So far we have been concerned with the problems of assessing projects with a view to making simple accept-reject decisions. However, in most cases, the values of some variables are dependent on the values of others. A decision tree illustrates this dependence by following a chain of events to some final outcome.

A decision tree consists of a series of rods and branches. Each alternative course of action under consideration is represented by a main branch which, in turn, may have subsidiary branches for related chance events that appear in chronological order. In other words, the tree diagram charts the paths that lead to possible consequences.

Let us take as a simple example the decision to launch a new product. It is very likely that sales of a new product in the second year will be influenced by the level of sales in the first year. We assume that management estimates that sales in the first year have a 0.6 probability of being excellent, let us say 7000 units, and 0.4 probability of being good, let us say 5000 units. The probability distribution of sales in the second year will vary depending on whether the first year achieved excellent or good results.

The sum of the probabilities of each event is still the sum of the separate probabilities, but the probabilities of each of the six possible outcomes is the product of their individual probabilities.

Application of decision tree analysis

Decision tree analysis has a number of important applications to the solution of management problems. It has an important role to play in the study of investment projects which depend upon a series of events occurring in the right order, and which depend also upon the factors involved behaving according to forecast.

We shall take as our specific example the application of decision tree analysis to the problem of budgeting. In all budget planning exercises, there is a strong element of probability forecasting. It is possible to examine the probable outcome of the elements of the budget, and this process is known as probabilistic budgeting. The objective of budgeting is to produce an operational plan for the planning period, and if a change occurs in an important variable, such as the price of raw materials, the budget may have to be revised. Such a revision involves a laborious effort to reconstruct the budget, and in practice because of the time element required, it is normal to construct budgets on the basis of a limited number of assumptions. Traditional budgeting, therefore, does not provide a flexible instrument which may be adjusted to meet changing conditions. At best, they are flexible merely as regards varying sales and production levels. Moreover, such budgets are determined after considering only a limited number of alternative possibilities, which are aimed at attaining satisfactory rather than optimal targets.

With the advent of the computer, it is now possible to prepare in advance a number of budgets to meet all foreseeable contingencies, so that if they occur it is possible immediately to select the appropriate course of action to take.

Having worked out a total possible number of budgets, it would be too time-wasting in practice to go to the length of envisaging all the probable combinations of events which might occur and to plan budgets accordingly. It is advisable, therefore, to establish a cut-off point below which contingencies are not planned for, and in our example, we might select the probability of 0.024 as the cut-off point. Let us consider the effect of imposing such a cut-off point by reference to the frequency distribution of the probabilities.

As a result of selecting the cut-off point at the probability of 0.024, we can omit 18 budgets from the reckoning with a loss of coverage of only 0.2 of the total budgets which may be needed to meet all eventualities. As the table above shows, there is a probability of 0.8, or 4 chances out of 5, that the nine budgets included in part of the frequency distribution above the cut-off point will occur. Management may concentrate, therefore on the preparation of these nine budgets only.

Probabilistic budgeting increases dramatically the effectiveness of budgetary control systems, for as circumstances change the firm is ready to meet these changes. As we have noted above, probabilistic budgeting is concerned with planning for contingencies well in advance of their occurrence, so that the firm may alter its plan should these contingencies materialize. As a result, the firm is in a better position to make the best use of its resources; its plans include the possibility of shifts in resource allocations should circumstances change. Equally important is the manner in which probabilistic budgeting enables management to understand the manner in which changes affect the firm and alter the inter-relationship of its various elements.

The certainty equivalent method of risk appraisal

We may consider the aforementioned example in terms of the distribution of the probabilities around the mean (X) so that the standard deviation (a) may be used to represent the entire probability distribution. Students of statistics will appreciate that the standard deviation represents a measure of dispersion around the mean. The dispersion around the mean is much smaller for Project B than it is for Project A. The mean return is defined for this purpose as the expected return, that is £5000 for both projects. The standard deviation... see: The Certainty Equivalent Method of Risk Appraisal