A method for a robust optimization of joint product and supply chain design

Transcription

1 DOI /s A method for a robust optmzaton of jont product and supply chan desgn Bertrand Baud-Lavgne Samuel Bassetto Bruno Agard Receved: 10 September 2013 / Accepted: 21 March 2014 Sprnger Scence+Busness Meda New York 2014 Abstract Ths paper proposes a method for fndng a robust soluton to the problem of jont product famly and supply chan desgn. Optmzng product desgn and the supply chan network at the same tme brngs substantal benefts. However, ths approach nvolves decsons that can generate uncertantes n the long term. The challenge s to come up wth a method that can adapt to most possble envronments wthout strayng too far from the optmal soluton. Our approach s based on the generaton of scenaros that correspond to combnatons of uncertan parameters wthn the model. The performance of desgns resultng from these scenaro optmzatons are compared to the performance of each of the other desgn scenaros, based on ther probablty of occurrence. The proposed methodology wll allow practtoners to choose a sutable desgn, from the most proftable to the most relable. Keywords Robust desgn Supply chan Product famly Mxed lnear programmng Introducton Most companes functon n a complex and unstable envronment, whch makes accurate forecastng dffcult. At the same tme, stayng compettve requres keepng producton B. Baud-Lavgne S. Bassetto B. Agard (B) CIRRELT, Département de Mathématques et Géne Industrel, École Polytechnque de Montréal, C.P. 6079, succ. Centre-Vlle, Montreal, Québec H3C 3A7, Canada e-mal: [email protected] B. Baud-Lavgne e-mal: [email protected] S. Bassetto e-mal: [email protected] costs to a mnmum. Optmzaton offers a soluton to ths dlemma, however t calls for long term decson makng based on market forecasts. How can practtoners deal wth fluctuatons n parameters, such as demand, the prce of raw materals, and transportaton costs, n ther effort to optmze a supply network? It had once been thought that ncreasng the level of commonalty n the platform product would provde the necessary leverage to reduce producton and dstrbuton costs on a large product famly (Jao et al. 2007; We et al. 2007). Ths paper extends the jont product and supply chan desgn model proposed by Baud-Lavgne et al. (2011) wth a robust desgn methodology. We frst present the model, and then explan and dscuss the concept of robustness n ths context. In Robust methodology for jont product and supply chan desgn secton, we propose our robust desgn methodology, and apply t n Experments to llustrate the methodology secton n an academc case study. The state of the art Smultaneous product and supply chan desgn After decades of research on supply network desgn, practtoners have ntegrated mathematcal methods nto supply chan optmzaton (Shapro 2001), and the emphass must now move to cost reducton and the fulfllment of customer needs. The benefts of smultaneous product and supply chan desgn has been hghlghted by Baud-Lavgne et al. (2012). However, ths approach, n whch decsons on standardzaton are made for a product famly through product optmzaton followed by supply chan optmzaton, s now consdered to be sub optmal. Several modelng hypotheses have been studed recently as solutons to the problem of jont

2 product and supply chan desgn. From the perspectve of customer needs and how they are met by functonaltes, t s possble to model the problem wth a generc bll of materals (Lamothe et al. 2006; Zhang et al. 2010; Shahzad and Hadj- Hamou 2013). In ths approach, the composton of the entre product famly s determned at the same tme as the supply chan network s desgned, n order to lower procurement, producton, and dstrbuton costs whle maxmzng profts. Modular product desgn has been studed by Schulze and L (2009) and Hadj Khalaf et al. (2010), who were lookng at fndng the rght modules to allow specfc fnal products to be manufactured at lower cost. The ntegraton of complex blls of materals allows manufacturers to adapt to most ndustral ssues, but the models produced are dffcult to solve. Scenaro modelng, proposed by ElMaraghy and Mahmoud (2009), has made solvng them easer, thanks to a lmted number of decson varables and smple constrants. The dffculty s to temze each combnaton of blls of materals, whch can be consdered as a combnatoral problem for complex products. Chen (2010) consders blls of materals wth an unlmted number of levels and wth substtuton possbltes. Wth ths approach, modelng s flexble, as there are many decson varables. However, such models are also dffcult to solve. Robustness assessment n supply chan desgn In ths paper, we focus on classcal robustness, whch enables us to evaluate some behavors of a system that are characterzed by uncertanty. Accordng to Klb et al. (2010), there are three types of uncertanty: randomness, chance, and deep uncertanty. Wth randomness, the parameters are not known precsely, but rather as a range wth some probabltes; wth chance, there s a possblty that one or more unexpected events wll occur; wth deep uncertanty, t s mpossble to determne what s possble. Robustness s a broad term n optmzaton, whch can be appled to the mathematcal model, the algorthm, or the soluton. A model s robust when t can adapt to several confguratons; a robust algorthm s amed at fndng a good soluton (or an optmal one) n a mnmum amount of tme and n most cases. There are three types of soluton robustness: classcal robustness, responsveness, and reslence. Classcal robustness apples to a soluton that gves good results wthout dependng on the actual envronment. For example, Pan and Nag (2010) propose a supply chan desgn whch deals wth demand uncertanty and opportuntes. Responsveness measures the capacty of the supply chan to react approprately when there s some randomness n the process (Parvaresh et al. 2012; Meepetchdee and Shah 2007). Reslence evaluates the ablty of the organzaton to return to normal operaton followng a major breakdown (Chlderhouse and Towll 2004). We suggest two ways of dealng wth randomness n supply chan desgn optmzaton, usng stochastc models or determnstc models. Wth the frst opton, the problem s modeled wth stochastc parameters and solved wth analytcal methods see the revew of ths opton provded by Pedro et al. (2009). For example, an optmzaton method has been presented by Mohammad Bdhand and Mohd Yusuff (2011), whch consders parameters based on statstcal rules, and Krstanto et al. (2013) propose a method for modular product desgn takng nto account the uncertanty nherent n future evoluton. The second opton s to model the problem wth determnstc parameters and defne several possble scenaros. Shmzu et al. (2011), for example, propose a mult-objectve algorthm whch consders a number of scenaros at each step, whle Chan et al. (2006) experment wth an algorthm that allows order due dates to be tracked. In the next secton, we present a methodology for smultaneously desgnng a famly of products and ts supply chan wth the randomness condton usng the concept of the scenaro. Robust methodology for jont product and supply chan desgn The desgn Ths paper extends a model proposed by Baud-Lavgne et al. (2011). It enhances the model of Chen (2010) byusng fewer decson varables, and extends the concept of standardzaton used n Baud-Lavgne et al. (2012) by means of substtuton. Substtuton ncludes product standardzaton (.e. upgradng one part by replacng t wth another part wth more functonalty or of better qualty), externalzaton (.e. buyng the part drectly from a subcontractor), and changng the operaton sequence (.e. proposng another order of operatons n the sequence, whch nvolves dfferent sub assembles). In ths model, product substtuton s consdered through product transformaton,.e. exchangng one part for an equvalent one. The man hypothess underlyng ths model s that the demand s known and the company has to meet t. Demand can be an uncertan parameter n ths new model. The problem s modeled as a mxed lnear program wth flow and cost constrants. Substtuton possbltes are ncluded at each level of the bll of materals (BOM), and apples to components, sub assembles, and products. The product famly and the supply chan are optmzed smultaneously, based on a cost mnmzaton target. Frst, we defne the followng sets and ndces: P: products; p, q P R P: raw materals or suppled components M P: manufactured products/sub-assembles

3 F P: fnshed products P p P: products, sub-assembles, or components that can substtute for p N : network nodes;, j N S N : supplers U N : producton centers D N : dstrbuton centers C N : customers T : technologes; t T. A technology s a generc method of producton that s needed to manufacture a product. A technology s characterzed by certan capacty optons. T p T : technologes needed by product p, p M F O: capacty optons; o O O t O: capacty optons for technology t General parameters: g pq : quantty of q n p. q can be a component or a subassembly. g represents the BOM, p M F, q R M, d p : demand for product p by customer, p F, C l pt : processng tme requred by product p on technology t, p M F, t T The decson varables are as follows. A p s the quantty of p manufactured at producton center. B p s a bnary varable that s equal to one f producton center s used to manufacture product p, zero otherwse. S pq s the quantty of p that substtutes q n producton center. j defnes the flow of p between to j. T p j and L j are bnary varables. The frst one s equal to one when the flow of p from to j s strctly postve, and the second one s equal to one when at least one p uses the arc from to j, zero otherwse. O l s the quantty of capacty opton l nstalled at producton center. Z s a bnary varable that s equal to 1 f the node s used. Each varable s assocated wth ts proper cost. For the bnary varables, that cost s fxed, and only pad f the Table 1 Decson varables (DV) and ther assocated costs DV Doman Cost Quantty of p produced at A p R α p Producton of p at B p {0, 1} β p Quantty of p that s substtuted for q at S pq R σ pq Flow of p between and j j R φ p j Use of flow of p between and j T p j {0, 1} τ p j Use of axs between and j L j {0, 1} λ j Number of optons o at O l N ω l Use of node Z {0, 1} ζ varable s set to 1. For contnuous varables, t s a unt cost. The decson varables and the costs are presented n Table 1. The mathematcal model s as follows. The objectve functon (1) mnmzes procurement, producton and transportaton fxed and varable costs. Mn (A p α p N p P + S qp N p P q P p + N j N \{} p P + L j λ j N j N \{} + O o ωo N o O + N + B p β p ) σ qp ( ) j φ p j + T p j τ p j Z ζ (1) Constrants (2) to(6) are flow constrants. The sources are the component flows from the supplers to the producton centers, the snks are the fnal product flows to customers. Constrant (2) consders the flow of each product assembly manufactured at each producton center. A p + j + = j U\{} j U\{} j + S qp q P p q M F g qp A q + S pq q/p P q U, p M (2) Constrant (3) consders the flow of each component at each producton center. j + j (S U)\{} = j U\{} j + S qp q P p q M F g qp A q + S pq q/p P q U, p R (3) Constrant (4) consders the flow of each component from each suppler. A p = j U j S, p R (4) Constrant (5) consders the flow of each fnal product to each dstrbuton center. j U D\{} j = j D C\{} j D, p F (5)

4 Constrant (6) consders the flow of each fnal product from each producton center. A p + j = j U j D C\{} j U, p F (6) Constrant (7) ensures the customer s demands have been satsfed. j D j + S qp = S pq + d p q P p q/p P q C, p F Constrant (8) ensures that B p s set to 1 f a producton of p occurs. It also ensures that fxed costs are pad when a component s provded by a suppler or when an assembly s manufactured at a center. Amax p s the upper bound of A p U. A p (7) B p A p max S U D, p P (8) Constrant (9) ensures that Z s set to 1 f producton center s used. B p Z S U D, p P (9) Constrant (10) defnes the capacty of each technology needed at a center. l pt A p O o co U, t T (10) p/u P p o O t Constrant (11) ensures that T p j ssetto1fthearcfrom to j s used by at least one product p. A p max s the upper bound of T p j. j T p j A p max N, j N \{}, p P (11) Constrant (12) ensures that L j s set to 1 f at least one product uses the arc from to j. T p j L j N, j N \{}, p P (12) Constrant (13) lmts the number of substtuted products to be used at the producton center n whch they were created. q P p S qp q M\p g qp A q + j C j U, p P (13) Robust desgn methodology From the defntons of robustness presented n Robustness assessment n supply chan desgn secton, we consder as robust a supply chan network desgn that can adapt to all plausble future scenaros n ths paper, a scenaro s a set of parameters correspondng on normal condtons as well as major dsruptons by provdng a soluton that s close to optmal for each scenaro (the proxmty concept wll be defned at the end of ths secton). In order to fnd a robust desgn, a methodology n three steps s followed, as shown n Fg. 1. In the frst step, possble scenaros are generated and ther optmal desgns calculated. In the second step, the robustness of each desgn s assessed for each scenaro. In the fnal step, a decson s made on the most robust desgn. Step 1. Scenaro generaton The methodology proposed here s based on the generaton of scenaros that reflect the parameters of the problem. All the uncertan parameters (e.g. demands, transportaton costs, labor rates, ) are tested wthn a range of levels, dependng on the probablty of occurrence of each scenaro and ts mpact on the soluton. When the number of varables and levels s not too hgh, a factoral combnaton can be computed to generate the scenaros. Each scenaro has a probablty of occurrence equal to the product of multplyng the ndvdual probablty levels. When the combnatoral exploson s too hgh, the number of levels has to be reduced for varables that don t have a strong mpact on the soluton. Two methods can be used to assess the nfluence of a varable on the output: desgn of experment (Taguch 1986) and data mnng (Dan 2009). A varable wth a major nfluence should be tested precsely. We do not address ths problem n depth here. Then, the optmal desgn (desgn(l) n Fg. 1) s computed for each scenaro to determne the nvestments t needs and ts objectve value (obj(l) n Fg. 1) for each scenaro (). The nvestment n the optmal desgn n Step 2 refers to mnmal nvestment constrants, and the objectve value s the base on whch to assess the robustness of the desgn for each scenaro. In Fg. 1, scenaro generaton s based on the varaton of two parameters on two levels Four scenaros are generated and solved. Step 2. Robustness assessment Here, the robustness of each desgn appled to each scenaro s evaluated. Choosng a desgn nvolves some nvestments. We consder that once a desgn s chosen, nvestments are made. When the real scenaro s known, the

5 Fg. 1 Robust desgn methodology supply chan can change, but there wll be some nvestments that have already been made (e.g. at producton centers and n terms of equpment acquston). So, we proceed wth a new optmzaton to determne the most effcent supply chan, takng nto account the nvestment that has already been made, as some decson varables were fxed n Step 1 (e.g. nodes opened, producton lnes organzed and specfc equpment bought, and transportaton arranged). In ths step, each scenaro s solved agan for each desgn created n Step 1, ncludng the extra constrants that correspond to the nvestments made n each desgn (Cont (l) n Fg. 1). Constrant (14) apples to producton centers, (15) to producton lne nvestments, (16) to specfc equpment and (17) to arrangng transportaton. Z Sol k (Z ) U (14) B p Sol k (B p ) S U D, p P (15) O o Sol k (O o ) U, o O (16) L j Sol k (L j ) N, j N \{}, (17) The objectve value generated n Step 2 s then compared to the optmal soluton of the orgnal scenaro (obj(l)). The robustness of a soluton k on a scenaro l s assessed by formula (18), whch represents the gap between the objectve value of the optmal soluton when consderng scenaro k and the optmal soluton. robustness k (l) = obj k(l) obj(l) obj(l) (18) A value of 0 means perfect robustness, and the hgher the value, the less robust the soluton. Note that robustness k (k) = 0 and robustness k (l) 0,.e. the soluton s optmal f the effectve scenaro s the one that has been scheduled, otherwse t s worse. Step 3. Desgn selecton A desgn has to be chosen from all the scenaros generated based on ther robustness n all of them. Several crtera can be used to classfy the desgns:

6 Table 2 Results example to llustrate Steps 2 and 3 Prob. Desgn 1 Desgn 2 Desgn 3 Desgn 4 Scenaro Scenaro Scenaro Scenaro Mnmax Average robustness Mnumum standard devaton Least beyond the threshold (20 %) Best values are gven n bold Producton centers Supplers Dstrbuton centers Customers Fg. 2 Geographcal poston of the potental nodes n the case study Mnmax: a desgn that mnmzes the possble loss for a worst-case scenaro. Average robustness: a desgn that s close, on average, to the optmal soluton for each scenaro. Mnmum standard devaton: a desgn that s more stable than the others for each scenaro. It can be used to dfferentate between two desgns wth the same mnmax or the same average robustness. Least beyond the threshold: a desgn that lmts unacceptable solutons. Ths threshold has to be fxed. Dependng on the acceptable rsk, the decson favors desgns n the mnmax and mnmum standard devaton categores, n order to avod the worst-case scenaros, and desgns n the average robustness category for the best expected value. Table 2 llustrates Steps 2 and 3 of the methodology wth four scenaros. Once Step 1 has been completed, four scenaros wll have been generated, each wth a probablty of occurrence (20 % for scenaro 1, 30 % for scenaro 2 and so on ). The model s optmzed for each of these scenaros, and yelds four desgns. To assess the robustness of these desgns, Step 2 s appled, gvng the robustness of each desgn on each scenaro. Step 3 aggregates the results and assesses each desgn. Desgn 1 has the best average robustness (38 %); however, ts standard devaton s hgh (41 %), ts maxmum possble loss s 150 %, and two scenaros are beyond the threshold, whch was set at 20 %. Desgn 2 has smlar results, but wth a better maxmum loss (100 %) and standard devaton (38 %), and more results beyond the threshold. Desgn 3 s the least rsky, as t has the lowest maxmum loss and the mnmum standard devaton, but ts expected value s above that of Desgns 1 and 2. Desgn 4 has poor results because of ts maxmum loss of 1,000 %, caused by scenaro 1 whle the other scenaros are well predcted, and so ths s the best desgn wth respect to the threshold crteron. The complexty of the algorthm s O(n 2 ), wth n beng the number of scenaros. Ths means that a reasonable number of scenaros has to be consdered, as the resoluton of a sngle problem s not neglgble. For example, Baud-Lavgne et al. (2011) experment wth a resoluton tme of around 1 second for cases wth 10 producton centers and 20 parts, 5 mn for 10 producton centers and 100 parts, and 30 mn for 15 producton centers and 150 parts. Experments to llustrate the methodology Experments were conducted nvolvng soluton of the MILP presented n The desgn secton wth ILOG CPLEX 12.5 Java lbrares on a laptop wth a Intel Core2Duo CPU at 2.26 GHz and 4 GB RAM. Fg. 3 Root BOM for the product famly n the case study

7 Fg. 4 Bll of materals for three products Table 3 Cost characterstcs of the case study Fxed costs Value Table 5 Results of the Pareto-optmal solutons Prob. Desgn4(%) Desgn7(%) Per (axe, product) $200 Per (component, suppler) $1,000 Per (product, producton center) $50,000 Per suppler $5,000 Per producton center $200,000 Per DC $10,000 Table 4 Uncertan parameters n the case study Parameter Tested values Logstcal costs 50/100/150$ /m 3 Labor costs Unts 1, 2: 15/20/25 unts 3, 4: 5/10/15 $ /h Demand 50/100/150 % The case study was taken from the generator proposed n Baud-Lavgne et al. (2011), wth two markets areas, four producton centers (two per market), and two dstrbuton Mnmax Average robustness Mnmum standard devaton Least beyond the threshold (1 %) Best values are gven n bold centers (Fg. 2); a famly of nne products wth a three level BOM. Each product s an nstance of the root BOM presented n Fg. 3, wth 8 assembles and 13 components. Each component can be present or not and has dfferent qualty level possbltes. These combnatons defne products sold by the company. Fgure 4 llustrates three products P 1, P 2 and P 3, of the nne n the product famly. The sze of the problem s small, n order to speed up the experment. Table 3 shows the parameters used n ths case study. Step 1. Scenaro generaton Scenaros are generated from varaton of the followng parameters as descrbed 0.06 max avg std dev Threshold sol2 sol4 sol6 sol8 sol10sol12sol14sol16sol18sol20sol22sol24sol26sol28sol30sol32sol34sol36sol38sol40sol42sol44sol46sol48sol50sol52sol54sol56sol58sol60sol62sol64sol66sol68sol70sol72sol74sol76sol78sol80 sol1 sol3 sol5 sol7 sol9 sol11sol13sol15sol17sol19sol21sol23sol25sol27sol29sol31sol33sol35sol37sol39sol41sol43sol45sol47sol49sol51sol53sol55sol57sol59sol61sol63sol65sol67sol69sol71sol73sol75sol77sol79sol81 Fg. 5 Average, standard devaton and maxmum of robustness for each desgn

8 Producton centers Supplers Dstrbuton centers Customers Fg. 6 Relatve geographcal locaton of the supply chan components n Desgn 4 n Table 4: demand (3 levels), transportaton costs (3 levels), and labor costs (3 levels n two unts). These parameters and ther levels are determned to llustrate our method. In an ndustral case study, a systematc methodology should be used to determne these scenaros, as seen n Robust desgn methodology secton. Ths leads to 81 scenaros, resultng from the combnaton of each level and each parameter. Then, each scenaro s solved, resultng n 81 desgns. Step 2. Robustness assessment The robustness of the 81 desgns s assessed by determnng ther robustness relatve to that of each of the 81 scenaros constraned by these desgns. Step 3. Desgn selecton Results for all the desgns are llustrated n Fg. 5. For each desgn on the X-axs, the robustness results for all the scenaros are aggregated on the four crtera, followng Step 3 of the methodology: measurng maxmal loss, average loss, standard devaton (left Y-axs) and threshold (rght Y-axs). Results of the Pareto optmal solutons, Desgn(4) and Desgn(7), are presented n Table 5. The supply chan of the two desgns s dentcal and s presented n Fg. 6. Fg. 8 Actual bll of materals of product P 3 n Desgn 7 Concluson The actual bll of materals of the two desgns are presented n Fgs. 7 and 8. They are dentcal at the followng ponts: Product P 1 s produced as-s and s used nstead of 5 others products (P 5 to P 9 ); product P 2 s close to the orgnal one, only component 6.1 has been standardzed by component 6.2; t defers for product 3, as Desgn (4) standardzes B2 byb1 contaned n P1, and Desgn (7) standardzes B2 by B4 contaned n P4 (not llustrated here). Based on the mnmax crteron, Desgn 7 would be chosen, because ts maxmum loss value s the lowest of all the desgns. Nevertheless, t has a hgh average robustness value, but a low standard devaton. Ths means that Desgn 7 s a relable soluton, whch can adapt to all possble scenaros, but falls short of the optmal soluton. By contrast, Desgn 4 has a very low average robustness value, but ts standard devaton s a bt hgher than that of Desgn 7 and ts worst result s nearly twce as hgh. Ths paper has addressed the problem of robustness n the jont product famly and supply chan desgn problem. The proposed methodology allows us to fnd a desgn that best suts all the possble parameter varatons. Dependng on Fg. 7 Actual bll of materals of product P 2 and P 3 n Desgn 4

9 the rsk the company s able to take, several decson plans are proposed, based on four ndcators. Its am s to choose the desgn that s ether the most proftable or the most relable, or a good balance of the two. The optmzaton method proposed n ths paper can be used to explore a wde range of desgns wth lttle confguraton requred. However, the computaton tme s hgh, n the range of the square of the number of solutons explored. There are two possble ways to shorten ths tme. The frst s to reduce the number of scenaros by changng the way they are generated. The desgn of experment or data mnng methods could be an mportant step n dong so. The second s to create desgns that are not based on scenaro optmzaton. If we know that none of the predcted scenaros wll occur, newer, better fttng desgns can be created from several scenaros usng genetc algorthm. References Baud-Lavgne, B., Agard, B., & Penz, B. (2011). A MILP model for jont product famly and supply chan desgn. 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