Schumann, a modeling framework for supply chain management under uncertainty

Case Study

Schumann, a modeling framework for supply chain managementunder uncertainty

L.F. Escudero a,b,*, E. Galindo a, G. Garc�õa a, E. G�omez a, V. Sabau a

a IBERDROLA Ingenier�õa y Consultor�õa, Avda. de Burgos 8b, 28036 Madrid, Spainb DEIO, Mathematical Science School, Universidad Complutense de Madrid, Madrid, Spain

Received 1 October 1998; accepted 1 October 1998

Abstract

We present a modeling framework for the optimization of a manufacturing, assembly and distribution (MAD)

supply chain planning problem under uncertainty in product demand and component supplying cost and delivery time,

mainly. The automotive sector has been chosen as the pilot area for this type of multiperiod multiproduct multilevel

problem, but the approach has a far more reaching application. A deterministic treatment of the problem provides

unsatisfactory results. We use a 2-stage scenario analysis based on a partial recourse approach, where MAD supply

chain policy can be implemented for a given set of initial time periods, such that the solution for the other periods does

not need to be anticipated and, then, it depends on the scenario to occur. In any case, it takes into consideration all the

given scenarios. Very useful schemes are used for modeling balance equations and multiperiod linking constraints. A

dual approach splitting variable scheme is been used for dealing with the implementable time periods related variables,

via a redundant circular linking representation. Ó 1999 Elsevier Science B.V. All rights reserved.

Keywords: Supply chain planning; Stochastic parameters; Implementable periods; Scenario analysis; Nonanticipativity

principle; 2-stage decision making

1. Introduction

Decision making is inherent to all aspects ofindustrial, business and social activities. In all ofthem, di�cult tasks must be accomplished. One ofthe most reliable decision support tools availabletoday is Optimization, a ®eld at the con¯uence ofMathematics and Computer Science. The purpose

of the ®eld is to build and solve e�ectively realisticmathematical models of the situation under study,allowing the decision makers to explore a hugevariety of possible alternatives. As reality is com-plex, many of these models are large (in terms ofthe number of decision variables), and stochastic(there are parameters whose value cannot becontrolled by the decision maker and are uncer-tain). The last fact makes the problems di�cult totackle, yet its solution is critical for many leadingorganizations in ®elds such as supply chain plan-ning among many other areas.

European Journal of Operational Research 119 (1999) 14±34www.elsevier.com/locate/orms

* Corresponding author. Tel.: +34 1 383 31 80; fax: +34 1 383

33 11; e-mail: [email protected]

0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 7 - 2 2 1 7 ( 9 8 ) 0 0 3 6 6 - X

Manufacturing, Assembly and Distribution(MAD) Supply Chain Management is concernedwith determining supply, production and stocklevels in raw materials, subassemblies at di�erentlevels of the given Bills of Material (BoM), endproducts and information exchange through(possibly) a set of factories, depots and dealercentres of a given production and service networkto meet ¯uctuating demand requirements. If re-sources can be acquired as needed and plant ca-pacity is in®nitely expandable and contractible atno cost, then the optimal production scheduleconsists of producing end products according tothe demand schedule, and producing and trans-porting subassemblies exactly when needed as in-put to the next assembly process. However, inmany supply chain systems, the supply of someraw materials is tightly constrained, with longproduction and/or procurement lead times. Thedemand for products ¯uctuates, both in totalvolume and in product mix. As a result, just-in-time production is not usually feasible, and whenfeasible, may result in poor utilization of the sup-ply chain. Four key aspects of this problem areidenti®ed as time, uncertainty, cost and customerservice level. In these circumstances, the supplychain management optimisation consists of de-ciding on the best utilization of the available re-sources in suppliers, factories, depots anddealerships given the di�erent scenarios for thestochastic parameters along the planning horizon.

Problems with the characteristics given aboveare transformed into mathematical optimizationmodels. Often there are tens of thousands ofconstraints and variables for a deterministic sit-uation. The problems can be modeled as large-scale linear programs. Given today's OperationsResearch state-of-the-art tools, deterministic lo-gistics scheduling optimization problems shouldnot present major di�culties for not very large-scale problem solving, at least. However, it haslong been recognized (Beale, 1955; Dantzig, 1955)that traditional deterministic optimization is notsuitable for capturing the truly dynamic behaviorof most real-world applications. The main reasonis that such applications involve data uncertain-ties which arise because information that will beneeded in subsequent decision stages is not

available to the decision maker when the decisionmust be made. MAD supply chain planning ap-plications, such as those that this work dealswith, exhibit uncertain product demand as well asuncertain procurement and production availabil-ity, supply costing and lag time and others. Ad-ditionally, the problem has a large-scale naturethat makes it di�cult, even in its deterministicversion.

The aim of this work is to present a novelmodeling approach for the MAD supply chainplanning optimization problem under uncertaintyfor very large-scale instances. Although the schemehas been primarily designed for tackling MADsupply chain planning problems in the automotivesector, the approach has a far more reaching ap-plication to the very broad supply chain area thatdeals with multiperiod, multiproduct and multi-level types of problems in manufacturing, assem-bly and distribution.

The paper is organized as follows. Section 2presents the MAD supply chain planning problemto solve. Section 3 gives the notation and themeaning of the main parameters and variables.Section 4 presents a concept-oriented mathemati-cal representation of the model. Section 5 intro-duces our modeling framework to treat theuncertainty via scenario analysis. Sections 6 and 7give the parameters and variables as well as theimplementation-oriented mathematical represen-tation of the deterministic equivalent model for thestochastic version of the problem.

2. Problem description

2.1. Current state-of-the-art

A global multinational player (e.g., in the au-tomotive sector) would ideally like to take businessdecisions which span sourcing, manufacturing,assembly and distribution. Thus, a company withmultiple suppliers at di�erent levels of the BoMproduction plants and multiple markets may seekto allocate demand quantities to di�erent plantsover the next month, next quarter or next yeartime horizon. Its objective is to minimize the sumof manufacturing, assembly and distribution sup-

L.F. Escudero et al. / European Journal of Operational Research 119 (1999) 14±34 15

plying costs associated with satisfying customerdemands. Alternatively, if product mix is allowedto vary, the company may seek to maximize netrevenue and/or market share among others. Veryfrequently the implication for planning is to min-imize work-in-progress across the supply network.The plants and selected suppliers and dealers aredescribed as particular entities with respect to: itsdirect, indirect and overhead costs; its resourcesincluding machine capacities, labour and rawmaterials; and its recipes for producing productsfrom raw materials and other resources. A modelfor decision support should capture the ¯ow ofproducts and information from the plants throughdepots to the markets.

Traditional MAD Supply Chain Managementoptimisation models develop production plansthat minimize material procurement, inventoryholding and labor costs given time varying de-mands. See in Afentakis et al. (1984), Dzielinskiand Gomory (1965), Florian and Klein (1971) andLasdon and Terjung (1971) examples of thesemodels where the demand is assumed to be knownor modelled deterministically; see also Goyal andGunasekaran (1990) for a good survey and addi-tional references.

A capacity planning system is presented inKekre and Kekre (1985) to explicitly model thework-in-progress and lead times and to combine itwith a discrete time mathematical programmingmodel with deterministic time varying demands. Atactical planning model has been suggested inGraves and Fine (1988) to evaluate capacityloading under varying demand conditions. Theinterrelations between capacity loading, produc-tion lead times and work-in-progress have beenhighlighting in Karmarkar (1987, 1989). See inEppen et al. (1989) an excellent discussion on ca-pacity planning based on a scenario approach, butthe emphasis is on longer range decisions regard-ing facility selection for manufacturing. Modelsfor global chain management optimisation inmanufacturing have been presented in Cohen andLee (1989), Cohen and Moon (1991) and Shapiro(1993). Finally, see in Cheng and Miltenburg(submitted) a motivation for a hierarchical ap-proach to the production planning of BoM, givenits complexity and large-scale dimensions.

The Supply Chain Management problems havebeen cast in the form of deterministic mathe-matical optimisation models and many real in-stances have been computationally solved. Butdesigning and implementing a sound genericmodel which closely couples the strategic plan-ning and tactical logistic decisions as well ascaptures the time phasing and the uncertaintyelements of the supply chain remains a challeng-ing task. See in Escudero (1994b) and Escuderoand Kamesam (1995) and Escudero et al. (1993) aprevious work on modeling the supply chainmanagement optimisation under uncertainty. It isbased on a scenario approach that uses the non-anticipativity principle (Rockafellar and Wets,1991; Wets, 1989) and it is very amenable fordecomposition schemes, see Escudero et al. (toappear), Escudero and Salmer�on (1998), Vladi-mirou (to appear) and Vladimirou and Zenios(1997) among others. See also the contribution tothe subject made by Baricelli et al. (1996) andEscudero (1994a).

Finding the right decision support tools is oneof the most technologically challenging problemsthat operators and decision makers face today.Several approaches, based on di�erent mathe-matical methods, are being pursued with the sameaim of optimizing part of or the full problem ofsupply chain planning. Stochastic optimization viascenario analysis is a powerful methodology thatwe propose for the MAD supply chain problemsolution; see Alvarez et al. (1994), Birge (1985),Birge et al. (1996), Birge and Louveaux (1988,1997), Dempster and Gassmann (1990), Dempsterand Thompson (to appear), Escudero et al. (1993),Escudero and Salmer�on (1998), Gassmann (1990),Kall and Wallace (1994), Mulvey et al. (1995),Mulvey and Vladimirou (1991), Mulvey andRuszczynski (1992), Ruszczynski (1993), VanSlyke and Wets (1969), Wets (1988, 1989) amongothers. At this point in time, we know of no suc-cessful system that has been developed to solve thetype of real-life problems as described in this work.The available sequentially based alternatives cansolve the deterministic version of the full problemfor a given scenario, or a stochastic version in-volving a very small number of scenarios. Even thedeterministic version very frequently does not treat

16 L.F. Escudero et al. / European Journal of Operational Research 119 (1999) 14±34

the problem as a whole, given the dimensions andcomplexity of the problem.

The aim of this work is the study of the ap-proach which today appears most promising,based on stochastic optimisation, since interestingresults have already been obtained, either on asmall scale or on parts of the problem, exploitingthe decomposable nature of the representation ofthe stochastic model. In this regard it is importantto note that the mathematical procedures of choiceto handle the uncertainty in the optimisationmodel, namely the use of Augmented Lagrangianand Benders Decomposition schemes, are partic-ularly well suited for adaptation to a distributed orparallel computation environment. See Escuderoand Kamesam (1993, 1995), Escudero et al. (toappear), Escudero and Salmer�on (1998) and Higleand Sen (1996) among others.

2.2. Problem elements

A planning horizon is a set of (consecutive andinteger) time periods of non-necessarily equallength. An end product is the ®nal output of themanufacturing network. A subassembly is aproduct that is assembled by the manufacturingnetwork and, together with other components, isused to produce another product. (External de-mand and/or procurement for subassemblies isalso allowed). By the term product we will referto both end products and subassemblies. ItsBoM is a concern of the system decision-making.Let us use the term component to describe anypart number (i.e., a raw component or a subas-sembly) that is required for the production. Wewill name raw component to a component whoseBoM is not a concern of the system decision-making (i.e., the supply is only from outsidesources). A transferable component is a compo-nent whose available volume at the end of anytime period can be transferred to the next one(e.g., materials, subassemblies). A non-transfer-able (raw) component is a component whosevolume that is not used in a given time periodcannot be transferred to the next one (e.g., en-ergy, machine and labour time, etc.). The stockof a product or a (transferable) raw component

is the available volume at the end of a given timeperiod.

The cycle time (i.e., lead time) of a product isthe set of (consecutive and integer) time periodsthat are required for its completion, from the re-lease of the product in the assembly line until it isavailable for use.

The Bill of Materials (BoM) of a product is thestructuring of the set of components that are re-quired for its manufacturing/assembly. Note: Asubassembly is a product that belongs to the BoMof some other product(s). On the contrary, a rawcomponent is not assembled by the network and,then, it is supplied from outside sources only.

A production period is a time period in theproduct's cycle time. We assume that each com-ponent in a BoM is only required in one speci®cproduction period (e.g., the ®rst week from a two-week cycle time).

There are two components' supply modes,namely, standard and expediting modes. Thesupply of raw components by using the standardmode has a maximum volume allowed per timeperiod. If more supply is needed it is possible touse the expediting mode by paying an added pen-alty. Note: Procurement is the standard and ex-pediting supply modes for raw components.Subassemblies can also be supplied either by thestandard mode (i.e., in-house production) or bythe expediting mode (e.g., third parties, extrashifts). In the second case an added penalty mustbe paid. Note: Production is the standard supplymode and procurement is the expediting supplymode for subassemblies.

High-tech products are subject to design andengineering changes and, then, the set of compo-nents used in a product (e.g., its BoM) may changeduring the planning horizon. An E�ective PeriodsSegment (EPS) of a component in a given productis a set of (consecutive and integer) time periodsde®ned by the earliest period and the latest periodwhere the component can be used in the givenproduct. Engineering changes (EC) are the mostfrequent reason for having an EPS that is smallerthan the length of the planning horizon. Note thatthe avoidance of assembling a product with ob-solete components does not prevent its use forsatisfying external demand or as a component in


other BoM's. The using of obsolete products canbe prevented by forcing a zero stock at the ap-propriate time period. A Mandatory E�ective Pe-riods Segment (MEPS) of a component in a givenproduct does not allow to assemble the compo-nent, nor using the old (obsolete) product. Note 1:An EC is mandatory if the product with the oldtechnology is not allowed to be used after the EPS,even if the product was assembled in advance.Note 2: A mandatory EC's must be performed incascade up to end products.

Let us de®ne backlog of a product at the end ofa time period as the (non-negative) di�erence be-tween the cumulated demand and shipment up tothat period. Multiple external demand sources fora product (either end product or a subassembly)are allowed.

A practical model should allow the replacementof some components in a BoM by using othercomponents. Let us term prime component to thecomponent listed in a BoM. The other compo-nents that can be used (in the standard mode) willbe termed alternate components. Multiple alter-nates for a prime component are allowed. Thealternates for a prime component may dependupon both the component's and the product'scosts and availability. Each alternate has its ownunit usage, e�ective periods segment, procurement/production cost and fallout. In any case, a primesubassembly can be substituted by another sub-assembly or by a raw component. A raw compo-nent can be substituted by another raw componentonly. Note that in-house production and vendorsourcing can be modeled by using the alternatemechanism.

As an illustration, see that the decision on theprocurement of a raw component from di�erentsuppliers, to di�erent depots and the supplying ofa subassembly from di�erent depots can be mod-eled by using the alternate components mecha-nism. Additionally, the utilization of the capacityof non-dedicated manufacturing lines can bemodeled by using the concept of alternate non-transferable (raw) components. See that the releaseof a given product can be diverted to di�erentlines, each line has its own capacity (e.g., timeavailability), and a processing time is given foreach product and line. Another interesting appli-

cation of the alternate components mechanism isthe piecewise representation of convex productioncost functions.

The model should allow to assign products toso-called product groups. An aggregate capacityconstraint (weighted Product Going Rate, PGR)for any group (usually, a manufacturing line) canbe considered per time period. So-called rawcomponent groups are also allowed, such that thetotal amount that is used per group and time pe-riod can be bounded; as an illustration, raw com-ponents from the same supplier or from the samegeographical area can be handled by using thistype of functionality.

Note that the concept of non-transferable (raw)components allows to consider resources such asmachine capacity, tool and manpower availability,etc. The following data can be given for eachproduct and resource: unit usage, production pe-riod, EPS (representing, e.g., equipment change-over), alternate resources, etc.

Note that single-level production requires thatthe components are assembled sequentially alongthe cycle time of the product. On the contrary, amultilevel production allows that subsets of com-ponents to be assembled independently and, then,the production resources can be better utilized.

3. Data representation

3.1. Sets

T set of time periods in the planninghorizon

J set of productsJE � J set of end productsJS � J set of subassemblies (JE \ JS �£

and J � JE [ JS)JSD � JS set of subassemblies with external

demandPG set of product groupsJh � J set of products that belong to

group h, for h 2 PGDSj set of (external) demand sources for

product j, for j 2 JE [ JSD, suchthat DSj \DSj0 �£ for j,j0 2 JE [ JSD j j 6� j0


3.2. Constraint related parameters

Demand

Bill of Material (BoM)

Dd;t demand from external source d at timeperiod t, for d 2 DS; t 2 T

MBd;t maximum backlog on demand source dthat is allowed at time period t, ford 2 DS; t 2 T

ld delivery lag time, i.e., number of timeperiods to deliver the end product orsubassembly with external demand dafter its completion, for d 2 DS

dd lost (expected) demand fraction of non-served cumulated demand at any timeperiod for external demand d, ford 2 DS

qd unit lost demand penalty at any timeperiod for external demand d, ford 2 DS

rd;t unit backlog weight at time period t fordemand source d, for d 2 DS; t 2 T

DS [DSj for all j 2 JE [ JSDI set of componentsIR � I set of raw components, such that

IR \ JS �£ and I � IR [ JSIT � IR set of transferable raw components

(Note: IR ÿ IT is the set of non-transferable raw components)

RG set of raw component groupsIRh � IR set of raw components that belong

to group h, for h 2 RGIj � I set of prime components in the

BoM of product j, for j 2 JIi;j � I set of components that are alter-

nates to prime component i in theBoM of product j, for i 2 I , j 2 J .

cj cycle time of product j, for j 2 Jpi;j production time period in the cycle

time of product j where primecomponent i is needed, fori 2 Ij; j 2 J (Note 1: 16 pi;j6 cj;Note 2: Alternate components havethe same production period as therelated prime component)

oi;j o�set of component i in the cycletime of product j, for i 2 Ij; j 2 J . Itgives the number of time periodsbefore the completion of product jwhose BoM includes component i,such that oi;j � cj ÿ pi;j

ai;j amount of prime component i thatis needed per unit of product j, fori 2 Ij; j 2 J

ci;j fallout of prime component i in theBoM of product j

ai;j net amount of prime component ithat is needed per unit of product j,for i 2 Ij; j 2 J , whereai;j � ai;j �1ÿ 0:001ci;j�

�ls

i number of time periods to supplyraw component i to its depot byusing the standard mode, for i 2 IR

lei number of time periods to supply

component i to its depot by usingthe expediting mode, for i 2 I (Note1: The order should be performed ls

ior le

i time periods in advance to thetime period where the component isrequired; Note 2: It is assumed thatls

i � 0 and lei � 0 for i being any

non-transferable (raw) component)li;j number of time periods required to

deliver component i from its depotfor product j, for i 2 Ij; j 2 J andi 2 If ;j; f 2 Ij; j 2 J

s�ÿ�i;j ; s��i;j earliest time period and latest timeperiod, respectively (i.e., e�ectiveperiods segment), where primecomponent i or any of its alternatescan be used in the BoM of productj, for i 2 Ij; j 2 J(Note: pi;j6 s�ÿ�i;j 6 s��i;j 6 jT j ÿ oi;j)

MEPSi;j ¯ag for the mandatory e�ectiveperiods segment of prime compo-nent i in the BoM of product j, fori 2 Ij; j 2 J , such that MEPSi;j � 1if the ¯ag is on and, otherwise, it is 0

bi;jf amount of component f that is

needed per unit of product j pro-vided that it substitutes prime com-ponent i, for f 2 I i;j, i 2 Ij; j 2 J


Technological constraints

Components availability

Production and stock levels

3.3. Cost function related parameters

uj;h capacity usage of resources allocated toproduct group h per unit of product j,for j 2 Jh; h 2 PG

ri;h unit usage of raw component i in grouph, for i 2 IRh; h 2 RG

MXi;t0 maximum amount of raw com-ponent i that can be orderedfrom outside sources at (thebeginning of) time period t byusing the standard mode, fori 2 IR; t � 1; . . . ; jT j ÿ ls

iMEi;t maximum amount of compo-

nent i that can be ordered fromoutside sources at (the begin-ning of) time period t by usingthe expediting mode, fori 2 I ; t � 1; . . . ; jT j ÿ le

iMRh;t, mRh;t maximum and minimum usage

that is allowed for raw compo-nent group h at time period t,for h 2 RG; t 2 T

ui;jf fallout of alternate component f

for prime component i in theBoM of product j

bi;jf net amount of component f that

is needed per unit of product j,provided that it substitutes primecomponent i, forf 2 I i;j; i 2 Ij; j 2 J , wherebi;j

f � bi;jf � =�1ÿ 0:001ui;j

f �si;j�ÿ�

f ; si;j��f earliest time period and latest

time period, respectively, wherecomponent f can substitute primecomponent i in the BoM ofproduct j, forf 2 I i;j; i 2 Ij; j 2 J . (Note:s�ÿ�i;j 6 si;j�ÿ�

f 6 si;j��f 6 s��i;j )

MZj;t maximum release volume that isallowed for product j at timeperiod t, for j 2 J ; t 2 T

(1 ÿ uj;t) yield of product j provided that itis made available at time period tfor 0 < uj;t6 1, for j 2 J ; t 2 T

MPh;t00 , mPh;t maximum and minimum releasecapacity that is allowed forproduct group h at time period t,for t 2 T ; h 2 PG

Vj;t external input volume for prod-uct or (transferable) raw compo-nent j, at (the beginning of) timeperiod t, for j 2 J [ IT ; t 2 T .Note: The volume V for the ®rsttime periods can represent theavailability of the product or(transferable) raw componentthat was a work-in-progress atthe beginning of the time horizon.On the other hand it can repre-sent the initial stock for periodt� 1

MSj;t0 , mSj;t maximum and minimum volumeof product or (transferable) rawcomponent j that is allowed tokeep in stock at (the end of) timeperiod t, for j 2 J [ IT; t 2 T .Note: MSj;sj � mSj;sj � 0 forsj � s��i;j � oi;j ; i 2 Ij; j 2 J andMEPSi;j � 1; as a consequence,MSl;sl � msl;sl � 0 where l 2 Jis any node in the tree whose rootnode is component i and theleaves are the end-productswhose BoM (given by primecomponents) include (directly orindirectly) componenti; sl � sk � ok;l and k is a node inthe tree such that k 2 Il. Note:The zero stock ®xing process isinterrupted whenever s�ÿ�k;l > sk

PCj;t unit production cost for product jand time period t, for j 2 J , t 2 T .Note: No procurement cost isincluded for the components in itsBoM; it is already consideredbelow


3.4. Optimization variables

Note. For the deterministic model (see Sec-tion 4), the X- and E-variables may also refer tothe (beginning of) time period where the compo-nent is available for depot. However, in the sto-chastic setting (see Section 7) the related timeperiod is important, since the component's envi-ronment may di�er from the ordering time periodup to the receiving time period.

3.5. Lag indices and operators

The X-, E-, Z-, ZP-, ZA- and Y-variables' e�ecton latter time periods to the indexed period can betaken into account by instrumenting the followingidentities to be used for time period s in the model

PAi;jf ;t unit extra production cost for

product j and time period t, due toa substitution of prime componenti by alternate component f, forf 2 I i;j; i 2 Ij; j 2 J ; t 2 T . Note:No procurement cost is includedfor the alternate component; it isconsidered below

Hj;t unit holding cost for product or(transferable) raw component jand time period t, for j 2 J [ IT,t 2 T . Note: This element can alsobe used for discouraging building-ahead in case of tie, by consideringHj;t as a stock penalization; in thiscase, a typical value can be10 ´ 10ÿ4

SCsi;t; SCe

i;t unit procurement cost for rawcomponent i and time period t byusing the standard and expeditingmodes, respectively, for i 2 IR,t 2 T

SCei;t unit procurement cost for subas-

sembly i and time period t (byusing the expediting mode), fori 2 JS, t 2 T

Zj;t volume of product j that is madeavailable at (the end of) time period t,by using the (production) standardmode for shipment, production orstock, for j 2 J , t 2 T . Note: The endproducts are made available only forshipment or stock. The subassembliesare made available for production,shipment (if there is external demand)or stock.

ZPi;jt volume of product j that is made

available at (the end of) time period t,by using the (production) standardmode, whose prime component i wasnot substituted by any alternate com-ponent, although it could be fori 2 Ij; j 2 J ; t 2 T . Note: I i;j 6�£

ZAi;jf ;t volume of product j that is made

available at (the end of) time period t,by using the (production) standardmode, whose prime component i issubstituted by alternate component f,for f 2 I i;j, i 2 Ij, j 2 J , t 2 T . Note 1:ZPi;j

t , ZAi;jf ;t do not exist for I i;j �£

where i 2 Ij, j 2 J . Note 2: ZPi;jt ,

ZAi;jf ;t6Zj;t for I i;j 6�£, i 2 Ij, j 2 J ,

t 2 TXi;t volume of raw component i that is

ordered at (the beginning of) timeperiod t by using the (procurement)standard mode, for i 2 IR, t 2 T

Ei;t volume of (raw or subassembly) com-ponent i that is ordered at (the begin-ning of) time period t by using the(procurement) expediting mode, fori 2 I , t 2 T

Yd;t volume of served demand for externalsource d that is being shipped at (theend of) time period t, for d 2 DS, t 2 T

Ld;t lost demand from external source d attime period t, for d 2 DS, t 2 T

Bd;t backlog volume for external demandsource d at (the end of) time period t,for d 2 DS, t 2 T

Sj;t volume of product or (transferable) rawcomponent j to keep in stock at (the endof) time period t, for j 2 J [ IT, t 2 T


below, for s 2 T . (Note: This e�ect is due to the l-lag time interval among other reasons.)

ss � sÿ lsi 8i 2 IR

�procurement standard mode�;

se � sÿ lei 8i 2 I

�procurement expediting mode�;

sr � sÿ cj � 1 8j 2 J �product release�;

sp � s� li:j � oi;j 8i 2 Ij; j 2 J

�prime component supply�;

sa � s� li;j � of ;j 8i 2 If ;j; f 2 Ij; j 2 J

�alternate component supply�;

sd � sÿ ld 8d 2 DS

�external demand delivery�:

Let c1; c2; c3 denote the operators forj 2 J ; s 2 T , such that

c1 � 1 for s�ÿ�i;j 6 s� li;j6 s��i;j ;0 otherwise;

�for i 2 Ij,

c2 � 1 for sf ;j�ÿ�i 6 s� li;j6 sf ;j��

i ;0 otherwise;

�for i 2 If ;j; f 2 Ij,

c3 � 1 for sf ;j�ÿ�i 6 sÿ of ;j6 sf ;j��

i ;0 otherwise;

�for i 2 If ;j; f 2 Ij.

4. A concept-oriented deterministic model

4.1. Objective functions

Option 1: Optimizing the system resources us-age by minimizing the total production costs (by

using prime and alternate components in theproducts' BoM), and the procurement standardand expediting costs as well as lost demand pe-nalization.

z�1 � minXj2J

Xs2T

PCj;sZj;s

�Xj2J

Xs2T

Xi2Ij

Xf2Ii;j

c3PAi;jf ;sZAi;j

f ;s

�Xi2IR

Xs�1;T s

SCsi;sXi;s

Xi2I

�X

s�1;T e

SCei;sEi;s

�Xd2DS

Xs2T

qdLd;s; �1�

where c3 is given in Section 3.5, and

T s � jT j ÿ lsi ; T e � jT j ÿ le

i �2�

subject to Eqs. (5)±(24).Option 2: Optimizing the market opportunities

by minimizing the maximum weighted productbacklog.

z�2 � min z=z P rd;sBd;s 8d 2 DS; s 2 T ; �3�subject to Eqs. (5)±(24).

Option 3: Optimizing the market opportunitiesby minimizing the total weighted product backlogand lost demand penalization.

z�3 � minXd2DS

Xs2T

rd;sBd;s� � qdLd;s� �4�

subject to Eqs. (5)±(24).

4.2. End-product balance equations

The balance equations for the production,shipment and stock of end products are givenby the constraints (5). These constraints state that,for given end product and time period, the stockvolume at the end of the previous period plus theexternal input volume at the beginning of thecurrent period plus the net production volume thathas been completed at the given period mustequate the shipment volume plus the stock volumeat the end of the current period.


Sj;sÿ1 � Vj;s � uj;sZj;s �X

d2DSj

Yd;s � Sj;s

8j 2 JE; s 2 T : �5�Note. Sj;sÿ1 does not exist for s� 1 given Vj;s.

4.3. Subassembly balance equations

The balance equations for the production,shipment (internal as well as external demandsatisfaction) and stock are given by the constraints(6). These constraints state that, for given subas-sembly and time period, the stock volume at theend of the previous period plus the external inputvolume at the beginning of the current period plusthe subassembly supply volume at the beginning ofthe current period by using the expediting modeplus the net production volume at the end of thecurrent period must equate the component's vol-ume that is shipped at the beginning of the currentperiod to satisfy internal needs given by the BoMof other products plus the shipment volume at theend of the current period to satisfy external de-mand plus the stock volume at the end of thecurrent period.

Si;sÿ1 � Vi;s � Ei;se � ui;sZi;s

�X

j2J=i2Ij:Ii;j�£

c1ai;jZj;sp �X

j2J=i2Ij:I i;j 6�£

c1ai;jZPi;jsp

�X

f2Ij;j2J :i2If ;j

c2bf ;ji ZAf ;j

i;sa �X

d2DSi

Yd;s � Si;s

8i 2 JS; s 2 T ; �6�where, se, sp,sa, c1, c2 are given in Section 3.5.

Note 1. c1� 0 does not allow to use primecomponent (subassembly) i in the BoM of productj. It is the case where time period s does not belongto the E�ective Periods Segment (EPS) for usingsubassembly i in the BoM of product j.

Note 2. c26 c1. See that c2� 0 does not allowalternate component i to substitute prime com-ponent f in the BoM of product j, due to EPSreasons.

The balance equations as given by the con-straints (6) do not consider the synchronisation ofthe stock balance's input and output. So, the fol-

lowing constraint is required for the subassembly's¯ow at the beginning of the given time period.

Si;sÿ1 � Vi;s � Ei;se

ÿX

j2J=i2Ij:Ii;j�£

c1ai;jZj;sp

ÿX

j2J=i2Ij:Ii;j 6�£

c1ai;jZPi;jsp

ÿX

f2Ij;j2J :i2If ;j

c2bf ;ji ZAf ;j

i;sa P 0

8i 2 JS; s 2 T : �7�

4.4. External demand balance equations

The balance equations for the external demandserviceability are given by the constraints (8).These constraints state that, for given externaldemand source and time period, the backlog vol-ume at the end of the previous time period plus thedemand volume at the given period must equatethe shipment volume to satisfy the external de-mand plus the lost (non-served) demand volumeduring the period plus the backlog volume at theend of the given period.

Bd;sÿ1 � Dd;s � Yd;sd � Ld;s � Bd;s

8d 2 DS; s 2 T ; �8�where sd is given in Section 3.5, such that the lostdemand Ld;s can be expressed.

Ld;s � dd Bd;sÿ1

ÿ � Dd;s ÿ Yd;sd

�P 0: �9�

4.5. Raw component balance equations

The balance equations for the procurement,utilization and stock of (transferable and non-transferable) raw components are given by theconstraints (10). These constraints state that, forgiven component and time period, the stock vol-ume (only for transferable components) at the endof the previous time period plus the external inputvolume at the beginning of the current period(only for transferable components) plus the sup-ply volume at the beginning of the current period


(by using both the standard and expeditingmodes) must equate the component's volume thatis shipped at the beginning of the current periodto satisfy internal needs given by the BoM ofrelated products plus the stock volume (only fortransferable components) at the end of the givenperiod.

Si;sÿ1 � Vi;s � Xi;ss � Ei;se

�X

j2J=i2Ij:Ii;j�£

c1ai;jZj;sp

�X

j2J=i2Ij:Ii;j 6�£

c1ai;jZPi;jsp

�X

f2Ij;j2J :i2If ;j

c2bf ;ji ZAf ;j

i;sa � Si;s

8i 2 IR; s 2 T : �10�where ss, se, sp, sa, c1, c2 are given in Section 3.5.

Note. Vi;s; Si;s do not exist and ss� se� s for i 2IRÿ IT (i.e., non-transferable components).

4.6. Prime component substitution balance equa-tions

For a product to be made available at a giventime period and each of its prime components withsubstitution capabilities in the related BoM, theseconstraints state that the product volume wherethe prime component is not substituted plus theproduct volume where it is substituted by other(alternate) components must equate the productvolume to be made available at the given period.

Zj;s �ZPi;js �

Xf2I i;j

c3ZAi;jf ;s8s

� s�ÿ�i;j � oi;j; . . . ; s��i;j � oi;j;

i 2 Ij=I i;j 6�£; j 2 J ; �11�where c3 is given in Section 3.5.

4.7. Components and products group bounding

The maximum and minimum usage that is al-lowed for a given raw component group at a giventime period can be expressed.

mRh;s6Xi2IRh

Xj2J=i2Ij:Ii;j�£

c1ri;hai;jzj;sp

24�

Xj2J=i2Ij:I i;j 6�£

c1ri;hai;jZPi;jsp

�X

f2Ij;j2J :i2If ;j

c2ri;hbf ;ji ZAf ;j

i;sa

356MRh;s

8h 2 RG; s 2 T ; �12�where, sp, sa, c1, c2 are given in Section 3.5.

The maximum and minimum release capacitythat is allowed for a given product group at a giventime period can be expressed.

mPh;sr 6Xj2Jh

uj;hZj;s6MPh;sr 8h 2 PG; s 2 T ;

�13�where sr is given in Section 3.5.

4.8. Bounds on the variables

Maximum product release volume

06 Zj;s6MZj;sr 8s � cj; . . . ; jT j; i 2 J ; �14�where sr is given in Section 3.5.

06ZPi;js 8s � s�ÿ�i;j � oi;j; . . . ; s��i;j � oi;j;

i 2 Ij=I i;j 6�£; j 2 J ; �15�

06ZAi;jf ;s 8s � si;j�ÿ�

f � oi;j; . . . ; si;j��f � oi;j;

f 2 I i;j; i 2 Ij; j 2 J : �16�Maximum component procurement volume

06Xi;s6MXi;s 8s � 1; . . . ; jT j ÿ lsi ; i 2 IT

8s � Xi; . . . ;Ci; i 2 IRÿ IT; �17�

06Ei;s6MEi;s 8s � 1; . . . ; jT j ÿ lei ;

i 2 JS [ IT 8s � Xi; . . . ;Ci;

i 2 IRÿ IT; �18�


where

X1 � min minj2J=i2Ij

s�ÿ�i;j

n�ÿ li;j

o;

minf2Ij;j2J=i2If ;j

sf ;j�ÿ�i

nÿ li;j

o�; �19�

Ci � max maxj2J=i2Ij

s��i;j

n�ÿ li;j

o;

maxf2Ij;j2J=i2If ;j

sf ;j��i

nÿ li;j

o�: �20�

Maximum and minimum product and (trans-ferable) raw component stock volume

mSj;s6 Sj;s6MSj;s 8s 2 T ; j 2 J [ IT: �21�Maximum product backlog, and nonnegative

character of the variables

06Bd;s6MBd;s 8s 2 T ; d 2 DS; �22�

06 Yd;s 8s � 1; . . . ; jT j ÿ ld ; d 2 DS: �23�

4.9. Preventing unnecessary product and componentstock

The following penalization can be included inthe objective functions (1), (3) and (4) to preventunnecessary build-ahead:

a �X

j2J[IT

Xs2T

Hj;sSj;s: �24�

See constraints (5), (6) and (10) and note that(24) discourages building for stock in case of tie.

Note. See that model (1)±(24) is an LP repre-sentation of a multiperiod multilevel multiproductproblem. Given the problem's dimensions, thecurrent representation may have a very large-scalesize. Anyway the products linking is due to theBoMs communality, and the periods linking is dueto the transferable raw components and productstock from one period to the next one, the cycletime for each product, the components delivery lagtime for products and the products delivery lag timeto external demand sources (in case the lag time ismore than one time period). So, the problem's el-ements are very much time period inter-related.

4.10. An implementation-oriented variables reduc-tion scheme

Given the dimensions of the deterministicmodel (1)±(24) and the type of its constraints (6),(7), (9)±(11), one can need to reduce the number ofvariables. In particular, the variables Sj;s 8j 2J [ IT, Ld;s 8d 2 DS and ZPi;j

s 8i 2 Ij=I i;j 6�£; j 2J for all s 2 T are not explicitly required to solvethe model. As an example, let the following genericsystem:

V1 � A1X1 � S1;

V2 � S1 � A2X2 � S2;

V3 � S2 � A3X3 � S3;

mSt6 St6MSt; t � 1; 2; 3;

�25�

where Xt and St are vectors of variables, Vt is avector of data, At a conformable constraint matrixand mSt and MSt are vectors of constants for t� 1,2, 3. It is easy to see that Eq. (25) can be replacedby the system

mS1 ÿ V1 6A1X1 6MS1 ÿ V1;mS2 ÿ V1 ÿ V2 6A1X1 � A2X2 6MS2 ÿ V1 ÿ V2;mS3 ÿ V1 ÿ V2 ÿ V3 6A1X1 � A2X2 � A3X3 6MS3 ÿ V1 ÿ V2 ÿ V3;

�26�

and, in general,Xs�1;t

AsXs6MSt ÿXs�1;T

Vs; t � 1; 2; 3; �27a�

mSt ÿXs�1;t

Vs6Xs�1;t

AsXs; t � 1; 2; 3; �27b�

and, so, the S-variables are not required at theprice of increasing the constraint matrix density.

Note 1. Constraint (27a) for a given t is re-dundant provided that 9s � t � 1; . . . ; jT j suchthat MSt P MSs ÿ

P#�t�1;s V# for t � 1; . . . ;

jT j ÿ 1. So, the constraints (27a) for the set of timeperiods ftg such that t: MSt < MSs ÿ

P#�t�1;s V#

for all s=t < s are only required.Note 2. Constraint (27b) for a given t is re-

dundant, provided that 9s � 1; . . . ; t ÿ 1 such thatmSs P mSt ÿ

P#�s�1;t V# for t � jT j; . . . ; 2. So, the

constraints (27b) for the set of time periods ftgsuch that t: mSs < mSt ÿ

P#�s�1;t V# for all s=s < t

are only required.


Assume that AtXt above can be decomposed as

AtXt � A1t X 1

t ÿ A2t X 2

t ; t � 1; 2; 3; �28�such that

Vt � Stÿ1 ÿ A2t X 2

t P 0; t � 1; 2; 3: �29�It is easy to see that (29) can be presented as fol-lows by eliminating St:Xs�1;t

Vs �X

s�1;tÿ1

A1sX 1

s ÿXs�1;t

A2sX 2

s P 0; t � 1; 2; 3:

�30�

5. Uncertainty

5.1. General approach

The model described in the previous section canbe compacted in the following model structuring:

min cTzs:t: Az � p;

z P 0;�31�

where c is the vector of the objective function co-e�cients, A the m ´ n constraint matrix, p theright-hand side (r.h.s.) m-vector and z the n-vectorof the decision variables to optimise. It must beextended in order to deal properly with uncer-tainty on the values of some parameters. We mayemploy a technique so-called scenario analysis,where the uncertainty is modelled via a set ofscenarios, say G. Dempster (1988) among others.

In our MAD Supply Chain problem, the mainstochastic parameters are the production andprocurement costs as well as the product demand.It will mean that the vector c of the objectivefunction coe�cients and the r.h.s. p in model (31)should be represented by cg and pg, for, 8g 2 G,respectively. We also introduce a weight, say wg, torepresent the likelihood that the decision makerassigns to scenario g, for g 2 G.

One way to deal with the uncertainty is to ob-tain the solution x that best tracks each of thescenarios, while satisfying the constraints for eachscenario. This can be achieved by obtaining a so-lution that minimizes a norm of the weighted up-

per di�erence between the proposed solution andthe optimal solution value for each scenario. Theresulting model does not increase the number ofvariables in the original representation, but nowthere are mjGj constraints. Unfortunately, thisrepresentation does not preserve the structure ofthe deterministic model (31), and the objectivefunction is no longer linear; see in Escudero(1994b) some procedures to overcome this di�-culty. Models of this form are known as scenarioimmunization models, or SI models for short, seeDantzig (1985) and Dembo (1991), Infanger (1994)and Mulvey et al. (1995) among others.

As an alternative goal, we could minimize theexpected value of the objective function; in thiscase model (31) becomes

minXg2G

wgcgT

z

s:t: Az � pg 8g 2 G; z P 0:

�32�

Note that Eq. (32) gives an implementable policybased on the so-called simple recourse scheme.Note that the whole vector of decision variables isanticipated at (the beginning of) time period t� 1.

5.2. Non-anticipative policies

Let T denote the set of time periods over thetime horizon, T1 is the set (so-called ®rst stage) ofthe ®rst time periods from set T whose relatedparameters are deterministic and T2 � T ÿ T1 (theso-called second stage), where the realizations ofthe parameters related to set T2 are considered viathe set of scenarios G; henceforth, this partitionwill be termed a 2-stage time period framework.

The SI models do anticipate decisions in z thatfor the 2-stage environment may not be needed atthe ®rst stage. Very frequently the decisions for the®rst stage (i.e., the MAD supply chain planningdecisions to be implemented in the time periodsfrom set T1) are the only decisions to be madesince at the ®rst time period of the second stage(i.e., time period jT1j � 1) one may realize thatsome of the data has been changed, some scenariosvanish, etc. In this case, the models will be usuallyreoptimized in a rolling planning horizon mode.


When only spot decisions (i.e., decisions for the®rst stage) are to be made, the information aboutfuture uncertainty is only taken into account for abetter spot decision making. This type of scheme istermed partial recourse for jT1j > 1; otherwise, it istermed full recourse.

Let zgt denote the vector of the variables related

to time period t under scenario g for, t 2 T , g 2 G,and zg is the set of vectors zg

t 8t 2 Tf g.The so-called non-anticipative principle was in-

troduced in Rockafellar and Wets (1991) and Wets(1989) it states that if two di�erent scenarios, say, gand g0 are identical up to time period t on the basisof the information available about them at thattime period, then the values of the z variables mustbe identical up to time period t.

In our case, this condition guarantees that thesolution obtained from the model is not dependentat the ®rst stage on the information that is not yetavailable; the time periods from T1 and T2 aretermed implementable time periods and non-im-plementable time periods, respectively. In order tointroduce this condition in our approach, let Ndenote the set of solutions that satisfy the so-callednon-anticipativity constraints. That is,

z 2 N � zg zgt

��n� zg0

t 8g; g0 2 G; t 2 T1

o: �33�

So, the Deterministic Equivalent Model (DEM)for the partial recourse version of model (31) canbe expressed:

minXg2G

wgcgT

zg

s:t: Azg � pg8g 2 G; z 2 N ; zg P 0:

�34�

For the sake of a better analysis of model (34),let us introduce some additional notation. So, letthe z-vector of variables be partitioned into thevectors x and y, and let I denote the set of con-straints in the problem, by taking into consider-ation the structure exhibited by our deterministicmodel (4)±(24) with the modi®cation introduced inSection 4.10. Let xt denote the vector of variablesfor time period t for t 2 T1 (i.e., the set of timeperiods with deterministic parameters), and T i

xdenote the set of time periods whose related x-variables have non-zero coe�cients in constraintblock i, for, i 2 I , such that Ti

x � T1. The partition

of I is as follows: I � Ix [ Ixy , where Ix will denotethe set of constraint blocks where only the x-variables have non-zero coe�cients, and Ixy � I ÿIx will denote the set of constraint blocks where x-and y-variables will have non-zero elements. Let yg

t

denote the vector of variables for time period t fort 2 T2 (i.e., the set of time periods with stochasticparameters) under scenario g for g 2 G, and Ti

ydenote the set of time periods whose related y-variables have nonzero coe�cients in constraintblock i, for i 2 Ixy , such that Ti

y � T2.Model (34) has a nice structure that we may

exploit. Two approaches can be used to representthe non-anticipativity constraints (33). One ap-proach is based on a compact representation, whereEq. (33) is used to eliminate variables in Eq. (34)and, so, to reduce model size, such that there is asingle vector of variables for each time period fromset T1, but any special structure of the constraintsin Eq. (31) is destroyed.

However, given the structure of problem (1)±(24) and the dimensions of its real-life instances,the most attractive approach to represent the non-anticipativity constraints (33) is based on a dualsplitting variable representation that requires tosplit the x-variables into the new vector of vari-ables, say, xg

t 8g 2 G, such that Eq. (33) can berepresented by using the so-called redundant cir-cular linking scheme,

xgt ÿ xg�1

t � 0 8t 2 T1; g 2 G: �35�

(Note: The convention g � 1 � 1 is used forg � jGj:) The splitting variable representation ofmodel (34) can be expressed as follows:

minx;y

Xg2G

Xt2T1

agT

t xgt �

Xg2G

Xt2T2

bgT

t ygt

s:t:Xt2T i

x

A1itx

gt � p0

i 8i 2 Ix; g 2 G;

Xt2T i

x

A2itx

gt �

Xt2T i

y

Bgity

gt � pg

i 8i 2 Ixy ; g 2 G;

xgt ÿ xg�1

t � 0 8t 2 T1; g 2 G;

xgt P 0 8t 2 T1; g 2 G;

ygt P 0 8t 2 T2; g 2 G;

�36�


where agt � wgc0

t 8t 2 T1; g 2 G; c0t is the vector of

the x-related objective function coe�cients fortime period t for t 2 T1 ; bg

t � wgcgt is the vector of

the y-related objective function coe�cients fortime period t for t 2 T2 under secenario g forg 2 G; A1

it and A2it are the x-related constraint ma-

trices for time period t and constraint block i fort 2 T i

x , i 2 Ix and t 2 T ix , i 2 lxy , respectively; Bg

it isthe y-related constraint matrix for time period tand constraint block i for t 2 T i

y , i 2 Ixy underscenario g for g 2 G; and p0

i and pgi are the r.h.s for

constraint block i for i 2 Ix and i 2 Ixy under sce-nario g for g 2 G, respectively.

See that the dualization of the non-antic-ipativity constraints (35) allows jGj independentconstraint systems, namely, the subsystem fromEq. (36) related to each scenario. See in Section 7the splitting variable based stochastic version ofmodel (1)±(24) with the modi®cation introduced inSection 4.10.

6. Parameters and variables for the stochastic

approach

6.1. Constraint and objective functions relatedparameters

G is set of scenarios, T1 set of implementabletime periods, and T2 � Tÿ T1 set of non-imple-mentable time periods.

Remark. It is assumed that all parameters aredeterministic (i.e., known values) for time periodset T1, but the assumption can be very easilyremoved.

The uncertain parameters to be considered inthe model below are related to the production/procurement costs and availability, demand vol-ume and lost fraction, prime and alternate com-ponents' e�ective periods segment, product andcomponent external input and stock bounding,etc. These parameter types are scenario dependentelements and, then, the superindex g is required; asan example, Dg

dt gives the demand from externalsource d at time period t under scenario g, ford 2 DS, t 2 T, g 2 G, etc. On the other hand, note

that a given parameter indexed by time period thas the same value for all scenarios for t 2 T1.

6.2. Lag indices and operators

The X- and E-variables' e�ect on later timeperiods to the indexed period can be taken intoaccount by instrumenting the following identitiesto be used for time period s under scenario g in themodel below, for s 2 T, g 2 G. (Note: This e�ect isdue to the l-lag time interval among other rea-sons.)

ts � t ÿ ls;gi 8i 2 IR

�procurement standard mode�;te � t ÿ le;g

i 8i 2 I

�procurement expediting mode�:The operators c1, c2, c3 for j 2 J , s 2 T have in thestochastic version of the problem the same type ofequivalences as for the deterministic case (seeSection 3.5), but the superindex g for each sce-nario to occur is also required. Note that thes�ÿ�ij ; si;j�ÿ�

f ; si;j��f ; s��ij parameters are scenario de-

pendent elements and, then, the superindex g isrequired.

6.3. Variables

The same type of variables used in the deter-ministic model are to be used in the stochasticversion (by adding the superindex g to show theirrelationship with the scenario to occur), but theZP-, L- and S-variables are not required, see themodel below.

The variables, say x, with time period index, sayt, from the set T1 of implementable time periodswill have the same value under all scenarios. So,xg

t � xg�1t 8g 2 G; t 2 T1. On the other hand, a

compact representation of the problem may con-sider g � 0 for xg

t 8t 2 T1. In any case, the struc-ture of the implementation-oriented model (see inSection 4.10 the basic ideas) requires that most ofthe variables have non-zero elements in constraintsrelated to later time periods; so, either the variablesrelated to set T1 have a copy per scenario (in case of


a splitting variable representation) or most of thesevariables are to be included in the appropriatescenario related constraints for set T2 (in case of acompact representation).

7. An implementable-oriented 2-stage stochastic

splitting variable model

7.1. Objective functions

Option 1: Optimizing the system resource usageby minimizing the expected total production cost(by using prime and alternate components in theproducts' BoM) and the procurement standardand expediting cost as well as the lost demandpenalization.

z�1 � minXg2G

wgXj2J

Xs2T

PCgj;sZ

gj;s

"�Xj2J

Xs2T

Xi2Ij

Xf2I i;j

cg3PAi;jg

f ;s ZAi;jg

f ;s

�Xi2IR

Xs�1;T s

SCsg

i;sXgi;s �

Xi2I

Xs�1;T e

SCeg

i;sEgi;s

�Xd2DS

Xs�T

qddgdBg

d;s

ÿXd2DS

Xs�1;T d

qddgdY g

d;s

#; �37�

where cg3 is referred to in Section 6.2, and

T s � jT j ÿ ls;gi ; T e � jT j ÿ le;g

i ; T d � jT j ÿ ld

�38�subject to Eqs. (41)±(67). Note that dg

dDgd is a

constant.Option 2: Optimizing the market opportunities

by minimizing the expected maximum weightedproduct backlog.

z�2 � min zjz P wgrd;sBgd;s

8d 2 DS; s 2 T ; g 2 G �39�subject to Eqs. (41)±(67).

Option 3: Optimizing the market opportunitiesby minimizing the expected total weighted productbacklog and lost demand penalization.

z�3 � minXg2G

wgXd2DS

Xs2T

rd;s

�"� qdd

gd

�Bg

d;s

ÿXd2DS

Xs�1;T d

qddgdY g

d;s

#; �40�

where Td as given by Eq. (38), subject toEqs. (41)±(67).

7.2. End-product stock bounding

The end-product stock bounding (5) and (21) isrepresented by the constraints (41), based on theexpressions (27). They state that the stock volumefor given end-product and time period should beappropriately lower and upper bounded for eachscenario to consider.

mSgj;t ÿ

Xs�1;t

V gj;s6

Xs�1;t

uj;sr Zgj;s ÿ

Xs�1;t

Xd2DSj

Y gd;s

6MSgj;t ÿ

Xs�1;t

V gj;s 8j 2 JE; t 2 T ; g 2 G:

�41�See the remarks introduced in Section 4.10.

7.3. Subassembly stock bounding

The subassembly stock bounding (6) and (21) isrepresented by the constraints (42) and (43), basedon the expressions (27) and (30). These constraintsstate that the stock volume for given subassemblyand time period should be appropriately lower andupper bounded for each scenario to consider.

mSgi;t ÿ

Xs�1;t

V gj;s

6Xs�1;t

Egi;se �

Xs�1;t

ui;sZgi;s ÿ

Xs�1;t

Xj2J=i2Ij

cg1ai;jZ

gj;sp

�Xs�1;t

Xf2Ij;j2J=i2If ;j

cg2 ai;j

ÿ ÿ bf ;ji

�ZAf ;jg

i;sa

ÿXs�1;t

Xd2DSi

Y gd;s6MSg

i;t ÿXs�1;t

V gj;s

8i 2 JS; t 2 T ; g 2 G;

�42�


where sp, sa are given in Section 3.5 and se; cg1; cg

2

are referred to in Section 6.2. See the remarks in-troduced in Section 4.10.

ÿXs�1;t

Egi;se ÿ

Xs�1;tÿ1

ui;sZgi;s �

Xs�1;t

Xj2J=i2Ij

cg1ai;jZ

gj;sp

ÿXs�1;t

Xf2Ij;j2J=i2If ;j

cg2 ai;j

ÿ ÿ bf ;ji

�ZAf ;jg

i;sa

�X

s�1;tÿ1

Xd2DSi

Y gd;s6

Xs�1;t

V gi;s

8i 2 JS; t 2 T ; g 2 G: �43�

7.4. External demand balance equations

The balance equations (8) for the external de-mand serviceability with the expression (9) aregiven by the constraints (44), for given externaldemand and time period with the bounds (45) foreach scenario to consider.

ÿ 1ÿ ÿ dg

d

�Bg

d;sÿ1 � 1ÿ ÿ dg

d

�Y g

d;sd � Bgd;s

� 1ÿ ÿ dg

d

�Dg

d;s 8d 2 DS; s 2 T ; g 2 G; �44�where sd is given in Section 3.5, and

Y gd;sd ÿ Bg

d;sÿ16Dgd;s: �45�

7.5. Transferable raw component stock bounding

The raw component stock bounding (10) and(21) is represented by the constraints (46), based onthe expressions (27). These constraints state that thestock volume for given transferable raw componentand time period should be appropriately lower andupper bounded for each scenario to consider.

mSgi;t ÿ

Xs�1;t

V gj;s

6Xs�1;t

X gi;ss �

Xs�1;t

Egi;se ÿ

Xs�1;t

Xj2J=i2Ij

cg1ai;jZ

gj;sp

�Xs�1;t

Xf2Ij;j2J=i2If ;j

cg2 ai;j

ÿ ÿ bf ;ji

�ZAf ;jg

i;sa

6MSgi;t ÿ

Xs�1;t

V gj;s 8i 2 IT; t 2 T ; g 2 G;

�46�

where sp, sa are given in Section 3.5 and ss, se, cg1,

cg2 are referred to in Section 6.2. See the remarks

introduced in Section 4.10.

7.6. Non-transferable raw component balance equa-tions

The balance equations for the procurement andutilization of non-transferable raw components(10) are given by the constraints (47). These con-straints state that the supply volume for givencomponent and time period must equate thecomponent's volume that is delivered to satisfyinternal needs given by the BoM of related periodsfor each scenario to consider:

X gi;s � Eg

i;s ÿX

j2J=i2Ij

cg1ai;jZ

gj;sp

�X

f2Ij;j2J=i2If ;j

cg2 ai;j

ÿ ÿ bf ;ji

�ZAf ;jg

i;sa � 0

8i 2 IRÿ IT; s 2 T ; g 2 G;

�47�

where sp, sa are given in Section 3.5 and cg1, cg

2 arereferred to in Section 6.2.

7.7. Alternate components requirements

For a product to be made available at a giventime period and each of its prime components withsubstitution capabilities in the related BoM, thefollowing constraint type is required for each sce-nario to consider:

Zgj;s P

Xf2Ii;j

cg3ZAi;jg

f ;s

8s � Xi;jg; . . . ;Ci;jg

; i 2 Ij=I i;j 6�£; j 2 J ; g 2 G;

�48�where c3

d is referred to in Section 6.2, and

Xi;jg � minf2Ii;j

si;j�ÿ�gf

n� oi;j

o; �49�

Ci;jg � maxf2I i;j

si;j��gf

n� oi;j

o: �50�

See the balance equations (11).


7.8. Components and products group bounding

The maximum and minimum usage (12) that isallowed for a given raw component group at agiven time period can be expressed as follows foreach scenario to consider:

mRh;s6Xi2IRh

Xj2J=i2Ij

cg1ri;hai;jZ

gj;sp

24�X

f2Ij;j2J

cg2ri;h bf ;j

i

ÿ ÿ ai;j

�Zf ;jg

i;sa

356MRh;s

8h 2 RG; s 2 T ; g 2 G; �51�

where sp, sa are given in Section 3.5 and cg1, cg

2 arereferred to in Section 6.2.

The maximum and minimum release capacity(13) that is allowed for a given product group at agiven time period can be expressed as follows foreach scenario to consider.

mPh;sr 6Xj2Jh

uj;hZgj;s6MPh;sr

8h 2 PG; s 2 T ; g 2 G; �52�where sr is given in Section 3.5.

7.9. Bounds on the variables

Maximum product release volume

06 Zgj;s6MZj;sr 8s � cj; . . . ; jT j; j 2 J ; g 2 G;

�53�where sr is given in Section 3.5.

06ZAi;jg

f ;s 8s � si;j�ÿ�gf � oi;j; . . . ; si;j��g

f � oi;j;

f 2 I i;j; i 2 Ij; j 2 J ; g 2 G: �54�Maximum component procurement volume

06X gi;s6MXg

i;s

8s � 1; . . . ; jT j ÿ ls;gi ; i 2 IT; g 2 G;

8s � Xgi ; . . . ;Cg

i ; i 2 IRÿ IT; g 2 G; �55�

06Egi;s6MEg

i;s

8s � 1; . . . ; jT j ÿ le;gi ; i 2 JS [ IT; g 2 G

8s � Xgi ; . . . ;Cg

i ; i 2 IRÿ IT; g 2 G; �56�

where

XgI � min min

j2J=i2Ij

s�ÿ�g

i;j

n�ÿ li;j

o;

minf2Ij;j2J=i2If ;j

sf ;j�ÿ�gi

nÿ li;j

o�; �57�

Cgi � max max

j2J=i2Ij

s��g

i;j

n�ÿ li;j

o;

maxf2Ij;j2J=i2If ;j

sf ;j��gi

nÿ li;j

o�: �58�

Maximum product backlog, and non-negativecharacter of the variables

06Bgd;s6MBd;s 8s 2 T ; d 2 DS; g 2 G; �59�

06 Y gd;s 8s � 1; . . . ; jT j ÿ ld ; d 2 DS; g 2 G:

�60�

7.10. Splitting variable constraints

The Z-, ZA-, X-, E-, Y- and B-variables indexedwith time periods from the set T1 of implementabletime periods for the whole set of scenarios, requirethe following non-anticipativity constraints. Forall g 2 G, t 2 T1:

Zgj;t ÿ Zg�1

j;t � 0 8j 2 J ; �61�

ZAf ;jg

i;t ÿ ZAf ;jg�1

i;t � 0 8i 2 If ;j; f 2 Ij; j 2 J ;

�62�

X gi;t ÿ X g�1

i;t � 0 8i 2 IR; �63�

Egi;t ÿ Eg�1

i;t � 0 8i 2 I ; �64�


Y gd;t ÿ Y g�1

d;t � 0 8d 2 DS; �65�

Bgd;t ÿ Bg�1

d;t � 0 8d 2 DS: �66�

Note. By convention it is assumed that g � 1 �0 for g � jGj (circular redundant link).

7.11. Preventing unnecessary product and compo-nent stock

The following penalization can be included inthe objective functions (37), (39) and (40) to pre-vent unnecessary build-up:

a �Xg2G

wgXi2IT

Xt2T

Xs�1;t

Hi;sXgi;ss

24�X

i2JS[IT

Xt2T

Xs�1;t

Hi;sEgi;se

�Xj2J

Xt2T

Xs�1;t

Hj;suj;sZgj;s

ÿX

i2JS[IT

Xt2T

Xs�1;t

Xj2J=i2Ij

Hi;scg1ai;jZ

gj;sp

�X

i2JS[IT

Xt2T

Xs�1;t

Xf2Ij;j2J=i2If ;j

cg2Hi;s ai;j

ÿ ÿbf ;ji

�ZAf ;jg

i;ta

ÿXj2J

Xt2T

Xs�1;t

Xd2DSj

Hj;sYg

d;s

35 �67�

where sp, sa are given in Section 3.5 andss; se; cg

1; cg2 are referred to in Section 6.2, re-

spectively.

8. Conclusions

A modeling framework for MAD supply chainmanagement optimization under uncertainty hasbeen presented. The mathematical expressions ofcertain types of variables have been used to reducethe problem's dimensions. In spite of the con-straint matrix density increase, it seems to be avery good implementation-oriented model. How-

ever, the DEM for the 2-stage stochastic problemhas still such big dimensions that it is impracticalto solve it without using some type of decompo-sition approach. An intensive computational ex-perimentation is underway, via AugmentedLagrangian-based and Benders-based Decompo-sition approaches, by using real-life instances fromthe automotive sector for both sequential andparallel computing implementations.

Acknowledgements

This work has been partially supported by theEurope Commission within the ESPRIT programHPCN domaine, project ES26267.

References

Afentakis, P., Gavish, B., Karamarkar, U., 1984. Computa-

tional e�cient optimal solutions to the lot-sizing problem in

multistage assembly systems. Management Science 30, 222±

239.

Alvarez, M., Cuevas, C.M., Escudero, L.F., de la Fuente, J.L.,

Garc�õa, C., Prieto, F.J., 1994. Network planning under

uncertainty with application to hydropower generation.

TOP 2, 25±28.

Baricelli, P., Lucas, C., Mitra, G., 1996. A model for strategic

planning under uncertainty. TOP 4, 361±384.

Beale, E.M.L., 1955. On minimizing a convex function subject

to linear inequalities. Journal of Royal Statistics Society

17b, 173±184.

Birge, J.R., 1985. Decomposition and partitioning methods for

multistage linear programs. Operations Research 33, 1089±

1107.

Birge, J.R., Donohue, C.J., Holmes, D.F., Svintsistski, O.G.,

1996. A parallel implementation of the nested decomposi-

tion algorithm for multistage stochastic linear programs.

Mathematical Programming 75, 327±352.

Birge, J., Louveaux, F.V., 1988. A multicut algorithm for two-

stage stochastic linear programs. European Journal of

Operational Research 34, 384±392.

Birge, J., Louveaux, F.V., 1997. Introduction to Stochastic

Programming. Springer, Berlin.

Cheng, C.H., Miltenburg, J., submitted. A production planning

problem where products have alternate routings and bills of

material. European Journal of Operational Research.

Cohen, M.A., Lee, H.L., 1989. Resource deployment analysis

of global manufacturing and distribution networks. Journal

of Manufacturing and Operations Management 2, 81±104.


Cohen, M.A., Moon, S., 1991. An integrated plant loading

model with economics of scale and scope. European Journal

of Operational Research 50, 266±279.

Dantzig, G.B., 1955. Linear programming under uncertainty.

Management Science 1, 197±206.

Dantzig, G.B., 1985. Planning under uncertainty using parallel

computing. Annals of Operations Research 14, 1±17.

Dembo, R.S., 1991. Scenario optimisation. Annals of Opera-

tions Research 30, 63±80.

Dempster, M.A.H., 1988. On stochastic programming II:

Dynamic problems under risk. Stochastics 25, 15±42.

Dempster, M.A.H., Gassmann, H.I., 1990. Using the expected

value of perfect information to simplify the decision tree.

Abstracts 15th IFIP Conference on System Modeling and

Optimization. IFIP, Zurich, pp. 301±303.

Dempster, M.A.H., Thompson, R.T., to appear. EVPI-based

importance sampling solution procedures for multistage

stochastic linear programs on parallel MIMD Architectures.

Annals of Operations Research.

Dzielinski, G.D., Gomory, R., 1965. Optimal programming of

lot sizes, inventory and lot size allocations. Management

Science 11, 874±890.

Eppen, G.D., Martin, R.K., Schrage, L., 1989. A scenario

approach to capacity planning. Operations Research 37,

517±527.

Escudero, L.F., 1994a. CMIT: a capacitated multi-level implo-

sion tool for production planning. European Journal of

Operational Research 76, 511±528.

Escudero, L.F., 1994b. Robust decision making as a decision

making aid under uncertainty. In: Rios, S. (Ed.), Decision

Theory and Decision Analysis. Kluwer Academic Publish-

ers, Boston, MA, pp. 127±138.

Escudero, L.F., Kamesam, P.V., 1993. MRP modeling via

scenarios. In: Ciriani, T.A., Leachman, R.C. (Eds.), Opti-

misation in Industry. Wiley, New York, pp. 101±111.

Escudero, L.F., Kamesam, P.V., 1995. On solving stochastic

production planning problems via scenario modelling. TOP

3, 69±96.

Escudero, L.F., Salmer�on, J., 1998. 2-stage partial recourse for

stochastic programming: A comparison of di�erent model-

ing and algorithmic schemes. Working paper, IB-

ERDROLA Ingenier�õa y Consultor�õa, Madrid, Spain.

Escudero, L.F., de la Fuente, J.L., Garc�õa, C., Prieto, F.J., to

appear. A parallel computation approach for solving

multistage stochastic network problems. Annals of Opera-

tions Research.

Escudero, L.F., Kamesam, P.V., King, A.J., Wets, R.J.-B.,

1993. Production planning via scenario modelling. Annals

of Operations Research 43, 311±335.

Florian, M., Klein, M., 1971. Deterministic production plan-

ning with concave costs and capacity constraints. Manage-

ment Science 18, 12±20.

Gassmann, H.I., 1990. MSLiP: A computer code for the

multistage stochastic linear programming problem. Math-

ematical Programming 4, 407±423.

Goyal, S.K., Gunasekaran, A., 1990. Multi-stage production-

inventory systems. European Journal of Operational Re-

search 46, 1±20.

Graves, S.C., Fine, C.H., 1988. A tactical planning model for

manufacturing subcomponents of mainframe computers.

Journal of Manufacturing and Operations Management 2,

4±34.

Higle, J.L., Sen, S., 1996. Stochastic Decomposition. A

Statistical Method for Large-Scale Stochastic Linear Pro-

gramming. Kluwer Academic Publishers, Boston, MA.

Infanger, G., 1994. Planning under Uncertainty, Solving

Stochastic Linear Programs. Boyel & Fraser.

Kall, P., Wallace, S., 1994. Stochastic Programming. Wiley,

New York.

Karmarkar, U.S., 1987. Lot sizes, lead times and in process

inventories. Management Sciences 33, 409±423.

Karmarkar, U.S., 1989. Capacity loading and release planning

with work in progress and lead times. Journal of Manufac-

turing and Operations Management 2, 105±123.

Kekre, S., Kekre, S., 1985. Work in progress considerations in

job shop capacity planning. Working paper GSIA, Carnegie

Mellon University, Pittsburgh, PA.

Lasdon, L.S., Terjung, R.C., 1971. An e�cient algorithm for

multi-echelon scheduling. Operations Research 19, 946±969.

Mulvey, J.M., Vanderbei, R.J., Zenios, S.A., 1995. Robust

optimisation of large-scale systems: General modelling

framework and computations. Operations Research 43,

244±281.

Mulvey, J.M., Vladimirou, H., 1991. Solving multistage

stochastic networks: An application of scenario aggrega-

tion. Networks 21, 619±643.

Mulvey, J.M., Ruszczynski, A., 1992. A diagonal quadratic

approximation method for large-scale linear programs.

Operations Research Letters 12, 205±221.

Rockafellar, R.T., Wets, R.J.-B., 1991. Scenario and policy

aggregation in optimisation under uncertainty. Mathemat-

ics of Operations Research 16, 119±147.

Ruszczynski, A., 1993. Parallel decomposition of multistage

stochastic programs. Mathematical Programming 58, 201±

208.

Shapiro, J.F., 1993. Mathematical programming models and

methods for production planning and scheduling. In:

Graves, S.C., Rinnooy Kan, A.H.G., Zipkin, E. (Eds.),

Logistics of Production and Inventory. North-Holland,

Amsterdam, pp. 371±443.

Van Slyke, R., Wets, R.J.-B., 1969. L-shaped linear programs

with applications to optimal control and stochastic pro-

gramming. SIAM Journal on Applied Mathematics 17, 638±

663.

Vladimirou, H., to appear. Computational assessment of

distributed decomposition methods for stochastic linear

programs. European Journal of Operational Research.

Vladimirou, H., Zenios, S.A., 1997. Stochastic linear programs

with restricted recourse. European Journal of Operational

Research 101, 177±192.


Wets, R.J.-B., 1988. Large-scale linear programming tech-

niques. In: Ermoliev, Y., Wets, R.J.-B. (Eds.), Numerical

Techniques for Stochastic Optimization. Springer, Berlin,

pp. 65±93.

Wets, R.J.-B., 1989. The aggregation principle in scenario

analysis and stochastic optimisation. In: Wallace, S.W.

(Ed.), Algorithms and Model Formulations in Mathemat-

ical Programming. Springer, Berlin, pp. 92±113.


Schumann, a modeling framework for supply chain management under uncertainty

Education

Transcript of Schumann, a modeling framework for supply chain management under uncertainty