CN114415614B - Service class considered multiprocessor workpiece scheduling method - Google Patents

Service class considered multiprocessor workpiece scheduling method Download PDF

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CN114415614B
CN114415614B CN202210045808.6A CN202210045808A CN114415614B CN 114415614 B CN114415614 B CN 114415614B CN 202210045808 A CN202210045808 A CN 202210045808A CN 114415614 B CN114415614 B CN 114415614B
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CN114415614A (en
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王军强
孙涛
桑耀文
延爽
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Northwestern Polytechnical University
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    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B19/00—Programme-control systems
    • G05B19/02—Programme-control systems electric
    • G05B19/418—Total factory control, i.e. centrally controlling a plurality of machines, e.g. direct or distributed numerical control [DNC], flexible manufacturing systems [FMS], integrated manufacturing systems [IMS] or computer integrated manufacturing [CIM]
    • G05B19/41865—Total factory control, i.e. centrally controlling a plurality of machines, e.g. direct or distributed numerical control [DNC], flexible manufacturing systems [FMS], integrated manufacturing systems [IMS] or computer integrated manufacturing [CIM] characterised by job scheduling, process planning, material flow
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B2219/00—Program-control systems
    • G05B2219/30—Nc systems
    • G05B2219/32—Operator till task planning
    • G05B2219/32252—Scheduling production, machining, job shop
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
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    • Y02P—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN THE PRODUCTION OR PROCESSING OF GOODS
    • Y02P90/00—Enabling technologies with a potential contribution to greenhouse gas [GHG] emissions mitigation
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Abstract

The invention provides a multiprocessor work piece scheduling method considering service levels, and designs a dynamic programming algorithm and an approximation algorithm by considering the difference of flexible resource processing capacities of people, machines and the like and the requirements of workpieces on different processing resources and processing capacities, so as to realize a cooperative configuration scheme of 'make-up for the resources' and solve the technical problems of flexible resource optimization matching and multiprocessor work piece sequencing.

Description

Service class considered multiprocessor workpiece scheduling method
Technical Field
The invention relates to a multiprocessor workpiece cooperative scheduling problem, in particular to a multiprocessor workpiece scheduling method considering service level.
Background
The multiprocessor workpiece scheduling problem (Multiprocessor Job Scheduling Problem, MJSP) refers to that a workpiece can be processed in parallel by utilizing a plurality of processors at the same time, flexible resources such as people and machines are effectively utilized, the potential of man-machine cooperation is fully utilized, how to configure the machines and schedule the workpieces is decided, and the optimal matching and efficient cooperation of the people, the machines and the workpieces are realized, so that the optimal scheduling target is achieved. The machine and the workpiece form a many-to-one dynamic matching relation, the limit that one workpiece can only be processed by one machine at the same time in the classical scheduling is broken through, and the ordering problem related to the classical scheduling is expanded into the machine-workpiece matching and workpiece ordering optimization problem.
In practical production and service systems, multiprocessor workpieces are commonly used frequently. For example, the assembly process of the aero-engine is high in specialization degree and high in precision and technology density, a large number of industrial robots, special equipment and special tool fixtures are needed in the assembly process, a large number of technicians and fitters are needed, and the assembly precision is guaranteed and the assembly efficiency is improved through the cooperative assembly of flexible resources such as man-machine objects. It is apparent that the process of performing the assembly by a plurality of fitters and a plurality of special equipment in cooperation is represented as a machine-to-work machining mode. Meanwhile, the operation skills, operation levels, proficiency and cooperative modes of people and machines have significant influence on the assembly time, quality and precision. Then, how to embody factors such as operation skills, operation grades, proficiency and the like of people and machines in the multiprocessor work piece scheduling, further reasonably utilize production resources, fully release production capacity, further improve working efficiency, and finally realize efficient operation of an assembly line is a technical problem worthy of research.
Disclosure of Invention
According to the invention, actual people and machines are defined as generalized machines, skill levels of operators and machining capacity are uniformly extracted as service level parameters, the difference of the machining capacity of the generalized machines is considered according to the machining modes of the machines and workpieces in a multi-to-one mode, a multi-processor workpiece scheduling model is established to analyze the complexity of the problem, a dynamic planning algorithm and an approximation algorithm are designed, and the problems of machine-workpiece matching and workpiece sequencing optimization are solved, so that the utilization degree of assembly resources is improved, and the assembly efficiency of an aeroengine is improved.
The technical scheme of the invention is as follows:
a multiprocessor work piece scheduling method considering service level includes the following steps:
step 1: construction of multiprocessor work scheduling model qm|m considering service class j ,GoS|C max ：
obj.min C max (1.1)
S j ≥C j′ -M(2-x j,i,h -x j′,i,h-1 )j≠j′；i＝1,…,m；h＝1,…,n (1.5)
C j ＝S j +p j *m j /v j j＝1,…,n (1.6)
S i,h ≥C i,h-1 i＝1,…,m；h＝1,…,n (1.9)
S i,h ≤S j +M(1-x j,i,h )j＝1,…,n；i＝1,…,m；h＝1,…,n (1.10)
S i,h ≥S j -M(1-x j,i,h )j＝1,…,n；i＝1,…,m；h＝1,…,n (1.11)
S i,h ≥0i＝1,…,m；h＝1,…,n (1.12)
C max ≥C j j＝1,…,n (1.13)
x jih ＝{0,1}j＝1,2,…,n；i＝1,2,..,m；h＝1,2,..,n (1.14)
wherein ,xji 0-1 variable, if the workpiece J j In machine M i X during processing at position h of (2) jih =1; otherwise x jih =0; equation (1.1) is an objective function of the model, minimizing the maximum finishing time C max The method comprises the steps of carrying out a first treatment on the surface of the Formulas (1.2) and (1.3) represent a multiprocessor workpiece J j Needs m j Machining the workpiece, wherein the workpiece can be machined only when the service level of the workpiece is greater than the service level of the machine; formulas (1.4), (1.5) and (1.6) show that each machine can only process one workpiece at the same processing position, wherein M is a great positive number; the formulas (1.9), (1.10) and (1.11) represent that the machine for processing the same workpiece is started at the same time, wherein M is a very large positive number; formula (1.14) represents x jih Is a 0-1 variable; other constraints are additional constraints;
step 2: and designing a dynamic programming algorithm or a heuristic algorithm LG-LPT to solve a mathematical model of the multi-processor workpiece scheduling problem considering the service level, so as to obtain an optimal target.
Further, when constructing the multiprocessor workpiece scheduling model considering the service level:
given a workpiece set comprising n workpiecesMachine set with 3 parallel speed machinesMachine M 1 and M2 The service class of (1) is recorded as 1, the processing speed is set as 1, and the machine M 3 The service class of (2) is recorded as 2, and the processing speed is s (s<1) The method comprises the steps of carrying out a first treatment on the surface of the Defining k workpieces as k-processor workpieces, and workpiece J j The processing time of (2) is p j And the processing time of each workpiece varies with the processing machine, assuming that the 1-processor workpiece J is under the premise of meeting the processing constraint j In machine M 1 and M2 The processing time is l j 1, in machine M 3 The upper processing time is l j S; 2-processor workpiece J j In machine M 1 、M 2 The processing time is l j 2, in machine M 1 、M 3 Or M 2 、M 3 The processing time is l j 1+s; all workpieces arrive at the moment 0, the machine starts to process at the moment 0, the workpiece processing has no preparation time and the processing process can not be interrupted;
the optimization objective is the maximum processing time, and the investigated multiprocessor workpiece scheduling problem is expressed as:
Q2,1|m j ,GoS 2 |C max the mathematical model is as follows:
obj.min C max (1)
g(M i )·x ji ≤g(J j )j＝1,2,…,n；i＝{1,2,3} (4)
g(J j )≥m j j＝1,2,…,n (5)
(C j -p j )·x ji ＝S j j＝1,2,…,n；i＝{1,2,3} (6)
x ji ＝{0,1}j＝1,2,…,n (7)
wherein ,xji 0-1 variable, when the workpiece J j In machine M i Upper working, x ji =1, otherwise x ji =0; equation (1) is the objective function of the model, i.e. minimizing the maximum finishing time C max The method comprises the steps of carrying out a first treatment on the surface of the The expression (2) indicates that the multiprocessor workpiece requires a plurality of machines to be processed simultaneously; equation (3) shows that each machine can only process one workpiece at a time; equation (4) indicates that machining can be performed thereon only when the work service level is greater than the machine service level; equation (5) indicates that the service class of each work is not less than the number of machines required for processing the work; equation (6) indicates that all machines processing the same workpiece must start at the same time and end at the same time; formula (7) represents x ji Is a 0-1 variable;
step 2: and designing a dynamic programming algorithm or a heuristic algorithm LG-LPT to solve a mathematical model of the multi-processor workpiece scheduling problem considering the service level, so as to obtain an optimal target.
In step 2, the dynamic programming algorithm includes the following steps:
step 2.1: constructing a cost function:
by F (j, x) 1 ,x 2 ,x 3 ,x 4 ) To represent all at M 2 and M3 The sum of the processing lengths of the 2-processor workpiece to be processed is thatAnd for j.ltoreq.n.ltoreq.1, 0.ltoreq.x 1 +x 2 +x 3 +x 4 W is less than or equal to W; wherein the parameters y, x 1 、x 2 、x 3 X 4 Respectively and correspondingly represent the workpiece of the 2-processor at M 2 and M3 Length of working at M 1 and M2 Length of work on and 1-processor work piece at M 1 、M 2 and M3 A processing length of the upper part;
step 2.2: initializing a cost function:
step 2.3: iteration:
wherein ,
the three formulas respectively represent the workpiece J j From M 1 Machining, work J j From M 2 Machining and work J j From M 1 and M2 Processing; when x is 2 -p j <At 0, the first formula is discarded; x is x 3 -p j <At 0, the second formula is discarded; x is x 1 -p j <At 0, discard the third equation
The five formulas respectively represent the workpiece J j At M 1 Machining, work J j At M 2 Upper working, work J j At M 3 Upper working, work J j At M 1 and M2 Machining and work J j At M 2 and M3 Performing upper machining; when x is 2 -p j <At 0, discard the firstA formula (I); when x is 3 -p j <At 0, the second formula is discarded; x is x 4 -p j <At 0, the third formula is discarded; when x is 1 -p j <At 0, the fifth formula is discarded;
step 2.4: optimal target:
max{x 4 +F(n,x 1 ,x 2 ,x 3 ,x 4 ),max(x 2 ,F(n,x 1 ,x 2 ,x 3 ,x 4 )+x 3 )+x 1 and obtaining the optimal sequence through reverse backtracking.
In step 2, the heuristic algorithm LG-LPT comprises the following steps:
step 2.1: arranging all 2-processor workpieces in the machine M which completes them earliest 1 and M2 Performing upper machining;
step 2.2: for 1-processor workpieces, L are arranged in order of non-decreasing workpiece service level 1 ；
Step 2.3: for the 1-processor workpieces with the same service level, the workpieces are arranged according to the non-increasing sequence of the processing time, and the arrangement L is finally obtained for all the workpieces 2 ；
Step 2.4: according to the resulting sequence L 2 The work pieces are sequentially arranged on the machine that completed them earliest until all work pieces are arranged.
Advantageous effects
The invention provides a multiprocessor work piece scheduling method considering service levels, and designs a dynamic programming algorithm and an approximation algorithm by considering the difference of flexible resource processing capacities of people, machines and the like and the requirements of workpieces on different processing resources and processing capacities, so as to realize a cooperative configuration scheme of 'make-up for the resources' and solve the technical problems of flexible resource optimization matching and multiprocessor work piece sequencing.
Under the information of the workpiece and the machine shown in the table 1, according to the dynamic programming algorithm, the maximum finishing time of the workpiece is 6, and the scheduling Gantt chart is shown in fig. 2. According to the heuristic algorithm LG-LPT, the maximum finishing time of the workpiece is 6, and the scheduling Gantt chart is shown in figure 3. The optimal solution obtained through the heuristic algorithm LG-LPT solution and the dynamic programming algorithm is the same, the maximum finishing time of the workpiece is 6, and the performance of the heuristic algorithm LG-LPT algorithm meets the requirement of the worst performance ratio. Therefore, the invention can provide theoretical basis for assembly resource allocation and workpiece sequencing decision.
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.
Drawings
The foregoing and/or additional aspects and advantages of the invention will become apparent and may be better understood from the following description of embodiments taken in conjunction with the accompanying drawings in which:
FIG. 1 is a canonical Schedule Gantt chart;
FIG. 2 illustrates an example Gantt chart of a dynamic programming algorithm;
FIG. 3 illustrates an example Gantt chart of algorithm LG-LPT;
FIG. 4 algorithm LG-LPT Gantt chart;
FIG. 5 algorithm LG-LPT experimental results.
Detailed Description
The following detailed description of embodiments of the invention is exemplary and intended to be illustrative of the invention and not to be construed as limiting the invention.
The invention defines actual people and machines as generalized machines, uniformly extracts the skill level of operators and the machining capacity as Service level parameters (GoS), and the workpieces and the machines have corresponding Service levels, wherein the lower the Service level parameters are, the stronger the machining capacity is, and the lower the Service level parameters are, the higher the required machine capacity is. Only when the service level parameter of the workpiece is not lower than the service level parameter of the machine, the workpiece can be processed on the machine, and the matching mode of the machine and the service level of the workpiece is an important factor affecting the dispatching optimization. Further facing the matching and scheduling problems of processing tasks and flexible processing resources, fully considering the service grades of machines and workpieces, reasonably utilizing production resources, fully releasing production capacity, further improving working efficiency and finally realizing efficient operation of an assembly line.
In the invention, the problem of scheduling the multiprocessor workpieces considering the service level is explored by taking the assembly of the aero-engine as the background. Multiprocessor workpiece scheduling aims at improving the production capacity of an aircraft engine assembly line to achieve assembly efficiency maximization. Aiming at the machining mode of many-to-one machines and workpieces, the machining capability difference is considered, a multiprocessor workpiece scheduling model is established, the complexity of the problem is analyzed, a dynamic programming algorithm and an approximation algorithm are designed, the worst performance analysis of the algorithm is carried out, the maximum finishing time of the workpieces is minimized, the difficult problems of machine-workpiece matching and workpiece ordering optimization are solved, the utilization degree of assembly resources is improved, and the assembly efficiency of the aeroengine is improved.
Service class considered multiprocessor workpiece scheduling problem qm|m j ,GoS|C max The mathematical model is as follows
obj.min C max (1.1)
S j ≥C j′ -M(2-x j,i,h -x j′,i,h-1 )j≠j′；i＝1,…,m；h＝1,…,n (1.5)
C j ＝S j +p j *m j /v j j＝1,…,n (1.6)
S i,h ≥C i,h-1 i＝1,…,m；h＝1,…,n (1.9)
S i,h ≤S j +M(1-x j,i,h )j＝1,…,n；i＝1,…,m；h＝1,…,n (1.10)
S i,h ≥S j -M(1-x j,i,h )j＝1,…,n；i＝1,…,m；h＝1,…,n (1.11)
S i,h ≥0i＝1,…,m；h＝1,…,n (1.12)
C max ≥C j j＝1,…,n (1.13)
x jih ＝{0,1}j＝1,2,…,n；i＝1,2,..,m；h＝1,2,..,n (1.14)
wherein ,xji 0-1 variable, if the workpiece J j In machine M i X during processing at position h of (2) jih =1; otherwise x jih =0. Equation (1.1) is the objective function of the model, i.e. minimizing the maximum finishing time C max The method comprises the steps of carrying out a first treatment on the surface of the Formulas (1.2) and (1.3) represent a multiprocessor workpiece J j Needs m j Machining the workpiece, wherein the workpiece can be machined only when the service level of the workpiece is greater than the service level of the machine; formulas (1.4), (1.5) and (1.6) show that each machine can only process one workpiece at the same processing position, wherein M is a great positive number; the formulas (1.9), (1.10) and (1.11) represent that the machine for processing the same workpiece is started at the same time, wherein M is a very large positive number; formula (1.14) represents x jih Is a 0-1 variable; other constraints are additional constraints.
The multiprocessor work piece scheduling method considering the service level is given below, and the rationality of the multiprocessor work piece scheduling method is illustrated through simulation experiments.
1) Providing a mathematical programming model of the related problem;
2) Designing a pseudo polynomial time dynamic programming algorithm;
3) The approximate algorithm LG-LPT is also designed, and the performance of the approximate algorithm LG-LPT is analyzed to prove the worst performance ratio of the algorithm;
4) And (3) setting calculation example experiment simulation, and verifying the effectiveness of an approximation algorithm and the rationality of the method.
Step 1: a mathematical model of the multiprocessor workpiece scheduling problem that considers class of service.
The multiprocessor workpiece scheduling problem considering service class in the relevant machine environment can be described as: given a workpiece set comprising n workpiecesMachine set with 3 parallel speed machines ∈>Machine M 1 and M2 The service class of (1) is recorded as 1, the processing speed is set as 1, and the machine M 3 The service class of (2) is recorded as 2, and the processing speed is s (s<1). The work pieces with the number of machines k required for machining are defined as k-processor work pieces. Workpiece J j The processing time of (2) is p j And the processing time of each workpiece varies with the processing machine, assuming that the 1-processor workpiece J is under the premise of meeting the processing constraint j In machine M 1 and M2 The processing time is l j 1, in machine M 3 The upper processing time is l j S; 2-processor workpiece J j In machine M 1 、M 2 The processing time is l j 2, in machine M 1 、M 3 Or M 2 、M 3 The processing time is l j 1+s. All the workpieces arrive at the moment 0, the machine starts to process at the moment 0, the workpiece processing has no preparation time and the processing process can not be interrupted.
The optimization objective is the maximum processing time, and the investigated multiprocessor workpiece scheduling problem is expressed as:
Q2,1|m j ,GoS 2 |C max the mathematical model is as follows:
multiprocessor work-piece scheduling problem Q2,1|m j ,GoS 2 |C max The mathematical model is as follows
obj.min C max (1)
g(M i )·x ji ≤g(J j )j＝1,2,…,n；i＝{1,2,3} (4)
g(J j )≥m j j＝1,2,…,n (5)
(C j -p j )·x ji ＝S j j＝1,2,…,n；i＝{1,2,3} (6)
x ji ＝{0,1}j＝1,2,…,n (7)
wherein ,xji 0-1 variable, when the workpiece J j In machine M i Upper working, x ji =1, otherwise x ji =0. Equation (1) is the objective function of the model, i.e. minimizing the maximum finishing time C max The method comprises the steps of carrying out a first treatment on the surface of the The expression (2) indicates that the multiprocessor workpiece requires a plurality of machines to be processed simultaneously; equation (3) shows that each machine can only process one workpiece at a time; equation (4) indicates that machining can be performed thereon only when the work service level is greater than the machine service level; equation (5) indicates that the service class of each work is not less than the number of machines required for processing the work; equation (6) indicates that all machines processing the same workpiece must start at the same time and end at the same time; formula (7) represents x ji Is a 0-1 variable.
Step 2: and designing a dynamic programming algorithm to solve a mathematical model of the multiprocessor workpiece scheduling problem considering the service level.
Firstly, the property of an optimal scheduling scheme for the multiprocessor workpiece scheduling problem considering the service level is described, and then a dynamic programming algorithm is designed according to the optimal property.
Properties of the optimal scheduling scheme: let S * Representing the problem Q2,1|m j ,GoS 2 |C max Is an optimal schedule of S * Can be converted into the canonical schedule as shown in fig. 1 without changing the target value C max Is of a size of (a) and (b). From this, the maximum finishing time of the Gantt chart obtained by the standard scheduling scheme is equal to the maximum finishing time of the optimal scheduling scheme. This illustrates that the canonical schedule can get the optimal maximum completion time in the face of the multiprocessor workpiece scheduling problem considering service class.
The optimal scheduling scheme may change to the cause of the specification: the machine and the workpiece have only two service classes, machine M 1 and M2 Service class 1, machine M 3 The service class is 2. 1-processor workpieces of class 1 may alternatively use machine M 1 Or M 2 Processing; 2-processor workpieces of class 1 can only be processed by machine M 1 and M2 Processing together; 1-processor workpieces with grade 2 can be processed by any machine; a grade 2-processor workpiece may be machined by any two machines. For any optimal schedule, if a 2-processor workpiece of grade 2 is selected to be machined by two machines of different grades, i.e. by M 1 and M3 Working up or by M 2 and M3 Processing, converting all the materials into M 2 and M3 Processing and can move the start time to the front end of the scheduling scheme, and simultaneously, M 1 and M2 The machined 2-processor work piece is moved behind all 1-processor work pieces. Due to M 1 and M2 The service class and the processing speed are all consistent, and the position exchange performed above does not affect the maximum finishing time C of the machine max Is of a size of (a) and (b).
Dynamic programming algorithm: problem Q2,1|m j ,GoS 2 |C max Any optimal schedule S of (1) * Completely by the parameters y, x 1 、x 2 、x 3 X 4 Determination of the respective representation of the 2-processor workpiece at M 2 and M3 Length of working at M 1 and M2 Length of work on and 1-processor work piece at M 1 、M 2 and M3 And a working length. The invention gives the problem Q2,1|m j ,GoS 2 |C max Is provided.
Step 2.1: (construction cost function) by F (j, x 1 ,x 2 ,x 3 ,x 4 ) To represent all at M 2 and M3 The sum of the processing lengths of the 2-processor workpiece to be processed is thatAnd for j.ltoreq.n.ltoreq.1, 0.ltoreq.x 1 +x 2 +x 3 +x 4 ≤W。
Step 2.2: (initialization)
Step 2.3: (iterative procedure)
wherein ,
the three formulas respectively represent the workpiece J j From M 1 Machining, work J j From M 2 Machining and work J j From M 1 and M2 And (5) processing. Note that when x 2 -p j <At 0, the first formula is discarded; x is x 3 -p j <At 0, the second formula is discarded; x is x 1 -p j <At 0, the third equation is discarded.
The five formulas respectively represent the workpiece J j At M 1 Machining, work J j At M 2 Upper working, work J j At M 3 Adding upWork piece J j At M 1 and M2 Machining and work J j At M 2 and M3 And (5) performing upper processing. Note that when x 2 -p j <At 0, the first formula is discarded; when x is 3 -p j <At 0, the second formula is discarded; x is x 4 -p j <At 0, the third formula is discarded; when x is 1 -p j <At 0, the fifth equation is discarded.
Step 2.4: (optimal solution): optimal target
max{x 4 +F(n,x 1 ,x 2 ,x 3 ,x 4 ),max(x 2 ,F(n,x 1 ,x 2 ,x 3 ,x 4 )+x 3 )+x 1 And obtaining the optimal sequence through reverse backtracking.
Time complexity analysis:
aiming at the problem Q2,1|m j ,GoS 2 |C max The dynamic programming algorithm may be applied to the dynamic programming algorithm in the range of O (nW 4 ) Obtaining an optimal solution in time, whereinBecause j is not less than 1 and not more than n,0 is not less than x 1 ,x 2 ,x 3 ,x 4 W is not more than and therefore the number of states O (nW 4 ). For a given j, x 1 ,x 2 ,x 3 ,x 4 It takes O (n) time to calculate the cost function. Thus, the time complexity of the algorithm is O (nW 4 )。
Consider the following three correlators, example of 6 workpieces, M 1 、M 2 and M3 The service levels of (2) are 1, 1 and 2, respectively, the machining time is 1, 1 and 0.5, respectively, and the work machining time, the service level and the required number of machines are shown in table 1.
Table 1 example workpiece parameters
According to the dynamic programming algorithm, the workpiece J 1 From M 1 and M2 Machining, work J 2 Select from M 2 and M3 Machining, work J 3 、J 4 J 5 From M 1 Machining, work J 6 From M 2 Processing, so that the maximum finishing time of the workpiece is C max =max {4, max (5, 4+1) +1} =max {4,6} =6. The scheduling scheme obtained by the dynamic programming algorithm is shown in fig. 2.
For the multiprocessor work-piece scheduling problem in the related machine environment, a heuristic algorithm can be designed to relate to two sub-problems, namely, how to allocate machine resources, how to sort work-pieces, and aiming at the problem Q2,1|m j ,GoS 2 |C max The heuristic algorithm LG-LPT is designed.
Step 3.1: arranging all 2-processor workpieces on the machine for which they were first finished, i.e. at M 1 and M2 Processing;
step 3.2: for 1-processor workpieces, L are arranged in order of non-decreasing workpiece service level 1 ；
Step 3.3: for the 1-processor workpieces with the same service level, the workpieces are arranged according to the non-increasing sequence of the processing time, and the arrangement L is finally obtained for all the workpieces 2 ；
Step 3.4: according to the resulting sequence L 2 The work pieces are sequentially arranged on the machine that completed them earliest until all work pieces are arranged.
Also for the work piece processing time, service class and number of machines required shown in Table 1, according to heuristic LG-LPT, 2-processor work piece J is first of all 1 And workpiece J 2 Arranged at M 1 and M2 Processing, arranging J according to LG-LPT rule by using residual 1-processor workpiece 5 ≤J 6 ≤J 3 ≤J 4 Arranging the workpieces in this sequence on the machine which completes them earliest, the workpieces J 5 Select from M 1 Machining, work J 6 And workpiece J 4 From M 2 Machining, work J 3 From M 3 Processing, thus the maximum finishing time is C max ＝max{6,6,4}＝6。
The resulting scheduling scheme according to the algorithm LG-LPT is shown in fig. 3.
Heuristic LG-LPT performance analysis is given below:
aiming at the problem Q2,1|m j ,GoS 2 |C max The worst performance ratio of the heuristic algorithm LG-LPT is And s is<1, i.e
First, for this problem, a minimum counterexample (J, M) is constructed, i.e., the workpiece set J contains the least number of workpieces in all counterexamples, such that the solution generated by the algorithm LG-LPT satisfies
Second, the maximum finishing time that can be obtained for this counterexample is obtained from the last workpiece J due to the minimal number of workpieces n And (3) determining. If for J 1 ,J 2 ,…,J j ,j<n, if the worst performance ratio generated by the algorithm LG-LPT does not satisfy the unsatisfied formula (8), another counterexample (J ', M) can be constructed, and the work piece set J' = { J 1 ,J 2 ,…,J j The number of workpieces contained is strictly less than (J, M). As shown in fig. 4.
The lower bound of the optimal target value of the problem can be obtained, namely
Let F i Representing the arrangement of workpieces J n Previous machine M i The last workpiece J n Determining the maximum finishing time, i.e
Is adjusted to
s i ·f-s i ·F i ≤p n (12)
According to the LG-LPT algorithm step, the last workpiece J n For a 1-handler workpiece, two cases are discussed below.
Case 1 workpiece J n For 2-processor workpieces, the system can only process one 2-processor workpiece at a time, in which case
f＝f *
Case 2 workpiece J n 1-handler workpieces, in this case, are classified into the following two cases according to the service class of the workpiece.
Case 2.1 if workpiece J n Service class 1, then the system does not have a class 2 1-handler workpiece, and i is summed from 1 to 2 for equation (12)
Is adjusted to
From the equation (9) and the equation (10), it is possible to obtain
From equation (14) and equation (15), it is possible to obtain
2·f≤p n +(2+s)f * ≤f * +(2+s)f * (16)
Thus (2)
Case 2.2 if workpiece J n Class of service of 2, i is summed from 1 to 3 for equation (12)
Is adjusted to
From the equation (9) and the equation (10), it is possible to obtain
From the equation (19) and the equation (20), it is possible to obtain
(2+s)·f≤2·p n +(2+s)f * ≤2·f * +(2+s)f * (21)
Thus (2)
In conclusion, the method comprises the steps of,
to measure the performance of the heuristic LG-LPT, for any one instance I, use V H (I) The solution obtained by the algorithm LG-LPT is represented by OPT (I), the optimal solution is represented by LB (I) according to the formula (10), and the lower bound of the optimal solution is represented by:
the relative deviation dev (H) of the algorithm H is defined as follows:
for any one example I, LB (I). Ltoreq.OPT (I) holds, so the solution obtained by algorithm H does not deviate more than dev (H) from the optimal solution, i.e.)
In order to verify the rationality of the proposed LG-LPT algorithm, experiments of different machining speed coefficients are set, and the experimental parameters of the calculation are shown in table 2 and are based on the number of workpieces and the machine M 3 Speed s, 10×4=40 sets of examples are designed, for each set of examples, 100 examples are randomly generated, so there are 4000 examples in total.
Table 2 simulation experiment parameters
FIG. 5 includes four curves, each representing a batch machine M 3 The maximum deviation change at the machining speed s= 0.2,0.4,0.6,0.8. The abscissa in each subgraph represents the number of workpieces, from 100 to 1000, and the ordinate represents the heuristic deviation dev (H) from the optimal solution lower bound. Each data point is the average of 100 example deviations. The smaller the deviation, the better the heuristic performance.
By means of simulation experiments, the error magnitudes of the heuristic LG-LPT and the optimal solution lower bound are compared, and the following conclusion can be obtained from FIG. 5:
1) The maximum deviation between the heuristic algorithm LG-LPT and the lower boundary of the optimal solution is not more than 30%, and the average deviation is in a descending trend along with the increase of the scale of the calculation example;
2) Comparing the sub-graphs, when the scale of the examples is the same, machine M 3 The smaller the processing speed s, the smaller the average deviation;
3) Intra-subgraph comparison, when machine M 3 When the processing speed s of the workpiece is the same, the average deviation gradually decreases as the scale of the example increases.
As can be seen from the theoretical results, for the problem Q2,1|m j ,GoS 2 |C max The worst performance ratio of the heuristic algorithm LG-LPT is not more than 2 at maximum. According to experimental results, the maximum deviation of the heuristic algorithm LG-LPT is not more than 30%, and gradually decreases as the scale of the calculation example increases.
Although embodiments of the present invention have been shown and described above, it will be understood that the above embodiments are illustrative and not to be construed as limiting the invention, and that variations, modifications, alternatives, and variations may be made in the above embodiments by those skilled in the art without departing from the spirit and principles of the invention.

Claims (4)

1. A multiprocessor work-piece scheduling method considering service level is characterized in that: the method comprises the following steps:
step 1: construction of multiprocessor work scheduling model qm|m considering service class j ,GoS|C max ：
obj.minC max (1.1)
S j ≥C j′ -M(2-x j,i,h -x j′,i,h-1 )j≠j′；i＝1,…,m；h＝1,…,n (1.5)
C j ＝S j +p j *m j /v j j＝1,…,n (1.6)
S i,h ≥C i,h-1 i＝1,…,m；h＝1,…,n (1.9)
S i,h ≤S j +M(1-x j,i,h )j＝1,…,n；i＝1,…,m；h＝1,…,n (1.10)
S i,h ≥S j -M(1-x j,i,h )j＝1,…,n；i＝1,…,m；h＝1,…,n (1.11)
S i,h ≥0i＝1,…,m；h＝1,…,n (1.12)
C max ≥C j j＝1,…,n (1.13)
x jih ＝{0,1}j＝1,2,…,n；i＝1,2,..,m；h＝1,2,..,n (1.14)
wherein ,xji 0-1 variable, if the workpiece J j In machine M i X during processing at position h of (2) jih =1; otherwise x jih =0; equation (1.1) is an objective function of the model, minimizing the maximum finishing time C max The method comprises the steps of carrying out a first treatment on the surface of the Formulas (1.2) and (1.3) represent a multiprocessor workpiece J j Needs m j Machining the workpiece, wherein the workpiece can be machined only when the service level of the workpiece is greater than the service level of the machine; formulas (1.4), (1.5) and (1.6) show that each machine can only process one machine at the same processing positionA workpiece, wherein M is a very large positive number; the formulas (1.9), (1.10) and (1.11) represent that the machine for processing the same workpiece is started at the same time, wherein M is a very large positive number; formula (1.14) represents x jih Is a 0-1 variable; other constraints are additional constraints;
step 2: designing a dynamic programming algorithm to solve a mathematical model of the multi-processor workpiece scheduling problem considering the service level to obtain an optimal target;
the dynamic programming algorithm comprises the following steps:
step 2.1: constructing a cost function:
by F (j, x) 1 ,x 2 ,x 3 ,x 4 ) To represent all at M 2 and M3 The sum of the processing lengths of the 2-processor workpiece to be processed is thatAnd for j.ltoreq.n.ltoreq.1, 0.ltoreq.x 1 +x 2 +x 3 +x 4 W is less than or equal to W; wherein the parameters y, x 1 、x 2 、x 3 X 4 Respectively and correspondingly represent the workpiece of the 2-processor at M 2 and M3 Length of working at M 1 and M2 Length of work on and 1-processor work piece at M 1 、M 2 and M3 A processing length of the upper part;
step 2.2: initializing a cost function:
step 2.3: iteration:
wherein ,
the three formulas respectively represent the workpiece J j From M 1 Machining, work J j From M 2 Machining and work J j From M 1 and M2 Processing; when x is 2 -p j <At 0, the first formula is discarded; x is x 3 -p j <At 0, the second formula is discarded; x is x 1 -p j <At 0, discard the third equation
The five formulas respectively represent the workpiece J j At M 1 Machining, work J j At M 2 Upper working, work J j At M 3 Upper working, work J j At M 1 and M2 Machining and work J j At M 2 and M3 Performing upper machining; when x is 2 -p j <At 0, the first formula is discarded; when x is 3 -p j <At 0, the second formula is discarded; x is x 4 -p j <At 0, the third formula is discarded; when x is 1 -p j <At 0, the fifth formula is discarded;
step 2.4: optimal target:
max{x 4 +F(n,x 1 ,x 2 ,x 3 ,x 4 ),max(x 2 ,F(n,x 1 ,x 2 ,x 3 ,x 4 )+x 3 )+x 1 and obtaining the optimal sequence through reverse backtracking.
2. A multiprocessor work-piece scheduling method considering service level is characterized in that: the method comprises the following steps:
step 1, constructing a multiprocessor work piece scheduling model qm|m considering service level as set forth in step 1 of claim 1 j ,GoS|C max ；
Step 2: designing a heuristic algorithm LG-LPT to solve a mathematical model of the multi-processor work piece scheduling problem considering the service level to obtain an optimal target;
the heuristic algorithm LG-LPT comprises the following steps:
step 2.1: arranging all 2-processor workpieces in the machine M which completes them earliest 1 and M2 Performing upper machining;
step 2.2: for 1-processor workpieces, L are arranged in order of non-decreasing workpiece service level 1 ；
Step 2.3: for the 1-processor workpieces with the same service level, the workpieces are arranged according to the non-increasing sequence of the processing time, and the arrangement L is finally obtained for all the workpieces 2 ；
Step 2.4: according to the resulting sequence L 2 The work pieces are sequentially arranged on the machine that completed them earliest until all work pieces are arranged.
3. A multiprocessor work-piece scheduling method considering service level is characterized in that: the method comprises the following steps:
step 1: constructing a multiprocessor workpiece scheduling model considering service level:
given a workpiece set comprising n workpiecesMachine set with 3 parallel speed machinesMachine M 1 and M2 The service class of (1) is recorded as 1, the processing speed is set as 1, and the machine M 3 The service class of (2) is recorded as 2, and the processing speed is s (s<1) The method comprises the steps of carrying out a first treatment on the surface of the Defining k workpieces as k-processor workpieces, and workpiece J j The processing time of (2) is p j And the processing time of each workpiece varies with the processing machine, assuming that the 1-processor workpiece J is under the premise of meeting the processing constraint j In machine M 1 and M2 The processing time is l j 1, in machine M 3 The upper processing time is l j S; 2-processor workpiece J j In machine M 1 、M 2 The processing time is l j 2, in machine M 1 、M 3 Or M 2 、M 3 The processing time is l j 1+s; all workpieces arrive at the moment 0, the machine starts to process at the moment 0, the workpiece processing has no preparation time and the processing process can not be interrupted;
the optimization objective is the maximum processing time, and the investigated multiprocessor workpiece scheduling problem is expressed as:
Q2,1|m j ,GoS 2 |C max the mathematical model is as follows:
obj.min C max (1)
g(M i )·x ji ≤g(J j ) j＝1,2,…,n；i＝{1,2,3} (4)
g(J j )≥m j j＝1,2,…,n (5)
(C j -p j )·x ji ＝S j j＝1,2,…,n；i＝{1,2,3} (6)
x ji ＝{0,1} j＝1,2,…,n (7)
wherein ,xji 0-1 variable, when the workpiece J j In machine M i Upper working, x ji =1, otherwise x ji =0; equation (1) is the objective function of the model, i.e. minimizing the maximum finishing time C max The method comprises the steps of carrying out a first treatment on the surface of the The expression (2) indicates that the multiprocessor workpiece requires a plurality of machines to be processed simultaneously; equation (3) shows that each machine can only process one workpiece at a time; equation (4) indicates that machining can be performed thereon only when the work service level is greater than the machine service level; formula (5) shows that the service grade of each workpiece is not less than the required processing machine for the workpieceThe number of devices; equation (6) indicates that all machines processing the same workpiece must start at the same time and end at the same time; formula (7) represents x ji Is a 0-1 variable;
step 2: designing a dynamic programming algorithm to solve a mathematical model of the multi-processor workpiece scheduling problem considering the service level to obtain an optimal target;
the dynamic programming algorithm comprises the following steps:
step 2.1: constructing a cost function:
by F (j, x) 1 ,x 2 ,x 3 ,x 4 ) To represent all at M 2 and M3 The sum of the processing lengths of the 2-processor workpiece to be processed is thatAnd for j.ltoreq.n.ltoreq.1, 0.ltoreq.x 1 +x 2 +x 3 +x 4 W is less than or equal to W; wherein the parameters y, x 1 、x 2 、x 3 X 4 Respectively and correspondingly represent the workpiece of the 2-processor at M 2 and M3 Length of working at M 1 and M2 Length of work on and 1-processor work piece at M 1 、M 2 and M3 A processing length of the upper part;
step 2.2: initializing a cost function:
step 2.3: iteration:
wherein ,
the three formulas are divided intoRespectively represent the work J j From M 1 Machining, work J j From M 2 Machining and work J j From M 1 and M2 Processing; when x is 2 -p j <At 0, the first formula is discarded; x is x 3 -p j <At 0, the second formula is discarded; x is x 1 -p j <At 0, discard the third equation
The five formulas respectively represent the workpiece J j At M 1 Machining, work J j At M 2 Upper working, work J j At M 3 Upper working, work J j At M 1 and M2 Machining and work J j At M 2 and M3 Performing upper machining; when x is 2 -p j <At 0, the first formula is discarded; when x is 3 -p j <At 0, the second formula is discarded; x is x 4 -p j <At 0, the third formula is discarded; when x is 1 -p j <At 0, the fifth formula is discarded;
step 2.4: optimal target:
max{x 4 +F(n,x 1 ,x 2 ,x 3 ,x 4 ),max(x 2 ,F(n,x 1 ,x 2 ,x 3 ,x 4 )+x 3 )+x 1 and obtaining the optimal sequence through reverse backtracking.
4. A multiprocessor work-piece scheduling method considering service level is characterized in that: the method comprises the following steps:
step 1: construction of a service level considered multiprocessor workpiece scheduling model Q2,1|m as set forth in step 1 of claim 3 j ,GoS 2 |C max ；
Step 2: designing a heuristic algorithm LG-LPT to solve a mathematical model of the multi-processor work piece scheduling problem considering the service level to obtain an optimal target;
the heuristic algorithm LG-LPT comprises the following steps:
step 2.1: arranging all 2-processor workpieces in the machine M which completes them earliest 1 and M2 Performing upper machining;
step 2.2: for 1-processor workpieces, L are arranged in order of non-decreasing workpiece service level 1 ；
Step 2.3: for the 1-processor workpieces with the same service level, the workpieces are arranged according to the non-increasing sequence of the processing time, and the arrangement L is finally obtained for all the workpieces 2 ；
Step 2.4: according to the resulting sequence L 2 The work pieces are sequentially arranged on the machine that completed them earliest until all work pieces are arranged.
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Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN107967558A (en) * 2017-11-20 2018-04-27 西北工业大学 Consider the input and output decision-making technique of parallel batch processing machine cost
CN113805545A (en) * 2021-09-14 2021-12-17 大连理工大学 Flexible flow shop combined scheduling rule generation method considering batch processing
WO2022000924A1 (en) * 2020-07-01 2022-01-06 北京工业大学 Double-resource die job shop scheduling optimization method based on ammas-ga nested algorithm

Patent Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN107967558A (en) * 2017-11-20 2018-04-27 西北工业大学 Consider the input and output decision-making technique of parallel batch processing machine cost
WO2022000924A1 (en) * 2020-07-01 2022-01-06 北京工业大学 Double-resource die job shop scheduling optimization method based on ammas-ga nested algorithm
CN113805545A (en) * 2021-09-14 2021-12-17 大连理工大学 Flexible flow shop combined scheduling rule generation method considering batch processing

Non-Patent Citations (1)

* Cited by examiner, † Cited by third party
Title
考虑运输的柔性流水车间多处理器任务调度的混合遗传优化算法;轩华;王潞;李冰;王薛苑;;计算机集成制造系统(第03期);全文 *

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