B.1 The LP relationships that follow were formulated by Richard Martin at the Long Beach Chemical Company. Which ones are invalid for use in a linear programming problem, and why?
Maximize = 6X1 + 12 X1X2 + 5X3
Subject to: 4X1X2 + 2X3 ≤ 70
7.9X1 – 4X2 ≥ 15.6
3X1 + 3X2 + 3X3 ≥ 21
19X2 – 13 X3 = 17
-X1-X2 + 4X3 = 5
4X1 + 2X2 + 3 √X3 ≤ 80
B.5 The Attaran Corporation manufactures two electrical products: portable air conditioners and portable heaters. The assembly process for each is similar in that both require a certain amount of wiring and drilling. Each air conditioner takes 3 hours of wiring and 2 hours of drilling. Each heater must through 2 hours of wiring and 1 hour of drilling. During the next production period, 240 hours of wiring time are available and up to 140 hours of drilling time may be used. Each air conditioner sold yields a profit of $25. Each heater assembled may be sold for a $15 profit.
Formulate and solve this LP production-mix situation, and find the best combination of air conditioners and heaters that yields the highest profit.
B.7 Green Vehicle, Inc. manufactures electric cars and small delivery trucks. It has just opened a new factory where the C1 car and the T1 truck can both be manufactured. To make either vehicle, processing in the assembly shop and in the paint shop are required. It takes 1/40 of a day and 1/60 of a day to paint a truck of type T1 and a car of type C1 in the paint shop, respectively. It takes 1/50 of a day to assemble either type of vehicle in the assembly shop. A T1 truck and a C1 car yield profits of $300 and $220, respectively, per vehicle sold. Formulate and solve an LP to maximize profit for the company.
Define the objective function and constraint equations.
Graph the feasible region.
What is a maximum profit daily production plan at the new factory?
How much profit will such a plan yield, assuming whatever is produced is sold?
B.11
Each coffee table produced by Kevin Watson Designers nets the firm a profit of $9. Each bookcase yields a $15 profit. Watson’s firm is small and its resources limited. During any given production period (of 1 week), 10 gallons of varnish and 12 lengths of high-quality redwood are available. Each coffee table requires approximately 1 gallon of varnish and 1 length of redwood. Each bookcase takes 1 gallon of varnish and 2 lengths of wood.
Formulate Watson’s production-mix decision as a linear programming problem, and solve. How many tables and book-cases should be produced each week? What will the maximum profit be?
B.27
Doug Turner Food Processors wishes to introduce a new brand of dog biscuits composed of chicken- and liver-flavored biscuits that meet certain nutritional requirements. The liver-flavored biscuits contain 1 unit of nutrient A and 2 units of nutrient B; the chicken-flavored biscuits contain 1 unit of nutrient A and 4 units of nutrient B. According to federal requirements, there must be at least 40 units of nutrient A and 60 units of nutrient B in a package of the new mix. In addition, the company has decided that there can be no more than 15 liver-flavored biscuits in a package. If it costs 1¢ to make 1 liver-flavored biscuit and 2¢ to make 1 chicken-flavored, what is the optimal product mix for a package of the biscuits to minimize the firm’s cost?
a) Formulate this as a linear programming problem.
b) Solve this problem graphically, giving the optimal values of all variables.
c) What is the total cost of a package of dog biscuits using the optimal mix?
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