The Porsche Club of America sponsors driver education events that provide high-performance driving instruction on actual race tracks. Because safety is a primary consideration at such events, many owners elect to install roll bars in their cars. Deegan Industries manufactures two types of roll bars for Porsches. Model DRB is bolted to the car using existing holes in the car's frame. Model DRW is a heavier roll bar that must be welded to the car's frame. Model DRB requires 20 pounds of a special high alloy steel, 40 minutes of manufacturing time, and 60 minutes of assembly time. Model DRW requires 25 pounds of the special high alloy steel, 100 minutes of manufacturing time, and 40 minutes of assembly time. Deegan's steel supplier indicated that at most 36,000 pounds of the high-alloy steel will be available next quarter. In addition, Deegan estimates that 2,000 hours of manufacturing time and 1,800 hours of assembly time will be available next quarter. The profit contributions are $200 per unit for model DRB and $280 per unit for model DRW. The linear programming model for this problem is as follows:Max 200DRB + 280DRW s.t. 20DRB + 25DRW ≤ 36,000 Steel available 40DRB + 100DRW ≤ 120,000 Manufacturing minutes 60DRB + 40DRW ≤ 108,000 Assembly minutes DRB, DRW ≥ 0 The computer solution is shown below.Optimal Objective Value = 388800.00000VariableValueReduced CostDRB600.000000.00000DRW960.000000.00000 ConstraintSlack/SurplusDual Value10.000008.8000020.000000.60000333600.000000.00000 VariableObjectiveCoefficientAllowableIncreaseAllowableDecreaseDRB200.0000024.0000088.00000DRW280.00000220.0000030.00000 ConstraintRHSValueAllowableIncreaseAllowableDecrease136000.000007636.363646000.000002120000.0000024000.0000048000.000003108000.00000Infinite33600.00000(a) What is the optimal solution and the total profit contribution (in $)?DRB=600, DRW=960, total profit= 388800(b) Another supplier offered to provide Deegan Industries with an additional 500 pounds of the steel alloy at $2 per pound. Should Deegan purchase the additional pounds of the steel alloy? Explain.Yes, the dual value for steel available is 8.8. Each pound of steel will increase profits more than the $2 per pound that the supplier is offering(c) Deegan is considering using overtime to increase the available assembly time. What would you advise Deegan to do regarding this option? Explain.Constraint 3 has a slack. Increasing the number of hours of assembly time will not improve profits.(d) Because of increased competition, Deegan is considering reducing the price of model DRB such that the new contribution to profit is $175 per unit. How would this change in price affect the optimal solution? Explain.The objective coefficient range for model DRB shows a lower limit of $112. Thus, the optimal solution will not change and the new value will be $________?(e) If the available manufacturing time is increased by 500 hours, will the dual value for the manufacturing time constraint change? Explain.The allowable increase is ________ minutes, so the value for the constraints will change.** In d I have found the lower limit but the upper limit value (224) is incorrect with my calculations and in e I used 30000 (500 hours=30000 minutes) however, this is not the correct output.

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Description: The Porsche Club of America sponsors driver education events that provide high-performance driving instruction on actual race tracks. Because safety is a primary consideration at such events, many owners elect to install roll bars in their cars. Deeg

Answered: (d) Because of increased competition,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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The Porsche Club of America sponsors driver education events that provide high-performance driving instruction on actual race tracks. Because safety is a primary consideration at such events, many owners elect to install roll bars in their cars. Deegan Industries manufactures two types of roll bars for Porsches. Model DRB is bolted to the car using existing holes in the car's frame. Model DRW is a heavier roll bar that must be welded to the car's frame. Model DRB requires 20 pounds of a special high alloy steel, 40 minutes of manufacturing time, and 60 minutes of assembly time. Model DRW requires 25 pounds of the special high alloy steel, 100 minutes of manufacturing time, and 40 minutes of assembly time. Deegan's steel supplier indicated that at most 36,000 pounds of the high-alloy steel will be available next quarter. In addition, Deegan estimates that 2,000 hours of manufacturing time and 1,800 hours of assembly time will be available next quarter. The profit contributions are $200 per unit for model DRB and $280 per unit for model DRW. The linear programming model for this problem is as follows:

Max	200DRB	+	280DRW
s.t.
	20DRB	+	25DRW	≤	36,000	Steel available
	40DRB	+	100DRW	≤	120,000	Manufacturing minutes
	60DRB	+	40DRW	≤	108,000	Assembly minutes
			DRB, DRW	≥	0

The computer solution is shown below.

Optimal Objective Value = 388800.00000

Variable	Value	Reduced Cost
DRB	600.00000	0.00000
DRW	960.00000	0.00000

Constraint	Slack/Surplus	Dual Value
1	0.00000	8.80000
2	0.00000	0.60000
3	33600.00000	0.00000

Variable	Objective Coefficient	Allowable Increase	Allowable Decrease
DRB	200.00000	24.00000	88.00000
DRW	280.00000	220.00000	30.00000

Constraint	RHS Value	Allowable Increase	Allowable Decrease
1	36000.00000	7636.36364	6000.00000
2	120000.00000	24000.00000	48000.00000
3	108000.00000	Infinite	33600.00000

(a) What is the optimal solution and the total profit contribution (in $)?

DRB=600, DRW=960, total profit= 388800

(b)
Another supplier offered to provide Deegan Industries with an additional 500 pounds of the steel alloy at $2 per pound. Should Deegan purchase the additional pounds of the steel alloy? Explain.

Yes, the dual value for steel available is 8.8. Each pound of steel will increase profits more than the $2 per pound that the supplier is offering

(c) Deegan is considering using overtime to increase the available assembly time. What would you advise Deegan to do regarding this option? Explain.

Constraint 3 has a slack. Increasing the number of hours of assembly time will not improve profits.

(d) Because of increased competition, Deegan is considering reducing the price of model DRB such that the new contribution to profit is $175 per unit. How would this change in price affect the optimal solution? Explain.

The objective coefficient range for model DRB shows a lower limit of $112. Thus, the optimal solution will not change and the new value will be $________?

(e) If the available manufacturing time is increased by 500 hours, will the dual value for the manufacturing time constraint change? Explain.

The allowable increase is ________ minutes, so the value for the constraints will change.

** In d I have found the lower limit but the upper limit value (224) is incorrect with my calculations and in e I used 30000 (500 hours=30000 minutes) however, this is not the correct output.

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