The Design of Production-Distribution Networks: A Mathematical Programming Approach

Abstract

This text proposes a mathematical programming approach to design international production-distribution networks for make-to-stock
products with convergent manufacturing processes. Various formulations of the elements of production-distribution network
design models are discussed. The emphasis is put on modeling issues encountered in practice which have a significant impact
on the quality of the logistics network designed. The elements discussed include the choice of an objective function, the
definition of the planning horizon, the manufacturing process and product structures, the logistics network structure, demand
and service requirements, facility layouts and capacity options, product flows and inventory modeling, as well as financial
flows modeling. Major contributions from the literature are reviewed and a number of new formulation elements are introduced.
A typical model is presented, and the use of successive mixed-integer programming to solve it with commercial solvers is discussed.
A more general version of the model presented and the solution method described were implemented in a commercial supply chain
design tool which is now available on the market.

Full text

Page 1

The design of production-distribution
networks: A mathematical
programming approach
Alain Martel
(alain.martel@centor.ulaval.ca)
August 2004
Working Paper DT-2004-AM-2
To appear in:
J. Geunes and P.M. Pardalos (eds.), Supply Chain Optimization, Kluwer Academic Publishers, 2005.
Network Organization Technology Research Center (CENTOR),
Université Laval, Québec, Canada, G1K 7P4
© Alain Martel, Centor, 2004

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DT-2004-AM-2
1
The design of production-distribution networks:
A mathematical programming approach
Alain Martel
CENTOR Research Center, Université Laval, Canada, G1K 7P4
Abstract
This text proposes a mathematical programming approach to design international
production-distribution networks for make-to-stock products with convergent manufacturing
processes. Various formulations of the elements of production-distribution network design
models are discussed. The emphasis is put on modeling issues encountered in practice which
have a significant impact on the quality of the logistics network designed. The elements
discussed include the choice of an objective function, the definition of the planning horizon, the
manufacturing process and product structures, the logistics network structure, demand and
service requirements, facility layouts and capacity options, product flows and inventory
modeling, as well as financial flows modeling. Major contributions from the literature are
reviewed and a number of new formulation elements are introduced. A typical model is
presented, and the use of successive mixed-integer programming to solve it with commercial
solvers is discussed. A more general version of the model presented and the solution method
described were implemented in a commercial supply chain design tool which is now available
on the market.
Keywords
logistics network design, Supply chain engineering, Location-allocation problems, Capacity
planning, Technology selection, Mathematical programming.

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The design of production-distribution networks: A mathematical programming approach
Context
How many production and distribution centres should a company have to satisfy the demand
of its targeted markets? Where should they be located and what should their mission be? What
supply sources should they use? What technologies should they install for production, storage,
shipping and receiving? Which sub-contractors and public warehouses should they do business
with? What means of transportation should they choose? All of these questions are related to
strategic and tactical logistics network design issues, which are critical for the success of
modern manufacturing and distribution companies. This text proposes a mathematical
programming approach to analyze several of these logistics network design issues.
The exact nature of the logistics network design problems encountered in practice depends
very much on the industrial context in which they occur. For example:
!
The design problem to solve for a high volume consumer goods manufacturer is very
different than the problem found in a highly customized make-to-order products industry or
in a slow moving repair parts distribution context. In a make-to-stock industry, the order-to-
delivery time depends on the positioning of finished goods inventories but, in a make-to-
order context, it depends on manufacturing lead times and on the depth of penetration of
customer orders in the supply chain, i.e. on the positioning of semi-finished product or raw-
material inventories.
!
When manufacturing resource acquisition, deployment and/or allocation decisions are
considered, the nature of the production process must also be taken into account. In some
industries, manufacturing processes are divergent: several products are made from a
common raw material (e.g. pulp and paper industry, meat industry, etc.). In other sectors the
manufacturing processes are convergent: several raw-materials and components are
assembled into finished products. In some industries, the manufacturing processes may even
include feedback loops.
!
Networks covering several countries lead to much more complex design problems than
single-country networks. Factors such as exchange rates, transfer prices, duties and income
taxes must then be taken into account.

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The design of production-distribution networks: A mathematical programming approach
The detailed discussion of all these variants is beyond the scope of this paper. In what follows
our coverage focuses on the design of international production-distribution networks for make-
to-stock products with convergent manufacturing processes.
As can be seen, logistics network design problems, as defined here, integrate several sub-
problems which have been treated separately in the literature: capital investment planning for
the acquisition of new capacity, technology selection, facility location and manufacturing-
distribution resource allocation problems. Capacity expansion problems are usually posed as
multi-year capital investments problems under uncertainty (Freidenfelds, 1981; Luss, 1982).
The financial planning aspects of the problem, such as real options (Trigeorgis, 1996), are
predominant in the analysis and the logistics aspects are highly aggregated. Technology
selection problems can be seen as an extension of capacity planning where there are several
alternative capacity types available (Fine, 1993, Paquet et al., 2004). At the other extreme,
resource allocation problems deal with detailed plant loading and inventory placement decisions
under the assumption that the plant/warehouse network configuration is fixed (Glover et al.,
1979; Cohen and Moon, 1991; Mazzola and Schantz, 1997). They often consider a single year
planning horizon divided into several seasons. The literature on basic discrete location models
(Francis et al., 1992; Daskin, 1995; Sule, 2001) concentrates on single period, single echelon,
geographical deployment problems. A lot of the effort in this field has been devoted to finding
efficient solution methods for a set of well defined problems. Some extensions to classical
facility location problems are reviewed by Revelle et al. (1996) and by Owen et al. (1998). An
abundant literature exists on location, capacity acquisition and technology selection problems.
An integrated review of the early work done in these fields is found in Verter and Dincer (1992).
Supply chain design models incorporate elements of all the sub-problems discussed previously.
Geoffrion and Powers (1995) and Shapiro et al. (1993) discuss the evolution of strategic supply
chain design models and Vidal and Goetschalckx (1997) present many of these models. Shapiro
(2001) provides an excellent coverage of several supply chain modeling issues.

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The design of production-distribution networks: A mathematical programming approach
In this paper, various formulations of the elements of production-distribution network
design models are discussed. The emphasis is put on modeling issues encountered in practice
which have a significant impact on the quality of the logistics network designed. The elements
discussed include the choice of an objective function, the definition of the planning horizon, the
manufacturing process and product structures, the logistics network structure, demand and
service requirements, facility layouts and capacity options, product flows and inventory
modeling, as well as financial flows modeling. Major contributions from the literature are
reviewed and a number of new formulation elements are introduced. A typical model is
presented, and the use of successive mixed-integer programming to solve it with commercial
solvers is discussed. A more general version of the model proposed and the solution method
described were implemented in the Supply Chain Studio, a commercial supply chain design tool
sold by Modellium. This tool was used to optimize the production-distribution network of
several multinational companies, including Domtar, one of the largest Pulp and Paper Company
in North-America.
Modeling approach
Performance evaluation
Although most of the logistics network design models presented in the literature adopt a
total system cost minimization objective, this does not necessary lead to the creation of a
competitive advantage. Low cost is an order winning criteria valued by several customers but it
is not the only one (Hill, 1999). Delivery time, quality and flexibility are other valued criteria
which are affected by the logistics activities and resources of the firm. In a make-to-stock
industry, for example, the order-to-delivery time depends on the positioning of finished goods
inventories in the logistics network and it is a criteria as important as cost for the evaluation of
network designs. As explained by Porter (1985), it is the additional value given by customers to
such an order winning criterion that creates a competitive advantage. Figure 1 illustrates the
cost accumulation process and the impact of inventory positioning on customer delivery times

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The design of production-distribution networks: A mathematical programming approach
for a simple multi-echelon (stage) supply chain. As can be seen, costs accumulate as the
products pass through the procurement, production and distribution stages, and value is added
when the finished products are purchased by customers. The cost of support activities can be
interpreted here as all the non-logistic costs incurred by the firm. The response time depends on
whether the customers are served from a local or regional warehouse or from a production-
distribution center or, more generally, on the distance between a customer and its supply facility.
When delivery time is shorter, more revenues are generated through a price premium and/or an
increased market share. Total system cost, maximum delivery time and total revenue figures are
therefore associated with any logistics network design.
Figure 1 : Costs, value added and delivery time in the supply chain
In order to evaluate the performance of various designs, their cost and delivery time can be
plotted on a graph, as shown in Figure 2a). The non-dominated designs are located on an
efficient-frontier, and any of these designs could constitute a good solution for a firm
(Rosenfield, 1985). However, if the impact of delivery time on prices and on demand, and thus
on total revenue, is taken into account, as shown in Figure 2b), the design maximizing the value
added (Total revenue - Total logistics network cost - Cost of support activities) by the logistics
a) Value Chain
b) Delivery time
Inventory:Inventory:Operations: Operations:External entities:External entities:
TransportationTransportation
Handling HandlingRaw materialsRaw materialsManufacturing Manufacturing
CustomersCustomersWork in processWork in process
Vendors Vendors
Finished products Finished products
Assembly Assembly
Warehousing/retailingWarehousing/retailing
MM
ww
AA
VV
CC
Production-distribution Center Production-distribution Center Production-distribution Center
Total revenueTotal revenue
Total logistic network cost
Value addedValue added
Cost
(value)(value)
Cost of support activitiesCost of support activities
Raw
material
costscosts
Costs of primaryactivitiesCosts of primaryactivities
Stages/EchelonsStages/Echelons
VV
CC
ww
MMAA
ww
Total logistic network cost
Cost
Raw
material
Production-distribution CenterProduction-distribution CenterProduction-distribution Center
Local
stock stock
Stages/EchelonsStages/Echelons
Make to
stockstock
Regional
stock stock
Assemble
to orderto order
Make to
order order
MT: Maximum delivery time
accepted by marketaccepted by market
CCVV
ww
ww
MM
AA
Source
to orderto order
Delivery
timetime
Local
Make to
Regional
Assemble
Make to
MT: Maximum delivery time
Source
Delivery

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The design of production-distribution networks: A mathematical programming approach
network can be identified. Ideally, the objective to pursue should therefore be to find the
logistics network design maximizing net revenues. In an international context, since different
countries have different taxation levels, one should rather seek to maximize after tax global net
revenues in a reference currency. Unfortunately, it is not always possible in practice to model
the impact of delivery time on price and demand. When this is the case, one should at least
sketch the efficient frontier by finding the designs minimizing total system costs for a set of
predetermined delivery times.
Figure 2 : Performance evaluation methods
Despite the fact that an abundant literature exists on the impact that quality and flexibility
may have on competitiveness, little work has been done to explicitly incorporate them as
performance criterion in logistics network design models. By associating different technologies
to different quality levels, quality can often be treated in a way similar to delivery times. Some
dimensions of flexibility, such as operational flexibility in global networks under exchange rate
risk (Kogut and Kulatilaka, 1994; Huchzermeier and Cohen, 1996), have been studied, but more
research is needed on the incorporation of the various dimensions of flexibility into network
design models. The model presented in what follows seeks to maximize after tax net revenues,
taking the impact of delivery times on revenues into account.
a) Cost-delivery frontier b) Value added maximization
MTMT
Total
logistic
network
costcost
Delivery timeDelivery time
Efficient
frontierfrontier
Logistic
network
designs designs
Total
logistic
network
Efficient
Logistic
network
Total revenueTotal revenue
MTMT
Value
Value
added
added
added
added
Delivery time Delivery time
Total logistic
network costnetwork cost
Cost
(Value)(Value)
Value
Value
Total logistic
Cost

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The design of production-distribution networks: A mathematical programming approach
Planning horizon and uncertainty
In capital intensive industries, capacity expansion decisions may require the explicit
consideration of a planning horizon including as much as ten years (Everett et al., 2000, 2001).
On the other end, when product supply and/or demand is seasonal, decisions on production and
inventory levels for each network location must be made on a quarterly basis or even on a
monthly basis. This means that the number of planning periods in logistics network design
models could be very large. In addition to the explosion of problem size, using a long planning
horizon makes the gathering of meaningful information on the future business environment
extremely difficult. Some approach to reduce this complexity must therefore be used in practice.
To clarify this issue, let us first make a distinction between the notions of season and period.
In most design models, 0-1 variables are associated with capacity acquisition and deployment
decisions and continuous variables to resource allocation decisions (production and inventory
levels, network flows). A multi-period model is concerned with the change of state of the
network structure (number, location, technology and capacity of facilities) over the long term
(typically several one year periods). A multi-season model is concerned with the change of
mission of the network resources during a planning period (typically months or quarters during
a year). Several formulations presented as multi-period models in the literature are in fact
single-period multi-season models (Cohen et al., 1989; Arntzen et al., 1995; Dogen and
Goetschalckx, 1999). Multi-period models usually concentrate on capacity investment decisions
and they limit themselves to single echelon network structures (Shulman, 1991, Everett et al.,
2000; Bhutta et al., 2003). Following the pioneering work of Pomper (1976), some authors have
also proposed multi-period scenario based stochastic programming models (Eppen et al., 1989;
Ahmed et al., 2001; Everett et al., 2001).
Most of the models published in the literature are deterministic single-period mathematical
programs (Geoffrion and Graves, 1974; Brown et al., 1987; Cohen and Lee, 1989; Cohen and
Moon, 1990; Pirkul and Jayaraman, 1996; Lakhal et al., 2001; Vidal and Goetschalckx, 2001;
Cordeau et al., 2002; Paquet et al., 2004). It is understood, however, that since the acquisition

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The design of production-distribution networks: A mathematical programming approach
and deployment decisions have long-term effects, their analyses must span multiple periods and
the model must be either run sequentially over some finite time horizon or, when size permits,
expanded to incorporate multiple time periods directly (Cohen and Lee, 1989). Also, the fact
that the future is uncertain requires the examination of several scenarios with respect to the
firm’s strategic options and the evolution of its internal and external environment (Shapiro,
2001) or, when size permits, the transformation of the model into a multi-stage stochastic
program with recourse (Birge and Louveaux, 1997). Keeping this in mind, the approach
presented in what follows yields deterministic multi-season logistics network design models.
The following set is used to denote the planning horizon:
T= Seasons of the planning horizon (t ∈ T).
Modeling process and product structures
In order to arrive at a general production-distribution network design model for a given
industrial context, a generic conceptual model of the manufacturing process of the industry must
first be elaborated. Such a conceptual model treats products and production stages in an
aggregate manner to capture the essence of the manufacturing process, but without concern for
operational details (Shapiro, 2001). It can take the form of an activity network or of a bill-of-
materials, as illustrated in Figure 3 (Lakhal et al., 1999). In these conceptual models, products
are grouped into product families and some activities may be an amalgam of several operations.
It is common to use process network representations in process manufacturing environments
such as petro-chemicals, food, pulp and paper, pharmaceutical, etc. (Brown et al., 1987; Dogan
and Goetschalckx, 1999; Philpott and Everett (2001); Vila et al., 2003). In such contexts,
associated with each activity are a number of methods (recipes) that describe how inputs are
transformed into outputs using different potential technologies. In discrete parts manufacturing
industries, however, a bill-of-materials representation is usually more adequate (Cohen and
Moon, 1990; Arntzen et al., 1995; Paquet et al., 2004). This is the approach taken in this paper.
More specifically, the following product structure modeling assumptions are made. Products
are classified in families p ∈ P requiring the same type of production capacity or supplied by the

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The design of production-distribution networks: A mathematical programming approach
same vendors. Products available only from external suppliers are considered as raw material
(RM) and other products can be manufactured in the network plants. The manufactured
products (MP) are sub-assemblies (SA) or make-to-stock (MS) finished products. The semi-
finished products can come partly from external suppliers and partly from the network plants.
The aggregated bill-of-materials, illustrated in Figure 3b), is an acyclic directed graph. The
number associated with the edge (p’p) of the bill-of-materials graph indicates the quantity of the
product p’ needed to make one product of type p. It is assumed that the vertices of this graph are
numbered in topological order, i.e. that for each edge (p’p), we have p > p’.
Figure 3 : Potential logistics Network
A technology is defined by the set of products it can manufacture/store, and it is assumed
that the bill-of-materials is independent of the technology used. As illustrated in Figure 3b), the
capacity required to produce one product can be provided by either flexible or dedicated
technologies. Dedicated technologies are associated with only one product family, but flexible
technologies can be used to make several product families. Similarly, the capacity needed to
stock the finished products can be provided by a set of potential storage technologies. When a
facility is used, a technology for the reception and shipping of products must also be
implemented. To simplify, it is assumed that this technology can be used for any products and
that its capacity can be expressed adequately in terms of the facility outflows.
a) Manufacturing Process Network
Activity 1Activity 1
Activity 2Activity 2
Activity 5 Activity 5
Activity 4 Activity 4
Activity 3 Activity 3
p1p1
p2p2
p3p3
p3 p3
p4 p4
p5 p5
p6p6
p6p6
p7p7
111111111111111111111111111111111111111111111111222222222222222222222222222222222222222222222222333333333333333333333333333333333333333333333333
111111111111111111111111111111111111111111111111
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111111111111111111111111111111111111111111111111
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333333333333333333333333333333333333333333333333
444444444444444444444444444444444444444444444444
111111111111111111111111111111111111111111111111
888888888888888888888888888888888888888888888888999999999999999999999999999999999999999999999999
444444444444444444444444444444444444444444444444555555555555555555555555555555555555555555555555
666666666666666666666666666666666666666666666666777777777777777777777777777777777777777777777777
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333333333333333333333333343434343434343433434344334433443343434433443344
777777777777777777777777
888888888888888888888888
999999999999999999999999
10101010101010101010 1010101010101010101010101010
111111111111111111111111111111111111111111111111
1313131313131313131313131313 131313131313131313 13
1212121212121212121212121212121212121212121212121414141414141414141414141414141414141414 14141414
555555555555555555555555
666666666666666666666666
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Products (p ∈P)
Dedicated Technologies
Flexible Technologies
Bill-of-material
Bill-of-
Bill-of-materials
Bill-of-Bill-of-
Bill-of-materialBill-of-material
Bill-of-Bill-of-
Bill-of-materialsBill-of-materials
p ∈RM
∈RM
∈RM
∈RM
p ∈RM
∈RM
p ∈RM
∈RM
∈RM
∈RM
p ∈RM
∈RM
p ∈RM
p ∈RM
∈RM
∈RM
∈RM
∈RM
∈RM
∈RM
p ∈RM
p ∈RM
∈RM
∈RM
p ∈RM
p ∈RM
∈RM
∈RM
∈RM
∈RM
∈RM
∈RM
p ∈RM
p ∈RM
∈RM
∈RM
p ∈MP
∈MP
∈MP
∈MP
p ∈MP
∈MP
∈MP
∈MP
p ∈MP
∈MP
∈MP
∈MP
p ∈MP
∈MP
∈MP
∈MP
p ∈MP
p ∈MP
∈MP
∈MP
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∈MP
∈MP
∈MP
p ∈MP
p ∈MP
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∈MP
p ∈MP
p ∈MP
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∈MP
∈MP
∈MP
∈MP
p ∈MP
p ∈MP
∈MP
∈MP
∈MP
∈MP
∈MP
p ∈FP
∈FP
∈FP
∈FP
p ∈FP
∈FP
∈FP
∈FP
p ∈FP
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∈FP
∈FP
p ∈FP
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p ∈FP
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p ∈FP
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p ∈FP
p ∈FP
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∈FP
∈FP
∈FP
∈FP
∈FP
p ∈FP
p ∈FP
∈FP
∈FP
∈FP
∈FP
∈FP
Raw
MaterialsMaterialsMaterialsMaterialsMaterialsMaterialsMaterialsMaterialsMaterialsMaterialsMaterialsMaterials
Finished ProductsFinished ProductsFinished ProductsFinished ProductsFinished ProductsFinished ProductsFinished ProductsFinished ProductsFinished ProductsFinished ProductsFinished ProductsFinished Products
Manufactured
ProductsProductsProductsProductsProductsProductsProductsProductsProductsProductsProductsProducts
Bill-of-
RMRMRMRMRMRMRMRMRMRMRM
RawRawRawRawRawRawRawRawRawRawRaw
ManufacturedManufacturedManufacturedManufacturedManufacturedManufacturedManufacturedManufactured ManufacturedManufacturedManufactured
22222
2 2 2 2 2 2 2 22 2 2 2 2 2 2 22 2 2 2 22 2 2 2 2 2 2 2
2 2 2
2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 22 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 33 3 3 3 3 3 3 33 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3
Products (p ∈P)
Dedicated Technologies
Flexible TechnologiesFlexible TechnologiesFlexible TechnologiesFlexible TechnologiesFlexible Technologies
Products (p ∈P)Products (p ∈P)
Dedicated TechnologiesDedicated TechnologiesDedicated TechnologiesDedicated Technologies
Products (p ∈P)Products (p ∈P)
b) Bill-of-materials with Potential Technologies

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The design of production-distribution networks: A mathematical programming approach
The notation used to model product structures and technologies is the following:
P
RM
MP
SA
gpp’
KW
KM
KS
qpk
wp
=Product families (p ∈ P).
=Raw material families (
=Manufactured product families, i.e. sub-assemblies and finished products (
= Sub-assemblies families (
=Quantity of product p needed to make one product p’.
= Receiving/shipping/handling technologies (k ∈ KW).
=Production technologies (k ∈ KM).
=Storage technologies (k ∈ KS).
=Technology k capacity consumption rate per unit of product p.
=Average weight of family p products in standard weight units.
).
).
).
Network optimization model structure
The structure of logistics networks can be represented by a directed graph. The network
nodes correspond to supply sources, to existing facilities, to sites where it would be possible to
build or buy a production or distribution center, to the facilities of potential partners (sub-
contractors, public warehouses, 3PL consolidation centers, etc.) or to demand zones. The
network arcs represent the flow of products between the nodes. The specification of the
structure of the network and of the mission of its facilities is an important strategic decision.
Two approaches to the problem are found in the literature, as illustrated in Figure 4. A popular
modeling approach has been to assume a priori that a multi-echelon structure is required
(Geoffrion and Graves, 1974; Cohen and Lee, 1989; Pirkul and Jayaraman, 1996; Vidal and
Goetschalckx, 2001). This limits the mission of the facilities to a predetermined role (e.g.
intermediate product plant, final product plant, distribution center) and it forbids product flows
between facilities on the same echelon. In some contexts, this approach can be far from optimal.
In practice, the same facility often has multiple roles: a production-distribution center may
produce both intermediate and finished products and serve as a shipping point to some
customers; a warehouse close to a supplier may serve as a central warehouse for this supplier’s
products, but as a local distribution center for other products, etc. For this reason, other authors
do not impose any a priori echelon structure and expect the optimization model to determine the
best structure and mission for the facilities (Arntzen et al., 1995; Paquet et al., 2004).
RMP
⊂
MPP
⊂
SAMP
⊂

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The design of production-distribution networks: A mathematical programming approach
Two approaches are also used in the literature to model flows in the network. One of them
associates decision variables to paths in the network (Geoffrion and Graves, 1974; Martel. and
Vankatadri, 1999). This is particularly appropriate for multi-echelon distribution networks.
However, for this approach to work, the product flowing on the path must not change between
the first node and the last node, which cannot hold for production facilities. For this reason,
models incorporating more than one production echelon either associate decision variables to
the arcs of the network (Arntzen et al., 1995, Dogan and Goetschalckx, 1999; Vidal and
Goetschalckx, 2001; Cordeau, 2002, Paquet et al., 2004), or they use a hybrid approach (Cohen
and Moon, 1990).
Figure 4 : Potential logistics Network
The model presented in this paper is an arc-based formulation for the general logistics
network illustrated in Figure 4a). Three types of nodes, located in several countries, are present
in the network: external vendors (v∈V), internal potential facility sites (s∈S) and demand zones
(d∈D). A list of potential internal sites (S) must be identified a priori and classified as either
production-distribution center sites (Spd) or distribution center sites (Sd). This list usually
includes the location of the current facilities, of public warehouses or sub-contractors which
could be included in the network, of existing facilities which could be purchased or rented, and
of lands where a new facility could be constructed. It is possible also to limit the mission given
a) General logistics networkb) Multi-echelon network
Raw Material
VendorsVendors
Intermediate
Product Plants Product Plants
FinalFinal
Product PlantsProduct Plants
DCs /DCs /
Warehouses Warehouses
Demand
ZonesZones
Raw Material
Intermediate
Demand
Fournisseurs
Sources
FournisseursFournisseurs
SupplySources (v ∈ ∈ ∈ ∈ V)
Fournisseurs
Sources
FournisseursFournisseurs
SupplySources (v ∈ ∈ ∈ ∈ V)
res(v∈ ∈ ∈ ∈
(v ∈ ∈ ∈ ∈V)
res res(v∈ ∈ ∈ ∈
res(v∈ ∈ ∈ ∈
(v ∈ ∈ ∈ ∈V)
resres(v∈ ∈ ∈ ∈
Demand Zones (d∈ ∈ ∈ ∈D) Demand Zones (d∈ ∈ ∈ ∈D)
To thenetwork facilities To thenetwork facilities
Distribution Centers
Production-DistributionCenters
To thedemandzonesTo thedemandzones
(s ∈ ∈ ∈ ∈ S)(s (s ∈ ∈ ∈ ∈ S)(s
(d(d D)
D)
D)
D)
D)
D)
S)S)
Internal
Lanes
Demand
Lanes
Supply
Lanes
Distribution Centers
Production-DistributionCenters
Internal
Lanes
Demand
Lanes
Supply
Lanes

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The design of production-distribution networks: A mathematical programming approach
to potential sites by restricting the set of production (KMs) and storage (KSs) technologies
which can be implemented in a site, or the set of products (Ps) which can be produced/stored in
a site. The network arcs are associated with transportation lanes. Three types of arcs are
distinguished: supply arcs, internal arcs and demand arcs. The internal arcs adjacent to a site s
are defined by the set of origins of its inbound arcs () and the set of destinations of its
outbound arcs (). Similar node input and output sets are defined for supply and demand arcs.
A continuous decision variable Fpnst is associated with the flow of a product p on lane (n,s) in
season t. Given that a real logistics network may include several hundred thousand arcs,
defining these sets and flow variables in practice is not trivial and it requires the use of an
automated arc generation mechanism.
The customer ship-to locations are grouped into demand zones (D). The definition of these
demand zones depends on the product-markets (M) of the company and on the geographical
dispersion of ship-to points (Ballou, 1994). It is assumed that the company operates national
divisions in several countries o ∈O, and that each of these divisions covers a set of distinct
product-markets m ∈ Mo constituted of several demand zones d ∈ Dm. A market is
characterized by a distinct price and service policy. It is assumed that the products shipped to a
demand zone can come from more than one distribution center. This is common today because
companies tend to operate centralized selling organizations independent of the DC’s. Modifying
the model however to enforce single DC sourcing is not difficult. Similarly, vendors in close
geographic proximity who provide products in the same family can be aggregated into a supply
source (V). It is assumed that the seasonal quantity of product which can be supplied by a
vendor is bounded.
The following sets, indices, parameters and variables are required to define a potential
logistics network:
S
O
So
=Potential network sites (s ∈ S).
= Countries of the network sites (o ∈O, o(s) = country of site s).
=Potential sites in country o.
Sps
i
Sps
o

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The design of production-distribution networks: A mathematical programming approach
Sd
Spd
Sdmax=Upper bound on the number of distribution centers in the network.
Spdmax=Upper bound on the number of production-distribution centers in the network.
Vp
=Vendors of raw material p ∈ RM or of manufactured product p ∈ MP.
bpvt
=Upper bound on the quantity of raw material p which can be supplied by vendor v in
season t.
=Set of potential sites (output destinations) which can receive product p from node n.
=Set of potential sites (input sources) which can ship product p to site s.
Mo
= Potential product-markets in country o (m ∈ M = ∪o∈OMo).
Dm
= Demand zones in product-market m (d ∈ D = ∪m∈MDm).
Do =Demand zones in country o (Do = ∪m∈MoDm).
m(d)= Product-market of demand zone d.
=Set of demand zones (output destinations) which can receive product p from node s.
Ps
=Products which can be manufactured/stocked on site s.
Pks
=Products which can be manufactured/stocked with technology k on site s.
KMps=Production technologies which can be used to manufacture product p on site s
( = ∪pKMps).
KSps =Storage technologies which can be used to stock product p on site s
( = ∪pKSps).
Fpnst
=Flow of product p between node nVp
∈
= Potential distribution center sites (
= Potential production-distribution center sites (
).
).
and site s during season t.
The essence of the logistics network design problem boils down to finding an optimal
mapping of the product/activity structure onto the potential network structure.
Modeling demand, prices and customer service
Although most of the models available in the literature assume that demand is given and not
affected by the logistics network design, this is clearly not realistic. As explained earlier,
demand depends on logistics outputs such as delivery times, and the market may be prepared to
pay a price premium to obtain these outputs. To take this into account, it is assumed that the
company has a choice of marketing policies i ∈ Im for each of its product-markets m (Vila et al,
2004). A marketing policy i ∈ Im is characterized by the price Ppdit the market is prepared to pay
for each product p ∈ P in the demand zones d ∈ Dm during seasons t ∈ T. It is also characterized
by a maximum delivery time and possibly by other value criteria. These value criteria are
SdS
⊂
SpdS
⊂
Spn
Sps
o
i
Dps
o
KMps
KM KMs
,⊂
KSps
KS KSs
,⊂
Sps
i
∪

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The design of production-distribution networks: A mathematical programming approach
related to the network design by defining the set of sites in the potential network which
could deliver the value characteristics of marketing policy i ∈ Im(d), for each product p. It is
further assumed that the largest demand the company can expect for product p in demand
zone d, when marketing policy i ∈ Im(d) is used, can be estimated, and that the company has
minimum market penetration objectives xpdt for each of its product-markets.
In this context, the following notation is required to model the demand:
Im
Spdi
= Marketing policies considered for market m (i ∈ Im).
=Set of potential sites (input sources) which can ship product p to demand zone d,
when marketing policy i ∈ Im(d) is selected.
=Amount received for the sales of product p to demand zone d in season t when
marketing policy i ∈ Im(d) is used (in the demand zone country currency).
=Lower bound on the flow of product p to demand zone d in season t imposed by the
market penetration objectives of the company.
=Upper bound on the flow of product p to demand zone d in season t imposed by the
largest market share the company can expect when marketing policy i ∈ Im(d) is used.
=Binary variable equal to 1 if marketing policy i ∈ Im is used for market m and to 0
otherwise.
Fpsdit
=Flow of product p between site s and demand zone d during season t, when marketing
policy i ∈ Im(d) is selected.
Ppdit
xpdt
Parallel arcs are defined between the network sites s and the demand zones d to model the
flow of products Fpsdit under the different marketing policies i ∈ Im(d). Using these flow
variables and the marketing policy selection variables , it is seen that the seasonal sale
targets of the company must respect the following demand and policy selection constraints:
t ∈T, p ∈P, d ∈D, i ∈Im(d)
(1)
m ∈M(2)
Modeling facility layouts and capacity options
The technical and economic characteristics of the facilities which could be operated on the
network sites can be specified with a facility layout. The facility layout concept is illustrated
Spdi
i
xpdit
i
xpdit
Ymi
M
Ymi
M
xpdtYm d ( )i
M
Fpsdit
sSpdi
i
∈∑
xpditYm d ( )i
M
≤≤
Ymi
M
iIm
∈
∑
1
≤