A Mixed Integer Linear Programming Method for the Joint Scheduling of Multiple AGVs and Multiple Machines

Abstract

A mixed-integer linear programming model for the joint scheduling of multiple AGVs and multiple machines, which relates to the technical field of workshop scheduling, and particularly belongs to a mixed-integer linear programming model for the joint scheduling of multiple AGVs and multiple machines in an automated electrode foil production workshop. It includes Step 1: Define the model parameters and decision variables of the mixed-integer linear programming model, and take the minimum value Minimise C of the maximum completion time required for the production of the measured electrode foil max , as the objective value output by the mixed-integer linear programming model; Step 2: On the basis of determining the optimization objective, perform linear constraints on the mixed-integer linear programming model in terms of time and AGV capacity; Step 3: For the established mixed-integer linear programming model for the scheduling of multiple AGVs and multiple machines in the automated electrode foil production workshop, use the GUROBI solver for verification and obtain a Gantt chart according to the example. The present invention can make full use of the maximum carrying capacity of the AGV and has the positive effect of improving the production efficiency of the electrode foil.

Description

Technical Field

[0001] The present invention relates to the technical field of workshop scheduling, and in particular, belongs to a mixed integer linear programming method for the joint scheduling of multiple AGVs and multiple machines in an automated electrode foil production workshop. Background Art

[0002] In the upcoming Industrial 4.0 era, the automatic guided vehicle (AGV) is widely used in the manufacturing industry as a flexible and efficient transportation device. In an automated electrode foil production workshop, AGVs are used to transport foil rolls. The entire production line can be described as follows: all the foil rolls to be processed are stored in the material warehouse and need to be transported by AGVs to the machines for processing. After processing, the AGVs transport the finished products to the finished product warehouse. In order to improve the production efficiency of the automated electrode foil production workshop, it is necessary to ensure a reasonable distribution of the foil rolls in the workshop so that processing can be completed in the shortest possible time, which requires formulating an efficient scheduling solution. During the production process of electrode foils, two scheduling problems are involved: one is which AGV to assign to transport the foil rolls, and the other is which machine to assign the foil rolls to for processing, that is, the AGV scheduling problem and the parallel machine scheduling problem. Considering the above two problems comprehensively is the only way to truly improve the efficiency of an automated electrode foil production plant, that is, to consider the joint scheduling of AGVs and parallel machines.

[0003] In the field of workshop scheduling research, the minimization of the makespan is usually taken as the research goal. That is, a batch of workpieces are assigned to different machines for processing. The time when the last workpiece processed on each machine is completed is the makespan of that machine. Then, the maximum makespan among all the machines is the makespan of this batch of workpieces. Taking the minimization of the makespan as the goal is to formulate an efficient machine scheduling plan to improve the production efficiency of the entire production workshop. In the research on the scheduling of an automated electrode foil production workshop, the shorter the time for the AGV to transport the foil roll to the processing machine, the earlier the machine starts processing, and thus the shorter the production cycle from the transportation of the foil roll from the warehouse until the processing is completed. Therefore, considering the transportation time of the AGV in the scheduling of the intelligent production line is more in line with the actual production needs. However, in the current scheduling solutions for automated electrode foil production workshops, most studies only consider the scheduling problem of machines, that is, only consider how to more reasonably allocate the foil rolls to the machines to obtain the minimization of the makespan, and ignore the transportation time of how to transport the foil rolls to the machines. Therefore, establishing a problem model with the minimization of the makespan with transportation time as the optimization goal is an urgent problem to be solved in the current research on the joint scheduling of multiple AGVs and multiple machines in an automated electrode foil production workshop. Summary of the Invention

[0004] The objective of the present invention is to provide a mixed-integer linear programming method for the joint scheduling of multiple AGVs and multiple machines, so as to solve the problem of how to meet the optimization objective of minimizing the maximum completion time of electrode foil production in line with actual production needs.

[0005] A mixed-integer linear programming method for the joint scheduling of multiple AGVs and multiple machines provided by the present invention is characterized by including the following steps:

[0006] Step 1, define the model parameters and decision variables of the mixed-integer linear programming model, and take the minimum value Minimise C of the maximum completion time required for the production of the measured electrode foil max , as the objective value output by the mixed-integer linear programming model;

[0007] Step 2, on the basis of determining the optimization objective, perform linear constraints on the mixed-integer linear programming model in terms of time and AGV capacity;

[0008] Step 3, for the established mixed-integer linear programming model of the multi-AGV and multi-machine scheduling in the automated electrode foil production workshop, use the GUROBI solver for verification and obtain a Gantt chart according to the example.

[0009] Furthermore, the model parameters include the following parameters:

[0010] J - the number of foil rolls;

[0011] j, j′ - the numbers of the foil rolls, where j, j′ ∈ [0, J];

[0012] M - the number of machines;

[0013] m, m ' - the numbers of the machines, where m, m′ ∈ [1, M];

[0014] K - the number of AGVs;

[0015] k - the number of the AGV, where k ∈ [1, K];

[0016] x m 、y m - the position coordinates of machine m;

[0017] t u - the unloading time, t u =15s;

[0018] p j - the processing time of foil roll j;

[0019] Q - the maximum load capacity of the AGV;

[0020] v - the speed of the AGV;

[0021] The maximum number of trips that a P-AGV can make, where P = J;

[0022] The p-th trip of p-AGV transportation;

[0023] The total number of foil rolls transported by an N-AGV in a certain trip;

[0024] The n-th position where an n-AGV transports in a certain trip;

[0025] h - An infinitely large number;

[0026] T m,m' - The transportation time of an AGV from machine m to machine m';

[0027] The decision variables include the following variables,

[0028] t j,m - It has two values, 0 and 1. If foil roll j is assigned to machine m with number for processing, t j,m = 1; if foil roll j is not assigned to machine m with number for processing, that is, assigned to a machine other than machine m for processing, t j,m = 0;

[0029] u j,j',m - It has two values, 0 and 1. If both foil rolls numbered j and j' are arranged to be processed on machine m with number, and foil roll j' is the successor of j, that is, foil roll j' is processed after j, then u j,j',m = 1; if either one of the foil rolls numbered j and j' is not assigned to machine m with number, or both foil rolls are assigned to machine m with number, but foil roll j' is processed before j, then u j,j′,m = 0;

[0030] w j,k,p,n - It has two values, 0 and 1. If foil roll j numbered is arranged at the n-th position of the p-th trip of the AGV trolley numbered k, then w j,k,p,n = 1 if the following three conditions are met: (1) Foil roll j is not transported by the AGV numbered k; (2) Foil roll j is not transported by the AGV numbered k in the p-th trip; (3) Foil roll j is not at the n-th position of the AGV numbered k in the p-th trip, then w j,k,p,n = 0;

[0031] z k,p,n - It has two values, 0 and 1; if the n-th position of the p-th trip of the AGV trolley numbered k is the last position to place the foil roll, then z k,p,n = 1; if there is still a foil roll at the n + 1 position, then z k,p,n = 0.

[0032] Furthermore, in step 1, the maximum completion time required for electrode foil processing is calculated by the following formulas (1) - (5),

[0033]

[0034] C max ≥c j' , j' = 1, 2, …, J (5)

[0035] where formula (1) represents the transportation time T of the AGV from machine m to machine m', m,m' ; formula (2) represents that when foil rolls j and j' are transported by the same AGV, and the order of foil roll j

[0036] is before foil roll j', the arrival time A of foil roll j' at the processing machine m j',m ; formula (3) represents the start time S of foil roll j' to be processed on machine m j',m ; formula (4) represents the completion time c of foil roll j' j' ; formula (5) represents the maximum completion time C required for electrode foil processing max .

[0037] Furthermore, when allocating machines for unprocessed foil rolls, it is necessary to ensure the integrity of a foil roll, that is, a foil roll can only be processed on one machine at the same time, and once a foil roll starts processing, it must be processed until completion without interruption. The constraint conditions are shown in formula (6); when two unprocessed foil rolls are processed on the same machine, they are processed in sequence. The constraint conditions are shown in formulas (7) - (8),

[0038]

[0039] Furthermore, a virtual foil roll is set before the first foil roll and after the last foil roll in the foil roll sequence to be processed on a certain machine. The virtual foil roll is denoted as foil roll 0. Then each virtual foil roll has a unique predecessor and a unique successor on machine m. The constraint conditions are shown in formulas (9) - (10),

[0040]

[0041] Furthermore, at any given moment, a foil roll can only be transported and distributed at one position during one trip of an AGV, from the foil roll warehouse to the processing machine without being unloaded midway. The constraint conditions are shown in formula (11); there must be a sequential order for two unprocessed foil rolls to be transported by the same AGV, that is, at most one foil roll can be placed at a certain position during a certain trip of a certain trolley. The constraint conditions are shown in formula (12); if there is no foil roll at the nth position during the pth trip of AGV trolley numbered k, then there cannot be a foil roll at the next position of this trip either. The constraint conditions are shown in formula (13); if AGV trolley numbered k does not transport a foil roll during the pth trip, then it cannot transport a foil roll in the next trip (that is, there is no next trip). The constraint conditions are shown in formula (14); it is necessary to determine whether the last position (the Nth position) during the pth trip of AGV trolley numbered k is the last position where the foil roll is placed and whether there is only a virtual foil roll in the last trip. The constraint conditions are shown in formulas (15)-(16).

[0042]

[0043] Furthermore, when the foil roll is being processed on the machine, the completion time of the foil roll is determined according to its processing sequence. If foil roll j' is the first foil roll being processed on the machine, the constraint conditions are shown in formula (17); if foil roll j' is the processing successor of j, the constraint conditions are shown in formula (18). The constraint conditions regarding the completion time of foil roll j, its arrival time, unloading time, and processing time are shown in formula (19),

[0044]

[0045] c j ≥A j,m +t u +p j , j = 1, 2, …, J (19).

[0046] Furthermore, the time when the foil roll arrives at the processing machine includes the transportation time and unloading time of the AGV. The constraint conditions are shown in formulas (20)-(21).

[0047]

[0048] A j',m' ≥A j,m +t u +T m,m'

[0049] +(t j,m +t j',m' +w j,k,p,n +w j',k,p,n+1-4)·h(21)

[0050] where j' = 1, 2, …, J, j = 1, 2, …, J, j' ≠ j, m = 1, 2, …, M, m' = 1, 2, …, M, k = 1, 2, …, K, p = 1, 2, …, P, n = 1, 2, …, N - 1.

[0051] Furthermore, the makespan of the target value is greater than or equal to the completion time of any one foil roll, and the constraint condition is shown in formula (22).

[0052] C max ≥ c j , j = 1, 2, …, J (22).

[0053] Furthermore, the GUROBI solver is used for verification, and the specific process includes the following.

[0054] Obtain the numbers, weights, and processing times of each foil roll task, as well as the maximum load capacity of the AGV.

[0055] Input the above data into the GUROBI solver to obtain the optimal production scheduling sequence.

[0056] A mixed-integer linear programming method for the joint scheduling of multiple AGVs and multiple machines provided by the present invention establishes a problem model with the minimum makespan with transportation time as the optimization target for an automated electrode foil production line, and is verified by the GUROBI solver. In the joint scheduling problem of AGVs and machines in an automated electrode foil production workshop, by considering the AGV utilization rate, the maximum carrying capacity of the AGV during transportation is fully utilized, which has the positive effect of improving the production efficiency of electrode foils. Description of the Drawings

[0057] Figure 1 It is a scenario diagram of the joint scheduling of multiple AGVs and multiple machines in the matrix workshop constructed by the present invention.

[0058] Figure 2 It is a schematic diagram of the present invention.

[0059] Figure 3 It is a parameter setting table of 6 foil rolls in the small-scale example of the present invention.

[0060] Figure 4 It is a Gantt chart of the solution result of the GUROBI solver of the present invention. Detailed Embodiment

[0061] Such as Figure 1-2As shown, a mixed-integer linear programming method for the joint scheduling of multiple AGVs and multiple machines provided by the present invention is applicable to the automated electrode foil production line in a matrix production workshop. Operating equipment, such as electrolysis and foil roll drying machines, is arranged according to the Figure 1 matrix distribution shown. All unprocessed aluminum foil rolls are placed in the material warehouse and need to be transported by AGVs to the machines for processing. The AGVs transport the finished electrode foils to the finished product warehouse. The objective value of the present invention is to minimize the makespan, that is, to calculate the time when the last foil roll is processed on all machines, without considering the stage of transporting the finished products back to the finished product warehouse.

[0062] As Figure 2 shown, the specific implementation steps of the present invention are as follows:

[0063] Step 1, define the model parameters and decision variables of the mixed-integer linear programming model, and take the minimum value Minimise C max of the makespan required for the measured electrode foil processing as the objective value output by the mixed-integer linear programming model. Specifically, the model parameters include the following parameter settings:

[0064] J - the number of foil rolls;

[0065] j, j' - the numbers of the foil rolls, where j, j' ∈ [0, J];

[0066] M - the number of machines;

[0067] m, m' - the numbers of the machines, where m, m' ∈ [1, M];

[0068] K - the number of AGVs;

[0069] k - the number of the AGV, where k ∈ [1, K];

[0070] x m 、y m - the position coordinates of machine m;

[0071] t u - the unloading time, t u = 15s;

[0072] p j - the processing time of foil roll j;

[0073] Q - the maximum load capacity of the AGV;

[0074] v - the speed of the AGV;

[0075] P - the maximum number of trips that the AGV can make, where P = J;

[0076] p - the p-th trip of the AGV transportation;

[0077] Total number of foil rolls transported by N-AGV in a certain trip;

[0078] n-AGV arrives at the nth position during a certain trip;

[0079] h - An infinitely large number;

[0080] T m,m' - Transportation time of AGV from machine m to machine m';

[0081] The decision variables include the following variable settings,

[0082] t j,m - It has two values, 0 and 1. If foil roll j is assigned to be processed on the machine numbered m, then t j,m = 1; if foil roll j is not assigned to the machine numbered m for processing, that is, it is assigned to a machine other than machine m for processing, then t j,m = 0;

[0083] u j,j',m - It has two values, 0 and 1. If both foil rolls numbered j and j' are arranged to be processed on the machine numbered m, and foil roll j' is the successor of j, that is, foil roll j' is processed after j, then u j,j',m = 1; if one of the foil rolls numbered j and j' is not assigned to the machine numbered m, or both foil rolls are assigned to the machine numbered m, but foil roll j' is processed before j, then u j,j',m = 0;

[0084] w j,k,p,n - It has two values, 0 and 1. If foil roll j is arranged at the nth position of the pth trip of the AGV numbered k, then w j,k,p,n = 1 if the following three conditions are met: (1) Foil roll j is not transported by the AGV numbered k; (2) Foil roll j is not transported by the AGV numbered k in the pth trip; (3) Foil roll j is not at the nth position of the pth trip of the AGV numbered k, then w j,k,p,n = 0;

[0085] z k,p,n - It has two values, 0 and 1; if the nth position of the pth trip of the AGV numbered k is the last position to place the foil roll, then z k,p,n = 1; if there is still a foil roll at the n + 1 position, then z k,p,n = 0.

[0086] In addition, in step 1, the maximum completion time required for electrode foil processing is calculated by the following formulas (1) - (5),

[0087]

[0088]

[0089] C max ≥c j' , j' = 1, 2, …, J (5)

[0090] Among them, formula (1) represents the transportation time T of the AGV from machine m to machine m'. m,m' ; formula (2) represents the arrival time A of coil j' at the processing machine m when coils j and j' are transported by the same AGV and the order of coil j is one before that of coil j'. j',m ; formula (3) represents the start time S of coil j' for processing on machine m. j',m ; formula (4) represents the completion time c of coil j'. j' ; formula (5) represents the maximum completion time C required for electrode foil processing. max .

[0091] Step 2, on the basis of determining the optimization objective, linear constraints are imposed on the mixed-integer linear programming model in terms of time and AGV capacity. Specifically, the linear constraint conditions are as follows.

[0092] (1) Coil j can only be arranged to be processed on one machine, that is,[[]]

[0093]

[0094] (2) If coil j' is processed on machine m, then coil j' has a unique predecessor (not itself) on machine m. That is to say, on machine m, the coil j adjacent to and in front of coil j' is unique. If coil j' is the first coil to be processed on this machine, then its predecessor is the virtual coil 0, that is,[[]]

[0095]

[0096] (3) If coil j is processed on machine m, then coil j has a unique successor (not itself) on machine m. That is to say, on machine m, the coil j' adjacent to and behind coil j is unique. If coil j is the last coil to be processed on this machine, then its successor is the virtual coil 0, that is,[[]]

[0097]

[0098] (4) Constraints on the predecessors and successors of the virtual coil (coil 0). Coil 0 has a unique predecessor (which can be itself, indicating that there is no coil processed on machine m, that is, u 0,0,m = 1) on machine m, that is,[[]]

[0099]

[0100] (5) The foil roll 0 has a unique successor on the machine m (which can be itself, indicating that there is no foil roll being processed on the machine m, i.e., u 0,0,m = 1), that is,

[0101]

[0102] (6) The foil roll j can only be arranged at a certain position in a certain trip of a certain trolley, that is,

[0103]

[0104] (7) At most one foil roll can be placed at a certain position in a certain trip of a certain trolley, that is,

[0105]

[0106] (8) Since the maximum load capacity Q of each AGV is 200, to ensure that the single - trip transportation of the AGV is not overloaded, that is, the sum of the weights of the foil rolls at all positions in the p - th trip of the AGV numbered k does not exceed Q, that is,

[0107]

[0108] (9) If there is no foil roll at the n - th position in the p - th trip of the AGV numbered k, then there cannot be a foil roll at the next position in this trip. For example, if this AGV only transports 3 foil rolls in this trip and the fourth position is the virtual foil roll 0, then there cannot be a foil roll at the fifth position either. When sorting, it is in the order of 1, 2, 3, not 1, 2, 4, that is,

[0109] (10) If the AGV numbered k does not transport any foil roll in the p - th trip, then it cannot transport any foil roll in the next trip, that is,

[0110]

[0111] (11) If the foil roll j' is the first foil roll to be processed on the machine, then the completion time of the foil roll j' is greater than or equal to its processing time p j , that is,

[0112]

[0113] When the foil roll j' is the first foil roll to be processed on the machine, then At this time c j' ≥ p j' .

[0114] (12) If the foil roll j' is the processing successor of j, then the completion time of the workpiece j' is greater than or equal to the completion time of the foil roll j plus the processing time p of the foil roll j', that is, j , that is,

[0115] When the foil roll j' is the processing successor of j, then c j' ≥c j +p j' .

[0116] (13) The completion time of the foil roll j is greater than or equal to the arrival time of the foil roll j plus the unloading time plus the processing time of the foil roll j, that is,

[0117] c j ≥A j,m +t u +p j , j = 1, 2, …, J,

[0118] The processing starts after the first foil roll arrives at the machine. At this time, c j =A j,m +t u +p j ; The remaining foil rolls are divided into two cases: one is that the foil roll arrives, but the previous foil roll has not been processed yet and needs to wait. At this time, c j >A j,m +t u +p j ; The other is that the previous foil roll has been processed and the machine is idle. Then c j =A j,m +t u +p j .

[0119] (14) If the foil roll j is the foil roll in the first position of the first trip of the AGV cart k, then the arrival time of the foil roll j is greater than or equal to the time for the AGV cart k to reach the machine for processing the workpiece j from the origin plus the unloading time, that is,

[0120] Because if the foil roll j is the foil roll in the first position of the first trip of the AGV cart k, then t j,m =1, that is, A j,m ≥T m,m' +t u .

[0121] (15) If the foil roll j is at the n-th position in the p-th transportation of the AGV cart k, and j' is at the (n + 1)-th position in the p-th transportation of the AGV cart k, then the arrival time of the foil roll j' is greater than or equal to the arrival time of the foil roll j plus the unloading time plus the time for the AGV cart to travel from the machine processing the foil roll j to the machine processing the foil roll j', that is,

[0122] A j′,m′ ≥A j,m +t u +T m,m′ +(t j,m +t j′,m′ +w j,k,p,n +w j',k,p,n+1 -4)·h where j' = 1, 2, …, J, j = 1, 2, …, J, j' ≠ j, m = 1, 2, … M, m′ = 1, 2, …, M,

[0123] k = 1, 2, …, K, p = 1, 2, …, P, n = 1, 2, …, N - 1.

[0124] (16) Determine whether the last position (the N-th position) in the p-th transportation of the AGV cart numbered k is the last position for placing the foil roll, that is,

[0125]

[0126] Determine whether the other positions except the last position in the p-th transportation of the AGV cart numbered k are the last positions for placing the foil roll, that is,

[0127] If the position n is the last one, then So z k,p,n = 1.

[0128] (18) If the foil roll j is sorted at the last 1 position in the p-th transportation of the AGV cart k, and j' is sorted at the 1st position in the (p + 1)-th transportation of the AGV cart k, then the arrival time of the workpiece j' is greater than or equal to the arrival time of the workpiece j + the unloading time + the time for the AGV cart to return to the origin from the machine processing the foil roll j + the time for the AGV cart to travel from the origin to the machine processing the foil roll j', that is, where j' = 1, 2, …, J, j = 1, 2, …, J, j' ≠ j, k = 1, 2, …, K, p = 1, 2, …, P - 1, n = 1, 2, …, N.

[0129] (19) Target value C maxIt refers to the time when the last foil roll is processed on all machines. Therefore, it should be greater than or equal to the completion time of any foil roll, that is,

[0130] C max ≥c j , j = 1, 2, …, J.

[0131] Step 3: For the mixed-integer linear programming model of multi-AGV and multi-machine scheduling in the established automated electrode foil production workshop, use the GUROBI solver for verification and obtain a Gantt chart based on the example.

[0132] To verify the correctness of the MILP model, the present invention is verified through specific examples. Considering time limitations and the search for the solution space, in the specific embodiments of the present invention, a small-scale example - 6 foil rolls is adopted, and the GUROBI solver is used for verification. The verification process includes: obtaining the numbers, weights, and processing times of each foil roll task, as well as the maximum load capacity of the AGV; inputting the above data into the GUROBI solver to obtain the optimal production scheduling sequence. The present invention is further described in detail through the simulation content. The scheduling model of the small-scale example with a foil roll task scale of 6 in this experiment adopts a maximum load capacity of 200 KG for the AGV. It is assumed that there are 4 available machines and 2 available AGVs in the matrix workshop. The specific settings of the parameter values of the 6 foil roll tasks are as Figure 3 shown. Then, through the GUROBI solver, it can be obtained that foil roll 1 and foil roll 4 are processed on machine 1, foil roll 3 and foil roll 2 are processed on machine 2; foil roll 5 is processed on machine 3, and foil roll 2 and foil roll 6 are processed on machine 4. AGV1 transports foil rolls (1, 3, 4), and the delivery sequence is (3, 4, 1). AGV2 transports foil rolls (2, 5, 6), and the delivery sequence is (6, 5, 2). From Figure 4 the Gantt chart of the solution result obtained by the GUROBI solver shown, it can be seen that the waiting time of the machine in the Gantt chart is 0, indicating that the GUROBI solver has found the optimal solution, thus proving the correctness of the mixed-integer linear programming model of multi-AGV and multi-machine scheduling in the automated electrode foil production workshop proposed by the present invention, which meets the actual production needs.

Claims

1. A mixed integer linear programming method for the joint scheduling of multiple AGVs and multiple machines, characterized in that It includes the following steps: Step 1, define the model parameters and decision variables of the mixed-integer linear programming model, and take Minimise C, which is the minimum value of the maximum completion time required for the production of the measured electrode foil, as the objective value output by the mixed-integer linear programming model; max , as the objective value output by the mixed-integer linear programming model; Step 2: Based on the determined optimization objective, perform linear constraints on the mixed-integer linear programming model in terms of time and AGV capacity; Step 3: For the established mixed-integer linear programming model of multi-AGV and multi-machine scheduling in the automated electrode foil production workshop, use the GUROBI solver for verification and obtain the Gantt chart according to the example. Among them, The model parameters include the following parameters: The number of J - foil rolls; The numbers of j, j' - foil rolls, where j, j' ∈ [0, J]; The number of M - machines; The numbers of m, m' - machines, where m, m' ∈ [1, M]; The number of K - AGVs; The number of k - AGV, where k ∈ [1, K]; x m 、y m - Coordinate position of machine m; t u - Unloading time, t u = 15 s; p j - Processing time of the foil roll j; The maximum load capacity Q of the AGV; The speed v of the AGV; The maximum number of trips P that the AGV can transport, where P = J; The p - th trip of the AGV transportation; The total number of foil rolls N transported by the AGV in a certain trip; The n - th position where the AGV transports to in a certain trip; A number h that is infinitely large; T m,m' - The transportation time of the AGV from machine m to machine m'; The decision variables include the following variables: t j,m - It has two values, 0 and 1. If the foil roll j is assigned to the machine numbered m for processing, t j,m = 1; if the foil roll j is not assigned to the machine numbered m for processing, that is, assigned to a machine other than machine m for processing, t j,m = 0; u j,j',m - It has two values, 0 and 1. If the foil rolls numbered j and j' are both arranged to be processed on the machine numbered m, and the foil roll j' is the successor of j, that is, the foil roll j' is processed after j, then u j,j',m = 1; If one of the foil rolls numbered j and j' is not assigned to the machine numbered m, or both foil rolls are assigned to the machine numbered m, but the foil roll j' is processed before j, then u j,j',m = 0; w j,k,p,n - It has two values, 0 and 1. If the foil roll numbered j is arranged at the nth position in the pth transportation of the AGV car numbered k, then w j,k,p,n = 1 if the following three conditions are met: (1) The foil roll j is not transported by the AGV numbered k; (2) The foil roll j is not transported by the AGV numbered k in the pth transportation; (3) The foil roll j is not at the nth position in the pth transportation of the AGV numbered k, then w j,k,p,n = 0; z k,p,n - It has two values, 0 and 1; if the nth position in the pth transportation on the AGV trolley numbered k is the last position to place the foil roll, then z k,p,n = 1; if there is still a foil roll in the n+1th position, then z k,p,n = 0; The maximum completion time required for electrode foil processing is calculated by the following formulas (1) - (5); C max ≥ c j' , j' = 1, 2, …, J (5) Among them, formula (1) represents the transportation time T of the AGV from machine m to machine m'. m,m' ; formula (2) represents the arrival time A of coil j' at the processing machine m when coils j and j' are transported by the same AGV and the order of coil j is one before that of coil j'. j',m ; formula (3) represents the start time S of processing coil j' on machine m j',m ; formula (4) represents the completion time c of coil j'. j' ; formula (5) represents the maximum completion time C required for electrode foil processing max ; Using the GUROBI solver for verification specifically includes the following process: Obtain the numbers, weights, and processing times of each foil roll task, as well as the maximum load capacity of the AGV; Input the above data into the GUROBI solver to obtain the optimal production scheduling sequence.

2. The mixed integer linear programming method for joint scheduling of multiple AGVs and multiple machines according to claim 1, further characterized in that When allocating machines for unprocessed foil rolls, it is necessary to ensure the integrity of a foil roll, that is, a foil roll can only be processed on one machine at the same time, and once a foil roll starts processing, it must be processed until completion and cannot be stopped. The constraint conditions are shown in formula (6); When two unprocessed foil rolls are processed on the same machine, they are processed in sequence. The constraint conditions are shown in formulas (7) - (8); 3. A mixed integer linear programming method for the joint scheduling of multiple AGVs and multiple machines according to claim 1, further characterized in that A virtual foil roll is set before the first foil roll and after the last foil roll in the foil roll sequence that needs to be processed on a certain machine. The virtual foil roll is denoted as foil roll 0. Then each virtual foil roll has a unique predecessor and a unique successor on machine m. The constraint conditions are shown in formulas (9) - (10); 4. The mixed integer linear programming method for the joint scheduling of multiple AGVs and multiple machines according to claim 1, further characterized in that A foil roll can only be transported and distributed at one position of a certain trip of one AGV at a time, transported from the foil roll warehouse to the processing machine without being unloaded midway, and the constraint conditions are shown in formula (11); there must be a sequence for two unprocessed foil rolls to be transported by the same AGV, that is, at a certain position of a certain trip of a certain trolley arranged, at most only 1 foil roll can be placed, and the constraint conditions are shown in formula (12); if there is no foil roll at the nth position of the pth trip of the AGV numbered k, then there cannot be a foil roll at the next position of this trip, and the constraint conditions are shown in formula (13); if the AGV numbered k does not transport a foil roll in the pth trip, then it cannot transport a foil roll in the next trip, and the constraint conditions are shown in formula (14); judge whether the last position of the pth trip of the AGV numbered k, that is, the Nth position, is the last position to place the foil roll and whether there is only a virtual foil roll in the last trip, and the constraint conditions are shown in formulas (15)-(16).

5. The hybrid integer linear programming method for joint scheduling of multiple AGVs and multiple machines according to claim 1 is further characterized in that, When the foil roll is processed on the machine, the completion time of the foil roll is determined according to its processing sequence. If the foil roll j' is the first foil roll processed on the machine, the constraint conditions are shown in formula (17); if the foil roll j' is the processing successor of j, the constraint conditions are shown in formula (18), and the constraint conditions for the completion time of the foil roll j, the arrival time of the foil roll, the unloading time, and the processing time of the foil roll are shown in formula (19). c j ≥A j,m +t u +p j , j = 1, 2, …, J (19).

6. A mixed integer linear programming method for joint scheduling of multiple AGVs and multiple machines according to claim 1, further characterized in that The time when the foil roll arrives at the processing machine includes the transportation time and unloading time of the AGV, and the constraint conditions are shown in formulas (20)-(21). A j',m' ≥A j,m +t u +T m,m' +(t j,m +t j',m' +w j,k,p,n +w j',k,p,n+1 -4)·h (21) Among them, j' = 1, 2, …, J, j = 1, 2, …, J, j' ≠ j, m = 1, 2, …, M, m' = 1, 2, …, M, k = 1, 2, …, K, p = 1, 2, …, P, n = 1, 2, …, N - 1.

7. A mixed-integer linear programming method for the joint scheduling of multiple AGVs and multiple machines according to claim 1, further characterized in that, The target value of the maximum completion time is greater than or equal to the completion time of any foil roll, and the constraint conditions are shown in formula (22). C max ≥c j , j = 1, 2, …, J (22).

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