This document discusses facility location decisions and methods for analyzing location strategies. It begins with an overview of what can be located, such as plants, warehouses, retail outlets, and key questions to consider around location. Common methods for solving single and multiple facility location problems are then presented, including the center-of-gravity (COG) method and optimization approaches. The document concludes with examples of applying COG and discussing other techniques like simulation and weighted checklists for analyzing retail location decisions.

1. 13-1
Facility Location Decisions
Chapter 13
CR (2004) Prentice Hall, Inc.

2. 13-2CR (2004) Prentice Hall, Inc.
Facility Location in Location
Strategy
PLANNING
ORGANIZING
CONTROLLING
Transport Strategy
• Transport fundamentals
• Transport decisions
Customer
service goals
• The product
• Logistics service
• Ord . proc. & info. sys.
Inventory Strategy
• Forecasting
• Inventory decisions
• Purchasing and supply
scheduling decisions
• Storage fundamentals
• Storage decisions
Location Strategy
• Location decisions
• The network planning process
PLANNING
ORGANIZING
CONTROLLING
Transport Strategy
• Transport fundamentals
• Transport decisions
Customer
service goals
• The product
• Logistics service
• Ord . proc. & info. sys.
Inventory Strategy
• Forecasting
• Inventory decisions
• Purchasing and supply
scheduling decisions
• Storage fundamentals
• Storage decisions
Location Strategy
•Location decisions
• The network planning process

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Location Overview
What's located?
• Sourcing points
− Plants
− Vendors
− Ports
• Intermediate points
− Warehouses
− Terminals
− Public facilities (fire, police, and ambulance
stations)
− Service centers
• Sink points
− Retail outlets
− Customers/Users

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Location Overview (Cont’d)
Key Questions
• How many facilities should there be?
• Where should they be located?
• What size should they be?
Why Location is Important
• Gives structure to the network
• Significantly affects inventory and
transportation costs
• Impacts on the level of customer service to
be achieved

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When to Analyze Location
• Changing service requirements
• Partnerships
• Shifting locations (customer/supplier)
• Changing corporate ownership
• Cost pressure
• Global markets

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Nature of Location Analysis
Manufacturing (plants & warehouses)
Decisions are driven by economics. Relevant costs
such as transportation, inventory carrying, labor, and
taxes are traded off against each other to find good
locations.
Retail
Decisions are driven by revenue. Traffic flow and
resulting revenue are primary location factors, cost is
considered after revenue.
Service
Decisions are driven by service factors. Response
time, accessibility, and availability are key dimensions
for locating in the service industry.

7. 13-7CR (2004) Prentice Hall, Inc.
Methods of Solution
• Single warehouse location
– Graphic
– Grid, or center-of-gravity, approach
• Multiple warehouse location
– Simulation
– Optimization
– Heuristics
Location Overview (Cont’d)

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-Finding solution can be challenging
-But with the advent of fast PCs, it is
more widely used these days
-Model formulation
Optimization Method

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Method appraisal
• A continuous location method
• Locates on the basis of transportation costs alone
The COG method involves
• Determining the volumes by source and destination
point
• Determining the transportation costs based on
$/unit/mi.
• Overlaying a grid to determine the coordinates of
source and/or destination points
• Finding the weighted center of gravity for the graph
COG Method

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COG Method (Cont’d)
∑
∑
∑
∑ ==
i ii
i iii
i ii
i iii
RV
YRV
Y,
RV
XRV
X
where
Vi = volume flowing from (to) point I
Ri = transportation rate to ship Vi from (to) point i
Xi,Yi = coordinate points for point i
= coordinate points for facility to be located
Y,X

11. 13-11CR (2004) Prentice Hall, Inc.
COG Method (Cont’d)
Example Suppose a regional medical warehouse is to be
established to serve several Veterans Administration hospitals
throughout the country. The supplies originate at S1 and S2 and
are destined for hospitals at H1 through H4. The relative locations
are shown on the map grid. Other data are: Note rate is a
per mile cost
Point
i
Prod-
ucts Location
Annual
volume,
cwt.
Rate,
$/cwt/
mi. Xi Yi
1 S1 A Seattle 8,000 0.02 0.6 7.3
2 S2 B Atlanta 10,000 0.02 8.6 3.0
3 H1 A & B Los
Angeles
5,000 0.05 2.0 3.0
4 H2 A & B Dallas 3,000 0.05 5.5 2.4
5 H3 A & B Chicago 4,000 0.05 7.9 5.5
6 H4 A & B New York 6,000 0.05 10.6 5.2

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COG Method (Cont’d)
Map scaling factor, K

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COG Method (Cont’d)
Solve the COG equations in table form
i Xi Yi Vi Ri ViRi ViRiXi ViRiYi
1 0.6 7.3 8,000 0.02 160 96 1,168
2 8.6 3.0 10,000 0.02 200 1,720 600
3 2.0 3.0 5,000 0.05 250 500 750
4 5.5 2.4 3,000 0.05 150 825 360
5 7.9 5.5 4,000 0.05 200 1,580 1,100
6 10.6 5.2 6,000 0.05 300 3,180 1,560
1,260 7,901 5,538

14. 13-14
COG Method (Cont’d)
Now,
X = 7,901/1,260 = 6.27
Y = 5,538/1,260 = 4.40
This is approximately Columbia, MO.
The total cost for this location is found by:
where K is the map scaling factor to convert
coordinates into miles.
∑ −+−= i iiii YYXXKRVTC 22
)()(
CR (2004) Prentice Hall, Inc.

15. 13-15CR (2004) Prentice Hall, Inc.
COG
COG Method (Cont’d)

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COG Method (Cont’d)
2,360,882Total
660,4920.056,0005.210.66
196,6440.054,0005.57.95
160,7330.053,0002.45.54
561,7060.055,0003.02.03
271,8250.0210,0003.08.62
509,4820.028,0007.30.61
TCRiViYiXii
Calculate total cost at COG
22
1
4.40)(7.36.27)(0.6)(500)8,000(0.02TC −+−=

17. 13-17CR (2004) Prentice Hall, Inc.
Note The center-of-gravity method does not necessarily
give optimal answers, but will give good answers if there are
a large numbers of points in the problem (>30) and the
volume for any one point is not a high proportion of the total
volume. However, optimal locations can be found by the
exact center of gravity method.
∑
∑
∑
∑ ==
i iii
i iiiin
i iii
i iiiin
/dRV
/dYRV
Y,
/dRV
/dXRV
X
where
22
)Y(Y)X(Xd
n
i
n
ii
−+−=
and n is the iteration number.
COG Method (Cont’d)

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Solution procedure for exact COG
COG Method (Cont’d)
1) Solve for COG
2) Using find di
3) Re-solve for using exact formulation
4) Use revised to find revised di
5) Repeat steps 3 through 5 until there is no
change in
6) Calculate total costs using final coordinates
Y,X
Y,X
Y,X
Y,X

19. 13-19CR (2004) Prentice Hall, Inc.
• A more complex problem that most firms have.
• It involves trading off the following costs:
− Transportation inbound to and outbound from the facilities
− Storage and handling costs
− Inventory carrying costs
− Production/purchase costs
− Facility fixed costs
• Subject to:
− Customer service constraints
− Facility capacity restrictions
• Mathematical methods are popular for this type of problem
that:
− Search for the best combination of facilities to minimize
costs
− Do so within a reasonable computational time
− Do not require enormous amounts of data for the analysis
Multiple Location Methods

20. 13-20
Multiple COG
•Formulated as basic COG model
•Can search for the best locations for a selected number of
sites.
•Fixed costs and inventory consolidation effects are handled
outside of the model.
A multiple COG procedure
•Rank demand points from highest to lowest volume
•Use the M largest as initial facility locations and assign
remaining demand centers to these locations
•Compute the COG of the M locations
•Reassign all demand centers to the M COGs on the basis
of proximity
•Recompute the COGs and repeat the demand center
assignments, stopping this iterative process when there is
no further change in the assignments or COGs
CR (2004) Prentice Hall, Inc.

21. 13-21CR (2004) Prentice Hall, Inc.
•Location of truck maintenance terminals
•Location of public facilities such as offices, and
police and fire stations
•Location of medical facilities
•Location of most any facility where transportation
cost (rather than inventory carrying cost and
facility fixed cost) is the driving factor in location
•As a suggestor of sites for further evaluation
Examples of Practical COG
Model Use

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• A method used commercially
- Has good problem scope
- Can be implemented on a PC
- Running times may be long and memory
requirements substantial
- Handles fixed costs well
- Nonlinear inventory costs are not well
handled
• A linear programming-like solution procedure
can be used (MIPROG in LOGWARE)
Mixed Integer Programming

23. 13-23
Location by Simulation
CR (2004) Prentice Hall, Inc.
•Can include more variables than typical algorithmic
methods
•Cost representations can be precise so problem can
be more accurately described than with most
algorithmic methods
•Mathematical optimization usually is not
guaranteed, although heuristics can be included to
guide solution process toward satisfactory solutions
•Data requirements can be extensive
•Has limited use in practice

24. 13-24
Commercial Models for Location
Features
•Includes most relevant location costs
•Constrains to specified capacity and customer
service levels
•Replicates the cost of specified designs
•Handles multiple locations over multiple echelons
•Handles multiple product categories
•Searches for the best network design
CR (2004) Prentice Hall, Inc.

25. CR (2004) Prentice Hall, Inc.
Commercial Models (Cont’d)
13-46

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Retail Location
Methods
• Contrasts with plant and warehouse location.
- Revenue rather than cost driven
- Factors other than costs such as parking, nearness to competitive
outlets, and nearness to customers are dominant
• Weighted checklist
- Good where many subjective factors are involved
- Quantifies the comparison among alternate locations

27. CR (2004) Prentice Hall, Inc.
A Hypothetical Weighted Factor Checklist for a
Retail Location Example
a
Weights approaching 10 indicate great importance.
b
Scores approaching 10 refer to a favored location status.
(1)
Factor
Weight
(1 to 10)
a
Location Factors
(2)
Factor Score
(1 to 10)
b
(3)=(1)×(2)
Weighted
Score
8 Proximity to competing stores 5 40
5 Space rent/lease
considerations 3 15
8 Parking space 10 80
7 Proximity to complementary
stores 8 56
6 Modernity of store space 9 54
9 Customer accessibility 8 72
3 Local taxes 2 6
3 Community service 4 12
8 Proximity to major
transportation arteries 7 56
Total index 391
13-48

28. 13-28CR (2004) Prentice Hall, Inc.
Retail Location (Cont’d)
• Huff's gravity model
- A take-off on Newton's law of gravity.
- "Mass" or retail "variety" attracts customers, and the distance from
customers repels them.
- The basic model is:
E PC
S T
S T
Cij ij i
j ij
a
j ij
a
j
i
= =
∑
/
/
where
Eij = expected demand from population center i that will be attracted to
retail location j
Pij = probability of customers from point i traveling to retail location j
Ci = customer demand at point i
Sj = size of retail location j
Tij = travel time between customer location i and retail location j
n = number of competing locations j
a = an empirically estimated parameter

29. CR (2004) Prentice Hall, Inc.
Retail Location (Cont’d)
Example of Huff's method
Two shopping centers (RA and RB ) are to attract customers from C1, C2, and C3.
Shopping center A has 500,000 square feet of selling area whereas center B
has 1,000,000. The customer clusters have a buying potential of $10, $5, and
$7 million respectively. The parameter a is estimated to be 2. What is the sales
potential of each shopping center?
Solution matrix
Custo-
mer i
Time from
Customer i
to Location j
Tij
2
S Tj ij/
2 P
S T
S T
ij
j ij
j ijj
=
∑
/
/
2
2
E P Cij ij i
=
A B A B A B A B A B
C1 30.0 56.6 900 3200 555 313 0.64 0.36 $6.4 $3.6
C2 44.7 30.0 2000 900 250 1111 0.18 0.82 0.9 4.1
C3 36.0 28.3 1300 800 385 1250 0.24 0.76 1.7 5.3
Total shopping center sales ($ million) $9.0 $13.0
13-50

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0 10 20 30 40 50 60 70 80
0
10
20
30
40
50
60
70
80Y
XTime (minutes)
Time(minutes)
C2
C1
C3
RB
RA
Retail Location (Cont’d)

31. 13-31CR (2004) Prentice Hall, Inc.
Retail Location (Cont’d)
Location-Allocation Model
Mixed-Integer Programming Example – p.592