Problem $16.$ Bearland Manufacturing produces $4$ different types of wood paneling. Each type of paneling is made by gluing and pressing together a different mixture of pine and oak chips. The following table summarizes the required amount of gluing, pressing, and mixture of wood chips required to produce a pallet of $50$ units of each type of paneling:

Resources Required per Pallet of Paneling Type

Paneling Type Tahoe Pacific Savannah Aspen

Glue $($quarts$)505010050$

Pressing $($hours$)5015010050$

Pine chips $($pounds$)500400300200$

Oak chips $($pounds$)500750250500$

Assume the company has $6,000$ quarts of glue; $7,500$ hours of pressing capacity; $30,000$ pounds of pine chips; and $62,500$ pounds of oak chips available in the next production cycle. Further assume that each pallet of Tahoe, Pacific, Savannah, and Aspen panels can be sold for profits of $$450,$ $$1,150,$ $$800,$ and $$400,$ respectively. Finally, for marketing purposes, the company wants to produce at least $4$ pallets of each type of paneling.

a$)$ Formulate an LP model for this problem.

b$)$ Create a spreadsheet mode for this problem, and solve it using Solver.

c$)$ What is the optimal solution?