Duals of LP:

Consider the linear programming problem given below:

max z= 4y1+2y2−y3

Subject to constraints

y1+2y2≤6

y1−y2+2y3=8

y1,y2≥0,y3

x1+x2≤5

2x1+x2≤7

2x2+x3≥6

x1+x3=4

x1≤0,x2,x3 (unrestricted in sign(urs))

As the above linear programming problem is a normal min problem, and therefore, perform the following steps to convert it into a normal form.

Step1:

Multiply each ≤ constraint by -1 as follows,

y1+2y2≤6

−(y1+2y2)≥−6

−y1−2y2≥−6

Step 2:

Replace the equality constraint y1−y2+2y3=8 by two inequality constraints as,

y1−y2+2y3≥8

y1−y2+2y3≤8

Now convert ≤ constraint ≥ constraint as,

−y1+y2−2y3≥8

Step 3:

Now replace each urs variable y1 by yi=yi'−yi'' as,

y3=y3'−y3''

Here,

y3',y3''≥0

Now the standard min LP is,

min w= 4y1+2y2−y3'+y3''

Subject to constraints,

−y1−2y2≥−6

y1−y2+2y3'−2y3''≥8

−y1+y2−2y3'+2y3''≥−8

y1,y2,y3',y3''≥0

As the above linear programming problem is a normal min problem, it can be solved with the tabular approach as follows

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Chapter 6 Solutions

Student Suite Cd-rom For Winston's Operations Research: Applications And Algorithms

Ch. 6.1 - Prob. 1P Ch. 6.1 - Prob. 2P Ch. 6.1 - Prob. 3P Ch. 6.1 - Prob. 4P Ch. 6.1 - Prob. 5P Ch. 6.2 - Prob. 1P Ch. 6.2 - Prob. 2P Ch. 6.3 - Prob. 1P Ch. 6.3 - Prob. 2P Ch. 6.3 - Prob. 3P

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

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