Location
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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. 0. 1. 2. 3. 4. 5. 6. Where to locate?.

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Location

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

1-Dimensional Intuition

Customers

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

Where to locate?


1 dimensional intuition1

1-Dimensional Intuition

Customers

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

Where to locate?


Location

1-Dimensional Intuition

Customers

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

Where to locate?


Location

1-Dimensional Intuition

Customers

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

Where to locate


Location

What about “Weight”

Weight 3

Customers

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

Where to locate?


Location

What about “Weight”

Weight 2

Customers

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

Where to locate?


2 dimensional location

2-Dimensional Location

D

5

D

4

Manhattan Metric or L1 norm

Distance 4 + 5 = 9


2 dimensions

7

7

4

4

1

1

2

2

5

5

3

3

8

8

9

9

10

10

6

6

2-Dimensions

  • If all the points are the same “weight”

D

Y

D

D

D

Where to locate?

X


Location

D

D

D

D

7

7

4

4

1

1

2

2

5

5

3

3

8

8

9

9

10

10

6

6

2-Dimensions

  • If all the points are the same “weight”

Y

Where to locate?

X


2 dimensions1

D

D

D

D

D

4

7

1

7

4

2

5

1

3

2

5

3

8

9

8

9

6

10

10

6

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

  • 1/d1 + 1/d2 + …1/d4

  • Repeat until movement is small

Y

X


Location

D

D

D

D


Weights and rates

“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)


Over emphasis

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

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

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 centers1

Locate Distribution Centers


Mixed integer linear programming

Mixed Integer Linear Programming

  • Does guarantee the quality of the solution

  • Computationally more demanding

  • More Flexible

  • Technically more demanding


Example 13 5 page 498

Example 13.5 page 498

  • 2 Products

  • 3 Customers

  • Single sourcing

  • 2 warehouses


An mip model

An MIP Model


Heuristics

Heuristics

  • Speed the MIP solution

  • Reduce computational demands

  • More interactive

  • No guarantee of optimality


Some heuristics

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

Successive Elimination

  • Illustrate with our MIP example

  • Replace the computationally demanding MIP with sequence of LPs


An mip model1

An MIP Model


Successive elimination1

Successive Elimination

  • Both $ 3,150,000

  • Remove 1 $ 3,050,000

  • Remove 2Not Feasible

  • With more choices, continue as long as costs reduce…

  • Does not always find optimum


Successive approximation

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

Covering Models

  • Each site “covers” some customers

  • Select a best set of sites that cover all customers


Western airlines

Western Airlines


Solver model

Solver Model


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