Production workshop-oriented constrained multi-row facility layout method

Abstract

The invention provides a production workshop-oriented constrained multi-row facility layout method, which comprises the following steps of: aiming at a multi-row facility layout problem, establishing a mixed integer programming model taking minimization of material handling cost as an optimization target, and considering a specified row positioning constraint and a sorting constraint of facilities in the model; the invention provides a heredity-based hyper-heuristic algorithm to efficiently solve the above model, the algorithm has a reward mechanism, the solution space can be effectively explored, and a problem-specific heuristic rule is developed in the algorithm to generate a high-quality initial solution, so that the problem of the mixed integer programming model can be effectively solved, and the problem of the mixed integer programming model can be effectively solved. According to the algorithm, a genetic algorithm and a low-level heuristic operator are adopted to act on a problem domain on the aspect of an advanced algorithm, the algorithm performance can be effectively improved, and finally, it is proved that a constraint type multi-row facility layout scheme finally obtained based on the method can effectively reduce the enterprise workshop material carrying cost and improve the productivity by solving a large number of standard calculation examples.

Description

Technical Field

[0001] The present invention relates to the technical field of facility layout, and particularly to the strategy optimization of layout. Specifically, it is a constrained multi-row facility layout method for a production workshop with the goal of minimizing material handling costs. Background Art

[0002] With the breakthrough of key technologies and the rapid development of intelligent manufacturing, the global manufacturing industry is accelerating its transformation towards intelligence and digitization. In order to conform to this development trend and gain an advantage in the fierce market competition, the manufacturing industry needs to make reasonable transformation adjustments to the workshop layout, improve production efficiency, and reduce production costs, so as to improve economic benefits. In this process, a reasonable facility layout plays a crucial role. By optimizing the facility layout, the material handling cost can be effectively reduced by 10% to 30%, the production cycle can be shortened, the material flow speed can be increased, and thus the economic benefits of the enterprise can be significantly improved.

[0003] The multi-row facility layout problem involves allocating a set of facilities to different rows and determining their order and exact positions. It has wide applications in manufacturing systems, semiconductor manufacturing workshops, and service offices. Constraints play an important role in factory manufacturing. In some cases, constraints must be considered, such as the designated positions of specific facility points and the sequencing relationships between facility points. For example, in the hospital facility layout, the emergency room is usually located near the first-floor exit to facilitate the rapid entry of patients in case of emergencies. In part processing, some processes are prioritized. In a flexible workshop, the facilities with processing sequences should satisfy the precedence relationships of arranging the facilities, and the unconstrained facilities should be reasonably arranged. In order to better meet the actual needs, the constrained multi-row facility layout problem considers the designated row positioning constraints and sequencing constraints between facilities, and a new model of the constrained multi-row facility layout problem is developed with the optimization goal of minimizing the material handling cost to meet the needs of the actual manufacturing system.

[0004] In addition, the multi-row facility layout problem is an NP-hard problem, and it is difficult to obtain an optimal solution in polynomial time. To solve the proposed constrained multi-row facility layout problem, an efficient genetic algorithm-based hyper-heuristic method is designed. First, a new solution representation scheme is designed according to the problem attributes. Second, the genetic algorithm is selected as the high-level heuristic algorithm, and a reward mechanism is designed as the selection operation of the genetic algorithm to replace the conventional selection operation in the genetic algorithm, and the better-performing low-level heuristic operators are adaptively selected to act on the problem domain. The results show that the designed method can effectively reduce the material handling cost and optimize the resource allocation. Summary of the Invention

[0005] To solve the above problems, the main object of the present invention is to provide a constrained multi-row facility layout method for a production workshop, based on the following assumptions:

[0006] (1) All facilities occupy different rectangles of known size. The side length of the facility parallel to the x-axis is the length of the facility.

[0007] (2) The material interaction point of each facility is located at the center point of the facility. The unit distance MHC between facility points is a known quantity.

[0008] (3) No space is allowed between adjacent facilities in the same row.

[0009] (4) The facilities in each row are arranged starting from a common point at the leftmost end of the row, and the horizontal coordinate of the leftmost facility in the layout is zero.

[0010] (5) Ignoring the facility width, the aisle width is set to 2.

[0011] (6) The layout of the facilities is not restricted by the area size or other factors.

[0012] In addition, this constrained multi-row facility layout method has the following constraint situations and the constrained facilities are known:

[0013] (1) Specified row positioning constraint: Place the facility at a predetermined position in the specified row. For example, large facilities that are not easy to move must be placed at the specified position.

[0014] (2) Sorting constraint: The facilities have a relative positional relationship with other facilities according to their attributes or special relationships with other facilities. For example, facilities with a processing sequence should satisfy the priority relationship for arranging facilities in a flexible workshop. In this article, it is assumed that the sorting constraint considers the order relationship of facility coordinates.

[0015] This constrained multi-row facility layout method includes the following steps:

[0016] S100. Establish a mixed-integer programming model with minimizing the material handling cost as the optimization objective for the multi-row facility layout problem, considering the specified row positioning constraint and sorting constraint of the facilities in the model;

[0017] S200. Solve the mixed-integer programming model to obtain a constrained multi-row facility layout plan;

[0018] As a specific implementation manner of the present invention, step S2 uses a hyper-heuristic method based on a genetic algorithm to solve the mixed-integer programming model; the hyper-heuristic method based on the genetic algorithm includes a high-level strategy and low-level heuristic operators, and the solution is composed of a low-level heuristic vector and a facility sequence vector; the high-level uses a genetic algorithm, and uses a reward mechanism as the selection mechanism of the genetic algorithm, adaptively and selectively applying the low-level heuristic operators to the low-level heuristic operator vector to update the low-level heuristic operator vector, and then, the operators in the low-level heuristic vector act on the facility sequence vector in sequence to update the facility sequence vector, and uses the Monte Carlo acceptance criterion as the movement acceptance criterion of the genetic algorithm.

[0019] As a specific implementation manner of the present invention, when using the hyper-heuristic method based on the genetic algorithm to solve, it specifically includes the following steps:

[0020] S210. Initialize parameters, including population size P, maximum number of iterations Max_It, maximum number of iterations LMax_It of low-level heuristic operators, crossover probability Pc, mutation probability Pm, initial scores Sc of each low-level heuristic operator, number L of low-level heuristic operators, constraint matrices Am and Bk of facilities;

[0021] S220. Generate an initial solution in a real number coding manner based on heuristic rules and form an initial population, and screen the optimal solution in the initial population based on the objective value of each individual;

[0022] S230. Randomly select a low-level heuristic operator for calculation for each individual in the initial population, update the optimal solution and score the selected low-level heuristic operator according to the reward mechanism, and normalize the cumulative scores of all low-level heuristic operators to update Sc;

[0023] The selection rule of the reward mechanism is: if the optimal solution is improved before and after the update, the score of the selected low-level heuristic operator increases, otherwise the score of the selected low-level heuristic operator decreases;

[0024] S240. Select an individual from the initial population, and select low-level heuristic operators based on the screening mechanism to form the low-level heuristic operator vector of this individual, so as to update the low-level heuristic operator vector of this individual;

[0025] Screening mechanism: In odd-numbered iterations, select the four heuristic operators with the highest percentage to form the low-level heuristic vector; in even-numbered iterations, select the low-level heuristic operators according to probability by roulette to obtain the low-level heuristic vector;

[0026] S250. Cross and mutate the low-level heuristic vectors obtained in the selection process in S240 to obtain new low-level heuristic vectors, so as to update the low-level heuristic vectors of this individual. Process the facility sequence vector of this individual according to the order of the low-level heuristic vectors in this individual to update this individual. During the update process, each operator acts on the facility sequence vector in turn. When each operator updates the facility sequence vector, heuristic rule judgment is performed. If the facility sequence does not meet the constraint layout conditions, the operation of this operator is performed again until the facility sequence meets the constraint layout conditions. Then, output the improved solution and perform the exponential Monte Carlo acceptance criterion judgment, update the optimal solution, and update the corresponding operator scores according to the reward mechanism.

[0027] S260. Repeat steps S240 and S250 to traverse the individuals in the initial population.

[0028] S270. Repeat S240 - 260 for iteration until the number of iterations is greater than Max_It. Output all current optimal solutions as the calculation results and end the algorithm.

[0029] S280. Output the optimal solution as the constraint-based multi-row facility layout plan.

[0030] Beneficial effects:

[0031] The present invention designs a constraint-based multi-row facility layout method for a production workshop. Considering that constraints play an important role in factory manufacturing in the production workshop layout, in many cases, the specified row constraints and sorting constraints between facilities must be considered. For example, once some large facilities are positioned, it is inconvenient to move their positions; in part processing, some processes are prioritized; in a flexible workshop, facilities with processing sequences should meet the precedence relationship of arranging facilities, and unconstrained facilities should be reasonably arranged. The basic multi-row facility layout form can no longer meet the actual needs of the current manufacturing system. Therefore, the constraint-based multi-row facility layout form better meets the production needs in the manufacturing field, thereby effectively reducing the material handling cost and improving the production efficiency.

[0032] The hyper-heuristic method based on the genetic algorithm proposed by the present invention redesigns a new encoding and decoding based on the characteristics of the problem to be solved, and designs a heuristic rule based on a specific problem, effectively combining the genetic algorithm of the high-level strategy with the low-level heuristic operator to strengthen the search ability of the algorithm, and comparing the algorithm with the solution results of the exact solver CPLEX and comparing it with different algorithms based on a large number of benchmark examples to prove its effectiveness and efficiency. Description of the Drawings

[0033] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings required for use in the embodiments will be briefly introduced below.

[0034] Figure 1 It is a schematic diagram of the constrained multi-row facility layout problem;

[0035] Figure 2 It is a schematic diagram of the encoding and decoding of the solution;

[0036] Figure 3 It is a schematic diagram of single-point crossover and two-point mutation;

[0037] Figure 4 It is a schematic diagram of the low-level heuristic operator for the facility sequence;

[0038] Figure 5 It is a schematic diagram of the low-level heuristic operator for the breakpoint sequence;

[0039] Figure 6 It is a flowchart of the hyper-heuristic algorithm based on the genetic algorithm;

[0040] Figure 7 It is a box plot comparing the algorithm solution results;

[0041] Figure 8 It is a comparison graph of different algorithm iterations. Detailed implementation manners

[0042] The following will clearly and completely describe the specific implementation manners of the present invention in combination with examples. Obviously, the described examples are only a part of the embodiments of the present invention, rather than all of the embodiments.

[0043] The following further describes the present invention in combination with embodiments:

[0044] In the following embodiments, unless otherwise specifically stated, the operations are conventional operations in the art.

[0045] Embodiment

[0046] The constrained multi-row facility layout method is based on the basic multi-row facility layout problem and combines the actual layout area situation. It involves placing multiple facilities orderly on different rows of a given two-dimensional layout area and seeking to minimize the material handling cost under a non-overlapping facility layout that meets the corresponding constraint conditions. The constraint conditions are divided into two categories: (1) Specified row positioning constraint: Place the facility at a predetermined position in the specified row. (2) Sorting constraint: The facilities have a relative position relationship with other facilities according to their attributes or special relationships with other facilities. In the present invention, it is assumed that the sorting constraint considers the order relationship of the facility coordinates.

[0047] Such as Figure 1As shown. In this problem, each rectangle represents a facility, and the material handling points of all facilities are located at the center points of the facilities. Facility k has a specified row positioning constraint. Therefore, regardless of the arrangement of other facility points, this facility should be placed at a fixed position (assuming the fixed position of facility k is the first position in the first row). The sorting constraint indicates that facilities i and j have a position sorting constraint situation, and these facilities must be placed in a specific order, that is, the horizontal coordinate of facility i is in front of that of facility j. In the present invention, this model takes into account the generalization of constrained facilities and can apply this model to add specified row positioning and sorting constraints to any facility.

[0048] Establish a mathematical model:

[0049] Objective function:

[0050]

[0051] Constraint conditions:

[0052]

[0053] dx ij ≥x i -x j , 1 ≤ i < j ≤ n (3)

[0054] dx ij ≥x j -x i , 1 ≤ i < j ≤ n (4)

[0055]

[0056] dy ij ≥y i -y j , 1 ≤ i < j ≤ n (6)

[0057] dy ij ≥y j -y i , 1 ≤ i < j ≤ n (7)

[0058]

[0059]

[0060]

[0061]

[0062]

[0063]

[0064]

[0065]

[0066]

[0067]

[0068]

[0069] The objective function of the mixed-integer linear programming model for the constrained multi-row facility layout problem for the production workshop is Equation (1), and the constraint conditions are Equations (2)-(19). Among them, Equation (1) represents minimizing the material handling cost, where dx ij and dy ij are the logistics interaction distances between facilities i and j on the x-axis and y-axis. Formulas (2) and (5) are used to calculate the horizontal and vertical coordinates of each facility to determine the exact location of the facility. Formulas (3) and (4) and Formulas (6) and (7) are used to calculate the material handling distances between facilities i and j in the x-direction and y-direction based on the coordinates of the facilities. Formula (8) is the non-overlap constraint between facilities in the same row. It ensures that two facilities are not assigned to the same position in a given row, thus preventing overlap and ensuring that each facility occupies a unique position. Formula (9) determines that each facility can only be assigned to one row. It enforces the constraint on assigning facilities to rows, thus ensuring that all facilities are placed in the specified rows. Formulas (10)-(12) are used to calculate the decision variables α ik and q ijk values. These formulas are crucial for determining the optimal location of facilities in a row. Formulas (13) and (14) limit the ranges of the variables α ik and q ijk . Formula (15) calculates and determines the sorting based on the coordinates of facilities i and j, representing the sorting constraint. Formulas (16)-(19) are the specified row positioning constraints, which are used to determine the positioning facilities that need to be assigned to the specified rows in the model. Formulas (16) and (17) determine the position p i of the facilities with specific row constraints through the relative position relationship between the facilities. Formulas (18) and (19) determine the assignment of the functions with specific row constraints to specific positions on the specified row.

[0070] Step S2 uses a hyper-heuristic method based on a genetic algorithm to solve the mixed-integer programming model; the hyper-heuristic method based on the genetic algorithm includes a high-level strategy and low-level heuristic operators, and the solution consists of a low-level heuristic vector and a facility sequence vector; the high-level uses a genetic algorithm, with a reward mechanism as the selection mechanism of the genetic algorithm, adaptively and selectively applying low-level heuristic operators to the low-level heuristic operator vector to update the low-level heuristic operator vector, and then, the operators in the low-level heuristic vector act on the facility sequence vector in sequence to update the facility sequence vector, and the Monte Carlo acceptance criterion is used as the movement acceptance criterion of the genetic algorithm.

[0071] When using the hyper-heuristic method based on the genetic algorithm to solve, it specifically includes the following steps:

[0072] S210. Initialize parameters, including population size P, maximum number of iterations Max_It, maximum number of iterations of low-level heuristic operators LMax_It, crossover probability Pc, mutation probability Pm, initial scores Sc of each low-level heuristic operator, number of low-level heuristic operators L, constraint matrices Am and Bk of facilities;

[0073] S220. Generate an initial solution in real number coding based on heuristic rules and form an initial population, and screen the optimal solution in the initial population based on the objective value of each individual;

[0074] S230. Randomly select a low-level heuristic operator for each individual in the initial population for calculation, update the optimal solution and score the selected low-level heuristic operator according to the reward mechanism, and normalize the cumulative scores of all low-level heuristic operators to update Sc;

[0075] The selection rule of the reward mechanism is: if the optimal solution is improved before and after the update, the score of the selected low-level heuristic operator increases, otherwise the score of the selected low-level heuristic operator decreases;

[0076] S240. Select an individual from the initial population, and select low-level heuristic operators based on the screening mechanism to form the low-level heuristic operator vector of this individual, thereby updating the low-level heuristic operator vector of this individual;

[0077] Screening mechanism: In odd-numbered iterations, select the four heuristic operators with the highest percentage to form the low-level heuristic vector; in even-numbered iterations, select low-level heuristic operators according to probability by roulette wheel to obtain the low-level heuristic vector;

[0078] S250. Cross and mutate the low-level heuristic vectors obtained in the selection process in S240 to obtain new low-level heuristic vectors, so as to update the low-level heuristic vectors of this individual. Process the facility sequence vector of this individual according to the order of the low-level heuristic vectors in this individual to update this individual. During the update process, each operator acts on the facility sequence vector in turn. When each operator updates the facility sequence vector, heuristic rule judgment is performed. If the facility sequence does not meet the constraint layout conditions, the operation of this operator is carried out again until the facility sequence meets the constraint layout conditions. Then, output the improved solution and perform the exponential Monte Carlo acceptance criterion judgment, update the optimal solution, and update the corresponding operator scores according to the reward mechanism.

[0079] S260. Repeat steps S240 and S250 to traverse the individuals in the initial population.

[0080] S270. Repeat S240 - 260 for iteration until the number of iterations is greater than Max_It, output all current optimal solutions as the calculation results, and end the algorithm.

[0081] S280. Output the optimal solution as the constraint multi-row facility layout plan.

[0082] In implementation, the following operations can be performed:

[0083] I. Encoding and decoding

[0084] In the genetic algorithm (GA), the solution representation scheme includes two components: the low-level heuristic operator (LLH) vector and the facility sequence vector. In the genetic-based hyper-heuristic algorithm (GA-HH), GA is applied as the high-level strategy, and crossover and mutation directly act on the LLH vector. The LLH vector contains various operations, which are used to update the facility sequence vector and directly act on the facility sequence. In the LLH vector, L i represents one of the LLHs (i ∈ [1, 7]), and each LLH selected for application corresponds to a solution of the constraint multi-row facility layout model. The facilities in the facility sequence vector are encoded using integer encoding, which is based on two one-dimensional arrays representing the problem solution, that is, each solution contains a facility permutation array and a breakpoint array. The facility permutation array is used to determine the permutation order of all facilities, and the breakpoint array is used to determine the number of facilities in each row. For specific examples, see Figure 2 .

[0085] Suppose the decoded sequence s = [43519786236], and the breakpoint array is

[36] ; therefore, the layout scheme is as follows. The order of the facilities is divided into three rows. The facilities in the first row are arranged in the order of

[435] , the facilities in the second row are arranged in the order of

[197] , and the facilities in the third row are arranged in the order of

[862] .

[0086] II. High-Level Strategies

[0087] In the present invention, GA is used as an advanced heuristic algorithm. GA includes the following main stages:

[0088] (1) Initialization

[0089] The solution includes a vector of LLH and a vector of facility sequences. The LLH vector consists of 7 LLHs, and these operators follow a uniform distribution. The facility sequence vector consists of a facility arrangement and an array of breakpoints. The array of breakpoints is randomly generated, and each breakpoint is an integer between 1 and the problem size n. Additionally, the value of each consecutive breakpoint in the array must be greater than the previous breakpoint. This ensures that the array correctly divides the facility sequence into multiple-line layouts, where each line of the facility arrangement is clearly defined by the array of breakpoints. The array of breakpoints is updated by LLH. If the initial solution is randomly generated, it may not be a feasible solution to the constrained multi-line facility layout problem. Therefore, the generated initial solution is considered to satisfy the row positioning and sorting constraints specified in the model. Based on this, problem-specific heuristic rules are developed for a specific problem to ensure that each initial solution meets the requirements. The pseudocode for generating the initial solution is shown in Algorithm 1. P represents the population size. After generating a feasible initial solution, the same initial score is assigned to each LLH, and a specific LLH is randomly selected to provide a basis for subsequent selection.

[0090] (2) Fitness Function

[0091] For GA, it is better to have a higher fitness value so as to have more selection opportunities when generating new chromosomes. The fitness function represents the quality of an individual or a solution. The fitness function has different definitions for different problems. The objective function can be used as the fitness function to search for the maximum value of the solution. The inverse function of the objective function can be used as the fitness function to search for the minimum value of the solution. In the present invention, the optimization objective is to minimize the material handling cost, and the inverse function is used to represent the fitness value. The formula for the fitness function is shown in Formula (20).

[0092]

[0093] Where F is the optimization objective function for the constrained multi-line facility layout problem. F1 is the fitness function.

[0094] Table 1 Pseudocode for Generating the Initial Solution

[0095]

[0096] (3) Selection

[0097] Selecting suitable individuals is the core of this selection process. Therefore, it is important to select suitable individuals for offspring production. In the present invention, after each iteration, the LLH is rewarded, which has a positive impact on the search process. In addition, the LLH that has no positive impact will be penalized. An equal score is assigned to each reward or penalty. In the new iteration, the scores of all LLHs are normalized. In addition, roulette is used to select the LLH operator. To avoid missing the current best LLH, we represent the probability of selecting the LLH in even iterations in formula (21). In odd iterations, GA-HH selects the four LLHs with the highest percentage scores and solves them in order. For the individuals in the problem domain, the best individual is retained in each iteration.

[0098] p h = Score h / ΣScore i , i = 1, 2,..., H (21)

[0099] h is the selected LLH and H is the set of LLHs.

[0100] (4) Crossover

[0101] In the crossover phase, the advanced heuristic directly performs crossover on the LLH vector without considering other factors in the solution. In the present invention, GA-HH performs single-point crossover on the selected LLH vector. A point in the LLH vector is randomly selected, and the elements after this point are exchanged with the LLH vectors of other solutions. For a specific example, see Figure 3 .

[0102] (5) Mutation

[0103] In the mutation phase, the advanced heuristic directly acts on the LLH vector without considering other factors in the solution. In the present invention, GA-HH is used to perform two-point mutation on the selected LLH vector. Two operators in the LLH vector are randomly selected, and the selected operators are randomly assigned to the other two operators. For a specific example, see Figure 3 .

[0104] (6) Generate a new solution

[0105] After selection, the LLH vector is updated using crossover and mutation. Subsequently, the facility sequence and breakpoint array are updated using a series of updated LLHs, and a new solution is generated after determining the Monte Carlo criterion.

[0106] (7) Monte Carlo

[0107] In the present invention, according to the characteristics of the solutions generated for the constrained multi-row facility layout problem, the exponential Monte Carlo is selected as the move acceptance criterion. This solution has the property of a negative exponential ratio between the acceptance probability and the objective value, tending towards worse solutions. When this criterion is used during the iterative process of the algorithm, each time a neighborhood solution of the current solution is generated, after comparing the quality of the two solutions, the better solution is accepted, and with a certain probability p m the worse solution is accepted. As shown in formula (22):

[0108] p m = e -δ (22)

[0109] where δ = F(s c ) - F(s0), F(s c ) is the neighborhood solution generated by GA-HH for each selection of LLH based on the constrained multi-row facility layout problem model, and F(s0) is the current solution selected during each iteration. GA-HH determines whether the neighborhood solution generated during this iteration is acceptable according to the size of p m relative to the random number. If p m is greater than the random number between [0, 1], the neighborhood solution is accepted, and the probability of accepting a worse solution decreases as δ increases.

[0110] III. Low-level heuristic operators

[0111] LLH constructs a link between the high-level strategy and the problem domain. Each LLH can generate a unique neighborhood solution that other LLHs may not be able to generate. However, continuously using the same LLH will cause the algorithm to quickly enter the local optimal state. Therefore, 7 simple and efficient LLHs are designed to solve the constrained multi-row facility layout problem model, as Figure 4 and Figure 5 shown, and the specific descriptions are as follows.

[0112] (1) Modified facility sequence vector operator

[0113] 1) Two-point exchange operator: Randomly exchange the positions of 2 different facilities in the specified stage sequence array.

[0114] 2) Reverse order operator: Randomly select a random facility sequence in the specified stage sequence array and reverse it.

[0115] 3) Partially mapped crossover operator: Randomly select the start and end positions of several sequences in 2 sequence arrays, and exchange the selected sequence positions. If there are facility conflicts after the exchange, the conflicting facilities are eliminated according to the mapping relationship.

[0116] 4) Order Crossover Operator: Randomly select the start and end positions of several sequences in one sequence array, put the sequence into a new sequence, find the position of the selected sequence in the other sequence array, and put the remaining sequences into the new sequence in order.

[0117] (2) Modified Breakpoint Vector Operator

[0118] 1) Left Breakpoint Change Operator: Except for the leftmost element, randomly select a position in the breakpoint array, add 1 to its own value and subtract 1 from the left breakpoint value, which is used when the number of facility rows is 4 or 5.

[0119] 2) Right Breakpoint Change Operator: Except for the rightmost element, randomly select a position in the breakpoint array, add 1 to its own value and subtract 1 from the right breakpoint value, which is used when the number of facility rows is 4 or 5.

[0120] 3) Random Breakpoint Value Change Operator: Randomly select two positions in the breakpoint array, add 1 to one value and subtract 1 from the other value.

[0121] IV. Algorithm Steps

[0122] The flow of the proposed genetic-based hyper-heuristic algorithm is as Figure 6 shown.

[0123] To further illustrate the effect of the present invention, a specific example is used to illustrate it below.

[0124] The following is the specific program test environment of this embodiment:

[0125] The simulation computing environment is an Intel(R) Core(TM) i5-9400 CPU @ 2.90GHz, 16GB RAM, Windows 10 operating system, and MATLAB R2022a is used for programming and running.

[0126] Since there is no relevant literature on solving the constrained multi-row facility layout problem, the exact solver CPLEX is used to accurately solve the model. In the present invention, the facility constraint rules are formulated as follows: Facility 4 has a specific row positioning constraint and is located at the first position in the first row; Facilities 5 and 6 have a relationship constraint, and the horizontal coordinate value of the material handling point of Facility 5 is greater than that of Facility 6. CPLEX is used to accurately solve the model to verify the correctness of the model, and its solution results are compared with the algorithm solution results to verify the rationality of the algorithm. In the algorithm verification part, the proposed GA-HH is also used to solve the gapless multi-row facility layout problem, and the problem solution results are compared with the results of relevant literature to verify the solution performance of GA-HH. Finally, the performance of the proposed GA-HH is analyzed according to the benchmark examples commonly used for the facility layout problem.

[0127] As the scale increases, CPLEX cannot determine the optimal solution within a reasonable time. For the sake of solving efficiency, the solving time is limited to 18,000 s, and the feasible solution of this node is regarded as the solving result. In the present invention, in order to verify the rationality and effectiveness of the algorithm, CPLEX, GA-HH and GA are used to solve the benchmark instances of scale 9-49. Currently, there is no dataset that can be directly used for the constrained multi-row facility layout problem. Therefore, the test instances are generated using well-known benchmark instances and supplemented with facility constraints. The evaluation criteria are the solving time t, the objective function value Obj, and the standard deviation SD. Obj is the solving result, and t and SD are the average running time and standard deviation of the algorithm running 10 times respectively. The algorithm parameters after multiple data tests and parameter debugging are shown in Table 2. In addition, in order to enhance the stability of the algorithm calculation results and reduce the random error of the solutions, each example is solved 10 times, and the best value is obtained. The calculation results are shown in Table 3. The crossover probability and mutation probability of GA and GA-HH are 0.85 and 0.15 respectively. gap is the percentage deviation between the upper and lower bounds in the exact solver.

[0128] Table 2 Algorithm parameters

[0129]

[0130] Table 3 Solving results of CPLEX exact solver, GA and GA-HH

[0131]

[0132]

[0133] As shown in Table 3, when solving the scale of 9-13, the gap of CPLEX is 0, which indicates that CPLEX provides the best solution at this time. However, as the facility scale gradually increases, CPLEX can no longer find the optimal solution within 18,000 s, and due to the exponential increase in the solving difficulty, the gap value exceeds 70% (scale 30-49).

[0134] As shown in Table 3, the results of GA-HH are consistent with those of CPLEX when solving benchmark instances with a scale of 9-13, and the results of GA are consistent with those of CPLEX when solving benchmark instances with a scale of 9-11, verifying the effectiveness and correctness of the algorithm in solving. As the scale of the benchmark test instance (Am15-sko_49_01) increases, the objective values obtained by GA-HH and GA are better than those obtained using CPLEX, and the solving time is shorter (more than 100 times). In addition, compared with GA, GA-HH achieves better solutions in 12 out of 16 groups of benchmark instances, the solutions of the other 4 groups are the same, and the solving time of 16 groups of benchmark instances is shorter, verifying that the proposed GA-HH has good solving performance. In terms of the SD value, among 16 groups of benchmark test instances, the SD values of 12 groups are better than that of GA, and the SD values of 1 group of benchmark test instances are the same, verifying that GAHH has higher stability.

[0135] For the constrained multi-row facility layout problem, since specified row positioning and sorting constraints are added to the basic multi-row facility layout, more infeasible solutions may be generated during the algorithm solving process. Although the reduction of the solution space will reduce the solving time of the exact solver CPLEX, compared with solving the basic multi-row facility layout, the design of the algorithm will increase a large number of judgments and spend more solving time.

[0136] To further verify the performance of the algorithm proposed in the present invention, a comparative experiment of a hybrid metaheuristic algorithm and the state-of-the-art genetic-based hyperheuristic algorithm (GA-HH1) was designed. Among them, the hybrid metaheuristic algorithm is a combination of the grey wolf optimization and genetic algorithm (GWO-GA). To adapt to the characteristics of the constrained multi-row facility layout problem model proposed in the present invention, the designed problem-specific heuristic rules are added to the above comparative algorithms GWO-GA and GA-HH1 to obtain the correct solutions. Each instance was run 10 times, and the results are shown in Table 4.

[0137] Table 4 Solving Results of GA-HH, GWO-GA and GA-HH1

[0138]

[0139]

[0140] On the premise that the difference in solving time is relatively small, GA-HH obtains the smallest objective value in all 7 instances. GAHH1 obtains three objective values equal to those of GA-HH. GWO-GA obtains four objective values equal to those of GA-HH. In addition, the SD value of GA-HH is smaller than that of the other two algorithms, which also proves that GA-HH has better performance and stability when solving.

[0141] To verify the generality and efficiency of the proposed algorithm, GA-HH is used to solve the gapless multi-row layout problem model, where m is 3, 4, and 5. The gapless multi-row layout problem is in a given two-dimensional layout area, with the number of rows m ≥ 3, no space between two adjacent facilities, and the rows are arranged from the same starting point on the leftmost side. Each algorithm is solved 10 times; the best calculation result and the average solution time are selected. The calculation results are shown in Table 5.

[0142] Table 5 Comparison of the solution results of GA-HH and the literature

[0143]

[0144] Table 4 shows that GA-HH proposed in the present invention has the same result after 10 calculations in terms of solution quality, and the SD value is 0. The solution result is consistent with the optimal solution obtained in the literature, but GA-HH is far superior to all 16 groups of benchmark instances in terms of solution time, verifying the generality and efficiency of GA-HH in solving the gapless multi-row facility layout problem.

[0145] To further verify the performance of GA-HH, 8 instances are selected. GA-HH and GA are applied to solve each instance 10 times, and box plots are drawn based on the solution results, as Figure 7 shown. RPD is the deviation of each solution result from the minimum objective value, and the calculation formula is RPD = (F - F min ) / F min × 100%, where F represents the current objective value in each calculation, and F min represents the minimum objective value in the solution process of the algorithm. The upper and lower margins, upper and lower box lines, solid line, dashed line, and + in the box plot represent the upper and lower quartiles, maximum and minimum values, median, average value, and data outliers of the solution results, respectively.

[0146] When solving small-scale instances (9 - 13), the RPD of GA-HH is 0. The solution result of GA-HH is the same as that of the solver, indicating that the algorithm can determine the optimal solution each time. For medium and large-scale benchmark test instances, the height of the rectangle box of GA-HH is lower than that of GA in most cases, indicating higher solution stability, further verifying that GA-HH has good solution performance. The black lines of the rectangle boxes of GA-HH are all lower than those of GA, indicating that the data deviation of the GA-HH solution is smaller, further verifying the superior performance of GA-HH.

[0147] In addition, to further analyze the performance of the proposed algorithm, instance Am15 is selected, and GWO-GA, GA, and GA-HH1 are used to solve the constrained multi-row facility layout problem and compared with GA-HH proposed in the present invention. The iterative convergence of different algorithms is plotted for comparison. The number of iterations is 100. In Figure 8Among them, the target value of GA-HH has an ascending part because the Monte Carlo move acceptance criterion is adopted during the algorithm operation, which will accept worse solutions with a certain probability. Therefore, it jumps out of the local optimum, which also shows the effectiveness of the Monte Carlo operation. Similarly, GA-HH and GA find better values than GWO-GA and GA-HH1, and the convergence rate of GA-HH is faster than that of GA, and it is superior to GWO-GA and GA-HH1. This once again proves that GA-HH has good solving performance.

[0148] As described above, it is only the preferred specific implementation mode 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 those skilled in the art within the technical scope disclosed in the embodiments of the present invention should be covered by the protection scope of the present invention. Therefore, the protection scope of the present invention should be subject to the protection scope of the claims.

Claims

1. A constraint-based multi-row facility layout method for a production workshop, characterized in that, It includes the following steps: S100. Establish a mixed-integer programming model aiming at minimizing the material handling cost for the multi-line facility layout problem. The model takes into account the designated row positioning constraints and sorting constraints of facilities. S200. Solve the mixed-integer programming model to obtain a constrained multi-line facility layout plan. The mathematical model is as follows: Objective function: The constraint conditions are: dx ij ≥x i -x j ,1 ≤ i < j ≤ n dx ij ≥ x j -x i , 1 ≤ i < j ≤ n dy ij ≥y i -y j , 1 ≤ i < j ≤ n dy ij ≥y j -y i , 1 ≤ i < j ≤ n q ijk +q jik ≥α ik +α jk -1, 1 ≤ i, j ≤ n, i ≠ j, q ijk +q jsk +q sik ≤2, 1≤i<j≤n, s≠j, s≠i, α ik ∈{0,1}, q ijk ∈ {0, 1}, 1 ≤ i, j ≤ n, i ≠ j, p i = m, m ∈ N In the formula, n is the number of facilities. i, j, and s are facility numbers, where i, j, s, m ∈ N and N is the set of facilities, N = {1, 2, …, n}; f ij The material flow between facilities i and j; dx ij Is the coordinate distance between facilities i and j in the x direction; dy ij Is the coordinate distance between facilities i and j in the y direction; x i 、x j Are the abscissa values of the center points of facilities i and j respectively; y i 、y j Are the ordinate values of the center points of facilities i and j respectively; l i 、l j 、l s Are the lengths of facilities i, j, and s respectively; w is the width of the passage; p i Is the location index of facility i; K is the set of rows, K = {1, 2, …, k}, where k is the row number; A m Is the set of facilities assigned to location m; B k Is the set of facilities assigned to row k; q (·)(·)k Is a decision variable. The first two symbols in the subscript are facility numbers, where the first symbol is the number of the first facility and the second symbol is the number of the second facility. For example, if both the first facility and the second facility are assigned to row k and the first facility is arranged to the left of the second facility, then q (·)(·)k = 1, otherwise q (·)(·)k = 0. For instance, if facilities i and j are assigned to row k and facility i is arranged to the left of facility j, then q ijk = 1, otherwise q ijk = 0; α ik Is a decision variable. If facility i is assigned to row k, then α ik = 1, otherwise α ik = 0; M is a sufficiently large constant, taking the sum of the lengths of all facilities.

2. The constraint-based multi-line facility layout method for a production workshop according to claim 1, wherein In step S2, a hyper-heuristic method based on the genetic algorithm is used to solve the mixed-integer programming model. The hyper-heuristic method based on the genetic algorithm includes a high-level strategy and a low-level heuristic operator. The solution is composed of a low-level heuristic vector and a facility sequence vector. The high level uses the genetic algorithm, takes the reward mechanism as the selection mechanism of the genetic algorithm, adaptively and selectively applies the low-level heuristic operator to the low-level heuristic operator vector to update the low-level heuristic operator vector. Then, the operators in the low-level heuristic vector act on the facility sequence vector in sequence to update the facility sequence vector. The Monte Carlo acceptance criterion is used as the movement acceptance criterion of the genetic algorithm.

3. The constraint-based multi-line facility layout method for a production workshop according to claim 2, wherein, When using the hyper-heuristic method based on the genetic algorithm to solve, it specifically includes the following steps: S210. Initialize parameters, including population size P, maximum number of iterations Max_It, maximum number of iterations of the low-level heuristic operator LMax_It, crossover probability Pc, mutation probability Pm, initial scores Sc of each low-level heuristic operator, number of low-level heuristic operators L, and constraint matrices A m and B k ; S220. Generate an initial solution in real number coding based on heuristic rules and form an initial population, and screen the optimal solution in the initial population based on the objective value of each individual. S230. Randomly select a low-level heuristic operator for calculation for each individual in the initial population, update the optimal solution and score the selected low-level heuristic operator according to the reward mechanism, and normalize the cumulative scores of all low-level heuristic operators to update Sc. The selection rule of the reward mechanism is: if the optimal solution is improved before and after the update, the score of the selected low-level heuristic operator increases, otherwise the score of the selected low-level heuristic operator decreases. S240. Select an individual from the initial population, and select low-level heuristic operators based on the screening mechanism to form the low-level heuristic operator vector of this individual, so as to update the low-level heuristic operator vector of this individual. Screening mechanism: In odd-numbered iterations, select the four heuristic operators with the highest percentage to form the low-level heuristic vector; in even-numbered iterations, select the low-level heuristic operator according to the probability by roulette wheel to obtain the low-level heuristic vector. S250. Cross and mutate the low-level heuristic vector obtained in the selection process of S240 to obtain a new low-level heuristic vector to update the low-level heuristic vector of this individual. Process the facility sequence vector of this individual according to the order of the low-level heuristic vector in this individual to update this individual. During the update process, each operator acts on the facility sequence vector in turn. When each operator updates the facility sequence vector, heuristic rule judgment is performed. If the facility sequence does not meet the constrained layout conditions, the operation of this operator is performed again until the facility sequence meets the constrained layout conditions. Then output the improved solution and perform the exponential Monte Carlo acceptance criterion judgment, update the optimal solution and update the corresponding operator scores according to the reward mechanism. S260. Repeat steps S240 and S250 to traverse the individuals in the initial population. S270. Repeat S240 - 260 for iteration until the number of iterations is greater than Max_It, output all current optimal solutions as the calculation result, and end the algorithm. S280. Output the optimal solution as the constrained multi - row facility layout plan.

4. The constraint-based multi-line facility layout method for a production workshop according to claim 3, characterized in that: The encoding of the initial solution consists of two parts, namely the encoding of the low - level heuristic operator and the encoding of the facility sequence; the encoding of the low - level heuristic operator is a sequence composed of each heuristic operator, and each number in the sequence corresponds to an operator; the encoding of the facility sequence includes a facility sequence array and a breakpoint array. The facility sequence array is used to determine the arrangement order of all facilities, and the breakpoint array is used to represent the positions of the breakpoints. The decoding of the facilities divides the facility sequence into rows according to the facility sequence array and the breakpoint array.

5. The constraint-based multi-line facility layout method for a production workshop according to claim 3, wherein: The described low - level heuristic operators are specifically: Operators for modifying the facility sequence array: two - point swap operator, reverse operator, partially mapped crossover operator, insertion mutation operator; Operators for modifying the breakpoint array: change left breakpoint operator, change right breakpoint operator, randomly change breakpoint value operator.

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