Location Where to put facilities? Transportation costs Rates and distances Volumes to be moved Other issues Market presence (speed to market) Fixed costs

1-Dimensional Intuition Customers -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 Where to locate?

1-Dimensional Intuition Customers -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 Where to locate?

1-Dimensional Intuition Customers -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 Where to locate?

1-Dimensional Intuition Customers -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 Where to locate

What about “Weight” Customers Weight 3 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 1 2 3 4 5 6 Where to locate?

What about “Weight” Customers Weight 2 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 1 2 3 4 5 6 Where to locate?

2-Dimensional Location 5 D 4 Manhattan Metric or L1 norm Distance 4 + 5 = 9

2-Dimensions If all the points are the same “weight” D Y D D D 1 2 3 4 5 6 7 8 9 10 Y D D D 1 2 3 4 5 6 7 8 9 10 Where to locate? X

2-Dimensions If all the points are the same “weight” Y D 1 2 3 4 5 6 7 8 9 10 Y 1 2 3 4 5 6 7 8 9 10 Where to locate? X

2-Dimensions Euclidean Distance or L2 norm Successive Approximations: X = Average of X’s Y = Average of Y’s Calculate distances d1, d2, ... X = X1/d1+X2/d2+…X4/d4 1/d1 + 1/d2 + …1/d4 Y = Y1/d1+Y2/d2+…Y4/d4 Repeat until movement is small D 1 2 3 4 5 6 7 8 9 10 Y 1 2 3 4 5 6 7 8 9 10 X

D D D D

“Weights and Rates” If there are associated with each location Different volumes V1, V2, …, V4 Different transportation rates R1, R2, …, R4 associated with each location Replace Xi with ViRiXi Replace Yi with ViRiYi (Not when calculating distances)

Over Emphasis Useful for getting in the neighborhood One or two iterations generally does this Ignores lots of (important) details Availability and cost of sites Actual transportation network Reality of freight rates Non-linear Often relatively insensitive to distance (LTL) Dynamics of demand

Locating Many Facilities Select a number of locations Guess at initial positions Assign Customers to those locations Repeat: Calculate best location to serve assigned customers Calculate best customers to serve from those locations

Locate Distribution Centers Based on Ford Auto Dealerships in Canada Parts distribution 4 Distribution Centers Consider only distance to dealerships Ignore volume (to keep it simple) Illustrate approach Compare with “Actual”

Locate Distribution Centers

Mixed Integer Linear Programming Does guarantee the quality of the solution Computationally more demanding More Flexible Technically more demanding

Example 13.5 page 498 2 Products 3 Customers Single sourcing 2 warehouses

An MIP Model

Heuristics Speed the MIP solution Reduce computational demands More interactive No guarantee of optimality

Some Heuristics Multiple Center of Gravity method for each number Evaluate other costs after the fact Inventory Fixed Costs Etc. Successive Elimination Successive Approximation

Successive Elimination Illustrate with our MIP example Replace the computationally demanding MIP with sequence of LPs

An MIP Model

Successive Elimination Both $ 3,150,000 Remove 1 $ 3,050,000 Remove 2 Not Feasible With more choices, continue as long as costs reduce… Does not always find optimum

Successive Approximation Calculate an imputed cost per unit based on anticipated volume through each warehouse Solve an LP to determine best volumes at these rates Repeat Calculate imputed costs per unit based on volumes Calculate best volumes at imputed costs

Covering Models Each site “covers” some customers Select a best set of sites that cover all customers

Western Airlines

Solver Model