Matchpoint Company produces 3 types of tennis balls: Heavy Duty, Regular, and Extra Duty, with a profit contribution of $24, $12, and $36 per gross (12 dozen), respectively.
The linear programming formulation is: Max. 24x1 + 12x2 + 36x3 Subject to: .75x1 + .75x2 + 1.5x3 < 300 (manufacturing) .8x1 + .4x2 + .4x3 < 200 (testing) x1 + x2 + x3 < 500 (canning) x1, x2, x3 > 0 where x1, x2, x3 refer to Heavy Duty, Regular, and Extra Duty balls (in gross). The LINDO solution is on the following page.
a) How many balls of each type will Matchpoint product?
b) Which constraints are limiting and which are not? Explain.
c) How much would you be willing to pay for an extra man-hour of testing capacity? For how many additional man-hours of testing capacity is this marginal value valid? Why?
d) By how much would the profit contribution of Regular balls have to increase to make it profitable for Matchpoint to start producing Regular balls?
e) By how much would the profit contribution of Heavy Duty balls have to decrease before Matchpoint would find it profitable to change its production plan?
f) Matchpoint is considering producing a low-pressure ball, suited for high altitudes, called the Special Duty. Each gross of Special Duty balls would require 1 ? and ž man-hours of manufacturing and testing, respectively, and would give a profit contribution of $33 per gross.
Special Duty balls would be packed in the same type of cans as the other balls.
Should Matchpoint produce any of the Special duty balls?
Explain; provide support for your answer. Max 24x1 + 12x2 + 36x3 Subject to .75x1 + .75x2 + 1.5x3 <300 .8x1 + .4x2 + .4x3 <200 x1 + x2 + x3 < 500 end
LP OPTIMUM FOUND AT STEP 2 OBJECTIVE FUNCTION VALUE
1) 8400.000 VARIABLE VALUE REDUCED COST X1 200.000000 0.000000 X2 0.000000 8.000000 X3 100.000000 0.000000 ROW SLACK OR SURPLUS DUAL PRICES
2) 0.000000 21.333334
3) 0.000000 10.000000
4) 200.000000 0.000000
NO. ITERATIONS= 2 RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE X1 24.000000 48.000000 6.000000 X2 12.000000 8.000001 INFINITY X3 36.000000 12.000000 24.000000 RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE
2 300.000000 450.000000 112.500000
3 200.000000 120.000000 120.000000
4 500.000000 INFINITY 200.000000