Question $3$: Engineering with Algorithms There are $n$ components and assembly line $j$ takes $a[i,j]>0$ minutes to assemble the $i$ th component, where $1in$ and $1jk.$ The components need to be assembled sequentially. We can assemble the $i$ th component in one line and then we can transfer it to a different line for the assembly of the $i+1$ th component. It takes $t>0$ minutes to transfer the product from one line to a different line. Given $nk$ array $a$ of assembly times and transfer time $t,$ you would like to compute the minimum possible time it will take to assemble the product.

For example if $n=4,k=3,t=1\mathrm{.}2$ and

$a=\left[\begin{array}{cccc}3\mathrm{.}1& 2\mathrm{.}9& 1\mathrm{.}5& 1\mathrm{.}3\\ 2\mathrm{.}8& 2\mathrm{.}4& 2\mathrm{.}0& 1\mathrm{.}1\\ 1\mathrm{.}4& 2\mathrm{.}5& 3\mathrm{.}5& 1\mathrm{.}7\end{array}\right]$

the minimum time to assemble the product will be $1\mathrm{.}4+2\mathrm{.}5+1\mathrm{.}2+1\mathrm{.}5+1\mathrm{.}3=7\mathrm{.}9$ minutes by assembling the fir two components on the third line and then transferring to the first line to assemble the final two components.

Develop an efficient algorithm for this problem and analyze its runtime. You do not need to prove the corre ness of your algorithm. Do not write pseudo$-$code, a succinct description and formulation of the key compone of your algorithm will suffice.