## 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**is 1 if and only if route 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 | - | - | - | - |

**binary variable, where it is 1 if vehicle l 1 connecting plant and is used**

Table 3

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**binary variable, where it is 1 if vehicle connecting and customer 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**quantity transported from plant to using vehicle

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 |