facility location decisions
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
13-1
Facility Location Decisions
Chapter 13CR (2004) Prentice Hall, Inc.
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
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Facility Location in Location Strategy
PL
AN
NIN
G
OR
GA
NIZ
ING
CO
NT
RO
LL
ING
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
PL
AN
NIN
G
OR
GA
NIZ
ING
CO
NT
RO
LL
ING
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 OverviewWhat'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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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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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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Some Location Theory/Practice
Early economic analysis•Bid rent curves•Weber’s isodapanes•Weber’s classification of industries•Hoover’s tapered transport rates•Agglomeration
Mathematical approaches•Light analysis
-Chart, compass, ruler techniques-Spreadsheets-Checklists
•Continuous location methods•Mathematical programming
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Bid Rent Curve
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Web
er’s
Iso
dap
anesVariable spacing
can mean nonlinear
transportation costs
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Weber’s Classification of Industries
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Hoover’s Transport Curves
YY
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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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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
$/unit/mi. Overlaying a grid to determine the coordinates of
source and/or destination points Finding the weighted center of gravity for the graph
COG Method
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COG Method (Cont’d)
i ii
i iii
i ii
i iii
RV
YRVY,
RV
XRVX
where
Vi = volume flowing from (to) point I
Ri = transportation rate to ship Vi from (to) point i
Xi,Yi = coordinate points for point i
= coordinate points for facility to be locatedY,X
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COG Method (Cont’d)Example Suppose a regional medical warehouse is to be established to serve several Veterans Administration hospitals throughout the country. The supplies originate at S1 and S2 and are destined for hospitals at H1 through H4. The relative locations are shown on the map grid. Other data are: Note rate is a
per mile costPointi
Prod-ucts Location
Annualvolume,
cwt.
Rate,$/cwt/
mi. Xi Yi1 S1 A Seattle 8,000 0.02 0.6 7.32 S2 B Atlanta 10,000 0.02 8.6 3.03 H1 A & B Los
Angeles5,000 0.05 2.0 3.0
4 H2 A & B Dallas 3,000 0.05 5.5 2.45 H3 A & B Chicago 4,000 0.05 7.9 5.56 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 ViRiYi1 0.6 7.3 8,000 0.02 160 96 1,1682 8.6 3.0 10,000 0.02 200 1,720 6003 2.0 3.0 5,000 0.05 250 500 7504 5.5 2.4 3,000 0.05 150 825 3605 7.9 5.5 4,000 0.05 200 1,580 1,1006 10.6 5.2 6,000 0.05 300 3,180 1,560
1,260 7,901 5,538
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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 convertcoordinates into miles.
i iiii YYXXKRVTC 22 )()(
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COGCOG 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
TCRiViYiXiiCalculate total cost at COG
221
4.40)(7.36.27)(0.6)(500)8,000(0.02TC
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Note The center-of-gravity method does not necessarilygive optimal answers, but will give good answers if there area large numbers of points in the problem (>30) and thevolume for any one point is not a high proportion of the totalvolume. However, optimal locations can be found by theexact center of gravity method.
i iii
i iiiin
i iii
i iiiin
/dRV
/dYRVY,
/dRV
/dXRVX
where
22 )Y(Y)X(Xdn
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 COG2) Using find di
3) Re-solve for using exact formulation4) 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,XY,X
Y,X
Y,X
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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 problemthat:
Search for the best combination of facilities to minimizecosts
Do so within a reasonable computational time Do not require enormous amounts of data for the analysis
Multiple Location Methods
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Location Cost Trade-Offs
Number of warehouses
Cos
t
Production/purchaseand order processing
Inventory carryingand warehousing
Warehousefixed
Inbound andoutboundtransportation
Total cost
00
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Multiple COGFormulated as basic COG modelCan 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 procedureRank demand points from highest to lowest volumeUse the M largest as initial facility locations and assign
remaining demand centers to these locationsCompute the COG of the M locationsReassign all demand centers to the M COGs on the basis
of proximityRecompute the COGs and repeat the demand center
assignments, stopping this iterative process when there is no further change in the assignments or COGs
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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
13-28CR (2004) Prentice Hall, Inc.
Example
The database for the illustrated problem is prepared as fileILP02.DAT in MIPROG. The form is dictated by the problemformulation in the technical supplement of the location chapter.
The solution is to open only one warehouse (W2) with resultingcosts of:Category CostProduction $1,020,000Transportation 1,220,000Warehouse handling 310,000Warehouse fixed 500,000 Total $3,050,000
MIP (Cont’d)
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Plant P1
Production = $4/cwt.Capacity =60,000 cwt.
Plant P2
Production = $4/cwt.Capacity =Unrestricted
$0/cwt.
$2/cwt.
$4/c
wt.
$5/cwt.
Customer C1
50,000 cwt.
Customer C2
100,000 cwt.
Customer C3
50,000 cwt.
$4/cwt
$3/cwt.
$5/cwt.
$2/c
wt.
$1/cwt.
$2/cwt
Warehouse W1
Warehouse W2
Handling = $2/cwt.
Handling = $1/cwt.
Plant P1
Production = $3/cwt.Capacity =50,000 cwt.
Plant P2
Production = $2/cwt.Capacity =Unrestricted
$0/cwt.
$2/cwt.
$4/c
wt.
$5/cwt.
Customer C1
20,000 cwt.
Customer C2
30,000 cwt.
Customer C3
60,000 cwt.
$3/cwt
$2/cwt.
$4/cwt.
$3/c
wt.
$2/cwt.
$3/cwt
Warehouse W1
Warehouse W2
Handling = $2/cwt.
Handling = $1/cwt.
Product 1
Product 2
Fixed = $100,000Capacity = 110,000 cwt.
Fixed = $500,000Capacity = Unrestricted
MIP (Cont’d)
A Multiple Product Network Design Problem
The situation
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Less complicated than previous example, butbased on integer programming
Locates on basis of transportation costs andfacility fixed costs
Locations are restricted to the node (e.g.,demand) points in the problem
Method finds the optimal location of M facilities at a time
P-Median Location Method
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Example Five incinerators are to be located for toxic chemicalreclamation. The cost to move chemicals from 12 market areas tothe incinerators is $0.0867/cwt./mi. The areas, the chemicalvolume, and fixed costs to operate an incinerator are:
Market
Annualvolume,
cwt.
Fixedoperating
cost, $ Market
Annualvolume,
cwt.
Fixedoperating
cost, $Boston MA 30,000 3,100,000 Chicago IL 240,000 2,900,000New York NY 50,000 3,700,000 Minneapolis MN 140,000 --Atlanta GA 170,000 1,400,000 Phoenix AZ 230,000 1,100,000Baltimore MD 120,000 -- Denver CO 300,000 1,500,000Cincinnati OH 100,000 1,700,000 Los Angeles CA 40,000 2,500,000Memphis TN 90,000 -- Seattle WA 20,000 1,250,000
P-Median (Cont’d)
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The database for this problem is PMED02.DAT inLOGWARE.
Answer
No. Facility name Volume Assignednode numbers
1 New York NY 200,000 1 2 4 2 Atlanta GA 260,000 3 6 3 Chicago IL 480,000 5 7 8 4 Phoenix AZ 270,000 9 11 5 Denver CO 320,000 10 12 Total 1,530,000
Total cost: $24,739,040.00and graphically shown…
P-Median (Cont’d)
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P-Median (Cont’d)
Repeating the analysis for a different number of incinerators can find the optimal number of incinerators as well.
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Selective Evaluation Using COG
Note : Inventorycosts and fixedcosts increase withmore warehouses
User selects the locations or the number of locations to be evaluated. Analysis is repeateduntil the optimum is found. Inventory value can be easily added as in the following example
where I NT = $6, ,000 000 . At 25% per year, the inventory carrying cost is CC = 0.25 I T .
Number ofWarehouses
TransportationCost, $
FixedCost, $
InventoryCost, $
TotalCost, $
1 41,409,628 2,000,000 1,500,000 44,909,6282 25,989,764 4,000,000 2,121,320 32,111,0843 16,586,090 6,000,000 2,598,076 25,184,1664 11,368,330 8,000,000 3,000,000 22,368,3305 9,418,329 10,000,000 3,354,102 22,772,4316 8,032,399 12,000,000 3,674,235 23,706,6347 7,478,425 14,000,000 3,968,627 25,447,0528 2,260,661 16,000,000 4,242,641 22,503,3029 948,686 18,000,000 4,500,000 23,448,686
10 0 20,000,000 4,743,416 24,743,416
Note : Transport cost declinewith more warehouses
Example
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Shows effect of inventory policy on location. Presence of local optima make it difficult to find the optimum solution in complex problems.
Total Cost Curve for Selective Evaluation Example
05,000,000
10,000,00015,000,00020,000,00025,000,00030,000,00035,000,00040,000,00045,000,000
To
tal c
ost
, $
1 2 3 4 5 6 7 8 9 10
Number of warehouses
Local optimumOptimum
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Guided Linear ProgrammingSearches 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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Plant P1
Production = $4/cwt.Capacity =60,000 cwt.
Plant P2
Production = $4/cwt.Capacity =Unrestricted
$0/cwt.Customer C1
50,000 cwt.
Customer C2
100,000 cwt.
Customer C3
50,000 cwt.
$5/cwt.
$2/c
wt.
Warehouse W1
Warehouse W2
Handling = $2/cwt.Capacity = 60,000 cwt.Fixed = $100,000
Handling = $1/cwt.Capacity = UnrestrictedFixed = $400,000
Inventory carrying cost =100(Throughput)0.7
Guided Linear Programming (Cont’d)The situation
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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 1100,000/200,000 = $0.50/cwt.
Warehouse 2400,000/200,000 = $2.00/cwt.
Allocate inventory carrying costs. Assume that throughput for each product isequally divided between warehouses.
Each warehouse
100{[200,000/2)0.7]/(200,000/2)} = $3.2
Build the cost matrix for the transportation problem of LP. Note its specialstructure.
Total customer demand
No. of whses
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Matrix of Cell Costs and Solution Values for the First Iteration in the Example Problem Warehouses Customers
W1 W2 C1 C2 C3
Plant &warehousecapacities
P1
4a
60,0009 99
b99 99
60,000
P2
8 6140,000
99 99 99999,999
c
W1
0 99 9.7d
8.760,000
10.760,000
W2
99b
0 8.2e
50,0007.2
40,0008.2
50,000 999,999c
Warehousecapacity &customerdemand 60,000 999,999
c50,000 100,000 50,000
Plants
Warehouses
a Production plus inbound transport rates, that is, 4 + 0 = 4.b Used to represent an infinitely high cost.c Used to represent unlimited capacity.d Inventory carrying, warehousing, outbound transportation, and fixed rates, that is, 3.2 + 2 + 4 + 0.5 = 9.7.e 3.2 + 1 + 2 + 2.0 = 8.2. 13-39
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Guided Linear Programming (Cont’d) Solve as an LP problem, transportation type. Note the results in the
solution matrix. Recompute the per-unit fixed and inventory costs from the first solution
results as follows.
1 + 2 + 3.57 + 2.86 = 9.43
Ware-house
Per-unit Fixed Cost,$/cwt.
Per-Unit Inventory Carrying Cost,$/cwt.
W1 $100,000/60,000 cwt.= 1.67
$100(60,000 cwt.)0.7/60,000 cwt.= 3.69
W2 $400,000/140,000 cwt.= 2.86
$100(140,000 cwt.)0.7/140,000 cwt.= 2.86
Place these per-unit costs along with transportation costs in thetransportation matrix. The portion of the matrix that is revised is:
C1 C2 C3
W1 11.36a 10.36 12.36W2 8.72b 7.72 8.72
a 2 + 4 + 1.67 + 3.69 = 11.36b
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The revised solution shows that all volume should be produced at plant 2and 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 thecell costs of the LP problem. The solution is:
Guided Linear Programming (Cont’d)
The iterative process is repeated until there are no changes. The answeris to have only warehouse 2 open.
Cost typeWarehouse 1
0 cwt.Warehouse 2200,000 cwt.
Production $0 200,0004 = $800,000Inbound transportation 0 200,0002 = 400,000Outbound transportation
0
50,0002 = 100,000100,0001 = 100,000 50,0002 = 100,000
Fixed 0 400,000Inventory carrying 0 100(200,000)0.7 = 513,714Handling 0 200,0001 = 200,000
Subtotal $0 $2,613,714
Total $2,613,714
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Location by Simulation
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•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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Sh
yco
n/M
affe
i Sim
ula
tio
n
Start
Preprocessingprogram
Read inall customerorder data
and locations
Volumeshipment
orders
Orders filledthrough ware-
housing system
Testprogram
Read inwarehouse
locationconfiguration
to be evaluated
Read infreight rates,
warehousing costs,taxes, etc.
Cost of warehouselocation configuration
Is anotherrun desired?
Stop
Yes
No
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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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Co
mm
erci
al M
od
els
(Con
t’d)
Plants/vendors/
ports/sources
Level 2warehousesor stocking
points
Level 1warehousesor stocking
points
Customers/demandcenters/
sinks
Supply
Demand
Production/purchase costs
Inventory &warehousing
costsInventory &
warehousingcosts
Transportation costs
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Commercial Models (Cont’d)
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Dynamic Location
Retail Location
Methods
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 presentvalue over the planning horizon
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
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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)FactorWeight
(1 to 10)a Location Factors
(2)
Factor Score(1 to 10)b
(3)=(1)(2)
WeightedScore
8 Proximity to competing stores 5 405 Space rent/lease
considerations 3 15
8 Parking space 10 807 Proximity to complementary
stores 8 56
6 Modernity of store space 9 549 Customer accessibility 8 723 Local taxes 2 63 Community service 4 128 Proximity to major
transportation arteries 7 56 Total index 391
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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 P C
S T
S TCij ij i
j ija
j ija
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
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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 Bhas 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 salespotential of each shopping center?
Solution matrix
Custo- mer i
Time from Customer i
to Location j Tij
2
S Tj ij/ 2
PS T
S Tij
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
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0 10 20 30 40 50 60 70 800
10
20
30
40
50
60
70
80Y
XTime (minutes)
Tim
e (m
inut
es)
C2
C1
C3
RB
RA
Retail Location (Cont’d)
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Retail Location (Cont’d)
Factors such as (1) no. of parking spaces, (2) store size, (3) no. of items in store, etc.
Regression analysis- Model is of the form:
Sales = b0 + b1(F1) + b2(F2) + … + bn(Fn)
and Fs are factors relating to location