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mthsc810-lecture02

# mthsc810-lecture02 - Linear programming MthSc 810...

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MthSc 810: Mathematical Programming Lecture 2 Pietro Belotti Dept. of Mathematical Sciences Clemson University August 30, 2011 Reading for today: Chapters 1.1, 1.2, 1.3 Reading for Sep. 1: Chapters 1.4, 2.1, 2.2 Linear programming The optimization problem, with n variables and m constraints: min n i = 1 c i x i s . t . n i = 1 a ji x i = b j j = 1 , 2 . . . , m x i 0 i = 1 , 2 . . . , n is called a LinearProgramming (from now on LP) problem. They are often written in matricial form: min c x s . t . A x = b x 0 A is the ( m × n ) coefficient matrix , b is the right-hand side vector , and c is the objective coefficient vector . Standard form of an LP The problem in the previous slide is said to be written in standard form. An LP in the previous slide is the intersection between an affine subspace ( A x = b ) and the first orthant ( x 0 ). Any LP not in standard form can be rewritten with equations only: for any inequality a x b , introduce a new slack variable s and rewrite the inequality as a x + s = b s must be nonnegative, of course: s 0 If a variable x i is unrestricted in sign (u.r.s.), that is, it is not constrained to be nonnegative or nonpositive, we can replace it with x + i - x i with x + i 0 and x i 0, thus obtaining an LP in standard form. Maximization problems They are not so different from their minimization counterpart. max { f ( x ) : x F } = - min {- f ( x ) : x F } we take the opposite of the objective function only . Example: max { 2 x - 3 : x [ 4 , 5 ] } = - min {- 2 x + 3 : x [ 4 , 5 ] } 7 = - ( - 7 )

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Example: Transportation problem A large manufacturing company produces liquid nytrogen in five plants spread out in East Georgia Each plant has a monthly production capacity Plant i 1 2 3 4 5 Capacity p i 120 95 150 120 140 There are seven retailers in the same area Each retailer has a monthly demand to be satisfied Retailer j 1 2 3 4 5 6 7 Demand d j 55 72 80 110 85 30 78 Transportation between any plant i
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mthsc810-lecture02 - Linear programming MthSc 810...

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