Location

1
Location

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

2
1-Dimensional Intuition
Customers
0
-1
-2
-3
-4
-5
5
4
3
2
1
6
-6
Where to locate?
3
1-Dimensional Intuition
Customers
0
-1
-2
-3
-4
-5
5
4
3
2
1
6
-6
Where to locate?
4
1-Dimensional Intuition
Customers
0
-1
-2
-3
-4
-5
5
4
3
2
1
6
-6
Where to locate?
5
1-Dimensional Intuition
Customers
0
-1
-2
-3
-4
-5
5
4
3
2
1
6
-6
Where to locate
6
What about Weight
Weight 3
Customers
0
-1
-2
-3
-4
-5
5
4
3
2
1
6
-6
Where to locate?
7
What about Weight
Weight 2
Customers
0
-1
-2
-3
-4
-5
5
4
3
2
1
6
-6
Where to locate?
8
2-Dimensional Location
D
5
D
4
Manhattan Metric or L1 norm Distance 4 5 9
9
2-Dimensions

• If all the points are the same weight

D
Y
D
D
D
Where to locate?
X
10
2-Dimensions

• If all the points are the same weight

Y
Where to locate?
X
11
2-Dimensions

• Euclidean Distance or L2 norm

• Successive Approximations
• X Average of Xs
• Y Average of Ys
• Calculate distances d1, d2, ...
• X X1/d1X2/d2X4/d4
• 1/d1 1/d2 1/d4
• Y Y1/d1Y2/d2Y4/d4
• 1/d1 1/d2 1/d4
• Repeat until movement is small

Y
X
12
D
D
D
D
13
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14
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15
Weights and Rates

• If there are
• 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)

16
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

17
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

18
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

19
Locate Distribution Centers
20
Mixed Integer Linear Programming

• Does guarantee the quality of the solution
• Computationally more demanding
• More Flexible
• Technically more demanding

21
Example 13.5 page 498

• 2 Products
• 3 Customers
• Single sourcing
• 2 warehouses

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An MIP Model
24
Heuristics

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

25
Some Heuristics

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

26
Successive Elimination

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

27
An MIP Model
28
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

29
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

30
Covering Models

• Each site covers some customers
• Select a best set of sites that cover all
  customers

31
Western Airlines
32
Solver Model