Question

(8 points) M&D Chemicals produces two products that are sold as raw materials to companies manufacturing bath soaps and laundry detergents. Let A=number of gallons of product A B=number of gallons of product B M&D’s objective is to satisfy all requirements at a minimum total cost. Production costs are $2 per gallon for product A and $3 per gallon for product B. The objective function together with the constraints is given below:

Objective function:

                                     Min 2A+3B

Subject to (s.t)

1A >= 125 Demand for product A

1A + 1B >= 350 Total production

2A + 1B <= 600 Processing time

A >= 0 and B >= 0

Excel Solver was used to product the outputs below.

Using the above outputs,

(a) (2 points) What are the optimal values of the decision variables?

(b) (2 points) What is the value of the optimal solution?

(c)(2 points) Which constraint(s) has a slack variable?

(d) (2 points)Which constraint(s) has a surplus and what is the value(s) of the surplus?

M&D Chemicals Available Operation Demand for A Total production Processing time cost per gallon 1 1 2 0 125 350 Model Decision Variables Number Produced 250 100 minimum total cost 800 Used(LHS) Available(RHS) Contraints Demand for A Total production Processing time 250 350 125 350 Microsoft Excel 15.0 Answer Report Worksheet: [M&D Chemical Example.xlsx]Sheetl Report Created: 8/30/2016 3:48:44 PM Result: Solver found a solution. All Constraints and optimality conditions are satisfied. Solver Engine Engine: Simplex LP Solution Time: 0 Seconds Iterations: 3 Subproblems: 0 Solver Options Max Time Unlimited, Iterations Unlimited, Precision 0.000001, Use Automatic Scaling Max Subproblems Unlimited, Max Integer Sols Unlimited, Integer Tolerance 1%, Assume NonNegative Objective Cell (Min) Cell SB$17 minimum total costA Name Original Value 0 Final Value 800 Variable Cells Cell SB$15 Number Produced A SC$15 Number Produced B Name Original Value Final Value Integer 0 0 250 Contin 100 Contin Constraints Cell ŞB$21 Demand for A Used(LHS) ŞB$22 Total production Used(LHS) ŞB$23 Processing time Used(LHS) Cell Value Status Slack 250 ŚB$21>-SD$21 Not Binding 125 0 0 Name Formula 350 $B$22>-SD$22 Binding 600 $B$23

Answer 1

(a)

The optimal values of the decision variables are as follows:

Number of gallons of product A produced = 250.

Number of gallons of product B produced = 100.

(b)

The minimum total cost of the products produced = $ 800 (Optimal Solution).

(c)

The constraint 2A + 1B   $\leq$   600 (Processing time) has a slack variable and the value of the slack variable is zero since it is a binding constraint.

(d)

The constraints of “Demand for product A” and “Total production” (which are ‘greater than or equal to’ constraints) have surpluses.

The surplus value for constraint “Demand for product A” is 125.

The surplus value for constraint “Total production” is 0 (since the constraint is binding).