Solving Scheduling Deteriorating Jobs with Rate Modifying Activity. Yücel Yılmaz Őztűrkoğlu

Transcription

1 Solvg Schedulg Deteroratg Jobs wth Rate Modfyg Actvty by Yücel Yılmaz Őztűrkoğlu A dssertato submtted to the Graduate Faculty of Aubur Uversty partal fulfllmet of the requremets for the Degree of Doctor of Phlosophy Aubur, Alabama May 9, 20 Keywords: schedulg, rate-modfyg actvty-deteroratg obs, sgle mache Copyrght 200 by Yücel Yılmaz Őztűrkoğlu Approved by Robert L. Bulf, Char, Professor Emertus of Idustral ad System Egeerg Jorge Valezuela, Assocate Professor of Idustral ad System Egeerg Emmett J. Lodree, Assstat Professor of Idustral ad System Egeerg

2 Abstract Tradtoal schedulg problems assume that the processg tme of a gve ob s always fxed. However, the processg tmes may chage real dustral applcats. To reflect the real world, we use two ew pheomeos ths study. The frst oe s kow as the deterorato ob where the ob processg tmes are defed by fuctos of ther startg tmes ad postos the sequece. Ad, the other oe s rate-modfyg-actvty s a actvty whch affects ad chages the producto rate of the maches. Ths study ams to determe the work sequece, the umber of breaks ad ther postos wth the work sequece whle cosderg the deterorato obs ad rate-modfyg-actvtes. I ths dssertato, three dfferet mathematcal models are troduced to solve the schedulg problems. The frst model addresses a classcal sgle mache problem wth makespa ad total flow tme obectves to cosder deteroratg obs ad rate-modfygactvty. The secod model troduces a sgle worker schedulg problem whch cosders workers physologcal factors whle determg the break tme. The thrd model troduces a b-crtera schedulg model wth deteroratg obs ad rate-modfyg-actvtes o a sgle mache. Also, several dfferet heurstc algorthms are developed for large-szed problems. For each mathematcal model ad heurstc, depedet expermets are performed to aalyze the effectveess of these algorthms.

3 Ackowledgmets Frst ad foremost I would lke to thak Dr. Robert L. Bulf for hs gudace, support edless patece ad beg my father Aubur. Wthout hm ad hs cotuous gudace, I ever would have see the ed of ths study. Specal thaks go to Dr. Jorge Valezuela ad Dr. Emmett J. Lodree for ther valuable suggestos ad opos. Also, I would lke to thak Dr. Alce E. Smth for her support throughout my doctoral study. My scere apprecato s also exteded to all my freds who made my years Aubur memorable. I would lke to thak to my parets, Dr. Zek Ylmaz ad Dr. Caa Ylmaz for ther ulmted love ad cotued support; to my brother Dr. Yetk Z. Ylmaz ad my husbad Omer Ozturkoglu. Fally, my thaks go to my bggest motvator to complete ths study, my lttle prcess Duru. I dedcate ths dssertato to the memory of my ucle Dr. Chagr Cftc ad my gradma, Hale A. Cftc, who truly beleved me.

4 Table of Cotets Abstract... Ackowledgmets... Lst of Tables... v Chapter. Itroducto.... Backgroud Maor Cotrbutos Orgazato of the Dssertato... 4 Refereces... 5 Chapter 2. Sgle Mache Schedulg Wth Deteroratg Jobs ad RMA... 6 Abstract Itroducto Lterature Revew Problem Defto The Mathematcal Model Fudametal Propertes ad Specal Cases A Polyomal Alg. for Makespa wth Idetcal Processg Tmes The Heurstc Algorthms Heurstc for Makespa Heurstc for Total Completo Tme v

5 7. Computatoal Results Summary Refereces...37 Chapter 3. Schedulg Jobs to Cosder Physologcal Factors Abstract Backgroud The Mathematcal Model Complexty Result Heurstc Algorthm Numercal Example of Heurstc Summary Refereces Chapter 4. A B-Crtera Sgle Mache Schedulg wth Rate-Modfyg-Actvty Abstract Itroducto Lterature Revew Problem Descrpto Crtero Total Completo Tme Total Tardess Maxmum Tardess Number of Tardy Jobs Computatoal Expermets... 70

6 5. Data Geeratos B-crtera Approach Secodary Obectve Approach Weghted Method Coclusos Refereces Chapter 5. Coclusos v

7 Lst of Tables Chapter 2 Table Complexty Results of He et al. (2005)... Table 2 Average Ru Tme (sec.) for Total Completo Tme Table 3 Average Ru Tme (sec.) for Makespa Table 4 Comparso of Heurstc ad Mathematcal Model for Makespa Table 5 Comparso of Heurstc ad Mathematcal Model for Total Completo Tme Table 6 Comparso of Ru Tmes of Heurstc for Makespa wth 50 Jobs... 3 Table 7 Comparso of Ru Tmes of Heurstc for Total Completo Tme wth 50 Jobs Table 8 Results of Heurstc 2 for Total Completo Tme Table 9 Number of Rmas versus Factors for Total Completo Tme Obectve Table 0 Total Ru Tme versus Factors for Total Completo Tme Obectve Chapter 3 Table Work-Rest Problems Table 2 The Parameters of the Problem Table 3 Parameter Settgs for Recovery Rate Table 4 Parameter Settgs for Usage Rate Table 5 Parameter Settgs for Cumulatve Chages Table 6 Parameter Settgs for Acceptable Lmts Table 7 Average Ru Tme for Total Completo Tme (sec.)... 50

8 Table 8 Average Ru Tme for Makespa (sec.) Table 9 Basc Iformato for the Numercal Examples Table 20 Expermetal Factors Table 2 Comparso of Heurstc ad Mathematcal Model for Makespa Table 22 Comparso of Heurstc ad Mathematcal Model for Total Completo Tme Chapter 4 Table 23 The Parameter of the Problem... 7 Table 24 Average Ru Tme (sec.) for / Table 25 Average Ru Tme (sec.) for / ( ) p, rm/ Tmax U ( ) p, rm/ T F Table 26 Weghted Method for / ( ), rm/ f ( T, F) Table 27 Weghted Method for / ( ), rm/ f ( T, U) p p max v

9 CHAPTER INTRODUCTION. Backgroud The competto betwee compaes the same dustry has become vtal due to the recesso the atoal ad the teratoal ecoomy. Nowadays, the cost of producto ad the respose to the customers requremets are becomg more mportat o a daly bass for may of the dustres to be successful the market place. So, ths leads them to redesg ther processes ad obs. Hece, reducg the completo tme of the product becomes oe of the most mportat factors ob desg. I ths study, to be able to reduce the completo tme of the products, we focus o the sequecg ad schedulg of obs, as well as some beeft actvtes. I both maufacturg ad servce dustres, obs mght have varable processg tmes. Delayg a ob may result addtoal tme to complete t, such as cleag drty dshes the loger they wat, the harder they are to clea. These kds of obs are kow as deteroratg obs the lterature; f processed later they take more tme tha whe processed earler. I other words, deteroratg obs are tasks whch eed more tme ad effort to complete the process tha whe they are doe earler. Gupta ad Gupta (988), ad Browe ad Yechal (990) tated studes o schedulg deteroratg obs. Some examples of deteroratg obs are searchg for a eemy uder growg darkess, treatg a patet uder worseg health codtos or producg steel uder decreasg ove temperature (Mosheov 996).

10 If a ob deterorates, there s a actvty called a rate-modfyg actvty (rma) whch recovers the lost tme of processg for a ob because of the deterorato. Lee ad Leo (200) frst troduced the rma whch chages the producto rate of a processor such as maches ad workers. I the schedulg lterature, a rma s defed as a mateace or repar actvty of a mache. So, whe a rma s take, there s some mprovemet o processg of that ob,.e., ts processg tme decreases. I ths study we deal wth two smlar problems of schedulg a set of deteroratg obs o a sgle mache. We propose three dfferet models whch reflect real lfe stuatos whch the processg tme of a ob creases or decreases depedg o ts tal processg tme ad other actvtes such as mateace or repar of a mache, break for workers, etc. I our frst model, we schedule deteroratg obs o a mache ad a rma as a mateace actvty. I our secod model we schedule tasks processed by a huma worker where the rma as a break gve to the worker order to let hm/her recover. Sce our processor s a worker, we cosder some physologcal factors whch may affect processg of the ob ad cause deterorato of the ob. I the last model, we have b-crtera obectves to satsfy both customers ad maagers wth real lfe assumptos. Therefore, maagers should cosder more tha oe measure whe they try to fd the best schedule for ther producto process. The geeral problem we deal wth ths study s to determe the best work sequece, the umber of rmas ad the posto of each rma for our specfed model assumptos ad obectves. 2. Maor Cotrbutos Ths study dffers from exstg studes several ways. Cotrbuto. We develop the frst mathematcal model to schedule deteroratg obs ad 2

11 rma to reflect real dustral stuatos. Aother dfferece of the frst model s our processg tmes. Specfcally, we cosder olear deterorato whch depeds o the posto. Up utl ow, all other research papers have cosdered tme-depedet deterorato. Cotrbuto 2. Aother mportat cotrbuto of ths research s that we cosder physologcal factors of workers whe schedulg the obs. There has bee o research whch smultaeously cosdered deteroratg obs ad rma uder some physologcal costrats. Except for Lodree ad Geger (200), to fd a optmal sequece posto for a rma, ergoomc factors have bee mostly gored schedulg problems. So, ths research acts as a brdge whch coects schedulg wth ergoomc ssues. Thus, ths study s a mportat mlestoe. Cotrbuto 3. I our secod model, we assume that obs deterorate because of worker fatgue. We defe rate modfyg actvtes as a restg perod of workers. All other studes use rma as mateace or repar actvty of a mache. As workers tre, the ob processg tme creases. I the exstg lterature, obs deterorate because of the watg tme. Hece, as that kd of ob wats the queue to be processed, ts processg tme creases. Cotrbuto 4. The lterature s rch for ob schedulg problems whe we cosder ether deteroratg obs or rma.but these are ot drectly related to customer ad maager satsfacto at the same tme. We propose the frst b-crtera model to cosder rma ad deteroratg obs smultaeously. I our research the followg three maor questos are addressed to mmze the completo tme of all gve obs: ) How should obs be scheduled? 3

12 2) How may rmas are eeded? 3) Where are these rmas the schedule? These questos are addressed for the problem wth ad wthout physologcal factors. 3. Orgazato of the Dssertato The rest of the dssertato s orgazed as follows. I Chapter II, we preset a mathematcal model for schedulg deteroratg obs wth rma. A exteso of the model, whch cosders physologcal factors s dscussed Chapter III. I Chapter IV, we dscuss a b-crtera schedulg model. Cocluso ad future work are provded Chapter V. 4

13 Refereces. Browe S. ad Yechal U. (990). Schedulg deteroratg obs o a sgle processor. Operatos Research, 38, Gupta, J.N.D. ad Gupta, S.K. (988). Sgle faclty schedulg wth olear processg tmes. Computers ad Idustral Egeerg, 4, Lee, C.Y. ad Leo, V.J. (200). Mache schedulg wth a rate-modfyg actvty. Europea Joural of Operatoal Research, 28, Lodree, E. J. ad Geger, C.D. (200). A ote o the optmal sequece posto for ratemodfyg actvty uder smple lear deterorato. Europea Joural of Operatoal Research, 20 (2), Mosheov, G. (996). Λ-Shaped polces to schedule deteroratg obs. Joural of the Operatoal Research Socety, 47,

14 CHAPTER 2 SINGLE MACHINE SCHEDULING WITH DETERIORATING JOBS AND RATE- MODIFYING-ACTIVITIES Abstract I ths study, we exame the schedulg a set of deteroratg obs o a sgle processor. We propose a model whch reflects real lfe stuatos whch the processg tme of obs chage depedg o ts tal processg tme ad actvtes, such as mache mateace or a break for workers. A rate-modfyg-actvty (rma) s a mateace actvty gve to a mache to restore t to ts orgal state. The geeral problem we deal wth ths study s to determe the ob sequece, the umber of rmas, ad rma postos wth the work sequece. We formulate a uque teger program to solve ths model for makespa ad total completo tme obectves. We also propose effcet heurstc algorthms for solvg large sze problems for both makespa ad total completo tme obectves. Polyomal algorthms for several specal cases are derved.. Itroducto I the last four decades, schedulg researchers have prmarly cocetrated o problems wth a stadard set of assumptos. Oe of these assumptos s that processg tmes of the obs are costat. But realty, processg tmes may chage due to varous factors such as deterorato ad wear pheomea. 6

15 A deteroratg ob ca be defed as a ob whch takes more tme whe processed later tha whe processed earler. I our study, a crease processg tmes s caused by mache deterorato. After a whle, the mache eeds more effort to accomplsh the task, ths s descrbed as the deterorato rate of obs. Costat speed of maches or fxed processg tmes ca be chaged by ratemodfyg actvtes (rma). The rma was frst troduced by Lee ad Leo (200). The processg tmes of the obs vary depedg o whether a ob s scheduled before or after the rma because the rma lets the mache or worker recover. After maches are mataed, they ted to have dfferet speeds tha before. I our study we defe a rma as a mateace actvty durg whch the mache stops for a gve perod of tme. After a rma s completed, the capablty of the mache s expected to retur to ormal. I our study, the schedulg model we propose cludes both deteroratg obs ad rma. More specfcally, we develop a mathematcal model to determe ob sequeces ad placemet of breaks uder makespa ad total completo tme. We follow the three feld otato troduced by Graham et al. (979) to descrbe schedulg problems. Ths otato s, where deotes the worker/mache codto, dcates the characterstcs of the problem ad shows the performace measure. Hece, we study p ( ) p, rm C ad p ( ) p, rm Cmax. I ths otato p,, rm ad C represet actual processg tme, deterorato rate, rma ad completo tme of obs respectvely. The remader of ths chapter s orgazed eght sectos. I Secto 2, a lterature revew s gve. The problem defto s Secto 3. A mathematcal model s preseted Secto 4. I Sectos 5, a algorthm for makespa s preseted. Heurstcs for both 7

16 obectves are gve Secto 6. Computatoal results are gve the Secto 7. Fally, the coclusos are preseted the last secto. 2. Lterature Revew Classcal mache schedulg problems have bee wdely studed by may researchers. Recetly, researchers have started to gve more atteto to schedulg problems wth dfferet characterstcs cludg deteroratg obs, learg effects or rate-modfyg actvtes. Makespa, total completo tme, total weghted completo tme, maxmum lateess, maxmum tardess ad umber of tardy obs are the most commoly studed performace measures. Schedulg deteroratg obs was frst troduced by Gupta ad Gupta (988), ad Browe ad Yechal (990). Gupta ad Gupta (988) troduced a schedulg model wth varable processg tme of a ob whch s a polyomal fucto of ts tal processg tme. The Browe ad Yechal (990) metoed deteroratg obs; ther processg tmes crease whle they awat servce. They cosdered that a processor loses ts effcecy wth a certa rate as soo as t fshes ts operato. I ther model, all obs are avalable at the begg wth ther tal processg tmes. If the processg of a ob s delayed, the the requred tme to process that ob creases learly based o ts tal processg tme. They costructed a schedulg problem to mmze the makespa for obs o a sgle mache. Mosheov (99) cosdered the problem of mmzg total completo tme of obs wth dfferet deteroratg rates ad foud that the optmal sequece of ths problem s V-shaped. V-shaped schedulg dcates that obs are arraged descedg order of growth rate f they are placed before the mmal growth rate ob, ad ascedg order f placed after t. (Mosheov (99). Ca et al. (998) developed a fully polyomal tme approxmato 8

17 scheme to mmze makespa for deteroratg obs. Also Kubak ad Vede (998) vestgated the computatoal complexty of makespa uder deterorato. They developed a heurstc ad brach-ad-boud algorthm for the problem. Kovalyov ad Kubak (998) preseted a fully polyomal approxmato scheme for a sgle mache schedulg problem to mmze makespa of deteroratg obs. Cheg ad Dg (2000) studed a sgle mache to mmze makespa wth deadles ad creasg rates of processg tmes. They foud that both problems are solvable by a dyamc programmg algorthm. Bachma ad Jaak (2000) cosdered a sgle mache schedulg problem mmzg maxmum lateess uder lear deterorato. They preseted two heurstcs ad proved that the maxmum lateess problem s NP hard. Bachma et al. (2002) showed that total weghted completo tme s NP hard for sgle mache schedulg whch the ob processg tmes are decreasg lear fuctos depedet o ther start tmes. These models all have the processg tme of a ob as a fucto of ts start tme. Noe of these works are vald for posto based deterorato. Rma s a pheomeo schedulg problems appearg the last decade. I the schedulg lterature, rma s defed as a mateace or repar actvty whch mproves the codto of the mache. Q et al (997) cosdered a problem where multple mateace actvtes eed to be scheduled wth obs o a sgle mache. Also Lee ad Che (2000) scheduled obs ad mateace actvtes to mmze total completo tme o a set of obs o parallel maches. They proposed brach-ad boud algorthms for solvg medum szed problems. Lee ad Leo (200) troduced a dfferet perspectve of schedulg mateace actvtes. They studed a schedulg problem wth mateace actvtes whch s commoly foud electroc assembly les. Oe of the ma decsos ther model addresses s 9

18 whether to stop the mache ad fx the problem or to let t work wth a lower producto rate. Hece, processg tmes of obs may chage after a mateace actvty. They also studed varous performace measures cludg makespa, total completo tme, total weghted completo tme ad maxmum lateess. They developed polyomal algorthms for solvg problems of mmzg both makespa ad total completo tme. I addto, they developed pseudo-polyomal algorthms to solve the total weghted completo tme problem. They used the start tme of the mateace actvty as a decso varable ther model. However, ther model does ot clude the possblty of mache breakdows. Lee ad L (200) studed sgle mache schedulg problems volvg repar ad mateace actvtes whch they also called rma. They focused o two types of processg cases whch are resumable ad oresumable. Ther obectve fuctos sought to mmze the expected makespa, total expected completo tme, maxmum expected lateess, ad expected maxmum lateess respectvely. If they decde ot to do mateace actvtes for the problem of mmzg the expected makespa ad the total expected completo tme, they obtaed these terestg results: ) whe the cumulatve dstrbuto fucto of x, F(x), whch dcates that mache breaks dow f there s o mateace actvty, s cocave the the sequece of the obs s SPT (shortest processg tme) order; ) f F(x) s covex, the the sequece of the obs s LPT (largest processg tme) order. He et al. (2005) studed a sgle mache to mmze makespa ad total completo tme of obs. They assumed that a rma s ot always vald because a actvty eeds some addtoal resources such as operators ad equpmet. These resources may ot be avalable all the tme. Thus, they cosdered the problem wth a restrcted rma. If the rma must be performed, they called t madatory (ma.); otherwse, t s called optoal (opt.). They 0

19 aalyzed the computatoal complexty of both makespa ad total completo tme. Table presets ther complexty results. Table. Complexty Results of He et al. (2005) / rm, opt / C NP hard f all Pseudo-polyomally solvable max max O( ms ) m tme 2 / rm, ma/ C NP hard f all FPTAS rug O ( m / ) C / rm, opt / NP hard f all, C Ope f / rm, ma/ NP hard f all, Ope f Ope geeral ad pseudopolyomally solvable for the agreeable rate case O ms ) tme ( m Ope geeral ad pseudopolyomally solvable for the agreeable rate case O ms ) tme ( m To mmze makespa, they preseted a pseudo-polyomal tme algorthm ad a fully polyomal tme approxmato scheme (FPTAS). To mmze the total completo tme, they proposed a pseudo-polyomal algorthm as a specal case. Whe they fxed the start tmes for the rma, avalablty costrats ca be appled. There has bee lttle research that smultaeously cosdered tme-depedet processg tmes ad rma. Lodree ad Geger (200) tegrated tme depedet processg tmes ad rma for assgg a sgle rma to a posto. They showed that a sgle rma should be serted the mddle of the optmal ob sequece to mmze makespa. To the best of our kowledge, except for the recet study of Lodree ad Geger (200), the schedulg problem wth the effects of deterorato ad rma has ot bee studed the lterature. We also dffer that our deterorato depeds o ob posto rather tha

20 start tme. 3. Problem Defto The problem we study ths paper s to schedule a set of depedet obs J J J J,..., ad oe or more rma s for a sgle mache. All obs are avalable for, 2 processg at all tmes. The mache (worker) ca do oly oe ob at a tme. Each ob has a deterorato rate α whch reflects a delay tme (worker s fatgue) from processg obs. We assume that the deterorato rate α has the same effect o processg tmes of dfferet obs ad t chages the processg tme of the ob olearly based o ts posto. Let us defe model parameters ad varables as follows: Model Parameters: s the umber of obs to be sequeced dcates the posto umber whch s from to k dcates the posto umber whch s from 0 to (k=0 s tal posto) dcates the ob umber whch s from to 0, s the costat deterorato rate of obs whe delayed by oe posto. q s the fxed perod of tme to perform a rma. p s the tal processg tme of ob before deterorato. p s the processg tme of ob f doe postos after a rma or the tal posto,.e. p p (3.) 2

21 Model Varables: x k y 0 0 f ob s the th posto after a rma otherwse f a rma s doe before posto, otherwse zero whch s doe ust before posto k, C Completo tme of the ob posto. C max Completo tme of the ob the last posto. I addto, our model assumptos are gve as the followg: There s oly oe mache (worker). The deterorato of a ob depeds o ts posto. Jobs are o-preemptve. After a rma, obs revert to ther tal/base processg tme p. That meas the mache (worker) recovers completely after a rma (00% recovery). Deterorato process s the same after a rma. I our research the followg three questos are addressed to mmze the completo tme or makespa of all gve obs: ) How should obs be scheduled? 2) How may rmas are eeded? 3) Where are rmas the schedule? 3

22 4. The Mathematcal Model As metoed before, we cosder two performace measures: total completo tme ad makespa. Hece, depedetly our obectve fucto s to mmze each of those performace measures our models. Mmze Z C,.., (total completo tme) or Mmze Z Cmax, Cmax C,.., (makespa) The related costrats wth our model are gve as follows. C p x 0 (4.) C C p k x,, k q y 2,..., k (4.2) k0 x,..., (4.3) k k 0 x k,..., (4.4) x k 2...,,...,,..., k (4.5) k y 4

23 x 0,,...,,..., k 0,..., (4.6) k y 0, k 2,..., (4.7) k C 0,..., (4.8) I costrat (4.), the completo tme of the ob posto s equal to the processg tme of the ob assged to posto. Before the frst posto, there s o rma ( y 0 ). I costrat (4.2), the completo tme of the ob posto s equal to the completo tme of the ob posto plus the processg tme of the ob assged to posto plus the rma tme f assged. I costrat (4.3), each ob s assged to exactly oe posto. I costrat (4.4), each posto s scheduled for oly oe ob. Costrat (4.5) requres a rma to be doe the related posto f obs are scheduled after rma ad to cotrol the sequece of the rma. Costrats (4.6), (4.7) ad (4.8) show that the varables should be bary ad o-egatve. 5. Fudametal Propertes ad Specal Cases I ths secto, we develop some fudametal propertes ad develop polyomal algorthms for the ut processg tme problem. Frst, we exame the problem wthout rmas Theorem ad the develop the result wth a rma Theorem 2. Theorem. For / p p / C max, LPT sequecg mmzes the makespa. 5

24 Proof: Suppose schedule S mmzes makespa ad s ot LPT order. The there must be a par of obs S, say ob ad ob, wth ob mmedately after ob the k th ad st k postos, ad p p after ob. Let C(A) ad C(B) Now cosder the schedule. Let B be the set of obs before ob, ad A the set of obs be the sum, of processg tmes sets A ad B respectvely. ' S, where ' S the same as S except ob ad have bee terchaged ad the sets of obs A ad B are the same posto both schedules. S,,, ' ad ',,, ' The makespa for S s; S where ad ' deote partal sequeces. C S CB p p CA C ad the makespa for 2 B p p CA (5.) ' S s; C ' S CB p p CA C 2 B p p CA (5.2) Subtractg equato (5.) from (5.2) we get C ' 2 S CS p p 0 Ths mples the makespa of ' S s smaller tha S, whch cotradcts the assumpto that S was optmal. Therefore, a optmal soluto must be LPT., Theorem 2. For / p p rm / C max the sequece of the obs a optmal soluto s always LPT order betwee ay gve par of rmas. 6

25 7 Proof: Let B C be the total completo tme of the obs before gve rma ad A C be the total completo tme of the obs after gve secod rma. Now cosder the schedule },,,,,, { ' rma k rma S (SPT order) ad ' ',,,,,, rma k rma S (LPT order) where ad ' deote partal sequeces. The sets of obs A ad B are the same posto both schedules. We assume that schedule S s a optmal schedule. ' ',,,,,, rma k rma S. Let us assume that, k p p p ad after rma the frst ob assged s the th ) ( posto. The m s defed as a durato of mateace actvty (rma). The makespa for S s; (5.3) ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 max max max max A C m p p p m B C S C p p p m B C S C p p m B C S C p m B C S C k k ad the makespa for ' S s; (5.4) ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ' 2 max ' max ' max ' max A C m p p p m B C S C p p p m B C S C p p m B C S C p m B C S C k k k k Subtractg, equato (5.3) to (5.4) we get; 0 ' max max k p p S C S C The schedule ' S s better tha schedule S whch cotradcts the assumpto that S was optmal. Ths meas that betwee gve two rma, the schedule s always LPT order, lke schedule },,,, { ' ' k S.

26 5.. A Polyomal Algorthms for Makespa wth Idetcal Processg Tmes, I ths secto, we gve a polyomal algorthm for / p p rm / Cmax whch s a specal case wth p p. Suppose we have a gve umber of rmas, say m. The ay schedule ca be dvded to m groups of obs scheduled betwee rmas ad the begg ad of the schedule. Let the rmas be scheduled before postos k k,..., k, 2 m. Let d r m ad k d, k k k d d 2,..., m r m r,..., m A schedule wth the m rmas before postos k,,..., m s called a balaced schedule. Ths mples the umber of obs each of the m groups s as equal as possble, ether havg d or d obs. If r 0 schedule s perfectly balaced., each group has exactly d obs ad we say the Theorem 3. Balaced schedules are optmal for / p p, rm m / max C. Proof: We wll assume a ubalaced schedule s optmal ad show a cotradcto. The makespa for a schedule wth rmas at k,...,, k2 km s; C k k2 k km km max p( ) q p( )... q p( ) A balaced schedule has m r groups wth d obs ad r groups wth d+ obs. A ubalaced schedule must ether have more tha r groups wth d+ obs or at least oe 8

27 group wth d+2 obs. Assume a ubalaced schedule of the frst type has mmal makespa. Wthout loss of geeralty; let the group before k cosst of d- obs ad the group after have d+ obs. The cotrbuto to makespa of these two groups wll be; M ( S) d p p( ) d p( ) p( )... p( ) d 2 p p( )... p( ) d Now let ' S be a detcal schedule to S except oe ob betwee k ad k2 s moved betwee k ad k. The M ( S ' ) d p( ) p( ) p p( )... p( ) d d p p( )... p( ) d ad ' d d M ( S) M ( S ) p( ) ( ) 0, So S caot be optmal. The other case ca be show o-optmal the same way. To solve / p p, rm m possble value of m. Ths s doe Algorthm. / C max, we create a balaced schedule for each Algorthm Step 0. Set m 0, C. max Step. Create a balaced schedule wth m rmas ad optmal makespa C ( m) If C m) C, C C ( m) max ( max max max max If m, stop Step 2. Set m m, ad go to Step. 9

28 Step requres at most O() operatos ad t wll be repeated at most tmes, so the algorthm s O ( 2 ). 6. The Heurstc Algorthms Although our mathematcal model, whch we dscussed the prevous secto, ca solve some problems a reasoable ru tme, larger problems (e.g. >50) are dffcult to solve wthout cosderable computatoal effort. Therefore, we propose heurstc algorthms to solve large problems. 6.. Heurstc for Makespa For problems wth a large umber of obs, Heurstc provdes a soluto very close to the optmal solutos. Let us defe; b s m Job wth largest processg tme wth the set of uassged obs. Job wth smallest processg tme wth the set of uassged obs. Potetal posto umber after rma or tal posto. J [] Job posto y 0 f a rma s placed before posto otherwse p Processg tme of ob posto as k p p C Completo tme of the ob posto The proposed heurstc frst apples the LPT rule to sort the obs. The ob whch has 20

29 the largest processg tme s assged to posto. The calculate the cremetal amout of processg tme of the frst ob whch has smallest processg tme. If ths amout s greater tha the rma tme (q), the do a rma ad assg the ext largest ob to the curret posto mmedately after the rma. Otherwse do ot do a rma ad assg the smallest ob to the curret posto wthout dog a rma. The procedure for the proposed heurstc for makespa problem s stated as follows: Heurstc Step. Order the obs descedg order (LPT) of processg tmes p. set b max p,.., Step 2. Assg the ob wth largest processg tme to posto ad o rma at the begg y 0, b, s Set C p, J[ ] b b b, s ad k 2. Step 3. Calculate; D p s p [ k] s[ k] If D q go to Step 4 y, b J[ k ] Set C C p b q k k, b b Go to Step 5. 2

30 Step 4. y 0, s J[ k ] Set C C p, s s s, m m ad k k Step 5. If k, Stop the algorthm, Makespa C [ ],otherwse go to Step 3. Ths algorthm s O ( log ). It s well kow that Step, ca be completed O ( log ). I Step 2, there s oly oe terato ad ths terato ca be completed O(). There are at most - teratos step 3 ad 4 ad go through oce for each ob. Each terato ca be completed O ( ). Lastly, Step 5 ca be completed O () Heurstc for Total Completo Tme The heurstc s developed for the problem of total completo tme wth gve multple rma s. I the proposed heurstc, let J be a set of obs, J J, J,... J } ad R be a set of { 2 m rmas, R R, R,... R } represets the posto of the last ob before the rma. Let S { 2 m represet a schedule of obs ad m rmas o a sgle mache, S {( J, J 2,... J )( R, R2,... Rm )}. CT m s the total processg tme of the assged obs the part m ad assume { CT, CT2,..., CT m } s the separate total processg of obs each part. The steps of the developed heurstc are gve below: 22

31 Heurstc 2 Step : Splt S to m parts so that each part cludes a set of obs before each gve cosecutve rma. S { S, S2,... Sm }. Ad assume m{ CT, CT2,..., CT m } s the separate total completo of obs each part. Step 2: Order obs based o LPT rule. Step 3: Assg oe ob, whch has the largest processg tme the uscheduled lst of obs, to each part. S { J},{ J 2},{ J m} Step 4: Calculate total completo tme of the each part. CT p, CT 2 p2, CT m pm Step 5: To assg the other obs each part, frst select m{ CT, CT2,..., CTm} ad the start to assg the ext ob to m{ct }. S { J, J m4},{ J 2, J m3},{ J m, J m2} Ad calculate ew total completo tme of the each part wth addg deterorated processg tmes. CT p p, m4, CT2 p2 p, m3, CTm pm p, m2 Step 6: Repeat Step 5 utl all obs are assged. Hece, the postos of the rma are aturally determed based o the scheduled obs each part. The best schedule s the S *, whch gves the smallest flowtme. 23

32 7. Computatoal Results I ths secto, we coduct three expermets to aalyze the effectveess of our model. Expermet focuses o the computatoal tme to solve the proposed mathematcal model for mmzg makespa ad total completo tme. I Expermet 2, we compare the computatoal effectveess of all of the heurstc algorthms wth the proposed mathematcal model. I Expermet 3, a expermetal desg s bult to estmate the relatoshp betwee parameters of the problem. To coduct our aalyses o the proposed models, we detfy four expermetal factors: deterorato rate ( ), RMA tme (q), mea processg tme (M) ad varace of processg tme (V). Also, we specfy three levels for each factor. So ths s a 3 4 expermetal desg wth sx two-factor, four three-factor ad oe four-factor teractos. The defed levels of those factors the expermets are: ) : 0.02, 0.04 ad ) q : 5, 0 ad 5 3) M : 20, 40 ad 80 4) V : 0.20, 0.40 ad 0.80 Usg the varace ad mea of processg tme, we produced a terval for each combato; [8, 22], [0, 30], [, 40], [36-44], [20-60], [-80], [72-88], [40-20] ad [- 60]. We tested our models for 50 obs, ad 80 staces (0 replcatos) for each expermet. 24

33 Expermet : Performace of the Mathematcal Model The proposed mathematcal model s coded usg AMPL ad solved by CPLEX 9. o a computer wth a Petum IV 2.8 GHz processor ad GB of RAM. Te replcatos of each of the combato were ru for each performace measure. The average computatoal tme ( secods) of each of the combato for each obectve s gve Table 2 ad Table 3, respectvely. Table 2. Average Ru Tme (sec.) for Total Completo Tme Jobs umb. () Det. Rate ( ) Rma tme (q) Ave. Tme (sec.)

34 Table 3. Average Ru Tme (sec.) for Makespa Jobs umb. Det. Rate Rma tme () ( ) (q) Ave. Tme (sec.) The effcacy of the model s based o the average ru tme secods. The average tme for 50 obs for total completo tme s secods ad for the makespa t s secods. As see the tables, as the rma tme creases, ru tme for both models crease. We expect ths result because creasg the rma tme meas fewer rmas ad obs deterorate more. For example, f the rma tme s very small compared to deterorato, we expect the model to have may rmas. Whe the deterorato rate creases wth the fxed rma tme, ru tme for both models decreases for the same reaso. Also, the problem of mmzg total completo tme requres less tme tha that of makespa. Expermet 2: Performace of the Heurstc Algorthms The put parameters are the same as Expermet. The heurstc algorthms were coded Java. The soluto qualty percetage error s calculated by usg the followg 26

35 equato: e F F F * 00 h opt / opt I ths equato, F h s the measure of the heurstc model ad F opt s the measure of the optmal soluto obtaed by the mathematcal model. Whe we tested our models for 0 replcatos for each specfc set of codtos, we obtaed the average percetage errors of the heurstc models, whch are gve the Table 4. 27

36 Table 4. Comparso of Heurstc ad Mathematcal Model for Makespa Ave. Error of Heurstc (%) Mea Rages of Process Tme α RMA tme [8,22] [0,30] [,40] [36,44] [20,60] [,80] [72,88] [40,20] [,60] The average percetage error of all 80 staces s calculated as 2.06 % wth the worst stace 5.85% for makespa. We also ra Heurstc for the total completo tme obectve. 28

37 Table 5. Comparso of Heurstc ad Mathematcal Model for Total Completo Tme Ave. Error of Heurstc (%) Mea Rages of Process Tme α RMA tme [8,22] [0,30] [,40] [36,44] [20,60] [,80] [72,88] [40,20] [,60] The average percetage error of all 80 staces s calculated as.85 % wth the worst stace 4.39% for total completo tme. If the deterorato rate s fxed ad the rma durato creases, average percetage error creases. Whe the rma tme creases, 29

38 schedulg the obs based o the order results a larger error tha schedulg the ob by vestgatg each ob separately. O the other had, f the durato of the rma s fxed ad the deterorato rate rses, the average percetage error decreases. I ths case, obs are deteroratg more because of the creasg deterorato rate wth fxed rma tme. The dfferece betwee the requred addtoal tme due to the deterorated ob ad the rma tme coverges. Hece, the posto of the rma becomes less crtcal. Addtoally, whe the varato of the mea of processg tme creases, the average percetage error of all staces that group creases. Table 6 ad Table 7 shows the comparso of the computatoal tmes for both the proposed models for makespa ad total completo tme obectves. By comparg the heurstc model wth the mathematcal model, t s clear that the heurstc model requres much less computato tme tha the proposed mathematcal model. 30

39 Table 6. Comparso of Ru Tmes of Heurstc for Makespa wth 50 Jobs Rages of Process Tme [8,22] [36,44] [72,88] [0,30] [20,60] [40,20] [,40] [,80] [,60] Α Mathematcal Model (sec.) rma tme=0 Heurstc Model (sec.)

40 Table 7. Comparso of Ru Tmes of Heurstc for Total Comp. Tme wth 50 Jobs rma tme=0 Rages of Process Tme Α Mathematcal Heurstc Model Model (sec.) (sec.) [8,22] [36,44] [72,88] [0,30] [20,60] [40,20] [,40] [,80] [,60] Therefore, as the umber of obs creases, the computatoal tme for the mathematcal model creases dramatcally comparso to the heurstc model. For example, we tested both models wth 00 obs wth 5 replcatos. We used a deterorato 32

41 rate of 0.08, the rma tme of 5, ad the rages of processg tme s [,60] wth mea 80. Based o the results, the average percetage error s 0.369%. However, whle the averages ru tme for the mathematcal model s secods, the heurstc model s oly 46.5 secods. Ths shows that the heurstc model has reasoable error wth very short computatoal tme for a large umber of obs compared to the mathematcal model. Table 8. Results of Heurstc 2 for Total Completo Tme Ave. Error of = 50 Heurstcs (%) Rma Tme Det. Rate Num. of Rma Heurstc The results of the average percetage error for mmzg total completo tme to Heurstc 2 are preseted Table 8. Heurstc 2 gves very close to optmal solutos whle the average percetage error s 0.65%. Expermet 3: Expermetal Desg To estmate the effects of put factors of the model, a expermetal desg s bult. The put parameters are the same as prevous expermets. Table 9 shows the result of the expermetal desg for the problem of mmzg completo tme. 33

42 Table 9. Number of Rmas Versus Factors for Total Completo Tme Obectve Source D.F. SS MS F P Det. Rate (α) <0.05 sg. RMA Tme (q) <0.05 sg. Mea (M) <0.05 sg. Varace (V) <0.05 sg. α *q <0.05 sg. α *M <0.05 sg. α *V <0.05 sg. q *M <0.05 sg. q * V <0.05 sg. M* V <0.05 sg. α *q*m <0.05 sg. α *q*v <0.05 sg. α *M*V <0.05 sg. q *M*V <0.05 sg. α *q*m*v <0.05 sg. Error Total R 2 = 99.22% Table 9 clearly dcates that 99.22% of the varato the umber of rmas, whch s a ma varable of the model, s explaed by all factors. Table 0 shows that 54.46% of the varato total ru tme of the model s explaed by all factors. Ths s the evdeced by the fact that the teracto of some factors s ot sgfcatly dfferet. 34

43 Table 0. Total Ru Tme Versus Factors for Total Completo Tme Obectve Source D.F. SS MS F P Det. Rate (α) <0.05 sg. RMA Tme (q) <0.05 sg. Mea (M) <0.05 sg. Varace (V) <0.05 sg. α *q <0.05 sg. α *M ot sg. α *V <0.05 sg. q *M ot sg. q * V sg. M* V <0.05 sg. α *q*m <0.05 sg. α *q*v ot sg. α *M*V <0.05 sg. q *M*V <0.05 sg. α *q*m*v <0.05 sg. Error Total R 2 = 54.46% Accordg to the expermetal desg results, there are sgfcat dffereces amog deterorato rate, rma tme, mea processg tmes, varace of mea processg tmes ad ther teractos at the level of sgfcace of If oe of the factors s chaged, the optmal umber of rma ad ther sequece posto chages also. Furthermore, the chage the teractos betwee the model factors affects the result to a lesser degree. 8. Summary Ths paper vestgates a schedulg problem wth deteroratg obs ad ratemodfyg-actvty smultaeously. Frst, we preset a mathematcal model wth the obectve of mmzg the makespa ad total completo tme. Our model ca decde the sequece 35

44 whch obs should be scheduled, how may rmas to use, f ay, ad where to sert them the schedule. We show that, as the umber of obs creases the computatoal tme to solve the problem creases dramatcally for the mathematcal model. We provde polyomal tme algorthms for ut processg tme for makespa to solve the problem optmally. Several theoretcal proofs are proposed to solve the problem wth dfferet specal cases. We propose heurstcs for both makespa ad total completo tme. The, we preset some computatoal expermets. Accordg to the expermetal results, the performace of the proposed model ad the heurstcs are qute satsfactory. Also, the heurstc models gve a reasoable error for 50 obs whe compared to the mathematcal model. These heurstc algorthms are able to costruct ear optmal solutos wth much less computatoal tme for large umber of obs ( 50 ). 36

45 Refereces. Bachma, A., Jaak, A. ad Kovalyov, M.Y. (2002). Mmzg the total weghted completo tme of deteroratg obs. Iformato Processg Letters, 8(2), Bachma, A. ad Jaak, A. (2000). Mmzg maxmum lateess uder lear deterorato. Europea Joural of Operatoal Research, 26(3), Browe S. ad Yechal U. (990). Schedulg deteroratg obs o a sgle processor. Operatos Research, 38, Bureau of Labor Statstcs, US Dept of Labor (2009). Geerated report-number of ofatal occupatoal ures ad llesses volvg days away from work by selected dustry. Retreved from 5. Ca, J.Y. ad Ca, P. (998). O a schedulg problem of tme deteroratg obs. Joural of Complexty, 4, Cheg, T. C.E. ad Dg, Q. (2000). Sgle mache schedulg wth deadles ad creasg rates of processg tmes. Acta Iformatca, 36, He Y., J, M. ad Cheg, T.C.E. (2005). Schedulg wth a restrcted rate-modfyg actvty. Naval Research Logstcs, 52,

46 8. Graham, R.L., Lawler, E.L., Lestra, J.K. ad Rooy, K. A.H.G. (979). Optmzato ad approxmato determstc sequecg ad schedulg: a survey. Aual Dscrete Math., 5, Gupta, J.N.D. ad Gupta, S.K. (988). Sgle faclty schedulg wth olear processg tmes. Computers ad Idustral Egeerg, 4, Kovalyov, Y. M. ad Kubak, W. (998). A fully polyomal approxmato scheme for mmzg makespa of deteroratg obs. Joural of Heurstcs, 3, Kubak W. ad Vede, S. (998). Schedulg deteroratg obs to mmze makespa. Naval Research Logstcs, 45, Lee, C. Y. ad Che, Z. L. (2000). Schedulg of obs ad mateace actvtes o parallel maches. Naval Research Logstcs, 47, Lee, C.Y ad L, C.S. (200). Sgle-mache schedulg wth mateace ad repar rate-modfyg actvtes. Europea Joural of Operatoal Research, 35, Lee, C.Y. ad Leo, V.J. (200). Mache schedulg wth a rate-modfyg actvty. Europea Joural of Operatoal Research, 28,

47 5. Lodree, E. J. ad Geger, C.D. (200). A ote o the optmal sequece posto for a ratemodfyg actvty uder smple lear deterorato. Europea Joural of Operatoal Research, 20 (2), Mosheov, G. (99). V-Shaped polces to schedulg deteroratg obs. Operato Research, 39, Mosheov, G. (996). Λ-Shaped polces to schedule deteroratg obs. Joural of the Operatoal Research Socety, 47, Q, X., Che, T. ad Tu, F. (997). Schedulg wth the mateace o a sgle mache. Workg Paper, Departmet of Computer System Sceces, Naka Uversty, Cha. 39

48 CHAPTER 3 SCHEDULING JOBS TO CONSIDER PHYSIOLOGICAL FACTORS Abstract I ths paper, we study schedulg obs ad breaks for a sgle worker. The processg tmes of obs creases as the worker tres. I ths study, we assume that covetoal mache schedulg models do t always work for humas. So whle determg the break tme, we cosder the workers physologcal factors. The two obectves cosdered are total flow tme ad makespa. A exact mathematcal model s preseted wth some physologcal costrats. We prove the problem s NP hard by a reducto from Equal-Sze-Partto. Thus, we develop a heurstc algorthm to solve large problems. Numercal examples are preseted for uderstadg ad aalyzg the performace of the mathematcal model ad the heurstc.. Backgroud I classcal schedulg problems, the optmal sequece of obs o a sgle mache s equvalet to the optmal sequece of obs performed by a sgle worker. But ths s ot the case real lfe because the algorthm gores costrats related to a huma s physologcal factors ad lmtatos. Workers get tred both physcally ad metally whle they are dog ther ob. So, ths stuato causes reduced performace ad productvty of the workers. Fatgue s oe mportat reaso for decreasg performace. It ca be caused by 40

49 exteded workg hours, adequate rest perods ad usutable workg codtos. Accordg to Fatgue Maagemet System Gudeles (FMSG), a fatgued worker s ablty to perform ther task may be lost or mpared. Fatgued workers ca have; Reduced motvato Decreased speed of task Icrease memory errors Iablty to cocetrate Icorrect acto Koz (998) has declared that fatgue creases expoetally wth tme. Therefore, t s mportat to get rest before the fatgue level becomes too hgh. To prevet fatgue, workers eed adequate rest perods durg work perods. Breaks are desged to provde tme for workers to overcome the fatgue arsg from the work. Restg tme s classfed by Koz ad Johso (2004) as formal breaks (luch, coffee), formal break (terruptos, trag) ad mcro breaks (short pauses of a mute). Whe a break s gve, the worker s performace s expected to crease due to recovery. The recovery depeds o how fatgued the worker s whe the rest begs ad the legth of the rest. If the legth of the break s small, the cremetal amout of recovery s less tha the cremetal amout of tme. For example, 4 breaks of 5 mutes are ofte more useful tha break of 20 mutes for two reasos; fatgue wll ot be creased as much ad the recovery wll be greater. I ths study, we use rate- modfyg actvtes (rma) as a kd of restg actvty for workers. Rma was frst troduced the lterature by Lee ad Leo (200). They defed rma as a actvty whch alters the producto rates of maches. The rma plays a mportat 4

50 role work-rest schedulg the huma factors lterature. The ma dea of work-rest schedulg s to obta the umber, place ad durato of rest perods. Whle determg those decsos, productvty, safety ad comfort are cosdered. I the lterature, the obectves of the work-rest problem are to mmze fatgue ad to provde recovery of the worker whle ot reducg productvty. Boucse ad Thum (997) foud that short breaks are more effectve promotg recovery from both metal ad emotoal recovery. Table shows several researchers who studed the work-rest problem for varous physologcal varables such as heart rate, blood pressure, oxyge uptake ad electromyography (EMG) sgals. They foud that restg tme s mportat to recover from fatgue. Table. Work-Rest Problems Veltma ad Gallard (993) Heart rate, blood pressure Heart rate varablty (HRV), EMG sgals, electro Boucse (993) dermal actvty Imbeau et al. (995) Heart rate, Maxmum aerobc capacty (% VO 2 max ) Wu ad Wag (2002) % VO 2 max, Heart Rate Twar ad Gte (2006) Heart rate Hse et al. (2009) % VO 2 max Our schedulg model s based o oe developed by Ozturkoglu et al. (20). Ther model s to determe the work sequece, the umber of breaks ad the place of each break wthout cosderg the physologcal factors of the worker. I ths study, we cosder the huma characterstcs whle determg the work sequece, the umber of breaks ad the optmal break schedule. Ths study dffers from exstg research several ways. Frst, we defe ratemodfyg actvtes as a restg perod for workers; whereas all other studes use rate- 42

51 modfyg actvtes as mache mateace ad repar tme. Therefore, a rate modfyg actvty s a break gve to workers order to let them recover. Secod, there s o research that cosdered the workers physologcal factors whe schedulg the obs. Thrd, our study, deterorated obs are caused by fatgue of the worker, but the lterature, obs deterorate whle watg to process. Thus, tredess or fatgue of the worker s descrbed as the deterorato rate of obs. Whe we cosder the dustral sde, ths study ca provde sght o creasg worker effcecy to assged ob tasks. To the best of our kowledge, there s o research that proposed a task sequecg approach whch combes deteroratg obs, rma ad cosders the huma characterstcs. The chapter s orgazed as follows. I Secto 2, we propose a mathematcal model ad test t computatoally. I Secto 3, we determe the computatoal complexty of the problem whch s troduced Secto. We preset a heurstc algorthm for larger problems ad expermetal results of the heurstc sectos 4 ad 5 respectvely. Lastly, we offer a summary Secto The Mathematcal Model Ths model s motvated by the problem of maual order pckg actvtes warehousg systems. The problem s to determe the sequece whch the orders should be pcked mmum tme ad wth acceptable levels of huma physologcal factors. I addto, breaks ca be scheduled to allow workers to recover durg the order pckg sequece. We cosder sequecg orders (obs) o a sgle worker (processor) wth varyg processg speed due to the effects of varous ergoomc factors. We assume that a break tme (rma) ca be scheduled to mprove the worker state ay tme after the frst task s 43