Facility Location Decisions
Experience teaches that men are so much
governed by what they are accustomed to see and
practice, that the simplest and most obvious
improvements in the most ordinary occupations are
adopted with hesitation, reluctance, and by slow
graduations.
Alexander Hamilton, 1791
Chapter 13
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13-1
CONTROLLING
Customer
service goals
• The product
• Logistics service
• Ord. proc. & info. sys.
Transport Strategy
• Transport fundamentals
• Transport decisions
PLANNING
Inventory Strategy
• Forecasting
• Inventory decisions
• Purchasing and supply
scheduling decisions
• Storage fundamentals
• Storage decisions
ORGANIZING
Facility Location in Location
Strategy
Location Strategy
• Location decisions
• The network planning process
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13-2
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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13-3
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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13-4
Location Overview (Cont’d)
Methods of Solution
 Single warehouse location
– Graphic
– Grid, or center-of-gravity, approach
 Multiple warehouse location
– Simulation
– Optimization
– Heuristics
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13-5
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.
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13-6
Weber’s Classification of
Industries
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13-7
Agglomeration
Based on the observation that the output of one
industry is the input of another. Customers for an
industry’s products are the workers of those
industries. Hence, suppliers, manufacturers, and
customers group together, especially where
transportation costs are high. Historically, the
growth of the auto industry showed this pattern.
Today, the electronics industry (silicon valley) has a
similar pattern although it is less obvious since the
product value is high and transportation costs are a
small portion of total product price.
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13-8
COG Method
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
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13-9
COG Method (Cont’d)
VR X

X
 VR
i
i
i
i
i
i
i
VR Y

,Y 
 VR
i
i
i
i
i
i
i
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
X, Y = coordinate points for facility to be located
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13-10
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
Annual Rate,
Point
i
1 S1
2 S2
3 H1
Products
A
B
A&B
Location
Seattle
Atlanta
Los
Angeles
4 H 2 A & B Dallas
5 H 3 A & B Chicago
6 H 4 A & B New York
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volume, $/cwt/
cwt.
mi.
8,000
0.02
10,000
0.02
5,000
0.05
3,000
4,000
6,000
0.05
0.05
0.05
Xi
0.6
8.6
2.0
Yi
7.3
3.0
3.0
5.5
7.9
10.6
2.4
5.5
5.2
13-11
COG Method (Cont’d)
Map scaling factor, K
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13-12
COG Method (Cont’d)
Solve the COG equations in table form
i
1
2
3
4
5
6
Xi
0.6
8.6
2.0
5.5
7.9
10.6
Yi
Vi
Ri
7.3 8,000 0.02
3.0 10,000 0.02
3.0 5,000 0.05
2.4 3,000 0.05
5.5 4,000 0.05
5.2 6,000 0.05
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V iR i
160
200
250
150
200
300
1,260
V i R iX i
96
1,720
500
825
1,580
3,180
7,901
V iR i Y i
1,168
600
750
360
1,100
1,560
5,538
13-13
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:
TC  i Vi Ri K ( X i  X ) 2  (Yi  Y ) 2
where K is the map scaling factor to convert
coordinates into miles.
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13-14
COG Method (Cont’d)
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COG
13-15
COG Method (Cont’d)
TC1  8,000(0.02)(500) (0.6  6.27)2  (7.3  4.40)2
Calculate total cost at COG
i
Xi
Yi
Vi
Ri
TC
1
0.6
7.3
8,000
0.02
509,482
2
8.6
3.0
10,000
0.02
271,825
3
2.0
3.0
5,000
0.05
561,706
4
5.5
2.4
3,000
0.05
160,733
5
7.9
5.5
4,000
0.05
196,644
6
10.6
5.2
6,000
0.05
660,492
Total 2,360,882
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13-16
COG Method (Cont’d)
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.
n
X 
V R X /d
V R /d
i i
i i
i
i
i
i
i
n
,Y 
V R Y /d
V R /d
i i
i i
i
i
i
i
i
where
n
n
d i  (X i  X ) 2  (Yi  Y ) 2
and n is the iteration number.
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13-17
COG Method (Cont’d)
Solution procedure for exact COG
1)
2)
3)
4)
5)
Solve for COG
Using X ,Y find di
Re-solve for X ,Y using exact formulation
Use revised X ,Y to find revised di
Repeat steps 3 through 5 until there is no
change in X ,Y
6) Calculate total costs using final coordinates
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13-18
Multiple Location Methods
 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
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13-19
Location Cost Trade-Offs
Cost
Total cost
Warehouse
fixed
Inventory carrying
and warehousing
Production/purchase
and order processing
Inbound and
outbound
transportation
0
0
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Number of warehouses
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
13-21
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Examples of Practical COG
Model Use
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
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13-22
Mixed Integer Programming
 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)
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13-23
MIP (Cont’d)
Example
The database for the illustrated problem is prepared as file
ILP02.DAT in MIPROG. The form is dictated by the problem
formulation in the technical supplement of the location chapter.
The solution is to open only one warehouse (W2) with resulting
costs of:
Category
Cost
Production
$1,020,000
Transportation
1,220,000
Warehouse
handling
310,000
Warehouse fixed
500,000
Total
$3,050,000
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13-24
The situation
Product 1
wt
/c
$4
$0/cwt.
t.
/cw
$5
Plant P1
Production = $4/cwt.
Capacity =
60,000 cwt.
$3
/c
Warehouse W1
wt
.
.
t
w
/c
$1
Warehouse W2
Customer C2
100,000 cwt.
wt.
wt.
c
$2/
$2
/cw
t
Fixed = $100,000
Capacity = 110,000 cwt.
Customer C3
50,000 cwt.
Fixed = $500,000
Capacity =
Unrestricted
Handling = $2/cwt.
/cw
$5
/c
$2
Warehouse W2
t
wt.
$4
/cw
t.
$3
/cw
Customer C2
30,000 cwt.
/c
$4
wt.
Plant P2
Production = $2/cwt.
Capacity =
Unrestricted
wt
.
.
t
w
Handling = $1/cwt.
c
$2/
A Multiple
Product
Network
Design
Problem
$2
/c
Warehouse W1
t.
Plant P1
Production = $3/cwt.
Capacity =
50,000 cwt.
Customer C1
20,000 cwt.
$3
/c
$0/cwt.
wt
/c
$3
wt.
Product 2
MIP
(Cont’d)
/c
$5
$4
/cw
t.
Handling = $1/cwt.
Plant P2
Production = $4/cwt.
Capacity =
Unrestricted
Customer C1
50,000 cwt.
$2
/cw
t.
Handling = $2/cwt.
Customer C3
60,000 cwt.
13-25
Mixed Integer programming
13-26
13-27
Guided Linear Programming
Searches for the best locations.
A typical problem is shown in the next figure and can be configured as
a transportation problem of linear programming in terms of its variable
costs.
 The inventory and warehouse fixed costs are computed as per-unit
costs, but this depends on the warehouse throughput. They are
placed along with the variable costs in the linear programming
problem.
 The linear programming problem is solved. This gives revised
demand assignments to warehouses. Some may have no
throughput and are closed.
 Fixed costs and inventory carrying costs are recomputed based on
revised warehouse throughputs.
 When there is no change in the solution from one iteration to the
next, STOP.
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13-28
Guided Linear Programming (Cont’d)
The situation
Warehouse W1
Plant P2
Production = $4/cwt.
Capacity =
Unrestricted
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t. Warehouse W2
w
c
/
$2
Handling = $1/cwt.
Capacity =
Unrestricted
Fixed = $400,000
$3
/cw
t.
.
wt
c
/
$1
$2
/c
wt
Inventory carrying cost =
100(Throughput)0.7
Customer C1
50,000 cwt.
Customer C2
100,000 cwt.
.
cwt
$5/
$4/
cw
t.
t.
/cw
$5
Plant P1
Production = $4/cwt.
Capacity =
60,000 cwt.
$2/
cwt
.
Handling = $2/cwt.
Capacity = 60,000 cwt.
Fixed = $100,000
wt
c
/
4
$
$0/cwt.
Customer C3
50,000 cwt.
13-29
Guided Linear Programming (Cont’d)
Example
 Use data from the network figure.
 Allocate the fixed costs. Assume initially that each warehouse handles the
entire volume. Hence,
Warehouse 1
100,000/200,000 = $0.50/cwt.
Warehouse 2
400,000/200,000 = $2.00/cwt.
 Allocate inventory carrying costs. Assume that throughput for each product is
equally divided between warehouses.
Each warehouse
Total customer demand
0.7
100{[200,000/2) ]/(200,000/2)} = $3.2
No. of whses
 Build the cost matrix for the transportation problem of LP. Note its special
structure.
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13-38
Matrix of Cell Costs and Solution Values for the First
Iteration in the Example Problem
Warehouses
Plants
P1
W1
a
4
60,000
W2
8
6
140,000
99
99
9.7
P2
Warehouses
W1
0
b
W2
Warehouse
capacity &
customer
demand
Customers
C1
b
99
9
C2
99
C3
99
Plant &
warehouse
capacities
60,000
99
0
60,000
999,999
c
c
999,999
d
8.2
50,000
8.7
60,000
7.2
40,000
8.2
50,000
50,000
100,000
50,000
e
99
99
10.7
60,000
c
999,999
a
Production plus inbound transport rates, that is, 4 + 0 = 4.
Used to represent an infinitely high cost.
c
Used to represent unlimited capacity.
d
Inventory carrying, warehousing, outbound transportation, and fixed rates, that is,
3.2 + 2 + 4 + 0.5 = 9.7.
e
3.2 + 1 + 2 + 2.0 = 8.2.
b
13-39
Guided Linear Programming (Cont’d)
 Solve as an LP problem, transportation type. Note the results in the
solution matrix.
 Recompute the per-unit fixed and inventory costs from the first solution
results as follows.
Ware- Per-unit Fixed Cost,
house $/cwt.
W1 $100,000/60,000 cwt.
= 1.67
W2 $400,000/140,000 cwt.
= 2.86
Per-Unit Inventory Carrying Cost,
$/cwt.
0.7
$100(60,000 cwt.) /60,000 cwt.
= 3.69
0.7
$100(140,000 cwt.) /140,000 cwt.
= 2.86
 Place these per-unit costs along with transportation costs in the
transportation matrix. The portion of the matrix that is revised is:
W1
W2
a
b
C1
C2
a
11.36 10.36
b
8.72
7.72
C3
12.36
8.72
2 + 4 + 1.67 + 3.69 = 11.36
1 + 2 + 3.57 + 2.86 = 9.43
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13-40
Guided Linear Programming (Cont’d)
The revised solution shows that all volume should be produced at plant 2
and all customers served from warehouse 2.
 Once the throughputs are known, they are used to compute the cost
summaries shown in the solution table. Actual costs are used, not the
cell costs of the LP problem. The solution is:
Cost type
Production
Inbound transportation
Outbound transportation
Fixed
Inventory carrying
Handling
Subtotal
Total
Warehouse 1
Warehouse 2
0 cwt.
200,000 cwt.
$0
200,0004
= $800,000
0
200,0002
= 400,000
50,0002
= 100,000
100,0001
= 100,000
0
50,0002
= 100,000
0
400,000
0.7
0
100(200,000) = 513,714
0
200,0001
= 200,000
$0
$2,613,714
$2,613,714
 The iterative process is repeated until there are no changes. The answer
is to have only warehouse 2 open.
13-33
Location by Simulation
•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
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13-34
Read in
all customer
order data
and locations
Preprocessing
program
Volume
shipment
orders
Orders filled
through warehousing system
Read in
freight rates,
warehousing costs,
taxes, etc.
Test
program
Read in
warehouse
location
configuration
to be evaluated
Cost of warehouse
location configuration
Is another
run desired?
Yes
Shycon/Maffei Simulation
Start
No
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Stop
13-35
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
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13-36
Level 2
warehouses
or stocking
points
Level 1
warehouses
or stocking
points
Customers/
demand
centers/
sinks
Supply
Demand
Production/
purchase costs
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Inventory &
warehousing
costs
Transportation costs
Inventory &
warehousing
costs
Commercial Models (Cont’d)
Plants/
vendors/
ports/
sources
13-45
Commercial Models (Cont’d)
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13-46
Dynamic Location
 The general long-range nature of the location problem
- Network configurations are not implemented immediately
- There are fixed charges associated with moving to a new
configuration
 We seek to find a set of network configurations that minimizes the present
value over the planning horizon
Retail Location
 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
Methods
 Weighted checklist
- Good where many subjective factors are involved
- Quantifies the comparison among alternate locations
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13-39
A Hypothetical Weighted Factor Checklist for a
Retail Location Example
(1)
Factor
Weight
a
(1 to 10) Location Factors
8
Proximity to competing stores
5
Space rent/lease
considerations
8
Parking space
7
Proximity to complementary
stores
6
Modernity of store space
9
Customer accessibility
3
Local taxes
3
Community service
8
Proximity to major
transportation arteries
Total index
a
b
(2)
(3)=(1)(2)
Factor Score
b
(1 to 10)
5
Weighted
Score
40
3
10
15
80
8
9
8
2
4
56
54
72
6
12
7
56
391
Weights approaching 10 indicate great importance.
Scores approaching 10 refer to a favored location status.
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13-48