$(30$ points, plus $15$ points possible extra credit$)$ Heretofore $we$ have used a formula $to$ compute

the schedule score $as$ the negative $of$ the IPQ value divided $by$ the average fab out rate over the

target cycle time $to$ fab out plus one shift. This $is$ exact $if$ the target fab outs are constant over

this time, but $an$ approximation otherwise. Suppose $we$ wish $to$ refine the computation $of$ the

schedule score $to$ more precisely account for a time$-$varying fab out rate. The schedule score

should express how many days early $or$ late $to$ the fab out schedule would the downstream WIP

$beat$ the end $of$ the upcoming shift $ifno$ more WIP $of$ a particular product$-$step $is$ processed

during the shift.

Assume the target fab out schedule $is$ characterized $by$ a constant target rate for each product $in$

each work week, possibly changing rate $at$ the work$-$week boundaries. For simplicity $of$

exposition, assume $100\%$ line yields.

Initially, $we$ consider the computation $of$ ideal production quantities and schedule scores $at$ the

start $of$ a production shift. $We$ have the following set$-up$:

Notation

Indices

i work week. $We$ assume the first work week $of$ fab$-$out commits $is$ numbered $i=1.$

$l$ product

$j$ process step; $j_l$ denotes the last process step for fabricating product $l$

Parameters

$A{W}_{lj}=$ actual WIP $of$ product $lat$ step $j$

$AD{W}_{lj}={\displaystyle \underset{k=J+1}{\overset{j_l}{}}}A{W}_{lk}=$ actual WIP $of$ product $l$ downstream from step $j$

$t_i=$ time $in$ days from now $(time0)$ until the end $of$ week $i,i=1,2,3,$dots,$I{t}_{\text{I}}$ must $beat$ least $as$ large $as$ the longest target cycle time for any product.$t_{0}is$ set $to0.$ The value $of{t}_1$ could $be0\mathrm{.}5,1,1\mathrm{.}5,2,$dots,$6\mathrm{.}5,or7,$ depending $on$ where $in$ the

week the upcoming production shift lies. The value $of{t}_2$ will $be{t}_1+7(assuming$ the second

work week has $7$ days$),$ the value $of{t}_3$ will $be{t}_1+14,$ and $soon.$

For a given $t,$ let $i^{-}(t)$ denote the largest value $of$ i such that $ti+1$fol$(i)=illjllF{O}_l(t_i)=l{t}_{i}iF{O}_l(t_i)iF{O}_l(t_i)={\displaystyle \underset{k=1}{\overset{i}{}}}f{o}_l(t_k)F{O}_l(t_0)=0TCTF{O}_{lj}=jljlIP{Q}_{lj}=jlS{S}_{lj}=jljt{t}_i,i.e.,t$ falls $in$ work week $i+1.$

fol$(i)=$ unfulfilled fab$-$out commits $in$ week i for product $l.$ These commits express desired wafer

track$-$outs $of$ product $l$ from step $jl.We$ allow for the case that different weeks have different fab$-$

out targets for product $l.We$ assume target output $isat$ a uniform rate over the work week.

$F{O}_l(t_i)=$ cumulative target fab$-$outs, expressed $in$ wafers, for product $l$ due $by$ time $t_i(the$ end $of$

week $i).$ Note that $F{O}_l(t_i)is$ simply the sum $of$ weekly target fab outs $up$ through week $i,i.e.,$

$F{O}_l(t_i)={\displaystyle \underset{k=1}{\overset{i}{}}}f{o}_l(t_k)$

For convenience, set $F{O}_l(t_0)=0.$

$TCTF{O}_{lj}=$ the total target cycle time $(expressedin$ days$)$ from completion $of$ step $jon$ product $l$

$to$ the completion $of$ step $jl.$

Variables $(tobe$ computed$)$

$IP{Q}_{lj}=$ the ideal production quantity $in$ the upcoming shift for step $jon$ product $l$

$S{S}_{lj}=$ the number $of$ days early $(if$ positive$)or$ late $(if$ negative$)$ production $at$ step $jon$ product $l$

will $beat$ the end $of$ the shift $ifno$ wafers are tracked out $of$ step $jin$ the upcoming shift.

$(a)(5$ points$)$ For the assumption $of$ constant target output rates within each work week, express

the cumulative target fab$-$outs due $atan$ arbitrary point $tin$ continuous time $(measuredin$ days $(b)(5$ points$)$ Consider the figure below. Add dotted lines $to$ show $IP{Q}_{lj}on$ the vertical axis and

$to$ show $-S{S}_{lj}on$ the horizontal axis. Explain.

$(c)(10$ points$)$ Provide formulas $to$ compute $IP{Q}_{lj}$ and $S{S}_{lj}$ precisely. You may assume target fab

outs for product $l$ are positive $in$ every week.

$(d)(5$ points$)As$ lots are dispatched during the shift, the IPQ should $be$ updated. Suppose a wafer

quantity $W_{lj}of$ step $iin$ the process flow $to$ fabricate product $lis$ dispatched. How should the

values $ofIP{Q}_{lj},A{W}_{lj},$ and $AD{W}_{lj}be$ updated?

$(e)(5$ points$)$ For a dispatch $of{W}_{lj}$ wafers $as$ above, how should the schedule score $S{S}_{lj}be$

updated?

$(e)(15$ points extra credit $-$ hard$)If$ there are one $or$ more weeks $in$ which there $isno$ output

commitment for product $l,$ then the cumulative target output curve will have "flat spots" $init.$

When $AD{W}_{lj}or(TCTF{O}_{lj}+0\mathrm{.}5)is$ located along such a flat spot $in$ the cumulative target output

curve, ambiguity arises $asto$ expressing how late $or$ how early $is$ the production $to$ date. Provide

a procedure $to$ compute the schedule score $in$ this case, and explain how $it$ should $be$ updated

after a dispatch

and fractions thereof from the start $of$ the upcoming shift$)as$ follows: