LOOK-AHEAD TECHNIQUES FOR MICRO-OPPORTUNISTIC JOB SHOP SCHEDULING

Transcription

1 LOOK-AHEAD TECHNIQUES FOR MICRO-OPPORTUNISTIC JOB SHOP SCHEDULING Norman Sadeh March 99 CMU-CS-9-02 Submtted n parta fufment of the requrements for the degree of Doctor of Phosophy Schoo of Computer Scence Carnege Meon Unversty Pttsburgh, Pennsyvana 523 Copyrght 99 Sadeh Ths research was supported, n part, by the Defense Advanced Research Projects Agency under contract #F C-000, and n part by grants from McDonne Arcraft Company and Dgta Equpment Corporaton. The vews and concusons contaned n ths document are those of the author and shoud not be nterpreted as representng the offca poces, ether expressed or mped, of the Defense Advanced Research Projects Agency, McDonne Arcraft, or Dgta Equpment Corporaton.

2 To my parents, Dense and Leon, and my wfe, Patrca.

3 Acknowedgements I woud ke to express my grattude to the many peope who heped make ths dssertaton possbe. In partcuar, I woud ke to thank: My parents and my wfe for ther ove and support throughout a these years; My advsor, Mark Fox, for hs gudance, enthusasm and unfang support; The other members of my thess commttee, Tom Mtche, Tom Morton, Judea Pear, and Steve Smth, for many hepfu comments on ths research; A present and past members of the CORTES project for many stmuatng dscussons - Speca thanks to Bob Frederkng, who vounteered to proofread parts of ths dssertaton; A my frends and coeagues at the Center for Integrated Manufacturng Decson Systems and n the Schoo of Computer Scence for havng made these past four years such a unque experence; Les Gasser for hs frendshp and support durng my earer graduate fe n Los Angees. I woud aso ke to acknowedge the fnanca support I receved drecty or ndrecty from DARPA, Dgta Equpment Corporaton, and McDonne Dougas, whe a student at Carnege Meon, as we as the fnanca support of the Began Amercan Educatona Foundaton durng my frst year n the U.S..

4 Abstract Schedung deas wth the aocaton of resources over tme to perform a coecton of tasks. Schedung probems arse n domans as dverse as manufacturng, computer processng, transportaton, heath care, space exporaton, educaton, etc. Schedung probems are convenenty formuated as Constrant Satsfacton Probems (CSPs) or Constraned Optmzaton Probems (COPs). A genera paradgm for sovng CSPs and COPs rees on the use of backtrack search. Wthn ths paradgm, the schedung probem s soved through the teratve seecton of a subprobem and the tentatve assgnment of a souton to that subprobem. Because most schedung probems are NP-compete, even fndng a souton that smpy satsfes the probem constrants coud requre exponenta tme n the worst case. Ths dssertaton demonstrates that the granuarty of the subprobems seected by the backtrack search procedure crtcay affects both the effcency of the procedure and the quaty of the resutng souton. A so-caed mcroopportunstc search procedure s deveoped, n whch subprobems can be as sma as a snge operaton. Look-ahead technques are presented that constanty track the evouton of so-caed botteneck resources. These ook-ahead technques enabe the scheduer to take advantage of the fne granuarty of ts search procedure by opportunstcay revsng ts schedung strategy as bottenecks shft from one part of the probem space to another. More specfcay, two varatons of the job shop schedung probem are successvey studed:. The frst varaton s one n whch operatons have to be performed wthn non-reaxabe tme wndows. Heurstcs to gude a mcro-opportunstc scheduer are presented that are shown to outperform both generc CSP heurstcs as we as specazed heurstcs deveoped for smar schedung probems. 2. The second part of ths work deas wth the factory schedung probem. A mcro-opportunstc factory scheduer, caed MICRO-BOSS, s descrbed that expcty accounts for both tardness and nventory costs. MICRO- BOSS s shown to outperform severa competng schedung technques. Expermenta resuts aso ndcate that schedue quaty deterorates as the granuarty of the search procedure ncreases, thereby suggestng the superorty of a mcroopportunstc approach to job shop schedung over coarser search procedures such as those mpemented n ISIS, OPT, and OPIS. They aso ndcate that the abty of the mcro-opportunstc approach to constanty revse ts search strategy s nstrumenta n effcenty sovng probems n whch some operatons have to be performed wthn nonreaxabe tme wndows.

5 Chapter Introducton.. Overvew Ths dssertaton s concerned wth dynamcay revsng a search procedure to focus on the most crtca decson ponts and the most promsng decsons at these ponts. Heurstcs to redrect search are presented n the context of the job shop schedung doman that have yeded mportant ncreases n both search effcency and schedue quaty over a varety of competng technques. Schedung deas wth the aocaton of resources over tme to perform a coecton of tasks [Baker 74, Rnnooy Kan 76, French 82]. Schedung probems arse n many domans. In the manufacturng doman, tasks, often referred to as jobs, correspond to parts or batches of parts that need to be processed on a set of machnes [Muth 63, Johnson 74, Graves 8, Sver 85, Rodammer 88]. In hosptas, tasks are patents and resources are nurses, hospta beds or medca equpment requred to treat the patents. Schedung probems arse n schoos, where tasks are casses and resources can be teachers, cassrooms, and students [Fedman 89, Dhar 90]. Other exampes ncude transportaton-reated probems (e.g. troop transportaton, arport termna schedung, tran schedung [Fukumor 80], etc.), computer schedung probems (e.g. CPU schedung [Peterson 85]), space teescope schedung [Muscettoa 89, Johnston 90], appontment schedung [Godsten 75], etc. Schedung probems are convenenty formuated as ether Constrant Satsfacton Probems (CSPs) or Constraned Optmzaton Probems (COPs). A CSP s defned by a set of varabes and a set of constrants that restrct the vaues that can smutaneousy be assgned to these varabes [Montanar 7, Mackworth 77, Dechter 88]. A COP s a CSP wth an objectve functon to be optmzed subject to the probem constrants [Papadmtrou 82, Nemhauser 88, Fox 89, Dechter 90]. A genera paradgm for sovng CSPs and COPs rees on the use of backtrack search [Waker 60, Goomb 65, Btner N. SADEH

6 2 INTRODUCTION 75]. Wthn ths paradgm, the schedung probem s soved through the teratve seecton of a subprobem and the tentatve assgnment of a souton to that subprobem. If n the process of budng a souton, a parta souton s reached that cannot be competed wthout voatng a probem constrant, one or severa earer assgnments need to be undone. Ths process of undong earer assgnments s caed backtrackng. It deterorates the effcency of the search procedure, and hence ncreases the tme requred to come up wth a souton. Because of ts mathematca structure [Garey 79], the genera verson of the schedung probem studed n ths dssertaton (known as the job shop schedung probem) can potentay requre very arge amounts of backtrackng. Tradtonay, schedung technques have deat wth the backtrackng ssue by transformng the mathematca structure of the probem, and aowng some constrants to be reaxed as needed. Ths approach s commony used n factory schedung, where rather than requrng that a jobs be competed by ther due dates, job due dates are reaxed as much as necessary n order to effcenty come up wth a schedue. Whe producng effcent schedung procedures, ths approach often resuts n fary poor soutons. In probems such as space teescope schedung, where some due dates cannot be reaxed (e.g. schedung the operatons requred to take snapshots of an ecpse), ths approach smpy does not work. Instead, ths dssertaton nvestgates new schedung technques, whch, short of guaranteeng backtrack-free search, provde quaty schedues whe generay mantanng search (.e. backtrackng) at a ow eve. Addtonay, these technques are capabe of deang wth schedung probems n whch constrants such as due dates are not aways reaxabe. A key feature of the search technques that w be dscussed es n ther abty to dynamcay adapt durng search. In the Artfca Integence (AI) terature, ths abty to dynamcay revse the search procedure has been termed opportunstc search [HayesRoth 79, Erman 80, Stefk 8a]. Whe earer opportunstc scheduers have reed on coarse probem decompostons, ths work presents a so-caed mcro-opportunstc schedung approach that aows for much fner subprobems. It s shown that the extra fexbty of a mcro-opportunstc schedung procedure can be expoted to constanty redrect the schedung effort towards those resources that are key to be the most dffcut to schedue (so-caed botteneck resources). Look-ahead technques are descrbed that hep the mcro-opportunstc scheduer dentfy crtca subprobems and promsng soutons to these subprobems. These technques are shown to aow the mcro-opportunstc scheduer to perform partcuary we both wth respect to search effcency and schedue quaty. LOOK-AHEAD TECHNIQUES FOR MICRO-OPPORTUNISTIC JOB SHOP SCHEDULING

7 OVERVIEW 3 More specfcay, two studes of the mcro-opportunstc schedung paradgm are successvey presented:. The frst study s concerned wth a varaton of the generc Job Shop CSP, n whch operatons have to be performed wthn non-reaxabe tme wndows (e.g. non-reaxabe due dates and reease dates). The study ndcates that often generc CSP heurstcs are not suffcent to gude the search for a souton to ths probem. Ths s because these heurstcs fa to propery account for constrant tghtness and for the connectvty of the constrant graph. Instead, a probabstc mode of the search space s ntroduced. New heurstcs are deveoped based on ths mode, that are shown to sgnfcanty speedup search. 2. The second part of ths work deas wth the factory schedung probem. Ths optmzaton probem aows us to smutaneousy study both the quaty and effcency performance of the mcro-opportunstc approach. A mcro-opportunstc factory schedung system, caed MICRO-BOSS (Mcro-Botteneck Schedung System) s descrbed that attempts to smutaneousy reduce both tardness and nventory costs. A arge scae computatona study ndcates that the mcro-opportunsm embedded n MICRO-BOSS enabes the scheduer to outperform a varety of competng schedung technques, both from Operatons Research and Artfca Integence, under a wde range of schedung condtons. The study aso ndcates that schedue quaty deterorates as the granuarty of the subprobems used n the search procedure ncreases, thereby suggestng the superorty of a mcro-opportunstc approach over coarser search procedures such as those mpemented n the ISIS [Fox 83], OPT [Godratt 80, Jacobs 84, Fox 87], and OPIS [Smth 86a, Ow 88a] schedung systems. The baance of ths ntroducton gves a more forma defnton of the job shop schedung probem and revews reevant work both n job shop schedung and n Constrant Satsfacton/Constraned Optmzaton. N. SADEH

8 4 INTRODUCTION.2. The Job Shop Schedung Probem The job shop schedung probem requres schedung a set of jobs on a fnte set of resources. Each job s a request for the schedung of a set of operatons accordng to a process pan (often referred to as process routng n the manufacturng doman) that specfes a parta orderng among these operatons. In order to be successfuy performed, each operaton requres one or severa resources (e.g. a machne, a human operator, a set of fxtures), for each of whch there may be severa aternatves (e.g. severa machnes of the same type). Operatons are atomc: once started they cannot be nterrupted. In the smpest stuaton, each operaton has a fxed duraton, and each resource can ony process one operaton at a tme. In manufacturng, jobs typcay have reease dates, before whch they cannot start (e.g. because the raw materas requred to process a job are not schedued to arrve before that date) and due dates by whch, deay, they shoud be competed. These dates are generay provded by a master schedung modue [Sver 85]. In make-to-order envronments, aso referred to as open shops, due dates correspond to devery dates of customer orders. In make-to-stock envronments, aso referred to as cosed shops, reease and due dates are artfcay generated to reduce the compexty of the probem, prevent the shop from beng overfooded wth nventory, and avod stockouts (.e. avod runnng out of fnshed-goods to meet customer demand). Job shop schedung s a Constrant Satsfacton Probem (CSP) or Constraned Optmzaton Probem (COP). The varabes of the probem are the operaton start tmes, and the resources assgned to each operaton, when there s a choce. The constrants of the probem ncude precedence constrants specfed by the process routngs and capacty constrants that prevent resources from beng aocated to more operatons than they can process at one tme (resource capacty). Job reease dates and due dates are constrants that restrct the domans of acceptabe operaton start tmes. Addtona constrants may further restrct these domans such as constrants n factory schedung that requre some operatons to be performed over a snge shft. Smar constrants are often found n a varety of other schedung probems, n whch the domans of ega operaton start tmes can be made up of dsjont tme ntervas (e.g. [Muscettoa 89]). Fgure - depcts a sma job shop probem wth four jobs. Each operaton s represented by a box abeed by a trpe consstng of the name of the operaton (e.g. O ), LOOK-AHEAD TECHNIQUES FOR MICRO-OPPORTUNISTIC JOB SHOP SCHEDULING

9 THE JOB SHOP SCHEDULING PROBLEM 5 j O 2 R O 6 2 R 2 O 4 2 R 3 O 5 2 R 4 O 3 3 R 5 j 2 O 2 7 R O 2 5 R 2 2 j 3 O 3 R 3 2 R 3 R 2 O 3 2 O 3 3 j 4 O 4 3 R 4 O R 2 precedence constrant capacty constrant Fgure -: A smpe job shop probem wth 4 jobs. Each node s abeed by the operaton that t represents, ts duraton, and the resource that t requres. the duraton of that operaton (e.g. 2), and ts resource requrement (e.g. R ). The arrows represent precedence constrants. For nstance, job j requres 5 operatons O, O 2,..., O 5. O has to be performed before O 2, O2 before O 4, etc. The other arcs n the graph represent capacty constrants. In ths exampe, each resource s assumed to be capabe of processng ony one operaton at a tme. A capacty constrant between two operatons expresses that these two operatons cannot overap n tme. Ceary there s a capacty constrant between each par of operatons that requre the same resource (.e. there s a cque of capacty constrants for each resource). For nstance, the cque for resource R nvoves operatons O 2, O 2, O3 and O 2. Typcay, each job aso has a reease date and a due date, whch are not represented n Fgure -. N. SADEH

10 6 INTRODUCTION In some schedung probems, t s suffcent to come up wth a souton that satsfes the probem constrants (job shop CSP). Often however, not a soutons are equay preferred. When ths s the case, job shop schedung becomes a COP wth an objectve functon to optmze. Dfferent schedung domans generay enta dfferent schedung crtera. There exsts however a sma set of crtera that pervades the schedung terature. Because these metrcs w be referred to ater on n ths work, they are now brefy defned. A more compete set of schedung metrcs s descrbed n [Rnnooy Kan 76] aong wth equvaence reatons between them. Tardness: Job tardness s defned as the amount of tme a job competes past ts due date. The tota (average) tardness of a schedue s the tota (average) job tardness n that schedue. Fowtme: Job fowtme s the tme spent by a job n the shop whe beng processed. It s the ength of the tme nterva that spans from the reease of the job to ts competon. The tota (average) fowtme of a schedue s the tota (average) job fowtme n that schedue. In factory schedung, average job fowtme s an ndcator of the tme requred to produce a part. A schedue wth a arger average fowtme w resut, on the average, n more parts sttng n the shop watng to be competed. In other words, there w be more n-process nventory (aso referred to as work-n-process or workn-progress). Earness: Job earness s defned as the amount of tme a job competes before ts due date. The tota (average) earness of a schedue s the tota (average) job earness n that schedue. In factory schedung probems where parts competed before ther due dates have to wat to be shpped, average job earness s a measure of fnshed-goods nventory. Job tardness shoud not be confused wth job ateness, whch s negatve when a job competes before ts due date. By defnton the tardness of a job competng before ts due date s aways zero. LOOK-AHEAD TECHNIQUES FOR MICRO-OPPORTUNISTIC JOB SHOP SCHEDULING

11 THE JOB SHOP SCHEDULING PROBLEM 7 Makespan: The makespan of a schedung probem s the ength of the tme nterva that spans from the start tme of the frst reeased job to the competon tme of the ast competed job. Ths measure s approprate n project schedung, where there s a fnte number of jobs to be carred out. In manufacturng domans, where new jobs arrve every day or every week, t s more natura to speak n terms of throughput (.e. the number of jobs processed per unt of tme) or n terms of resource utzaton (.e. the fracton of ts tme that a resource s actve). These three metrcs are equvaent n the sense that a reducton n the makespan of a schedue produces a proportonate ncrease n throughput and resource utzaton [Rnnooy Kan 76]. There are many other possbe ways to evauate the quaty of a schedue. In factory schedung, each of the measures defned above ooks ony at one mportant source of costs n the producton process. In Chapter 4, t w be argued that, nstead of reyng on any one of these measures, one shoud attempt to smutaneousy account for the dfferent costs hdden behnd these measures. More forma defntons of the schedung probems studed n ths thess are provded at the begnnng of Chapters 3 and 4. The next secton brefy revews the state of the art n schedung and CSP/COP..3. Reated Work The frst part of ths secton revews the state of the art n job shop schedung. Rather than attemptng to gve a comprehensve survey of the fed, the revew focuses on a coupe of recent technques cosey reated to the work presented n ths dssertaton. In the process, the revew attempts to pont out shortcomngs of these technques and motvates the mcro-opportunstc approach nvestgated n ths thess. The second part of the secton summarzes reevant work n CSP/COP. Weaknesses of current CSP/COP technques are brefy dentfed that need to be remeded n order to successfuy appy the CSP/COP paradgm to rea fe probems such as job shop schedung. N. SADEH

12 8 INTRODUCTION.3.. The State of the Art n Job Shop Schedung Job shop schedung s an NP-hard probem [Garey 79, Lawer 82]. Whe mathematca programmng technques deveoped n Operatons Research have proved partcuary usefu for aggregate pannng [Lawrence 84], they are overwhemed by the combnatora number of dscrete varabes2 requred to represent job shop schedung probems [Nemhauser 88]. More generay, wth the excepton of a coupe of one-, two-, and three-machne schedung probems, for whch there exst effcent agorthms [Rnnooy Kan 76], a attempts to guarantee an optma souton have faed. Instead, job shop schedung probems have tradtonay been soved usng prorty dspatch rues [Baker 74, Panwakar 77, French 82]. These are oca decson rues of the greedy type that bud schedues va a forward smuaton of the shop. Because these rues ack a goba vew of the shop, they usuay bud up arge amounts of nventory n front of botteneck resources 3. More recenty, wth the advent of more powerfu computers, a coupe of more sophstcated schedung technques have been deveoped [Godratt 80, Fox 83, Ow 85, Adams 88, Ow 88a, Morton 88]. The frst and by far most pubczed of these technques s the one deveoped by Eyahu Godratt and hs coeagues n the ate seventes and eary eghtes wthn the context of the OPT factory schedung system [Jacobs 84, Fox 87] 4. Among other thngs, ths system emphaszed the need to dstngush between botteneck and non-botteneck machnes. In OPT, bottenecks drve the entre schedue as they determne the throughput of the pant. More specfcay, a modue caed SERVE produces an nta nfnte capacty schedue by workng backwards from the job due dates. Ths nta schedue heps detect potenta bottenecks. The OPT modue tsef s then caed upon to generate a forward fnte capacty schedue that optmzes the utzaton of these bottenecks. The 2 A smpe job shop probem wth n jobs and m resources of unary capacty, n whch each job needs to be processed by each of the m resources, produces m cques of n () 2 capacty constrants. Because these capacty constrants are dsjunctve, they each transate nto a bnary varabe n a Mxed Integer Programmng mode. 3 Informay, a botteneck s a resource whose utzaton s expected to be cose to or arger than ts avaabe capacty. 4 See aso [Godratt 86] for a vey descrpton of the phosophy behnd OPT. LOOK-AHEAD TECHNIQUES FOR MICRO-OPPORTUNISTIC JOB SHOP SCHEDULING

13 RELATED WORK 9 resutng botteneck schedues are passed back to the SERVE modue, whch schedues the non-botteneck operatons whe tryng to mnmze nventory. At about the same tme, the ISIS factory schedung system deveoped by Mark Fox and hs team frst demonstrated the potenta of AI modeng and heurstc search technques to hep sove producton schedung probems [Fox 83, Smth 86b]. For the frst tme, rather than reyng on a smpfed mode of the shop, ISIS attempted to dea wth the fu range of constrants and objectves encountered n the manufacturng doman. Unfortunatey, the overa performance of the system was somewhat mtgated by the rgdty of ts search procedure, whch requred jobs to be schedued one by one (socaed job-centered approach). Whe ths search procedure was partcuary effcent at reducng nventory, t had probems optmzng utzaton of botteneck resources. As a resut, a new system, caed OPIS, was deveoped by Steve Smth, Peng S Ow, and ther coeagues [Smth 86a, Smth 86b, Ow 88a]. In OPIS, the noton of botteneck resource was pushed one step further, as t was recognzed that new bottenecks can appear durng the constructon of the schedue. The OPIS scheduer combnes two schedung perspectves: a resource-centered perspectve for schedung botteneck resources, and a job-centered perspectve to schedue non-botteneck operatons on a job by job bass. Rather than reyng on ts nta botteneck anayss, OPIS typcay repeats ths anayss each tme a resource or a job has been schedued. Ths abty to detect the emergence of new bottenecks durng the constructon of the schedue and revse the current schedung strategy has been termed opportunstc schedung [Ow 88a]. Nevertheess, the opportunsm n ths approach remans mted n the sense that t typcay requres schedung an entre botteneck (or at east a arge chunk of t) before beng abe to swtch to another one. For ths reason, such schedung technques shoud n fact be caed macro-opportunstc. In reaty, bottenecks do not necessary span over the entre schedung horzon. Moreover they tend to shft before beng entrey schedued. A scheduer that can ony schedue entre resources w not be abe to take advantage of these consderatons. Often t w overconstran ts set of aternatves before havng worked on the subprobems that w most crtcay determne the quaty of the entre schedue. Ths n turn w often resut n poorer soutons. A more fexbe approach woud aow to qut schedung a resource as soon as another resource s dentfed as beng more constranng. In fact, n the presence of mutpe bottenecks, one can magne a technque that constanty shfts attenton from one botteneck to another rather than focusng on the optmzaton of a N. SADEH

14 0 INTRODUCTION snge botteneck at the expense of others. For these reasons, t seems desrabe to nvestgate a more fexbe approach to schedung, or a mcro-opportunstc approach, n whch the evouton of bottenecks s contnuousy montored durng the constructon of the schedue, and the probem sovng effort constanty redrected towards the most serous botteneck. In ts smpest form, ths mcro-opportunstc approach resuts n an operaton-centered vew of schedung, n whch each operaton s consdered an ndependent decson pont and can be schedued wthout requrng that other operatons usng the same resource or beongng to the same job be schedued at the same tme. Ths s the approach adopted n ths thess. An aternatve approach for deang wth the emergence of new bottenecks has been recenty proposed by Adams, Baas and Zawack [Adams 88] (See aso [Dauzere-Peres 90]). Ths approach, known as the Shftng Botteneck Procedure (SBP), sequences resources one by one, whe contnuousy reoptmzng the schedue of resources sequenced earer. SBP has aowed for the producton of schedues wth near-optma makespan for probems wth up to 500 operatons. Attempts to generaze the procedure to account for due dates and more compex objectves seem to have been ess successfu so far [Serafn 88]. It shoud be ponted out that the dea of contnuousy reoptmzng the current parta schedue s not ncompatbe wth the mcro-opportunstc approach 5. The SCHED-STAR schedung modue deveoped by Morton, Lawrence, Rajagopoan and Kekre takes yet another approach to deang wth botteneck resources [Morton 88]. Rather than reyng on a smpfed objectve functon, ths prce-based factory scheduer accounts drecty for both tardness and nventory costs n order to mnmze the net present vaue of cash fows n the pant. Based on these exogenous costs, mpct resource prces are derved va nterna smuaton that refect resource contenton n functon of tme. These prces are used to determne job reeases and attrbute prortes to competng jobs at each machne. The MICRO-BOSS factory schedung system descrbed n Chapter 4 aso uses tardness and nventory costs to hep dentfy botteneck resources. The desgn of SCHED-STAR prevents however the system to take fu advantage of ts botteneck anayss. In partcuar, the system buds schedues va a forward smuaton of the shop. As a resut, reease decsons are aways made before 5 By ony schedung those operatons that appear to be most crtca, a mcro-opportunstc approach shoud n fact aow for more effectve reoptmzaton procedures. LOOK-AHEAD TECHNIQUES FOR MICRO-OPPORTUNISTIC JOB SHOP SCHEDULING

15 RELATED WORK botteneck sequencng decsons. Ths s contrary to the essons taught by OPT 6. SCHED-STAR makes up for ths potenta weakness by teratng ts smuaton, usng ts prevous schedue to derve new resource prces and generate new schedues. It s not cear how effectve ths teratve approach can be and how dependent t s on the abty to guess a good nta schedue. The man purpose of ths bref revew was to emphasze the need for a mcroopportunstc schedung approach n order to produce better schedues. As w be demonstrated n ths thess, the extra fexbty of the mcro-opportunstc search procedure advocated here, not ony heps produce better schedues but aso enabes a schedung system to dea more effectvey wth operatons that need to be performed wthn non-reaxabe tme wndows (e.g. non-reaxabe reease dates and due date constrants). More comprehensve revews of the job-shop schedung terature can be found n [Baker 74, Rnnooy Kan 76, French 82, Lawer 82]. For recent surveys of the producton schedung terature, the reader s referred to [Graves 8] and [Rodammer 88]. Conventona approaches to producton pannng and schedung are dscussed at ength n [Johnson 74, Hax 84, Sver 85]. The second part of ths revew deas wth earer work n CSP/COP. It ncudes references to other appcatons of the CSP paradgm to job shop schedung probems Reevant Work n CSP/COP The genera CSP s NP-compete [Garey 79]. Technques for sovng the genera CSP extend the depth-frst backtrack search procedure [Waker 60, Goomb 65, Btner 75, Pear 84], n whch a souton s ncrementay but by nstantatng one varabe (or more generay one subprobem) after another. Every tme a varabe s nstantated, a new search state s created, where new constrants are added to account for the vaue assgned to that varabe. If a parta souton s but that cannot be competed, the current search state s sad to be a deadend. The system needs to backtrack to an earer ess compete souton, and try aternatve varabe assgnments. Search typcay stops 6 Indeed, by frst sequencng botteneck machnes, systems ke OPT determne how much can be produced by the pant. Botteneck schedues are then used to determne when to reease jobs wthout starvng the botteneck and wthout budng excess nventory. N. SADEH

16 2 INTRODUCTION when a frst souton has been found, or when a aternatves have been tred wthout success. In the atter case, the CSP s sad to be nfeasbe. Because the genera CSP s NP-compete, backtrack search may requre exponenta tme n the worst case. Research n CSP has produced four types of technques that can hep mprove the average effcency of the basc backtrack search procedure [Dechter 9]:. Consstency Enforcng (Checkng) Technques: These technques are meant to prune the search space by emnatng oca nconsstences that cannot partcpate n a goba souton [Mackworth 85]. Ths s done by nferrng new constrants and addng them to the current probem formuaton. If durng ths process the doman of a varabe becomes empty, a deadend stuaton has been dentfed. Consstency enforcng technques can be apped ether before or durng search. In genera, achevng hgher eves of consstency reduces backtrackng. There s however a tradeoff [Harack 80, Mackworth 85, Nade 88] between the amount of computaton spent enforcng consstency and the savngs acheved n the actua search. Parta consstency enforcng agorthms have been cassfed accordng to the degree of consstency that they acheve between varabes. In partcuar, consstency enforcng agorthms that acheve consstency among subsets of k varabes are sad to enforce k-consstency [Freuder 82]. In genera k-consstency agorthms have a compexty exponenta n k. 2. Varabe and Vaue Orderng Heurstcs: These heurstcs are concerned wth the order n whch varabes are nstantated and vaues assgned to each varabe. A good varabe orderng s one that starts wth the varabes that are the most dffcut to nstantate. By frst nstantatng these crtca varabes, one hopes to avod wastng a ot of tme budng parta soutons that cannot be competed. A good vaue orderng heurstc s one that eaves open as many optons as possbe to the remanng unnstantated varabes (.e. a so-caed east constranng vaue orderng heurstc). These heurstcs are meant to reduce the chances of backtrackng and ts cost, when t cannot be avoded. Both theoretca and expermenta studes show LOOK-AHEAD TECHNIQUES FOR MICRO-OPPORTUNISTIC JOB SHOP SCHEDULING

17 RELATED WORK 3 that varabe and vaue orderng heurstcs can sgnfcanty reduce search [Harack 80, Purdom 83, Stone 86, Dechter 88, Zabh 88, Dechter 89a, Fox 89]. 3. Deadend Recovery Technques: These technques hep decde whch earer assgnments to undo n order to recover from a deadend. The smpest such strategy s known as chronoogca backtrackng. It conssts n undong the ast assgnment, and tryng another one (f there s one eft). More sophstcated deadend recovery strateges have been desgned that attempt to go back to the source of faure and undo one or severa of the assgnments that prevent the current parta souton from beng successfuy competed [Staman 77, Doye 79, Gaschng 79, Dechter 89b]. Technques have aso been deveoped that attempt to "earn" from deadends by abstractng from these stuatons a set of parta assgnments that are nconsstent and shoud therefore be avoded n the future 7 [Dechter 89b]. 4. Herarchca Reformuaton Technques: These are technques to automatcay defne abstractons n a CSP. If carefuy chosen, such abstractons have been known to sgnfcanty reduce search [Sacerdot 74, Sussman 80, Stefk 8b, Fox 86]. Wth the excepton of [Dechter 89c, Knobock 9], very tte forma work has been done n ths area. At the tme ths research started, a good dea of expermenta resuts reported n the CSP terature had been obtaned on toy probems such as N-queens 8. The probems used n these experments generay nvoved 0 to 20 varabes, each wth at most 0 to 20 vaues. How the CSP paradgm woud scae up on harder and arger probems such as job shop schedung remaned an open ssue. Ths thess demonstrates that whe the CSP paradgm tsef scaes up pretty we, the partcuar heurstcs that have been proposed so far n the terature are often too weak to produce good resuts. Chapter 3 ponts to the 7 Ths ast technque can aso be seen as a form of dynamc consstency enforcement. 8 The N-queens probem requres postonng N queens on an N N chess board so that they cannot attack each other accordng to chess rues [Kratchck 42]. Ths probem s not NP-hard [Yagom 64, Abramson 89]. N. SADEH

18 4 INTRODUCTION shortcomngs of popuar varabe and vaue orderng heurstcs. A new probabstc mode s aso ntroduced that aows for the defnton of more powerfu varabe and vaue orderngs. Ths mode was nfuenced by the work of Bernard Nade [Nude 83, Nade 86a, Nade 86b, Nade 86c], who hmsef generazed a probabstc mode ntroduced earer by Harack and Eott [Harack 80]. In hs work, Nade dentfed a sma set of measures that he used to seect between aternatve search orderngs based on compexty estmates. A key measure n Nade s work s that of constrant satsfabty, namey the number of ways n whch a constrant can be satsfed. The probabstc estmates deveoped by Nade soey account for these constrant satsfabtes and the ways n whch constrants n a probem are connected to each other 9. It seems however that, n hard probems ke job-shop schedung, measures of constrant satsfabty are not suffcent. The dffcuty n satsfyng a constrant, whch w from now on be referred to as the tghtness of that constrant, crtcay depends on the specfc ways n whch that constrant nteracts wth other probem constrants,.e. the tghtness of a constrant s generay determned by the specfc pars of vaues aowed by that constrant and the other nteractng constrants. For nstance, n job shop schedung, nteractons between precedence constrants can be very dfferent from nteractons between capacty constrants, even f these constrants form smar constrant graphs and have smar satsfabtes. Recenty severa appcatons of CSP technques to job shop schedung probems have been reported n the terature [Conot 88, LePape 88, Dncbas 88, Keng 89, Burke 89, Prosser 89, Eeby 89, Johnston 90, Bade 90, Mnton 90]. Of partcuar reevance to ths dssertaton s the work of Napng Keng [Keng 89], who deveoped a par of varabe and vaue orderng heurstcs that attempts to account for nteractons between capacty constrants. Hs heurstcs are descrbed n more deta n Chapter 3. Expermenta resuts are aso presented ndcatng that, athough more powerfu than generc CSP heurstcs, Keng s heurstcs st fa to account for some mportant constrant nteractons. Recenty, Mnton proposed a so-caed repar heurstc approach to job-shop schedung [Mnton 90]. Wthn ths approach, each varabe s ntay assgned a tentatve vaue. Ths nta assgnment s then refned n order to get rd of a constrant voatons. The repar heurstc suggests to frst work on the varabe whose current tentatve assgnment 9 The term "connected" refers to a graphca representaton of a CSP wth bnary constrants, known as the constrant graph of the probem. In a constrant graph, each varabe s represented by a node, and bnary constrants are represented by arcs between two nodes. LOOK-AHEAD TECHNIQUES FOR MICRO-OPPORTUNISTIC JOB SHOP SCHEDULING

19 RELATED WORK 5 voates the argest number of constrants, and to repace ths assgnment wth one that mnmzes the number of remanng confcts. Whe ths technque has performed partcuary we on the N-queens probem, ts performance on more dffcut probems, such as job-shop schedung, remans to be assessed. A potenta drawback of Mnton s technque s ts reance on a snge tentatve assgnment to dentfy crtca varabes and promsng vaues for these varabes. Indeed, parta soutons produced by the procedure may n fact bear tte resembance to earer tentatve assgnments. As a consequence, the ook-ahead provded by the repar heurstc w generay be mted to a coupe of search states, rather than appyng to the entre probem. So far very tte work has been done to extend the CSP paradgm to dea wth optmzaton probems 0. Whe mathematca programmng technques, both contnuous and dscrete, have provded eegant soutons to many COPs, there reman probems on whch these technques have had very tte mpact so far. As ponted out earer, job shop schedung beongs to ths cass of more dffcut probems. The work reported n Chapters 4 and 5, wthn the context of the MICRO-BOSS factory scheduer, ndcates that the CSP paradgm can sometmes provde a vabe aternatve to tradtona Mxed Integer Programmng technques. Extendng the CSP approach to dea wth optmzaton probems s far from trva. Good varabe and vaue orderng heurstcs to fnd a feasbe souton w often perform poory n the presence of an objectve functon. Earer experments reported n [Sadeh 89a, Sadeh 90], n the context of job shop schedung probems, ndcate that, athough partcuary effectve to reduce search, east constranng vaue orderng heurstcs, such as those advocated by Keng [Keng 89], tend to produce poor soutons. Lookng for a good souton generay requres more constranng vaue orderngs, and may therefore resut n more backtrackng. Ths w generay requre more powerfu consstency enforcng mechansms (see Chapter 2), and varabe orderng heurstcs that account for the bas of the vaue orderng heurstc towards the seecton of better vaues (see Chapter 4). 0 An excepton s the work of Rna Dechter, Av Dechter, and Judea Pear who dentfed a cass of COPs wth acycc constrant graphs that can be soved to optmaty n poynoma tme usng a dynamc programmng technque [Dechter 90]. N. SADEH

20 6 INTRODUCTION.4. Summary of Contrbutons The man contrbutons of ths dssertaton can be summarzed as foows: A Mcro-opportunstc Approach to Job Shop Schedung: Whe earer scheduers such as OPT, ISIS, and OPIS have reed on coarse probem decompostons, ths dssertaton presents a mcro-opportunstc approach to job shop schedung that aows for much fner subprobems. It s shown that the extra fexbty of ths approach can be expoted to constanty redrect the schedung effort towards those botteneck operatons that appear to be the most crtca. Expermenta resuts ndcate that the abty of the mcroopportunstc approach to constanty revse ts search procedure aows for sgnfcant ncreases n schedue quaty and s nstrumenta n effcenty sovng probems n whch some operatons have to be performed wthn non-reaxabe tme wndows (e.g. non-reaxabe reease and due dates). Appcaton of the CSP Paradgm to Job Shop Schedung/A Probabstc Mode of the Search Space: Ths thess demonstrates that, whe the CSP paradgm (.e. combnng consstency enforcng technques wth varabe and vaue orderng heurstcs) scaes up to arger and harder probems such as job shop schedung, the partcuar heurstcs that had been proposed earer are not suffcent for probems such as job shop schedung. Ths s because these heurstcs fa to propery account for constrant tghtness and for the connectvty of the constrant graph. Instead, a new probabstc mode of the search space s defned that aows for the defnton of more powerfu varabe and vaue orderng heurstcs. These heurstcs have aowed for the souton of schedung probems wth non-reaxabe tme wndows that coud not be soved effcenty by pror technques. Extenson of the CSP Paradgm to dea wth COPs: Ths dssertaton aso extends the CSP paradgm to dea wth job shop schedung as an optmzaton probem. Whe east constranng vaue orderng heurstcs used to sove CSPs are partcuary good at reducng backtrackng, they typcay fa to provde good soutons. Instead, more constranng vaue LOOK-AHEAD TECHNIQUES FOR MICRO-OPPORTUNISTIC JOB SHOP SCHEDULING

21 SUMMARY OF CONTRIBUTIONS 7 orderng heurstcs are requred. Ths n turn requres the use of stronger consstency enforcng mechansms and more accurate varabe orderng heurstcs n order to mantan backtrackng at a ow eve. Ths thess descrbes such mechansms wthn the framework of the job shop schedung probem. In partcuar, a new varabe orderng heurstc s descrbed that dentfes crtca operatons as those nvoved n mportant tradeoffs. The resutng scheduer mpements a two-step optmzaton procedure. In the frst step, reservaton assgnments are optmzed wthn each jobs, and crtca operatons are dentfed as those whose good reservatons confct most wth the good reservatons of other operatons. Reservatons for the crtca operatons are then ranked accordng to ther abty to mnmze the costs ncurred by the operaton tsef and the operatons wth whch t competes. The MICRO-BOSS Factory Schedung System: One of the most tangbe contrbuton of ths thess s certany the MICRO-BOSS factory schedung system tsef, whch s abe to dea expcty wth tardness costs, n-process nventory costs, and fnshed-goods nventory costs. Ths system has outperformed a varety of competng schedung technques under a wde range of schedung condtons. MICRO-BOSS ntroduces the noton of botteneck operaton, whch s drecty formazed n terms of tardness and nventory costs and accounts for earer schedung decsons. A more compete summary of contrbutons s provded n Chapter Thess Outne Chapter 2 descrbes the mcro-opportunstc search procedure studed n ths dssertaton wth a speca emphass on consstency enforcng mechansms. Chapter 3 deas wth a generc verson of the job shop CSP, n whch operatons may requre severa resources for whch there can be aternatves, and some operatons have to be performed wthn one or severa non-reaxabe tme wndows. The chapter starts by revewng some popuar varabe and vaue orderng heurstcs, and expans why these heurstcs typcay fa when apped to job shop schedung. A probabstc mode of the search space s then N. SADEH

22 8 INTRODUCTION ntroduced that aows for the defnton of more powerfu varabe and vaue orderng heurstcs. Expermenta resuts presented at the end of ths chapter demonstrate that these new varabe and vaue orderng heurstcs outperform a varety of other CSP heurstcs (both generc heurstcs and specazed CSP heurstcs desgned for job shop schedung). Chapter 4 deas wth job shop schedung as a COP. Ths chapter presents the ook-ahead technques deveoped wthn the context of the MICRO-BOSS factory scheduer to hep the system decde whch operaton to schedue next and whch reservaton to assgn to that operaton. Severa expermenta studes are reported n Chapter 5 that demonstrate the superorty of MICRO-BOSS over both tradtona prorty dspatch rues and coarser opportunstc schedung technques. Chapter 6 concudes ths dssertaton wth a set of fna remarks. LOOK-AHEAD TECHNIQUES FOR MICRO-OPPORTUNISTIC JOB SHOP SCHEDULING

23 9 Chapter 2 The Mcro-opportunstc Search Procedure 2.. Overvew In the mcro-opportunstc (or operaton-centered) approach studed n ths dssertaton, each operaton s consdered an ndependent decson pont. Any operaton can be schedued at any tme, f deemed approprate by the scheduer. There s no obgaton to smutaneousy schedue other operatons upstream or downstream wthn the same job, nor s there any obgaton to schedue other operatons competng for the same resource. The mcro-opportunstc scheduer proceeds accordng to the generc backtrack search procedure by teratvey seectng an operaton to be schedued and a reservaton to be assgned to that operaton (.e. a start tme and a specfc resource for each resource requrement for whch there are severa aternatves). Every tme an operaton s schedued, a new search state s created, where new constrants are added to account for the reservaton assgned to that operaton. If an unschedued operaton s found to have no possbe reservatons eft, a deadend state has been reached: the system needs to backtrack (.e. t needs to undo some earer reservaton assgnments n order to be abe to compete the schedue). If the search state does not appear to be a deadend, the scheduer moves on and ooks for a new operaton to schedue and a reservaton to assgn to that operaton. Ths process goes on unt a operatons have been schedued, or unt the schedung probem has been found to be nfeasbe. Because job shop schedung s NP-compete, ths procedure coud requre exponenta tme n the worst case. In practce, as demonstrated by the expermenta studes presented n ths thess, t s generay possbe to mantan the average compexty of the procedure at a very ow eve whe producng quaty schedues. Ths s acheved by ntereavng search wth the appcaton of consstency enforcng technques and a set of ook-ahead technques that hep decde whch operaton to schedue next (so-caed varabe orderng heurstc) and whch reservaton to assgn to that operaton (so-caed vaue orderng heurstc). N. SADEH

24 20 THE MICRO-OPPORTUNISTIC SEARCH PROCEDURE. Consstency Enforcement: Consstency enforcng technques are used to prune possbe reservatons (of unschedued operatons) that have become unavaabe due to earer reservaton assgnments. By constanty nferrng new constrants resutng from earer schedung decsons, these technques reduce the chance of seectng reservaton assgnments that are nconsstent wth earer schedung decsons. Ths reduces the chances of backtrackng. Addtonay, by aowng for the eary detecton of deadend states, these technques mt the amount of work wasted n the exporaton of frutess aternatves. In other words these technques reduce both the frequency and the amount of backtrackng. 2. Look-ahead Anayss: The purpose of the ook-ahead anayss s to hep dentfy crtca operatons and promsng reservatons for these operatons. Operaton crtcaty and reservaton goodness are measures that are not ntrnsc to a probem. They depend on earer schedung decsons (.e. they change from one search state to another), and on the objectve to be optmzed. In ths dssertaton, two varatons of the job shop schedung probem are successvey studed, one n whch the ony concern s to come up wth a feasbe souton as fast as possbe (job shop CSP), and one n whch the objectve s to effcenty come up wth as good a souton as possbe (job shop COP). In the job shop CSP, a crtca operaton s one whose reservatons are key to become unavaabe f other operatons were schedued frst. By schedung these operatons frst, the scheduer avods budng parta schedues that t w not be abe to compete ater on. Smary a good reservaton, n the job shop CSP, s one that eaves enough room to other unschedued operatons so that the schedue can be competed wth mnma backtrackng. In the case of the job shop COP, the notons of operaton crtcaty and reservaton goodness appear to be more compex. Ths dssertaton suggests that, n COPs, crtca varabes (e.g. crtca operatons n job shop schedung) are varabes partcpatng n mportant tradeoffs and promsng vaues for these varabes are vaues that optmze these tradeoffs. An mportant tradeoff s one that crtcay mpacts the quaty of the entre souton. By frst optmzng the most LOOK-AHEAD TECHNIQUES FOR MICRO-OPPORTUNISTIC JOB SHOP SCHEDULING

25 OVERVIEW 2 mportant tradeoffs n a probem, the system can ater use the soutons to these tradeoffs to hep sove the remander of the probem. Indeed, once crtca tradeoffs have been worked out, the remander of the probem tends to become more decouped, and hence easer to optmze. Chances of backtrackng tend to smutaneousy subsde as we. A system, that does not attempt to work out crtca tradeoffs frst, runs the rsk overconstranng ts set of aternatves before havng worked on the subprobems that w mpact most the quaty of the entre souton. Ths thess descrbes a unfyng framework n whch measures of reservaton reance (.e. the reance of an operaton on the avaabty of a reservaton) and resource contenton are computed to dentfy crtca operatons and promsng reservatons for these operatons. Ths framework s apped to both the CSP and COP versons of the job shop schedung probem studed n ths dssertaton. In job shop CSPs, the reance of an operaton on the avaabty of a specfc reservaton (n a search state) s defned as a functon of the number of aternatve reservatons st avaabe to that operaton (n that search state). Operatons wth a sma number of aternatve reservatons eft are sad to rey more on each one of ther remanng possbe reservatons. In job shop COPs, the reance of an operaton on a reservaton s defned as a functon of the expected mert of assgnng that reservaton to the operaton compared to the mert of aternatve reservatons st avaabe to the operaton, and compared to dfferences n mert between the reservatons st avaabe to other operatons. In other words, operatons wth ony a sma fracton of ther remanng reservatons expected to resut n a hgh vaue of the objectve are sad to hghy rey on ths sma fracton of good reservatons. Once reservaton reance has been evauated, crtca resource/tme ntervas are dentfed as hghy reed upon resource/tme ntervas, and crtca operatons as those that most heavy rey on the avaabty of these crtca resource/tme ntervas. Whe measures of reservaton reance depend on whether the schedung probem s a CSP or a COP, the procedure used to measure resource contenton and dentfy crtca operatons remans the same. N. SADEH

26 22 THE MICRO-OPPORTUNISTIC SEARCH PROCEDURE The next secton gves a more forma descrpton of the top-eve procedure embedded n the mcro-opportunstc approach The Search Procedure Concretey, the mcro-opportunstc search procedure starts n a search state n whch no operaton has been schedued yet, and proceeds accordng to the foowng steps:. If a operatons have been schedued then stop, ese go on to 2; 2. Appy the consstency enforcng procedure; 3. If a deadend s detected then backtrack (.e. seect an aternatve f there s one eft and go back to, ese stop and report that the probem s nfeasbe), ese go on to step 4; 4. Perform the ook-ahead anayss: evauate the reance of each unschedued operaton on the avaabty of ts remanng possbe reservatons, and measure resource contenton over tme; 5. Seect the next operaton to be schedued (so-caed operaton orderng heurstc): seect the operaton that rees most on the most contended (.e. most hghy reed upon) resource/tme nterva; 6. Seect a promsng reservaton for that operaton (so-caed reservaton orderng heurstc) 7. Create a new search state by addng the new reservaton assgnment to the current parta schedue. Go back to. In ths search procedure, the so-caed opportunstc behavor resuts from the abty of the scheduer to constanty revse ts search strategy and redrect ts effort towards the schedung of the operaton that appears to be the most crtca n the current search state. Ths degree of opportunsm dffers from that dspayed by other approaches where the schedung entty s an entre resource or an entre job [Ow 88a],.e. where an entre resource (or at east a arge chunk of t) or an entre job (or at east a arge porton of t) needs to be schedued before the scheduer s aowed to revse ts current schedung strategy. The resuts reported n ths dssertaton were obtaned usng a smpe chronoogca LOOK-AHEAD TECHNIQUES FOR MICRO-OPPORTUNISTIC JOB SHOP SCHEDULING