Intro to Transportation Problem

One of the key applications of linear

programming is in the area of shipping of goods and
services from a number of supply points known as Figure 33. Dummy Demand and Supply Illustration
"origins" or "sources" to several demand points
known as "receivers" or "destinations". The goal is An example is presented in tables 26 and
to select the best (usually least cost and/or time) 27. On the one hand, left side of the table shows
transportation routes between supply and demand the unbalanced transportation table. On the other
points in order to satisfy the required quantity of hand, right side of the table shows the converted
goods or services at each destination points , with balanced transportation table. This conversion is a
available quantity of goods or services at each prerequisite to solving transportation problems.
supply points. Another objective could be Notice that the cost of transportation of the dummy
maximizing profit contribution. demand or supply is zero.

The transportation algorithms help to Table 26. Unbalanced Transportation Problem -

minimize the total cost of transporting a Greater Supply
homogeneous commodity (product) from supply
centers to demand centers. One basic
transportation algorithm has been developed for
solving a transportation problem is known as
Stepping Stone Method.

Types of Transportation Problems

Transportation problems (TP) can be categorized

as follows:

(a) Balanced - a TP problem where demand units

equals to the supply units. Table 25 is an example.

Table 25. Balanced Transportation Problem

(b) Unbalanced - a TP problem where demand units

is not equal to the supply units. A "dummy" is used
to convert an unbalanced table to a balanced one. It
is something we pretend to exists, although in
reality it does not. There are two kinds of dummy
namely, dummy demand and dummy supply. Refer
to figure 33.
 Table 27. Unbalanced Transportation Problem - Table 28. Northwest Corner Rule Illustration
Greater Demand

Steps are enumerated below:

(1) Start by selecting the cell in the most

“Northwest” corner of the table. Warehouse 1A is
located at the northwest corner cell hence the first
allocation is placed there. We will assign the
maximum amount to this cell that is allowable
based on the requirements and the capacity
constraints, hence, satisfying first the demand of
the 1st project.
(2)

(a) Exhaust the capacity from each row before

moving down to another row. If allocation made in
Northwest Corner Rule Step 1 is equal to the supply available at first
warehouse (a1, in first row), then move vertically
There are several methods that can be used to
down to the cell (2, 1), i.e., second row and first
obtain an initial basic feasible solution. However, we
column. Apply Step 1 again, for next allocation.
will focus on the Northwest Corner Rule. This
method starts at the northwest-corner cell of the
table. Refer to table 28 for an illustration. The (b) Exhaust the requirement from each column
allocations (158, 16, 168, 36, and 143) pertains to before moving right to another column. If allocation
the stones while the unused cells pertains to made in Step 1 is equal to the demand of the first
the waters. receiving point (b1 in first column), then move
horizontally to the cell (1, 2), i.e., first row and
second column. Apply Step 1 again for next
allocation.

\
(c) If a1 = b1, allocate a1 or b1 and move diagonally
to the cell (2, 2).

(3) Continue the procedure step by step until an

allocation is made in the south-east corner cell of
the transportation table. Check to make sure that
the capacity and requirements are met. The number
of used cells must be equal to the number of rows plus
the number of columns minus one. In table 28 we
have 3 sources and 3 destinations so the equation Initial Transportation Cost Computation:
will be ((3+3)-1) resulting to 5 occupied cells.
Occupied
Computation Amount(P)
Cells
Northwest Corner Rule (Initial Transportation Cost) 1A 158 * 4 632
Problem: 2A 16*16 256
The PRS Cement Supply Company has received a 2B 168*24 4032
contract to supply cement to three new road
projects located at three various locations. Project 3B 36*16 576
A, B, and C need 174, 204, and 143 truckloads
respectively.
3C 143*24 3432
PRS has three cement warehouses located in three
various places. Warehouse 1, 2, and 3 have 158, Total P 8928
184, and 179 truckloads available respectively. Cost
of transportation from the warehouse to the Following our computation above, the initial
projects are: from warehouse 1 to Project A, B, C: transportation cost is P8928. The question is can
P4, P8, and P8 per truckload respectively. From we still reduce this? Let's try to solve.
warehouse 2: P16, P24, P16 and from warehouse 3:
P8, P16, and P24, respectively. The objective is to Transportation Problem Solving - Balanced
design a plan of distribution that will minimize the
We will be using the software in computing for
cost of transportation.
the minimum transportation cost in the problem of
Solution: PRS Cement Supply Company. You have to
understand the problem very well for you to draw
The problem can be drawn into a table as shown your correct initial transportation table. Let me
in table 28 (same table in previous page), let us refresh you with our initial transportation table
compute our initial transportation cost. with transportation cost of P8928.
Table 28. Northwest Corner Rule Illustration
Table 28. Northwest Corner Rule
Illustration

Please watch the video below for the process

of solving transportation problem.

Play media comment.

 Figure 34 shows the final transportation result Figure 35. Unbalanced Transportation Problem
of PRS Cement Supply Company. Encoding

Figure 36 shows the final transportation result

of the unbalanced problem using the software.

Figure 34. PRS Cement Supply Company Final

Result

Result of the software revealed 6008 final

transportation cost. To compute for the Figure 36. Unbalanced Transportation Problem Sample
transportation savings we have to deduct the final Result
transportation cost from the initial transportation
Result of the software revealed 2424 final
cost. That will be P8928 - P6008 = P2920.
transportation cost.
Transportation Problem Solving - Unbalanced
Transportation Problem Solving - Practice Problem
Try using the software in computing for the
LM Santos Electronics company
minimum transportation cost in the unbalanced
manufactures laptops at three sites: 1, 2 and 3 with
transportation problem presented in table 26. Let
100, 300, and 300 capacities respectively. It
me refresh you with our initial transportation table.
distributes through provincial warehouses located in
Luzon (L), Visayas (V), and Mindanao (M) with
requirements of 300, 200, and 200 respectively.
Shipping cost from each site to each warehouses is
presented in table 29.

Table 29. LM Santos Electronics company Shipping

Cost

Source: Table 26. Unbalanced Transportation Problem

- Greater Supply

You have to encode the transportation table

which includes the dummy demand (or supply).
Please refer to Figure 35.

Let's understand the problem. Try to create

the initial transportation table before clicking on the
next page.
Transportation Problem Solving - Practice Problem Initial Transportation Cost Computation:
Solution

For the initial transportation table, we Occupied Cells Computation Amount(P)

have to place the sites (sources) as the row header
and the warehouse (destinations) as the column
1L 100 * 5 500
header. Then indicate the total supply and demand
as shown in table 30.
2L 200*8 1600
Table 30. LM Santos Electronics company Problem
Understanding
2V 100*4 400

3V 100*7 700

3M 200*5 1000

Total P 4200

Figure 39 shows the final transportation table with

a shipping list using the software.
Start by encoding your created
transportation table. Please refer to Figure 37.

Figure 37. LM Santos Electronics company Initial

Transportation table encoding

Figure 38 shows the initial transportation table

of the problem using the software looking at the
Iterations.
Figure 39. Final Transportation Table Result for LM
Santos Electronics company

The result of the software revealed 3900 final

transportation cost. Savings is 4200 - 3900 = 300.

Lesson Synthesis / Generalization

The transportation problem is teaching us how

to bring products from sources to destinations that
will result to the minimum cost. This is very
Figure 38. Initial Transportation Table from promising for businesses considering the high
Iterations LM Santos Electronics company amount of fuel in the country. It really matters
gaining knowledge on this concept to help achieve
company goals.