OPTIMIZING SUPPLY NETWORK STRUCTURES UNDER ECONOMIES OF SCALE

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1 OPTIMIZING SUPPLY NETWORK STRUCTURES UNDER ECONOMIES OF SCALE Alain Martel and Uday Vanatadri CENTOR Research Center FSA, Université Laval, Québec, Canada., G1K 7P4 Tel: ( , Fax: ( , ABSTRACT This paper is cncerned with a class f lcatin-allcatin prblems invlving several prduct families, several echelns, capacity expansin r cntractin pssibilities by technlgy type at the facilities and ecnmies f scale at the facilities. In the first part f the paper, we describe the cntext and the nature f these supply netwr ptimizatin prblems. We then present the ptimizatin mdel and the scenari imprvement apprach develped t slve it. Briefly, the mdel prpsed is a large-scale, nn-linear, nn-separable, mixed-integer prgram and it is slved with a successive mixed-integer linear prgramming algrithm where slutins are cnstrained within trust regins. 1.0 Cntext Supply chains are netwrs f lgistic and manufacturing activities starting with raw material surcing and ending with the distributin f finished gds t marets. The perfrmance f a supply chain fr a given prduct-maret critically depends n tw interdependent strategic issues: 1 the netwr structure, i.e. the number, lcatin, missin, technlgy and capacity f the facilities f the firms invlved; 2 the material flw planning and cntrl prcesses used. This paper addresses the first f these tw issues: the ptimizatin f supply netwr structures. This is an area where academic researchers have played a leadership rle in practice (Arntzen et al, 1995; Geffrin & Pwers, 1995; Shapir et al, This is an extremely cmplex prblem, and its slutin usually invlves the use f simplified mdels with extensive sensitivity and scenari analysis. The dminant apprach t date has invlved the frmulatin f a single perid (static mathematical prgramming mdel and the successive use f this mdel ver several years, under a number f prbable scenaris, t arrive at a strategic plan. A recent review f static netwr design mdels is fund in (Vidal & Getschalcx, Althugh the mdels available yield difficult large scale mixed integer prgrams, they typically neglect imprtant issues. Sme f the aspects which are prly cvered include facility lcatin decisins n mre than ne echeln, ecnmies f scale, inventry level dependence n delivery times, internatinal factrs and trade-ffs between lcatin, capacity and technlgy issues. The mdels by Chen et al. (1989 and by Arntzen et al. (1995 appear as the mst cmprehensive t date. In Chen s wr, the mdels are nt slved ptimally: they are used t explre pssible slutins by fixing subsets f 0-1 variables. In Arntzen s wr, ecnmies f scale are nt explicitly cnsidered. Given the size f the prblems invlved, a user friendly scenari imprvement apprach seems mre liely t give high payffs in the shrt term than a glbal ptimisatin apprach. This paper presents a mdeling framewr and a slutin methd based n successive linear prgramming which supprt this apprach. 2.0 Mdeling Framewr In this sectin we discuss the supply chain cncepts cvered in ur mdeling framewr and we define the basic terminlgy and ntatin used in the paper. The type f supply system OPTIMIZING SUPPLY NETWORK STRUCTURES UNDER ECONOMIES OF SCALE January

2 studied is illustrated in Figure 1. The multi-lcatin prductin-distributin netwrs cnsidered are defined ver a set f ndes, N, rganized int echelns. The first echeln ndes are cnceptual and they are assciated t prduct families (p P. All the ther ndes represent gegraphical lcatins. The last echeln lcatins are demand znes (z Z and the intermediate ndes crrespnd t either existing r ptential sites (s S f prductin (m M S r distributin (w W S facilities. Specialized strage technlgies (t T and prjects t mdify the capacity f existing facilities can be taen int accunt. The arcs adjacent t the prduct family ndes are assciated t prcurement activities and all the ther arcs represent transprtatin activities. Direct shipments are pssible, that is, sme arcs may cut acrss several echelns. Currently, the framewr supprts tw types f supply systems: 1 Pure distributin systems, that is, netwrs in which the intermediate ndes are rganized int an arbitrary number f echelns with facilities such as natinal warehuses, reginal warehuses and stres, and with pssible direct shipments; 2 Prductin-distributin systems which, in additin t the abve can, at the secnd echeln, have parallel prductin facilities. In what fllws we explain briefly the cncepts and ntatin prpsed. A mre detailed discussin is fund in Martel (1998. All the prducts included in a prduct family p P must have quantity, space and weight characteristics measurable in the same units. The territry cvered is divided int a set f demand znes with nwn centrid crdinates. We define: Z p Set f the znes in which prduct p is sld ( Z. d pz Demand fr prduct p in zne z during the planning perid cnsidered. The size f the demand znes defined can vary significantly, depending n the demand density f the varius regins f the territry cvered. The planning perid cnsidered is usually a base-year and, withut lst f generality, we assume it is a year in the rest f the paper. Z p FIGURE 1. Fur Echeln Supply Netwr Structure Prduct families Prcurement Warehuse/Plant sites Lng-haul transprtatin new site Supply chain i p P t p P t Direct supply chain j T pw T ps s H ps T psw (a G s, A s ps (p P (b t s, B t s (b ts, B ts (s S Warehuse/Stre sites w H pw G pw (a w, A w Lcal delivery Demand znes p p' z T pwz (z Z p (z Z p d pz OPTIMIZING SUPPLY NETWORK STRUCTURES UNDER ECONOMIES OF SCALE January

3 In mst studies, the supply netwr cnsidered already uses a set f facilities and the cnstructin, acquisitin r renting f facilities n new sites may be cnsidered. The gegraphical lcatin f existing and ptential sites is assumed t be nwn. Ndes in the netwr are assciated t sites and decisins must be made n whether existing r ptential sites will be used. This may invlve using current facilities, clsing current facilities r pening new facilities. We distinguish between manufacturing and warehusing facilities. Manufacturing facilities incur prductin csts based n thrughput and their capacity is usually expressed in terms f a maximal annual thrughput (ften based n the prductin level f a bttlenec peratin by prduct family. This is natural since, in this cntext, prducts are clustered int families by manufacturing technlgy. Warehusing facilities incur handling csts based n thrughput as well as strage and inventry hlding csts based n average inventry levels. Their capacity culd be related t handling cnsideratins but, in mst cases, it is limited rather by space availability. Mrever, several technlgies may be required t stre all the prducts. Fr example, sme prduct families may have t be stred in a refrigerated area and thers utside in the yard, and hence, prduct families must be gruped accrding t the type f strage technlgy they require, and capacity restrictins must be expressed by strage technlgy types. Finally, lwer bunds are usually impsed n facilities thrughput, by technlgy, t avid lw activity peratins. We define: P t Set f the prduct families using strage technlgy t ( P. u p b ts L ts a s y s Average size f ne family p prduct (expressed in the flw units used fr the technlgy required by p, say, a standard cntainer, a pallet, m 3,... Level f technlgy (r prduct family t capacity available at site s when its facility is pen (if s M, then t P. Minimum level f technlgy (r prduct family t capacity that must be used at site s facility if it is t be pened r ept pen as the case may be (if s M, then t P. The fixed cst f pening, r eeping pen, a facility n site s, including ut-fpcet csts as well as pprtunity csts, and taing financing, incme tax and salvage factrs int accunt. Binary decisin variable equal t 1 if a facility is pened at site s and t 0 therwise. In mst applicatins, in additin t pening r clsing facilities, ne can mdify existing nes. This may invlve expanding the capacity f sme f the technlgies required, r rerganizing the facility which invlves decreasing the capacity f sme technlgies and increasing the capacity f thers. Here we cnsider ne mdificatin prject per existing facility, but the apprach is easily extensible t several alternative prjects per facility. We define: B ts The amunt f technlgy (r prduct family t capacity added (r remved if B ts < 0 at site s when the mdificatin prject cnsidered is realized (if s M, then t P. Clearly, we must have B ts b ts. A s The cst f the mdificatin prject cnsidered fr the facility n site s. Y s Binary decisin variable equal t 1 if a facility mdificatin prject is undertaen at site s and t 0 therwise (s S. Clearly, A s is relevant nly fr sites which already have a facility in place. P t OPTIMIZING SUPPLY NETWORK STRUCTURES UNDER ECONOMIES OF SCALE January

4 Three types f arcs are fund in Figure 1. Prcurement arcs (p, s start with a prduct family nde p and finish with a facility site nde s, and they give the initial prduct value. This value includes the purchase price, the rder cst and the inbund transprtatin cst and it depends n the agreements with suppliers (quantity discunts and n prcurement plicies. Lng-haul transprtatin arcs (s, w start with a facility site s (plant r warehuse and finish with a warehuse w. The nature f the csts n these arcs depend n whether the lt sizes are fixed (truc lad r ptimized. Lcal delivery arcs (s, z start with a facility site s and finish with a demand zne z. They are largely respnsible fr the time required t satisfy custmer demands. It shuld be nted that sme f the transprtatin means used n a given arc may be shared by several prduct families. We define: T ps Average value (in $/unit f the family p prducts (r f the materials required t prduce p if s is a plant site which are purchased frm an utside supplier by the facility n site s. T psn Unit cst f flw (in $/unit f family p prducts n the arc between lcatins s and n. A path thrugh the supply netwr defines what we call a supply chain. Every supply chain starts with a prduct family p, it ends with a demand zne z and it has at least ne site in between. If the supply netwr includes plants, there is always ne manufacturing site m in a chain and it always fllws the prduct family nde. In ther wrds, in a pure distributin system, a chain i is defined by the set f ndes C i { p,, z} and in a prductin-distributin system by the set { pm,,, z}. We define: I C i Set f all the feasible supply chains in the netwr (i I. Ordered set f the ndes n the path f supply chain i in the supply netwr. p i (z i Prduct family (Demand zne at the rigin (end f supply chain i, i.e. the first (last element f C i. x i C i Flw f family p i prducts in supply chain i during the base-year (i I. The prcurement lead time and value f a prduct at a given lcatin depends n the supply chain used t prcure it. The lead time f a prduct at nde n is the ttal time elapsed between the mment an rder is placed and the mment the prduct is available fr use. This includes the transit time assciated t the inbund arc, but als all the time used at the supply nde befre the shipment is made. The value f a prduct at nde n includes the initial value n the prcurement arc (the first arc f the supply chain used, plus all the prductin, handling, strage and transprtatin csts added in the preceding ndes and arcs n the chain. We define: τ ni Average lead time f family p i prducts at nde n when they are prcured thrugh chain i. v ni Average value f family p i prducts (in $/unit at nde n when they are prcured thrugh chain i. Nte that, as seen in Figure 2, when tw prducts f the same family arrive at nde n thrugh different chains, they d nt have the same lead time and value. It is therefre necessary t use a mdeling apprach which memrizes the paths prducts fllw in the supply netwr. Sme f the csts n site ndes are independent f flws (siting and prject csts but mst f them depend n sme aggregatin f the flws n the supply chains ging thrugh the nde (prductin, warehusing and inventry hlding csts. The mdeling f these csts requires the definitin f the fllwing flw aggregatin variables: OPTIMIZING SUPPLY NETWORK STRUCTURES UNDER ECONOMIES OF SCALE January

5 FIGURE 2. Dependence f Value and Lead Time n Chain Used Direct chain j C j {p, n, z} v nj v ni! v nj τ n nj z p v ni τ τ si v si ni s Chain i passing thrugh warehuse s C i {p, s, n, z} X ps V ps Ttal flw f family p prducts thrugh site s during the base-year, i.e. X ps x i { i {,} ps C i }, p P, s S (1 Ttal value f family p prducts flwing thrugh site s during the base-year, i.e. V ps { i {,} ps C i } v si x i, p P, s S (2 Observe that the rati f these tw quantities /X ps, yields the average value f the family p prducts flwing thrugh site s during the base-year. p ps Cumulative lead time f prducts p flwing thrugh site s during the base-year, i.e. Τ ps { i {,} ps C i } τ si x i, p P, s S (3 The rati /X ps, yields the average lead time f the family p prducts flwing thrugh site s during the base-year. This lead t the definitin f the fllwing nde csts functins: g ps (X ps family p prducts prductin cst functin, if s is a manufacturing site, r warehusing cst functin if s is a warehusing site. h pw (X pw, V pw, Τ pw annual inventry hlding cst functin fr family p prducts held at warehuse w. As thrughput increases, it is usually pssible t use mre perfrming technlgies, t imprve methds and t adjust layuts, s that ecnmies f scale can be realized. Hence, we assume that g ps is a cncave functin f the ttal prduct flw (X ps. Similarly, frm inventry thery, it can be shwn that inventry hlding csts are cncave functins f ttal prduct flw (X ps in the nde, average prduct value (V pw /X pw and average lead times (Τ pw /X pw. As can be seen, when these crucial factrs are taen int accunt, g ps and h pw becme cmplex implicit nn-linear and nn-separable functins f x i, i { i {, p w} C i }. This seriusly cmplicates the cmputatin f the value f a prduct at different pints n a supply chain. Cnsider the supply chain i given as an example in Figure 3. This chain includes a manufacturing nde m and a warehusing nde w. The unit flw csts n the arcs are linear, but the unit flw csts n the ndes, btained by dividing the manufacturing, warehusing and hlding cst functins by the ttal flw f family p prducts thrugh the relevant ndes, are implicit nn-linear functins nt nly f x i, but als f the flw variables assciated t all the ther chains mving prducts p thrugh ndes m r w. OPTIMIZING SUPPLY NETWORK STRUCTURES UNDER ECONOMIES OF SCALE January

6 FIGURE 3. Supply Chain Unit Flw Csts Chain i: Arc csts: Nde csts: p m w z T pm T pmw T pwz g pm ( X pm X pm g pw ( X pw h pw ( X pw, V pw, Τ pw X pw X pw Let, c i Ttal linear cst per unit flw thrugh supply chain i. G ps H pw g ps X ps h pw ( X pw, V pw, Τ pw X pw Site s unit manufacturing/warehusing cst fr family p prduct. Unit inventry cst fr prducts p at warehuse w. When all the nde csts are nn-linear, c i is the sum f the unit csts assciated t all the arcs in chain i. When sme nde unit csts are linear, they can als be included in c i. Prduct values at a given nde are the sum f all the arc and nde unit csts incurred n a chain t get t that specific nde. Since G ps and H pw are functins f the flw variables, this illustrates the fact that the prduct value v ni at the different ndes n in the chain is als a functin f the flw variables. Mrever, H pw is a functin f V pw which in turn is a functin f v wi and x i (relatin 2, fr all the chains i prcuring prducts p t warehuse w. Hence, the functins H pw are even mre cmplex than initially indicated. This quicly becmes intractable and sme simplificatins are required t arrive at a practical slutin apprach. It is wrth nting that when ne wants t cmpute prduct values and chain csts fr a given scenari, the cmplex relatinships described abve d nt create any prblems. When x [x i ] is nwn, X ps and Τ ps can be calculated fr each intermediate nde s, prvided that the calculatins are dne sequentially echeln by echeln, starting with echeln tw. This is required t mae sure that the prduct values v si are available when V ps is calculated with (2. This fact will be explited in the slutin prcedure prpsed belw. The apprach we prpse here t tae ecnmies f scale int accunt prperly is t estimate prduct values v si, a priri, based n an average behavir scenari, which ensures that relatin (2 remains linear, but t include nly linear unit csts in the chain cst c i, and t incrprate the nn-linear cst functins g ps and h pw explicitly in the ptimizatin mdel. Let x be a scenari generated by evenly spreading the flw f all the prducts ging thrugh an echeln as much as pssible, withut vilating a capacity cnstraint. Then, prceeding as indicated abve, the unit prductin, warehusing and inventry csts can be calculated fr this scenari. These csts are dented by and. G ps H ps Fr the Figure 3 example, with this apprach, the prduct values and the linear unit chain csts wuld be calculated as fllws: G pm G pw H pw v mi T pm, v wi v mi + + T pmw, v zi v wi T pwz c i T pm + T pmw + T pwz OPTIMIZING SUPPLY NETWORK STRUCTURES UNDER ECONOMIES OF SCALE January

7 3.0 General Supply Netwr Optimizatin Mdel T simplify the ntatin, let; c ps The fllwing nn-linear, nn-separable, mixed-integer prgramming mdel captures the essence f the supply netwr ptimizatin prblem: NLP min c i x i + a s y s + A s Y s + c ps (5 subject t the flw aggregatin relatins (1, (2 and (3, the demand cnstraints: the capacity cnstraints: i I g ps h the mdificatin prject realizatin cnditins: s S x { i {,} pz C i i } g ps if s M + ps if s W d pz p P (4 p P, z Z p (6 L pm y m x i b pm y m + B pm Y m, m M, p P { i {, pm} C i } L tw y w u p x i b tw y w + B tw Y w, w W, t T p P t { i {, pw} C i } (7 (8 y s Y s 0, s S (9 and t the nn-negativity cnstraints y s, Y s {0, 1}, s S ; x i 0, i I ( Slving the Mdel by Scenari Imprvement In what fllws, is a feasible scenari index, x [ ], y [ y s ], Y [ Y s ] and s is the site cst functin Linear apprximatin f s arund a feasible scenari (x, y, Y Let and s be, respectively, the value f and f s fr scenari. Let als the peratr dente a gradient, s that ( X is the partial derivative f c ps with respect t X ps evaluated at. Similarly, we have ( c ps V and ( c ps Τ. At the rigin, the hyperplane defined by these gradients has the fllwing value: x i s a s y s + c ps p P c ps (11 a s s p P [( Xps X + ( Vps V + ( Τps Τ ] c ps c ps c ps (12 OPTIMIZING SUPPLY NETWORK STRUCTURES UNDER ECONOMIES OF SCALE January

8 It shuld be clear that in the vicinity f, the value f the cncave functin s is clsely apprximated by the fllwing linear functin: In ur scenari imprvement apprach, when a nde is pen, the nn-linear functin (11 is replaced by the linear apprximatin (13. This des nt wr, hwever, when a facility is clsed, i.e. when y s 0. Althugh a zer flw fr a facility is feasible, the slutins in the vicinity f (0, 0, 0 may nt be feasible because f the lwer bunds impsed n factry thrughput and n warehuse space utilizatin. Fr this reasn, under null flws, the gradients evaluated at (0, 0, 0, d nt prvide an adequate basis fr the linear apprximatin f s. c ps Initial Linear Apprximatin Prblem Let x be an initial feasible average behavir scenari generated by trying t spread flws equally between all the facilities f a given echeln. Such a scenari is easily btained by slving a linear prgram with has the same cnstraints as NLP and the bjective f minimizing deviatins frm a pre-calculated equal share slutin. Fr such a scenari, we have y 1, and hence the apprximatin (13 can be used fr all the facilities. When s is replaced by s, the nn-linear prgram frmulated abve becmes a linear mixed-integer prgram, and the aggregatin variables defined by the relatins (1, (2 and (3 can be eliminated by substitutin. This yields the fllwing mixed-integer prgram: MIP + 1 min c (14 i xi + ( a sys + A s Y s subject t cnstraints (6 t (10, where 0 and where, s a sys + [( c ps Xps X + ( c ps Vps V +( c ps Τps Τ ] c i c i + p P s S C i i I s S Trust Regin and Scenari Imprvement Prgram Let the index f the feasible scenari btained by slving prgram MIP be 1. There is n guarantee that is in the vicinity f fr all p and s and hence, the slpe f s at may be significantly different frm its slpe at, i.e. there is n guarantee that the apprximatin used is gd at the slutin btained. This has tw implicatins: 1. If the slpes used are replaced by the slpes at the slutin fund, fr the ndes with psitive flws, and the prgram is reslved, a better feasible slutin shuld be btained. 2. A better slutin will be btained in as much as the slpes at the new pint fund are clse t the slpes used in the prgram. The nly way t be certain that this will ccur is t limit the search fr a better slutin t a regin in the vicinity f the previus slutin. This leads t the definitin f a trust regin arund a given feasible scenari. Since the cntinuus part f the cncave functins s are defined ver, and since these three aggregatin variables depend n the same flw variables namely x i, i { i {,} p s C i }, defin- [( c pi s X + v si ( c pi s V + τ si ( c pi s Τ ] (13 (15 OPTIMIZING SUPPLY NETWORK STRUCTURES UNDER ECONOMIES OF SCALE January

9 ing trust regins fr X ps is sufficient. Nte that the cnstraints (7 are als defined n the same sets f flw variables. This suggest that an efficient way t define trust regins simply invlves, 1 adjusting the lwer and upper bunds fr the manufacturing sites, L pm, b pm and B pm, adequately and replacing (8 in MIP by L pm ( φy m x i b pm ( φy m + B pm ( φy m, m M, p P { i {, p m} C i } and, 2 fr the warehusing sites, adding the cnstraints X pw ( φy w x i X pw ( φ, w W, p P { i {, pw} C i } (16 (17 where L ps ( φ, b pm ( φ, B pm ( φ, X pw ( φ and X pw ( φ are the bund values required t define a prper trust regin arund scenari. The resulting mathematical prgram is dented MIP(φ. In these expressins, φ [ 01, ] is a cefficient used t restrict the search t a trust regin f width φx ps n each side f X ps. Fr the manufacturing sites, these bunds are cmputed as fllws: b pm ( φ max(l pm, ( 1 φx pm, ( φ min(b pm, ( 1 + φx pm and B pm ( φ min(b pm + B pm, ( 1 + φx pm b pm ( φ Fr the warehusing sites they are cmputed as fllws: Scenari Imprvement Algrithm ( φ ( 1 φx pw and X pw ( φ ( 1 + φx pw. Let be the value f the ecnmic functin (5 fr scenari, and S be the set f all the sites which have a psitive flw in scenari. Imprved scenaris are fund by slving a series f MIP defined by the algrithmic prcedure included in Figure 4. Our experimental results with this apprach t date shw that very gd slutins can be btained in 4 r 5 iteratins fr mderate size prblems. Mre extensive testing is required t reach firm cnclusins, but the apprach seems very prmising. 5.0 References X pw L pm (1 Arntzen, B. et al (1995, Glbal Supply Chain Management at Digital Equipment Crpratin, Interfaces, 21-1, (2 Chen, M. and H. Lee (1989, Resurce Deplyment Analysis f Glbal Manufacturing and Distributin Netwrs, J. Mfg. Oper. Mgt., 2, (3 Chen, M. and S. Mn (1990, Impact f Prductin Scale Ecnmies, Manufacturing Cmplexity, and Transprtatin Csts n Supply Chain Facility Netwrs, J. Mfg. Oper. Mgt., 3, (4 Geffrin, A. and G. Graves (1974, Multicmmdity Distributin System Design by Benders Decmpsitin, Man. Sci., 20-5, OPTIMIZING SUPPLY NETWORK STRUCTURES UNDER ECONOMIES OF SCALE January

10 FIGURE 4. Successive Mixed Integer Prgramming Algrithm 1. Define the chain sets C i, and cmpute an average behavir scenari (x, y, Y. Cmpute the c i, v si,, c ps, and. Set φ 1, S S and 0, and chse scalars ε, 0 < ρ 1 < ρ 2 < 1 and µ > Fr all i I cmpute. 3. Fr all m M and p P cmpute L pm ( φ, b pm ( φ and B pm ( φ. Fr all w W and p P cmpute ( φ and X pw ( φ. Slve MIP(φ btaining the scenari (x +1, y +1, Y +1 with and S +1. Set Fr all s S and p P cmpute. Cmpute and the imprvement in the scenari cst 1. If 1 < ε stp, Otherwise cmpute the apprximatin quality rati r ( If r 0 set φ φ/µ and -1, and g t step 3. c i X pw Fr all s cmpute c ps, p P, as well as and. Fr all s S set all, c ps and a s t scenari (-1 values. If 1 r < ρ 1 set φ µφ, Otherwise if 1 r > ρ 2 set φ φ/µ. Return t step 2. S a s + 1 s a s (6 Geffrin, A. and R. Pwers (1995, 20 years f strategic distributin system design: An evlutinary perspective, Interfaces, frthcmming. (7 Martel, A. (1998, Cnceptin et gestin de chaînes lgistiques, Ntes de curs, FSA, Université Laval, Québec, Canada. (8 Mn, S. (1989, Applicatin f Generalized Benders Decmpsitin t a Nnlinear Distributin System Design Prblem, Nav. Res. Lg., 36, (9 Rbinsn Jr, P. and Swin, M. (1994, Reasn based slutins and the cmplexity f distributin netwr design prblems, EJOR, 76, (10 Shapir, J., V. Singhal and S. Wagner (1993, "Optimizing the Value Chain", Interfaces, 23-2, (11 Vidal, C. and M. Getschalcx (1997, Strategic prductin-distributin mdels: A Critical Review with Emphasis n Glbal Supply Chain Mdels, EJOR, 98, (12 Zhang, J., N. Kim and L. Lasdn (1985, An Imprved Successive Linear Prgramming Algrithm, Man. Sci., 31-10, OPTIMIZING SUPPLY NETWORK STRUCTURES UNDER ECONOMIES OF SCALE January