Question $1$: $[5$ points$]$

Consider an electronic system consisting of three components, each of which must work for the system to function. The reliability of the system can be improved by installing several parallel units in one or more of the components. The following table gives the probability that the respective components will not function if they consist of one, two, three, or four parallel units, along with the cost $($in hundreds of dollars$)$ of installing one, two, or three parallel units in the respective components $($labeled as Comp. $1,2,3,$ and $4)$ are given in the following table:

$\backslash$table$[[,$Probability of not functioning,Cost,$],[\backslash$table$[[$Parallel$],[$Units$]],$Comp. $1,$Comp. $2,$Comp. $3,$Comp. $1,$Comp. $2,$Comp. $3],[1,0\mathrm{.}5,0\mathrm{.}4,0\mathrm{.}3,2,1,2],[2,0\mathrm{.}4,0\mathrm{.}3,0\mathrm{.}2,3,3,4],[3,0\mathrm{.}3,0\mathrm{.}2,0\mathrm{.}15,4,4,5],[4,0\mathrm{.}2,0\mathrm{.}2,0\mathrm{.}1,6,6,7]]$

Because of budget limitations$,$ a maximum of AED $1200$ can be expended.

Formulate a dynamic programming problem to determine how many parallel units should be installed in each of the three components to minimize the probability that the system will not work. $($Do not solve$)$

Write explicitly the recursive function for the second stage of your formulation and set the calculation table. $($Do not calculate$/$solve$)$