第3讲-作业车间

西安交通大学管理学院第二讲作业车间调度授课教师：高杰西安交通大学管理学院工业工程系11/13/2015Job-shop Scheduling Problem1.Introduction2.JSP Mathematical / Graph Models2.1 Integer Programming Model2.2 Linear Programming Model2.3 Disjunctive Graph Model3.Conventional Heuristics for JSP3.1 Priority Dispatching Heuristics3.2 Shifting Bottleneck Heuristic3.3 Randomized Dispatching Heuristics1. Introduction☐Determining the operation sequences on the machines in order to minimize the makespan.⏹There are n different jobs to be scheduled on mdifferent machines.⏹Each job is composed of a set of operations and theoperation order of each job is prespecified.⏹Each operation is characterized by the requiredmachine and the fixed processing time.☐Makespan: find the sequence of jobsminimizing the maximum flow time.1. IntroductionAssumptions for JSP models1.There are a set of jobs and a set of machines.2.Each job consists of sequence of operations.3.Each machine can processes at most one operation at a time.4. A operation does not visit the same machines twice.5.The processing time of each operation has been determined.6.There are no precedence constraints among operations ofdifferent jobs.7.Any operation cannot be interrupted.8.Neither release times nor due dates are specified.Simple Example of JSPFor showing a feasible schedule, there are two Gantt charts : machine Gantt chart and job Gantt chart312332312232111323351233321Job 321Job Operations Operations Sequence Machine TimeProcessing M M M J J M M M J J M M M J J Example of a 3-Job 3-machine ProblemFig.13.15Machine Gantt chartO 33O 13O 23O 22O 12O 32O 31O 11O 21M 1M 2M 3121110987654321time O 13O 12O 11O 33O 31O 32O 22O 23O 21J 1J 2J 3121110987654321Fig.13.16Job Gantt chart timeJob-shop Scheduling Problem1. Introduction2.JSP Mathematical / Graph Models2.1 Integer Programming Model2.2 Disjunctive Graph Model2.3 Mathematical Model (n-job m-machine)2.4 Simple Example of JSP3. Conventional Heuristics for JSP4. Genetic Algorithms for JSP5. Gen-Tsujimura-Kubota’s Method6. Cheng-Gen-Tsujimura’s Method☐Notations⏹Indicesi , j : index of jobs , i , j = 1,2,…nh , k : index of machines, h , k = 1,2,…m⏹Parameters n : total number of jobsm : total number of machinesp jk : processing time of job j on machine k ⏹Decision Variables2.1 Integer Programming ModelM k M h . . .time o o jk t jk p jk t ikt ih o ih ik p ik M M Fig. Time chart for model⎩⎨⎧=otherwise0, machine on job precedes job if 1, k j i x ijk ⎩⎨⎧=otherwise0, job for machine on that precedes machine on processing 1, i k h a ihk t jk : completion time of job j on machine k2.1 Integer Programming Model☐IP formulation is discussed by Baker (1974).☐The objective function is to minimize the makespan.☐Constraint (2) ensuring the processing sequence of operationsfor each job corresponds to the prescribed order.☐Constraint (3) ensuring each machine can process only one jobat a time.{}()()()()()5,...,2,1,,...,2,1,,1or 04,...,2,1,,...,2,1,03,...,2,1,,,...,2,1)1(2,...,2,1,,,...,2,1)1(t.s.1max max min ,,11m k n j i x m k n i t m k h n i p x M t t m k h n i t a M p t t t ijk ik jk ijk ik jk ih ihk ik ik ik n i m k M ==≥==≥==≥-+-==≥-+-⎭⎬⎫⎩⎨⎧=≤≤≤≤⏹Baker, K.: Introduction to Sequencing and Scheduling , John Wiley,1974.2.3Disjunctive Graph Model⏹Balas, E.: "Machine sequencing via disjunctive graphs: an implicitenumeration algorithm”, Operations Research,Vol. 17, pp. 941-957, 1969.☐Disjunctive graph model for JSP:G= (N, A,E)N: Set of nodes representing all operationsA: Set of arcs connecting consecutive operations of the same jobE : Set of disjunctive arcs connecting operations⏹To be processed by the same machine, a disjunctive arc can besettled by either of its two possible orientations.☐A feasible schedule: Order of the operations on each machine.⏹To settle the orientation of the disjunctive arcs such that:1. the resulting graph is acyclic (there are no precedence conflictsbetween operations) and2. the length of the maximum weight path between the start and endnodes, i.e., makespan, is minimal2.3 Disjunctive Graph Model12345678910☐The nodes of N= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} correspond to operations, where nodes 0 and 10 are the dummy operations. The conjunctive arcs of A={(1,2), (2,3), (4,5), (5,6), (7,8), (8,9)} to precedence constraints on operations of same jobs.☐The disjunctive arcs :E1={(1,5), (5,9), (9,1)} to operations to be performed on machine 1E2={(4,2), (2,8), (8,4)} to operations to be performed on machine 2E3={(7,3), (3,6), (6,7)} to operations to be performed on machine 3Job-shop Scheduling Problem1. Introduction2. JSP Mathematical / Graph Models3.Conventional Heuristics for JSP3.1 Priority Dispatching Heuristics3.2 Shifting Bottleneck Heuristic3.3 Randomized Dispatching Heuristics3.4 Comparison with Different Heuristics4. Genetic Algorithms for JSP5. Gen-Tsujimura-Kubota’s Method6. Cheng-Gen-Tsujimura’s Method☐The relationship among active, semiactive, and nondelay schedules isshown as follows:All feasible schedules SemiactiveActiveNondela y☐When considering a regular performance measure , the optimal schedule iswithin the set of active schedules . The nondelay schedules are smaller than the active schedules , but there is no guarantee that the former will contain an optimum .*Opt .⏹Baker, k.,Introduction to Sequencing and Scheduling , John and Wiley & Sons, New York, 1974.⏹French, S , “Sequencing and Scheduling,”In Mathematics and Its Applications : Ellis Horwood Limited, 1982. ⏹Gen, M. and Cheng, R., Genetic Algorithms & Engineering Design , John Wiley & Sons, 1997.☐Priority rules are probably the most frequently applied heuristics for solving JSP because of their ease of implementation and their low time complexity.☐The algorithms of Giffer and Thompson [OR -1960]can be considered as the common basis of all priority-rule-based heuristics.3.1Priority Dispatching Heuristicsd completede be could operation which at time earlist the started be could operation which at time earlist the given a to ing correspond stage at operation e schedulabl of set the operationsscheduled containing schedule partial a t i t i t t t S i φS i σPS t S t PS ∈=∈===,☐Notation⏹Giffler, B. and G. Thompson : "Algorithms for solving production scheduling problems”, Operations Research , Vol. 8, No. 4, pp. 487-503, 1960.☐Shifting bottleneck heuristic is probably the most powerful procedure among all heuristics for JSP.It sequences the machines one by one, successively,taking each time the machine identified as a bottleneck among machines not yet sequenced.Every time after a new machine is sequenced, all previously established sequences are locally re-optimized.☐Algorithm: Shifting Bottleneck Heuristicstep 1.Identify a bottleneck machine m among the machines k∈M, M0 and sequence it optimally. Set M0←M0⋃{m}.step 2.Re-optimize the sequence of each critical machine k∈M0in turn,while keeping the other sequences fixed. Then if M0=M, stop; otherwise go to step 1.where, M0 is the set of machine already sequenced.o 1,1o 1,2o 1,3o 1,4o 2,1o 2,2o 2,3o 2,4o 3,1o 3,2o 3,3o 3,4o 4,1o 4,2o 4,3o 4,4M 1M 2M 3M 4t0130120110100908070605020304010140150Figure. Illustration of Bottleneck Shifting☐Rules of five heuristic methods:⏹Colorni A., M. Dorigo: “Ant system for Job-shop Scheduling”, JORBEL-BelgianJournal of Operations Research, Statistics and Computer Science, 34, 1, pp.39-53, 1994.1. SPT: Select the operation with the shortest processing time(SPT).2. LPT: Select the operation with the longest processing time(LPT).3. LRT: Select the operation belonging to the job with the longestremaining processing time(LRT).4. SRT: Select the operation belonging to the job with the shortestremaining processing time(SRT).5. LRM: Select the operation belonging to the job with the longestremaining processing time excluding the operation underconsideration (LRM).1. SPT (shortest processing time)select the operation with the shortest processing time. We firstly add some new definition for the procedure.VariablesO l: a set of operations {o ikj}at the l th steppred(oikj ):the predecessor of the operation oikjsucc(oikj ):the successor of the operation oikjN i: number of operations for job iN:total number of operations S:schedule list for N operations∑==niiN N1Procedure of SPT rule: Generate a feasible scheduleprocedure :SPT rule to generate a feasible schedule endoutput else then if argmin to for begin:output :input );( list schedule };\ );(succ \ )(succ ;)()( };{ {;)( };,)(pred {)( list schedule operations of number total ,time processing l S o O O o o O O φo o l S l S O o p o N l φS i φo o O l S N p ikj l l ikjikj l l ikj ikj l ikj ikj ikj ikj ikj ikj *+**+***←⋃←≠⋃←∈←=←∀=←11111Example of SPT ruleTable 3.2 Processing Time, Machine Sequenceo 111o 123o 323o 312o 222o 233o 211o 132o 331)22: ,18: ,8:()9: ,20: ,15:()12: ,21: ,16:(331323312323322221121321231111o o o J o o o J o o o J ===J i =(o ikj : P ikj , …)M 1M 3M 2M 1M 2M 3M 2M 3M 1p ikj123Operations J 1162112J 215209J 381822M j 123Example of SPT rulel O lp ikjo ikj *S (l )1{o 111, o 211, o 312}p 111=16, p 211=15, p 312=8o 312{o 312}2{o 111, o 211, o 323}p 111=16, p 211=15, p 323=18o 211{o 312, o 211}3{o 111, o 222, o 323}p 111=16, p 222=20, p 323=18o 111{o 312, o 211, o 111}4{o 123, o 222, o 323}p 123=21, p 222=20, p 323=18o 323{o 312, o 211, o 111, o 323}5{o 123, o 222, o 331}p 123=21, p 222=20, p 331=22o 222{o 312, o 211, o 111o 323, o 222}6{o 123, o 233, o 331}p 123=21, p 233=9, p 331=22o 233{o 312, o 211, o 111, o 323, o 222, o 233}7{o 123, o 331}p 123=21, p 331=22o 123{o 312, o 211, o 111, o 323, o 222, o 233, o 123}8{o 132, o 331}p 132=12, p 331=22o 132{o 312, o 211, o 111, o 323, o 222, o 233, o 123, o 132}9{o 331}p 331=22o 331{o 312, o 211, o 111, o 323, o 222, o 233, o 123, o 132, o 331}ScheduleS ={o 312, o 211, o 111, o 323, o 222, o 233, o 123, o 132, o 331}Trace table by SPT ruleo 111o 123o 323o 312o 222o 233o 211o 132o 331Example dataOperationsp ikj123J 1162112J 215209J 381822Example of SPT rule77}35 ,77 ,65,44 ,53 ,62 ,13 ,51 ,8max{ },,,,,,,,max{}{max 331132123233222323111211312,,====t t t t t t t t t t t ikj jk i M Schedule : S ={o 312, o 211, o 111, o 323, o 222, o 233, o 123, o 132, o 331}S ={o 312(t 312-t 312F ), o 211(t 211-t 211F ), o 111(t 111-t 111F ), o 323(t 323-t 323F ), o 222(t 222-t 222F ), o 233(t 233-t 233F ),o 123(t 123-t 123F ), o 132(t 132-t 132F ), o 331(t 331-t 331F )}= {o 312:(0-8), o 211(0-15), o 111(15-31), o 323(8-26), o 222(15-35),o 233(35-44), o 123(44-65), o 132(65-77), o 331(31-53)}J 1J 2J 3Makespan : 77 (units)M 1M 2M 353o 312o 111o 323o 222o 233o 211o 123o 132o 331time7760653515202631447040852. LPT : select the operation with the longest processing time .procedure : LPT rule to generate a feasible schedule endoutput else then if argmax to for begin:output :input ;)( list schedule };\ );(succ \ )(succ ;)()( };{ {;)( };,)(pred { )( list scheduleoperations of number total , time processing l S o O O o o O O φo o l S l S O o p o N l φS i φo o O l S N p ikj l l ikj ikj l l ikj ikj l ikj ikj ikj ikj ikj ikj *+**+***←⋃←≠⋃←∈←=←∀=←11111Example of LPT ruleScheduleTrace table by LPT rulelO lp ikjo ikj *S (l )1{o 111, o 211, o 312}p 111=16, p 211=15, p 312=8o 111{o 111}2{o 123, o 211, o 312}p 123=21, p 211=15, p 312=8o 123{o 111, o 123}3{o 132, o 211, o 312}p 132=12, p 211=15, p 312=8o 211{o 111, o 123, o 211}4{o 132, o 222, o 312}p 132=12, p 222=20, p 312=8o 222{o 111, o 123, o 211, o 222}5{o 132, o 233, o 312}p 132=12, p 233=9, p 312=8o 132{o 111, o 123, o 211, o 222, o 132}6{o 233, o 312}p 233=9, p 312=8o 233{o 111, o 123, o 211, o 222, o 132, o 233}7{o 312}p 312=8o 312{o 111, o 123, o 211, o 222, o 132, o 233, o 312}8{o 323}p 323=18o 323{o 111, o 123, o 211, o 222, o 132, o 233, o 312, o 323}9{o 331}p 331=22o 331{o 111, o 123, o 211, o 222, o 132, o 233, o 312, o 323, o 331}S ={o 111, o 123, o 211, o 222, o 132, o 233, o 312, o 323, o 331}o 111o 123o 323o 312o 222o 233o 211o 132o 331Example dataOperationsp ikj123J 1162112J 215209J 3818224 Comparison with Different HeuristicsExample of LPT rule100}001 ,87 ,8 ,06 ,36 ,15 ,13 ,73 ,16max{ }, , , , , , , ,max{}{max 331323312233132222211123111,,====t t t t t t t t t t t ikj jk i M Schedule : S ={o 111, o 123, o 211, o 222, o 132, o 233, o 312, o 323, o 331}S ={o 111(t 111-t 111F ), o 123(t 123-t 123F ), o 211(t 211-t 211F ), o 222(t 222-t 222F ), o 132(t 132-t 132F ), o 233(t 233-t 233F ),o 312(t 312-t 312F ), o 323(t 323-t 323F ), o 331(t 331-t 331F ) }={o 111(0-16), o 123(16-37), o 211(16-31), o 222(31-51), o 132(51-63)o 233(51-60), o 312(0-8), o 323(60-78), o 331(78-100)}Makespan : 100(units)M 1M 2M 316313751859095100timeo 3317860706325o 132o 233o 312o 3238o 111o 123o 211o 22245J 1J 2J 3procedure : LRT rule to generate a feasible schedule3. LRT : select the operation belonging to the job with thelongest remaining processing time.endoutput else then if argmax to for begin:output :input );( list schedule } ;\ );(succ \ )(succ ;)()( };{ {;)( }; ,)(pred { )( list schedule ,l S o O O o o O O φo o l S l S O o r o N l φS i φo o O l S N r ikjl l ikjikj l l ikj ikj l ikj ikj ikjikj ikj ikj *+**+***←⋃←≠⋃←∈←=←∀=←11111 operations of number total time processing remaining r ikj : remaining processing time of J i corresponding tooperation o ikj jN i j k i ikj ikj i p p p r ,,,1,,+++=+ M jIn this case we assume all of remaining operations for job i are assigned on the same machine M j .ikjp p i,N ijtimet i,k +1,j+1o t i,N i , jo ikjo i,k +1,jt ikjp i,k +1,j …i,N i jTime chart for modelExample of LRT ruleScheduleTrace table by LRT rulelO lr ikjo ikj *S (l )1{o 111, o 211, o 312}r 111=49, r 211=44, r 312=48o 111{o 111}2{o 123, o 211, o 312}r 123=33, r 211=44, r 312=48o 312{o 111, o 312}3{o 123, o 211, o 323}r 123=33, r 211=44, r 323=40o 211{o 111, o 312, o 211}4{o 123, o 222, o 323}r 123=33, r 222=29, r 323=40o 323{o 111, o 312, o 211, o 323}5{o 123, o 222, o 331}r 123=33, r 222=29, r 331=22o 123{o 111, o 312, o 211, o 323, o 123}6{o 132, o 222, o 331}r 122=12, r 222=29, r 331=22o 222{o 111, o 312, o 211, o 323, o 123, o 222}7{o 132, o 233, o 331}r 133=12, r 233=9, r 331=21o 331{o 111, o 312, o 211, o 323, o 123, o 222, o 331}8{o 132, o 233}r 132=12, r 233=9o 132{o 111, o 312, o 211, o 323, o 123, o 222, o 331, o 132}9{o 233}r 233=9o 233{o 111, o 312, o 211, o 323, o 123, o 222, o 331, o 132, o 233}S ={o 111, o 312, o 211, o 323, o 123, o 222, o 331, o 132, o 233}o 111o 123o 323o 312o 222o 233o 211o 132o 331Example dataOperationsp ikj123J 1162112J 215209J 381822Example of LRT ruleSchedule : S ={o 111, o 312, o 211, o 323, o 123, o 222, o 331, o 132, o 233}63}06 ,36 ,35 ,15 ,74 ,62 ,13 ,8 ,16max{ }, , , , , , , ,max{}{max 233132331222123323211312111,,====t t t t t t t t t t t ikj jk i MS ={o 111(t 111-t 111F ), o 312(t 312-t 312F ), o 211(t 211-t 211F ), o 323(t 323-t 323F ), o 123(t 123-t 123F ),o 222(t 222-t 222F ), o331(t 331-t 331F ), o132(t 132-t 132F ), o233(t 233-t 233F ) }={o 111(0-16), o 312(0-8), o 211(16-31), o 323(8-26), o 123(26-47),o 222(31-51), o 331(31-53), o 132(51-63), o 233(51-60)}Makespan : 63(units)Gantt chart of the scheduleM 1M 2M 38165147o 323o 123o 111o 312o 211o 222o 331o 132o 23331354063time6026J 1J 2J 3Example dataOperationsp ikj 123J 1162112J 215209J 381822procedure : SRT rule to generate a feasible schedule4. SRT : select the operation belonging to the job with theshortest remaining processing time.r ikj : remaining processing time of J i correspondingto operation o ikj jN i j k i ikj ikj i p p p r ,,,1,,+++=+ M jIn this case we assume all of remaining operations for job i are assigned on the same machine M j .ikjp p i,N i jtimet i,k +1,j+1o t i,N i , jo ikjo i,k +1,jt ikjp i,k +1,j …i,N i jTime chart for modelendoutput else then if argmin to for begin:output :input ; )( list schedule };\ );(succ \ )(succ ;)()( };{ {;)( };,)(pred { )( list schedule N operations of number total time remaining l S o O O o o O O φo o l S l S O o r o N l φS i φo o O l S r ikj l l ikj ikj l l ikj ikj l ikj ikj ikjikj ikj ikj *+**+***←⋃←≠⋃←∈←=←∀=←11111 ,Example of SRT ruleScheduleTrace table by SRT rulelO lr ikjo ikj *S (l )1{o 111, o 211, o 312}r 111=49, r 211=44, r 312=48o 211{o 211}2{o 111, o 222, o 312}r 111=49, r 222=29, r 312=48o 222{o 211, o 222}3{o 111, o 233, o 312}r 111=49, r 233=9, r 312=48o 233{o 211, o 222, o 233}4{o 111, o 312}r 111=49, r 312=48o 312{o 211, o 222, o 233, o 312}5{o 111, o 323}r 111=49, r 323=40o 323{o 211, o 222, o 233, o 312, o 323}6{o 111, o 331}r 111=49, r 331=22o 331{o 211, o 222, o 233, o 312, o 323, o 331}7{o 111}r 111=49o 111{o 211, o 222, o 233, o 312, o 323, o 331, o 111}8{o 123}r 123=33o 123{o 211, o 222, o 233, o 312, o 323, o 331, o 111, o 123}9{o 132}r 132=12o 132{o 211, o 222, o 233, o 312, o 323, o 331, o 111, o 123, o 132}S ={o 211, o 222, o 233, o 312, o 323, o 331, o 111, o 123, o 132}o 111o 123o 323o 312o 222o 233o 211o 132o 331Example dataOperationsp ikj123J 1162112J 215209J 381822Example of SRT rule97}79 ,58 ,46 ,84 ,62 ,8 ,44 ,53 ,15max{ }, , , , , , , ,max{}{max 132123111331323312233222211,,====t t t t t t t t t t t ikj jk i M Schedule : S ={o 211, o 222, o 233, o 312, o 323, o 331, o 111, o 123, o 132}S = {o 211(t 211-t 211F ), o 222(t 222-t 222F ), o 233(t 233-t 233F ), o 312(t 312-t 312F ),o 323(t 323-t 323F ), o 331(t 331-t 331F ), o 111(t 111-t 111F ), o 123(t 123-t 123F ), o 132(t 132-t 132F ) }={o 211(0-15), o 222(15-35), o 233(35-44), o 312(0-8), o 323(8-26)o 331(26-48), o 111(48-64), o 123(64-85), o 132(85-97)}Makespan 97(units)Gantt chart of the scheduleM 1M 2M 3848o 211o 222o 233o 312o 323o 331o 111o 123o 13215202630354044556064707580859097timeJ 1J 2J 3Example dataOperationsp ikj 123J 1162112J 215209J 381822procedure : LRM rule to generate a feasible schedule5. LRM : select the operation belonging to the job with theshortest remaining processing time.e ikj : remaining processing time of J i correspondingto operation o ikj M jIn this case we assume all of remaining operations for job i are assigned on the same machine M j .ikjp p i,N i jtimet i,k +1,j+1o t i,N i , jo ikjo i,k +1,jt ikjp i,k +1,j …i,N i jjN i j k i ikj i p p e ,,,1,,++=+ endoutput else then if argmin to for begin:output :input ; )( list schedule ;\ );(succ \ )(succ ;)()( };{ ;)( };,)(pred { )( list schedule operations of number tatal exclude for time processing remainning l S o O O o o O O φo o l S l S O o e o Nl φS i φo o O l S N p J e ikj l l ikj ikj l l ikj ikj l ikj ikj ikjikj ikj ikj ikj *+**+***←⋃←≠⋃←∈←=←∀=←111i 11 Time chart for modelExample of LRM ruleScheduleTrace table by LRM rulelO le ikjo ikj *S (l )1{o 111, o 211, o 312}e 111=33, e 211=29, e 312=40o 312{o 312}2{o 111, o 211, o 323}e 111=33, e 211=29, e 323=22o 111{o 312, o 111}3{o 123, o 211, o 323}e 123=12, e 211=29, e 323=22o 211{o 312, o 111, o 211}4{o 123, o 222, o 323}e 123=12, e 222=9, e 323=22o 323{o 312, o 111, o 211, o 323}5{o 123, o 222, o 331}e 123=12, e 222=9, e 331=0o 123{o 312, o 111, o 211, o 323, o 123}6{o 132, o 222, o 331}e 132=0, e 222=9, e 331=0o 222{o 312, o 111, o 211, o 323, o 123, o 222}7{o 132, o 233, o 331}e 132=0, e 233=0, e 331=0o 331{o 312, o 111, o 211, o 323, o 123, o 222, o 331}8{o 132, o 233}e 132=0, e 233=0o 132{o 312, o 111, o 211, o 323, o 123, o 222, o 331, o 132}9{o 233}e 233=0o 233{o 312, o 111, o 211, o 323, o 123, o 222, o 331, o 132, o 233}S ={o 312, o 111, o 211, o 323, o 123, o 222, o 331, o 132, o 233}o 111o 123o 323o 312o 222o 233o 211o 132o 331Example dataOperationsp ikj123J 1162112J 215209J 381822Example of LRM ruleSchedule : S ={o 312, o 111, o 211, o 323, o 123, o 222, o 331, o 132, o 233}S ={o 312(t 312-t 312F ), o 111(t 111-t 111F ), o 211(t 211-t 211F ), o 323(t 323-t 323F ), o 123(t 123-t 123F ), o 222(t 222-t 222F ), o 331(t 331-t 331F ), o 132(t 132-t 132F ), o 233(t 233-t 233F ) }={o 312(0-8), o 111(0-16), o 211(16-31), o 323(8-26), o 123(26-47)o 222(31-51), o 331(31-53), o 132(51-63), o 233(51-60)}63}06 ,36 ,35 ,15 ,74 ,62 ,13 ,16 ,8max{ }, , , , , , ,,max{}{max 233132331222123323211111312,,====t t t t t t t t t t t ikj jk i M Makespan : 63(units)Gantt chart of the scheduleM 1M 2M 38165147o 323o 123o 111o 312o 211o 222o 331o 132o 23331354063time6026J 1J 2J 3Example dataOperationsp ikj123J 1162112J 215209J 381822ComparisonTable 13.22Comparison results with different heuristic method Heuristic method Schedule Makespan t M SPT{o312, o211, o111, o323, o222, o233, o123, o132, o331}77LPT{o111, o123, o211, o222, o132, o233, o312, o323, o331}100LRT{o111, o312, o211, o323, o123, o222, o331, o132, o233}63SRT{o211, o222, o233, o312, o323, o331, o111, o123, o132}97LRM{o312, o111, o211, o323, o123, o222, o331, o132, o233}63A quick comparison of the solutions reveals that the heuristic Conclusion:methods LRT and LRM outperform other heuristic methods withlower makespan.☐ComparisonTable 13.23Comparison results with different heuristic method by different dataThe comparisons of the solutions (using different size ofproblems) reveal that the heuristic methods LRT and LRM outperform other heuristic methods with lower makespan.Conclusion:Heuristic methodsMakespan t M3x34x46x610x510x1015x10SPT 773188392018561719LPT 1003447988918601519LRT 633376773514511314SRT 9734094121021241806LRM633595776314931313⏹The data come from OR lib. (/jeb/orlib/jobshopinfo.html )。