Cohen Chemicals, Inc., produces two types of photo-developing fluids. The first, a black-and-white picture chemical, costs Cohen$2,500 per ton to produce. The second, a color photo chemical, costs $3,000 per ton.

Based on an analysis of current inventory levels and outstanding orders, Cohens production manager has specified that at least 30 tons of the black-and-white chemical and at least 20 tons of the color chemical must be produced during the next month. In addition, the manager notes that an existing inventory of highly perishable raw material needed in both chemicals must be used within 30 days. To avoid wasting the expensive raw material, Cohen must produce a total of at most 60 tons of the photo chemicals in the next month.

Let X₁= number of tons of the black-and-white picture chemical produced

X₂=the number of tons of the color photo chemical produced

To find a solution Cohen Chemicals problem, we need to solve the formulated minimization problem below:

Min 2500X₁+3000X₂

X₁>=30 Constraint1

X₂>=20 Constraint 2

X₁+X₂<=60 Constraint3

X₁, X₂ >=0

(a) What is the optimal value of the decision variable X1?

(b) What is the optimal value of the decision variable X2?

(c) Find the optimal solution to this optimization problem.

(d) What is the value of the slack variable associated with constraint 3?