Facility Location
Location
and Distribution
Distribution
System Planning
Planning
Thomas
Thomas L.
L. Magnanti
Magnanti
Today’s
Today’s Agenda
Agenda
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Why
Why study
study facility
facility location?
location?
Issues
Issues to
to be
be modeled
modeled
Basic
Basic models
models
� � Fixed
Fixed charge
charge problems
problems
� � Core
Core uncapacitated
uncapacitated and
and capacitated
capacitated
facility
facility location
location models
models
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Large-scale
Large-scale application
application (Hunt(HuntWesson
Wesson Foods)
Foods)
Logistics
Logistics Industry
Industry
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U.S.
U.S. logistics
logistics industry:
industry: $900
$900 billion
billion -- almost
almost
double
double the
the size
size of
of the
the high-tech
high-tech industry:
industry: >> 10
10
percent
percent of
of the
the U.S.
U.S. gross
gross domestic
domestic product
product
11
11 per
per cent
cent of
of Singapore's
Singapore's GDP
GDP with
with aa growth
growth of
of
99 per
per cent
cent in
in year
year 2000
2000
Singapore
Singapore Logistics
Logistics Enhancement
Enhancement &
&
Applications
Applications Programme
Programme (LEAP)
(LEAP) 2001
2001
Global
Global logistics:
logistics: $3.43
$3.43 trillion
trillion
1998,
1998, U.S.
U.S. trucking
trucking industry
industry revenues
revenues just
just
under
under $200
$200 billion
billion
7.7
7.7 million
million trucks
trucks carried
carried over
over 11 trillion
trillion ton
ton
miles
miles of
of freight
freight
Singapore
Singapore Retail
Retail 21
21 Plan
Plan
Basic
Basic Issue
Issue
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Where
Where to
to locate
locate and
and how
how to
to size
size
facilities?
facilities?
How
How to
to meet
meet customer
customer demands
demands from
from
the
the facilities?
facilities?
� � Which
Which facility
facility (facilities)
(facilities) serve
serve each
each
customer?
customer?
� � How
How much
much customer
customer demand
demand is
is met
met by
by
each
each facility?
facility?
Facilities might be warehouses, retail outlets,
wireless bay stations, communication concentrators
Some
Some Elements
Elements of
of Cost
Cost &
& Service
Service
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Transportation
Transportation Costs
Costs
� � Vehicles,
Vehicles, Drivers,
Drivers, Fuel
Fuel
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Warehousing
Warehousing
� � Facility
Facility Construction/Rental,
Construction/Rental, Handling
Handling
Costs,
Costs, Inventory
Inventory
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Customer
Customer Service
Service
� � Service
Service Time,
Time, Single
Single Sourcing
Sourcing
System
System Trade-offs
Trade-offs
Transportation
Costs
-
Fixed
Costs
+
Effect of More Facilities
System
System Trade-offs
Trade-offs
Costs
-
Service
+
Effect of “Individualized”
Service (e.g., Direct Shipments)
Nature
Nature of
of Costs
Costs
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Fixed
Fixed Costs
Costs
� � Facility
Facility construction/rental
construction/rental
� � Vehicle
Vehicle purchases
purchases &
& rentals
rentals
� � Personnel
Personnel (drivers,
(drivers, managers)
managers)
� � Fixed
Fixed overhead
overhead
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Variable
Variable Costs
Costs
� � Inventory,
Inventory, handling,
handling, fuel
fuel
Optimization
Optimization Applications
Applications
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Hunt-Wesson
Hunt-Wesson Foods
Foods saves
saves over
over $1
$1 million
million
per
per year
year
Restructuring
Restructuring North
North America
America operations,
operations,
Proctor
Proctor and
and Gamble
Gamble reduces
reduces plants
plants by
by
20%,
20%, saving
saving $200
$200 million/year
million/year
Many,
Many, many
many others
others (e.g.,
(e.g., supplying
supplying parts
parts
to
to plants)
plants)
Facility
Facility Location
Location Challenge
Challenge
19
19
Modeling
Modeling Issue
Issue
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How
How do
do we
we model
model
“lumpiness”
“lumpiness” of
of the
the
costs
costs (e.g.,
(e.g., fixed
fixed
costs)?
costs)?
How
How do
do we
we model
model
logical
logical conditions
conditions
(e.g.,
(e.g., choice
choice of
of
warehouse
warehouse locations)?
locations)?
Modeling
Modeling Fixed
Fixed Costs
Costs
Flow x > 0
Flow z > 0
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Incur
Incur fixed
fixed cost
cost F
F if
if either
either xx >> 00 or
or zz >> 00
Suppose
3/2 (demand
(demand limitation)
limitation)
Suppose xx ++ zz << 3/2
Model
Minimize Fy + other terms
subject to y = 1 if either x > 0 or z > 0
Three
Three Models
Models (LP
(LP Relaxations)
Relaxations)
Model 1
x + z < 3/2
x < 1, z < 1
x + z < 2y
x > 0, z > 0
0<y<1
Forcing
Constraints
Weak
Strong
Model 3
x + z < 3y/2
x < 1, z < 1
x < y, z < y
x > 0, z > 0
0<y<1
Model 2
x + z < 3/2
x < 1, z < 1
x < y, z < y
x > 0, z > 0
0<y<1
Geometry
Geometry (Weak
(Weak Model)
Model)
y
x + z < 3/2
Feasible
Feasible region
region
with x<1,
x<1, y<1,
y<1,
y with
z<1,
z<1, xx ++ z<
z< 3/2
3/2
Feasible
points
z
x
z
x
x + z < 2y
intersects at
y = 3/4
Geometry
Geometry (Improved
(Improved Model)
Model)
x + z < 3/2
y
z
z
x
y
x
xx << y,
y, zz << yy
intersects
intersects at
at
xx == zz == yy == 3/4
3/4 &
& at
at
yy == 11 w/
w/ xx or
or zz =1
=1
Geometry
Geometry (Strong
(Strong Model)
Model)
y
y
z
z
x
x
Exact
Exact representation!
representation!
xx ++ zz << 3y/2,
3y/2,
xx << y,
y, zz << yy
intersects
intersects at
at yy == 11
Core
Core (Uncapacitated)
(Uncapacitated) Facility
Facility Location
Location
Minimize Fixed + Routing Costs
Subject to
� Meet customer
demand from facilities
� Assign customer only to
open facility
Parameters:
Parameters:
Core
Core Facility
Facility Location
Location Model
Model
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Demand
Demand d
dii for
for each
each customer
customer ii
Fixed
Fixed cost
cost F
Fjj for
for each
each facility
facility location
location jj
Cost
of routing
routing all
all customer
customer ii
Cost ccijij of
demand
demand to
to facility
facility jj == per
per unit
unit cost
cost times
times
demand
demand d
dii
Decisions:
Decisions: Core
Core Facility
Facility Location
Location
Model
Model
��
Where
Where do
do we
we locate
locate facilities?
facilities?
� � yyjj == 11 ifif we
we locate
locate facility
facility at
at location
location jj
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Fraction
Fraction of
of service
service that
that customer
customer ii
receives
receives from
from facility
facility jj (x
(xijij ))
Network
Network Representation
Representation
33 Customers,
Customers, 44 Facilities
Facilities
d11
1
d22
2
d33
3
Customers
xijij
yjj
1
2
3
4
Facilities
Facility
Facility Location
Location Costs
Costs
c11
x 11 + c12
x 12
11 11
12 12
+ c13
x 13 + c14
x 14
13 13
14 14
+. . .
+ c31
x 31 + c32
x 32
31 31
32 32
+ c33
x 33 + c34
x 34
33 33
34 34
+ F11y11 + F22y22 + F33y33 + F44y44
Routing
Routing
Costs
Costs
Fixed
Fixed
Costs
Costs
Constraints:
Constraints: Tabular
Tabular Representation
Representation
C
u
s
t
o
m
e
r
s
Facilities (Locations)
x11
x12
x13
x14
11
12
13
14
x21
x22
x23
x24
21
22
23
24
x31
x32
x33
x34
31
32
33
34
=1
=1
=1
x11
x 12 ≤ y2,2, x13
x 14 ≤ y44
11 ≤ y1,
1, 12
13 ≤ y3,
3, 14
x21
x 22 ≤ y2,2, x23
x 24 ≤ y44
21 ≤ y1,
1, 22
23 ≤ y3,
3, 24
x31
x 32 ≤ y2,2, x33
x 34 ≤ y44
31 ≤ y1,
1, 32
33 ≤ y3,
3, 34
Model
Model
(Uncapacitated
(Uncapacitated Facilities)
Facilities)
Minimize ΣiiΣjj cijijxijij + Σjj Fjjyjj
Subject to
Σjj xijij = 1
xijij ≤ yjj
xijij ≥ 0
yjj = 0 or 1
}
for all customers i
for all customers i
and facilities j
for all facilities j
Modeling
Modeling Variations
Variations
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Open
Open at
at most
most three
three of
of facilities
facilities 1,
1, 66 and
and
8-11
8-11
� � yy11 ++ yy66 ++ yy88 ++ yy99 ++ yy10
+ y 11 ≤≤ 33
10 + y11
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Assign
Assign each
each customer
customer to
to aa single
single facility
facility
� � xx11
integer, x 12 integer,
integer, etc.
etc.
11 integer, x12
Modeling
Modeling Variations
Variations
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Open
Open aa facility
facility at
at location
location 33 only
only if
if we
we open
open
one
one at
at location
location 77
� � yy33 ≤≤ yy77
Note: Power of using integer
variables to model
logical restrictions
Modeling
Modeling Enhancements
Enhancements
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Multiple
Multiple products
products
Facility
Facility capacities
capacities and
and operating
operating
ranges
ranges (min
(min and
and max
max throughput
throughput if
if
open)
open)
Multi-layered
Multi-layered distribution
distribution networks
networks
Service
Service restrictions
restrictions
� � Single
Single sourcing
sourcing
� � Timing
Timing of
of deliveries
deliveries
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Inventory
Inventory positioning
positioning and
and control
control
Alternate
Alternate Model
Model (Uncapacitated
(Uncapacitated Facilities)
Facilities)
Minimize ΣiiΣjj cijijxijij + Σjj Fjjyjj
Subject to
Σjj xijij = 1
Σii xijij ≤ nyjj
xijij ≥ 0
yjj = 0 or 1
for all customers i
for all facilities j
for all pairs i,j
for all facilities j
Alternate
Alternate Model
Model (Uncapacitated
(Uncapacitated Facilities)
Facilities)
Minimize ΣiiΣjj cijijxijij + Σjj Fjjyjj
Subject to
Σjj xijij = dii
Σii xijij ≤ ((Σ
Σii dii)yjj
xijij ≥ 0
yjj = 0 or 1
for all customers i
for all facilities j
for all pairs i,j
for all facilities j
Model
Model (Capacitated
(Capacitated Facilities)
Facilities)
Minimize ΣiiΣjj cijijxijij + Σjj Fjjyjj
Subject to Kj
Σjj xijij = dii
for all customers i
xijij ≤ diiyjj
for all i, j pairs
Σii xijij ≤ CAPjj yjj for all facilities j
xijij ≥ 0
for all i, j pairs
yjj = 0 or 1 for all facilities j
Tabular
Tabular Representation
Representation
C
u
s
t
o
m
e
r
s
Locations
x11
x12
x13
x14
= d11
11
12
13
14
x21
x22
x23
x24
= d22
21
22
23
24
x31
x32
x33
x34
= d33
31
32
33
34
≤
≤
≤
≤
K11y11 K22y22 K33y33 K44y44
plus cell constraints xijij ≤ diiyjj
Network
Network Representation
Representation
yjj
xijij
d11
1
d22
2
d33
3
Customers
1
1
K11
2
2
K22
3
3
K33
4
4
Facilities
K44
Solution
Solution Approaches
Approaches
‰
‰ Heuristics
Heuristics
� � Add,
Add, drop,
drop, and/or
and/or exchange
exchange
� � Linear
Linear programming
programming relaxation
relaxation
� � Bounding
Bounding (Lagrangian
(Lagrangian relaxation)
relaxation)
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Optimization
Optimization methods
methods
� � Large-scale
Large-scale mixed
mixed integer
integer programming
programming
� � Benders
Benders decomposition
decomposition
� � Lagrangian
Lagrangian relaxation
relaxation (e.g.,
(e.g., dualize
dualize capacity
capacity
constraints
constraints to
to give
give uncapacitated
uncapacitated facility
facility
location
location subproblem
subproblem
Hunt-Wesson Foods
Ingredients
Ingredients
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Multiple
Multiple products
products
Multiple
Multiple plants
plants
Many
Many DCs,
DCs, many
many customers
customers
Site
Site selection
selection and
and sizing
sizing
Customer
Customer service
service levels
levels
Complex
Complex costs
costs
Flows
�
i �
�
�
�
j
�
�
�
�
14 plants
�
k
�
�
45 DC Choices
121 Customer Zones
17 Product Groups p
Data
Data Preprocessing
Preprocessing
��
49
49 product-plant
product-plant combinations
combinations
� � (from
(from 14x17
14x17 == 238)
238)
��
682
682 DC-customer
DC-customer zone
zone
combinations
combinations
� � (from
(from 45x121
45x121 == 5,445
5,445 possibilities)
possibilities)
Data
Data Preprocessing
Preprocessing
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23,513
23,513
product-plant-DC-customer
product-plant-DC-customer
combinations
combinations
� � (from
(from 49x682
49x682 == 33,418
33,418 possibilities)
possibilities)
System
System Requirements
Requirements
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Data
Data easy
easy to
to acquire
acquire
Inexpensive/quick
Inexpensive/quick to
to run
run
Easily
Easily updated
updated
User-oriented
User-oriented
Flexible
Flexible (what
(what if
if capabilities)
capabilities)
Measurable
Measurable benefits
benefits
Indices
Indices
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p
p == products
products
ii == plants
plants
jj == distribution
distribution centers
centers
kk == customer
customer zones
zones
Plant
DC
Customer Zone
Decision
Decision Variables
Variables
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xxpijk
= amount of product p shipped
pijk = amount of product p shipped
from
from plant
plant ii to
to customer
customer zone
zone kk
through
through DC
DC jj
zzjj == 11 if
if DC
DC jj open
open
yyjkjk == 11 if
if DC
DC is
is sole
sole source
source of
of customer
customer
zone
zone kk
Constraints
Σjkjk xpijk
pijk ≤ Spi
pi
Σii xpijk
= Dpk
y jk
pijk
pk jk
Σjj yjkjk = 1
Vjjzjj ≤ Σpk
D pkyjkjk ≤ Vjjzjj
pk pk
xpijk
pijk ≥ 0
zjj,yjkjk = 0 or 1
+ Configuration Constraints on y,z
Objective
Objective
Σpijk
c pijkxpijk
pijk pijk
pijk Transportation Cost
+ Σjj fjjzjj
Fixed DC Cost
+ Σjj vjjΣpk
D pkyjkjk Variable DC Cost
pk pk
Model
Model Development
Development
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Aggregation
Aggregation of
of Data
Data
Preselection
Preselection of
of Certain
Certain Decisions
Decisions in
in
Large
Large Applications
Applications
Choice
Choice of
of Models
Models
Why integer
instead of
linear
programming?
Power
Power of
of Integer
Integer Programming
Programming
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Fixed
Fixed costs
costs
Bounding
Bounding ## of
of facilities
facilities
Precedence
Precedence constraints
constraints
Mandatory
Mandatory service
service constraints
constraints
� � Sole
Sole sourcing
sourcing
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Service
Service timing
timing
Stages
Stages in
in Model
Model Development
Development
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Probationary
Probationary analysis
analysis
Analyzing
Analyzing results
results
� � Sensitivity
Sensitivity analysis
analysis
� � What
What ifif analysis
analysis
� � Priority
Priority analysis
analysis
Today’s
Today’s Lessons
Lessons
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Facility
Facility location
location and
and distribution
distribution
important
important in
in practice
practice
Geometry
Geometry of
of fixed
fixed cost
cost modeling
modeling
Model
Model choice
choice is
is important
important in
in problem
problem
solving
solving
� � Strong
Strong vs.
vs. weak
weak forcing
forcing constraints
constraints
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Optimization
Optimization models
models are
are able
able to
to
solve
solve large-scale
large-scale practical
practical problems
problems