Herarchcal Fores anageen wh Ancpaon : an applcaon o accal-operaonal plannng negraon Beaudon, D., Frare, J.-., LeBel, L. ars 2007 (revsed) Workng Paper D-2005-JF-7 Research Consoru n e-busness n he Fores Producs ndusr (FOR@C) nerunvers research Cener on Enerprse Neworks, Logscs and ransporaon (CRREL), Unversé Laval, Québec, Canada CRREL, 2007
Herarchcal Fores anageen wh Ancpaon : an applcaon o accaloperaonal plannng negraon Beaudon, D,3,4, Frare, J.-. 2,3,4, LeBel, L,3,4 Faculé de foresere e géoaque, Unversé Laval Québec, Canada 2 École Polechnque de onréal, Québec, Canada 3 Research Consoru n e-busness n he fores producs ndusr (FOR@C), 4 Cenre ner-unversare de Recherche sur le Réseaux d Enreprse, Logsque e de ranspor (CRREL). Absrac hs paper exanes he proble of harves capac plannng a a accal level n a conex of procureen acves ousourced o ndependen conracors. Annual capac plannng allows planners o deerne he nuber of conracors he need o hre per perod hroughou he ear and o defne he duraon of her conracs. n pracce, hs process usuall nvolves he analss of hsorcal daa regardng he operaonal use of capac and aggregaed deand forecas, he oupu of whch hen serves o plan harves operaons. Alhough hs for of herarchcal plannng reduces he coplex of he ask, he decoposon no sub-probles ha us be successvel resolved can lead o nfeasbl or poor use of harvesng capac. he specfc proble addressed here resdes n how one can consder he operaonal pac of harvesng decsons aken a he accal level n order o ensure a plan s feasbl a he operaonal level. We presen a accal plannng process usng an ancpaon funcon based on Schneewess generc herarchcal coordnaon echans. he ancpaon funcon corresponds o a sequencng and equpen ransporaon proble. We also presen and es a xed-neger odel and a heursc soluon procedure o solve he ancpaed proble. he ancpaon approach we propose
appears o be a vald ehod o beer negrae ke operaonal-level decsons no accal plans, especall wh regards o he possbl of harvesng each block over several perods. he ancpaon approach allows accal plannng o accoun for operaonal crera.
nroducon he use of aheacal odels o deal wh wood procureen probles daes back o he earl 960s. Snce hen, a large bod of odels has been developed o address varous aspecs of he wood procureen proble. Over he ears, ncreased requreens fro he ndusres, he general publc and governen for raw aeral, coodes, recreaon, conservaon and preservaon have greal ncreased he coplex of he resulng fores anageen plannng proble (Wenraub and Davs 996). Researchers have approached hese ncreasngl coplex probles wh wo lnes of hnkng: hrough he use of large onolhc odels or b eans of herarchcal decoposon. On one sde, Bran and rupa (993) and Schneewess (999) denf ls n huan cognon, aheacs, and copuaonal power as an peden o solvng large-scale probles as a sngle en. Along he sae lne, Bare and Feld (986) hghlgh severe laons of onolhc odels of ver large denson: () he are oo poorl undersood and oo cosl n ers of seup e, soluon e and user sklls o be of uch value o presen or fuure fores plannng effors; and (2) he do no adequael address he dfferen, hough relaed, probles of fores plannng: sraegc (allocaon), accal (schedulng), and operaonal (pleenaon) probles. On he oher hand, cnaughon e al. (2000) usf he use of a onolhc approach because of he conssenc allows he planner o acheve beween he resuls of decson odels defned a wo herarchcal levels. Whle he auhors presen a large odel ha negraes boh sraegc and accal aspecs of fores harvesng, a full-negraed, real-sze proble reans e o be solved. he prar reason of hs laon relaes o he cobnaoral naure and he resulng sze of he proble. Even f a large odel could be solved, he cenralzed approach o fores anageen plannng does no properl represen he proble as encounered n pracce. ndeed, cenralzed approaches do no ake no
accoun he fac ha decsons a dfferen levels ofen coe fro dfferen persons. Furherore, he do no consder ha decsons are no aken a he sae frequenc nor e, bu raher n a successve anner, soees spaced ou b weeks or even onhs. As pu b Wenraub and Davs (996) he challenge s «o recognze and negrae dfferen decsonakers who have dfferen probles and obecves bu are hopelessl bound ogeher n a cuulave effec herarchcal proble». Herarchcal plannng Herarchcal producon plannng (HPP) as o splf coplex plannng probles. Hax and eal (975) nroduced he dea of HPP b paronng he decson process no sub-probles coverng dfferen e horzons. nforaon s aggregaed and dsaggregaed hrough he varous herarchcal levels. Herarchcal analss refers o he organzaon of nforaon for akng decsons a dfferen levels when he qual/accurac of he decsons ade a one level depends upon decsons or nforaon a oher levels (Boland 2003). Levels a be defned eporall or spaall where he scope of he hgher level full encopasses he scope of he lower level (Haes 982). n hs conex, eal (984) suarzes soe of he advanages of he herarchcal plannng approach: () reduces proble coplex b separang he no sub-probles and aggregang daa a hgher decson levels; (2) s easer o undersand b provdng a good organzaonal f; and (3) reduces unceran b posponng decsons as long as possble. n he conex of fores anageen, Gunn (996) pons ou ha he use of a deernsc odel on a rollng plannng horzon and replannng represen a good heursc procedure for dealng wh fores anageen plannng under unceran condons. However, HPP has s drawbacks. ndeed, HPP nvolves solvng a se of probles n a sequenal anner. Such an approach can lead o sub-opal, nconssences and even o nfeasbl. he degree of sub-opal depends upon he qual of he coordnaon
schee used o lnk ogeher he decson levels. nconssences a arse because of conflcng obecves a dfferen plannng levels, whle nfeasbl usuall resuls fro nforaon aggregaon (Gelders and Van Wassenhove 98) and he loss of coheson beween odels and real. Zork-Schalla (200) adds ha «d-er plannng uses less dealed and dfferen nforaon han shor-er plannng, because dealed daa s no e avalable a he e ha d-er plannng decsons need o be ade. Ye he d-er decsons should be such ha he shor-er decsons can be aken n lne wh overall operaonal obecves». o usf he use of a onolhc odel o overcoe he lack of coheson beween he resuls a dfferen levels of a herarchcal approach, cnaughon e al. (2000) refer o he paper of Daus and Nelson (993). he auhors provde an exaple of a proble where longer harves schedules were developed usng aspaal, sraa-based forulaons and spaal block schedulng forulaons. he susaned elds esaed b he spaal forulaons were n all cases lower han hose esaed b he aspaal forulaons b a range of 2% o 29%. hese resuls rase he crcal queson of how o oban conssenc beween he resuls of decson odels defned a wo or ore herarchcal levels. he conclude ha for conssenc, regulaons governng he spaal dsrbuon of harves uns should also be ncorporaed no he long-er plannng process where susanable harves levels are calculaed. hs suggess ha lower-level pacs ancpaed durng decson akng a a hgher level us be beer odeled and assessed. Adequae procedures us be defned o creae beer lnks beween levels. n order o address hs ssue, Schneewess (2003) proposes a general herarchcal fraework whch as o brng conssenc beween herarchcal levels whle respecng he dsrbued naure of plannng probles. hs fraework also allows for he explc consderaon of he pac a a gven level of decsons aken a a lower-level hrough he use
of ancpaon echanss. Schneewess and Zer (2004) conduced an exensve quanave analss of operaonal coordnaon echanss n he conex of herarchcal plannng. he concluded ha he use of ancpaon echanss resuls n sgnfcan proveen over he pure op-down herarchcal process. hs paper apples Schneewess fraework o a large-scale fores procureen plannng proble. he an conrbuons of hs paper nclude: () a descrpon of he wood procureen proble herarchcal decoposon; (2) a herarchcal negraon echans of he proble; (3) an ancpaon odel for he sequencng and equpen ransporaon proble; and (4) a heursc procedure for he ancpaon operaonal odel. he reander of hs paper follows he followng fora: he frs secon descrbes he applcabl of Schneewess fraework o wood procureen plannng, followed b a descrpon of a herarchcal coordnaon echans and s relaonshp o he accal wood procureen plannng process. hen follows a sequencng and equpen ransporaon cosancpaon odel along wh a heursc soluon procedure, and a perforance evaluaon of he heursc soluon procedure. Fnall, a dscusson on he search for opal and prospecve rearks conclude hs paper. Applcaon o wood procureen plannng n wood procureen plannng probles, one of he obecves pursued b accal level plannng nvolves seng he requred producon capaces. Alhough fores copanes plan and anage fores operaons, he ofen sub-conrac he execuon of hese operaons. Capac seng hus allows copanes o denf how an conracors o hre hroughou he ear, o specf workng perods, and o defne he lengh of he conracs bndng he conracors o he fores copan. Consequenl, fro he fores copan s pon of vew, capac seng does no nvolve oblzng large aouns of s own resources o purchase equpen.
Beaudon e al. (n press) presen a xed-neger prograng odel whch as a supporng he wood procureen accal decsons of a ul-facl copan. hs odel allows for wood exchange beween copanes. Furherore, he aeral flow hrough he suppl chan s drven b boh a deand o sasf (Pull sraeg) and a arke echans (Push sraeg), enablng he planner o ake no consderaon boh wood freshness and he noon of qual relaed o he age of harvesed ber. hs accal odel does no explcl address he capac seng decson. Raher, suggess ha once planners selec a plan for pleenaon fro a se of canddae plans, harvesng capac requreens can be evaluaed n regard o he producon arges per perod proposed. However, arges se b aggregaed producon plans a he accal level consran operaonal plannng. Unforunael, nfeasbl a occur for a couple of reasons. Frs, harvesng decsons a he accal level depend on aggregaed capac fgures. Also, se up es (ovng equpen fro a block o anoher), lo szng (ne capac requreens depend on he volue harvesed on a block each e harvesng occurs) and harves block sequencng decsons all se fro operaonal level decsons. herefore, he proble resdes n how o adequael consder he pac of fuure operaonal harvesng decsons on he accal level, and how o ensure ha a accal plan reans feasble a an operaonal level. he nex secon oulnes he heorecal background exploed o propose a soluon o hs proble. Generc herarchcal coordnaon echans As prevousl explaned, coordnaon echanss are requred n order o overcoe he an probles of HPP. n order o do so, Schneewess general herarchcal fraework proposes a no so pure op-down approach ha akes no accoun he plcaon of cascaded decsons n a herarchcal plannng conex. Fgure depcs he srucure of he herarchcal
plannng sse referred o as a accal-operaonal Dsrbued Decson akng (DD) sse. NSER FGURE he accal-operaonal DD sse nvolves a wo-level decson odel, respecvel he op and base-level decsons. A decson odel s defned b s sse of crera C (.e., obecve funcon) and s decson feld A (.e., se of consrans). he accal odel B B B B corresponds o = ( C, A ) and ( C, A ) = represens he operaonal odel. nforaon saus and he e a whch decsons us be ade rean poran, so 0 and B denoe he respecve nforaon saus a 0 and 0. Coordnaon beween he accal and operaonal levels proposed b he auhor s acheved b reacve plc ancpaon, eanng ha onl par of he operaonal level s ancpaed as a boo-up nfluence and an nsrucon as a op-down sgnal. Before decson akng occurs a he accal level, he decson-aker ancpaes he base-level s decson reacon o a poenal accal decson (.e. N) hrough an ancpaon funcon AF(N). n urn, b negrang he oupu of hs funcon n hs decson process, he decson-aker can be nfluenced. hs process s called reacve ancpaon because he ancpaon s assessed hrough a funcon ha provdes an esae of how he base-level would reac f subed o such an nsrucon (.e. he poenal accal decson). ore specfcall, AF(N) s deerned hrough he use B B B B B of an ancpaon base-odel ˆ ( Cˆ, Aˆ, ˆ ) ˆ =. he op creron C can hus be broken down no wo crera, C and C B. he forer represens he prvae creron whch corresponds o he obecve funcon of he accal proble, whle he laer represens he op-down creron whch corresponds o he obecve funcon of he operaonal odel. he op-down creron s he par of he op-creron whch explcl akes no accoun he operaonal level and depends on he ancpaon funcon. For furher deals on
Schneewess herarchcal coordnaon echans, he readers are referred o Schneewess (2003). We address a wofold challenge n applng such a fraework o he specfc conex of accal wood procureen plannng. Frs, we consder he negraon of he ancpaon odel nfluence no he op-level decson odel, and second, he desgn of he ancpaon odel. he nex secon addresses boh. accal wood procureen plannng n a wood procureen conex, accal plannng negraes harvesng, ransporaon and nvenor (sandng, roadsde and log ards) decsons over he nex ear. he an purposes of accal plannng nclude seng producon arges and requred producon capaces per perod. accal plannng process Whle wood procureen plannng has grown n coplex, he ndusr sll plans wh led aheacal prograng suppors. Such an nuve and anual process pcall leads o wo shorcongs: () he nabl o consder alernave plans for pleenaon due o he prohbve aoun of e requred o develop a plan; and (2) he dffcul of assessng he perforance of plans subeced o sochasc condons. n Beaudon e al. (n press), he auhors propose a accal plannng process o overcoe hese wo shorcongs. Fgure 2 aps hs plannng process ono Schneewess fraework. NSER FGURE 2 he op-level decson odel ( ) ncorporaes several coponens (a scenaro generaor, a accal wood procureen plannng odel, and a rule-based sulaor). he base-level decson odel corresponds o all decsons ha us be ade a he operaonal level. hs ncludes sequencng and equpen ransporaon decsons, he dealed allocaon of producs o blocks, he selecon of buckng paerns for each block, ec. Fnall,
he ancpaon odel of he operaonal level ncorporaes onl he operaonal decsons ha os nfluence he accal level. n bref, hese decsons concern sequencng and equpen ransporaon, for whch, he cos-ancpaon odel wll be explaned n he nex secon. ogeher, he op-level and he ancpaon of he base-level decson odels depced n Fgure 2 consue a ul-crera decson-akng process o suppor a decson-aker n selecng a accal plan o pleen. negraon of he ancpaon and he op level decson odels he overall accal plannng process sars b creang a predefned nuber of scenaros S (defned b he planner) based on randol generaed values for he unceran paraeers, for each perod consdered n he odel (Scenaro generaor). For each scenaro s S, he planner deerne he opal plan a s (referred o as a canddae plan) b solvng a deernsc xed-neger progra (accal wood procureen plannng odel). Each canddae plan hen coes under furher analss. Frs, he planner sulaes each canddae plan a s a s whn dfferen scenaros (Ruled-based sulaor). hs analss provdes nforaon on he prvae crera ( ) a s C s of he op level. Nex, each canddae plan a s s subed as an nsrucon N o he ancpaon odel ( ˆ B ) n order o ancpae he sequencng and equpen ransporaon cos AF ( N ), as well as oher nforaon on he op-down crera B C s, such as he feasbl of he canddae plan a s. n each of hese analses, he planner gahers sascs n order o help resolve he resulng ul-crera accal decson proble. For furher deals on he accal plannng process dscussed above and s coponens, we refer he reader o Beaudon e al. (n press). Operaonal ancpaon he ancpaon offers a eans hrough whch he decson-aker akes no accoun he pac of hs decsons on a lower level. he odelng decsons aken a he desgn sage of
he ancpaon odel pac he qual of he nforaon provdes. he ancpaon operaonal odel shares a odelng relaonshp wh boh he accal ( ) and operaonal B ( ) odels. s desgn nvolves a process of analss and deducon (Fgure 3). NSER FGURE 3 We defne herarchcal levels whle n he desgn sage of such a herarchcal producon plannng sse. For each level, we denf anageen obecves. he obecves a each level us lne up wh he overall obecve of he organzaon. he goal pursued b he accal level should dcae he consuens of he ancpaon odel, all he whle represenng an accurae enough assessen of he nfluence of he operaonal decson level. n order o do so, he decson-aker us frs a he accal level denf he coponens of he operaonal decson level ha os nfluence hs decsons. n pracce, n order o avod havng o ancpae operaonal decsons, foresers defne decson rules o splf he plannng process. For exaple, he largel use he rule of no-preepon of blocks durng harvesng (.e., never parall harves blocks). Such a rule reduces he need o ancpae he cos of ransporng harvesng equpen beween blocks, because resuls n a cos reducon a he operaonal level due o less ransporaon of equpen. hs also splfes he schedulng proble. However, ls he flexbl offered a he accal level b elnang he possbl of harvesng onl par of a block. Furherore, no allowng preepon (.e., abl o harves a block over several perods) conrbues a a accal level o poor capac deploen whch can ranslae no () ncreased nvenor of unneeded producs, (2) shorages of needed producs, (3) value loss hrough fbre degradaon and (4) los sales opporunes. n oher words, no allowng preepon accords ore porance o equpen ransporaon cos han o coss relaed o nvenores balances, value loss and los sales opporunes.
Consequenl, a sse analss s requred o denf he operaonal feaures ha have he os pac on he accal level. Adequae crera o ancpae us hence be denfed based on he obecves of he operaonal and accal decsons levels. hese crera do no have o cover he enre operaonal proble. n he exaple enoned above, because blocks are harvesed enrel, harvesng decsons ake he for of bnar varables. n he accal wood procureen odel proposed n Beaudon e al. (n press), hese decsons appear as connuous varables, whch plcl allows foresers o grasp he benefs of harves block preepon. However, hs pracce resuls n an ncrease n he nuber of equpen ransporaons beween blocks. Alhough laons can be posed on he nuber of perods over whch harvesng can occur on a gven block and he nuber of blocks on whch harvesng can occur durng a gven perod, such a pracce exers a defne pac on a achne s avalable producon e. hus becoes necessar o ake hs facor, as well as s cos, no accoun for he selecon of a accal plan. ore specfcall, when he decson-aker consders a canddae accal plan, he needs o consder boh s feasbl wh regard o harvesng capac and he equpen ransporaon cos nvolved n pleenng he proposed harves arges. hese wo crera reflec he pac of accal decsons on he operaonal level. Consequenl, he ancpaon odel we have desgned s no nended for he dealed plannng of operaonal acves, whch nvolves dealed se buckng paern selecon, aong ohers. Hence, we ancpae onl par of he operaonal level n order o assess he os srcl relevan nforaon for he accal decson-aker. hus, n he conex of he proble on hand, he ancpaon odel as o nze he oal equpen ransporaon cos n pleenng he accal canddae plan. gnorng harves cos n he ancpaon odel wll no ranslae no a schedule ha groups achnes o ceran blocks for he sake of reducng he equpen ransporaon cos for wo reasons. Frs, a a accal level, harves
coss are alread accouned for per pe of achne n a specfc block and a a gven perod. he resulng accal canddae plan hus alread provdes nforaon relaed o harves capac ulsaon per pe of achne and perod. n herarchcal plannng, accal decsons are forwarded o he operaonal level for her pleenaon. Consequenl, a he operaonal level, he planner does no reassess he pe of achne assgned o each block and he perods over whch harvesng wll occur. Secondl, for a gven achne harvesng a gven block, seasonal or onhl harves cos varaons can be observed and have also been accouned for n he developen of he accal canddae plans. A he operaonal level, he ng of he harves whn he e frae covered b a accal perod does no pac he cos of he acv. n general, he analss of he feaures of he operaonal decson level allows he decson process desgner o denf hose havng he os pac on he nforaon needed o address he decson proble. Consequenl, dependng on he requred nforaon, oher operaonal crera a be accouned for n he ancpaed proble. Sequencng and equpen ransporaon cos-ancpaon odel hs secon proposes a specfc ancpaon odel of a fr s sequencng and equpen ransporaon cos-ancpaon odel ( ˆ B ). Frs, we nroduce daa ses, followed b he paraeers and varables used o forulae he odel. Fnall, we presen he odel forulaon. Ses : he se of harvesng blocks ( =, K, ) : he se of achnes ( =, K, ) R : he se of rounds whn perod ( r =, K, ) : he se of perods ( =, K, ) R
n HPP, wo eporal feaures defne each level: he e horzon and he perod. he e horzon defnes he nerval over whch he decsons exend, whle he perod represens he nerval of e afer whch he decsons coe under reconsderaon. he hgher he level, he longer he horzon and he perod. Snce he se producon arges orgnang fro he accal level serve a he operaonal level, wo defnons of perod are requred. n he reander of hs paper, he er perod refers o a accal perod, whle round refers o he sequence of achne-block allocaons over e such ha each perod ncludes several rounds. Paraeers S : Sar block of achne a he begnnng of he plannng horzon. V : Volue o be harvesed on block durng perod. D : Capac of achne durng perod. α : δ : Accepable dfference n oal volue harvesed b each achnes. Poron of lowbed (fla deck raler) oal e no avalable for equpen ransporaon. L : oal lowbed capac durng perod. : Requred e o ove achne fro block o block durng perod. C : Cos o ove achne fro block o block durng perod. P : Producv of achne on block durng perod. N r : axu nuber of achnes on block durng round r of perod. Decson varables Fgure 4 suarzes decson varables and her relaonshps wh one anoher. NSER FGURE 4 x r : e spen b achne harvesng on block durng round r of perod.
r : oherwse 0,. perod of durng round o block oves fro block achne f, r odel [] R r r C n Subec o: Capac consrans [2] D x R r R r r r +, [3] ( ) L R r r δ [4] R r N r r,, Suppl consrans [5] V x P R r r =, [6] ( ) V x P R r r, α [7] R r P V D x r r,,,, n Flow consrans [8] R r r,, [9] S = where,, 0
[0.] r = ( r+ ),, r < R, [0.2] ( ),, R = < + Non-negav consrans [] x 0,, r R, r [2] { 0,},,, r R, r Obecve funcon he obecve funcon as o nze he oal ancpaed equpen ransporaon cos. Copanes ncur equpen ransporaon coss whenever he us use a lowbed o ove equpen fro one block o anoher. n he case where subsequen blocks le close o one anoher, operaors a drve he achnes whou ncurrng exra coss, alhough ovng e us be aken no accoun. Consrans Capac consrans Equaons [2] and [3] represen, respecvel, achne and lowbed capaces. he planner us consder ndvdual achne s capaces n order o deerne a sequence of blocks o harves and o snchronze he ng of her dsplaceens. Equaon [2] also ensures ha e spen harvesng and ovng does no exceed he achne s avalable e. For he lowbed, aggregaed capac s consdered raher han ndvdual capac snce no lowbed schedulng s aeped (eq. [3]). Due o he operaonal laons posed b he harvesng blocks sze as well as safe reasons, equaon [4] ls he nuber of achnes on a block a an gven e. Suppl consrans he sarng pon for he ancpaed proble nvolves a ls of argeed volues o be harvesed per block for ever perod consdered. Equaon [5] ensures ha equpen spends
enough e on he blocks o reach hese arges. Equaon [6] allows for a relavel unfor dsrbuon of he workload beween conracors. Equaon [7] oulnes he seup forcng consran: f here exss an posve producon for achne on block a round r of perod, a seup s enforced (ranspor achne o block ). n order o srenghen he forulaon, we l he producon b boh he axu possble producon e wh he avalable capac and he axu e o harves he argeed volue on he block. Flow consrans Snce a achne canno work on ore han one block a a e, equaon [8] serves o render ndvsble. Also, he locaon of he achnes a he begnnng of he plannng horzon wll have an pac on her subsequen desnaons as he odel wll a o nze equpen ransporaon cos whch relaes o ovng dsances. Equaon [9] denfes he nal locaon of he equpen. Fnall, equaons [0.] and [0.2] represen nra- and ner-perod flow conservaon consrans and ensure ha equpen can be oved fro a block onl f drven or delvered here prevousl. he sequencng and equpen ransporaon cos-ancpaon proble elds a largescale xed-neger lnear proble. Bnar varables correspond o ovng decsons and connuous varables descrbe harvesng e. Heursc procedure he proble a hand corresponds o a schedulng proble wh sequence-dependen seup es, one of he os dffcul pes of schedulng probles. A one-achne sequencedependen seup schedulng proble s equvalen o a ravelng-salesan proble (SP) and s NP-hard (Pnedo 995). Sequence-dependen seup schedulng of a ul-achne and ul-producon sage sse creaes an even greaer challenge. Parallel achnes schedulng proble (PSP) dae back o he lae 950 s (cnaughon 959 and Hu 96). Cheng and Sn (990) provded a sae-of-he-ar of schedulng approaches unl 990 on parallel
achnes schedulng. ore recenl okooff (200) copleened he revew wh new developens on PSP. he proble s solvable b usng a coercal solver drecl wh a led nuber of perods. n vew of he dffcul of fndng he opal soluon o a real-sze proble, a sple heursc procedure has been developed o solve he sequencng proble. Heursc soluon procedure A heursc soluon procedure was proposed n an aep o fnd a good qual soluon n a reasonable aoun of e. he proposed heursc ses fro e decoposon. he e decoposon ehod consss n dvdng a large e horzon no several saller perods where schedulng probles can be solved effcenl (Wu and eraperou 2003). he heursc akes use of he soluon procedure depced n Fgure 5. NSER FGURE 5 For a gven orgnal proble, he soluon procedure begns b nalzng sub-proble p o zero. he procedure solves a seres of n sub-probles sequenall where n corresponds o he nuber of perods n he orgnal accal proble. Usng resuls fro he acual subproble p, consrans are propagaed o p+ n order o ensure ha he endng locaon of a achne becoes s sarng poson for he nex sub-proble. Heursc hree heurscs underwen esng for he sequencng and equpen ransporaon cosancpaon proble. he an dfferences beween hese heurscs resde n he plannng horzon covered b he sub-probles and he naure of he decson varables. Hereafer, we presen onl he bes perforng heursc. For furher deals on he wo oher heurscs and her perforance evaluaons, we refer he reader o Beaudon e al. (2005). he ul-perod sequencng and equpen ransporaon cos-ancpaon proble s decoposed b paronng he plannng horzon no n overlappng, dependen sub-
probles. Le represen he curren perod consdered no sub-proble p, p =,2,..., n, where n corresponds o he nuber of perods consdered n he orgnal proble. Le p represen he frs perod consdered no sub-proble p. he range of perods assgned o subproble p corresponds o {, +} p p. For each sub-proble, he frs perod consdered corresponds o he curren perod, hus p = p =. Varables correspondng o ovng decsons are of pe neger. hs forulaon eravel solves he sub-probles b consderng he pac of he ovng decsons for he subsequen perod. hs odfcaon faclaes copuaons whle consderng fuure dsplaceen needs. Fro he opal soluon of each sub-proble, onl he soluon of he curren perod s used n he soluon of he orgnal proble. Sub-probles are solved o opal usng he odel prevousl presened (equaons []-[2]). Heursc perforance evaluaon wo copuaonal experens were conduced o evaluae he perforance of he heursc. hrough hese experens, we copared soluons found wh he heursc wh hose obaned hrough: () drec solvng of sall nsances of es probles; and (2) lower bound calculaon obaned b Lagrangean relaxaon wh a subgraden opzaon schee. Drec solvng used sandard branch-and-bound echnque. For he copuaonal experens, we consdered hree harvesng sses, each coposed of a processor and a forwarder. No possbl exss of usng exra sses, as capac deernaon occurs a he accal level. Whn he accal plannng process, he planner gahers sascs regardng plans feasbl. eanwhle, n order o evaluae he perforance of he heursc n er of s abl o fnd soluons close o opal, we se harvesng and lowbed capaces n order o avod an nfeasbl.
All copuaons were perfored wh CPLEX 9. on a.27 GHz Penu 3 personal copuer wh.83 GB of RA o solve he xed-neger probles drecl and hrough he heursc soluon procedure. he aheacal odel s pleened n he Opzaon Proble Language (OPL) of log and he heursc soluon procedure as well as he Lagrangan relaxaon n OPLscrp. For he frs experen, we developed 30 sall nsances of es probles wh he nuber of perods and he nuber of blocks o be harvesed per perod randol seleced fro unfor dsrbuon [, 6] and [0, 5], respecvel. We also developed he levels of harvesng o occur on he denfed blocks fro a unfor dsrbuon [2000, 6000]. he soluons found b solvng he xed-neger progra presened prevousl served o benchark he soluon found b he heursc. Le C H and C P represen he coss found b he heursc and he xed-neger progra, respecvel. able suarzes he perforance of he heursc. NSER ABLE he average requred e o solve he es probles o opal equaes o 89.7 nues, he nu e,.4 nues, and he axu, over 240 nues - he e l posed o CPLEX for he experen. he average e o solve he sae es probles wh he heursc equaes o 3.3 nues, he nu e, onl 0. nue, and he axu e, 6.0 nues. he average cos devaon s.8%, he nu devaon, 0.0%, and he axu devaon, 4.8%. able clearl ndcaes ha he heursc can fnd reasonabl good soluons n a shor perod of e. Fndng he opal soluon b drecl solvng he xed-neger progra, however, reans praccal. Several of he sall nsances of es probles exceeded he e l of four hours. For he second experen, we developed 30 es probles n a slar fashon wh he nuber of perods and he nuber of blocks o be harvesed per perod randol seleced
fro unfor dsrbuon [6, 26] and [3, 6], respecvel. We also deerned he levels of harvesng o occur on he denfed blocks fro a unfor dsrbuon [2000, 6000]. We copued Lower bounds hrough Lagrangean relaxaon wh a subgraden opzaon schee. he Lagrange relaxaon reforulaon of he orgnal proble dualzes he nerperod flow balancng consran [0.2] n he obecve funcon []. A coplee descrpon of he lower bound evaluaon procedure appears n he Appendx. Le C H and C LB represen he coss found b he heursc and he copued lower bound respecvel. able 2 suarzes he perforance of he heursc. NSER ABLE 2 he average e o solve he probles wh he heursc equaes 33. nues, he nu e, 2.6 nues, and he axu e, 53.7 nues. he average cos devaon, 6.%, he nu devaon,.6%, and he axu devaon,.5%. Ancpaon and ls of opal he ancpaon approach proposed n hs paper nvolves a wo-sep procedure because op-level nsrucons are nroduced as consrans n he ancpaon odel of he operaonal level. he resuls of hs ancpaon hen re-ener he accal ul-crera decson proble. Because hese resuls represen an ancpaon of wha operaonal plannng would reseble f each of he canddae accal plans were pleened, he need for an opal soluon becoes unnecessar for wo reasons. he frs relaes o he saus of nforaon. ore specfcall, when operaonal plannng occurs, he nforaon requred o produce a plan a dffer fro he avalable nforaon when conducng accal plannng. An opal soluon of he ancpaon decson odel hus lkel becoes sub-opal. he second reason nvolves he e fraework dfferenal of accal and operaonal plannng. Operaonal plannng occurs ndeed several es whn one accal perod. he resulng plan pleened a execuon e hus represens he concaenaon of an paral
operaonal plans (he frs perods beween wo plannng ccles). Consequenl, even he opal soluon of he ancpaon odel would no full represen he operaonal plannng dnacs wh s abl o recover fro perurbaons. hs becoes even ore coplcaed when he operaonal plannng horzon covers ore han one accal plannng ccle (.e., perods) for whch accal decsons have no e been ade. Fgure 6 llusraes he neracons and e fraework dfferenal of hese plannng levels. NSER FGURE 6 Consequenl, n he conex of herarchcal plannng wh ancpaon, he search for an opal ancpaon decson sees raher rrelevan. sees ore poran o consder he ancpaon no as an opzaon proble bu raher as an nforaon gaherng process o help evaluae how decsons ha are aken a one level pac lower levels abl o reach he se producon arges. Alhough proven useful n he soluon approach proposed n hs paper, rases oher quesons such as how o evaluae he level of qual of an ancpaon and how o prove hs qual over e. Concluson Wood procureen plannng reans b naure a coplex process. HPP s known for reducng proble coplex b paronng he proble no sub-probles ha are solved n a sequenal anner. Such approach can lead o sub-opal, nconssences and even o nfeasbl. We have seen how Schneewess odelng fraework, akng use of ancpaon, can operae n he conex of accal wood procureen plannng n order o lessen he shorcongs of HPP whle respecng he dsrbued naure of he plannng proble. ndeed, hs approach provdes he flexbl needed o nclude several ke decsons aken a one level bu havng he poenal o greal nfluence a plan a a dfferen level. We presen a ul-denson odelng approach eplong accal harves plannng wh preepon and operaonal sequencng and equpen ransporaon. he approach can
also serve o ancpae oher operaonal feaures. he approach can be used advanageousl n plannng a hgher levels ncorporang a broad range of probles. A fr s sequencng and equpen ransporaon cos-ancpaon proble has been presened as a xed-neger odel. hs ancpaon odel s no nended for he acual plannng of operaonal acves. We ancpaed par of he operaonal level n order o gaher nforaon relevan o he decson-aker a a accal level. hs nforaon reveals s value n a accal plannng process n he evaluaon of he pac of accal decsons on he operaonal level. he sequencng and equpen ransporaon cos-ancpaon odel reans solvable wh a coercal solver f consderng a led nuber of perods. n vew of he dffcul and he relevance of fndng he opal soluon o hs proble, we have also esed a heursc soluon procedure based on e decoposon. he perforance of he heursc soluon procedure has been evaluaed b coparsons wh copued lower bounds obaned hrough Lagrangean relaxaon. he copuaonal resuls show ha he oal equpen ransporaon cos averages 6.% above he lower bound. he search for an opal ancpaon decson sees raher rrelevan n he lgh of laons posed b he nforaon aser and he asnchronous plannng n he varous plannng levels. o lessen he shorcongs resulng fro he nforaon aser, unceran could be accouned for n he ancpaed operaonal odel nsead of usng he presened deernsc approach. Sulang he pleenaon of each canddae accal plan over a deerned nuber of unceran operaonal scenaros could provde ore valuable nforaon o he decson-aker seekng o selec a canddae accal plan for pleenaon. Acknowledgeens
hs work was funded b he Research Consoru n E-Busness n he Fores Producs ndusr (FOR@C) and suppored b he nerunvers Research Cener on Enerprse Neworks, Logscs and ransporaon (CRREL). References Bare, B., Feld, R. 986. An evaluaon of FORPLAN fro an operaons research perspecve. Presened a he sp. FORPLAN: An evaluaon of a Fores Plannng ool. GR R-40. n: Wenraub, A., Cholak, A. 99. A herarchcal Approach o Fores Plannng. Fores Scence, 37(2):439-460. Beaudon, D., LeBel, L, Frare, J.-. accal suppl chan plannng n he fores producs ndusr hrough opsaon and scenaro-based analss. n press, Canadan Journal of Fores Research. Beaudon, D., Frare, J.-., LeBel L.G., 2005. Harves block sequencng and equpen ransporaon snchronzaon, Cenre CENOR, Unversé Laval, Docuen de raval D- 2005-JF-7 Bran, G.R., rupa, D. 993. Herarchcal Producon Plannng. n: Zork-Schalla, A.J. 200. odelng of Decson akng Processes n Suppl Chan Plannng Sofware: a heorecal and eprcal assessen of 2 radearx. Ph.D. hess, Endhoven Unvers Press. Boland,. 2003. Herarchcal Plannng n Foresr. ALAS/SFOR Proec echncal Repor, 7p. Cheng,.C.E., Sn, C.C.S. 990. A sae of he ar revew of parallel achne schedulng research. European Journal of Operaonal Research. 47:27-292. Daus, D., and Nelson, J. 993. Spaal reducon facors for sraa-based harves schedules. For. Sc. 39:52-65.
Fsher,.L. 98. he Lagrangean Relaxaon ehod for Solvng neger Prograng Probles. anageen Scence, 27:-8. Gelders, L.F., Van Wassenhove, L.N. 98. Producon Plannng: A Revew. European Journal of Operaonal Research, 7:pp.0-0. Gunn, E.A. 996. Herarchcal Plannng Processes n Foresr: A Sochasc Prograng- Decson Analc Perspecve. n Proceedngs: Herarchcal Approaches o Fores anageen n Publc and Prvae Organzaons. Peawawa Naonal Foresr nsue, nforaon Repor P-X-24, pp.85-97. Haes, Y.Y. 982. odellng of large-scale sses n a herarchcal-ulobecve fraework, n: Sudes n he anageen Scences and Sses, 7: -7. Hax, A.C., eal, H.C. 975. Herarchcal negraon of producon plannng and schedulng, n: Sudes n he anageen Scences,.A. Gesler ed., Logscs, Norh Holland Aercan Elsever. Hu,.C. 96. Parallel sequencng and assebl lne probles. Operaons Research. 9:84-948. cnaughon, R. 959. Schedulng wh deadlnes and loss funcons. anageen Scences. 6:-2. cnaughon, A., Rönnqvs,., Ran, D. 2000. A odel whch negraes sraegc and accal aspecs of fores harvesng..j.d. Powell and Scholes (Eds.), Sse odellng and Opzaon: ehods, heor and Applcaons, Kluwer Acadec Publshers. Pp.89-207. eal, H.C. 984. Pung Producon Decsons Where he Belong. Harvard Busness Revew. 62(2):02-. okooff, E. 200. Parallel achne Schedulng Probles: A Surve, Asa-Pacfc Journal of Operaonal Research. 8:93-242.
Pnedo,. 995. Schedulng heor, and Sses. Englewood Clffs, NJ: Prence Hall. Schneewess, C. 999. Herarches n Dsrbued Decson akng, Berln, Sprnger. Schneewess, C. 2003. Dsrbued Decson akng, 2 nd Ed. Sprnger-Verlag Berln, Geran. 528p. Schneewess, C. and Zer, K. 2004. Herarchcal coordnaon echanss whn he suppl chan. European Journal of Operaonal Research, 53: 687-703. Wenraub, A., Davs, L. 996. Herarchcal Plannng n Fores Resource anageen: Defnng he Densons of he Subec Area. n Proceedngs: Herarchcal Approaches o Fores anageen n Publc and Prvae Organzaons. Peawawa Naonal Foresr nsue, nforaon Repor P-X-24, pp.2-4. Wu, D., eraperou, G. 2003. Decoposon approaches for he effcen soluon of shorer schedulng probles. Copuers and Checal Engneerng, 27: 26-276. Zork-Schalla, A.J. 200. odelng of Decson akng Processes n Suppl Chan Plannng Sofware: a heorecal and eprcal assessen of 2 radearx. Ph.D. hess, Endhoven Unvers Press.
Appendx Lagrangean relaxaon consss n absorbng (dualzng) he boundng consrans no he obecve funcon and n solvng he resulng proble. n he Lagrange relaxaon reforulaon of he orgnal proble, he ner-perod flow balancng consran [0.2] s dualzed n he obecve funcon [] wh dual ulplers λ unresrced n sgn. [3] ( ) + + = R R r r C n λ Afer rearrangng he ers n he obecve funcon, he Lagrange proble becoes: [4] ( ) + = = R R r r C 2 n λ λ s.. [2]-[0.], []-[2]. he Lagrange proble decoposes no separae sub-probles for each perod : For =: [5] R R r r C n λ s.. [2]-[0.], []-[2]. For < < : [6] ( ) + R R r r C n λ λ s.. [2]-[8], [0.], []-[2]. For = :
[7] ( ) + R r r C n λ s.. [2]-[8], [0.], []-[2]. he Lagrange proble s solved hrough several eraons and he Lagrange dual prces λ are updaed b a sandard subgraden opzaon schee forulaed n [8]. [8] ( ) S R,, < + = + + λ λ φ φ φ φ φ Le φ λ be he dual prces a eraon Φ and le ( ) φ φ r x, be he opal soluon for he Lagrange proble a eraon Φ. he opal obecve value of [4] for he Lagrange proble a eraon Φ s ( ) φ λ v. n he calculaon of he sep sze S (eq. [9]), UB s he bes-known upper bound for he orgnal proble []-[2] and π s nall se o wo and s decreased whenever ( ) φ φ r x, has faled o prove n a specfed nuber of eraons. For furher deals on Lagrangean relaxaon, we refer he reader o Fsher (98). [9] ( ) ( ) ( ) = + = R v UB S φ φ φ λ π 2 For he calculaon of he sep sze as defned b equaon [9], UB o full sze probles are provded b he heursc and π s nall se o 2 and s decreased whenever no proveen occurred n he las 30 eraons. he soppng creron for he subgraden opzaon schee was se o 200 eraons.
able Perforance of he heursc soluon procedure copared wh he opal soluons found b branch and bound. e requred o fnd he e requred o o solve he ( CH CP) CP*00% heursc soluon n nues P n nues n ax ean n ax ean n ax ean 0.0 4.8.8 0. 6.0 3.3.4 240.0* 89.7 *he e requred o fnd he opal soluon exceeds he e l of 4 hours se for solvng he P b CPLEX.
able 2 Perforance of he heursc soluon procedure copared wh copued lower bounds. ( ) % e requred o fnd he CH CLB CLB *00 heursc soluon n nues n ax ean n ax ean,6,5 6, 2,6 53,7 33,
Fgure accal-operaonal DD sse ( C, A, ) 0 AF ( N ) N( a ) ˆ B B B ( Cˆ, Aˆ, ˆ ) B 0 N * = N B B B ( C, A, ) B * ( a )
Fgure 2 accal plannng process Scenaro generaor denf probabl dsrbuon funcons accal wood procureen plannng odel Forulae P proble Generae S scenaros Solve S probles Rule-based sulaor Sulae S plans n S- scenaros S canddae plans ( C, A, ) 0 AF( N) Copue sascs for each canddae plans Sequencng & equpen ransporaon cos ancpaon odel Solve S probles Operaonal odel N * = N a * ( ) N( a ) ˆ B B B ( Cˆ, Aˆ, ˆ ) B 0 B B B ( C, A, ) B
Fgure 3 Desgn relaonshp Herarchcal producon plannng sse ( C, A, ) 0 Need for ancpaon Ancpaon odel developen accal-operaonal DD sse ( C, A, ) 0 AF ˆ N B B B ( Cˆ, Aˆ, ˆ ) B 0 B B B ( C, A, ) B Analss of he operaonal feaures ha os pac accal plannng N * B B B ( C, A, ) B
Fgure 4 Sequencng and equpen ransporaon proble r = r = 2 r =ψ r = R S, =, =, r=,, =, =, r= 2,, =, =, r= ψ,, =, =, r= R, x, =, r=, x, =, r= 2, x x, =, r= ψ,, =, r= R, S, =, = 2, r=, 2 2 2 2 S x,=2,r=, S, =, =Ω, r=, Ω Ω Ω Ω x,=ω,r=, S, =, =, r=, x, =,r =,
Fgure 5 Flow char of he heursc soluon procedure nalze sub-proble: p=0 p=p+. Se paraeers for sub-proble p. Propagae localzaon consrans o p=p+. Solve sub-proble p. Sore soluon of sub-proble p. NO p=n? YES negrae soluons {p, p+,, n}.
Fgure 6 Rollng plannng horzon n herarchcal plannng accal Operaonal Execuon pleened decson varables Ancpaed fuure decsons varables