Construct a mathematical model in the form of a linear programming problem.  Then solve the problem by the simplex, dual problem, or big M methods.

A company manufactures outdoor furniture consisting of regular chairs, rocking chairs, and chaise lounges. Each piece of furniture passes through three different production departments: fabrication, assembly, and finishing. Each regular chair takes 1 hour to fabricate, 2 hours to assemble, and 3 hours to finish. Each rocking chair takes 2 hours to fabricate, 2 hours to assemble, and 3 hours to finish. Each chaise lounge takes 3 hours to fabricate, 4 hours to assemble, and 2 hours to finish. There are 2,500 labor-hours available in the fabrication department, 3,000 in the assembly department, and 3,500 in the finishing department. The company makes a profit of $17 on each regular chair, $24 on each rocking chair, and $31 on each chaise lounge.

(A) How many chairs of each type should the company produce in order to maximize profit? What is the maximum profit? 

(B) Discuss the effect on the optimal solution in part (A) if the profit on a regular chair is increased to $25 and all other data remain the same.

(C) Discuss the effect on the optimal solution in part (A) if the available hours on the finishing department are reduced to 3,000 and all other data remain the same.