3. Matchpoint Company produces 3 types of tennis balls: Heavy Duty, Regular, and
ID: 461197 • Letter: 3
Question
3. 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.
How many balls of each type will Matchpoint product?
Which constraints are limiting and which are not? Explain.
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?
By how much would the profit contribution of Regular balls have to increase to make it profitable for Matchpoint to start producing Regular balls?
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?
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 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
Explanation / Answer
1. How many balls of each type will Matchpoint product?
It should produce 200 Heavy Duty, 0 Regular, and 100 Extra Duty balls.
2. Which constraints are limiting and which are not? Explain.
Manufacturing and testing constraints are limiting, as increasing them will lead to change in optimal solution and increasing in profit. Canning is not a limiting constraint, and increasing it will not have any benefit.
3. How much would you be willing to pay for an extra man-hour of testing capacity?
$ 10 . refer to dual price, unit increase in testing capacity will lead to $ 10 increase in total profit contribution.
4. For how many additional man-hours of testing capacity is this marginal value valid? Why?
These marginal values are valid up to 120 hours of increase in testing capacity, i.e. for total testing capacity up to 320 hours. This is evident from the allowable increase in the right hand side ranges.
5. By how much would the profit contribution of Regular balls have to increase to make it profitable for Matchpoint to start producing Regular balls?
Profit contribution of regular balls has to increase by at least $ 8 to make it profitable for matchpoint to start producing regular balls. This is evident from allowable increase of obj coefficient ranges.
6. 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?
Profit Contribution of heavy duty has to decrease by $ 6 before matchpoint would find it profitable to change its production plan. This is evident from allowable decrease in obj coefficient ranges.
7. 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.
With the given resource constraints, matchpoint should not produce any of the Special duty balls, because it does not have any spare capacity in manufacturing or testing. And its profit contribution per man-hour of manufacturing (33/1.5 = 22) or testing (33/0.75 = 44) is less than the profit contribution of extra duty balls, and contribution per mfg hour is less than that of heavy duty balls (24/.75 = 32). Therefore matchpoint should rather produce more of extra duty or heavy duty balls if there is an increase in capacity.
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