DECISION MODELING WITH MICROSOFT EXCEL Copyright 2001 Prentice Hall DECISION Chapter 8 ANALYSIS Part 2

DECISION ANALYSIS Decision Trees: Marketing Cellular Phones decision tree A decision tree is a graphical device for analyzing decisions under risk (models in which the decisions and the probabilities on the states of nature are specified). Decision trees are especially useful when there is a sequence of decisions. We will use TreePlan ( ), a spreadsheet add-in, which draws decision trees in a spreadsheet. Bayes’ Theorem will be used to incorporate new information into the process.

ALTERNATE MARKETING AND PRODUCTION STRATEGIES The design and product-testing phase has just been completed for Sonorola’s new line of cellular phones. Three major alternatives are being considered on a marketing and production strategy for this product: A 1. Aggressive (A) Major commitment from the firm Major capital expenditure Large inventories of all models Major global marketing campaign

B 2. Basic (B) Move current production to Osaka C 3. Cautious (C) Use excess capacity on existing phone lines to produce new products SW Management decides to categorize the level of demand as either strong (S) or weak (W). Modify current line in Tokyo Inventories for only most popular items Only local or regional advertising Minimum of new tooling Production satisfies demand Advertising at the discretion of local dealer

Here is the spreadsheet model: =SUMPRODUCT(B5:C5,$B$1:$C$1) Net profits measured in millions of dollars. Managements best estimate of the probability of a strong or weak market. B The optimal decision if you are risk-indifferent is to select B which yields the highest expected payoff.

CREATING A DECISION TREE This marketing model can also be represented by a decision tree. In decision trees: square node A square node represents a point at which a decision must be made. Each line (branch) leading from the square represents a possible decision. Each line (branch) leading from the circle represents a possible outcome. circular node A circular node represents an event (a situation when the outcome is not certain).

In order to create the decision tree in Excel, the TreePlan add-in must be loaded. To do this, double-click on Treeplan.xla. After loading the add-in, select the Decision Tree option from the Tools pull-down menu.

In the resulting dialog, click on New Tree. By default, a tree is displayed with 2 decision nodes. To add another node, click on the decision node and hit Ctrl-t to bring up a menu in which you can select the Add Branch option.

After labeling the three branches, replace the terminal node with a random event node by clicking on the terminal node and hitting Ctrl-t to bring up the menu from which you will select Change to event node and two branches.

Here is the resulting decision tree: By default, the probabilities for each of the 2 random events are 0.5.

Repeat the last few steps for the remaining decisions. The resulting decision tree is: Initial decision node. Choose from three alternatives. Event node with states of nature branches. Terminal positions Terminal node (since it is not followed by another node)

APPENDING THE PROBABILITIES AND TERMINAL VALUES Now we must append some additional information in order to use this decision tree to find the optimal decision. terminal value Assign the terminal value (the return associated with each terminal position). Additionally, probabilities will be assigned to each branch emanating from each circular node.

First change the probabilities from 0.5 to: =B1 =C1

Next, change the terminal values: =B5 =C5 =B6 =C6 =B7 =C7

Using a decision tree to find the optimal solution is called “solving the tree.” FOLDING BACK folding back To solve a decision tree, one works backward (i.e., from right to left) by folding back the tree. First the terminal branches are folded back by calculating an expected value for each terminal node. For example, Expected terminal value = 30(0.45) + (-8)(0.55) = 9.10

Once the expected terminal values are calculated, management must choose the alternative that yields the highest expected terminal value. Basic Of the three expected values, choose 12.85, the branch associated with the Basic strategy. 2 This decision is indicated in the TreePlan by the number 2 in the decision node.

DECISION ANALYSIS Before implementing the Basic strategy, the Board of Directors insists that a market research study first be performed. The corporate marketing research group at Sonorola headquarters in Tokyo will perform the research study and will report on whether the study is encouraging (E) or discouraging (D). The new information should be taken into account before making a decision. Sensitivity Analysis EXPECTED RETURN AS A FUNCTION OF THE PROBABILITY FOR A STRONG MARKET

OBTAINING REVISED PROBABILITIES BASED ON NEW INFORMATION Once the expected terminal values are calculated, management must choose the alternative that yields the highest expected terminal value. Thus, ER(A) = 30P(S) – 8[1-P(S)] = P(S) This expected return is a linear function of the probability that the market response is strong. ER(B) = 20P(S) + 7[1-P(S)] = P(S) For alternatives B and C, the expected returns are ER(C) = 5P(S) + 15[1-P(S)] = P(S)

Now, plot these functions in Excel using the Data Table command. To do this, first change the probability in cell C1 to a formula (=1-B1). Next, enter 0 in cell A10, go to Edit – Fill – Series and select these settings.

Here is the resulting spreadsheet: =D5=D6=D7 Now, highlight these cells and click on Data - Table

In the resulting dialog, enter B1 as the Column Input cell and click OK to calculate the expected returns from the three different strategies for P(S) values between 0 and 1. Now, use this data to create a graph of the expected return (ER) as a function of P(S):

For a given P(S), determine which alternative is best. For example, let P(S) = 0.45, then the highest ER is given by alternative B. What about when P(S) = 0.8? Sonorola should select the Basic strategy if < P(S) < 0.6

Conditional Probability: given Conditional Probability: For two events A and B, the conditional probability [P(A|B)], is the probability of event A given that event B will occur. DECISION ANALYSIS Lets expand on the previous example with new information concerning the likelihood of the uncertain events. given For example, P(E|S) is the conditional probability that marketing gives an encouraging report given that the market is in fact going to be strong. Decision Trees: Incorporating New Information A MARKET RESEARCH STUDY FOR CELLULAR PHONES

If marketing were perfectly reliable, P(E|S) = 1. However, marketing has the following “track record” in predicting the market: P(E|S) = 0.6 P(D|S) = 1 - P(E|S) = 0.4 P(D|W) = 0.7 P(E|W) = 1 - P(D|W) = 0.3

Calculating the Posterior Probabilities: Calculating the Posterior Probabilities: Suppose that marketing has come back with an encouraging report. Knowing this, what is the probability that the market is in fact strong [P(S|E)]? prior probabilities Note that probabilities such as P(S) and P(W) are initial estimates called a prior probabilities. posterior probabilities Conditional probabilities such as P(S|E) are called posterior probabilities. The domestic tractor division has already estimated the prior probabilities as P(S) = 0.45 and P(W) = Now, use Bayes’ Theorem to determine the posterior probabilities.

P(E|S)P(E|W)P(D|S)P(D|W)=B12/$D12=SUM(B12:B13)=SUM(B12:C12)=B3*B$8

The previous spreadsheet makes it easy to calculate the sensitivity of the posterior probabilities to the reliabilities and the prior probabilities. The Data Table command can be used to generate all posterior probabilities of a strong market [P(S|E) and P(S|D)] for values of the prior probability P(S) between 0 and 1 in increments of 0.1.

To do this, in cell I3, enter an initial value of 0, then use the Fill – Series option to fill the remaining cells.

Next, enter the formulas for the quantities that we want to track [P(S|E) and P(S|D)] for each of the values of P(S). =B19=B20

Finally, highlight the area below and click Data – Table: In the resulting table, enter B8 as the Column input cell and click OK.

Here is the resulting table: As the prior prob. of a strong market increases, so does the posterior prob. of a strong market given either an encouraging or a discouraging test result. Given an encouraging test result, the posterior prob. of a strong market is > the prior probability. Given a discouraging test result, the posterior prob. is < the prior (except when P(S) = 0 or 1.

Now, represent the model with a decision tree: INCORORATING POSTERIOR PROBABILITIES IN THE DECISION TREE I II IV V VI III VII VIII IX E D A B C A B C S W S W S W S W S W S W P(E) P(D) P(W|E) P(S|E) P(S|D) P(W|E) P(W|D) P(S|D) Note that the tree is built in the chronological order in which information becomes available and decisions are required:

TreePlan is used to build the tree in Excel. Here is the first half of the tree:

Here is the second half of the tree:

From the previous results, the expected return of performing the market test and making optimal decisions after determining the results is: THE EXPECTED VALUE OF SAMPLE INFORMATION ER = (0.435) (0.565) = Remember that if the market test is not performed, the optimal decision is to select B, the basic strategy with an expected return of Performing the market test increases Sonorola’s expected return by – = $0.61 million. expected value of sample information This value is called the expected value of sample information (EVSI).

In general, EVSI = - maximum possible expected return with sample information maximum possible expected return without sample information The EVSI is an upper bound of how much one would be willing to pay for this particular sample information.

EVPI = 30(0.45) + 15(0.55) –  8.90 The EVPI is the expected value of perfect information: Thus, perfect information will bring an expected increase of $8.90 million over the previous expected return. Expected return for decision A Expected return for decision C P(S)P(W) Expected return for optimal strategy (B) This is the maximum possible increase in the expected return that can be obtained from new information.

The expected value of sample information (EVSI = $0.61 million) is the increase in the expected return that was obtained with the information produced by the market test. The expected value of perfect information (EVPI) is $8.9 million. Since EVPI = 8.9 and EVSI = 0.61, we see that the market test is not very effective. If it were, the value for EVSI would be much closer to EVPI. As the probabilities of correct sample information increase, EVSI approaches EVPI. When P(E|S) = 1.00 and P(D|W) = 1.00, then EVSI = EVPI.

DECISION ANALYSIS Sequential Decisions: To Test or Not to Test The value in performing the market research test depends (in part) on how Sonorola uses the information generated by the test. The value of performing the test depends in part on how Sonorola uses the information generated by the test. sequence sequential decision model The value of an initial decision depends on a sequence of decisions and uncertain events that will follow the initial decision. This is called a sequential decision model.

ANALYZING SEQUENTIAL DECISIONS This is the type of situation that decision trees are designed to handle. Here is the upper half: This half represents the decision to perform the test (assuming a cost of $500,000).

Here is the lower half: Here, we do not perform the test so there is no cost. The optimal strategy is to conduct the test, and if the result of the test is Encouraging, then choose an Aggressive campaign. If the result is Discouraging, choose a Cautious campaign.

THE IMPACT OF UTILITIES It is simple to incorporate utilities in a decision tree. Consider the following utilities of all possible payoffs: A graph of utility vs. payoff would show that Sonorola is risk-averse.

To incorporate the utilities into the decision tree, replace the payoffs with their respective utilities and fold back the tree as before. Here is the upper half of the tree.

The optimal decision is still to test. However, if the test is encouraging, alternative B is chosen because it has a higher expected utility. Here is the lower half of the tree.

OTHER FEATURES OF TREEPLAN Clicking on the Options feature in TreePlan will result in the following dialog. TreePlan has a built-in exponential utility function (choose Use exponential Utility Function). Choosing this will cause Tree Plan to assume a risk-averse utility function and calculate the utilities for the given cash flows already on the tree. Tree Plan defaults to a Maximize Profit approach to folding back the tree.

SENSITIVITY OF THE OPTIMAL DECISION TO PRIOR PROBABILITIES Often, we want to see how sensitive the optimal decision is to various parameter values. This spreadsheet reproduces the previous graphical analysis.

We can graph the expected utility of Test and No- Test by creating a table in Excel using the Data – Table option. Expected utility of No-Test decision Expected utility of Test decision Whenever the two curves cross, the optimal decision changes.

DECISION ANALYSIS Management Decision Theory A typical management decision has the following characteristics: 1. It is made once and only once. 2. The return depends on an uncertain event that will occur in the future and we have no historical information about this future event. We know about related events that may tell something about the likelihood of the outcomes, but we cannot perform an experiment to provide estimates of the probabilities.

The following conceptual framework is recommended: 1. For each decision, determine the utility of each possible outcome. 2. Determine the probability of each possible outcome. 3. Calculate the expected utility of each decision. 4. Select the decision with the largest expected utility. Probabilitiesutilities best judgment and taste of the manager Probabilities and utilities are not known but are subjective and represent the best judgment and taste of the manager.

ASSESSING SUBJECTIVE PROBABILITIES equivalent lottery consistent framework An equivalent lottery concept gives one a consistent framework for quantifying both probabilities and utilities. The values obtained through this process are personal and a matter of judgment, and thus by definition, will vary from person to person. What do we gain from assessing probability and utility separately? Separating the assessments of probabilities and utilities forces a manager to give appropriate and separate consideration to each before combining the two to determine the final decision.

DECISION ANALYSIS Notes on Implementation Decision analysis involves assigning probabilities and utilities to possible outcomes and maximizing expected utility. This approach is applied to highly complex models that are typically sequential in nature. There are four parts: 1. Structuring the model 2. Assessing the probability of the possible outcomes 3. Determining the utility of the possible outcomes 4. Evaluating alternatives and selecting a strategy

ROLE OF PERSONAL JUDGMENT Decision analysis does not provide a completely objective analysis of complicated models. However, experience has shown that the framework provided by decision analysis has been useful. Many qualitative and nonobjective factors are involved in all decision making, but the important role of decision analysis is to make it consistent, not just “objective” and devoid of any subjective judgments.