Consider the Part Type Selection problem we covered in class. This problem helps us

determine which parts we will produce in a new FMS$.$ The problem assumes that we have

candidate parts $i=1,2,$dotsN, that the total savings achieved by producing part $i$ in the FMS

is $s_i$ and that we know that there is a 'key bottleneck' in the FMS$,$ which has $P$ available time

units. Each part i requires a total processing time $p_i$ in the 'key bottleneck'.

In class we saw that a way to solve this problem is the following Greedy Heuristic:

Step $1$: Order parts $[1]$ to $[$N$]$ such that

$\frac{s_{[1]}}{p_{[1]}}\frac{s_{[2]}}{p_{[2]}}$dots$\frac{s_{[N]}}{p_{[N]}}$

Define remaining available time $T=P$

Step $2$: For $i=1$dotsN select part i and add the part to the FMS if $s_i>0iT=T-p_{i}P,QRi{p}_i,q_i{r}_i{p}_i$ and $s_i>0.$ Each

time you add a new part $i,$ update the remaining available time $asT=T-p_i.$

Now suppose you had $3$ 'key bottlenecks' $(thatis3$ resources that each might $be$ the

bottleneck depending $on$ the parts you choose $to$ produce $in$ the $FMS).$ These resources

have $P,Q$ and $R$ time units available respectively. And each part i requires a total processing

time $p_i,q_i$ and $r_{i}in$ each $of$ the three, respectively. Modify the algorithm above $to$ solve the

$3-$bottleneck version $of$ the problem.

TIP: $In$ Step $1$ you need $to$ consider the processing times $at$ all $3$ stations $todo$ the

ordering. $In$ step $2,$ you must keep track $of$ the remaining capacity $at$ each station

separately.