$5.$ Referring to the situation in problem $4,$ now let$$s consider the case where step n$+1$ is actually a test performed on the generic product, and there are bin splits of the generic product at step n$+1$ into the specific products i $=1,2,...,$ N$.$ On average, a fraction ai of the generic product is classified as a bin of quality corresponding to specific product i$.$ Assume there are no substitution possibilities among bins, i$.$e$.,$ each bin corresponds one$-$to$-$one with one specific product and cannot be used for anything else.

$($a$)$ Provide a formula or method for computing IPQ$0,$n$+1,$ the ideal amount of the generic product to be tested at step n$+1$ this shift.

$($b$)$ Now suppose that some bins are suitable for several specific products, and that input to step n$+2$ of a specific product might by sourced from more than one bin generated at step n$+1.$ But once a bin is allocated to a specific product, it assumes the identity of that specific product in all downstream steps and cannot be subsequently re$-$allocated.

Let Sij denote the amount of bin i that we allocate to specific product j this shift $($defined only for feasible i$-$j combinations$).$

Let IIi denote the initial inventory of unallocated bin i at the start of the shift.

Provide a method for computing IPQ$0,$n$+1$ in this case.