Location decisions are strategic decisions. The reasons for location decisions Growth –Expand existing facilities –Add new facilities Production Cost.

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


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)


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


(2,2)

(8,5)

(5,4)

(3,5)

800

900 100

200

Solution


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


(0,0)

(6,3)

(3,2)

(1,3)

800

900 100

200

Solution


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


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


x11 = Volume of product sent from P1 to W1









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


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

Location decisions are strategic decisions. The reasons for location decisions Growth –Expand existing facilities –Add new facilities Production Cost.

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