Problem
Suppose that a lumberyard has a supply of 10-ft boards, which are cut into 3-ft, 4-ft, and 5-ft boards according to customer demand. The 10-ft boards can be cut into several sensible patterns, each in such a way that there the leftover material is less than 3-ft. The lumberyard just received an order for 90 3-ft boards, 60 4-ft boards, and 60 5-ft boards.
1. Determine all sensible patterns the lumberyard may use to cut the 10-ft boards.
2. The lumberyard would like to use as few 10-ft boards as possible in meeting the demand. Write out a mathematical formulation for the problem as an integer linear program (ILP).
3. Setup/solve this ILP using Microsoft Excel with Analytic Solver (or Open Solver) AND using Gurobi Python (or Pyomo). What is the optimal number of each pattern, and what is the minimum number of boards to cut?