Linear Programming

1. 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. 24×1 + 12×2 + 36×3

Subject to: .75×1 + .75×2 + 1.5×3 < 300 (manufacturing)

.8×1 + .4×2 + .4×3 < 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 ½ and ¾ 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 24×1 + 12×2 + 36×3
Subject to
.75×1 + .75×2 + 1.5×3 <300
.8×1 + .4×2 + .4×3 <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