Distributed Operations Planning in the Lumber Supply Chain: Models and Coordination

Transcription

1 Disribued Oeraions Planning in he Lumber Suly Chain: Models and Coordinaion Jonahan Gaudreaul Pascal Forge Jean-Marc Frayre Alain Rousseau Sohie D Amours February 2009 CIRRELT

2 Disribued Oeraions Planning in he Lumber Suly Chain: Models and Coordinaion Jonahan Gaudreaul 1,2,*, Pascal Forge 1,3, Jean-Marc Frayre 1,2, Alain Rousseau 1, Sohie D Amours 1,3 1 Ineruniversiy Research Cenre on Enerrise Neworks, Logisics and Transoraion (CIRRELT) 2 Mahemaics and Indusrial Engineering Dearmen, École Polyechnique de Monréal, C.P. 6079, succursale Cenre-ville, Monréal, Canada H3C 3A7 3 Déaremen de génie mécanique, Universié Laval, Pavillon Adrien-Poulio, 1065, avenue de la médecine, Québec, Canada G1V 0A6 Absrac. Agen-based echnology rovides a naural aroach o model suly chain neworks. Each roducion uni, reresened by an agen, is resonsible or lanning is oeraions and uses communicaion o coordinae wih he ohers. In his aer, we sudy a sowood lumber suly chain made o hree lanning unis (sawing uni, drying uni and inishing uni). We deine he roblems and roose agen-seciic mahemaical models o lan and schedule oeraions. Then, in order o coordinae hese lans beween he hree agens, we roose dieren coordinaion mechanisms. Using hese develomens, we show how an agen-based simulaion ool can be used o inegrae lanning models and evaluae dieren coordinaion mechanisms. Keywords. Disribued oeraions lanning, coordinaion, oimizaion, agen, suly chain, lumber. Acknowledgemens. This work was unded by he FORAC Research Consorium and he Naural Sciences and Engineering Research Council o Canada (NSERC). The auhors would also like o acknowledge he imressive work o many FORAC research assisans who have worked on he develomen o he agen-based laorm. Resuls and views exressed in his ublicaion are he sole resonsibiliy o he auhors and do no necessarily relec hose o CIRRELT. Les résulas e oinions conenus dans cee ublicaion ne relèen as nécessairemen la osiion du CIRRELT e n'engagen as sa resonsabilié. * Corresonding auhor: Jonahan.Gaudreaul@cirrel.ca Déô légal Bibliohèque e Archives naionales du Québec, Bibliohèque e Archives Canada, 2009 Coyrigh Gaudreaul, Forge, Frayre, Rousseau, D Amours and CIRRELT, 2008

3 Disribued Oeraions Planning in he Lumber Suly Chain: Models and Coordinaion 1. INTRODUCTION Canadian lumber comanies are conroned wih he need o reengineer he way hey manage and lan heir suly chain oeraions. Suly chains are global neworks o organizaions where maerial and inormaion low in many direcions wihin and across organizaional boundaries hrough comlex business neworks o suliers, manuacurers and disribuors, o he exernal cusomers. These organizaions can be ar o he inernal suly chain, which consiss o members o he same comany, or ar o he exernal suly chain, which includes members o dieren comanies. They mus all cooerae o exchange maerials and inormaion o maximize cusomer saisacion a he lowes ossible cos. In ha regard, he lumber suly chain is similar o oher indusries: lumber maerial lows rom ores conracors, o sawing aciliies, o value-added mills (reerred o as secondary ransormaion), and hrough he many channels o disribuors and wholesalers o inally reach he markes. However, lumber oeraional lanning reresens a major challenge. Unlike he radiional manuacuring indusry which has a convergen roduc srucure (i.e., assembly), he lumber indusry needs o maser indusry-seciic oeraional rocesses. These are characerized by: (1) a divergen roduc srucure (i.e., rees are broken down ino many roducs), (2) he highly heerogeneous naure o is raw maerial and (3) radically dieren lanning roblems mus be solved by each roducion cener. Disribued lanning is an ineresing aroach or suly chain oeraional lanning since i enables he use o seciic oimizaion sraegies and inormaion available only locally. Unlike cenralized lanning aroaches, which generally canno ake ino accoun seciic oeraional deails, disribued lanning makes i ossible o creae deailed models o seciic lanning roblems. Building on Gaudreaul e al. (2008), his aer aims o roose lanning models or he lumber roducion unis and o comare dieren coordinaion mechanisms. In Secion 2, we resen a descriion o he sowood lumber roducion rocesses and lanning sraegy used by raciioners. Nex, in Secion 3, a lieraure review is rovided on lumber lanning, suly chain and coordinaion. Then, we roose in Secion 4 a disribued lanning sysem, including seciic lanning models or each roducion uni and coordinaion mechanisms o ensure coherence beween agens. Secion 5 resens he exerimenaion o he models and coordinaion mechanisms in an indusrial alicaion. We show how agenbased ools can be used o comare hem. Finally, Secion 6 concludes he aer. 2.1 Producion Unis 2. SOFTWOOD LUMBER PRODUCTION AND PLANNING This secion inroduces he hree dieren roducion unis involved in sowood lumber roducion: (1) he sawing aciliy, where logs are cu ino various sizes o rough ieces o lumber; (2) he drying aciliy, which reduces he lumber moisure conen and (3) he inishing aciliy, where lumber is laned (suraced), rimmed and sored. Figure 1 resens he dieren roducion unis. Figure 1. Producion unis and roducs in a lumber suly chain CIRRELT

4 Disribued Oeraions Planning in he Lumber Suly Chain: Models and Coordinaion This aer does no look ino he log suly roblem. In racice, ores is harvesed by enrereneurs resonsible or elling rees and crosscuing hem ino logs. We consider ha log suly o he sawing uni is known, which is he curren racice in he indusry Sawing Uni Logs oen remain or long eriods in a sawmill yard beore being rocessed. They are sored in huge los according o cerain hysical characerisics (secies, lengh, average diameer, ec.), each lo reresening a seciic class o logs. Logs are hen broken down ino various sizes o rough ieces o lumber. Dieren dimensions o lumber will be obained a he same ime rom a single log, which is called co-roducion. Mos o he ime, sawmills have access o daa regarding as roducion, allowing hem o orecas he execed quaniies o he dieren yes o lumber o be rom a seciic quaniy o logs o a given log class. This inormaion deines a roducion marix (Figure 2). Arcs show he quaniy o each ye o lumber execed when sawing a given volume rom a seciic log class. Figure 2. Examle o a roducion marix In mos sawmills, he roducion line can be se u in dieren modes; each seu is associaed o a seciic roducion marix. This gives he roducion manager some conrol over he roducion ouu mix. Cerain log classes may be incomaible wih cerain seus. For examle, in mos sawmills ir and sruce canno be rocessed in he same roducion shi; hey are associaed wih dieren seus. Thereore, he lanning decisions he roducion manager mus make are he ollowing: (1) decide how he lan will be se u or each roducion shi, and (2) decide which quaniies o each log class o consume a each roducion shi. Once logs are sawn, green ieces o lumber are assembled ino bundles o he same dimension (2 x3, 2 x4, ec.) and lengh (8-oo, 12-oo, 16-oo, ec.), and generally o he same secies (sruce, ir, ec.) in order o be dried Drying Uni Lumber drying is a ransormaion oeraion which aims a decreasing he lumber moisure conen in order o mee cusomer requiremens. These requiremens are usually seciied by indusry sandards, alhough some cusomers may require seciic levels o moisure conen. Sowood lumber drying is a raher comlex rocess o carry ou. I akes days and i is done in baches wihin large kiln dryers. Bundles o lumbers o dieren lengh can be dried in he same bach (e.g. 8-oo and 16-oo), bu lumbers mus be o he same dimension and secies (alhough here are some exceions). A bundle mus be assembled as a recangular rism illing he kiln dryer almos enirely. There are many consrains relaed o he sabiliy o his sacking. For hese reasons, each sawmill deines is own se o loading aerns ha can be used. Figure 3 shows wo examles o loading aerns. CIRRELT

5 Disribued Oeraions Planning in he Lumber Suly Chain: Models and Coordinaion Figure 3. Examles o small loading aerns (acual kilns and aerns conain a ew hundred bundles) Under cerain circumsances, secial secions o he wood yard may be used o erorm air drying. Air drying, which recedes kiln drying, may ake several weeks bu allows he reducion o he drying ime in he kiln. Air drying also lays a role in increasing he overall qualiy o he inished roduc (obained aer inishing). For a given bach o green lumber, here are dieren ossible alernaive oeraions ha can be used or air-drying and kiln-drying. Figure 4 resens an examle o our ossible alernaive combinaions o oeraions. For air drying, hese are mosly diereniaed according o heir duraions. For kiln oeraions, hey are dieren w.r.. air emeraure, humidiy arameers, and duraion. The lanning decisions or his roducion uni are he ollowing: (1) wha drying aciviies o erorm, (2) wha loading aern o use, and (3) when o erorm hem. Figure 4. Examle o drying rocesses available or a given bach o lumber Finishing Oeraions A he inishing aciliy, lumbers are irs laned (or suraced). They are hen sored according o heir grade (i.e. qualiy) wih resec o he residual moisure conen and hysical deecs. Lumber may be rimmed in order o roduce a shorer lumber o a higher grade and value. This rocess is usually oimized by hardware o roduce roducs wih he highes value, wih no consideraion or he acual cusomer demand. This causes he roducion o mulile roduc yes a he same ime (co-roducion) rom a single roduc ye in inu (divergence). I is imoran o noe ha he co-roducion canno be avoided rom a lanning oin o view; i is embedded wihin he ransormaion rocess. I is common o obain more han 20 dieren yes o roducs rom a single roduc. The execed roducs mix o obain rom a bach deends on he drying rocess used. Thereore, in he lanning models inroduced hereaer, we consider he ouu roduc associaed wih each o he drying rocesses (ah in Figure 4) as a dieren kind o inu or he inishing rocess. There is also a seu cos each ime he aciliy rocesses a dieren dimension (e.g. rom 2 x3 o 2 x6 ). Consequenly, mos sawmills allow such a seu only beween roducion shis as hey reer camaigns (a bach o roducs o he same dimension bu variable lengh) wih a duraion o more han one shi. To sum u, he decisions ha mus be aken in order o lan he inishing oeraions are he ollowing: (1) which camaign o realize (i.e. which lumber dimensions), (2) when and or how long and (3) or each camaign, which quaniies o each lengh o rocess. Figure 6 shows a simle examle o a roducion lan, including he camaigns (2 x3, 2 x6 and 2 x4 ) and he ime sen on each lengh. CIRRELT

6 Disribued Oeraions Planning in he Lumber Suly Chain: Models and Coordinaion Figure 5. Producion lan or a inishing line or six consecuive roducion shis 2.2 Lumber Producion Planning Due o he highly heerogeneous naure o he resource and he inheren comlexiy o orecasing roducion hroughu, he dominan hinking in he Norh American lumber indusry is o roduce he maximum volume wih he available resource. This can be ideniied as a ush roducion mode, where demand rom seciic cliens is no aken ino accoun. Producion is oriened owards large baches o ake advanage o economy o scale, resuling in large invenories, low lexibiliy and low agiliy. The roducion manager has as main objecive o eed he roducion line coninuously, in order o maximize he roducion rae and he hroughu. He also ries o orecas he quaniy o ouu roducs as recisely as ossible. This way, he can give clear inormaion o he sales dearmen abou wha will be available and when. While his lanning aroach has he advanage o maximizing he hroughu value, i does no ake clien needs ino accoun. On-hand invenory can be dieren rom wha inal cliens really wan, leading o missed sales ooruniies. Also, when unoreseen evens occur (such as kiln dryer breakdown, unlanned mainenance or ower ouage) roducion can be very dieren rom wha has been orecased. In hese cases, i becomes diicul o ulill romised deliveries. Using heir curren lanning ools (generally in-house sreadshee alicaions), i is diicul or roducion managers o adjus a roducion lan ha rooses a soluion o such roblems. Also, hey canno ry mulile alernaive lanning scenarios. This is why some Canadian lumber roducers are invesigaing he ossibiliy o evolving rom a ush roducion mode o a ull mode. In order o design such a ull roducion sysem, some characerisics o he roblem mus be aken ino accoun. For examle, i akes days o roduce a bach o green lumber ha is ready o be dried because o co-roducion a he sawing uni. The relaively large size o kiln dryers and he consrain o dry similar roducs ogeher also imoses some imoran roducion lead imes. Furhermore, due o co-roducion a he inishing uni, a single bach o green lumber o be dried and inished conribues o ulill many cusomer orders or dieren roduc yes. Also, because he volume o each cusomer order or a seciic roduc is usually larger han he amoun wih one single bach, many baches are usually needed o ulill one aricular need. These seciic issues have raised he need or a ighly inegraed rocess lanning and scheduling (Barak e al, 2002). 3.1 Lumber Producion Planning 3. LITERATURE REVIEW Few auhors have worked on he seciic roblem o sowood lumber roducion lanning. Among hem, Maness e al (1993) have roosed a mixed rogramming model ha simulaneously deermines he oimal bucking and sawing olicies based on demand and inal roduc rice (inegraion o sem bucking and log sawing). This model was laer modiied o handle several eriods (Maness e al, 2002). These works ocus on he ideniicaion o new cuing aerns/olicies. Taking a more global view o he suly chain, Singer e al (2007) recenly resened a model or oimizing lanning decisions in he sawmill indusry. They modeled a simliied inernal suly chain, including wo ransormaion sages and wo invenory sages. The objecive was o demonsrae how collaboraion can benei he arners, by ranserring imber and using he comeiive advanages o each. Oher ineresing sudies have been resened abou he inegraed suly chain lanning in he wood urniure indusry (Ouhimmou e al, 2005) and in he OSB anel indusry (Feng e al, 2008). 3.2 Suly Chains Planning Suly chain oeraions lanning is a comlex issue. Comanies usually deal wih his by imlemening and using inormaion and decision suor sysems, which address various lanning asks. Some comanies also ado jus-in-ime aroaches o conrol he ace o roducion and relenishmen. When organizaional unis are ar o he same comany, cenralized inormaion and lanning sysems are someimes used. Gahering inormaion in a cenralized managemen CIRRELT

7 Disribued Oeraions Planning in he Lumber Suly Chain: Models and Coordinaion sysem and redisribuing lans can ensure synchronizaion and oimizaion o lans. Decision suor sysems, such as Advanced Planning and Scheduling (APS) sysems are sohisicaed ses o decision suor alicaions using oeraional research (OR) echniques o ind soluions o comlex lanning roblems (Frayre, 2002). APS sysems are considered by many as sae o he ar manuacuring and suly chain lanning and scheduling racices. The reader is reerred o Sadler (2005) and Sadler e al (2005) or a horough descriion o APS. However, even in an inernal suly chain, he lanning roblem is comlex and diicul o handle. In ac, currenly available soware soluions generally do no rovide he necessary suor o nework organizaions and are clearly insuicien in lanning and coordinaing aciviies in heerogeneous environmens (Sadler, 2005). Planning, scheduling and radiional conrol mechanisms are insuicienly lexible o reac o raid changes in roducion modes and clien needs (Maurana e al, 1999). In oher words, radiional sysems have no been develoed o work in decenralized, dynamic and heerogeneous environmens, like suly chains. Collaboraion and coordinaion mechanisms are needed o ensure synchronizaion and consisency hroughou he suly chain. This has oened he way o an enirely new research domain, where researchers are ineresed in coordinaion and decision-making beween suly chain arners o oimize he suly chain erormance (Srader e al, 1998). 3.3 Coordinaion in Suly Chains An imoran managemen challenge in suly chains is he need or arners o erorm dieren lanning asks locally and o simulaneously manage heir inerdeendencies. Suly chain coordinaion has been sudied by several auhors. Bhanagar e al (1993) diereniae beween he iner-uncion coordinaion, reerred o as he general coordinaion roblem, and he muli-lan coordinaion o he same uncion. The general coordinaion roblem is usually subdivided ino hree classes o coordinaion roblems, namely, suly and roducion lanning, roducion and disribuion lanning, and invenory and disribuion lanning. Thomas and Griin (1996) resen a review o he lieraure concerned wih he coordinaion o hese uncions, while Bhanagar e al (1993) ocus on issues concerning he muli-lan coordinaion roblem. This work ocuses on he muli-lan coordinaion roblem and rooses hree oeraions lanning models linked by heir maerial low variables (i.e., delivery and order variables), and coordinaion mechanisms o make sure he resuling oeraions lans are coheren wih each oher. 4. DISTRIBUTED PLANNING FOR THE LUMBER SUPPLY CHAIN In his secion, we resen a disribued-aps sysem or he lumber suly chain. The local roblems associaed o each roducion uni (sawing, drying and inishing) are modeled and solved searaely. This seciic srucure is neiher unique nor oimal: i would be ossible o design a more cenralized srucure wih a single agen resonsible or he lanning o all roducion oeraions. However, due o he comlexiy and he seciiciies o hose roblems, i seems diicul o ake advanage o a single generic lanning algorihm, as i is likely ha only aggregaed inormaion could be handled wih such an agen. Also, disribued lanning allows relicaing he naural ineracions exising beween hese roducion unis. Firs, dieren Mixed Ineger Programming (MIP) models or he oeraions lanning and scheduling o he hree roducion ceners are roosed. These models can be used individually or collecively. Then, we roose dieren coordinaion mechanisms o ensure he coherence and easibiliy o he roducion lans. 4.1 Planning and Scheduling Models In order o lan individually or collecively he oeraions o he dieren roducion ceners, lanning and scheduling models have been develoed. The ollowing subsecions resen he oimizaion models develoed seciically or he sawing uni, he drying uni and he inishing uni Sawing Model The ollowing deines he roosed lanning and scheduling model or he sawing uni. Each rocessing aciviy is modeled as an associaion beween a quaniy o logs o consume, execed roducion, and machine usage. More han one rocessing aciviy can be used during he same roducion shi, bu wih some limiaions imosed by seu consrains (see Secion 2.1.1). Ses T he number o eriods in he lanning horizon. The index reers o he eriods = 1,,T; P se o roducs ; CIRRELT

8 Disribued Oeraions Planning in he Lumber Suly Chain: Models and Coordinaion P roducs ha can be (raw roducs). P roducs ha can be. P P ; P and P P. Please noe ha subses. M se o machines m. A se o rocessing aciviies a A available o he uni; F each F deines a mode in which he lan can be se u o oerae; a F Parameers P are non inersecing a seciies he modes F F such ha he lan can execue he rocessing aciviy a. An aciviy may be comaible wih many modes and many aciviies can be comaible wih he same mode. i,0 invenory o roduc P in sock a he beginning o he lanning horizon; i holding cos (er eriod) or roduc ; s, suly or roduc d, demand or roduc w v a P rovided a he beginning o eriod ; P he lan is execed o deliver by he end o eriod ; backorder cos ha occurs when one uni o roduc P is lae or one eriod; variable cos associaed o erorming a, including cos o raw maerial, i i alies; φ a, volume o raw maerial P each ime a is execued. A single rocessing aciviy a can consume dieren roduc yes a he same ime; ρ a, quaniy o roduc P each ime a is execued. A single rocessing aciviy can roduce many roduc yes a he same ime. δ caaciy o machine m M (number o ime unis) used each ime a is execued; am, c m, available caaciy o machine or eriod (number o ime unis). Variables QC, oal volume o roduc QP, oal volume o roduc P during eriod ; P during eriod ; I + volume o roduc, P in sock a he end o eriod ; I, cumulaed demand or roduc P no saisied a he end o eriod (ha is, backorder); I volume o roduc P ha would be in sock i he cumulaed demand was saisied. This variable can, ake negaive values. I is inroduced in order o simliy low consrains ormulaion. See consrain (1.7) or he relaion beween I,, I +, and I, ; Y, binary variable equals o 1 i he lan is se u in mode a eriod ; 0 oherwise; X a, usage o rocessing aciviy a during eriod (coninuous variable). Objecive uncion Wihin he indusrial alicaion considered in his aer i is imracical o consider demand as a hard consrain (lae deliveries are ineviable). Consequenly, we ry o minimize he cos o hese backorders. We also ake ino accoun invenory cos and variable roducion cos: T T T + Min w I, + i I, + v X a a, (1.1) = 1 P = 1 a A = 1 P Producion consrains The roduc consumion and roducion o he lan are relaed o he number o imes each rocessing aciviy is used: QC, = ( X a, φa, ) P, = 1,..., T (1.2) a A φ > a, 0 CIRRELT

9 Disribued Oeraions Planning in he Lumber Suly Chain: Models and Coordinaion QP, a A ρa, > 0 ( X a, ρa, ) = P, = 1,..., T (1.3) A each eriod o he lanning horizon, he sawing line can be se u in only one mode and hus can only use he rocessing aciviies comaible wih ha mode: Y, 1 = 1,..., T (1.4) F 0 Xa, Y, a A, = 1,..., T (1.5) a F where is a signiicanly large number The number o imes each rocessing aciviy is execued is consrained by he caaciy o each machine: ( X a, δ am, ) cm, m M, = 1,..., T (1.6) a A Flow consrains Consrain (1.7) and (1.8) ogeher wih he objecive uncion allow he comuaion o backorder level. O course, no backorder is allowed or raw roducs (1.9). + I = I I P, = 1,...,T (1.7) I,,, 0; I 0 P, = 1,...,T (1.8) +,, I = P, = 1,...,T (1.9), 0; Consrains (1.10) and (1.11) esablish he relaion beween invenory, suly and consumion o logs. I = i + s QC P (1.10),1,0,1,1 I = I + s QC P, = 2,...,T (1.11),, 1,, Consrains (1.12) and (1.13) esablish he relaion beween invenory, demand and roducion. I = i d P (1.12),1,0,1 I = I + QP d P, = 2,...,T (1.13),, 1, 1, Model imlemenaion and resoluion For real indusrial roblems, his model was easily solved using he MIP solver ILOG CPLEX 9.1. Near oimal soluions can be ound in a ew minues Drying Model Drying is a muli-sage rocess (see Secion 2.1.2). In he roosed model we chose no o model direcly he alernaive combinaion o aciviies (ahs in Figure 4). Insead, we modeled he individual aciviies. The connecion beween he aciviies (i.e. a valid recedence relaionshi beween wo aciviies) is enorced by he socks level consrains or inermediary roducs (an inermediary roduc needed by an aciviy, mus irs be by is redecessor). Figure 6 resens he main idea involved in his model. We have dieren aciviy yes a A. Each ye o aciviy can be execued on a comaible machine m Ma M, and has a seciied duraion δ a. The arameers φ a, and ρ a, seciy he consumion and roducion or roducs P. In his conex, building a lan can be seen as deciding which aciviies o erorm, when o do hem and which machines o use. A soluion can be reresened by a Gan char o aciviies (see Figure 6). Each ye o aciviy can be insered as many imes as needed in he lan. Insered aciviies have an imac on roduc invenories ( I, ) by increasing or decreasing i since hey roduce and consume dieren roducs. Demand rom he inishing uni ( d, ) also inluences roduc invenories. These kinds o models are reerred o as a imeable models or ime-line models (Barak, 1999a; 1999b; 2002). CIRRELT

10 Disribued Oeraions Planning in he Lumber Suly Chain: Models and Coordinaion Figure 6. Illusraion o he drying model Ses This drying model uses noaion similar o he sawing model. The ollowing ses have he same meaning in boh models: M. Because drying rocesses allow inermediary roducs eriods ( ) T, roducs ( ) P, P, P and machines ( ) which can be boh and, P and P may now inersec. Consequenly, in order o simliy he resenaion o low consrains, we will consider each roduc P as a resource ha can be,, sulied and shied. The ollowing deines oher ses used by he drying model: A se o all yes o drying aciviies a; consume consume A subse o aciviies which consume he roduc. A A ; roduce roduce A subse o aciviies which roduce he roduc. A A; A m subse conaining all yes o aciviies ha can be rocessed on machine m M. A A; m M = m M a A. M a subse o machines ha can carry ou aciviy a. { } a m Parameers The ollowing arameers have he same meaning as in he sawing model alhough hey are now deined or every roduc P : suly ( s, ), demand ( d, ), iniial invenory and coss ( i,,,0 i w ), maerial consumion and roducion o aciviies ( φa,, ρ a, ). However, we will suose ha d, is equal o zero or all roducs ha can be. The arameer v a also has he same meaning as in he sawing model. In addiion, we deine hese arameers: c m, 1, i machine m is available during eriod, 0 oherwise; δ a number o consecuive eriods needed o realize aciviy a. Variables The ollowing variables have he same meaning as in he sawing model alhough hey are now deined or every roduc P : QC,, QP,, I +,, I,, I,. In addiion, we deine his decision variable: X Binary decision variable aking value 1 i an aciviy o ye a sars on machine m a eriod, 0 oherwise. I is ( am, ), deined or each coule ( am, ) a A m. CIRRELT

11 Disribued Oeraions Planning in he Lumber Suly Chain: Models and Coordinaion Objecive uncion The objecive uncion is similar o he one in he sawing model: T T T + Min w I, + i I, + v X a ( am, ), P = 1 = 1 ( am, ) a A = 1 m (2.1) Producion consrains The consumion consrain (2.2) deines QC, as being he oal consumion o aciviies saring during eriod and consuming roduc. The sum is comued only or he coule o aciviies a and machines m or which m can carry ou a, and a consumes. Consumion or roducs ha are never is se o zero. QC, = X ( am, ), φa, P, = 1,...,T (2.2) ( am, ) a A m a Aconsume The roducion consrain (2.3) is he counerars o he revious consrain. I ses ha oal roducion is he sum o wha is by aciviies a ending during eriod (i.e. hose saring a eriod δ a + 1) and roducing roduc. QP, = X ( am, ), δa + 1 ρa, a A m ( am, ) a roduce P, = 1,...,T (2.3) A δ a The caaciy consrain (2.4) ses ha he number o aciviies running on a machine m a eriod mus be smaller han or equal o 1. For each ye o aciviy a, here is an insance running a eriod i one has sared in he inerval[ δ + 1,..., ]. a a Am τ= max -δa + 1, 1 X c ( am, ), τ m, m M, = 1,...,T (2.4) Flow consrains Flow consrains are similar o hose o he sawing model, wih he exceion ha some roducs can boh be and : + I = I I P, = 1,...,T (2.5) I,,, 0; I 0 P, = 1,...,T (2.6) +,, I = P, = 1,...,T (2.7), 0; I = i + s QC d P (2.8),1,0,1,1,1 I = I + s + QP QC d P, = 2,...,T (2.9),, 1,, 1,, Model imlemenaion and resoluion For real indusrial-size roblems we were no able o obain good easible soluions in reasonable ime. Consequenly, we roose a simle greedy heurisic o solve his roblem. Firs, a lis o he available drying rocesses mus be esablished a riori (i.e. each ah in Figure 4). Then, he lan is incremenally (saring wih an emy lan) by erorming he ollowing ses: 1. Comue he value o he objecive uncion or he curren lan. 2. Inser ino he lan he rocess ha will mos reduce he value o he objecive uncion. Each rocess is evaluaed by erorming hese ses: a. Comue he earlies dae or which one o he roducs by he rocess is backordered. b. By considering his as being he due dae, backward schedule he aciviies o he rocess using jus-in-ime lanning. c. Comue he imrovemen o he objecive uncion. 3. Go back o se 1 (he algorihm sos when i becomes imossible o inser a rocess reducing he value o he objecive uncion in se 2). CIRRELT

12 Disribued Oeraions Planning in he Lumber Suly Chain: Models and Coordinaion Finishing Model In racice, he hree ses o he inishing rocess (i.e., laning, soring and rimming) are erormed on a single roducion line. For lanning uroses, his line can be considered as a single machine, whose roducion rae is equal o ha o he machine which is he boleneck on he line. As saed in Secion 2.1.3, a inishing roducion lan is a sequence o camaigns (see Figure 5). Each one has a roduc amily associaed o i (e.g. 2 x4 ), which is relaed o a seciic seu or he lan. During he camaign, dieren yes o roducs corresonding o he amily can be rocessed (e.g. 2 x4-8, 2 x4-10 ). However, hey mus be rocessed in a seciic order. I he ollowing model, binary decision variables seciy how he lan is seu a each eriod. A seu cos ( ζ ) mus be accouned each ime here are wo consecuive eriods wih a dieren seu (ha is, each ime a new camaign begins). Oher decision variables in he model reresen he quaniies o each lengh (e.g. 8, 10 ) o rocess a each eriod (raher han he quaniy o rocess a each camaign as imosed by he roblem). To comensae or his relaxaion, a consrain saes ha all he consumion o he camaign akes lace a is beginning, and is roducion a he end. On he oher hand, we have o mainain wo dieren invenories: one in he yard (similar o he sawing and drying roblems) and one in he lan. Ses This inishing model uses noaion similar o revious models. The ollowing ses have he same meaning: T, P, P, and P. Similar o he sawing model, each F deines a mode in which he lan can be se u o oerae. Each mode corresonds o a roduc amily (e.g. 2 x4 ). In addiion, we deine he ollowing ses: P roducs ha can be when he lan is se u in mode. P P ; P inished roducs ha can be when he lan is se u in mode. P P ;, F P. FP se o coules ( ) ( ) ( ) Parameers The ollowing arameers have he same meaning as in he sawing model: s,, d,, i,0, i, and w. In addiion, we deine hese arameers: ρ volume o roduc P when one uni o roduc ' P is while he lan (, ' ), (, ) is se u in mode. Deined or coules (, ' ) FP ; δ ime needed o consume one uni o roduc (, ) coules (, ) FP ; v cos o rocessing one uni o roduc c caaciy o he lan (number o ime uni) or eriod ; ζ cos o erorming a seu change. P when he lan is se u in mode. Deined or P. Deined or coules (, ) r FP ; Variables The ollowing variables have he same meaning as in he sawing model. Variable Y, ideniy in which mode he lan is se u. The variables I +,, I, and I, corresonds o he invenory in he yard. Variable QC, corresonds o a quaniy ranserred in he lan a he beginning o a camaign. Variable QP, o a quaniy ranserred rom he lan o he yard a he end o he camaign. In addiion, we deine hese variables: BS 1, i he lan is se u or mode a eriod and i was no he case a eriod -1. 0, oherwise. Said, oherwise, his variable is equal o 1 i a camaign using mode begins a eriod ; BS 1, i a camaign (o any mode) begins a eriod. 0, oherwise; BE 1, i a camaign (o any mode) ends a eriod. 0, oherwise. UC volume o raw roduc o rocess a eriod while he lan is se u in mode. The roduc mus be (, ),, FP. already in he lan o be rocessed. Deined or coules ( ) CIRRELT

13 Disribued Oeraions Planning in he Lumber Suly Chain: Models and Coordinaion UP volume o roduc a eriod. The roduc remains in he lan and is no available o saisy, demand unil he end o he camaign. I will be released and hus be considered a roducion o he lan only when he camaign is over. Deined or P ; UI, volume o roduc in he lan a he end o eriod. Deined or P. Objecive uncion The objecive uncion is similar o he one in he sawing model alhough in his roblem he seu coss are aken ino accoun. Producion coss deend boh on how he lan is se u and he raw maerial. T T T T + Min w I, + i I, + ζ BS + v (, ' ) UC(, ' ), (3.1) = = = 1 = 1 ( ) P 1 P 1, ' FP Producion consrains Firs, he lan can only be se u in one mode a a ime (3.2) and a roduc can be rocessed () only i he lan is conigured in a comaible mode (3.3). Y, 1 = 1,...,T (3.2) F (, ) FP, = 1,...,T (3.3) 0 UC ( ) ( Y,,, ) where is a signiicanly large number. Consrains (3.4) o (3.6) sae a bach can sar or end only i he lan is se u in a comaible mode. Consrain (3.7) ensures ha a seciic bach will run unil i has ended. BS = Y F (3.4) BS BE,1,1 Y F, = 1,...,T (3.5),, Y F, = 1,...,T (3.6),, Y = Y BE + BS F, = 2,...,T (3.7),, 1, 1, Variables BS and BE resecively ake value 1 i and only i a bach is saring (3.8) or ending (3.9) a eriod. BS BE = F = F BS BE,, = 1,...,T (3.8) = 1,...,T (3.9) Raw roducs can ener he lan only a he beginning o a comaible camaign (3.10). Finished roducs are released only a he end o he camaign (3.11). No roducs can be le inside he lan a he end o he camaign (3.12). QC QP UI BS,, F P,, F P P, = 1,...,T (3.10) BE P, = 1,...,T (3.11) BE P, = 1,...,T (3.12), 1 ( ) The ollowing are he low conservaion consrains or he raw roducs inside he lan: UI,1 = QC,1 ( UC(, ),1) P (3.13) F (, ) FP, =, 1 +, ( (, ), ) F (, ) FP UI UI QC UC, 0 P, = 1,...,T (3.14) UI P, = 1,...,T (3.15) CIRRELT

14 Disribued Oeraions Planning in he Lumber Suly Chain: Models and Coordinaion These are he low conservaion consrains or he inished roducs inside he lan: UI = UP QP P (3.16),1,1,1 UI = UI + UP QP P, = 2,...,T (3.17),, 1,,, 0 UI P, = 1,...,T (3.18) These consrains esablish he relaion beween consumion and roducion inside he lan: UP, = UC(, ') ρ, (, ' ), P, = 1,...,T (3.19) (, ' ) FP Finally, he caaciy o he inishing line mus be reseced: ( UC( ) δ,, (, ) ) (, ) FP c = 1,...,T (3.20) Flow consrains (yard) The consrains or he roduc low in he yard are he same as in he sawing model. Thereore, we reuse consrains (1.7) o (1.13) in order o esablish he relaions beween i,0, I,, I +,, I,, s,1, d,, QC,1 and QP,. Model imlemenaion and resoluion As or he drying model, we were no able o obain good easible soluions in reasonable ime or real indusrial roblems. Again, we roosed a simle greedy heurisic o solve his roblem. Each ime he inishing line is available, we sar a camaign or he roduc amily F or which i is mos urgen o sar roducion (i.e. he amily wih he smalles irs eriod wih unsaisied orders minus execed roducion ime ). Because o seu coss, we wan his camaign o have he longes ossible duraion (i.e. saisying as many uure orders as ossible). However, he camaign mus be over beore he nex delivery dae. We also need o leave room or he roducion o oher amilies. Here is he deailed seudocode: 1. Le be he irs eriod or which he inishing line and some raw maerial are available. 2. For each mode F : a. Le be he irs eriod where an order or a roduc roducion lan. b. Le us suose a camaign ending a e 1 (wihou considering raw maerial availabiliy). Le needed roducion duraion. c. I here is no raw maerial 3. Sor he modes in increasing order o s. P is no saisied according o he curren = ha allows saisying he demand or all P available a s, increase 4. Considering he modes F or which we have some raw maerial wih he smalles s. Inser a camaign or his mode ino he lan: P a eriod s be he sar ime o his camaign according o he s unil some is ( e remains unchanged). P available a eriod, selec he one a. I begins a. b. I has he longes ossible duraion (according o raw maerial availabiliy) bu wihou resassing (nex delivery dae or curren amily) or he nex s in he vecor o se 3 (saring dae o a camaign or he nex mos urgen amily). c. The roducs/quaniies are esablished as ollows: i. Quaniies o P needed o saisy he nex unsaisied order or his amily. ii. I room is available, he quaniies or he nex order o his amily, ec. 5. Go back o se Coordinaion Mechanisms In a disribued lanning sysem, i is necessary o deloy a coordinaion mechanism beween he dieren roducion unis in order o inegrae he dieren lans so as o make sure hey are coheren wih each oher (in erms o maerial CIRRELT

15 Disribued Oeraions Planning in he Lumber Suly Chain: Models and Coordinaion availabiliy) bu also o guaranee a cerain level o collecive erormance. Such a sraegy deines which inormaion is ransmied rom one agen o he oher, when i is ransmied and wha he sequence used o roagae he inormaion is. The mos common class o coordinaion mechanisms (boh in lieraure and indusrial racice) can be described as hierarchical. In his aroach here is a sequence (naurally deined or seciied in a long-erm agreemen) seciying he order in which he arners mus lan heir oeraions. Schneeweiss (2003) describes many roblems using real indusrial alicaions o illusrae he challenge o disribued decision making. In his secion, we describe hree dieren coordinaion mechanisms, which are (1) usream lanning, (2) wo-hase lanning and (3) boleneck-irs lanning. Then, a modiicaion o he las wo mechanisms is roosed in order o ush alernaive roducs Usream Planning The mos common hierarchical aroach is reerred o as usream lanning (Bhanagar e al, 1993; Dudek e al, 2005). Agens lan heir oeraions one aer he oher, beginning wih he agen ha is closes o he cusomer (righ-hand side agen in Figure 7). Knowing demand rom he exernal cusomer, his agen lans is aciviies. This allows ideniying he suly need o he roducion uni (in he model, is suly arameer s, is ransormed ino a variable). This suly need is hen ranserred o is sulier and becomes demand or he laer ( d, ). All his resuoses ha each agen is always able o saisy any demand. O course, his assumion canno be me in all conexs. This is why i canno be used in our alicaion and is urhermore no imlemened. Figure 7. Usream lanning Two-hase Planning One varian o his aroach, aricularly relevan in a rocess indusry wih srong suly consrains, is o aly wo lanning hases: one usream and he oher downsream (see Figure 8). This aroach involves each agen wice. The agen irs makes a emorary lan o comue is suly needs and sends his inormaion o is sulier. In urn, he sulier ries o saisy his demand and resonds wih a suly lan ha does no necessarily mee all demand (e.g., some deliveries may be lanned o be lae or some roducs can be relaced by subsiues). When inormed o he suly graned by is sulier, he iniial agen has o revise is roducion lan in order o accoun or suly consrains. The ask low orms a loo wih wo hases: one usream where demand is enaively roagaed and he oher downsream where inal suly is roagaed. Figure 8. Two-hase lanning Boleneck-irs Planning A runcaed version o he wo-hase lanning aroach is illusraed in Figure 9. The exernal cusomer demand is ransmied direcly o he drying agen insead o going hrough he inishing agen. This modiicaion is insired by he roosiion rom Goldra e al (1992) o lan he roducion boleneck irs. In he lumber suly chain, he drying CIRRELT

16 Disribued Oeraions Planning in he Lumber Suly Chain: Models and Coordinaion roducion cener is oen he boleneck because o he invesmen needed o deloy kiln dryers. Kiln drying duraion can san rom 12 o 60 hours, immobilizing and having a remendous imac on he lexibiliy o he suly chain. Figure 9. Boleneck-irs lanning Because demand or inished roducs is ransmied o he drying agen, i mus have a deiniion o he inishing rocesses. This is done by adding he inishing aciviies in he grah o aciviies (Figure 4). By doing so, he drying agen has a simliied reresenaion o he inishing uni rocesses Pushing alernaive roducs In our conex, inernal demand (demand rom one uni o anoher) canno be considered as a hard consrain. When a roducion uni canno saisy all he demand o a arner, i migh be useul o roose alernaive roducs o his arner. Wih his oion in mind, a simle modiicaion o he revious mechanisms (wo-hase and boleneck-irs) is roosed. In he downsream hase, he uni uses available roducion caaciy o roduce alernaive roducs and ush hem o he nex uni (hoing i will be able o use hem). To do his, we simly run he lanning model o he agen anoher ime: we remove he already used caaciy and raw maerial, and rovide he model high demand or oher roducs. In he objecive uncion, hese roducs are weighed ( ) w using heir resecive execed marke value. 5. INDUSTRIAL APPLICATION This secion resens work ha was done wih an indusrial arner in order o validae he models (Secion 5.1). We hen show how an agen-based simulaion laorm was used in order o sudy he imac o dieren coordinaion mechanisms. The lanning models were embedded in an agen-based laorm (Secion 5.2) and simulaions were done in order o evaluae he coordinaion mechanisms according o he suly chain erormance (Secion 5.3). Finally, Secion 5.4 discusses relaed work. 5.1 Process Modeling and Indusrial Validaion In order o carry ou validaion o he roosed lanning models, we develoed a case sudy wih a lumber comany which includes roducion rocesses, roducs, orders, on-hand invenory, selling rices, resource coss, orecased suly, caaciy and work-in-rocess invenories. Processes were modeled in collaboraion wih he comany s roducion manager. Cusomer daa and on-hand invenory daa were exraced rom he arner s ERP sysem. Finally, he arner s sales eam rovided he daa abou roduc rices and resource coss. Each lanning model was assessed using real indusrial daa in an o-line lanning mode. Each week, he arner s roducion manager sen us udaed roducion daa, which we used o generae a roducion lan. The roducion manager gave eedback concerning he qualiy and easibiliy o he lans. This ineracive validaion hase allowed us o review and adjus he lanning arameers as well as he lanning models. This validaion rocess ook abou one year and many correcions were made o he models. 5.2 Agen-based Planning Plaorm In order o develo an Advanced Planning and Scheduling (APS) sysem or he lumber suly chain, he FORAC Research Consorium o Universié Laval (Québec, Canada) has develoed an agen-based lanning laorm. In brie, his laorm aims o address: (1) he abiliy o lan and coordinae oeraions hroughou he suly chain; and (2) he abiliy o analyze he dynamics and simulae dieren suly chain scenarios hrough simulaion. I allows he user, wheher a roducion manager or a researcher, o evaluae and comare dieren lanning models, coordinaion mechanisms or suly chain CIRRELT

17 Disribued Oeraions Planning in he Lumber Suly Chain: Models and Coordinaion coniguraions, according o user seciied erormance measures. Essenially, each lanning model is embedded wihin a soware agen ha has he caaciy o maniulae daa, solve is lanning model, and exchange inormaion wih oher agens according o coordinaion mechanisms. To allow seciicaion o coordinaion mechanisms, he laorm relies on he conces o conversaion roocols ha are commonly used in muli-agen sysems. The ineresed reader is reerred o Frayre e al (2007) or a more horough descriion o he design seciicaions and uncions o his laorm. 5.3 Simulaion We simulaed he coordinaion mechanism using he agen-based lanning laorm described reviously. A virual suly chain was buil, based on he indusrial case. The lanning agens (i.e., sawing, drying and inishing) were se u according o hese daa. The sudied case has a oal o 393 yes o roducion aciviies and 114 roducs, including 45 inished roducs available o he exernal cusomer. Orders were generaed using a robabilisic demand generaor (Lemieux e al, 2008), or a 50-day lanning horizon. This generaor creaed random demand, according o redeermined seings such as disribuion uncions, minimum/maximum limis and seasonaliy as well. However, we did no simulae he relenishmen o he sawing uni; we considered unlimied log suly. Table 1 resens he erormance o our coordinaion mechanisms (wo-hase lanning and boleneck-irs; wih or wihou he oion o ushing alernaive roducs). The erormance indicaors used o comare he mechanisms are: (a) he ercenage o lae deliveries, (b) he average ardiness or lae deliveries and (c) he average ardiness er ordered uni (ha is, a mulilied by b ). Resuls show a clear advanage o boleneck-irs lanning over wo-hase lanning. This is due o he advanage o lanning roducion o he boleneck (here he drying agen) irs, which ses he oal caaciy o he enire suly chain. Also, ushing alernaive roducs allowed an addiional reducion o backorders or boh coordinaion mechanisms. The hybrid aroach o boleneck-irs lanning wih he abiliy o ush alernaive roducs gave he bes resuls in his case. Two-hase lanning Saisy demand Saisy demand +ush al. roducs Boleneck-irs lanning Saisy demand Saisy demand +ush al. roducs Lae deliveries (%) % % % % Average ardiness (days) er lae uni volume delivered Average ardiness (days) er uni volume ordered Table 1. Coordinaion mechanisms evaluaion 5.4 Relaed Work The revious secion showed how such mehodology can be used o sudy he imacs o dieren coordinaion mechanisms on a suly chain. However, he resuls should no be used as generic rescriions or he lumber indusry as hey are seciic o his case sudy. The bes coordinaion mechanism o use deends on various acors; moving he boleneck or changing he mix o alernaive roducs can have a major imac on he erormance o he mechanisms. As an illusraion o his, Cid-Yanez e al (To aear) used he laorm and he lanning model inroduced here in order o sudy he imac o moving uward he oin u o where demand inormaion is ransmied (i.e., he decouling oin, called he ush-ull limi; beyond ha oin agen jus ush roducs). Nine suly chain coniguraions were invesigaed under six dieren environmenal condiions in order o carry ou a comlee mixed level design o 54 simulaion runs. Each o hese coniguraions was a combinaion o a decouling oin osiion and a caaciy level commied o conracs wih cusomers. Similarly, he environmenal condiions were designed as a combinaion o suly ye (i.e., log diameer disribuion) and lumber marke rices. CIRRELT

18 Disribued Oeraions Planning in he Lumber Suly Chain: Models and Coordinaion Figure 10. Sudied decouling oin osiions This sudy demonsraed he advanage in erms o logisic erormance o seing u he decouling oin osiion as ar uward as ossible. However, his also illusraed he cos o such a racice in erms o oal value creaion. In oher words, seing u he models resened in his aer so as o maximize cusomer demand saisacion insead o simly roducing he mos valuable roducs a each sage and ushing hem downward, has he eec o decreasing he overall value o oal roducion. Consequenly, making a commimen o regularly mee a cusomer s demand, is a service ha mus be aid hrough adding a remium o he marke rice. 6. CONCLUSION This aer described he roducion lanning roblems and models or hree roducion unis in he Norh American sowood lumber indusry. The models can be used individually or in a disribued suly chain conex. Dieren coordinaion mechanisms have been described. We showed how he lanning models can be inegraed in an agen-based lanning laorm in order o evaluae and comare he coordinaion mechanisms. Dieren imrovemens can be made o he models and coordinaion mechanisms. For examle, an agen could exloi inormaion abou arner s reerence regarding alernaive roducs. Also, i migh be ineresing o use simulaion ools o evaluae he robusness o he sysem in a conex where unexeced evens haen. Finally, oher coordinaion mechanisms are sudied in order o oer alernaives o he ones resened in his aer. Forge e al (2008) have roosed an inelligen lanning agen model, called muli-behavior agen, which can ada is coordinaion mechanisms according o seciic saes in is environmen. Also, Gaudreaul e al (To aear) have roosed a disribued lanning algorihm based on disribued consrain oimizaion, where agens are requesed o submi mulile local lans in order or he collecive o ind he bes arrangemens o lans. 7. REFERENCES 1. Barak, R. (1999a). Conceual Models or Combined Planning and Scheduling. Proceedings o he CP99 Worksho on Large Scale Combinaorial Oimizaion and Consrains. Alexandria, USA, Barak, R. (1999b). On he Boundary o Planning and Scheduling: A Sudy. Proceedings o he Eigheenh Worksho o he UK Planning and Scheduling Secial Ineres Grou (PlanSIG), Mancheser, UK, Barak, R. (2002). Viso ShoFloor: on he edge o lanning and scheduling. Proceedings o he Princiles and Pracice o Consrain Programming Inernaional Conerence, LNCS#2470, Ihaca, USA, Bhanagar, R., Chandra, P. and Goyal, S.K. (1993). Models or muli-lan coordinaion. Euroean Journal o Oeraional Research, 67(2): Cid-Yanez, F., Frayre J.M., Leger, F. and Rousseau, A. Agen-Based Simulaion and Analysis o Demand-driven CIRRELT

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