Key words. stochastic scheduling, flow shop, job shop, open shop, .... these
results, see Pinedo and Schrage (1982) and, more recently, Pinedo (1983). Milch
.
?
SIAM J. APPL. MATH. Vol. 44, No. 4, August 1984
INEQUALITIES
AND BOUNDS
1984 SocietyforIndustrialand Applied Mathematics 015
IN STOCHASTIC
MICHAEL PINEDOt
AND
SHOP SCHEDULING*
RICHARD WEBERt
Abstract.In thispaper,stochasticshop models withm machinesand n jobs are considered.A job has to be processedon all m machines,whilecertainconstraintsare imposedon the order of processing.The effect ofthevariability oftheprocessingtimeson theexpectedcompletiontimeofthelastjob (themakespan) and on the sumof the expectedcompletiontimesof all jobs (the flowtime)is studied.Bounds are obtained are New Better (Worse) than Used in forthe expectedmakespanwhen the processingtimedistributions Expectation. Key words. stochasticscheduling,flowshop,job shop,open shop,exponentialdistribution, makespan, flowtime
1. Introduction.Consider a shop with m machinesand n jobs. Any given job requiresprocessingon each one of the m machinesand all jobs are available for processingat t=O. The mannerin whichthe jobs are routedthroughthe systemis and fixedand dependson theparticularshopmodelunderconsideration. predetermined The processingtimeof job j on machinei is a randomvariableXij withdistribution of the randomvariablesX1ll , Xmnmayhave one of the Gij.The jointdistribution followingtwo forms: (i) The mn processingtimesX11, *, Xmnare mutuallyindependent. (ii) The m processingtimesof a job on the m machinesare identical,but the processingtimeof any givenjob on a machineis independentof the processingtime of any otherjob on that machine,i.e. X1j= . = Xmj= X withdistributionGj for j=1, r , n and Xi and Xk are mutuallyindependentifj1 k. In whatfollows,thesetwo cases are called, respectively, the independentand the on the expected equal case. In thispaper,the effectof the processingtimes'variability completiontime of the last job (the makespan) and on the sum of the expected completiontimesof all jobs (the flowtime) is studied. Four shop models are considered,namelyflowshops withan unlimitedstorage space in between the machines,flow shops with no storage space in between the machines,job shops and open shops. A shortdescriptionof these modelsfollows. intermediate (I) Flow shopswithunlimited storage.The n jobs are to be processed machinesbeing the m with order of the on the machines processingon the different on to first machine same forall jobs. Each job has be processed 1, thenon machine 2, etc. The sequence in whichthe jobs go throughthe systemis predetermined;job 1 has to go firstthroughthe system,followed by job 2, etc. There is an infinite intermediate storagein betweenanytwoconsecutivemachines;ifmachinei + 1 is busy whenjob j is completedon machinei, job j is storedin betweenmachinesi and i + 1. Preemptionsare not allowed and a job maynot "pass" anotherjob whilewaitingfor a machine. (II) Flow shopswithno intermediate storage.This model is similarto the previous model. The onlydifference lies in the factthatnow thereis no intermediatestorage * Received by the editorsMarch 8, 1983, and in revisedformOctober 5, 1983. This researchwas supportedin part by the Officeof Naval Research under contractN00014-80-k-0709 and in part by the National Science FoundationundergrantEC5-8115344. t Departmentof IndustrialEngineeringand Operations Research,Columbia University,New York, New York 10027. t Control and Management Systems Division, Engineering Department, Cambridge University, Cambridge,England. 869
870
MICHAEL
PINEDO
AND RICHARD
WEBER
in betweenthe machines.This mayhave the followingeffect:Job j, aftercompleting itsprocessingon machinei, maynot leave machinei ifjob j-1 is stillbeingprocessed on machine i + 1. Job j + 1 cannot start then its processingon machine i. This phenomenonis called blocking. (III) Jobshops.Only job shops withtwo machinesare considered.Some of the jobs, say jobs 1,... , p, have to be processed firston machine 1 and afterwardson machine2 (job 1 goingfirst,followedby job 2, etc.). The remainingq ( = n- p) jobs on machine1 (job p + 1 going have to be processedfirston machine2 and afterwards storagein between followedbyjob p + 2, etc.). There is an unlimitedintermediate first, the two machines,so no blockingwill occur. The policyunderwhichthe jobs are to ,p and underthispolicyjobs 1, be processedon thetwo machinesis predetermined (p + 1, *. , n) musthave completedtheirprocessingon machine1 (2) beforeany one of jobs p+1, * *, n (1, * , p) is allowed to starton machine1 (2). It is clear thatif p is either0 or n, thisjob shop reduces to a two machineflowshop withunlimited intermediatestorage. (IV) Open shops. Only two machineopen shops are considered.The order in whicha job is to be processedon the two machinesis now immaterial.There is an unlimitedintermediate storage,so no blockingwilloccur.Onlypoliciesare considered whichalways give priorityto jobs whichhave not yet receivedprocessingon either one of the two machines. In the literaturethese models have been dealt withextensively.The researchin the past has been aimed mainlyat findingjob sequences and policies thatminimize criteriasuch as the expected makespanand the expectedflowtime.For a surveyof Pinedo (1983). Milch theseresults,see Pinedo and Schrage(1982) and, morerecently, and Waggoner (1972) studiedthe two machinejob shop where the two processing timesof any givenjob are independentexponentiallydistributedwithmean one and obtained a closed formexpressionforthe expectedmakespan. A summaryof the resultsfollows.Section2 discussesa formof stochasticdominance based on variabilityordering.The effectof the processingtimesvariabilityon the expectedmakespanand on the expectedflowtimeis studiedforthe first,second and thirdmodels describedabove. In ? 3, closed formexpressionsfor the expected makespan are presentedforthe firstthreemodels when the processingtimesof any given job on the various machinesare i.i.d. exponentialwith mean one. Furthermore, bounds are obtained for the expected makespan when the processingtimes of any given job on the various machinesare independentand NBUE (NWUE) with mean one. Section 4 repeats the work of ? 3 for the equal case. In ? 5, the equal and the independentcases of the two machine open shop are considered. Again, closed formexpressionsare obtained when the processingtime distributions are exponentialwithmean one and bounds are obtained for when theyare NBUE (NWUE). is used. S m,c,k denotesa shop. If the S is The followingnotationand terminology an F, the shop is a flowshop; if it is a J a job shop, and if it is an 0 an open shop. The subscriptm denotesthenumberofmachines.Ifthec is an i (e), thentheprocessing timesare distributedaccordingto the independent(equal) case. The k indicatesthe size of theintermediate storage;it is omittediftheshop is an open shop or a job shop. The timejob j leaves the systemis denoted by Cj; the makespanand the flowtime are respectivelydenoted by Cmaxand L Cj. The time epoch at which job j leaves machinei is denoted by Tij.The makespanand flowtimeof shop Sm,c,kare denoted ifit is clear fromthecontextwhichshop by Cmax(Sm,c,k) and L Cj(Sm,c,k), respectively; is beingconsidered,the argumentSm,c,kis omitted.When all processingtimedistribu.
.
.
INEQUALITIES ANDBOUNDSIN STOCHASTIC SHOPSCHEDULING
871
tionsare exponentialwithmean one, thisis indicatedby an asterisk,e.g., C*ax(Sm,c,k) or Cmax. 2. Preliminaries.The randomvariable Y1 withdistribution F1 is said to be more variablethanthe randomvariable Y2 withdistribution F2 if
0
h(x) dF1(x)
h(x) dF2(x)
100
0
forall functionsh thatare increasingconvex. This formof stochasticdominancehas been used repeatedlyin the literature(see Bessler and Veinott (1966), Stoyan and Stoyan (1969), Niu (1981), Whitt (1980) and their references)and is writtenas Y, > Y2. If E( Y1) = E( Y2), then Y1 is more variablethan Y2 if and onlyif h(x) dF1(x)
{
h(x) dF2(x)
forall functionsh whichare convex,not necessarilyincreasing. A randomvariable Y1 is said to be NBUE (NWUE) if E(Y1-tl
Y1> t)'--(i')E(Y1)
forall t_-:O.
NBUE (NWUE) standsforNew Better(Worse) thanUsed in Expectation. LEMMA 1. Let E( Y1) = E( Y2) and let Y1 be an exponentialrandomvariable.If Y2 is NBUE (NWUE), thenY2 < ) Y1. Proof.See Marshalland Proshan (1972). LEMMA 2. Let Yi,Zi, i = 1, , n be independent randomvariables.Then Yi < Zi for all i = 1, ** , n if and only if h( Y1, * Yn) < h(Z1, Zn) for all increasing convexfunctions. Proof.See Besslerand Veinott(1966). ConsidertheshopFm,c,k,m = 2, 3,.*.*, c e, i, k=0, 1,2, * * *, and the shop J2,, c=e, i. LEMMA 3. In theshopsFm,c,k and J2, thetimeepochTij,themakespanCmax and theflowtimelCj are functionswhichare increasingconvexin Xi1. Proof.Consider firstFm,i,o.For the firstjob that goes throughthe systemthe followingholds. i i = 1,* ,m. Til =LXn1, ,
1=1
Thisis clearlyan increasingconvexfunction. For job j, j = 2, Tij = max (Tl,j-_1+ Xlj, T2,J_1),
j=-2, ***, n,
Tij =max (Ti-1j +Xij, Ti+,,j-,),
i=2,*
,
n,thefollowingholds.
*,m, j=2,***,n.
It follows by induction that for Fm,i,o the time epoch Tij, the makespan Cmax(= Tmn)
and the flowtimeE Cj are functionswhichare increasingconvexin Xij. The proofof the lemmaforFm,e,o is similar. The resultforFm,i,k can be shownby assumingk dummymachinesin between any two real machines.The processingtimesof the n jobs on a dummymachineare assumedto be zero. Note thatwithn jobs the shop Fm,i,n-I behaves just like the shop Fmi,oo.The proofforFm,e,k is similar. In J2,ijob j, j = 2,.*. , p, startsits processingon machine2 at max (T1j, T2,j1). Therefore T2j =max (T1j, T2,J11)+X2j, j=2,***. p.
872
MICHAEL PINEDO AND RICHARD WEBER
Note that T1 = X11 and T21= max (X11,
n
\
X2j
YE
j=p+l
+X21.
A similarexpressioncan be formulatedfor the departuretimesof jobs p+ 1, , n from machine 1. It follows then by inductionthat T2j,j= 1, * , p, and T1j,j= p + 1, * * *, n,are functions thatare increasingconvexinXi1.Now Cmax = max (T2p, T n) This provesthe lemmaforJ2,i.Again, the proofof the lemmaforJ2,e is similar. Now, considertwo shops of the same type,the typebeingone of theFm,c,k shops or one of the J2,,shops. A distinctionis made betweenthese two shops throughthe use of a primeand a double prime,forexample,F',i os) and F'j,i,o. The processingtime distributions in one shop are not identicalto the processingtimedistributions in the othershop: The processingtimeof job j on machine i in the first(second) shop is denotedby Xt' (X'5) and its distribution by G{, (G'l). All otherquantitiesof interest in thetwoshopsreceivea primeand a double primeas well.In thesubsequenttheorem and corollariesshops Fm,c,k and J, are comparedwithshops F'm,c,k respecJn2cand tively.The resultsfollow immediatelyfromLemmas 1, 2 and 3, and are therefore presentedwithoutproofs. THEOREM 1. If
Xtl < Xl',
i-1,*
Tl' < T'l,
i=l
,m
j= 1,*
,n
then ,m.
j=l ,99..n,
C' max
