Location, Routing, and Inventory
Read this article. In it, a model is presented to help determine the number of distribution centers, their locations, and capacity among other factors. Among the 15 assumptions presented, which do you feel are most important and least important?
Solution method
Optimization is a mathematical procedure to
determine devoting the optimal allocation to scarce resources, and it
helps to get the best result from the model solution. In this paper, we
consider five examples, and then they are solved by the Lingo 9.0
software to show that this model works well. This software is a
comprehensive tool designed to make building and solving linear,
nonlinear, and integer optimization models faster, easier, and more
efficient. It provides a completely integrated package that includes a
powerful language for expressing optimization models, a full featured
environment for building and editing problems, and a set of fast
built-in solvers. Objective functions have been normalized
between zero and one. In other words, they have been without any
dimension (i.e., scaleless). By using the following formula, these
objectives are converted to a single objective function, where
and
are the normalized forms of
and
objective functions.
To
minimize deviations from the ideal, this function is reduced. As the
first objective function () is more important than the second
objective function (
) in the given problem, the coefficients of the
above formula are considered in the form of
and
.
This
problem is implemented by this software, and a global optimal solution
is obtained. The computational results are shown in Tables 1, 2, 3, 4,
and 5.
Table 1 is 1 if distribution center
is opened with capacity level
. and 0 otherwise
DC1 | DC2 | |||
---|---|---|---|---|
Capacity 1 | Capacity 2 | Capacity 1 | Capacity 2 | |
Example 1 | 1 | 0 | 0 | 1 |
Example 2 | 0 | 1 | 1 | 0 |
Example 3 | 0 | 1 | 1 | 0 |
Example 4 | 1 | 0 | 0 | 1 |
Example 5 | 0 | 1 | 1 | 0 |
Table 2
Route 1 | Route 2 | Route 3 | Route 4 | Route 5 | Route 6 | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Truck | Airplane | Truck | Airplane | Truck | Airplane | Truck | Airplane | Truck | Airplane | Truck | Airplane | |
Example 1 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | - | - | - | - |
Example 2 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | - | - | - | - |
Example 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
Example 4 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
Example 5 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | - | - | - | - |
Table 3
Plant 1 | Plant 2 | |||||||
---|---|---|---|---|---|---|---|---|
DC1 | DC2 | DC1 | DC2 | |||||
Train | Airplane | Train | Airplane | Train | Airplane | Train | Airplane | |
Example 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
Example 2 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 |
Example 3 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 |
Example 4 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
Example 5 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
Table 4
DC1 | DC2 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Demand 1 | Demand 2 | Demand 3 | Demand 1 | Demand 2 | Demand 3 | |||||||
Truck | Airplane | Truck | Airplane | Truck | Airplane | Truck | Airplane | Truck | Airplane | Truck | Airplane | |
Example 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
Example 2 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
Example 3 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
Example 4 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
Example 5 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
Table 5
Plant 1 | Plant 2 | |||||||
---|---|---|---|---|---|---|---|---|
DC1 | DC2 | DC1 | DC2 | |||||
Train | Airplane | Train | Airplane | Train | Airplane | Train | Airplane | |
Example 1 | 4 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
Example 2 | 0 | 0 | 2 | 0 | 8 | 0 | 0 | 0 |
Example 3 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 2 |
Example 4 | 5 | 0 | 0 | 6 | 0 | 0 | 0 | 0 |
Example 5 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 1 |