1-hw116 - MATHEMATICS 116, FALL 2007 CONVEXITY AND...

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Unformatted text preview: MATHEMATICS 116, FALL 2007 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Assignment #1 Last modified: September 26, 2007 Due Thursday, Sept. 27 at class 1. The Math 116 bakery has started producing chocolate goods. A batch of fudge squares uses 3 bars of chocolate, 1 pound of sugar, 1 pound of flour. A batch of chocolate cake uses 1 bar of chocolate, 1 pound of sugar, 2 pounds of flour. Let x = number of batches of fudge bars, y = number of batches of cake. These quantities need not be integers, but they cannot be negative. The supplier delivers 15 chocolate bars, 7 pounds of sugar, and 12 pounds of flour. Draw a diagram showing the convex region of the xy-plane that is consistent with all the constraints. Let a and b be the price of a batch of fudge squares and a batch of chocolate cake respectively. Using the graph, determine values of x and y that max- imize the revenue in each of the following cases. For each case, determine the maximum possible revenue, and specify what ingredients are left over. You do not have to show any algebra. Identify any case where the solution is not unique. (a) a = 6, b = 1 (b) a = 3, b = 2 (c) a = 2, b = 3 (d) a = 1, b = 5 (e) a = 2, b = 2 2. Here is a planning problem like the biofuels problem discussed in lecture where the inventory costs are negative....
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1-hw116 - MATHEMATICS 116, FALL 2007 CONVEXITY AND...

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