1. Introduction. Scheduling Theory


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1 . Itroductio. Itroductio As a idepedet brach of Operatioal Research, Schedulig Theory appeared i the begiig of the 50s. I additio to computer systems ad maufacturig, schedulig theory ca be applied to may areas icludig agriculture, health care ad trasport. Operatioal Research Mathematical Programmig Combiatorial Optimisatio Schedulig Theory Schedulig deals with the problems of optimal arragemet, sequecig ad timetablig. Schedulig is a decisiomakig process of allocatig limited resources to activities over time. Resources (machies): machies at a workshop, ruways at a airport, crews at a costructio site, processig uits i a computig eviromet. Tasks (obs): operatios i a workshop, takeoffs ad ladigs, stages at a costructio proect, computer programs. A schedule is a ob sequece determied for every machie of the processig system. Stadard schedulig requiremets say: a ob caot be processed by two or more machies at a time, a machie caot process two or more obs at the same time. Depedig o the type of schedulig system, specific costraits should be satisfied (obs may be released at differet times, there may be allowed preemptio of obs by other obs, etc.).
2 . Itroductio Gatt chart is a horizotal bar chart that graphically displays the time relatioships betwee the differet tasks i a proect: the Xaxis represets the time, the Yaxis represets machies, a colour ad/or patter code may be used to idicate operatios of the same ob. M M M 3 M time Schedulig theory covers more tha 0,000 differet models. The models are specified accordig to threefield classificatio α β γ where α specifies the machie eviromet, β specifies the ob characteristics, ad γ determies the optimality criterio. Machie eviromet Sigle stage systems Multistage systems If there is a sigle machie (m=), each ob should be processed by that machie exactly oce. If there are several parallel machies {M, M,, M m }, each ob ca be processed by ay machie. α = P sigle (dedicated) machie p p i processig time of detical parallelmachies, = p o machie i ob processig time of ob Each ob should be processed o each machie from the set {M, M,, M m }. All machies are differet. F α = J O flow shop, ob is processed first o machie, the o machie,..., ad fially o machiem ob shop, eachob has its ow route to follow ope shop, each ob ca be processed by the machiesi a arbitrary order
3 . Itroductio Job eviromet There are obs N={,,}. Processig time of ob o machie i is p i. If there is a sigle machie, the processig time of ob does ot deped o machie umber ad it is deoted by p. For ob there may be give also r release time (the time the ob arrives at the system), d due date (the time the ob is promised to the customer), w weight (the importace of ob ). Preemptio (pmt) implies the processig of ay ob ca be iterrupted ad resumed later. Optimality criterio The schedule ca be characterised by startig or completio times of all operatios of the obs. The obective is to costruct a schedule, that miimizes a give obective fuctio F. Usually fuctio F depeds o ob completio times C, =,,, where C is the completio time of the last operatio of ob. The most commo obective fuctios are Makespa Total completio time Total weighted completio time C max = max {C =,,} = = w C = C C = w C Other obective fuctios deped o due dates d. We defie for each ob L = C d E = max{0, d C } T = max{0, C d } 0 if C d U = otherwise lateess of ob earliess, tardiess, uit pealty., The correspodig obective fuctios ca be defied as follows: Maximum lateess Total (weighted) tardiess Total (weighted) umber of late obs L max = max {L =,,} = T ( = w T ) = U ( = w U ) 3
4 . Itroductio Examples: ) r, pmt L max is the problem of fidig a preemptive schedule o oe machie for a set of obs with give release times such that maximum lateess is miimised. ) P p = C max is the problem of schedulig obs with uit processig times o m idetical parallel machies such that the makespa is miimised. 3) J3 p i = C max is the problem of miimisig maximum completio time i a threemachie ob shop with uit processig times. Iclass exercise : Cosider a schedulig problem with readers ad two books. Classify the followig schedulig models: Machies Jobs Obective α β γ Two volumes of oe book Fiish readig as soo as possible F C max Two volumes of oe book Miimise the cost of late book retur Two differet (idepedet) books Fiish readig as soo as possible Two differet (idepedet) books (each reader has its ow readig sequece ) Fiish readig as soo as possible Iclass exercise : Classify the examples of schedulig problems: Publishig idustry: typesettig, actual pritig, bidig, packagig. Differet items have differet processig times depedig o the book size, the umber of copies, etc. The obective is to produce all items as soo as possible. Clothig idustry: cuttig, sewig, pressig, packig. The obective is to produce all items as soo as possible. Steel mills: differet rods or girders pass through the set of rollers i their ow orders with their ow temperatures ad pressure settigs. The obective is to produce all items as soo as possible. Repair of cars i a garage: replace tires, repair gear box, check brakes, repair headlights, etc. The obective is to repair all cars i a garage as soo as possible. Completig several pieces of CW so that the maximum lateess is miimised. CW is released at time r, requires p days for completio ad has a due date d. Revisio schedule: startig o 3//004, revise the material of modules by their exam dates. Revisio time for module is p. F4 C max? Literature review for FYP should be based o library books. Book ca be read i p days ad it should be retured by its due date d. The library charges 30p per day o each overdue book. The obective is to miimise the total fie. 4
5 . Itroductio Complexity Hierarchy There is a certai complexity hierarchy amog schedulig systems ad obective fuctios as show i the diagrams below. The arrows go to harder problems. Σw T Σw U J Σw C ΣT ΣU P F O ΣC L max C max C max reduces to L max by settig d =0 for all. ΣC reduces to Σw C by settig w = for all. ΣC reduces to ΣΤ by settig d =0 for all. Σw C reduces to Σw Τ by settig d =0 for all. I this course we demostrate most schedulig methods ad techiques. Most algorithms will be illustrated usig schedulig software systems LEKIN ad LiSA. LiSA Library of Schedulig Algorithms, OttovoGuericke Uiversität, Magdeburg, Germay, Proect Leader: Professor Heidemarie Bräsel, LEKIN Educatioal Schedulig System, Ster School of Busiess, New York Uiversity, Proect Leader: Professor Michael Piedo, The followig books are cosidered to be stadard texts o the topic:. J.Blazewicz, K.H. Ecker, E.Pesch, G.Schmidt, J. Weglarz Schedulig Computer ad Maufacturig Processes, Spriger, 996, 00.. P. Brucker Schedulig Algorithms, Spriger, 995, 998, S. Frech Sequecig ad schedulig, Ellis Horwood, M. Piedo Schedulig: Theory, Algorithms, ad Systems, Pretice Hall, 995, M. Piedo Operatios Schedulig with Applicatios i Maufacturig ad Services, Irwi/McGraw Hill,
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