Suppose that a malfunction which causes a certain system to break down and become inoperative can occur in either of two different parts of the system, part A or part B. Suppose also that when the system does become inoperative, it is not known immediately whether the malfunction causing the breakdown has occurred in part A or in part B. It is assumed that the repair procedures are quite different for the two different parts. Therefore, when a breakdown occurs in the system, one of the following three decisions must be chosen: Decision d1: The repair procedure for a breakdown in part A is activated immediately. If the malfunction causing the breakdown actually occurred in part B. then the cost of this decision in terms of unnecessary labor and lost time is $1000. If the malfunction actually occurred in part A, then this decision leads to the repair of the malfunction in the most efficient manner and the cost is regarded as zero. Decision d 2 : The repair procedure for a breakdown in part B is activated immediately. If the malfunction actually occurred in part A, then the cost of this decision is $3000. If the malfunction occurred in part B, then the cost is again regarded as zero. Decision dj:
A test is applied to the system that will determine with certainty whether the malfunction has occurred in part A or in part B. The cost of applying this test is $300.
(a) If 75 percent of all malfunctions occur in part A and only 25 percent occur in part B, what is the Bayes decision when the system breaks down?
(b) Suppose that the breakdown in the system is always caused by a defect in one of 36 similar components, all of which are equally likely to be defective. H 4 of these components are used in part A and the other 32 components are used in part B. what is the Bayes decision when the system breaks down?