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71 Consider the following linear program maximize 5x + 3y 5x 2y 0 x+y 7 x 5 x 0 y 0
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Plot the feasible region and identify the optimal solution 72 Duckwheat is produced in Kansas and Mexico and consumed in New York and California Kansas produces 15 shnupells of duckwheat and Mexico 8 Meanwhile, New York consumes 10 shnupells and California 13 The transportation costs per shnupell are $4 from Mexico to New York, $1 from Mexico to California, $2 from Kansas to New York, and $3 and from Kansas to California Write a linear program that decides the amounts of duckwheat (in shnupells and fractions of a shnupell) to be transported from each producer to each consumer, so as to minimize the overall transportation cost 73 A cargo plane can carry a maximum weight of 100 tons and a maximum volume of 60 cubic meters There are three materials to be transported, and the cargo company may choose to carry any amount of each, upto the maximum available limits given below Material 1 has density 2 tons/cubic meter, maximum available amount 40 cubic meters, and revenue $1,000 per cubic meter Material 2 has density 1 ton/cubic meter, maximum available amount 30 cubic meters, and revenue $1,200 per cubic meter Material 3 has density 3 tons/cubic meter, maximum available amount 20 cubic meters, and revenue $12,000 per cubic meter Write a linear program that optimizes revenue within the constraints 74 Moe is deciding how much Regular Duff beer and how much Duff Strong beer to order each week Regular Duff costs Moe $1 per pint and he sells it at $2 per pint; Duff Strong costs Moe $150 per pint and he sells it at $3 per pint However, as part of a complicated marketing scam, the Duff company will only sell a pint of Duff Strong for each two pints or more of Regular Duff that Moe buys Furthermore, due to past events that are better left untold, Duff will not sell Moe more than 3,000 pints per week Moe knows that he can sell however much beer he has Formulate a linear program for deciding how much Regular Duff and how much Duff Strong to buy, so as to maximize Moe s pro t Solve the program geometrically 75 The Canine Products company offers two dog foods, Frisky Pup and Husky Hound, that are made from a blend of cereal and meat A package of Frisky Pup requires 1 pound of cereal and 15 pounds of meat, and sells for $7 A package of Husky Hound uses 2 pounds of cereal and 1 pound of meat, and sells for $6 Raw cereal costs $1 per pound and raw meat costs $2 per pound It also costs $140 to package the Frisky Pup and $060 to package the Husky Hound A total of 240,000 pounds of cereal and 180,000 pounds of meat are available each month The only production bottleneck is that the factory can only package 110,000 bags of Frisky Pup per month Needless to say, management would like to maximize pro t (a) Formulate the problem as a linear program in two variables
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(b) Graph the feasible region, give the coordinates of every vertex, and circle the vertex maximizing pro t What is the maximum pro t possible 76 Give an example of a linear program in two variables whose feasible region is in nite, but such that there is an optimum solution of bounded cost 77 Find necessary and suf cient conditions on the reals a and b under which the linear program max x + y ax + by 1 x, y 0 (a) Is infeasible (b) Is unbounded (c) Has a unique optimal solution 78 You are given the following points in the plane: (1, 3), (2, 5), (3, 7), (5, 11), (7, 14), (8, 15), (10, 19) You want to nd a line ax + by = c that approximately passes through these points (no line is a perfect t) Write a linear program (you don t need to solve it) to nd the line that minimizes the maximum absolute error, max |axi + byi c|
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79 A quadratic programming problem seeks to maximize a quadratric objective function (with terms like 3x2 or 5x1 x2 ) subject to a set of linear constraints Give an example of a quadratic program 1 in two variables x1 , x2 such that the feasible region is nonempty and bounded, and yet none of the vertices of this region optimize the (quadratic) objective 710 For the following network, with edge capacities as shown, nd the maximum ow from S to T , along with a matching cut
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