Geoscience Reference
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receiving it from other locations. The outbound flow results from using the product
as a raw material to manufacture other commodities, distributing the item to other
facilities, or serving demand in case the location is a customer zone. Inequali-
ties (
16.3
), resp. (
16.4
), guarantee that the usage of manufacturing, resp. handling,
resources does not exceed the available capacity. Constraints (
16.5
)-(
16.6
) stipulate
that capacity expansions must be within given limits. Constraints (
16.7
)rule
the maximum amount of unsatisfied demand. Inequalities (
16.8
)-(
16.11
) ensure
that procurement, production, and distribution activities only occur at operating
facilities. A sufficiently large constant
is used in these constraints which can
be adjusted depending on each specific situation. Typically,
M
is replaced by
the maximum quantity that can be processed by a facility with respect to all
product types. Finally, constraints (
16.12
) are non-negativity conditions for the
logistics operations in non-selectable locations, while constraints (
16.13
) are binary
requirements for the location variables.
Although the above problem is NP-hard, being a generalization of the simple
plant location problem (see Krarup and Pruzan
1983
), Melo et al. (
2008
) could solve
medium and large-sized randomly generated instances to optimality with general
purpose optimization software within reasonable time. To analyze the quality of the
MILP formulation, the linear relaxation bound was also compared with the optimal
solution of the tested instances. In general, a relatively small gap could be observed.
These findings have important practical implications, since managers often need to
base their decisions on the results of several scenarios. Hence, for a company to
be able to perform “what-if” analysis and thereby identify good quality (or even
optimal) solutions with an acceptable level of computational effort is a major step
towards better decision support.
M
16.2.3
Special Cases and Model Extensions
Historically, researchers have focused relatively early on the design of distribution
systems with at most two facility layers (e.g., plants and warehouses). In these
simple networks, decisions were mostly confined to facility location and distribution
operations. The contribution by Geoffrion and Graves (
1974
)issuchanexample.
In recent years, the trend has been towards the development of more comprehensive
models that integrate location decisions with supplier selection, production plan-
ning, technology acquisition, inventory management, transportation mode selection,
and vehicle routing, just to mention some important logistics functions considered
in this area (see Melo et al.
2009
for a comprehensive review). In many cases, the
proposed models combine strategic decisions (e.g., location and capacity choices)
with tactical decisions (e.g., inventory and transportation management) or even
operational decisions (e.g., vehicle routing). Usually, the interplay of different
planning levels can only be captured at the cost of increased model complexity.
This will be illustrated in Sect.
16.4
by three applications.
