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1 Open Archve Toulouse Archve Ouvere (OATAO) OATAO s an open access reposory ha collecs he work of Toulouse researchers and makes freely avalable over he web where possble. Ths s an auhor-deposed verson publshed n: hp://oaao.unv-oulouse.fr/ Eprns ID: 5300 To lnk o hs arcle: DOI:0.06/j.jpe URL: hp://dx.do.org/0.06/j.jpe To ce hs verson: Argues, Chrsan and Lopez, Perre and Haï, Alan The energy schedulng problem: Indusral case-sudy and consran propagaon echnques. (203) Inernaonal Journal of Producon Economcs, vol. 43 (n ). pp ISSN Any correspondence concernng hs servce should be sen o he reposory admnsraor: saff-oaao@np-oulouse.fr

2 The energy schedulng problem: ndusral case sudy and consran propagaon echnques Absrac Ths paper deals wh producon schedulng nvolvng energy consrans, ypcally elecrcal energy. We sar by an ndusral case-sudy for whch we propose a wo-sep neger/consran programmng mehod. From he ndusral problem we derve a generc problem, he Energy Schedulng Problem (EnSP). We propose an exenson of specfc resource consran propagaon echnques o effcenly prune he search space for EnSP solvng. We also presen a branchng scheme o solve he problem va ree search. Fnally, compuaonal resuls are provded. Keywords: Producon schedulng, energy consrans, consran propagaon, energec reasonng Inroducon Conex of he sudy Snce he las wo decades, hard combnaoral problems, manly n schedulng, have been he arge of many approaches combnng Operaons Research and Arfcal Inellgence echnques [3]. These approaches are generally focused on consran sasfacon as a general paradgm for represenng and solvng effcenly such problems [23]. A he hear of hese approaches, a panel of conssency enforcng echnques s used o dramacally prune he search space. Therefore, propagaon echnques dedcaed o resource and me consraned schedulng problems, vewed as specal nsances of Consran Sasfacon Problems (CSPs), have been developed o speed up he

3 search for a feasble schedule or o deec early an nconssency. For nsance he energec reasonng [8], he cornersone of he presen sudy, has enabled he jon negraon of boh resource and me consrans n order o preven he combnaorcs of solvng conflcs beween acves n compeon for lmed resources. Furhermore, s sll of neres o search for propagang novel ypes of consrans accordng o real-world problems. The new envronmenal consrans, bu also he ncrease of he energy cos, should promp us o consder as a crucal and promsng ssue o look no he problems of emssons, wases, and power consumpon opmzaon n producon schedulng [24]. Real-me (processor) schedulng heory has ofen addressed energy consrans. Indeed, energy consumpon managemen s a crcal ssue n compuer sysems, neworks and embedded sysems where many (on-lne) algorhmc problems are rased and well suded [4]. However, complexy s a major dffculy for he negraon of energy consrans o producon schedulng and he leraure on he subjec s raher sparse. For example, producon schedulng for seel manufacurng has been suded, bu few papers focus on energy cos [7]. Ths generally leads o he developmen of heurscs. For example, [4] propose a herarchcal approach for schedulng a seel plan subjec o a global lmaon on he power suppled o he furnaces. [2] use a decomposon approach o solve a seel manufacurng schedulng problem wh mulple producs. Fnally, o he bes of our knowledge, parcular sudes focused on consran propagaon echnques for energy consderaons have been unexplored. Problem saemen As we wll see laer, he producon problem under sudy s defned as a new problem called he energy schedulng problem (EnSP). The EnSP s a generalzaon of he cumulave schedulng problem (CuSP) self an exenson of he parallel machne shedulng problem (PMSP). In a PMSP, a ask j has o be processed on one machne among a se of m machnes. The CuSP s an exenson of he PMSP where each ask needs a subse k < m (k ) of machnes. Furhermore, he ndusral problem we sudy n hs paper nvolves furnaces ha can be modeled by parallel machnes. Parallel machne schedulng has been wdely suded [6], especally because appears as a relaxaon of more complex shop or projec schedulng problems, lke he hybrd flow shop schedulng problem or he resource-consraned projec schedulng problem. Several mehods have been proposed o solve hs problem. In [5], a column generaon sraegy s proposed. [8] propose a lnear program and an effcen heursc for large-sze nsances for he resoluon of prory consrans and famly seup mes problem. [22] solve he problem wh a ree search mehod. [6] compare wo dfferen branchng sschemes and several ree search sraeges for he problem wh heads and als for makespan mn- 2

4 mzaon. In [], a consran programmng-based approach s proposed o mnmze he weghed number of lae jobs. In [2], a hybrd Ineger/Consran Programmng approach s proposed o solve a mnmum-cos assgnmen problem. Among he varans presened n he laer, he mos effecve sraegy s o combne a gh and compac, bu approxmae, mxed neger lnear programmng (MILP) formulaon wh a global consran esng sngle machne feasbly. Many varans or exensons of he CuSP have been consdered, for whch feasbly ess and adjusmen rules have been ssued, based for example on he energec reasonng [8]. Paper objecves & organzaon The objecve of hs paper s wofold. Frs, we presen n Secon 2 an ndusral case-sudy nvolvng energy consrans and objecves lnked o elecrc power consumpon, and a wo-sep consran programmng and mxedneger lnear programmng framework o solve, as well as a frs se of compuaonal expermens. Second, n Secon 3, we focus on he energy par of he ndusral problem, ssueng a generc problem, he Energy Schedulng Problem (EnSP). To enhance he prevous approach, we propose a formal descrpon for he propagaon of energy consrans based on an exenson of he energec reasonng. In Secon 4, we presen domnance rules and praccal assumpons n order o reduce he search space, a branchng scheme o solve he problem va ree search, as well as compuaonal resuls. Secon 5 hghlghs he conclusons of he paper and proposes some fuure research drecons. 2 A wo-sep approach for he ndusral problem In hs secon, we presen an ndusral case-sudy where energy consrans have a grea mporance n schedulng. A wo-sep approach was developped o solve he problem. 2. Indusral case-sudy The addressed problem comes from a ppe-manufacurng plan. The plan s dvded n hree man deparmens: foundry, drawng mll, and ppe-ubng. In hese deparmens, melng and heang processes use a huge quany of energy: elecrcy, naural gas, and seam. Elecrcy expenses accoun for more han half he annual energy coss for he plan. The elecrcy bll s based on he cos of he energy consumed and on penales for power overrun, n reference o a subscrbed maxmal power. The sudy focuses on he foundry where meal s meled n nducon furnaces and hen 3

5 cas n ndvdual blles. Non-regular power consumpon peaks occur and cause hgh elecrcy blls. To cope wh hs problem, equpmens such as power cuers and relays can be nsalled a small cos o avod peaks, bu hey cause producon shudowns ha are no desred. Consequenly, producon schedulng needs o consder energy consumpon as a cenral elemen n order o manan he producon a he curren level. The foundry has fve smlar lnes of producon o perform he melng jobs. From a schedulng vew-pon, hs facly can easly be recognzed as a parallel machne problem. However, a parculary of he problem s ha melng jobs have varable duraons ha depend on he power gven o he furnace, consraned n a range [P mn,p max ] by physcal and operaonal consderaons. Melng of job ends when an amoun E of energy has been suppled. Producon schedulng deermnes he assgnmen and sequencng of he jobs on he furnaces, and he sarng/fnshng daes of hese jobs ha allow o supply he requred energy whle respecng he power lms and he me wndows. The goal s o mnmze he energy bll, wh energy and overrun coss evaluaed perodcally, every ffeen mnues. We proposed a wo-sep Consran Programmng / Mxed Ineger Lnear Programmng approach o solve hs problem, consderng addonal consrans ha may nfluence he energy consumpon, as human resource avalably for loadng and unloadng he furnaces. Ths approach s descrbed n he followng. Furher deals can be found n []. 2.2 Overvew of he solvng mehod As menoned n Secon 2., we wan o schedule melng jobs whose duraon depends on he power gven o he furnace. Acually, a job s composed of hree sequenal pars: loadng, heang, and unloadng (see Fg. ). The duraons of loadng and unloadng are known (dl and du), bu heang duraon depends on he followng condons: melng duraon depends on he power gven o he furnace, n a range [P mn,p max ]; when melng s complee, he emperaure mus be hold n he furnace unl an operaor s ready o unload. Fgure : Job descrpon and correspondng operaor s asks. 4

6 The goal s o mnmze he cos of he schedule, dependng on he energy consumed and on penales when he overall power n he foundry exceeds a gven subscrbed value. Varous mxed neger lnear models have been developed for hs problem. Frs, a dscree me model has been proposed [25], bu he huge number of bnary varables made mpossble o hold realsc problems. A connuous me model allowed he reducon of he number of bnary varables [9], bu he resoluon was sll very long. Fnally, a decomposon of he problem led o much more accepable compuaon mes []. The man prncple of he wo-sep approach s shown n Fg. 2. n CP Model Assgnmen Sequencng assgn(, f) seq(, 2 ) MILP Model Schedulng Energy duraon() Fgure 2: Two-sep approach. Durng he frs sep, sequencng of jobs on he furnaces s performed wh fxed job duraons,.e., we consder ha he power gven o he furnace s known for each job. Snce may happen ha no feasble soluon exss consderng he me wndows, due dae volaon s admed and he objecve s o mnmze he maxmum ardness. Hence he problem resors o a parallel machne problem wh machne avalably, release daes, and ardness creron. The resul of hs sep s he assgnmen and sequencng of job on furnace f. Durng he second sep, he jobs are scheduled,.e., operaon sarng and fnshng daes are fxed, whle he power seng of each furnace durng each nerval deermnes he duraon of each job. Job assgnemen and sequencng are nhered from Sep so assgn(,f) and seq(, 2 ) are consdered as daa a Sep 2. The objecve funcon s he energy and overrun cos mnmzaon wh an addonal erm o penalze due dae volaons. Then we close he loop by usng a Sep he new job duraons gven by Sep 2. The process s nerruped f he objecve funcon of Sep 2 s no beer han he one of he prevous eraon, and f he ardness s no mproved. Alhough hs wo-sep approach may no gve he opmal soluon, expermenaon gves very good resuls wh a hghly reduced processng me. 5

7 2.3 Schedulng model Sep corresponds o solvng an almos sandard parallel machne schedulng problem. We propose a consran programmng approach o ackle hs problem. A commercal consran programmng modelng language and solver (IBM ILOG OPL 6.3/CP Opmzer 2.3) s used. The OPL language provdes hgh level prmves o model schedulng componens. Job loadng, melng and unloadng, and operaors unavalables are defned as asks (ype nerval n OPL) specfyng for each of hem he me wndows and he duraon. Furhermore, oponal asks are assocaed o each loadng, melng, and unloadng asks o model he furnace assgnmen problem, so ha here exss an oponal ask per loadng, melng, and unloadng operaon and canddae furnace. For he frs eraon, we consder ha he furnace power s se o P max o fx he nal melng duraons o her mnmal values. Once wren n OPL, he parallel machne problem can be solved by he IBM ILOG CP Opmzer, a commercal consran programmng solver embeddng precedence and resource consran propagaon echnques and an effcen self-adapng large neghborhood search mehod dedcaed o schedulng problems [5]. A me lm s se and he bes soluon found whn he me lm s reurned. 2.4 Energy model In he second sage of he proposed heursc, an MILP model s used o se precse job poson and power supply whle keepng he job sequences found n he frs sage. Job posons are gven by melng sarng and fnshng mes, represened as connuous varables. The schedulng consrans of hs connuous model are: s dl rel () f s +E /P max (2) f s +E /P mn (3) s 2 dl 2 f +du M( seq(,2)) (4) where () locaes he loadng sar me afer he release dae, (2) and (3) se he bounds of melng duraon, and job sequencng s gven by (4) accordng o he bnary values seq from Sep. The me horzon s dvded no nervals of unform duraon D = 5 mn. These nervals are used o deermne he overall energy consumpon and power requremen 6

8 on each nerval. Bnary varables are used o denfy he nervals n whch energy s suppled o he furnace for a gven job. Durng he melng of job, an amoun of energy em,u s suppled a an nerval u. I s he negraon of he power gven o he furnace over he melng duraon dm,u n hs nerval. Our model uses energy and duraon as varables, bu s no necessary o represen explcly he power, consdered as a consan over he melng duraon for each nerval (see Fg. 3). Furnace heang Melng Holdng P max Power P mn em, em,2 em,3 P hold dm, dm,2 dm,3 dh,3 dh,4 Inervals Fgure 3: Energy supply by nerval: melng and holdng. Melng duraon dm,u, for nervals u where melng occurs, s beween 0 and D. Melng s performed whou nerrupon and he sum of he melng duraons of a job s equal o f s, he duraon of he melng operaon. For each nerval, he amoun of energy provded o a job (5) depends on he melng duraon and he suppled power n [P mn,p max ]. The melng ends when he requred energy quany E s reached (6). P mn.dm,u em,u P max.dm,u (5) em,u = E (6) u Consrans o defne he holdng energy, accounng for operaors unavalably, are defned n a smlar way. For a gven nerval, he energy consumpon s he sum of melng and holdng energy on every job. The mean power s equal o hs energy dvded by nerval duraond. I s compared o he subscrbed powerp o deec power overruns. The objecve funcon s he sum of he energy and power overrun coss for all he nsances. The due daes can be volaed bu ardness s hghly penalzed n order o seek 7

9 for a feasble fnal soluon. Hence he heursc does no sop f, for a gven eraon, he MILP problem has no soluon ha sasfes he due daes. 2.5 Expermenal resuls 2.5. Soluon seps on an llusrave nsance Table shows he soluon seps for an llusrave problem nsance of 36 jobs on 6 furnaces (furher deals are gven n []). Full MILP approach (connuous-me model) and wo-sep approach resuls are compared. All he ess have been performed on a SUN Sunfre server wh four Quad-Core AMD Operon(m) 2.5 GHz processors. Parallel CPLEX 2. s used o solve he MILP problems. A 30 s me lm s se for Sep of he approach. The ables gve he maxmum ardness (T max ), he sum of power overruns (Over.) and of holdng duraons (Hold), and he compuaon me. Table : Illusrave nsance solved wh MILP and wo-sep approaches. T max Over. Hold Tme MILP Two-sep T max Over. Hold Tme Sep Sep Sep Sep Sep Sep The MILP model s solved o opmaly n more han 20 mnues. Compared o hs solvng me, he wo-sep approach s very fas. A he frs sep, he mehod gves a soluon wh ardness, due o he nal values. The assgnmen and sequencng varables are sen o Sep 2, and a frs soluon s gven. The objecve value s hgh because of he huge penaly gven o ardness. A he second eraon, a soluon wh ardness s found agan by he CP solver a Sep, bu Sep 2 hen gves a soluon wh only a holdng duraon greaer han 0. Noe ha s he opmal soluon. A hrd eraon s 8

10 performed. As nohng s mproved, he process ends. The overall solvng duraon s less han 30 seconds, and no eraon me lm has been reached Resuls on randomly generaed problem nsances A se of 00 problem nsances wh 36 jobs and 6 furnaces were generaed, nspred by he ndusral case-sudy. Among hese, 47 were found feasble by solvng o opmaly he full MILP connuous-me model. Table 2 summarzes he resuls of full MILP and wo-sep approaches for he 47 feasble nsances. MILP solvng me says hgh so ha usng hs model would be dffcul n a suaon wh hundreds of jobs. Some nsances have overrun or holdng duraons n her opmal soluon. Table 2: Comparson of he approaches: mean values on 47 feasble nsances. T max Over. Hold Tme Ier. Opm. MILP % Two-sep % The wo-sep approach s very fas, wh a mean solvng me less han 0 seconds. Only one nsance among 47 has no been solved o opmaly. Mos of he nsances have been solved n one eraon Improvemens The OPL modelng language gves he opporuny o defne a job duraon as a range. Thus, he melng nerval varables can be defned as a range [E j /P max, E j /P mn ], leng he solver deermne he adequae duraon. To hs am, he objecve funcon of Sep s modfed n order o penalze melng operaons wh a duraon close o her mnmum value, because means ha he furnace s se o a hgh power and could lead o an overrun. Expermenaons showed ha he modfed objecve funcon s no represenave enough of he problem o gve he rgh assgnmen and sequencng resuls. Ths clams for a real energy handlng n he consran programmng sep. Therefore, we presen n he nex secon an exenson for he Energy Schedulng Problem (EnSP) of he energec reasonng, an approach o solve he CuSP n consran programmng. 9

11 3 Energec reasonng 3. The schedulng problem under energy consrans In he followng, we nroduce he energy schedulng problem (EnSP). We frs presen he relaed cumulave schedulng problem (CuSP). Then we presen he EnSP. Fnally we show how we can model our ndusral applcaon schedulng problem as an assocaon of an EnSP and a CuSP. 3.. The cumulave schedulng problem The CuSP s an exenson of he classcal parallel machne problem, also called he mulprocessor ask problem and denoed by P rel,due ;sze n he well-known hree feld schedulng noaon [7]. An nsance of he CuSP can be defned as follows: a se of n acves A = {,2,...,n} s o be processed whou nerrupon on a gven resource of capacy P. To each acvy are assocaed s resource requremen (sze) p, s release dae rel, s deadlne due, and s duraon d (noe ha capacy and resource requremens are assumed o be consan over he plannng horzon). A sandard parallel machne problem can be modeled as a CuSP where acves requre only one resource un. The CuSP can be saed as follows. Acvy sar me (s ) and fnsh me (f = s +d ) have o belong o he me wndow [rel,due ]. Acves can be smulaneously processed accordng o he sasfacon of he cumulave consran: A p P, for every me pon, where p = p f s < f and p = 0 oherwse The energy schedulng problem The energy schedulng problem (EnSP) akes as npu a se ofnacvesa = {,2,...,n} havng o be processed whou nerrupon usng an energy resource of capacy (.e., avalable power) P. Insead of beng defned hrough s duraon d and resource demand p, each acvy s defned hrough s requred energy E and s mnmum and maxmum resource requremens P mn and P max such ha he allocaed resource uns (provded power) has o reman beween hese wo values. Noe here ha for praccal movaons, we consder ha changes n he power allocaed o an acvy only occur a dscree me perods of duraon δ. The EnSP consss n fndng a sar mes rel, a compleon mef due and a 0

12 power allocaon p such ha P mn p P max for [s,f ] andp = 0 oherwse. The global power lmaon consran s wren A p P for any me perod. We consder boh p and d = f s as dscree varables. Las, an energy requremen consran E δ. f =s p holds for each acvy,.e., he energy brough o mus be a leas E. We remark ha enforcng equaly would yeld o possbly nfeasble soluons n he case where he remanng energy o be brough o an acvy a a gven me perod s srcly lower han P mn. Consequenly, n accordance wh praccal cases, we consder he energy brough o an acvy can be larger han he requred one. Consder a problem nsance of 3 acves wh P = 5 and δ =. Oher daa are gven n Table 3. Table 3: Example daa E P mn P max rel due Fg. 4 dsplays a feasble soluon for he problem. One can observe ha here s no soluon for whch all he acves have a recangular shape. P = rel rel 3 rel 2 due 3 due due 2 Fgure 4: Soluon of an EnSP Dscusson / Relaed works Clearly he CuSP canno be used o model he EnSP snce acves are no necessarly of recangular shape (see Secon 3). In fac, he EnSP can be defned as a relaxaon of he (connuous) CuSP. Indeed, we oban he CuSP by seng P mn = P max = p.

13 However n [2], oher relaxaons of he CuSP are consdered. The fully elasc relaxaon corresponds o a parcular EnSP where P mn = 0 and P max = P. Hence alhough he feasbly ess and adjusmen rules proposed for he fully elasc CuSP hold for he EnSP, hey may no capure all he srucure of he EnSP snce he fully elasc CuSP s self a relaxaon of he EnSP. The parally elasc relaxaon resrcs elascy by enforcng regulary consrans of he changes nvolvng nomnal p. Namely, we have P mn = 0 and P max = P as for he fully elasc case, bu for any nerval [rel,] he relaon τ=rel p τ p.( rel ) mus hold. We do no have such regulary consrans n he EnSP, hence he parally elasc CuSP and he EnSP are no comparable n erms of complexy. Anoher relaed exenson of he CuSP has been proposed n [9], amng a consderng an acvy as a sequence of consecuve subasks such ha he resource consumpon of each subask s gven by a funcon of he subask duraon. In our case he consumpon of an acvy a a me perod s a decson varable. Fnally, n he dscree me-resource rade-off model [20], he duraon of each acvy s no predeermned, bu changes as a dscree non-ncreasng funcon of he amoun of renewable resources assgned o. Ths s very smlar o he concep of malleable ask frequenly encounered n parallel processor sysems. A malleable ask may be execued by several processors smulaneously and he processng speed of a ask s a nonlnear funcon of he number of processors allocaed o [3]. However, n hese cases he acves sll have a recangular shape. 3.2 Classcal energec reasonng for he CuSP In he energec reasonng for schedulng, he dea s o propose a smar way for smulaneously consderng me and resource consrans n a unque reasonng. In ha conex, he energy s genercally defned as he produc of a me duraon by a resource quany. As an llusraon, we can say ha he problem of schedulng n acves of duraon d, =..n n an amoun p, =..n usng a gven resource avalable n a consan amoun P over a me horzon of duraon s somorphc o he placemen problem of n recangles of surface area p.d, =,...,n, n a recangle of surface area P.. To presen he energec reasonng, one mus consder a workng me nerval, an avalable energy and a oal consumed energy over hs nerval. Le [, 2 ] be a reference me nerval. Bounds of he nerval are arbrarly chosen bu hey also can be fxed o parcular mes. Over [, 2 ] and for a resource of capacy 2

14 Fgure 5: Consumpon of fve acves. P, he avalable energy s defned as P.( 2 ). We denoe by w(,, 2 ) he consumpon of acvy (.e., how long uses he resource) over [, 2 ]. Two cases mus be dsngushed:. [s,f ] [, 2 ] = w(,, 2 ) = 0; 2. [s,f ] [, 2 ] w(,, 2 ) = p.(mn(f, 2 ) max(s, )). In Fg. 5, srped areas represen he consumpon of each acvy from o 5 beween and 2. One s usually especally neresed n compung he lower and upper bounds of he consumpon: for he consumpon of acvy over nerval [, 2 ], we mgh derve from above equaons he mnmum and he maxmum consumpons. The relevan noon for our purpose s obvously he mnmum consumpon, also called he mandaory consumpon: when ryng o check wheher before j s feasble, we nend o ake no accoun ha anoher acvy k wll necessarly consume he resource, beween s and f j, for a leas some me T. Therefore we wll no consder anymore he maxmum consumpon n he remander of he paper. The mandaory consumpon of an acvy s denoed by w(,, 2 ). To compue, he acvy has o be shfed o s lef and rgh umos posons on s me wndow [rel,due ], reanng he mnmum value of all nersecons beween such posons and he reference nerval. One hen ges: he lef-shfed consumpon: w L (,, 2 ) = p.max{0,mn(d, 2,rel +d )} 3

15 Fgure 6: Mandaory consumpon of fve acves. he rgh-shfed consumpon: w R (,, 2 ) = p.max{0,mn(d, 2, 2 due +d )}. The mandaory consumpon of acvy s hen: w(,, 2 ) = mn{w L (,, 2 ),w R (,, 2 )} = p.max{0,mn(d, 2,rel +d, 2 due +d )}. On he same bass as example (Fg. 5), Fg. 6 shows he mandory consumpon (srpped areas) of he 5 asks where a me wndow s now assocaed wh each of hem. From hs defnon, yelds a sasfably es (global nconssency rule) whch ncludes oal mandaory consumpon over he se of acves A: Propery CuSP feasbly es. If [, 2 ] s.. A w(,, 2 ) > P.( 2 ), hen no feasble soluon exss for he CuSP. In [2], he se of relevan nervals [, 2 ] s characerzed and an O(n 2 ) algorhm s provded o perform he feasbly ess over all hese nervals. From hs sasfably es, we can now propose local conssency rules o derve me wndows adjusmens for a specfed ask. LeSL(,, 2 ) = P.( 2 ) j A\{} w(j,, 2 ) be he maxmum avalable energy (.e., he slack) for processng over [, 2 ]. Propery 2 CuSP me-bound adjusmens. Release dae adjusmen. If an acvyverfes: [, 2 ] s.. w L (,, 2 ) > SL(,, 2 ), hen a vald lower bound of he compleon me of can be deduced and hen mpacs s release dae as follows: rel max{rel, 2 SL(,, 2 )/p }. Deadlne adjusmen. Symmercally, f an acvyverfes: [, 2 ] s.. w R (,, 2 ) > SL(,, 2 ), hen a vald upper bound of he sar me of can be deduced and hen mpacs s deadlne as follows: due mn{due, +SL(,, 2 )/p }. In [2], an O(n 3 ) algorhm s provded o perform all he me-bound adjusmens over he relevan nervals. 4

16 3.3 Energec reasonng for he EnSP A frs basc feasbly rule s o check wheher here s enough me n each acvy me wndow o brng he energy requres when he maxmum power s allocaed o he acvy. Namely, hs basc feasbly es can be wren as follows: Propery 3 EnSP basc feasbly es. If, for an acvy, P max.(due rel ) < E, he EnSP s nfeasble. In wha follows we consder hs condon s fulflled for each acvy. To exend he energec reasonng, he basc queson o answer s: Gven an nerval [, 2 ], wha s he mandaory consumpon e(,, 2 ) of each acvy? Obvously f rel 2 or due, e(,, 2 ) = 0. Le us consder now ha rel < 2 and due >. As for he sandard energec reasonng, he mandaory consumpon of each acvy n [, 2 ] s aaned eher when he acvy sars a s release dae or when ends by s due dae. When rel < 2, he relevan cases are dsplayed n Fg. 7. To compue e(,, 2 ) we need o compue he maxmum energy e (, ) consumed by before, as well as he maxmum energy e + (, 2 ) consumed by afer 2. We have: where: e (, ) = mn{e,max(0,p max.( rel ))} e + (, 2 ) = mn{e,max(0,p max.(due 2 ))}. I follows ha he mnmal energy consumpon of nsde [, 2 ] verfes e(,, 2 ) v or equvalenly: v = mn{e e (, ),E e + (, 2),P mn.( 2 )} v = mn{e mn(e,p max.max(0, rel,due 2 )),P mn.( 2 )}. Because of he mnmal resource requremen P mn, we canno have e(,, 2 ) < P mn f e(,, 2 ) > 0. Furhermore he requred work E has o be performed nsde he me wndow [rel,due ]. Thus, n he case where s necessary o consume P mn.( 2 ) nsde he nerval, we have o check wheher consumng he maxmal energy ousde he nerval s suffcen o brng he requred energy E. The case where P mn.( 2 ) s 5

17 P P max P mn rel P max P P P max P mn P mn rel 2 rel 2 due P P P max P max P mn P mn rel P 2 rel P due 2 P max P mn P max P mn P rel 2 P rel due 2 P max P max P mn rel 2 P mn rel 2 due Fgure 7: Dfferen cases for lef-shfed consumpon w L (,, 2 ). 6

18 no a suffcen energy amoun because of me wndow ghness s dsplayed a he rgh boom of Fg. 7. Hence we se: e(,, 2 ) = 0 f v = 0, and e(,, 2 ) = max(p mn,v,e e (, ) e + (, 2)) oherwse. Ths yelds he followng feasbly es: Propery 4 EnSP feasbly es. If [, 2 ] s.. A e(,, 2 ) > P.( 2 ), hen no feasble soluon exss for he EnSP. As for he CuSP, lesl(,, 2 ) = P.( 2 ) j A\{} e(j,, 2 ) denoe he maxmum avalable energy (.e., he slack) for processng over [, 2 ]. We oban me-bound adjusmens consderng he wo exreme cases for an acvy. Consder e L (,,2) he mnmal energy consumpon of n [, 2 ] when s lef shfed (.e., s = rel ). We have e L (,, 2 ) x where: or equvalenly: and, we have: x = mn{e e (, ),P mn.( 2 )} x = mn{e mn(e,p max.max(0, rel )),P mn.( 2 )} e L (,, 2 ) = 0 f x = 0, and e L (,, 2 ) = max(p mn,x,e e (, ) e + (, 2)) oherwse. Symmercally, consder e R (,,2) he mnmal energy consumpon of n [, 2 ] when s rgh shfed (.e., f = due ). We have e L (,, 2 ) y where: or equvalenly: and, we have: y = mn{e e + (, 2),P mn.( 2 )} y = mn{e mn(e,p max.max(0,due 2 )),P mn.( 2 )} e R (,, 2 ) = 0 f y = 0, and 7

19 e R (,, 2 ) = max(p mn,y,e e (, ) e + (, 2)) oherwse. We oban he followng me-bound adjusmens: Propery 5 EnSP me-bound adjusmens. Release dae adjusmen. If an acvyverfes: [, 2 ] s.. e L (,, 2 ) > SL(,, 2 ), hen he release dae can be updaed as follows: rel max{rel, 2 SL(,, 2 )/P mn }. Deadlne adjusmen. Symmercally, f an acvyverfes: [, 2 ] s.. e R (,, 2 ) > SL(,, 2 ), hen he deadlne can be updaed as follows: due mn{due, +SL(,, 2 )/P mn }. As he EnSP adms he CuSP as specal case, s a pror dffcul o enumerae he nervals o be consdered. Indeed, from [2], we know ha a par of he relevan nervals for he CuSP s such ha = rel + d and/or 2 = due d for some acvy. For he EnSP, excep when P mn = P max (whch corresponds o he CuSP case), we have no a fxed acvy duraon bu a se of possble duraons from E /P max o E /P mn For he sake of smplcy we resrc he consdered nervals o he Caresan produc O O 2, where O = {rel A} and O 2 = {due A}.. We can llusrae he adjusmens performed n Fg. 4 example. Consder nerval [, 2 ] = [2,5]. We have e(,, 2 ) = e L (,, 2 ) = 2, e(2,, 2 ) = e R (2,, 2 ) = 7 and e(3,, 2 ) = e L (3,, 2 ) = e R (3,, 2 ) = 6. Noe he confguraon dsplayed n Fg. 4 acually corresponds o he mnmal consumpon of he hree acves n [, 2 ] = [2,5]. Consder he case where acvy s rgh shfed. We have e R (,, 2 ) = 7 (same confguraon as he one dsplayed for acvy 2). Snce e(3,, 2 ) + e(2,, 2 ) = 3 he slack for acvy n [, 2 ] s SL(,, 2 ) = 5 3 = 2. Snce e R (,, 2 ) > SL(,, 2 ), he deadlne of acvy can be updaed accordng o Propery 5 by seng due 2+2/ = 4. 4 Solvng he EnSP 4. Domnance rules and praccal assumpons for he EnSP The followng properes are consdered. Propery 6 (Domnance Rule) Acve schedules. 8

20 Acve schedules are domnan for he EnSP. Consder a soluon S o he EnSP such ha here s an acvy sarng a me s and a me perod < s such ha here s a feasble soluon S seng s = whou changng he schedule of oher acves. The search space can be obvously reduced o he se of soluons for whch no such propery holds. Propery 7 (Praccal assumpon) Power change. The search s resrced o schedules for whch, for any acvy, changes n he allocaed power only occur on acvy release daes, or compleon mes. Alhough we dd no prove hs assumpon s domnan, makes sense n pracce o resrc he daes where he power allocaed o a ask s changed only when somehng happens,.e. when a new ask s ready for beng processed or when a ask complees 4.2 Branchng scheme A smple branchng scheme based on me ncremenaon can be derved from he domnance rules and praccal assumpons presened n Secon 4.. Each node corresponds o a decson me pon nally se o = mn A rel. For each acvy he requred energy E s progressvely decreased and all acves are scheduled when E = 0 for all acves. A each node, assocaed wh a decson me, acves are paroned no he followng subses. The sared acves are such ha he decson o sar he acvy has been aken a some ancesor node (a a me pon < ) bu no decson has been aken ye for he curren decson pon and E > 0. The compleed acves are such ha f and E = 0. The avalable acves are such ha rel bu no sar decson has been aken ye for hese acves. The processed acves are such ha he decson o process he acvy a me wh some resource amoun p has already been aken and E > 0. The unavalable acves verfy rel > and E > 0. The posponed acves are hose seleced for beng scheduled laer (see branchng scheme below). A each node an acvy eher sared or avalable s seleced for beng ncluded n he processed se (or n he posponed se for he avalable acves). The acvy wh he smalles due dae s seleced frs and, n case of es, he acvy wh he mos remanng energy (E ) s seleced. Le Q and R denoe he se of sared and processed acves, respecvely. If Q, p = P j Q\{ } Pmn j j R p j denoes he avalable power for a me. If Q, he avalable power for s p = P j Q Pmn j j R p j. 9

21 If p > P mn, a par of can be scheduled a me. A chld node s generaed for,mn(p,pmax p [P mn )] correspondng o an allocaon of power p = p o a me. An addonal chld node, only for avalable acves, corresponds o posponng acvy o a decson pon > such ha s eher equal o he mnmum beween he smalles possble compleon me of an acvy of R and he smalles release dae of unavalable acves, srcly greaer han. Ths me pon s unknown a hs sep snce se R s under consrucon, herefore he acves are jus marked as posponed whou any oher updae. If no acvy can be seleced for beng scheduled a, we have dfferen reasons. If all acves are n he compleed se, he search succeeds. If all acves are eher processed, posponed, unavalable or avalable bu whou enough resource capacy, he search mus connue from he nex decson me pon se o he smalles release dae or compleon me of processed acves greaer han. A hs me we check wheher he new decson pon s sll compable wh he due dae of he avalable acves. We also check wheher here remans unposponed acves. Oherwse, he schedule s clearly no sem-acve. If one due dae canno be sasfed or f he schedule s no more sem-acve, a falure occurs and he node s pruned. Oherwse, decson me pon s updaed. The processed acves are ransferred eher o he compleed se or o he sared se. The posponed acves are moved o he avalable se. The unavalable acves such ha rel are moved o he avalable se. The acvy selecon process sars agan and he process s eraed unl an acvy s seleced for beng processed, or a falure occurs. We llusrae he branchng process on he Fg. 4 problem nsance. The developped nodes are dsplayed n Fg. 8. For he roo node where = 0, acvy s n he avalable se whle acves 2 and 3 are n he unavalable se. We branch o he second node (Fg. 8.a) by selecng acvy for beng scheduled a maxmal power. Acvy s ncluded n he processed se. A me = 0 no oher acvy s avalable. Tme s se of he nex decson pon = 2. Acves 2 and 3 are ncluded n he avalable se and acvy s ransferred no he sared se. The hrd node (Fg. 8.b) selecs acvy 3 wh power p = 2 as he acvy wh he smalles due dae for beng nsered n he processed se. Acves and 2 can boh be processed a me 2 and have he same due dae bu acvy 2 has he mos remanng work. So he fourh node (Fg. 8.c) selecs 2 for beng processed a me = 2 wh he maxmal avalable power akng accoun of processed and sared acves p = 2. For he ffh node (Fg. 8.d), acvy can now be processed a me = 2 wh s mnmal power p =. No acvy s avalable anymore a 20

22 me = 2, so we proceed o he nex me pon correspondng wh he compleon me of acvy a me = 4 and acves 2 and 3 are now boh n he sared se whle s pu no he compleed se. For he sxh node (Fg. 8.e), acvy 3 s sll seleced wh power p = 2 as has he smalles due dae. Then, he sevenh node (Fg. 8.f) selecs acvy 2 wh he maxmal avalable power p = 3. Snce all acves are n he processed se, he me pon s ncreased o he compleon me = 5 of acvy 3 and acvy 2 s ncluded n he sared se. For he egh node (Fg. 8.g), acvy 2 s seleced wh he maxmal power p = 5 and compleed a me = 6. For hs example no backrackng has been necessary. 4.3 Compuaonal expermens In hs secon, we llusrae on randomly generaed problem nsances he neres of he proposed energec reasonng echnques. Usng he same branchng scheme, we compare he energec reqsonng feasbly condons and adjusmens wh he fully elasc ones [2]. Recall ha he fully elasc relaxaon of he EnSP (or he CuSP) consders ha, a any me, asks can be alloed any resource amoun beeen 0 and P, provded he oal resource amoun s equal o E. Bapse [2] proved ha hs relaxaon s equvalen o he well-known preempve one machne problem wh release daes and due daes, and proposed feasbly condons and adjusmens based on hs propery. Clearly, he fully elasc echnque yelds weaker relaxaons bu, gven a lmed CPU me, he queson s o known wheher he sronger adjusmens brough by energec reasonng compensae or no he addonal compuaon requremens. We have coded he algorhms n C++ and he resuls have been obaned on an Inel Code 2 Duo processor. Insances have been generaed accordng o he followng framework. The resource avalably s se o P = 0. For each ask, he requred energy E has been generaed n U[,2.5 P]. The mnmum power P mn he maxmum power P max follows dsrbuon U[P mn s randomly generaed n U[0,0.25 E ] whle generaed nu[0,o.5 n], due daes are generaed nu[rel + E /P max n].,2 P mn ]. Release daes rel are,rel + E /P mn + We presen he resuls on a frs se of 20 nsances wh 20 asks each. Then, o es how mehods scale, we gve he resuls on 9 nsances wh 25 asks and 0 nsances wh 30 asks. In Table 4, we provde he resuls of wo ree search mehods on he 20 ask nsances, 2

23 b a c d e f g posponed = 0, = 3 posponed 2 posponed = 2, = 3 = 2, = 2 = 2, = 3 2 = 4, = 3 = 4, = = 5, = Fgure 8: Generaed nodes for he EnSP example 22

24 he frs one wh energec reasonng feasbly ess and me-bound adjusmens appled a each node, and he second one wh fully leasc feasbly ess and me-bound adjusmens appled a each node. The obaned resul (Soluon found, No soluon or Tme ou), he CPU me n seconds, and he number of nodes n he search ree are provded for each par nsance / mehod. CPU me has been lmed o 400s. Table 4: Compared resuls on EnSP nsances wh 20 asks Energec reasonng Fully elasc Insance Soluon Tme (s) #Nodes Soluon Tme (s) #Nodes EnSP20_ Soluon Found Soluon Found EnSP20_2 Soluon Found Soluon Found EnSP20_3 Soluon Found Soluon Found EnSP20_4 Soluon Found Soluon Found EnSP20_5 Soluon Found Soluon Found ENSP20_6 Soluon Found Soluon Found ENSP20_7 Tme ou - - Tme ou - - ENSP20_8 Soluon Found Soluon Found ENSP20_9 Tme ou - - Tme ou - - ENSP20_0 Tme ou - - Tme ou - - ENSP20_ No soluon 0.00 No soluon ENSP20_2 No soluon Tme ou - - ENSP20_3 Tme ou - - Tme ou - - ENSP20_4 Tme ou - - Tme ou - - ENSP20_5 Tme ou - - Tme ou - - ENSP20_6 Soluon Found Soluon Found ENSP20_7 Soluon Found Soluon Found ENSP20_8 Tme ou - - Tme ou - - ENSP20_9 No soluon No soluon 0.00 ENSP20_20 Tme ou - - Tme ou - - The resul show ha he energec reasonng-based mehod solves (fnds a soluon or proves nfeasbly) 2 nsances ou of 20 whle he fully elasc-based mehod solves nsances. The fac ha only a lle more han half of he nsances are solved underlnes he dffculy of he problem. On one nsance (ENSP20_2) he enegec reasonng was able o prove nfeasbly a he roo node, whle he fully-elasc mehod reaches he me lm. On he easy nsances (less han 5 nodes) he fully-elasc and he energy 23

25 reasonng-based mehods oban he same number of nodes bu he fully elasc mehod s faser (alhough hese nsances are solved by boh mehods n much less han one second). However on he hard nsances (more han 0000 nodes), he energec reasonng-based mehod obans sgnfcanly smaller CP mes (almos en mes faser for ENSP20_7). Table 5 presens he resuls on he 25 and 30 ask nsances. These resuls corroborae he ones obaned for he 20 ask nsances, excep ha a larger number of unsolved nsances s obaned. There s also an nsance (ENSP25_4) proved nfeasble a he roo node by energec reasonng whle he fully elasc rules deduces nohng. The requred CPU me for fndng a soluon s hgly reduced by usng energy reasonng on nsance ENSP25_7. Table 5: Compared resuls on EnSP nsances wh 25 and 30 asks Energec reasonng Fully elasc Insance Soluon Tme (s) #Nodes Soluon Tme (s) #Nodes EnSP25_ Tme ou - - Tme ou - - EnSP25_2 Tme ou - - Tme ou - - EnSP25_3 Tme ou - - Tme ou - - EnSP25_4 No soluon Tme ou - - EnSP25_5 Soluon Found Soluon Found ENSP25_6 Soluon Found Soluon Found ENSP25_7 Soluon Found Soluon Found ENSP25_8 Tme ou - - Tme ou - - ENSP25_9 Tme ou - - Tme ou - - ENSP30_ Tme ou - - Tme ou - - ENSP30_2 No soluon Tme ou - - ENSP30_3 Tme ou - - Tme ou - - ENSP30_4 Tme ou - - Tme ou - - ENSP30_5 Tme ou - - Tme ou - - ENSP30_6 Soluon Found Soluon Found ENSP30_7 Tme ou - - Tme ou - - ENSP30_8 Tme ou - - Tme ou - - ENSP30_9 Tme ou - - Tme ou - - ENSP30_0 Soluon Found Soluon Found In concluson, despe he problem dffculy, he resuls show generally he superory 24

26 of he approach ncorporang energec reasonng, boh for he number of nodes and he CPU me. 5 Concluson Fuure work We presened he energy schedulng problem (EnSP), an exenson of he cumulave schedulng problem o represen energy requremens of acves. We showed hs model s well-adaped o a parallel machne schedulng ndusral conex wh elecrc power lmaons. We proposed a wo-sep Ineger/Consran programmng approach o solve he ndusral problem. Ths approach exhbed he need for a furher refnemen n consderng specfcally he energy consrans. We proposed an exenson of he sandard energec reasonng scheme for he EnSP ha was no covered by prevous works on hs subjec. Fnally we draw he scheme of a ree search mehod based on domnance rules and praccal assumpons. Compuaonal expermens llusrae he neres of energec reasonng. Furher work wll conss n exendng he compuaonal experence n order o consoldae he way o parameerze he applcaon of energec reasonng n a solvng procedure. One of our objecves would hen be o negrae he proposed energy consran propagaon reasonng n he ndusral problem solvng mehod. References [] Ph. Bapse, A. Jougle, C. Le Pape, and W. Nujen. A consran-based approach o mnmze he weghed number of lae jobs on parallel machnes. UTC Techncal Repor 2000/288, 288, [2] Ph. Bapse, C. Le Pape, and W. Nujen. Sasfably ess and me-bound adjusmens for cumulave schedulng problems. Annals of Operaons Research, 92: , 999. [3] J. Blazewcz, M. Machowak, J. Weglarz, M.Y. Kovalyov, and D. Trysram. Schedulng malleable asks on parallel processors o mnmze he makespan. In Models and Algorhms for Plannng and Schedulng Problems, Ph. Bapse, J. Carler, A. Muner, A.S. Schulz (Eds), Annals of Operaons Research, 29(-4):65 80,

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