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).
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    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.
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    (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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