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Alternative solution to an assignment problem
**

*R. Sivakumar*

The CA Final examination on cost management, which was held on November
6, 2007, was once again based on the previous pattern, that is, as
existed prior to November 2006. Unlike the previous two exams, that is,
in November 2006 and May 2007, wherein the mix was almost in line with
the expected allocation of marks between cost accounting and operations
research, the focus this time around was more on cost accounting.

The overall ratio between cost accounting and operations research
was found to be 78:22, as against the promised ratio of 70:30 the ICAI
(Institute of Chartered Accountants of India). To this extent the ICAI
has not been consistent, but considering the overall nature of the
questions, students would have been very satisfied as none of them was
tough. In fact the questions were a straight lift from the study
material and past examinations papers. Any student who had prepared on
these lines should get at least 60 per cent. Overall, the paper does
not conform to the standard befitting a Final examination. What follows
is a detailed analysis of the questions.

The allocation of marks among the topics is presented in Tables 1 and 2.

Alternative solutions

Question 4(a) on assignment was as follows: A company has four zones
open, and four marketing managers available for assignment. The zones
are not equal in their sales potentials. It is estimated that typical
marketing manager operating in each zone would bring in the following
annual sales (Table 3). The four marketing managers are also different
in ability. It is estimated that, working under the same conditions,
their yearly sales would be proportionately as shown in Table 4.

*Required:* If the criterion is maximum expected total sales, find the optimum assignment and the maximum sales.

*Alternative 1 — intuitive method:* The above question
on assignment in an interesting one and a similar problem finds a place
in the study material. The same has been solved in the usual manner by
applying the assignment technique and the solution is obtained in six
iterations.

This is one of the classic problems on the topic and the simpler way
of solving the same is by the intuitive method. That is, assign the
best marketing manager to the richest zone and the next best marketing
manager to the second best zone and so on — Zone East to Manager M,
Zone West to Manager N, and so on.

This is so because of the specific property: “If the assignment is
on a proportion and if it is noticed that the elements are on a
descending order the maximum occurs in the diagonal from the North-west
corner and the minimum occurs from North-east corner.” For the problem
at hand, the initial matrix is as shown in Table 5. The final
assignment by the usual procedure will be as shown in Table 6. From
this it can be seen that the solution lies in the diagonal elements
from the North-west corner which exactly matches with the property
stated above and also the solution through the intuitive method.

*Alternative 2 — application of transportation technique:*
Yet another alternative is to consider this as a transportation
problem, wherein the very first iteration will result in the optimum
assignment, with degeneracy.

Under Vogel’s Approximation Method the first allocation will be to
Manager M to Zone East who has the highest difference and then to
Manager N to Zone West who has the second highest difference, and so
on.

Any assignment problem solved by transportation technique is always
a degenerate case and the optimal solution will always be with
degeneracy. In the above case, the number of allocation falls short of
the total allocation of 7 and the same is resolved by placing the zero
allocation in the cells having the sales value of 70, 40 and 24
respectively or 64, 42 and 25 respectively. The optimal solution
however will be the same as computed through the assignment technique.

To conclude, if the elements of the matrix conform to the property
specified, the optimal solution will always be the diagonal elements
and interestingly in these kinds of problems application of North-west
corner rule will also lead to the same allocation.