Page 304 - DMGT523_LOGISTICS_AND_SUPPLY_CHAIN_MANAGEMENT

Unit 13: Logistics Design and Operational Planning

Mathematical Programming Notes

Mathematical programming methods, which are classified as optimization techniques, are one
of the most widely used strategic and tactical logistics planning tools. Linear programming, one of
the most common techniques used for location analysis, selects the optimal supply chain design
from a number of available options while considering specific constraints. House and Karrenbauer
provided a long-standing definition of optimization relevant to logistics:
An optimization model considers the aggregate set of requirements from the customers, the
aggregate set of production possibilities for the producers, the potential intermediary points,
the transportation alternatives and develops the optimal system. The model determines on an
aggregate flow basis where the warehouses should be, where the stocking points should be,
how big the warehouses should be and what kinds of transportation options should be
implemented.

To solve a problem using linear programming, several conditions must be satisfied. First, two
or more activities or locations must be competing for limited resources.

Example: Shipments must be capable of being made to a customer from at least two
locations.

Second, all pertinent relationships in the problem structure must be deterministic and capable
of linear approximation. Unless these enabling conditions are satisfied, a solution derived from
linear programming, while mathematically optimal, may not be valid for logistical planning.
While linear programming is frequently used for strategic logistics planning, it is also applied
to operating problems such as production assignment and inventory allocation. Within
optimization, distribution analysts have used two different solution methodologies for logistics
analysis.

One of the most widely used forms of linear programming for logistics problems is network
optimization. Network optimization treats the distribution channel as a network consisting of
nodes to identify production, warehouses, and markets and arcs reflecting transportation links.
Costs are incurred for handling goods at nodes and moving goods across arcs. The network
model objective is to minimize the total production, inbound and outbound transportation
costs subject to supply, demand, and capacity constraints.
Beyond the basic considerations for all analytical techniques, network optimization has specific
advantages and disadvantages that both enhance and reduce its application for logistics analyses.
Rapid solution times and ease of communication between specialists and non-specialists are the
primary advantages of network models. They may also be applied in monthly, rather than
annual, time increments, which allows for longitudinal or across-time analysis of inventory
level changes. Network formulations may also incorporate fixed costs to replicate facility
ownership. The results of a network model identify the optimum set of distribution facilities
and material flows for the logistics design problems as it was specified for the analysis.
The traditional disadvantages of network optimization have been the size of the problem that
can be solved and the inclusion of fixed cost components. The problem size issue was of particular
concern for multistage distribution systems such as those including suppliers, production
locations, distribution centres, wholesalers, and customers. While problem size is still a concern,
advancements in solution algorithms and hardware speed have significantly improved network
optimization capabilities. The fixed cost limitation concerns the capability to optimize both
fixed and variable costs for production and distribution facilities. There have been significant
advancements in overcoming this problem through the use of a combination of network
optimization and mixed-integer programming.

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