The energy scheduling problem: industrial case study and constraint propagation techniques

(1)

HAL Id: hal-00522387

https://hal.archives-ouvertes.fr/hal-00522387

Submitted on 30 Sep 2010

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires

The energy scheduling problem: industrial case study and constraint propagation techniques

Christian Artigues, Pierre Lopez, Alain Hait

To cite this version:

Christian Artigues, Pierre Lopez, Alain Hait. The energy scheduling problem: industrial case study and constraint propagation techniques. International Journal of Production Economics, Elsevier, 2013, 143 (1), pp.13-23. �10.1016/j.ijpe.2010.09.030�. �hal-00522387�

(2)

and onstraint propagation tehniques

1

CNRS;LAAS; 7avenueduColonelRohe,F-31077Toulouse,Frane

2

UniversitédeToulouse;UPS,INSA,INP,ISAE; LAAS;F-31077Toulouse,Frane

3

UniversitédeToulouse;InstitutSupérieurdel'Aéronautiqueetdel'Espae;10avenueE. BelinB.P.

54032;F-31055artigueslaas.fr, lopezlaas.fr, alain.haitisae.fr

Abstrat

This paper deals withprodution sheduling involving energy onstraints, typi-

allyeletrial energy. We start by anindustrialase-study for whih we proposea

two-step integer/onstraint programmingmethod. From theindustrialproblem we

derive a generi problem, theEnergy Sheduling Problem (EnSP). We propose an

extension of spei resoureonstraint propagationtehniquesto eiently prune

the searh spaeforEnSP solving. Wealso present abranhingsheme tosolve the

problemvia treesearh. Finally,omputationalresults areprovided.

Keywords: Produtionsheduling, energyonstraints, onstraint propagation,ener-

geti reasoning

1 Introdution

Contextofthestudy Sinethelasttwodeades,hardombinatorialproblems,mainly

in sheduling, have been the target of many approahes ombining Operations Researh

and Artiial Intelligene tehniques [13℄. These approahes are generally foused on

(3)

problems [23℄. At the heart of these approahes, a panel of onsisteny enforing teh-

niques isused todramatiallyprune the searhspae. Therefore, propagationtehniques

dediated to resoure and time onstrained sheduling problems, viewed as speial in-

stanesofConstraintSatisfationProblems (CSPs),havebeendevelopedtospeedup the

searh for a feasible shedule or to detet early an inonsisteny. For instane the ener-

getireasoning [8℄, the ornerstoneofthe present study,has enabledthejointintegration

of both resoure and time onstraints in order to prevent the ombinatoris of solving

onits between ativities inompetitionfor limited resoures.

Furthermore, itis stillof interest tosearhfor propagating noveltypes of onstraints

aording to real-world problems. The new environmental onstraints, but also the in-

reaseoftheenergyost, shouldpromptustoonsiderasaruialandpromisingissueto

look intothe problems ofemissions,wastes, and poweronsumptionoptimizationinpro-

dution sheduling[24℄. Real-time (proessor) sheduling theory has often addressed en-

ergyonstraints. Indeed, energyonsumptionmanagementisaritialissue inomputer

systems,networks andembeddedsystems wheremany(on-line)algorithmiproblems are

raised andwell studied[14℄. However, omplexity isa major diulty forthe integration

of energy onstraints toprodutionshedulingand the literature onthe subjet israther

sparse. Forexample,produtionshedulingforsteelmanufaturinghas beenstudied, but

few papers fous on energy ost [17℄. This generally leads to the development of heuris-

tis. Forexample,[4℄propose ahierarhial approahforshedulinga steelplantsubjet

to a global limitation on the power supplied to the furnaes. [12℄ use a deomposition

approah to solve a steel manufaturing sheduling problemwith multiple produts. Fi-

nally, to the best of our knowledge, partiular studies foused on onstraint propagation

tehniques for energy onsiderationshave been unexplored.

Problem statement As we will see later, the prodution problem under study is de-

ned as a new problem alled the energy sheduling problem (EnSP). The EnSP is a

generalization of the umulative sheduling problem (CuSP) itself an extension of the

parallelmahinesheduling problem(PMSP).In aPMSP,a taskj ^has ^to^be ^proessed ^on

one mahine among aset of m ^mahines. ^The ^CuSP îsân êxtension ôf ^the ^PMSP^where

eah task needs a subset k < m ⁽k 6= 1⁾ ^of ^mahines. ^Furthermore, the industrialprob- lem we study in this paper involves furnaes that an be modeled by parallel mahines.

Parallel mahine sheduling has been widely studied[6℄, espeiallybeause it appears as

a relaxation of more omplex shop or projet sheduling problems, like the hybrid ow

shop shedulingproblemor theresoure-onstrained projet shedulingproblem. Several

(4)

proposed. [18℄ propose alinear programand an eient heuristifor large-sizeinstanes

for the resolution of priority onstraints and family setup times problem. [22℄ solve the

problem with a tree searh method. [16℄ ompare two dierent branhing sshemes and

several tree searh strategies for the problem with heads and tails for makespan mini-

mization. In [1℄, a onstraint programming-based approah is proposed to minimize the

weightednumberoflatejobs. In[21℄,ahybridInteger/ConstraintProgrammingapproah

isproposed tosolvea minimum-ostassignmentproblem. Amongthe variants presented

in the latter, the most eetive strategy is toombine a tight and ompat, but approx-

imate, mixed integer linear programming (MILP) formulation with a global onstraint

testing single mahine feasibility. Many variants or extensions of the CuSP have been

onsidered, for whih feasibility tests and adjustment rules have been issued, based for

example onthe energeti reasoning [8℄.

Paper objetives & organization The objetive of this paper is twofold. First, we

present in Setion 2 anindustrial ase-study involvingenergy onstraints and objetives

linked toeletripoweronsumption,andatwo-steponstraintprogrammingandmixed-

integer linear programming framework to solve it, as well as arst set of omputational

experiments. Seond, inSetion3,wefous onthe energy partof theindustrialproblem,

issueinga generiproblem,the Energy ShedulingProblem(EnSP).Toenhane the pre-

viousapproah,weproposeaformaldesriptionforthe propagationofenergy onstraints

based on an extension of the energeti reasoning. In Setion 4, we present dominane

rules and pratialassumptions in order to reduethe searh spae, abranhing sheme

tosolvethe problemviatreesearh,aswellasomputationalresults. Setion5highlights

the onlusions of the paper and proposes some future researh diretions.

2 A two-step approah for the industrial problem

Inthis setion,we presentanindustrialase-study whereenergy onstraintshaveagreat

importanein sheduling. A two-step approah was developped tosolvethe problem.

2.1 Industrial ase-study

The addressed problem omes froma pipe-manufaturingplant. The plant isdivided in

three main departments: foundry, drawing mill,and pipe-tubing. In these departments,

(5)

steam. Eletriity expenses aount for more than half the annual energy osts for the

plant. The eletriity bill is based on the ost of the energy onsumed and onpenalties

for power overrun, inreferene toa subsribed maximal power.

Thestudyfousesonthefoundrywheremetalismeltedinindutionfurnaesandthen

ast in individual billets. Non-regular power onsumption peaks our and ause high

eletriity bills. To opewith this problem,equipmentssuh aspower uttersand relays

anbeinstalledatsmallosttoavoidpeaks,buttheyauseprodutionshutdownsthatare

not desired. Consequently, prodution shedulingneeds to onsider energy onsumption

as aentralelement inorder to maintainthe produtionatthe urrent level.

The foundry has ve similar lines of produtionto perform the meltingjobs. From a

shedulingview-point,thisfailityaneasilybereognizedasaparallelmahineproblem.

However, a partiularityof the problemis thatmelting jobshave variabledurations that

dependonthe powergiven tothe furnae, onstrainedinarange [Pmin, Pmax] ^by ^physial

and operationalonsiderations. Melting of job i ênds ^when ân âmount Ei ôf ênergy ^has

been supplied. Prodution sheduling determines the assignment and sequening of the

jobs on the furnaes, and the starting/nishing dates of these jobs that allow to supply

the required energy while respeting the powerlimitsand the time windows. The goalis

to minimize the energy bill, with energy and overrun osts evaluated periodially, every

fteen minutes.

We proposed a two-step Constraint Programming / Mixed Integer Linear Program-

ming approah to solve this problem, onsidering additional onstraints that may inu-

ene the energy onsumption, as human resoure availability for loading and unloading

the furnaes. This approah is desribed in the following. Further details an be found

in [11℄.

2.2 Overview of the solving method

As mentioned in Setion 2.1, we want to shedule melting jobs whose duration depends

onthe power given tothe furnae. Atually, ajob is omposed of three sequentialparts:

loading,heating,and unloading (see Fig. 1). The durations of loadingand unloading are

known (dl ând du^), ^but ^heating^duration ^depends ôn^the ^followingônditions:

• ^melting^duration^depends ^on^the ^power^given^to^the^furnae, ⁱⁿ^a^range[Pmin, Pmax]^;

• ^when ^melting ^is ^omplete, ^the temperature must be hold in the furnae until an operator isready tounload it.

(6)

The goalis to minimizethe ost of the shedule, depending on the energy onsumed

and onpenaltieswhenthe overall power inthe foundry exeedsagiven subsribed value.

Various mixed integer linear models have been developed for this problem. First, a

disretetimemodelhasbeenproposed[25℄,butthehugenumberofbinaryvariablesmade

it impossible to hold realisti problems. A ontinuous time model allowed the redution

of the number of binary variables [9℄, but the resolution was still very long. Finally, a

deomposition of the problemled tomuh more aeptable omputationtimes [11℄. The

main prinipleof the two-step approah is shown in Fig.2.

Figure2: Two-step approah.

During the rst step, sequening of jobs on the furnaes is performed with xed job

durations, i.e., we onsider that the power given to the furnae is known for eah job.

Sine it may happen that no feasible solution exists onsidering the time windows, due

dateviolationisadmittedandtheobjetiveistominimizethemaximumtardiness. Hene

theproblemresortstoaparallelmahineproblemwithmahineavailability,releasedates,

and tardiness riterion. The result of this step is the assignment and sequening of job i

onfurnae f^.

During the seond step, the jobs are sheduled, i.e., operation starting and nishing

dates are xed, while the power setting of eah furnae during eah interval determines

(7)

assign(i, f) ând seq(i1, i2) âre ônsidered âs ^data ât ^Step ^2. ^The ôbjetive ^funtion îs

the energy and overrun ost minimization with an additionalterm to penalize due date

violations.

Thenwelosethe loopbyusing atStep1the newjobdurations given byStep 2. The

proess isinterruptedif the objetive funtionof Step 2is not better thanthe one of the

previousiteration,and if thetardiness is notimproved. Althoughthis two-step approah

may not give theoptimal solution,experimentation givesvery goodresults with ahighly

redued proessing time.

2.3 Sheduling model

Step 1 orresponds to solving an almost standard parallel mahine sheduling problem.

We propose a onstraint programming approah to takle this problem. A ommerial

onstraint programming modeling language and solver (IBM ILOG OPL 6.3/CP Opti-

mizer 2.3)is used. The OPLlanguageprovideshigh level primitivesto modelsheduling

omponents.

Jobloading,meltingand unloading,andoperatorsunavailabilitiesare denedastasks

(typeinterval in OPL) speifyingforeah ofthem the time windows and the duration.

Furthermore,optional tasksare assoiated toeahloading, melting,and unloading tasks

tomodelthe furnaeassignment problem,sothat there exists anoptionaltask perload-

ing, melting, and unloading operation and andidate furnae. For the rst iteration,we

onsider that the furnae power isset toPmax ^to^x ^the^initial^melting^durations ^to^their

minimalvalues.

One written inOPL, the parallel mahine probleman be solved by the IBM ILOG

CP Optimizer, a ommerial onstraint programming solver embedding preedene and

resoure onstraint propagation tehniques and an eient self-adapting large neighbor-

hood searh method dediated to sheduling problems [15℄. A time limit is set and the

best solutionfound within the time limitis returned.

2.4 Energy model

In the seond stage of the proposed heuristi, an MILP modelis used to set preise job

position and power supply while keeping the job sequenes found in the rst stage. Job

positions are given by melting starting and nishing times, represented as ontinuous

(8)

sti−dli ≥reli ⁽¹⁾

f ti ≥sti+Ei/Pmax ⁽²⁾

f ti ≤sti+Ei/Pmin ⁽³⁾

sti2−dli2 ≥f ti1 +dui1−M(1−seq(i1, i2)) ⁽⁴⁾

where(1)loatesthe loadingstarttime afterthe releasedate, (2)and(3)set the bounds

ofmeltingduration,and jobsequening isgiven by(4)aordingtothe binaryvalues seq

fromStep 1.

The time horizon is divided into intervals of uniform duration D = 15 ^min. ^These

intervals are used to determine the overall energy onsumption and power requirement

on eah interval. Binary variables are used to identify the intervals in whih energy is

suppliedtothe furnae fora given job. During the meltingof jobi^, ânâmountôf ênergy emi,u îs ^suppliedât ânîntervâl u^. Ît îs ^the integration of the power given to the furnae

over the melting duration dmi,u ⁱⁿ ^this întervâl. Ôur ^model ûses ênergy ând ^duration

as variables, but it is not neessary to represent expliitly the power, onsidered as a

onstant overthe meltingdurationfor eahinterval (see Fig. 3).

Figure3: Energy supply by interval: meltingand holding.

Melting duration dmi,u^, ^for întervals u ^where ^melting ôurs, îs ^between ⁰ ând D^.

Melting isperformedwithoutinterruptionand the sum of the meltingdurations of ajob

isequaltof ti−sti^,^the ^durationôf^the ^meltingôperation. ^Fôrêah întervâl, ^theâmount

(9)

in [Pmin, Pmax]^. ^The ^meltingênds ^when ^the ^required ênergy ^quantity Ei îs ^reahed ^(6).

Pmin.dmi,u ≤emi,u ≤Pmax.dmi,u ⁽⁵⁾

X

u

emi,u =Ei ⁽⁶⁾

Constraints to dene the holding energy, aounting for operators unavailability, are

dened in a similar way. For a given interval, the energy onsumption is the sum of

meltingandholdingenergyoneveryjob. The meanpowerisequaltothis energydivided

byintervaldurationD^. Îtîsômpared^to^the^subsribed^powerP ^to^detet^powerôverruns.

The objetive funtion is the sum of the energy and power overrun osts for all the

instanes. Theduedatesanbeviolatedbuttardinessishighlypenalizedinordertoseek

fora feasiblenal solution. Henethe heuristidoesnot stop if,for agiven iteration,the

MILP problem has nosolution that satisesthe due dates.

2.5 Experimental results

2.5.1 Solution steps on an illustrative instane

Table 1 shows the solution steps for an illustrative problem instane of 36 jobs on 6

furnaes (furtherdetails are given in [11℄). FullMILP approah (ontinuous-time model)

and two-step approah results are ompared. All the tests have been performed on a

SUNSunre serverwithfourQuad-CoreAMD Opteron(tm)2.5GHzproessors. Parallel

CPLEX 12.1 is used to solve the MILP problems. A 30 s time limit is set for Step 1 of

the approah.

The tables give the maximum tardiness (Tmax^), ^the ^sum ^of ^power ^overruns ^(Over.)

and of holdingdurations (Hold), and the omputationtime.

Table 1: Illustrative instane solved with MILP and two-step approahes.

Tmax ^Over. ^Hold ^Time

MILP 0 0 53.8 1206.8

(10)

Two-step Tmax ^Over. ^Hold ^Time

Step 1 30 - - 0.11

Step 2 30 0 25.7 15.48

Step 1 30 - - 0.11

Step 2 0 0 53.8 6.44

Step 1 0 - - 0.09

Step 2 0 0 53.8 5.22

The MILP model is solved tooptimalityin more than 20minutes. Compared to this

solving time, the two-step approah is very fast. At the rst step, the method gives a

solutionwithtardiness, duetotheinitialvalues. Theassignmentandsequeningvariables

are sent to Step 2, and a rst solution is given. The objetive value is high beause of

the huge penalty given to tardiness. At the seond iteration, a solution with tardiness

is found again by the CP solver at Step 1, but Step 2 then gives a solution with only a

holdingdurationgreaterthan 0. Notethat itisthe optimalsolution. A third iterationis

performed. Asnothingisimproved,the proess ends. Theoverall solving durationis less

than 30seonds, and noiteration time limithas been reahed.

2.5.2 Results on randomly generated problem instanes

A set of 100 problem instanes with 36 jobs and 6 furnaes were generated, inspired by

the industrial ase-study. Among these, 47 were found feasible by solving to optimality

the full MILP ontinuous-time model. Table 2 summarizes the results of full MILP and

two-step approahes for the 47 feasible instanes. MILP solving time stays high so that

using this model would be diult ina situation with hundreds of jobs. Some instanes

haveoverrun or holdingdurations in their optimalsolution.

Table 2: Comparison of the approahes: mean values on47 feasible instanes.

Tmax Ôver. ^Hold ^Time Îter. Ôptim.

MILP 0 38.2 4.0 5397 - 100%

Two-step 0.13 38.2 4.6 8.7 1.1 97.8%

The two-step approah is very fast, with a mean solving time less than 10 seonds.

Only one instane among 47 has not been solved to optimality. Most of the instanes

havebeen solved inone iteration.

(11)

The OPL modeling language gives the opportunity to dene a job duration as a range.

Thus, the melting interval variables an be dened as a range [Ej/Pmax, Ej/Pmin]^,

lettingthe solverdeterminethe adequateduration. Tothis aim,the objetivefuntionof

Step 1is modied in order to penalize melting operations with a duration lose to their

minimum value, beause it means that the furnae is set to a high power and it ould

lead toanoverrun. Experimentationsshowed that the modiedobjetive funtionis not

representativeenoughoftheproblemtogivetherightassignmentandsequeningresults.

Thislaims forarealenergy handlinginthe onstraintprogrammingstep. Therefore, we

present in the next setion an extension for the Energy Sheduling Problem (EnSP) of

the energeti reasoning, anapproah to solve the CuSP inonstraint programming.

3 Energeti reasoning

3.1 The sheduling problem under energy onstraints

Inthefollowing,weintroduetheenergyshedulingproblem(EnSP).Werstpresentthe

related umulative sheduling problem (CuSP). Then we present the EnSP. Finally we

show how we an model our industrial appliation sheduling problem as an assoiation

of anEnSP and aCuSP.

3.1.1 The umulative sheduling problem

TheCuSP isanextensionof thelassialparallelmahineproblem,alsoalledthe multi-

proessor taskproblemand denoted by P|reli, duei;sizei|− ⁱⁿ^the ^well-known ^three ^eld

sheduling notation [7℄. An instane of the CuSP an be dened as follows: a set of n

ativities A = {1,2, . . . , n} îs ^to ^be ^proessed ^without interruption on a given resoure of apaity P^. ^Tô êah âtivity i âre âssoiated îts ^resoure requirement (size) pi^, îts

release date reli^, îts ^deadline duei^, ând îts ^duration di ^(note ^that âpaity ând ^resoure

requirementsare assumed tobeonstant overthe planninghorizon). A standard parallel

mahine problem an be modeled as a CuSP where ativities require only one resoure

unit.

The CuSP an bestated as follows. Ativity i ^start ^time ⁽sti⁾ ând ^nish ^time ⁽f ti = sti+di⁾^have ^to^belong ^to^the ^time ^window[reli, duei]^. Âtivities ân ^be simultaneously proessed aording to the satisfation of the umulative onstraint:

P

i∈Apit ≤ P^, ^for