Dual Problem of an LPP

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Dual Problem of an LPP Given a LPP (called the
primal problem), we shall associate another LPP
called the dual problem of the original (primal)
problem. We shall see that the Optimal values of
the primal and dual are the same provided both
have finite feasible solutions. This topic is
further used to develop another method of solving
LPPs and is also used in the sensitivity (or
post-optimal) analysis.
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Definition of the dual problem
Given the primal problem (in standard form)
Maximize
subject to
3
the dual problem is the LPP
Minimize
subject to
4
If the primal problem (in standard form) is
Minimize
subject to
5
Then the dual problem is the LPP
Maximize
subject to
6
We thus note the following
1. In the dual, there are as many (decision)
variables as there are constraints in the
primal. We usually say yi is the dual
variable associated with the ith constraint of
the primal. 2. There are as many constraints in
the dual as there are variables in the primal.
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3. If the primal is maximization then the dual is
minimization and all constraints are ? If
the primal is minimization then the dual is
maximization and all constraints are ?

• In the primal, all variables are ? 0 while in the
  dual all the variables are unrestricted in sign.

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5. The objective function coefficients cj of the
primal are the RHS constants of the dual
constraints. 6. The RHS constants bi of the
primal constraints are the objective function
coefficients of the dual. 7. The coefficient
matrix of the constraints of the dual is the
transpose of the coefficient matrix of the
constraints of the primal.
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Problem 4(a) Problem Set 4.1A Page 119 Write
the dual of the LPP
Maximize
subject to
10
Thus the primal in the standard form is
Maximize
subject to
11
Hence the dual is
Minimize
subject to
12
Problem 4(b) Problem Set 4.1A Page 119 Write
the dual of the LPP
Minimize
subject to
13
Thus the primal in the standard form is
Minimize
subject to
14
Hence the dual is Maximize
subject to
15
Problem 4(c) Problem Set 4.1A Page 119 Write
the dual of the LPP
Maximize
subject to
16
Thus the primal in the standard form is
Maximize
subject to
17
Hence the dual is
Minimize
subject to
18
From the above examples we get the following SOB
rules for writing the dual
Label Maximization Minimization
Constraints
Variables
Sensible ? form
? 0
Odd form
unrestricted
Bizarre ? form
? 0
Variables
Constraints
Sensible ? 0
? form
Odd unrestricted
form
Bizarre ? 0
? form
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Theorem The dual of the dual is the primal
(original problem).
Proof. Consider the primal problem (in standard
form)
Maximize
subject to
20
The dual problem is the LPP
Minimize
subject to
21
Case (i) All cj ? 0. Then the dual problem in
the standard form is the LPP
Minimize
subject to
22
Hence its dual is the LPP
Maximize
subject to
23
Which is nothing but the LPP
Maximize
subject to
This is the primal problem.
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Case (ii) All cj ? 0 except c1. Then the dual
problem in the standard form is the LPP
Minimize
Subj-ect to
25
Hence its dual is the LPP
Maximize
subject to
26
Which is nothing but the LPP
Maximize
subject to
Putting
27
This is nothing but the LPP
Maximize
subject to
This is the primal problem. Cases where other cj
are ? 0 are similarly treated.