In this exercise we consider a component with a uniform $(=$ homogeneous$)$ failure distribution. We assume that the part may fail from the beginning. Without loss of generality, we assume that the maximum lifetime is $1.$ Hence the failure distribution $F(t)$ is given by $F(t)=t,0t1.$ The component is part of an important machine. As soon as the component fails it has to be replaced by a new one.

For this component we investigate two maintenance policies, the failure$-$based policy and the age policy with parameter $,01.$ The costs for a planned replacement of the component are denoted by $C_p.$ The costs for an unplanned replacement are equal to $C_u.$ We assume that $C_u>C_p>0.$

The average costs under the failure$-$based policy are denoted by $g_{fb}.$ The average costs under the age policy with parameter $$ are denoted by $g_{\text{a}\text{g}\text{e}}().$ The parameter for which these costs are minimized is denoted by ${}^{**}$ and $g_{\text{a}\text{g}\text{e}}^{**}=g_{\text{a}\text{g}\text{e}}({}^{**}).$ The ratio $\frac{g_{fb}-g_{\text{a}\text{g}\text{e}}^{**}}{g_{fb}}$ denotes the relative savings that are obtained under the optimal age policy in comparison to the failure$-$based policy. We are interested in how large these savings are as a function of the ratio $\frac{C_u}{C_p}.$