Operations Research : Applications and Algorithms
Operations Research : Applications and Algorithms
4th Edition
ISBN: 9780534380588
Author: Wayne L. Winston
Publisher: Brooks Cole
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Chapter 6.7, Problem 2P

Explanation of Solution

a.

Duel of the LP:

Maximize, z=-2x1-x2+x3

Subject to the constraints,

x1+x2+x33x2+3x32x1+x3=0

And x1,x2,x30

In the above LP formulation, all constraints can be converted to type by multiplying both sides by -1. Then the LP becomes,

Maximize, z=-2x1-x2+x3

Subject to the constraints,

x1+x2+x33

Explanation of Solution

b.

Optimal solution of Dual of the LP:

Minimize, z=3x12x2

Subject to the constraints,

x1+x32

In the above, b1=-2<0

The LP problem is converted to canonical form by adding surplus, slack, and artificial variables in appropriate places.

  • The user should add slack variable S1, as the constraint -1 is of type “”.
  • The user should subtract the surplus variable S2 and add artificial variable A1, as the constraint -2 of type “”.
  • The user should add slack variable S3, as the constraint -3 is of type “”.

After introducing slack, surplus, artificial variables

Minimize, Z=3x1-2x2+0x3'+0x3''+0S1+0S2+0S3+MA1

Subject to the constraints,

x1x3'+x3''+S1=2x1+x2S2+A1=1x13x2+x3'x3''+S3=1

And x1,x2,x3',x3'',S1,S2,S3,A10

Iteration-1 Cj3-200000M

Min Ratio

BCBXBx1x2x3'x3''S1S2S3A1XBx2
S102-10-111000
A1M1-1(1)000-10111=1
S3011-31-10010
-MA2101/20-1/201 2 
Z=M Zj-MM000-M0M 
  ZjCj-M-3M+2000-M00 

Positive maximum ZjCj is M+2 and its column index is 2

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Chapter 6 Solutions

Operations Research : Applications and Algorithms

Ch. 6.3 - Prob. 4PCh. 6.3 - Prob. 5PCh. 6.3 - Prob. 6PCh. 6.3 - Prob. 7PCh. 6.3 - Prob. 8PCh. 6.3 - Prob. 9PCh. 6.4 - Prob. 1PCh. 6.4 - Prob. 2PCh. 6.4 - Prob. 3PCh. 6.4 - Prob. 4PCh. 6.4 - Prob. 5PCh. 6.4 - Prob. 6PCh. 6.4 - Prob. 7PCh. 6.4 - Prob. 8PCh. 6.4 - Prob. 9PCh. 6.4 - Prob. 10PCh. 6.4 - Prob. 11PCh. 6.4 - Prob. 12PCh. 6.4 - Prob. 13PCh. 6.5 - Prob. 1PCh. 6.5 - Find the duals of the following LPs: Ch. 6.5 - Prob. 3PCh. 6.5 - Prob. 4PCh. 6.5 - Prob. 5PCh. 6.5 - Prob. 6PCh. 6.6 - Prob. 1PCh. 6.6 - Prob. 2PCh. 6.7 - Prob. 1PCh. 6.7 - Prob. 2PCh. 6.7 - Prob. 3PCh. 6.7 - Prob. 4PCh. 6.7 - Prob. 5PCh. 6.7 - Prob. 6PCh. 6.7 - Prob. 7PCh. 6.7 - Prob. 8PCh. 6.7 - Prob. 9PCh. 6.8 - Prob. 1PCh. 6.8 - Prob. 2PCh. 6.8 - Prob. 3PCh. 6.8 - Prob. 4PCh. 6.8 - Prob. 5PCh. 6.8 - Prob. 6PCh. 6.8 - Prob. 8PCh. 6.8 - Prob. 9PCh. 6.8 - Prob. 10PCh. 6.8 - Prob. 11PCh. 6.9 - Prob. 1PCh. 6.9 - Prob. 2PCh. 6.9 - Prob. 3PCh. 6.10 - Prob. 1PCh. 6.10 - Prob. 2PCh. 6.10 - Prob. 3PCh. 6.11 - Prob. 1PCh. 6.11 - Prob. 3PCh. 6.11 - Prob. 4PCh. 6.12 - Prob. 5PCh. 6.12 - Prob. 6PCh. 6.12 - Prob. 7PCh. 6 - Prob. 1RPCh. 6 - Prob. 2RPCh. 6 - Prob. 3RPCh. 6 - Prob. 4RPCh. 6 - Prob. 5RPCh. 6 - Prob. 6RPCh. 6 - Prob. 7RPCh. 6 - Prob. 8RPCh. 6 - Prob. 9RPCh. 6 - Prob. 10RPCh. 6 - Prob. 11RPCh. 6 - Prob. 13RPCh. 6 - Prob. 14RPCh. 6 - Prob. 15RPCh. 6 - Prob. 17RPCh. 6 - Prob. 18RPCh. 6 - Prob. 19RPCh. 6 - Prob. 20RPCh. 6 - Prob. 21RPCh. 6 - Prob. 22RPCh. 6 - Prob. 25RPCh. 6 - Prob. 29RPCh. 6 - Prob. 33RPCh. 6 - Prob. 34RPCh. 6 - Prob. 35RPCh. 6 - Prob. 36RPCh. 6 - Prob. 37RP
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