20:
416:
The flow shop contains n jobs simultaneously available at time zero and to be processed by two machines arranged in series with unlimited storage in between them. The processing time of all jobs are known with certainty. It is required to schedule n jobs on machines so as to minimize makespan. The
225:-th operation. Jobs can be executed in any order, however. Problem definition implies that this job order is exactly the same for each machine. The problem is to determine the optimal such arrangement, i.e. the one with the shortest possible total job execution makespan.
220:
Operations within one job must be performed in the specified order. The first operation gets executed on the first machine, then (as the first operation is finished) the second operation on the second machine, and so on until the
355:
As presented by Garey et al. (1976), most of extensions of the flow-shop-scheduling problems are NP-hard and few of them can be solved optimally in O(nlogn); for example, F2|prmu|C
335:
281:
187:
159:
703:
579:
217:-th machine. No machine can perform more than one operation simultaneously. For each operation of each job, execution time is specified.
95:-th machine. No machine can perform more than one operation simultaneously. For each operation of each job, execution time is specified.
594:
696:
233:
The sequencing problem can be stated as determining a sequence S such that one or several sequencing objectives are optimized.
861:
189:" is a 3-machines flow-shop problem with unit processing times, where the goal is to minimize the maximum completion time.
79:– the total length of the schedule (that is, when all the jobs have finished processing). In the specific variant known as
742:
732:
856:
689:
851:
737:
102:
where there is strict order of all operations to be performed on all jobs. Flow-shop scheduling may apply as well to
727:
758:
830:
804:
794:
712:
122:
43:
799:
574:
Malakooti, B (2013). Operations and
Production Systems with Multiple Objectives. John Wiley & Sons.
115:
592:
Garey, M. R.; Johnson, D. S.; Sethi, Ravi (1976). "The complexity of flowshop and jobshop scheduling".
366:
Taillard provides substantial benchmark problems for scheduling flow shops, open shops, and job shops.
294:
240:
763:
624:
Johnson, S. M. (1954). "Optimal two-and three-stage production schedules with setup times included".
547:
31:
825:
773:
552:
383:
99:
39:
809:
165:
652:
575:
410:
387:
360:
664:
633:
603:
535:
379:
344:
134:
35:
19:
413:
but for general case there is no algorithm that guarantee the optimality of the solution.
375:
118:
order of the jobs on the resources is the same for each subsequent step of processing.
845:
668:
103:
681:
514:(i) The job in set1 go first in the sequence and they go in increasing order of p
374:
The proposed methods to solve flow-shop-scheduling problems can be classified as
417:
Johnson's rule for scheduling jobs in two-machine flow shop is given below.
107:
637:
789:
607:
76:
534:
A detailed discussion of the available solution methods are provided by
456:
are processing times of job j on machine 1 and machine 2 respectively.
75:
machines with varying processing power, while trying to minimize the
18:
685:
343:
detailed discussion of performance measurement can be found in
110:
designs. A special type of flow-shop scheduling problem is the
71:
of varying processing times, which need to be scheduled on
531:
This type schedule is referred as SPT(1)–LPT(2) schedule.
448:
is the processing time of job i on machine 2. Similarly, p
129:
in the first field. For example, the problem denoted by "
123:
three-field notation for optimal-job-scheduling problems
521:(ii) The jobs in set2 follow in decreasing order of p
297:
243:
168:
137:
46:. In a general job-scheduling problem, we are given
818:
782:
751:
720:
444:is the processing time of job i on machine 1 and p
329:
275:
181:
153:
213:-th operation of the job must be executed on the
91:-th operation of the job must be executed on the
420:In an optimal schedule, job i precedes job j if
174:
697:
8:
467:be the processing time of job j on machine 1
570:
568:
704:
690:
682:
653:"Benchmarks for basic scheduling problems"
98:Flow-shop scheduling is a special case of
619:
617:
474:the processing time of job j on machine 2
321:
308:
296:
267:
254:
242:
173:
167:
142:
136:
657:European Journal of Operational Research
492:Form set2 containing all the jobs with p
482:Form set1 containing all the jobs with p
564:
229:Sequencing performance measurements (γ)
125:, the flow-shop variant is denoted by
7:
525:(LPT). Ties are broken arbitrarily.
626:Naval Research Logistics Quarterly
595:Mathematics of Operations Research
351:Complexity of flow-shop scheduling
14:
409:can be solved optimally by using
359:can be solved optimally by using
330:{\displaystyle \sum (w_{i})T_{i}}
276:{\displaystyle \sum (w_{i})F_{i}}
205:jobs. Each job contains exactly
112:permutation flow-shop scheduling
511:Form the sequence as follows:
314:
301:
260:
247:
16:Class of computational problem
1:
651:Taillard, E. (January 1993).
669:10.1016/0377-2217(93)90182-M
83:, each job contains exactly
878:
508:may be put in either set.
459:For Johnson's algorithm:
182:{\displaystyle C_{\max }}
23:Flow Shop Ordonnancement
831:Truthful job scheduling
783:Optimization objectives
862:Engineering management
713:Optimal job scheduling
638:10.1002/nav.3800010110
394:Minimizing makespan, C
331:
277:
183:
155:
154:{\displaystyle p_{ij}}
44:optimal job scheduling
24:
478:Johnson's algorithm:
332:
291:(Average) Tardiness,
278:
237:(Average) Flow time,
184:
156:
114:problem in which the
42:. It is a variant of
22:
608:10.1287/moor.1.2.117
548:Open-shop scheduling
295:
241:
166:
135:
81:flow-shop scheduling
32:optimization problem
28:Flow-shop scheduling
857:Workflow technology
826:Interval scheduling
553:Job-shop scheduling
384:heuristic algorithm
100:job-shop scheduling
40:operations research
852:Optimal scheduling
819:Other requirements
743:Unrelated machines
733:Identical machines
327:
273:
179:
151:
25:
839:
838:
580:978-1-118-58537-5
500:, the jobs with p
388:genetic algorithm
193:Formal definition
106:facilities as to
64:, ...,
869:
752:Multi-stage jobs
738:Uniform machines
706:
699:
692:
683:
673:
672:
648:
642:
641:
621:
612:
611:
589:
583:
572:
380:branch and bound
370:Solution methods
336:
334:
333:
328:
326:
325:
313:
312:
282:
280:
279:
274:
272:
271:
259:
258:
209:operations. The
188:
186:
185:
180:
178:
177:
160:
158:
157:
152:
150:
149:
121:In the standard
87:operations. The
36:computer science
877:
876:
872:
871:
870:
868:
867:
866:
842:
841:
840:
835:
814:
778:
747:
716:
710:
679:
677:
676:
650:
649:
645:
623:
622:
615:
591:
590:
586:
573:
566:
561:
544:
524:
517:
507:
503:
499:
495:
489:
485:
473:
466:
455:
451:
447:
443:
437:
433:
429:
425:
408:
404:
399:
397:
376:exact algorithm
372:
358:
353:
317:
304:
293:
292:
288:
263:
250:
239:
238:
231:
195:
169:
164:
163:
138:
133:
132:
69:
63:
56:
17:
12:
11:
5:
875:
873:
865:
864:
859:
854:
844:
843:
837:
836:
834:
833:
828:
822:
820:
816:
815:
813:
812:
807:
802:
797:
792:
786:
784:
780:
779:
777:
776:
771:
766:
761:
759:Parallel tasks
755:
753:
749:
748:
746:
745:
740:
735:
730:
728:Single machine
724:
722:
721:One-stage jobs
718:
717:
711:
709:
708:
701:
694:
686:
675:
674:
663:(2): 278–285.
643:
613:
602:(2): 117–129.
584:
563:
562:
560:
557:
556:
555:
550:
543:
540:
529:
528:
527:
526:
522:
519:
515:
509:
505:
501:
497:
493:
490:
487:
483:
476:
475:
471:
468:
464:
453:
449:
445:
441:
435:
431:
427:
423:
411:Johnson's Rule
406:
402:
398:
395:
392:
371:
368:
361:Johnson's Rule
356:
352:
349:
341:
340:
337:
324:
320:
316:
311:
307:
303:
300:
289:
286:
283:
270:
266:
262:
257:
253:
249:
246:
230:
227:
194:
191:
176:
172:
148:
145:
141:
67:
61:
54:
15:
13:
10:
9:
6:
4:
3:
2:
874:
863:
860:
858:
855:
853:
850:
849:
847:
832:
829:
827:
824:
823:
821:
817:
811:
808:
806:
803:
801:
798:
796:
793:
791:
788:
787:
785:
781:
775:
772:
770:
767:
765:
762:
760:
757:
756:
754:
750:
744:
741:
739:
736:
734:
731:
729:
726:
725:
723:
719:
714:
707:
702:
700:
695:
693:
688:
687:
684:
680:
670:
666:
662:
658:
654:
647:
644:
639:
635:
631:
627:
620:
618:
614:
609:
605:
601:
597:
596:
588:
585:
581:
577:
571:
569:
565:
558:
554:
551:
549:
546:
545:
541:
539:
537:
532:
520:
513:
512:
510:
491:
481:
480:
479:
469:
462:
461:
460:
457:
440:. Where as, p
439:
418:
414:
412:
405:and F3|prmu|C
393:
391:
389:
385:
381:
377:
369:
367:
364:
362:
350:
348:
346:
338:
322:
318:
309:
305:
298:
290:
284:
268:
264:
255:
251:
244:
236:
235:
234:
228:
226:
224:
218:
216:
212:
208:
204:
201:machines and
200:
192:
190:
170:
162:
146:
143:
139:
128:
124:
119:
117:
113:
109:
105:
101:
96:
94:
90:
86:
82:
78:
74:
70:
60:
53:
49:
45:
41:
37:
33:
29:
21:
768:
678:
660:
656:
646:
632:(1): 61–68.
629:
625:
599:
593:
587:
533:
530:
477:
458:
430:} < min{p
421:
419:
415:
400:
373:
365:
354:
342:
232:
222:
219:
214:
210:
206:
202:
198:
196:
130:
126:
120:
111:
97:
92:
88:
84:
80:
72:
65:
58:
51:
47:
27:
26:
285:Makespan, C
846:Categories
810:Throughput
559:References
197:There are
116:processing
104:production
805:Tardiness
795:Earliness
769:Flow shop
764:Open shop
536:Malakooti
401:F2|prmu|C
345:Malakooti
299:∑
245:∑
108:computing
800:Lateness
790:Makespan
774:Job shop
715:problems
542:See also
538:(2013).
386:such as
378:such as
347:(2013).
77:makespan
57:,
578:
496:> p
486:< p
30:is an
518:(SPT)
470:and p
463:Let p
452:and p
422:min{p
50:jobs
576:ISBN
382:and
339:....
38:and
665:doi
634:doi
604:doi
504:= p
407:max
403:max
396:max
357:max
287:max
175:max
131:F3|
34:in
848::
661:64
659:.
655:.
628:.
616:^
598:.
567:^
523:2j
516:1j
506:2j
502:1j
498:2j
494:1j
488:2j
484:1j
472:2j
465:1j
454:2j
450:1j
446:2i
442:1i
436:2i
434:,p
432:1j
428:2j
426:,p
424:1i
390:.
363:.
705:e
698:t
691:v
671:.
667::
640:.
636::
630:1
610:.
606::
600:1
582:.
438:}
323:i
319:T
315:)
310:i
306:w
302:(
269:i
265:F
261:)
256:i
252:w
248:(
223:m
215:i
211:i
207:m
203:n
199:m
171:C
161:|
147:j
144:i
140:p
127:F
93:i
89:i
85:m
73:m
68:n
66:J
62:2
59:J
55:1
52:J
48:n
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
