Solving Constrained Flow-Shop Scheduling. Problems with Three Machines

Transcription

1 It J Cotemp Math Sceces, Vol 5, 2010, o 19, Solvg Costraed Flow-Shop Schedulg Problems wth Three Maches P Pada ad P Rajedra Departmet of Mathematcs, School of Advaced Sceces, VIT Uversty, Vellore , Ida pada61@redffmalcom Abstract A ew method s proposed to obta a optmal schedulg sequece for flow-shop schedulg problems volvg trasportato tme, break dow tme ad weghts of jobs (costraed flow-shop schedulg problems) wth 3-maches The proposed method s very smple ad easy to uderstad ad also, provdes a mportat tool for decso makers whe they desg a schedulg for costraed flow-shop schedulg problems wth 3 maches The proposed method s llustrated wth help of umercal examples Mathematcs Subject Classfcato: 90B35, 68M20 Keywords: Flow-shop schedulg problem; Trasportato tme; Weghts of jobs; Optmal sequece 1 Itroducto Now-a-days, the decso makers for the maufacturg plat must fd a way to successfully maage resources order to produce products the most effcet way the complex maufacturg settg, wth multple les of products, each requrg may dfferet steps ad maches for completo Also, they eed to desg a producto schedule that promotes o-tme delvery as well as mmzes the flow tme of a product Out of these cocers grew a area of studes kow as the schedulg problems I the schedulg problem, oe of the cetral tasks hgh-level sythess s the problem of determg the order whch the operatos the behavoural descrpto wll execute It volves solvg for the optmal schedule uder varous objectves, dfferet mache evromets ad characterstcs of the jobs The umber of possble schedules of the flow-shop m schedulg problem volvg jobs ad m maches s (!) The optmal soluto

2 922 P Pada ad P Rajedra to the problem s to fd the sequece of jobs o each mache order to complete all the jobs o all the maches the mmum total tme provded each job s processed o maches 1, 2, 3,, m that order The geeral flow-shop schedulg problem s NP-hard The schedulg problem practcally depeds upo three mportat factors amely, job trasportato tme whch cludes loadg tme, movg tme ad uloadg tme etc, relatve mportace of a job over aother job ad breakdow mache tme (due to the falure of electrc curret, the o-supply of raw materal or other techcal terruptos) These three factors were separately studed by may researchers [ 1-3, 5-8] Chadramoul [4] proposed a heurstc algorthm for flow-shop schedulg problem wth 3-maches volvg trasportato tme, break dow tme ad weghts of jobs to fd a optmal or ear optmal sequece I ths paper, we propose a ew method for flow-shop schedulg problems volvg trasportato tme, break dow tme ad weghts of jobs ( costraed flow-shop schedulg problems) wth 3-maches to obta a optmal sequece The proposed method s very smple ad easy to uderstad ad also, provdes a mportat tool for decso makers whe they desg a schedule for costraed flow-shop schedulg problems wth 3-maces Wth the help of the umercal examples, the proposed method s llustrated 2 Mache flow-shop problem Cosder the followg costraed flow-shop problem wth 3-maches whch ca be stated as follows: (a) Let - job be processed through three maches A, B ad C the order ABC (b) Let deote the job S where S s a arbtrary sequece (c) All jobs are avalable for processg at tme zero (d) Let each job be completed through the same producto stage, that s, ABC, other words, passg s ot allowed the flow shop (e) Let A, B ad C deote the processg tme of job o the mache A, B ad C respectvely (f) Let t ad g deote the trasportato tme of job from A to B ad from B to C respectvely (g) Let job be assged wth a weght w accordg to ts relatve mportace for performace the gve sequece (h) The performace measure studed weghted mea flow tme defed by F w = 1 = f f = 1, where f s flow tme of th job

3 Solvg costraed flow-shop schedulg problems 923 () Let the break dow terval ( a, b) s already kow to us, that s, determstc ature The break dow terval legth b a whch s kow The, our am s to fd out the optmal sequece of jobs so as to mmze the total elapsed tme The above stated problem (P) the tabular form may be stated as follows: Maches wth Processg tmes ad trasportg tmes Weght of s A t B g C W 1 A 1 1 B 1 g 1 C 1 W 1 2 A 2 2 B 2 g 2 C 2 W 2 M M M M M M M A t B g C W Let us assume that the problem (P) satsfes ay oe of the followg structural codtos volvg the processg tme ad trasportato tme of jobs hold Structural codtos: 1 Mmum { A } Maxmum { t + B} or Mmum C + g } Maxmum t + B } { { 2 Mmum { A + t } Maxmum { B } or Mmum C + g } Maxmum B } { 3 Mmum { A + t } Maxmum { B + g } or Mmum C } Maxmum B + g } { { { 4 Mmum{ A } Maxmum { t + B + g} or Mmum C } Maxmum t + B + g } { { 3 New proposed method We, ow troduce a ew method for fdg a optmal sequece to the problem (P) The ew proposed method proceeds as follows Step 1 : Reduce the gve problem (P) to two maches flow-shop problem by troducg two fctos maches, G ad H whose mache processg tmes G ad H, 1,2,, are gve below: =

4 924 P Pada ad P Rajedra (a) If the structural codto (1) s satsfed, the G = A + t + B ad H = t + B + g + C (b) If the structural codto (2) s satsfed, the G = A + t + B ad H = B + g + C (c) If the structural codto (3) s satsfed, the G = A + t + B + g ad H = B + g + C ad (d) If the structural codto (4) s satsfed, the G = A + t + B + g ad H = t + B + g + C The tabular form of the reduced problem s gve below: Mache wth processg tmes G H Weght of s W 1 G 1 H 1 W 1 2 G 2 H 2 W 2 M M M M G H W Step 2 Compute Mmum ( G, H) a If Mmum ( G, H ) = G, the defe G = G w ad H = H b If Mmum ( G, H) = H, the defe G = G ad H = H + w Step 3 Formulate a ew reduced schedulg problem volvg two maches as follows: 1 2 Maches wth processg tmes G H G 1 H 1 w1 w1 G 2 H 2 w w 2 M M M G H 2 w w where G ad H are obtaed from the Step 2

5 Solvg costraed flow-shop schedulg problems 925 Step 4 Determe the optmal sequece to the ew reduced schedulg problem obtaed the Step 3 ad also, the total elapsed tme for the gve problem (P) by usg Johso s algorthm ( If the mache processg tmes of a job are equal, put the job at last the optmal sequece ) Step 5 Idetfy the effect of break-dow terval ( a, b) o dfferet jobs Step 6 Modfy the gve problem usg the ew mache processg tmes A, B ad C whch are obtaed from oe of the followg cases () If the break-dow terval ( a, b) has o effect o job, at the tme of processg the maches A, B ad C, the A = A, B = B ad C = C () If the break-dow terval ( a, b) has affected o job, at the tme of processg the maches A, B ad C, the A = A + ( b a), B = B + ( b a) ad C = C + ( b a) Step 7 Usg the modfed schedulg problem ad the optmal sequece of the gve problem obtaed Step 4, determe the total elapsed tme ad weghted mea-flow tme 4 Numercal Examples The proposed method s llustrated by the followg examples Example 1: Cosder the followg costraed flow-shop schedulg problem of 4-jobs o 3-maches wth processg tmes, trasportato tmes ad the weghts of jobs: Maches wth processg tmes ad trasportg tmes Weght of s A t B g C W gve that the break-dow terval (a, b) = (18, 25) Now, sce Mmum { A + t } = 9 7 = Maxmum { B }, the structural codto (2) s satsfed The, usg the Step 1 to the Step 4 of the proposed method, we obta that (2,1,4,3) s a optmal sequece for the gve problem Now, the total elapsed tme for the optmal sequece (2,1,4,3) s calculated as follows:

6 926 P Pada ad P Rajedra Maches wth processg tmes ad trasportg tmes Weght of s A t B g C W Therefore, the total elapsed tme s 47 hrs Now, 2, 1 ad 3 have bee affected by the break dow terval (18, 25) o the optmal sequece (2, 1, 4, 3) Now, usg the Step 6 of the proposed method, we modfy the processg tmes for affected jobs ad we obta the followg ew schedulg problem the tabular form: Maches wth processg tmes ad trasportg tmes Weght of s A t B g C W Now, the elapsed tme for the optmal sequece (2,1,4,3) after applyg the breakdow tme s calculated as follows: Maches wth processg tmes ad trasportg tmes Weght of s A t B g C W x5 + (50 8)x3 + (56-28)x2 + (61-40)x4 The mea weghted flow tme = = Hece, the total elapsed tme s 61 hrs ad the mea weghted flow tme s 304 hrs

7 Solvg costraed flow-shop schedulg problems 927 Example 2: Cosder the followg costraed flow-shop schedulg problem of 4-jobs o 3-maches wth processg tmes, trasportato tmes ad the weghts of jobs: Maches wth processg tmes ad trasportg tmes Weght of s A t B g C W gve that the break-dow terval (a, b) = (18, 22) Now, sce Mmum { A + t } = 8 7 = Maxmum { B }, the structural codto (2) s satsfed The, usg the Step 1 to the Step 4 of the proposed method, we obta that (3,2,1,4) s a optmal sequece for the gve problem Now, the total elapsed tme for the optmal sequece (3,2,1,4) s calculated as follows: Maches wth processg tmes ad trasportg tmes Weght of s A t B g C W Therefore, the total elapsed tme s 49 hrs Now, 3, 2 ad 1 have bee affected by the break dow terval (18, 22) o the optmal sequece (3,2, 1, 4) Now, usg the Step 6 of the proposed method, we modfy the processg tme for affected jobs ad we obta the followg ew schedulg problem the tabular form: Maches wth processg tmes ad trasportg tmes Weght of s A t B g C W

8 928 P Pada ad P Rajedra Now, the elapsed tme for the optmal sequece (3,2,1,4) after applyg the breakdow tme s calculated as follows: Maches wth processg tmes ad trasportg tmes t B g C A Weght of s W The Mea weghted flow tme = 24x4 + (39 9)x5 + (45-15)x3 + (51-29)x = 2786 hrs Hece the total elapsed tme s 51 hrs ad the mea weghted flow tme s 2786 hrs Note: The Example 2 ca ot be solved usg the algorthm gve [4] sce both structural codtos gve [4] are ot satsfed 5 Cocluso The ew method provdes a optmal schedulg sequece for costraed flow-shop schedulg problems of 4-jobs o 3-maches Ths method s very easy to uderstad ad apply ad also, wll help maagers the schedulg related ssues by adg them the decso makg process ad provdg a optmal schedulg sequece a smple ad effectve maer Determg a best schedule for gve sets of jobs ca help decso makers effectvely to cotrol job flows ad to provde a soluto for job sequecg We have a pla to exted the proposed method to costraed flow-shop problems wth m-maches REFERENCES [1] KRBaker, Itroducto of Sequecg ad Schedulg, Joh Wley ad Sos, New York, 1974 [2] SP Basal, Resultat job restrcted two mache flow-shop problem, IJOMAS, 2 (1986), [3] R Bellma, Mathematcal aspects of schedulg theory, J Soc Idust Appl Math, 4 (1956),

9 Solvg costraed flow-shop schedulg problems 929 [4] A B Chadramoul, Heurstc approach for -job, 3-mache flow-shop schedulg problem volvg trasportato tme, breakdow tme ad weghts of jobs, Mathematcal ad Computatoal Applcatos, 10 (2005), [5] JRJackso, A exteso of Johso s results o job schedulg, Nav Res Log Quar, 3 (1956), [6] SMJohso, Optmal two ad three stage producto schedule wth set-up tmes cluded, Nav Res Log Quar, 1 (1954), [7] SMyazak ad NNshyama, Aalyss for mmzg weghted mea flow tme flow-shop schedulg, JOR Soc of Japa, 23 (1980), [8] RG Parker, Determstc schedulg theory, Chapma ad Hall, New York, 1995 Receved: October, 2009