Sngle-Faclty Schedulng over Long Tme Horzons by Logc-based Benders Decomposton Elvn Coban and J. N. Hooker Tepper School of Busness, Carnege Mellon Unversty ecoban@andrew.cmu.edu, john@hooker.tepper.cmu.edu Abstract. Logc-based Benders decomposton can combne mxed nteger programmng and constrant programmng to solve plannng and schedulng problems much faster than ether method alone. We fnd that a smlar technque can be benefcal for solvng pure schedulng problems as the problem sze scales up. We solve sngle-faclty non-preemptve schedulng problems wth tme wndows and long tme horzons that are dvded nto segments separated by shutdown tmes (such as weekends). The objectve s to fnd feasble solutons, mnmze makespan, or mnmze total tardness. 1 Introducton Logc-based Benders decomposton has been successfully used to solve plannng and schedulng problems that naturally decompose nto an assgnment and a schedulng porton. The Benders master problem assgns jobs to facltes usng mxed nteger programmng (MILP), and the subproblems use constrant programmng (CP) to schedule jobs on each faclty. In ths paper, we use a smlar technque to solve pure schedulng problems wth long tme horzons. Rather than assgn jobs to facltes, the master problem assgns jobs to segments of the tme horzon. The subproblems schedule jobs wthn each tme segment. In partcular, we solve sngle-faclty schedulng problems wth tme wndows n whch the objectve s to fnd a feasble soluton, mnmze makespan, or mnmze total tardness. We assume that each job must be completed wthn one tme segment. The boundares between segments mght therefore be regarded as weekends or shutdown tmes durng whch jobs cannot be processed. In future research we wll address nstances n whch jobs can overlap two or more segments. Logc-based Benders decomposton was ntroduced n [2, 8]. Its applcaton to assgnment and schedulng va CP/MILP was proposed n [3] and mplemented n [9]. Ths and subsequent work shows that the Benders approach can be orders of magntude faster than stand-alone MILP or CP methods on problems of ths knd [1, 7, 4 6, 10, 11]. For the pure schedulng problems consdered here, we fnd that the advantage of Benders over both CP and MILP ncreases rapdly as the problem scales up.
2 The Problem Each job j has release tme, deadlne (or due date) d j, and processng tme p j. The tme horzon conssts of ntervals [z,z +1 ] for =1,...,m. The problem s to assgn each job j a start tme s j so that tme wndows are observed (r j s j d j p j ), jobs run consecutvely (s j + p j s k or s k + p k s j for all k j), and each job s completed wthn one segment (z s j z +1 p j for some ). We mnmze makespan by mnmzng max j {s j +p j }. To mnmze tardness, we drop the constrant s j d j p j and mnmze j max{0,s j + p j d j }. 3 Feasblty When the goal s to fnd a feasble schedule, the master problem seeks a feasble assgnment of jobs to segments, subject to the Benders cuts generated so far. Because we solve the master problem wth MILP, we ntroduce 0-1 varables y j wth y j =1when job j s assgned to segment. The master problem becomes y j =1, all j Benders cuts, relaxaton y j {0, 1}, all, j The master problem also contans a relaxaton of the subproblem, smlar to those descrbed n [4 6], that helps reduce the number of teratons. Gven a soluton ȳ j of the master problem, let J = {j ȳ j =1} be the set of jobs assgned to segment. The subproblem decomposes nto a CP schedulng problem for each segment : } r j s j d j p j, all j J z s j z +1 p j (2) dsjunctve ({s j j J }) where the dsjunctve global constrant ensures that the jobs assgned to segment do not overlap. Each nfeasble subproblem generates a Benders cut as descrbed below, and the cuts are added to the master problem. The master problem and correspondng subproblems are repeatedly solved untl every segment has a feasble schedule, or untl the master problem s nfeasble, n whch case the orgnal problem s nfeasble. Strengthened nogood cuts. The smplest Benders cut s a nogood cut that excludes assgnments that cause nfeasblty n the subproblem. If there s no feasble schedule for segment, we generate the cut j J y j J 1, all (3) The cut can be strengthened by removng jobs one by one from J untl a feasble schedule exsts for segment. Ths requres re-solvng the th subproblem repeatedly, (1)
but the effort generally pays off because the subproblems are much easer to solve than the master problem. We now generate a cut (3) wth the reduced J. The cut may be stronger f jobs less lkely to cause nfeasblty are removed from J frst. Let the effectve tme wndow [ r j, d j ] of job j on segment be ts tme wndow adjusted to reflect the segment boundares. Thus r j = max {mn{r j,z +1 },z }, dj = mn {max{d j,z },z +1 } Let the slack of job j on segment be d j r j p j. We can now remove the jobs n order of decreasng slack. 4 Mnmzng Makespan Here the master problem mnmzes µ subject to (1) and µ 0. The subproblems mnmze µ subject to (2) and µ s j + p j for all j J. Strengthened nogood cuts. When one or more subproblems are nfeasble, we use strengthened nogood cuts (3). Otherwse, for each segment we use the nogood cut µ µ 1 j J(1 y j ) where µ s the mnmum makespan for subproblem. These cuts are strengthened by removng jobs from J untl the mnmum makespan on segment drops below µ. We also strengthen the cuts as follows. Let µ (J) be the mnmum makespan that results when n jobs n J are assgned to segment, so that n partcular µ (J )=µ. Let Z be the set of jobs that can be removed, one at a tme, wthout affectng makespan, so that Z = {j J M (J \{j}) =M }. Then for each we have the cut µ µ (J \ Z ) 1 (1 y j ) j J \Z Ths cut s redundant and should be deleted when µ (J \ Z )=µ. Analytc Benders Cuts. We can develop addtonal Benders as follows. Let J = {j J r j z } be the set of jobs n J wth release tmes before segment, and let J = J \ J. Let ˆµ be the mnmum makespan of the problem that remans after removng the jobs n S J from segment. It can be shown as n [6] that µ ˆµ p S + max{ d j } mn{ d j } (4) j J j J where p S = j S p j. Thus f jobs n J are removed from segment, we have from (4) a lower bound on the resultng optmal makespan ˆµ. If jobs n J are removed, there s nothng we can say. So we have the followng Benders cut for each : µ µ p j (1 y j ) + max{d j } mn{d j } µ j J j J j J (1 y j ) (5) j J
when one or more jobs are removed from segment, µ 0 when all jobs are removed, and µ µ otherwse. Ths can be lnearzed: µ µ j j ( w max j J p j (1 y j ) w µ (1 y j) µ q, q 1 y j,j J ) {d j } mn{d j } j J j J j J (1 y j ), w max j J {d j } mn{d j } j J 5 Mnmzng Tardness Here the master problem mnmzes τ subject to (1), and each subproblem mnmzes j J τ j subject to τ j s j + p j d j and τ j 0. Benders cuts. We use strengthened nogood cuts and relaxatons smlar to those used for mnmzng makespan. We also develop the analytc Benders cuts τ ˆτ τ ( r max + ) + p l d j (1 y j ), f r max + p l z +1 j J l J l J ˆτ τ 1 j J(1 y j ), otherwse where the bound on ˆτ s ncluded for all for whch τ > 0. Here τ s the mnmum tardness n subproblem, r max = max{max{r j j J },z }, and α + = max{0,α}. 6 Problem Generaton and Computatonal Results Random nstances are generated as follows. For each job j, r j, d j r j, and p j are unformly dstrbuted on the ntervals [0, αr], [γ 1 αr, γ 2 αr], and [0, β(d j r j )], respectvely. We set R = 40 m for tardness problems, and otherwse R = 100 m, where m s the number of segments. For the feasblty problem we adjusted β to provde a mx of feasble and nfeasble nstances. For the remanng problems, we adjusted β to the largest value for whch most of the nstances are feasble. We formulated and solved the nstances wth IBM s OPL Studo 6.1, whch nvokes the ILOG CP Optmzer for CP models and CPLEX for MILP models. The MILP models are dscrete-tme formulatons we have found to be most effectve for ths type of problem. We used OPL s scrpt language to mplement the Benders method. Table 1 shows the advantage of logc-based Benders as the problem scales up. Benders faled to solve only four nstances, due to nablty to solve the CP subproblems.
Table 1. Computaton tmes n seconds (computaton termnated after 600 seconds). The number of segments s 10% the number of jobs. Tght tme wndows have (γ 1,γ 2,α)=(1/2, 1, 1/2) and wde tme wndows have (γ 1,γ 2,α)=(1/4, 1, 1/2). For feasblty nstances, β =0.028 for tght wndows and 0.035 for wde wndows. For makespan nstances, β = 0.025 for 130 or fewer jobs and 0.032 otherwse. For tardness nstances, β = 0.05. Tght tme wndows Wde tme wndows Feasblty Makespan Tardness Feasblty Makespan Tardness Jobs CP MILP Bndrs CP MILP Bndrs CP MILP Bndrs CP MILP Bndrs CP MILP Bndrs CP MILP Bndrs 50 0.91 8.0 1.5 0.09 9.0 4.0 0.05 1.3 1.1 0.03 7.7 2.5 0.13 13 3.5 0.13 1.3 1.1 60 1.1 12 2.8 0.09 18 5.5 0.14 1.8 1.5 0.05 12 1.6 0.94 29 5.7 0.11 2.3 1.4 70 0.56 17 3.3 0.11 51 6.7 1.3 3.9 2.1 0.13 17 2.3 0.11 39 6.2 0.16 3.0 1.9 80 600 21 2.8 600 188 7.6 0.86 6.0 4.5 600 24 5.0 600 131 7.3 1.9 6.4 5.0 90 600 29 7.5 600 466 10 21 11 4.6 600 32 9.7 600 600 8.5 5.9 9.5 11 100 600 36 12 600 600 16 600 11 2.0 600 44 9.7 600 600 19 600 24 22 110 600 44 20 600 600 17 600 600 600 600 49 17 600 600 24 600 600 600 120 600 62 18 600 600 21 600 15 3.3 600 80 15 600 600 23 600 12 3.1 130 600 68 20 600 600 29 600 17 3.9 600 81 43 600 600 31 600 18 3.9 140 600 88 21 600 600 30 600 600 600 600 175 27 600 * 35 600 600 14 150 600 128 27 600 600 79 600 386 8.5 600 600 43 160 600 408 82 600 600 34 600 174 5.2 600 600 53 170 600 192 5.9 600 600 37 600 172 5.9 600 600 600 180 600 600 6.6 600 * 8.0 600 251 6.5 600 600 56 190 600 600 7.2 600 * 8.5 600 600 7.3 600 * 78 200 600 600 8.0 600 * 85 600 600 8.2 600 * 434 MILP solver ran out of memory. References 1. I. Harjunkosk and I. E. Grossmann. Decomposton technques for multstage schedulng problems usng mxed-nteger and constrant programmng methods. Computers and Chemcal Engneerng, 26:1533 1552, 2002. 2. J. N. Hooker. Logc-based benders decomposton. Techncal report, CMU, 1995. 3. J. N. Hooker. Logc-Based Methods for Optmzaton: Combnng Optmzaton and Constrant Satsfacton. Wley, New York, 2000. 4. J. N. Hooker. A hybrd method for plannng and schedulng. Constrants, 10:385 401, 2005. 5. J. N. Hooker. An ntegrated method for plannng and schedulng to mnmze tardness. Constrants, 11:139 157, 2006. 6. J. N. Hooker. Plannng and schedulng by logc-based benders decomposton. Operatons Research, 55:588 602, 2007. 7. J. N. Hooker and G. Ottosson. Logc-based benders decomposton. Mathematcal. Programmng, 96:33 60, 2003. 8. J. N. Hooker and H. Yan. Logc crcut verfcaton by Benders decomposton, pages 267 288. Prncples and Practce of Constrant Programmng: The Newport Papers, MIT Press (Cambrdge, MA, 1995), 1995. 9. V. Jan and I. E. Grossmann. Algorthms for hybrd MILP/CP models for a class of optmzaton problems. INFORMS Journal on Computng, 13(4):258 276, 2001. 10. C. T. Maravelas and I. E. Grossmann. A hybrd MILP/CP decomposton approach for the contnuous tme schedulng of multpurpose batch plants. Computers and Chemcal Engneerng, 28:1921 1949, 2004. 11. C. Tmpe. Solvng plannng and schedulng problems wth combned nteger and constrant programmng. OR Spectrum, 24:431 448, 2002.