A construction engineering project scheduling method based on multi-objective dynamic crane optimization

Abstract

The application provides a construction engineering project scheduling method based on multi-target dynamic multi-objective optimization, first, an optimization model with the minimum time limit, the minimum cost and the minimum resource fluctuation as the targets is established, wherein the objective function covers the total time limit, the total cost and the resource fluctuation variance, and the practical constraints such as the task dependency relationship, the resource upper limit and the variable limit are considered, second, an improved multi-target dynamic multi-objective optimization algorithm is designed, the algorithm ensures the feasibility and diversity of the solution through a dynamic elite reservation mechanism, a mixed position updating rule, a three-layer constraint processing mechanism and a double-threshold convergence monitoring system, finally, the multi-target dynamic multi-objective optimization algorithm is used to solve the construction engineering project problem, and a construction project scheduling scheme is obtained. The application can realize the automatic and intelligent optimization of the project scheduling process, improve the project management efficiency and scientificity under the premise of meeting the resource limit and the task logic relationship, and collaboratively optimize the project time limit, the total cost and the stability of the resource use.

Description

Technical Field

[0001] This invention relates to the interdisciplinary field of intelligent optimization algorithms and computer-aided engineering project management, and in particular to a scheduling method for construction engineering projects based on multi-objective dynamic optimization. Background Technology

[0002] With the acceleration of global urbanization and the expansion of construction project scale, optimizing the organizational design of construction projects, a core aspect of engineering project management, has become increasingly critical. Traditional project management methods (such as Gantt charts and critical path methods) rely heavily on human experience, resulting in problems such as low resource allocation efficiency, difficulty in coordinating conflicts among multiple objectives such as schedule, cost, and quality, and a lack of dynamic adjustment capabilities to cope with uncertainties.

[0003] In recent years, metaheuristic algorithms (such as genetic algorithms, particle swarm optimization, simulated annealing and its multi-objective variant NSGA-II, multi-objective gray wolf optimization, and multi-objective particle swarm optimization) have been introduced into this field. These algorithms perform global searches by simulating natural phenomena, supporting custom multi-objective weights and generating Pareto solution sets to overcome the local optima trap of traditional methods. Existing research has integrated non-dominated sorting into algorithms such as Rao-1 and Rao-2 to solve multi-objective time-cost trade-offs in the scheduling of small and medium-sized construction projects, developed particle swarm optimization-based models to solve static and dynamic resource-constrained problems in construction projects, and utilized NSGA-II and other algorithms to solve multi-objective models that include customer satisfaction.

[0004] However, existing algorithms still face challenges such as slow convergence and susceptibility to local optima in high-dimensional, nonlinear, and dynamically disturbed construction scenarios. While commercial tools support basic resource balancing, they rely on heuristic rules and have limited multi-objective optimization capabilities. Platforms integrating Building Information Modeling (BIM) support visual scheduling but lack embedded metaheuristic optimizers for generating Pareto fronts, requiring customized algorithm extensions. In particular, existing Resource Constrained Project Scheduling Problem (RCPSP) models often fail to adequately consider resource fluctuation objectives, while time-cost trade-off models tend to ignore dynamic adjustments, and most commercial tools lack the ability to generate Pareto solution sets. Furthermore, while existing dynamic optimization algorithms demonstrate superior robustness in high-dimensional problems, they are single-objective algorithms and lack multi-objective collaborative optimization capabilities. Summary of the Invention

[0005] To address the aforementioned problems, this invention proposes a construction project scheduling method based on multi-objective dynamic optimization, which can effectively solve the problem of multi-objective scheduling optimization for construction projects under resource-constrained conditions.

[0006] A construction project scheduling method based on multi-objective dynamic optimization, the method comprising:

[0007] Step S1, Problem modeling of the construction project. Problem modeling includes optimization objectives and constraints. Optimization objectives include schedule compression, cost control and resource fluctuation. Constraints include task dependencies, resource limits and variable limits.

[0008] Step S2: Construct a multi-objective dynamic optimization algorithm. The improvement strategies of the multi-objective dynamic optimization algorithm include a dynamic elite solution retention mechanism and memory mechanism, an improved archive update strategy, a convergence monitoring and early termination mechanism, and an improved constraint handling mechanism.

[0009] Step S3, Time complexity analysis of the multi-objective dynamic optimization algorithm, the time complexity of each iteration includes the time complexity of non-dominated sorting, the time complexity of file maintenance and the time complexity of position update;

[0010] Step S4: Use a multi-objective dynamic optimization algorithm to solve the construction project problem and obtain a construction project scheduling scheme.

[0011] Furthermore, in step S1, there are a total of N activities, and each activity i has two decision variables. and The decision variables are defined as follows:

[0012] ;

[0013] in, Indicates the start time of activity i. This represents the labor allocation for activity i, and N represents the number of activities. It represents the set of real numbers.

[0014] The objective functions corresponding to schedule compression, cost control, and resource fluctuation in the optimization objectives need to minimize the total project duration objective, the total cost objective, and the resource fluctuation objective, respectively. Minimizing the total project duration objective is equivalent to minimizing the total duration, which is defined as the maximum value among the completion times of all activities. , where d i Indicates the duration of activity i;

[0015] The total cost of a project includes direct and indirect costs. It is quantified as the sum of human resource expenditures and fixed management fees for each task. Human resource expenditures include basic labor costs and additional labor costs. The total cost is defined by three elements: basic labor costs, additional labor costs, and total fixed costs, as shown below:

[0016]

[0017]

[0018]

[0019] Among them, C l c represents the basic labor cost. l C represents the unit labor cost. el This represents the additional labor cost when the number of workers assigned to each activity exceeds a threshold w. threshold When applying a penalty coefficient β, C to surplus labor... f This represents the total fixed cost. This represents the fixed administrative cost for a single task; therefore, the total cost is expressed as:

[0020] ;

[0021] Resource fluctuations are defined as the variance of the total labor force. The total labor force on day t is calculated as follows: ,in, The total number of days the project lasts, if and only if hour ,otherwise , Indicates rounding up;

[0022] The resource fluctuation target is defined as follows: ,in, , representing the average daily labor force.

[0023] Furthermore, the constraints in step S1 are as follows:

[0024] Task dependencies are defined as follows: For any two tasks, i.e., activities i and j in the sequence constraint graph, if task j completes before task i starts, then a completion-start FS dependency exists.

[0025]

[0026] in, This indicates the start time of task j. The duration of task j is represented by the adjacency matrix, and the dependencies between tasks are represented by the adjacency matrix. It means that, among them, This means that task i should only begin after task j has been completed. If so, then task i and task j are not adjacent;

[0027] The resource ceiling is set at a fixed constant W, meaning the total labor force does not exceed this constant. ;

[0028] The variable is bounded by the start time s of task i. i Has an upper limit ub s and lower limit lb s The size of the workforce w allocated to task ii Has an upper limit ub w and lower limit lb w In the objective function, all violations of constraints are combined into a penalty term: for dependency violations, the violation severity is multiplied by a penalty coefficient α and added to the total completion time objective; for resource limit violations, the daily excess amount ub is applied to exceed the limit. w Multiplying by a penalty coefficient α and adding it to the total cost objective, the penalty objective function is expressed as:

[0029]

[0030]

[0031] Where n represents the same number as N, i.e., the total number of tasks;

[0032] Therefore, the multi-objective optimization problem of construction engineering projects can be expressed as:

[0033] .

[0034] Furthermore, the memory mechanism in step S2 is as follows: in each iteration, for each individual in the population, the algorithm selects an elite solution with a set probability from the elite pool to directly replace the individual's current position.

[0035] The dynamic elite solution retention mechanism in step S2 is as follows: if the memory mechanism is not triggered, one of the following two update strategies is randomly selected with a 50% probability:

[0036] The first strategy is based on random perturbation updates: a number is randomly selected from [0,1]. When the random number exceeds 0.5, the first strategy is activated. Its implementation depends on the individual's position in the population and can be divided into three cases:

[0037] The first case is: if the index of an individual in the population is an integer multiple of one-tenth of the population size, then the position update formula and the calculation method for the random perturbation vector are as follows:

[0038]

[0039]

[0040] in, This represents the updated position of the i-th individual, where i∈[0,N]. The mean position of the current population is represented by the built-in MATLAB function mean(X), calculated using the following formula: , This indicates the position of the i-th individual in the current population during the last iteration. Let represent the random perturbation vector, and c represent an adaptive parameter that gradually decreases with the number of iterations, where D represents the number of decision variables. This represents the upper bound of the decision variable. Indicates element-wise multiplication;

[0041] The second scenario is: if the current individual index modulo 10 has a remainder of 1, the position update formula is: ,in, The first solution in the elite reference point is selected according to the following rules: prioritize the solution with the higher non-dominated level, and if they are in the same level, choose the solution with the larger crowding distance.

[0042] The third case is: In all other cases, the position update formula is: When i is 1, The initialization location;

[0043] The second strategy is an update based on physical heuristics: A number is randomly selected from [0,1]. When the random number does not exceed 0.5, there is a 50% probability of choosing one of the following two update methods based on physical heuristics:

[0044] The first method for updating rules inspired by physics is: ,in, It is a random vector that follows a Lévy distribution, which introduces a step-size-based random perturbation; Let be an adaptive parameter that decreases as the number of iterations increases, where , It is a random vector. The calculation method is as follows:

[0045]

[0046] unifrnd is a function in MATLAB that generates random arrays that follow a continuous uniform distribution. For the quality of oat seeds, This represents the deflection coefficient during the rotation of oat seeds. The length of the primary awn of oats. For dynamically adjusted boundary vectors: ,in Indicates the maximum number of iterations;

[0047] The second method for updating rules inspired by physics is: ,in, It is a random vector based on the physical pendulum model, defined as follows:

[0048]

[0049] in, The elastic coefficient of the main awn of oat seeds; This represents the current position of individual i. The angle between the seed trajectory and the ground; It is the gravitational acceleration constant; For an adaptive parameter, , Represents an integer randomly selected from [0, current iteration number]; For another dynamically adjusted boundary vector: .

[0050] Furthermore, the improved archive update strategy in step S2 specifically includes:

[0051] Step 1: Dynamically unpack and merge sets, then archive. and the current population Merged into candidate solution set middle: ;

[0052] in, This represents the candidate solution set for the current decision variable. Indicates the current save file size. This represents all archives from the initial state to the current state corresponding to the current decision variable. This represents the population corresponding to the current decision variable. This indicates that two matrices and Perform vertical splicing. This represents the solution set of the current objective function. This indicates the current objective function archive. This represents the population corresponding to the current objective function;

[0053] Step 2: Use the dual dominance detection algorithm to identify non-dominated solutions. The dominance relationship is defined as follows: For each candidate solution in the solution set... traverse all other solutions To check Whether by What is controlled is represented as:

[0054]

[0055] in, , It is a dominance relation judgment function used to determine the solution. Is the solution dominant? , Indicates the first The solution is at the th solution. The value of the objective function Indicates the first The solution is at the th solution. The value of the objective function It is the number of optimization targets;

[0056] Step 3, Crowded Distance Pruning: When the number of non-dominated solutions exceeds the file size, crowded distance pruning is performed for each objective. The solutions within the current non-dominated front are sorted in ascending order of fitness value. The distance difference between adjacent solutions is calculated and normalized. This operation is repeated for each objective, and the results are accumulated. , The normalized sum of the neighbor distances between all objectives in the solution, i.e., the crowding distance, is expressed as:

[0057]

[0058]

[0059] in, This represents the value of the (i+1)th solution on the m-th objective function. This indicates that in the current non-dominated layer, the first... The maximum value of each objective function. This indicates that in the current non-dominated layer, the first... The minimum value of an objective function. This represents the crowding distance of the solution that minimizes the current objective function value. This represents the crowding distance of the solution with the largest current objective function value.

[0060] Furthermore, the convergence monitoring and early termination mechanism in step S2 includes two strategies:

[0061] Strategy 1: Monitor the improvement progress to quantify the optimization progress of each generation of the optimal solution, and calculate the... The first generation of all individuals in the population Objective function Minimum fitness value:

[0062] , Represents the i-th individual The function value on the k-th objective function, 1≤i≤N;

[0063] The improvement magnitude is calculated as follows:

[0064]

[0065] in, Let M represent the fitness value of the population in generation t, and M represent the number of objective functions.

[0066] Strategy 2: Introduce a stagnation counter to track algebras that show no significant improvement for consecutive periods. The stagnation criterion is defined as follows:

[0067]

[0068] in, Indicates the first All individuals in the first generation of the population The optimal value of each objective function relative to the improvement of the previous generation. This represents a pre-set positive threshold, indicating a critical value for significant improvement. When the improvement is less than this threshold, it is considered that there is no significant improvement.

[0069] At the same time, a maximum allowed number of stagnant algebras is set as the condition for triggering termination.

[0070] Furthermore, the improved constraint processing mechanism in step S2 includes boundary truncation, temporal constraint repair, and resource constraint repair;

[0071] Boundary truncation: Vectorized operations are used to handle variable boundaries, represented as:

[0072]

[0073] in, It is a vector of decision variables, representing the current position of the individual. Indicates the upper bound of a variable. Indicates the lower bound of a variable. Indicates element-wise multiplication. Represents the logical NOT operation. Represents a logical OR operation;

[0074] Timing constraint repair: The forward propagation repair strategy ensures the start time of the repaired task. This will not affect subsequent tasks, thus implicitly handling chained task dependencies:

[0075]

[0076] in, This represents a new timing constraint. This indicates the old timing constraints;

[0077] Resource constraint repair: Associated with the resource penalty term in the objective function, the repair strategy is defined as follows:

[0078]

[0079] in, This indicates new resource constraints. This represents the old resource constraints. This indicates the maximum available amount of resources.

[0080] Furthermore, the time complexity of each iteration of the multi-objective dynamic optimization algorithm is... ,in, This represents the time complexity of position updates. Let O(kN²) be the average time complexity of archive maintenance, O(kN²) be the time complexity of non-dominated sorting, k be the number of objective functions, N be the number of solutions, and D be the number of decision variables.

[0081] This invention proposes a construction project scheduling method based on multi-objective dynamic optimization. First, an optimization model is established with the objectives of minimizing construction period, cost, and resource fluctuation. The objective function encompasses the total construction period, total cost, and resource fluctuation variance, and considers realistic constraints such as task dependencies, resource limits, and variable boundaries. Second, an improved multi-objective dynamic optimization algorithm is designed. This algorithm ensures the feasibility and diversity of solutions through a dynamic elite retention mechanism, a hybrid position update rule, a three-layer constraint processing mechanism, and a dual-threshold convergence monitoring system. Finally, the improved multi-objective dynamic optimization algorithm is used to solve the constructed construction project problem, obtaining a construction project scheduling scheme. Specifically, this invention has the following beneficial technical effects:

[0082] An innovative fusion of bio-inspired algorithms and multi-objective optimization: The dynamic wheat optimization algorithm mechanism, which combines the dual diffusion strategy of rolling (local development) and ejection (global exploration) of oat seeds in a humid environment, is combined with multi-objective optimization techniques (non-dominated sorting, dynamic archives, etc.) to form a multi-objective dynamic wheat optimization algorithm specifically for multi-objective scheduling problems in construction projects. The algorithm has strong global exploration and local development capabilities, fast convergence speed, and the Pareto solution set obtained is superior to existing mainstream multi-objective optimization algorithms in terms of convergence, uniformity of distribution, and breadth.

[0083] Efficient hybrid search strategy: The Levy flight and dynamic boundary adjustment strategies are introduced to enhance global search. A hybrid position update rule combining elite reference points and random perturbations is designed to prevent premature convergence, effectively balancing exploration and development.

[0084] Strong feasibility assurance: It proposes a three-layer constraint processing mechanism of boundary truncation, time dependency repair, and resource over-limit repair. Closely combined with the characteristics of the problem, it ensures that the generated scheduling scheme fully meets all real-world constraints (task logic, resource limits), which is superior to the traditional penalty function method. Through the three-layer constraint processing mechanism, it can ensure that all output scheduling schemes meet the task logic dependency and resource limit conditions, and can be directly used to guide actual project planning.

[0085] Support scientific decision-making: Provides a series of non-dominated solutions (Pareto fronts) that clearly show the trade-offs between schedule, cost, and resource fluctuations, enabling project managers to choose the most suitable scheduling scheme (such as shortest schedule, lowest cost, minimum resource fluctuation, or a compromise scheme) according to their preferences.

[0086] Automation and intelligence: It reduces reliance on human experience, realizes the automation and intelligent optimization of the project scheduling process, helps to improve the efficiency and scientific nature of project management, and reduces the risk of project delays and cost overruns;

[0087] Excellent performance: In the simulation results of the algorithm, the Pareto front obtained by the multi-objective dynamic optimization algorithm on the standard test function (ZDT1,2,3) basically coincides with the real Pareto front, which strongly verifies its good performance.

[0088] This invention can collaboratively optimize the stability of project duration, total cost, and resource usage while meeting resource constraints and task logic relationships, thereby achieving automated and intelligent optimization of the project scheduling process and improving project management efficiency and scientific rigor. Attached Figure Description

[0089] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0090] Figure 1 This is a timing constraint diagram of activity dependencies in an embodiment of the present invention;

[0091] Figure 2 This is a flowchart of the multi-objective dynamic microphone optimization algorithm provided in the embodiments of the present invention;

[0092] Figure 3 This is a Pareto front distribution map in a two-dimensional target space provided in an embodiment of the present invention;

[0093] Figure 4 This is a Pareto front and current population distribution map in a three-dimensional target space provided in this embodiment of the invention;

[0094] Figure 5 This is the construction project scheduling scheme with the lowest cost provided by the embodiments of the present invention;

[0095] Figure 6 This is a construction project scheduling scheme with the shortest construction period provided by an embodiment of the present invention;

[0096] Figure 7This is a construction project scheduling scheme with minimal resource fluctuations provided by an embodiment of the present invention;

[0097] Figure 8 This is a compromise construction project scheduling scheme provided by the embodiments of the present invention;

[0098] Figure 9 This describes the convergence of the multi-objective dynamic optimization algorithm provided in this embodiment of the invention on the ZDT1 function;

[0099] Figure 10 This describes the convergence of the multi-objective dynamic optimization algorithm provided in this embodiment of the invention on the ZDT2 function;

[0100] Figure 11 This is the convergence result of the multi-objective dynamic optimization algorithm provided in this embodiment of the invention on the ZDT3 function. Detailed Implementation

[0101] 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.

[0102] The Animated Oat Optimization Algorithm (Animated Oat Optimization Algorithm for short) is a novel metaheuristic algorithm whose computational model differs from traditional bio-inspired algorithms. Inspired by the alternating rolling and ejection motion of wild oat seeds in a hygroscopic medium, it establishes a two-layer collaborative computational architecture: global search and local optimization. This design organically unifies exploratory optimization and precise convergence mechanisms within the same operational framework through biomimetic principles. The algorithm comprises four stages: initialization, biological parameter mapping, exploration, and development.

[0103] The resource-constrained project scheduling problem is a typical NP-hard problem in combinatorial optimization. It aims to allocate execution time for project activities under time and resource constraints to achieve specific objectives, such as minimizing completion time, balancing resource usage, or maximizing net present value. Its core constraints include time constraints, resource constraints, and the objective function.

[0104] Timing constraints are activity dependencies, requiring that subsequent tasks can only begin after the preceding tasks have been completed. For example... Figure 1As shown, the activity network on the arrows typically models these temporal relationships. Nodes represent events (1, 2, 3, 4, ... P), while arrows represent activities, with direction indicating priority. An activity executes only when all its predecessor events have completed. Virtual events 0 and P+1 represent the start and end states of the project, respectively; start event 0 has no predecessor events, and end event P+1 has no successor events. The total project duration is equal to the earliest start time of the only virtual activity Q+1 associated with end event P+1.

[0105] Resource constraints mean that resource consumption in each activity cycle should not exceed available capacity. Furthermore, for activities between events, there are the following four relationships:

[0106] SS: Subsequent events can only begin after the previous event has started;

[0107] FF: Subsequent events can only be completed after the previous event has been completed;

[0108] FS: Subsequent events can only begin after the previous event has been completed;

[0109] SF: Subsequent events can only be completed after the preceding event has begun.

[0110] The main objective of the objective function is to minimize the completion time (i.e., the longest completion time of all activities).

[0111] Existing dynamic optimization algorithms are single-objective algorithms and do not support multi-objective optimization. Although some scholars have improved classic single-objective algorithms into multi-objective algorithms (such as multi-objective particle swarm optimization), they suffer from slow convergence under resource constraints. Regarding the limitations of resource-constrained project scheduling problem models, resource fluctuation objectives are generally not considered, and time-cost trade-off models ignore real-time dynamic adjustments. In terms of engineering applicability, most existing tools rely on heuristic rules and lack the ability to generate Pareto fronts. Therefore, the multi-objective dynamic optimization algorithm proposed in this invention combines the biological mechanisms of dynamic optimization algorithms and utilizes a dynamic elite library to resolve multi-objective conflicts.

[0112] This invention proposes a construction project scheduling method based on multi-objective dynamic optimization. The core of this method is to improve and extend the dynamic optimization algorithm, forming a multi-objective dynamic optimization algorithm, or simply multi-objective dynamic optimization algorithm.

[0113] The multi-objective dynamic optimization algorithm of this invention adapts the dynamic optimization algorithm into a multi-objective optimization framework. It handles conflicting objectives through non-dominated sorting and a dynamic elite solution retention strategy. The algorithm stores Pareto optimal solutions in a dynamic external archive and uses a crowding distance mechanism to ensure the diversity of the solution set. In each iteration, the current population and elite solution set are merged, hierarchically sorted using fast non-dominated sorting, and filtered based on crowding distance to retain high-quality solutions. This framework effectively balances the synergistic optimization of completion time, cost, and resource fluctuations, thereby overcoming the limitations of single-objective algorithms in generating Pareto fronts.

[0114] A construction project scheduling method based on multi-objective dynamic optimization includes the following steps:

[0115] Step S1, problem modeling, which includes objective function and constraints.

[0116] The problem model in this invention is a resource-constrained project scheduling model with continuous resource allocation and limited resource availability. Resource fluctuation penalties are calibrated based on contractor surveys assessing overtime productivity losses. To compress the schedule and minimize total construction time, optimized planning is needed while meeting resource constraints. The specific optimization principles are as follows: Starting from the earliest start time of all activities, the resource usage of each activity is verified one by one. When resource conflicts occur, it is assumed that the start time of the corresponding activity will be delayed by one day to eliminate resource over-allocation. This daily incremental adjustment conforms to industry practice of daily resource reallocation in construction scheduling, minimizing disruptions while ensuring resource utilization for each shift. This process is repeated until resource constraints are met for all periods, ultimately yielding the project duration. Regarding cost control, project cost is directly related to benefits. To maximize benefits, time-related objectives, cost-related objectives, and resource fluctuations must be considered simultaneously. This three-objective problem must adhere to the following rules:

[0117] The logical relationship between tasks and task durations remains unchanged in network planning;

[0118] The resource intensity for each task remains unchanged, and resources are allocated rationally;

[0119] The activities should be kept continuous and interruptions are strictly prohibited.

[0120] There are a total of N activities, and each activity i (1≤i≤N) has two decision variables s. i and w i The decision variables are defined as follows:

[0121]

[0122] Among them, s i w represents the start time of activity i.i This represents the allocation of labor for activity i.

[0123] This invention addresses three optimization objectives: schedule compression, cost control, and resource fluctuation. The corresponding objective functions need to minimize the total project duration, the total cost, and resource fluctuation, as detailed below:

[0124] Minimize total duration objective: Total duration is defined as the maximum of the completion times of all activities, i.e. , where d i This indicates the duration of activity i.

[0125] Minimizing Total Cost Objective: The total cost of a project comprises two parts: direct costs and indirect costs. Generally, there is an inverse relationship between activity duration and direct costs, as shortening duration requires increasing the amount of machinery and labor allocated to the relevant activities, thereby increasing the project's direct costs. Conversely, direct costs decrease as the project duration increases. However, under certain conditions, increasing direct costs may not directly achieve duration compression. When duration compression reaches a threshold constrained by work surface limitations or project structural characteristics, additional expenditures cannot further increase resource input to achieve the time objective. To simplify this complex duration-related constraint and align with the human resource component of the resource fluctuation minimization objective, this invention quantifies total cost as the sum of human resource expenditure (i.e., variable costs, including basic labor costs and additional labor costs) and fixed administrative costs for each task (defined as the sum of direct and indirect costs, with each task set to the same fixed administrative cost). This model balances computational flexibility with real-world cost drivers, and the total cost comprises three elements: basic labor costs, additional labor costs, and total fixed costs.

[0126]

[0127]

[0128]

[0129] Among them, C l c represents the basic labor cost. l C represents the unit labor cost. el This represents the additional labor cost, where the number of workers allocated to each activity exceeds a threshold w. threshold When applying a penalty coefficient β, C to surplus labor... f This represents the total fixed cost. This represents the fixed administrative cost for a single task. Therefore, the total cost is expressed as:

[0130] .

[0131] Minimize resource fluctuation objective: Resource fluctuation is defined as the variance of the total labor force. The total labor force on day t is calculated as follows: ,in, The total number of days the project lasts, if and only if hour ,otherwise , This indicates rounding up to the nearest integer.

[0132] Resource fluctuations are defined as follows: ,in, , representing the average daily labor force.

[0133] The constraints are as follows: There are generally three constraints in modeling construction organization design problems: task dependency, resource limit, and variable limit.

[0134] Task dependencies are defined as follows: For any two tasks (i.e., activities in the sequence constraint graph) i and j, if task j should be completed before task i starts, then a completion-start (FS) dependency exists.

[0135]

[0136] in, This indicates the start time of task j. The duration of task j is represented by the adjacency matrix, and the dependencies between tasks are represented by the adjacency matrix. It means that, among them, This means that task i should only begin after task j has been completed. If If so, then task i and task j are not adjacent.

[0137] The resource ceiling is that the total labor force should not exceed a fixed constant W. .

[0138] The variable bound is specifically: the start time s of task i. i Has an upper limit ub s and lower limit lb s The size of the workforce w allocated to task i i Has an upper limit ub w and lower limit lb w In the objective function, all violations of constraints are combined into a penalty term: for violations of dependencies, the degree of violation (i.e., the time deviation between the actual start time and the completion time of its predecessor task) is multiplied by a penalty coefficient α and added to the total completion time target; for violations of resource constraints, the daily excess amount ub for exceeding the limit is added. w Multiply by a penalty coefficient α and add to the total cost objective. Therefore, the penalty objective function is expressed as:

[0139]

[0140]

[0141] Where n represents the same number as N, i.e., the total number of tasks.

[0142] This multi-objective optimization problem can be formulated as follows:

[0143]

[0144] The above constitutes the complete mathematical model of the problem. The model assumes stable resource pricing and excludes external disturbances (e.g., environmental changes, supply chain risks), and is applicable to projects with predictable labor and fixed task management costs.

[0145] Step S2: Construct a multi-objective dynamic optimization algorithm.

[0146] The process of the multi-objective dynamic microphone optimization algorithm constructed in this invention is as follows: Figure 2 As shown, the algorithm first needs to initialize the population, parameters, non-dominated sorting, elite pool, and constraints. Then, it checks the current iteration number. If the current iteration number is less than the maximum iteration number, it updates the algorithm parameters, calculates the objective function, and then implements an improved strategy. The improved strategy includes a dynamic elite solution retention and memory mechanism, an improved archive update strategy, a convergence monitoring and early termination mechanism, and an improved constraint handling mechanism. When the convergence stalls for more than 20 generations, the algorithm is terminated early to obtain the optimal Pareto front and output the Pareto solution set.

[0147] The dynamic elite solution retention and memory mechanisms in the improved strategy specifically include:

[0148] The multi-objective Oat Optimization algorithm of this invention reconstructs the position update strategy of the Animated Oat Optimization algorithm (AOO) by introducing an Elite Reference Point (ERP) and a memory mechanism (elite pool). This framework utilizes Pareto front solutions obtained through non-dominated sorting to guide the evolutionary direction of the population, ensuring convergence to the high-quality solution space. The current Pareto optimal solution is stored by applying fast non-dominated sorting and crowding distance filtering to each generation of the population. Simultaneously, the memory mechanism preserves historical elite solutions across generations through the elite pool to prevent the loss of high-quality solutions. Then, based on the elite solutions and combined with a physics-inspired update formula, new solutions are generated.

[0149] The memory mechanism operates as follows: In each iteration, for each individual in the population, the algorithm selects an elite solution with a set probability from the elite pool to directly replace the individual's current position. In this embodiment, the set probability is 10%. This mechanism is beneficial to the preservation of population diversity and introduces superior genetic information from the evolutionary process.

[0150] The dynamic elite solution retention mechanism is as follows: if the memory mechanism is not triggered, one of the following two update strategies is randomly selected with a 50% probability:

[0151] The first strategy is based on random perturbation updates: a number is randomly selected from [0,1]. When the random number exceeds 0.5, the strategy is activated. Its implementation depends on the individual's position in the population and can be divided into three cases:

[0152] The first case is as follows: If the index of an individual in the population is an integer multiple of one-tenth of the population size, then the position update formula and the method for calculating the random perturbation vector are given as follows:

[0153]

[0154]

[0155] in, This represents the updated position of the i-th individual, where i∈[0,N]. The mean position of the current population is represented by the built-in MATLAB function mean(X), calculated using the following formula: , This indicates the position of the i-th individual in the current population during the last iteration. Let represent the random perturbation vector, and c represent an adaptive parameter that gradually decreases with the number of iterations, where D represents the number of decision variables. This indicates the upper bound of the decision variable.

[0156] The second scenario is: if the current individual index modulo 10 has a remainder of 1, the position update formula is:

[0157]

[0158] in, The first solution in the elite reference point is selected according to the following rules: prioritize solutions at the top of the non-dominated hierarchy, and if they are in the same hierarchy, choose solutions with a large crowding distance.

[0159] The third case is: In all other cases, the position update formula is: When i is 1, The initialization location.

[0160] The second strategy is an update based on physics-inspired rules: A number is randomly selected from [0,1]. When the random number does not exceed 0.5, there is a 50% probability that one of the following two physics-inspired update methods will be chosen:

[0161] The first physics-inspired update method is: ,in, It is a random vector that follows a Lévy distribution, which introduces a step-size-based random perturbation; Let be an adaptive parameter that decreases as the number of iterations increases, where , It is a random vector. The calculation method is as follows:

[0162]

[0163] Among them, unifrnd is a function in MATLAB that can generate random arrays that follow a continuous uniform distribution. For the quality of oat seeds, This represents the deflection coefficient during the rotation of oat seeds. The length of the primary awn of oats. For dynamically adjusted boundary vectors: ,in This indicates the maximum number of iterations.

[0164] The second physics-inspired update method is: ,in, It is a random vector based on the physical pendulum model, defined as follows:

[0165]

[0166] in, The elastic coefficient of the main awn of oat seeds; This represents the current position of individual i. The angle between the seed trajectory and the ground; It is the gravitational acceleration constant; For an adaptive parameter, , Represents an integer randomly selected from [0, current iteration number]; For another dynamically adjusted boundary vector: .

[0167] The improved archive update strategy specifically includes:

[0168] As a Pareto front-based multi-objective optimizer, the multi-objective dynamic optimization algorithm strictly relies on its external archive to store non-dominated solutions. Archive maintenance involves two core steps: first, sorting all solutions to the non-dominated front (complexity O(kN²), where k is the number of objectives); second, within each front, retaining solutions with larger crowding distances (measuring neighbor sparsity) to ensure consistency along the Pareto boundary. This dual mechanism guarantees convergence to the true Pareto front while maintaining solution diversity.

[0169] This invention proposes an enhanced archive update strategy to maintain the non-dominated solution set. When a new solution is added to the archive, this strategy considers not only the non-dominated relationship but also controls the archive size to ensure consistent solution distribution. Specifically, when the archive reaches its capacity, the algorithm prioritizes retaining non-dominated solutions; within the same non-dominated front, solutions with smaller crowding distances are eliminated through a screening process. Based on the NSGA-II (a multi-objective genetic algorithm) framework, this strategy introduces a dynamic merging mechanism and an efficient pruning algorithm, significantly improving the convergence and distribution of the solution set. Therefore, it ensures the diversity and wide distribution of solutions in the archive, thus better approximating the entire Pareto front. The strategy includes the following three steps:

[0170] Step 1: Dynamically unpack and merge sets, then archive. and the current population Merged into candidate solution set middle: ;

[0171] in, This represents the candidate solution set for the current decision variable. Indicates the current save file size. This represents all archives from the initial state to the current state corresponding to the current decision variable. This represents the population corresponding to the current decision variable. This indicates that two matrices and Perform vertical splicing. This represents the solution set of the current objective function. This indicates the current objective function archive. This represents the population corresponding to the current objective function.

[0172] Step 2: The dual dominance detection algorithm is used to identify non-dominated solutions, thereby achieving rapid non-dominated screening. The dominance relationship is defined as follows: for each candidate solution in the solution set... traverse all other solutions To check Whether by What is controlled is represented as:

[0173]

[0174] in, , It is a dominance relation judgment function used to determine the solution. Is the solution dominant? , Indicates the first The solution is at the th solution. The value of the objective function Indicates the first The solution is at the th solution. The value of the objective function It refers to the number of optimization targets.

[0175] Step 3, Crowd Distance Pruning: When the number of non-dominated solutions exceeds the archive size, crowd distance pruning is performed to maintain diversity. The crowd distance of all solutions is initialized to zero. For each objective dimension, all solutions are sorted according to their fitness values. Boundary solutions (those with the minimum and maximum fitness values) are assigned infinite crowd distances to ensure their preservation. Intermediate solutions are assigned a crowd distance equal to the normalized Euclidean distance between adjacent solutions in the sorted list. Finally, the solution with the largest crowd distance is selected until the archive capacity is reached. For each objective... The solutions within the current non-dominated front are sorted in ascending order of fitness value. The distance difference between adjacent solutions is calculated and normalized to eliminate the influence of dimensionality. This operation is repeated for each objective, and the results are accumulated. , The normalized sum of the neighbor distances between all objectives in the solution is called the crowding distance.

[0176]

[0177]

[0178] in, This represents the value of the (i+1)th solution on the m-th objective function. This indicates that in the current non-dominated layer, the first... The maximum value of each objective function. This indicates that in the current non-dominated layer, the first... The minimum value of an objective function. This represents the crowding distance of the solution that minimizes the current objective function value. This represents the crowding distance of the solution with the largest current objective function value.

[0179] Furthermore, to address the issue of inconsistent target dimensions, a relative scaling factor ϵ is introduced into the denominator during the normalization operation, thereby avoiding division by zero errors and enhancing the algorithm's robustness. Through pre-sorting, the solution set is independently sorted for each target; when adding a new solution or removing an existing one, only the dominance state and crowding distance of the affected solution are updated using incremental calculations, thus reducing the algorithm's computational complexity.

[0180] The convergence monitoring and early termination mechanism specifically includes:

[0181] To improve computational efficiency, the multi-objective dynamic optimization algorithm incorporates a dual-threshold convergence monitoring system. This system automatically terminates the iteration when a steady state is reached by evaluating the improvement of the objective function in real time. Specifically, if the improvement of all objective functions remains below a preset threshold for several consecutive generations, the algorithm is considered converged and the iteration process is terminated early. This mechanism avoids unnecessary iterations, saves computational resources, and ensures the quality of the solution. The dual-threshold convergence monitoring system includes two core strategies:

[0182] Strategy 1:

[0183] Monitor the improvement progress to quantify the optimization progress of each generation of the optimal solution, and calculate the... The first generation of all individuals in the population Objective function (Objective function corresponding) , , The minimum fitness value (i.e., the optimal value in the current generation):

[0184] , Represents the i-th individual The function value on the k-th objective function, 1≤i≤N.

[0185] The formula for calculating the improvement magnitude is as follows:

[0186]

[0187] in, Let M represent the fitness value of the population in generation t, and M represent the number of objective functions.

[0188] Strategy Two:

[0189] A stagnation counter is introduced to track algebras that show no significant improvement. By introducing an adaptive threshold tailored to the problem characteristics, invalid computations are reduced while ensuring the quality of the solution. The stagnation criterion is defined as follows:

[0190]

[0191] in, Indicates the first All individuals in the first generation of the population The optimal value of each objective function relative to the improvement of the previous generation. This represents a pre-set positive threshold, indicating a critical value for significant improvement. When the improvement is less than this threshold, it is considered that there is no significant improvement.

[0192] At the same time, a maximum allowed number of stagnant algebras is set as the condition for triggering termination.

[0193] The improved constraint handling mechanism specifically includes:

[0194] To address the complex constraints in resource-constrained project scheduling, this invention proposes a three-layer constraint handling mechanism. This mechanism not only handles conventional boundary constraints but also incorporates specific repair strategies for task dependency and resource constraints. The constraint handling mechanism includes boundary truncation, temporal constraint repair, and resource constraint repair.

[0195] Boundary truncation: Vectorized operations are used to handle variable boundaries, represented as:

[0196]

[0197] in, It is a vector of decision variables, representing the current position of an individual (a set of solutions). Indicates the upper bound of a variable. Indicates the lower bound of a variable. Indicates element-wise multiplication. Represents the logical NOT operation. Represents a logical OR operation.

[0198] Timing constraint repair: The forward propagation repair strategy ensures the start time of the repaired task. This will not affect subsequent tasks, thus implicitly handling chained task dependencies (e.g., task i should start after task j is completed):

[0199]

[0200] in, This represents a new timing constraint. This indicates the old timing constraints.

[0201] Resource constraint repair: Associated with the resource penalty term in the objective function, the repair strategy is defined as follows:

[0202]

[0203] in, This indicates new resource constraints. This represents the old resource constraints. This indicates the maximum available amount of resources.

[0204] A multi-objective-guided position update strategy reconstructs the traditional single-objective optimization update mechanism through a hybrid framework based on dynamic elite solutions. This strategy integrates multi-objective optimization information into a physically inspired model and uses the Pareto front solution obtained through non-dominated sorting as a benchmark, injecting a memory mechanism to achieve cross-generational reuse of historical elite solutions. This achieves a complementary fusion of stochastic perturbations and the physical model through multi-strategy collaboration.

[0205] Step S3: Time complexity analysis of the multi-objective dynamic optimization algorithm.

[0206] The time complexity of each iteration of the multi-objective dynamic optimization algorithm is mainly determined by the following three operations:

[0207] Non-dominated sorting: For k objectives and N solutions, all solutions in the population and archive need to be compared pairwise to determine the dominance relationship. Each comparison needs to check all k objective functions to determine whether one solution dominates another. Therefore, for N solutions, a total of about N² comparisons are required, with each comparison involving k objectives. Thus, the time complexity of non-dominated sorting is O(kN²).

[0208] Archive maintenance: Sort by crowd distance. Since it is based on merge sort, the average time complexity of archive maintenance is O(n log n). ;

[0209] Position Update: The position update of each individual (N in total) involves D decision variables. The update operation includes physical model calculations, random perturbations, etc. Each variable requires constant time operation. Therefore, the total complexity is proportional to the product of N and D. The time complexity of position update is O(N). .

[0210] Therefore, the time complexity of each iteration of the multi-objective dynamic optimization algorithm is O(n). .

[0211] Step S4: The constructed multi-objective dynamic optimization algorithm is used to solve the modeled construction project problem to obtain the construction project scheduling scheme.

[0212] This embodiment uses the construction project problem model constructed in step S1 and the multi-objective dynamic optimization algorithm constructed in step S2 to verify the engineering feasibility of a construction case of a small residential project in Chengdu, thereby obtaining a construction project scheduling plan.

[0213] The experimental environment is shown in Table 1. To achieve a balance between computational manageability and real-world relevance, the core construction process was abstracted into 10 key procedures, as shown in Table 2. The duration and dependencies of each procedure conform to the national construction engineering progress standard GB / T50901-2013, and resource constraints are determined based on the actual manpower allocation records provided by a construction company in Sichuan Province undertaking the project.

[0214] Table 1. Experimental Hardware and Software Environment:

[0215]

[0216] Table 2 Project Configuration Table:

[0217]

[0218] The reasons for choosing this case are as follows: its moderate complexity (including 10 processes) can fully reflect the trade-offs between multiple objectives and does not oversimplify the actual scenario; this case focuses on the typical challenges of resource-constrained urban projects (such as tight schedules and a shortage of skilled workers).

[0219] The decision variable vector of the model in this invention is set to 20 dimensions, including the start time of 10 processes and the number of workers assigned to each process. The population size is set to 150, and the archive size is set to 200. To ensure the rationality of the optimization results and the full utilization of human resources, the start time of each process is constrained to the interval [0, 500] days, and the number of workers is constrained to the interval [10, 50].

[0220] The objective function is configured based on the three-objective optimization model proposed in this invention. For this model, the basic labor cost per worker is set at $200, the fixed cost per construction step is $50,000, the project delay penalty coefficient α is 30,000, the excess labor cost coefficient β is 100, and the worker number threshold w is set. threshold The weighting coefficients for the time-cost-resource fluctuation trade-off in the compromise solution are set to 0.4, 0.4, and 0.2, respectively. The simulation results after 25, 50, and 100 iterations are shown in Table 3.

[0221] Table 3 Simulation Experiment Results:

[0222]

[0223] The main optimization objectives are highlighted in bold in the table. Since the resource fluctuation objective has a relatively low weight in this problem, a separate minimum resource fluctuation solution is not listed in the table, but its optimization result is reflected in the compromise solution. After 25 iterations, the optimization effect is significant, and the result is very close to the final objective. With the increase in the number of iterations, the main optimization objectives in the non-dominated solution are further improved, and some secondary objectives are also improved.

[0224] In this engineering case, the proposed multi-objective dynamic optimization algorithm MOAOO achieved incremental optimization for all solution types after 100 iterations: the shortest duration solution reduced the project duration by 18.2% while keeping the cost below $6.12 million; the lowest cost solution maintained a competitive cost level throughout, even though the duration was extended to 685 days; and the compromise solution demonstrated ideal balance, with a stable duration of 279 days, a cost of $1.34 million, and a 72.7% reduction in resource fluctuations. This proves its practical advantages in real-world scheduling, as extreme solutions often require significant compromises in terms of time or budget.

[0225] The algorithm’s outstanding performance in engineering cases is due to its innovative multi-objective framework: it effectively balances three conflicting objectives through dynamic elite archives and non-dominated sorting, and combines a three-layer constraint processing mechanism to ensure the feasibility of the generated schedule.

[0226] In this embodiment, the convergence criterion is defined as follows: when the improvement of the main optimization objective of each solution is less than 1% for 20 consecutive generations, the algorithm converges on average in 118 generations after 30 independent runs. Figures 3-8 The simulation results of this embodiment at generation 100 are shown. Figure 3 The distribution of the Pareto front in the two-dimensional target space at the 100th generation is presented. Figure 4 This demonstrates the relationship between the Pareto front and the current population distribution in the three-dimensional target space. Figures 5 to 7 The construction task initiation plan and worker scheduling plan were presented under three scenarios: lowest cost, shortest construction period, and minimum resource fluctuation. Figure 8 The paper presents a compromise solution for both the construction task initiation plan and the worker scheduling plan. In actual engineering practice, decision-makers have adopted the compromise solution as the benchmark schedule. This is a mature strategy that has been verified by decades of field observation, and subsequent optimizations (such as robustness enhancement, safety supervision optimization, or multi-standard trade-off analysis) are systematically implemented on this basis.

[0227] The outstanding performance of the multi-objective dynamic optimization algorithm in engineering cases directly verifies the effectiveness of its innovative framework. The algorithm's novel hybrid position update rule enables it to efficiently explore complex solution spaces, while its innovative three-layer constraint handling mechanism ensures the feasibility of all generated solutions. Notably, the multi-objective dynamic optimization algorithm achieves a 72.7% reduction in resource fluctuations while maintaining a balance between project duration and cost objectives. This fully demonstrates the practical application value of the multi-objective extension of the dynamic optimization algorithm.

[0228] To verify the optimization performance of the multi-objective optimization algorithm in multi-objective problems, three functions (ZDT1, ZDT2, and ZDT3) were selected from the ZDT benchmark problem set for testing. All problems in this set are bi-objective optimization problems. ZDT1 has 30 decision variables and a convex Pareto front, used to verify the algorithm's ability to approximate the global Pareto front, achieve a uniform distribution of solutions, and maintain stability under high-dimensional variables. ZDT2 also has 30 decision variables, but its Pareto front is non-convex, specifically used to test the algorithm's ability to handle concave fronts, avoid solution gaps in the front center region, and escape flat regions. ZDT3 also has 30 decision variables, and its Pareto front is discontinuous, setting local optimum traps through discrete segments. This function is used to verify the algorithm's ability to find all discontinuous front segments, capture discontinuous edges, and ensure a balanced distribution of solutions among segments. The population size was set to 400, the file size to 200, the maximum number of iterations per function to 300, the problem dimension to 30, the variable boundary to [0,1], and the number of objective functions to 2. Under the above parameter configuration, the convergence of the multi-objective dynamic microphone optimization algorithm on these three functions is as follows: Figures 9-11 As shown, the black dots represent the Pareto fronts obtained by the multi-objective dynamic optimization algorithm, and the red lines represent the true Pareto fronts. It can be observed that on all three test functions, the Pareto fronts obtained by the multi-objective dynamic optimization algorithm largely coincide with the true Pareto fronts, effectively verifying the optimization performance of the algorithm on different problems.

[0229] The construction project scheduling method based on multi-objective dynamic optimization provided by this invention is mainly applied to the project planning and resource scheduling stages in fields such as construction engineering and civil engineering. It is used to collaboratively optimize the stability of project duration, total cost and resource usage under the premise of meeting resource constraints and task logic relationships, realize the automation and intelligent optimization of the project scheduling process, help improve project management efficiency and scientificity, and reduce the risk of project delays and cost overruns.

[0230] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A construction project scheduling method based on multi-objective dynamic optimization, characterized in that, The method includes: Step S1, Problem modeling of the construction project. Problem modeling includes optimization objectives and constraints. Optimization objectives include schedule compression, cost control and resource fluctuation. Constraints include task dependencies, resource limits and variable limits. Step S2: Construct a multi-objective dynamic optimization algorithm. The improvement strategies of the multi-objective dynamic optimization algorithm include a dynamic elite solution retention mechanism and memory mechanism, an improved archive update strategy, a convergence monitoring and early termination mechanism, and an improved constraint handling mechanism. Step S3, Time complexity analysis of the multi-objective dynamic optimization algorithm, the time complexity of each iteration includes the time complexity of non-dominated sorting, the time complexity of file maintenance and the time complexity of position update; Step S4: Use the multi-objective dynamic optimization algorithm to solve the construction project problem and obtain the construction project scheduling scheme; In step S1, there are a total of N activities, and each activity i has two decision variables. and The decision variables are defined as follows: ； in, Indicates the start time of activity i. This represents the labor allocation for activity i, and N represents the number of activities. Represents the set of real numbers; The objective functions corresponding to schedule compression, cost control, and resource fluctuation in the optimization objectives need to minimize the total project duration objective, the total cost objective, and the resource fluctuation objective, respectively. Minimizing the total project duration objective is equivalent to minimizing the total duration, which is defined as the maximum value among the completion times of all activities. , where d i Indicates the duration of activity i; The total cost of a project includes direct and indirect costs. It is quantified as the sum of human resource expenditures and fixed management fees for each task. Human resource expenditures include basic labor costs and additional labor costs. The total cost is defined by three elements: basic labor costs, additional labor costs, and total fixed costs, as shown below: ； ； ； Among them, C l c represents the basic labor cost. l C represents the unit labor cost. el This represents the additional labor cost when the number of workers assigned to each activity exceeds a threshold w. threshold When applying a penalty coefficient β, C to surplus labor... f Represents total fixed costs. This represents the fixed administrative cost for a single task; therefore, the total cost is expressed as: ； Resource fluctuations are defined as the variance of the total labor force. The total labor force on day t is calculated as follows: ,in, The total number of days the project lasts, if and only if hour ,otherwise , Indicates rounding up; The resource fluctuation target is then defined as: ,in, , representing the average daily labor force; The improved constraint handling mechanism in step S2 includes boundary truncation, temporal constraint repair, and resource constraint repair. Boundary truncation: Vectorized operations are used to handle variable boundaries, represented as: ； in, It is a vector of decision variables, representing the current position of the individual. Indicates the upper bound of a variable. Indicates the lower bound of a variable. Indicates element-wise multiplication. Represents the logical NOT operation. Represents a logical OR operation; Timing constraint repair: The forward propagation repair strategy ensures the start time of the repaired task. This will not affect subsequent tasks, thus implicitly handling chained task dependencies: ； in, This represents a new timing constraint. This indicates the old timing constraints; Resource constraint repair: Associated with the resource penalty term in the objective function, the repair strategy is defined as follows: ； in, This indicates new resource constraints. This represents the old resource constraints. This indicates the maximum available amount of resources.

2. The construction project scheduling method based on multi-objective dynamic optimization according to claim 1, characterized in that, The specific constraints in step S1 are as follows: Task dependencies are defined as follows: For any two tasks, i.e., activities i and j in the sequence constraint graph, if task j completes before task i starts, then a completion-start FS dependency exists. ； in, This indicates the start time of task j. The duration of task j is represented by the adjacency matrix, and the dependencies between tasks are represented by the adjacency matrix. It means that, among them, This means that task i should only begin after task j has been completed. If so, then task i and task j are not adjacent; The resource ceiling is set at a fixed constant W, meaning the total labor force does not exceed this constant. ; The variable bound is: the start time s of task i. i Has an upper limit ub s and lower limit lb s The size of the workforce w allocated to task i i Has an upper limit ub w and lower limit lb w In the objective function, all violations of constraints are combined into a penalty term: for dependency violations, the violation severity is multiplied by a penalty coefficient α and added to the total completion time objective; for resource limit violations, the daily excess amount ub is applied. w Multiplying by a penalty coefficient α and adding it to the total cost objective, the penalty objective function is expressed as: ； ； Where n represents the same number as N, i.e., the total number of tasks; Therefore, the multi-objective optimization problem of construction engineering projects can be expressed as: 。 3. The construction project scheduling method based on multi-objective dynamic optimization according to claim 1, characterized in that, The memory mechanism in step S2 is as follows: In each iteration, for each individual in the population, the algorithm selects an elite solution with a set probability from the elite pool to directly replace the individual's current position. The dynamic elite solution retention mechanism in step S2 is as follows: if the memory mechanism is not triggered, one of the following two update strategies is randomly selected with a 50% probability: The first strategy is based on random perturbation updates: a number is randomly selected from [0,1]. When the random number exceeds 0.5, the first strategy is activated. Its implementation depends on the individual's position in the population and can be divided into three cases: The first case is: if the index of an individual in the population is an integer multiple of one-tenth of the population size, then the position update formula and the calculation method for the random perturbation vector are as follows: ； ； in, This represents the updated position of the i-th individual, where i∈[0,N]. The mean position of the current population is represented by the built-in MATLAB function mean(X), calculated using the following formula: , This indicates the position of the i-th individual in the current population during the last iteration. Let represent the random perturbation vector, and c represent an adaptive parameter that gradually decreases with the number of iterations, where D represents the number of decision variables. This represents the upper bound of the decision variable. Indicates element-wise multiplication; The second scenario is: if the current individual index modulo 10 has a remainder of 1, the position update formula is: ,in, The first solution in the elite reference point is selected according to the following rules: prioritize the solution with the higher non-dominated level, and if they are in the same level, choose the solution with the larger crowding distance. The third case is: In all other cases, the position update formula is: When i is 1, The initialization location; The second strategy is an update based on physical heuristics: A number is randomly selected from [0,1]. When the random number does not exceed 0.5, there is a 50% probability of choosing one of the following two update methods based on physical heuristics: The first method for updating rules inspired by physics is: ,in, It is a random vector that follows a Lévy distribution, which introduces a step-size-based random perturbation; Let be an adaptive parameter that decreases as the number of iterations increases, where , It is a random vector. The calculation method is as follows: ； unifrnd is a function in MATLAB that generates random arrays following a continuous uniform distribution. For the quality of oat seeds, This represents the deflection coefficient during the rotation of oat seeds. The length of the primary awn of oats. For dynamically adjusted boundary vectors: ,in Indicates the maximum number of iterations; The second method for updating rules inspired by physics is: ,in, It is a random vector based on the physical pendulum model, defined as follows: ； in, The elastic coefficient of the main awn of oat seeds; This represents the current position of individual i. The angle between the seed trajectory and the ground; It is the gravitational acceleration constant; For an adaptive parameter, , Represents an integer randomly selected from [0, current iteration number]; For another dynamically adjusted boundary vector: .

4. The construction project scheduling method based on multi-objective dynamic optimization according to claim 1, characterized in that, The improved archive update strategy in step S2 specifically includes: Step 1: Dynamically unpack and merge sets, then archive. and the current population Merged into candidate solution set middle: ; in, This represents the candidate solution set for the current decision variable. Indicates the current save file size. This represents all archives from the initial state to the current state corresponding to the current decision variable. This represents the population corresponding to the current decision variable. This indicates that two matrices and Perform vertical splicing. This represents the solution set of the current objective function. This indicates the current objective function archive. This represents the population corresponding to the current objective function; Step 2: Use the dual dominance detection algorithm to identify non-dominated solutions. The dominance relationship is defined as follows: For each candidate solution in the solution set... traverse all other solutions To check Whether by What is controlled is represented as: ； in, , It is a dominance relation judgment function used to determine the solution. Is the solution dominant? , Indicates the first The solution is at the th solution. The value of the objective function Indicates the first The solution is at the th solution. The value of the objective function It is the number of optimization targets; Step 3, Crowded Distance Pruning: When the number of non-dominated solutions exceeds the file size, crowded distance pruning is performed for each objective. The solutions within the current non-dominated front are sorted in ascending order of fitness value. The distance difference between adjacent solutions is calculated and normalized. This operation is repeated for each objective, and the results are accumulated. , The normalized sum of the neighbor distances between all objectives in the solution, i.e., the crowding distance, is expressed as: ； ； in, This represents the value of the (i+1)th solution on the m-th objective function. This indicates that in the current non-dominated layer, the first... The maximum value of each objective function. This indicates that in the current non-dominated layer, the first... The minimum value of an objective function. This represents the crowding distance of the solution that minimizes the current objective function value. This represents the crowding distance of the solution with the largest current objective function value.

5. The construction project scheduling method based on multi-objective dynamic optimization according to claim 1, characterized in that, The convergence monitoring and early termination mechanism in step S2 includes two strategies: Strategy 1: Monitor the improvement progress to quantify the optimization progress of each generation of the optimal solution, and calculate the... The first generation of all individuals in the population Objective function Minimum fitness value: , Represents the i-th individual The function value on the k-th objective function, 1≤i≤N; The improvement magnitude is calculated as follows: ； in, Let M represent the fitness value of the population in generation t, and M represent the number of objective functions. Strategy 2: Introduce a stagnation counter to track algebras that show no significant improvement for consecutive periods. The stagnation criterion is defined as follows: ； in, Indicates the first All individuals in the first generation of the population The optimal value of each objective function relative to the improvement of the previous generation. This represents a pre-set positive threshold, indicating a critical value for significant improvement. When the improvement amount is less than this threshold... At that time, it was believed that there was no significant improvement; At the same time, a maximum allowed number of stagnant algebras is set as the condition for triggering termination.

6. The construction project scheduling method based on multi-objective dynamic optimization according to claim 1, characterized in that, The time complexity of each iteration of the multi-objective dynamic optimization algorithm is O(n). ,in, This represents the time complexity of position updates. The average time complexity of archive maintenance, is the time complexity of non-dominated sorting, k is the number of objective functions, N is the number of solutions, and D is the number of decision variables.

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