The Hindu Business Line : A probe into `degeneracy'

ANY student appearing for the CA examination would normally expect the questions to be set from the study material and, at the same time, forming part of the syllabus.

However, it would be unfair to the students if advanced level of understanding is required to tackle the questions.

This exactly was the situation faced by the students who appeared for the November 2005 cost management examination. The question that put them off was the problem on linear programming.

Before examining the problem, it will be necessary to look at the scope of the operations research subject in the CA curriculum.

Operations research was first introduced in 1973 and was offered as an optional subject carrying 100 marks. The first examination was conducted in November 1976. Till November 1995 it was a separate paper carrying 100 marks and was part of Combination B, along with systems analysis and data processing, and cost accounting subjects. Most students opted for this combination since it was easy to score vis-à-vis the other two combinations on offer.

From May 1996 till November 2002, operations research, with the modified syllabus, was combined with systems analysis and data processing and was a compulsory paper in Group II with the portions carrying 50 marks. Under the current syllabus (November 2002 onwards), it is a part of cost management with the portions being restricted to six topics and constituting 30 per cent of the paper.

One of the topics, namely, linear programming, is a favourite of most paper-setters. An analysis of the past question papers from November 1976 till date reveals that questions on linear programming have appeared in almost all the examinations. Till May 1995, the focus was on solving the problem after formulating the same. Though the method was not specified, except on a few occasions, the answers provided by the ICAI were mostly through the simplex method.

From May 1996 till November 2002, the requirement was only to formulate the problem, and solving the same, on a few occasions, using the graphical method. Students were never tested on the simplex method. However, with the introduction of the current syllabus, questions based on the simplex method appeared in the November 2003 as well as in the recently concluded November 2005 exam, wherein the following problem was asked:

Three grades of coal, A, B and C, contain phosphorus and ash as impurities.

In a particular industrial process, fuel up to 100 tonnes (maximum) is required which could contain ash not more than 3 per cent and phosphorus not more than 0.03 per cent. It is desired to maximise the profit while satisfying these conditions. There is an unlimited supply of each grade. The percentage of impurities and the profits of each grades are as shown in Table 1.

Formulate the linear programming (LP) model and solve it using the simplex method to determine optimal product mix and profit.

Observations

Compared to the problems asked so far, this is the first on degeneracy, a situation wherein one or more of the basic variables become zero at some stage of the calculations. However, in the given problem, it can be noticed that "zero" finds a place in the initial iteration itself. Further, the coefficients of the variables are both positive and negative.

On preparation of the initial simplex table, the ratios that are computed for replacing any one of the rows are found to be 0, 0 and 100. As per the study material /textbooks on the subject (including advanced text books), the student is taught to choose that ratio which is positive and also least among all the ratios. A ratio with zero is not to be considered.

Hence, most of the students would have chosen the ratio of 100 and would have replaced that row accordingly.

In that case, the subsequent iteration would look like an optimal solution but it is not so since it does not satisfy the requirement of non-negativity, that is, variable S2 has a negative quantity (Table 2). This point would have confused the students.

If, on the other hand, if a student proceeds further by choosing any one of the other two ratios consisting of zero he would have noticed that iterations 1 and 2 result in the same Z value, that is, zero. At this point, the student would still be clueless since as per the simplex method every subsequent iteration should result in an improved Z value. A situation of this kind clearly indicates the point of degeneracy.

However, if a student were to go beyond the second iteration he would have found the optimal solution in the fourth iteration (Table 2).

Considering all the questions on simplex method from 1976 onwards, the latest one does not seem justified. Some of the limitations in the questions are hard to comprehend.

In solving the problem, the student would have found it quite difficult to compute the required four iterations to arrive at the optimal solution. In the process, the situation of degeneracy would have had to be resolved.

On a perusal of the study material, it can be noticed that degeneracy has not been discussed, though it was included in the earlier materials.

One is not able to understand the intention behind asking such a question. One may argue that in a professional examination exceptions do form a part. However, from the students' standpoint, expectation that a problem on linear programming should be more applications oriented with reference to the subject of cost management is more justified.

How many of the students would have attempted and done the problem correctly is for the ICAI to find out. And, perhaps, it is its responsibility to apply Clause 39(2) of the Regulation in the interest of the institution.

Formulation: Let x1, x2 and x3 be the tonnes of grades A, B and C coal respectively.

Objective function - Maximize Z=12X1+15X2+14X3

Subject to: 1) -X2+2X3 =0, 2) -X1+X2=0, 3) X1+X2+X3=100 (X1, X2, X3 {gt}=0)