3-Stage Specially Structured Flow Shop Scheduling to Minimize the Rental Cost Including Transportation Time, Job Weightage and Job Block Criteria

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1 -Stage Specally Structured Flow Shop Schedulng to Mnmze the Rental Cost Includng Transportaton Tme, Job Weghtage and Job Block Crtera Deepak Gupta Shash Bala * Payal Sngla Sameer Sharma. Department of Mathematcs, M.M.Unversty,Mullana,Ambala,Inda. Department of Mathematcs, M.P.College for women, Mand Dabwal,. Department of Mathematcs, Baba Fard College of Engneerng & Technology, Bathnda, Punjab, Inda. Department of Mathematcs, D.A.V. College, Jalandhar, Punjab, Inda * E-mal of the correspondng author: shashgarg97@gmal.com Abstract Ths artcle descrbe the development of a new heurstc algorthm whch guarantees an optmal soluton for specally structured flow shop schedulng problem wth n-s,- machnes, to mnmze the rental cost under specfed rental polcy wth transportaton tme, weghtage and block crtera. Further the processng tmes are not merely random but bear a well defned relatonshp to one another. Most of lterature emphaszed on mnmzaton of dle tme/ make span. But mnmzaton of make span may not always lead to mnmze rental cost of machnes. Objectve of ths work s to mnmze the rental cost of machnes under a specfed rental polcy rrespectve of make span. Keywords: rental polcy, Job weghtage, block, transportaton tme, utlzaton tme.. Introducton Schedulng can be defned as the allocaton of resources over a perod of tme to perform a collecton of tasks. The goal s to specfy a schedule that specfy when and on whch machne each s to be executed. A varety of approaches have been developed to solve the problem of schedulng. Schedulng problems are common occurrence n our daly lfe e.g. orderng of s for processng n a manufacturng plant, programs to be run n a sequence at a computer center etc. Such problems exst whenever there s an alternatve choce n whch a number of s can be done. Now-a-days, the decson makers for the manufacturng plant have nterest to fnd a way to successfully manage resources n order to produce products n the most effcent way. They need to desgn a producton schedule to mnmze the flow tme of a product. The number of possble schedules n a flow shop schedulng problem nvolvng n-s and m-machnes s ( n! ) m. The optmal soluton for the problem s to fnd the optmal or near optmal sequence of s on each machne n order to mnmze the total elapsed tme. Majorty of research n schedulng assumes transportaton tme (loadng tme, movng tme and unloadng tme) from one machne to another as neglgble or ncluded n processng tme.but n some real lfe stuatons transportaton tme has great mpact on the performance measure, separate consderaton s needed. The basc concept of equvalent for a block has been ntroduced by Maggu and Dass (977). All the schedulng models begnnng from Johnson s work n 95 upto 980, there s no reference of weghtage n the lterature. The weghts of a show ts relatve prorty over some other s n a schedulng model. Hgher the weght of a has, the more mportant t becomes for processng n comparson wth other s n the operatng schedule. The schedulng problems wth weght arse when nventory costs for s are nvolved. Further the schedulng problem whch does not nvolve weght of s called smple or unweghted schedulng problem, whereas the schedulng problem nvolvng weght of s s referred to as weghted schedulng problem. The frst research concerned to the flow shop schedulng problem was proposed by Johnson (95). Johnson descrbed an algorthm to mnmze make span for the n-s - stage flow shop schedulng problem. Smth (970) consdered mnmzaton of mean flow tme for n s,m-machnes. Gupta, J.N.D. (975) gave an algorthm to fnd the optmal schedule for specally structured flow shop schedulng. Gupta (0]) studed specally structured two stage flow shop problem to mnmze the rental cost of the machnes under pre-defned rental polcy n whch the probabltes have been assocated wth processng tme. Maggu & Das (977) consder a two machne flow shop problem wth block ctera. Ghansham(978) studed three machne sequencng problem wth equvalent s for blocks for n-s. Anup(00) dscussed two stage flow shop schedulng wth ordered block. Heydar (00)dealt wth a flow shop schedulng problem where n s are processed n two dsjont blocks n a strng conssts of one block n whch order of s s fxed and other block n whch order of s s arbtrary. Myazak.(980) mnmzed weghted mean rental cost n two stage flow shop schedulng n whch the processng tme are assocated wth probabltes ncludng block. Chen and Lee (00)studed the transportaton machne schedulng. Khodadad (0) developed a new heurstc for three machne flow shop schedulng. Maggu and Dass (98), studed the concept of transportaton tme and

2 equvalent for a block. Sngh,T.P.(99) gave heurstc approach to solve two stage flow shop schedulng problem wth random processng and transportaton tme under group restrcton on s. Gupta (0) studed -Stage specally structured flow shop schedulng to mnmze the rental cost ncludng weghtage. Bagga, P.C. (969) solved sequencng problem n a rental stuaton. In ths paper we presents a specally structured flow shop schedulng model to mnmze the utlzaton tme of the machnes and hence ther rental cost under specfed rental polcy wth transportaton tme, block and weghtage. Most of the work emphasze on mnmzaton of make span. Here we have dscussed the algorthm whch shows that mnmzaton of make span does not always lead to mnmze rental cost of machnes.. Practcal Stuaton Manufacturng ndustres are the backbone n the economc structure of a naton, as they contrbute to ncreasng G.D.P. / G.N.P. and provdng employment. Productvty can be maxmzed, f the avalable resources are utlzed n an optmzed manner. Optmzed utlzaton of resources can only be possble f there s a proper schedulng system makng schedulng a hghly mportant aspect of a manufacturng system. The practcal stuaton may be taken n a paper mll, sugar factory and ol refnery etc. where varous qualtes of paper, sugar and ol are produced wth relatve mportance.e. weght n s, hence weghtage of s s sgnfcant. Transportaton becomes sgnfcant when the machnes on whch s are to be processed are placed at dfferent places. Due to unavalablty of funds n real lfe n hs startng career one has to be taken the machnes on rent. To start a ftness centre many machnes lke tread mll, laser har removal equpment, sk cabn, cardovascular, stretches, free weghts, ellptcal, cycles, rowers, plate loaded & benches, mult staton, one does not want to nvest huge money by buyng all the machnes, nstead he prefer to take the machnes on rent. By rentng one can stay fnancally afloat more easly and stll manage to procure the best hgh technology for customers.. Notatons S : Sequence of s,, n S k : Sequence obtaned by applyng Johnson s procedure, k =,,, M j : Machne j, =,, a j : Processng tme of th on machne M j. t j (S k ) : Completon tme of th of sequence S k on machne M j I j (S k ) : Idle tme of machne M j for n the sequence S k. T,j k :Transportaton tme of th from j th machne to k th machne. U j (S k ) : Utlzaton tme for whch machne M j s requred R(S k ) : Total rental cost for the sequence S k of all machne. C j : Rent cost per unt tme of machne M j. w : Weght of th s. β : Equvalent for -block (k, m). Defnton Completon tme of th on machne M j s denoted by t j and s defned as: t j = max (t -, j, t, j- + T,j- j )+a j for j. 5. Rental Polcy The machnes wll be taken on rent as and when they are requred as and when they are no longer requred..e. the frst machne wll be taken on rent n the startng of the processng the s and nd machne wll be taken on rent at tme when st s completed on the st machne and transported to nd machne and rd machne wll be taken on rent when st s completed on nd machne and transported to rd machne. 6. Problem Formulaton Let some s (,,..,n) are to be processed on three machnes M j (j=,,) under the specfed rental polcy (P). Let a j be the processng tme of th on j th machne and T j k be the transportaton tme of th from j th machne to k th machne. w be the weght of th. Let β be equvalent for block (k,m). Our am s to fnd the sequence {S k } of the s whch mnmze the rental cost of all the machnes. The mathematcal model of the problem n matrx form can be tabulated as n table.

3 Mathematcally, the problem s stated as: Mnmze Subject to constrant: Rental Polcy (P) Our objectve s to mnmze rental cost of machnes whle mnmzng the utlzaton tme. 7. Assumptons Jobs are ndependent to each other. Let n s be processed through three machnes M, M, M n order M M M. Machne break down s not consdered. Pre-empton s not allowed. Ether max (a + t ) mn (a + t ) max (a + t ) mn (a + t ) Or both. 8. Algorthm The algorthm to mnmze the rental cost s as follows: Step : Convert the problem nto two machnes problem. Let G and H be fcttous machnes havng G and H as ther processng tmes : G = a + t + a + g H = t + a + g + a Step : If mn (G, H ) = G Then G = G + w, H = H w w If mn (G, H ) = H Then G = G, H = H + w W w Step : Take equvalent β = (k,m) and calculate processng tme G β and H β on the gude lnes of Maggu & Dass (977) as follows: G β = G k + G m mn (G m,h m ) H β = H k + H m mn (G m, H m ) Step : Defne a new reduced problem wth processng tme as n Step & Step. Step 5: Apply Johnson s (95) technque and obtan the optmal schedule of gven s. Let the sequence be S. Step 6: Obtan other sequences by puttng nd, rd,..n th s of sequence S n the st poston and all other s of S n same order. Let these sequences be S,S S n-. Step 7: Compute A, U (S k ), U (S k ) and For all possble sequences S k (k =,,, n) Step 8: Fnd mn R(S k ); k=,,.,n. let t be mnmum for the sequence S p, then sequence S p wll be the optmal sequence wth rental cost R(S p ). 9. Numercal Illustraton Consder 5 s and machnes flow shop problem n whch processng tmes ncludng transportaton tme and weghtage are gven n the Table-.Let β be equvalent for block (,5). The rental cost per unts tme for machnes M, M and M are unts, 5 unts and unts respectvely. Our objectve s to obtan optmal a sequence of s wth mnmum possble rental cost of the machnes.

4 Jobs Machne M T Machne M T Machne M Weght I a t a t a w Table - Soluton: Check condtons: Here max (a + t ) mn (a + t ) satsfes. Therefore, as per Step the processng tme for two fcttous machnes G and H are shown n table As per Step : The new reduced problem wth weghted flow tmes G & H, s as n table - As per step : The Processng tme for the equvalent on fcttous machnes s shown n table - 5 As per Step 5: Obtanng the sequence wth mnmum makespan s S : 5. As per Step 6: Other feasble sequences whch may corresponds to mnmum rental cost are : S = 5 S = 5 S = 5 From n out tables for these sequences, we have: For S : CT(S ) = 77; U (S ) = 5; U (S ) = ; R(S ) =956 For S : CT(S ) = 77; U (S ) = 5; U (S ) = ; R(S ) =9 For S : CT(S ) = 77; U (S ) = ; U (S ) = 99; R(S ) =798 For S : CT(S ) = 9; U (S ) = 8; U (S ) = 06; R(S ) =7 Therefore mn R{S K } = R(S ) = 7 unts. Therefore mnmum rental cost s 7 unts and s for the sequence S. Hence the sequence S = 5 s optmal sequence wth mnmum rental cost. But total elapsed tme / completon tme for S s not mnmum. 0. Concluson The algorthm proposed n ths paper to mnmze the rental cost of machnes gven an optmal sequence havng mnmum rental cost of machnes rrespectve of total elapsed tme. The algorthm proposed by Johnson s (95) to fnd an optmal sequence to mnmze the makespan/ total elapsed tme s not always corresponds to mnmum rental cost of machnes under a specfed rental polcy. Further the work can be extended by ntroducng parameters lke, break down nterval, set-up etc.. References Anup (00), On two machne flow shop problem n whch processng tme assumes probabltes and there exsts equvalent for an ordered block, JISSOR, XXIII (-),-. Bagga, P.C. (969), Sequencng n a rental stuatons, Jr. of Canadan Operaton Research Socety, 7,5-5. Chander, S., Rajendra, K. & Deepak, C. (99), An Effcent Heurstc Approach to the schedulng of s n a flow shop, European Journal of Operaton Research, 6,8-5. Chen, Z.L. and Lee, C.Y. (00), Machne schedulng wth transportaton consderaton, USA Journal of schedulng,, -. Das, G., (978), Equvalent s for blocks for n-, -machne, sequencng problems, PAMS, -, 5-0. Gupta D., Shash B., Sharma S. (0), To Mnmze The Rental Cost For - Stage Specally Structured Flow Shop Schedulng wth Job Weghtage Internatonal Journal of Engneerng Research and Applcatons (IJERA), ( ), Gupta, D., Sharma, S. and Bala,S. (0), Specally Structured Two Stage Flow Shop Schedulng To Mnmze the Rental Cost, Internatonal Journal of Emergng trends n Engneerng and Development, ( ),06-5. Gupta, J.N.D.(975), Optmal Schedule for specally structured flowshop, Naval Re search Logstc,

5 (), Heydar, A. P. (00), On flow-shop schedulng problem wth processng of s n a strng of dsjont blocks fxed under s & arbtrary order s, JISSOR, XXIV,9-. Johnson, S.M. (95), Optmal two and three stage producton schedule wth setup tme ncluded, Nav. Res. Log. Quart.(),6-68. Khodadad, A. (0), Solvng constrant flow shop schedulng problems wth three machnes, Internatonal Journal of Academc Research, (),8-0. Maggu & Das (98), On n x sequencng problem wth transportaton tme of s, Pure and Appled Mathematka Scences, -6. Maggu, P. L. and Das, G. (977), Equvalent s for block n sequencng, Opsearch, 5, Maggu, P. L., Das G. and Kumar R., (98), Equvalent s wth transportaton tme sequencng, J. O. R. Soc. Of Japan, (). Maggu, P.L., Yadav S.K., Sngh T.P. and Dev, Arjun (98), Flow shop schedulng problems nvolvng weght and transportaton tme, PAMS XX, -,5-58. Myazak, S., Nshyama,N.(980), Analyss for mnmzng weghted mean Mnmzng rental cost n two stage flow shop, the processng tme assocated wth probabltes ncludng block, Reflectons de ERA, (), Sngh, T P, Gupta, D. (005), Mnmzng rental cost n two stage flow shop,the processng tme assocated wth probables ncludng block, Reflectons de ERA, (), Sngh, T.P. (99), n x flow shop schedulng wth random processng and transportaton tme under group restrcton on s, Journal of Informaton Scences,5(). Smth, R. A. and Dudek, R.A. (967), A general algorthm for soluton of n-, M-machne sequencng problem of the flow shop,opns. Res. 5( 7). Szware, W.(977), Specal cases of the flow shop problems, Naval Research Logstc, (), 8-9. Vjay Sngh (0), Three machnes flow shop schedulng problems wth total rental cost, Internatonal referred journal II, Table : The mathematcal model of the problem Jobs Machne M T Machne M T Machne M Weght I a t a g a w a t a g a w a t a g a w a t a g a w n a n t n a n g n a n w n Table : The processng tme G & H for two fcttous machnes G and H Jobs G H Weght G H w Table : New reduced problem wth weghted flow tmes G & H, Jobs G H

6 Table 5: The Processng tme for the equvalent on fcttous machnes s: Jobs G H β Table 6: In-Out table for the optmal sequence S s as follows: Machne A Machne B Machne C Table 7: In-Out table for the optmal sequence S s as follows: Machne A Machne B Machne C Table 8:In-Out table for the optmal sequence S s as follows: Machne A Machne B Machne C In Out Table 9: The In-Out table for the optmal sequence S s as follows: Machne A Machne B Machne C

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