A logistics firm has identified four operations, which are to be conducted in succession, for an order to be processed. The tolerances (in hours) are shown in Table 9-9. Assume that the tolerances are independent of each other and that the time in each phase is normally distributed. Further assume that for each operation, the processes are barely capable.
(a) Find the natural tolerance limits for order completion time.
(b) If the company sets a goal of 23.5 hours, what proportion of the orders will fail to satisfy this goal?
(c) Find an appropriate capability index and comment.
(d) Using a methods study, the company has improved operation 3 to a mean time of 7.0 hours. What proportion of the orders will now meet the goal?

Operation Mean Time (hours) Tolerance (hours) 640.6 0.6 80.8 5 0.3