Stability Contract in the Forest Products Supply Chain: Case Study for a Quebec Region

Transcription

1 Sably Conrac n he Fores Producs Supply Chan: Case Sudy for a Quebec Regon Berrand Hellon, Sophe d Amours, Nada Lehoux, Faben Mangone, Bernard Penz Laboraore G-SCOP n o av. Félx Valle, GRENOBLE, France ISSN : X July 2013 Se nerne : hp://

2

3 Sably Conrac n he Fores Producs Supply Chan: Case Sudy for a Quebec Regon Berrand Hellon, Sophe D Amours, Nada Lehoux, Faben Mangone and Bernard Penz G-SCOP, Unversé de Grenoble / Grenoble INP / UJF Grenoble 1 / CNRS 46 avenue Félx Valle, Grenoble Cedex 01, France. FORAC, CIRRELT, Unversé Laval Pavllon Adren-Poulo 1065, av. de la Médecne Québec (QC) Canada G1V 0A6 Absrac In hs paper an ndusral case ncludng a papermll and s hree supplers (sawmlls) s suded. Sawmlls produce lumber and chps from wood, and he paper mll needs hese chps o make paper. These sawmlls assgn a lower prory o he chps marke even hough he paper mll s her man cusomer. The focus of he research s herefore on securng he paper mll supply by creang benefcal conracs for boh sakeholders. Indusral consrans are aken no accoun, leadng o separae conrac desgns. Conracs are hen esed on varous nsances and compared o a cenralzed model ha opmzes he oal prof of he supply chan. Resuls show ha he decenralzed prof wh separae conracs s 99.3% he cenralzed prof, for a fxed demand varance. Dfference beween cenralzed and decenralzed prof slghly ncreases wh he varance, o reach 3% for a varance of 50%. 1 Inroducon Members of he Canadan fores ndusry agree ha he wood marke s currenly experencng s wors crss for a long me. The US s he man fnal cusomer for all Canadan wood producs. Thus he subprme crss n 2008 had a dsasrous mpac on Norh Amercan nvesmens. In parcular, he US lumber demand fell n he followng years, causng a lumber prce decrease on marke from 450$ (CAD) (near he years 2000) o 298$ (CAD) (2009) [9]. The crss s no he only reason explanng hs prce decrease. The recen compeon of emergen counres such as Chle, Brazl and Chna s also hghlghed. In hs work, we sudy an ndusral case concernng fores produc companes locaed n he Côe-Nord regon n Quebec, Canada. Our neres s focused on he las paper mll of hs regon, and s hree man supplers, whch are sawmlls. Sawmlls operaons and he paper makng process are lnked ogeher bu ypcally managed ndependenly, leadng o a prof wase. Furhermore, he paper mll s 1

4 he only purchaser for all he wood chp produced by nearby sawmlls. The shung down of hs paper mll would have unpredcable consequences. In order o ensure he collaboraon beween he man sakeholders, a workng group gaherng all he fores companes has been creaed [14] o rehnk abou her busness model. Every week he group mees o dscuss abou how o sasfy he paper mll demand. Each sawmll mus share he nformaons abou he volume of chps o send o he paper mll. These nformaons nclude he chps freshness and densy, whch are relaed o he ree speces. As he sawmlls have only one cusomer, hey have o adap her producon plannng unl he global paper mll needs are sasfed [14]. Based on hs conex, we propose he use of benefcal conracs for he sakeholders n order o secure he paper mll supples. In hs way, becomes possble o delver he volume of chps needed whle beer coordnang nework operaons. The relaonshp beween decson makers and supplers becomes one of he mos mporan ssues of he supply chan. To be profable, supply chan acves need o be beer coordnaed, necessang sronger neracons beween sakeholders [5]. However, examples of poor collaboraons ha have dsasrous consequences are gven by Thomas and Grffn [24]. The so-called Bullwhp effec s a good example of wha can happen whou any nformaon sharng. Dejonckheere e al. [8] suded from a mahemacal and sascal pon of vew. Lee e al. [19] poned ou he Bullwhp effec n he MIT beer game. De souza e al. [7] made a large expermen o assess he facors leadng o unsuccessful collaboraons, hghlghng he mporance of nformaon sharng. Collaboraon beween sakeholders n supply chan s besdes a huge subjec of neres. Huang e al. [18] presened a revew n whch hey conclude ha he number of papers abou collaboraon exploded beween 1996 and Camarnha-maos e al. [4] proposed dfferen classes of collaborave neworks reflecng ndusry s realy. Sahn and Robnson [22] suggesed a revew ncludng many ndusral references. Many auhors have also poned ou he fac ha collaboraons mus be guded n order o be profable for each supply chan member [20]. For example, Prahnsky and Benon [21] showed ha f auomove frms demonsrae ncreased wllngness o share nformaon, he suppler s commmen o he relaonshp also ncreases. The decson maker should have all he sakeholders s nformaon o beer opmze a supply chan. However hs s usually operaonally unrealsc [16]. Consequenly, when knowledge s combned, deermnng he key nformaons ha have o be shared as well as he prof ha may occur have been even more suded durng he pas years. Dyer and Chu [11] [12] suded nformaon sharng n he car ndusry, and conclude ha frms sgnal her own rusworhness hrough a wllngness o share nformaon. Daa and Chrsopher [6] showed he mporance of nformaon sharng beween supply chan members o beer face unceranes. Some papers suded 2

5 he consequences of forecasng error (Zhao and Xe [26]) and nformaon sharng (Yu e al. [25]) n real case sudes. Snce s an mporan sep n collaboraon mplemenaon, praconers and academcs heavly suded he conrac desgn. Elmaghraby [15] provded an overvew of he conrac compeon n he manufacurng supply chan whle Cachon [3] descrbed dfferen ypes of conrac as coordnaon mechansms for he supply chan. More recenly, rus has been suded as an mporan par of he suppler-realer relaonshp. Eckerd and Hll [13] modeled he relaon beween buyers and supplers, from an ehcal pon of vew. Blomqvs e al. [2] found from several case sudes ha every me he parnershp comes o end, a rus rule mus have been broken. Trus has also been defned and deeply suded by Doney and Cannon [10]. Researcher as Selnes [23] demonsraed ha enhanced communcaon conrbues sgnfcanly o cusomers sasfacon. The sze of he sakeholders may also have an mpac on collaboraon creaon and managemen, leadng o specfc leadershp and ownershp models [1]. The conex suded n hs research concerns he neracon beween hree sawmlls and one paper mll. The paper mll raw maeral s he chps suppled by he sawmlls. In parcular, when producng lumber from wood, sawmlls generae a he same me chps ha can be combned wh chemcals o produce pulp and hen paper. The paper mll requres a large amoun of he wood chps produced by he sawmlls, bu he laers usually focus on her core busness. As a resul, sawmlls make plannng decsons n order o ensure lumber qualy raher han chps qualy. Chps delvered are herefore varable n erms of volume and qualy, leadng o hgher paper producon coss. The purpose of hs paper s herefore o secure he paper mll s supply by creang benefcal conracs for boh sakeholders. The paper s organzed as follows: Secon 2 descrbes he problem and nroduces mahemacal noaons. Secon 3 exposes he formulaons used n he paper, for boh cenralzed and decenralzed cases. Secon 4 ncludes he cenralzed model and a cos analyss. The decenralzed model and he conrac desgn are suded n Secon 5. Conracs are also valdaed by several expermens on many mul-perods problems wh normally dsrbued demands. Manageral mplcaons and conclusons are gven n Secon 6. 2 Problem descrpon The case suded ncludes wo knds of sakeholders, a paper mll and hree sawmlls (see Fgure 1). All he prces, coss and varables n hs paper are based on hs ndusral conex and expressed n m 3. 3

6 Fores Trucks Sawmll 1 Sawmll 2 Trucks Lumber cusomer Paper cusomer Trucks Paper mll Sawmll 3 Fgure 1: Case sudy Sawmll: A each perod, for he sawmll, wood are delvered by rucks from he fores, a prce W P. Then he sawmll produces from wood boh lumber and chps. The coss for processng wood nclude harvesng, ransporaon and sawmllng coss. A each perod, for he sawmll, lumber are sold o a specfc cusomer a prce P B, for a maxmum demand of d,, whereas chps are sold o he paper mll a prce P C. The volume of lumber produced by he sawmll s consraned by a maxmum capacy K. The chps produced by he sawmll has a gven qualy Q. Ths qualy s mporan for he paper mll and akes value n he real un nerval [0, 1]. A he end of each perod, boh lumber and chps can be sored a he sawmll, nvolvng a cos HO and HS, respecvely. Paper mll: Chps are delvered by rucks from he sawmll o he paper mll. However, hese ransporaon coss are ncluded n he chps prce P C. A each perod, he paper mll uses hese chps o produce paper a a cos of T C, sellng o a specfc cusomer a prce P P, for a maxmum demand of dp. A he end of each perod, boh paper and chps can be sored a he paper mll. The paper mll cos reflecs chps purchasng and sorage, as well as paper producon and sorage coss (P C, HC, T C and HP, respecvely). Sawng capaces: Sawmll capaces are gven and known. In parcular, he capacy of S 1 and S 3 are he same, and hey are boh wce he capacy of S 2. As a resul, hey can be subsued n he model as K, 0.5K and K, as showed n Table 3. 4

7 Chps qualy As sad before, he chps produced by he sawmlls have a gven qualy, whch s assocaed wh he sawmll. Ths qualy akes value n he real un nerval [0, 1] and depends of he humdy, densy, and he ree speces. The values exposed here have been gven by he paper mll. Consran MCP: The sawmll process wll conduc o boh a ceran volume of lumber as man produc and a mnmum amoun of chps as co-produc. Ths mnmum s denoed as Mnmum Chps Proporon (MCP) n he res of hs paper. We denoe as exsng chps he chps ha have been produced by he MCP consran. The chps ha have been produced beyond he MCP consran are denoed as addonal chps. Consran MCQ: To be effecve, he paper producon requres a mnmum chps qualy. More precsely, he average qualy of chps mus be a leas a gven value. Ths value s denoed as Mnmum Chps Qualy (MCQ) n he res of hs paper. Varables defnon: For convenence, he same varables are used n all he models descrbed n hs paper. For each perod : Wood arrvng a he sawmll s noed W,. Two producs are hen produced, lumber and chps. A a sawmll, produced, sored and sold lumber are noed ZZ,, IO, and Z,, respecvely. Chps are noed XP,, aferwards hey can be sored (IS, ), and hen delvered o he paper mll (X, ). Chps can also be sored a he paper mll (IC, ), and hen used n paper producon (XT, ). A he paper mll, produced, sored and sold paper s noed Y Y,, IP, and Y,, respecvely. The general process s summarzed n Fgure 2. Sawmlls, paper mll and cusomer demands are defned n Table 1. Consdered coss are presened n Table 2. Noe ha n he models, all coss are consan. Indusral consrans and specal noaons are lsed n Table 3. Varables are dsplayed n Table 4. Table 1: Noaon for he mahemacal models P S 1, S 2, S 3 d, dp paper mll Sawmlls demand for lumber of he sawmll a he perod demand for paper of he paper mll a he perod 5

8 X 1, W 1, XP 1, IS 1, IC 1, S 1 ZZ 1, IO 1, Z 2, d 1, XT 1, X 2, W 2, S 2 XP 2, IS 2, IC 2, XT 2, Y Y Y P IP dp ZZ 2, IO 2, Z 1, d 2, XT 3, X 3, W 3, XP 3, IS 3, IC 3, S 3 ZZ 3, IO 3, Z 3, d 3, Fgure 2: Supply chan modellng Table 2: Coss W P Wood producon cos per m 3 of S 105$/m 3 P C Chp prce per m 3 purchased by P a S 56$/m 3 P B Lumber prce per m 3 sold o sasfy d, a S, a perod 142$/m 3 P P Paper prce per m 3 sold o sasfy dp a P, a perod 250$/m 3 HO Lumber sorage cos per m 3 a S 0.38$/m 3 HC Chp sorage cos per m 3 a P 0.13$/m 3 HS Chp sorage cos per m 3 a S 0.13$/m 3 HP Paper sorage cos per m 3 a P 0.58$/m 3 T C Paper producon cos per m 3 by P, usng chps 156$/m 3 Table 3: Indusral consrans Q 1 Chps qualy produced by he sawmll S 0.6 Q 2 Chps qualy produced by he sawmll S 0.75 Q 3 Chps qualy produced by he sawmll S 0.98 MCQ Chps qualy requred o ransform chps no paper 0.82 MCP Mnmum chps proporon produced by sawmll. 10% K 1 Sawng Capacy of he sawmll S 1 K K 2 Sawng Capacy of he sawmll S 2 0.5K K 3 Sawng Capacy of he sawmll S 3 K 6

9 Table 4: Varables W, XP, X, XT, Z, ZZ, Y Y Y IO, IS, IC, IP L, Lp Amoun of wood produced by S, a perod Amoun of chps produced by S, a perod Amoun of chps provded by S o P, a perod Amoun of chps uses o make paper by P, orgnally provded by S, a perod Amoun of lumber sold by S, a perod, n response o demand d, Amoun of lumber produced by S, a perod Amoun of paper sold by P, a perod, n response o demand dp Amoun of paper produced by P, a perod Amoun of lumber sored by S, a he end of he perod Amoun of chps sored by S, a he end of he perod Amoun of chps sored by P, orgnally provded by S, a he end of he perod Amoun of paper sored a P, a he end of he perod Amoun of chps los for he sawmll and he paper mll, a perod Amoun of chps los for he paper mll, a perod 3 Mahemacal formulaons In hs secon, he formulaons for boh he cenralzed and he decenralzed models are presened. The decenralzed formulaon s used o es conracs, whle he cenralzed formulaon s an upper bound for he prof of he whole supply chan. The laer plays he role of a reference for assessng conracs deermned va he decenralzed model. The cenralzed model opmzes he supply chan prof, modellng he sysem as a sngle decson maker. The decenralzed model assumes ha each sakeholders wans o opmze s own prof.e. each acors s a decson maker. The sum of all sakeholders prof s sad o be he prof of he decenralzed model. The prof generaed by he cenralzed and he decenralzed model can hen be compared. 3.1 Cenralzed lnear program C The cenralzed model C ams a maxmzng he whole supply chan prof.e., he sum of all acors prof, for all producs (wood, lumber and paper). Snce chps are necessarly produced durng he sawmllng process (MCP consran), sawmlls are allowed o hrow away a par of hem (e.g., f here s no demand for hs co-produc or f he qualy obaned s oo poor o be used n oher processes). Consequenly, chps flow equaons nclude some addonal varables : L, and Lp, whch are he chps los for he sawmll and he paper mll, respecvely. C can be defned as follows: 7

10 max T lumber sale paper sale lumber and chps sorage a sawmlls {}}{{}}{{}}{ T T T (Z, P B ) + (Y P P ) (IO, HO + IS, HS ) (W, W P ) } {{ } wood producon s.. ( T S ) (IC, HC ) + IP HP }{{} paper and chps sorage a he paper mll T (XT, T C) }{{} pulp and paper producon (1) Z, d, S, T (2a) W, K S, T (2b) W, = XP, + ZZ, S, T (2c) W, MCP XP, S, T (2d) XP, + IS, 1 + L, = IS, + X, S, T (2e) ZZ, + IO, 1 = IO, + Z, S, T (2f) Y dp T (3a) XT, = Y Y T (3b) (XT, Q ) Y Y MCQ T (3c) X, + IC, 1 + Lp = IC, + XT, S, T (3d) Y Y + IP 1 = IP + Y T (3e) X,, XP,, W,, IO,, IS,, Z,, ZZ, R S, T (4a) X,, IP,, IC,, XT,, Y,, Y Y,, R S, T (4b) The objecve funcon s defned by (1) and ams a maxmzng he sales of lumber and paper, whle mnmzng all producon, ransformaon and sorage coss. Consrans from (2a) o (2f) are he sawmll consrans. The sawmlls canno sell more lumber han he marke demand (2a). The sawmllng process s consraned by he sawmll capacy (2b). There s no wase nor produc creaon durng he sawmllng process (2c). Snce s mpossble o only ge lumber from rees (.e., dvergen process), a mnmum amoun of chps have o be produced (2d). Equaon (2e) concerns he sawmll chp flow consran, afer sawmllng. Equaon (2f) s he sawmll lumber flow consran, afer sawmllng. 8

11 Consrans from (3a) o (3e) reflec he paper mll consrans. The paper mll canno sell more paper han he marke demand (3a). One m 3 of chps s used for producng one m 3 of paper (3b). Paper qualy mus be a a mnmum gven qualy (3c). Equaon (3d) s he paper mll npu chp flow consran, before he producon process. Equaon (3e) s he paper mll oupu paper flow consran, afer he paper makng process. The range of values for sawmlls and paper mll varables are defned by consrans (4a) and (4b), respecvely. Ths lnear program assumes ha here s a sngle decson maker, amng a maxmzng he oal prof. The nex secon proposes decenralzed lnear programs, o opmze plannng decsons of each sakeholder. 3.2 Sawmll decenralzed lnear program Each sawmll res o maxmze s own prof generaed from boh lumber and chps sale. Therefore, from he sawmll pon of vew, he relevan consran o consder are consrans (2a) o (2f), plus he varables defnon consran (4a). In order o opmze s prof, a sngle sawmll has o solve he followng mahemacal problem: max }{{} wood producon lumber sale {}}{ T (Z, P B ) + chps sale {}}{ T X, P C T T (W, W P ) (IO, HO + IS, HS ) s.. (2a), (2b), (2c), (2d), (2e), (2f), (4a) } {{ } lumber and chps sorage (5) The objecve (5) res o maxmze he prof.e., lumber and chps sales, mnus wood producon and sorage coss. Here s consdered ha he paper mll buys all chps produced by he sawmlls. In a followng secon he relevance of such a model s dscussed. 3.3 paper mll decenralzed lnear program In a decenralzed supply chan, he paper mll focuses on mprovng s own prof. In hs case, he relevan consran o ake no accoun are consran (3a) o (3e), as well as he varables defnon consran (4b). In order o opmze s prof, he paper mll has o solve he followng mahemacal problem: 9

12 paper sale chps purchasng {}}{{}}{ T T max + (Y P P ) o, P C ( T S ) (IC, HC ) + IP HP } {{ } paper and chps sorage (3a), (3b), (3c), (3d), (3e), (4b) T (XT, T C) }{{} pulp and paper producon (6) The objecve (6) ams a maxmzng he prof.e., paper sales mnus coss relaed o chps purchase and sorage coss and paper producon. Here s consdered ha he chps ordered by he paper mll have been produced and are avalable for sale. In a followng secon he relevance of such a model s dscussed. 3.4 Dscusson on prof The profably of he dfferen producs s frs analysed, usng values gven n Table 2. For ha purpose, all he producs producon profs (lumber, chps and paper) are calculaed consderng a gven sakeholder. The consdered sakeholders are he supply chan (f he model s cenralzed), he sawmll and he paper mll. When lookng a he whole supply chan, an nerval for he paper prof s compued. The lower bound s se as he paper s fully produced from addonal chps. The upper bound s compued based on he hypohess ha he paper s fully produced from exsng chps. The paper prof generaed from addonal chps can be calculaed as follows : prof = paper sale wood processng pulp and paper producon = = -11$/m 3 Sawmllng when paper s he only produc s no profable for he supply chan. The paper prof generaed from exsng chps can be calculaed as follows : prof = paper sale pulp paper producon = = 94$/m 3 The MCP consran forces o produce a mnmum amoun of chps. Consderng ha he MCQ s sasfed, producng paper from exsng chps s profable. We can herefore esmae ha profably for producng paper from one m 3 s beween 94$ and 11$. Remark : consder ha MCQ s sasfed. To be profable for he supply chan, he paper mus be made wh a gven mnmum proporon of exsng chps (.e. produced by he MCP consran). Ths value can be calculaed. Consder x 10

13 he mnmum proporon of exsng chps. prof = exsng chps cos x + addonal chps cos (1 x) paper prof = 0 x (1 x) 94 x The exsng chps proporon s = 10.5%. Ths means ha for each 0.105m3 of exsng chps, s profable o produce an addonal 0.895m 3 of chps o make 1m 3 of paper. The producs profably for he sawmlls can be esmaed as follows : Prof for chps : prof = chps sale wood processng = = -49$/m 3 Wood only used for chps producon s no profable for he sawmlls. In a decenralzed model, whou any ncenve, a sawmll has no neres o produce more chps han he MCP consran. Prof for lumber, consderng he MCP consran prof = lumber sale (1 MCP ) wood processng = % 105 = 22.8$/m 3 Focusng on sawmllng o produce he maxmum amoun of lumber s profable, even whou he chps sales. If we hen look a he paper mll, he prof for paper can be esmaed as follows: prof = paper sale chps purchase paper producon = = 38$/m 3 Consderng ha he MCQ s sasfed, chps are profable for he paper mll. In a decenralzed model, paper mll has neres o produce and sell as much as paper as possble, whle he MCQ holds. Snce n hs sudy, cos are known and consan, he above properes hold n he whole paper. 4 Cenralzed model analyss Ths secon nvesgaes an unque decson maker usng he cenralzed model. The specal case consderng consan demand s also suded. 4.1 Consan demands All he demands are now supposed o be consan. Takng hs assumpon allows o undersand he properes ha hold n he cenralzed case. In hs conex, here s no neres o sore any produc. In fac, he problem C can be seen as a successon of dencally and separaed mono-perod problems. Snce coss are known and consan, he properes of he Secon 3.4 sll hold. The key decson for he supply chan s o know wha opmal quany of chps should be used for 11

14 paper, for each qualy. To calculae ha, a mahemacal program can be wren, focusng on chps varables. I s a mono-perod mahemacal program called P L Mahemacal program P L P L s found ou by smplfcaon of C. As sad prevously, here s no neres o sore any produc, whch leads o an elmnaon of all he nvenory varables. Snce he resulng mahemacal program s mono-perod, he flow equaons can be smplfed n varables equvalences. For nsance, he flow consran 3d whch s : X, + IC, 1 IC, + XT, S, T I can be urned no : X, XT, S In fac, s a mono-perod problem, and herefore he paper mll canno sore chps, has no neres o buy addonal quany, whch are desned o be hrown away. As a resul : X, = XT, All hese smplfcaons lead o keep he followng varables, XP, and X,. These varables represen he quany of chps a sawmll should produce, and he quany of chps ha should be delvered o he paper mll. Afer hese varable elmnaons, he focus s on he objecve funcon. Dscardng all he sorage coss, consderng mono-perod problem, he objecve funcon of C becomes : max lumber sale {}}{ (Z, P B ) + wood producon pulp paper producon paper sale {}}{{}}{{}}{ Y P P (W, W P ) (XT, T C) n whch Y, and XT, can be urned no X, (for he reason explaned above). Furhermore W, can be replaced by XP,, because W P s he cos for producng any addonal chps (beyond he MCP). Also, he erm S (Z, P B ) can be dscarded because lumber demand s no lnked o chps flows. Ths leads o anoher objecve funcon, whch s : max prof wh exsng chps {}}{ X, (P P T C) cos o produce addonal chps {}}{ XP, W P The consrans o consder are herefore he MCP, he MCQ, he sawmll capacy, and he paper demand. For a sngle perod, he mono-perod mahemacal program P L s defned below: 12

15 s.. max X, (P P T C) { } MCP XP, mn 1 MCP d,, MCP K (X, Q ) XP, W P (7) S (8a) (X, MCQ) (8b) XP, K S (8c) X, dp (8d) X, XP, S (8e) X,, XP, R + S (8f) The amoun of lumber produced by a sawmll s bounded eher by s capacy, or by he lumber demand. Consderng he MCQ, he mnmum amoun of chps produced by a sawmll s a fracon of eher s capacy or he lumber demand. Ths mnmum amoun s gven by he consran (8a). Even f hs consran s no lnear, can be ransformed no a lnear consran, addng an addonal varable. Thus hs mahemacal program can be wren as a lnear program. Consran (8b) (MCQ) verfes ha he qualy of he chps used s a leas he mnmum paper qualy. The chps produced are also bounded by he capacy of he sawmlls (Consran (8c)). The consran (8d) bounds he amoun of chps used o he paper demand. Fnally, consrans (8e) and (8f) ensure ha he varables are well defned and lnked. The mahemacal program P L gves he opmal amoun of chps produced and used for a sngle perod. Even n a mul-perod problem, n he case where demands are consan, he opmal amoun of chps produced and used can be provded eher by P L or C. 5 Decenralzed model wh conracs In hs secon, dfferen decenralzed scenaros are nvesgaed. Demands are no consan anymore and may vary. All he problems consdered are herefore mulperods. However, he mono-perod resuls above are used o desgn conracs a he end of hs secon. Ths decenralzed problem D ams a maxmzng each sakeholder s own prof. In hs purpose, a each perod, each sakeholder successvely opmzes s own plannng on a rollng horzon of H perods,.e. from o + H 1. 13

16 5.1 Imbalance of exreme decenralzed cases The frs wo scenaros show ha n a decenralzed case, he lack of regulaon or conrac leads o a prof loss for he supply chan. A new varable o, s nroduced o reflec he amoun of chps ordered by he paper mll a he sawmll, a perod. The wo scenaros are : A decenralzed supply chan where he paper mll s domnan,.e. he paper mll chooses he quany of chps o buy from he sawmlls (Algorhm 1). A decenralzed supply chan where he sawmlls are domnan,.e. he sawmlls choose he quany of chps o produce, and hen he paper mll chooses o buy an amoun lesser or equal han he avalable amoun of chps (Algorhm 2). Algorhm 1 A decenralzed algorhm wh domnan paper mll for each perod : do The paper mll opmzes s plannng on he rollng horzon H. Then sends H orders o, for he nex H perods for each sawmll, =1 o, K. for each sawmll : do The sawmll opmzes s own plannng regardng he amoun of chps X = o, o provde o he paper mll. end for end for Le he paper mll be he domnan (Algorhm 1). A each perod, he paper mll orders a ceran amoun of chps o, ha he sawmll has o sasfy (he capacy of he sawmll s respeced,.e. =1 o, K ). A unque chps producon s no profable for he sawmlls, so any chps quany ordered beyond he MCP consran can be a prof wase. The paper s profable for he paper mll, so dependng on he paper demand, a large quany of chps could be ordered. Moreover, snce he paper prof for he whole supply chan s also negave (see Secon 3.4), he supply chan prof overall decreases. If he sawmlls are domnan (Algorhm 2), hey produced a gven amoun of chps. Aferwards, he paper mll chooses wha quany o buy. As sad prevously, sawmlls mus respec he MCP consran, forcng hem o produce a mnmum quany of chps. Also, any quany of chps produced beyond he MCP s a prof wase for a sawmll. Thus he sawmlls wll only produced he mnmum quany of chps mandaory. Indeed here s no ncenve for he sawmlls o produce more han he MCP. 14

17 Algorhm 2 A decenralzed algorhm wh domnan sawmlls for each perod : do for each sawmll : do On a rollng horzon of H perods, he sawmll opmzes s plannng, producng an amoun of chps XP,. The paper mll buys a quany of chps X, o he paper mll, X, XP,. end for On a rollng horzon of H perods, he paper mll opmzes s plannng based on he X, provded by he sawmlls. end for Remnd ha he MCQ has o be respeced when paper s produced, nvolvng ha ceran quany of chps may be hrown away. However, would be profable for he whole supply chan o produce addonal hgh qualy chps o produce more paper, usng exsng chps (see Secon 3.4). Ths suaon does no allow addonal chps producon, leadng o a non-opmal global soluon. To conclude, boh hese exreme decenralzed suaons are no convenen. In he nex secon, a decenralzed scenaro wh conrac s nvesgaed. 5.2 Conrac desgn The nex scenaro proposed a more balanced and flexble decenralzed supply chan dynamc. The prevous mahemacal program P L defnes he opmal quany of chps o order, bu only n he case where demands are known and consan. Consderng a mul-perod problem wh varyng demands, conracs have o be fxed and consan on he whole horzon n order o gude he dfferen sakeholders n he decsons hey make. Hellon e al. [17] developed he sably conracs for securng a realer s supply whle ensurng a benefcal relaonshp for he sakeholders (realer and supplers). These sably conracs nally consraned he realer order n wo ways : by defnng mnmum and maxmum bounds on orders amouns (denoed L and U, respecvely); by defnng dynamc me wndows for each orders. Based on Hellon e al. [17] work, sably conracs are used for beer sasfyng paper mll needs. However, snce each perod corresponds o a possble delvery, he me dscrezaon does no allow o defne any dynamc me wndows. Consequenly, he sably conrac mus defne L and U for each sawmll. Then, for each perod, L o, U. L and U values are compued accordng o he sawmlls capacy and lumber demand. Sawmlls mus also provde a quany 15

18 of chps X, such as X, = o,. Algorhm 3 presens how he supply chan works n he mul-perod decenralzed conex. Algorhm 3 The decenralzed procedure D for each sawmll : do The paper mll and he sawmll agree on boh L and U bounds. end for for each perod : do The paper mll opmzes s plannng on he rollng horzon H, calculang he values o,, {... + H 1}, such as L o, U. The paper mll sends H orders o, a each sawmll. for each sawmll : do The sawmll opmzes s own plannng akng no accoun he paper mll s orders. The sawmll provdes an amoun of chps X, o he paper mll, such as X, = o,, for every H nex perods. end for end for Remark : Snce demands are no consan n hs secon, he average demand s used o calculae he conrac parameers. The average demand of paper s noed dp. Furhermore, he average demand for lumber a each sawmll s noed d. The oupu values of P L are he opmal quanes of chps o order so as o sasfy he demand for a sngle perod, a mul-perod problem when demands are consan. However, n a decenralzed mul-perod problem wh varyng demands, hs can be seen as a lower bound of chps for he paper mll a each perod, whou decreasng he prof of he supply chan. Then he oupu values X of P L can be assgned o L. Defnon : mn s he mnmum of chps produced by he sawmll, accordng o s lumber marke and s capacy. { } MCP mn = mn 1 MCP d,, MCP K Remnd ha f low qualy chps are avalable, s profable for he supply chan o produce addonal hgh qualy chps o make paper (see Secon 3.4). Thus he upper bound U should nclude all chps produced by he sawmll, accordng o he MCP consran. The lkely nsuffcen qualy of hs mx of chps mus be mproved by producng addonal hgh qualy chps, n order o reach he requred qualy (MCQ). Formally, U defnes for each sawmll he mnmum quany of chps sasfyng : 16

19 U mn S; S U Q S U MCQ. Ths can be calculaed by a smlar mahemacal program han P L. Snce he demands are no longer consan, would be profable o order more chps o laer sasfy a large paper demand. Furhermore, U values can be compued usng P L, bu he consran (8d) ensurng ha he amoun of chps provded does no exceed he paper demand for one perod, mus be dscarded. Ths new mahemacal program s called P U and s defned as follows : s.. (8a) (8b) (8c) (8e) (8f) max X, (P P T C) XP, W P The values for U are compued based on he oupu values X of hs mahemacal program. Remark : Snce he only dfference beween P U and P L s he consran (8d), f dp s large, he oupu from P U and P L are he same. Formally, f dp S U, he consran (8d) has no mpac on he formulaon and consequenly L = U, S. To summarze, for each sawmll, he values for L and U can be compued usng P L and P U, respecvely. Moreover, he soluons of he mahemacal programs P L and P U presen some properes : Propery 1. Opmal soluon of P U deermnes he value U o s mn, excep for a se of chps S (whch only nclude he chps wh he bes qualy). The soluon follows he form below. Sor all he sawmlls by Q n descendng order : {1;... ; k 1}, U = K U k = k 1 =1 U (MCQ Q ) Q k MCQ {k + 1;... ; S}, U = mn Remark : he proof explans how o fnd k. Proof. The proof s gven n he appendx. The propery 1 and s proof leads o a smple procedure R U (Algorhm 4) whch compues he opmal soluon of P U. For each sawmll he value of U can be assgned va he oupu of he procedure R U. 17

20 Algorhm 4 procedure R U compung he opmal soluon of P U for each from 1 o S do calculae L k as f = k k s he frs ha sasfes L k K k end for calculae L k. P L presens some properes as well. The MCQ s a consran n he formulaon, and snce all he chps have he same prce, an nfny of opmal soluons exss. However hese soluons do no have he same fnal qualy of chps, because MCQ s a consran, no an objecve. The purpose s o fnd, among he opmal soluons, he soluon wh he bes qualy of paper. The followng propery (propery 2) defnes he srucure of an opmal soluon for P L, he one wh he bes qualy of paper. Propery 2. Sor all sawmlls by Q n descendng order. The opmal soluon of P L whch maxmzes he average paper qualy s he followng form: {1;... ; k 1}, L = K L k = dp(mcq Qm)+(Qm Q )( k 1 =1 K + m 1 =k+1 mn ) Q k Q m {k + 1;... ; m 1}, L = mn L m = {dp j S j m L j} {m + 1;... ; S}, L = 0 Remark : he proof explans how o fnd k and m. Proof. The proof s gven n he appendx. Propery 2 and s proof leads o a smple procedure R L (Algorhm 5) whch compues he opmal soluon of P L. For each sawmll, he value of L can be assgned o he oupu of he procedure R L. For each sawmll, he conrac s creaed by assgnng a L and U he oupus of R L and R U, respecvely. 5.3 Expermens In hs secon, decenralzed procedure D wh sably conracs s compared o he cenralzed procedure C, usng nsances wh demands followng a normal dsrbuon. For each sawmll, sably conracs are based on parameers L and U compued by procedures R L and R U 18

21 Algorhm 5 procedure R L compung he soluon wh bes paper qualy among he opmal soluons of P L for each from 1 o S do calculae L k as f = k k s he frs ha sasfy L k K k end for for each from k + 1 o S do wh he value of L k, calculae he fnal paper qualy end for keep he m ha maxmze he paper qualy. calculae L k. calculae L m. Dfferen groups of nsances are generaed, and hey dffer by lumber and paper average demands. The capaces of he sawmlls, whch are K, 0.5K and K, are fxed o 90, 45 and 90, respecvely. The lumber demand can be much larger han he paper demand, and nversely. Also, he capacy of he sawmlls can be sgnfcan or no. Tha leads o 4 groups, as shown n Table 5, ha encompass he average demand for each sakeholder or produc. In each group, 20 dfferen nsances are generaed. Table 5: Insance parameers Group d 1, d 2, d 3, dp G G G G These nsances are esed n wo expermens. Frs, he prof of each sakeholders s evaluaed usng a gven fxed varance (20%) (Secon 5.3.1). The global prof s hen nvesgaed for a varance varyng, from 5% o 50%, wh a sep of 5 (secon 5.3.2) Comparson of each sakeholder s prof usng fxed varance For each group, he varance s fxed a 20% of he average demand. Resuls are dsplayed n Table 6. Resuls shows ha he prof obaned wh he proposed mehod are close o he ones generaed usng he cenralzed mehod (lower han 1%). Ceran values are even beer n he proposed mehod han wh he cenralzed approach, whch can be explaned by a dfferen prof dsrbuon. However, all values are close o her opmum. The overall prof of he decenralzed supply chan s 99.3% he 19

22 Table 6: Supply chan member s prof wh fxed varance Group G1 G2 G3 G4 Scenaro Average prof for he whole group Chps proporon P S 1 S 2 S 3 Toal S 1 S 2 S 3 C D C D C D C D cenralzed prof. In Table 6, poor local resuls (more han 1% lower han he cenralzed prof) are dsplayed n bold. Those resuls correspond o he cases where he paper demand s low for sawmlls havng he bes chps qualy. Decenralzed decson makng for groups G2 and G4 s very close o he opmal. Consderng ha hese wo groups face a large paper demand, conracs seems o be parcularly effecve n ha case. The las hree columns of he able dsplay he proporon of chps produced for each sawmll. When he value s greaer han 0.1, addonal chps have o be produced. The chps proporon of sawmll S 3 s 0.16 for G2 and G4, whch are he wo groups facng a large demand for paper Comparson of he whole supply chan prof when he varance s ncreased The nex expermen ams a comparng he prof of he whole supply chan generaed from he cenralzed and decenralzed model usng an ncreasng varance. In parcular, he varance sars a 5, up o 50, and ncreases by a sep of 5. For each varance and each group, 100 nsances are solved usng C and D. In oal, 800 nsances are solved. For each group of nsances, he prof dfference ncreases wh he varance. Concernng he decenralzed model, conracs are generaed based on he average demand for paper and lumber. However, he average demand are less and less ndcave as varance ncreases. For a varance of 50%, he prof dfference almos reach 3% for he wors groups G1 and G3. On he oher hand, consderng he group G2 and G4, even wh he larges varance, he prof dfference beween he cenralzed and he decenralzed sysem s a 1% and 0.5%, respecvely. In summary, he expermens based n sakeholder s prof show ha he supply chan prof s correcly dsrbued among he sakeholders because her profs 20

23 Fgure 3: Comparson of he whole supply chan prof, for each group of nsances 3,0% 2,5% 2,0% ,5% 1,0% 0,5% 0,0% are close beween he cenralzed and he decenralzed model. The expermen usng an ncreasng varance demonsraes ha sably conracs can opmze he supply chan prof, even wh large varances on he demands. However, o be effecve, hese conracs need a good assessmen of he curren average demand of he marke. Furhermore, each change n he average demand should nvolve a modfcaon of he conrac erms o ensure far dsrbuon of he supply chan prof. By usng sably conracs, fores produc companes of he Côe-Nord regon could herefore boh beer respond o he paper mll demand whle mprovng coordnaon beween supply chan operaons. 6 Concluson In hs paper, we nvesgae he case of a paper mll and s hree supplers, whch are sawmlls. All hese companes are locaed n he Côe-Nord regon, n Quebec, Canada. The purpose s on securng he paper mll supples by creang benefcal conracs for all sakeholders. The paper mll and he sawmlls are modeled, wh her coss, capaces, and chps qualy. Two ndusral consrans are consdered, reflecng he dvergen producon process and he qualy requremens for paper producon, leadng o a specfc problem. These consrans are hen used o desgn parcular conracs beween he paper mll and each sawmll. Two conexs are presened and compared : he cenralzed envronmen and he decenralzed decson makng process. Lumber and chps marke are consdered n he models o ake no accoun ndusry s realy. Algorhm used o creae he specfc conracs are provded, leadng o a praccal soluon. An expermenal sudy shows ha he prof of he decenralzed model, managed by he conracs, 21

24 s 99.3% he cenralzed prof. Furhermore, each ndvdual prof under conrac s close o he cenralzed opmal soluon. Anoher expermen shows he effcency of he sably conracs, even for larges demand varances. The expermens showed ha f a good assessmen of he average demand (for boh paper and lumber marke) s conduced, he sably conracs could be effecve for he whole supply chan, as well as for each sakeholder. The key of he problem s herefore o ge a good assessmen of he fuure average demand whle deermnng and negoang he conracs erms effcenly. In ha purpose, he Côe-Nord sakeholders should share he necessary nformaons o ge he bes possble demand forecas. Ths case sudy has served well o propose a conrac desgn mehodology. Generalzaon of he mehodology o dvergen process ndusres such as hose found n refnery or agrculural ndusry could be done. References [1] J. F. Audy, N. Lehoux, S. D Amours, and M. Rönnqvs. A framework for an effcen mplemenaon of logscs collaboraons. Inernaonal Transacons n Operaonal Research, 19(5): , [2] K. Blomqvs, P. Hurmelnna, and R. Seppänen. Playng he collaboraon game rgh-balancng rus and conracng. Technovaon, 25: , [3] G. P. Cachon. Supply chan coordnaon wh conracs. Handbooks n operaons research and managemen scence, 11: , [4] L. M. Camarnha-Maos, H. Afsarmanesh, N. Galeano, and A. Molna. Collaborave neworked organzaons - conceps and pracce n manufacurng enerprses. Compuers & Indusral Engneerng, 57(1):46 60, [5] H. K. Chan and F. T. S. Chan. A revew of coordnaon sudes n he conex of supply chan dynamcs. Inernaonal Journal of Producon Research, 48(10): , [6] P. P. Daa and M. G. Chrsopher. Informaon sharng and coordnaon mechansms for managng uncerany n supply chans: A smulaon sudy. Inernaonal Journal of Producon Research, 49(3): , [7] R. De Souza, Z. Song, and C. Lu. Supply chan dynamcs and opmzaon. Inegraed Manufacurng Sysems, 11: , [8] J. Dejonckheere. Measurng and avodng he bullwhp effec: a conrol heorec approach. European Journal of Operaonal Research, 147(3): ,

25 [9] B. Del Degan and M. Vncen. Impac des cous d opéraon sur la valeur de la redevance e les cours d approvsonnemen en bos. Mnsère des ressources naurelle du Québec, [10] P. M. Doney and J. P. Cannon. An examnaon of he naure of rus n buyer-seller relaonshps. Journal of Markeng, 61(2):35 51, [11] J. H. Dyer. Effecve nerfrm collaboraon: How frms mnmze ransacon coss and maxmze ransacon value. Sraegc Managemen Journal, 18(7): , [12] J. H. Dyer and W. Chu. The role of rusworhness n reducng ransacon coss and mprovng performance: Emprcal evdence from he Uned Saes, Japan, and Korea. Organzaon Scence, 14(1):57 68, [13] S. Eckerd and J. A. Hll. The buyer-suppler socal conrac: nformaon sharng as a deerren o unehcal behavors. Inernaonal Journal of Operaons and Producon Managemen, 32: , [14] M. Elleuch, N. Lehoux, L. Lebel, and S. Lemeux. Collaboraon enre les aceurs pour accrore la profablé : éude de cas dans l ndusre foresère. 9h Inernaonal Conference on Modelng, Opmzaon and Smulaon, Bordeaux, June 2012, 6-8. [15] W. J. Elmaghraby. Supply conrac compeon and sourcng polces. Manufacurng and Servce Operaons Managemen, 2(4): , [16] I. Gannoccaro and P. Ponrandolfo. Supply chan coordnaon by revenue sharng conracs. Inernaonal Journal of Producon Economcs, 89(2): , [17] B. Hellon, F. Mangone, and B. Penz. Sably conracs beween suppler and realer: a new lo szng model. Les Cahers Lebnz, (201), [18] G. Q. Huang, J. S. K. Lau, and K. L. Mak. The mpacs of sharng producon nformaon on supply chan dynamcs: a revew of he leraure. Inernaonal Journal Producon Research, 41: , [19] H. L. Lee, V. Padmanahan, and S. Whang. Informaon dsoron n a supply chan: he bullwhp effec. Managemen Scence, 43(4): , [20] N. Lehoux, S. D Amours, and A. Langevn. Iner-frm collaboraons and supply chan coordnaon: revew of key elemens and case sudy. Producon Plannng & Conrol, (1):1 15, [21] C. Prahnsk and W. C. Benon. Suppler evaluaons: communcaon sraeges o mprove suppler performance. Journal of Operaons Managemen, 22(1):39 62,

26 [22] F. Sahn and E. P. Robnson. Flow coordnaon and nformaon sharng n supply chans: revew, mplcaons, and drecons for fuure research. Decson Scences, 33: , [23] F. Selnes. Anecedens and consequences of rus and sasfacon n buyerseller relaonshps. European Journal of Markeng, 32(1): , [24] D. J. Thomas and P. M. Grffn. Coordnaed supply chan managemen. European Journal of Operaonal Research, 94:1 15, [25] Z. Yu, H. Yan, and Cheng T. C. E. Benefs of nformaon sharng wh supply chan parnershps. Indusral Managemen and Daa Sysems, 101: , [26] X. Zhao and J. Xe. Forecasng errors and he value of nformaon sharng n a supply chan. Inernaonal Journal of Producon Research, 40: , Appendx Proof. of propery 1. Ths proof s presened n hree pars. Frs, we demonsrae ha here always exss a soluon ha conans only one ndex k, such as Uk s a fraconal value,.e. Uk > mn k and Uk < K k. Then, we show how o calculae Uk. Fnally (wh he sawmlls sored by Q n descendng order), we prove ha hs formula appled on each such as < k leads o U > K. Prelmnary remark : Q k > MCQ, because here s no reason o produce chps below he MCQ. Say ha exss an opmal soluon such as here s wo fraconal value, say U a and U b, wh a < b. In hs case we can reallocae he value α = U b mn b from b o a. If U a + α K a, assgn a U a he value K a, reassgn he res n U b and hus here s only one fraconal value, whch s U b. Oherwse, f U a + α < K a, here s only one fraconal value, whch s U a, and he chps qualy of he new soluon s beer. The value U k s found below : U Q = U MCQ =1 =1 S S U Q + U k Q k = U MCQ + U k MCQ k k 24

27 U k = S k U (MCQ Q ) Q k MCQ If hs formula s used on a gven, such as < k, hs means ha U k = mn k, and U < K. In hs case, here s a chps qualy loss and he only way o sasfy he MCQ s o have a U > K, whch s a conradcon. Thus, all have o be red, from 1 o S, and he frs sasfyng U K means = k. Proof. of propery 2. Ths proof s presened n hree pars. Frs, we show ha he opmal soluon of P L whch maxmzes he average paper qualy adms a mos one m and one k, such as 0 < L m < mn m and mn k < L k < K k. Then, we demonsrae how o compue L m and L k. Fnally (wh he sawmlls sored by Q n descendng order), we show how o fnd m and k. Say ha exss an opmal soluon whch maxmzes he average paper qualy such as here are wo ndexes, say k and k, wh k < k. These ndexes boh sasfy mn k < L k < K k and mn k < L k < K k. By reallocang a small value from k o k, he average paper qualy ncreases, whch s a conradcon. Smlarly, here are wo oher ndexes, say m and m, wh m < m, 0 < L m < mn m and 0 < L m < mn m. By reallocang a small value from m o m, he average paper qualy ncreases, whch s a conradcon. The purpose of he value of L m s o mee he paper demand dp. In he case where m exss (see he hrd par of he proof): L m = dp j S j m L j The purpose of he value of U k s o maxmze he paper qualy : L Q = L MCQ (9) Noe ha [1;... ; k 1], L = K and [k + 1;... ; m 1], L = mn and L m = dp S m and [m + 1;... ; S], L = 0. k 1 (9) K Q + L k Q k + =1 k 1 = K MCQ + L k MCQ + =1 m 1 =k+1 m 1 =k+1 k 1 mn Q + Q m (dp K L k =1 k 1 mn MCQ + MCQ(dp K L k =1 m 1 =k+1 m 1 =k+1 mn ) mn ) 25

28 Afer developmen and smplfcaons: k 1 = K Q =1 m 1 =k+1 Afer rearrangemen and facorng: (9) L k Q k L k Q m k 1 mn Q dpq m + K Q m + =1 m 1 =k+1 mn Q m + dpmcq (9) L k = dp(mcq Q m) + (Q m Q )( k 1 =1 K + m 1 =k+1 mn ) Q k Q m The hrd par of he proof follows : If hs formula s used on a gven, such as < k, hs means ha U k = mn k, and L < K. In hs case, here s a chps qualy loss and he only way o sasfy he MCQ s o have a L > K, whch s a conradcon. Thus, all have o be red, from 1 o S, and he frs sasfyng L K means = k. The value for L k s dependen of he m chosen. The formula for L k guaranees ha consderng a gven m, L k maxmzes he qualy of paper. Thus for each possble m (beween k + 1 and S) L k has o be calculaed. The fnal m s he one ha maxmzes he qualy,.e. S =1 L Q. 26

29 Les cahers Lebnz on pour vocaon la dffuson des rappors de recherche, des sémnares ou des projes de publcaon sur des problèmes lés au mahémaques dscrèes. Responsables de la publcaon : Nada Brauner e András Sebő ISSN : X - c Laboraore G-SCOP