Optimization Method for Two-Shift Multitask Scheduling Considering Switching Time

Abstract

The present invention proposes an optimization method for two-shift multi-task scheduling considering switching time, taking two-shift multi-task processing as the research object and considering the actual problem of the impact of workpiece switching between different shifts on production objectives. By establishing a workpiece switching model, analyzing the relationship between the switching time of workpiece switching and shifts, designing a pseudo-polynomial time dynamic programming algorithm, a multi-task scheduling decision scheme is obtained. 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 under the two-shift working system, and also provide a research model for reference for working modes such as three-shift.

Description

Technical Field

[0001] The present invention relates to the optimization problem of multi - task scheduling decision - making, and specifically to a two - shift multi - task scheduling optimization method considering the switching time. Background Technique

[0002] 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. Two Shift Pattern means that the working hours can be divided into day shift and night shift, and workers can choose to process workpieces during the day shift or the night shift. 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. Traditional production scheduling research focuses on seeking optimization solutions, but due to the complex technology and many constraint conditions in the production workshop, the optimization space is small and the optimization difficulty is large.

[0003] In actual production and service systems, the two - shift work system is widely and frequently applied. Usually, day - shift workers handle day - shift tasks and night - shift workers handle night - shift tasks. In particular, when tasks need to be continuously processed and completed as early as possible, there are the following processing scenarios: during the assembly process of parts, workers in each shift need to record the task processing situation. When the next shift processes the same task, a preparation time is required to confirm the status of the workpiece, clean the tools, and then proceed with the next step of processing. This time is defined as the switching time of the workpiece. 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 work system to minimize its scheduling objective, consider different processing methods of multi - tasks, analyze the influence relationship between task switching and switching time, obtain the multi - task scheduling scheme, and provide theoretical support for the multi - task scheduling optimization of the two - shift work system in the workshop, which is also the problem to be solved by the present invention.

[0006] The technical solution of the present invention is as follows:

[0007] A two - shift multi - task scheduling optimization method considering the switching time, comprising the following steps:

[0008] Step 1: For the two - shift multi - task scheduling problem considering the switching time, construct a multi - task scheduling model in which tasks switch once in the odd - even processing intervals: the optimization objective of the model is to minimize the sum of completion times;

[0009] 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 the switching time in step 1 and obtain the optimal schedule.

[0010] Furthermore, the two-shift multi-task scheduling problem considering switching time is:

[0011] Given a set of n artifacts J = {J1, J2, ..., J n +, workpiece J j The processing time is p j , the construction period is d j ,remember

[0012] The time of a single machine is divided into odd intervals and even intervals; 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 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;

[0013] Arrange the workpiece for processing on the machine. There are three processing modes for the workpiece: 1. The workpiece is processed only in the odd-numbered interval; 2. The workpiece is processed only in the even-numbered interval; 3. The workpiece is processed by switching between the odd and even intervals. The workpiece processed in the first two processing modes is called a non-switching workpiece, and the workpiece processed in the third processing mode is called a switching workpiece.

[0014] 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, the processing of the workpiece can be interrupted, the workpiece can choose to switch once in adjacent time intervals, and other workpieces are not allowed to interrupt during the switching process. When the workpiece switches from the current interval to an adjacent interval, a switching time s is generated in the adjacent interval, s is a positive real number, and the workpiece switches at most once. Denote p j ≤min{τ o ,τ e +.

[0015] Furthermore, the two-shift multi-task scheduling problem considering the switching time is NP-hard, and the properties of the optimal scheduling of the problem are as follows:

[0016] Property 1: Each non-switching workpiece that is completed only in odd or even numbers satisfies the SPT rule, and the machine has no idle time;

[0017] Property 2: Each workpiece that is continuously processed and switched in odd and even intervals satisfies the SPT rule, and the machine has no idle time.

[0018] Furthermore, the dynamic programming algorithm based on the properties of the optimal solution in step 2 is as follows:

[0019] Let (j, t o , k o , k e ) 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 k o represents the sum of the number of switching times in the odd intervals; the parameter k e represents the sum of the number of switching times in the even intervals; F(j, t o , k o , k e ) represents minimizing the sum of the total completion times of the corresponding partial schedules, where j, k o , k e = 0, 1, …, n, In addition, mark τ j,o as the idle time of the current interval when the workpiece J j is completed in the odd interval; τ j,e is the idle time of the current interval when the workpiece J j is completed in the even interval, j = 0, 1, …, n;

[0020] The specific steps of the dynamic programming algorithm are as follows:

[0021] 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 ;

[0022] Step 2.2: Determine the boundary conditions:

[0023]

[0024] Step 2.3: Algorithm iteration:

[0025] For j = k o , k e = 0, 1, …, n; 0 < ∈ j,o < τ o , o < ∈[[ID=..24]] j,e < τ e .

[0026]

[0027] Among them,

[0028]

[0029] Condition 1: s < ∈ o < p j and

[0030]

[0031] Condition 2: s < ∈ e < p j and

[0032] Step 2.4: Obtain the optimal objective Z * = min{F(j, t o , k o , k e ) | 0 ≤ t o ≤ P, k o = 0, 1, …n, k e = 0, 1, …, n}, and obtain the optimal sorting through reverse tracing.

[0033] 1. According to the two-shift multi-task scheduling optimization method considering switching time described in claim 4, characterized in that: the dynamic programming algorithm is used to solve the multi-task scheduling problem of two-shift considering switching time, and the time complexity of obtaining the optimal scheduling is O(n 3 P), where the state variable t o has at most P possibilities, k o has at most n possibilities, k e 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 production goals. By establishing a workpiece switching model, analyzing the relationship between the switching time of workpiece switching and shifts, designing a dynamic programming algorithm with pseudo - polynomial time, a multi - task scheduling decision scheme is obtained. 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 provide a research model for reference for working modes such as three - shift systems.

[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 understood through the practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

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

[0038] Figure 1 : Schematic diagram of Case 1 of the optimal property of the algorithm;

[0039] Figure 2 : In Case 3 of the optimal property of the algorithm Schematic diagram;

[0040] Figure 3 : In Case 3 of the optimal property of the algorithm Schematic diagram;

[0041] Figure 4 : Schematic diagram of the workpiece sorting in Example 1. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0042] The present invention proposes a two - shift multi - task scheduling problem, considering a two - shift multi - task scheduling model where tasks require switching time when switching between different shifts, and gives a multi - task decision scheme by designing a dynamic programming algorithm with pseudo - polynomial time. And through examples for verification, the rationality of the method of the present invention is further verified.

[0043] The two - shift multi - task scheduling problem considering switching time 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 , 2τo +τ e ,[2(τ o +τ e ),2(τ o +τ e )+τ o ,… are denoted as [a(τ o +τ e ), a(τ o +τ e ) + τ o , where 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 workpieces to be processed on the machine. There are three processing methods for the workpieces: 1. The workpiece is only processed in the odd interval; 2. The workpiece is only processed in the even interval; 3. The workpiece switches and is processed in both the odd and even intervals. The workpieces processed by the first two methods are called non-switch jobs, and the workpieces processed by the third method are called switch jobs. 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 workpiece processing process can be interrupted. The workpiece can choose to switch once in adjacent time intervals (odd-even period or even-odd period). The switching process is not allowed to be interrupted by other workpieces. When the workpiece switches from the current interval to the adjacent interval, a switching time s is generated in the adjacent interval. s is a positive real number. We assume that the workpiece switches at most once, and denote p j ≤min{τ o , τ e}. In summary, we denote the workpiece switching once model as "alt-switch-ONCE" (alternate Switch-Once).

[0044] Step 1: Consider the multi-task scheduling model of tasks switching once in odd and even processing intervals and prove the properties of the optimal solution.

[0045] The optimization objective is to minimize the sum of completion times, that is The multi-task scheduling problem considering the task switching once between odd and even processing intervals is expressed as: 1|alt-switch-ONCE|∑C j . It has been proven that the problem is NP-hard. The properties of the optimal solution of the problem are as follows:

[0046] In the problem 1|alt-switch-ONCE|∑C j , any optimal schedule satisfies the following two properties:

[0047] Property 1: Each non-switching workpiece that is only completed in the odd (or even) number satisfies the SPT rule, and the machine has no idle time.

[0048] Property 2: Each workpiece that is continuously processed and switched between odd and even intervals satisfies the SPT rule, and the machine has no idle time.

[0049] 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:

[0050] Case 1: Workpiece J i and workpiece J j both switch from the odd interval to the even interval.

[0051] Case 2: Workpiece J i and workpiece J j both switch from the even interval to the odd interval.

[0052] 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.

[0053] We assume that there exists a sequence σ = (π1, J j , π, J i , π2), where π1, π, π2 respectively represent three subsequences in the sorting σ, and the processing times of workpiece J i and workpiece J j satisfy: p i < p j . Since workpiece J i and workpiece J j are two adjacent switching workpieces, so the subsequence π only includes non-switching workpieces and is non-empty. Now swap the positions of workpiece J i and workpiece J j to get a new sequence σ' = (π1, J i , π, Jj , π2), where the workpiece J i and the workpiece J j are both switched workpieces. Otherwise, the case where the workpiece J i is a non-switched workpiece has been considered in Property 1.

[0054] Case 1: It can be observed that C i (σ′) < C j (σ), where C i (σ′) and C j (σ) represent the completion time of the workpiece J i in the sequence σ′ and the completion time of the workpiece J j in the sequence σ respectively, and both include a switching time s. Define C j (σ) - C i (σ′) = ∈ as the lead of the workpiece J i in the sequence σ′ relative to the workpiece J j in the sequence σ. It can be obtained that C j (σ′) = C i (σ) + ∈. Since both the workpiece J i and the workpiece J j are workpieces switched from the odd interval to the even interval, so π is non-empty. In the subsequence π of the sequence σ′, assume that the non-switched workpiece is completed in the even interval. Since the workpiece J i and the workpiece J j are exchanged in position, so in the new sequence σ′, the amount by which the non-switched workpiece is completed earlier is ∈, so it can be obtained that ∑C π (σ′) < ∑C π (σ); if no non-switched workpiece is completed in the even interval, then there is a first workpiece J k in the sequence π2 that will be processed before the workpiece J j and after the workpiece J i . In this case, ∑C π (σ′) = ∑C π (σ), as shown in Figure 1 . In summary, it can be obtained that: It can be inferred that the sequence σ′ is superior to the sequence σ.

[0055] Case 2 is similar to Case 1.

[0056] Case 3: If it can be obtained that C j (σ′) < C i (σ), from p i < p j it can be obtained that C i (σ′) < C j (σ). Thus, it can be seen As Figure 2 shown. If because C i (σ′) < C j (σ), then for the non-switching jobs in the self-sequence π of sequence σ′, their processing in the even processing intervals will be advanced. Therefore, it can be obtained that ∑C π (σ′) < ∑C π (σ). In this case, C j (σ′) = C i (σ), as Figure 3 shown. In summary, it can be inferred that sequence σ′ is superior to sequence σ.

[0057] Q.E.D.

[0058] Step 2: Solve using a dynamic programming algorithm based on the properties of the optimal solution.

[0059] Based on the properties of the optimal solution, we give the following dynamic programming algorithm. Let (j, t o , k o , k e ) 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 k o represents the sum of the number of switching times in the odd intervals; the parameter k e represents the sum of the number of switching times in the even intervals. F(j, t o , k o , k e ) represents minimizing the total completion time sum of the corresponding partial schedules, where j, k o , k e = 0, 1, …, n, In addition, we mark ∈ j,o as the idle time of the current interval when job J j is completed in the odd interval; ∈ j,e as the idle time of the current interval when job J j is completed in the even interval, j = 0, 1, …, n.

[0060] Therefore, the state vector (j, t o , k o , k e ) can be obtained from the previous state vectors, and the following four cases may be considered:

[0061] (1) Job J j is completed only in the odd interval as a non-switching job. In this case, the state vector (j, t o , k o, k e ) obtained from the state vector (j - 1, t o -p j , k o , k e ) and there is where the second to fourth terms on the right - hand side of the equation represent the completion time of workpiece J j . At this time, i = 1, 2, …, j - 1.

[0062] (2) Workpiece J j is completed only in even intervals as a non - switching workpiece. In this case, the state vector (j, t o , k o , k e ) is obtained from the state vector (j - 1, t o , k o , k e ) and there is where the second to fifth terms on the right - hand side of the equation represent the completion time of workpiece J j . At this time, i = 1, 2, …, j - 1.

[0063] (3) Workpiece J j is completed in odd - even intervals as a switching workpiece. The workpiece starts in the odd interval and is completed in the even interval. In this case, the state vector (j, t o , k o , k e ) is obtained from the state vector (j - 1, t o s -∈ o , k o , k e - 1) and there is F(j, tk o , k o , k e ) = F o-e (j, t o , k o , k e ), where

[0064] Condition 1: s <∈ o < p j and ensures that workpiece J j can be used as a switching workpiece to switch from the odd interval to the even interval. The second to fourth terms on the right - hand side of the equation represent the completion time of workpiece J j . At this time, ∈ j,e = τ e -(s + pj -∈ o ), ∈ i,o = 0, i = 1, 2, …, j - 1.

[0065] (4) Workpiece J j As a switching workpiece, it is completed in the even - odd interval. The workpiece starts in the even interval and is completed in the odd interval. In this case, the state vector (j, t o , k o , k e ) is obtained from the state vector (j - 1, t o -(p j -∈ e ), k o -1, k e ), and F(j, t o , k o , k e ) = F e-o (j, t o , k o , k e ), where

[0066] Condition 2: s < ∈ e < p j and ensures that workpiece J j can be used as a switching workpiece to switch from the even to the odd interval. The second to fourth terms on the right - hand side of the equation represent the completion time of workpiece J j . At this time, ∈ j,o = τ o -(s + p j -∈ e ), ∈ i,e = 0, i = 1, 2, …, j - 1.

[0067] Dynamic programming algorithm DP - Alt C 1 :

[0068] Step a: (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 .

[0069] Step b: (Boundary conditions)

[0070]

[0071] Step c: (Iterative process)

[0072] For j = ko , k e = 0, 1, …, n; 0 < ∈ j,o <τ o , o < ∈ j,e <τ e .

[0073]

[0074] Among them,

[0075]

[0076] Condition 1: s < ∈ o <p j and

[0077]

[0078] Condition 2: s < ∈ e <p j and

[0079] Step d: (Optimal solution): Optimal objective Z * = min{F(j, t o , k o , k e ) | 0 ≤ t o ≤ P, k o = 0, 1, … n, k e = 0, 1, …, n}, and the optimal sorting is obtained through reverse backtracking. >>

[0080] Step 3. Time complexity analysis

[0081] Problem 1|alt-switch-ONCE|∑C j The optimal solution can be obtained by the dynamic programming algorithm DP-Alt C 1 in O(n 3 P) time. Since the algorithm uses the property of the optimal solution structure and compares all possible state spaces, it generates an optimal schedule. Step a requires O(nlogn) to implement the sorting process. In step c, for the state variable t o there are at most P possibilities; k o there are at most n possibilities; k e there are at most n possibilities. Therefore, it can be implemented in O(n 3 P) time, which is also the time complexity of the entire algorithm.

[0082]

Example 1

[0083] Table 1 Initial processing time of jobs

[0084]

[0085] According to the dynamic programming algorithm DP-Alt C 1 The optimal scheduling plan obtained is: J1 is processed only in the odd intervals; J2 is processed in both odd and even intervals and switches from the odd interval to the even interval; J3 is processed in both even and odd intervals and switches from the even interval to the odd interval. The optimal objective value is Z * = F(2, 12, 1, 1) = 40, as specifically Figure 4 shown.

[0086] The following details the embodiments of the present invention. The embodiments are exemplary and are intended to explain the present invention and should not be construed as limiting the present invention.

[0087] The job-machine information is as follows: Single-machine environment, the length of the odd time interval is τ o = 10; the length of the even interval is τ e = 8, and the job switching time is s = 1. The initial processing lengths of the jobs (p1, p2, p3) = (6, 6, 7).

[0088] 1) From the boundary conditions F(0, 0, 0, 0) = 0 and F(j, k o , k e , t o ) = +∞, for t o ≠ 0, 1, …, 19, and ∈ 0,o = 0, ∈ 0,e = 0. The jobs are sorted according to the SPT rule as J1 → J2 → J3.

[0089] 2) For j = 1, k o = 0, 1, k e = 0, 1, t o = 0, 1, …, 6, we can calculate through the iterative equation to obtain In this state, ∈ 1,e = 2.

[0090] Similarly, all possible state vectors can be obtained: In this state, ∈ 1,o = 4.

[0091] 3) For j = 2, k o= 0, 1, 2, k e = 0, 1, 2, t o = 0, 1, …, 12. We can calculate F(2, 0, 0, 0) = F(1, 0, 0, 0) + 32 = 48 through the iterative equation. At this time, ∈ 2,o = 0, ∈ 2,e = 6.

[0092] Similarly, all possible state vectors can be obtained: F(2, 6, 0, 0) = min{F(1, 0, 0, 0) + 5, F(1, 6, 0, 0) + 16} = 22. At this time, ∈ 2,o = 4, ∈ 2,e = 2.

[0093] For t o = 10, k o = 0, k e = 1, because s < ∈ 1,o = 4 < p2, and We can obtain At this time, ∈ 2,o = 0, ∈ 2,e = 5.

[0094] Similarly, we can get F(2, 12, 0, 0) = F(1, 6, 0, 0) + 20 = 26. At this time, ∈ 2,o = 8, ∈ 2,e = 0.

[0095] [[ID=--]]4) For j = 3, k o = 0, 1, 2, 3, k e = 0, 1, 2, 3, and t o = 0, 1, …, 19. We can calculate F(3, 0, 0, 0) = F(2, 0, 0, 0) + 49 = 81 through the iterative equation. Similarly, all possible state vectors can be obtained:

[0096] F(3, 6, 0, 0) = F(2, 6, 0, 0) + 33 = 55;

[0097] F(37, 0, 0) = F(2, 0, 0, 0) + 7 = 55;

[0098] F(3, 10, 0, 1) = F(2, 10, 0, 1) + 30 = 49;

[0099] F(3, 12, 0, 0) = F(2, 12, 0, 0) + 17 = 43;

[0100] F(3, 12, 1, 1) = F(2, 10, 0, 1) + 21 = 40; It should be noted that there seems to be an unclear or incorrect expression in the original text where "For j = 3, k" is written in an unusual way. I've translated it as is while keeping the possible error in the original text. You may want to double-check the original content for accuracy.

[0101] F(3, 13, 0, 0) = F(2, 6, 0, 0) + 21 = 43;

[0102] F(3, 19, 0, 0) = F(2, 12, 0, 0) + 27 = 53.

[0103] 5) The optimal objective value is Z * = min{F(3, k o , k e , t o ) | 0 ≤ t o ≤ 41, k o = 0, 1, 2, 3, k e = 0, 1, 2, 3,} = F(3, 12, 1, 1) = 40. By backtracking, the optimal sorting plan can be obtained: J1 is processed only in the odd intervals; J2 is processed in both odd and even intervals and switches from the odd interval to the even interval; J3 is processed in both even and odd intervals and switches from the even interval to the odd interval.

[0104] The above example shows that in a two - shift working system, when the staff between different shifts process tasks continuously, a switching time is required. How to allocate tasks to minimize the total completion time. It provides a theoretical basis for the enterprise to coordinate the multi - task scheduling optimization decision of the two - shift working system, and also provides a research model for reference in other working modes such as three - shift systems.

[0105] Although the embodiments of the present invention have been shown and described above, it can 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 purposes of the present invention.

Claims

1. A two-shift multi-task scheduling optimization method considering switching time, characterized in that: It includes the following steps: Step 1: For the two-shift multi-task scheduling problem considering the switching time, construct a multi-task scheduling model where tasks switch once in the odd and even processing intervals. The optimization objective of the model is to minimize the sum of completion times; The two-shift multi-task scheduling problem considering the switching time is as follows: Given a set of jobs \(J=\{J_1,J_2,\ldots,J_n\}\) containing \(n\) jobs, n where the processing time of job \(J_i\) is \(p_i\), j and its due date is \(d_i\). Let j be denoted as j and Divide the time of a single machine into odd intervals and even intervals; 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 , where 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 )],… denoted as [b(τ o +τ e )-τ e ,b(τ o +τ e )], where b = {1, 2, …} is the even 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 in the first two methods are called non-switching workpieces, and the workpieces processed in 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, the processing of the workpiece can be interrupted, the workpiece can choose to switch once in adjacent time intervals, and other workpieces are not allowed to interrupt during the switching process. When the workpiece switches from the current interval to an adjacent interval, a switching time s is generated in the adjacent interval, s is a positive real number, and the workpiece switches at most once, denoted as p j ≤ min{τ o , τ e}; 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 the switching time 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 , k o , k e ) be the state vector of the set of feasible partial schedules \(\{J_1, J_2, \ldots, J j \}\), where the parameter \(t o \) represents the sum of the initial processing times of the jobs in the odd intervals; the parameter \(k o \) represents the sum of the number of switching times in the odd intervals; the parameter \(k e \) represents the sum of the number of switching times in the even intervals; \(F(j, t o , k o , k e )\) represents minimizing the sum of the total completion times of the corresponding partial schedules, where, In addition, denote \(\in j,o \) as the idle time of the current interval when job \(J j \) is completed in the odd interval; \(\in j,e \) is the idle time of the current interval when job \(J j \) is completed in the even interval, \(j = 0, 1, \ldots, n\); 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, Step 2.4: Obtain the optimal objective Z * = min{F(j, t o , k o , k e ) | 0 ≤ t o ≤ P, k o = 0, 1, … n, k e = 0, 1, …, n}, and obtain the optimal sorting through reverse backtracking.

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

3. The two-shift multi-task scheduling optimization method considering switching time according to claim 1, characterized in that: The dynamic programming algorithm is used to solve the multi-task scheduling problem of two-shift operation considering the switching time, and the time complexity of the optimal scheduling is O(n 3 P), where the state variable t o has at most P possibilities, and k o has at most n possibilities, and k e has at most n possibilities.

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