NCSU Mean Variance Net Profit & Background Profit Excel Worksheet

Question Description

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A coffee shop just installed a new coffee maker to cover their ever-increasing de- mand of organic coffee. They have calculated that, due to increased productivity, the coffee shop will produce a profit of C = 2 USD per hour. However, the coffee maker is subject to unpredictable breakdowns, with the number of breakdowns X(t) during an interval of t hours follows a Poisson dis- tribution with parameter ? = t. The cost of breakdowns is given by (X^2(t) ? X(t)). The coffee maker can be restored to full performance by a preventive maintenance. Management wishes to choose the time t between scheduled maintenance sessions so that the expected profit.

(a) Write an expression for the expected net profit from this coffee maker.

(b) Plot the expected net profit function for values of t between 0 and 30.

(c) Using Excel Solver to help, compute the value of t that maximizes expected net profit, and computationally confirm it is indeed a maximizer.

(d) Now assume ? = 20. Compute the probability of more than 20 failures in t = 20 hours. Use the appropriate Excel functions to verify your calculation.

(e) Repeat (d) using a Normal Approximation of Poisson distribution parameters.

(f) Write an expression for the variance of the net profit from this machine.




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