CN107437121A - Handle the production process control method of either simplex part simultaneously suitable for more machines - Google Patents

Handle the production process control method of either simplex part simultaneously suitable for more machines Download PDF

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CN107437121A
CN107437121A CN201610353532.2A CN201610353532A CN107437121A CN 107437121 A CN107437121 A CN 107437121A CN 201610353532 A CN201610353532 A CN 201610353532A CN 107437121 A CN107437121 A CN 107437121A
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李新宇
肖胜强
高亮
陈鹏
陈羊幸
余傲蓉
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Huazhong University of Science and Technology
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Abstract

The invention discloses a kind of production process control method for handling either simplex part simultaneously suitable for more machines, structure including carrying out multiparallel-processor problem model according to workshop feature first, in model comprising more machines simultaneously, the processing situation of independent process either simplex part;Model solution method is built secondly based on DABC algorithms, to minimize Maximal Makespan and minimize the equipment investment cost work pieces process sequence optimal as goal seeking and the machine resources method of salary distribution;Finally, production control result scheme is issued and stored, the workshop data information being related to is preserved and managed.The present invention can solve the problem that with more machines while handle the production control problem of single work pieces process scene, consider work pieces process order to distribute with machine resources, model solution efficiency is improved by improving DABC algorithms, the production process control program after optimization drastically increases productivity ratio.

Description

Handle the production process control method of either simplex part simultaneously suitable for more machines
Technical field
The invention belongs to plant manufacturing process control technology field, in particular it relates to which a kind of controlling of production process method for handling either simplex part simultaneously for more machines, handles the processing of single workpiece suitable for Discrete Production Workshop with more machines simultaneously.
Background technology
For production process control method, in the prior art in the presence of some ripe schemes, can preferably solve related process control problem in Workshop Production, obtain preferable technique effect.Such as a kind of Workshop Production control method based on improved adaptive GA-IAGA is proposed in Chinese invention patent 201210012320.X, to maximize all product customer satisfactions and maximize min-satisfaction degree as production control targe, the complicated Workshop Production control problem for solving machined parameters uncertainty and thering is dynamic disturbance event to occur.A kind of decision search genetic method is proposed in Chinese invention patent 201310202922.6, to minimize Maximal Makespan and minimize weighted delay temporal summation as target, solves the multistage hybrid flowshop production control problem containing two kinds of distinct device types of discrete machine and batch processor simultaneously.In addition, some technical process control schemes for being directed to specific different production type workshops in the prior art also be present, such as, a kind of iron and steel enterprise's production control and management system and method is disclosed in Chinese invention patent 201510261796.0, a kind of information monitoring and management system for hydraulic support welding shop is disclosed in the specification of Chinese invention patent 201510592591.0.
It is above-mentioned that some solution methods are proposed to general production process control problem with different Intelligent Optimization Techniques and method in the prior art, but, from the point of view of technical process control technical method, it is often in the majority for classical standard problem, it has been short of for technical side's rule of specific production and processing scene;In addition, then laying particular stress on the structure with workshop information system in terms of production process control system, it is made inadequate in terms of intelligent control technology method.Further, since workshop type and the mode of production are numerous, still there is many other types of workshop to lack more effective production control method.
Particularly, in actual manufacture process, in workpiece body is larger, machine is processed around workpiece production scene, exist more machines simultaneously,It is independentProcess the mode of production of same workpiece, the welding of such as large-sized structural parts, for this mode of production, how optimized control its production technology, so that production control more precision, improving resource utilization simultaneously reduces Workshop Production cost, currently without effective solution, the problem of being still current urgent need to resolve.
The content of the invention
For the disadvantages described above or Improvement requirement of prior art, the invention provides a kind of generation technical process control method for handling either simplex part simultaneously for more machines, it is by a kind of multiparallel-processor problem control mode and is based on improved discrete artificial bee colony method, so that its production control process lowest optimization, it is inadequate to solve above-mentioned generation control precision, the technical problem of low production efficiency.
To achieve the above object, it is proposed, according to the invention, provide a kind of generation technical process control method for handling either simplex part simultaneously for more machines, it comprises the following steps:
(1) more machines are established while handle the production process Controlling model of either simplex part, are specifically included:
(1.1) optimization aim of setting production skill process control model, i.e., to minimize Maximal Makespan and minimize machine input cost as target, it is respectively:
minf1=maxTi,j (1)
Wherein, Ti,jFor workpiece j process i completion date;pi,jFor workpiece j process i process time;M is process collection and M={ 1,2 ..., m }, wherein m are natural number;I is process numbering and i ∈ M;N is workpiece collection and N={ 1,2 ..., n }, wherein n are natural number;J is workpiece numbering and j ∈ N;MiIt is the machine collection in i-th of process;R is machine serial number and r ∈ Mi, the y if workpiece j in process i is processed on machine ri,j,r=1, otherwise yi,j,r=0, μi,jFor workpiece j on process i service machine number,C be increase additional machine penalty factor, C ∈ [0,1];
(1.2) according to workshop actual conditions, following constraint is set:
Wherein, constraint (3) determines the unit one in each stage, constrain (4) and represent that workpiece has determination sequence precedence relationship, constraint (5) is the constraint of the parallel processor sum to be come into operation to any time each process, wherein L is to can be used for processing machine sum, it is constant, Φi , tIt is the machine quantity that process i comes into operation in the t minutes of processing, it is 0-1 variable bounds to constrain (6) and constraint (7);
(2) above-mentioned model is solved using artificial bee colony algorithm, obtains solving result, specifically include:
(2.1) Population Size P, number of machines limitation Lm are set, abandons solution limit L a and Local Search probability P s, generation initializes the machine assignment matrix μ of each workpiece on the job sequence permu and each operation of population and calculates individual ind fitness value πind, wherein ind represents i-th nd individual, each individual include a length be Number of Jobs be n job sequence permu, size be m × n machine assignment matrix μ and corresponding fitness value πind, fitness value is the inverse of Maximal Makespan;
(2.2) the honeybee stage is employed, to individual ind=1 ..., P in population, is repeated below process:
1 insertion of (2.2.1) random selection τ/exchange operation, generates a neighborhood, the wherein ∈ of τ 1 { 1,2 };
(2.2.2) generates random number rand1, if rand1 < Ps, enterRow is localThe new individual of search generation simultaneously calculates the fitness value of new individual;
(2.2.3) retains the larger individual of fitness value by comparing new individual and original ideal adaptation angle value;
(2.3) the honeybee stage is followed, generates random number rand2, ifWherein πq(q=1 ..., ind) is the fitness value of q-th of individual, chooses the i-th nd and employs honeybee conduct to follow honeybee and be repeated below process:
2 insertions of (2.3.1) random selection τ/exchange operation, generate a neighborhood, the wherein ∈ of τ 2 { 1,2,3 };
(2.3.2) entersRow is localThe new individual of search generation simultaneously calculates the fitness value of new individual, and Local Search is exactly that an individual is carried out to generate a new individual after random operation, and it is algorithm ripe in the industry to solve as individual, Local Search.
(2.3.3) chooses larger individual be used as of fitness value and employs honeybee by comparing new individual and original ideal adaptation angle value;
(2.4) the honeybee stage is investigated, if the fitness value that max (Bas) > La, wherein Bas is individual ind adds up not improve number, then this is chosen and employs honeybee as investigation honeybee and be repeated below process:
(2.4.1) is using the new individual of DestrConstr algorithms generation as investigation honeybee;Wherein, the process of DestrConstr algorithms is:Take out individual ind job sequence permu and machine assignment matrix μ, three numberings are removed from permu at random and obtain remaining sequence permu0, successively by three removal numbering insetion sequence permu0 and retain fitness value maximum sequence, obtain new sequence and machine assignment matrix;
(2.4.2) generates new machine assignment matrix μ 1 as N μ > Lm, and wherein N μ are total machine quantities;
(2.4.3) calculates the fitness value of new individual and by comparing new individual and original ideal adaptation angle value, and μ is replaced with μ 1 if the fitness value of new individual is bigger;
(2.5) parameters such as record is found at present optimum individual and corresponding workpiece sequencing, machine assignment matrix;
(2.6) stop if end condition meets, otherwise go to (2.2) step.
In general, by the contemplated above technical scheme of the present invention compared with prior art, have the advantages that:
(1) in method of the invention, it can solve the problem that with more machines while handle the production process control problem of single work pieces process scene;Consider work pieces process order to distribute with machine resources, obtained production task control program is more reasonable;
(2) in method of the invention, the optimization of the consideration of time and machine resources distribution makes production process control more precision, so as to improve resource utilization, reduces Workshop Production cost;
(3) in method of the invention, problem solving efficiency is improved by improving DABC algorithms, optimization production control program drastically increases productivity ratio.
Brief description of the drawings
Figure 1It is being used for more machines while handling the implementation frame of the controlling of production process method of either simplex part for one embodiment of the inventionFigure
Figure 2It is being used for more machines while handling the flow of the controlling of production process method of either simplex part for one embodiment of the inventionFigure
Figure 3It is the result for being used for more machines while handling the controlling of production process method of the either simplex part signal of one embodiment of the inventionFigure
Embodiment
In order to make the purpose , technical scheme and advantage of the present invention be clearer, below in conjunction withAccompanying drawingAnd embodiment, the present invention will be described in further detail.It should be appreciated that the specific embodiments described herein are merely illustrative of the present invention, it is not intended to limit the present invention.It is mutually combined in addition, as long as technical characteristic involved in each embodiment of invention described below does not form conflict can each other.
According to a kind of production process control method for handling either simplex part simultaneously for more machines constructed by one embodiment of the invention, referenceFigure 1-Figure 2, including the solution procedure of the structure of production process Controlling model and model, detailed description given below:
1) structure of production process Controlling model
Such asAccompanying drawing 1Shown technical scheme implements frameFigure, customer order information, technology characteristics information, available devices information and material information are collected according to the concrete condition in workshop first.Wherein, customer order information determines the type of production Controlling model, typically provides order demand information by sales department;Technology characteristics information is provided by technique department, determines work station quantity, processing machine feature and constraint etc. in production Controlling model;Available devices information is provided by equipment department, and total number of machines, the machine assignment constraint etc. of production control problem are understood by facility information;Material information is provided by material department, and the preparation of the workpiece machining state with can be known by material information.Model construction process is as follows:
(1.1) defined variable
M=1,2 ..., m }:Process collection, all workpiece flow through the process collection of specified order successively, and m is natural number
Mi:The machine collection of i-th of process, i ∈ M
Φi,t:Parallel processor quantity on t minute processes i
L:Processing machine sum
N=1,2 ..., n }:Workpiece collection, n are natural number
pi,j:Process time s of the workpiece j in process i
si,j:Workpiece j is in the sequence that process i is processed unrelated time
Ti,j:Completion dates of the workpiece j in process i
C:Increase the penalty factor of additional machine, C ∈ [0,1]
xi,j,k:If in process i workpiece j immediately process before workpiece k if xi,j,k=1, otherwise xi,j,k=0
yi,j,r:The y if workpiece j in process i is processed on machine ri,j,r=1, otherwise yi,j,r=0
μi,j:The service machine number on workpiece j in process i,
(1.2) optimization aim of setting model
Maximal Makespan is minimized using (1) and (2) minimize machine input cost and are used as optimization aim:
minf1=maxTi,j (1)
Wherein, Ti,jIt is completion dates of the workpiece j in process i, is determined by formula (8);Target (2) represents increased cost after increase machine input, is determined by the process time of each operation and the machine quantity of machine assignment.
(1.3) according to workshop actual conditions, following constraint is set:
Constraint (3) determines the unit one in each stage.
Constrain (4) and represent that workpiece has determination sequence precedence relationship.
Constraint (5) is the constraint of the parallel processor sum to be come into operation to any time each process, wherein L be can processing machine sum, be constant, Φi , tIt is the machine quantity that process i comes into operation in the t minutes of processing.
Constrain (6), constraint (7) is decision variable xi,j,k、yi,j,r0-1 variable bounds.
(1.4) completion date of the workpiece on every procedure is determined, as shown in formula (8).
2) using improvement DABC Algorithm for Solving production Controlling models
This programme is using such asAccompanying drawing 2The model for handling either simplex part processing scene simultaneously with more machines of shown DABC Algorithm for Solving part 1 structure.Detailed step is:
(2.1) Population Size P, number of machines limitation Lm are set, abandons solution limit L a and Local Search probability P s;
(2.2) generation initialization population, generation initialize the machine assignment matrix μ of each workpiece on the job sequence permu and each operation of population and calculate individual ind fitness value πind, wherein ind represents i-th nd individual, each individual include a length be Number of Jobs be n job sequence permu, size be m × n machine assignment matrix μ and corresponding fitness value πind
(2.3) the honeybee stage is employed, to individual ind=1 ..., P, is repeated:
(2.3.1) random selection τ 1 (∈ of τ 1 { 1,2 }) secondary insertion/exchange operation, generates a neighborhood, devises four kinds of insertions/exchange operation operator here:
M1:To being carried out when former generation once adjacent to insertion operation
M2:To being operated when former generation carries out once neighbouring exchange
M3:To carrying out a radom insertion operation when former generation
M4:To being operated when former generation carries out once random exchange
(2.3.2) generates random number rand1, if rand1 < Ps, enterRow is localThe new individual of search generation simultaneously calculates the fitness value of new individual;
(2.3.3) retains the larger individual of fitness value by comparing new individual and original ideal adaptation angle value;
(2.4) the honeybee stage is followed, generates random number rand2, ifWherein πq(q=1 ..., ind) is the fitness value of q-th of individual, chooses the i-th nd and employs honeybee conduct to follow honeybee and be repeated below process:
(2.4.1) random selection τ 2 (∈ of τ 2 { 1,2,3 }) secondary insertion/exchange operation, generates a neighborhood;
(2.4.2) entersRow is localThe new individual of search generation simultaneously calculates the fitness value of new individual;
(2.4.3) chooses larger individual be used as of fitness value and employs honeybee by comparing new individual and original ideal adaptation angle value;
(2.5) the honeybee stage is investigated, if the fitness value accumulation that max (Bas) > La, wherein Bas is individual ind does not improve number, then this is chosen and employs honeybee as investigation honeybee and be repeated below process:
(2.5.1) is using the new individual of DestrConstr algorithms generation as investigation honeybee;Wherein, the process of DestrConstr algorithms is:Take out individual ind job sequence permu and machine assignment matrix μ, three numberings are removed from permu at random and obtain remaining sequence permu0, successively by three removal numbering insetion sequence permu0 and retain fitness value maximum sequence, obtain new sequence and machine assignment matrix;
(2.5.2) generates new machine assignment matrix μ 1 as N μ > Lm, and wherein N μ are total machine quantities;
(2.5.3) calculates the fitness value of new individual and by comparing new individual and original ideal adaptation angle value, and μ is replaced with μ 1 if the fitness value of new individual is bigger;
(2.6) parameters such as record is found at present optimum individual and corresponding workpiece sequencing, machine assignment matrix;
(2.7) stop if end condition meets, otherwise go to (2.3) step.
In this programme, sexual satisfaction has been forced in constraint (3), (4), (6), (7) in initialization of population, (5) are constrained in 2.4.1 parts, the part is a circulation in fact, new machine assignment matrix is generated as N μ > Lm, untill meeting to constrain (5).
The above method of the present invention can also include the process of the issue of production control program and data information management, and it is specially:
3) issue of control result scheme is produced
It is the gunter illustrated example that this programme proposes multiparallel-processor problem as shown in Figure 3.In figure, process time block is divided into longitudinal several pieces by white stripes, and multimachine problem is represented with this.For example, in the 5th process of workpiece 3, there are 3 machines while handle workpiece, so there are 2 informal vouchers during the grey block in this stage is represented in figure, the number of machines for representing processing simultaneously is 3.The Gantt chart shows that Maximal Makespan is 305min.It should be pointed out that although multiparallel-processor model has shorter Maximal Makespan compared to conventional model, total machine is also increased simultaneously and is used, machine assignment matrix is:
Machine assignment matrix is specific to the solution of multiparallel-processor problem, naturally it is also possible to which it is 1- matrixes to think the machine assignment matrix in traditional problem.Machine assignment matrix herein reflects machine demand of each workpiece in each process, and this is an important reference factor in the quality that production control program result is discussed.
By such asAccompanying drawing 3Shown production control program visualization technique, production management personnel can clearly obtain the order to be processed of planned order, and each workpiece in machine type, quantity needed for each operation processing.
4) data base administration
Data are to carry out the driving force of production control and management, and workshop data mainly includes two aspect contents:On the one hand it is the data related to production Controlling model feature, such as customer order information, technology characteristics information, available devices information, model can be embodied based on this kind of data;On the other hand it is the data related to production process, such as time, process time, such data are the object of production process control method processing proposed by the invention, and workpiece is abstracted in manufacture process, and its result is to produce the generation of control program.Need to be acquired some data before carrying out producing Controlling model structure, store, need to issue result data after production control program generation, store.
For the implementation process of the present invention is expanded on further, explained below by way of the embodiment of welding shop production process.
1) structure of welding shop model
In embodiment, the first step is according to shop characteristic produce the structure of control problem model.Welding production process is more machines proposed by the invention while handles the typical case scene of either simplex part processing scene.In large-sized structural parts welding, welding process is carried out around workpiece, and more welding machines can weld same workpiece simultaneously, namely the problem of Multi-computer Processing either simplex part.Consider a typical welding shop, the workshop there are 5 master operations, or referred to as should homogeneously there be a number of machine, frock clamp and worker etc. in 5 stages, each stage.All workpiece pass sequentially through this 5 procedure with identical order.For the welding shop of box-like beam, 5 procedures are:The splicing of widget, the splicing of big part, interior seam welding, encapsulation and outer seam welding.
The production Controlling model of the present embodiment can directly quote foregoing model.
2) using improvement DABC Algorithm for Solving embodiments
Separately design 10 × 5,30 × 5 and 60 × 5 three examples to be analyzed, to embody the effect of the present invention, compared with two algorithms of DABC and GA.It is 50 to set population scale, and algorithm iteration number is 1000, and machine quantity is constrained to 7, and it is 3 to abandon the solution limit, and Local Search probability is 0.2.
Calculated, analyzed according to aforementioned algorism step, obtain result such asFollowing table 1It is shown:
Table 1DABC and GA algorithms simulation result
Table 1Result show that the problem of for different scales, no matter comparing in terms of production efficiency or machine input, DABC algorithms of the present invention have significant advantage compared to GA.
3) issue of control result scheme is produced
By taking the problem of 10 × 5 scale as an example, after it produces control result schemes generation, work pieces process sequence is:
8→10→1→6→5→2→7→3→9→4
Corresponding machine assignment matrix is:
According to the above results, you can produce and control accordingly.
Those skilled in the art is readily appreciated that; the foregoing is merely illustrative of the preferred embodiments of the present invention; it is not intended to limit the invention, all any modification, equivalent and improvement made within the spirit and principles of the invention etc., should be included in the scope of the protection.

Claims (2)

1. a kind of generation technical process control method for handling either simplex part simultaneously for more machines, it comprises the following steps:
(1) more machines are established while handle the generation technical process control model of either simplex part, are specifically included:
The optimization aim of setting generation technical process control model, i.e., thrown with minimizing Maximal Makespan and minimizing machine Enter cost as target, be respectively:
min f1=maxTi,j (1)
Wherein, Ti,jFor workpiece j process i completion date;pi,jFor workpiece j process i process time;M is work Sequence collection and M={ 1,2 ..., m }, wherein m are natural number;I is process numbering and i ∈ M;N is workpiece collection and N={ 1,2 ..., n }, Wherein n is natural number;J is workpiece numbering and j ∈ N;MiIt is the machine collection in i-th of process;R be machine serial number and r∈Mi, the y if workpiece j in process i is processed on machine ri,j,r=1, otherwise yi,j,r=0, μi,jIt is workpiece j in process The upper service machine numbers of i,C be increase additional machine penalty factor, C ∈ [0,1];
(2) above-mentioned model is solved using artificial bee colony algorithm, obtains solving result, specifically include:
(2.1) Population Size P, number of machines limitation Lm are set, abandons solution limit L a and Local Search probability P s, generation Initialize the machine assignment matrix μ of each workpiece on the job sequence permu and each operation of population and calculate individual ind's Fitness value πind, i-th nd individual of wherein ind expressions, each individual is that Number of Jobs is n comprising a length Job sequence permu, size be m × n machine assignment matrix μ and corresponding fitness value πind
(2.2) the honeybee stage is employed, to individual ind=1 ..., P in population, is repeated below process:
1 insertion of (2.2.1) random selection τ/exchange operation, generates a neighborhood, the wherein ∈ of τ 1 { 1,2 };
(2.2.2) generates random number rand1, if rand1 < Ps, carries out the new individual of Local Search generation and calculates The fitness value of new individual;
(2.2.3) retains the larger individual of fitness value by comparing new individual and original ideal adaptation angle value;
(2.3) the honeybee stage is followed, generates random number rand2, ifWherein πq(q=1 ..., ind) be The fitness value of q-th of individual, choose the i-th nd and employ honeybee conduct to follow honeybee and be repeated below process:
2 insertions of (2.3.1) random selection τ/exchange operation, generate a neighborhood, the wherein ∈ of τ 2 { 1,2,3 };
(2.3.2) carries out the new individual of Local Search generation and calculates the fitness value of new individual;
(2.3.3) chooses larger individual be used as of fitness value and employs honeybee by comparing new individual and original ideal adaptation angle value;
(2.4) the honeybee stage is investigated, if the fitness value accumulation that max (Bas) > La, wherein Bas is individual ind does not improve Number, then choose this and employ honeybee as investigation honeybee and be repeated below process:
(2.4.1), as investigation honeybee, is specially using the new individual of DestrConstr algorithms generation:Take out individual ind's Job sequence permu and machine assignment matrix μ, three numberings are removed from permu at random and obtain remaining sequence Permu0, the numbering insetion sequence permu0 for successively removing three simultaneously retain the maximum sequence of fitness value, obtain new Sequence and machine assignment matrix;
(2.4.2) generates new machine assignment matrix μ 1 as N μ > Lm, and wherein N μ are total machine quantities;
(2.4.3) calculates the fitness value of new individual and by comparing new individual and original ideal adaptation angle value, if new individual Fitness value bigger μ is then replaced with μ 1;
(2.5) parameters such as record is found at present optimum individual and corresponding workpiece sequencing, machine assignment matrix;
(2.6) stop if end condition meets, otherwise go to (2.2) step.
A kind of 2. generation technical process control side for handling either simplex part simultaneously for more machines according to claim 1 Method, wherein, it is described establish more machines while handle either simplex part generation technical process control model the step of in also include set Fixed following constraint:
<mrow> <munder> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>&amp;Element;</mo> <mi>N</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>&amp;ForAll;</mo> <mi>i</mi> <mo>&amp;Element;</mo> <mi>M</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <munder> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>&amp;Element;</mo> <mo>{</mo> <mn>0</mn> <mo>}</mo> <mo>&amp;cup;</mo> <mi>N</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>&amp;Element;</mo> <mi>N</mi> <mo>&amp;cup;</mo> <mo>{</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>}</mo> </mrow> </munder> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mo>&amp;ForAll;</mo> <mi>k</mi> <mo>&amp;Element;</mo> <mi>N</mi> <mo>,</mo> <mo>&amp;ForAll;</mo> <mi>i</mi> <mo>&amp;Element;</mo> <mi>M</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <munder> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>&amp;Element;</mo> <mi>M</mi> </mrow> </munder> <msub> <mi>&amp;Phi;</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>t</mi> </mrow> </msub> <mo>&amp;le;</mo> <mi>L</mi> <mo>,</mo> <mo>&amp;ForAll;</mo> <mi>t</mi> <mo>&amp;Element;</mo> <mo>&amp;lsqb;</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>T</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>&amp;rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>;</mo> <mo>&amp;ForAll;</mo> <mi>j</mi> <mo>&amp;Element;</mo> <mo>{</mo> <mn>0</mn> <mo>}</mo> <mo>&amp;cup;</mo> <mi>N</mi> <mo>,</mo> <mo>&amp;ForAll;</mo> <mi>k</mi> <mo>&amp;Element;</mo> <mi>N</mi> <mo>&amp;cup;</mo> <mo>{</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>}</mo> <mo>,</mo> <mo>&amp;ForAll;</mo> <mi>i</mi> <mo>&amp;Element;</mo> <mi>M</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>y</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>;</mo> <mo>&amp;ForAll;</mo> <mi>i</mi> <mo>&amp;Element;</mo> <mi>M</mi> <mo>,</mo> <mo>&amp;ForAll;</mo> <mi>j</mi> <mo>&amp;Element;</mo> <mi>N</mi> <mo>,</mo> <mo>&amp;ForAll;</mo> <mi>r</mi> <mo>&amp;Element;</mo> <msub> <mi>M</mi> <mi>i</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
Wherein, constraint (3) determines the unit one in each stage, and constraint (4) represents that workpiece is present and determines that sequence is first Relation afterwards, constrain the constraint that (5) are the parallel processor sums to be come into operation to any time each process, wherein L It is that can be used for processing machine sum, is constant, ΦI, tIt is the machine quantity that process i comes into operation in the t minutes of processing, It is 0-1 variable bounds to constrain (6) and constraint (7).
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