# mathematics help - 35760

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Case Study 19.2Chi U. Ikoku continues his World Oil article on decision analysis in the petroleum industry (see Case Study 19.1 attached) with a discussion of decision trees and their importance in solving decision problems much more complex than the examples and exercises of chapter 19. Nontrivial decision problems consist of not one but a sequence of major choices or decisions that must be made. Writes Ikoku quotIn a complex decision problem involving a long sequence of alternatives, a formal procedure for decision analysis is necessary to array the alternatives so that economic ramifications of each are clearly delineated. This formal array also promotes effective internal communication. The decision tree analysis fits this criterion.quot To illustrate, Ikoku presented the following example of a drilling venture evaluation.

XYZ Enterprises has a nontransferable short-term option to drill on a certain plot of land. Two recent dry holes elsewhere have reduced XYZs liquid assets to \$130,000 and John Doe, president and principal stockholder, must decide whether XYZ should exercise its option (i.e., drill) or allow it to expire. Complicating the decision problem is the fact that Doe may pay to have a seismic test run in the next few days, and then, depending on the results, decide whether to drill. Thu, Doe has three possible alternatives or actions from which to choose:

a1:Drill immediately.

a2:Pay to haave a seismic test run. then decide whether to drill.

a3:Let the opinion expire (1.e., do not run the seismic test and do not drill).

XYZ can have the seismic test performed for a fee of \$30,000 and the well can be drilled for \$100,000. XYZ usually sells the rights of any oil discovered. A major oil company has promised to purchase all of the oil rights for \$400,000.

What is the oil company\'s decision, using the expected payoff criterion?

This decision problem is structered in the form of a decsion tree, as shown in Figure 19.4. As before, the square denotes a decision fork and the circle denotes a chance fork. Notice the probabilities assigned to the states of nature, Strike Oil, Do not strike oil (given in parentheses under the appropiate fork), vary according to whether the seismic test is run, and if run, whether the test is favorable. Also note that the objective variable in this decision problem is Net liquid as sets (in dollars).

The expected payoff strategy as applied to decision trees that involve a sequence of choices to be made requires the decision maker to work through the decision tree \"backward\" (i.e., from right to left), computing at each chance fork (circle) the corresponding expected payoff. The at each decision fork (square), choose the action with the maximum expected payoff (thereby eliminating from further consideration all other alternatives at that particular fork). Repeat the procedure until only a single action or branch remains. This is the optimal expected payoff strategy.

(a) From Figure 19.4, compute the expected payoff for action a1: Drill immediately.

(b) From Figure 19.4(attached), compute the expected payoff for the action a3: Do not run seismic test, do not drill.

(c) The expected payoff for the action, a2: Run seismic test, then decide whether to drill, is not computed as easily as for action a1 and a3 because action a2 involves two different chance forks-the fork corresponding to the result of the seismic test Favorable or Unfavorable) and the fork corresponding to the result of drilling (Oil or No oil). The first step in computing the expected payoff for a2 is to compute the expected payoff at each of the two rightmost chance forks in the Run seismic test branch of the tree. Place these expected values above the corresponding chance fork symbol (circle) in Figure 19.4. [For example, the expected payoff for the top, rightmost is (\$400,000)(.85) + (\$0) (1.5) = \$340,000.]

(d) The second step is to determine the optimal action at each decision fork (square) in the Run seismic test branch by comparing the expected payoffs of the two actions Drill and Do not drill. Choose the action with the largest expected payoff and eliminate the alternative branch from further consideration. [For example, the optimal action corresponding to the top, rightmost decision fork (square) os Drill, because the expected payoff for this action, namely \$340,000, is larger than the expected payoff (\$100,000) corresponding to the action Do not drill. Thus, we can eliminate the action Test favorable, Do not drill from further consideration.]

(e) If you have performer the second step in part d correctly, there should remain only two \"clear\" paths or options available to the decision maker in the upper portion of the tree: the Test favorable, Drill option and the test unfavorable, Do not drill option. To compute the expected payoff for action a2: Run seismic test, multiply the expected payoffs of these optimal actions by their corresponding state of nature (Favorable or Unfavorable) probabilities and sum these two values. [Hint: The correct expected payoff is (\$340,000) (.6) + (\$100,000) (.4) = \$244,000.]

(f) Now that you have computed the expected payoffs for each of the three actions, apply the expected payoff criterion in the usual manner, that is choose the action with the maximum expected payoff. What is the oil company\'s decision?