A company manufacturing copper releases three pollutants into the air: A, B, and C. The state has just passed a law that mandates that the company's plant must reduce its emis- sions of A, B, and C by at least 15, 10, and 20 units, respectively. The company can reduce its emissions of A, B, and C by using three pollution abatement processes: x1, x2, and x3. Use of 1 unit of x1 will reduce pollutants A, B, and C by 1, 1, and 2 units, respec- tively. Use of 1 unit of x2 will reduce pollutants A, B, and C by 3, 0, and 1 units, respec- tively. That is, only pollutants A and C can be reduced by using x2; pollutant B cannot be reduced by using x2. Finally, use of 1 unit of X3 will reduce pol- lutants A, B, and C by 0, 2, and 0 units, respectively. That is, only pollutant B can be reduced under pollution control device x3. Each unit of x1, x2, and x3 costs the company
$4, $2, and $3, respectively. The objective of the company is to install the least expen- sive combination of pollution control devices (x1, x2, and x3) that also satisfies the state's requirement that A, B, and C be reduced by at least 15, 10, and 20 units, respectively.
a. State this problem in standard form.
b. Restate the objective function of this problem in such a way that it is technically a maximization problem, with a solution identical to a minimization problem given the constraints stated in part a.
c. Solve this problem using the simplex method.
d. What is the optimal solution to this problem?