dual1
TRANSCRIPT
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Dual Problem of an LPP
Given a LPP (called the primal problem), weshall associate another LPP called the dual
problem of the original (primal) problem. We
shall see that the Optimal values of the primaland dual are the same provided both have
finite feasible solutions. his topic is further
used to develop another method of solvingLPPs and is also used in the sensitivit! (or
post"optimal) anal!sis.
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Definition of the dual problem
Given the primal problem (in standard form)
#a$imi%enn
xcxcxcz +++= ...&&''
subect to'' ' '& & ' '
&' ' && & & &
' ' & &
' & ' &
...
...
.
.
...
, , ..., ), , ,..., )
n n
n n
m m mn n m
n m
a x a x a x b
a x a x a x b
a x a x a x b
x x x b b b
+ + + =
+ + + =
+ + + =
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the dual problem is the LPP
#inimi%e mmybybybw +++= ...&&''
subect to
'' ' &' & ' '
'& ' && & & &
' ' & &
' &
...
...
.
....
, ,..., unrestricted in sign
m m
m m
n n mn m n
n
a y a y a y c
a y a y a y c
a y a y a y c
y y y
+ + +
+ + +
+ + +
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*f the primal problem (in standard form) is
#inimi%e nnxcxcxcz +++= ...&&''
subect to'' ' '& & ' '
&' ' && & & &
' ' & &
' & ' &
...
...
.
....
, ,..., ), , ,..., )
n n
n n
m m mn n m
n m
a x a x a x b
a x a x a x b
a x a x a x b
x x x b b b
+ + + =
+ + + =
+ + + =
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hen the dual problem is the LPP
#a$imi%e mmybybybw +++= ...&&''
subect to
'' ' &' & ' '
'& ' && & & &
' ' & &
' &
...
...
.
.
...
, ,..., unrestricted in sign
m m
m m
n n mn m n
n
a y a y a y c
a y a y a y c
a y a y a y c
y y y
+ + +
+ + +
+ + +
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'. *n the dual, there are as man! (decision)variables as there are constraints in the
primal.
We usuall! sa! yiis the dual variableassociated with the ithconstraint of the
primal.
&. here are as man! constraints in the dual as
there are variables in the primal.
We thus note the following+
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. *f the primal is ma$imi%ation then the
dual is minimi%ation and all constraintsare
*f the primal is minimi%ation then the dualis ma$imi%ation and all constraints are
-. *n the primal, all variables are while in
the dualall the variables are
unrestricted in sign.
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. he obective function coefficients cjof
the primal are the RHS constantsof the
dual constraints.
/. he RHS constants biof the primal
constraints are the objective functioncoefficientsof the dual.
0. he coefficient matrix of the constraintsof the dual is the transpose of the
coefficient matrix of the constraintsof
the primal.
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Problem '
Write the dual of the LPP
&' & xxz +=
subect to
),
&
&
&'
&'
&'
+
+
xx
xx
xx
#a$imi%e
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hus the primal in the standard form is+
-&' ))& xxxxz +++=subect to
,,,
&&
-&'
-&'
&'
=++
=
xxxx
xxx
xxx
#a$imi%e
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1ence the dual is+
&' & yyw +=
subect to
' &
' &
'
&
' &
&
&
)
)
, unrestricted in sign
y y
y y
y
y
y y
+
+
),)
&' yy
Minimize
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Problem &
Write the dual of the LPP
&' / xxz +=
subect to
,,
-
&/
&'
&'
&'
++
+
xxx
xxx
xxx
#inimi%e
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hus the primal in the standard form is+
.-&' )))/ xxxxxz ++++=
subect to
,,,,
-&/
-&'
&'
-&'
=++
=+
xxxxx
xxxx
xxxx
#inimi%e
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1ence the dual is+
Maximize&' & yyw
+=
subect to
' &
' &
' &
'
&
' &
/ /
-
)
))
, unrestricted in sign
y y
y y
y y
yy
y y
+
+
+
), &' yy
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Problem
Write the dual of the LPP
&' xxz +=
subect to
' &
' &
' &
&
/, unrestricted in sign
x x
x x
x x
+ =
=
#a$imi%e
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#a$imi%e++
+= &&'' xxxxz
subect to
),,,
/
&&
&&''
&&''
&&''
=+
=+
++
++
++
xxxx
xxxx
xxxx
hus the primal in the standard form is+
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Minimize&' / yyw +=
subect to
' &
' &
' &
' &
' &
& '
& '
'
'
, unrestricted in sign
y y
y y
y y
y y
y y
+
+
'& &' =+ yy
'&' = yy
1ence the dual is+
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2rom the above e$amples we get the
following Brules for writing the dual+
Label #a$imi%ation #inimi%ation
Constraints ariables
Sensible form !dd 3 form unrestricted"i%arre form
ariables ConstraintsSensible form!dd unrestricted 3 form
"i%arre
form
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#heorem$he dual of the dual is the primal
(original problem).
%roof. 4onsider the primal problem (in standardform)
#a$imi%e nnxcxcxcz +++= ...&&''
subect to '' ' '& & ' '
&' ' && & & &
' ' & &
' &
...
...
.
.
...
, ,..., )
n n
n n
m m mn n m
n
a x a x a x b
a x a x a x b
a x a x a x b
x x x
+ + + =
+ + + =
+ + + =
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he dual problem is the LPP
#inimi%e mmybybybw +++= ...&&''
subect to'' ' &' & ' '
'& ' && & & &
' ' & &
' &
...
...
.
....
, ,..., unrestricted in sign
m m
m m
n n mn m n
n
a y a y a y c
a y a y a y c
a y a y a y c
y y y
+ + +
+ + +
+ + +
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4ase (i)+ 5ll c. hen the dual problem in
the standard form is the LPP
#inimi%e
n
mmmm
ttt
ybybybybw
)...))
...
&'
''''
++++
++= ++
subect to
),...,,,,,...,,
...
.
.
...
...
&'''
''''
&&&&''&''&
''''''''''
=++
=++
=++
++
++
++
++
nmm
nnmmnmmnnn
mmmm
mmmm
tttyyyy
ctyayayaya
ctyayayaya
ctyayayaya
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1ence its dual is the LPP
#a$imi%enn
xcxcxcz +++= ...&&''
subect to '' ' '& & ' '
'' ' '& & ' '
' ' & &
' ' & &
' &
' &
...
...
.
...
...
), ),..., )
, ,..., unrestricted in sign
n n
n n
m m mn n m
m m mn n m
n
n
a x a x a x b
a x a x a x b
a x a x a x b
a x a x a x b
x x x
x x x
+ + +
+ + +
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Which is nothing but the LPP
#a$imi%e nnxcxcxcz +++= ...
&&''
subect to
),...,,
...
.
.
...
...
&'
&&''
&&&&&'&'
''&'&'''
=+++
=+++
=+++
n
mnmnmm
nn
nn
xxx
bxaxaxa
bxaxaxa
bxaxaxa
his is the primal problem.
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4ase (ii)+ 5ll c e$cept c'. hen the dual
problem in the standard form is the LPP
#inimi%en
mmmm
ttt
ybybybybw
)...))
...
&'
''''
++++
++= ++
6ub"
ectto
),...,,,,,...,,
...
.
.
......
&'''
''''
&&&&''&''&
''''''''''
=++
=++
=++++
++
++
++
++
nmm
nnmmnmmnnn
mmmm
mmmm
tttyyyy
ctyayayaya
ctyayayayactyayayaya
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1ence its dual is the LPP
#a$imi%e nnxcxcvcz +++= ...&&''
subect to '' ' '& & ' '
'' ' '& & ' '
' ' & &
' ' & &
' &
' &
...
....
...
...
), ),..., )
, ,..., unrestricted in sign
n n
n n
m m mn n m
m m mn n m
n
n
a v a x a x b
a v a x a x b
a v a x a x b
a v a x a x b
v x x
v x x
+ + +
+ + +
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Which is nothing but the LPP
#a$imi%e nnxcxcvcz +++= ...
&&''
subect to
),...,,)...
.
.
...
...
&'
&&''
&&&&&'&'
''&'&'''
=+++
=+++
=+++
n
mnmnmm
nn
nn
xxvbxaxava
bxaxava
bxaxava
Putting '' vx =
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his is nothing but the LPP
#a$imi%e nnxcxcxcz +++= ...&&''
subect to'' ' '& & ' '
&' ' && & & &
' ' & &
' &
...
...
.
.
...
, ,..., )
n n
n n
m m mn n m
n
a x a x a x b
a x a x a x b
a x a x a x b
x x x
+ + + =
+ + + =
+ + + =
his is the primal problem. 4ases where
other c are are similarl! treated