Consider a modification to the rod-cutting problem in which, in addition to a value pi for each rod, there is handling cost ci that is one and a half times the length of the rod cut plus a flat fee of 3 (i.e., the handling cost to cut a rod of length 5 is 7.5+3 =10). The revenue generated is the sum of the value of the pieces cut minus the sum of handling costs of the cuts.

Operations Research : Applications and Algorithms
4th Edition
ISBN:9780534380588
Author:Wayne L. Winston
Publisher:Wayne L. Winston
Chapter8: Network Models
Section8.5: Minimum-cost Network Flow Problems
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6. Consider a modification to the rod-cutting problem in which, in addition to a value pi for each rod, there is handling cost ci that is one and a half times the length of the rod cut plus a flat fee of 3 (i.e., the handling cost to cut a rod of length 5 is 7.5+3 =10). The revenue generated is the sum of the value of the pieces cut minus the sum of handling costs of the cuts. Provide a dynamic programming approach to solve this problem given an initial rod of length n with potential cut lengths of 1, 2, 5, and 7 [Adapted from ITA, pg. 370, 15.1-3] length of cut (i) 1 2 5 7 value (pi) 5 7 30 45 handling cost (ci) 4.5 6 10.5 13.5
Expert Solution
Step 1

Algorithm:

MODIFIED-CUT-ROD(p,n,c)

            let r[0..n] be a new array

            r[0]=0

            for j=1 to n

                        q=p[j]

                        for i=1 to j-1

                                    q=max(q,p[i]+r[j-i]-c)

                        r[j]=q

            return r[n]

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