CN107704985A - A kind of differential evolution Flexible Workshop Optimization Scheduling of dynamic strategy - Google Patents

A kind of differential evolution Flexible Workshop Optimization Scheduling of dynamic strategy Download PDF

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CN107704985A
CN107704985A CN201710683897.6A CN201710683897A CN107704985A CN 107704985 A CN107704985 A CN 107704985A CN 201710683897 A CN201710683897 A CN 201710683897A CN 107704985 A CN107704985 A CN 107704985A
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张贵军
王文
周晓根
王柳静
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Abstract

A kind of differential evolution Flexible Workshop Optimization Scheduling of dynamic strategy, discrete scheduling problem is converted into by tractable continuous problem using process and machine bilayer coded system, then using each the distance between individual degree of crowding for weighing population, and then the stage residing for evaluation algorithm；Then, for each stage the characteristics of adaptively selected corresponding Mutation Strategy；Further boosting algorithm performance, finally solves optimal scheduling scheme by performance indications of Maximal Makespan.The differential evolution Flexible Workshop Optimization Scheduling of corresponding Mutation Strategy come the partitioning algorithm stage and then is selected based on individual crowding in population the invention provides a kind of.

Description

A kind of differential evolution Flexible Workshop Optimization Scheduling of dynamic strategy
Technical field
The present invention relates to a kind of Flexible Workshop, intelligence manufacture, differential evolution algorithm, Optimized Operation field, more particularly to It is a kind of differential evolution Flexible Workshop Optimization Scheduling of dynamic strategy.
Background technology
Currently, the tide of industry 4.0 that development intelligence manufacture is core has been started in world wide, China complies with scientific and technological change Leather, it is proposed that the developing goal of " made in China " realizes manufacturing industry intelligence manufacture, finally realizes manufacturing industry digitlization, network Change, be intelligent.This means：Conventional commercial, less the unification production model of the production line balance of kind will no longer conform to work as Modern manufacturing idea of development.In addition, with social progress and the improvement of people's living standards, the market demand is from relative stable type Progressively turn to dynamic multiple changing type.The market demand and enterprise's productive prospecting are shown as：The competition in market simultaneously carries changeable Property and unpredictability, the upgrading period of product increasingly accelerate, customer is more diversified to product demand so that the demand tendency of product Customerization.Under this market environment with keen competition, Business survival and possess competitiveness whether depend on enterprise can have Within the shorter construction cycle, produce lower cost, better quality different cultivars product ability, to use most short life The production cycle responds to changes in market demand, and the resource including factory building, equipment and manpower is obtained most effective land productivity With reaching the purpose of enterprise production and management ability global optimization.It can be seen that flexible manufacturing starts to occupy increasingly in manufacture system Important position.Manufacturing development level is the important embodiment of national economy strength, as intelligence manufacture constantly promotes, with more Flexible manufacturing system (Flexible Manufacturing System, FMS) based on kind, the small lot mode of production is progressively Into the people visual field, wherein key technology and core content of the workshop Optimized Operation as the flexible manufacturing production schedule, improving Production efficiency, reduce production cost etc. and serve vital effect.Flexible job shop scheduling problem (Flexible Job Shop Scheduling Problem, FJSP) it is one of traditional work workshop scheduling problem extension, it allows one Operation can be by any machine processing in a given set；Because there is the distribution of extra machine, it compares job shop Scheduling problem is more complicated.
Differential evolution algorithm (DE) is a kind of emerging evolutionary computation technique.Differential evolution algorithm is that a kind of simulation biology enters The stochastic model of change, by iterating so that those individuals for adapting to environment have been saved.Meanwhile the distinctive notes of DE The ability of recalling allows it dynamically to track current search situation, to adjust its search strategy, has stronger global convergence ability And robustness, and need not be by the characteristic information of problem, can not using conventional mathematic programming methods suitable for solving some Optimization problem in the complex environment of solution.The scheduling problem (Job Shop Scheduling Problem, JSP) one in workshop As refer to how under the limited resources of production and device constraints, arrange out effective and reasonable production process and processing set It is standby so that goal-selling is optimal；The problem is typically multiple constraint, multiple target, random uncertain.But present flexible car mostly Between Optimization Scheduling solving speed it is slow and reliability is low, it is difficult to find optimal scheduling mode.
Therefore, existing Flexible Workshop Optimized Operation mode Shortcomings are, it is necessary to improve.
The content of the invention
In order to overcome, the solving speed of existing Flexible Workshop Optimized Operation mode is slow and reliability is low, is difficult to find that optimal row Production mode, it is based on individual crowding in population the invention provides one kind and is made a variation accordingly come partitioning algorithm stage and then selection The differential evolution Flexible Workshop Optimization Scheduling of strategy.
The present invention solves the technical method that its technical problem uses：
A kind of differential evolution Flexible Workshop Optimization Scheduling of dynamic strategy, the described method comprises the following steps：
A1, by Flexible Workshop production scheduling problem be converted into mathematical formulae description；N kinds workpiece is processed on m platform machines, Every kind of workpiece includes niProcedure；The object function of required optimization is：
F (x)=min (maxCi)
Wherein, CiRepresent that workpiece i completes the time of processing；I represents workpiece sequence number, i=1,2,3 ..., n；J represents workpiece Process, j=1,2,3 ..., ni；K represents the sequence number of machine, k=1,2,3 ..., m；PijkFor workpiece i jth procedure in machine Process time on device k, PabkFor workpiece a process time of the b procedures on machine k, a=1,2,3 ..., n, b=1, 2,3,...,ni；XijkRepresent that workpiece i jth procedure is processed on machine k；Xi(j-1)lBefore the jth procedure for representing workpiece i One procedure is processed on machine l, l=1,2,3 ..., m；RijegkRepresent workpiece i jth procedure e g procedures same Processed on one machine k, and process j first with process g, process time of the jth procedure on machine k, e=1,2,3 ..., ni, g=1,2,3 ..., ni；CijkFor workpiece i deadline of the jth procedure on machine k, CabkFor workpiece a b roads Deadline of the process on machine k；
A2, initialization：Population scale N is setP, memory algebraically LP；
A3, Job Scheduling is converted into by row square using the double-deck coding method based on processing machine corresponding to process and process Battle array X, i.e., scheduling X=[x feasible to one1,x2,…,xD]T, D=2d, d represent the summation of current process, make its first half X1=[x1,x2,…,xd]ΤThe coding of process scheduling is represented, i.e., from first process of unit one to a last workpiece Last process according to integer 1,2,3 ..., d mode counts；Latter half X2=[xd+1,xd+2,…,xD]ΤRepresent work The machine of the corresponding processing of sequence, i.e. xd+lRepresent that process l can be in the xth of processing machine collectiond+lProcessed on machine；
A4, based on coded system described in A3, population is initialized, initial population individual scale is Np, to first half Sub-sequence carries out NpIt is secondary randomly ordered, per first half sequence of the minor sort as an individual；Latter half sequence passes through each work Ordered pair answers machine collection to randomly generate；
A5, the average distance calculated in current population between each individual are：
Wherein, xji,G、xjk,GG is represented respectively ties up element for the jth of i-th in population and k-th of individual；
A6, the dist of every generation is normalized according to average distance dist maximum and minimum value and worked as The crowding of preceding population：
Wherein, average distance obtains minimum value dist between individualmin=0；Dist maximum distmaxInitial value be first The average distance of beginning population, and during evolution, if the dist of certain generation is more than distmax, then current dist conducts are taken distmax；
A7, according to population crowdingChange, corresponding Mutation Strategy is selected to different phase；
Wherein, φ represents the stage residing for algorithm, random number between rand (0,1) represents 0 to 1, S1Algorithm is represented to be in Global detection phase, S2Represent algorithm and be in the Local Search stage；If meet S1Condition, then into A8；It is unsatisfactory for then entering A9；
A8, randomly select a strategy from strategy in following 3 and enter row variation：
DE/rand/1
DE/rand/2
DE/current-to-rand/1
Wherein, G is evolutionary generation, r1、r2、r3、r4、r5∈{1,2,…,Np, and r1≠r2≠r3≠r4≠r5≠ i, vji,G Element is tieed up for the jth of i-th of variation individual in population for G, Respectively G generations R in population1、r2、r3、r4、r5Individual jth dimension element, K=0.5 is the fixation of DE/current-to-rand/1 strategies Gain constant；Represent the gain constant of i-th of individual in G generations；
A9, one strategy of random selection enters row variation from following 3 kinds of strategies：
DE/best/1
DE/best/2
DE/current-to-best/1
Wherein,It is G for the 0.5N in populationpThe jth of the optimum individual randomly selected in rand (0,1) individuals Tie up element；
A10, crossover operation generation test vector u is carried out by formula (7)ji,G+1：
Wherein, uji,G+1Represent that G+1 ties up for the jth of i-th of test vector in population,For crossover probability, It can be automatically updated per a generation, jrandFor 0 to the random integers between D, the dimension of D problem of representation；Represent i-th in G generations Individual crossover probability,Represent withFor average, with 0.1 for error normal state Distribution random numbers, as G ＜ LP,As G ＞ LP,New individual caused by previous generation is taken to be successfully reserved As it is of future generation individual whenAverage value, while when remember algebraically reach LP when, whenever produce a new generationThen delete An earliest generationThere may be the intangibility for being unsatisfactory for constraints, it is necessary to not meeting constraints after crossover operation Test vector is handled, if the test vector through A10 step operations does not meet coding rule, returns to A7, can until producing Row solution；
A11, decoding operate, first, it is the job sequence based on workpiece process that it is Sequence Transformed, which will to test individual first half,； Secondly, the order of manufacturing procedure on every machine is determined by latter half sequence；Finally, every procedure of all workpiece is determined Between at the beginning of on processing machine and completion date；Based on job sequence and process constraint to each operation to allow to process earliest Time is processed one by one, calculates every machine last work deadline, takes its Maximal Makespan to be solved to be corresponding；
A12, by comparing the adaptive value of object vector in test vector and current population select optimum individual
Wherein, xi,G+1Represent i-th individuals of the G+1 for population, ui,GAnd xi,GRepresent G for i-th in population respectively For i-th of individual of population, f (x) is object function by variation individual and G；
A13, judge whether to meet end condition, if it is satisfied, then preserving result and exiting, otherwise return to A5.
Further, in the A8,N (0.5,0.3) represents that average is 0.5, standard deviation 0.3 Normal distribution random number.
The present invention technical concept be：Using it is each it is the distance between individual weigh the degree of crowding of population, and then judge Stage residing for algorithm；Then, for each stage the characteristics of adaptively selected corresponding Mutation Strategy；Secondly, with maximum complete man-hour Between be target, using process and machine bilayer coded system, optimization obtains optimal scheduling scheme.
Beneficial effects of the present invention are mainly manifested in：Stage according to residing for the degree of crowding of individual come evaluation algorithm, and Multiple suitable Mutation Strategies are set in each stage, so as to randomly choose different Mutation Strategies in each iteration in each stage To produce new individual, the performance of algorithm is further improved.
Brief description of the drawings
Fig. 1 is a kind of flow chart of the differential evolution Flexible Workshop Optimization Scheduling of dynamic strategy.
Fig. 2 is that a kind of differential evolution Flexible Workshop Optimization Scheduling of dynamic strategy is put down to 4 × 6FJSP Optimization Solutions Change curve after equal range normalization.
Fig. 3 is a kind of tune of the differential evolution Flexible Workshop Optimization Scheduling to 4 × 6FJSP Optimization Solutions of dynamic strategy Spend Gantt chart.
Embodiment：
The present invention is further described below in conjunction with the accompanying drawings.
A kind of 1~Fig. 3 of reference picture, the differential evolution Flexible Workshop Optimization Scheduling of dynamic strategy, comprises the following steps：
A1, by Flexible Workshop production scheduling problem be converted into mathematical formulae description；N kinds workpiece is processed on m platform machines, Every kind of workpiece includes niProcedure；The object function of required optimization is：
F (x)=min (maxCi)
Wherein, CiRepresent that workpiece i completes the time of processing；I represents workpiece sequence number, i=1,2,3 ..., n；J represents workpiece Process, j=1,2,3 ..., ni；K represents the sequence number of machine, k=1,2,3 ..., m；PijkFor workpiece i jth procedure in machine Process time on device k, PabkFor workpiece a process time of the b procedures on machine k, a=1,2,3 ..., n, b=1, 2,3,...,ni；XijkRepresent that workpiece i jth procedure is processed on machine k；Xi(j-1)lBefore the jth procedure for representing workpiece i One procedure is processed on machine l, l=1,2,3 ..., m；RijegkRepresent workpiece i jth procedure e g procedures same Processed on one machine k, and process j first with process g, process time of the jth procedure on machine k, e=1,2,3 ..., ni, g=1,2,3 ..., ni；CijkFor workpiece i deadline of the jth procedure on machine k, CabkFor workpiece a b roads Deadline of the process on machine k；
A2, initialization：Population scale N is setP, memory algebraically LP；
A3, Job Scheduling is converted into by row square using the double-deck coding method based on processing machine corresponding to process and process Battle array X, i.e., scheduling X=[x feasible to one1,x2,…,xD]T, D=2d, d represent the summation of current process, make its first half X1=[x1,x2,…,xd]ΤThe coding of process scheduling is represented, i.e., from first process of unit one to a last workpiece Last process according to integer 1,2,3 ..., d mode counts；Latter half X2=[xd+1,xd+2,…,xD]ΤRepresent work The machine of the corresponding processing of sequence, i.e. xd+lRepresent that process l can be in the xth of processing machine collectiond+lProcessed on machine；
A4, based on coded system described in A3, population is initialized, initial population individual scale is Np, to first half Sub-sequence carries out NpIt is secondary randomly ordered, per first half sequence of the minor sort as an individual；Latter half sequence passes through each work Ordered pair answers machine collection to randomly generate；
A5, the average distance calculated in current population between each individual are：
Wherein, xji,G、xjk,GG is represented respectively ties up element for the jth of i-th in population and k-th of individual；
A6, the dist of every generation is normalized according to average distance dist maximum and minimum value and worked as The crowding of preceding population：
Wherein, average distance obtains minimum value dist between individualmin=0；Dist maximum distmaxInitial value be first The average distance of beginning population, and during evolution, if the dist of certain generation is more than distmax, then current dist conducts are taken distmax；
A7, according to population crowdingChange, corresponding Mutation Strategy is selected to different phase；
Wherein, φ represents the stage residing for algorithm, random number between rand (0,1) represents 0 to 1, S1Algorithm is represented to be in Global detection phase, S2Represent algorithm and be in the Local Search stage；If meet S1Condition, then into A8；It is unsatisfactory for then entering A9；
A8, randomly select a strategy from strategy in following 3 and enter row variation：
DE/rand/1
DE/rand/2
DE/current-to-rand/1
Wherein, G is evolutionary generation, r1、r2、r3、r4、r5∈{1,2,…,Np, and r1≠r2≠r3≠r4≠r5≠ i, vji,G Element is tieed up for the jth of i-th of variation individual in population for G, Respectively G generations R in population1、r2、r3、r4、r5Individual jth dimension element, K=0.5 is the fixation of DE/current-to-rand/1 strategies Gain constant；Represent the gain constant of i-th of individual in G generations；
A9, one strategy of random selection enters row variation from following 3 kinds of strategies：
DE/best/1
DE/best/2
DE/current-to-best/1
Wherein,It is G for the 0.5N in populationpThe jth of the optimum individual randomly selected in rand (0,1) individuals Tie up element；
A10, crossover operation generation test vector u is carried out by formula (7)ji,G+1：
Wherein, uji,G+1Represent that G+1 ties up for the jth of i-th of test vector in population,For crossover probability, It can be automatically updated per a generation, jrandFor 0 to the random integers between D, the dimension of D problem of representation；Represent i-th in G generations Individual crossover probability,Represent withFor average, with 0.1 for error normal state Distribution random numbers, as G ＜ LP,As G ＞ LP,New individual caused by previous generation is taken to be successfully reserved As it is of future generation individual whenAverage value, while when remember algebraically reach LP when, whenever produce a new generationThen delete An earliest generationThere may be the intangibility for being unsatisfactory for constraints, it is necessary to not meeting constraints after crossover operation Test vector is handled, if the test vector through A10 step operations does not meet coding rule, returns to A7, can until producing Row solution；
A11, decoding operate, first, it is the job sequence based on workpiece process that it is Sequence Transformed, which will to test individual first half,； Secondly, the order of manufacturing procedure on every machine is determined by latter half sequence；Finally, every procedure of all workpiece is determined Between at the beginning of on processing machine and completion date；Based on job sequence and process constraint to each operation to allow to process earliest Time is processed one by one, calculates every machine last work deadline, takes its Maximal Makespan to be solved to be corresponding；
A12, by comparing the adaptive value of object vector in test vector and current population select optimum individual
Wherein, xi,G+1Represent i-th individuals of the G+1 for population, ui,GAnd xi,GRepresent G for i-th in population respectively For i-th of individual of population, f (x) is object function by variation individual and G；
A13, judge whether to meet end condition, if it is satisfied, then preserving result and exiting, otherwise return to A5.
Further, in the A8,N (0.5,0.3) represents that average is 0.5, standard deviation 0.3 Normal distribution random number.
The present embodiment is using 4 × 6FJSP test functions of classics as embodiment, a kind of differential evolution flexibility car of dynamic strategy Between Optimization Scheduling, wherein comprising the steps of：
A1, by Flexible Workshop production scheduling problem be converted into mathematical formulae description；4 kinds of workpiece are processed on 6 machines, Every kind of workpiece includes 3 procedures；The object function of required optimization is：
F (x)=min (maxCi)
Wherein, CiRepresent that workpiece i completes the time of processing；I represents workpiece sequence number, i=1,2,3 ..., n；J represents workpiece Process, j=1,2,3 ..., ni；K represents the sequence number of machine, k=1,2,3 ..., m；PijkFor workpiece i jth procedure in machine Process time on device k, PabkFor workpiece a process time of the b procedures on machine k, a=1,2,3 ..., n, b=1, 2,3,...,ni；XijkRepresent that workpiece i jth procedure is processed on machine k；Xi(j-1)lBefore the jth procedure for representing workpiece i One procedure is processed on machine l, l=1,2,3 ..., m；RijegkRepresent workpiece i jth procedure e g procedures same Processed on one machine k, and process j first with process g, process time of the jth procedure on machine k, e=1,2,3 ..., ni, g=1,2,3 ..., ni；CijkFor workpiece i deadline of the jth procedure on machine k, CabkFor workpiece a b roads Deadline of the process on machine k；
A2, initialization：Population scale N is setP=30, memory algebraically LP=20；
A3, Job Scheduling is converted into by row square using the double-deck coding method based on processing machine corresponding to process and process Battle array X, i.e., scheduling X=[x feasible to one1,x2,…,x24]T, D=24, d represent the summation of current process, make its first half X1=[x1,x2,…,x12]ΤThe coding of process scheduling is represented, i.e., from first process of unit one to a last workpiece Last process according to integer 1,2,3 ..., 12 mode counts；Latter half X2=[x13,x14,…,x24]ΤRepresent work The machine of the corresponding processing of sequence, i.e. x13Represent that process 1 can be in the xth of processing machine collection13Processed on machine；
A4, based on coded system described in A3, population is initialized, initial population individual scale is 30, to first half Sub-sequence carries out 30 randomly ordered, every first half sequences of the minor sort as an individual；Latter half sequence passes through each work Ordered pair answers machine collection to randomly generate；
A5, the average distance calculated in current population between each individual are：
Wherein, xji,G、xjk,GG is represented respectively ties up element for the jth of i-th in population and k-th of individual；
A6, the dist of every generation is normalized according to average distance dist maximum and minimum value and worked as The crowding of preceding population：
Wherein, average distance obtains minimum value dist between individualmin=0；Dist maximum distmaxInitial value be first The average distance of beginning population, and during evolution, if the dist of certain generation is more than distmax, then current dist conducts are taken distmax；
A7, according to population crowdingChange, corresponding Mutation Strategy is selected to different phase；
Wherein, φ represents the stage residing for algorithm, random number between rand (0,1) represents 0 to 1, S1Algorithm is represented to be in Global detection phase, S2Represent algorithm and be in the Local Search stage；If meet S1Condition, then into A8；It is unsatisfactory for then entering A9；
A8, randomly select a strategy from strategy in following 3 and enter row variation：
DE/rand/1
DE/rand/2
DE/current-to-rand/1
Wherein, G is evolutionary generation, r1、r2、r3、r4、r5∈{1,2,…,Np, and r1≠r2≠r3≠r4≠r5≠ i, vji,G Element is tieed up for the jth of i-th of variation individual in population for G, Respectively G generations R in population1、r2、r3、r4、r5Individual jth dimension element, K=0.5 is the fixation of DE/current-to-rand/1 strategies Gain constant；Represent the gain constant of i-th of individual in G generations；
A9, one strategy of random selection enters row variation from following 3 kinds of strategies：
DE/best/1
DE/best/2
DE/current-to-best/1
Wherein,It is G for the 0.5N in populationpThe jth of the optimum individual randomly selected in rand (0,1) individuals Tie up element；
A10, crossover operation generation test vector u is carried out by formula (7)ji,G+1：
Wherein, uji,G+1Represent that G+1 ties up for the jth of i-th of test vector in population,For crossover probability, It can be automatically updated per a generation, jrandFor 0 to the random integers between D, the dimension of D problem of representation；Represent i-th in G generations Individual crossover probability,Represent withFor average, with 0.1 for error normal state Distribution random numbers, as G ＜ LP,As G ＞ LP,New individual caused by previous generation is taken to be successfully reserved As it is of future generation individual whenAverage value, while when remember algebraically reach LP when, whenever produce a new generationThen delete An earliest generationThere may be the intangibility for being unsatisfactory for constraints, it is necessary to not meeting constraints after crossover operation Test vector is handled, if the test vector through A10 step operations does not meet coding rule, returns to A7, can until producing Row solution；
A11, decoding operate, first, it is the job sequence based on workpiece process that it is Sequence Transformed, which will to test individual first half,； Secondly, the order of manufacturing procedure on every machine is determined by latter half sequence；Finally, every procedure of all workpiece is determined Between at the beginning of on processing machine and completion date；Based on job sequence and process constraint to each operation to allow to process earliest Time is processed one by one, calculates every machine last work deadline, takes its Maximal Makespan to be solved to be corresponding；
A12, by comparing the adaptive value of object vector in test vector and current population select optimum individual
Wherein, xi,G+1Represent i-th individuals of the G+1 for population, ui,GAnd xi,GRepresent G for i-th in population respectively For i-th of individual of population, f (x) is object function by variation individual and G；
Whether A13, discriminant function evaluation number reach 6000 times, if it is satisfied, then preserving result and exiting, otherwise return A5。
Further, in the A8,N (0.5,0.3) represents that average is 0.5, and standard deviation is 0.3 Normal distribution random number.
Described above is the excellent effect of optimization that one embodiment that the present invention provides shows, it is clear that the present invention is not Above-described embodiment is suitable only for, without departing from essence spirit of the present invention and the premise without departing from content involved by substantive content of the present invention Under can do many variations to it and be carried out.

Claims (1)

  1. A kind of 1. differential evolution Flexible Workshop Optimization Scheduling of dynamic strategy, it is characterised in that：Methods described includes following Step：
    A1, by Flexible Workshop production scheduling problem be converted into mathematical formulae description；N kinds workpiece is processed on m platform machines, it is every kind of Workpiece includes niProcedure；The object function of required optimization is：
    <mrow> <mtable> <mtr> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>min</mi> <mrow> <mo>(</mo> <mi>max</mi> <mi> </mi> <msub> <mi>C</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>C</mi> <mrow> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>C</mi> <mrow> <mi>i</mi> <mrow> <mo>(</mo> <mi>j</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>l</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>P</mi> <mrow> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <mo>&amp;GreaterEqual;</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>X</mi> <mrow> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>X</mi> <mrow> <mi>i</mi> <mrow> <mo>(</mo> <mi>j</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>C</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>k</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>C</mi> <mrow> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>P</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>k</mi> </mrow> </msub> <mo>&amp;GreaterEqual;</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>X</mi> <mrow> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>X</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>R</mi> <mrow> <mi>i</mi> <mi>j</mi> <mi>a</mi> <mi>b</mi> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
    Wherein, CiRepresent that workpiece i completes the time of processing；I represents workpiece sequence number, i=1,2,3 ..., n；J represents the work of workpiece Sequence, j=1,2,3 ..., ni；K represents the sequence number of machine, k=1,2,3 ..., m；PijkFor workpiece i jth procedure in machine k On process time, PabkFor workpiece a process time of the b procedures on machine k, a=1,2,3 ..., n, b=1,2, 3,...,ni；XijkRepresent that workpiece i jth procedure is processed on machine k；Xi(j-1)lRepresent the previous of workpiece i jth procedure Procedure is processed on machine l, l=1,2,3 ..., m；RijegkRepresent workpiece i jth procedure e g procedures same Processed on platform machine k, and process j elder generations and process g, process time of the jth procedure on machine k, e=1,2,3 ..., ni, g =1,2,3 ..., ni；CijkFor workpiece i deadline of the jth procedure on machine k, CabkFor workpiece a b procedures Deadline on machine k；
    A2, initialization：Population scale N is setP, memory algebraically LP；
    A3, Job Scheduling is converted into by column matrix X using the double-deck coding method based on processing machine corresponding to process and process, Scheduling X=[x i.e. feasible to one1,x2,…,xD]T, D=2d, d represent the summation of current process, make its first half X1= [x1,x2,…,xd]ΤThe coding of process scheduling is represented, i.e., from first process of unit one to a last workpiece most The latter process according to integer 1,2,3 ..., d mode counts；Latter half X2=[xd+1,xd+2,…,xD]ΤRepresent process institute The machine of corresponding processing, i.e. xd+lRepresent that process l can be in the xth of processing machine collectiond+lProcessed on machine；
    A4, based on coded system described in A3, population is initialized, initial population individual scale is Np, to first half sequence Carry out NpIt is secondary randomly ordered, per first half sequence of the minor sort as an individual；Latter half sequence is corresponding by each process Machine collection randomly generates；
    A5, the average distance calculated in current population between each individual are：
    <mrow> <mi>d</mi> <mi>i</mi> <mi>s</mi> <mi>t</mi> <mo>=</mo> <mfrac> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>N</mi> <mi>p</mi> </msub> </munderover> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> <msub> <mi>N</mi> <mi>p</mi> </msub> </munderover> <msqrt> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mi>i</mi> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mi>k</mi> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> <mrow> <msub> <mi>N</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>N</mi> <mi>p</mi> </msub> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>/</mo> <mn>2</mn> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
    Wherein, xji,G、xjk,GG is represented respectively ties up element for the jth of i-th in population and k-th of individual；
    A6, the dist of every generation is normalized according to average distance dist maximum and minimum value and currently planted The crowding of group：
    <mrow> <mover> <mrow> <mi>d</mi> <mi>i</mi> <mi>s</mi> <mi>t</mi> </mrow> <mo>&amp;OverBar;</mo> </mover> <mo>=</mo> <mfrac> <mrow> <mi>d</mi> <mi>i</mi> <mi>s</mi> <mi>t</mi> <mo>-</mo> <msub> <mi>dist</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>dist</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>dist</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
    Wherein, average distance obtains minimum value dist between individualmin=0；Dist maximum distmaxInitial value be initial kind The average distance of group, and during evolution, if the dist of certain generation is more than distmax, then current dist conducts are taken distmax；
    A7, according to population crowdingChange, corresponding Mutation Strategy is selected to different phase；
    <mrow> <mi>&amp;phi;</mi> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> <mo>,</mo> </mrow> </mtd> <mtd> <mtable> <mtr> <mtd> <mrow> <mi>i</mi> <mi>f</mi> </mrow> </mtd> <mtd> <mrow> <mi>r</mi> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>&gt;</mo> <mover> <mrow> <mi>d</mi> <mi>i</mi> <mi>s</mi> <mi>t</mi> </mrow> <mo>&amp;OverBar;</mo> </mover> </mrow> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>S</mi> <mn>2</mn> </msub> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>o</mi> <mi>t</mi> <mi>h</mi> <mi>e</mi> <mi>r</mi> <mi>w</mi> <mi>i</mi> <mi>s</mi> <mi>e</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
    Wherein, φ represents the stage residing for algorithm, random number between rand (0,1) represents 0 to 1, S1Represent algorithm and be in global spy Survey stage, S2Represent algorithm and be in the Local Search stage；If meet S1Condition, then into A8；It is unsatisfactory for then entering A9；
    A8, randomly select a strategy from strategy in following 3 and enter row variation：
    DE/rand/1
    <mrow> <msub> <mi>v</mi> <mrow> <mi>j</mi> <mi>i</mi> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>jr</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mi>i</mi> <mi>G</mi> </msubsup> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>jr</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>jr</mi> <mn>3</mn> </msub> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow>
    DE/rand/2
    <mrow> <msub> <mi>v</mi> <mrow> <mi>j</mi> <mi>i</mi> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>jr</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mi>i</mi> <mi>G</mi> </msubsup> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>jr</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>jr</mi> <mn>3</mn> </msub> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>F</mi> <mi>i</mi> <mi>G</mi> </msubsup> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>jr</mi> <mn>4</mn> </msub> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>jr</mi> <mn>5</mn> </msub> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow>
    DE/current-to-rand/1
    <mrow> <msub> <mi>v</mi> <mrow> <mi>j</mi> <mi>i</mi> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>jr</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>+</mo> <mi>K</mi> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>jr</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mi>i</mi> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>F</mi> <mi>i</mi> <mi>G</mi> </msubsup> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>jr</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>jr</mi> <mn>3</mn> </msub> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow>
    Wherein, G is evolutionary generation, r1、r2、r3、r4、r5∈{1,2,…,Np, and r1≠r2≠r3≠r4≠r5≠ i, vji,GFor G ties up element for the jth of i-th of variation individual in population, Respectively G is for population In r1、r2、r3、r4、r5Individual jth dimension element, K=0.5 is the fixed gain of DE/current-to-rand/1 strategies Constant；Fi GRepresent the gain constant of i-th of individual in G generations；
    A9, one strategy of random selection enters row variation from following 3 kinds of strategies：
    DE/best/1
    <mrow> <msub> <mi>v</mi> <mrow> <mi>j</mi> <mi>i</mi> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>x</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>G</mi> </mrow> <mrow> <mi>p</mi> <mi>b</mi> <mi>e</mi> <mi>s</mi> <mi>t</mi> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>F</mi> <mi>i</mi> <mi>G</mi> </msubsup> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>jr</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>jr</mi> <mn>3</mn> </msub> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow>
    DE/best/2
    <mrow> <msub> <mi>v</mi> <mrow> <mi>j</mi> <mi>i</mi> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>jr</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mi>i</mi> <mi>G</mi> </msubsup> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>G</mi> </mrow> <mrow> <mi>p</mi> <mi>b</mi> <mi>e</mi> <mi>s</mi> <mi>t</mi> </mrow> </msubsup> <mo>-</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>jr</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>F</mi> <mi>i</mi> <mi>G</mi> </msubsup> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>jr</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>jr</mi> <mn>3</mn> </msub> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow>
    DE/current-to-best/1
    <mrow> <msub> <mi>v</mi> <mrow> <mi>j</mi> <mi>i</mi> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mi>i</mi> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mi>i</mi> <mi>G</mi> </msubsup> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>G</mi> </mrow> <mrow> <mi>p</mi> <mi>b</mi> <mi>e</mi> <mi>s</mi> <mi>t</mi> </mrow> </msubsup> <mo>-</mo> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mi>i</mi> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>F</mi> <mi>i</mi> <mi>G</mi> </msubsup> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>jr</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>jr</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow>
    Wherein,It is G for the 0.5N in populationpThe jth dimension member of the optimum individual randomly selected in rand (0,1) individuals Element；
    A10, crossover operation generation test vector u is carried out by formula (7)ji,G+1：
    <mrow> <msub> <mi>u</mi> <mrow> <mi>j</mi> <mi>i</mi> <mo>,</mo> <mi>G</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <msub> <mi>v</mi> <mrow> <mi>j</mi> <mi>i</mi> <mo>,</mo> <mi>G</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mtable> <mtr> <mtd> <mrow> <mi>i</mi> <mi>f</mi> </mrow> </mtd> <mtd> <mrow> <mi>r</mi> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>&amp;le;</mo> <msubsup> <mi>C</mi> <mrow> <mi>R</mi> <mi>i</mi> </mrow> <mi>C</mi> </msubsup> <mi>o</mi> <mi>r</mi> <mi>j</mi> <mo>=</mo> <msub> <mi>j</mi> <mrow> <mi>r</mi> <mi>a</mi> <mi>n</mi> <mi>d</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mi>i</mi> <mo>,</mo> <mi>G</mi> </mrow> </msub> </mtd> <mtd> <mrow> <mi>o</mi> <mi>t</mi> <mi>h</mi> <mi>e</mi> <mi>r</mi> <mi>w</mi> <mi>i</mi> <mi>s</mi> <mi>e</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
    Wherein, uji,G+1Represent that G+1 ties up for the jth of i-th of test vector in population,It is each for crossover probability Dai Junhui is automatically updated, jrandFor 0 to the random integers between D, the dimension of D problem of representation；I-th was represented in G generations The crossover probability of body,Represent withFor average, with 0.1 for error normal distribution Random number, as G ＜ LP,As G ＞ LP,New individual caused by previous generation is taken to be successfully reserved conduct When of future generation individualAverage value, while when remember algebraically reach LP when, whenever produce a new generationThen delete earliest A generationThere may be the intangibility for being unsatisfactory for constraints, it is necessary to test to not meeting constraints after crossover operation Vector is handled, if the test vector through A10 step operations does not meet coding rule, returns to A7, until producing feasible solution；
    A11, decoding operate, first, it is the job sequence based on workpiece process that it is Sequence Transformed, which will to test individual first half,；Its It is secondary, the order of manufacturing procedure on every machine is determined by latter half sequence；Finally, determine that every procedure of all workpiece exists Between at the beginning of on processing machine and completion date；Based on job sequence and process constraint to each operation to allow processing earliest when Between be processed one by one, calculate every machine last work deadline, take its Maximal Makespan to be solved to be corresponding；
    A12, by comparing the adaptive value of object vector in test vector and current population select optimum individual
    <mrow> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>G</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <msub> <mi>u</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>G</mi> </mrow> </msub> </mtd> <mtd> <mtable> <mtr> <mtd> <mrow> <mi>i</mi> <mi>f</mi> </mrow> </mtd> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>u</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>&amp;le;</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>G</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>G</mi> </mrow> </msub> </mtd> <mtd> <mrow> <mi>o</mi> <mi>t</mi> <mi>h</mi> <mi>e</mi> <mi>r</mi> <mi>w</mi> <mi>i</mi> <mi>s</mi> <mi>e</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
    Wherein, xi,G+1Represent i-th individuals of the G+1 for population, ui,GAnd xi,GRepresent G for i-th of variation in population respectively For i-th of individual of population, f (x) is object function by individual and G；
    A13, judge whether to meet end condition, if it is satisfied, then preserving result and exiting, otherwise return to A5.
    Further, in the A8, Fi G=N (0.5,0.3), N (0.5,0.3) represent that average is 0.5, and standard deviation is 0.3 just State distribution random numbers.
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