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