lp1 - 1 Math 1b Practical March 6 2009 Intoduction to...

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1 Math 1b Practical, March 6, 2009 Intoduction to Linear Programming — Part 1 A linear programming problem (LP problem) is one that asks for the minimum or maximum of some linear function (the ‘objective function’) of variables x 1 ,x 2 ,...,x n over a domain deFned by linear inequalities and equations (linear ‘constraints’) on x . Example A. Maximize 2 x 1 + x 2 + x 4 subject to 3 x 1 +2 x 2 x 3 2 x 4 =1 1 + x 3 x 4 =2 1 0 2 0 3 0 4 0 . Example B. Minimize x + y + z subject to x y z 5 + y 4 z 7 0 ,y 0 ,z 0 . Example C. Minimize x + y + z subject to x + y 2 y +3 z 11 , 7 x y + z 3 , 2 x + y + z 3 . Often the constraints include the requirement that the variables be nonnegative, but that is not required in general. Practical examples arise in many areas. Suppose a factory makes products A, B, and C. Each requires ‘cutting’, ‘folding’, and ‘packaging’. The times, in person-hours, required for each product to be processed are listed below, along with the total number of person-hours available in each department, and also the proFt per unit of the products. ABC available cutting 10 5 2 2000 folding 3 9 4 1500 packaging 1 1 2 400 proFt 10 15 20 You are entrusted to maximize proFts. You decide to make x units of A, y units of B, and z units of C. You might want x, y, z to be integers, but let’s ignore that. But you
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This note was uploaded on 09/25/2010 for the course MATH 1bPRAC taught by Professor Wilson during the Winter '09 term at Caltech.

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lp1 - 1 Math 1b Practical March 6 2009 Intoduction to...

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