Method for optimizing repetitive items

Abstract

The application provides an optimization method of repetitive projects, defines the free float of processes in the repetitive projects under a linear scheduling framework, analyzes the float characteristics of divergent and convergent process pairs, and constructs a robust evaluation index suitable for the repetitive projects; an optimization model with the double objectives of minimizing the total cost of the projects and maximizing the scheduling robustness is established, and the model is converted into a single objective sub-problem by using an epsilon-constraint algorithm; a GA-CP hybrid algorithm combining a genetic algorithm and a constraint programming is designed for solving, the genetic algorithm is responsible for global search and decision variable dimension reduction, and the constraint programming is responsible for local fine optimization, so that a Pareto optimal solution set is efficiently obtained under the premise of meeting the duration constraint. The application can provide an economic and stable progress plan for the repetitive engineering projects, significantly improves the anti-interference ability of the scheduling scheme in an uncertain environment, and is suitable for the engineering fields of roads, tunnels, high-rise buildings, wind power and photovoltaic power.

Description

Technical Field

[0001] This invention relates to a method for optimizing the dual objectives of project cost and scheduling robustness in repetitive projects under uncertain environments. Background Technology

[0002] Repetitive projects refer to engineering projects where the workplace is divided into multiple units, and each process is repeatedly carried out within each unit. Examples include roads, tunnels, bridges, high-rise buildings, oil and gas pipelines, wind farms, and photovoltaic power stations. These projects mostly fall under infrastructure engineering, broadly involving transportation, energy, and other fields closely related to national economic and social development. Repetitive projects are typically characterized by large investment scales and long construction periods, making project planning and scheduling management particularly important.

[0003] In uncertain environments, project scheduling methods are mainly classified into three categories: stochastic project scheduling, fuzzy project scheduling, and robust project scheduling. Stochastic project scheduling is based on probability theory, assuming that uncertain parameters follow a known probability distribution and deriving the distribution function using historical data or statistical methods to model uncertainty. Fuzzy project scheduling, on the other hand, is based on fuzzy set theory, describing uncertain parameters through fuzzy membership functions to handle information with fuzziness or subjectivity. However, both methods have limitations in practical applications: stochastic scheduling relies on precise probability distributions, while fuzzy scheduling requires constructing reasonable membership functions. Both have high requirements for data quality and prior knowledge, high computational complexity, and limited applicability in scenarios where data is scarce or uncertainty is difficult to model accurately.

[0004] Chinese invention patent application No. 202511692467.1 discloses an intelligent matching system based on business operation features and key processes, comprising: a data acquisition module for collecting data, including historical work order data and multi-source business plan data, wherein the multi-source business plan data includes order attributes, equipment status, and process constraints; the historical work order data includes task steps, risk points, and control standards; a business feature construction module for extracting business feature vectors based on the multi-source business plan data using semantic analysis, wherein the business feature vectors include risk features, resource features, and temporal features of multiple single-line business plans; a process graph generation module for constructing a directed process graph based on historical work order data using the critical path method, and inputting the business feature vectors into the directed process graph to identify key processes; a work plan generation module for generating multi-objective work plans based on the key processes using temporal constraint programming and genetic algorithms, distributing the work plans to workers, and obtaining the execution progress of the work plans; and a monitoring module for using reinforcement learning algorithms to compare the execution progress with the work plans to determine the completion degree of the work plans. The above system did not use robustness metrics.

[0005] Chinese invention patent application No. 202410842502.2 discloses a multi-timescale operation optimization method for a hydro-solar-storage microgrid, comprising the following steps: First, establishing a day-ahead stochastic optimization method for a hydro-solar-storage microgrid based on uncertain scenarios. This method is executed by a computer and includes the following steps: Collecting historical data on hydropower and photovoltaic output and preprocessing the data; Establishing a generative adversarial network model for generating renewable energy scenarios, using the processed historical hydro-solar data and random noise as model inputs, with the network's generator outputting typical uncertain scenario values ​​and their probabilities for hydro-solar power; Based on typical scenarios of hydropower and photovoltaic output, and with the objective of minimizing the total scheduling cost of the hydro-solar-storage microgrid system, constructing a day-ahead stochastic chance-constrained optimization model for the hydro-solar-storage microgrid. The model constraints include power balance constraints, energy storage system constraints, electricity purchase and sale constraints, and tracking output constraints; The construction of the day-ahead stochastic chance-constrained optimization model for the hydro-solar-storage microgrid includes a genetic algorithm, which, based on a reliability evaluation function and an evolutionary strategy based on heuristics, outputs the day-ahead stochastic optimization scheduling results for the hydro-solar-storage microgrid. Based on the day-ahead stochastic optimization scheduling results of the aforementioned hydro-solar-storage microgrid, a multi-timescale operation optimization method for the hydro-solar-storage microgrid is generated, encompassing day-ahead, intraday, and real-time scales. Intraday rolling optimization includes: determining whether the day-ahead scheduling strategy can track power output within four hours based on ultra-short-term predicted values ​​of hydropower and photovoltaic power output; if not, constructing an intraday rolling optimization model with the objective of minimizing power variation compared to the day-ahead scheduling strategy. Model constraints include intraday power balance constraints, intraday energy storage system constraints, intraday power purchase and sale constraints, and intraday power tracking constraints. The day-ahead scheduling plan is adjusted using the aforementioned genetic algorithm to obtain the intraday scheduling plan. Finally, real-time rolling optimization of the hydro-solar-storage microgrid is performed. Based on the current measured photovoltaic and hydropower power and energy storage state of charge, it is determined whether the current energy storage operation strategy can track the reported power of hydro-solar-storage. If not, within the energy storage system operation constraints, an optimization model is constructed to minimize the difference between the combined hydro-solar-storage power output and the target power output, resulting in a real-time scheduling plan. The final real-time scheduling results of the hydro-solar-storage microgrid are then output. The above method does not employ robustness indicators.

[0006] In contrast, robust project scheduling draws on the robustness concept from control theory, focusing on assessing the project plan's ability to withstand uncertainties and generating scheduling schemes with strong anti-interference capabilities. This method does not rely on the probability distribution or fuzzy membership function of uncertain parameters, making it more widely applicable; its computational complexity is relatively low, making it suitable for large-scale or complex problems; the resulting solutions exhibit good robustness and stability, especially suitable for high-risk or high-uncertainty scenarios; and it has lower data requirements, showing a significant advantage in environments where data is difficult to obtain or uncertainty modeling is challenging. Therefore, robust project scheduling, as a fundamental scheduling strategy to address uncertainty, demonstrates significant research potential and is gradually becoming an important research direction in the field of project scheduling under uncertain environments. Summary of the Invention

[0007] In view of this, the present invention provides a method for solving problems based on a genetic algorithm-constraint programming (GA-CP) hybrid algorithm, which effectively solves the problems existing in the prior art mentioned above.

[0008] Specifically, this invention provides an optimization method for repetitive projects, comprising the following steps: Step 1: Constructing a robustness index for repetitive projects, wherein the robustness index is defined based on the time interval and free float of work pairs in the first and last units of a linear schedule; Step 2: Establishing a cost-robust bi-objective trade-off optimization model for repetitive projects, wherein the model aims to minimize the total project cost and maximize the project robustness, and satisfies constraints on project deadline, work priority, and work team number; Step 3: Using an ε-constraint algorithm, the bi-objective trade-off optimization model is transformed into a single-objective optimization sub-model with robustness as the objective and project cost as the constraint; Step 4: Using a GA-CP hybrid algorithm combining genetic algorithm and constraint programming to solve the single-objective optimization sub-model to obtain the optimal schedule.

[0009] According to the optimization method for repetitive projects of the present invention, preferably, the mathematical expression of the robustness index is: , in, n For the number of processes, For process i The set of subsequent processes, For process i With process j The instability coefficient between them Let be the utility function of the time interval. For process i With process j The free time difference between them.

[0010] According to the optimization method for repetitive items of the present invention, preferably, the utility function It is a piecewise linear function used to handle non-integer time intervals, defined as: , in, for x The decimal part.

[0011] According to the optimization method for repetitive projects of the present invention, preferably, the total project cost includes direct costs and indirect costs, wherein direct costs include labor costs, material costs and equipment costs, and indirect costs are proportional to the total project duration.

[0012] According to the optimization method for repetitive projects of the present invention, preferably, the ε-constraint algorithm is implemented as follows: the total project cost is... PC As a constraint, its upper and lower bounds are defined. By gradually adjusting the value of ε, a series of single-objective robustness maximization subproblems are solved, thereby constructing the Pareto front.

[0013] According to the optimization method for repetitive projects of the present invention, preferably, the solution process of the GA-CP hybrid algorithm includes: using a genetic algorithm to solve the model, running it multiple times to obtain multiple schedule plans and the number of work teams for each process; inputting the number of work teams as known parameters into the constraint programming solver, solving the remaining time buffer variable and the sub-process start time variable to obtain a feasible solution; comparing the robustness index of each feasible solution, and outputting the optimal schedule plan.

[0014] According to the optimization method for repetitive projects of the present invention, preferably, the genetic algorithm is used to handle the work team number variable among discrete decision variables, and the constraint programming is used to handle the time buffer variable and the sub-process start time variable.

[0015] According to the optimization method for repetitive projects of the present invention, preferably, the process priority relationship constraint is based on the trapezoidal motion rule, and only needs to be expressed at the first and last units, specifically as follows: , in, Each is a process i At the start time of the first and last units, For process i The unit construction period.

[0016] The optimization method for repetitive projects according to the present invention is preferably applicable to the following types of repetitive projects: roads, tunnels, bridges, high-rise buildings, oil and gas pipelines, wind farms, or photovoltaic power stations.

[0017] According to the optimization method for repetitive projects of the present invention, preferably, the solution tool of the GA-CP hybrid algorithm is modeled using the ILOG OPL language and calls CPLEX V12.8 as the constraint programming solver.

[0018] This invention provides an optimization method for repetitive projects. Under a linear scheduling framework, it defines the free float of processes within the repetitive project, analyzes the float characteristics of divergent and convergent process pairs, and constructs a robustness evaluation index suitable for repetitive projects. An optimization model with the dual objectives of minimizing total project cost and maximizing scheduling robustness is established, and the ε-constraint algorithm is used to transform it into a single-objective subproblem. A GA-CP hybrid algorithm combining genetic algorithm and constraint programming is designed for solving the problem. The genetic algorithm handles global search and dimensionality reduction of decision variables, while constraint programming handles local fine-grained optimization, efficiently obtaining the Pareto optimal solution set while satisfying schedule constraints. This invention can provide economical and stable schedule plans for repetitive engineering projects, significantly improving the anti-interference ability of scheduling schemes in uncertain environments, and is applicable to engineering fields such as roads, tunnels, high-rise buildings, and wind and solar power. Attached Figure Description

[0019] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings used in the description of the embodiments or the prior art will be discussed below. Obviously, the technical solutions described in conjunction with the accompanying drawings are only some embodiments of the present invention. For those skilled in the art, other embodiments and their accompanying drawings can be obtained based on the embodiments shown in these drawings without creative effort.

[0020] Figure 1 Examples of divergent and convergent process pairs given under the inventive concept are shown, wherein Figure 1 (a) and Figure 1 (b) shows different scenarios.

[0021] Figure 2 The diagram shows the time difference utility function in the CPM network according to the concept of this invention.

[0022] Figure 3 The flowchart of the solution process is shown, which combines the genetic algorithm and the constraint programming algorithm using the GA-CP hybrid algorithm under the concept of this invention.

[0023] Figure 4 A flowchart illustrating the solution of the bi-objective optimization problem using the constraint algorithm under the concept of this invention is shown.

[0024] Figure 5 The simulation results under different scenarios provided by the present invention are shown. Detailed Implementation

[0025] The technical solutions of various embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. Based on the embodiments described in 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.

[0026] Unless otherwise stated, the term "unit" and "sub-process" in this invention have the same meaning.

[0027] The specific concept of the present invention will be introduced below based on existing technology.

[0028] As mentioned in the previous article, existing technologies have many shortcomings in the field of scheduling repetitive projects.

[0029] First, most existing robustness metrics are based on the Critical Path Method (CPM) framework. However, the structure of a Linear Bypass (LOB) scheduling graph differs fundamentally from that of a CPM network graph, and the LOB scheduling method is more suitable for complex engineering environments. Therefore, it is necessary to redesign corresponding robustness optimization metrics and models based on the characteristics of LOB scheduling.

[0030] Secondly, existing literature has introduced robust scheduling theory into the scheduling of repetitive projects for the first time. These literatures have established scheduling models under the LOB framework, with the goal of minimizing the total number of work teams employed for all activities under a given time constraint, thereby improving the stability of the schedule. However, such models still have the following limitations: (1) the robustness metrics they use are still based on CPM networks and are not applicable to the LOB scheduling framework; (2) the model only considers solution robustness (i.e., the stability of the solution) and fails to achieve a trade-off between quality robustness and solution robustness; (3) the project cost is not optimized because a reduction in the number of work teams does not equate to a reduction in the total project cost.

[0031] Therefore, the technical concept of this invention aims to construct a multi-objective robust optimization model to achieve a multi-objective trade-off between project cost and robustness under deadline constraints. Simultaneously, this invention designs a genetic algorithm (GA) based on constrained programming (CP), namely the GA-CP algorithm, to solve the above model. This invention can provide economical and stable schedule planning for practitioners of repetitive projects and support their quantitative decision-making.

[0032] In accordance with the above-mentioned technical objectives and overall technical concept, the technical solutions adopted in this invention will be described in detail below.

[0033] The first step in this invention is to construct robustness indicators for repetitive projects, which will be described in detail below.

[0034] In constructing robustness metrics, the first step is to calculate the free float in LOB scheduling.

[0035] Specifically, in LOB (Linear Path) scheduling, since the same process is repeatedly executed in multiple units, the free float concept in the traditional CPM (Critical Path Method) network needs to be redefined to adapt to the characteristics of repetitive projects.

[0036] In CPM networks, free float refers to the maximum time a current process can be delayed without affecting the earliest start time of subsequent processes. However, in LOB scheduling, because processes are repeatedly executed across multiple units, free float needs to be redefined at the sub-process level. Specifically, the free float of a sub-process is the maximum time a current sub-process can be delayed without affecting the earliest start time of subsequent sub-processes (including sub-processes corresponding to the next work team within the same process, and subsequent sub-processes within the same unit).

[0037] Next, we will introduce the sub-operation relationships in LOB scheduling. In LOB scheduling, an operation is typically executed repeatedly by multiple work teams in different units. For sub-operations corresponding to different work teams under the same operation, since they are executed consecutively by the same work team, the time delays between them do not affect the overall schedule. Therefore, these sub-operations should not be included in the calculation of free float. In contrast, subsequent sub-operations within the same unit are the key objects for free float calculation. If a sub-operation is delayed, it may affect the start time of subsequent sub-operations within the same unit, thereby affecting the overall project schedule.

[0038] Subsequently, the free float is formally calculated. Directly applying the free float definition from the CPM network to LOB scheduling would significantly reduce or even eliminate the free float for most sub-processes, because the relationships between sub-processes in LOB scheduling are more complex. In this invention, the free float calculation under LOB scheduling should consider the location of the sub-process within its unit and its relationship with subsequent sub-processes within the same unit. Specifically, the free float calculation follows two principles: first, when calculating free float, only the earliest start time of subsequent sub-processes within the same unit needs to be considered; second, sub-processes corresponding to different work teams within the same process do not participate in the free float calculation because these sub-processes are executed continuously by the same work team and do not have mutual constraints.

[0039] In the LOB scheduling optimization process, this invention allows for the adjustment of the slope of processes that meet the conditions while keeping the slope of other processes unchanged. Specifically, this is achieved by increasing or decreasing the number of work teams in the corresponding processes. To ensure that the adjustment of the number of work teams in key processes does not interfere with the normal execution of other processes, the concept of free float in LOB scheduling has been proposed in this field. In LOB scheduling, the constraint relationship between process pairs is mainly divided into the following two types: (1) divergent type, such as Figure 1 (a) process i With process h As shown, its characteristic is that the time interval between the two processes in the first unit constitutes the priority relationship between them; (2) convergent type, such as Figure 1 (b) process i With process h As shown, its characteristic is that the time interval between the two processes at the last unit constitutes the priority relationship between them. Figure 1 (a) and Figure 1 (b) Examples of divergent and convergent process pairs are given respectively.

[0040] In this invention, the free float of a process in a LOB is defined as the time it takes for its sub-processes to complete to be delayed without affecting the number of subsequent sub-processes belonging to other processes. Based on this definition, in the process... i unit j Sub-process ( The free float of a twentieth ...

[0041] in, Indicate process i The collection of subsequent processes, Indicate process i The sub-process is not in The maximum amount that can be delayed in subsequent processes.

[0042] For the process i and its subsequent processes h , Indicates that they are in the unit j The time interval at the point. If the process... i and process h The last unit between them m time interval Greater than them in the first unit 1 time interval ,like Figure 1 As shown in (a), these two processes exhibit a divergent state. In this scenario, The value of can be calculated as follows:

[0043] Conversely, if process i and its subsequent processes h It exhibits a convergent state, such as Figure 1 As shown in (b). In this case, if its sub-process If a specific work team performs its last sub-process, then the time interval is equal to the corresponding time interval; otherwise, only a portion of the time interval is equal. To offset the effect of its delayed end time. At this time, The expression is:

[0044] In summary, as shown above, the execution process is as follows: i and its subsequent processes h The two scheduling diagrams represent the same total duration (e.g., 10). However, in these two different cases, the free float between the two divergent operation pairs ( ,like Figure 1 a) greater than the free time between two convergent process pairs ( ,like Figure 1 b). Therefore, when the process i When the end time of the last sub-process is delayed, Figure 1 Process A h The risk of delay in the start time of the sub-process is higher than Figure 1 The lower value in b.

[0045] The construction of robustness metrics will be further described below.

[0046] When performing LOB scheduling for engineering projects, the preceding analysis shows that the free float between processes increases with the time interval between the first and last units. Further research reveals that increasing the time interval at the last unit has a more significant impact on free float. This is because this operation generates more divergent process pairs, thus providing greater flexibility and robustness to the scheduling process. Based on these findings, this invention proposes a robustness index suitable for repetitive projects to quantify the robustness of scheduling schemes and provide theoretical decision support for project managers.

[0047] The robustness metric proposed in this invention comprehensively considers the following key factors: (1) Distribution of time intervals: focusing on the impact of the time interval between the first and last units on free float; (2) Generation of divergent work pairs: maximizing the number of divergent work pairs by increasing the time interval at the last unit, thereby improving scheduling flexibility; (3) Optimization of free float: optimizing free float by reasonably allocating time intervals under the premise of meeting project constraints. The robustness metric (Robustness Measure) RM The mathematical expression for ) is as follows:

[0048] In the above formula, n represents the number of processes; The time difference utility function is used to quantify the contribution of time intervals to scheduling robustness. Its specific form can be defined according to project requirements, such as a linear function, an exponential function, or other nonlinear functions. Indicate process i and its subsequent processes h The weighted instability coefficient of the time interval between processes is used to reflect the impact of the dynamic changes in the time interval between processes on scheduling stability. The larger the value, the more significant the impact of the time interval between processes on scheduling robustness.

[0049] To avoid unnecessary time buffering in certain processes, the time difference utility function in a CPM network is typically expressed exponentially. ,in Indicate process i The total amount of free time available, This is the attenuation coefficient, used to control the sensitivity of the utility function to free time difference. However, in practical applications, the time interval utility function cannot be simply used as the attenuation coefficient. and express.

[0050] For example, in some cases, It may not be an integer, in Figure 1 In (b), To effectively solve this problem, this paper will use the formula... and Transform into a piecewise linear function ,like Figure 2 As shown.

[0051] For every positive integer There is a corresponding utility value. Based on the properties of piecewise functions, the utility index for non-integer time intervals can be easily calculated. Assume... Correspondingly The utility function can be expressed as: .

[0052] in, Integer time interval The corresponding utility value, Integer time interval The corresponding utility value, Time interval The corresponding decimal part.

[0053] At this time, when Then 2 ≤ 7 / 3 ≤ 3. According to the definition of a piecewise function above, Further calculations show that... .

[0054] The following section will further introduce the establishment of the robust-cost trade-off optimization model in this invention.

[0055] Let set A ={1,2,...,n} represents the set of repetitive project processes. U ={1,2,...,m} represents the set of process units for repetitive items. d i This indicates the unit duration. k i This indicates each process. i ∈ A The maximum number of available work teams. Let there be a binary variable. x ik Indicate process i Whether to hire k A work team, when x ik =1 indicates the first i Employment for each process k A work team, otherwise x ik =0; continuous variable S i,j Indicate process i In unit j The start time, of which Represents the last process n In the last unit m The start time.

[0056] The first objective of the model proposed here is to minimize project costs. The costs, including direct and indirect costs of all processes, can be expressed as:

[0057] The first part represents the project's direct costs (DC), which consist of labor costs, material costs, and equipment costs; the second part represents the project's indirect costs (IC), which are usually related to the project's total duration and can be expressed as the project's total duration (…). S n,m + d n The product of the unit indirect cost (ICR) and the unit indirect cost (ICR). LC ik MC ik EC ik Each represents a process. i Hired k The labor, material, and equipment costs of each work team, and the process i Number of hired work teams k i Relevant. An increase in the number of work teams leads to an increase in the direct costs of the process. Therefore, LC ik MC ik EC ik The calculation formula is as follows:

[0058]

[0059]

[0060] Among them, coefficient These represent the growth rates of labor, material, and equipment costs when an additional work team is added, respectively. Indicate process i Workload in each unit. Indicate process i Material cost ratio, Indicate process i Labor cost rate Indicate process i Equipment cost ratio.

[0061] The model provided in this invention also needs to consider maximizing the robustness of the project. Robustness measures for typical repetitive projects have been proposed above. However, the formulas above cannot be used directly because of the time intervals. For unknown continuous variables, which may be decimals, the utility function... The computation is difficult. To solve this problem, the above formula can be restated as the following equivalent robustness optimization objective and its corresponding constraints:

[0062]

[0063]

[0064]

[0065] Among them, if the process i and its subsequent processes h The time interval of its first (or last) unit is greater than or equal to ( t-1 Timing, binary variables Equals 1 if it is equal to 1, otherwise equals 0; continuous variable These are intermediate variables used to calculate the process. i The time interval between the current process and its successor process in the first (or last) unit; parameters It is a positive integer representing the upper limit of the time interval.

[0066] To better understand the above expression, examples will be provided below. Assume... and In the above formula, it is made Substituting this result into the formula above, we can calculate: .

[0067] This perfectly matches the time difference utility function defined in the figure, verifying the correctness and effectiveness of the model.

[0068] Based on the trapezoidal motion rule, the process i and its subsequent processes h The priority relationship between them only needs to remain unchanged in the first and last units to guarantee that it will be satisfied in all units. Therefore, the process... i With process h The priority relationship between them can be expressed as follows:

[0069]

[0070] Set up process i Start time at the first unit For virtual processes n+1 This represents the project deadline, and its start time in both the first and last units is fixed at the given deadline. For example, the start time of a virtual process can be expressed as:

[0071] Assuming a given process n This is the last actual process in the project, and also a virtual process. n+1 This is the subsequent process. To ensure the project meets the predetermined deadline D, the following constraints must be met:

[0072] in, These represent the last actual process. n At the start time of the first and last units; These represent the last actual process. n The project duration for the first and last units; These represent virtual processes. n+1 The start time of the first and last units, with a fixed value being the deadline. .

[0073] The above constraints ensured the success of the final actual process. n The completion time shall not exceed the project deadline. This constraint provides an important time boundary condition for project scheduling optimization, ensuring the feasibility and rationality of the project plan.

[0074] According to the rules of trapezoidal work team movement, the process i The start times of the first and last units must meet the following constraints:

[0075] in, Indicate process i The indicator for measuring the progress of a process is calculated using the following formula: .

[0076] Process i The delay in the completion of a sub-process may lead to the process h The subsequent sub-processes are delayed. If this occurs, the process... h The direct costs may increase due to additional labor and equipment costs. To comprehensively consider this cost factor and generate a stable project scheduling plan (e.g., robustness of the solution), the present invention also defines procedures. i and its subsequent processes h The instability coefficient between them is calculated using the following formula:

[0077] coefficient These are the rates for labor costs and equipment costs for each process; it should also be noted that... .

[0078] Based on the above formula, it is recommended to allocate longer time intervals to processes with higher labor and equipment cost rates that follow. In this invention, the project deadline is assumed to be the last process. n The subsequent process is considered a process. n+1 The virtual process. Then, this article sets... It is a larger value, which allows for a longer time buffer between the completion date of the last process and the project deadline, ensuring the project's completion rate (i.e., qualitative robustness).

[0079] In summary, maximizing robustness and minimizing project cost within the LOB framework can be expressed as:

[0080]

[0081]

[0082]

[0083]

[0084]

[0085]

[0086]

[0087]

[0088]

[0089]

[0090]

[0091]

[0092]

[0093] The following describes the steps for solving the model.

[0094] The above model is a multi-objective optimization model, therefore its optimal solution is a Pareto optimal solution set consisting of multiple individual solutions that have no dominant relationship with each other. E-constraint algorithm, as one of the main methods for solving bi-objective optimization problems, can obtain the exact Pareto solution set within a finite number of steps. Therefore, this paper first uses the ε-constraint algorithm to transform the above model into a robust single-objective sub-model with cost constraints, and then designs an improved genetic algorithm based on constraint programming for solving it. The GA-CP hybrid algorithm, combining genetic algorithm and constraint programming algorithm, is used for solving the problem. The solution process is as follows: Figure 3 As shown.

[0095] Next, we will introduce the ε-constraint algorithm in detail.

[0096] The basic idea of ​​the ε-constraint algorithm is to transform one of the objectives into constraints, solve multiple single-objective problems separately, and thus establish the Pareto front. Based on this, this paper sets the main optimization objective as project robustness. RM Total project cost PC As constraints, the following single-objective optimization problem is obtained. :

[0097]

[0098] in, These represent the lower and upper bounds of the total project cost, respectively. Internal change The value of can yield multiple Pareto optimal solutions.

[0099] In order to calculate They need to be calculated separately. RM and PC Ideal Point And the worst point (NadirPoint) ,in This can be obtained by solving the following two optimization problems: :

[0100] :

[0101] This can be obtained by solving the following two optimization problems: :

[0102] :

[0103] Therefore, we can obtain PC lower bound Upper Realm The solution process of the bi-objective optimization problem based on the ε-constraint algorithm in this paper is as follows: Figure 4 As shown.

[0104] exist Figure 4 First, the minimum and maximum total project cost are calculated (i.e., the cost values ​​corresponding to the ideal point and the worst point). Then, the Pareto solution is initialized.

[0105] After transforming the bi-objective problem (minimizing project cost and maximizing robustness) into a single-objective (maximizing robustness) optimization problem, it is necessary to iteratively change the boundary value ε of the cost constraint to generate a series of Pareto front solutions.

[0106] In each iteration, the step size is incremented for ε. It is at the lower bound of the total project cost. With the upper realm between.

[0107] Specifically, set step size In each iteration, let Set the cost constraint to Then solve the corresponding robustness maximization subproblem.

[0108] With step size By gradually increasing the value of ε, a series of non-dominated Pareto solutions can be obtained, thus forming a complete Pareto front, and the results can be determined. Whether it is valid or not.

[0109] If true, then solve the ε-constrained objective optimization problem P( ), and its optimal solution Add the solution to set F and return to determine the step size. The choice of step size directly affects the fineness of the Pareto front: the smaller the step size, the more solutions are generated and the smoother the front is, but the computational cost increases; the larger the step size, the higher the solution efficiency, but some optimal solutions may be missed.

[0110] If this condition is not met, the dominated point is removed from the set F, and the Parato optimal solution F is obtained.

[0111] Based on the above discussion, the GA-CP hybrid algorithm will be further introduced below.

[0112] Since the transformed sub-model is a mixed-integer linear programming model, it has a complex structure and a large number of decision variables. Preliminary experiments show that it is difficult to obtain the optimal solution of the model using the constrained programming (CP) algorithm in a finite time. Compared with CP, the solution obtained by the genetic algorithm (GA) is of lower quality and has poorer stability.

[0113] The core idea of ​​the GA-CP hybrid algorithm of this invention is: let GA be responsible for the global "search" to find promising regions; let CP be responsible for the local "utilization" to perform a fine search in the regions found by GA to ensure the feasibility and high quality of the solution.

[0114] Therefore, this paper will use a hybrid algorithm combining genetic algorithm (GA) and constraint programming (CP) to solve the model.

[0115] The model in this paper mainly involves three types of discrete decision variables, namely the number of work teams ( ), time buffer ( ) and the start time of each sub-process ( Once the first two types of decision variables are determined, the last type of decision variable can be determined.

[0116] Based on the characteristics of the model, the core technical concept of the GA-CP algorithm of this invention is as follows.

[0117] First, GA is used to process a class of decision variables ( The model is dimensionality reduced, and then the processed class of decision variables is used as known parameters in the computational logic (CP). Leveraging CP's strong search capabilities, the remaining variables are solved within an acceptable timeframe. ).

[0118] This hybrid algorithm achieves a division of labor and collaboration, resulting in efficiency higher than the sum of its parts. GA continuously provides new and diverse starting points to CP, preventing the search from stalling. CP, on the other hand, ensures that local optima can be found from each starting point. This mechanism allows the algorithm to both escape local optima and perform fine-tuning, thus increasing the likelihood of finding a globally optimal or near-global optimal solution.

[0119] Based on the above technical concept, the specific steps of the present invention are described as follows.

[0120] In the first step, the project parameters are input, and GA is used to solve the problem. However, due to the instability of GA, the process may be slow. N The optimal solution and corresponding value are obtained in ten iterations (e.g., ten times). N A schedule.

[0121] In the second step, the results obtained in the first step are...N The number of work teams in each schedule is taken as a known parameter, and then substituted into the CP (Work Plan) to solve for the results. N There are one feasible solution, and the corresponding result is... N A feasible schedule.

[0122] In the third step, compare the results obtained from the second step. N The quality of each feasible solution is evaluated, and the highest quality feasible solution and its corresponding schedule are output.

[0123] The concept and implementation steps of this invention have many advantages over the prior art.

[0124] Firstly, it significantly improves solution efficiency. Because the genetic algorithm performs an extensive search in the search space, it can find a good initial solution to the problem. Using this initial solution as input to the IBM ILOG CPLEX Optimization Studio model can further reduce the search space, thereby improving solution efficiency.

[0125] Secondly, it significantly improves the quality of the solution. Genetic algorithms are heuristic algorithms that can search for the global optimum among multiple local optima. This invention, by inputting the results of the genetic algorithm into the IBM ILOG CPLEX OptimizationStudio model, optimizes the problem from a global perspective, thereby improving the quality of the solution.

[0126] Furthermore, this invention can handle complex problems and has broad compatibility. Specifically, when some problems may be difficult to solve using traditional integer programming or constraint programming methods, the hybrid optimization method, which combines the advantages of different algorithms, can appropriately solve these problems.

[0127] For example, when dealing with large-scale problems, using IBM ILOG CPLEX Optimization Studio to find the optimal solution may be less suitable for the following reasons: High computational resource requirements: IBM ILOG CPLEX Optimization Studio models typically require significant computational resources to solve large-scale optimization problems. Under conditions of limited computational resources, this method may not be suitable for finding the optimal solution. As the problem size increases, the complexity of the IBM ILOG CPLEX Optimization Studio model also increases. This may lead to excessively long solution times or the need for more advanced computing technologies.

[0128] In addition, the present invention can use ILOG OPL language as the model formula language. First, the results of the genetic algorithm are input into the IBM ILOG CPLEX Optimization Studio model, and CPLEX V12.8 is used as the general solver for the constraint programming algorithm to solve the model.

[0129] In fact, this invention defines two main types of processes based on the LOB framework. Through quantitative analysis and comparative study of the free float of the two types of processes, the results show that divergent processes exhibit significant advantages in dealing with uncertainties encountered during the implementation of engineering projects.

[0130] Furthermore, based on the above, this invention proposes a new method for calculating free float under the LOB framework, and constructs a robustness evaluation index system suitable for the LOB scheduling framework.

[0131] Moreover, this invention uses the ε-constraint algorithm to solve real-world engineering cases. First, the ε-constraint algorithm is used to transform the above model into a robust optimization single-objective sub-model with cost constraints. Then, an improved genetic algorithm based on constraint programming, GA-CP, is designed to solve the model.

[0132] To verify the effectiveness of the optimization method for repetitive projects proposed in this invention, the following examples are provided for verification using actual highway engineering cases.

[0133] The highway used in this embodiment is the Keshiketeng to Ulanbutong (Inner Mongolia-Hebei border) section of the G1611 Danxi Expressway. This expressway is 105.58 km long and comprises 15 work processes, each consisting of 10 identical units. The constraint relationship between preceding and following work processes is an "end-start" constraint. The maximum number of work teams per work process is 3. The project contract period is 410 days, and the unit indirect cost is 25,000 yuan / day. The unit duration, labor cost, machinery cost, material cost, and instability weight for each work process are shown in Table 1. Direct costs are derived from actual engineering data, and instability weights are calculated using the aforementioned instability weight formula, as shown in the last row of the actual engineering case in Table 1 below: To verify the effectiveness of the robustness index proposed in this invention, a comparative experiment was set up in this embodiment. Three scheduling schemes were generated using two robustness indices proposed in existing literature and the robustness index proposed in this invention, which were respectively denoted as scheduling 1, scheduling 2 and scheduling 3.

[0134] This embodiment employs a Monte Carlo simulation experiment to simulate potentially harmful interruption events that may occur during the implementation of an engineering project. The simulation experiment considers the following three uncertainties: the total number of interruptions, the duration of each interruption, and the location of each interruption. The total number of interruptions is set to three levels: 10, 30, and 50; the duration of each interruption is set to no more than 20% or 40% of the unit's duration; the total number of simulations is 10,000. The above factors combine to form six test scenarios, as detailed in Table 2.

[0135] Table 1. Actual Engineering Cases

[0136] Table 2 Key Factor Combinations

[0137] This embodiment uses the following two evaluation metrics to measure the stability of each scheduling scheme: (1) Degree of deviation from the construction period index Defined as the percentage of actual completion time deviating from planned completion time, calculated using the following formula:

[0138] in, This represents the actual average completion date of the project. This is the planned completion date for the project. R 1 The larger the value, the worse the stability of the scheduling scheme during actual execution.

[0139] (2) Cost inspection indicators R 2 Defined as the average cost of the delay after an interruption occurs, calculated using the following formula: , in, Let N be the cost incurred for each delay in the project during the k-th simulation, and N be the number of simulations. The larger the value, the worse the project scheduling plan's ability to withstand uncertain risks.

[0140] The simulation results are shown in Table 3 and Figure 5 As shown.

[0141] Table 3 Simulation Results

[0142] The experimental results show that, in all six test scenarios and a total of 60,000 random experiments, the scheduling method 3 proposed in this invention has the lowest deviation between the actual completion time and the planned completion time, as well as the lowest delay cost. Specifically, scheduling method 3... R 1The value was significantly lower than that of Schedule 1 and Schedule 2 in all six scenarios, with a maximum deviation of only 7.76%, while the maximum deviations of Schedule 1 and Schedule 2 reached 13.09% and 20.59%, respectively; meanwhile, the value of Schedule 3 was significantly lower. R 2 The value is also the lowest across all scenarios, demonstrating optimal economic robustness.

[0143] It is worth noting that although the actual completion time deviation of Schedule 1 was lower than that of Schedule 2, its delay cost was higher. This comparison reveals that existing robustness indicators have inherent limitations: they either focus on time stability or on controlling delay costs, failing to comprehensively consider the overall performance of scheduling schemes in terms of both stability and economy.

[0144] In summary, the robustness index and optimization method based on the GA-CP algorithm proposed in this invention have significant advantages in the following aspects: (1) better overall performance, achieving the lowest time deviation and the lowest delay cost, overcoming the contradiction of "high stability but high cost" or "low cost but poor stability" in traditional methods; (2) stronger risk resistance, with a stronger buffering capacity for uncertain factors, minimal fluctuation in completion time, reflecting better plan executability and reliability; (3) improved economic robustness, not only maintaining progress stability when dealing with disturbances, but also significantly reducing economic losses caused by delays, achieving a double guarantee of economic robustness.

[0145] This embodiment fully verifies the effectiveness and superiority of the method proposed in repetitive engineering projects.

[0146] The above description is merely an exemplary embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. An optimization method for repetitive projects, characterized in that, Includes the following steps: Step 1: Construct robustness metrics for repetitive projects, which are defined based on the time interval and free float of the process pair at the first and last units in a linear schedule; Step 2: Establish a cost-robust dual-objective trade-off optimization model for repetitive projects. The dual-objective trade-off optimization model aims to minimize the total project cost and maximize the project robustness, while satisfying constraints on project deadline, work process priority, and number of work teams. Step 3: Use the ε-constraint algorithm to transform the bi-objective trade-off optimization model into a single-objective optimization sub-model with robustness as the objective and project cost as the constraint; Step 4: The GA-CP hybrid algorithm, which combines genetic algorithm and constraint programming, is used to solve the single-objective optimization sub-model to obtain the optimal schedule plan.

2. The method according to claim 1, characterized in that, The mathematical expression for the robustness index is: ， in, n For the number of processes, For process i The set of subsequent processes, For process i With process j The instability coefficient between them Let be the utility function of the time interval. For process i With process j The free time difference between them.

3. The method according to claim 2, characterized in that, The utility function It is a piecewise linear function used to handle non-integer time intervals, defined as: ， in, The decimal part.

4. The method according to claim 1, characterized in that, The total project cost includes direct costs and indirect costs. Direct costs include labor costs, material costs, and equipment costs, while indirect costs are proportional to the total project duration.

5. The method according to claim 1, characterized in that, The ε-constraint algorithm is implemented as follows: Total project cost PC As a constraint, its upper and lower bounds are defined. By gradually adjusting the value of ε, a series of single-objective robustness maximization subproblems are solved, thereby constructing the Pareto front.

6. The method according to claim 1, characterized in that, The solution process of the GA-CP hybrid algorithm includes: A genetic algorithm is used to solve the model, and the algorithm is run multiple times to obtain multiple schedules and the number of work teams for each process. The number of work teams is used as a known parameter and input into the constraint programming solver to solve for the remaining time buffer variable and the sub-process start time variable to obtain a feasible solution. Compare the robustness indices of each feasible solution and output the optimal schedule plan.

7. The method according to claim 6, characterized in that, The genetic algorithm is used to handle the number of work teams among the discrete decision variables, and the constraint programming is used to handle the time buffer variable and the sub-process start time variable.

8. The method according to claim 1, characterized in that, The process priority relationship constraint is based on the trapezoidal motion rule, and only needs to be expressed at the first and last units. The specific expression is as follows: , in, Each is a process i At the start time of the first and last units, For process i The unit construction period.

9. The method according to claim 1, characterized in that, The method is applicable to the following types of repetitive projects: roads, tunnels, bridges, high-rise buildings, oil and gas pipelines, wind farms, or photovoltaic power stations.

10. The method according to any one of claims 1-9, characterized in that, The solution tool for the GA-CP hybrid algorithm is modeled using the ILOG OPL language and calls CPLEX V12.8 as the constraint programming solver.

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