasks conflict with each other and no other tasks can be added. This is because within a maximal clique, each employee can only choose one task to perform, providing a crucial constraint for subsequent optimization. The system begins its search, starting with task A. It finds that A conflicts with B and D. Checking if B and D also conflict, it confirms this, thus finding a maximal clique {A, B, D}, where these three tasks are pairwise conflicting and each employee can only choose one. Continuing with task C, which conflicts with B, E, and G, it finds that E and G also conflict, thus finding a maximal clique {C, E, G}. Continuing the search, it finds a maximal clique {E, G, H}. It also finds a maximal clique {B, D, F}. Ultimately, the system found four maximal cliques: maximal clique 1 is {task A, task B, task D}, maximal clique 2 is {task B, task D, task F}, maximal clique 3 is {task C, task E, task G}, and maximal clique 4 is {task E, task G, task H}.

[0326] After step S3 is completed, the key data passed to step S4 is the list of these four maximal cliques. This list tells S4 that in maximal clique 1 {A, B, D}, each employee can accept at most one task; in maximal clique 2 {B, D, F}, each employee can accept at most one task; in maximal clique 3 {C, E, G}, each employee can accept at most one task; and in maximal clique 4 {E, G, H}, each employee can accept at most one task.

[0327] (III) Execution process and output of step S4

[0328] S4 receives four maximal cliques from S3, along with the original employee and task information. S4's task is to transform the decision problem of "whether to assign a certain employee to a certain task" into an optimization model that can be solved mathematically.

[0329] The first stage of S4 is defining the decision variables. The system defines a binary variable x[employee, task] for each combination of employee and task. When x[Zhang San, task A] = 1, it means Zhang San is assigned to task A; when x[Zhang San, task A] = 0, it means Zhang San is not assigned to task A. Since there are 5 employees and 8 tasks, a total of 40 decision variables are defined. Once the values ​​of these 40 variables are determined, the allocation scheme is completely determined.

[0330] The second stage of S4 is to transform the maximal cliques from S3 into constraints. This is the key connection point between S3 and S4. After receiving the four maximal cliques from S3, S4 immediately transforms them into conflict constraints. The transformation rule is as follows: For maximal clique 1 {A, B, D}, the system generates one constraint x[i, A] + x[i, B] + x[i, D] ≤ 1 for each employee i. Specifically, this expands to x[Zhang San, A] + x[Zhang San, B] + x[Zhang San, D] ≤ 1, x[Li Si, A] + x[Li Si, B] + x[Li Si, D] ≤ 1, and so on, for a total of 5 constraints. These constraints ensure that Zhang San can only accept at most one task from A, B, or D, and Li Si can only accept at most one task from A, B, or D. The same applies to other employees. Similarly, 5 constraints are generated from maximal clique 2 {B, D, F}, 5 constraints from maximal clique 3 {C, E, G}, and 5 constraints from maximal clique 4 {E, G, H}. Four maximal cliques multiplied by five employees generate a total of 20 conflict constraints. This is the process of directly converting the maximal cliques found in S3 into the constraints in the S4 model.

[0331] The third stage of S4 involves adding other business constraints. Besides conflict constraints, the system also needs to generate personnel quantity constraints to ensure that the required number of personnel for each task is met. For example, x[Zhang San, A] + x[Li Si, A] + x[Wang Wu, A] + x[Zhao Liu, A] + x[Sun Qi, A] = 1 indicates that task A requires 1 person. There are 8 constraints for 8 tasks. The system generates task locking constraints. For example, task C is locked to Zhao Liu, therefore x[Zhao Liu, C] = 1, and the decision variables for other employees regarding task C are all 0. The system generates task score constraints to ensure that the total score of the assigned employees meets the task requirements. For example, 80 × x[Zhang San, A] + 85 × x[Li Si, A] + 75 × x[Wang Wu, A] + ... ≥ 80 indicates that task A requires the total score of employees to be no less than 80 points. There are 8 constraints for 8 tasks. The system generates work duration constraints to ensure that each employee's total work time for the day does not exceed 8 hours. For example, 2 + 1 × x[Zhang San, A] + 1 × x[Zhang San, B] + ... ≤ 8 means that Zhang San has already worked 2 hours, and the time added to the newly assigned task cannot exceed 8 hours. There are a total of 5 constraints for 5 employees. The system also generates skill matching constraints based on the list of assignable employees. For example, if Zhang San is not in the list of available tasks for task F, then x[Zhang San, F] = 0.

[0332] The fourth stage of S4 is setting optimization goals. The system employs a two-tiered hierarchical objective function. The first-tier goal is the core business objective, which includes maximizing the number of tasks completed and maximizing the number of pre-assigned tasks executed. The number of tasks completed equals the sum of all employees' decision variables for all tasks. The number of pre-assigned tasks is represented by multiplying the priority of pre-assigned tasks by 100. For example, if task A is pre-assigned to Zhang San, the objective function includes the term 100×x[Zhang San, A]. The second-tier goal is auxiliary optimization objectives, including minimizing the variance of employee scores (making the task scores obtained by each employee as balanced as possible), minimizing the variance of employee working hours (making the working hours of each employee as balanced as possible), and minimizing the use of virtual employees (using real employees as much as possible).

[0333] After step S4 is completed, the data passed to step S5 is a complete MIP optimization model. This model includes 40 decision variables x[i, j], approximately 45 constraints (including 20 conflict constraints from the 4 maximal cliques in S3, 8 personnel quantity constraints, 8 task score constraints, 5 work duration constraints, and several locking and pre-assignment constraints), and two levels of 5 optimization objective functions. This model is essentially a mixed integer programming problem; the variables are integers (0 or 1), the constraints are linear, and the objective function is also linear. However, due to the integer nature of the variables, the problem has high solution complexity and cannot be solved quickly manually, requiring a professional optimization solver.

[0334] (iv) Execution process and output of step S5

[0335] S5 receives a complete MIP optimization model from S4, containing 40 decision variables, 45 constraints, and 5 optimization objectives. S5's task is to call the Gurobi optimization solver to find the set of x[i,j] values ​​that satisfy all constraints and optimize the objective function.

[0336] The first stage of S5 involves configuring the solver and inputting the model. The system inputs the model from S4 into the Gurobi solver, configuring the solver parameters including a maximum solver time of 300 seconds and a relative error tolerance of MIPGap=0.11, setting the priority to first-level objective priority. This configuration stage completes the conversion from the abstract mathematical model to the solver's executable format. The system sequentially numbers all decision variables, converts constraints into a coefficient matrix A and a right-hand side vector b, and converts the objective function into a coefficient vector c, forming a standard matrix representation.

[0337] The second stage of S5 is to perform the first round of optimization of the core objective. The solver uses a branch and bound algorithm to search for feasible solutions that satisfy all constraints and optimize the first-level objective function. Internally, the solver maintains a branch tree, where each node represents a partial solution. The upper bound of each node is calculated through linear relaxation, and a pruning strategy is used to eliminate branches that are unlikely to produce the optimal solution. During the search process, the solver tries various allocation schemes. For example, if it tries the scheme "x[Zhang San, A]=1, x[Li Si, B]=1, x[Zhao Liu, C]=1,..." and finds that it might violate the constraint of maximal clique 1 (if Zhang San connects to A and then wants to connect to B), it discards the scheme. If it continues trying the scheme "x[Zhang San, A]=1, x[Li Si, B]=1, x[Zhao Liu, C]=1, x[Wang Wu, D]=1,..." and finds that it satisfies all constraints, it records the scheme and calculates its objective function value. After several seconds of searching, the solver finds the first-level optimal solution. For example, x[Zhang San, A]=1 indicates that Zhang San is responsible for task A, x[Li Si, G]=1 indicates that Li Si is responsible for task G, x[Wang Wu, D]=1 indicates that Wang Wu is responsible for task D, x[Zhao Liu, C]=1 and x[Zhao Liu, F]=1 indicates that Zhao Liu is responsible for tasks C and F, x[Sun Qi, E]=1 indicates that Sun Qi is responsible for task E, and all other decision variables are 0. The first-level result shows that 6 tasks (A, C, D, E, F, G) are completed, the pre-assigned task A has a 100% execution rate, and the uncompleted tasks are B and H.

[0338] The third stage of S5 involves a second round of optimization of the secondary objective. The system fixes the first-level results and adds new constraints requiring at least 6 tasks to be completed and a pre-allocation execution rate of 100%. Based on this, the second-level objective is optimized. The solver continues searching for a solution that minimizes the variance of employee scores, the variance of working hours, and the use of virtual employees while maintaining the first-level objective. After several seconds of searching, the optimal solution for the second level is found. The allocation scheme may be the same as or slightly adjusted from the first level. Final statistics show that Zhang San is responsible for task A (score 80 points, total working hours 3 hours today), Li Si is responsible for task G (score 85 points, total working hours 2 hours today), Wang Wu is responsible for task D (score 75 points, total working hours 4.5 hours today), Zhao Liu is responsible for tasks C and F (score 90 points, total working hours 2 hours today), and Sun Qi is responsible for task E (score 78 points, total working hours 3.5 hours today). The employee score variance is 32.5, the working hour variance is 0.85, and zero virtual employees are used, all reaching a relatively optimal level.

[0339] The fourth stage of S5 is result verification. The system comprehensively verifies the solution results, checking for conflict constraints (Zhang San is only responsible for A and does not conflict with B and D, Zhao Liu is responsible for C and F but they are not in the same maximal clique), personnel number constraints (one person is assigned to each task), score constraints (employees achieve the required score for each task), time constraints (each employee's working time is ≤8 hours), locking constraints (task C was indeed assigned to Zhao Liu), and pre-assignment constraints (task A was indeed assigned to Zhang San). After all verifications pass, the solution is confirmed as a feasible optimal solution.

[0340] The fifth stage of S5 is generating the allocation scheme. The system converts the combination of x[i,j]=1 into actual allocation records, generates a final allocation scheme table, and lists the tasks assigned to each employee, the task score obtained, and the working time. At the same time, it generates a list of incomplete tasks, marking that task B (MU5678 arrived, low priority, insufficient personnel) and task H (JD6677 arrived, low priority, conflict with tasks E and G) could not be allocated. An execution statistics report is generated, showing a task completion rate of 75% (6 / 8), a pre-allocated execution rate of 100% (1 / 1), and a locked task execution rate of 100% (1 / 1).

[0341] Step S5 outputs this complete allocation plan to the scheduling center, where the scheduler can directly issue task instructions. Thus, from inputting information on 8 tasks and 5 employees to outputting the optimal allocation plan, the entire technical chain completes a closed loop.

[0342] Example 3

[0343] The present invention also provides an intelligent scheduling system for airport ground staff based on the Bron-Kerbosch maximal clique algorithm. The system is used to implement the method described in Embodiment 1. The system includes: a data acquisition module, a task splitting module, a conflict identification module, an optimization modeling module, and a solution module.

[0344] The data acquisition module is used to obtain employee and task information from external systems, and to perform data preprocessing and virtual employee supplementation; the external systems include: human resource management system and flight operation system;

[0345] The task splitting module is used to split a task into several sub-tasks based on the manpower requirements of the task; and to generate a set of executable employees for each sub-task based on employee ability constraints, time constraints and dynamic priority assignment algorithms.

[0346] The conflict identification module is used to construct a conflict matrix based on time overlap and competition for employee resources; based on the conflict matrix, the Bron-Kerbosch algorithm is used to identify the maximal clique set of conflict tasks.

[0347] The optimization modeling module is used to set decision variables, construct constraints, and set multi-objective hierarchical optimization functions based on the maximal clique set to obtain a multi-objective mixed integer programming model.

[0348] The solver module is used to perform hierarchical optimization of multi-objective mixed-integer programming models using a mixed-integer programming solver, and to extract and output employee task allocation schemes.

[0349] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made to the technical solutions of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A method for intelligent scheduling of airport ground staff based on the Bron-Kerbosch maximal clique algorithm, characterized in that, The method includes: S1: Obtain employee and task information from external systems, and perform data preprocessing and virtual employee supplementation; the external systems include: human resource management system and flight operation system; S2: Divide the task into several sub-tasks according to the manpower requirements of the task; generate a set of executable employees for each sub-task based on employee ability constraints, time constraints and dynamic priority assignment algorithm; S3: Construct a conflict matrix based on time overlap and competition for employee resources; based on the conflict matrix, use the Bron-Kerbosch algorithm to identify the maximal clique set of conflict tasks; S4: Based on the maximal clique set, decision variables are set, constraints are constructed, and a multi-objective hierarchical optimization function is set to obtain a multi-objective mixed integer programming model; S5: Use a mixed-integer programming solver to perform hierarchical optimization of the multi-objective mixed-integer programming model, and extract and output the employee task allocation scheme; In S3, the method for constructing the conflict matrix includes: For any two subtasks, a two-condition judgment is performed. First, it is determined whether the time intervals overlap, that is, whether the end time of subtask i is greater than the start time of subtask j and whether the end time of subtask j is greater than the start time of subtask i. Second, the employee sharing situation is checked, that is, whether there is an intersection of the sets of allocable employees. When the time overlaps and employee sharing is simultaneously true, it is marked as a conflict in the conflict matrix. In S3, the method for identifying the maximal clique set of conflict tasks using the Bron-Kerbosch algorithm includes: The recursive backtracking strategy of the Bron-Kerbosch algorithm is adopted to maintain three sets: the current clique R, the candidate set P, and the exclusion set X. The algorithm introduces a pivot optimization strategy to start from an empty clique, and gradually expand the current clique by selecting the node with the most neighbors. At each time, a candidate node is selected to be added to the current clique, and the candidate set is updated to the nodes adjacent to that node. When both the candidate set and the exclusion set are empty, a maximal clique is output. Among them, time overlap refers to the time windows of two tasks intersecting; employee resource competition refers to the set of executable employees of two tasks intersecting.

2. The method according to claim 1, characterized in that, In S1, employee information includes employee identifier, working time window, qualification information, work area authorization, and performance data; task information includes task identifier, time window, manpower requirements, constraints, and task personnel allocation details.

3. The method according to claim 1, characterized in that, In S1, the method for supplementing virtual employees includes: When a lock is assigned but employee information is missing, the virtual employee automatic compensation mechanism is activated. By automatically creating placeholder employee records, each lock constraint can find a corresponding employee index.

4. The method according to claim 1, characterized in that, In S2, the dynamic priority assignment algorithm is as follows: based on the pre-allocation of tasks and the minimum manpower requirements, the priority of each sub-task is intelligently adjusted; specifically, it includes: The system calculates the pre-allocation status, including the number of locked and semi-locked tasks; it also calculates the manpower gap, i.e., the number of sub-tasks that still need to be guaranteed to meet the minimum manpower requirements; it assigns execution priorities: the priority of pre-allocated sub-tasks is increased to 100 times their original priority; to meet the minimum manpower requirements, the priority of the corresponding number of remaining sub-tasks is increased to 100 times in sequence; other ordinary sub-tasks retain their original priorities.

5. The method according to claim 1, characterized in that, In S2, the conditions for generating the executable employee set include: the employee's qualifications meet the task requirements, the employee's work area covers the task location, and the employee's time window includes the task's time window.

6. The method according to claim 1, characterized in that, In S4, the decision variable is set as follows: a binary variable x[i,j] is defined for each combination of employee and subtask. When x[i,j]=1, it means that employee i is assigned to subtask j, otherwise it means that it is not assigned. Constraint Construction: Receive a list of maximal cliques and generate a set of conflict constraints for all subtasks in each maximal clique. Specifically, for each maximal clique {T1, T2, ..., Tk}, generate the constraint x[i, T1] + x[i, T2] + ... + x[i, Tk] ≤ 1 for each employee i, ensuring that any employee can be assigned to at most one subtask in the maximal clique. In addition to conflict constraints, generate personnel quantity constraints to ensure that the personnel requirements of each subtask are met, generate task score constraints to ensure that the total score of the assigned employees reaches the task requirements, generate working time constraints to ensure that the daily working time of each employee does not exceed the upper limit, and generate locking and pre-assignment constraints to ensure that the determined assignment relationship is executed. Objective function setting: A two-level hierarchical structure is adopted. The first level objective includes maximizing the number of tasks completed and maximizing the number of pre-assigned tasks executed. The second level objective includes minimizing the variance of employee scores, minimizing the variance of working hours, and minimizing the number of virtual employees used.

7. The method according to claim 6, characterized in that, In S5, the method of using a mixed integer programming solver to perform hierarchical optimization of a multi-objective mixed integer programming model and extracting and outputting the employee task allocation scheme includes: using a two-round iterative solution strategy; In the first round of solving, the constructed multi-objective mixed integer programming model MIP is input into the mixed integer programming solver Gurobi. The solution parameters are configured, including a time limit of 300 seconds and a relative error tolerance of MIPGap=0.

11. The solver uses a branch and bound algorithm to search for feasible solutions that satisfy all constraints and make the first-level objective function optimal. During the search process, the solver maintains a branch tree, with each node representing a partial solution. The upper bound of each node is calculated through linear relaxation, and a pruning strategy is used to eliminate branches that cannot produce the optimal solution, thereby obtaining the optimal value of the first-level objective. In the second round of solving, new constraints are added to the model, requiring that the first-level objective value cannot be lower than the optimal value obtained in the first round of solving. Then, the second-level objective function is optimized. After the solution is completed, the values ​​of all decision variables are extracted from the solver, the satisfaction of all constraints is verified, and the variable combination of x[i,j]=1 is converted into the actual employee-task allocation record, generating an allocation scheme report and an unfinished task list.

8. An intelligent scheduling system for airport ground staff based on the Bron-Kerbosch maximal clique algorithm, the system being used to implement the method described in any one of claims 1-7, characterized in that, The system includes: a data acquisition module, a task decomposition module, a conflict identification module, an optimization modeling module, and a solution module; The data acquisition module is used to acquire employee and task information from external systems, and to perform data preprocessing and virtual employee supplementation; wherein, the external systems include: human resource management system and flight operation system; The task splitting module is used to split a task into several sub-tasks according to the manpower requirements of the task; and to generate a set of executable employees for each sub-task based on employee ability constraints, time constraints and dynamic priority assignment algorithm. The conflict identification module is used to construct a conflict matrix based on time overlap and employee resource competition as judgment conditions; and to identify the maximal clique set of conflict tasks based on the conflict matrix using the Bron-Kerbosch algorithm. The optimization modeling module is used to set decision variables, construct constraints, and set multi-objective hierarchical optimization functions based on the maximal clique set to obtain a multi-objective mixed integer programming model. The solution module is used to perform hierarchical optimization of a multi-objective mixed integer programming model using a mixed integer programming solver, and to extract and output the employee task allocation scheme.

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