Linear programming and decision making

Linear programming is a mathematical technique which seeks to make the best use of a firm's limited resources to meet chosen objectives, which in accounting terms may take the form of the maximization of profits or the minimization of costs. In those situations, for example, where a manufacturer has a limited plant capacity, the level and cost of output will be determined by such capacity.

The approach to the solution of a problem under linear programming consists, firstly, of formulating the problem in simple algebraic terms. There are two aspects to the problem, the first being the wish to maximize profits and the second being the need to recognize the production limits. The two aspects may be stated algebraically as follows:

(a) The objective is to maximize the contribution to fixed overheads and profit. This objective is called the objective function, and may be expressed thus:

Maximize C = 15A + lOB

where C is the total contribution and A and B being the total number of units of the two products which must be manufactured to maximize the total contribution.

(b) The constraints on production arising from the machine capacity limits of the Milling Department (600 hours) and of the Grinding Department (400 hours) may also be expressed in algebraic terms as follows:

5A � HB ZC 600

2A+2B 400

The first inequality states that the total number of hours used on milling must be equal to or less than 600 hours; the second inequality states that the total hours used on grinding machines must be equal to or less than 400 hours.

It is possible to solve the problem by means of a graph showing the manufacturing possibilities for the two departments.

It is noteworthy, also, that the linear programming approach to the best product combination mix gives a solution which is more profitable than the one which relates the contribution margin to the machine capacity limits, which we discussed on page 559, and which suggested that only product B should be made so that 200 units of B would be manufactured to yield a contribution of £2000.

We have so far only discussed simple cases involving at the maximum only two resource constraints. In real life, a firm may be faced with more than two constraints, but mathematical techniques exist for coping with larger numbers of limits. The Simplex Method, for example, which is based on matrix algebra may be employed in such cases and it is ideally suited for solutions using a computer.

Decision making in the face of limiting factors

In the examples which we have examined so far, the selection of alternative courses of action has been made on the basis of seeking the most profitable result. Business enterprises are limited in the pursuit of profit by the fact that they have limited resources at their disposal, so that quite apart from the limitation on the quantities of any product which the market will buy at a given price, the firm has its own constraints on the volume of output. Hence at a given price, which may be well above costs of production, the firm may be unable to... see: Decision Making in the Face of Limiting Factors