$4.(20$ points$)$ Product A is processed through the first n steps of a process flow. At step n$+1,$ there are two possible recipes that can be applied. Recipe #$1$ is used if we desire the output to be product A$1.$ Recipe #$2$ is used if we desire the output to be product A$2.$ Once step n$+1$ is performed, WIP is committed to be either product A$1$ or product A$2.$ All WIP that is downstream from step n$+1$ is product$-$specific and cannot be re$-$assigned to the other product. The company does not wish to hold any inventory in front of step n$+1,$ but it desires to maintain the flexibility to not pre$-$designate WIP of product A to be either A$1$ or A$2$ before performing step n$+1.$ The company would like to calculate IPQ scores for product A for the steps $1,2,...,$ n$.$

$($a$)$ Let$$s consider a specific numerical example. Suppose the total WIP of A$1$ is $225$ wafers, and the total WIP of A$2$ is $150$ wafers. $($These figures include any WIP currently being processed through step n$+1.)$ The target cycle time from completion of step n$+1$ to fab out is $10$ days for both A$1$ and A$2.$ The target cycle time for product A from fab start to the completion of step n$+1$ is $40$ days. Time to transfer lots between steps is negligible. The target fab out schedule for product A$1$ is $25$ wafers per day for the next $60$ days, and the target fab out schedule for product A$2$ is $12\mathrm{.}5$ wafers per day for the next $60$ days. You may assume that line yield losses are negligible. You may also assume that actual fab outs due before time $0$ of each specific product equal the target fab outs for that product due before time $0.$ A shift lasts $0\mathrm{.}5$ days, and we are at the start of a shift. The WIP of product A awaiting processing at step n$+1$ is $25$ wafers. The target cycle time for step n on product A is $0\mathrm{.}6$ days. What should be the IPQ values for step n$+1?$

What should be the IPQ value for step n$?$

$($b$)$ Now let$$s generalize this situation as follows: N products are produced in the same process flow. There is only one generic product that is input to the process flow, and the first n steps of the process flow are performed on this generic product. WIP at these steps can be assigned to any of the N final products. When we perform step n$+1,$ the WIP must be assigned to a specific final product. All WIP that is downstream from step n$+1$ is product$-$specific and cannot be re$-$assigned to a different product.

As before, you may assume that line yield losses are negligible. You may also assume that actual fab outs due before time $0$ of each specific product equal the target fab outs for that product due before time $0.$

Let WIPi $=$ total amount of WIP of specific product i $($this is downstream from step n$+1$ or currently undergoing processing at step n$+1).$

Let WIP$0,$k $=$ total amount of WIP of the generic product that is at step k$,$ k $=1,2,...,$ n$,$ n$+1.$

Let TFOi$($t$)=$ target fab$-$out rate at time t for specific product i$.$

Let TCTFOk $=$ target cycle time from completion of step k to fab out $($same for all products$).$

Let h $=$ length of a production shift.

Provide a formula for computing IPQi,n$+1,$ the ideal production quantity of specific product i at step n$+1.$

$($c$)$ Now let$$s extend the IPQ concept to a generic product that feeds multiple specific products. The general idea of an IPQ is it expresses a production target for the end of the shift that will keep up with the target fab$-$out schedule and the target cycle time to fab out. Develop a formula for computing IPQ$0,$k$,$ the ideal production quantity of the generic product at step k$,$ for k $=1,2,...,$ n$.$