One component of a system's life-cycle cost is the cost of system failure. Failure costs can be reduced by designing a more reliable system. A simplified expression for system life-cycle cost, C, can be written as a function of the system's failure rate:
C = Cl / λ + CR ∙ λ ∙ t.
Here, Cl = investment cost ($ per hour per failure),
CR = system repair cost,
λ = system failure rate (failures/operating hour),
t = operating hours.
a. Assume that Cl, CR, and t are constants. Derive an expression for k, say k*, that optimizes C.
b. Does the equation in Part (a) correspond to a maximum or minimum value of C? Show all work to support your answer.
c. What trade-off is being made in this problem?