Consider a modification of the rod-cutting problem in which, in addition to a price pi for each rod, each cut incurs a fixed cost of c. The revenue associated with a solution is now the sum of the prices of the pieces minus the costs of making the cuts. Give a dynamic-programming algorithm to solve this modified problem.

(a) Write a recurrence for the solution; pay attention to how sub-problems are addressed.

(b) Write psuedocode for a brute force algorithm based on your recurrence

(c) Write psuedocode for a memoized algorithm based brute force solution

(d) Write psuedocode for a bottom-up dynamic programming solution