A method for solving joint optimization of aisle layout and job shop scheduling

Abstract

A method for solving the joint optimization of aisle layout and job shop scheduling, relates to the technical field of workshop layout optimization, and comprises the following steps: establishing a mathematical model for minimizing the total logistics cost between equipment and minimizing the maximum completion time in two stages, iteratively screening out the optimal layout plan after the cross-variation fusion of the population generated in the previous stage, associating the optimal layout plan with the obtained population after the cross-variation fusion of the population generated in the latter stage, iteratively calculating until the iteration condition is satisfied, and outputting the optimal joint solution; compared with the research method in the prior art that studies the layout problem and the scheduling problem as two separate problems and fails to consider the influence of the transportation time on the completion time under different layouts, the present invention can jointly investigate the two problems, always consider the influence of the layout on the transportation time in the process of scheduling optimization, and obtain a better scheduling completion time.

Description

Technical Field

[0001] The present invention relates to the technical field of workshop layout optimization, and in particular to a method for solving the joint optimization of aisle layout and job shop scheduling. Background Art

[0002] When designing a high-efficiency, low-cost manufacturing system, we often encounter some engineering optimization problems of layout planning and production scheduling. From a production perspective, in the optimized layout of some factories in the United States, material handling costs account for 50% of the total operating costs, and at least 30% of these transportation costs can be saved through layout planning and design. Production scheduling optimizes the manufacturing process from the perspective of delivery time, energy saving and other goals, and almost every batch of parts manufacturing requires scheduling. These two problems cannot be ignored for manufacturing companies to reduce costs, and are often solved as two independent problems in current applications and research. They can usually be summarized as the corridor allocation problem (CAP) and the job-shop scheduling problem (JSP). Among them, the definition of CAP is to arrange several equipment of the same width and different length along two horizontal lines parallel to the x-axis of the Cartesian coordinate system. How to arrange the equipment can minimize the logistics interaction between the equipment. This layout problem has two significant characteristics. First, there is no gap or overlap between adjacent facilities in the same row. Second, the starting point on the left side of the two rows of facilities starts from the coordinate axis x=0 and is arranged to the right. The initial definition of JSP is how to set the processing plan to minimize the processing time for a number of workpieces that need to be processed and a number of processing equipment, and each workpiece has a specific processing sequence between each process. From a computational perspective, these two optimization problems are very complex combinatorial optimization problems, and both are NP-hard problems, which makes it more difficult to jointly optimize the two, and traditional planning methods will undoubtedly be more difficult to solve. Therefore, it is urgent to use advanced intelligent optimization algorithms to examine the joint application of these two types of problems, so as to optimize workshop production layout planning problems from more angles. Summary of the invention

[0003] In view of this, the main purpose of the present invention is to provide a method for solving the joint optimization of aisle layout and job shop scheduling, taking the logistics interaction frequency and transportation time as the connection points to form a combined problem of aisle layout and job shop scheduling, improving the whole process and layout integrity of studying such scheduling problems, and making the optimization process more comprehensive.

[0004] The technical solution of the present invention is a method for solving the joint optimization of aisle layout and job shop scheduling, comprising the following steps:

[0005] Step S1: Divide the algorithm optimization process into two stages, and establish a mathematical model for the first stage with the goal of minimizing the total logistics cost between devices.

[0006] Step S2: Establish the mathematical model of the second stage with the goal of minimizing the maximum completion time.

[0007] Step S3: Determine the crossover rate Pc, mutation rate Pm, mutation coding ratio Lm, and the population size nind of the first stage in the two stages respectively CAP 、The second stage population size nind JSP , and the maximum number of iterations in the first stage iter_max CAP and the maximum number of iterations in the second stage iter_max JSP ;

[0008] Step S4: Perform the first stage of the algorithm optimization. First, the number of chromosomes generated by encoding and decoding is nind CAP The first-stage initial population is obtained by performing crossover and mutation operations on the first-stage initial population to obtain the first-stage crossover offspring population and the first-stage mutation offspring population;

[0009] Step S5: The crossover offspring population and the mutation offspring population of the first stage are merged with the initial population of the first stage. The target values ​​of all individuals in the merged population are calculated according to the mathematical model of the first stage. The target values ​​of all individuals are used as fitness values ​​for roulette sampling to obtain nind CAP The sampled population of chromosomes;

[0010] Step S6: Use the sampled population in step S5 to replace the first-stage initial population in step S4 and return to step S4, which is recorded as increasing the number of iterations by one, i.e. iter=iter+1, and perform iterative operations in the order of the steps starting from step S4, and loop until the number of iterations iter>iter_max CAP The iteration is terminated when , and the minimum value in the sampled population at this time is output as the optimal layout solution, completing the first stage of optimization;

[0011] Step S7: Perform the second stage of the algorithm optimization. First, the number of chromosomes generated by encoding and decoding is nind JSP The second-stage initial population is obtained by performing crossover and mutation operations on the second-stage initial population to obtain the second-stage crossover offspring population and the second-stage mutation offspring population;

[0012] Step S8: Fusing the second-stage crossover offspring population and the second-stage mutation offspring population with the second-stage initial population, and combining the fused population with the optimal layout solution obtained in step S6, respectively associating the optimal layout solution with any individual in the fused population to obtain a combined population;

[0013] Step S9: Calculate the joint target value of all individuals in the joint population according to the second-stage mathematical model, sort the joint target values, and then use the sorted joint target values ​​as fitness values ​​for roulette sampling to obtain nind JSP The sampled population of chromosomes;

[0014] Step S10: Use the sampled population in step S9 to replace the second-stage initial population in step S7 and return to step S7, and record it as an increase in the number of iterations, i.e., iter=iter+1. At the same time, perform iterative operations in the order of the steps starting from step S7, and loop until the number of iterations iter>iter_max JSP The iteration is terminated when , and the minimum value of the joint objective value at this time is output as the optimal joint solution.

[0015] The technical effects of the present invention are:

[0016] Compared with the prior art method that studies layout problems and scheduling problems as two separate problems and fails to consider the impact of transportation time on completion time under different layouts, the present invention can jointly investigate the two problems, always consider the impact of layout on transportation time during the scheduling optimization process, and obtain a better scheduling completion time. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] Figure 1 It is a flow chart of the algorithm of the present invention;

[0018] Figure 2 It is the basic principle of the two-stage optimization strategy in the present invention;

[0019] Figure 3 is a plan view of the integer encoding in the present invention;

[0020] Figure 4 It is the 3×3 case coding Gantt chart in the present invention;

[0021] Figure 5 It is a schematic diagram of the CAP cross-repair process in the present invention;

[0022] Figure 6 It is a schematic diagram of the JSP cross repair process in the present invention;

[0023] Figure 7 It is a schematic diagram of the 3-opt mutation operation in the present invention; DETAILED DESCRIPTION

[0024] The present invention will be further described in detail below in conjunction with the embodiments.

[0025] 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 combination with 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.

[0026] Example:

[0027] The present invention proposes a method for solving the joint optimization of aisle layout and job shop scheduling, comprising the following steps:

[0028] Step S1: The algorithm optimization process is divided into two stages, and the mathematical model of the first stage is established with the goal of minimizing the total logistics cost between devices. The objective function of the mathematical model is shown in formula (1):

[0029]

[0030] Where i, j are device numbers; n is the number of devices; f ij is the logistics volume from device i to device j, where i,j∈I + , I + ={0,1,2,…,n}; d ij is the distance between the logistics interaction point of equipment i and equipment j, where i,j∈I + , I + ={0,1,2,…,n}; w is the width of the aisle;

[0031] Its constraints are shown in equations (2)-(12):

[0032]

[0033] x 0 =0 (3)

[0034] d ij ≥x i -x j ,(0≤i<j≤n) (4)

[0035] d ij ≥x j -x i ,(0≤i<j≤n) (5)

[0036] d ji ≥x i -x j ,(0≤i<j≤n) (6)

[0037] dji ≥x j -x i ,(0≤i <j≤n) (7)

[0038]

[0039]

[0040]

[0041]

[0042] α ij ∈{0,1},(i,j∈I,i≠j) (12)

[0043] Among them, x i , x j is the coordinate value of the mechanical equipment; l i is the length of equipment i along the aisle; l j is the length of equipment j along the aisle; l k is the length of device k along the aisle; n is the number of devices; i, j, k are the device numbers; α ki is a binary variable. If facility k is to the left of facility i and in the same row, then α ki =1, otherwise α ki =0; α kj is a binary variable. If facility k is to the left of facility j and in the same row, then α kj =1, otherwise α kj =0;x 0 is the coordinate value of the main warehouse. The raw material warehouse and the finished product warehouse are considered to be in the same warehouse, and the coordinate value of the warehouse is x 0 ;d ij is the distance between the logistics interaction point of equipment i and equipment j, where i,j∈I + ;I,I + is a set of device numbers, I = {1, 2, ..., n}, I + ={0,1,2,…,n};α ij is a binary variable. If device i is to the left of device j and both are in the same row, then α ij =1, otherwise α ij =0; α ji is a binary variable. If device j is to the left of device i and both are in the same row, then α ji =1, otherwise α ji =0; α ik is a binary variable. If facility i is to the left of facility k and in the same row, then α ik =1, otherwise α ik =0; α jkis a binary variable. If facility j is to the left of facility k and in the same row, then α jk =1, otherwise α jk =0;

[0044] Step S2: With the goal of minimizing the maximum completion time, a mathematical model for the second stage is established. The objective function of the mathematical model is shown in formula (13):

[0045] F 2 =minC max (13)

[0046] Among them, C max is the maximum completion time of the workpiece;

[0047] The constraints are shown in equations (14)-(25):

[0048]

[0049]

[0050] s ai ≥0, (i∈I + , a∈J) (16)

[0051]

[0052]

[0053]

[0054]

[0055]

[0056]

[0057] tt ija ≥0,(i,j∈I + ,i≠j,a∈J) (23)

[0058] β abi ∈{0,1},(i∈I,a,b∈J + ) (twenty four)

[0059] o ija ∈{0,1},(i,j∈I + ,a∈J) (25)

[0060] Where a, b are workpiece numbers; m is the number of workpieces; s ai is the start processing time of workpiece a on equipment i; s ajis the start processing time of workpiece a on equipment j; p ai is the processing time of workpiece a on equipment i; tt ija is the time required to transport workpiece a from equipment i to equipment j; M is a positive number, and M>1000000; o ija is a binary variable, indicating the processing order of each workpiece. If equipment j processes a immediately after equipment i, then o ija =1, otherwise o ija =0;s bi is the start processing time of workpiece b on equipment i; β abi is a binary variable, indicating the processing order of workpieces on each device. If workpiece b is processed before workpiece a on device i, then β abi =1, otherwise β abi =0; J,J + is a set of workpiece numbers, J = {1, 2, ..., m}, J + ={0,1,2,…,m};β 0bi is a binary variable, indicating that if workpiece b is the first workpiece processed on equipment i, then β 0bi =1, otherwise β 0bi =0;β a0i is a binary variable, indicating that if workpiece a is the last workpiece processed on equipment i, then β a0i =1, otherwise β a0i =0;tt i0a is the transportation time of the workpiece from the i device to the finished product warehouse as the 0th process after the workpiece is processed on the device i through the last process; d ij is the distance between the logistics interaction point of equipment i and equipment j, where i,j∈I + ; w is the aisle width; v is the speed of the transport equipment.

[0061] In the established two-stage mathematical model, equations (1) and (13) represent the two objectives of the two-stage optimization. The aisle layout model in the first stage is optimized to minimize the total logistics cost F1 between equipment, and the workshop scheduling model in the second stage is optimized to minimize the maximum completion time F2. The two models can communicate information using the logistics flow table f and the equipment distance table d respectively. The communication method is as follows: Figure 2 shown.

[0062] In the first stage model, formula (2) calculates the coordinates of each equipment logistics interaction point along the x-axis; formula (3) means setting the coordinates of equipment No. 0 to 0, that is, the coordinates of the raw material warehouse and the finished product warehouse are 0; formulas (4)-(7) represent the calculation of the transportation distance of the interaction points between processing equipment along the x-axis. This transportation distance is the absolute value of the difference between the equipment coordinates. The formulas here are all expressed in the form of absolute value linearization; formula (8) means that two adjacent equipment do not overlap; formulas (9)-(12) are used to constrain the decision variable α so that its value corresponds to a reasonable arrangement plan and eliminates unreasonable decision space.

[0063] In addition, in the second stage model, formula (14) represents the close processing relationship between different processes of the same workpiece, ensuring that when equipment i processes workpiece a before equipment j, the start time of workpiece a on equipment j is greater than the start time of workpiece a on equipment i; formula (15) represents the processing relationship between different workpieces of the same equipment, ensuring that when workpiece a is processed before workpiece b on equipment i, the start time of workpiece b processed by equipment i is greater than the start time of workpiece a processed by equipment i; formula (16) indicates that the start time is a non-negative number, which limits the range of the start time; formulas (17)-(20) and formula (24) jointly limit the value range of β, so that the scheduling plan is in line with the actual situation. Wherein, equations (17) and (18) respectively indicate that there is only one first process and one last process processed by equipment i, while equations (19) and (20) respectively indicate that there is only one preceding process and one following process for each process; equation (21) indicates that the maximum time consumed for transporting all workpieces to the finished product warehouse after the last process is completed is the maximum completion time; equation (22) indicates the transportation time between two equipments, and the constant v indicates the speed of the handling equipment. If you want to express the transportation time more accurately, the constant v can be expanded to v ij Matrix, that is, the transportation speeds between different devices are different, to more finely express the transportation time between devices; Formula (23) indicates that the transportation time is a non-negative number; Formula (25) represents the 0-1 variable of the processing order of workpiece a, equipment 0 represents the raw material warehouse and the finished product warehouse, o 0ja 1 means that equipment j is the first process for processing workpiece a, that is, the raw materials are transported from the raw material warehouse to equipment j, and o i0a A value of 1 indicates that equipment i is the last step in processing workpiece a, that is, the finished product is transported from equipment i to the finished product warehouse. This matrix is ​​a known quantity.

[0064] Step S3: Determine the crossover rate Pc, mutation rate Pm, mutation coding ratio Lm, and the population size nind of the first stage in the two stages respectively CAP 、The second stage population size nind JSP , and the maximum number of iterations in the first stage iter_max CAP and the maximum number of iterations in the second stage iter_max JSP ;

[0065] In order to balance the solution time and solution quality, the maximum number of iterations in the two stages is set according to the problem scale and trial calculations.

[0066] Step S4: Perform the first stage of the algorithm optimization. First, the number of chromosomes generated by encoding and decoding is nind CAP The first-stage initial population is obtained by performing crossover and mutation operations on the first-stage initial population to obtain the first-stage crossover offspring population and the first-stage mutation offspring population;

[0067] Coding and decoding is a mapping method between the coding set and the feasible solution set in the algorithm, so that the coding can correspond to the feasible solution. By observing the characteristics of the feasible solutions of CAP and JSP, it is easy to find that the coding methods of the two problems are quite different, and each has a different meaning. Therefore, a two-stage coding method is used in this algorithm, and the coding also uses their own operation rules when crossover and mutation generate new individuals.

[0068] The first section of the code adopts the integer coding method widely used in CAP, with the device number as the gene bit, and a non-repeating arrangement of numbers is a layout method. For example, the layout scheme of the coding sequence [7,1,9,2,5,8,4,3,6] on the plane is as follows Figure 3 As shown, when the auxiliary code t=5, it means that the number of devices in the first row of the aisle is 5, that is, the first row of devices is [7, 1, 9, 2, 5], and the rest are the second row of devices [8, 4, 3, 6].

[0069] The second section of the code adopts the process-based coding method that is widely used in JSP. The workpiece number is the gene position of the chromosome. The order of different workpiece numbers indicates the order of processing on the same equipment, thus forming a scheduling plan. Taking a 3×3 job scheduling as an example, the meaning of the scheduling plan corresponding to the coding sequence [3,1,2,1,2,3,3,2,1] is: the order of the same workpiece indicates the order of the workpiece process, such as the 3 in the first position of the code means that the process 1 of workpiece 3 is processed first, and the 3 in the seventh position of the code means that the process 3 of workpiece 3 is processed last; the order of different workpieces indicates the processing order on the equipment, such as the process 1 of the second workpiece 1 is before the process 2 of the sixth workpiece 3, and both processes are processed on M1, then workpiece 1 is processed before workpiece 3 on equipment 1. See below for a more intuitive scheduling. Figure 4 .

[0070] In addition, since the initial population in the first stage is the CAP initial population, the CAP code is cross-repaired using the partial mapping cross-repair method. The specific process is as follows: Figure 5 shown.

[0071] The mutation operations in the two stages are calculated in proportion. For example, in a joint case, the number of codes in the two stages is not equal and the total number of scheduling codes is greater than the total number of layout codes. Therefore, the number of mutation codes is calculated in proportion. If the mutation code ratio value L m is 0.1, then L=ceil(L m ×num), ceil is the rounding up function, and num is the total number of codes in a section. For example, if you need to layout 6 facilities and schedule 36 processes (6 workpieces × 6 processes per workpiece), the total number of codes for the two problems, num, is 6 and 36 respectively. The number of mutations required for the layout code is ceil(5×0.1)=1. If the number is less than 2 and cannot be mutated, the number is set to 2, and the number of mutations required for the scheduling code is ceil(36×0.1)=4. Then the layout code uses the 2-opt operator for mutation operation, while the scheduling code uses the 8-opt operator for mutation operation. Figure 7 This is a schematic diagram of the 3-opt operation. The codes that need to be mutated are 2, 1, and 3 in the dotted lines. In order of left and right, 2 is moved to the gene position of 1, 1 is moved to the gene position of 3, and 3 is moved to the gene position of 1 to obtain new offspring. It can be seen that 3-opt is a mutation of three gene positions, and the corresponding 2-opt, 4-opt, etc. represent the mutation processes of two gene positions and four gene positions respectively, that is, the k-opt operation is deduced by analogy, where the value of k is L.

[0072] Step S5: The crossover offspring population and the mutation offspring population of the first stage are merged with the initial population of the first stage. The target values ​​of all individuals in the merged population are calculated according to the mathematical model of the first stage. The target values ​​of all individuals are used as fitness values ​​for roulette sampling to obtain nind CAP The sampled population of chromosomes;

[0073] Step S6: Use the sampled population in step S5 to replace the first-stage initial population in step S4 and return to step S4, which is recorded as increasing the number of iterations by one, i.e. iter=iter+1, and perform iterative operations in the order of the steps starting from step S4, and loop until the number of iterations iter>iter_max CAP The iteration is terminated when , and the minimum value in the sampled population at this time is output as the optimal layout solution, completing the first stage of optimization;

[0074] Step S7: Perform the second stage of the algorithm optimization. First, the number of chromosomes generated by encoding and decoding is nind JSP The second-stage initial population is obtained by performing crossover and mutation operations on the second-stage initial population to obtain the second-stage crossover offspring population and the second-stage mutation offspring population;

[0075] The initial population in the second stage is the JSP initial population, and the JSP coding sequence is repaired using the counting method. Figure 6 , code sub-generation C 1 If there are too many workpieces 2 and too few workpieces 3, the number of workpiece processes and their respective position information are recorded, and the workpiece number with insufficient workpieces in any process is replaced and repaired with the workpiece number with excessive workpieces in any process according to the quantity and position index.

[0076] Step S8: Fusing the second-stage crossover offspring population and the second-stage mutation offspring population with the second-stage initial population, and combining the fused population with the optimal layout solution obtained in step S6, respectively associating the optimal layout solution with any individual in the fused population to obtain a combined population;

[0077] Step S9: Calculate the joint target value of all individuals in the joint population according to the second-stage mathematical model, sort the joint target values, and then use the sorted joint target values ​​as fitness values ​​for roulette sampling to obtain nind JSP The sampled population of chromosomes;

[0078] Step S10: Use the sampled population in step S9 to replace the second-stage initial population in step S7 and return to step S7, and record it as an increase in the number of iterations, i.e., iter=iter+1. At the same time, perform iterative operations in the order of the steps starting from step S7, and loop until the number of iterations iter>iter_max JSP The iteration is terminated when , and the minimum value of the joint objective value at this time is output as the optimal joint solution.

[0079] The following is a comparison of the solution effect of this embodiment and a demonstration of the effect in actual application:

[0080] Program testing environment:

[0081] The processor is Intel(R) Core(TM) i5-9500, main frequency is 3.00GHz, memory is 8GB, the operating system is Windows 10, the mathematical model solving software is IBM ILOG CPLEX Optimization Studio 12.10.0.0 (referred to as CPLEX) exact solver, and the heuristic algorithm programming environment is MATLAB R2020b.

[0082] 1. Comparison of solution effects:

[0083] When solving jointly, the difficulty of solving the JSP problem is greater than that of solving the CAP problem, so the focus of solving the JSP problem is placed on the JSP problem. In the CPLEX precise calculation, for large-scale cases where the exact value cannot be calculated in a suitable time, the relative mixed integer programming gap tolerance (Relative Mixed Integer Programming Gap Tolerance, referred to as Gap) indicator is used to measure the quality of the solution. The Gap calculation is obtained by querying the technical documentation of the solver CPLEX as shown in formula (26).

[0084]

[0085] In formula (26), best integer is the best integer target, best bound is the target of the remaining best nodes, when the value is less than the set Gap value, the mixed integer optimization calculation will stop, and the default value of Gap is 1×10 -4 , if it is less than the default value, it means that the value sought is the global optimal solution.

[0086] Both are the scheduling times obtained under the same scheduling scheme. The latter is the method proposed in which the layout is considered when calculating the scheduling time. The former is the traditional method, where the two are calculated separately and then the transportation time generated by the layout is added to obtain the latter. We introduce the indicator up to highlight the optimization effect, and its formula is as follows (27)

[0087]

[0088] The data symbols in Table 2 are now explained. 1 and F 2 They represent the CAP target value and JSP target value in the joint problem respectively; F uni represents the target value of the original JSP optimization alone. The specific results are shown in Table 1. Opt is the world-recognized F uni The optimal value or the best value that can be obtained; F uni2 Indicates that in 2 Under the same layout scheme, F uni The dispatching scheme corresponding to the target value is substituted into the joint dispatching model for solution, that is, the optimal dispatching scheme alone is added with the transportation time to obtain F uni2 , F 2 and F uni2 They represent the joint optimization with layout considered and the separate optimization without layout considered and then adding transportation time; Up is F 2 and F uni2 The effect parameter for comparing the two is calculated as shown in formula (27). This value shows the maximum completion time improvement ratio of the original scheduling scheme and the joint scheduling scheme under the same layout (that is, the same transportation time) conditions.

[0089] Table 1 Original JSP separate solution results

[0090]

[0091] Note: The CPLEX calculation time T efficiency values ​​in this table are

[0092] Keep the integer place;

[0093] Table 2 CPLEX solution joint case results

[0094]

[0095]

[0096] Note: 1. The effective digits of the solution time T and Gap value data in this table are the same as those in Table 7;

[0097] 2. The valid values ​​of Up in this table should be rounded to two decimal places.

[0098] Table 3 lists the algorithm runtime parameter settings. In a joint case, n is the total number of devices; m is the total number of workpieces; n ... JSP 、nind CAP are the total number of populations in each stage; iter_max JSP 、iter_max CAP are the maximum number of iterations in each stage; P c is the crossover probability, P c =0.2; P m is the mutation probability, P m =0.4; L m is the variation coding ratio, L m =0.1; the value of CAP auxiliary code t is set to floor(n / 2-2) and floor(n / 2), where floor is a rounding function.

[0099] Table 3 TSGA algorithm calculation joint case parameter settings

[0100]

[0101] The symbols in Table 4 are explained below. Each case is calculated 10 times to minimize the randomness of the algorithm. 1cplex 、F 2cplex They are respectively F in Table 4 1 and F 2 ; F 1 、F 2 is the best value among the 10 results; F 1mean 、F 2mean is the average of 10 results; T 1 , T2 The average time taken to run the algorithm 10 times; SD 1 , SD 2 10F 1 、F 2 The standard deviation of is used as a measure of the discreteness of the results obtained by the algorithm 10 times.

[0102] Table 4 TSGA algorithm verification solution joint case

[0103]

[0104]

[0105] Note: 1. The algorithm operation time T in this table 1 and T 2 Keep two decimal places;

[0106] 2. Standard deviation SD in this table 1 and SD 2 Keep two decimal places.

[0107] 3. Two-phase Genetic Algorithm (TSGA)

[0108] Compare F in Table 4 2cpelx and F 2 It can be seen that the two-stage genetic algorithm can optimize the proposed problem, and the results obtained are cross-correlated with the model results obtained by CPLEX, proving the correctness of the mixed integer programming model built above. Different joint cases have different complexities, so the optimization results are different, resulting in some cases where the algorithm finds the same or similar solutions as CPLEX, while some solutions are quite different; T 1 The running time of the SD algorithm is also volatile, and the longer the solution time is, the greater the volatility; 1 and SD 2 The values ​​are all small, indicating that the parameter settings are reasonable and the final solution has good stability.

[0109] It can be seen that after numerous case calculations, it is fully proved that the proposed new problem has certain value in finding a better layout and scheduling, which shows that the two-stage method in the present invention has a good solution effect for this type of problem.

[0110] 2. Practical application cases:

[0111] In order to further explore the feasibility and effectiveness of applying the proposed joint optimization problem in reality, this section demonstrates the details of an application model by solving a real workshop case. The process is to collect the processing time and equipment data of a workshop, simplify the data, and then substitute the processed data into the model mentioned above to solve and obtain the joint optimization result.

[0112] A company is equipped with a processing workshop for some parts of aviation structural parts. Its processed parts and processing information are shown in Table 4. The process number in Table 4 is the processing order of the workpieces, J represents different workpieces to be processed, and the listed time is the number of hours required for a process. The processing flow of this company is a typical mechanical manufacturing processing workshop, involving processing technologies such as CNC vertical lathe 1, end face turning 2, boring 3, groove turning 4, milling plane 5, outer circle turning 6, surface treatment 7, inspection 8, bench work 9, heat treatment 10. For the convenience of representation, the process processing area is simplified to M in sequence. 1 、M 2 、M 3 、M 4 、M 5 、M 6 、M 7 、M 8 、M 9 、M 10 , the raw material warehouse and finished product warehouse are M 0 , the process sequence is simplified to O 1 , O 2 , O 3 , O 4 , O 5 , O 6 , O 7 , O 8 , O 9 The dimension data of some common processing areas in production workshops are shown in Table 6. The dimensions of each processing area are almost the same in width, but the lengths vary greatly. The width difference is ignored and the length is used as the layout data. The width of the aisle layout channel is set to 3m.

[0113] Two conclusions can be drawn from the collected raw processing data in Table 5. First, the workpiece process processing does not conform to some assumptions of job shop scheduling, such as not all equipment needs to process each part, and not all parts have the same number of processes and are processed once by all equipment. Instead, it is a sub-problem of the ideal job shop scheduling problem, which requires some processing of the raw data; second, considering the complexity of the actual situation, the transportation time will fluctuate greatly due to many factors such as the difference in transportation equipment and loading and unloading time.

[0114] Based on the above two observations, the collected data is first formatted to obtain standardized data that can be applied to the model and listed in Table 7. The processing method is to assume that all workpieces are to be processed on all equipment, and the process time of the process that does not exist in the actual workpiece is set to 0, so that the process has no effect on the maximum completion time; secondly, different transportation equipment speeds are set for comparative calculation to explore the impact of different transportation times on the optimization results. The calculation results are shown in Table 8.

[0115] Table 5 Workpiece process information table

[0116]

[0117]

[0118] Note: 1. “—” in the table means that the workpiece does not have this process;

[0119] 2.The unit of process processing time in this table is hour.

[0120] Table 6 Processing area dimensions

[0121]

[0122] Table 7 Workpiece processing time and processing sequence table

[0123]

[0124] Table 8 CPLEX solution example results table

[0125]

[0126]

[0127] Note: The speed unit in the table is m / h. This speed applies to all images.

[0128] The comprehensive speed is calculated based on the factors affecting transportation time;

[0129] The results of the example in Table 8 prove again that the joint optimization proposed in the present invention has a better optimization effect. The scheduling time F obtained by joint optimization under most transportation speeds is 2 are all smaller than their independent optimization F uni2 The greater the proportion of transportation time to processing time, the more significant the effect of joint optimization.

[0130] 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 method for solving the joint optimization of aisle layout and job shop scheduling, characterized in that: The following steps are involved: Step S1: The algorithm optimization process is divided into two stages, and the mathematical model of the first stage is established with the goal of minimizing the total logistics cost between devices. The objective function of the mathematical model is shown in formula (1): Where i, j are device numbers; n is the number of devices; f ij is the logistics volume from device i to device j, where i,j∈I + , I + ={0,1,2,…,n}; d ij is the distance between the logistics interaction point of equipment i and equipment j, where i,j∈I + , I + ={0,1,2,…,n}; w is the width of the aisle; Its constraints are shown in equations (2)-(12): x0＝0 (3) d ij ≥x i -x j ,(0≤i<j≤n) (4) d ij ≥x j -x i ,(0≤i<j≤n) (5) d ji ≥x i -x j ,(0≤i<j≤n) (6) d ji ≥x j -x i ,(0≤i<j≤n) (7) a ij ∈{0,1},(i,j∈I,i≠j) (12) Among them, x i , x j is the coordinate value of the mechanical equipment; l i is the length of equipment i along the aisle; l j is the length of equipment j along the aisle; l k is the length of equipment k along the aisle; n is the number of equipment; i, j, k are the equipment numbers; α ki is a binary variable. If facility k is to the left of facility i and in the same row, then α ki =1, otherwise α ki =0; α kj is a binary variable. If facility k is to the left of facility j and in the same row, then α kj =1, otherwise α kj =0; x0 is the coordinate value of the total warehouse; d ij is the distance between the logistics interaction point of equipment i and equipment j, where i,j∈I + ;I,I + is a set of device numbers, I = {1, 2, ..., n}, I + ={0,1,2,…,n};α ij is a binary variable. If device i is to the left of device j and both are in the same row, then α ij =1, otherwise α ij =0; α ji is a binary variable. If device j is to the left of device i and both are in the same row, then α ji =1, otherwise α ji =0; α ik is a binary variable. If facility i is to the left of facility k and in the same row, then α ik =1, otherwise α ik =0; α jk is a binary variable. If facility j is to the left of facility k and in the same row, then α jk =1, otherwise α jk =0; Step S2: With the goal of minimizing the maximum completion time, a mathematical model for the second stage is established. The objective function of the mathematical model is shown in formula (13): <h2 style=";text-align:left;direction:ltr">F2 = minC<h2 style=";text-align:left;direction:ltr"> max <h2 style=";text-align:left;direction:ltr"> (13) Among them, C max is the maximum completion time of the workpiece; The constraints are shown in equations (14)-(25): s ai ≥0,(i∈I + ,a∈J) (16) tt ija ≥0,(i,j∈I + ,i≠j,a∈J) (23) b abi ∈{0,1},(i∈I,a,b∈J + ) (24) the ija ∈{0,1},(i,j∈I + ,a∈J) (25) Where a, b are workpiece numbers; m is the number of workpieces; s ai is the start processing time of workpiece a on equipment i; s aj is the start processing time of workpiece a on equipment j; p ai is the processing time of workpiece a on equipment i; tt ija is the time required to transport workpiece a from equipment i to equipment j; M is a positive number, and M>1000000; o ija is a binary variable, indicating the processing order of each workpiece. If equipment j processes a immediately after equipment i, then o ija =1, otherwise o ija =0;s bi is the start processing time of workpiece b on equipment i; β abi is a binary variable, indicating the processing order of workpieces on each device. If workpiece b is processed before workpiece a on device i, then β abi =1, otherwise β abi =0; J,J + is a set of workpiece numbers, J = {1, 2, ..., m}, J + ={0,1,2,…,m};β 0bi is a binary variable, indicating that if workpiece b is the first workpiece processed on equipment i, then β 0bi =1, otherwise β 0bi =0;β a0i is a binary variable, indicating that if workpiece a is the last workpiece processed on equipment i, then β a0i =1, otherwise β a0i =0;tt i0a is the transportation time of the workpiece from the i device to the finished product warehouse as the 0th process after the workpiece is processed on the device i through the last process; d ij is the distance between the logistics interaction point of equipment i and equipment j, where i,j∈I + ; w is the width of the aisle; v is the speed of the transport equipment; Step S3: Determine the crossover rate Pc, mutation rate Pm, mutation coding ratio Lm, and the population size nind of the first stage in the two stages respectively CAP 、The second stage population size nind JSP , and the maximum number of iterations in the first stage iter_max CAP and the maximum number of iterations in the second stage iter_max JSP ; Step S4: Perform the first stage of the algorithm optimization. First, the number of chromosomes generated by encoding and decoding is nind CAP The first-stage initial population is obtained by performing crossover and mutation operations on the first-stage initial population to obtain the first-stage crossover offspring population and the first-stage mutation offspring population; Step S5: The crossover offspring population and the mutation offspring population of the first stage are merged with the initial population of the first stage. The target values ​​of all individuals in the merged population are calculated according to the mathematical model of the first stage. The target values ​​of all individuals are used as fitness values ​​for roulette sampling to obtain nind CAP The sampled population of chromosomes; Step S6: Use the sampled population in step S5 to replace the first-stage initial population in step S4 and return to step S4, which is recorded as increasing the number of iterations by one, i.e. iter=iter+1, and perform iterative operations in the order of the steps starting from step S4, and loop until the number of iterations iter>iter_max CAP The iteration is terminated when , and the minimum value in the sampled population at this time is output as the optimal layout solution, completing the first stage of optimization; Step S7: Perform the second stage of the algorithm optimization. First, the number of chromosomes generated by encoding and decoding is nind JSP The second-stage initial population is obtained by performing crossover and mutation operations on the second-stage initial population to obtain the second-stage crossover offspring population and the second-stage mutation offspring population; Step S8: Fusing the second-stage crossover offspring population and the second-stage mutation offspring population with the second-stage initial population, and combining the fused population with the optimal layout solution obtained in step S6, respectively associating the optimal layout solution with any individual in the fused population to obtain a combined population; Step S9: Calculate the joint target value of all individuals in the joint population according to the second-stage mathematical model, sort the joint target values, and then use the sorted joint target values ​​as fitness values ​​for roulette sampling to obtain nind JSP The sampled population of chromosomes; Step S10: Use the sampled population in step S9 to replace the second-stage initial population in step S7 and return to step S7, and record it as an increase in the number of iterations, i.e., iter=iter+1. At the same time, perform iterative operations in the order of the steps starting from step S7, and loop until the number of iterations iter>iter_max JSP The iteration is terminated when , and the minimum value of the joint objective value at this time is output as the optimal joint solution.

2. The method for solving the joint optimization of aisle layout and job shop scheduling according to claim 1, characterized in that: The encoding method of the initial population in the first stage is integer encoding method.

3. The method for solving the joint optimization of aisle layout and job shop scheduling according to claim 1, characterized in that: The encoding method of the initial population in the second stage is a process-based encoding method.

4. The method for solving the joint optimization of aisle layout and job shop scheduling according to claim 1, characterized in that: The crossover method for the initial population in the first stage described in step S4 is a partial mapping crossover method.

5. The method for solving the joint optimization of aisle layout and job shop scheduling according to claim 1, characterized in that: The crossover method of the second stage initial population described in step S7 is the counting method.

6. The method for solving the joint optimization of aisle layout and job shop scheduling according to claim 1, characterized in that: The mutation operation is a k-opt operation mutation.

7. The method for solving the joint optimization of aisle layout and job shop scheduling according to claim 1 is characterized by: The crossover rate Pc=0.2 and the mutation rate Pm=0.4 of the two stages described in step S3.

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