Q$4.$ Consider the following optimization problem that is coded in CBC$.$

from pulp import$*$

Factory $=[A,B]$

$Ma11=["$M$1","$M$2","$M$3","$M$4"]$

Prod $=\{A$:$1000,B$:$2000\}$

Sales M$1"$: $1500,M2$:$600,M3$:$700,M4$:$200$

cost $,$

$"$B$"$: $\{"$M$1"$:$3,"$M$2"$:$1,"$M$3"$:$3,"$M$4"$:$2\}\}$

prob $=$ LpProblem$("$Final Exam", LpMinimize$)$

$x=$ LpVariable.dicts $("$x$",($Factory$,$ Mall$),$ lowBound $=0,$

upBound $=$ None, cat $=$ const. LpInteger$)$

prob $+=1$ pSum$($Cost $[f][s]**x[f][s]$ for $f$ in Factory for $s$ in

Mal$1)$

for $f$ in Factory:

prob t$=$ IpSum$($X$[$f$][$s$]$ for $s$ in Mall Prod$[$f$]$

for $s$ in Mall:

prob for $f$ in Factory$)>=$Sales$[$s$]$

We are trying to model the sales distribution channel of a supply chain. There are two factories $(A,B)$ and four Malls $($M$1,$M$2,$M$3,$M$4).$ On the left you can see the Python code of the optimization problem for CBC library. Please solve the following questions.

a$.$ Write the mathematical model of the first for loop that is written in bold and italic.

b$.$ Assume factory A can serve only $M1$ and M$3.$ Write the necessary mathematical equation for this constraint.