Location

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Chapter 8
Location
Dr. Ardavan Asef-Vziri Operations and Supply
Chain Management March 2001
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Nature of Location Decisions

• Location decisions are strategic decisions.
• The reasons for location decisions
• Growth
• Expand existing facilities
• Add new facilities
• Production Cost
• Depletion of Resources

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Factors Affecting Plant location
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Regional Raw Material
1-Location of raw material Raw material oriented
factories weight of input gtgtgt weight of output

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Regional Raw Material

These types of plants tends to be closer to
the raw material resources. Indeed row
material or any other important input.
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Regional Market
2-Location of market Market oriented
plants Space required for output gtgtgt
space required for
input. Car manufacturing, Appliances
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Regional Labor, Water, Electricity
3-Labor, water, Electricity Availability of
skilled labor, productivity and wages, union
practices Availability of water Blast furnace
requires a high flow of water Availability of
electricity Aluminum plant strongly depends on
availability and cost of electricity, it
dominates all other inputs.
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Community
1-Quality of Life Cost of living, housing,
schools, health care, entertainment, church 2-
Financial support Tax regulations, low rate
loans for new industrial and service plants
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Site
1-Land Cost of land, development of
infrastructure. 2-Transportation Availability
and cost of rail road, highways, and air
transportation. 3-Environment Environmental
and legal regulations and restrictions
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Decentralization
Small is beautiful Instead of a single huge
plant in one location, several smaller plants in
different locations
Decentralization based on product Decentralizatio
n based on geographical area Decentralization
based on process
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Decentralization based on Product
Each product or sub-set of products is made in
one plant Each plant is specialized in a narrow
sub-set of products. Lower operating costs due
to specialization.
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Decentralization Based on Geographical Area
Each plant is responsible for a geographical
region, Specially for heavy or large products.
Lower transportation costs.
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Decentralization Based on Process
Car industry is an example. Different plants
for engine, transmission, body stamping,
radiator. Specialization in a process results
in lower costs and higher quality. Since volume
is also high, they also take advantage of economy
of scale. However, coordination of production
of all plants becomes an important issue and
requires central planning and control
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Trends in Global Locations

• Foreign producers locating in U.S.
• Made in USA
• Currency fluctuations
• Just-in-time manufacturing techniques
• Focused factories
• Information highway

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BEP in Location Analysis

• Cost-volume Analysis
• Determine fixed and variable costs
• Plot total costs
• Determine lowest total costs

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Example

• Fixed and variable costs for four potential
  locations

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Solution
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Graphical Solution
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Center of Gravity Single Facility Location
Center of gravity is a method to find the
optimal location of a single facility The single
facility is serving a set of demand centers or
It is being served by a set of supply
centers The objective is to minimize the total
transportation Transportation is Flow Distance
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Examples of Single Facility Location Problem
There are a set of demand centers in different
locations and we want to find the optimal
location for a Manufacturing Plant or a
Distribution Center (DC) or a Warehouse to
satisfy the demand of the demand centers or There
are a set of suppliers for our manufacturing
plant in different locations and we want to find
the optimal location for our Plant to get its
required inputs The objective is to minimize
total Flow Distance
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Center of Gravity Single Facility Location
Suppose we have a set of demand points. Suppose
demand of all demand points are equal. Suppose
they are located at locations Xi, Yi Where is
the best position for a DC to satisfy demand of
these points Distances are calculated as
straight line not rectilinear. There is another
optimal solution for the case when distances are
rectilinear.
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Optimal Single Facility Location
The coordinates of the optimal location of the
DC is
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Example
We have 4 demand points. Demand of all demand
points are equal. Demand points are located at
the following locations
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Example
Where is the optimal location for the center
serving theses demand points
(8,5)
(3,5)
(5,4)
(2,2)
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Solution
Where is the optimal location for the center
serving theses demand points
(2,2)
(8,5)
(5,4)
(3,5)
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Solution
The optimal location for the center serving
theses demand points
(8,5)
(3,5)
(5,4)
(2,2)
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Center of Gravity Single Facility Location
Suppose we have a set of demand points. Suppose
they are located at locations Xi, Yi Demand of
demand point i is Qi. Now where is the best
position for a DC to satisfy demand of these
points Again the objective is to minimize
transportation.
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Optimal Single Facility Location
The coordinates of the optimal location of the
DC is
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Example
Where is the optimal location for the center
serving theses demand points
900
100
(8,5)
(3,5)
200
(5,4)
800
(2,2)
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Solution
Where is the optimal location for the center
serving theses demand points
800 (2,2)
900 (3,5)
200 (5,4)
100 (8,5)
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Solution
Where is the optimal Y location for the center
serving theses demand points
800 (2,2)
900 (3,5)
200 (5,4)
100 (8,5)
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Solution
The optimal location for the center serving
theses demand points
(100)
(900)
(200)
(800)
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Example
Where is the optimal location for the center
serving theses demand points
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Solution
Where is the optimal location for the center
serving theses demand points
800 (0,0)
900 (1,3)
100 (6,3)
200 (3,2)
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Solution
Where is the optimal location for the center
serving theses demand points
800 (0,0)
900 (1,3)
100 (6,3)
200 (3,2)
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Solution
The optimal location for the center serving
theses demand points is at the same location
(100)
(900)
(200)
(800)
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The Transportation
Problem
Dr. Ardavan Asef-Vaziri
Industrial Engineering
Mar 2003
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The Transportation Problem
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Transportation problem Narrative representation

There are 3 plants, 3 warehouses. Production of
Plants 1, 2, and 3 are 300, 200, 200
respectively. Demand of warehouses 1, 2 and 3
are 250, 250, and 200 units respectively. Transpo
rtation costs for each unit of product is given
below
Warehouse 1 2 3 1 16 18 11 Plant
2 14 12 13 3 13 15 17
Formulate this problem as an LP to satisfy demand
at minimum transportation costs.
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Transportation problem I decision variables
x11
300
1
1
x12
250
x13
x21
2
200
2
x22
250
x23
x31
x32
3
3
200
200
x33
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Transportation problem I decision variables
x11 Volume of product sent from P1 to W1 x12
Volume of product sent from P1 to W2 x13
Volume of product sent from P1 to W3 x21
Volume of product sent from P2 to W1 x22
Volume of product sent from P2 to W2 x23
Volume of product sent from P2 to W3 x31
Volume of product sent from P3 to W1 x32
Volume of product sent from P3 to W2 x33
Volume of product sent from P3 to W3 We want to
minimize Z 16 x11 18 x12 11 x13 14 x21
12 x22 13 x23 13 x31 15 x32 17 x33
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Transportation problem I supply and demand
constraints
x11 x12 x13 300 x21 x22 x23
200 x31 x32 x33 200 x11 x21 x31
250 x12 x22 x32 250 x13 x23 x33
200 x11, x12, x13, x21, x22, x23, x31, x32,
x33 ? 0
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Origins
s1 s2 si sm

• We have a set of ORIGINs
• Origin Definition A source of material
• - A set of Manufacturing Plants
• - A set of Suppliers
• - A set of Warehouses
• - A set of Distribution Centers (DC)
• In general we refer to them as Origins

There are m origins i1,2, ., m Each origin i
has a supply of si
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Destinations
d1 d2 di dn
We have a set of DESTINATIONs Destination
Definition A location with a demand for
material - A set of Markets - A set of
Retailers - A set of Warehouses - A set of
Manufacturing plants In general we refer to them
as Destinations
There are n destinations j1,2, ., n Each
origin j has a supply of dj
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Transportation Model Assumptions

• Total supply is equal to total demand.
• There is only one route between each pair of
  origin and destination
• Items to be shipped are all the same
• for each and all units sent from origin i to
  destination j there is a shipping cost of Cij per
  unit

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Cij cost of sending one unit of product from
origin i to destination j
C11
1
1
C12
C21
2
C22
2
C2n
C1n
i
j
n
m
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Xij Units of product sent from origin i to
destination j
X11
1
1
X12
X21
2
2
X22
X2n
X1n
i
j
n
m
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The Problem
The problem is to determine how much material is
sent from each origin to each destination, such
that all demand is satisfied at the minimum
transportation cost
1
1
2
2
i
j
n
m
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The Objective Function
1
1
If we send Xij units from origin i to
destination j, its cost is Cij Xij We want to
minimize
2
2
i
j
n
m
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Transportation problem I decision variables
x11
200
1
1
x12
150
x13
x21
2
200
2
x22
250
x23
x31
x32
3
3
200
200
x33
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Transportation problem I supply and demand
constraints
x11 x12 x13
200
x21 x22 x23
200
x31
x32 x33 200 x11
x21 x31
150 x12
x22 x32
250 x13
x23
x33 200
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Transportation Problem Solution Algorithms

• Transportation Problem is a special case of LP
  models.
• Each variable xij appears only in rows i and
  mj. Furthermore, The coefficients of all
  variables are equal to 1 in all constraints.
• Based on these properties, special algorithms
  have been developed. They solve the
  transportation problem much faster than general
  LP Algorithms. They only apply addition and
  subtraction
• If all supply and demand values are integer,
  then the optimal values for the decision variable
  will also come out integer. In other words, we
  use linear programming based algorithms to solve
  an instance of integer programming problems.

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Data for the Transportation Model
Supply
Supply
Supply
Demand
Demand
Demand

• Quantity demanded at each destination
• Quantity supplied from each origin
• Cost between origin and destination

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Data for the Transportation Model
Supply Locations
Waxdale
Brampton
Seaford
Min.
Milw.
Chicago
Demand Locations
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Our Task
Our main task is to formulate the problem. By
problem formulation we mean to prepare a tabular
representation for this problem. Then we can
simply pass our formulation ( tabular
representation) to EXCEL. EXCEL will return
the optimal solution. What do we mean by
formulation?
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Supply
D -3
D -2
D -1
20
O -1
40
O -2
50
O -3
110
Demand
20
60
30
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Excel
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Excel
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Excel
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Excel
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Excel
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Excel
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Assignment Solve it using excel
We have 3 factories and 4 warehouses. Production
of factories are 100, 200, 150 respectively. Deman
d of warehouses are 80, 90, 120, 160
respectively. Transportation cost for each unit
of material from each origin to each destination
is given below. Destination 1 2 3 4
1 4 7 7 1 Origin 2 12 3 8 8 3 8 10 16 5
Formulate this problem as a transportation problem
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Excel Data