Question

The Jean-Pierre Amor Company manufactures two lines of designer yard gates, called model A and model B. Every gate requires blending a certain amount of steel and zinc; the company has available a total of 25,000lb of steel and 6,000lb of zinc. Each model A requires a mixture of 150lb of steel and 20lb of zinc, and each yields a profit of $90. Each model B gate requires 100lb of steel and 30lb of zinc and can be sold for a profit of $70. Formulate an LP for this problem to find the best production mix of yard gates by answering the questions below.

a) What are the decision variables? How many decision variables are there? Clearly explain the decision variables in English and provide the mathematical notation used in the algebraic formulation of the decision problem.
b) What is the objective function? Clearly explain the objective function in English and provide the
mathematical formulation.
c) What are the constraints of the problem? Clearly explain each constraint in English and provide the mathematical formulation.
d) Are there any sign or type restrictions in your formulation? Why or why not? Clearly explain the sign and type restrictions (if any).

Answer 1

The decision variables are the number of model A gates produced and the number of model B gates produced. The objective function is to maximize the profit. The constraints are the available steel and zinc.

Step-by-step explanation:

a) The decision variables in this problem are the number of model A gates produced and the number of model B gates produced. Let's denote the number of model A gates as 'x' and the number of model B gates as 'y'. So, the decision variables are x and y.

b) The objective function is the function that we want to maximize or minimize. In this case, we want to maximize the profit. The objective function can be expressed as: Profit = 90x + 70y, where x is the number of model A gates and y is the number of model B gates.

c) The constraints of the problem are the limitations or restrictions that we need to consider. In this case, the constraints are the available steel and zinc. The constraint for steel is: 150x + 100y ≤ 25000. The constraint for zinc is: 20x + 30y ≤ 6000.

d) Yes, there are sign and type restrictions in this formulation. The decision variables x and y must be non-negative, as the company cannot produce a negative number of gates. The constraints also have type restrictions, as they are inequalities that limit the amount of steel and zinc used.