A manufacturer produces an item consisting of two components, which must both work for the item to function properly. The cost of returning one of the items to the manufacturer for repairs is α dollars, the cost of inspecting one of the components is β dollars, and the cost of repairing a faulty component is φ dollars. She can ship each item without inspection with the guarantee that it will be put into perfect working condition at her factory in case it does not work; she can inspect both components and repair them if necessary; or she can randomly select one of the components and ship the item with the original guarantee if it works, or repair it and also check the other component.
(a) Construct a table showing the manufacturer’s expected losses corresponding to her three “strategies” and the three “states” of Nature that 0, 1, or 2 of the components do not work.
(b) What should the manufacturer do if a = $ 25.00, θ = $10.00, and she wants to minimize her maximum expected losses?
(c) What should the manufacturer do to minimize her Bayes risk if a = $ 10.00, β = $ 12.00, φ = $30.00, and she feels that the probabilities for 0, 1, and 2 defective components are, respectively, 0.70, 0.20, and 0.10?