A multi-objective parallel bilateral disassembly line balance optimization method based on constraint programming

Abstract

A multi-objective parallel bilateral disassembly line balance optimization method based on constraint programming, which relates to the technical field of disassembly line balance optimization, comprises the following steps: obtaining the original data of the disassembly line according to the characteristics of the types of waste products and the disassembly process, and establishing an original database; constructing a constraint programming model for the multi-objective parallel bilateral disassembly line problem, wherein the objective function includes weighted line length, idle balance index, hazard demand index and disassembly line energy consumption cost, and the constraint conditions include takt time constraint, task priority relationship constraint, task area constraint and workstation opening constraint; inputting waste product information according to database data, and solving the constraint programming model according to the dictionary optimization method to obtain non-inferior disassembly solutions on multiple Pareto frontiers. The present invention can effectively help remanufacturing and recycling enterprises to formulate reasonable and efficient disassembly line production plans, reduce enterprise equipment costs, and improve the safety and efficiency of disassembly operations.

Description

Technical Field

[0001] The invention relates to the technical field of disassembly line balance optimization, in particular to a multi-objective parallel bilateral disassembly line balance optimization method based on constraint programming. Background Art

[0002] Green design and green manufacturing are the development trends of contemporary manufacturing. Their goal is to minimize the harm to the environment caused by products during the process of design, production, transportation, use and scrapping. Disassembly is an important part of the green manufacturing process. Remanufacturing is achieved by selectively separating the required components, parts and materials from discarded products for recycling and reuse. The state has issued relevant policies that clearly point out that the regulations on the elimination of old products should be strictly implemented, and efforts should be made to improve the recycling and disassembly system of waste products. The disassembly line is an optimal operating method for disassembling and properly disposing of parts of different properties in waste products in the form of an assembly line. In view of the complexity brought about by the differences in the characteristics of different disassembled products and parts, there is a common load imbalance in each workstation of the disassembly line, which greatly increases the difficulty of planning. Therefore, how to stably and efficiently obtain the optimal disassembly solution has become the key to the optimization of the disassembly line.

[0003] For large waste products, such as cars and refrigerators, it is difficult to change their direction during disassembly due to their own characteristics. For such products with disassembly direction constraints, bilateral disassembly is often adopted, which can effectively reduce the walking distance of workers and improve disassembly efficiency. At present, most bilateral disassembly line balance theories are based on a single disassembly line model, while there may be multiple production lines in actual disassembly enterprises, and multiple end-of-life products can be processed on different production lines. These multiple bilateral disassembly lines can be combined into a parallel bilateral disassembly line. Compared with multiple independent disassembly lines, parallel disassembly lines allow two lines to form a common workstation and can process tasks on two adjacent lines at the same time. Practice has proved that this structure helps to improve production efficiency while reducing floor space. However, such problems are rarely mentioned and solved. In addition, most of the existing mathematical programming models for disassembly line balance problems are NP-hard problems, so the existing solutions are mostly heuristic algorithms. Traditional intelligent optimization algorithms can only provide partial optimal solutions when used to solve disassembly line balance problems, and cannot prove the optimality of the solutions. Therefore, seeking an efficient and accurate solution method for solving disassembly line optimization problems has become an urgent problem to be solved in the remanufacturing industry. Summary of the invention

[0004] In view of this, the main purpose of the present invention is to provide a multi-objective parallel bilateral disassembly line balance optimization method based on constraint programming, which can effectively solve the process allocation and optimization problem of parallel bilateral disassembly lines with disassembly direction constraints, improve disassembly efficiency and increase corporate profits. This method is suitable for various transformation scenarios such as production line expansion, production line transformation and process optimization of remanufacturing enterprises.

[0005] The technical solution of the present invention is a multi-objective parallel bilateral disassembly line balance optimization method based on constraint programming, comprising the following steps:

[0006] Step S1: Collect the original data of the disassembly line to form a data database, and use the constraint programming language to establish a linear programming model with the goals of minimizing the weighted line length of the disassembly line, minimizing the idle balance index, minimizing the hazard demand index, and minimizing the disassembly cost.

[0007] Step S2: Input the data database consisting of the original data into the docplex.CP module of the CPLEX solver, use the CPLEX solver, combine the dictionary optimization method to optimize multiple objectives of the constraint programming model, solve the constraint programming model, and obtain multiple non-inferior disassembly solutions under different priorities.

[0008] The technical effects of the present invention are:

[0009] 1. The present invention studies the balance optimization problem of parallel bilateral disassembly lines. Parallel bilateral disassembly lines can disassemble a variety of waste products at the same time, greatly improving the disassembly efficiency and reducing the waiting time compared to the traditional serial disassembly method. At the same time, it takes into account the shared workstations between multiple bilateral disassembly lines, effectively increasing the production flexibility of the disassembly lines and improving the disassembly efficiency.

[0010] 2. The method used in the present invention effectively improves disassembly efficiency and reduces disassembly time and cost by optimizing the disassembly line settings and process allocation, thereby increasing the company's profits. At the same time, by considering the hazard demand index and the energy consumption cost of the disassembly line, the method can also reduce potential risks and improve energy utilization efficiency, further increasing the company's profits.

[0011] 3. The parallel bilateral disassembly line balancing problem studied in this invention is applicable to a variety of remanufacturing scenarios, including production line expansion, production line transformation, process optimization and other transformation scenarios of remanufacturing enterprises. Whether it is expanding the scale of production lines, upgrading equipment or optimizing process flows, this method can be flexibly applied and adjusted and optimized according to the needs of specific scenarios.

[0012] 4. The present invention constructs a constraint programming model for the first time to solve the parallel bilateral disassembly line problem. This method is an accurate method. Compared with the traditional heuristic algorithm, it can obtain the proven optimal multi-objective Pareto frontier solution; compared with the traditional mathematical programming model, it has fewer constraints and the model solving speed is faster.

[0013] 5. The constraint programming method proposed in the present invention provides multiple non-inferior disassembly solutions. By using the dictionary optimization method to solve the constraint programming model, the present invention can obtain multiple non-inferior disassembly solutions with different priorities. This provides decision makers with multiple options and allows them to make trade-offs and decisions based on actual conditions, further improving the quality and adaptability of the disassembly solutions. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings required for use in the embodiments are briefly introduced below.

[0015] Figure 1 This is a flowchart of a multi-objective parallel bilateral disassembly line balance optimization method based on constraint programming in an embodiment of the present invention;

[0016] Figure 2 It is a schematic diagram of a parallel bilateral disassembly line;

[0017] Figure 3 It is a diagram of AND / OR precedence relationship;

[0018] Figure 4 It is a diagram of the dictionary optimization algorithm. DETAILED DESCRIPTION

[0019] The present invention will be further described in detail below in conjunction with the embodiments and drawings.

[0020] In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. The described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0021] Example:

[0022] See also Figure 1 The present invention proposes a multi-objective parallel bilateral disassembly line balance optimization method based on constraint programming, comprising the following steps:

[0023] Step S1: Collect the original data of the disassembly line to form a data database. The original data includes the disassembly time of each task, the AND / OR priority relationship matrix between tasks, the cycle time of each disassembly line, the side information of the task, the hazard coefficient and the demand index. Use the constraint programming language to establish a linear programming model with the goals of minimizing the weighted line length of the disassembly line, minimizing the idle balance index, minimizing the hazard demand index, and minimizing the disassembly cost.

[0024] In the present invention, the bilateral disassembly line is a line that sets up disassembly workstations on both sides of the conveying device, and the left and right workstations form a combined workstation. After the disassembly starts, the product to be disassembled enters from the input end through the conveyor belt, and stops between each combined workstation in turn according to the beat time. The left and right workstations disassemble the product according to the assigned disassembly tasks. When assigning disassembly tasks, tasks that can only be disassembled on the left can only be assigned to the left workstation, and the disassembly direction is recorded as L; tasks that can only be disassembled on the right can only be assigned to the right workstation, and the disassembly direction is recorded as R; tasks that can be disassembled on both sides are recorded as E. The parallel bilateral disassembly line places multiple bilateral disassembly lines in parallel, and the two sides of two adjacent disassembly lines are combined into a common workstation. The workers at the shared workstation need to handle the disassembly tasks of the disassembly lines on both sides. The schematic diagram of the relevant concepts in the embodiment of the present invention is shown in the figure. Figure 2 shown.

[0025] Disassembly tasks have AND / OR priority constraints. Subsequent tasks can only be assigned after the previous tasks are assigned. Disassembly tasks also have hazard and demand attributes. Tasks with hazards and demands need to be assigned as early as possible. A reasonable allocation plan requires opening fewer workstations to maintain a balanced load and lower operating costs. Figure 3 The figure below is a disassembly priority diagram of 10 disassembly tasks. Completing one task can disassemble one part. The numbers in the circle represent the disassembly tasks, the letters outside the circle represent the disassembly directions of the tasks, and the arrows represent the disassembly relationships between tasks. Taking Task 7 of the P8 product as an example, its AND predecessor task is Task 8. Only after the disassembly of AND predecessor task 8 is completed can Task 7 be assigned for disassembly. Taking Task 8 of the P10 product as an example, its OR predecessor tasks are Tasks 2 and 3. Only after the disassembly of one of the OR predecessor tasks 2 or 3 is completed can Task 8 be assigned for disassembly.

[0026] Each disassembly task must comply with the time constraints within the combined workstation. The task should be completed within the assigned combined workstation cycle time. If a task is assigned to one side of the combined workstation, and some of its immediate predecessor tasks are assigned to the other side of the same combined workstation, the task must wait until the previous task and its immediate predecessor tasks assigned to the same side of the combined workstation are all disassembled before it can start disassembly. Among waste products, parts with recycling value are often defined as demand parts, while parts that are harmful to the environment are defined as hazardous parts. For the requirements of green and environmentally friendly manufacturing, these two types of products generally need to be dismantled first to avoid hazardous tasks increasing the risk factor of the production line and reducing the risk of damage to demand tasks.

[0027] At the same time, the present invention considers the following assumptions:

[0028] 1) Arrange the task allocation relationship of two or more bilateral disassembly lines at the same time, and each line is allowed to disassemble different products;

[0029] 2) The disassembly information of the waste product is known and the product is not defective;

[0030] 3) The cycle times of different disassembly lines are unified into a common cycle time through the least common multiple method;

[0031] 4) Ignore the workers’ walking time and the conveyor belt’s moving time.

[0032] The present invention constructs the constraint programming model variables including:

[0033] Interval variable, representing the execution interval of task i on line h. Length is the execution time of task i, ranging from [0, CT max(n h )], where n is the total number of parts in a product,

[0034]

[0035] Based on the above analysis, the objective functions of the present invention can be constructed as follows:

[0036] Minimize the weighted length of the disassembly line:

[0037]

[0038] In formula (1), is an interval variable, representing the time window of task i on the p side of the combined workstation k on the disassembly line h; λ1 and λ2 are line length weighting coefficients. Objective F1 represents minimizing the weighted line length, where the weighted line length is the weighted sum of the disassembly line length and the number of open workstations. The disassembly line length is the longest unit of a single bilateral disassembly line.

[0039] Minimize idle balance indicator:

[0040]

[0041] In formula (2), CT is the common cycle time, and the target F2 represents the minimization of the idle balance index to ensure that the task load on each side workstation is balanced and full.

[0042] Minimize harm requirements indicators:

[0043]

[0044] In formula (3), is the working time window of task i; is the hazard coefficient of task i; is the demand index of task i; startof is the constraint programming syntax, indicating the starting point of the interval variable. Objective F3 represents minimizing the hazard demand index, that is, reducing the residence time of the hazard task on the production line and reducing potential risks. At the same time, the demanded tasks are removed as early as possible to reduce the probability of damage caused by the previous process.

[0045] Minimize disassembly costs:

[0046]

[0047] In formula (4), C1, C2, C3 and C4 represent the unit cost of opening the workstation, the unit time cost of ventilation and lighting of the workstation, the unit time cost of tool energy consumption and the unit processing cost of hazardous tasks, respectively. represents the operation time of task i on disassembly line h. Objective F4 represents minimizing the cost of the disassembly line, including the construction cost of the workstation, the operating cost of lighting and ventilation, the energy consumption cost of tool operation, and the additional processing cost of hazardous tasks.

[0048] The constraints of the constraint programming model include five types of constraints: non-overlapping constraints of in-line tasks, non-overlapping constraints of shared workstation tasks, task direction restriction constraints, task AND / OR priority relationship constraints, and public takt time constraints, as follows:

[0049] In-line tasks do not overlap constraints:

[0050]

[0051] In formula (5), alternative(a, A) is a global constraint unique to constraint programming, which means that in set A, there is only one interval variable a' that is equal to interval variable a, that is, the start and end time of both are the same; [] represents a set, TaskD i h Indicates the directions in which task i on line h can be assigned, including 1, 2 and [1, 2], 1 represents the left side and 2 represents the right side; this constraint means that each task must be assigned to a certain direction of a workstation on the line to which it belongs.

[0052] Shared workstation tasks do not overlap constraints:

[0053]

[0054] In formula (6), nooverlap(A) is a global constraint unique to constraint programming, which means that interval variables in set A cannot overlap. This constraint means that tasks assigned to the p side of paired workstation k on line h cannot overlap, that is, each workstation can only process one task at a time.

[0055] Task direction restriction constraints:

[0056]

[0057] In formula (7), the constraint indicates that the tasks assigned to the shared workstations formed by line h-1 and line h cannot overlap, where p=1 represents the left side and p=2 represents the right side.

[0058] Task AND / OR precedence constraints:

[0059]

[0060] In formula (8), the endof(a) function represents the end point of interval variable a, the startof(a) function represents the starting point of interval variable a, and ANDPT h represents the set of tasks with AND predecessor tasks on line h, ANDP h (i) represents the AND predecessor task of task i on line h.

[0061] Common takt time constraints Five types of constraints:

[0062]

[0063] In formula (9), the any(A) function indicates that at least one expression in the logical expression set A is true, ORPT h represents the set of tasks with OR predecessor tasks on line h, ORP h (i) represents the OR predecessor task of task i on line h.

[0064] Step S2: Input the data database consisting of the original data into the docplex.CP module of the CPLEX solver, use the CPLEX solver, combine the dictionary optimization method to optimize multiple objectives of the constraint programming model, solve the constraint programming model, and obtain multiple non-inferior disassembly solutions under different priorities.

[0065] The present invention uses a dictionary optimization method to optimize multiple objectives of the constraint programming model according to priority. The specific operation process is as follows: Figure 4As shown. After solving the highest priority goal minF1 = objective1, F1≤objective is added as a constraint to the subsequent model, and the hot start method is used to execute the secondary priority goal to obtain minF2 = objective2. The hot start method is: after each solution is completed, it is necessary to input the partial solution information of this time into the next solution process to provide a better initial solution for the problem, which can reduce the amount of calculation for the solver to find the initial feasible solution and speed up the solution. Similarly, F2≤objective2 is continued to be added as a constraint to the subsequent model solution, and the corresponding operations are used to solve the remaining optimization goals.

[0066] Based on the obtained model, the docplex.CP module in the CPLEX solver is used to find the optimal solution of the model. The so-called optimal solution means minimizing the value of the fitness function under the premise of satisfying the constraints. Then the optimal solution is output as the best disassembly plan. The solution algorithm of the docplex.CP module in the CPLEX solver includes the following steps:

[0067] Step 1: Initialize the solution environment and input relevant disassembled product data, including the type, quantity, and disassembly time of the disassembled products;

[0068] Step 2: Construct a constraint programming model, which includes the highest priority objective function F1 and constraint conditions;

[0069] Step 3: Initial constraint transfer. According to the constraint programming model, the values ​​in the variable value range that do not meet the constraint conditions are pruned and the solutions that do not meet the constraint conditions are deleted.

[0070] Step 4: Build a search tree, select a search strategy based on the solution space, and combine the variable value ranges to generate a search space, that is, determine the variable value range of the child node based on the variable value of the current node;

[0071] Step 5: Use the search strategy to traverse the search tree in the search space to search for a solution. Start from the root node and search each leaf node in each branch. The search strategy can be depth-first search, breadth-first search, minimum remaining value first, etc.

[0072] Step 6: Constraint transfer in search. During the search process, according to the variable value determined by the current node, the constraint condition is passed to the next node for further pruning.

[0073] Step 7: Check the current node and check whether all variables on the node have determined values. If all variables have determined values, the search for the current node is completed and the target value and variable value on the node are obtained. Otherwise, the assignment fails and returns to Step 7 to continue searching for the next node.

[0074] Step 8: Update the current optimal solution, compare the current optimal solution with the target value of the current node, and if the node target value is better than the current optimal value, update the optimal solution;

[0075] Step 9: Check whether all nodes on the search tree have been traversed or the termination condition has been reached. If the termination condition has been reached, jump to Step 10. Otherwise, use the backtracking algorithm to continue searching for the next node, that is, return to Step 6.

[0076] Step 10: Output the optimal solution of the primary objective function or the solution that meets the requirements F1 = objective1.

[0077] Step 11: Add a new constraint F1≤objective1 to the original model, set the objective function to the secondary optimization target F2, and solve the new model using the hot start method: that is, input the information of the first solution result into the solver, including the search process information and the optimal solution, and repeat Steps 3 to 10 until the optimal solution of the secondary optimization target F2=objective2 is obtained.

[0078] Step 12: Repeat step 11 to optimize each objective in order of priority until all objectives are optimally solved, and output the optimal allocation plan and the corresponding multi-objective Pareto solution.

[0079] The following is a demonstration of the actual operation of this embodiment:

[0080] Program testing environment:

[0081] The code running environment of the numerical experiments in this paper is: Inter(R)Core(TM)i5-9400CPU@2.90GHz.

[0082] This example uses Python 3.7 to develop the relevant constraint programming algorithm, and uses the docplex.CP package in the CPLEX solver to solve the constraint programming model. At the same time, the mixed integer programming model is solved using the docplex.MP package, and the waste product cases with task scales of 18 to 32 are selected for experiments, and the MILP and CP models are used to solve them respectively. The experimental results are shown in Table 1.

[0083] Table 1 Experimental results

[0084]

[0085] It can be seen from Table 1 that the constraint programming model used in the present invention can obtain the optimal solution to the problem of all cases in a shorter time, while the conventional mixed integer programming model cannot obtain the optimal solution of some objective values ​​in the 2P10-2P22 case, and the model solution time is longer than that of the constraint programming model, indicating that the constraint programming model has good superiority in solving the parallel bilateral disassembly line balance problem.

[0086] The above is only a preferred specific embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or substitutions that can be easily thought of by a person skilled in the art within the technical scope disclosed in the embodiments of the present invention should be included in the protection scope of the present invention. Therefore, the protection scope of the present invention should be based on the protection scope of the claims.

Claims

1. A multi-objective parallel bilateral disassembly line balance optimization method based on constraint programming, characterized in that: The following steps are involved: Step S1: Collect the original data of the disassembly line to form a data database, and use the constraint programming language to establish a linear programming model with the goals of minimizing the weighted line length of the disassembly line, minimizing the idle balance index, minimizing the hazard demand index, and minimizing the disassembly cost. The objective functions are: Minimize the weighted length of the disassembly line: Minimize idle balance indicator: Minimize harm requirements indicators: Minimize disassembly costs: in, is an interval variable, representing the time window of task i on the p side of the combined workstation k on the disassembly line h; λ1 and λ2 are line length weighting coefficients; CT is the common cycle time; is the working time window of task i; is the hazard coefficient of task i; is the demand index of task i; startof is the constraint programming syntax, indicating the starting point of the interval variable; C1, C2, C3 and C4 represent the unit cost of opening the workstation, the unit time cost of ventilation and lighting of the workstation, the unit time cost of tool energy consumption and the unit processing cost of the hazardous task respectively; represents the disassembly time of task i on disassembly line h; And the constraint conditions established by constraint programming language that the objective function needs to satisfy are: In-line tasks do not overlap constraints: Shared workstation tasks do not overlap constraints: Task direction restriction constraints: Task AND / OR precedence constraints: Common takt time constraints Five types of constraints: In the above formulas, H represents the set of disassembly lines, K represents the number of combined workstations, and I h represents the set of tasks on the disassembly line h, is the working time window of task i and task j, is an interval variable. alternative(a, A) is a global constraint unique to constraint programming, indicating that there is only one interval variable a' in set A that is equal to interval variable a, that is, the start and end times of both are the same; [] represents a set. Indicates the direction in which task i on line h can be assigned; nooverlap(A) is a global constraint unique to constraint programming, indicating that interval variables in set A cannot overlap; endof(a) function indicates the end point of interval variable a, startof(a) function indicates the starting point of interval variable a, ANDPT h represents the set of tasks with AND predecessor tasks on line h, ANDP h (i) represents the AND predecessor task of task i on line h; the any(A) function indicates that at least one expression in the logical expression set A is true, ORPT h represents the set of tasks with OR predecessor tasks on line h, ORP h (i) represents the OR predecessor task of task i on line h; Step S2: Input the data database consisting of the original data into the docplex.CP module of the CPLEX solver, use the CPLEX solver, combine the dictionary optimization method to optimize multiple objectives of the constraint programming model, solve the constraint programming model, and obtain multiple non-inferior disassembly solutions under different priorities.

2. The multi-objective parallel bilateral disassembly line balance optimization method based on constraint programming according to claim 1, characterized in that: The solution algorithm based on the docplex.CP module in the CPLEX solver includes the following steps: Step 1: Initialize the solution environment and input relevant disassembled product data, including the type, quantity, and disassembly time of the disassembled products; Step 2: Construct a constraint programming model, which includes the highest priority objective function F1 and constraint conditions; Step 3: Initial constraint transfer: according to the constraint programming model, prune the values ​​in the variable value range that do not meet the constraint conditions and delete the solutions that do not meet the constraint conditions; Step 4: Build a search tree, select a search strategy based on the solution space, and combine the variable value ranges to generate a search space, that is, determine the variable value range of the child node based on the variable value of the current node; Step 5: Use the search strategy to traverse the search tree in the search space to search for solutions, starting from the root node and searching each leaf node in each branch. The search strategies include depth-first search, breadth-first search, and minimum remaining value first. Step 6: Constraint transfer in search. During the search process, according to the variable value determined by the current node, the constraint condition is passed to the next node for further pruning. Step 7: Check the current node and check whether all variables on the node have determined values. If all variables have determined values, the search for the current node is completed and the target value and variable value on the node are obtained. Otherwise, the assignment fails and returns to Step 7 to continue searching for the next node. Step 8: Update the current optimal solution, compare the current optimal solution with the target value of the current node, and if the node target value is better than the current optimal value, update the optimal solution; Step 9: Check whether all nodes on the search tree have been traversed or the termination condition has been reached. If the termination condition has been reached, jump to Step 10. Otherwise, use the backtracking algorithm to continue searching for the next node, that is, return to Step 6. Step 10: Output the optimal solution of the primary objective function or the solution that meets the requirements F1 = objective1; Step 11: Add a constraint F1≤objective1 to the original model, set the objective function to the secondary optimization objective F2, and solve the new model using the hot start method: that is, input the information of the first solution result into the solver, including the search process information and the optimal solution, and repeat Steps 3 to 10 until the optimal solution of the secondary optimization objective F2=objective2 is obtained; Step 12: Repeat step 11 to optimize each objective in order of priority until all objectives are optimally solved, and output the optimal allocation plan and the corresponding multi-objective Pareto solution.

3. The multi-objective parallel bilateral disassembly line balance optimization method based on constraint programming according to claim 2 is characterized by: The termination conditions in Step 9 include a maximum time limit and a node assignment failure limit; wherein the maximum time limit is: when the solution time reaches the set maximum time, the solution is terminated; the node assignment failure limit is: when the node assignment failure number reaches the maximum value, the solution is terminated.

4. The multi-objective parallel bilateral disassembly line balance optimization method based on constraint programming according to claim 1, characterized in that: The dictionary optimization method is as follows: a solution order of four objectives is given, and after instructing the CPLEX solver to solve the highest priority objective minF1=objective1, F1≤objective is added as a constraint to the subsequent model, and a hot start method is used to execute the secondary priority objective solution to obtain minF2=objective2. Similarly, F2≤objective2 is continued to be added as a constraint to the subsequent model solution, and the corresponding operation is used to solve, and so on until minF1, minF2, minF3, and minF4 are obtained. Among them, the hot start method is: after each solution is completed, the partial solution information of that time needs to be input into the next solution process.

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