Iteratoal Joural of Egeerg Research ad Developmet e-issn: 2278-067X, p-issn: 2278-800X,.jerd.com Volume 3, Issue 5 (August 2012), PP. 72-77 Specally Structured To Stage Flo Shop Schedulg To Mmze the Retal Cost, Processg Tme, Set up Tme Are Assocated Wth Ther Probabltes Icludg Job Block Crtera Ad Job Weghtage Deepak Gupta 1, Shash Bala 2, Payal Sgla 3 1 Prof. & Head, Departmet of Mathematcs, M.M.Uversty, Mullaa, Ambala,Ida 2 Departmet of Mathematcs, M. P. College for Wome, Mad Dabal,Ida 3 Departmet of Mathematcs, M.M.Uversty,Mullaa, Ambala,Ida Abstract The preset paper s attempt to develop a e heurstc algorthm, a alteratve to the tradtoal algorthm proposed by Johso s (1954) to fd the optmal sequece to mmze the utlzato tme of the maches ad hece ther retal cost for to stage specally structured flo shop schedulg uder specfed retal polcy hch processg tmes ad set up tme are assocated th ther respectve probabltes, cludg job block crtera. Further jobs are attached th eghts to dcate ther relatve mportace. The proposed method s very smple ad easy to uderstad ad also provde a mportat tool for the decso maker. Algorthm s justfed by umercal llustrato. Keyords Specally structured flo shop schedulg. Retal polcy, Processg tme, eghtage of jobs, Set up, Job block. I. INTRODUCTION Schedulg theory deals th formulato ad study of varous schedulg models. Some dely studed classcal models comprse sgle mache, parallel mache, flo shop schedulg, job shop schedulg, ope shop schedulg etc. The objectve of flo shop schedulg problem s to fd a permutato schedule that mmzes the maxmum completo tme of a sequece. Schedulg has become a major feld th operato research th several hudred publcatos appearg each year. The majorty of schedulg research assumes set up as eglgble or part of processg tme. Whle ths assumpto adversely effects soluto qualty for may applcato hch requre explct treatmet of setup. Johso [9] frst of all gave a method to mmze the make spa for -jobs, to mache schedulg problems. Gupta J.N.D. [7] gave a algorthm to fd the optmal schedule for specally structured flo shop schedulg. Gupta [5] studed specally structured flo shop problem to mmze the retal cost of the mache uder predefed retal polcy hch the probabltes have bee assocated th processg tme th eghtage of jobs ad job block crtero. Yoshda ad Htom [17] further cosdered the problem th set up tme. The basc cocept of equvalet job for a job block has bee troduced by Maggu & Das [10]. Sgh T.P. ad Gupta Deepak [13] studed the optmal to stage producto schedule hch processg tme ad set up tme both ere assocated th probabltes cludg job block crtera. The ork as developed, Chader Shekhera [3], Bagga & P.C. [2] ad Gupta Deepak et. al [14] by cosderg varous parameters. Ths paper s a attempt to exted the study made by Gupta & Sgla [5] by troducg set up tme separated from processg tme. Thus the problem dscussed ths paper become der ad very close to practcal stuato maufacturg/ process dustry. We have obtaed a algorthm hch gves mmum possble retal cost hle mmzg total utlzato tme. II. PRACTICAL SITUATION The practcal stuato of specally structured flo shop schedulg occur our day to day orkg, bakg, offces, educatoal sttutos, factores ad dustral cocer e.g I a readymade garmet maufacturg plat hch has maly to maches. vz, cuttg mache ad seg mache, hch the processg tme of jobs o 2 d mache(seg mache) ll alays be greater tha the processg tme of jobs o frst mache(cuttg mache). Moreover dfferet qualty of garmet are to be produced th relatve mportace.e. eght of jobs become sgfcat. Varous practcal stuatos occur real lfe he oe has got the assgmet but does ot have oe s o mache or does ot have eough moey to purchase mache. Uder such crcumstaces the mache has to be take o ret order to complete the assgmet. Retal of varous equpmets s a affordable ad quck soluto for a busessma, a maufacturer or a compay, hch presetly costraed by the avalablty of lmted fuds due to recet global ecoomc recesso. Retg eables savg orkg captal, gves opto for havg the equpmet ad allos up-gradato to e techology. Further the prorty of oe job over the other may be sgfcat due to some urgecy or demad of oe partcular type of job over other. Hece the job block crtera become mportat. 72
Specally Structured To Stage Flo Shop Schedulg To Mmze the Retal Cost, Processg III. NOTATIONS S : Sequece of jobs 1, 2, 3,., S k : Sequece obtaed by applyg Johso s procedure, k = 1, 2, 3, ------ r. M j : Mache j, j= 1,2. a j : Processg tme of th job o mache M j s j : Set up tme of th job o mache M j p j :Probablty assocated to the processg tme a j q j : Probablty assocated to the processg tme s j A j : Expected processg tme of th job o mache M j S j : Expected set up tme of th job o mache M j t j (S k ) : Completo tme of th job of sequece S k o mache M j : eght of th job. β : Equvalet job for job-block(k, m) G : eghted flo tme of th job o mache M 1. H : eghted flo tme of th job o mache M 2. U j (S k ) : Utlzato tme for hch mache M j s requred. Cj : Real cost per ut tme of j th mache. R(S k ) : Total retal cost for the sequece S k of all mache IV. DEFINITION Completo tme of th job o mache M j s deoted by t j ad s defed as: t j = max (t -1,,j,+ S -1,,j, t,,j-1 ) + A j ; j 2. here Aj=Expected processg tme of th job o j th mache. S j = Expected set up tme of th job o j th mache. V. RENTAL POLICY (P) The maches ll be take o ret as ad he they are requred ad are retured as ad he they are o loger requred..e. the frst mache ll be take o ret the startg of the processg the jobs, 2 d mache ll be take o ret at tme he 1 st job s completed o the 1 st mache. VI. PROBLEM FORMULATION Let some job ( = 1,2,..,) are to be processed o to maches M j ( j = 1,2) uder the specfed retal polcy P. Let A j & S j respectvely be the expected processg tme ad set up tme of th job o j th mache. Let be eght of the th job. β = (k, m) be equvalet job for job block (k,m).our am s to fd the sequece S k of jobs hch mmze the retal cost of the maches hle mmzg the utlzato tme of maches.the mathematcal model of the problem matrx form ca be stated as: Jobs Mache M 1 Mache M 2 Weght of jobs a 1 p 1 s 1 q 1 a 2 p 2 s 2 q 2 1 a 11 p 11 s 11 q 11 a 12 p 12 s 12 q 12 1 2 a 21 p 21 s 21 q 21 a 22 p 22 s 22 q 22 2 3 a 31 p 31 s 31 q 31 a 32 p 32 s 32 q 32 3 - - - - - - - - - - a 1 p 1 s 1 q 1 a 2 p 2 S 2 q 2 Table -1 Mathematcally, the problem s stated as: Mmze U 2 (S k ) ad hece R S A C U S C Mmze k 1 1 j k 2 1 Subject to costrat: Retal Polcy (P)..e. our objectve s to mmze utlzato tme of mache ad hece retal cost of maches. A A If 1 2 VII. THEOREM for all, j, j, the k 1, k 2.k s a mootocally decreasg sequece, here Proof: Let A 1 A j2 for all, j, j.e., max A 1 m A j2 for all, j, j 1. 1 2 73
Specally Structured To Stage Flo Shop Schedulg To Mmze the Retal Cost, Processg Let 1 1 2 Therefore, e have k 1 =A 11 Also k 2 = A 11 + A 21 A 12 = A 11 + (A 21 A 12 ) A 11 ( A 21 A 12 ).. k 1 k 2 No, k 3 = A 11 + A 21 + A 31 A 12 A 22 = A 11 + A 21 A 12 + (A 31 A 22 )= k 2 + ( A 31 A 22 ) k 2 ( A 31 A 22 ) Therefore, k 3 k 2 k 1 or k 1 k 2 k 3. Cotug ths ay, e ca have k 1 k 2 k 3. k, a mootocally decreasg sequece. Corollary 1: The total retal cost of maches s same for all the sequeces, f A 1 A 2, for all, j, j. T( S) A k A A Proof: The total elapsed tme 2 1 2 11. It mples that uder retal polcy P the total elapsed tme o mache M 2 s same for all the sequeces thereby the retal cost of maches s same for all the sequeces. VIII. THEOREM If A 1 A j2 for all, j, j, the K 1, K 2. K s a mootocally creasg sequece, here Proof: Let 1 1 2 74 1. 1 2 Let A 1 A j2 for all, j, j.e., m A 1 max A j2 for all, j, j Here k 1 = A 11 k 2 = A 11 + A 21 A 12 = A 11 + (A 21 A 12 ) k 1 ( A 21 A j2 ) Therefore, k 2 k 1. Also, k 3 = A 11 + A 21 + A 31 A 12 A 22 = A 11 + A 21 A 12 + (A 31 A 22 ) = k 2 + (A 31 A 22 ) k 2 ( A 31 A 22 ) Hece, k 3 k 2 k 1. Cotug ths ay, e ca have k 1 k 2 k 3. k, a mootocally creasg sequece. Corollary 2: The total elapsed tme of maches s same for all the possble sequeces, f A 1 A j2 for all, j, j. Proof: The total elapsed tme 1 1 T( S) A k A A A A A A A A 2 2 1 2 1 2 2 1 2 Therefore total elapsed tme of maches s same for all the sequeces. IX. ASSUMPTIONS 1. Jobs are depedet to each other. Let jobs be processed thorough to maches M 1 ad M 2 order M 1 M 2 2. Mache breakdo s ot cosdered. 3. Pre-empto s ot alloed. 4. Weghted flo tme must has the follog structural relato.e. Ether G H or G H for all X. ALGORITHM Step 1: Calculate the expected processg tmes, A j = a j p j ; S j = s j q j Step 2: Compute 1 A = A 1 S 2 A 2 = A 2 S 1 Step 3: Calculate eghted flo shop tme G & H as follo If m (A 1,A 2) A 1 ( A 1 ) A 2 The G = H = Ad (A,A ) A If m 1 2 2
Specally Structured To Stage Flo Shop Schedulg To Mmze the Retal Cost, Processg The G = A 1 H = ( A 2 ) Step 4: Take equvalet job β = (k,m) ad calculate processg tme G β ad H β o the gude les of Maggu & Dass (1977) as follos: G β = G k + G m m (G m,h k ) H β = H k + H m m (G m,h k ) Step 5: Defe a e reduced problem th processg tme G & H as defed Step 3 ad jobs (k,m) are replaced by sgle equvalet job β th processg tme G β & H β as defed step 4. Step 6: Check the structural relatoshp Ether G H or G H, for all f the structural relato hold good go to Step 6 other se modfed the problem.. Step 7: If J 1 J the put J 1 o the frst posto ad J as the last posto ad go to step 10 otherse go to step 8. Step 8: Take the dfferece of processg tme of job J 1 o M 1 from job J 2 (say) havg ext maxmum processg tme o M 1 call ths dfferece as G. also take the dfferece of processg tme of job J o M 2 from job J -1 (say) havg ext mmum processg tme o M 2. Call the dfferece as G 2. Step 9: If G 1 G 2 put J o the last posto ad J 2 o the frst posto otherse put J 1 o 1 st posto ad J -1 o the last posto. Step 10: Arrage the remag (-2) jobs betee 1 st job & last job ay order, thereby e get the sequeces S 1, S 2 S r. Step 11: Compute - out table for ay oe (say S 1 ) of the sequece S 1, S 2,..S. Step 12: Compute the total completo tme CT(S 1 ). Step 13: Calculate utlzato tme U 2 of 2 d mache here U 2 (S 1 ) = CT(S 1 ) A 1 (S 1 ); Step 14: Fd retal cost here C 1 & C 2 are the retal cost per ut tme of 1 st & 2 d mache respectvely. XI. NUMERICAL ILLUSTRATION Cosder 5 jobs, 2 maches problem to mmze the retal cost. The processg tmes set up tmes ad eght jobs are gve the follog table. Let β = (2,4) as equvalet job block. The retal cost per ut tme for maches M 1 ad M 2 are 10 uts ad 7 uts respectvely. Jobs Mache M 1 Mache M 2 Weght of jobs I a 1 p 1 s 1 q 1 a 2 p 2 s 2 q 2 W 1 140 0.2 3 0.3 90 0.2 2 0.3 1 2 160 0.3 4 0.2 110 0.1 3 0.1 2 3 130 0.2 2 0.1 70 0.2 1 0.2 3 4 180 0.2 6 0.2 80 0.2 5 0.2 1 5 220 0.1 5 0.2 50 0.3 4 0.2 2 Table :2 Soluto : As per step 1: The expected processg tme & expected set up tmes for maches M 1 ad M 2 are as follo Jobs Mache M 1 Mache M 2 Weght of jobs A 1 S 1 A 2 S 2 W 1 28.0 0.9 18.0 0.6 1 2 48.0 0.8 11.0 0.3 2 3 26.0 0.2 14.0 0.2 3 4 36.0 1.2 16.0 1.0 1 5 22.0 1.0 15.0 0.8 2 Table : 3 75
Specally Structured To Stage Flo Shop Schedulg To Mmze the Retal Cost, Processg As per step 2: Expected flo tme for to maches M 1 ad M 2 as follo : Jobs Mache M 1 Mache M 2 Weght A 1 A 2 1 27.4 17.1 1 2 47.7 10.2 2 3 25.8 13.8 3 4 35.0 14.8 1 5 21.2 14.0 2 Table : 4 As per step 3: Weghted flo tme for maches M 1 ad M 2 as follo : Jobs Mache M 1 Mache M 2 I G H 1 27.40 18.1 2 23.85 6.1 3 08.60 5.6 4 35.00 15.8 5 10.60 8.0 Table : 5 As per step 5: the e reduced problem become as uder: Jobs Mache M 1 Mache M 2 I G H 1 27.40 18.1 Β 52.74 15.8 3 08.60 6.0 5 10.60 8.0 Table : 6 Here, G H for all. As per step 7 max G = 52.74 hch s for job 2.e. J 1 =2 Ad m H = 5.6 hch s for job 3.e. J = 3. Sce J 1 J. e put J 1 = 2 o the frst posto Ad J = 3 o the last posto Therefore the optmal sequeces are S 1 = 2 1 5 3 = 2 4 1 5 3. S 2 = 2 5 1 3 = 2 4 5 1 3 Due our structural codtos the total elapsed tme s same for all these 2 possble sequeces S 1, S 2 ; say for S 1 = 2 4 1 5 3 s : Jobs Mache M 1 Mache M 2 I I-Out I-Out 2 0.0 48.0 48.0 59.0 4 48.0 84.8 84.8 100.8 1 86.0 114.0 114.0 132.0 5 114.9 136.9 136.9 151.9 3 137.9 163.9 163.9 177.9 Table : 7 Therefore, the total elapsed tme = CT(S 1 ) = 177.9 uts Utlzato tme of mache M 2 = U 2 (S 1 ) = 177.9 48.0 = 129.9 uts Also 1 =163.9 uts. 1 A Therefore the total retal cost for each of the sequece (S k ); k = 1, 2 s R(S k ) = 163.9 10 + 129.9 5 = 1639 + 649.5 = 2288.5 uts. XII. REMARKS a. If e solve the same problem by Johso s methods e get the optmal sequece as S= 1 2 4 5 3. The out flo table s: Jobs Mache M 1 Mache M 2 I - Out I - Out 3 0 28.0 28.0 46.0 76
Specally Structured To Stage Flo Shop Schedulg To Mmze the Retal Cost, Processg 4 28.9 76.9 76.9 87.9 2 77.7 113.7 113.7 129.7 5 114.9 136.9 136.9 151.9 1 137.9 163.9 163.9 177.9 Therefore, the total elapsed tme = CT(S) = 177.9 uts Utlzato tme of mache M 2 = U 2 (S) = 149.9 uts Also 1 =163.9 uts. 1 A Therefore the total retal cost s R(S k ) = 163.9 10 + 149.9 5 = 1639 + 750.5 = 2389.5 uts. b. Equvalet job formato s assocatve ature e block ((k, m)) = ((k)m, ) c. The equvalet job formato rule s o commutatve.e. block (k, m) (m, k). d. If set up tmes of each mache s eglgble small, the results are smlar as [5]. XIII. CONCLUSION The algorthm proposed here for specally structured to stage flo shop schedulg problem setup tme separated from processg tme, th eghtage of jobs cludg job block crtero s more effcet as compared to the algorthm proposed by Johso(1954) to fd a optmal sequece to mmze the utlzato tme of the maches ad hece ther retal cost. The study may further be exteded by cosderg varous parameters lke breakdo effect, trasportato tme etc. REFERENCES [1]. Aup (2002), O to mache flo shop problem hch processg tme assumes probabltes ad there exsts equvalet for a ordered job block, JISSOR, Vol. XXIII No. 1-4, pp. 41-44. [2]. Bagga P C (1969), Sequecg a retal stuato, Joural of Caada Operato Research Socety, Vol.7, pp.152-153. [3]. Chadrasekhara R (1992), To-Stage Floshop Schedulg Problem th Bcrtera O.R. Soc.,Vol. 43, No. 9, pp.871-84. [4]. Gupta Deepak & Sharma Sameer (2011), Mmzg retal cost uder specfed retal polcy to stage flo shop, the processg tme assocated th probabltes cludg breakdo terval ad Job-block crtera, Europea Joural of Busess ad Maagemet, Vol. 2, No. 3, pp.85-103. [5]. Gupta D., Shash B ad Sgla P (2012), Mmzg retal cost for specally structured to stage flo shop schedulg, processg tme assocated th probabltes cludg eghtage of jobs ad job block crtero, Iteratoal Joural of Research Maagemet, Vol. 3, No. 2, pp. 79-92. [6]. Gupta J N D (1975), Optmal Schedule for specally structured flo shop, Naval Research Logstc, Vol.22, No.2, pp. 255-269. [7]. Igall E ad Schrage L (1965), Applcato of the brach ad boud techque to some flo shop schedulg problems, Operato Research, Vol.13, pp.400-412. [8]. Johso S M (1954), Optmal to ad three stage producto schedule th set up tmes cluded, Naval Research Logstc, Vol.1, No.1, pp. 61-64. [9]. M. Dell Amco, Shop problems th to maches ad tme lags, Oper. Res. 44 (1996) 777 787. [10]. Maggu P L ad Das G (1977), Equvalet jobs for job block job schedulg, Opsearch, Vol. 14, No.4, pp. 277-281. [11]. Nara L Bagga P C (2005), To mache flo shop problem th avalablty costrat o each mache, JISSOR, Vol. XXIV 1-4, pp.17-24. [12]. Pada & Rajedra (2010), Solvg Costrat flo shop schedulg problems th three maches, It. J. Cotemp. Math. Sceces, Vol.5, No. 19, pp.921-929. [13]. Sgh T P (1985), O 2 shop problem volvg job block. Trasportato tmes ad Break-do Mache tmes, PAMS, Vol. XXI, pp.1-2. [14]. Sgh T P, K Rajdra & Gupta Deepak (2005), Optmal three stage producto schedule the processg tme ad set tmes assocated th probabltes cludg job block crtera, Proceedgs of Natoal Coferece FACM- 2005,pp. 463-492. [15]. Sgh, T P, Gupta Deepak (2005), Mmzg retal cost to stage flo shop, the processg tme assocated th probables cludg job block, Reflectos de ERA, Vol 1, No.2, pp.107-120. [16]. Szare W (1977), Specal cases of the flo shop problems, Naval Research Logstc, Vol.22, No.3, pp. 483-492. [17]. Yoshda ad Htom (1979), Optmal to stage producto schedulg th set up tmes separated, AIIE Trasactos, Vol.11, No. 3,pp. 261-269. 77