Decision Trees Any problem

Decision Trees Any problem that can be presented in a decision table can also be graphically illustrated in a decision tree. All decision trees are similar in that they contain decision nodes or decision points and state-of-nature nodes or state-of-nature points:

· A decision node from which one of several alternatives may be chosen

· A state-of-nature node out of which one state of nature will occur In drawing the tree, we begin at the left and move to the right.

Thus, the tree presents the decisions and outcomes in sequential order. Lines or branches from the squares (decision nodes) represent alternatives, and branches from the circles represent the states of nature. Figure 3.2 gives the basic decision tree for the Thompson Lumber example.

First, John decides whether to construct a large plant, a small plant, or no plant. Then, once that decision is made, the possible states of nature or outcomes (favorable or unfavorable market) will occur.

The next step is to put the payoffs and probabilities on the tree and begin the analysis. Analyzing problems with decision trees involves five steps:

Five Steps of Decision Tree Analysis 1. Define the problem.

2. Structure or draw the decision tree.

3. Assign probabilities to the states of nature.

4. Estimate payoffs for each possible combination of alternatives and states of nature.

5. Solve the problem by computing EMVs for each state-of-nature node. This is done by working backward, that is, starting at the right of the tree and working back to decision nodes on the left. Also, at each decision node, the alternative with the best EMV is selected.

The final decision tree with the payoffs and probabilities for John Thompson’s decision situation is shown in Figure 3.3. Note that the payoffs are placed at the right side of each of the tree’s branches.

The probabilities are shown in parentheses next to each state of nature. Begin-ning with the payoffs on the right of the figure, the EMVs for each state-of-nature node are then calculated and placed by their respective nodes.

The EMV of the first node is $10,000. This represents the branch from the decision node to construct a large plant. The EMV for node 2, to construct a small plant, is $40,000. Building no plant or doing nothing has, of course, a payoff of $0.

The branch leaving the decision node leading to the state-of-nature node with the highest EMV should be chosen. In Thompson’s case, a small plant should be built. . A MORE COMPLEX DECISION FOR THOMPSON LUMBER—SAMPLE INFORMATION When sequential decisions need to be made, decision trees are much more powerful tools than decision tables.

Let’s say that John Thompson has two decisions to make, with the second decision dependent on the outcome of the first. Before deciding about building a new plant, John has the option of conducting his own marketing research survey, at a cost of $10,000.

The information from his survey could help him decide whether to construct a large plant or a small plant or not to build at all. John recognizes that such a market survey will not provide him with perfect information, but it may help quite a bit nevertheless.

All outcomes and alternatives must be considered

John’s new decision tree is represented in Figure 3.4. Let’s take a careful look at this more complex tree. Note that all possible outcomes and alternatives are included in their logical sequence.

This is one of the strengths of using decision trees in making decisions. The user is forced to examine all possible outcomes, including unfavorable ones. He or she is also forced to make decisions in a logical, sequential manner.

Examining the tree, we see that Thompson’s first decision point is whether to conduct the $10,000 market survey. If he chooses not to do the study (the lower part of the tree), he can con-struct a large plant, a small plant, or no plant.

This is John’s second decision point. The market will be either favorable (0.50 probability) or unfavorable (also 0.50 probability) if he builds. The payoffs for each of the possible consequences are listed along the right side.

As a matter of fact, the lower portion of John’s tree is identical to the simpler decision tree shown in Figure 3.3. Why is this so? The upper part of Figure 3.4 reflects the decision to conduct the market survey. State-of-Most of the probabilities are conditional probabilities. nature node 1 has two branches. There is a 45% chance that the survey results will indicate a fa-vorable market for storage sheds.

We also note that the probability is 0.55 that the survey results will be negative. The derivation of this probability will be discussed in the next section.

The rest of the probabilities shown in parentheses in Figure 3.4 are all conditional probabilities or posterior probabilities (these probabilities will also be discussed in the next section).

For example, 0.78 is the probability of a favorable market for the sheds given a favorable result will be either favorable (0.50 probability) or unfavorable (also 0.50 probability) if he builds. The payoffs for each of the possible consequences are listed along the right side. As a matter of fact, the lower portion of John’s tree is identical to the simpler decision tree shown in Figure 3.3.

Why is this so? The upper part of Figure 3.4 reflects the decision to conduct the market survey. State-of nature node 1 has two branches. There is a 45% chance that the survey results will indicate a fa-vorable market for storage sheds. We also note that the probability is 0.55 that the survey results will be negative.

The derivation of this probability will be discussed in the next section.

Most of the probabilities are conditional probabilities. The rest of the probabilities shown in parentheses in Figure 3.4 are all conditional probabilities or posterior probabilities (these probabilities will also be discussed in the next section). For example, 0.78 is the probability of a favorable market for the sheds given a favorable result

from the market survey. Of course, you would expect to find a high probability of a favorable market given that the research indicated that the market was good. Don’t forget, though, there is a chance that John’s $10,000 market survey didn’t result in perfect or even reliable information.

Any market research study is subject to error. In this case, there is a 22% chance that the market for sheds will be unfavorable given that the survey results are positive. We note that there is a 27% chance that the market for sheds will be favorable given that John’s survey results are negative.

The probability is much higher, 0.73, that the market will actually be unfavorable given that the survey was negative.

The cost of the survey had to be subtracted from the original payoffs. Finally, when we look to the payoff column in Figure 3.4, we see that $10,000, the cost of the marketing study, had to be subtracted from each of the top 10 tree branches. Thus, a large plant with a favorable market would normally net a $200,000 profit. But because the market study was conducted, this figure is reduced by $10,000 to $190,000.

In the unfavorable case, the loss of $180,000 would increase to a greater loss of $190,000. Similarly, conducting the survey and building no plant now results in a -$10,000 payoff.

We start by computing the EMV of each branch. With all probabilities and payoffs specified, we can start calculating the EMV at each state-of-nature node. We begin at the end, or right side of the decision tree, and work back toward the origin. When we finish, the best decision will be known.