Transcript Chapter 1, Heizer/Render, 5th edition

Operations
Management
Transportation Models
Quantitative Methods for Managerial Decision-Making
ACN 309-5
Outline
 TRANSPORTATION MODELING
 DEVELOPING AN INITIAL SOLUTION
The Northwest-Corner Rule
 The Intuitive Lowest-Cost Method

 THE STEPPING-STONE METHOD
 SPECIAL ISSUES IN MODELING
Demand Not Equal to Supply
 Degeneracy

Quantitative Methods for Managerial Decision-Making
ACN 309-5
Learning Objectives
After you read these notes, you should be able to
Identify or Define:
Transportation modeling
 Facility location analysis

Explain or be able to use:
Northwest-corner rule
 Stepping-stone method

Quantitative Methods for Managerial Decision-Making
ACN 309-5
Transportation Problem
DesMoines
(100 unit
capacity)
Albuquerque
(300 units
required)
Cleveland
(200 units required)
Boston
(200 units
required)
Evansville
(300 units
capacity)
Fort Lauderdale
(300 units capacity)
Quantitative Methods for Managerial Decision-Making
ACN 309-5
Transportation Problem
 How much should be shipped from several sources
to several destinations


Sources: Factories, warehouses, etc.
Destinations: Warehouses, stores, etc.
 Transportation models


Find lowest cost shipping arrangement
Used primarily for existing distribution systems
Quantitative Methods for Managerial Decision-Making
ACN 309-5
A Transportation Model Requires
 The origin points, and the capacity or supply per
period at each
 The destination points and the demand per period
at each
 The cost of shipping one unit from each origin to
each destination
Quantitative Methods for Managerial Decision-Making
ACN 309-5
Transportation Problem
Solution Steps
 Define problem
 Set up transportation table (matrix)


Summarizes all data
Keeps track of computations
 Develop initial solution

Northwest corner rule
 Find optimal solution

Stepping stone method
Quantitative Methods for Managerial Decision-Making
ACN 309-5
Transportation Costs
From
(Sources)
Albuquerque
To
(Destination)
Boston
Cleveland
Des Moines
$5
$4
$3
Evansville
$8
$4
$3
Fort
Lauderdale
$9
$7
$5
Quantitative Methods for Managerial Decision-Making
ACN 309-5
Transportation Table
Destination
Source
1
2
..
n
Supply
1
a1
2
a2
Quantity demanded or
required
:
:
m
Demand
am
b1
Quantitative Methods for Managerial Decision-Making
ACN 309-5
b2
bn
Transportation Table
Source
Cost of supplying
Destination
1 unit from sources to
2 destinations
..
n
1
Supply
1
x 11 c 11 x 12 c 12 . .
x 1n c 1n
a1
2
x 21 c 21 x 22 c 22 . .
x 2n c 2n
a2
:
m
Demand
:
:
:
:
:
x m1 c m1 x m2 c m2 . .
b1
Quantitative Methods for Managerial Decision-Making
ACN 309-5
b2
..
:
:
:
x mn c mn
bn
:
am
Transportation Table
Destination
1
2
1
x 11
x 12
2
x 21
.
from
x 22Quantity .supplied
x 2n
Source
sources to
: :
: destinations
:
:
:
m
x m1
x m2
b1
b2
Demand
..
..
:
Quantitative Methods for Managerial Decision-Making
ACN 309-5
..
n
Supply
x 1n
:
x mn
bn
a1
a2
:
:
am
Transportation Table
To Albuquerque
From
(A)
5
Des Moines
(D)
8
Evansville
(E)
9
Fort Lauderdale
(F)
Warehouse
300
Requirements
Quantitative Methods for Managerial Decision-Making
ACN 309-5
Boston
(B)
200
Cleveland
(C)
4
3
4
3
7
5
200
Factory
Capacity
100
300
300
700
Initial Solution Using the
Northwest Corner Rule
From
To Albuquerque
(A)
Des Moines
(D)
Evansville
(E)
100
200
Fort Lauderdale
(F)
Warehouse
Requirements
Quantitative Methods for Managerial Decision-Making
ACN 309-5
Cleveland
(C)
5
4
3
8
4
3
7
5
9
300
Boston
(B)
100
100
200
200
200
Factory
Capacity
100
300
300
700
The Stepping Stone Method
Select any unused square to evaluate
Begin at this square. Trace a closed path back
to the original square via squares that are
currently being used (only horizontal or vertical
moves allowed)
Place + in unused square; alternate - and + on
each corner square of the closed path
Calculate improvement index: add together the
unit cost figures found in each square containing
a +; subtract the unit cost figure in each square
containing a -.
Repeat steps 1-4 for each unused square
Quantitative Methods for Managerial Decision-Making
ACN 309-5
Stepping-Stone Method: Tracing a
Closed Path - Des Moines to Cleveland
From
To Albuquerque
(A)
Des Moines
(D)
Evansville
(E)
Fort Lauderdale
(F)
Warehouse
Requirements
+
100
200
Boston
(B)
5
4
3
+
8
-100
+
Quantitative Methods for Managerial Decision-Making
ACN 309-5
Factory
Capacity
Start
9
300
Cleveland
(C)
100
200
4
3
7
5
-200
200
100
300
300
700
The Intuitive Lowest Cost Method
Identify the cell with the lowest cost. Arbitrarily
break any ties for the lowest cost.
Allocate as many units as possible to that cell
without exceeding the supply or demand. Then
cross out that row or column (or both) that is
exhausted by this assignment.
Find the cell with the lowest cost from the
remaining cells.
Repeat steps 2 & 3 until all units have been
allocated.
Quantitative Methods for Managerial Decision-Making
ACN 309-5
Initial Solution Using the Intuitive LowestCost MethodSecond,
cross out
column C
From
To Albuquerque
(A)
Boston
(B)
Cleveland Factory
(C)
Capacity
Des Moines
(D)
Evansville
(E)
5
4
8
4
Fort Lauderdale
(F)
9
Warehouse
Requirements
200
100
3
100
3
7
5
300
300
Quantitative Methods for Managerial Decision-Making
ACN 309-5
200
200
100
First, cross
out top row
300
Third, cross
out row E
300
700
Specialized Methods
 Linear programming model
is difficult to formulate &
solve
 Special purpose methods



Are easier to formulate
Are faster to compute
Give integer solutions
 Methods



Stepping-stone
MODI
See your CD Tutorial
Quantitative Methods for Managerial Decision-Making
ACN 309-5
© 1995 Corel Corp.
Special Issues in the
Transportation Model
 Demand not equal to supply



Called ‘unbalanced’ problem
Add dummy source if demand > supply
Add dummy destination if supply > demand
 Degeneracy in Stepping Stone Method

Too few shipping routes (cells) used


Number of occupied cells should be: m + n - 1
Create artificially occupied cell (0 value)

Represents fake shipment
Quantitative Methods for Managerial Decision-Making
ACN 309-5
Transportation Table
Demand Not Equal Supply
To Albuquerque
From
(A)
5
Des Moines
(D)
8
Evansville
(E)
9
Fort Lauderdale
(F)
Warehouse
300
Requirements
Boston
(B)
200
Cleveland Dummy Factory
(C)
Capacity
4
3
0
250
4
3
0
300
7
5
0
200
150
New Des Moines capacity
Quantitative Methods for Managerial Decision-Making
ACN 309-5
300
700
Degeneracy
From
To Albuquerque
(A)
Des Moines
(D)
Evansville
(E)
100
200
Fort Lauderdale
(F)
Warehouse
Requirements
Boston
(B)
5
4
3
8
4
3
7
5
100
9
300
Quantitative Methods for Managerial Decision-Making
ACN 309-5
Cleveland
(C)
100
200
200
Factory
Capacity
100
300
200
700
Degeneracy - Continued
From
To Albuquerque
(A)
Des Moines
(D)
Evansville
(E)
100
200
Fort Lauderdale
(F)
Warehouse
Requirements
Boston
(B)
5
4
8
4
3
7
5
100
9
300
Quantitative Methods for Managerial Decision-Making
ACN 309-5
Cleveland
(C)
100
0
200
200
3
Factory
Capacity
100
300
200
700