Warehouse Location Problem

An online retail company in the US is planning to open warehouses in potential locations to optimize their logistics and service delivery. This project aims to solve the warehouse location problem using mathematical modeling and optimization techniques.

Problem Description

The company is considering opening warehouses in the following cities: Chicago, Atlanta, New York, St. Louis, Detroit, Cincinnati, Pittsburgh, Charlotte, and Boston. The goal is to minimize the total weighted distance between demand points and their nearest warehouse while considering budgetary constraints and service level requirements.

Data

Distance Matrix (in miles):

City	Chicago	Atlanta	New York	St. Louis	Detroit	Cincinnati	Pittsburgh	Charlotte	Boston
Chicago	0	720	790	297	283	296	461	769	996
Atlanta	720	0	884	555	722	461	685	245	1099
New York	790	884	0	976	614	667	371	645	219
St. Louis	297	555	976	0	531	359	602	715	1217
Detroit	283	722	614	531	0	263	286	629	721
Cincinnati	296	461	667	359	263	0	288	479	907
Pittsburgh	461	685	371	602	286	288	0	448	589
Charlotte	769	245	645	715	629	479	448	0	867
Boston	996	1099	219	1217	721	907	589	867	0

Demand at each location:

City	Demand
Chicago	2,870,000
Atlanta	572,000
New York	8,450,000
St. Louis	350,000
Detroit	901,000
Cincinnati	333,000
Pittsburgh	306,000
Charlotte	723,000
Boston	610,000

Objectives

1. Minimize Total Weighted Distance: The objective is to minimize the total weighted distance between demand points and their nearest warehouse, with weight being the demand of each location. The company can open only ( p ) warehouses due to budgetary limitations.

2. Maximize Service Distance: Ensure that no demand location is more than 500 miles away from a warehouse and maximize the total demand that can be covered within a 200-mile distance (high service distance).

Mathematical Model

Variables:

• ( x_{ij} ): Binary variable indicating whether a warehouse is placed in city ( i ) to serve city ( j ).
• ( y_i ): Binary variable indicating whether a warehouse is opened in city ( i ).

Parameters:

• ( d_{ij} ): Distance between city ( i ) and city ( j ).
• ( w_j ): Demand at city ( j ).
• ( p ): Number of warehouses to be opened.
• ( D_{\text{max}} ): Maximum distance a demand point can be from a warehouse.
• ( D_{\text{service}} ): High service distance.

Objective Function:

• Minimize total weighted distance: ( \sum_{i} \sum_{j} d_{ij} \cdot w_j \cdot x_{ij} ).

Constraints:

1. Each demand point must be assigned to exactly one warehouse: ( \sum_{i} x_{ij} = 1 \quad \forall j ).

2. A warehouse must be open in city ( i ) if it serves any city ( j ): ( x_{ij} \leq y_i \quad \forall i, j ).

3. The total number of warehouses must be exactly ( p ): ( \sum_{i} y_i = p ).

4. Maximum distance constraint: ( d_{ij} \cdot x_{ij} \leq D_{\text{max}} \quad \forall i, j ).

5. High service distance constraint: ( \sum_{i} \sum_{j: d_{ij} \leq D_{\text{service}}} w_j \cdot x_{ij} ).

Files

• warehouse_location.mod: The OPL model file.
• warehouse_location.dat: The data file containing distance matrix and demand values.