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 \((f_i)\) 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 \(f_{1}′\) and \(f_{2}′\) are the normalized forms of \(f_1\) and \(f_2\) objective functions.

\(min f = αf_{1}′+(1−α)f_2′\)

To minimize deviations from the ideal, this function is reduced. As the first objective function (\(f_1\)) is more important than the second objective function (\(f_2\)) in the given problem, the coefficients of the above formula are considered in the form of \(α = 0.7\) and \(1 − α = 0.3\).

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 \(U_{j}^{n}\) is 1 if distribution center \(j\) is opened with capacity level \(n\). 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 \(X_r\) is 1 if and only if route \(r\) is selected, and 0 otherwise

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 \(A_{ijl_1}\) binary variable, where it is 1 if vehicle l 1 connecting plant \(i\) and \(DC_j\) is used

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 \(B_{jkl_2}\) binary variable, where it is 1 if vehicle \(l_2\) connecting \(DC_j\) and customer \(k\) is used

  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 \(X_{ijl_1}\) quantity transported from plant \(i\) to \(DC_j\) using vehicle \(l_1\)

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