VARIATION OF TRANSPORTATION PROBLEMS
The objective is to maximize profit (Case Study Nifty Co.)
The Nifty Company specializes in the production of a single product, which it produces in
three plants. The product is doing very well, so the company currently is receiving more purchase
requests than it can fill. Plans have been made to open an additional plant, but it will not be ready
until next year.
For the coming month, four potential customers (wholesalers) in different parts of the
country would like to make major purchases. Customer 1 is the company’s best customer, so his
full order will be met. Customers 2 and 3 also are valued customers, so the marketing manager has
decided that, at a minimum, at least a third of their order quantities should be met. However, she
does not feel that Customer 4 warrants special consideration, and so is unwilling to guarantee any
minimum amount for this customer.
There will be enough units produced to go somewhat above these minimum amounts. Due
largely to substantial variations in shipping costs, the net profit that would be earned on each unit
sold varies greatly, depending on which plant is supplying which customer. Therefore, the final
decision on how much to send to each customer (above the minimum amounts established by the
marketing manager) will be based on maximizing profit.
The unit profit for each combination of a plant supplying a customer is shown in Table 3.4.
The rightmost column gives the number of units that each plant will produce for the coming month
(a total of 20,000). The bottom row shows the order quantities that have been requested by the
customers (a total of 30,000). The next-to-last row gives the minimum amounts that will be
provided (a total of 12,000), based on the marketing manager’s decisions described above.
The marketing manager needs to determine how many units to sell to each customer
(observing these minimum amounts) and how many units to ship from each plant to each customer
to maximize profit.
This problem is almost a transportation problem, since the plants can be viewed as sources
and the customers as destinations, where the production quantities are the supplies from the sources.
If this were fully a transportation problem, the purchase quantities would be the demands
for the destinations. However, this does not work here because the requirements assumption says
that the demand must be a fixed quantity to be received from the sources. Except for Customer 1,
all we have here are ranges for the purchase quantities between the minimum and the maximum
given in the last two rows of Table 3.4 below. In fact, one objective is to solve for the most
desirable values of these purchase quantities.
The marketing manager needs to determine how many units to sell to each customer
(observing these minimum amounts) and how many units to ship from each plant to each customer
to maximize profit.
Table 3.4 – Data for the Nifty Co. Problem
Unit profit
Customer
1
2
3
4
Production quantity
Plant
1
55
42
46
53
8,000
2
37
18
32
48
5,000
3
29
59
51
35
7,000
Minimum purchase 7,000 3,000 2,000
0
Requested purchase 7,000 9,000 6,000 8,000