Supply Chain Paper 1

8/20/2019 Supply Chain Paper 1

1/20

Design of Multi-echelon Supply Chain Networks under Demand

Uncertainty

P. Tsiakis, N. Shah, and C. C. Pantelides*

Centr e for Process Systems E ngi neeri ng, I mp eri al College of Science, T echnology and M edi cine,

L o n d o n S W 7 2 B Y , U n i t ed K i n g d o m

We consider the design of multiproduct, multi-echelon supply chain netw orks. The netw orksco m p rise a n u m b e r o f m an u factu rin g si te s at f ixe d lo catio n s, a n u m b e r o f ware h o u se s an ddistribution centers of unknown locations (to be selected from a set of potential locations), andfinally a number of customer zones a t f ixed loca tions. The syst em is modeled mat hemat ica lly a sa mixed-integer linear progra mming optimiza tion problem. The decisions t o be det erminedinclude the number, loca tion, and capacity of w a rehouses an d distribut ion centers to be set up,the t ra nsporta tion links tha t need t o be esta blished in the netw ork, a nd t he flows an d productionra tes of ma teria ls. The objective is t he minimizat ion of the t otal a nnua lized cost of th e netw ork,t a k i n g i n t o a ccou n t b ot h i n fr a s t r u ct u r e a n d op er a t i n g cos t s . A c a s e s t u d y i ll u st r a t e s t h eapplicability of such an integrated approach for production and distribution systems with orwithout product demand uncertainty.

1. Introduction

A supply chain is defined as a network of facilit iesthat performs the functions of procurement of materials,transformation of these materials into intermediate andfinished products, and distribution of these products tocustomers.1 A s imila r de fin it ion h a s bee n g ive n byB h a s k a ra n a n d L e u n g ,2 who describe the ma nufactur-in g s u p p ly c h a in a s a n in t e g ra t iv e a p p ro a c h u s e d t oma na ge the inter-related flows of products an d informa-tion a mong suppliers, manufa cturers, distributors, re-ta ilers, a nd customers.

A t y p ica l s u pp ly ch a in (s e e F ig u re 1) c omp ris essuppliers, production sites, storage facilities, and cus-tomers. It involves two basic processes tightly integratedwit h each other: (i) the production plan ning a nd inven-

tory control process, wh ich dea ls w ith m anufa cturing,storage, and their interfaces, and (ii) the distributionand logistics process, which determines how productsa re re t r ie ve d a n d t ra n s p ort e d f ro m t h e wa re h ou s e t oretailers.

Suppliers ar e at the sta rt of the supply chain provid-in g ra w ma t e ria l t o t h e ma n u f a ct u re rs. E a ch ma n u f a c-turer may have more than one supplier.

The ma nufacturing sites of interest t o this paper a remult ipurpose production plants where a wide ra nge ofproducts can be produced. The production capacity ofeach site is typically determined by the detailed sched-uling of each plan t .

Before being distributed to the customers, the f inal

p rod u ct s f r om p rod u ct i on p la n t s a r e s t or ed a t t w odist inct stages in the supply chain, namely, at majorwarehouses and at smaller distribution centers. Eachwa re h ou s e ma y be s u p plie d ma t e ria l f rom more t h a none manufacturing site. Similarly, a distribution centerc a n be s u p p lie d f ro m mo re t h a n o n e wa re h o u s e a l-though, for reasons of orga niza tional simplicity (“singlesourcing”), i t is often the case that each distribution

c e n t e r is s u p p lie d by o n ly o n e wa re h o u s e . B o t h t h e

mat erial storage a nd ha ndling capacit ies of wa rehousesa n d dist r ibu t ion ce n t ers a re l imit ed wit h in ce rt a inbounds.

At the end of the supply chain, there are the custom-e rs. U s u a l ly , e a ch cu s t ome r is a s s ig n ed t o a s in gledistribution center which supplies all of the requiredm a t e r ia l , a l t h ou g h t h i s m a y n ot a l w a y s b e t h e c a s e.Customers place their orders at distribution centerswhich pass this information to the upper levels until itgets to the suppliers. Thus, a ma in char acterist ic of thesupply chain is the f low of material from suppliers tocu s t ome rs a n d t h e cou n t e rflow of in forma t io n f romcustomers to suppliers.3

The places w here inventory is kept in the supply

chain a re called “echelons”. U sua lly the complexity of asupply chain is related to the number of echelons t ha tit incorporates.

The operat ion of supply cha ins is a complex t askbeca u s e of t h e la rg e -s ca le p h y sica l p rodu ct ion a n ddistribution netw ork flows, the uncerta inties associa tedwit h the externa l customer and supplier interfaces, a ndthe nonlinear dy nam ics a ssociat ed with internal infor-mation flows. In a highly competit ive environment, asupply chain should be managed in the most efficientway, with the objectives of ( i) m i n i m i z a t i o n of costs,delivery delays, inventories, and investment (ii) m a x i - mi za ti o n of deliveries, profit , return on investment(ROI), customer service level, a nd production.

The above tasks involve both strategic and opera-t ional decisions, w ith t ime horizons ra nging from sev-eral yea rs down to a few hours, respectively:1

1. L ocati on d ecisions consider the number, size, andphysical loca tion of production plant s, wa rehouses, a nddistribution centers.

2. Production decisions consider the products to bep rod u ce d a t e a ch p la n t a n d a l s o t h e a l l oca t i on ofsuppliers to plants, of plants to distribution centers, a ndof distribution centers to customers.

The deta iled production scheduling at each plantmust also be decided.

* To wh om correspondence should be a ddressed. Tel: (44)20-75946622. Fa x: (44) 20-75946606. E-ma il: c.pan telides @ic.ac.uk.

3585Ind. Eng. Chem. Res. 2001, 40 , 3585-3604

10.1021/ie0100030 CC C: $20.00 © 2001 American C hemical SocietyP ubl ish ed on Web 07/18/2001


2/20

3. Inventory decisions a re con ce rn ed wit h t h e ma n -agement of the inventory levels.

4. Transportation decisions include the tra nsporta tion

media to be used for and the size of each shipment ofma t e ria l .

As supply chains become increasingly global,5 a d-ditional a spects such a s differences in ta x regimes, dutydra wba ck an d a voidance, and fluctua tions in excha ngerat es also become important .

Section 2 of this paper presents a crit ical review ofp a s t wo rk o n s u p p ly c h a in mo de lin g a n d de s ig n . I nsection 3, the problem of optima l design of supply chainsis formulated as a mixed-integer linear programming(MILP ) model. The la tt er includes some of th e feat uresthat earlier models have failed to consider. Section 4e xt e nd s t h i s f or m u la t i on t o t a k e i nt o a c cou n t t h eu n ce rt a i n t y i n cu s t om er d em a n d s . A c a s e s t u dy i spresented in sect ion 5 t o illustra te t he a pplicability ofthe model.

2. Literature Review

In view of their importance in the modern economy,it is not surprising that supply chains have been on there se a rch a g e n da of a v a r iet y of bu s in es s a n d o t h e racademic disciplines for ma ny years.

The review presented in this section focuses on model-based m ethodologies tha t ca n provide q u a n t i t a t i v e sup-port to the design and operation of supply chains. Themajor desicions that need to be made in this contexthave already been described in the introduction of thispaper.

2.1. Heuristic-Based Approaches. Williams 6 pre-sents seven h euristic algorithms for scheduling produc-tion and distribution operat ions in supply chain net-wo rk s , c o mp a rin g t h e m wit h e a c h o t h e r a n d wit h adyna mic programm ing m odel. The objective is to deter-mine a minimum cost production and product distribu-tion schedule, satisfying th e product deman d, in a givendis t r ibu t io n n e t wo rk . I t is a s s u me d t h a t t h e de ma n drate is constant and that processing is instantaneous,with no delivery lags between facilit ies.

Th e r i sk s a r i s in g f r om t h e u s e o f h eu r is t ics i ndist r ibu t ion p la n n in g we re iden t i f ie d a n d discu s s edearly on by Geoffrion and Van Roy. 7 They pointed outthat heuristics (computerized or not) can lead to wrongdecisions. Three examples were presented in the area

of distribution planning considering questions such asthe number, size, and location of plant s a nd distr ibutionfacilities, the stocks, and the policies regarding inven-

tory a nd t ran sporta tion. All three exam ples demonstrat ethe fa ilure of “common sense” methods to come up w iththe best possible solution.

2.2. Mathematical Programming-Based Ap-proaches. The alternative to heurist ics is the use ofma thema tical models of supply chains. S uch models canbe classified into four cat egories:3 (1) determ inist ic, (2)stochast ic, (3) economic, a nd (4) simulat ion.

Workable and realistic models and algorithms beganto emerge in the mid-1970s with the emerging improve-ments in computa t ional capa bilit ies.

I n t h e ir p ion e erin g w ork , G e of f r ion a n d G ra v es8

present a model t o solve the problem of designing adistribution syst em w ith optimal locat ion of the inter-

mediate distribution facilit ies between plants an d cus-tomers. The objective is to minimize the total distribu-tion cost (including transportation cost and investmentcost), subject t o a num ber of constra ints s uch as supplyconstraints, demand constraints, and specification con-s t r a i nt s r eg a r d in g t h e n a t u r e of t h e p rob le m. Th eproblem is formulated as an MILP, which is solved usingB enders decomposition.

Wesolowsky and Truscott 9 p re s e n t a ma t h e ma t ic a lformulat ion for the mult iperiod locat ion-allocationproblem with relocation of facilities. They model a smalldistribution network comprising a set of facilities aimingto serve the demand at given points.

Williams 10 develops a dyna mic progra mming a lgo-rithm for simultaneously determining the production

a n d d is t r ib ut i on b a t ch s iz es a t e a ch n od e w i t h in asupply cha in network. The a verage cost is m inimizedover an infinite horizon.

Another determ inistic model presented by I shii et a l.11

aims to calculate the base stock levels and lead t imesassociated with the lowest cost solution for an integratedsupply chain in a f inite t ime horizon.

B r o w n e t a l .12 present an optimizat ion-based algo-ri t h m f o r a de c is io n s u p p o rt s y s t e m u s e d t o ma n a g ecomplex problems involving facility selection, equipmentlocat ion a nd utilizat ion, and ma nufacture and distribu-tion of products. They focus on opera tional issues suchas where each product should be produced, how muchshould be produced in each plant, a nd from w hich pla nt

Figure 1. Typical supply chain network.

3586 Ind. E ng. C hem. Res. , Vol. 40, No. 16, 2001


3/20

products sh ould be shipped to customers. S ome stra tegicissues a re a lso taken into account such as t he number,kind, and location of facilities (including plants).

B r e i t m a n a n d L u c a s13 present a modeling systemnam ed P LANETS (P roduction Location Analysis Net-wo rk Sy s t e m) . T h is p ro g ra m is u s e d t o de c ide wh a tproducts to produce and w hen, where, and h ow to makethese products. It also provides information on shippingallocat ions, capital spending schedules, and resourceusage.

An MINLP formulat ion is presented by Cohen andLee,14 seeking to maximize the total after-tax profit forthe manufacturing facilit ies and distribution centers.Man a gerial (resource and production) a s well a s logicalconsistency (feasibility, a vailability, a nd demands lim-its) constraints were a pplied. This w ork wa s extendedby Co h e n a n d M o o n ,15 who developed a constra inedoptimizat ion model, called PILOT, to investigate theeffects of various parameters on supply chain cost andto determine which m anufa cturing fa cilit ies a nd distri-bution centers should be established.

A two-phase approach was used by Newhart et al . 16

to design an optimal supply chain. First, a combinationof mathema tical progra mming a nd heurist ic models is

used to minimize th e number of product ty pes h eld ininventory throughout the supply chain. In the secondphase, a spreadsheet-based inventory model determinesthe minimum safety stock required to absorb demandand lead t ime fluctuations.

C h a n d r a 17 presents a model tha t plan s deliveries tocustomers based upon inventories, at wa rehouses anddistribution centers, a nd vehicle routes. In a lat er work,C h a n dr a a n d F is her 18 consider the coordination ofproduction a nd distribution planning.

Pooley 19 presents the results of an MILP formulationused by the Ault Foods company to restructure theirs u pp ly ch a in . Th e mode l a ims t o min imize t h e t o t a loperat ing cost of a production an d distribut ion netw ork.

The model is used to answer questions like the follow-ing: Where should th e division locate plan ts a nd depots(DCs)? How should the production be a llocated? How should the customers be served?

A rn t ze n e t a l .20 developed an MILP “global supplych a in mode l” (G S CM ) a imin g t o de t e rmin e : (1) t h enumber and locat ion of distribution centers, (2) cus-tomer-distribution center assignment, (3) number ofechelons, and (4) the product -plant assignment. Theob je ct i ve of t h e m od el i s t o m i ni m iz e a w e ig h t edcombination of total cost (including production, inven-tory, tra nsporta t ion, and fixed costs) and a ct ivity da ys.

Voudouris21 de ve lop ed a ma t h e ma t ic a l mode l de -signed to improve the efficiency and responsiveness ina supply chain. The target is to improve the flexibilityof the system. He identifies two types of manufacturingresources: act ivity r esources (manpower, w ar ehousedoors, packaging lines, forktrucks, etc.) and inventoryresources (volume of intermediate storage, warehousear ea, etc.). The objective function a ims t o represent t heflexibility of the plan t to a bsorb unexpected dema nds.

C a m m e t a l .22 developed an integer program mingm o d e l u s e d b y t h e P r o c t e r a n d G a m b l e c o m p a n y t odetermine the locat ion of distribution centers and toassign those selected to customer zones. A variety ofreasons (including the need to reduce transportat ioncosts, the introduction of new products, new qualitypolicies, the reduction in the life cycle of products, a ndrelat ively high number of plants) forced the company

to restructure their supply chain. The models developedwere a simple transportat ion model and an uncapaci-tated facility-locat ion problem, formulated as LP andMILP models, respectively.

P i r k ul a n d J a y a r a m a 23 studied a tri-echelon multi-commodity system concerning production, transporta-t ion , a n d dist r ibu t ion p la n n in g . Th e obje ct iv e is t ominimize the combined costs of establishing and operat-in g t h e p la n t s a n d t h e wa re h ou s es in a ddit ion t o a n yt ra n s p o rt a t io n a n d dis t r ibu t io n c o s t s f ro m p la n t s t owa re h ou s es a n d f rom wa re h ou s es t o cu s t ome rs. K e yde cis ion s a re wh e re t o lo ca t e wa re h ou s es a n d wh ichcustomers to assign to each w a rehouse. Another decisionincludes the selection of ma nufacturing plants an d th eira s s ig n me n t t o wa re h o u s e s . L a g ra n g ia n re la x a t io n isu s ed t o s o lv e t h e p roblem wh ic h is f ormu la t e d a s a nM I L P .

I n a la t e r wo rk , P irk u l a n d J a y a ra ma 24 present theP L A NWAR mode l t h a t s ee ks t o lo ca t e a n u mber ofproduction plants and distribution centers so that thet o t a l op era t in g cos t f or t h e dis t r ibu t ion n e t work isminimized. The netw ork consists of a potential set ofproduction plant s a nd distribution centers and a set ofgiven customers. This model ha s m any similarit ies to

the one presented by the sa me a uthors in 1996.Hindi and Pienkosz25 present a solution procedure for

large-scale, single-source, capacitated plant locat ionproblems (SSC P LP ). They present an MILP formula tionof the problem, with two types of decision variablesrelat ing t o the select ion of plants an d t o the a llocat ionof customers to plants, respectively.

I n a d d it ion t o t h e a b ov e w or ks , t h er e i s a l so asubsta ntia l literat ure on the problem of facility locat ion.Current et al .26 review the w ork in this a rea u p to 1990.They consider four ca tegories of objectives: (1) costminimization, (2) demand satisfaction, (3) profit maxi-mizat ion, an d (4) environmenta l concerns. Issues ofinterest a re a lso the number an d ty pes of facilities being

sited, their capacities, the nature of the problem (con-t i n uou s or d is cr et e ), t h e n a t u r e of t h e p a r a m et e r s(stochast ic or determinist ic), a nd finally t he na ture oft h e mode l ( st a t ic o r dy n a mic). M ore re ce n t wo rk isr e v i e w e d b y O w e n a n d D a s k i n , 27 w h i le S r i dh a r a n 28

reviews the solution methods for such problems.

This paper proposes a stra tegic planning model formulti-echelon supply chain networks, integrating com-ponents associated with production, facility locat ion,product transportat ion, and distribution. I t considersmult iproduct , mult i-echelon supply chain networksoperating under uncertainty. The network comprises an u mber of ma n u f a ct u rin g s i t e s, e a c h u s in g a s e t o fflexible, sha red resources for t he production of a num berof p rodu ct s . Th e ma n u f a c t u rin g s i t es a re a s s u me da l r ea d y t o ex is t a t g iv en l oca t i on s a n d s o d o t h ecustomer zones. The potential establishment of a num-ber of potential warehouses and distributions centersa t l oca t i on s t o b e s el ect e d f r om a s et of p os s ib lecandidat es is considered a s par t of the optimizat ion.

The a bove problem is formulat ed ma thema tically a sa n M I L P op t imiza t ion p roblem a n d is s o lv ed u s ingbranch-an d-bound techniques. Thus, in addit ion top rov id in g a s in g le m od el t h a t i n t eg r a t e s a l l o f t h easpects of the supply chain mentioned a bove, our aimis to obtain an optimal design of such systems. In view of t h e la rg e mon e y f lows in v olve d in s u pp ly ch a innetw orks, there ma y be substant ial differences betw eenoptima l a nd suboptimal solutions.

Ind. Eng . C hem. R es. , Vol. 40, No. 16, 2001 3587


4/20

Co mpa re d t o t h e ot h e r mo dels t h a t h a v e be en p re-sented in the lit era ture t o dat e (see Ta ble 1), our modelintegrates three dist inct echelons of the supply chain,within a single, ma thema tical program ming-based for-mula tion. Moreover, it ta kes into account t he complexityintroduced by the multiproduct nature of the productionfacilities, the economies of scale in transportation, andthe uncertainty inherent in the product demands.

3. Optimal Steady-State Design of Supply ChainNetworks

This w ork considers t he design of mult iproduct, m ulti-echelon production and distribution networks. As shownin Figure 2, the network consists of a number of existingmultiproduct ma nufacturing sites a t f ixed locat ions, an u mber of wa re h ou s es a n d dis t r ibu t ion ce n t ers ofunknown loca tions (to be selected from a set of ca ndida telocat ions), and finally a number of customer zones atfixed locat ions.

In genera l, each product can be produced at severa lplants at different locat ions. The production capa cityof each manufacturing site is modeled in terms of a set

of linear constraints relating the mean production ratesof the var ious products t o the a vaila bility of one or moreproduction resources. War ehouses a nd distributioncenters are described by upper and lower bounds ontheir material handling capacity. Warehouses can besupplied from more than one manufacturing site andc a n s u p p ly mo re t h a n o n e dis t r ibu t io n c e n t e r . E a c hdistribution center can be supplied by more than onewa rehouse. However, “single sourcing” constraints re-quiring tha t a distribution center be supplied by a singlewa rehouse (to be determined by the optimizat ion) arealso accommodated.

Each customer zone places demands for one or moreproducts. These demand s ma y be assum ed to be known

a priori; alterna tively, a number of dema nd scenarios,each w ith a given, nonzero probability, ma y be consid-e re d. A c u s t o me r ma y be s e rv e d by mo re t h a n o n edistribution center. Alternatively, single sourcing con-straints, according to which a customer zone must beserved by a single distribut ion center (to be determinedby the optimization), may be imposed.

The establishment of warehouses and distributioncenters incurs a f ixed infra structure cost . Operat ionalcosts include those associated with production, handlingof ma terial a t w arehouses a nd distribution centers, andtra nsporta t ion. Tra nsporta t ion costs a re a ssumed to bep ie ce wise l in e a r f u n ct ion s of t h e a c t u a l f low of t h eproduct from t he source sta ge to the dest inat ion sta ge,a n d t h e y ma y in c lu de t a x e s a n d du t ies .

The decisions to be determined include the number,locat ion, and capacity of warehouses and distributioncenters to be set up, the tra nsporta t ion links tha t needt o be e s t a bl is h e d in t h e n e t wo rk , a n d t h e f lo ws a n dp rodu ct ion ra t e s of ma t e ria ls . Th e obje ct iv e is t h eminimization of the total a nnua lized cost of the netw ork,taking into account both infrastructure and operat ingcosts.

This paper considers a stea dy-sta te form of the a boveproblem a ccording to which dema nds a re time-invaria nt(but possibly uncerta in) and a ll production a nd t ra ns-p ort a t ion f lows de t ermin ed by t h e op t imiza t ion a reconsidered to be t ime-averaged quantit ies. Section 3con s iders t h e de t ermin is t ic c a s e of k n own p rodu ct T

a b l e 1 .

S u p p l y C h a i n M o d e l s R e v

i e w e d i n T h i s W o r k

o p e r a t i o n a l d e c i s i o n s

s t r a t e g i c d e c i s i o n s

m o d e l t y p e

m o d e l f e a t u r e s

r e f

n o . o f

e c h e l o n s

c o n s i d e r e d a

h e u r i s t i c

d e t e r m i n i s t i c

s t o c h a s t i c

p r o d u c t i o n

m o d e l b

t r a n s p o r t a t i o n

c o s t m o d e l c

u n c e r t a i n t y

p r o d u c t i o n

a s s i g n m e n t

t o p l a n

t

i n v e n t o r y

l e v e l s

w a r e h o u s e - D C

a s s i g n m e n t /

l o c a t i o n

D C - c u

s t o m e r

a s s i g n

m e n t /

l o c a

t i o n

o b j e c t i v e

f u n c t i o n

6

n o e c h e l o n s

*

M P

C

*

c o s t

8

D C

*

C

*

*

*

c o s t

1 0

n o e c h e l o n s

*

M P

C

*

c o s t

1 2

n o e c h e l o n s

*

C

*

*

c o s t

1 4

P , W

*

M P

C

*

*

*

c o s t

1 5

n o e c h e l o n s

*

M P

C

*

c o s t

1 7

n o t c l e a r

*

C

*

c o s t

1 8

n o t c l e a r

*

C

*

c o s t

1 9

n o t c l e a r

*

C

*

*

c o s t

2 0

n o t c l e a r

*

C

*

*

*

c o s t

2 1

P

*

C

*

*

f l e x i b i l i t y

2 2

n o e c h e l o n s

*

C

*

c o s t

2 3

P , W

*

C

*

*

c o s t

2 4

W

*

C

*

*

c o s t

2 5

n o e c h e l o n s

*

C

*

*

c o s t

2 6

n o e c h e l o n s

*

C

*

*

c o s t

3 1

P , W

*

M P

C

*

*

c o s t

3 2

P , W

*

C

*

*

c o s t / r e s p o n s e

3 3

P

*

C

*

*

c u s t o m e r r e s p o n s e

3 4

P

*

M P

C

*

*

c o s t

3 5

P

*

M P

C

*

*

c o s t

p r o p o s e d m o d e l

P , W , D C

*

M P

V

*

*

*

*

*

c o s t

a

P l a c e s w h e r e i n v e n t o r y i s h e l d : P ,

p l a n t ; W , w a r e h o u s e ; D C , d i s t r i b u t i o n c e n t e r . b

S P : s i n g l e p r o d u c t . M P : m u l t i p r o

d u c t .

c

C : c o n s t a n t . V : v a r i a b l e .



5/20

demands, formulating the problem as an MILP. Section4 e x t en ds t h e f ormu la t ion t o t h e ca s e of u n ce rt a in

demands described in terms of multiple scenarios.3.1. Variables.3.1.1. Binary Variables.Four main

types of binary variables are defined:

3.1.2. Continuous Variables. This formulation usesa n u m be r of con t i nu ou s v a r i a b le s t o d es cr i be t h enetwork: (i) P i j is t he ra te of production of product i byplant j . (ii) Q ij m is the rate of flow of product i from plant

j to wa rehouse m . (iii) Q i m k is th e ra te of flow of producti from wa rehouse m to distribution center k . (iv) Q ik l isthe rate of flow of product i from distribution center k to customer zone l . These quant it ies are not generally

known a priori; in fact, their optimal va lues will dependstrongly on the product dema nds, th e production capac-

ity of the various plants, and the transportat ion costs.E a ch p rodu ct is s t ore d t wic e be fore re a ch in g t h e

customer, first in a wa rehouse and t hen in a distr ibutioncenter. The capacity of both has to be determined aspar t of the design of the distr ibution netw ork. Therefore,we in t rodu ce t h e f ol lowin g v a ria bles : (i) W m i s t h eca p a c i t y of wa re h ou s e m . (ii) D k i s t h e c a p a c i t y o fdistribution center k .

3.2. Constraints. 3.2.1. Network Structure Con-straints. A link betw een a wa rehouse m a nd a distribu-tion center k can exist only if warehouse m also exists:

I f i t is required that a certain distribution center beserved by a single w ar ehouse, then th is ca n be enforcedvia the constraint

wh e re K ss is the set of distribution centers for whichsingle sourcing is required.

If th e distribution center does not exist, then its linkswith wa rehouses can not exist either. This leads t o theconstraint

Figure 2. Overview of the supply chain netw ork.

X m k e Y m , ∀ m , k (1)

∑m

X m k ) Y k , ∀ k ∈ K ss

(2)

X m k e Y k , ∀ m , k ∉ K ss

(3)

Y m ) {1, i f th e war e h ou se at can d id a te posi t ion

m is t o be esta blished

0, ot h e r w is e

Y k ) {1, i f th e d istr ib u tion ce n ter at can d id ate

position k is t o be esta blished

0, ot h e r w i se

X m k )

{

1, i f w a r e hou s e m is to supply dist ribution

center k

0, ot h e r w i se

X kl ) {1, if distribution center k is to supply

customer zone l

0, ot h e r w is e



6/20

We note tha t the a bove is writ t en only for distributioncenters tha t a re not single-sourced. F or single-sourceddistribution centers, constraint (2) already suffices.

Th e l in k bet we e n a dis t r ibu t ion ce n t er k a n d acustomer zone l w ill exist only if the distr ibution centeralso exists:

So me c us t ome r zo n es ma y be s u bje ct t o a s in gles ou rcin g c on s t ra in t re q u ir in g t h a t t h e y be s e rv ed byexactly one distribution center:

while L ss is the set of customer zones for which singlesourcing is required.

3.2.2. Logical Constraints for TransportationFlows. A flow of materia l i from plant j to warehousem can take place only if warehouse m exists:

A flow of materia l i from warehouse m to distribut ioncenter k ca n t a k e p la c e on ly i f t h e corre sp on din gconnection exists:

A f low of ma t e ria l i from distribution center k t ocustomer zone l ca n t ake pla ce only if th e correspondingconnection exists:

Va lu e s f or t h e u p pe r bou n ds Q i j m ma x, Q i m k ma x, a n d Q i k l ma x

appearing on the right-hand sides of constraints (6)-(8) can be obtained as described in section 3.4.

There is usually a m inimum total flow ra te of material(of whatever type) that is needed to just ify the estab-lishment of a transportation link between two locationsin the network. This consideration leads to constraintsof the form

concerning t he links between a wa rehouse m a n d adistribution center k a nd between a distribution centerk and a customer demand zone l , respectively.

3.2.3. Material Balances.The a ctual r at e of produc-tion of product i by plant j must equa l the total f low ofthis product from plant j to all warehouses m :

F o r s t e a dy -s t a t e op era t io n , t h e re is n o s t ock a c -cumulat ion or deplet ion and, therefore, the total rateof f low of each product i l ea v i ng a w a r eh ou s e or a

dis t r ibu t io n c e n t e r mu s t e q u a l t h e t o t a l ra t e o f f lo w entering this node of the supply chain network:

The total rate of f low of each product i received by

each customer zone l from th e distribution centers m ustbe equal t o the corresponding ma rket deman d:

3.2.4. Production Resource.An importa nt issue indesigning the distribution network is the a bility of them a n u fa c t ur i ng p la n t s t o cov er t h e d em a n d s of t h ecustomers as expressed through the orders receivedfrom t he w ar ehouses.

The ra te of production of each product at any plantcannot exceed certain limits. Thus, there is always ama x imu m p rodu ct ion ca p a c i t y f or a n y on e p rodu ct ;moreover, there is often a minimum production rate that

must be maintained while the plant is operat ing:

It is common in many manufacturing sites for someresources (equipment, utilities, manpower, etc.) to beused by several production lines and at different stagesof the production of each product . This shared usagelimits the a vailability of the resource that can be usedf or a n y on e p u rpos e a s e xp res s ed by t h e f ol lowin gconstraint :

The coefficient Fi je expresses the amount of resource e used by plant j to produce a unit amount of product i ,while R je re p re s e n t s t h e t o t a l ra t e o f a v a i la bi l i t y o fresource e a t p la n t j .

3.2.5. Capacity of Warehouses and DistributionCenters. The capacity of a warehouse m generally has

to lie between given lower and upper bounds, W m mi n a n d

W m ma x, provided, of course, tha t the w arehouse is a ctu-

ally established (i.e. , Y m ) 1):

Simila r con s t ra in t s a p p ly t o t h e ca p a c i t ie s o f t h edistribution centers:

We g en er a l ly a s s u m e t h a t t h e ca p a ci t ie s o f t h ewa re h o u s e s a n d t h e dis t r ibu t io n c e n t e rs a re re la t e dlinear ly to the flows of materials t hat they ha ndle. Thisis expressed via the constra ints

wh e re R i m a nd i k a re given coefficients.

X k l e Y k , ∀ k , l (4)

∑k

X kl ) 1, ∀ l ∈ Lss (5)

Q i j m e Q i jm ma x

Y m , ∀ i , j , m (6)

Q i m k e Q i m k ma x

X m k , ∀ i , m , k (7)

Q i k l e Q i kl ma x

X kl , ∀ i , k , l (8)

∑i

Q i m k g Q m k mi n

X m k , ∀ m , k (9)

∑i

Q i kl g Q kl mi n

X k l , ∀ k , l (10)

P i j ) ∑m

Q i jm , ∀ i , j (11)

∑ j

Q i jm )∑k

Q i m k , ∀ i , m (12)

∑m

Q i m k ) ∑l

Q i kl , ∀ i , k (13)

∑k

Q i k l ) D i l , ∀ i , l (14)

P i j mi n

e P i j e P i j ma x

, ∀ i , j (15)

∑i

Fi je P i j e R je , ∀ j , e (16)

W m mi n

Y m e W m e W m ma x

Y m , ∀ m (17)

D k mi n

Y k e D k e D k

ma xY

k , ∀ k (18)

W m g∑i ,k

Ri m Q i m k , ∀ m (19)

D k g∑i ,l

i k Q i kl , ∀ k (20)



7/20

3.2.6. Nonnegativity Constraints. All continuousvaria bles must be nonnegative:

3.3. Objective Function. In general, a distributionnetwork involves both capital and operating costs. Theformer ar e one-off costs associated with the establish-m en t of t h e i nf r a s t r uct u r e o f t h e n et w o r k a n d , i npart icular, i ts w ar ehouses and distribution centers. On

the other han d, opera t ing costs a re incurred on a dailybasis a nd a re a ssociat ed with the cost of production ofm a t e r ia l a t p la n t s , t h e h a n d li ng of m a t e ri a l a t w a r e-houses and distribution centers, and the transportationof material through the network.

3.3.1. Fixed Infrastructure Costs. The infrastruc-ture costs considered by our formulation are related tot h e e s t a bl ish me n t o f a wa re h ou s e o r a dist r ibu t ioncenter at a candidate location. These costs are expressedin the following objective function terms:

W e a s s u me t h a t t h e p ro du c t io n p la n t s a re a lre a dy

established. Therefore, we do not consider the capitalcost associated with their design and construction. Wealso ignore an y infrast ructure cost associated w ith th ecustomer zones.

3.3.2. Production Cost.The production cost incurreda t a p la n t j is a ssumed to be proport ional t o the ra te ofproduction of each product i , w i t h a con st a n t u ni t

production cost C i j P . Th e corre s pon din g t e rm in t h e

objective fun ction is of the form

3.3.3. Material Handling C osts at Warehouses

and Distribution Centers.Ha ndling costs can usua llybe approximat ed a s linear functions of the thr oughputof each product being handled. They can be expressedas follows:

3.3.4. Transportation Costs. We start by consider-in g g e ne rica l ly t h e t ra n s port a t ion cos t in cu rred intransport ing a certain f low Q o f a m a t e r ia l i n t h e a r cbetween any tw o nodes in the supply chain network, asshown in Figure 3. Usua lly the u n i t transportation costis a nonincreasing function of the r at e of flow, reflectingeconomies of scale. Thus, the total transportation cost

is a piecewise linear function of the flow of ma teria l, ofthe form shown in Figure 4. The possible range of t hetra nsporta t ion flow is divided into NR subranges, eachcorresponding to a different (and progressively lower)

unit transportat ion price C r T. The limits of interval r

∈[1, NR] are denoted as Q h r -1 a n d Q h r .We introduce a new set of bina ry va riables Z r :

The above definit ion can be effected via the linearconstraints:

Constraint (26) ensures tha t only one of the va riablesZ r (say, for r ) r *) takes a value of 1, with all othersbeing zero. Constr a int (25) then forces Q r to 0 for all r * r *, while a lso bounding Q r * in t h e ra n g e [Q h r *-1, Q h r *].F in a l ly, con s t ra in t (27) implie s t h a t Q ) Q r * a n d ,therefore, Q ∈ [Q h r *-1, Q h r *], as desired.

The a ctual t ra nsporta t ion cost is given by the linearexpression

We note that, because of constraints (25)-(27), only oneof the terms in the above summation will be nonzero,e ff ect in g a l in e a r in t erp ola t ion bet we e n t wo p oin t s(Q h r *-1, C h r *-1) and (Q h r *, C h r *) in F igure 4 (where r * is sucht h a t Z r * ) 1).

H a v in g e s t a bl is h ed h ow we ca n g e ne rica l ly mode lpiecewise linear transportation costs, we can apply thesema n ip u la t io n s t o t h e f lows of ma t e ria l s pe ci fica l lytaking place in the supply chain network. Although wecould a ssume tha t each product i under considerationhas a different transportat ion cost , in reality many ofthe products in a ny given supply chain ar e likely to bev ery s imila r t o e a c h ot h e r . Th u s , w e in t rodu ce t h econcept of a product fa mily a s a subset of the products,a l l o f w h ich h a v e t h e s a me u n it t ra n s p ort a t ion cos t s .We denote these families as I f , f ) 1, . . . , NF, where NFis t h e n u mber of f a mil ie s . E a ch p rodu ct belon g s t o

Figure3. Transportat ion of mat erial in an a rc of the supply chainnetwork.

P i j g 0, ∀ i , j (21)

Q i jm g 0, ∀ i , j , m (22)

Q i m k g 0, ∀ i , m , k (23)

Q i kl g 0, ∀ i , k , l (24)

∑m

C m W

Y m +∑k

C k D

Y k

∑i ,j

C i j P

P i j

∑i ,m

C i m WH(∑

j

Q i j m ) +∑i ,k

C i k D H (∑

m

Q i m k )

Figure 4. Tran sporta t ion cost a s a piecewise linear function oft h e m a t e r ia l f low .

Z r ) {1, if Q ∈ [Q h r -1, Q h r ]0, ot h e r w is e

Q h r -1Z r e Q r e Q h r Z r , ∀ r (25)

∑r )1

NR

Z r ) 1 (26)

Q ) ∑r )1

NR

Q r (27)

C ) ∑r )1

NR

[C h r -1Z r + (Q r - Q h r -1Z r )C h r - C h r -1

Q h r - Q h r -1]



8/20

exactly one family. Thus,

where NI is the t otal number of products.B e ca u s e a l l p rod u ct s i n a f a m i ly h a v e t h e s a m e

tra nsporta t ion cost , we can a pply the a bove manipula-t io n s t o t h e ir combined f l o w r a t e i n e a c h a r c o f t h es u pp ly ch a in n e t wo rk. O v e ra l l, t h e n , t h e t o t a l t ra n s -porta t ion cost incurred in t he netw ork is given by

subject to the following constraints:(i) For the tr an sporta t ion of ma terial betw een plant s

and warehouses:

(ii) For the t ran sporta t ion of ma terial between w ar e-

houses a nd distribution centers:

(iii) For the transportat ion of material between dis-tribution centers and customer zones:

3.3.5. Overall Objective Function. B y combiningthe cost terms derived in sect ions 3.3.1 an d 3.3.4, weobtain the total cost of the supply chain network whichis to be minimized by t he optimizat ion:

Th e a b ov e m i ni m iz a t i on i s s u bje ct t o a l l of t h e

c o n s t ra in t s p re s e n t e d in s e c t io n 3. 2 a s we ll a s c o n -straints (29)-(40).3.4. C alculation of Upper Bounds on Network

Flows. Constraints (6)-(8) involve the upper boundsQ i j m

ma x, Q i m k ma x, a n d Q i kl

ma x. Th e t i gh t n es s of t h e M I LPformulat ion and, consequently, the efficiency of itssolution will depend crucially on the quality of thesebounds.

To obtain est imates for these bounds, consider theflow of a product i a lon g t h e a rc A f B connecting tw onodes A and B in a network (see Figure 5). This flow,denoted by Q i AB , cannot exceed either the total flow ofproduct i entering node A or the total flow of product i leaving node B. Thus, we obtain the expression

Applying expression (42) to the flows t akin g pla ce int h e s u pp ly ch a i n n et w o r k, w e ob t a i n E q u a t i on 43

considers the production rate P i j

of product i a t p la n t j as a f low notiona lly entering plant node j . Similarly eq 45 c o n s ide rs t h e de ma n d ra t e D i l for product i a t acustomer zone l as a f ixed flow leaving customer zonenode l .

The a bove formulas n eed to be a pplied in a n itera tive

ma n n e r. I n i t ia l ly , we s et Q i j m ma x ) +∞, Q i m k

ma x ) +∞, a n dQ i kl

ma x ) +∞. Then we repeatedly apply eqs 40-42 untilnone of the above upper bounds changes.

4. Supply Chain Network Design underUncertainty in Product Demands

The formulation presented in section 3 assumes thatthe product demands, D i l , a r e c o n s t a n t a n d a i m s t o

∪f )1NF

I f ) {i | i ) 1, ..., NI}

I f ∩ I f ′ ) L, ∀ f , f ′ ∈ {1, ..., NF}, f * f ′

∑f,j,m

C fj m + ∑f,m,k

C fm k + ∑f,k,l

C fk l (28)

∑r )1

NR fj m

Z f j m r ) 1, ∀ f , j , m (29)

Q h f j m , r -1Z f j m r e Q f j m r e Q h f j m r Z f j m r ,∀ f , j , m , r ) 1, ..., NR fj m (30)

∑i ∈I f

Q i j m ) ∑r )1

NR fj m

Q f j m r , ∀ f , j , m (31)

C fj m ) ∑r )1

NR

[C h f j m , r -1Z f j m r + (Q f j m r -Q h f j m , r -1Z f j m r )

C h f j m r - C h f j m , r -1

Q h f j m r - Q h f j m , r -1], ∀ f , j , m (32)

∑r )1

NR fm k

Z f m k r ) 1, ∀ f , m , k (33)

Q h f m k , r -1Z f j m k e Q f m k r e Q h f m k r Z f m k r ,∀ f , m , k , r ) 1, ..., NR fm k (34)

∑i ∈I f

Q i m k ) ∑r )1

NR fm k

Q f m k r , ∀ f , m , k (35)

C fm k )

∑r )1

NR

[C h f m k , r

-1Z f m k r + (Q f m k r -

Q h f m k , r -1Z f m k r )C h f m k r - C h f m k , r -1

Q h f m k r - Q h f m k , r -1], ∀ f , m , k (36)

∑r )1

NR fk l

Z f k l r ) 1, ∀ f , k , l (37)

Q h f k l , r -1Z f k l r e Q f k l r e Q h f k l r Z f k l r ,∀ f , k , l , r ) 1, ..., NR fk l (38)

∑i ∈I f

Q i kl ) ∑r )1

NR fk l

Q f k l r , ∀ f , k , l (39)

C jk l ) ∑r )1

NR

[C h fkl ,r -1Z f k l r + (Q f k l r -Q h f k l , r -1Z f k l r )

C h f k l r - C h f k l , r -1

Q h f k l r - Q h f k l , r -1], ∀ f , m , k (40)

m in ∑m

C m W

Y m +∑k

C k D

Y k + ∑i ,j

C i j P

P i j +

∑i ,m

C imWH

(∑ j

Q i jm ) + ∑i ,k

C i k D H

(∑m

Q i m k ) + ∑f,j,m

C fj m +

∑f,m,k

C fm k + ∑f,k,l

C fk l (41)

Q i ABma x ) min ( ∑C∈INQ i CAma x, ∑C∈OU T

Q i B Cma x) (42)

Q i jm ma x ) min (P i j

ma x, ∑

k

Q i m k ma x

) ∀ i , j , m (43)

Q i m k ma x ) min (∑

j

Q i j m ma x

, ∑l

Q i kl ma x

) ∀ i , m , k (44)

Q i kl ma x ) min (∑

m

Q i m k ma x, D i l ) ∀ i , k , l (45)



9/20

design a production and distribution network capableof han dling t hem. We now proceed to consider t he case

where product dema nds a re not known exactly but a resubject to some uncertainty.A good description of the uncertaint ies that occur

throughout the entire supply chain network, a ffect ingits performance and generally its operat ion, has beengiven by Davis.29 He considers uncertaint y a rising fromsuppliers, man ufacturing, a nd customers. Suppliers canbe characterized through their past performance, andtheir r esponsiveness can be predicted w ith reasonableaccuracy. Manufacturing problems can be addressedu s in g r el ia b i li t y a n d m a i n t en a n ce a n a l y s is f or t h eequipment. Finally, customer dema nds involve uncer-t a in t y wh ic h n e e ds t o be a ddre s s e d v ia h ig h q u a l i t yforecasting methods.

At a more genera l level, Zimmerma nn 30 identifies the

sources of uncerta inty a s la ck of informa tion, complexityof information, conflict ing evidence, a mbiguity, andmeasurement errors.

4.1. Handling Uncertainty in Supply Chain Op-timization. Most of the factors affecting the operationof the supply cha in netw ork can be classified as eithershort-term fluctua tions or long-term tr ends. To a certa inextent, short-term fluctuations are captured implicitlyin s t e a dy - s t a t e mo de ls s u c h a s t h e o n e p re s e n t e d insection 3 by averaging each flow in the network over asufficiently long period of t ime. On the other han d,taking into account long-term variations necessitates amore direct approach.

M os t r es ea r c h on a d d r es s in g u n ce rt a i n t y ca n b e

dist inguished as two primary approaches, referred ast he p r ob a b i l i st i c a p p r o ach a n d t h e sce n a r i o p l a n n i n g approach . As argued by Zimmermann, 30 the choice ofthe appropriat e method is context-dependent , w ith nos in gle t h e ory bein g s u f ficien t t o mode l a l l k in ds ofuncertainty.

P robabilistic models consider the uncerta inty a spectsof the supply cha in trea t ing one or more para meters asra n do m v a ria bles wit h k n own p roba bili t y dis t r ibu -tions.28 This approach has been adopted by Cohen andLee,31 Svoronos and Zipkin,32 Lee and B illington,33 P y k eand Cohen,34 a n d L e e et a l .36

O n t h e o t h e r h a n d, s c e n a rio p la n n in g a t t e mp t s t oc a p t u re u n c e rt a in t y by re p re s e n t in g i t in t e rms o f amodera te num ber of discrete realizat ions of the stocha s-t ic quantit ies, constitut ing dist inct scenarios.37 E a c hcomplete rea lizat ion of all uncerta in pa ra meters givesris e t o a s ce n a rio.28 The objective is to f ind robustsolutions which perform well under al l scenarios. Insome applica tions, scena rio planning replaces forecast-in g a s a wa y o f t a k ing in t o a c cou n t p o t en t ia l c h a n g e sand trends in a business environment.

Th e se a re v a r iou s common a p p roa c h es t o robu s toptimization 38 seeking, for example, to optimize theexpected performance over all scenarios, to optimize theworst-case scenario, or to minimize th e expected orworst-case “regret” across all scenarios.

Mulvey39 uses scenario planning for formulat ing a ndsolving operational problems, while J enkins 40 employed

t h is a p proa c h t o a s s es s t h e e n viron me n t a l impa c t ofpossible disasters.

Mohamed 41 uses a scena rio approach to decide on thedesign of a production and distribution network thatoperates under varying exchange rates. A number ofscenarios for different exchange rates aim to determinethe production policy of the company, which operatesin more than two countries. One important issue thatarises in the context of the scenario planning approachis t h e in cre a s e in comp u t a t ion a l comp le xit y a s t h enumber of scenarios increases (see, e.g. , Cheung andPowell42). One approach towa rd a ddressing this concernis via the use of par allel computat ion.43 Alternatively,specialized solution techniques have been considered(e .g . , A h med a n d Sa h in idis44). Ve ry re ce n t ly , M ir-H a s s a n i et a l .45 ha ve proposed a heuristic a pproa ch forhan dling very lar ge numbers of scenarios.

4.2. Scenario Generation. In t his paper, we adopta scenario planning approach for handling the uncer-ta inty in product dema nds. A question th at needs to bead dressed in this context concerns the genera tion of thescenarios to be considered. It is, of course, possible toa s s u me t h a t t h e d e m a n d f or e a ch p rod u ct i n e a c hcustomer zone is an independent random parameter.

However, more realistically, demands for similar prod-u ct s wil l t e n d t o be c orre la t e d a n d wil l u l t ima t e ly becontrolled by a small n umber of major fa ctors such a seconomic growt h, political st a bility, competitor a ctions,and so on. This view is consistent with that of Mobasherie t a l . ,46 who describe scenarios as plausible possiblesta tes derived from the present st a te with considerat ionof potential ma jor industry events.

Th e ov er a l l a i m s h ou ld b e t o con s t r uct a s et ofscenarios representa t ive of both optimist ic a nd pes-simist ic situat ions within a risk analysis strategy. Ane xa m p le of s u ch a n a p pr oa c h i s t h e Tow e r s P e r insoftwa re tool described by Mulvey.40 A 12-step procedurefor generat ing appropriate scenarios and a discussionon the use of scena rio pla nning t echniq ues ar e presented

by Vanston et al .47From t he pra ctica l point of view, th e ma in conclusion

o f t h e a bo v e dis c u s s io n is t h a t t h e t o t a l n u mbe r o fscenarios tha t have to be considered is typically muchsmaller than what might be expected given the (oftenlarge) numbers of products and customer zones. In anycase, for th e purposes of this pa per, we will a ssume tha tt h e re is s o me s y s t e ma t ic wa y o f g e n e ra t in g p ro du c t

demand est imates D i l [s ] for a number of scenarios s ) 1,

. . . , NS.4.3. Mathematical Formulation. The formulat ion

of sect ion 3 needs to be modified to t ake into a ccountthe mult iple scenarios which are used to capture theuncertainty aspects (cf. section 4.1). Because we are still

a imin g a t a s i n g l e network design, the binary varia blesY m , Y k , X m k , a n d X kl and the capacity variables W k a n dD k rema in unchanged. However, the operating va riablesrelat ing t o production a nd t ra nsporta t ion flows will bedifferent depending on which demand scenario materi-a l izes . Th u s , we in t rodu ce a s u pe rscript [s ] o n t h e

corresponding variables which now become P i j [s ], Q i j m

[s ] ,

Q i m k [s ] , a n d Q i kl

[s ]. An y con s t r a in t t h a t i n vol ve s t h es evar iables must be enforced separa tely for each scena rio.For exa mple, constr a int (7) now becomes

The rest of the constraints in the formulat ion can be

Figure 5. Obta ining upper bounds for network f lows.

Q i m k [s ]

e Q i m k [s ],max

X m k , ∀ i , m , k , s ) 1, . .. , N S (7′)



10/20

derived in a simila r ma nner from those in section 3 andwil l n o t be re pe a t e d h e re . To a rr iv e a t a me a n in g fu l

objective function for th e optimiza tion, we a ssume th a tthe probability of scenario s occurring in pract ice isk n o wn a n d is de n o t e d by ψs . These probabilities willgenerally sat isfy

Our aim is to minimize the expected value of t he cost ofthe network taken over all of the scenarios. This leadsto t he modified objective function

(a) Scenario-Dependent Distribution Network Structure. T h e a bo v e dis c u s s io n wa s ba s e d o n t h eassumption that the structure of the distribution net-work (i.e. , the t ra nsporta tion links between w a rehouses,distribution centers, an d customer zones) wa s indepen-dent of the scenar io. In ma ny cases, this is unnecessarybecause the cost associated with the establishment of atra nsporta tion link is relat ively sm all. This is especiallyt h e c a s e wh e n t ra n s p o rt a t io n is o u t s o u rc e d t o t h ird

part ies. In such cases, we could allow the network oftransportat ion links to be different from one scenarioto anotherseffectively postponing the decision until thea c t u a l de ma n ds a re k n own .

The only change to the model formulation discussedearlier in this sect ion is that now var iables X m k a nd X kl also have a superscript [s ] to denote tha t th ey can ta kedifferent values for each scenario. For example, con-straint (7′) now becomes

while the objective function (47) remains unchanged.

5. A Case Study in the Steady-State Design of Supply Chain Networks

To i llu st ra t e t h e a p p lica bi l it y of t h e ma t h e ma t ica lformula tions presented in t his paper, we consider th reemanufacturing plants producing 14 different types ofproducts and located in three different European coun-tries, namely, the United Kingdom, Spain, and It aly (seeFigure 6). Each plant produces several products usinga number of shared production resources. However, nosingle plan t produces the entire ra nge of products.

P roduct dema nds a re such tha t Eur ope can be dividedinto 18 cust omer zones loca ted in 16 different count ries.We consider the establishment of enough distribution

centers (DCs) to cover the whole market. The distribu-tion centers can be located anywhere in 15 countriesand ar e to be supplied by a number of war ehouses, thelocat ion of which is to be decided a mong six candidat eplaces.

5.1. Problem Description. 5.1.1. ProductionPlants. The ma ximum production ca pacity of each plan t

with respect to each product (i .e. , the para meter P i j ma x

of our formulation) is given in Table 2. The correspond-ing minimum production ra tes a re ta ken to be zero (i.e.,

P i j mi n ) 0 , ∀ i , j ).The production of each product at each plant makes

use of shared equipment resources a s indicated in Ta ble3. The last column of this table lists the total rate ofa vaila bility of each resource (in hours of useful operat ionper week). These ar e th e par am eters R je in constra int(16) of our formulation.

The unit production costs for the products are listedin Table 4.

5.1.2. Warehouses and Distribution Centers. Theinfrast ructure costs for the esta blishment of the variouswarehouses and distribution centers under consider-a t io n a re l is t e d in T a ble 5; t h e s e h a v e a lre a dy be e na mortized a nd ar e expressed in £/w eek.

The w ar ehouses and distribution centers a re a ssumedt o h a v e m a x im u m m a t e r ia l h a n d li n g ca p a ci t ie s of14 000 an d 7000 t e/week, respectively. All minimumha ndling capa cities ar e set t o zero. The coefficients R im a nd i k relat ing t he capacity to the throughput of eachmaterial handled [cf. constraints (19) and (20)] are alltaken to be unity. The unit handling costs (includinglabor or other operat ing costs) are also listed in Table5; for t he purposes of this case study, w e assume t ha t

Table 2. Maximum Production Capacity of Each Plant for E ach Product

ma ximum production capa city (te/week)

pla n t P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 10 P 11 P 12 P 13 P 14

P L 1 158 2268 1701 1512 0 812 642 482 320 504 0 661 441 221P L 2 0 1411 1058 1328 996 664 664 0 0 0 530 496 330 0P L 3 972 778 607 540 0 416 416 312 208 0 403 0 270 0

∑s )1

NS

ψs ) 1 (46)

m in ∑m

C mW

Y m + ∑k

C kD

Y k + ∑s )1

NS

ψs (∑i , j

C i j P

P i j [s ] +

∑i ,m

C i m WH

(∑ j

Q i j m [s ] ) +∑

i ,k

C i k D H

(∑m

Q i m k [s ] ) + ∑

f,j,m

C fj m [s ] +

∑f,k,l

C fm k [s ] + ∑

f,k,l

C fk l [s ]) (47)

Q i m k [s ]

e Q i m k [s ],max

X m k [s ] , ∀ i , m , k , s ) 1, . . . , NS (7′′)

Figure 6. Europe-wide supply chain network design (candidatelocations for distribution centers not shown).



11/20

they a re the sa me for a ll products but ma y differ fromone location to another.

5.1.3. Transportation Costs. The transportat ioncosts are generally assumed to depend on the geographi-cal distances between the locat ions involved. For thep u rp o s e s o f t ra n s p o rt a t io n , t h e 14 p ro du c t s ma y beaggrega ted into the t hree families shown in Table 6.

The baseline unit costs for tra nsporta tion from plan tsto warehouses, warehouses to distribution centers, anddistribution centers to customer zones are shown inTables 7-9, respectively. These a pply to t ra nsporta tionflows up t o 40 te/w eek for ea ch fa mily. Above this limit ,it is possible to deploy a lterna tive models of tra nsporta -

tion tha t ha ve lower a verage unit costs. The ma gnitudesof these costs (relat ive to the corresponding baselinevalues) are shown in Table 10.

5.2. C ase Study 1: Deterministic Product De-mands. We start by considering a deterministic prob-lem aiming t o sat isfy t he demands shown in Table 11.

The optimal solution of the model of section 3 leadsto the network structure shown in Figure 7. The numberabove ea ch arc denotes the corresponding tota l ma terialf low (in t e /we e k), wh ile t h e n u mber n e xt t o e a chcustomer zone is the t otal dema nd a t this zone (also inte/w eek). Ta bles 12-15 present the costs and the flowsfor ea ch stage in the distribution cha in.

The ma rket is served by three wa rehouses and threedistr ibution centers loca ted close to the production sites.Th i s i s p ri m a r il y a r es u lt of t h e d em a n d p a t t er n sconsidered because the three countries which host themanufacturing plants are also the biggest customers.

The corresponding computat iona l sta t ist ics for thisproblem are summarized in Table 16. The MILP prob-lems were solved using the CP LEX v6.548 code embed-ded within oo M I L P (Tsiakis et al.49). As can been seenfrom the results of Table 16, computat ional demandsare relatively modest in this case despite the relativelyhigh number of integer varia bles. This can be a tt ributedprimarily to the low integra lity ga p of this formulat ion.

5.3. CaseStudy 2: Uncertain Product Demands.We now consider a case w ith three possible productdemand scenarios. The first one is the sa me as tha t used

Table 3. Utili zation and Availability Data for Shared Manufacturing Resources

sha red equ ipment resource utiliza tion coefficient (h/te)plantr esour ce P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 10 P 11 P 12 P 13 P 14

t ot a lresource

availability(h/w eek)

P L 1.E 1 0.2381 120P L 1.E 2 0.0463 0.0617 0.0694 105P L 1.E 3 01.634 0.2178 0.3268 105P L 1.E 4 0.2267 0.3401 0.6802 150P L 1.E 5 0.1292 105P L 1.E 6 0.6667 105P L 2.E 1 0.1984 0.2118 0.3174 105P L 2.E 2 0.0793 0.1054 0.1582 0.1582 105P L 3.E 3 0.0740 0.1000 120P L 3.E 2 0.1976 0.2222 165P L 3.E 3 0.1200 0.1543 0.3968 0.3968 0.5291 0.7936 120P L 3.E 4 0.2976 0.4444 120

Table 4. Unit Production Costs

unit product ion cost s (£/te)

pla n t P 1 P 2 P 3 P 4 P 5 P 6 P 7

P L 1 61. 27 61. 27 61. 27 61. 27 61. 27 61. 27 256. 90P L 2 59. 45 59. 45 59. 45 59. 45 59. 45 59. 45 268. 50P L 3 61. 44 61. 44 61. 44 61. 44 61. 44 61. 44 270. 80

unit product ion cost s (£/te)

pla n t P 8 P 9 P 10 P 11 P 12 P 13 P 14

P L 1 2 56 .9 0 2 56 .9 0 6 1. 27 2 56 .9 0 2 56 .9 0 2 56 .9 0 2 56 .9 0P L 2 2 68 .5 0 2 68 .5 0 5 9. 45 2 68 .5 0 2 68 .5 0 2 68 .5 0 2 68 .5 0P L 3 2 70 .8 0 2 70 .8 0 6 1. 44 2 70 .8 0 2 70 .8 0 2 70 .8 0 2 70 .8 0

Table 5. Fixed Infrastructure and Material HandlingCosts for Candidate Warehouses and DistributionCenters

wa rehou se/DCinfrastructure cost

(£/w eek )han dling cost

(£/t e)

W1 10000 4.25W2 5000 4.55W3 4000 4.98W4 6000 4.93W5 6500 4.06W6 4000 5.28D C 01 10000 4.25D C 02 5000 4.55D C 03 4000 4.98D C 04 6000 4.93D C 05 6500 4.85D C 06 4000 3.90D C 07 6000 4.06D C 08 4000 3.08D C 09 5000 6.00D C 10 3000 4.85D C 11 4500 4.12D C 12 7000 5.66D C 13 9000 5.28D C 14 5500 4.95D C 15 8500 4.83

Table 6. Division of Products into TransportationFamilies

fa m ily pr oduct s fa m ily pr oduct s

F 1 P 1-P 6, P 10 F 3 P 11-P 14F 2 P 7-P 9

Table 7. Baseline Unit Transportation Costs from Plantsto Warehouses

warehouses

pla n t W1 W2 W3 W4 W5 W6

Fa mily F 1 Pr oducts (£/te)P L1 1.24 58.56 62.30 26.16 17.44 36.13P L2 60.82 1.68 70.96 43.93 70.96 55.76P L3 76.16 79.21 1.52 54.83 68.54 41.12





12/20

for the deterministic case considered in section 5.2. Thedemands for the second and third scenarios are givenin Tables 17 and 18, respectively. We assume that allthree scenarios are equally probable, i.e. , ψ1 ) ψ2 ) ψ3) 1/3. O u r a im is t o de sig n a n e t work t h a t ca n h a n dleall three scenarios while minimizing the expected cost.

To gain some understa nding of the implications of thethree demand patt erns, we first consider each scenarioin isolation by formulating and solving the correspond-ing deterministic design problem as described in section3. The optima l netw ork structures a re shown in Figures7-9, respectively. The three network structures differin t he a llocat ion of products to plant s, w hich dependsstrongly on th e imposed dema nd pa tt erns. More specif-ically, in the first scenario, the I talian network needs

contribut ions from the U .K. an d Spa nish plant s to coverthe dema nds placed by its customers, while in the othertwo scenarios, i t is capable of supplying the requiredamounts out of its own production. The set of customersassigned to each distribution center is also different ineach scenario, which demonstrates the effects of trans-porta t ion costs on the network str ucture.

We now consider all three scenarios simultaneously,ma k in g u s e o f t h e s c e n a rio p la n n in g f o rmu la t io n o fsection 4. The optimal network structure obtained isshown in Figure 10. The three numbers above each arcdenote the t otal flow ra te (in te/week) of ma teria l overthis link for the three scenarios considered. S imilarly,the th ree numbers next to each customer zone are t hecorresponding total product demands.

Figure 7. Optima l network for determinist ic product demands.

Table 8. Baseline Unit Transportation Costs from Warehouses to Distribution Centers

distribution centers

w a reh ouse D C01 D C02 D C03 D C04 D C05 D C06 D C07 D C08 D C09 D C10 D C11 D C12 D C13 D C14 D C15

Fa mily F 1 Pr oducts (£/te)W1 0 74.40 76.13 25.96 69.21 29.41 17.30 117.66 44.99 1 10.74 76.13 12.11 39.79 64.02 60.56W2 58.85 0 62.96 45.16 1 09.49 69.80 67.06 94.44 90. 33 1 45.08 17.79 60.22 52. 01 108.12 72.54W3 72.83 76.14 0 49.66 94.35 99.32 62.90 43.04 79.45 1 29.12 94.35 62.90 33.10 104.29 28.14W4 28.54 62.78 57.08 0 87.52 58.98 32.34 106.55 62.78 1 35.09 72.30 22.83 19.02 89.42 49.47W5 16.51 73.58 57.06 2 5.52 48.05 37.54 0 93.10 25.52 84.09 78.08 7.50 30.03 45.05 42.04W6 69.52 67.78 34.76 17.38 78.21 69.52 3 4.76 79.95 59.09 121.66 78.21 3 3.02 0 83.42 2 7.80

Fa mily F2 P rodu cts (£/te)W1 0 75.28 77.03 26.26 70.02 29.76 17.50 119.04 45.51 1 12.04 77.03 12.25 40.26 64.77 61.27W2 60.87 0 65.12 46.71 1 13.25 72.20 69.36 97.68 93. 43 1 50.06 18.40 62.29 53. 79 111.84 75.03W3 90.75 94. 88 0 61.88 117.57 123.76 78.38 53.63 99. 00 160.89 117.57 78.38 41. 25 129.95 35.06W4 28.88 63.54 57.77 0 88.58 59.69 32.73 107.83 63.54 1 36.72 73.17 23.10 19.25 90.50 50.06W5 17.90 7 9.77 6 1.86 2 7.67 52.09 40.70 0 100.93 27.67 91.16 84.65 8.14 3 2.56 48.84 45.58W6 86.62 84.46 43.31 21.65 97.45 86.62 43.31 99.62 73.63 1 51.59 97.45 41.14 0 103.95 34.65

Fa mily F3 P rodu cts (£/te)W1 0 69.15 70.78 24.14 64.32 27.33 16.08 109.35 41.81 1 02.92 70.76 11.25 36.98 59.50 56.28W2 69.66 0 74.52 53.46 129.61 82.63 79.38 111.79 106. 93 171.74 21.06 71.28 61. 56 127.99 85.87W3 92. 01 96. 19 0 62. 73 119. 19 125. 47 79. 46 54. 37 100. 37 163. 11 119. 19 79. 46 41. 82 131. 74 35. 55W4 26.53 58.37 53.07 0 81.37 54.83 3 0.07 99.06 58.37 125.59 67.22 2 1.22 17.69 83.14 4 5.99W5 20.49 91.29 70.80 31.67 59.62 46.58 0 115.51 31.67 104.33 96.88 9.31 37.26 55.89 5 2.16W6 87.82 85.63 43.91 21.95 98.80 87.82 4 3.91 85.70 63.34 130.42 83.84 3 5.40 0 89.43 3 5.13



13/20


14/20

It is interesting to compare the expected costs of thenetwork to the costs that would be incurred if one were

t o f ix t h e s t ru ct u re o f t h e n e t wo rk t o t h o s e s h own inFigures 7-9, respectively, optimizing the operationaldecisions only. As can be seen from Table 19, the fixedstructure of Figure 7 h as the sa me expected cost overthe t hree demand scenarios as the optimal solution forthe three scenarios case. This is due to the fact that thecorresponding netw ork configurat ions are t he sa me (cf.Figures 7 and 10). On t he other ha nd, the networks ofFigures 8 and 9 result in more expensive operat ionstha n the optimal solution of Figure 10.

Figure 8. Optima l network str ucture for scenario 2.

Figure 9. Optima l network str ucture for scenario 3.

Table 10. Economies of Scale in T ransportation Costs

amount transported(te/w eek)

tra nsportat ion costrelat ive to the ba seline

0-40 1.0040-100 0.95

100-1000 0.891000-5000 0.80



15/20

Tables 20-23 present the costs and flows for eachsta ge in the distribution chain for each scena rio in theoptima l solution of the scena rio planning formulat ion.

Some computat ional results for this case are shownin Ta ble 24. A comp a ris on wit h t h e corre s pon din g

statistics for the deterministic design problem (cf. Table16) i n d ica t e s t h a t b ot h t h e i n t eg r a l it y g a p of t h eformulat ion and the performance of the branch-and-bound algorithm remain reasonable (e.g. , with respectto the number of nodes examined). However, th e overa llcomputat ional cost grows significantly because of theincrease in the number of constraints and variables.

6. Conclusions

The design of supply chain networks is a difficult taskbecause of the intrinsic complexity of the major sub-systems of these networks and the many interact ions

Figure 10. Optimal netw ork structure for a ll three scenarios considered simultaneously.

Table 11. Product Demands by Customer Zone

product dem an ds (te/w eek)

cust om er zon e P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 10 P 11 P 12 P 13 P 14

C Z01 18 0 0 106 0 252 0 0 43 0 0 0 70 34C Z02 0 99 55 203 76 0 30 0 0 0 20 0 0 0C Z03 0 155 50 266 0 66 17 0 0 0 0 0 15 0C Z04 15 150 126 0 0 0 5 27 0 0 0 25 0 0C Z05 0 0 92 0 0 0 0 0 21 0 0 0 50 0C Z06 0 0 50 0 0 68 0 0 20 0 0 0 10 10C Z07 0 114 0 140 0 0 0 0 34 0 0 0 68 0C Z08 0 50 0 45 40 23 0 0 5 0 0 0 0 0C Z09 0 50 0 0 0 0 52 0 7 0 0 0 0 0

C Z10 0 0 0 17 0 0 0 0 5 0 0 0 16 0C Z11 0 50 0 31 20 0 0 0 0 0 20 0 15 20C Z12 10 31 0 0 0 100 13 0 0 38 0 0 0 0C Z13 0 21 0 100 0 0 15 0 0 0 0 0 50 0C Z14 0 0 0 50 0 0 0 7 0 0 0 0 0 0C Z15 0 0 0 150 0 0 10 0 0 0 15 0 0 10C Z16 15 0 68 0 0 0 5 20 0 0 0 20 0 0C Z17 0 103 0 110 0 44 12 0 0 0 0 0 13 0C Z18 0 0 0 0 0 100 0 0 0 266 0 0 0 0

Table 12. Costs Associated with Manufacturing Plants

plantproduction(te/w eek )

man ufacturing cost(£/w eek )

U .K . 2729 295 516E S 659 57 201I T 1074 84 175

t ot a l 4462 436 892

Table 13. Costs Associated with Warehouses

warehouseinfrastructure cost

(£/w eek )throughput

(te/w eek )handling cost

(£/w eek )

U .K . 10 000 2624 11 152E S 5 000 639 2 907I T 4 000 1199 5 971t ot a l 19 000 4462 20 030

Table 14. Costs Associated with Distribution Centers

distributioncenter

infrastructure cost(£/w eek )

throughput(te/w eek )

handling cost(£/w eek )

U .K . 10 000 2624 11 152E S 5 000 639 2 907I T 4 000 1199 5 971

t ot a l 19 000 4462 20 030



16/20

a mo n g t h e s e s u bs y s t e ms , a s we ll a s e x t e rn a l f a c t o rss u ch a s t h e con s ide ra ble u n ce rt a in t y in p rodu ct de -mands. In the past, this complexity has forced much ofthe research in this area to focus on individual compo-nents of supply chain networks. Recently, however,at tention has increasingly been placed on the perfor-ma n c e , de s ig n , a n d a n a ly s is o f t h e s u p p ly c h a in a s awhole.

This paper has proposed a model based on a detailedma t h e ma t ic a l p ro g ra mmin g f o rmu la t io n t h a t a ims t oaddr ess some of t he complexity of the above problem.In particular, it considers flexible production facilitiesin which a number of products are produced, making

u s e o f s h a re d re s ou rce s . I t a ls o t a k e s in t o a ccou n tflexible tra nsporta t ion modes with economy-of-scaleeffects. Although the handling of uncertainty is dem-onstrated by considering uncertaint ies in product de-ma nds, other uncerta inties (e.g., in unit production a nd/or tra nsporta tion costs) ca n, in principle, be ha ndled ina simila r ma nner. Overa ll, it is hoped tha t th is approachca n l ea d t o a q u a n t i t a t i ve t ool t o s u pp or t s t r a t eg icplanning decisions in supply chain network design.

As is often the case with MILP-based formulat ions,one important issue is that of computational complexity,especially in th e context of scenario plann ing a pproachesfor t he ha ndling of uncertaint y. Clearly, t he identifica-

Table 15. Transportation Costs throughout the System

w a r eh ouse dist r ibut ion cen t er cust om er zon e

plantl oca t i on l oca t i on

flow (te/w eek )

cost a

(£/w e e k) loc a t ionflow

(te/w eek )cost a

(£/w e e k) loc a t ionflow

(te/w eek )cost a

(£/w eek )

U .K . U .K . 2634 3936 U .K . 2634 0.0 U .K . 523 1167F R n 198 5484S E 163 16152I R 158 6524NL 356 8136D K 109 7108

F I 38 6267NO 7 5408B E 192 3086C H 186 10360F R s 128 8314D E 366 17451

E S 20 2066I T 85 7778

E S E S 619 1380 E S 639 0.0 E S 483 1034P T 156 4802

I T 40 4258I T I T 1074 2046 I T 1199 0.0 I Tn 569 15235

G R 163 9351AU 185 7161I Ts 282 7872

Tot a l 4462 21464 4462 10769 4462 141002

a This is t he cost incurred for ma terial t o be tra nsported t o this dest inat ion.

Table 16. Computational Statistics for the Deterministic Design Problem

n o. of con st r a int s 17949 in t egr a lit y ga p 0.00%n o. of in t eger va r ia bles 4917 opt im a lit y m a r gin 0.1%n o. of con t in uous va r ia bles 11025 n o. of br a n ch -a n d-boun d n odes 2634opt im a l object ive va lue 677458 C P U t im e (S U N U lt r a S P AR C 60) 1082fully r ela xed L P object ive fun ct ion 677458 solver C P L E X6.5

Table 17. Product Demands by Customer Zone for Scenario 2



C Z01 18 0 0 506 0 452 0 0 43 0 0 0 120 34

C Z02 0 499 155 203 76 0 30 0 0 0 20 0 0 0C Z03 0 155 0 166 0 66 17 0 0 0 0 0 15 0C Z04 15 0 126 0 0 0 5 27 0 0 0 25 0 0C Z05 0 0 92 0 0 0 0 0 21 0 0 0 50 0C Z06 0 0 0 0 0 68 0 0 20 0 0 0 10 10C Z07 0 14 0 40 0 0 0 0 34 0 0 0 68 0C Z08 0 0 0 45 0 23 0 0 5 0 0 0 0 0C Z09 0 0 0 0 0 0 52 0 7 0 0 0 0 0C Z10 0 0 0 17 0 0 0 0 5 0 0 0 16 0C Z11 0 0 0 31 0 0 0 0 0 0 0 0 15 0C Z12 0 31 0 0 0 0 13 0 0 38 0 0 0 0C Z13 0 21 0 0 0 0 15 0 0 0 0 0 0 0C Z14 0 0 0 0 0 0 0 7 0 0 0 0 0 0C Z15 0 0 0 0 0 0 10 0 0 0 15 0 0 0C Z16 15 0 68 0 0 0 5 20 0 0 0 20 0 0C Z17 0 103 0 110 0 44 12 0 0 0 0 0 13 0C Z18 0 0 0 0 0 0 0 0 0 266 0 0 0 0



17/20

tion of the true underlying sources of such uncertainty(e.g. , m ajor polit ical an d economic events) is key ingenerat ing a representat ive but not excessive numberof scenarios.

Co mp le me n t a ry t o t h e de riv a t io n o f t h e min imu mp os s ible n u mbe r of s ce n a rios is t h e u s e of s pe cia ldecomposition techniques that exploit the special struc-ture of the “multiperiod” problem occurring in scenario-

based optimizat ion (see, for instance, Varvarezos andG ro s s ma n n50) . However, a complicat ion that arises inthe case of the problem considered in this paper is theexistence of discrete operational decisions (e.g., withrespect to transportation) within each of the periods. 51

This reduces the applicability of decomposition tech-niques that rely on the availability of gradient informa-tion from the subproblems describing each period. 52

Table 18. Product Demands by Customer Zone for Scenario 3



C Z01 38 0 0 206 0 252 0 0 4 3 30 0 0 120 34C Z02 0 199 155 103 76 0 30 0 0 0 20 0 0 0C Z03 0 155 0 80 0 66 17 0 0 0 0 0 15 0C Z04 15 0 126 0 0 0 5 27 0 0 0 25 0 0C Z05 0 0 92 100 0 0 0 0 21 0 45 0 50 0C Z06 0 0 0 0 36 68 15 0 20 30 0 0 10 10C Z07 0 14 0 80 0 0 0 40 34 0 0 0 68 0

C Z08 0 0 0 45 0 23 0 0 5 0 40 0 0 0C Z09 0 150 0 0 0 0 52 20 7 0 0 0 0 0C Z10 0 0 0 17 0 0 0 0 5 0 0 0 16 0C Z11 0 150 0 31 0 0 0 0 0 0 30 0 15 30C Z12 0 31 0 0 30 0 13 0 0 38 0 0 0 0C Z13 0 21 0 86 0 50 15 0 0 0 0 0 0 0C Z14 0 0 0 120 0 0 0 7 20 0 0 0 0 0C Z15 0 0 0 0 0 0 10 0 0 0 15 0 0 0C Z16 15 0 68 0 0 0 5 20 0 0 0 20 0 55C Z17 0 103 0 110 0 44 12 0 0 0 0 0 13 0C Z18 0 0 0 0 17 0 0 0 0 266 0 0 50 0

Table 19. Comparison of Network Structures

network structureminim um expected cost

o ve r t h r ee d em a n d s ce na r i os (£/w e ek ) d if fe r en ce f r om op t im a l (£/w e ek )

F igur e 7 1 957 046 0F igur e 8 1 977 571 20 525F igur e 9 1 962 922 5876F igur e 10 1 957 046 0

Table 20. Costs Associated with Manufacturing Plants

pr oduct ion (t e/w eek) m a n ufa ct ur in g cost (£)

pla n t scen a r io 1 scena r io 2 scen a r io 3 scen a r io 1 scen a r io 2 scen a r io 3

U .K . 2654 2378 2569 290 896 270 566 285 749E S 734 1029 1099 61 684 74 762 128 186I T 1074 835 763 84 175 67 304 83 685t ot a l 4462 4242 4460 436 755 412 632 497 620expect ed va lue 4383 448 555

Table 21. Costs Associated with Warehouses

t hr ough put (t e/w eek) h an dling cost (£/w eek)

w a r eh ou se W i nf ra s t ru ct u re cos t (£/w e ek ) s ce na r io 1 s cen a r io 2 s cen a r io 3 s ce na r io 1 s cen a r io 2 s cen a r io 3

U .K . 10 000 2624 2414 2828 11 152 10 983 12 019E S 5 000 639 1029 829 2 907 4 681 3 817I T 4 000 1199 799 793 5 971 3 955 4 073t ot a l 19 000 4462 4242 4460 20 030 19 619 18 680expect ed va lue 19 000 4383 19 423

Table 22. Costs Associated with Distribution Centers

t hr ou gh pu t (t e/w eek) h a nd lin g cost (£/w eek)

d i st r i bu t io n ce n te r i n fr a s t r u ct u r e co st (£/w e ek ) s ce n a r io 1 s ce n a r io 2 s ce na r i o 3 s ce na r i o 1 s ce na r i o 2 s ce na r i o 3

U .K . 10 000 2624 2414 2828 11 152 10 983 12 019E S 5 000 639 1029 829 2 907 4 681 3 817I T 4 000 1199 799 793 5 971 3 955 4 073t ot a l 19 000 4462 4242 4460 20 030 19 619 18 680expect ed va lue 19 000 4383 19 423



18/20

Notation

C h f j m r ) t ra ns port a t ion cos t for product s of fa mily f frompla nt j t o w a rehous e m over ra nge r of t ra ns port a t ionflow

C h f k m r ) transportation cost for products of family f fromw a rehous e m t o dis t r ibut ion cent er k over ra nge r oftransportation flow

C h f k l r ) t ra ns port a t ion cos t for product s of fa mily f fromdistribution center k to customer zone l over ra nge r oftransportation flow

C i m WH ) unit handling cost for product i a t w a r e h ou s e m

C i k DH ) unit ha ndling cost for product i a t dis t r ibut ioncenter k

C m W ) annualized fixed cost of establishing a warehouse atlocation m

C k D ) a nn ua lized f ixed cos t of es t a blis hing a dis t r ibut ioncenter at location k

C i j P ) unit production cost for product i a t p l a n t j

C h r ) t ra ns port a t ion cos t w hich corres ponds t o int erva l r C ij m ) actual transportation cost of product i from plant j

t o w a rehous e m C ik m ) actual transportation cost of product i f rom w a re-

house m to distribution center k C ik l ) actua l tr ansportat ion cost of product i from distribu-

tion center k to customer zone l

D k ma x

) maximum distribution center capacityD k mi n ) minimum distribution center capacity

D il ) demand for product i in customer zone l D k ) capacity of distribution center k e ) manufacturing resource (equipment, manpower, utili-

ties, etc.)f ) transportation product familiesi ) products j ) p la nt sK ss ) set of distribut ion centers requiring single sourcingk ) possible distr ibution centersL ss ) set of customer zones requ iring single sourcingl ) customer zonesm ) possible w ar ehousesNI ) number of products

NF ) number of famili

Documents

Supply Chain Paper 1