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Location decisions are strategic decisions.
• The reasons for location decisions
• Growth
– Expand existing facilities– Add new facilities
• Production Cost
• Depletion of Resources
Nature of Location Decisions
Multiple Plant Strategies
Community FactorsRegional Factors
Factors Affecting Plant location
Site Factors
1-Location of raw materialRaw material oriented factories; weight of input >>> weight of output
Iron ore Iron & SteelCoal
Alumina AluminumCokeElectricity
Steelplant
Aluminumplant
Regional; Raw Material
These types of plants tends to be closer to the raw material resources.
Indeed row material or any other important input.
Regional; Raw Material
2-Location of marketMarket oriented plants;
Space required for output >>> space required for input.
Car manufacturing, Appliances
Regional; Market
3-Labor, water, ElectricityAvailability 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.
Regional; Labor, Water, Electricity
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
Community
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
Site
Decentralization
Small is beautiful; Instead of a single huge plant in one location, several smaller plants in different locations
Decentralization based on product
Decentralization based on geographical area
Decentralization based on process
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.
Decentralization Based on Geographical Area
Each plant is responsible for a geographical region,
Specially for heavy or large products.
Lower transportation costs.
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
• Foreign producers locating in U.S.– “Made in USA”
– Currency fluctuations
• Just-in-time manufacturing techniques
• Focused factories
• Information highway
Trends in Global Locations
• Cost-volume Analysis
– Determine fixed and variable costs
– Plot total costs
– Determine lowest total costs
BEP in Location Analysis
Fixed and variable costs for four potential locations
Location FixedCost
VariableCost
ABCD
$250,000100,000150,000200,000
$11302035
Example
F i x e dC o s t s
V a r i a b l eC o s t s
T o t a lC o s t s
ABCD
$ 2 5 0 , 0 0 01 0 0 , 0 0 01 5 0 , 0 0 02 0 0 , 0 0 0
$ 1 1 ( 1 0 , 0 0 0 )3 0 ( 1 0 , 0 0 0 )2 0 ( 1 0 , 0 0 0 )3 5 ( 1 0 , 0 0 0 )
$ 3 6 0 , 0 0 04 0 0 , 0 0 03 5 0 , 0 0 05 5 0 , 0 0 0
Solution
800700600500400300200100
0
Annual Output (000)
8 10 12 14 166420
$(000)
A
BC
B SuperiorC Superior
A Superior
D
Graphical Solution
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
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 ora Distribution Center (DC) ora Warehouse to satisfy the demand of the demand centersorThere 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
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 whendistances are rectilinear.
Optimal Single Facility Location
The coordinates of the optimal location of the DC is
n
YY
n
XX
i
i
Example
We have 4 demand points.Demand of all demand points are equal.Demand points are located at the following locations
Example
Where is the optimal location for the center serving theses demand points
(2,2)
(8,5)
(5,4)
(3,5)
Solution
Where is the optimal location for the center serving theses demand points (2,2) (8,5)(5,4)(3,5)
44
5452
5.44
8532
n
YY
n
XX
i
i
Solution
The optimal location for the center serving theses demand points
(2,2)
(8,5)
(5,4)
(3,5)
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.
Optimal Single Facility Location
The coordinates of the optimal location of the DC is
i
ii
i
ii
Q
YQY
Q
XQX
Example
Where is the optimal location for the center serving theses demand points
(2,2)
(8,5)
(5,4)
(3,5)
800
900 100
200
Solution
Where is the optimal location for the center serving theses demand points
05.3100200900800
8)100(5)200(3)900(2)800(
X
Q
XQX
i
ii
800 : (2,2) 900 : (3,5)
100 : (8,5)
200 : (5,4)
Solution
Where is the optimal Y location for the center serving theses demand points
7.3100200900800
5)100(4)200(5)900(2)800(
Y
Q
YQY
i
ii
800 : (2,2) 900 : (3,5)
100 : (8,5)
200 : (5,4)
Solution
The optimal location for the center serving theses demand points
(800)
(100)
(200)
(900)
Example
Where is the optimal location for the center serving theses demand points
(0,0)
(6,3)
(3,2)
(1,3)
800
900 100
200
Solution
Where is the optimal location for the center serving theses demand points
05.1X100200900800
)6)(100()3)(200()1)(900()0)(800(
Q
XQX
i
ii
800 : (0,0) 900 : (1,3) 100 : (6,3) 200 : (3,2)
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)
7.1Y100200900800
)3)(100()2)(200()3)(900()0)(800(
Q
YQY
i
ii
Solution
The optimal location for the center serving theses demand points is at the same location
(800)
(100)
(200)
(900)
The Transportation Problem
D(demand)
D(demand)
D(demand)
S(supply)
S(supply)
S(supply)
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.
Transportation costs for each unit of product is given below
Transportation problem : Narrative representation
Warehouse1 2 3
1 16 18 11Plant 2 14 12 13
3 13 15 17
Formulate this problem as an LP to satisfy demand at minimum transportation costs.
Transportation problem I : decision variables
1
2
1
33
300x11
x12
2200
200 200
250
250
x13x21
x31
x22
x32
x23
x33
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
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
Origins
We have a set of ORIGINsOrigin 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
m
1
2
i
s1
s2
si
sm
There are m origins i=1,2, ………., m
Each origin i has a supply of si
Destinations
We have a set of DESTINATIONsDestination Definition: A location with a demand for material- A set of Markets- A set of Retailers- A set of Warehouses- A set of Manufacturing plantsIn general we refer to them as Destinations
n
1
2
j
d1
d2
di
dn
There are n destinations j=1,2, ………., n
Each origin j has a supply of dj
• 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
Transportation Model Assumptions
Cij : cost of sending one unit of product from origin i to destination j
m
1
2
i
n
1
2
jC1n
C12
C11
C2n
C22
C21
Xij : Units of product sent from origin i to destination j
m
1
2
i
n
1
2
jX1n
X12
X11
X2n
X22
X21
The Problem
m
1
2
i
n
1
2
j
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
The Objective Function
m
1
2
i
n
1
2
j
If we send Xij units
from origin i to destination j,
its cost is Cij Xij
We want to minimize
ijijXCZ
Transportation problem I : decision variables
1
2
1
33
200x11
x12
2200
200 200
250
150
x13x21
x31
x22
x32
x23
x33
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
Transportation Problem is a special case of LP models.
Each variable xij appears only in rows i and m+j. 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.
Transportation Problem Solution Algorithms
Supply
Demand
Supply Supply
Demand Demand
Data for the Transportation Model
• Quantity demanded at each destination
• Quantity supplied from each origin
• Cost between origin and destination
$600
$400
$300
$200
Waxdale Brampton Seaford
Min. Milw. Chicago
$700 $900$100
$700
$800
Supply Locations
Demand Locations
20 40 50
Data for the Transportation Model
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?
SupplyD -3D -2D -1
O -1
O -2
O -3
Demand 30 20 60
20
40
50
600 400 300
700 200 900
800 700 100
110
Excel
Excel
Excel
Excel
Excel
Excel
Assignment; Solve it using excel
We have 3 factories and 4 warehouses.Production of factories are 100, 200, 150 respectively.Demand of warehouses are 80, 90, 120, 160 respectively.Transportation cost for each unit of material from each origin to each destination is given below.
Destination1 2 3 4
1 4 7 7 1Origin 2 12 3 8 8
3 8 10 16 5
Formulate this problem as a transportation problem
Excel : Data