Method for analyzing influence of multi-processor job scheduling with increased number of machines

Abstract

The application provides a multi-processor workpiece scheduling influence analysis method with increased machine number, a resource expansion model with increased multi-processor machine number, considers the difference of the required machine number of the multi-processor workpiece and the change of the scheduling performance before and after the increase of the machine number, designs the optimal solution conversion scheme and heuristic rules before and after the increase of the machine number, reveals the influence law of the increase of the multi-processor machine number on the scheduling performance by analyzing the best case influence ratio of the optimal solution and the best case influence ratio of the approximate solution, and provides a theoretical basis for matching the appropriate machine number for the enterprise scheduling performance improvement.

Description

Technical Field

[0001] This invention relates to the field of workpiece scheduling technology, specifically to a method for analyzing the impact of increasing the number of machines on multiprocessor workpiece scheduling. Background Technology

[0002] Multiprocessor jobs are jobs that require processing by at least two machines simultaneously. A many-to-one dynamic matching relationship exists between machines and jobs, overcoming the uniqueness assumption of jobs in classical scheduling. The Multiprocessor Job Scheduling Problem (MJSP) is a problem that, under given constraints, sorts multiprocessor jobs and arranges them on machines in a specific order to optimize one or more objectives. Multiprocessors achieve collaborative configuration of machines and jobs by allocating processing resources and exploring the inherent relationship between processing bottlenecks and scarce resources.

[0003] In classic scheduling problems, the goal is to optimize certain objectives under fixed resource constraints. However, in practice, the amount of resources is not constant and can be increased under certain conditions. Resource augmentation problems refer to a class of problems in actual production that involve expanding resource capabilities such as the number of machines, machine speed, and machine capacity, while meeting production process requirements and the inherent structural requirements of the machines. By relaxing the resource constraints in the problem and investing additional resources, better scheduling performance can be achieved.

[0004] In actual production, reasonably increasing the number of multiprocessing machines greatly promotes equipment utilization and system productivity. Aero engines have highly complex structures and require high assembly precision. Taking the assembly of the casing as an example, workers need to use locating joints to position and align the upper and lower parts, and then multiple workers tighten the screws in sequence. Multiple assemblers and specialized equipment simultaneously assemble the casing, demonstrating a many-to-one processing method between machines and workpieces. By increasing the number of inspection equipment and technicians, screw tightening time can be reduced while ensuring locating precision, thereby improving the assembly efficiency of the casing. Furthermore, when the number of idle machines in the system does not match the number of machines required for the workpieces in the current order, the remaining machines will remain idle and cannot be effectively utilized. Figure 1As shown, the original system has 3 machines. For a batch of 2-processor workpiece orders, one machine is always idle. With the addition of a fourth machine, the order completion time is reduced by 50%. Here, people and machines are considered as machines in a broad sense, without considering the cost of purchasing or renting additional machines. For processing orders, the more machines in the system, the better the scheduling performance. However, the degree to which the increase in the number of machines improves the scheduling performance varies. It is crucial to characterize the performance improvement of a series of scheduling schemes when the number of machines increases to different values. Summary of the Invention

[0005] To address the impact of increasing the number of machines on the scheduling performance of multiprocessor tasks, this invention proposes a corresponding method. With the goal of minimizing the maximum completion time, it solves the problem of how to match the appropriate number of machines for different levels of scheduling performance improvement, thereby improving the utilization rate of product assembly resources.

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

[0007] The method for analyzing the impact of increasing the number of machines on multiprocessor job scheduling includes the following steps:

[0008] Step 1: Establish an analysis of the impact of increasing the number of machines on multiprocessor job scheduling problem Pm|m j ={1,k}|C max An integer programming model with the scheduling objective of minimizing the maximum completion time:

[0009] obj.

[0010] minC max

[0011] st

[0012]

[0013] C j =S j +p j j = 1, 2, ..., n

[0014] S j′ ≥C j -M(2-x j′i -x ji ),j <j′;j,j′=1,2,…,n;i=1,…,m

[0015] S i,j ≥C i,j-1 +p j-1 j = 1, 2, ..., n; i = 1, ..., m

[0016]

[0017] C max ≥C i i = 1, ..., m

[0018] x ji ={0,1},j=1,2,…,n; i=1,…,m

[0019] Define workpiece set Among them, workpiece J j For 1-processor workpiece or k-processor workpiece, p j For workpiece J j Processing time; let S j Indicates workpiece J j The start time of C j Indicates workpiece J j The completion time is given by $S$, where $j'$ is an index value greater than $j$. i,j Indicates machine M i upper workpiece J j The start time of C i,j-1 Indicates machine M i upper workpiece J j The start time of the previous workpiece, C i Indicates machine M i Completion time, C max Indicates the maximum completion time of the scheduling scheme; x ji x is a 0-1 variable; when workpiece j is processed on the i-th machine, x ji =1, otherwise x ji =0;m j Indicates workpiece J j The number of machines that need to be processed simultaneously;

[0020] Step 2: Analyze the impact analysis problem Pm|m j ={1,k}|C max The best-case impact ratio of the optimal solution is used to evaluate the potential improvement of the target value after increasing the number of machines.

[0021] Step 2.1: Determine the best-case impact ratio of the optimal solution:

[0022] Regarding the issue Pm|m j ={1,k}|C max Let m = αk + β, where make in μ and v are both positive integers satisfying μ≥1, 0≤v≤m-1. When the number of processing machines increases from m to v, the value of μ is determined by the condition μ ≥ 1 and v ≤ m-1. When the best-case scenario of the optimal solution has a greater impact than The upper bound is:

[0023]

[0024] Step 2.2: Evaluate the potential for improvement in the target value after increasing the number of machines:

[0025] Based on the current multi-processor workpiece production system, determine the initial number of machines m, the number of machines k required to process the workpieces, and the total number of machines after the planned increase in the number of machines. And based on m, k and Find the values ​​of μ, α, and β; then, based on the values ​​of μ, k, α, and β, substitute them into the best-case influence ratio expression for the optimal solution to obtain the upper bound of the best-case influence ratio for the current multiprocessor workpiece production system. This allows us to determine the optimal solution's influence ratio when the number of machines increases from m to... At that time, the maximum improvement in scheduling performance of the current multiprocessor workpiece production system;

[0026] Step 3: Design a rule for prioritizing workpieces with a large number of required processing machines and those with longer processing times (LS-LPT).

[0027] Based on the number of machines required for processing each workpiece, all workpieces are divided into two subsets T. 1 and T k T k This represents the set of all workpieces that require k machines to process simultaneously;

[0028] Workpiece set T 1 and T k All workpieces within are arranged according to the LPT rule;

[0029] Workpiece set T k The k-processor workpieces are sequentially arranged on the first idle machine starting from time zero;

[0030] Workpiece set T 1 The workpieces in the 1-processor are sequentially arranged on the first idle machine starting from time zero, until all workpieces are arranged for processing;

[0031] Step 4: Analyze the best-case impact ratio of the approximate solution to determine the appropriate number of machines to improve scheduling performance.

[0032] Step 4.1: Determine the best-case influence ratio of the approximate solution:

[0033] Regarding the issue Pm|m j ={1,k}|C max When the number of machines increases from m to At that time, the upper bound of the best-case impact ratio of the approximate scheduling scheme generated by the LS-LPT rules is:

[0034]

[0035] Step 4.2: Match the number of machines to improve scheduling performance:

[0036] Based on the current multiprocessor workpiece production system, α and β are calculated according to the initial number of machines m and k - the number of machines required to process the workpiece. This is then combined with the desired scheduling performance improvement value η.

[0037]

[0038] and

[0039]

[0040] Determine the number of machines after increasing the number of machines. The range of values ​​is obtained. The minimum value is used as the number of machines matched to improve scheduling performance.

[0041] Beneficial effects

[0042] This invention proposes an impact analysis method for multiprocessor job scheduling with an increased number of machines. It presents a resource expansion model for increasing the number of multiprocessors, considering the differences in the number of machines required for multiprocessor jobs and the changes in scheduling performance before and after the increase in machine count. The method designs optimal solution conversion schemes and heuristic rules before and after the increase in machine count. By analyzing the best-case impact ratio of the optimal solution and the best-case impact ratio of the approximate solution, it reveals the impact law of increasing the number of multiprocessors on scheduling performance, providing a theoretical basis for enterprises to match appropriate machine counts to improve scheduling performance.

[0043] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0044] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the description of the embodiments taken in conjunction with the following drawings, in which:

[0045] Figure 1 :2-Processing diagram of workpiece order;

[0046] Figure 2 Example 1: Gantt chart of optimal scheduling scheme; (a) 5 machines, (b) 6 machines;

[0047] Figure 3 When β = 0 Transform into case 1 of σ(m);

[0048] Figure 4 When 1≤β≤k-1 Transform into case 1 of σ(m);

[0049] Figure 5 When 1≤β≤k-1 Transform into case 2 of σ(m);

[0050] Figure 6 When 1≤β≤k-1 Transformed into case 3 of σ(m);

[0051] Figure 7 : Schematic diagram of scheduling scheme transformation under the following circumstances;

[0052] Figure 8 Case 1 of LS-LPT algorithm execution;

[0053] Figure 9 Workpiece J l Case 1.2①, for the workpiece in the 1-processor;

[0054] Figure 10 Workpiece J l Case 1.2②, for the workpiece in the processing machine;

[0055] Figure 11 Case 2 of LS-LPT algorithm execution. Detailed Implementation

[0056] This invention, based on the assembly context of multiple machines processing simultaneously, aims to minimize the maximum completion time and explores the impact analysis of increasing the number of multiprocessors. By analyzing the complexity of the problem, it designs optimal solution transformation schemes and heuristic rules before and after increasing the number of machines, obtaining the best-case impact ratio of the optimal solution and the best-case impact ratio of the approximate solution. This addresses the challenge of matching the appropriate number of machines to different levels of scheduling performance improvement, continuously adjusting the system's production capacity, and achieving efficient production line operation, thereby improving the utilization rate of aero-engine assembly resources.

[0057] The problem of analyzing the impact of increasing the number of machines on multiprocessor job scheduling for rigid multiprocessor jobs can be described as follows: Given a set of n jobs... and Let them represent the set of 1-processor jobs and the set of k-processor jobs, respectively, expressed in mathematical notation as follows: in Represents a set The first 1-processor workpiece in the process, Represents a set The first k-processor workpiece in the process, where k ≤ m; workpiece The processing time is workpiece The processing time is The processing times for all workpieces are not equal; the initial number of machines in the production system is m, and the machine set is denoted as m. After the number of machines increases, the number of machines in the production system will be: The set of machines is denoted as The i-th machine is denoted as M. i The number of machines required for multiprocessor workpiece processing is m. j For workpiece sets The workpiece in the image requires one machine for processing, i.e., m j =1; for workpiece sets The workpiece in the diagram requires k machines to process simultaneously, i.e., m. j =k. Assume that n given workpieces arrive at time 0, the machine begins processing at time 0, the workpieces have no preparation time, and the processing cannot be interrupted. The scheduling objective is to minimize the maximum completion time C. max To measure the impact of increasing the number of machines on scheduling performance improvement, the following metrics are used:

[0058] Best-case impact ratio of the optimal solution: For any instance I, the upper bound of the ratio of the optimal solution before and after increasing the number of multiprocessors, i.e.

[0059]

[0060] Best-case impact ratio of approximate solutions: For any instance I, the supremum of the ratio of approximate solutions before and after increasing the number of multiprocessors, for the approximate algorithm or heuristic rule A.

[0061]

[0062] Where C max (σ * (m),I) and Let σ represent the optimal scheduling scheme under instance I. * (m) and Maximum completion time; C max (σ A (m),I) and Let σ represent the approximate scheduling scheme under instance I. A (m) and The maximum completion time.

[0063] Specifically, this invention designs an optimal solution transformation scheme and heuristic rules before and after increasing the number of machines, analyzes the best-case impact ratio of the optimal solution and the approximate solution, and the main features of the method for analyzing the impact of increasing the number of machines on multiprocessor job scheduling are included in the following process:

[0064] 1) Propose a mathematical programming model for the relevant problem;

[0065] 2) Analyze the complexity of the problem;

[0066] 3) Analyze the best-case impact ratio of the optimal solution and evaluate the potential for improvement in the target value after increasing the number of machines;

[0067] 4) A heuristic rule system, LS-LPT, was designed;

[0068] 5) Analyze the impact ratio of the best case of the approximate solution to match the number of machines to improve scheduling performance.

[0069] Step 1: Analysis of the impact of increasing the number of machines on multiprocessor job scheduling using an integer programming model. The scheduling objective is to minimize the maximum completion time. The main focus is on analyzing the impact of increasing the number of multiprocessors on the scheduling objective. Based on the three-parameter representation proposed by Graham et al., the analysis problem of multiprocessor job scheduling can be expressed as: Pm|m j ={1,k}|C max Consider not distinguishing between sets of workpieces. and Define workpiece set Among them, workpiece J j For 1-processor workpiece or k-processor workpiece, p j For workpiece J j Processing time; let S j Indicates workpiece J j The start time of C j Indicates workpiece J j The completion time is given by $S$, where $j'$ is an index value greater than $j$. i,j Indicates machine M i upper workpiece J j The start time of C i,j-1 Indicates machine M i upper workpiece J j The start time of the previous workpiece, C i Indicates machine M i Completion time; C max This represents the maximum completion time of the scheduling scheme. Its integer programming model is specifically represented as follows:

[0070] obj.

[0071] minC max #(3)

[0072] st

[0073]

[0074] C j =S j +p j,j=1,2,…,n#(5)

[0075] S j′ ≥C j -M(2-x j′i -x ji ),j <j′;j,j′=1,2,…,n;i=1,…,m#(6)

[0076] S i,j ≥C i,j-1 +p j-1 ,j=1,2,…,n; i=1,…,m#(7)

[0077]

[0078] C max ≥C i ,i=1,…,m#(9)

[0079] x ji ={0,1},j=1,2,…,n; i=1,…,m#(10)

[0080] Objective function (3) indicates that the scheduling objective is to minimize the maximum completion time; constraint (4) indicates that the rigid multiprocessor job requires m j The machines process simultaneously; constraint (5) indicates that the completion time of workpiece j is equal to the sum of the start time and processing time of workpiece j; constraint (6) indicates that the start time of each workpiece on the same machine is not less than the completion time of any previous workpiece, where M is a very large positive number; constraint (7) indicates that the start time of each workpiece on the same machine is not less than the sum of the start time and processing time of the previous workpiece; constraint (8) indicates that the completion time of machine i is equal to the sum of the processing times of all workpieces on machine i; constraint (9) indicates that the maximum completion time is not less than the sum of the completion times of all workpieces on each machine; constraint (10) indicates that x ji x is a 0-1 variable; when workpiece j is processed on the i-th machine, x ji =1, otherwise x ji =0.

[0081] Example 1 illustrates a multiprocessor job scheduling model with an increased number of machines.

[0082] Example 1. The example consists of 8 workpieces and 5 machines, where k = 3. The number of machines required for processing the multiprocessor workpieces is m. j ={1,3}; workpiece index j = 1,2,…,8, machine index i = 1,2,…,5. The corresponding processing time and required number of machines for the workpiece are shown in Table 1.

[0083] Table 1 shows the processing time and number of machines required for the workpiece in Example 1.

[0084]

[0085] When the number of machines m = 5, there exists an optimal scheduling scheme as follows: Figure 2 As shown in (a), workpieces J8, J6, J7, and J5 are processed simultaneously on machines M1, M2, and M3 in sequence; workpieces J4 and J1 are processed sequentially on machine M4; and workpieces J3 and J2 are processed sequentially on machine M5. The maximum completion time is C. max (σ * (m),I)=8.

[0086] When the number of machines At that time, there exists an optimal scheduling scheme such as Figure 2 As shown in (b), workpieces J6, J7, and J5 are processed simultaneously on machines M1, M2, and M3 in sequence, while workpiece J8 is processed simultaneously on machines M4, M5, and M6. Then, workpieces J1 and J2 are processed sequentially on machine M4, workpiece J3 is processed on machine M5, and workpiece J4 is processed on machine M6. The maximum completion time is...

[0087] When the number of machines increases from m=5 to When the maximum completion time of the optimal scheduling scheme is C max (σ * (m),I)=8 decreased to

[0088] Step 2: Analyze the complexity of the problem.

[0089] When the number of machines required for the workpiece is k = 1, the problem Pm|m j ={1,k}|C max Equivalent to Pm||C in the case where the workpiece cannot be interrupted max Question: When m≥2, Pm||C max The problem can be reduced to a partitioning problem, and partitioning problems are NP-hard. Therefore, when m ≥ 2, the problem Pm||C max This is an NP-hard problem; this invention mainly studies a multiprocessor job scheduling model with an increased number of machines, where the number of machines increases... Considering that the number of machines can only be an integer, therefore Due to the problem Pm||C max If (m≥2) is a special case of the original problem, then the original problem Pm|m j ={1,k}|C max It is NP-hard.

[0090] Step 3: Analyze the best-case impact ratio of the optimal solution and evaluate the potential for improvement in the target value after increasing the number of machines.

[0091] Step 3.1: Analyze the best-case impact ratio of the optimal solution.

[0092] Regarding the issue Pm|m j ={1,k}|C max Let m = αk + β, where make in μ and v are both positive integers, satisfying μ≥1, 0≤v≤m-1. When the number of processing machines increases from m to... When, under the optimal scheduling scheme, the best-case impact ratio has an upper bound of .

[0093]

[0094] First, C max (σ * (m),I) and Simplify to C respectively max (σ * (m)) and The set of machines required for processing the workpiece is m. j ={1,k}, therefore, consider expressing the total number of machines before and after increasing the number of machines as a relationship between 1 and k. Substitute the initial number of machines m = αk + β into the expression for the total number of machines after increasing the number of machines. have to

[0095]

[0096] The range of values ​​for v is 0 ≤ v ≤ αk + β - 1, therefore we can obtain The range of values ​​is

[0097]

[0098] make The best-case impact ratio index can then be rewritten as:

[0099]

[0100] In equation (14), take The impact of the best-case scenario has been amplified.

[0101] Next, disregarding the number of workpieces, the present invention adopts the following proof approach: by constructing an intermediate scheduling scheme σ(m), a scheduling scheme σ is established. * (m) and The relationship between the two factors is used to determine the best-case influence ratio of the optimal solution. According to the possible values of β, the discussion will be carried out in two steps next.

[0102] Step 3.1.1: Consider that when the number of machines m is rounded to k with β = 0, at this time m = αk,

[0103] Divide the machines in the scheduling scheme into three parts: the first part contains one machine group which has μk machines; the second part has (α - 1) machine groups, each of which contains (μ + 1)k machines; the third part contains one machine group which has (k - 1) machines. It is now assumed that there are additionally m = αk machines, and every k machines are numbered sequentially as 1, 2, …, α respectively.

[0104] As shown in Figure 3 , all workpieces in the machine group of the first part of machines and the machine group of the third part of machines in the scheduling scheme are all allocated to the machine numbered 1, and then the workpieces on each machine group in the second part of machines are sequentially allocated to the machines numbered 2, …, α, so as to obtain the scheduling scheme σ(m).

[0105] In the scheduling scheme σ(m), the maximum load of the k machines numbered 1 is Among the remaining (α - 1)k machines with remaining numbers, the maximum load of each machine is Therefore

[0106]

[0107] Furthermore, since σ(m) is an approximate scheduling scheme for m = αk machines, then

[0108] C max (σ * (m))≤C max (σ(m))#(16)

[0109] Combining Equation (14), Equation (15) and Equation (16), we can obtain:

[0110]

[0111] Step 3.1.2: Consider that when the number of machines m is rounded to k, the range of β is 1≤β≤k-1. In order to better express the total number of machines in the system as a multiple relationship of k, define Discussions are divided into three cases according to different values of ω.

[0112] Case 1: After the number of machines is increased, the condition 0 < μβ + β - 1 < k is satisfied, that is At this time m = αk + β,

[0113] scheduling scheme The machines are divided into three parts: the first part contains α machine groups, each containing (μ+1)k machines; the second part contains (β-1) machine groups, each containing (μ+1) machines; and the third part contains one machine group containing μ machines. There are also m = αk + β machines. The first αk machines are numbered sequentially as 1, 2, ..., α; then the (β-1) machines are numbered sequentially as α+1, ..., α+β-1; finally, the remaining machine is numbered α+β.

[0114] like Figure 4 As shown, the scheduling scheme In the first part of the machines, all workpieces on each machine group are assigned to machines labeled 1, 2, ..., α. Then, the workpieces on each machine group in the second part of the machines are assigned to machines labeled α+1, ..., α+β-1. Finally, the workpieces on the machine groups in the third part of the machines are assigned to machines labeled α+β, thus obtaining the scheduling scheme σ(m).

[0115] In the scheduling scheme σ(m), for all machines labeled 1, 2, ..., α, the maximum load is... For all machines labeled α+1,…,α+β-1, the maximum load is... For the machine labeled α+β, its maximum load is Therefore, equation (15) is valid.

[0116] Combining equations (14) and (16), we can obtain

[0117]

[0118] In scenario 2, after the number of machines increases, the condition k ≤ μβ + β - 1 ≤ (α - 1)k is satisfied, i.e., 1 ≤ ω ≤ α - 1. In this case, m = αk + β. Where μβ+β-1-ωk <k。

[0119] scheduling scheme The machines are divided into four groups: the first group contains ω machine groups, each containing (μ+2)k machines; the second group contains (α-ω-1) machine groups, each containing (μ+1)k machines; the third group contains one machine group, containing (μ+1)k machines; and the fourth group contains one machine group, containing (μβ+β-1-ωk) machines. There are also m = αk+β machines. The first ωk machines are numbered sequentially as 1, 2, ..., ω; then the next (α-ω-1)k machines are numbered sequentially as ω+1, ..., α-1; then the next k machines are numbered as α; finally, the remaining machines are numbered sequentially as α+1, ..., α+β.

[0120] like Figure 5 As shown, the scheduling scheme In the first part of the machines, all workpieces on each machine group are assigned to machines numbered 1, 2, ..., ω. Then, workpieces on each machine group in the second part of the machines are assigned to machines numbered ω+1, ..., α-1. Next, workpieces on the machine groups in the third part of the machines are assigned to machines numbered α. Finally, workpieces on the machine groups in the fourth part of the machines are assigned to machines numbered α. No workpieces are placed on the β machines numbered α+1, ..., α+β, thus obtaining the scheduling scheme σ(m).

[0121] In the scheduling scheme σ(m), for all k machines labeled 1, 2, ..., ω, the maximum load is... For all k machines labeled ω+1,…,α-1, the maximum load is For k machines labeled α, the maximum load is... For the β machines labeled α+1,…,α+β, their load is 0, therefore

[0122] Combining equations (14) and (16), we can obtain

[0123] Note that for the (μβ+β-1-ωk) machines in the machine group of Part 4, the upper limit of their number is (k-1) machines; and the upper limit of the number of the β machines labeled α+1,…,α+β is also (k-1) machines. Since the specific values ​​of both cannot be determined, the maximum load that can be achieved by allocating all workpieces from the (μβ+β-1-ωk) machines to the β machines labeled α+1,…,α+β cannot be determined; however, allocating these (μβ+β-1-ωk) machines to the machines labeled α will not change the maximum load of the m labeled machines, thus not affecting the best-case effect ratio upper bound.

[0124] In scenario 3, after the number of machines increases, αk ≤ μβ + β - 1, i.e., α ≤ ω, and at this time m = αk + β. Where μβ+β-1-ωk <k。

[0125] Let ω = γ1α + γ2, where γ1 ≥ 1, 0 ≤ γ2 ≤ α - 1, then the number of machines after increasing the number of machines is:

[0126] scheduling scheme The machines are divided into three parts: the first part contains γ2 machine groups, each containing (μ+γ1+2)k machines; the second part contains (α-γ2) machine groups, each containing (μ+γ1+1)k machines; and the third part contains one machine group, containing (μβ+β-1-ωk) machines. There are also m = αk+β machines. The first γ2k machines are numbered sequentially as 1, 2, ..., γ2; the next k machines are numbered as γ2+1; and the remaining machines are numbered sequentially as α+1, ..., α+β.

[0127] like Figure 6 As shown, the scheduling scheme In the first part of the machines, all workpieces on each machine group are assigned to machines labeled 1, 2, ..., γ2. Then, the workpieces on the machine group in the second part of the machines are assigned to machines labeled γ2+1. Finally, the workpieces on the machine group in the third part of the machines are assigned to machines labeled α+1, ..., α+β, thus obtaining the scheduling scheme σ(m).

[0128] Since the relationship between (μβ+β-1-ωk) and β is unknown, we will discuss two cases as follows:

[0129] 1)μβ+β-1-ωk≤β.

[0130] At this point, in the scheduling scheme σ(m), the maximum load of all k machines labeled 1, 2, ..., γ2 is... For k machines labeled γ2+1, the maximum load is For β machines labeled α+1,…,α+β, the load is but Therefore, we can obtain

[0131] 2)μβ+β-1-ωk>β.

[0132] When all workpieces from (μβ+β-1-ωk) machines are assigned to the last β machine out of m machines, the following proof demonstrates... This illustrates that, in the case where μβ+β-1-ωk>β, the placement of workpieces on (μβ+β-1-ωk) machines on β machines will not affect the maximum completion time. Subtracting the two values ​​yields...

[0133]

[0134] Since αk≤μβ+β-1 and α≥1, i.e., k≤αk, therefore k≤μβ+β-1; and since Therefore, ω≥1, hence therefore

[0135] In conclusion, Right now Therefore, when μβ+β-1-ωk>β, all workpieces from (μβ+β-1-ωk) machines are assigned to the last β machines out of the m machines, the load on the β machines will not exceed [the maximum load]. therefore

[0136] Since this patent focuses on the optimal performance improvement after increasing the number of machines, when 1≤β≤k-1, considering cases 1, 2, and 3, the best-case impact ratio of the optimal solution is:

[0137] Based on the above analysis, the best-case impact ratio of the optimal solution for multiprocessor job scheduling in step 3.1 can be obtained.

[0138] Step 3.2: Evaluate the potential for improvement in the target value after increasing the number of machines.

[0139] Step 3.2.1: Based on the current multi-processor workpiece production system, determine the initial number of machines m, the number of machines k required to process the workpieces, and the total number of machines after the planned increase in the number of machines. And based on m, k and Find the values ​​of μ and β;

[0140] Step 3.2.2: Based on the values ​​of μ, k, and β, substitute them into the best-case influence ratio expression for the optimal solution.

[0141]

[0142] The best-case scenario impact of obtaining the optimal solution

[0143] Step 3.2.3: When the number of machines increases from m to... At that time, the maximum improvement in production system scheduling performance is

[0144] Step 4: Design heuristic rules.

[0145] Due to the problem Pm|m j ={1,k}|C max It is NP-hard, meaning that for any given instance, it is impossible to find an optimal algorithm to find the optimal scheduling scheme in polynomial time unless P=NP. Therefore, the design of the rule of prioritizing the largest number of processing machines and the longest processing time (Largest Size-Longest Processing Time, LS-LPT) is required.

[0146] Step 4.1: Based on the number of machines required for processing the workpieces, divide all workpieces into two subsets T. 1 and T k T k This represents the set of all workpieces that require k machines to process simultaneously;

[0147] Step 4.2: Assemble the workpiece set T 1 and T k All workpieces within the workpieces are arranged according to the LPT (Longest Processing Time) rule;

[0148] Step 4.3: Assemble the workpiece set T k The k-processor workpieces are sequentially arranged on the first idle machine starting from time zero;

[0149] Step 4.4: Set the workpiece T 1 The workpieces in the 1-processor are sequentially arranged on the first idle machine starting from time zero, until all workpieces are processed.

[0150] Step 5: Analyze the best-case impact ratio of the approximate solution to determine the number of matching machines to improve scheduling performance.

[0151] Step 5.1: Analyze the problem Pm|m j =k|C max The best-case effect ratio of the approximate solution.

[0152] To provide the question Pm|m j ={1,k}|C max The best-case impact ratio of the approximate solution is first considered when the number of workpieces in the 1-processor is 0. Then the above problem is transformed into Pm|m j =k|C max At this point, the heuristic rule LS-LPT is transformed into an LPT rule for processing k-processor workpieces; then, the above results are used to solve the problem Pm|m j ={1,k}|C max Perform the analysis.

[0153] For the problem Pm|m j =k|C max When the number of machines increases from m to When the best-case effect of the approximate solution generated by the LPT rule is greater than the upper bound, the effect is...

[0154]

[0155] First, assume there are several counterexamples such that when the number of machines increases from m to... At this point, the best-case impact ratio upper bound of the approximate scheduling scheme generated by the LPT algorithm does not satisfy the above theorem. Take the instance I1 with the fewest jobs among all these counterexamples, and let the number of jobs in instance I1 be n. k .

[0156] For instance I1, the multiprocessor workpiece Sort the workpieces according to a non-increasing sequence of processing times, and then use the LPT algorithm to sort the workpiece set. The workpieces are assigned to m identical machines. Let workpiece J... l σ LPT The last workpiece to be arranged in (m), workpiece J l The processing time is recorded as The start time of processing is recorded as

[0157] Secondly, under the LPT rule, the workpiece J with the shortest processing time... l It is the last workpiece to begin processing and also the last workpiece to be processed.

[0158] According to the LPT rule, the workpiece J with the shortest processing time... l It is the last workpiece to begin processing. Assume workpiece J... l If it is not the last workpiece to be processed, delete workpiece J. l We obtain counterexample I2. For counterexample I2, C max (σ LPT(m)) remains unchanged, remains unchanged or may become smaller. The influence of the best case of the approximate solution will not become worse, but at this time the instance I1 violates the principle of minimum number of jobs, so the counterexample I2 is not valid.

[0159] Next, let the initial number of machines m = αk + β, where it can be known through analysis that before , among all machines, there are only β (β < k) machines that will be idle, and all other machines are busy; otherwise, job J l can start processing before . Therefore, before job J l is arranged for processing, the average load on the m machines at this time is the upper bound of the start time of job J l start time , that is,

[0160]

[0161] therefore, the makespan under σ LPT (m) is

[0162]

[0163] when the number of machines is , the approximate scheduling scheme generated by the LPT algorithm is it can be known from equation (13) that furthermore, since jobs in the job set require k machines for simultaneous processing, we can obtain that the lower bound of is

[0164]

[0165] according to equations (19) and (20), we can obtain

[0166]

[0167] the job with the longest processing time among all jobs is denoted as p max , and the job with the shortest processing time is denoted as p min , then for equation (21), the discussion is carried out in two steps below.

[0168] Step 5.1.1: consider the case. Substituting into equation (21) gives

[0169]

[0170] therefore, we can obtain

[0171]

[0172] Step 5.1.2: Consider situation.

[0173] At this time, when the number of machines is At that time, due to Each machine can process at most one workpiece, scheduling scheme The optimal scheduling scheme and the maximum completion time The time taken to process the workpiece is determined by the workpiece with the longest processing time.

[0174] Known m = αk + β, that is Next, based on the optimal scheduling scheme Construct a new scheduling scheme σ(m), and then specify the target value of the scheduling scheme. and C max The best-case influence ratio of the approximate solution is given by the relationship between the magnitudes of (σ(m)).

[0175] like Figure 7 As shown, Figure 7 The optimal scheduling scheme for the middle right side The workpieces on the first αk machines are placed on the first αk machines of the scheduling scheme σ(m) for processing. The remaining β machines out of the machine count m do not have any workpieces placed on them. Then, similarly... Subsequent workpieces are sequentially placed on idle machines in scheduling scheme σ(m) in units of "αk machines" for processing. Each placement will result in β machines remaining idle. Furthermore, because scheduling scheme σ... LPT (m) involves placing k-processor workpieces on idle machines in descending order of processing time, while in scheduling scheme σ(m), all αk machines process the workpieces each time. The next batch of machine parts will be allocated after a certain time (some machines have already completed processing, but still need to wait until...). (Time interval), and the workpieces are also arranged to idle machines sequentially according to the LPT rules, therefore

[0176] In step 5.1.1, the following is obtained: In step 5.1.2, the following is obtained: because Furthermore, since αk+β-1≥v, it is impossible to determine the relationship between the magnitudes of the best-case influence ratios of the approximate solutions in the two cases.

[0177] Based on the above analysis, we can obtain the problem Pm|m j =k|Cmax The best case of the approximate solution has a greater impact than the ratio.

[0178] Next, we will address the issue Pm|m j ={1,k}|C max The following analysis is performed by executing the heuristic rule LS-LPT.

[0179] Step 5.2: Analyze the problem Pm|m j ={1,k}|C max The best-case effect ratio of the approximate solution.

[0180] Regarding the issue Pm|m j ={1,k}|C max When the number of machines increases from m to When the best-case impact ratio of the approximate scheduling scheme generated by the LS-LPT rules is [value missing], the upper bound is [value missing].

[0181]

[0182] First, construct a counterexample I1 with the minimum number of workpieces, that is, for example I1, when the number of machines increases from m to... At that time, the best-case influence of the approximate scheduling scheme generated by the LS-LPT algorithm does not satisfy equation (23) above the upper bound. In example I1, the number of workpieces is n (n = n1 + n). k The number of machines required for workpiece processing is 1 or k.

[0183] From the analysis in step 5.1, it is easy to conclude that for instance I1, under the LS-LPT algorithm, the workpiece J with the minimum processing time... l It is the last workpiece to begin processing and also the last to finish processing; its processing time is p. l .

[0184] Secondly, for instance I1, let C max (σ LS-LPT (m)) and This represents the approximate solution obtained by the LS-LPT algorithm before and after increasing the number of machines.

[0185] Next, the number of machines is increased to At that time, the target value The lower bound, that is

[0186]

[0187]

[0188] Meanwhile, for workpiece J with the shortest processing time l have

[0189]

[0190] According to the LS-LPT algorithm, the last workpiece J l Having determined the maximum completion time, we will now discuss it in two steps.

[0191] Step 5.2.1: Consider workpiece J l For k-processor workpieces.

[0192] The workpiece set contains 1-processor workpieces and k-processor workpieces. According to the LS-LPT algorithm, k-processor workpieces are scheduled first, followed by 1-processor workpieces. When the last workpiece J... l When dealing with k-processor jobs, according to the LS-LPT rule's operational steps, since both k-processor and 1-processor jobs are arranged according to the LPT rule, deleting 1-processor jobs from all machines will not change the target value C. max (σ LS-LPT The size of (m) is such that, since instance I1 is a counterexample with the fewest number of workpieces, i.e., there are no 1-processor workpieces, then when workpiece J l When the workpiece is a k-processor, the problem is equivalent to solving for Pm|m j =k|C max The problem is, when the number of machines increases from m to... When, find the best-case influence ratio of the approximate solution generated by the LPT rule.

[0193] According to step 5.1, we can obtain

[0194] Step 5.2.2: Consider workpiece J l For a 1-processor workpiece, we will discuss two cases based on whether the value of β is zero.

[0195] Case 1: β = 0, m = αk.

[0196] like Figure 8 As shown, let and Representing the sets of workpieces respectively and The maximum completion time when allocated to m machines. and These are the workpiece sets under the LS-LPT algorithm. and The last workpiece processed in the process, its processing time is denoted as follows: and The start dates are respectively and

[0197] according to Figure 8 Combining equation (19), we can obtain

[0198]

[0199] By combining equations (25) and (27), we can obtain

[0200]

[0201] For inequality (28), we will discuss two cases below.

[0202] Case 1.1

[0203] Will Substituting into equation (28), we can obtain

[0204]

[0205] Therefore, it can be obtained

[0206]

[0207] Case 1.2

[0208] When the number of machines is m, workpiece J l It is the last workpiece to begin processing and complete, and its completion time is equal to the maximum completion time C. max (σ LS-LPT (m)); when the number of machines is At that time, due to Each machine can process at most one workpiece. The optimal scheduling scheme and From set T k and set T 1 The first workpiece determines the outcome, and because workpiece J... l For a 1-processor workpiece, then

[0209] The following is based on the number of machines. Whether m is an integer multiple is discussed in the following two cases.

[0210] ① Where μ and α are both positive integers and μ≥1, α≥1.

[0211] Next, based on the optimal scheduling scheme Construct a new scheduling scheme σ(m), specifically as follows: Figure 9 As shown. Since m = αk, Figure 9 Optimal scheduling scheme The workpieces on the first αk machines are placed on the αk machines of the scheduling scheme σ(m) for processing, and then similarly... Subsequent workpieces are sequentially placed on the idle machines on the left side of the scheduling scheme σ(m) in units of "αk machines" for processing. Because... Therefore, we can know And because of the scheduling scheme σ LS-LPT (m) involves placing 1-processor workpieces and k-processor workpieces on idle machines in descending order of the number of machines required for processing and in descending order of processing time for the same type of workpieces. In the scheduling scheme σ(m), all αk machines are processed each time. The next batch of machine group workpieces will be allocated after a certain time (some machines have completed processing in advance, but still need to wait until...). (Time interval), and the workpieces are also arranged to idle machines sequentially according to the LS-LPT rules, therefore

[0212] ② Where μ, v, and α are all positive integers and μ ≥ 1, α ≥ 1, 1 ≤ v ≤ αk.

[0213] Similar to case ①, according to the optimal scheduling scheme Construct a new scheduling scheme σ(m). In the scheduling scheme σ(m), the number of workpieces processed on each machine and the optimal scheduling scheme are considered. The number of workpieces processed on each machine is equal, and The last remaining 1-processor job on v machines is placed in the last time interval of the scheduling scheme σ(m). Processing is carried out internally, due to the difference between scheduling scheme σ(m) and scheduling scheme σ LS-LPT (m) are all based on LS-LPT rules, therefore

[0214] In conclusion, when hour,

[0215] Case 2β≠0, m=αk+β.

[0216] In this case, the LS-LPT algorithm executes the Gantt chart as follows: Figure 11 As shown. Define F i For machine M i In workpiece J l The length has already been processed before the arrangement is determined. The following assumptions are given:

[0217]

[0218] First, in workpiece J lBefore being assigned, machine M αk+γ (1≤γ≤β) At least one workpiece has been processed. If in workpiece J l Before being assigned, machine M αk+γ (1≤γ≤β) The workpiece is not yet processed. Workpiece J l Arranged to machine M αk+γ Above, that is

[0219]

[0220] And because the total number of machines increases after the number of machines increases The number of machines is greater than the initial number of machines, therefore

[0221]

[0222] Equation (32) contradicts Equation (26).

[0223] Secondly, assuming workpiece J k In machine M αk+γ If processing is performed on (1≤γ≤β), then the following formula is given.

[0224] F αk+γ ≥p k ≥p l #(33)

[0225] For formula (30), summing i from 1 to (m-β) yields:

[0226]

[0227] Adjusting the left side of inequality (34) and combining it with equation (33), we get

[0228]

[0229]

[0230] And because Combining equation (25), we can obtain

[0231]

[0232] Finally, by combining equations (36) and (37), we can obtain...

[0233]

[0234] Equation (38) can be simplified to obtain

[0235]

[0236] Equation (39) contradicts equation (26), therefore

[0237]

[0238] Therefore, it can be obtained

[0239]

[0240] Since this invention focuses on the optimal performance improvement after increasing the number of machines, considering both Case 1 and Case 2, it can be seen that when workpiece J... l When the workpiece is a 1-processor, the best-case influence ratio of the approximate solution is...

[0241] Based on the analysis in steps 5.2.1 and 5.2.2, the best-case influence ratio of the approximate solution is... That is, the problem Pm|m j ={1,k}|C max The best case of the approximate solution has a greater impact than the ratio.

[0242] Step 5.3: Match the number of machines to improve scheduling performance.

[0243] Step 5.3.1: Based on the current multiprocessor workpiece production system, determine the initial number of machines m, k - the number of machines required to process the workpiece, and give the specific value η of the expected scheduling performance improvement. Based on the values ​​of m and k, calculate α and β.

[0244] Step 5.3.2: Based on the scheduling performance improvement value η, the following expression is obtained:

[0245]

[0246] Right now

[0247] Step 5.3.3: Based on the values ​​of μ, k, α, and β, combined with... The best-case influence ratio of the approximate solution to the expression

[0248]

[0249] This determines the number of machines after increasing the number of machines. The range of values ​​is obtained. The minimum value.

[0250] The following is an example that analyzes the relationship between orders and production system capacity, revealing potential for improvement in the production system for enterprises: Consider an instance with m = 6 machines and a batch of order workpieces. That is, k = 3, m j ={1,3}; Assuming the plan is to add 3 machines, that is, the total number of machines after the addition. To assess the potential for improvement in the target value after increasing the number of machines, the first step is to start with m=6... Given k=3, we can calculate μ=1 and β=0; secondly, based on the best-case influence ratio expression for the optimal solution, we can calculate ρ. * (6,9)=μ+1=2; The third step is to find that when the number of machines increases from 6 to 9, the scheduling performance of the multiprocessor workpiece production system is improved by a maximum of 200%.

[0251] Here's another example: when urgent orders occur, a company needs to further improve scheduling performance with its existing number of machines. The key is: how to find the minimum number of machines required to meet the scheduling performance requirements. Consider having m = 5 multiprocessors, and let the job set be... That is, k = 3, m j ={1,3}, assuming a 75% improvement in scheduling performance for the current multiprocessor job production system, i.e., η = 75%. To obtain the number of machines required to achieve the desired scheduling performance improvement, firstly, given m = 5 and k = 3, we have α = 1 and β = 2; secondly, based on the LS-LPT rules, and according to the scheduling performance improvement η = 75%, we obtain... Right now The third step is to The best-case influence ratio expression for the approximate solution is combined with that of the following: Find the total number of machines after increasing the number of machines. This indicates that the minimum number of machines required to be added is 7.

[0252] In summary, this invention can provide a theoretical basis for enterprise decisions on the number of machines to be deployed and for the allocation of workpiece processing machines.

[0253] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled 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 method for analyzing the impact of increasing the number of machines on multiprocessor job scheduling, characterized in that: Includes the following steps: Step 1: Analyze the impact of increasing the number of machines on multiprocessor job scheduling. An integer programming model with the scheduling objective of minimizing the maximum completion time: minC max st C j =S j +p j ,j=1,2,…,n S j′ ≥C j -M(2-x j′i -x ji ),j<j′;j,j′=1,2,…,n;i=1,…,m S i,j ≥C i,j-1 +p j-1 ,j=1,2,…,n;i=1,…,m C max ≥C i ,i=1,…,m x ji ={0,1},j=1,2,…,n;i=1,…,m Define workpiece set Among them, workpiece J j For 1-processor workpiece or k-processor workpiece, p j For workpiece J j Processing time; let S j Indicates workpiece J j The start time of C j Indicates workpiece J j The completion time is given by $S$, where $j'$ is an index value greater than $j$. i,j Indicates machine M i upper workpiece J j The start time of C i,j-1 Indicates machine M i upper workpiece J j The start time of the previous workpiece, C i Indicates machine M i The completion time, C max Indicates the maximum completion time of the scheduling scheme; x ji x is a 0-1 variable; when workpiece j is processed on the i-th machine, x ji =1, otherwise x ji =0;m j Indicates workpiece J j The number of machines that need to be processed simultaneously; Step 2: Analyze the impact analysis problem Pm|m j ={1,k}|C max The best-case impact ratio of the optimal solution is used to evaluate the potential improvement of the target value after increasing the number of machines. Step 2.1: Determine the best-case impact ratio of the optimal solution: Regarding the issue Pm|m j ={1,k}|C max Let m = αk + β, where make in μ and v are both positive integers satisfying μ≥1, 0≤ν≤m-1. When the number of processing machines increases from m to v, the value of μ is determined by the condition that μ ≥ 1 and 0 ≤ ν ≤ m-1. When the best-case scenario of the optimal solution has a greater impact than The upper bound is: Step 2.2: Evaluate the potential for improvement in the target value after increasing the number of machines: Based on the current multi-processor workpiece production system, determine the initial number of machines m, the number of machines k required to process the workpieces, and the total number of machines after the planned increase in the number of machines. And based on m, k and Find the values ​​of μ, α, and β; then, based on the values ​​of μ, k, α, and β, substitute them into the best-case influence ratio expression for the optimal solution to obtain the upper bound of the best-case influence ratio for the current multiprocessor workpiece production system. This allows us to determine the optimal solution's influence ratio when the number of machines increases from m to... At that time, the maximum improvement in scheduling performance of the current multiprocessor workpiece production system; Step 3: Design a rule for prioritizing workpieces with a large number of required processing machines and those with longer processing times (LS-LPT). Based on the number of machines required for processing each workpiece, all workpieces are divided into two subsets T. 1 and T k T k This represents the set of all workpieces that require k machines to process simultaneously; Workpiece set T 1 and T k All workpieces inside are arranged according to the LPT rule; Workpiece set T k The k-processor workpieces are sequentially arranged on the first idle machine starting from time zero; Workpiece set T 1 The workpieces in the 1-processor are sequentially arranged on the first idle machine starting from time zero, until all workpieces are arranged for processing; Step 4: Analyze the best-case impact ratio of the approximate solution to determine the appropriate number of machines to improve scheduling performance. Step 4.1: Determine the best-case influence ratio of the approximate solution: Regarding the issue Pm|m j ={1,k}|C max When the number of machines increases from m to At that time, the upper bound of the best-case impact ratio of the approximate scheduling scheme generated by the LS-LPT rules is: Step 4.2: Match the number of machines to improve scheduling performance: Based on the current multiprocessor workpiece production system, α and β are calculated according to the initial number of machines m and k - the number of machines required to process the workpiece. This is then combined with the desired scheduling performance improvement value η. and Determine the number of machines after increasing the number of machines. The range of values ​​is obtained. The minimum value is used as the number of machines matched to improve scheduling performance.

2. The method for analyzing the impact of increasing the number of machines on multiprocessor job scheduling according to claim 1, characterized in that: The problem Pm|m established in step 1 j ={1,k}|C max It is NP-hard.

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