A factory produces CPTs $($cheap plastic tat$).$

The profit from each CPT is $5\mathrm{.}00.$

Conversely if a CPT fails during the $100$ month warranty period, the net profit from the CPT is $0\mathrm{.}00($i$.$e$.$ all profit is refunded$)$

CPTs suffer from two types of faults:

Overuse faults: the probability of a given CPT having an overuse fault during month t is $0\mathrm{.}0001($t$^2/(1+$t$))$

Factory faults: the probability of a given CPT having a factory fault during month t

is $0\mathrm{.}01(1+((1-$t$)/(1+$ t$)))$

Contextual knowledge unnecessary for answering this question here

a$)$ Build a function remain$\_$prob$()$ which has no input, and which outputs the probability of a CPT surviving the first $100$ months without a fault. Run it so we can see the output.

b$)$ Build a function factory$\_$simulation$($n$0$::Number$)$ that simulates faults in n$0$ CPTs over $100$ months. EG n$0=100.$ The function should output a vector v$,$ where v$[$i$]$ is the number of remaining $($i$.$e$.$ unbroken$)$ CPTs on month i

$.$ Plot the output of this simulation as a scatter plot. EG with

scatter$(1$:$100,$ v$,$ xlabel$=$"days", ylabel $=$ "functioning CPTs$",$ ylims $=(0,100),$ label $=$ false$)$

The factory can invest in making better products.

If it invests $x$ per CPT in reducing factory faults, the factory fault probability scales by a factor $k(x)=1-20x.$

$.$In other words, the overuse fault probability becomes $k(x)*0\mathrm{.}0001(\frac{t^2}{1+t})$

If it invests $y$ per CPT in reducing overuse faults, the overuse fault probability also scales by the same factor $k(y)=1-20y$

Of course, no amount of investment can reduce the factors $k(x)$ and $k(y)$ below zero.

c$)$ Build a function profit$($x$,$y$)$ that calculates the expected profit for $100$ items if $x$

are invested in reducing factory faults, and $y$

are invested in reducing overuse faults.

d$)$ Let $p(x,y)$ be the mathematical representation of the profit$($x$,$y$)$ you did$/$not make in the last question. Suppose the factory is commited to investing exactly $10$

pence per CPT in fault reduction.

Write the optimisation problem the company has to solve to maximise profit. $($You don't need to write a mathematical expression for $p(x,y)$

What condition on the gradient$/$jacobian of $p$

should hold for a locally optimal allocation of investment?