Society for Business and Management Dynamics. Business Management Dynamics Vol.1, No.4, Oct 2011, pp.63-72

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1 Busess Maagemet Dyamcs Vol, No4, Oct 20, pp63-72 Solvg Sgle Mache Schedulg Problem Wth Commo Due Date Nord Haj Mohamad ad Fatmah Sad 2 Abstract he paper addresses a -job sgle mache schedulg problem wth commo due date to mmze the sum of total vetory ad pealty costs arless ad tardess are cosdered harmful to proftablty arless causes vetory carryg costs ad possble loss of product qualty, whle tardess causes loss of customer goodwll ad damage reputato as well as delay of paymet hus the schedulg problem of mmzg the total sum of earless ad tardess wth a commo due date o a sgle mache s mportat ad a compettve task the delvery of goods producto plats, ad t s kow to be NP-hard A smple, easy to uderstad/mplemet heurstc algorthm whch ca be performed maually for small problems ad computatoally feasble for large problems s preseted ad llustrated wth umercal example Key words: obs schedulg problem, due date, earless, tardess Avalable ole wwwbmdyamcscom ISSN: INRODUCION I today s complex dustral world, may busess problems that eed to be solved or optmzed are schedulg problems A maufacturg frm producg multple products, each requrg may dfferet processes ad mache facltes for completo, must fd a way to successfully maage resources the most effcet way possble he decso maker s faced wth a problem of desgg a producto or job schedule that promotes o tme delvery ad mmzes objectves such as the flow tme or completo tme of a product Out of these terests emerged a area of study kow as the schedulg problem A frequetly occurrg schedulg problem s oe of processg a gve umber of jobs or tasks o a specfed umber of maches or facltes hs class of problem also referred to by may as dspatchg or sequecg s categorzed as NP-hard he desre to process the jobs a specfc order to acheve some objectve fucto s what creates a problem that remas largely usolved he actual stuatos that gve rse to schedulg problems are wde ad vared hs cludes, for example, sgle mache schedulg problem, multple mache schedulg problem ad mapower schedulg problem A geeral jobs m maches schedulg problem ca be stated as follows Gve jobs to be processed o m maches the same techologcal order, the processg tme of job o mache j beg t j ( =, 2,, ; j =, 2,, m), t s desred to fd the order (schedule or sequece) whch these jobs should be processed o each of the m maches so as to optmze (mmze or maxmze) a well defed measure of some objectves (such as producto cost, umber of late jobs, etc) hs problem, geeral, gves (!) m possble schedules ve for problems as small as = m = 5, the umber of possble schedules s so large that a drect eumerato s ecoomcally mpossble (For = 5, m = 3, we have,728,000 possble schedules, whereas for = m = 5, we have x 0 0 possble schedules) However, for a smplfed verso where t s assumed that the order (or sequece) whch these jobs are processed o each mache s the same, the umber of feasble schedules reduces to! Sgle mache schedulg problems bear complex computatos ad the aalyss of such problems s mportat for a better uderstadg of the problem Amog sgle mache problems, those related to earless ad tardess s more mportat Completg jobs or tasks earler tha ther due dates should be dscouraged as much as completg jobs or tasks later tha ther due dates I real world, sce a customer expects to receve the product o a specfc date, schedulg based o the due date s also a mportat task the producto plag arless leads to vetory ad mateace carryg costs whle tardess leads to customer s dssatsfacto ad losg goodwll ad reputato Isttute of Mathematcal Sceces, Faculty of Scece, Uversty of Malaya, Lembah Pata, Kuala Lumpur, Malaysaelephoe : Fax: mal : ordhm@umedumy 2 Faculty of coomcs ad Admstrato, Uversty of Malaya, Kuala Lumpur, Malaysa fatmahs@umedumy

2 Busess Maagemet Dyamcs Vol, No4, Oct 20, pp63-72 LIRAUR RVIW ob schedulg or sequecg has a wde varety of applcatos, from desgg the product flow ad order a maufacturg faclty to modelg queues servce dustres It s deed a useful subject that s stll beg actvely researched Most researchers focused o specal cases whch stuatoal restrctos are mposed Moore (968) desged a algorthm that sequeces the jobs for the sgle-mache problem to mmze the umber of tardess jobs, whereas Gupta (969) proposed a geeral algorthm for the x m flowshop schedulg problem Brucker et al (999) showed that complex schedulg problems lke geeral shop problem ca be reduced to sgle-mache problem wth postve ad egatve tme-lags betwee jobs, solvable by a brach ad boud algorthm Schedulg multmache problems cosderg both earless ad tardess pealtes was surveyed by Lauff ad Werer (2004) whch corporated the just--tme (I) producto phlosophy A comprehesve revew of some earless ad tardess models ca be foud Baker ad Scudder (990) where seve dfferet objectve fuctos assocated wth the mmzato of varatos of job completo tmes from ther respectve due dates were detfed, cludg cases of olear pealtes avakkol-moghaddam et al (2005) cosder the commo due date problem wth the objectve of mmzg the sum of maxmum earless ad tardess costs usg a dle sert algorthm ad llustrated the effcecy of the proposed algorthm to 020 problems wth dfferet job szes A lear programmg approach to solve a fuzzy sgle mache schedulg problem s proposed by Kamalabad et al (2007) hs approach s applcable to just-tme systems, whch may frms face the eed to complete jobs as close as possble to ther due dates A recet study by Gupta (20) proposes a heurstc algorthm for small system wth dstct due dates uder fuzzy evromet I addto, the teger programmg method for solvg problems wth small sze of jobs was rased by Bskup ad Feldma (200) Roco ad Kawamura (200) proposed a brach-ad-boud algorthm for solvg sgle mache earless ad tardess schedulg problem hey troduced lower bouds ad prug that explot propertes of the problem Feldma ad Bskup (2003) studed sgle mache schedulg problems usg three meta-heurstcs approaches (evolutoary search, smulated aealg ad threshold acceptg) he applcato of these meta-heurstcs was demostrated by solvg 40 bechmark problems wth up to 000 jobs Several meta-heurstc algorthms for solvg sgle mache schedulg problems were aalyzed by Abtah ad aghavfard (2008) I ths paper, we cosder a smplfed verso of schedulg jobs wth commo due date o a sgle mache wth the objectve of mmzg the sum of total earless ad tardess pealtes Commo due date problems are relevat may real-lfe stuatos; for stace, f a customer orders a budle of goods whch has to be delvered at a specfed tme, f a frm has stalled a weekly bulk delvery to the wholesaler or a assembly evromet whch the compoets of a product should all be ready at the same tme to avod stagg delays (Yag ad Hsu, 200) Amog the poeers studyg commo due date problems were Kaet (98) ad Pawalker et al (982) All the jobs cosdered have a commo due date he objectve was to fd a optmal commo due date ad a optmal schedule whch mmzes the total earless, tardess ad due date costs Sce the, the problems have bee studed uder dfferet evromets Cheg et al (2004) studed a sgle mache due date assgmet schedulg wth deteroratg jobs hey provded some propertes ad a algorthm to solve the problem O(log) tme Later, Kuo ad Yag (2008) gave a cocse aalyss of the problem ad provded a smpler algorthm for the problem Comprehesve survey o ths topc s provded by Cheg ad Gupta (989), Baker ad Scudder (990) ad Gordo et al (2002) SAMN OF H PROBLM Oe of the most mportat objectves schedulg problem wth due dates s to mmze the sum of the earless ad tardess of jobs hs coforms to the I system (Ow ad Morto, 989) arless ad tardess cause pealtes creasg vetory cost ad losg customers respectvely arly jobs (completed before due dates) te up captal, crease the vetory level, take up scarce floor space, cause losses owg to deterorato, ad geerally dcate sub-optmal resource allocato ad utlzato, whereas late or tardy jobs (completed after due dates) result pealtes, such as loss of customers goodwll ad damaged reputato

3 Busess Maagemet Dyamcs Vol, No4, Oct 20, pp63-72 here are may real lfe problems that resemble sgle mache schedulg problem A typcal example s the laudry servce where orders (of dfferet szes) from customers arrve early morg, ad due dates are determed by pck-up tmes, ad pck-ups are made by customers If a due date (pck-up) s mssed, a specal delvery servce eeds to be bought by the laudry operator, the cost of whch s depedet of the tardess Other examples clude a sgle cotractor schedulg multple buldg/housg projects wth due date completo tmes ad producto of maufacturg goods wth dfferet processg tmes to meet delvery s deadles he geeral sgle-mache problem wth commo due date ca thus be formally stated as follows Gve jobs to be processed o a sgle mache, the processg tme of job beg t, =, 2,, It s assumed that all jobs are ready for processg at tme zero ad have the same commo due date (deadle) Also, o more tha oe job ca be processed at ay pot of tme ach job requres exactly oe operato ad ts processg tme p s kow If a job s completed before the due date, ts earless s gve by = d c where c s the completo tme of job Coversely, f a job s completed after the desred date, ts tardess s gve by = c d ach job has ts ow ut earless pealty α ad ut tardess pealty β he problem s to fd the order (schedule) whch these jobs should be processed so as to mmze the sum of total earless ad tardess costs It s also assumed that the due date s less tha the total processg tmes, a problem ofte referred to as restrcted due date problem A due date s called urestrctve f ts optmal value has to be calculated or f t s gve value does ot fluece the optmal schedule (Roco ad Kawamura, 200) If the gve due date s greater tha or equal to the sum of processg tmes of all jobs avalable, the problem s urestrctve (Feldma ad Bskup, 2003) Hall ad Poser (99) showed that ths schedulg problem s NP-hard eve wth α = β Ghosh ad Wells (994) addressed the urestrctve case whch α = β = for all jobs that ca be solved by a polyomal algorthm of O(log) complexty he restrctve due date problem s NP-hard eve wth α = β = (Hall et al 99) Due to ts complexty, may authors addressed ths problem usg heurstc ad metaheurstc approaches (Feldma ad Bskup, 2003; Ho et al 2005; Lao ad Cheg, 2007) he problem ca be mathematcally formulated as mmze : F = max( d c,0) max( c d,0)} () where c, =,2,,, s the completo tme of job, d s the commo due date, ad α ad β are the ut pealty costs assocated wth earless ad tardess respectvely o llustrate the problem, we cosder a smple 2-job ad 3-job examples wth α = α ad β = β, for all =,2,, xample (case = 2) Let ad 2 be two jobs wth processg tmes 3 ad 5 days respectvely Further, let the commo due date, d = 6 I other words, both jobs must be delvered o day 6 he problem ca be represeted as able able (case = 2, wth due date, d = 6) 2 t 3 5 here exst two schedules: S =, 2} ad S 2 = 2, } Schedule S mples job s processed frst, followed by job 2, whereas schedule S 2 mples the opposte, job 2 s processed frst, followed by job

4 Busess Maagemet Dyamcs Vol, No4, Oct 20, pp63-72 For schedule S =, 2}, we have 2 t 3 5 c d 3 6 = 3, 8 6 = 2 herefore, total earless ad tardess costs, F = 3α + 2β For schedule S 2 = 2, }, we have 2 t 5 3 c d 5 6 =, 8 6 = 2 herefore, total earless ad tardess costs, F 2 = α + 2β hus, F* = mmum (F, F 2) = F 2 = α + 2β for all α ad β he optmal decso s to schedule job 2 frst (whch s completed oe day before the due date, gvg a earless cost of α) followed by job (wth a delay of two days ad tardess cost of 2β) xample 2 (case = 3) Let, 2 ad 3 be three jobs wth processg tmes 3, 4 ad 6 days respectvely Further, let the commo due date, d = 9 I other words, all jobs must be delvered o day 9 he problem ca be represeted as able 2 able 2 (case = 3, wth due date, d = 9) 2 3 t here exst sx schedules: S =, 2, 3}, S 2 =, 3, 2}, S 3 = 2,, 3}, S 4 = 2, 3, }, S 5 = 3,, 2}, S 6 = 3, 2, } Schedule S mples job s processed frst, followed by job 2, ad fally job 3, whereas schedule S 2 mples job s processed frst, followed by job 3, ad fally job 2 All other schedules should be terpreted accordgly For schedule S =, 2, 3}, we have 2 3 t c d 3 9 = 6, 7 9 = 2, 3 9 = 4 herefore, total earless ad tardess costs, F = 8α + 4β For schedule S 2 =, 3, 2}, we have 3 2 t c d 3 9 = 6, 9 9 = 0, 3 9 = 4 herefore, total earless ad tardess costs, F 2 = 6α + 4β

5 Busess Maagemet Dyamcs Vol, No4, Oct 20, pp63-72 For schedule S 3 = 2,, 3}, we have 2 3 t c d 4 9 = 5, 7 9 = 2, 3 9 = 4 herefore, total earless ad tardess costs, F 3 = 7α + 4β For schedule S 4 = 2, 3, }, we have 2 3 t c d 4 9 = 5, 0 9 =, 3 9 = 4 herefore, total earless ad tardess costs, F 4 = 5α + 5β For schedule S 5 = 3,, 2}, we have 3 2 t c d 6 9 = 3, 9 9 = 0, 3 9 = 4 herefore, total earless ad tardess costs, F 5 = 3α + 4β For schedule S 6 = 3, 2, }, we have 3 2 t c d 6 9 = 3, 0 9 =, 3 9 = 4 herefore, total earless ad tardess costs, F 6 = 3α + 5β hus, F* = mmum (F, F 2, F 3, F 4, F 5, F 6) = F 5 = 3α + 4β for all α ad β he optmal decso s to schedule job 3 frst (whch s completed three days before the due date, gvg a earless cost of 3α) followed by job (whch s completed o tme), ad lastly by job 2 (wth a delay of four days ad tardess cost of 4β) From both examples, we observe that the optmal schedule appears to be depedet of the umercal values of α ad β hus wthout loss of geeralty ad for ease of computato, we ca cosder a case wth α = β = there are! schedules for jobs For 0 jobs, there are 0! = 3,628,800 possble schedules whch are ot maually feasble hus a algorthm capable of reducg the umber of eumeratos s much desred A HURISIC ALGORIHM Below, we preset a heurstc algorthm for solvg -job sgle mache schedulg problem wth commo due date he algorthm s smple to uderstad ad easy to mplemet Problem Statemet: Gve jobs to be processed o a sgle mache, the processg tme of job beg t, =, 2,, It s assumed that all jobs are ready for processg at tme zero ad have the same commo due date (deadle), D he problem s to fd the order (schedule) whch these

6 Busess Maagemet Dyamcs Vol, No4, Oct 20, pp63-72 jobs should be processed so as to mmze the sum of total earless ad tardess costs he commo due date, D s assumed to be less tha the total processg tme Step 0: Sort ad umber the jobs o-creasg order of processg tme t, ( =,2,, ) such that t t2 t t t t I geeral, we ca represet the jobs tabular form, t t t 2 t - t t + t Compute t total processg tme, 0 D, ad D Itroduce two empty sets, 0, S 0 ad S 0 Step : Cosder job wth processg tme t max( t,,2,, ) Set 0, ad 0 If, set S S 0 } ad S S0 If, set S S 0 } ad S S0 Step : Cosder job wth processg tme t, ( < < ) If prevous job, S ad t If prevous job, S t ad If, set S S } ad S S If, set S S } ad S S Iterato termates whe all jobs have bee assged to ether Schedulg decso Check (ad sort f ecessary) so that: S = jobs o-creasg order of processg tmes}, S = jobs o-decreasg order of processg tmes} S or he optmal schedule, S* = S, S } I other words, jobs are schedule accordg to ther sequece S, followed by S Note that the procedure oly volves eumeratos (teratos) as compared to! possble schedules We llustrate the above algorthm by cosderg a 0-job schedulg problem Illustratve example Cosder a 0-job schedulg problem gve by the table below (sorted o-creasg order of processg tmes), wth commo due date, D = 45 S Step 0: Compute t

7 total processg tme, t 02, 0 D , ad D 45 Itroduce two empty sets, 0 S 0 ad S 0 Step : Cosder job wth processg tme t 20 Set 57, ad S 0 0 S } = } ad S S0 = } Step 2: Cosder job 2 wth processg tme t 2 = 8 Prevous job, S compute t 37 ad S2 S 2} 2} ad S 2 S } Step 3: Cosder job 3 wth processg tme t 3 = 5 Prevous job, S compute 37, ad t S S }, } ad S S } Step 4: Cosder job 4 wth processg tme t 4 = 0 Prevous job, S compute t ad 27 S S }, } ad S S, } Step 5: Cosder job 5 wth processg tme t 5 = 9 Prevous job, S compute ad t S S },, } ad S S, } Busess Maagemet Dyamcs Vol, No4, Oct 20, pp Step 6: Cosder job 6 wth processg tme t 6 = 8 Prevous job, S compute t ad 7 S S },, } ad S S,, } Step 7: Cosder job 7 wth processg tme t 7 = 7 Prevous job, S compute ad t S S },,, } ad S S,, } Step 8: Cosder job 8 wth processg tme t 8 = 6 Prevous job, 7 S7 compute t7 ad

8 8 8 S8 S7 8} 2, 4, 6, 8}, ad S S,,, } Step 9: Cosder job 9 wth processg tme t 9 = 5 Prevous job, 8 S8 compute ad t S9 S8 9}, 3, 5, 7, 9}, ad S S,,, } Step 0: Cosder job 0 wth processg tme t 0 = 4 Prevous job, 9 S9 compute t9 ad S0 S9 0} 2, 4, 6, 8, 0}, ad S S,,,, } d of terato sce all jobs have bee assged to ether S 0 or S 0 Busess Maagemet Dyamcs Vol, No4, Oct 20, pp63-72 Observe that jobs S,,,, } are o-creasg order (as requred), but jobs , 3, 5, 7, 9} 0 9, 7, 5, 3, } S are ot the requred o-decreasg order hus, rewrte S whch s ow the requred o-decreasg order he optmal schedule s, therefore, gve by whch ca be tabulated as S* = S, S },,,, ;,,, }, , t c c D gvg, F* c D 92 (pealty days) For comparatve purposes, we compute a few selected schedules such as S,,,,,,,,, }, t c c D gvg F c D 320 (pealty days),

9 Busess Maagemet Dyamcs Vol, No4, Oct 20, pp63-72 S S, S },,,,,,,,, }, t c c D gvg F2 c D 342 (pealty days), S,,,,,,,,, }, t c c D gvg F3 c D 268 (pealty days) As ca be see, all the above schedules gve the sum of total earless ad tardess costs more tha 92 (pealty days) CONCLUSION I ths paper we have preseted a smple heurstc algorthm outlg the procedure for mmzg the sum of total earless ad tardess costs a -job sgle mache schedulg problem wth commo due date he algorthm volves oly teratos ad s computatoally ecoomcal for large problems ad maually feasble for small problems We llustrate for a 0-job problem However, t s assumed that the ut earless ad tardess costs are costat for all jobs What s preseted here s just the tp of a large ceberg Future research may focus o studyg smlar models mult-mache evromet ad try to detfy easly solvable specal cases he world of job-mache schedulg s almost edless For the last ffty years, more tha 200 papers o varous aspects of the problem have bee publshed the operatoal research ad maagemet scece lteratures he advet of just--tme system ad developmet supply cha maagemet, teret ad e-commerce has created ew ad complex schedulg problems to the exstg problems that we have just begu to uderstad RFRNCS Abtah, SM, ad aghavfard, M (2008) valuatg meta-heurstc algorthms for solvg restrcted sgle mache schedulg problems: a comparatve aalyss World Appled Sceces oural, 4(): Baker, KR ad Scudder, GD (990) Sequecg wth earless ad tardess pealtes: A revew Operatos Research, 38(): Bskup, D ad Feldma, M (200) Bechmarks for schedulg o a sgle-mache restrctve ad urestrctve commo due dates Computers ad Operatos Research, 28(8): Brucker, P, Hlbg, ad Hurk, (999) A brach ad boud algorthm for a sgle-mache schedulg problem wth postve ad egatve tme-lags Dscrete Appled Mathematcs, 94(): 77 99

10 Busess Maagemet Dyamcs Vol, No4, Oct 20, pp63-72 Cheg, C ad Gupta, MC (989) A survey of schedulg research volvg due-date determato decsos uropea oural of Operatoal Research, 38(2): Cheg, C, Kag, L ad Ng, C (2004) Due-date assgmet ad sgle mache schedulg wth deteroratg jobs oural of the Operatoal Research Socety, 55(2): Feldma, M ad Bskup, D (2003) Sgle-mache schedulg for mmzg earless ad tardess pealtes by meta-heurstc approaches Computers ad Idustral geerg, 44(2): Ghosh, PDB ad Wells, C (994) Solvg a geeralzed model for CON due date assgmet ad sequecg Iteratoal oural of Producto coomcs, 34(2): Gordo, VS, Proth, M ad Chu, C (2002) A survey of the state-of-art of commo due date assgmet ad schedulg research uropea oural of Operatoal Research, 39(): 25 Gupta, ND (969) A geeral algorthm for the x m flowshop schedulg problem he Iteratoal oural of Producto Research, 7: Gupta, S (20) Sgle mache schedulg wth dstct due dates uder fuzzy evromet Iteratoal oural of terprse Computg ad Busess Systems, (2): 9 Hall, NG ad Poser, M (99) arless-tardess schedulg problems, I: weghted devato of completo tmes about a commo due date Operatos Research, 39(5): Hall, NG, Kubak, GW ad Seth, SP (99) arless-tardess schedulg problems, II: weghted devato of completo tmes about a restrctve commo due date Operatos Research, 39(5): Ho, CM, Roco, DP ad Medes, AB (2005) Mmzg earless ad tardess pealtes a sgle-mache problem wth a commo due date uropea oural of Operatoal Research, 60(): Kamalabad, IN, Mrzae, AH ad avad, B (2007) A possblty lear programmg approach to solve a fuzzy sgle mache schedulg problem oural of Idustral ad Systems geerg, (2): 6 29 Kaet, (98) Mmzg the average devato of job completo tmes about a commo due date Naval Research Logstcs Quarterly, 28(4): Kuo, WH ad Yag, DL (2008) A ote o due-date assgmet ad sgle-mache schedulg wth deteroratg jobs oural of the Operatoal Research Socety, 59: Lauff, V ad Werer, F (2004) Schedulg wth commo due date, earless ad tardess pealtes for mult-mache problems: a survey Mathematcal ad Computer Modellg, 40(5): Lao, C ad Cheg, CC (2007) A varable eghborhood search for mmzg sgle mache weghted earless ad tardess wth commo due date Computers ad Idustral geerg, 52(4): Moore, M (968) A job, oe mache sequecg algorthm for mmzg the umber of late jobs Maagemet Scece, 5(): Ow, PS ad Morto, (989)he sgle mache early/tardy problems Maagemet Scece, 35(2): 77 9 Pawalker, S, Smth, M ad Sedma, A (982) Commo due-date assgmet to mmze total pealty for the oe mache schedulg problem Operatos Research, 30(2): Raco, DP ad Kawamura, MS (200) he sgle mache earless ad tardess schedulg problem: lower bouds ad a brach-ad-boud algorthm Computatoal ad Appled Mathematcs, 29(2): avakkol-moghaddam, R, Mosleh, G, Vase, M ad Azaro, A (2005)Optmal schedulg for a sgle mache to mmze the sum of maxmum earless ad tardess cosderg dle sert Appled Mathematcs ad Computato, 67(2): Yag, S ad Hsu, C (200) Sgle-mache schedulg wth due-date assgmet ad agg effect uder a deteroratg mateace actvty cosderato Iteratoal oural of Iformato ad Maagemet Sceces, 2(2):77 95