Linear_systems - OR 320/520 Optimization I Fall 2007 Prof....

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Unformatted text preview: OR 320/520 Optimization I Fall 2007 Prof. Bland A manufacturer has 80 tons of natural beef, 120 tons of natural pork, and 210 tons of recycled magazines on hand for the production of Grandmas All Natural Sausages during the current period. There are two types of sausages whose recipes and profitability are given below. (These recipes include only the ingredients in scarce supply and are calibrated in units called batches.) Type 1 Type 2 All Natural Beef (tons) 2 2 All Natural Pork (tons) 4 2 All Natural Magazines (tons) 3 6 Profit ($1000) 20 30 Grandma needs to determine a production program for this period to yield the maximum profit from the production and sale of type 1 and type 2 sausages. Systems of Linear Equations and Linear Optimization Now we will attempt to relate our understanding of how to solve systems of linear equations to linear programming: maximizing (or minimizing) a linear function in nonnegative variables subject to linear inequalities. To begin consider a system of m equations in n variables x 1 , . . . , x n e.g. with m = 3 , n = 5 2 x 1 + 2 x 2 + x 3 = 80 4 x 1 + 2 x 2 + x 4 = 120 3 x 1 + 6 x 2 + x 5 = 210 in general a 11 x 1 + . . . + a 1 n x n = b 1 . . . a m 1 x 1 + . . . + a mn x n = b m or, more compactly, Ax = b where A = a 11 . . . a 1 n . . . a m 1 . . . a mn , b = b 1 . . . b m x = x 1 . . . x n In our example A = 2 2 1 4 2 1 3 6 1 b = 80 120 210 x = x 1 x 2 x 3 x 4 x 5 Suppose we take [ A ....
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Linear_systems - OR 320/520 Optimization I Fall 2007 Prof....

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