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.

We say that a problem exhibits the optimal substructure property when optimal solutions to a problem incorporate optimal solutions to related subproblems, which we may solve independently. Suppose that in the rod$-$cutting problem above, we also had limit li on the number of pieces of length i that we are allowed to produce, for i $=1,2,...,$ n$.$ Show that the optimal$-$substructure property described no longer holds.