# Wolfram Mathematica

## Facility Location Problem

Determine the locations of new factories that minimize the transportation cost between supply warehouses.

This example demonstrates SecondOrderConeOptimization, which may be used to minimize a weighted sum of differences. The convexity of the distance functions allows the transformation to second-order cone constraints using the epigraph transformation.

A company wants to open new factories. The factories need raw materials from warehouses. Let be the distance between factory and warehouse and be the cost per unit distance to transport the raw materials. The objective is to minimize .

The new factories must be located such that , where is the position of the new factory and is the position of the warehouse .

As an example, assume that the company wants to open two new factories and has five warehouses.

The five warehouses are located at:

The cost of transporting materials from the warehouses to the new factories is shown.

Find the optimal distances between the new factories and the warehouses.

The new factory locations are closer to the warehouses where the transportation costs are higher.