Optimization Method for Two-Shift Multi-Task Scheduling Considering Different Processing Speeds

Abstract

The present invention proposes an optimization method for two-shift multi-task scheduling considering processing asynchronous speeds. Taking two-shift multi-task processing as the research object, it considers the actual problem of the impact of workpiece switching between different shifts on production goals. By establishing a model for workpiece switching, it studies two-shift multi-task scheduling with additional switching costs for different shifts with different processing speeds, analyzes the relationship between the switching costs of workpiece switching and shifts, designs a pseudo-polynomial time dynamic programming algorithm, and obtains a multi-task scheduling decision plan. The present invention is oriented to production practice, aiming to provide a theoretical basis for enterprises to coordinate the optimization decision of multi-task scheduling in the two-shift working system, and also provide a research model for reference for other types of working modes such as three-shift systems.

Description

Technical Field

[0001] The present invention relates to the optimization problem of multi - task scheduling decision, and specifically to a two - shift multi - task scheduling optimization method considering different processing speeds. Background Art

[0002] The two - shift pattern means that the working hours can be divided into day shift and night shift or shift A and shift B, and workers can choose to work on the day shift (shift A) or the night shift (shift B). Multitasking scheduling extends the processing method of a single task to that of multiple tasks, breaking through the limitation that a machine can only focus on arranging a single workpiece, and considering the diversity of processing methods for all workpieces and the complex requirements in the processing process. Production scheduling is the nerve center to ensure the orderly, stable, balanced and efficient operation of the production process, and is the key link restricting the performance of the production system.

[0003] In actual production and service systems, the two - shift working system is widespread and frequently applied. Usually, workers on the day shift (shift A) handle tasks on the day shift (shift A), and workers on the night shift (shift B) handle tasks on the night shift (shift B). In particular, for tasks that need to be continuously processed and completed as soon as possible, there are the following processing scenarios: Workers have different processing levels, such as multi - skilled workers and ordinary workers, full - time workers and part - time workers. Different workers have different work efficiencies and working hours. Tasks that need to be continuously processed in consecutive shifts are continuously processed by workers with different processing levels, which requires additional processing costs. This cost is defined as the switching cost. Summary of the Invention

[0004] Technical Problems to be Solved

[0005] The two - shift multi - task scheduling problem aims to optimize the multi - task scheduling scheme under the two - shift working system to minimize its scheduling objective, consider different processing methods of multiple tasks, analyze the influence relationship of task switching costs, obtain the multi - task scheduling scheme, and provide theoretical support for the multi - task scheduling optimization of the two - shift working system in the workshop, which is also the problem to be solved by this patent.

[0006] Technical Solutions

[0007] The present invention relates to the optimization problem of multi - task scheduling decision in the two - shift working system, and proposes a two - shift multi - task scheduling model with different processing speeds for different shift switches that requires additional switching costs. By analyzing the problem complexity and the nature of the optimal solution structure, an algorithm is designed to determine the multi - task decision scheme.

[0008] The technical solution of the present invention is as follows:

[0009] A two - shift multi - task scheduling optimization method considering different processing speeds: includes the following steps:

[0010] Step 1: For the two-shift multi-task scheduling problem considering processing speed differences, construct a two-shift multi-task scheduling model considering processing speed differences: The model optimization objective is to minimize the makespan sum and the total rush cost;

[0011] Step 2: Use a dynamic programming algorithm based on the properties of the optimal solution to solve the two-shift multi-task scheduling problem considering processing speed differences in Step 1 to obtain the optimal schedule.

[0012] Furthermore, the two-shift multi-task scheduling problem considering processing speed differences is as follows:

[0013] Given a set of jobs \(J=\{J_1,J_2,\cdots,J n \}\) containing \(n\) jobs, the processing time of job \(J j \) is \(p j \), and the due date is \(d j . Denote

[0014] Divide the time of a single machine into odd time intervals and even time intervals. Define \([0,\tau o ,[\tau o +\tau e ,2\tau o +\tau e , [2(\tau o +\tau e ), 2(\tau o +\tau e )+\tau o , \cdots\) and record it as \([a(\tau o +\tau e ), a(\tau o +\tau e )+\tau o \), where \(a = \{0,1,2,\cdots\}\) is the odd interval; define \([\tau o ,\tau o +\tau e ,[2\tau o +\tau e , 2(\tau o +\tau e ),[3\tau o +2\tau e , 3(\tau o +\tau e )], \cdots\) and record it as \([b(\tau o +\tau e )-\tau e , b(\tau o +\tau e )]\), where \(b = \{1,2,\cdots\}\) is the even interval;

[0015] Arrange the workpiece to be processed on the machine. There are three processing methods for the workpiece: 1. The workpiece is only processed in the odd intervals; 2. The workpiece is only processed in the even intervals; 3. The workpiece switches between the odd and even intervals and is processed to completion. The workpieces processed by the first two methods are called non-switching workpieces, and the workpieces processed by the third method are called switching workpieces;

[0016] The workpiece arrives at time 0, and the machine starts processing at time 0. There is no preparation time for the workpiece to start processing in each interval, and the processing of the workpiece can be interrupted. The machine runs at different speeds in different processing intervals. Among them, the processing speed of the machine in the odd intervals is v o , and the processing speed in the even intervals is v e ; Therefore, for workpiece J j the actual processing time in the odd intervals is p j / v o , and the actual processing time in the even intervals is p j / v e ; The workpiece can choose to switch back and forth between the odd and even intervals to complete the processing as soon as possible, and an urgent cost is generated.

[0017] Furthermore, the two-shift multi-task scheduling problem considering different processing speeds is NP-hard. The properties of the optimal scheduling of the problem are as follows:

[0018] Property 1: Each non-switching workpiece that is only processed in the odd or even intervals satisfies the SPT rule, and the machine has no idle time;

[0019] Property 2: Each switching workpiece that is continuously processed in the odd and even intervals satisfies the SPT rule, and the machine has no idle time.

[0020] Furthermore, the dynamic programming algorithm based on the properties of the optimal solution in step 2 is as follows:

[0021] Let (j, t o , w) be the state vector of the set of feasible partial schedules {J1, J2,..., J j}, where the parameter t o represents the sum of the initial processing times of the workpieces in the odd intervals; the parameter w represents the sum of the urgent costs of the partial schedule; F(j, t o , w) represents minimizing the sum of the total completion times and the total urgent costs of the corresponding partial schedule, where j = 0, 1,..., n, w = 0, φ,..., nφ;

[0022] The specific steps of the dynamic programming algorithm are as follows:

[0023] Step 2.1: Sort all the workpieces in the workpiece set J according to the SPT rule and re-number them to obtain the initial sequence: p1 ≤ p2 ≤... ≤ pn ;

[0024] Step 2.2: Determine the boundary conditions:

[0025]

[0026] Step 2.3: Algorithm iteration:

[0027] For j = w = 0, 1, …, n; w = 0, φ, 2φ, …, nφ;

[0028]

[0029] Wherein,

[0030] If the odd processing interval is idle first;

[0031] If the even processing interval is idle first;

[0032] Step 2.4: Obtain the optimal objective Z * = min{F(n, t o , w)|0 ≤ t o ≤ P, 0 ≤ w ≤ nφ}, and obtain the optimal sorting through reverse backtracking.

[0033] Furthermore, the dynamic programming algorithm is used to solve the two - shift multi - task scheduling problem considering different processing speeds, and the time complexity of obtaining the optimal scheduling is O(n 2 P), where the state variable t o has at most P possibilities, and w has at most n possibilities.

[0034] Beneficial effects

[0035] The present invention takes the two - shift multi - task processing as the research object, considering the actual problem of the impact of workpiece switching between different shifts on the production target. By establishing a workpiece switching model, studying the two - shift multi - task scheduling with different processing speeds and additional switching costs for different shift switches, analyzing the relationship between the switching cost of workpiece switching and shifts, designing a pseudo - polynomial time dynamic programming algorithm, and obtaining a multi - task scheduling decision scheme. The present invention is oriented to production practice, aiming to provide a theoretical basis for enterprises to coordinate the multi - task scheduling optimization decision of the two - shift working system, and also providing a research model for reference for other working modes such as the three - shift system.

[0036] Additional aspects and advantages of the present invention will be given in part in the following description, become apparent in part from the following description, or be learned through the practice of the present invention. Description of the Drawings

[0037] The above and / or additional aspects and advantages of the present invention will become apparent and be readily understood from the description of the embodiments in conjunction with the following drawings, in which:

[0038] Figure 1 Schematic diagram of workpiece sorting in Example 1. Detailed Description of the Invention

[0039] This patent proposes a multi-task scheduling problem for two-shift operation, studies a two-shift multi-task scheduling model with different processing speeds where additional switching costs are required for different shift changes, gives a multi-task decision-making scheme through designing a pseudo-polynomial time dynamic programming algorithm, and verifies it through examples, further verifying the rationality of the method of this patent.

[0040] The two-shift multi-task scheduling problem considering different processing speeds can be described as: Given a workpiece set J = {J1, J2, …, J n}, the processing time of workpiece J j is p j , the due date is d j , denote Divide the time of a single machine into odd periods and even periods, define [0, τ o , [τ o + τ e , 2τ o + τ e , [2(τ o + τ e ), 2(τ o + τ e ) + τ o , … denoted as [a(τ o + τ e ), a(τ o + τ e ) + τ o , a = {0, 1, 2, …} is the odd interval; define [τ o , τ o + τ e , [2τ o + τ e , 2(τ o + τ e ), [3τ o + 2τ e , 3(τ o + τ e)], … are denoted as [b(τ o + τ e ) - τ e , b(τ o + τ e ), where b = {1, 2, …} is the even interval. Arrange the jobs to be processed on the machine. There are three processing methods for the jobs: 1. The job is only processed in the odd interval; 2. The job is only processed in the even interval; 3. The job switches between the odd and even intervals and is processed to completion. The jobs in the first two processing methods are called non-switch jobs, and the jobs in the third processing method are called switch jobs. The jobs arrive at time 0, and the machine starts processing at time 0. There is no setup time for the jobs to start processing in each interval, and the job processing process can be interrupted. The machine runs at different speeds in different processing intervals. Specifically, the processing speed of the machine in the odd interval is v o , and the processing speed in the even interval is v e . Therefore, the actual processing time of job J j in the odd interval is p j / v o , and the actual processing time in the even interval is p j / v e . The job can choose to switch back and forth between the odd and even intervals to complete the processing as early as possible (ECT - Earliest Completion Time), and an urgent cost is generated. To sum up, we denote the job switching model in different speed intervals as "alt-speed-ECT" (alternate speed switch-ECT).

[0041] Step 1: Consider the two-shift multi-task scheduling model with different processing speeds and prove the properties of the optimal solution.

[0042] The optimization objective is to minimize the sum of the completion times and the total urgent cost, that is where is the total urgent cost. If job J j is a switch job, then φ j = φ; if job J j is a non-switch job, then φ j = 0. The multi-task scheduling problem studied, considering that the tasks continuously switch between the odd and even processing intervals with different speeds and generate urgent costs, can be expressed as: 1|alt-speed-ECT|∑C j + ∑φ j . It has been proven that the problem is NP-hard. The properties of the optimal solution of the problem are as follows:

[0043] In the problem 1|aalt-speed-ECT|∑Cj +∑φ j Among them, any optimal schedule satisfies the following two properties:

[0044] Property 1: Each non-switching workpiece that is only processed and completed at odd (or even) times satisfies the SPT rule, and the machine has no idle time.

[0045] Property 2: Each switching workpiece that is continuously processed in odd and even intervals satisfies the SPT rule, and the machine has no idle time.

[0046] The following only proves that Property 2 holds: It is easy to prove from the problem definition that the machine has no idle time during continuous processing. For any two adjacent switching workpieces J i and workpiece J j , there are three cases:

[0047] Case 1: Both workpiece J i and workpiece J j switch from the odd interval to the even interval.

[0048] Case 2: Both workpiece J i and workpiece J j switch from the even interval to the odd interval.

[0049] Case 3: Workpiece J i switches from the odd interval to the even interval, and workpiece J j switches from the even interval to the odd interval, and vice versa.

[0050] By contradiction, assume that there is an optimal sorting σ in which the processing times of two adjacent switching workpieces J i and workpiece J j satisfy: p i < p j . Now consider the new sequence σ′ obtained by swapping workpiece J i and workpiece J j . It is easy to get C j (σ′) = C i (σ), both including the same urgent cost 2φ. In addition, from p i < p j it can be obtained that C i (σ′) < C j (σ), where the number of switching pairs of workpiece J i in the odd and even intervals in the sequence σ′ will not exceed that of workpiece J j in the sequence σ. In summary, it contradicts the optimal sorting σ in the assumption, and the original problem is proved.

[0051] Step 2: A dynamic programming algorithm based on the properties of the optimal solution.

[0052] Based on the optimal solution property of Step 1, we present the following dynamic programming algorithm. Let (j, t o , w) be the state vector of the set of feasible partial schedules {J1, J2, …, J j}, where the parameter t o represents the sum of the initial processing times of the jobs in the odd intervals; the parameter w represents the sum of the rush costs of the partial schedule. F(j, t o , w) represents minimizing the sum of the total completion times and the total rush costs of the corresponding partial schedule, where j = 0, 1, …, n, w = 0, φ, …, nφ.

[0053] Therefore, the state vector (j, t o , w) can be obtained from the previous state vectors, considering the following three cases:

[0054] (1) Job J j is completed only in the odd intervals as a non-switching job. In this case, the state vector (j, t o , w) is obtained from the state vector (j - 1, t o - p j , w), and we have where the second and third terms on the right side of the equation represent the completion time of job J j .

[0055] (2) Job J j is completed only in the even intervals as a non-switching job. In this case, the state vector (j, t o , w) is obtained from the state vector (j - 1, t o , w), and we have where the second and third terms on the right side of the equation represent the completion time of job J j .

[0056] (3) Job J j is processed in the odd - even intervals as a switching job. In this case, the state vector (j, t o , w) is obtained from the state vector , and we have Let There are two cases: (i) If the odd processing interval is idle first. represents converting the idle time of the odd intervals in the occupied even intervals into the sum of the initial processing times of the jobs; represents converting the idle time of the last available even processing interval into the initial processing duration of the job; represents the number of a complete odd and an even interval that the job can occupy; Denote the remaining initial processing duration of the workpiece in the odd interval; Denote the remaining initial processing duration of the workpiece processed in the even interval. Similarly, (ii) if the even processing interval becomes idle first.

[0057] Dynamic programming algorithm DP-Alt C 2 :

[0058] Step 2.1: (Initialization) Sort all the workpieces in the workpiece set J according to the SPT (Shortest Processing Timefitst) rule and re-number them to obtain the initial sequence: p1 ≤ p2 ≤ … ≤ p n .

[0059] Step 2.2: (Boundary conditions)

[0060]

[0061] Step 2.3: (Iterative process)

[0062] For j = w = 0, 1, …, n; w = 0, φ, 2φ, …, nφ.

[0063]

[0064] where, (i) if the odd processing interval becomes idle first.

[0065] (ii) if the even processing interval becomes idle first.

[0066] Step 2. N4: (Optimal solution): Optimal objective Z * = min{F(n, t o , w)|0 ≤ t o ≤ P, 0 ≤ w ≤ nφ}, and obtain the optimal sorting through backward tracing.

[0067] Step 3: Time complexity analysis

[0068] Problem 1|alt-speed-ECT|∑C j can be solved by the dynamic programming algorithm DP-Alt C 2 in O(n 2The optimal solution is obtained within (P). Since the algorithm utilizes the property of the optimal solution structure and compares all possible state spaces generated, it produces an optimal schedule. Implementing the sorting process in Step 2.1 requires O(nlogn). In Step 2.3, for the state variable t o There are at most P possibilities; w has at most n possibilities. Therefore, Step 3 can be implemented within O(n 2 P) time, which is also the time complexity of the entire algorithm.

[0069]

Example 1

[0070] Table 1 Initial processing times of jobs

[0071]

[0072]

[0073] According to the dynamic programming algorithm DP-Alt C 2 The optimal scheduling plan obtained is: J1 is processed only in even intervals; J2 is processed only in odd intervals; J3 is processed continuously in both even and odd intervals and incurs an expediting cost. The optimal objective value is Z * = min{F(3, t o , w)|0 ≤ t o ≤ 41, 0 ≤ w ≤ 3 * 5} = F(3, 24, 5) = 61, as specifically shown in Figure 1 the following.

[0074] The following details the examples of the present invention. The examples implemented are illustrative and are intended to explain the present invention, and should not be construed as a limitation of the present invention.

[0075] The job-machine information is as follows: In a single-machine environment, the length of the odd time intervals is τ o = 3, and the machine speed in odd intervals is v o = 2; the length of the even intervals is τ e = 2, and the machine processing speed in even intervals is v e = 1, with an expediting cost of φ = 5. The initial processing lengths of the jobs (p1, p2, p3) = (10, 12, 19).

[0076] 1) From the boundary conditions F(0, 0, 0) = 0 and F(j, t o , w) = +∞, for t o ≠ 0, 1, …, 41, w < 0. The jobs are sorted according to the SPT rule as J1 → J2 → J3.

[0077] 2) For j = 1, t o = 0, 1, …, 10, w = 0, 5, we can calculate F(1, 0, 0) = 16 through the iterative equation. Similarly, all possible state vectors can be obtained: F(1, 6, 5) = 12, F(1, 10, 0) = 10.

[0078] 3) For j = 2, t o = 0, 1, …, 22, w = 0, 5, 10, we can calculate F(2, 0, 0) = 53 through the iterative equation. Similarly, all possible state vectors can be obtained: F(2, 6, 5) = 40, F(2, 10, 0) = 31, F(2, 12, 0) = 27, F(2, 12, 5) = 31, F(2, 12, 10) = 33, F(2, 18, 5) = 31, F(2, 22, 0) = 36.

[0079] 4) For j = 3, t o = 0, 1, …, 41, w = 0, 5, 10, 15, we can calculate F(3, 0, 0) = 121 through the iterative equation. Similarly, all possible state vectors can be obtained: F(3, 10, 0) = 80, F(3, 12, 0) = 74, F(3, 19, 0) = 77.5, F(3, 22, 0) = 67, F(3, 29, 0) = 65.5, F(3, 31, 0) = 67.5, F(3, 41, 0) = 86.5, F(3, 6, 5) = 96, F(3, 12, 5) = 78, F(3, 18, 5) = 69, F(3, 24, 5) = 61, F(3, 25, 5) = 72.5, F(3, 12, 10) = 80, F(3, 24, 10) = 65, F(3, 31, 10) = 73.5, F(3, 24, 15) = 67.

[0080] 5) The optimal objective value is Z * = min{F(3, t o , w)|0 ≤ t o ≤ 41, 0 ≤ w ≤ 3 * 5} = F(3, 24, 5) = 61. By backtracking, the optimal scheduling plan can be obtained: J1 is processed only in even intervals; J2 is processed only in odd intervals; J3 is processed continuously in both even and odd intervals and incurs rush costs.

[0081] The above example shows how to allocate tasks to minimize the total completion time and rush costs when different types of workers process multiple tasks in different shifts in a two-shift work system. It provides a theoretical basis for enterprises to coordinate the multi-task scheduling optimization decision in a two-shift work system and also provides a research model for reference in other work patterns such as three-shift systems.

[0082] Although the embodiments of the present invention have been shown and described above, it is to be understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those of ordinary skill in the art can make changes, modifications, substitutions, and variations to the above embodiments within the scope of the present invention without departing from the principles and spirit of the present invention.

Claims

1. A two-shift multi-task scheduling optimization method considering processing asynchronous speed, characterized in that: It includes the following steps: Step 1: For the two-shift multi-task scheduling problem considering processing different speeds, construct a two-shift multi-task scheduling model considering processing different speeds. The optimization objective of the model is to minimize the sum of the completion time and the total rush cost. The two-shift multi-task scheduling problem considering processing different speeds is as follows: Given a job set $J = \{J_1, J_2, \ldots, J_n\}$ containing $n$ jobs, n where the processing time of job $J_i$ is $p_i$ j and the due date is $d_i$. j Denote j by The time of a single machine is divided into odd time intervals and even time intervals, and [0,τ o ],[τ o +τ e ,2τ o +τ e ],[2(τ o +τ e ),2(τ o +τ e )+τ o ],…denoted as [a(τ o +τ e ),a(τ o +τ e )+τ o ], a={0,1,2,…} is an odd interval; define [τ o ,τ o +τ e ],[2τ o +τ e ,2(τ o +τ e )],[3τ o +2τ e ,3(τ o +τ e )],…denoted as [b(τ o +τ e )-τ e ,b(τ o +τ e )], b={1,2,…} is an even-numbered interval; Arrange the workpieces to be processed on the machine. There are three processing methods for the workpieces:

1. The workpiece is only processed in the odd intervals; 2. The workpiece is only processed in the even intervals; 3. The workpiece switches between the odd and even intervals and is processed to completion. The workpieces processed by the first two methods are called non-switching workpieces, and the workpieces processed by the third method are called switching workpieces. The workpiece arrives at time 0, the machine starts processing at time 0, there is no setup time for the workpiece to start processing in each interval, and the processing of the workpiece can be interrupted; the machine runs at different speeds in different processing intervals, where the processing speed of the machine in odd intervals is v o , and the processing speed in even intervals is v e ; Therefore, workpiece J j 's actual processing time in odd intervals is p j / v o , and the actual processing time in even intervals is p j / v e ; The workpiece can choose to switch back and forth between odd and even intervals to complete processing as soon as possible, and an urgent cost is incurred; Step 2: Use the dynamic programming algorithm based on the properties of the optimal solution to solve the two-shift multi-task scheduling problem considering processing different speeds in Step 1 to obtain the optimal schedule. The dynamic programming algorithm based on the properties of the optimal solution is as follows: Let (j, t o , w) be the state vector of the set of feasible partial schedules {J1, J2, …, J j}, where the parameter t o represents the sum of the initial processing times of the jobs in the odd intervals; the parameter w represents the sum of the rush costs of the partial schedule; F(j, t o , w) represents minimizing the sum of the total completion times and the total rush costs of the corresponding partial schedule, where, The specific steps of the dynamic programming algorithm are as follows: Step 2.1: Sort all the workpieces in the workpiece set J according to the SPT rule and re-number them to obtain the initial sequence: p1 ≤ p2 ≤ … ≤ p n ; Step 2.2: Determine the boundary conditions: Step 2.3: Algorithm iteration: For Among them, If the odd-numbered processing interval becomes idle first; If the even-numbered processing interval becomes idle first; Step 2.4: Obtain the optimal objective Z * = min{F(n, t o , w)|0 ≤ t o ≤ P, 0 ≤ w ≤ nφ}, and obtain the optimal sorting through reverse backtracking.

2. The two-shift multi-task scheduling optimization method considering processing different speeds according to claim 1, characterized in that: The two-shift multi-task scheduling problem considering processing different speeds is NP-hard, and the properties of the optimal schedule of the problem are as follows: Property 1: Each non-switching workpiece processed only in odd or even numbers satisfies the SPT rule, and the machine has no idle time; Property 2: Each switching workpiece continuously processed in odd and even intervals satisfies the SPT rule, and the machine has no idle time.

3. The two-shift multi-task scheduling optimization method considering different processing speeds according to claim 1, wherein: The dynamic programming algorithm is used to solve the two-shift multi-task scheduling problem considering processing asynchronous speed, and the time complexity of the optimal scheduling is O(n 2 P), where the state variable t o has at most P possibilities, and w has at most n possibilities.

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