A new type of airplane is to be purchased by the Air Force, and the number of spare engines to be ordered must be determined. The Air Force must order these spare engines in batches of five, and it can choose among only 15, 20, or 25 spares. The supplier of these engines has two plants, and the Air Force must make its decision prior to knowing which plant will be used. However, the Air Force knows from past experience that two-thirds of all types of airplane engines are produced in Plant A, and only one-third are produced in Plant B. The Air Force also knows that the number of spare engines required when production takes place at Plant A is approximated by a Poisson distribution with mean θ = 21, whereas the number of spare engines required when production takes place at Plant B is approximated by a Poisson distribution with mean θ = 24. The cost of a spare engine purchased now is $400,000, whereas the cost of a spare engine purchased at a later date is $900,000. Spares must always be supplied if they are demanded, and unused engines will be scrapped when the airplanes become obsolete. Holding costs and interest are to be neglected. From these data, the total costs (negative payoffs) have been computed as follows:

Determine the optimal alternative under Bayes€™ decision rule.

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State of Nature Alternative θ 21 θ 24 Order 15 Order 20 Order 25 1.155 x 107 1.012 x 107 1.047 x 10 1.414 x 10 1.207 x 10 1.135 x 10