JOB SCHEDULING WITH UNFIXED AVAILABILITY CONSTRAINTS

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1 JOB SCHEDUING WITH UNFIXED AVAIABIITY CONSTRAINTS Hoog Chui AU School of Computig Natioal Uivesity of Sigapoe 3 Sciece Dive 2, Sigapoe 7543 Phoe: lauhc@comp.us.edu.sg Chao ZHANG BuildFolio Techologies PTE TD 83 Sciece Pak D #3-4 Sigapoe 8258 Phoe: czhag@buildfolio.com ABSTRACT Stadad schedulig theoy assumes that all machies ae cotiuously available thoughout the plaig hoizo. I may maufactuig ad sevice maagemet situatios howeve, machies eed to be maitaied peiodically to pevet malfuctios. Duig the maiteace peiod, a machie is ot available fo pocessig obs. Hece, a moe ealistic schedulig model should take ito accout machie maiteace activities. I this pape, we study the poblem of ob schedulig with ufixed availability costaits o a sigle machie. We fist popose a pelimiay classificatio fo the schedulig poblem with ufixed availability costaits based o maiteace costaits, ob chaacteistics ad obective fuctio. We divide ou aalysis ito six cases. Fo each case, we peset complexity aalysis ad exact o appoximatio algoithms. We also peset esults of tabu seach o NP-had poblem istaces. Keywods: Maiteace, Availability, Schedulig.. INTRODUCTION Stadad machie schedulig liteatue assumes that machies ae available at all times. Howeve, i may maufactuig o sevice maagemet situatios, machies ae maitaied peiodically to pevet malfuctios. Duig the maiteace peiod, a machie is ot available fo pocessig obs. A moe ealistic schedulig model should take ito accout machie maiteace activities. The schedule of maiteace activities ca be detemied eithe befoe the schedulig of obs, o oitly with the schedulig of obs. I the fist case, the maiteace peiods ae kow ad fixed at the time whe obs ae to be scheduled. The poblem of schedulig obs with this type of maiteace educes to the poblem ofte efeed i the liteatue as schedulig with machie availability costaits because duig maiteace peiods the machie is ot available fo pocessig obs. ee ad ima [7] study a sigle machie poblem with a machie availability costait. ee [6] studies a paallel-machie poblem whee machies ae available statig fom diffeet times. ee ad ima [8] coside a two-paallel-machie poblem whee oe machie has a availability costait. Mosheiov [2] studies the same poblem by assumig that each machie is available duig a iteval. Note that all of these esults assume that the machie uavailability is kow ad fixed i advace. Schmidt [5, 6] studies a paallel machie poblem whee each machie has diffeet availability itevals. I [9], dyamic pogammig pocedues ae give fo two-machie flow shops with oe o-availability peiod ad o-peemptive opeatios. Espiouse & al. [4] have also cosideed o-peemptive tasks fo a o-wait flow shop poblem ad poposed heuistics with pefomace guaatee. I [], ee hadles the peemptive flow shop poblem with two machies ad oe uavailability peiod fist imposed o machie ad the o machie 2 with the makespa obective. ee poved that both poblems ae NP-had i the odiay sese, ad poposed heuistics with eo boudig aalysis. [] exteds the complexity esults foud i [] fo a geealized model whee a opeatio is completely (oesumable) o patially (esumable) estated if its executio is peempted by a maiteace task. He also developed heuistics with a eo boud aalysis. Blazewicz & al [] ivestigated the 2-machie poblem with abitay umbe of uavailability peiods o oe machie, ad poved that the poblem is stogly NP-had. Also, they poposed a bach ad boud algoithm fo the poblem with abitay umbe of uavailability peiods o both machies. Cheg ad Wag [2] ivestigated the poblem studied i [], i which a availability costait is imposed oly o the fist machie. They poposed a heuistic that impoves the wost-case eo boud povided by ee []. ee [9] cosides schedulig poblems with availability costaits ude diffeet machie cofiguatios ad pefomace measues. Accodig to the case, he has povided a polyomial algoithm to solve the poblem, o poved that the poblem was NP-had ad developed pseudo-polyomial algoithms o heuistics with a eo boud aalysis. Besides, ee itoduces a way of givig the eo boud fo o-esumable case. Istead of dealig with oesume case diectly, he cosides esumable case fist, ad uses it to obtai a eo boud fo o-esumable case. I ou expeiece with idusty applicatios, especially i semicoducto maufactuig, it is ofte obseved that a machie is idle while waitig fo maiteace pesoel to pefom pevetive maiteace, eve though obs ae waitig ad the machie has ot boke dow. This is due to lack of coodiatio betwee poductio plaig ad maiteace opeatios. Obviously, a caeful coodiatio betwee maiteace ad ob pocessig would esult i a bette oveall schedule. Hece the schedulig poblem to be studied i this pape is to detemie simultaeously whe to pefom each maiteace activity ad whe to pocess each ob so that a cetai pefomace measue is optimized. ittle eseach has bee doe i the liteatue fo the model of oitly schedulig maiteace 44

2 activities ad obs. Qi, Che, ad Tu [4] coducted a study o schedulig the maiteace o a sigle machie. I thei model, thee is a costait o the imum allowed cotiuously wokig time T of the machie. Gaves ad ee [5] cosideed a special case whee the sum of pocessig time is less tha 2T of this model. These studies addessed the poblem of maiteace, but oly i a limited way. They cosideed costaits oly o the imum legth of a available iteval, ad the obs scheduled ae without elease times ad deadlies costaits, which is iadequate fo solvig eal poblems. Oe ca imagie fo istace that it is ueasoable to schedule a maiteace immediately afte aothe oe. Hece, we eed to coside costaits o the miimum legth of a available iteval. I this pape, we will study the sigle machie poblem whee thee ae costaits eithe o the legth of available itevals o the umbe of obs completed befoe statig a ew maiteace, ad we eed to schedule both obs ad maiteace activities. We tem this the Job Schedulig with Ufixed Availability Costaits poblem (JSUA). We will povide a classificatio desiged to emulate the classical Job Schedulig Poblem, whee the thee key paametes ae maiteace popeties, ob chaacteistics ad obective fuctio. 2. PREIMINARIES Followig schedulig temiology, each poblem is defied o thee fields α/β/µ, whee α deotes the maiteace costait, β deotes the ob chaacteistic, ad u epesets the pefomace measue to be optimized. Each field has the followig possible cases: Fo α, Case : Costaits o the legth betwee two adacet maiteaces. Hee, we ae give two positive values A ad B, ad the legth betwee two adacet maiteaces must be betwee A ad B. Notatio: A-B/ _ / _ Fo β, Case : Case 2: Fo µ, Case : Case 2: obs do ot have elease time o deadlie. Notatio: _ / _ / _ obs have elease time. Notatio: _ / / _ legth of the schedule. Notatio: _/_ / MakeSpa umbe of obs that ca be completed i time i the available itevals. Notatio: _/_ / Fulfilmet Case 2: Costaits o the umbe of obs completed betwee two adacet maiteaces. Hee, we ae give a positive itege C, ad we must complete C obs withi a available iteval. Notatio: C/ _ / _ Case 3: Case 4: Case 3: obs have deadlie. Notatio: _ / d /_ obs have both elease time ad deadlie. Notatio: _ /- d /_ total tadiess of all the obs. Notatio: _/_/Total-tadiess The otatios used i this pape ae: s i : statig time of ith maiteace t i : ed time of ith maiteace u i : ob i,,, p : pocessig time of i u i i : elease time of ui d : deadlie of i u i c i : completio time of u i : legth of schedule i.e. makespa : optimal legth of schedule f : umbe of obs completed i time (i.e. fulfilled) f :optimal umbe of obs completed i time Q : maiteace duatio I this pape, we assume that the set of obs ae idepedet with detemiistic pocessig times to be scheduled o a sigle machie. All obs ae available at the begiig. Uless we metio that ob is peemptive, ob peemptio is ot pemitted. No two obs ca be scheduled at the same time o the sigle machie. Note that machie idle times ae allowed, hece the assumptios ae exactly the oes defied i Bake (Bake (974, p.) - assumptios C, C2, C3, C5). 3. THEORETICA ANAYSIS I this sectio, we peset theoetical esults fo the 6 JSUA poblem cases. 3. A-B/_ /MakeSpa Iput: obs with pocessig time p i ( i ), maiteace duatio Q,, two positive eal value A, B. Output: A schedule of miimum legth such that the legth of available itevals betwee adacet maiteace is betwee A ad B. Complexity Aalysis The Bi Packig Poblem is defied as follows: Give a fiite set U of N items; fo each item u i U a positive itege size s(u); positive itege C (called the bi capacity), patitio U ito disoit sets such that fo each set, the total sum of the sizes of the items i the set does ot exceed C ad the umbe of sets is miimized. Sice Bi-Packig poblem is a well-kow NP-had poblem ad a special case of A-B/_ /MakeSpa whee A=B, A-B/_ /MakeSpa is also NP-had poblem. Appoximatio Ratio 442

3 Heuistic (Best-Fit), defied as follows: Step : Set the legth of available iteval as B. Step 2: Sot all the obs i deceasig ode of pocessig time. Fo each ob i the ob list, place it ito the available iteval that will leave the least amout of oom left ove afte the item is placed i the bi. If it will ot fit ito cuet ay available iteval, ceate a ew bi fo it. Step 3: Afte all the obs have bee assiged, sca the available itevals. If thee is space left i a available iteval, we shift the schedule, while esuig that the legth of ay available iteval is at least A. Obseve that Heuistic takes O( log ) time i the wost case. Theoem : A-B/_ /MakeSpa ca be appoximated withi a facto of 2 i O( log ) time. Poof: Fist we obtai a schedule λ usig the Best-Fit heuistic. The i this schedule, if thee is ay available iteval, the legth of which is smalle tha B, shift the latte obs to the left, as though all the obs ae peemptive. Afte this step, we get a ew schedule λ 2 i which all the legths of the available itevals ae B (o space wasted), ad the legth of this schedule λ 2 is smalle tha o equal to whee is the optimal solutio of the o-peemptive case. O the othe had, we ca obtai a schedule λ fom λ 2. But i this case, istead of shiftig the ob to the ight, we ca get this ob out ad set a ew available iteval fo it at the ed of the schedule. We deote the schedule as λ 3. Clealy, λ 3 is o bette tha λ. The umbe of obs put at the ed is exactly the umbe of maiteaces i λ 2, ad the p i, A. Hece, the legth of the pat added at the legth of the available iteval set at the ed fo each of this kid of tady obs is ( ) ed of λ 2 is less tha that of λ 2. Hece, λ 3 is o wose tha 2, sice λ2 is the optimal solutio of peemptive case ad it is bette tha o equal to the optimal solutio of o-peemptive case. Afte movig those obs, whe we get λ3, we ca also shift the schedule to the left, because of the space aoud the maiteaces whee the obs ae slipped. But this schedule is still o bette tha λ. Hece, we ca see that, the solutio of the Best-Fit heuistic is much bette tha 2. (QED) It tus out that the above esult ca be futhe tighteed if we coside the followig two special cases: Case A=B: Sice this poblem is exactly the Bi Packig poblem which ca be appoximated withi.5 [7], it ca also be appoximated likewise. Case B-A p : If we use Best-Fit to solve A-B/_ /MakeSpa, we see that fo each available iteval betwee ay two adacet maiteaces, the legth is betwee value A ad value B, sice we ca always assig oe moe ob o cotiue pocessig the cuet ob afte the poit of A. Thus thee is o empty space befoe a ew maiteace stated. Ad fo the schedule achieved fom the Best-Fit heuistic the total sum of the legths of available itevals is always fixed because thee is o wasted space. Coollay : Whe B-A p, A-B/_ /MakeSpa ca be appoximated withi +, whee k is the atio of ad the sum + k p i of maiteace time i the optimal solutio. Poof: Agai, we use the Best-Fit heuistic to solve A-B/_ /MakeSpa. But fo λ, the sum of the legths of all the available itevals is exactly p i, sice thee is o empty space i each available iteval as metioed above. Fom migatig fom λ2 to λ 3, we ca easily see that the umbe of maiteaces is at most two times of that of the optimal solutio. The we kow the solutio achieved fom Best-Fit, whee p + 2 i p. Sice k i ( ) ( + 2 ) p + p, we get k i k i i = = + + k ( + ) p k i i = Note that i a typical maufactuig eviomet, the pocessig time of a machie is much lage tha its maiteace time. Hece, the appoximatio atio is vey close to (i.e. optimal). 3.2 A-B/ /MakeSpa Iput: obs with pocessig time p i ad elease time i, maiteace duatio Q,, two positive values A, B Output: A schedule of miimum legth such that the legth of available itevals must be betwee A ad B. Complexity Aalysis 443

4 This poblem is NP-had, sice A-B/_ /MakeSpa ca be teated as a special case of A-B//MakeSpa whee the elease times of all the obs ae zeo. Appoximatio Ratio Heuistic 2: We ca solve the coespodig poblem fo A-B/_ /MakeSpa by Heuistic fist, ad obtai the fial solutio by shiftig obs to the espective elease time. Although Heuistic 2 does ot seem vey efficiet (sice thee is wasted slot befoe its elease time), it is iteestig to obtai the followig appoximatio esult: Theoem 2: A-B//MakeSpa ca be appoximated withi a facto of +k, fo some k 2 i O( log ) time. Poof: et deote the optimal solutio fo the coespodig poblem A-B/_ /MakeSpa. et deote the solutio achieved by Heuistic fo A-B/_ /MakeSpa. et Sice = +, we get ( + ) ( + ) = Suppose k fo some k. Suppose, the Suppose ( ) ( ) k, the deote the solutio achieved by Heuistic 2. Clealy, ( ) + + k (, ) ( ) ( ) + k Hece, k A-B/ d /Fulfilmet. By Theoem, we kow that 2 k, ad hece k +, fo some k 2., Iput: obs with pocessig time pi ad deadlie di, maiteace duatio Q, two positive eal values A, B. Output: A schedule such that the legth of available itevals must be betwee A ad B, ad the umbe of obs completed i time is imized. Complexity Aalysis ike A-B/_/MakeSpa, based o the eductio fom bi-packig, A-B/ d /Fulfilmet is also NP-had. Appoximatio atio Fom Mooe-Hodgso s algoithm [3], we deive a ew algoithm: Algoithm 3 Step : Set the legth of each available iteval as B. Step : Sot the obs i EDD ode, ad place the obs ito the schedule as though all the obs ae peemptive. If thee exists a tady ob, amog the fist k obs fid the ob with lagest pocessig time, delete it fom the ob list ad assig it at ed of the sequece. Fom the above, we obtai a schedule λ. s Step 2: Fom λ, delete the obs stated befoe i, ad completed afte i of the ith maiteace. The shift the schedule to the left if thee is gap befoe ay maiteace, but esue that the legth of ay available iteval is at least A. We obtai a ew schedule λ2. We also use the obs emoved to fom a ew schedule λ 3. Step 3: Compae schedule λ 2 ad λ 3, ad choose the oe with moe obs i schedule. Theoem 3: A-B/d/Fulfillmet ca be appoximated withi (, k- ), whee k= B, i O( log ) time. 2 k p Poof: et f deote the solutio achieved fom algoithm 3, f deote the optimal solutio of o-peemptive case, ad deote the optimal solutio of peemptive case. Fom [9], we kow that the solutio of λ is f. t f 444

5 Note that, f f sice we ca shift those obs completed afte t of the o-peemptive poblem to the left to fill i the gap. Thus i f must have at least the same umbe of obs as f. We ca see that the complete time of the obs i eithe schedule λ2 o schedule λ 3 is smalle tha the coespodig complete time i schedule λ, the all the obs i schedule λ 2 ad λ 3 ca be completed i time. Sice ad f f 2 f f, we ca say f f. Futhemoe, let k= B, we ca say that fo each available iteval 2 ( p i ) of schedule λ, thee ae at least k obs. Ad afte we emove the obs slipped by maiteaces, we ca still have k- obs i each available iteval of λ 2. The, we ca get f k- f. Hece, the appoximatio atio of Heuistic 3 is ( k -, ). k k A-B/ -d /Total-tadiess Iput: obs with pocessig time pi, elease time i, ad deadlie di, maiteace duatio Q,, two positive eal values A, B. Output: A schedule such that the legth of available itevals must be betwee A ad B, ad the total tadiess is miimized. Complexity Aalysis I [3], the poblem of miimizig total tadiess o oe machie has bee poved to be NP-had, ad it ca be teated as a special case of A-B/ -d /Total-tadiess whee the values of A ad B is ifiitely lage ad the elease time of all the obs ae at the zeo poit. Thus, A- B/ -d /Total-tadiess is also NP-had. 3.5 C/ d /Fulfilmet Iput: obs with pocessig pocessig time pi ad deadlie d i, maiteace duatio Q, a positive value C Output: A schedule such that we have to pocess C obs, ad the umbe of tady obs is miimized. Heuistic 4: Step : sot the obs i EDD ode. Step : place the obs ito the schedule. If thee is a tady ob, amog the fist k obs fid the ob with lagest pocessig time, u k delete it fom the ob list ad assig it at ed of the sequece. Whe thee ae C obs i the cuet available iteval, stat a ew maiteace. Step 2: Repeat Step. Theoem 4: C/ d /Fulfilmet ca be solved optimally i i O( log ) time. Poof: Sice we idex the obs i EDD ode, the whe the ob u u is foud tady, we ca say befoe the cuet completio time t of thee exists at least oe tady ob o matte how to schedule the obs. Suppose thee is a schedule λ with o tady ob befoe t, hece u 2 u u 2 u u 2 u the deadlies of, u ae smalle tha t, the the obs, u u should be all fiished befoe t i schedule λ. Howeve, the sum of pocessig time of, equals t., ad suely the last ob i the schedule is a tady ob. This is a cotadictio with the assumptio. So we ca say thee exists at least oe tady ob befoe t. Hece, afte the step that the ob with the lagest pocessig time is take out, the schedule λ2 achieved is optimal cuetly, ad it leaves moe time fo the followig obs compaed with the schedules that ay othe ob is take out. We ca pefom the above pocess fo the followig obs, ad afte we schedule all the obs, the schedule λ 2 achieved is optimal. I this algoithm, the time complexity fo sotig i EDD ode is O( log ), the schedulig algoithm itself is liea. Hece, the time complexity is O( log ). 3.6 C/ /Makespa Iput: obs with pocessig time pi ad elease time i, maiteace duatio Q,, a positive value C. Output: A schedule such that C obs have bee pocessed, ad the legth of schedule is miimized. Aalysis The umbe of available itevals is c, implyig that the umbe of maiteaces is c -. Wheeve a ob is eleased, we iset it ito a queue. Whe the machie is available, fetch a ob fom the queue. If the queue is empty, we have to wait. Hece i each available iteval, o time is wasted. The total legth of available itevals is always optimal. Ad the umbe of maiteaces is always the same. Hece the sum of available itevals ad maiteace is optimal. 445

6 4. TABU SEARCH I this sectio, we peset esults of a tabu seach scheme fo solvig A-B/_/MakeSpa, A-B//MakeSpa ad A-B/-d total-tadiess. The basic ideas fo all thee poblems ae simila, except thei iitial solutio ad obective fuctio. Sice thee is o bechmak available fo the above poblems, we compae the esults with the stadad geedy appoach. We illustate ou esults with poblem istaces that ae geealized fom the OR ibay Bi-Packig poblem istaces by adomly geeatig the age (fom A to B) based o the bi sizes give by Bi-Packig poblem istaces. By applyig ou tabu seach scheme, we impove the esults achieved by the geedy algoithms discussed above. The followig chat summaizes the esults obtaied: % 8% 6% 4% 2% Befoe TS Afte TS case: A-B/_/MakeSpa case2: A-B/ /MakeSpa case3: A-B/-d/Total-tadiess % case case2 case3 Fom the above esults, we obseve that the geedy esults of A-B/-d/Total-tadies ca be impoved by 2% usig tabu seach. Fo A- B/_/MakeSpa, the esults ca be impoved by aoud %. Howeve, A-B//MakeSpa is ot impoved sigificatly by usig tabu seach. I fact, fo each istace of A-B//MakeSpa ad the coespodig istace of A-B/_/MakeSpa, the diffeece is that A- B//MakeSpa has oe moe costait: the elease time. This costait appaetly esticts the potetial of tabu seach of makig local impovemet sigificatly. 5. CONCUSION I this pape, we poposed a taxoomy that categoizes the JSUA poblem accodig to 3 divig factos we believe to be impotat. As this is the fist such taxoomy, we aticipate much futue eseach wok to efie the esults peseted hee. Refeeces [] Blazewicz J., Fomaowicz P., Kubiak W., Pzysuch M. ad Schmidt G., "Paallel bach ad boud algoithms fo the twomachie flow shop poblem with limited machie availability", Bulleti of the Polish Academy of Scieces, (2). [2] Cheg T. C. E. ad Wag G., "A impoved heuistic fo the two-machie flowshop schedulig with a availability costait", Ope Res ett, 26, , (2). [3] Du, J. ad eug, J. Y. T., Miimizig Total Tadiesso Oe Machie is NP-had. Math. Ope. Res, 5, pp (99). [4] Espiouse M.-., Fomaowicz P. ad Pez B., "Miimizig the makespa i the two-machie o-wait flow-shop", Computes & Idustial Egieeig, 37, 497-5, (999). [5] Gaves G. H. ad ee C.-Y., Schedulig maiteace ad semi-esumable obs o a sigle machie, Naval Reseach ogistics, 46, pp , (999). [6] ee C.-Y., Paallel machies schedulig with o-simultaeous machie available time, Discete Appl Math, 3, 53-6, (99). [7] ee C.-Y. ad ima S.D., Sigle machie flow-time schedulig with scheduled maiteace, Acta If, 29, , (992). [8] ee C.-Y. ad ima S.D., Capacitated two-paallel machies schedulig to miimize sum of ob completio times, Disc Appl Math 4, 2-222, (993). [9] ee C.-Y., "Machie schedulig with a availability costait", J. Global Optimizatio, 9, , (995). [] ee C.-Y., Miimizig the makespa i the two-machie flowshop schedulig poblem with a availability costait, Ope Res ett 2, 29-39, (997). [] ee C.-Y., "Two-machie flowshop schedulig with availability costaits", Euopea J. Opeatioal Reseach, 4, , (999). [2] Mosheiov G., Miimizig the sum of ob completio times o capacitated paallel machies, Math Comput Model, 2, 9-99, (994). [3] Mooe J. M., A ob, oe machie sequecig algoithm fo miimizig the umbe of late obs, Maagemet Sciece, 5:2-9, (968). [4] Qi X., Che T.,ad Tu F., Schedulig the maiteace o a sigle machie, J Ope Res Soc, 5, (999). [5] Schmidt G., Schedulig o semi-idetical pocessos, Z Ope Res 28, 53-62, (984). [6] Schmidt G., Schedulig idepedet tasks with deadlies o semi-idetical pocessos, J Ope Res Soc, 39, , (988). [7] Simchi-evi D., New wost-case esults fo the bi-packig poblem, Naval Res. ogistics, 4, (994). 446