Facility Location Decisions
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Classifying location decisions
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Driving force (critical factor - traffic, labor rates, emergency
facilities, obnoxious facilities)
Number of facilities
Discrete vs. continuous choices
Data aggregation
Time Horizon
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Facility Location
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Rent Curve - The rent of land is a decreasing function of the
distance to the market
Weight gaining vs. weight losing industries
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Weight losing should locate close to raw materials
Weight gaining should locate close to market
Tapered (concave) transportation costs
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The derivative of total transportation cost is non-increasing with the
distance to the market (holds for inbound and outbound costs)
Optimal solution will always locate either at raw materials or at
market (extreme point solution)
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Single Facility Location Model
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This model assumes a known set, I, of source and demand
points, each with known demand volumes, Vi, and
transportation rates, Ri.
The objective is to locate the facility at the point that minimizes
total transportation cost, TC:
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Let di denote the distance from the facility to demand point i.
Min TC  Vi Ri d i

iI
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subject to:di
 ( X i  X )2  (Yi  Y )2
The decision variables are X and Y , the coordinates of the facility
Xi, Yi denote the coordinates of demand point i.
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Single Facility Location Model
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Differentiating TC w.r.t. X and Y and setting the result
equal to zero gives the ‘center of gravity’:
 Vi Ri X i 
i 
d i 
X 
 Vi Ri 
i  di 
 Vi RiYi 
i 
d i 
Y 
 Vi Ri 
i  d i 
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Single Facility Location Model
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X and Y are given in terms of di, which
Note however, that
is a function of X and Y .
An algorithm that will converge to the optimal
X and Y .
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This continuous problem is often called the Weber problem
These problems are restrictive because they assume continuity
of location and straight-line distances
Also, only variable distance related costs are considered
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General Facility Location Model
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The general facility location problem considers the simultaneous
location of a number of facilities
Notation:
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I - Set of customers, indexed by i.
J - Set of facilities, indexed by j.
di - demand of customer i.
cij - cost of transporting a unit from facility j to customer i.
Fj - fixed cost of creating facility j.
xij - variable for flow from facility j to customer i.
Yj - binary variable that equals 1 if we create facility j, 0 otherwise
sj - capacity of facility j.
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Uncapacitated Facility Location Model
Formulation
Minimize
 F Y   c x
j
j
j
ij ij
i
j
subject to:
x
ij
 di ,
 iI
j
xij  d iY j ,
 i  I, j  J
xij  0, Y j  {0,1},  i  I , j  J
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Capacitated Facility Location Model
Formulation
Minimize
 F Y   c x
j
j
j
ij ij
i
j
subject to:
x
ij
 di ,
 iI
 sj,
 jJ
j
x
ij
i
xij  min{s j , d i }Y j ,  i  I , j  J
xij  0, Y j  {0,1},  i  I , j  J
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Location Decisions and Risk Pooling
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Suppose we must serve n independent markets with a single
product, and each market has average demand per period of D
and standard deviation  (we neglect lead times for simplicity)
Assume we have a service level policy to ensure that the
probability of not stocking out in each period equals 
Suppose we serve each market using a single inventory stocking
location.
The standard deviation of demand as seen by the single
stocking location in each period equals n
If demand is normally distributed, the safety stock required at
the single location equals z n
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Location Decisions and Risk Pooling
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Suppose, at the other extreme, we place an inventory stocking
location in each of the n markets
Each stocking location will need to hold z to meet the service
level requirement
The system-wide safety stock equals zn > z n
This example illustrates the risk-pooling effects of location
decisions
The more stocking locations we have, the more duplication in
safety stock we have
The single location, however, will incur the maximum
transportation costs, while n locations should minimize the
transportation costs
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Location Decisions and Risk Pooling
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The safety stock costs and transportation costs are at odds with
each other
We need to strike a balance between the two
Models for this decision are currently limited (Prof. Shen has
worked on a model that addresses this tradeoff)
However, this simple analysis can provide strong insights
If inventory costs dominate transportation costs (as in
expensive computing chips), we are driven to have less stocking
points; if transport costs dominate (as in coal), then we are
driven to have more stocking locations
One thing not included in the analysis is delivery lead time and
its impact on service levels – obviously more locations closer to
markets can respond much more quickly to customers
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Supply Chain Design Model
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The objective of this model is to determine the warehouse and
plant configuration that minimizes total costs for production and
distribution of multiple products.
Based on Geoffrion and Graves, 1974, “Multicommodity
distribution system design by Benders decomposition,”
Management Science, v20, n5. (see Tech. Suppl., Ch. 13)
Notation:
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i - index for commodities
j - index for plants
k - index for warehouses
l - index for customer zones
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Supply Chain Design Model
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Notation (continued):
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Sij - production capacity for commodity i at plant j.
Dil - demand for commodity i in customer zone l.
Vk , Vk - min and max total throughput for warehouse k.
fk - fixed part of annual costs for owning and operating warehouse
k.
vk - variable unit cost of throughput for warehouse k.
Cijkl - average unit cost of producing, handling, and shipping
commodity i from plant j through warehouse k to customer zone l.
Xijkl - amount of commodity i flowing from plant j through
warehouse k to customer zone l.
ykl - binary variable = 1 if warehouse k serves customer zone l, 0
otherwise
zk - binary variable = 1 if warehouse k is open, 0 otherwise.
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Supply Chain Design Model Formulation
Minimize
C
ijkl
X ijkl
ijkl


 
   f k zk  vk    Dil  ykl 
k 
l  i
 
subject to:
X
ijkl
 Dil ykl ,
 ikl
ijkl
 Sij ,
 ij
j
X
kl
y
kl
= 1,
l
k


Vk     Dil  ykl  Vk
l  i

X ijkl  0, ykl  {0,1}, zk  {0,1}
k
 ijkl
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Network Planning
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Network planning refers to assessing or reassessing the configuration
of facilities, commodities, and flows currently used to satisfy demand
Network planning data checklist:
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List of all products
Customer, stocking point, and source point locations
Demand by customer location
Transportation rates
Transit times, order transmittal times, and order fill rates
Warehouse rates and costs
Purchasing/production costs
Shipment sizes by product
Inventory levels by location, by product, control methods
Order patterns by frequency, size, season, content
Order processing costs and where they are incurred
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Network Planning
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Data Checklist (continued):
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Capital cost
Customer service goals
Available equipment and facilities and their capacities
Current distribution patterns (flows)
Note that many of these are decision variables
Accumulating these data usually results in improvements by uncovering
anomalies
We must decide our network design strategy:
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Specify minimum service levels
Specify shortage costs and minimize cost
Levels of acceptable aggregation of demand
Optimization vs. heuristic methods
Which areas require the most accuracy and attention?
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