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Chapter 8

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Chapter 8

Location

Dr. Ardavan Asef-Vziri Operations and Supply Chain Management March 2001

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

Factors Affecting Plant location

Regional Factors

Community Factors

Site Factors

Multiple Plant Strategies

Regional; Raw Material

1-Location of raw material

Raw material oriented factories;

weight of input >>> weight of output

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; Market

2-Location of market

Market oriented plants;

Space required for output >>>

space required for input.

Car manufacturing, Appliances

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.

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

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

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

Trends in Global Locations

- Foreign producers locating in U.S.
- “Made in USA”
- Currency fluctuations

- Just-in-time manufacturing techniques
- Focused factories
- Information highway

BEP in Location Analysis

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

Example

Fixed and variable costs for four potential locations

Solution

Graphical Solution

$(000)

800

700

600

500

400

300

200

100

0

D

B

C

A

A Superior

C Superior

B Superior

0

2

4

6

8

10

12

14

16

Annual Output (000)

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 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 forour 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 when

distances are rectilinear.

Optimal Single Facility Location

The coordinates of the optimal location of the DC is

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

(3,5)

(8,5)

(5,4)

(2,2)

Solution

Where is the optimal location for the center serving

theses demand points

(2,2)

(3,5)

(5,4)

(8,5)

Solution

The optimal location for the center serving

theses demand points

(3,5)

(8,5)

(5,4)

(2,2)

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

Example

Where is the optimal location for the center serving

theses demand points

900

100

(3,5)

(8,5)

200

(5,4)

800

(2,2)

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)

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)

Solution

The optimal location for the center serving

theses demand points

(900)

(100)

(200)

(800)

Example

900

100

(1,3)

(6,3)

200

(3,2)

800

(0,0)

Where is the optimal location for the center serving

theses demand points

Solution

Where is the optimal location for the center serving

theses demand points

800 : (0,0)

900 : (1,3)

200 : (3,2)

100 : (6,3)

Solution

Where is the optimal location for the center serving

theses demand points

800 : (0,0)

900 : (1,3)

200 : (3,2)

100 : (6,3)

Solution

The optimal location for the center serving

theses demand points is at the same location

(900)

(100)

(200)

(800)

The Transportation

Problem

Dr. Ardavan Asef-Vaziri Industrial Engineering Mar 2003

The Transportation Problem

D

(demand)

S

(supply)

S

(supply)

D

(demand)

D

(demand)

S

(supply)

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.

Transportation costs for each unit of product is given below

Warehouse

123

1161811

Plant 2141213

3131517

Formulate this problem as an LP to satisfy demand at minimum

transportation costs.

Transportation problem I : decision variables

x11

300

1

x12

1

250

x13

x21

200

2

x22

2

250

x23

x31

x32

3

200

3

x33

200

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

1

2

i

m

Origins

s1

s2

si

sm

- We have a set of ORIGINs
- Origin Definition: A source of material
- - A set of ManufacturingPlants
- - 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 i=1,2, ………., m

Each origin i has a supply of si

1

2

j

n

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 j=1,2, ………., n

Each origin j has a supply of dj

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

Cij: cost of sending one unit of product from origin i to destination j

C11

1

1

C21

C12

2

C22

2

C2n

C1n

i

j

n

m

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

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

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

Transportation problem I : decision variables

x11

200

1

x12

1

150

x13

x21

200

2

x22

2

250

x23

x31

x32

3

200

3

x33

200

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 Solution Algorithms

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.

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

Data for the Transportation Model

20

40

50

$300

$800

$600

$400

$700

$100

$700

$900

$200

30

20

60

Supply Locations

Waxdale

Brampton

Seaford

Min.

Milw.

Chicago

Demand Locations

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?

600

400

300

700

200

900

800

700

100

Supply

D -1

D -2

D -3

20

O -1

40

O -2

50

O -3

110

Demand

20

60

30

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.

Destination

1234

14771

Origin 212388

3810165

Formulate this problem as a transportation problem

Excel : Data