# Value of Imperfect Information

Topics: Conditional probability, Bayes' theorem, Decision theory Pages: 12 (3095 words) Published: November 9, 2011
Value of Imperfect Information

20

20.1 PRIOR PROBLEM BEFORE INFORMATION
Technometrics, Inc., a large producer of electronic components, is having some problems with the manufacturing process for a particular component. Under its current production process, 25 percent of the units are defective. The profit contribution of this component is \$40 per unit. Under the contract the company has with its customers, Technometrics refunds \$60 for each component that the customer finds to be defective; the customers then repair the component to make it usable in their applications. Before shipping the components to customers, Technometrics could spend an additional \$30 per component to rework any components thought to be defective (regardless of whether the part is really defective). The reworked components can be sold at the regular price and will definitely not be defective in the customers' applications. Unfortunately, Technometrics cannot tell ahead of time which components will fail to work in their customers' applications. The following payoff table shows Technometrics' net cash flow per component. Figure 20.1 Payoff Table Component Condition Good Defective Technometrics' Choice Ship as is Rework first +\$40 +\$10 -\$20 +\$10

What should Technometrics do? How much should Technometrics be willing to pay for a test that could evaluate the condition of the component before making the decision to ship as is or rework first?

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Chapter 20 Value of Imperfect Information

Figure 20.2 Solution of Prior Problem and EVPI
EVPI \$7.50 Ship as is \$25.00 No add'l info 1 \$25.00 EVUU Rework first \$10.00 \$10.00 1.00 Good \$40.00 Ship as is 2 \$32.50 0.75 "Good" 1 \$40.00 Rework first \$10.00 \$10.00 Perfect info \$32.50 EVPP Ship as is -\$20.00 0.25 "Defective" 2 \$10.00 Rework first \$10.00 \$10.00 1.00 Defective -\$20.00 -\$20.00 0.00 Good \$40.00 \$40.00 \$40.00 0.00 Defective -\$20.00 -\$20.00 \$40.00 -\$20.00 0.25 Defective -\$20.00 0.75 Good \$40.00 \$40.00

20.2 Imperfect Information

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Figure 20.3 EVPI Sensitivity to Prior Probability

EVPI Sensitivity to Prior Probability, P(Good)
\$16 \$14 \$12 \$10 EVPI \$8 \$6 \$4 \$2 \$0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Prior Probability, P(Good)

20.2 IMPERFECT INFORMATION
An engineer at Technometrics has developed a simple test device to evaluate the component before shipping. For each component, the test device registers positive, inconclusive, or negative. The test is not perfect, but it is consistent for a particular component; that is, the test yields the same result for a given component regardless of how many times it is tested. To calibrate the test device, it was run on a batch of known good components and on a batch of known defective components. The results in the table below, based on relative frequencies, show the probability of a test device result, conditional on the true condition of the component. Figure 20.4 Likelihoods Test Result Positive Inconclusive Negative Component Condition Good Defective 0.70 0.10 0.20 0.30 0.10 0.60

For example, of the known defective components tested, sixty percent had a negative test result. An analyst at Technometrics suggested using Bayesian revision of probabilities to combine the assessments about the reliability of the test device (shown above) with the original assessment of the components' condition (25 percent defectives).

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Chapter 20 Value of Imperfect Information

This method of assigning probabilities helps to keep the two uncertainties separate. The main event of interest is the condition of a component, either Good or Defective. The information event of interest is the result of the test, either Positive, Inconclusive, or Negative. Technometrics uses expected monetary value for making decisions under uncertainty. What is the maximum (per component) the company should be willing to pay for using the test device? When Technometrics has the option to test the component before deciding whether to ship as is or...

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