Problems:
Problem-1
The Southern Sporting Goods Company makes basketballs and footballs. Each product is produced from two resources- rubber and leather. The resource requirements for each product and the total resources available are as follows.
Resource Requirements per Unit
Product Rubber (lb) Leather ft^2)
Basketball 3 4
Football 2 5
Total resources available 500 lb 800 ft^2
Each basketball produced results in a profit of $12, and each football earns $16 in profit.
- Formulate a linear programming model to determine the number of basketballs, and footballs to produce in order to maximize profit.
- Transform this model into standard form.
Problem-2
For this problem, you will need to do problem 5. The solution is provided below.
(A) x1 = no. of basketballs
x2 = no. of footballs
maximize Z = 12x1+16x2
Subject to : 3x1+2x2≤500
4x1+5x2≤800
x1,x2≥0
(B) maximize Z = 12x1+16x2+0s1+0s2
subject to : 3x1+2x2+s1 = 500
4x1+5x2+s2 = 800
x1,x2,s1,s2 ≥0
a) Solve the model formulated for problem 5 graphically by providing the solution vertices, choose the optimum solution, and identify the amount of unused resources (i.e., slack) at each of the extreme points.
b) What would be the effect on the optimal solution if the profit for a basketball changed from $12 to $13? What would be the effect if the profit for a football changed from $16 to $15 (assume the profit for a basketball has been reset to $12)?
c) What would be the effect on the optimal solution if 500 additional pounds of rubber could be obtained? What would be the effect if 500 additional square feet of leather could be obtained? ( each question should be treated separately.)