QUANTITATIVE ANALYSIS
FOR
MANAGERS
TRANSPORTATION MODEL
Learning Objectives
When you complete this chapter, you should
be able
• to identify or define:
– Transportation modelling
• to explain or use:
–
–
–
–
Northwest-corner method
Least Cost method
Vogel’s Approximation method
Stepping-stone method
Outline
• Transportation Modelling
• Developing an Initial Solution
– The Northwest-Corner Method
– The Least-Cost Method
– The Vogel’s Approximation Method
• The Stepping-Stone Method
• Special issues in modelling
- Demand Not Equal to Supply
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
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
Transportation Problem Solutions
steps
• Define problem
• Set up transportation table (matrix)
– Summarizes all data
– Keeps track of computations
• Develop initial solution
– Northwest corner Method
– Vogel’s Approximation Method
• Find optimal solution
– Stepping stone method
Methods for finding Initial Solution
1.
North-West Corner Method (NWCM)
• Begin in the upper-left-hand corner of the
transportation table for a shipment and allocate
as many units as possible equal to minimum
between available capacity and requirement; i.e.
min (a1, b1).
• Allocate the maximum that is possible, min (100,
90) = 90. Now move horizontally to the second
column in the first row.
• Repeat the above steps
Example 1:
A company has three factories F1, F2 and F3 with production
capacity 100, 250 and 150 units per week respectively.
These units are to be shipped to four warehouses W1, W2,
W3 and W4 with requirement of 90, 160, 200 and 50 units
per week respectively. The transportation costs (in Rs) per
unit between factories and warehouses are given as follows:
Warehouse
F
a
c
t
o
r
y
W1
W2
W3
W4
Capacity
F1
30
25
40
20
100
F2
29
26
35
40
250
F3
31
33
37
30
150
Requirement
90
160
200
50
The total cost of transportation is obtained as
follows:
Route
From
F1
F1
F2
F2
F3
F3
To
Units
per unit
shipped X cost (Rs) = Total Cost
W1
W2
W2
W3
W3
F4
Total =
2.
The Least 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.
LCM
Total Transportation cost = Rs 15, 020
Condition:
Occupied shipping routes = (no. of rows +
no. of columns) – 1
= 3 + 4 –1
=6
This cost is less than the cost determined by
NWCM. Therefore, this method is preferred over
the NWCM.
3.
Vogel’s Approximation Method (VAM)
This method is preferred over the other two methods
because the initial basic feasible solution obtained is
either optimal or very close to the optimal solution.
• For each row and column, find the difference
between the two lowest unit shipping costs.
• Identify the row or column with the greatest
opportunity cost or difference.
• Assign as many units as possible to the lowestcost square in the row or in the column selected.
W1
W2
W3
W4
F1
30
25
40
20
100
F2
29
26
35
40
250
F3
31
33
37
30
150
90
160
200
50
500
Requirement
Capacity
Optimality Test
Stepping-Stone Method
This method starts with an evaluation of
each of the unoccupied cells to decide
whether it would be economical to introduce
any of these cells into the current solution.
The Stepping Stone Method
• Apply any of the three methods to obtain the
initial basic feasible solution
• Select any unused cell to be evaluated
• Begin at this cell. Trace a closed path back to the
original cell via cells that are currently being used
(only horizontal or vertical moves allowed)
• Place + in unused cell; alternate - and + on each
corner cell of the closed path
• Calculate improvement index: add together the
unit cost figures found in each cell containing a +;
subtract the unit cost figure in each cell
containing a -.
• Repeat steps 1-4 for each unused square
Stepping Stone Method contd.
• Check the sign of each of the net change in the unit
transportation costs. If all net changes are plus (+) or
zero, then an optimal solution has been achieved,
otherwise go to next step
• Select the unoccupied cell with most negative net change
among all unoccupied cells. If two minus values are
equal, select that one which will result in moving as many
units as possible into the selected unoccupied cell with
the minimum cost
• Assign the maximum unit that can be shipped on the new
route. This is done by looking at the closed paths (-) sign
and we select the smallest number found in the cells with
(-) signs and make the transfer
Example 2:
A company is spending Rs1000 on transportation of its
units from three plants to four distribution centres. The
availabilities and requirements of units with units cost of
transportations are given as:
Distribution Centres
D1
D2
D3
D4
Availabilities
P1
19
30
50
12
7
P2
70
30
40
60
10
P3
40
10
60
20
18
Requirements
5
8
7
15
What can be the maximum saving for the company by
optimum distribution?
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