Computer Repair

A microcomputer repairwoman has just been given
a failed machine by a good customer. She would like to fix it with the
least expenditure of money. One of four parts is probably failed. They
are A, B, C, and D, with the cost of replacement $100, $200, $30, and
$80, respectively. We use the letters A through D to represent the events
that the failure is caused by each of the four parts. Let E be the event
that something other than the four parts is causing the computer failure. 
Decision Node 
In
the following, we construct the decision tree that describes the details
of the decision process. The decision tree is the model for decision
analysis. We build the tree in steps, starting from the repairwoman's first
decision.
When the repairwoman first gets the machine she can repair the machine by replacing the motherboard at a cost of $500 or use test X to get a better idea what's wrong. This represents the first decision in the process, whether to perform the test. In the figure at the left, we begin to construct the decision tree. The decision tree is a graphical description of a sequential process and constitutes the major part of model of the decision process. The first decision is indicated by the rectangular node labeled 1. The symbol D1 interior to the node identifies the decision. Two arcs leave the node indicating the two possible decisions available at this point. The first decision has two results: either replace the motherboard or perform the X test. The labels on the arcs 1 and 2 correspond to these two possibilities. The cost associated with the test (50) is shown adjacent to the arc entering node 3. 
Terminal Node

On the figure the decision to replace the motherboard leads to a terminal node, shown as a black circle, labeled node 2. At terminal nodes the process stops, and the cost associated with the terminal state can be evaluated. The number adjacent to the node (500) is the cost associated with reaching this node, that is, the cost of replacing the motherboard. 
Chance Node 
The decision to perform test X leads to node 3. This is a chance node because the result of the test is uncertain. This kind of node is shown as a white circle. Test X can come up with three indications :failure probably due to A or B (arc 3), failure probably due to C or D (arc 4), or an indication to replace the motherboard (arc 5). The figure shows the three possible events associated with the experiment as arcs leaving the chance node. The numbers on the arcs leaving the node are the probabilities of the three test results. The events are mutually exclusive and constitute the entire range of possibilities, so their probabilities sum to 1. We describe how the probabilities are determine in a later section. The arcs terminate at nodes that either represent additional decisions or terminal events.

A Second Round of Decisions 
If Test X shows the first outcome, the repairwoman can either use test Y or replace the motherboard. If the test shows the second outcome, she can use test Z or replace the motherboard. If the test shows the third outcome, she must replace the motherboard. The figures show the detail associated with a second round of decisions. With the first outcome for test X, the repairwoman can either continue the fault isolation process with test Y or immediately replace the motherboard. This is shown as decision D2. The cost of test Y is $70. If test X results in the second outcome, she can continue the isolation with test Z or replace the motherboard. This is decision D3. The cost of test Z is $80. In both cases, the motherboard replacement costs $500. 
Outcomes for Tests Y and Z 
We assume for the example that test Y can accurately identify the cause of failure if it is due to component A or B. If either of these indications are observed, the faulty component is repaired. Every other cause (C, D, or E) is grouped into a third category. If the test does not indicate A or B, the motherboard is replaced. Similarly, test Z accurately identifies the fault if it is due to C or D. If the test does not indicate C or D, the fault must be in A, B or E. Rather than continue testing, the motherboard is replaced. The test results together with the appropriate probabilities and costs are shown in the figures. 
The Complete Tree

All the decision and chance nodes have been completely described and the tree is complete, as shown in the figure below. The tree is the model of the decision problem. It consists of numbered nodes and arcs. For a tree structure the number of nodes is always one more than the number of arcs. 