A company is facing heavy demand for one of its products. The existing manufacturing facility is presently working at full capacity on normal shifts The company had two options to meet the heavy demand: either instituting overtime (an option that will cost $2,000, or installing new machinery at a cost of $20,000. The choice between the options depends mainly on what happens to sales over the next two years. During the first year, management estimates that there is a 70 percent chance that sales will rise and a 30 percent chance that they will fall.

After one year of operation, management will be faced with another decision, which will depend on the action taken initially, the event during the year, and the projection of second-year sales.

If a new machine had been installed at time zero and sales had risen, then management could either install a second machine or institute overtime. This decision depends on the anticipated payoffs after two years. These depend on sales forecasts, which can be either high (20 percent chance), medium (70 percent chance), or low (10 percent chance). In the case that a second machine is installed, the anticipated payoffs are $80,000, $60,000, and $50,000, respectively.

However, if overtime is instituted, the profits are estimated to be $60,000 for high sales, $50,000 for medium sales, and $40,000 for low sales.

If a new machine had been installed initially and sales had fallen, then the decision maker would have no choice but to use the existing capacities to the fullest extent. The expected results are $50,000, $40,000, and $20,000, for high, medium and low sales, respectively.

If overtime had been instituted initially and sales had risen, then management would be faced with two alternatives: install a new machine or install a new machine and use overtime (simultaneously). The anticipated payoffs in case of installing a new machine are $60,000, $50,000, and $20,000, for high, medium and low sales, respectively. In case of both installing a new machine and using overtime, the payoffs will be $50,000, $40,000 and $20,000, for high, medium and low sales correspondingly.

If overtime has been instituted initially and sales had fallen, then management would have the only alternative – to institute overtime. The anticipated results are $40,000, $40,000 and $30,000 for high, medium and low sales, respectively.

The decision tree for the task is presented in Figure 4.

Figure 4. Decision tree

The task is to find the best course of action the company should take initially and at intermediate stages, knowing all the information above.

Evaluation of the tree

Using the procedure previously outlines, the expected values at all event points can be computed (starting from the right).

Next, the computation moves leftward, thus reaching decision points 4,5,6, and 7.

Point 4. At this decision point the alternative of a second machine ($63,000 - $20,000 = $43,000) is compared with the alternative of overtime ($51,000 - $2,000 = $49,000). Since the latter is more profitable, it is selected, and E of $49,000 is entered above point 4.

Point 5. There is only one alternative. The expected value of point 10 is entered above point 5.

Point 6. At this decision point there are two alternatives: install a new machine ($49,000 - $20,000 = $29,000) or overtime plus a new machine ($40,000 - $22,000 = $18,000). The first one is better, and so E of $29,000 is recorded at point 6.

Point 7. There is only one alternative at this point. The E value of point 13 less the $2,000 expense ($39,000 - $2,000 = $37,000) is recorded at point 7.

Computation of the left side of the tree:

For point 2:

E_{2} = 0.7 × $49,000 + 0.3 × $40,000 = $46,300

For point 3:

E_{3} = 0.7 × $29,000 + 0.3 × $37,000 = $31,400

Finally, decision point 1 is considered:

The expected value of installing a new machine is:

E = $46,300 - $20,000 = $26,300

The expected value of overtime is:

E = $31,400 - $2,000 = $29,400

Thus, it is better to plan now on overtime for an expected gain of $29,400.

The final decision, therefore, is to use overtime this year and, if sales rise, install a new machine the second year. If sales fall, however, the overtime should be continued.