LP Output Interpretation
1. The Cookie Store
at a small shopping center makes three types of cookies—chocolate chip, pecan
chip, and twists. The three main
ingredients are chocolate chips, pecans, and sugar. The store has 312 pounds of sugar, 125 pounds
of chocolate chips, and 50 pounds of pecans.
A
The linear programming model has been developed as given below for determining the number of batches of chocolate chip cookies (x1), pecan chip cookies (x2), and twists (x3) to make in order to maximize profit.
Max z = 20x1 + 25x2 +17x3 dollars
Subject to:
21x1 + 15x2 + 9x3 £ 312 pounds of sugar
10x1 + 5x2 £ 125 pounds of chocolate chips
x1 + 2x2 £ 50 pounds of pecan
x1 ³ 5 batches of chocolate chip cookies
x1, x2, x3 ³ 0
Solve the problem using Excel Solver and answer the following.
1) How many batches of each type of cookie should be made? What is the total profit?
2) Which constraints are binding? Why?
3) What does s2 represent? (Slack or surplus). Explain what it means.
4) How many pounds of pecans are used?
5) Should the company make more batches of chocolate chip cookies? Why or why not?
6) How profitable should the pecan chip cookies be in order to make some?
7) What would the new total profit be if there were 7 additional pounds of sugar?
8) What would happen to the optimal solution to this problem if the profit per batch of chocolate chip cookies were increased to $24? Why? What would the new total profit of the store be?
2. DY Chemicals blends its famous, private label insect spray from four individual compounds. Management would like to make the blend at as low a cost as possible while maintaining the requirements for chemical structure. The linear programming problem formulation is given below.
With xi = gallons of compound i in the blend; i =1, 2, 3, 4
Min z = 7x1 + 15x2 + 35x3 + 21x4 dollars
Subject to:
6x1 + 11x2 + 10x3 + 13x4 > 775 gallons of chemical 1
10x1 + 19x2 + 22x3 + 6x4 > 925 gallons of chemical 2
x1, x2, x3, x4 ³ 0
Solve the problem using Excel Solver and answer the following.
1) How much of each compound is used? What is the total cost?
2) Which constraints are binding? Why?
3) What costs would each of the non-used compounds have to have in order to be used in the blend?
4) What does s1 = 0 and s2 = 366.667 represent? (Slack or surplus). Explain what they mean.
5) How many gallons of chemical 2 are used in the process?
6) What would the new total cost be if the chemical 1 requirement were increased by 100 gallons?
7) Suppose the cost per unit for compound 1 were actually $7.25. How would the solution change (i.e., how much of each compound would be used) and why? How much would the total cost change?
Answers:
1. 1) x1 = 5, x2 = 0, x3 = 23, $491 2) 1 and 4. All the available sugar has been used up (no slack for the first constraint). Exactly 5 batches of chocolate chip cookies are made (no surplus for the fourth constraint) 3) Slack. There are 75 pounds of unused chocolate chips. 4) 5 pounds 5) No, the total profit would decrease by $19.67/batch 6) Above $28.33 7) $504.23 8) The optimal solution would stay the same, because $24 is within the objective coefficient range. $511
2. 1) x1 = 129.167, x2 = 0, x3 = 0, x4 = 0 $904.17 2) 1. Exactly 775 gallons of chemical 1 are used (no surplus) 3) For compound 2, less than $12.83, for compound 3, less than $11.67, for compound 4 less than $15.17 4) Surplus. Chemical 1 requirement is satisfied exactly. 366.667 gallons of chemical 2 are used above the minimum requirement of 925 gallons 5) 1291.667 gallons 6) $1020.87 7) It would not change, because $7.25 is within the objective coefficient range. The total cost would go up by $32.29