Production planning optimization in the wood remanufacturing mills using multi-stage stochastic programming

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roductio plaig oimizatio i the wood remaufacturig mills usig multi-stage stochastic programmig Rezva Rafiei 1, Luis Atoio De Sata-Eulalia 1, Mustapha Nourelfath 2 1 Faculté d admiistratio, Uiversité

1 roductio plaig oimizatio i the wood remaufacturig mills usig multi-stage stochastic programmig Rezva Rafiei 1, Luis Atoio De Sata-Eulalia 1, Mustapha Nourelfath 2 1 Faculté d admiistratio, Uiversité de Sherbrooke, Sherbrooke, Caada 2 Departmet of Mechaical Egieerig, Uiversité Laval, Quebec, Caada {rezva.rafiei@usherbrooke.ca, L.Sata-Eulalia@usherbrooke.ca, Mustapha.Nourelfath@gmc.ulaval.ca} Abstract. Wood remaufacturers grapple with several challegig characteristics. Oe of the most importat difficulties i this idustrial sector today cocers how to maitai promised service levels i highly dyamic market. To iclude the stochastic ature of demad i the productio plaig, multistage stochastic programmig models are proposed ad tested i this study. Our prelimiary results show that the solutio of a multi-stage model has the potetial to sigificatly improve the performace of traditioal plaig models at a relatively low cost. Keywords: multi-stage stochastic programmig, ucertai demad, co-productio, wood remaufacturig idustry 1 Itroductio This study is motivated by a real-scale case i the wood remaufacturig idustry, which trasforms pieces of lumber ito bed frame compoets. roductio plaig is quite a complex task i this cotext, sice from a give piece of lumber may types of products ca be while followig a diverget coproductio logic that caot be avoided. Moreover, a give compoet ca be usig differet alterative processes (cosumig differet types of lumber ad producig differet sets of co-products). These alterative processes are differetiated accordig to the productio yield, required iputs, set of coproducts, productio times, ad productio costs. Furthermore, the market is highly dyamic, havig a wide rage of products, with short order-cycle times, ad productio eeds to be plaed accordig to a maketo-order philosophy i a eviromet with ureliable demad. These characteristics cause complexity i productio plaig eviromets of such mills ad lead wood remaufacturers to dyamic plas. I this cotext, it is difficult to keep the promised service level at low costs. The literature reports that stochastic programmig approaches ca efficietly deal with ucertaity i productio plaig, especially whe dealig with highly variable demad. Most existig stochastic programmig models i productio plaig assume a two-stage model [1, 2]. I two-stage stochastic programmig, the decisio process takes place i two stages. I the first stage, actios tackle ucertaity ad i the secod stage the corrective actios are chose after the realizatio of the radom variables. Multistage stochastic programmig approaches have bee applied to deal with ucertai demad i several areas, such as capacity plaig [3, 4] ad lot sizig [5, 6], just to metio a few. Although stochastic programmig has also bee used i the area of plaig oimizatio i the forest products idustry [7, 8], it is still a ope research field i this idustrial sector. Accordig to recet research i the forest value chai by [9], represetig ucertaity through scearios for operatioal problems is essetial i forestry. The uderlyig ucertaity eeds to be well represeted i a maageable ad feasible model. The literature regardig productio plaig i the wood remaufacturig idustry is limited. A few researches propose lea maufacturig models [10], ad productio plaig models [11, 12, 13]. Eve though the iheret characteristics of softwood remaufacturig differetiate it from other plats i the forest products idustry ad make this sector eve more complex to maage, the applicatio of stochastic programmig has bee missed. Thus, the objectives of this paper are: 1) to propose a multi-stage stochastic programmig model for demad-drive productio plaig uder ucertai demad for the remaufacturig sector; 2) to perform some prelimiary tests i a real-scale idustrial cotext ad evaluate if this approach is really superior to the curret approaches ad estimate the possible performace gais; 3) to evaluate its computatioal costs i order to assess whether the ew approach ca be employed i a real idustrial eviromet. I order to do so, we suppose that demad ucertaity evolves durig the plaig horizo as discrete time stochastic - 1 -

2 process. As a result, the ucertaity is represeted through a sceario tree ad a objective fuctio is chose to represet the risk associated with the sequece of decisios to be made. The stochastic model aims to provide a productio pla that is techically possible to be implemeted while takig ito cosideratio the possible demad scearios ad deliverig a full recourse actio i the future. The remaider of the paper is as follows. I Sectio 2, we propose a multi-stage stochastic program for wood remaufacturig productio plaig with ucertai demad. Experimetatio ad computatioal results are preseted i Sectio 3. Sectio 4 cocludes the paper ad discusses possible future works. 2 A multi-stage stochastic program with ucertai demad Most practical decisio problems ivolve a sequece of decisios that respod to busiess coditios that evolve over time. Multi-stage stochastic programmig approach is proposed to address oimizatio models i multiple periods while the ucertaity is revealed, ad decisios are take at every stage. I the followig, we first describe the sceario tree, ad the provide mathematical formulatios for multi-stage stochastic programmig. 2.1 Modelig the ucertai demad We suppose that demad ucertaity evolves durig the plaig horizo as discrete time stochastic process. Sceario trees are commo structures to show how ucertaity ufolds over time as the possible sequeces of data are depicted. A commo sceario tree structure icludes regular sceario tree wherei all odes have the same umber of child odes. We cosider three possible market coditios for products demad; amely, High, Average, ad Low to geerate a regular triomial sceario tree where each ode has three child odes (braches) with High, Average, ad Low radom demads. Demad data is fitted by ormal distributio probability fuctio usig statistical tests. Thus, a three-poit discrete distributio ca be approximated by the Gaussia quadrature method [14]. To preset the radom demad as a sceario tree, the plaig horizo is divided ito stages. Each stage shows the step of time whe ew iformatio o the radom demad is available to decisio maker. Accordig to the case coditio, we also assume that the demads for all products are perfectly correlated ad have the same market coditio at each stage of the sceario tree. Moreover, it is supposed that the decisio maker is perfectly aware of the demad sceario at the start of each stage. As the availability of iformatio o the ucertai parameter at the start of each stage i the sceario tree is perfect, a full recourse actio is cosidered for this ucertai parameter i the multi-stage stochastic model. As we cluster the 54-period plaig horizo ito 4 stages, the first stage cosists of time period zero (curret time), the secod stage icludes periods 1-18, the third stage cosists of periods 19-36, ad fially the fourth stage icludes periods We suppose that if at stage i the market is boomig, the demad sceario for all products ca be expected to be High. If the market is steady, the demad sceario for all products ca be expected to be Average. I sluggish ad weak market coditios, the demad sceario for all products ca be expected to be Low. Such clusterig results i a sceario tree with 27 demad scearios ad 40 odes. 2.2 Multi-stage stochastic programmig model I this sectio we propose a multi-stage liear programmig model for productio plaig i wood remaufacturig mills. The origial determiistic model was proposed i [11]; however, we exted it through addig a differet oimizatio approach. The ew model (1A)-(9A) is i Appedix A Notatios The followig otatios are used for: Sets roducts p that ca be 2-2 -

3 roducts p that ca be T Set of periods i the plaig horizo, t T is a idex R Set of recipes r (A recipe is called a alterative process) ST Sceario tree, Nodes of the sceario tree, ST predec( ) t arameters c rt i redecessor of ode i the sceario tree Set of time periods correspodig to ode i the sceario tree (umber of periods i oe stage) roductio costs associated with usig recipe r i period t Ivetory holdig cost per uit of products bo Backorder cost per uit of product r c Capacity required for each recipe r per uit time i period t Available capacity for period t (umber of time uits) t ic p0 Ivetory of material s Supply of raw material ip p0 Ivetory of product pr The uits of raw material pr The quatity of product r pr Sellig price per uit of product i period t at the begiig of plaig horizo provided at the begiig of period t at the begiig of plaig horizo by recipe r by recipe r accordig to recipe r d ( ) Demad of product to be delivered by the ed of period t at ode of the sceario tree prob( ) robability of ode of the sceario tree Decisio variables X ( ) Cotrol variable - Number of times each recipe r should be ru i period t at ode of rt the sceario tree IC ( ) Ivetory size of raw material tree I ( ) State variable - Ivetory size of product the sceario tree BO ( ) State variable - Backorder size of product the sceario tree F ( ) Quatity of sold product p by the ed of period t at ode of the sceario by the ed of period t at ode of by the ed of period t at ode of by the ed of period t at ode of the sceario tree The multi-stage model A multi-stage stochastic model is formulated based o the sceario tree for the ucertai demad i this sectio. The cotrol variable of model (1)-(7) is productio pla X rt. The state variables of the pla are the ivetory quatity variable ( I ) ad the backorder quatity variable ( BO ). As we suppose that the decisio maker (plaer) is aware of which demad sceario is forced for the stage, the multi-stage stochastic model is a full recourse with regard to demad sceario. As a result, the decisio variables X rt, 3-3 -

4 the ivetory quatity variables I ad the backorder quatity variables BO for each ode of sceario tree are defied to preset the model. Maximize prob( ) ST t rr T T T prob( ) c X ( iv I ( ) bo BO ( )) rt rt ST t rr tt r R X ( ) c r rt t r F 1 pr rt r R pr IC ( ) IC ( ) s X ( ) (1) t t, ST (2) (3) p p, t t,, ST, I ( ) BO ( ) I ( ) BO ( ) X ( ) d ( ) 1 1 pr rt r R if t 1t predec( ) if t 1t (4) p p, t t, ST, if t 1t predec( ) if t 1t, T t F ( ) d ( ) p p, t t, ST (5) X ( ) 0, IC ( ) 0 rt p p, t t, ST, r R (6) I ( ) 0, F ( ) 0, BO ( ) 0 p p, t t, ST (7) The objective fuctio i (1) cosists i maximizig the expected profit, which is the differece betwee total reveue ad total costs. Costrait (2) guaratees that the total productio time does ot exceed the available time ad productio capacity. Costraits (3)-(4) esure flow equilibrium of raw materials ad fial products. Costrait (5) guaratees that sales do ot exceed customer demad. Costraits (6)-(7) eforce the o-egativity o the decisio variables. The decisio variables i the model (1)-(7) are idexed for each ode ad a set of periods i each stage deoted by t. To trasfer the ivetory ad backorder quatities from oe stage to aother oe, two variables are cosidered ( BO ad I ). These variables trasfer the edig values of ivetory ad backorder of the previous stage to the first period of relevat odes at the curret stage. I other words, whe the stage chages, the first period of the curret ode ( ) takes the iitial ivetory ad backorder quatities from the last period of the immediate predecessor ode ( ) i the previous stage. However, the other periods (exce the first period) at the curret ode ( ) i the give stage, the iitial ivetory ad backorder quatities derive from previous periods at the same ode ( ). Notice that machie recofiguratio (setup time) for switchig from oe recipe to aother is costat i this model. This assumio makes the model feasible to compute a exact result. However, cosiderig 4-4 -

5 biary decisio variables related to the setup time might cause ifeasibility to compute a exact result with a determiistic algorithm. Uder this coditio, a decompositio method i.e., Lagragia relaxatio or Beder decompositio would be adoed to solve the model. Aother poit is that alterative approaches to deal with this problem iclude Markov decisio processes ad Approximate Dyamic rogrammig. Such approximatio approaches may be useful for large problem istaces. 3 relimiary computatioal results A wood remaufacturig mill i Easter Caada is selected as a case study. The plaig horizo cosists of 54 periods. For radom demad data, we use ormal distributio accordig to statistical tests results. We also compare the solutios of three sceario trees for 4-stage, 3-stage ad 2-stage models to show the differeces betwee dyamic recourse models ad static resource oe. To describe the demad evolutio over the plaig horizo, four demad patters (D) are cosidered with the same mea but differet stadard deviatios (D1:5% mea, D2: 10% mea, D3: 20% mea, ad D4: 30% mea). Cosiderig oe plaig approach, demad patters (4 patters), sceario trees (3 trees), ad repetitive umerical experimets (30 rus), we geerate a total umber of 360 problems ad solve them. To solve the proposed model, CLEX ad OL 6.2 are used. All umerical experimets are coducted o a Itel Core i7-4700hq processor, 2.40GHz, 12 GB of RAM, ruig Microsoft Widows 8.1. I Table 1, we compare the solutio of a 4-stage stochastic programmig model to those of a 3-stage model, 2-stage model, ad determiistic equivalet model for the four demad patters with respect to the expected profit, the expected ivetory/backorder costs ad CU times. As ca be observed, the profits of the 2- stage models are less tha those of the multi-stage stochastic models ad by icreasig demad variability at each stage (from D1 to D4), the differeces betwee the profit of plas i multi-stage stochastic programmig models become bigger from the pla of two-stage stochastic models ad determiistic equivalet models. Moreover, by icreasig the umber of stages the ivetory/backorder costs also decrease. Alog with that, the table clearly idicate that i each stage, the icrease i demad variability (from D1 to D4) results i icrease the ivetory/backorder costs. The CU colum shows that the high quality of multi-stage stochastic model requires higher computatioal time compared to those of the twostage oes; however, it still has reasoable time. As a result, multi-stage models have better performace while demad variability icreases. Demad patter roductio plaig model Table 1: Results compariso of differet productio plaig models Expected profit Expected INV/BO costs CU time (secods) ossible gais (expected profit) ossible gais (Expected costs) D1 4-Stage S 111,581 48, Stage S 82,510 49, % 1% 2-Stage S 60,840 51, % 5% Determiistic 35,340 52, % 7% D2 4-Stage S 99,613 49, Stage S 76,448 51, % 4% 2-Stage S 52,826 52, % 6% Determiistic 32,660 52, % 7% D3 4-Stage S 91,075 51, Stage S 71,981 52, % 1% 2-Stage S 48,494 52, % 2% Determiistic 29,588 53, % 3% D4 4-Stage S 82,257 67, Stage S 64,843 73, % 8% 2-Stage S 42,879 80, % 16% Determiistic 27,985 84, % 21% 5-5 -

6 Figure 1 graphically presets some results of Table 1. Geerally, sice these prelimiary results idicate that the solutios of multi-stage stochastic models are better tha the determiistic models, this should come as o surprise. The expected profits of the 2-stage models are less tha those of the multi-stage stochastic models; however, their results are far superior to those of the determiistic models. I additio, it ca be oted that sigificat gais ca be obtaied i terms of expected profits (up to 216%) ad expected costs (up to 21%). Most importat, this is obtaied at a computatioal cost of less tha 14 miutes of CU time with a stadard machie, which makes this model fairly useful i a real-idustrial eviromet where decisio makers eed fast decisio support systems. We did ot regard more stages i the sceario trees because the differece betwee the 3-stage ad 4-stage models is ot very sigificat. Figure 1: Expected profits for differet plaig models 3.1 Quality of stochastic solutios To compare the value of the two-stage ad multi-stage models i differet demad patters, we evaluate the Value of Multi-stage Stochastic rogrammig (VMS) proposed by [15]. The VMS is defied as MS TS TS MS v v where v ad v are the oimum objective values of two-stage ad multi-stage models, respectively. Figure 2 shows the VMS i four differet demad patters. The observatio from Figure 2 is that the patter of demad distributio has a ifluece o the magitude of the VMS. I other words, as the variability of demad icreases at each stage, applyig a multi-stage stochastic model becomes more sigificat with respect to its maximizatio miimizatio objective. Value of multi-stage stochastic pogrammig (VMS) D1 D2 D3 D4 Demad patter Figure 2: Compariso of differet plaig models i terms of VMS 6-6 -

7 4 Coclusios We propose a multi-stage stochastic model to address the problem i a real-scale wood remaufacturig mill. The umerical results cofirm that the quality of the multi-stage stochastic model is better tha those of the determiistic equivalet ad two-stage models. Aother observatio is that, as the variability of demad icreases at each stage, the differece betwee the expected profit of the multi-stage stochastic model ad the determiistic ad two-stage stochastic models icreases ad this proves the sigificace of usig multi-stage uder icrease o demad variatios. As further extesios of this study, addig setup time costraits i the model ad solvig the model with a decompositio method ca be cosidered. For large problem istaces, Markov decisio processes ad Approximate Dyamic rogrammig could be cosidered as alterative approaches. Robust oimizatio ca be aother extesio for the productio plaig of the wood remaufacturig mills ivolvig challegig characteristics. Refereces 1. Bofill, A., Bagajewicz, M., Espua, A. ad uigjaer, L.: Risk maagemet i the schedulig of batch plats uder ucertai market demad. Idustrial ad Egieerig Chemistry Research, 43 (3), (2004). 2. Chu, Y., ad You, F.: Itegratio of Schedulig ad Dyamic Oimizatio of Batch rocesses uder Ucertaity: Two-Stage Stochastic rogrammig Approach ad Ehaced Geeralized Beders Decompositio Algorithm. Id. Eg. Chem. Res., 52 (47), pp (2013). 3. imetel, B.S., Mateus, G.R., Almeida, F.A.: Stochastic capacity plaig ad dyamic etwork desig. It. J. rod. Eco. 145 (1), (2013). 4. Stepha, H.A., Gschwid, T., Mier, S.: Maufacturig Capacity laig ad the Value of Multi-Stage Stochastic rogrammig uder Markovia Demad. Flexible Services ad Maufacturig Joural, 22(3 4), (2010). 5. Bradimarte,.: Multi-item capacitated lot-sizig with demad ucertaity. Iteratioal Joural of roductio Research, 44(15), (2006). 6. Huag, K., Ahmed, S.: The value of multi-stage stochastic programmig i capacity plaig uder ucertaity. Stochastic rogrammig E-rit Series, 15 (2008). 7. Kazemi, Z., M., Nourelfath, M., Ait-Kadi, D.: A multi-stage stochastic programmig approach for productio plaig with ucertaity i the quality of raw materials ad demad. Iteratioal Joural of roductio Research, 48(16), (2010). 8. Vila, D., Martel, A., Beauregard, R.: Takig market forces ito accout i the desig of productio-distributio etworks: A positio by aticipatio approach. Joural of Idustrial ad Maagemet Oimizatio, 3 (1), (2007). 9. Röqvist, M., D Amours,S., Weitraub, A., Jofre, A., Gu, E., Haight, R.G., Martell, D., Murray, A.T., Romero, C.: Operatios Research challeges i forestry: 33 ope problems. Aals of Operatios Research, 232(1), (2015). 10. irraglia, A., Saloi, D. ad Va Dyk, H.: Status of lea maufacturig implemetatio o secodary wood idustries icludig residetial, cabiet, millwork, ad pael markets. Bioresources, 4 (4), (2009). 11. Rafiei, R., Nourelfath, M., Gaudreault, J., Sata-Eulalia, L.A., Bouchard, M.: A periodic re-plaig approach for demad-drive wood remaufacturig idustry: a real -scale applicatio. Iteratioal Joural of roductio Research, 52 (14), (2014). 12. Rafiei, R., Nourelfath, M., Gaudreault, J., Sata-Eulalia, L.A., Bouchard, M.: Dyamic safety stock i a coproductio demad-drive wood remaufacturig mill: a case study. Iteratioal Joural of roductio Ecoomics, 165, (2015). 13. Rafiei, R., Nourelfath, M., Gaudreault, J., Sata-Eulalia, L.A., Bouchard, M.: Aalysis of ucotrollable supply effects o a co-productio demad-drive wood remaufacturig mill with alterative processes. INFOR: Iformatio Systems ad Operatioal Research, (2016) (I ress). 14. Millet, A.C. ad Rice, T.R.: Discrete approximatios of probability distributios. Maagemet Sciece, 29(3), (1983). 15. Huag, K., Ahmed, S.: The value of multi-stage stochastic programmig i capacity plaig uder ucertaity. Operatios Research, 57 (4), (2009)

8 Appedix: Determiistic liear programmig model T T T (1A) Maximize r F ( c X i I bo BO ) pr rt rt t rr t rr t X c r rt t t 1,..., r R IC ic s X 1 p0 p1 pr r1 r R IC IC s X 1 pr rt r R I BO ip X d p1 1 p0 pr r1 p1 r R I BO I BO X d 1 1 pr rt r R tt F d T (2A) p, t 1 (3A) p, t 2,..., T (4A), t 1 (5A) p, t 2,..., T (6A), t 1,..., T (7A) X, BO, I 0 rt rr,, t 1,..., T (8A) IC 0, t 1,..., T (9A) 8-8 -

Productivity depending risk minimization of production activities

Productivity depending risk minimization of production activities Productivity depedig risk miimizatio of productio activities GEORGETTE KANARACHOU, VRASIDAS LEOPOULOS Productio Egieerig Sectio Natioal Techical Uiversity of Athes, Polytechioupolis Zografou, 15780 Athes