The document provides an example of using linear programming to solve an optimization problem for a machine shop owner. It outlines maximizing contributions by determining the optimal number of units to produce of two products (A and B) given constraints on machine time, raw materials, and labor available. It presents the linear programming formulation and solution showing the feasible region and optimal point that maximizes total contribution. The document also provides another example of using the simplex method to solve a multi-variable optimization problem for a furniture company to maximize total contribution under constraints.