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Unformatted text preview: Economics 146 Fall, 2007 Linear Programming Duality Last revision: 11/14/07. A Small Example Revisited. Lets take another look at the Blending Model example given at the beginning of the notes on Linear Programming. Darwin was trying to blend two different pellet foods to feed his tortoises as cheaply as possible while providing (at least) the minimal nutritional requirements. Suppose that a peddler of nutritional pills approached him and offered to give him a way that he could meet the tortoises requirements without having to bother with buying and mixing pellet feeds. The peddlers idea is to to sell Darwin pills each one of which has exactly one unit of a nutrient. (He has three kinds of pills one for each nutrient.) Moreover, he will price the pills so that the cost to Darwin of replacing the nutritional content of each unit of pellet food by these pills will be no more than what he is currently spending. Of course, the peddler will set prices to maximize his own return. Let x 1 ,x 2 ,x 3 be the respective prices per pill of the first, second, and third type of nutrient pill. The peddler has a LP maximization problem: find x 1 ,x 2 ,x 3 so as to maximize 60 x 1 + 84 x 2 + 72 x 3 (1) subject to 3 x 1 + 7 x 2 + 3 x 3 10 (2) 2 x 1 + 2 x 2 + 6 x 3 4 (3) and x 1 , x 2 , x 3 (4) It is the relationship of Darwins problem and the peddlers problem that we want to investigate. The Dual of a Linear Program. Every linear programming problem is closely re- lated to an LP problem which is called the dual of the original problem. There are deep connections both mathematical and economic between an LP problem and its dual. Suppose we are given data an m n matrix A , an m-vector c and an n-vector b . Recall that an LP maximization problem in standard form is to find values of x 1 ,...,x m so as to maximize m X i =1 c i x i (5) subject to m X i =1 x i a ij b j for j = 1 ,...,n (6) and x i 0 for i = 1 ,...,m (7) 1 The dual of this LP maximization problem in standard form is to find y 1 ,...,y n so as to minimize n X j =1 b j y j (8) subject to n X j =1 a ij y j c i for i=1,. . . ,m (9) and y j 0 for j = 1 ,...,n (10) In matrix-vector notation we can write the standard form of the LP maximization problem as: find x R m so as to maximize c T x (11) subject to x T A b T (12) and x (13) In matrix-vector notation we can write the dual in standard form as: find y R n so as to minimize b T y (14) subject to Ay c (15) and y (16) Note that A , b , and c are common data for the original problem and its dual. The definition of the dual of an LP minimization problem in standard form is entirely analogous to that of the maximization problem....
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- Fall '07