please give correct answer only and give pseudocode for part$($b$).$

There is a large supply of a set of two types of rods $R,$ where each type $1$ rod is of length $r1$ with cost $c1$ and each type $2$ rod is of length $r2$ with cost $c2.$ You are given a set of $n$ required cut pieces $S=\{x1,x2,$dots,$xn\}$ where each cut$-$piece $$ is less than both $r1$ and $r2.$ You are required to select the required number of rods of each type to accommodate all the pieces of $S$ so that the total cost of rods used is minimum. Given that, if a rod is partly used, the full rod cost has to be paid, develop an optimal algorithm with the following:

$($a$)$ Present a suitable recursive definition to find the number of rods of each type. Analyze your recursive definition in terms of identical sub$-$problems produced, if any.

$($b$)$ Present both top$-$down and iterative Dynamic Programming based algorithms for the recursive definition. Analyze the time and space complexity of your algorithms.

$($c$)$ Show the working of the iterative algorithm on a non$-$trivial example of at least $15$ elements clearly indicating all storage elements memorized and used.