facility location decisions

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

13-2CR (2004) Prentice Hall, Inc.

Facility Location in Location Strategy

PL

AN

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GA

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CO

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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

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GA

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NT

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LL

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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


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


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


Methods of Solution

Single warehouse location

– Graphic

– Grid, or center-of-gravity, approach

Multiple warehouse location

– Simulation

– Optimization

– Heuristics

Location Overview (Cont’d)


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.


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


Bid Rent Curve


Web

er’s

Iso

dap

anesVariable spacing

can mean nonlinear

transportation costs

CR (2004) Prentice Hall, Inc.

13-10

Weber’s Classification of Industries



Hoover’s Transport Curves

YY


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.


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


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


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


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



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

13-18

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 )()(



COGCOG 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


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.



Solution procedure for exact COG


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


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

13-24

Location Cost Trade-Offs

Number of warehouses

Cos

t

Production/purchaseand order processing

Inventory carryingand warehousing

Warehousefixed

Inbound andoutboundtransportation

Total cost

00


13-25

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



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


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


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)

13-29

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


Plant P2


$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


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


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)

13-31


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)


P-Median (Cont’d)

Repeating the analysis for a different number of incinerators can find the optimal number of incinerators as well.

13-33


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

13-34

13-35

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



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.

13-37

Plant P1


Plant P2


$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



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

13-38

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


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

13-40

13-41

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

13-42

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

13-43

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


13-44

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



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

13-45


Commercial Models (Cont’d)

13-46


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


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

13-48


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


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

13-50


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)


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

facility location decisions

Documents

prentice hall

location strategycr

location overviewwhats

primary location factors

transportation costs

relevant costs

service factors

method appraisal