Facility Location - Handbook

Type: MILP (Mixed Integer Linear Programming)

This handbook explains the Facility Location sample problem in the LP Black Box platform.

This combines binary decisions (which facilities to open) with continuous decisions (how much to ship).

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The Problem

Scenario

A company wants to open warehouses to serve 3 customer regions. They can choose from 4 potential warehouse locations. Each warehouse has a fixed operating cost, and shipping costs vary by route.

Warehouse	Fixed Cost ($)	Capacity
Warehouse 1	1000	200
Warehouse 2	800	150
Warehouse 3	1200	250
Warehouse 4	900	180

Route	Shipping Cost per Unit
W1 → Customer A	5
W1 → Customer B	7
W1 → Customer C	6
W2 → Customer A	4
W2 → Customer B	8
W2 → Customer C	5
W3 → Customer A	6
W3 → Customer B	5
W3 → Customer C	7
W4 → Customer A	7
W4 → Customer B	6
W4 → Customer C	8

Demand: Customer A: 80, Customer B: 100, Customer C: 70

Your Goal

Minimize total cost (fixed + shipping) while meeting all demand.

The Variables

Variable	Type	Meaning
open1	Binary (0 or 1)	Open Warehouse 1?
open2	Binary (0 or 1)	Open Warehouse 2?
open3	Binary (0 or 1)	Open Warehouse 3?
open4	Binary (0 or 1)	Open Warehouse 4?
ship1A	Continuous ≥ 0	Units from W1 to Customer A
ship1B	Continuous ≥ 0	Units from W1 to Customer B
...	...	...

The Constraints

1. Demand Satisfaction (for each customer):

   • ship1A + ship2A + ship3A + ship4A = 80
   • ship1B + ship2B + ship3B + ship4B = 100
   • ship1C + ship2C + ship3C + ship4C = 70

2. Capacity (if warehouse not open, shipments must be 0):

   • ship1A + ship1B + ship1C ≤ 200 × open1
   • ship2A + ship2B + ship2C ≤ 150 × open2
   • ship3A + ship3B + ship3C ≤ 250 × open3
   • ship4A + ship4B + ship4C ≤ 180 × open4

3. Minimum Open Warehouses: Must open at least 2:

   • open1 + open2 + open3 + open4 ≥ 2

The Objective

Minimize: Fixed costs + Shipping costs

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Why MILP?

This is a Mixed Integer problem because:

• Binary variables: open1, open2, open3, open4 (yes/no decisions)
• Continuous variables: ship1A, ship1B, ... (quantities to ship)

The binary variables make this an MILP - much harder than pure LP, but enables realistic facility planning.

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How to Use

Step 1: Load the Sample

Select Facility Location from the dropdown.

Step 2: Solve

The solver finds which warehouses to open and how much to ship from each.

Step 3: Interpret Results

• Binary variables = 1 means "open this warehouse"
• Continuous variables show shipping quantities

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Try It Yourself

• Change the minimum warehouses constraint
• Add a constraint limiting total number of warehouses
• Increase demand and see how solution changes

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Real-World Applications

• Warehouse/Retail Location: Where to open new stores
• Distribution Network: Design supply chain
• Telecom Towers: Where to place cell towers
• Healthcare Facilities: Hospital/ clinic placement

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This demonstrates how MILP combines discrete decisions (where to invest) with continuous decisions (how much to allocate).