$11\mathrm{.}9$ Planmatics is undertaking a modernization program. The set of activities in Table $11\mathrm{.}16$ has been defined for refurbishing one of its

wave soldering machines. The AOA network is given in Figure $11\mathrm{.}5.$

a$.$ Find the critical path, total slacks, and free slacks.

b$.$ Find the probability of completion within $45$ days.

c$.$ Find the minimum cost increase to reduce the expected project duration by $1$ day.

d$.$ Find the minimum cost increase to reduce the expected project duration by $2$ days.

e$.$ Find the minimum project duration and the expected cost increase.