FFSP-AGV integrated scheduling optimization method based on mixed integer linear programming

Abstract

The invention relates to an FFSP-AGV integrated scheduling optimization method based on mixed integer linear programming, and the method comprises the following steps: S1, defining parameters and decision variables of a mixed integer linear programming model, taking the minimum production cycle as a scheduling target of an FFSP-AGV, carrying out the constraint of the model based on the target, and constructing the mixed integer linear programming model; s2, solving the constructed mixed integer linear programming model by using a CPLEX solver, and verifying the effectiveness of the model; and S3, outputting an optimal scheduling scheme, wherein the optimal scheduling scheme specifically comprises working procedure starting time, workpiece machining equipment selection, AGV transportation task allocation and arrival time. According to the invention, the scheduling of the flexible flow shop and the AGV is integrated, and the mixed integer linear programming model is constructed, so that the production cycle is shortened, the idle time of equipment is reduced, and the operation efficiency of the whole production system is improved.

Description

Technical Field

[0001] The invention belongs to the field of information technology, and in particular relates to a FFSP-AGV integrated scheduling optimization method based on mixed integer linear programming. Background Art

[0002] The flexible flow shop is a new type of production organization that can adapt to the needs of multi-variety and small-batch production and can quickly respond to dynamic changes in the market. Traditional assembly lines often find it difficult to cope with the needs of diversified production, while flexible flow shops provide a high degree of flexibility, allowing resource allocation to be quickly adjusted according to production conditions. The flexible flow shop scheduling problem (FFSP) is a flexible scheduling problem that is widely present in manufacturing enterprises. It aims to solve the two sub-problems of process assignment and process sequencing, and has been proven to be an NP-hard problem. However, the existing FFSP has not been fully studied and applied in the material transportation link and the use of AGV (automatic guided vehicle).

[0003] As an efficient material transportation tool, AGV has been widely used in manufacturing and logistics industries. In modern flexible production environments, AGV scheduling is crucial to improving the efficiency of production systems. AGV can automatically transport materials and products during the production process, and its path selection and scheduling strategy directly affect production efficiency. However, although AGV has been applied in many industries, the integrated scheduling optimization of flexible assembly lines and AGVs still faces challenges. Especially when performing comprehensive scheduling between multiple production lines and multiple equipment, existing research often focuses on separate production scheduling or AGV scheduling, and lacks integrated optimization research on the two. Summary of the invention

[0004] The purpose of the present invention is to provide a FFSP-AGV integrated scheduling optimization method based on mixed integer linear programming, which integrates the scheduling of flexible flow workshop production and AGV transportation, and shortens the production cycle, reduces equipment idle time, and improves the operating efficiency of the entire production system by constructing a mixed integer linear programming model.

[0005] In order to achieve the above object, the technical solution adopted by the present invention is: a FFSP-AGV integrated scheduling optimization method based on mixed integer linear programming, comprising the following steps: S1. Define the parameters and decision variables of the mixed integer linear programming model, take minimizing the production cycle as the scheduling goal of FFSP-AGV, and constrain the model based on this goal to construct a mixed integer linear programming model; S2. Use CPLEX solver to solve the constructed mixed integer linear programming model to verify the effectiveness of the model; S3. Output the optimal scheduling plan, which includes the process start time, equipment selection for workpiece processing, AGV transportation task allocation and its arrival time.

[0006] In step S1, the parameters of the mixed integer linear programming model include the following parameters:

[0007] I,h,b: artifact index, I,h,b∈I={1,…,n};

[0008] j,f: processing stage index, j,f∈J={1,…,s};

[0009] k, k′, k″: index of unrelated parallel machines at processing stage j, k, k′, k″∈M j ={1,...,l j};

[0010] n: total number of workpieces, n = |I|;

[0011] s: total number of processing stages, s = |J|;

[0012] l j : The total number of parallel machines in processing stage j, l j =|M j |;

[0013] r: total number of AGVs;

[0014] J i :workpiece i;

[0015] J h : Workpiece h;

[0016] G j : Processing stage j;

[0017] G f : Processing stage f;

[0018] O ij : Workpiece i, jth process;

[0019] O hf : Workpiece h, process f;

[0020] T ij :Transport workpiece i to execution O ij Transport tasks of parallel machines;

[0021] M jk : kth unrelated parallel machine at processing stage j;

[0022] PT ijk :Process O ij In parallel machine M jk Processing time on

[0023] The required transportation time between parallel machine k and parallel machine k′, and

[0024] L: A sufficiently large positive integer.

[0025] In step S1, the decision variables of the mixed integer linear programming model include:

[0026] X ihjk : 0-1 Boolean variable, when O ij Yes hj In parallel machine M jk The value is 1 if it is the immediate predecessor process, otherwise it is 0;

[0027] XF ijk : 0-1 Boolean variable, when O ij It is a parallel machine M jk It takes 1 when it is the first process on the process, otherwise it takes 0;

[0028] XL ijk : 0-1 Boolean variable, when O ij It is a parallel machine M jk The value is 1 if it is the last process on the process, otherwise it is 0;

[0029] Alpha ijk : 0-1 Boolean variable, when O ij In parallel machine M jk It takes 1 when processing, otherwise it takes 0;

[0030] 0-1 Boolean variable, when T ij It is T hf The value is 1 when the task is the previous transport task on the same AGV, otherwise it is 0;

[0031] 0-1 Boolean variable, when T ij If it is the first transport task on an AGV, it takes 1, otherwise it takes 0;

[0032] 0-1 Boolean variable, when T ij If it is the last transport task on an AGV, it takes 1, otherwise it takes 0;

[0033] C ijk : Continuous decision variable, representing process O ij In parallel machine M jk Completion time on

[0034] Continuous decision variable, indicating that workpiece i arrives at execution process O ij Parallel Machine Mjk time;

[0035] C max : Continuous decision variable, indicating the maximum completion time.

[0036] The optimization objective function of the mixed integer linear programming model is:

[0037] Define [P]Minimize C max (1)

[0038] The constraints are formulas (2)-(19):

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[0057] Before step S1, the characteristics of the FFSP-AGV system are analyzed to clarify the coupling relationship between the production tasks of the flexible assembly line and the transportation tasks of the AGV.

[0058] The beneficial effects of the present invention are as follows: based on the idea of ​​adjacent process modeling, the present invention constructs a mixed integer linear programming (MILP) model with the goal of minimizing the production cycle, introduces the objective function, decision variables and constraint equations, and codes them, hoping to become a general framework for FFSP-AGV integrated scheduling problems. Based on this code, the present invention also uses CPLEX software to solve different examples to verify the effectiveness of the proposed MILP model and analyze the complexity of the MILP model. By changing the number of AGVs, the law of diminishing marginal utility of AGV transportation in collaborative flexible flow workshop production is discovered and verified. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0060] Figure 1 It is a flow chart of the integrated scheduling optimization method of the present invention;

[0061] Figure 2 This is a schematic diagram of the FFSP-AGV process of the present invention;

[0062] Figure 3 Optimize results for CPLEX;

[0063] Figure 4 For Figure 3 The corresponding scheduling Gantt chart;

[0064] Figure 5 This is the effect of the number of AGVs on the production cycle when the number of workpieces is 15;

[0065] Figure 6 This is a diagram showing the effect of the number of AGVs on the production cycle when the number of workpieces is 10. DETAILED DESCRIPTION

[0066] The present invention will be further described in detail below in conjunction with the accompanying drawings and embodiments, but they are not intended to limit the invention in any way.

[0067] Embodiment 1: A FFSP-AGV integrated scheduling optimization method based on mixed integer linear programming, comprising the following steps: S1. Define the parameters and decision variables of the mixed integer linear programming model, take minimizing the production cycle as the scheduling goal of FFSP-AGV, and constrain the model based on this goal to construct a mixed integer linear programming model; S2. Use CPLEX solver to solve the constructed mixed integer linear programming model to verify the effectiveness of the model; S3. Output the optimal scheduling plan, which includes the process start time, equipment selection for workpiece processing, AGV transportation task allocation and its arrival time.

[0068] Before step S1, the characteristics of the FFSP-AGV system are analyzed to clarify the coupling relationship between the production tasks of the flexible assembly line workshop and the transportation tasks of the AGV. Figure 1 shown.

[0069] The problem characteristics of the FFSP-AGV system include: there are n workpieces in the loading area (LA) waiting for processing at the beginning, and they need to go through s processing stages in sequence to complete the processing; there are l j unrelated parallel machines and at least one stage has two or more parallel machines. ij Will be from l j One of the parallel machines is randomly selected for processing and the processing time is known. ij Indicates sending workpiece i to the next stage to perform process O ij Obviously, in the execution of process O ij Before, T ij It should be executed by one of the r identical AGVs. The specific process is as follows Figure 2 As shown. It should be noted that the AGV may need to perform unloaded transportation from the current position to the position where the workpiece is picked up, unless the two positions are the same. The transportation time is related to the distance (starting position) and has nothing to do with the workpiece being transported. The core scheduling goal of FFSP-AGV is to minimize the production cycle (Makespan), that is, the latest time for all workpieces to complete the last process. By reasonably allocating production tasks and transportation tasks, optimizing the number of AGVs and scheduling strategies to ensure the efficient operation of the entire system.

[0070] The FFSP-AGV system meets the following constraints:

[0071] 1. Workshop production scheduling related constraints

[0072] Workpiece processing constraints: Each workpiece must complete all processes in a predetermined order. It can only enter the next process after the previous process is completed. Each workpiece must first pass the previous process before entering the next stage.

[0073] Equipment allocation constraints: Each device can only process one workpiece at any time, each process can only be assigned to one device, one device can only process one workpiece at a time, and overlapping processing is not allowed.

[0074] Machine utilization constraints: All jobs must be allocated according to the available machine resources to minimize waiting time.

[0075] 2.AGV scheduling related constraints

[0076] AGV transport task sequence constraint: Each AGV can only perform the next task after completing the previous transport task. Each AGV can only perform transport tasks in sequence and can only receive the next task after completing the current task.

[0077] AGV load constraint: Each AGV can only transport one workpiece at a time. The workpiece can be processed only after completing the transportation task. An AGV can only transport one workpiece at a time.

[0078] Transportation path and time constraints: The time for AGV transportation depends on the distance from one workstation to another, and the transportation time will affect the scheduling of the entire system.

[0079] 3. Time Constraints

[0080] Maximum completion time constraint: The end time of the production cycle is defined as the time when the last workpiece completes all processes. The maximum completion time should be greater than or equal to the completion time of each workpiece in the last process.

[0081] Processing and transportation time constraints: Each process and each transportation has a fixed time, which needs to be taken into account during scheduling to ensure that the workpiece is completed within the specified time.

[0082] 4. Boolean variable constraints

[0083] Boolean decision variable constraints: used to represent the scheduling status of a workpiece, such as whether a workpiece is processed or transported at a specific time. Boolean variables are used to mark whether a workpiece is being processed on a certain device or whether a certain AGV is performing a transportation task.

[0084] Therefore, in the above steps, the parameters and definitions of the mixed integer linear programming model in step S1 are as follows:

[0085] I,h,b: artifact index, I,h,b∈I={1,…,n};

[0086] j,f: processing stage index, j,f∈J={1,…,s};

[0087] k, k′, k″: index of unrelated parallel machines at processing stage j, k, k′, k″∈M j={1,...,l j};

[0088] n: total number of workpieces, n = |I|;

[0089] s: total number of processing stages, s = |J|;

[0090] l j : The total number of parallel machines in processing stage j, l j =|M j ;

[0091] r: total number of AGVs;

[0092] J i : Workpiece i;

[0093] J h : Workpiece h;

[0094] G j : Processing stage j;

[0095] G f : Processing stage f;

[0096] O ij : Workpiece i, jth process;

[0097] O hf : Workpiece h, process f;

[0098] T ij :Transport workpiece i to execution O ij Transport tasks of parallel machines;

[0099] M jk : kth unrelated parallel machine at processing stage j;

[0100] PT ijk :Process O ij In parallel machine M jk Processing time on

[0101] The required transportation time between parallel machine k and parallel machine k′, and

[0102] L: A sufficiently large positive integer.

[0103] The decision variables include:

[0104] X ihjk :0-1 Boolean variable, when O ij Yes hj In parallel machine M jk The value is 1 if it is the immediate predecessor process, otherwise it is 0;

[0105] XF ijk :0-1 Boolean variable, when O ij It is a parallel machine M jk It takes 1 when it is the first process on the process, otherwise it takes 0;

[0106] XL ijk :0-1 Boolean variable, when O ij It is a parallel machine M jk The value is 1 if it is the last process on the process, otherwise it is 0;

[0107] Alpha ijk :0-1 Boolean variable, when O ij In parallel machine M jk It takes 1 when processing, otherwise it takes 0;

[0108] 0-1 Boolean variable, when T ij It is T hf The value is 1 when the task is the previous transport task on the same AGV, otherwise it is 0;

[0109] 0-1 Boolean variable, when T ij If it is the first transport task on an AGV, it takes 1, otherwise it takes 0;

[0110] 0-1 Boolean variable, when T ij If it is the last transport task on an AGV, it takes 1, otherwise it takes 0;

[0111] C ijk : Continuous decision variable, representing process O ij In parallel machine M jk Completion time on

[0112] Continuous decision variable, indicating that workpiece i arrives at execution process O ij Parallel Machine M jk time;

[0113] C max : Continuous decision variable, indicating the maximum completion time.

[0114] The optimization objective function of the mixed integer linear programming model is:

[0115] Define [P]Minimize C max (1)

[0116] The constraints are formulas (2)-(19):

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[0135] In this MILP model, inequality (2) ensures the correct value of the production cycle.

[0136] Constraint (3) aims to establish the relationship between the decision variables of the job sequence and machine assignment subproblems.

[0137] Constraint (4) ensures that each process will be processed by one parallel machine.

[0138] Constraint (5) ensures that the completion time of any process will be greater than or equal to its transport arrival time plus the required processing time.

[0139] Constraint (6) ensures that for each workpiece assigned to any parallel machine, each workpiece has only one immediate predecessor workpiece (for the first workpiece, this immediate predecessor workpiece will become an empty workpiece) and immediate successor workpiece (for the last workpiece, this immediate successor workpiece will become an empty workpiece).

[0140] Constraint (7) prevents any two workpieces at any processing stage from being processed simultaneously on the same parallel machine.

[0141] Constraint (8) ensures that at any time, any parallel machine in any processing stage processes at most one workpiece.

[0142] Constraints (9) and (10) ensure that the total number of the first transport tasks is at most equal to the total number of AGVs and equal to the total number of the last transport tasks.

[0143] Constraint (11) ensures that each transport task is either the first task of an AGV or has a preceding transport task before it.

[0144] Constraint (12) ensures that each transport task is either the last task of an AGV or has an immediate subsequent transport task after it.

[0145] Constraint (13) ensures that for the current process O hf (f≠1), when the assigned AGV is executing T hf When the AGV has a previous task, it needs to: (i) execute the previous task and transport the workpiece to its corresponding parallel processing machine; (ii) execute the previous task from O hf (f≠1) to pick up the current workpiece from the parallel machine position; (iii) transport the current workpiece to the execution machine O hf The parallel machine position.

[0146] Constraint (14) provides a supplementary explanation to constraint (13) when f = 1. The difference between constraint (14) and constraint (13) is whether the current workpiece needs to be picked up from the loading area (LA).

[0147] Constraints (15) and (16) are intended to establish the relationship between the transport arrival time and the completion time. Constraint (16) takes into account the special case that the current process is the first process.

[0148] Constraints (17) and (18) define Boolean decision variables.

[0149] Constraint (19) ensures the non-negativity of these variables.

[0150] Test results: In actual production applications, the FFSP-AGV problems studied often have different scales, including the complexity of the processing stage, the number of production equipment in each stage, and the number of AGVs available for investment. In order to avoid the influence of long test time on judgment due to the complexity of the example, a small-scale example is first used as the test object. Consider a flexible assembly line with two production stages, with 3 and 2 parallel machines in the first and second stages respectively. For the convenience of analysis, we assume that the parallel machines in each production stage are the same (that is, the time spent on processing the workpiece on all parallel machines in a certain production stage is the same). This example has 5 workpieces to be processed and 2 AGVs. The processing and transportation data are displayed in a matrix:

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[0154] Import the above data into the previously established MILP model, use CPLEX programming, and run the code to get the optimization results, such as Figure 3 As shown, the corresponding integrated scheduling Gantt chart is as follows Figure 4 shown.

[0155] from Figure 3 It can be seen that the MILP model is effectively solved in CPLEX, and the maximum completion time after optimization is 63; Figure 4 It can be found that the production and transportation links of workpieces in the flexible assembly line-AGV system are collaboratively optimized. Figure 4 In the figure, the rectangular blocks with different background shadows represent different processes / transportation tasks. Due to the limitation of the number of AGVs, the AGV will constantly switch back and forth between the three states of transportation, standby, and no-load. The entire production process does get the best scheduling solution. Therefore, it can be proved that the mixed integer linear programming model constructed by the present invention is accurate and effective in solving the FFSP-AGV integrated scheduling problem.

[0156] Embodiment 2:

[0157] Model complexity is a multi-dimensional concept, which involves multiple aspects such as the amount of calculation and the number of parameters. Model complexity directly determines the ability of the mathematical model to process data for computational solutions. There is a close relationship between the complexity of the algorithm and the scale of the problem. By analyzing the complexity of the model, we can evaluate the efficiency and memory usage of the algorithm, continuously optimize it, and select the most suitable algorithm to solve the problem. At the same time, model complexity analysis can also help find the bottleneck of the algorithm and optimize it. By deeply understanding the performance characteristics of the algorithm, more efficient algorithms can be designed and the efficiency of problem solving can be improved. Therefore, after ensuring the availability of the model, it is necessary to conduct a complexity analysis on the established model.

[0158] Model complexity is divided into many angles, including time complexity, space complexity, algorithm complexity, etc. This embodiment will mainly analyze the complexity of the proposed MILP model from the two angles of scale complexity and computational complexity. These can be directly checked in the property box of the CPLEX work interface. Among them, scale complexity has three evaluation indicators: the number of Boolean decision variables (NBV), the number of constraints (NC), and the number of continuous decision variables (NCV). These are directly determined by the number of parameters and variables in the model. Generally speaking, the larger the value of each parameter in the model, the larger the complexity scale, and accordingly, the more difficult the model is to solve. In the following calculations, the test case is represented by a combined nomenclature, such as J5-S2-M3-A2, which means that there are 5 workpieces to be processed (Job), 2 processing stages (Stage), and each stage has a maximum of 3 unrelated parallel machines (Machine) and 2 AGVs. The workpiece processing time obeys the uniform distribution on [5, 30], and the workpiece transportation time obeys the uniform distribution on [1, 6]. Finally, 6 groups of test cases were randomly generated, and the upper limit of CPLEX solution time was set to 900 seconds. The experimental results are shown in Table 1. Table 1 Complexity analysis of MILP model Note: “—” indicates that a feasible solution could not be obtained within 900 seconds.

[0159] It should be noted that, except for the first set of examples J5-S2-M3-A2, the other sets of data cannot obtain accurate solutions within the specified time. Therefore, the optimal production cycle obtained by the test is actually the result solved when the example reaches the set upper limit time, that is, the suboptimal solution when calculating 900 seconds. As can be seen from Table 1, the complexity of MILP varies with different examples. Specifically: NBV, NC and NCV increase nonlinearly with the increase of the example scale, and NC and NCV have the largest and smallest increases, respectively. Taking the test examples J10-S2-M3-A2 and J5-S2-M3-A2 as an example, the number of workpieces increased by 1.00 times, and NBV, NC and NCV increased by 2.41 times, 2.76 times and 0.98 times, respectively.

[0160] There are two evaluation indicators for computational complexity: optimal production cycle (FV) and CPU time (Times). Computational complexity determines whether the model can quickly and accurately obtain the optimal solution. As can be seen from Table 1, except for the first group of examples that obtained the optimal target value of 44 in 1.81 seconds, the other five groups of examples cannot enable CPLEX to obtain the corresponding optimal target value within 900 seconds, and J15-S4-M5-A2 cannot even obtain a feasible suboptimal solution. This shows that as the scale of the problem increases, the difficulty of solving the model increases exponentially, which is specifically reflected in the increase in computational complexity. In fact, the scale complexity index NBV and the computational complexity index Times in the MILP model are strongly correlated, and the two are causally related. It can be seen that the scope that this model can solve is limited. When dealing with larger-scale problems, it is often necessary to increase the CPU time or use a better algorithm to perform larger-scale operations.

[0161] The configuration of the number of AGVs is one of the key issues that need to be considered in the production and manufacturing activities in which AGVs participate in scheduling, and it is also the focus of the present invention. For a production and manufacturing system that relies on AGV scheduling to drive transportation, if the number of AGVs invested is too small, the transportation of workpieces will be slow, which will cause a lot of time and resources to be wasted, and the timeliness of the completion of production tasks will be difficult to guarantee; but the number of AGVs invested cannot be too many. Too many AGVs are not only costly, but may also increase energy consumption when idle and in standby mode, which will reduce the efficiency of the entire production system. Therefore, according to the production scale, investing the most appropriate number of AGVs is one of the important ways for manufacturing companies to reduce costs and increase efficiency. To this end, this embodiment will use J15-S2-M3 and J10-S2-M3 as basic data, and by changing the number of AGVs (from 1 to 10), study its performance impact on the integrated scheduling effect of FFSP-AGV. As mentioned above, in the MILP model constructed for FFSP-AGV, the processing equipment at each production stage is an unrelated parallel machine (UPM). In order to be fair, the processing equipment in each production stage is set to the same parallel machine (IPM), and the working time is averaged to cope with different production situations. The experimental results are recorded in Tables 2 and 3 and plotted in Figure 5 , Figure 6 middle. Table 2 Experimental results of the effect of AGV quantity change on production cycle (900s) Table 3 Experimental results of the effect of AGV quantity change on production cycle (900s)

[0162] From Table 2, Table 3 and Figure 5 , Figure 6It can be seen that, whether it is UPM or IPM, as the number of participating AGVs increases, the optimized target production cycle shows a downward trend. In other words, for a flexible assembly line workshop of a certain scale, using more AGVs can indeed effectively shorten the production cycle and improve production efficiency. At the same time, when the number of AGVs increases from 1 to 2, the production cycle decreases the most; as the number of AGVs continues to increase, this decrease is gradually shrinking. When the number of workpieces is 15, when the number of AGVs reaches 6, the maximum production cycle hardly changes. In another group of workpieces with a number of 10, when the number of AGVs is 3, the production cycle no longer decreases. The two sets of data conform to the law of diminishing marginal utility. Then, for the given J15-S2-M3 production conditions, the optimal number of AGVs is 4. When the number of AGVs invested is less than 4, by increasing the number of AGVs, faster transportation of workpieces can be achieved, thereby improving the operation rate of the entire workflow. However, when the number of AGVs exceeds 4, the production cycle is mainly limited by the production workshop itself, such as the number of production equipment, the processing time of a single workpiece, the travel time between stages, etc. For another group, J10-S2-M3, the optimal number of AGVs is reduced to 2. It can be expected that after the number of AGVs reaches the optimal number, the processing of each device reaches saturation. After excluding the distance factor, the working time ratio of each device will be infinitely close to 1. To further improve the processing rate, it is necessary to start from other links, such as increasing the number of processing equipment, reducing the processing time of each link, reducing the distance between adjacent processes, and increasing the running speed of the AGV. Combining the data of the two, it can be found that when the number of workpieces in a production task increases, the number of AGVs that achieve the optimal configuration will also increase. However, no matter how it increases, the final optimized production cycle will be greater than the task with a smaller number of workpieces. After further analysis, it can be found that the production cycle is roughly proportional to the increase in the number of workpieces itself. When the number of workpieces increases from 10 to 15, the optimized production cycle increases from 60 to 90, which is positively correlated with the increase in the number of workpieces. It is reasonable to speculate that at this time, the resource scheduling in the workshop is in a reasonable cycle. The increase in the number of workpieces will not change the production cycle itself, but will only increase the number of running rounds of the production cycle. This may also prove that after the number of AGVs reaches saturation, constraints on production equipment become the main reason limiting the improvement of production efficiency.

[0163] For certain numbers of AGVs, it can be found that the production cycle increases instead of decreases with the increase of the number of AGVs, which is inconsistent with common sense. In the J15 group, this phenomenon is more obvious than that in the J10 group. This may be due to the following reasons: (a) Unlike the traditional FFSP, the FFSP-AGV in this study is an integrated scheduling problem of collaborative workshop production and AGV transportation, which has more complex problem characteristics and constraints. As the number of workpieces increases, the model complexity increases; (b) The processing time and transportation time in the test case are set by chance, resulting in a bottleneck that is difficult to optimize in collaborative workshop production, causing the FFSP-AGV system to be blocked. These two reasons are fundamentally dependent on one factor: the computational complexity of the model. Different computational complexities result in different computational speeds and accuracy, and the test case used is relatively simple, which leads to a rapid increase in the computational complexity of the model after the number of AGVs increases. In a limited time, it is impossible to obtain a relatively accurate suboptimal solution. Therefore, based on J15-S2-M3, the running time was further increased to 1800s, the experiment was repeated, and a new set of data was obtained, as shown in Table 4. Table 4 Experimental results of the effect of AGV quantity change on production cycle (1800s)

[0164] As can be seen from Table 4, the occurrence of accidental data has been greatly reduced, and the production cycle also conforms to the trend of decreasing with the increase in the number of AGVs. It can be imagined that if the running time continues to increase, the accidental results will be reduced again, and the change trend of the final solution will be more in line with the actual situation.

[0165] At the same time, it can be found that in most cases, the production cycle of IPM is shorter than that of UPM. And the fewer AGVs, the more obvious this phenomenon will be. This is because UPM provides more equipment with different processing efficiencies, so that AGV will choose a specific set of machines when transporting workpieces, which indirectly reduces production efficiency. This is also similar to the actual situation. When there are differences in production capacity between multiple production equipment, production resources will be more concentrated on the side with shorter processing time, resulting in a decrease in the utilization rate of resources such as production scheduling. When the number of AGVs increases, almost all equipment works at the same time, diluting the gap caused by the selection of processing equipment. For real enterprises, under the condition of limited AGVs, it is also necessary to pay attention to the maintenance of each equipment and minimize the gap between the processing capabilities of different stations at the same stage.

[0166] The above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit the same. Those skilled in the art should understand that the specific implementation modes of the present invention may be modified or replaced by equivalents with reference to the above embodiments. Any modifications or equivalent replacements that do not depart from the spirit and scope of the present invention are within the scope of protection of the pending claims.

Claims

1. A FFSP-AGV integrated scheduling optimization method based on mixed integer linear programming, characterized by: The steps include: S1. Define the parameters and decision variables of the mixed integer linear programming model, take minimizing the production cycle as the scheduling goal of FFSP-AGV, and constrain the model based on this goal to construct a mixed integer linear programming model; S2. Use CPLEX solver to solve the constructed mixed integer linear programming model to verify the effectiveness of the model; S3. Output the optimal scheduling plan, which includes the process start time, equipment selection for workpiece processing, AGV transportation task allocation and its arrival time.

2. The FFSP-AGV integrated scheduling optimization method according to claim 1 is characterized in that: In step S1, the parameters of the mixed integer linear programming model include the following parameters: I,h,b: artifact index, I,h,b∈I={1,…,n}; j,f: processing stage index, j,f∈J={1,…,s}; k, k′, k″: index of unrelated parallel machines at processing stage j, k, k′, k″∈M j ={1,...,l j }; n: total number of workpieces, n = |I|; s: total number of processing stages, s = |J|; l j : The total number of parallel machines in processing stage j, l j =|M j |; r: total number of AGVs; J i :workpiece i; J h : Workpiece h; G j : Processing stage j; G f : Processing stage f; O ij : The j-th process of workpiece i; O hf : The fth process of workpiece h; T ij : Transport workpiece i to execution O ij Transport tasks of parallel machines; M jk : The kth unrelated parallel machine at processing stage j; PT ijk :Process O ij In parallel machine M jk Processing time on The required transportation time between parallel machine k and parallel machine k′, and L: A sufficiently large positive integer.

3. The FFSP-AGV integrated scheduling optimization method according to claim 2 is characterized in that: In step S1, the decision variables of the mixed integer linear programming model include: X ihjk :0-1 Boolean variable, when O ij Yes hj In parallel machine M jk The value is 1 if it is the immediate predecessor process, otherwise it is 0; XF ijk :0-1 Boolean variable, when O ij It is a parallel machine M jk It takes 1 when it is the first process on the process, otherwise it takes 0; XL ijk :0-1 Boolean variable, when O ij It is a parallel machine M jk The value is 1 if it is the last process on the process, otherwise it is 0; Alpha ijk :0-1 Boolean variable, when O ij In parallel machine M jk It takes 1 when processing, otherwise it takes 0; 0-1 Boolean variable, when T ij It is T hf The value is 1 when the task is the previous transport task on the same AGV, otherwise it is 0; 0-1 Boolean variable, when T ij If it is the first transport task on an AGV, it takes 1, otherwise it takes 0; 0-1 Boolean variable, when T ij If it is the last transport task on an AGV, it takes 1, otherwise it takes 0; C ijk : continuous decision variable, representing process O ij In parallel machine M jk Completion time on Continuous decision variable, indicating that workpiece i arrives at execution process O ij Parallel Machine M jk time; C max : Continuous decision variable, indicating the maximum completion time.

4. The FFSP-AGV integrated scheduling optimization method according to claim 3 is characterized in that: The optimization objective function of the mixed integer linear programming model is: Define [P]Minimize C max (1) The constraints are formulas (2)-(19):

5. The FFSP-AGV integrated scheduling optimization method according to claim 1 is characterized in that: Before step S1, the characteristics of the FFSP-AGV system are analyzed to clarify the coupling relationship between the production tasks of the flexible assembly line and the transportation tasks of the AGV.

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