ornotes2-2007

# ornotes2-2007 - SOLVING LINEAR PROGRAMMING PROBLEMS Three...

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SOLVING LINEAR PROGRAMMING PROBLEMS Three basic methods to solve LP problems: 1) Graphical: can be used when there are two decision variables. (Some people can solve three-dimensional graphical problems but I am not one of them.) 2) Matrix algebra: can theoretically be used on any size problem but become cumbersome quickly. 3) Computer algorithms: can do the mathematical calculations quickly, allowing us to solve problems with thousands of decision variables and constraints. A problem of diet (from Alpha Chiang, 2nd Edition) Assume a person's diet is to consist of only two foods, food I and food II, and that there are only three types of nutrients that must be considered: calcium, protein, and calories. Information on the two foods: Minimum Daily Food 1(lb) Food 2 (lb) Requirement Price \$0.60 \$1.00 Units calcium per pound 10 4 20 Units protein per pound 5 5 Calories per pound 2 6 12

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The problem is to minimize cost subject to the nutrition restrictions: Minimize z= \$0.60x1 + \$1.00x2 subject to: 10x1 + 4x2 20 5x1 + 5x2 20 2x1 + 6x2 12 x1, x2 0 The graphical solution: The points where the constraints cross each other or an axis are called "corner points."
direction of decreasing cost Isocost line X2 = Cost - 0.6 X1 We generate a family of parallel lines by changing the value of "Cost" and plotting the lines (dashed lines above). The solution to our problem will be at the "corner point" (3,1). Because the corner points "stick out" the solution to the LP will always be at one of these. Although, for any problem, the border of the feasible region has an infinite number of points, there are always a finite number of corner points, which makes solution of the LP problem possible.

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Graphing a Maximization Problem A firm produces 2 products, with a plant that has 3 production departments: cutting, mixing, and packaging. The equipment in each department can be used for 8 hours a day. Product 1 has a gross margin (returns above variable costs) of \$40 a ton and needs 1/2 hour of cutting, no mixing, and 1/3 hour of packaging per ton. product 2 has a gross margin of \$30 per ton and needs no
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## This note was uploaded on 11/15/2011 for the course AGEC 7100 taught by Professor Duffy,p during the Fall '08 term at Auburn University.

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ornotes2-2007 - SOLVING LINEAR PROGRAMMING PROBLEMS Three...

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