L16_Some problems illustrating the principles of duality

L16_Some problems illustrating the principles of duality -...

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Some problems illustrating the principles of duality

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In this lecture we look at some problems that uses the results from Duality theory (as discussed in Chapter 7).
Problem 7. Problem Set 4.2D Page 130 Consider the LPP Maximize subject to The following optimal tableau corresponds to specific values of b 1 and b 2 : 3 2 1 3 2 5 x x x z + + = 1 2 3 1 1 2 3 2 1 2 3 5 3 5 6 , , 0 x x x b x x x b x x x + + - -

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Basic z x 1 x 2 x 3 x 4 x 5 Sol z x 1 x 5 1 0 0 0 1 0 a b c 7 2 -8 d 1 -1 e 0 1 150 30 10 Determine the following: (a)The RHS values b 1 , b 2 (b)The entries a, b, c, d, e (c)The optimal solution of the dual.
(a) The optimal solution of the primal is B -1 b, where = - 1 B - 1 1 0 1 1 1 2 b b b   =   - +   Thus 2 1 b b Hence b 1 = 30, b 2 = 40 (b) 1 2 B A - = Hence b = 5, c = -10 b = b c   =     5 10     -   1 0 5 1 1 5     =     - -     B -1 b = 30 10      

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a = 1 2 2 2 2 B z c C B A c - - = - = 23 5 e = 0 as x 5 is a basic variable. (c) The optimal solution of the dual is [ ] 1 1 2 B y y C B - = = Note: The optimal solution of the dual is also given by [ d , e ] = [5, 0]. 1 4 4 4 4 B z c C B A c - - = - = d = [ ] 5 5 0 2 10   - =   -   [ ] 1 5 0 0 1   - =   -   [ ] 1 0 5 0 1 1   =   -   [ ] 5 0
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L16_Some problems illustrating the principles of duality -...

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