18.23 A food processor is considering the introduction of a new line of instant

Question

18.23 A food processor is considering the introduction of a new line of instant lunches. On a national basis, the company estimates a net profit of 50 million dollars if the product is highly succesful, a net profit of 20 million dollars if it is moderately successful, and a loss of 14 million dollars if it is not successful. If the company does not introduce the line, its research and development costs totaling 3 million dollars must be written off as a loss. Current estimates place the probability of high success at 0.1 and the probability of moderate success at 0.4. Prior to introducing it on a national level, the company could test market the line on a regional basis. The cost of such a test would be one million dollars. Although the test results would be significant, they would not be conclusive; the reliability of such a test is given by the conditional probabilities in Table 18-7. What should be the processor's decisions? Table 18-7 Test results will indicate High Moderate Success Success Success No Highly Successful 0.6 0.4 Moderately Sucesful0.2 0.6 0.2 Not Successful 0.1 0.3 0.6

Explanation / Answer

Probability of not successful = 1 - (0.1 + 0.4) = 0.5

Expected monetary value (EMV) = 50 * 0.1 + 20 * 0.4 - 14 * 0.5 = 6 million dollars

P(Test indicate high sucess) = P(High success) * P(Test indicate high sucess | High Success) + P(Moderate success) * P(Test indicate high sucess | Moderate Success) + P(No success) * P(Test indicate high sucess | No Success)

= 0.1 * 0.6 + 0.4 * 0.2 + 0.5 * 0.1 = 0.19

P(Test indicate moderate sucess) = P(High success) * P(Test indicate moderate sucess | High Success) + P(Moderate success) * P(Test indicate moderate sucess | Moderate Success) + P(No success) * P(Test indicate moderate sucess | No Success)

= 0.1 * 0.4 + 0.4 * 0.6 + 0.5 * 0.3 = 0.43

P(Test indicate no sucess) = P(High success) * P(Test indicate no sucess | High Success) + P(Moderate success) * P(Test indicate no sucess | Moderate Success) + P(No success) * P(Test indicate no sucess | No Success)

= 0.1 * 0 + 0.4 * 0.2 + 0.5 * 0.6 = 0.38

By Bayes theorem,

P(High Success | Test indicate high sucess) = P(Test indicate high sucess | High Success) * P(High success) / P(Test indicate high sucess)

= 0.6 * 0.1 / 0.19 = 0.3158

P(Moderate Success | Test indicate high sucess) = P(Test indicate high sucess | Moderate Success) * P(Moderate success) / P(Test indicate high sucess)

= 0.2 * 0.4 / 0.19 = 0.4211

P(No Success | Test indicate high sucess) = P(Test indicate high sucess | No Success) * P(No success) / P(Test indicate high sucess)

= 0.1 * 0.5 / 0.19 = 0.2632

P(High Success | Test indicate moderate sucess) = P(Test indicate moderate sucess | High Success) * P(High success) / P(Test indicate moderate sucess)

= 0.4 * 0.1 / 0.43 = 0.093

P(Moderate Success | Test indicate moderate sucess) = P(Test indicate moderate sucess | Moderate Success) * P(Moderate success) / P(Test indicate moderate sucess)

= 0.6 * 0.4 / 0.43 = 0.5581

P(No Success | Test indicate moderate sucess) = P(Test indicate moderate sucess | No Success) * P(No success) / P(Test indicate moderate sucess)

= 0.3 * 0.5 / 0.43 = 0.3488

P(High Success | Test indicate no sucess) = P(Test indicate no sucess | High Success) * P(High success) / P(Test indicate no sucess)

= 0 * 0.1 / 0.38 = 0

P(Moderate Success | Test indicate no sucess) = P(Test indicate no sucess | Moderate Success) * P(Moderate success) / P(Test indicate no sucess)

= 0.2 * 0.4 / 0.38 = 0.2105

P(No Success | Test indicate no sucess) = P(Test indicate no sucess | No Success) * P(No success) / P(Test indicate no sucess)

= 0.6 * 0.5 / 0.38 = 0.7895

Expected value for Test indicate high sucess = 50 * 0.3158 + 20 * 0.4211 - 14 * 0.2632 = 20.5272

Expected value for Test indicate moderate sucess = 50 * 0.093 + 20 * 0.5581 - 14 * 0.3488 = 10.9288

Expected value for Test indicate no sucess = 50 * 0 + 20 * 0.2105 - 14 * 0.7895 = -6.843

Now, if the test result will indicate High or moderately success, the company will go for the new line for the expected net profit of 20.5272 and 10.9288 million dollars respectively. If the test result indicate no success, the company will not go for new line and incur loss of 3 million dollars on research and development costs. So,

Expected value with sample information = P(Test indicate high sucess) * 20.5272 + P(Test indicate moderate sucess) * 10.9288 + P(Test indicate no sucess) * (-3)

= 0.19 * 20.5272 + 0.43 * 10.9288 - 0.38 * 3 = 7.4596

Expected value of sample information, EVSI = Expected value with sample information - EMV

= 7.4596 - 6 = 1.4596 million dollars

The cost of test is 1 million dollars. As, EVSI is greater than the cost of test, the processor company should go for test, and if the test indicates High or moderately success, the company will go for the new line for the expected net profit of 20.5272 and 10.9288 million dollars respectively. If the test result indicate no success, the company will not go for new line and incur loss of 3 million dollars on research and development costs.

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