Q$1.$ This problem is intended to evaluate the merits of a partially distributed inspection systems in which inspections are placed after processing steps that generate a high fraction defect rate. The processing sequence consists of eight operations with fraction defect rates for each operation as follows:

Three alternatives are to be compared: $($A$)$ fully distributed inspection, with an inspection after every operation; $($B$)$ partially distributed inspection, with inspections following operations $4$ and $8$ only; and $($C$)$ one final inspection station after operation $8.$ All inspections include sortations. In alternative $($B$),$ the inspection procedures are each designed to detect all of the defects for the preceding four operations. The cost of processing is $\backslash ($ C$\_\{$p r$\}=\backslash$$ $1\mathrm{.}50\backslash )$ for each of operations $1$ through $8.$ Inspection$/$sortation costs for each alternative are given in the table below. Compute total expected processing and inspection costs for the three cases. Assume batch size of $20,000$ parts.