The Hindu Business Line : Application of linear programming to blending problems

BLENDING is a frequently encountered problem. Typically different commodities are to be purchased, each having known characteristics and costs. The problem is to give a recipe showing how much of each commodity should be purchased and blended with the rest so that the characteristics of the mixture lie within specified bounds and the total purchase cost is minimised. Linear Programming offers a natural technique for optimising blending operations provided that the various quantities of interest blend linearly. It is by no means true that all physical quantities blend linearly and hence if linear programming is applied in a case of non-linear blending, it will yield only an approximate answer.

In many instances, however, the assumption on linearity has given answers which were good enough to make linear programming a very useful tool to study blending operations.

In general, blending problems refer to situations where a number of components are mixed together to yield one or more products. Usually, there are restrictions on the availability of raw materials, restrictions on the quality of the products, and perhaps restrictions on quantities of the products to be produced. However, there are usually many different ways in which the raw materials can be blended to form the final product while satisfying the various constraints. It is desired to carry out the blending operation so that some given objective function is optimised.

For example with respect to motor fuels, blending is a very important problem because a small percentage improvement in operations could mean crores of rupees to oil companies. Using linear programming many oil companies have saved substantial amounts of money. One oil company has worked with a gasoline blending problem involving eighty constraints. Any motor fuel must satisfy a number of quality requirements. Impurities, such as sulphur, must not exceed a given concentration, the vapour pressure must be within a certain range, and the octane number must be equal or exceed some minimum value.

In the recent Examination on Cost Management conducted by the ICAI (November 2005) the questions on Linear Programming related to blending. The formulation as well as the solution through Simplex Method has already been discussed (Business Line, December 19). However, a close look at the problem from the view of point of the concept of blending reveals certain limitations. Basically, the problem was a modification of a minimisation problem usually found in standard textbooks on OR (Organisation Research). The same is given below.

Three grades of coal A, B and C contain ash phosphorus as impurities. In a particular industrial process, fuel obtained by blending the above grades containing not more than 25 per cent ash and 0.03 per cent phosphorus is required. The maximum demand of fuel is 100 tonnes. Percentage impurities and costs of various grades of coal are as in the Table. Assuming that there is an unlimited supply of each grade of coal and there is no loss in blending, formulate the blending problem to minimise the cost. One can notice that on formulating the above problem all the constraints are all of the form = which is unusual for a minimisation problem. The solution of this problem unfortunately results in the minimum cost of zero which limits the concept of blending itself. However, if the constraints were to be of the form {gt}= the minimum cost equals Rs 27,000 for the above problem with grades A and B only forming part of the blending with 50 per cent proportion each.

In the examination, the above problem was modified by replacing the word `blending' with `maximisation', with changes in the per cent of ash. A fundamental question arises whether the problem lends itself to a direct problem on linear programming. In that case, the answer obtained will be absurd. Hence, it is certain that the problem can be solved only as a blending problem. Going by the concept of blending, sales can be effected only on the ultimate finished product and hence the cost of the various inputs forming part of the blending is to be minimised. Therefore, to this extent the problem asked in the examination is limited as there is no question of maximising the individual profits of each grade. If the intention is to test whether the student has understood the mechanics of a linear programming problem and the Simplex Method, the question set could have been more straight forward.

However, the question should lend itself to a proper application.

An amazing variety of problems lend themselves to solution by linear programming. The linear programming model is only an approximate representation of the real world.

However, representation is often good enough to yield useful results. In any field of science or engineering, it is rare, indeed to find a model which represents the real world exactly. The important question is whether the representation is accurate enough to provide valuable information. Naturally, one must not become overzealous and attempt to apply some model to situations where it is completely inapplicable. This is true for linear programming as well as for anything else (G. Hadley).

(The authors are faculty, SIRC of the ICAI.)