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Page 1
Edurev123
5. Duality
5.1 Construct the dual of the problem: Maximize ?? =?? ?? ?? +?? ?? +?? ?? subject to the
constraints ?? ?? +?? ?? +?? ?? =?? ,?? ?? ?? -?? ?? ?? +?? ?? ?? =?? ,-?? ?? ?? +?? ?? ?? -?? ?? ?? =?? and
?? ?? ,?? ?? ,?? ?? =?? .
(2010 : 12 Marks)
Solution:
The equation is Max. ?? =2?? 1
+?? 2
+?? 3
such that
?????????????????????????? 1
+?? 2
+?? 3
=6 or -?? 1
-?? 2
-?? 3
=-6????????????????????????????????????????????????(1)
?????????????+3?? 1
-2?? 2
+3?? 3
=3, which can be written as
????????????????3?? 1
-2?? 2
+3?? 3
=3?????????????????????????????????????????????????????????????????????????????????????????????????(2)
and ???????3?? 1
-2?? 2
+3?? 3
=3 or -3?? 1
+2?? 2
-3?? 3
=-3?????????????????????????????????????????(3)
and ?-4?? 1
+3?? 2
-6?? 3
=1 which can be written as
????????????-4?? 1
+3?? 2
-6?? 3
=1??????????????????????????????????????????????????????????????????????????????????????????????????(4)
????????????-4?? 1
+3?? 2
-6?? 3
=1 or 4?? 1
-3?? 2
+6?? 3
=-1????????????????????????????????????????????????(5)
Let ?? 1
,?? 2
,?? 3
,?? 4
and ?? 5
be dual variables.
? from (1), (2), (3), (4) and (5)
Dual of the given primal can be written as
Min. ?? =-6?? 1
+3?? 2
-3?? 3
+?? 4
-?? 5
Subject to
?-?? 1
+3?? 2
-3?? 3
-4?? 4
+4?? 5
=2
?-?? 1
-2?? 2
+2?? 3
+3?? 4
-3?? 5
=1
?-?? 1
+3?? 2
-3?? 3
-6?? 4
+6?? 5
=1
5.2 Solve the following linear programming problem by the simplex method. Write
its dual. Also write the optimal solution of the dual from the optimal table of given
problem:
Maximize :????????????????????????????????????? =?? ?? ?? -?? ?? ?
+?? ?? ??
subject to
?? ?? +?? ?? ?? -?? ?? ?? ?=?? -?? ?? +?? ?? ?? +?? ?? ?? ?=?? ?? ?? ,?? ?? ,?? ?? ?=??
(2015 : 20 Marks)
Page 2
Edurev123
5. Duality
5.1 Construct the dual of the problem: Maximize ?? =?? ?? ?? +?? ?? +?? ?? subject to the
constraints ?? ?? +?? ?? +?? ?? =?? ,?? ?? ?? -?? ?? ?? +?? ?? ?? =?? ,-?? ?? ?? +?? ?? ?? -?? ?? ?? =?? and
?? ?? ,?? ?? ,?? ?? =?? .
(2010 : 12 Marks)
Solution:
The equation is Max. ?? =2?? 1
+?? 2
+?? 3
such that
?????????????????????????? 1
+?? 2
+?? 3
=6 or -?? 1
-?? 2
-?? 3
=-6????????????????????????????????????????????????(1)
?????????????+3?? 1
-2?? 2
+3?? 3
=3, which can be written as
????????????????3?? 1
-2?? 2
+3?? 3
=3?????????????????????????????????????????????????????????????????????????????????????????????????(2)
and ???????3?? 1
-2?? 2
+3?? 3
=3 or -3?? 1
+2?? 2
-3?? 3
=-3?????????????????????????????????????????(3)
and ?-4?? 1
+3?? 2
-6?? 3
=1 which can be written as
????????????-4?? 1
+3?? 2
-6?? 3
=1??????????????????????????????????????????????????????????????????????????????????????????????????(4)
????????????-4?? 1
+3?? 2
-6?? 3
=1 or 4?? 1
-3?? 2
+6?? 3
=-1????????????????????????????????????????????????(5)
Let ?? 1
,?? 2
,?? 3
,?? 4
and ?? 5
be dual variables.
? from (1), (2), (3), (4) and (5)
Dual of the given primal can be written as
Min. ?? =-6?? 1
+3?? 2
-3?? 3
+?? 4
-?? 5
Subject to
?-?? 1
+3?? 2
-3?? 3
-4?? 4
+4?? 5
=2
?-?? 1
-2?? 2
+2?? 3
+3?? 4
-3?? 5
=1
?-?? 1
+3?? 2
-3?? 3
-6?? 4
+6?? 5
=1
5.2 Solve the following linear programming problem by the simplex method. Write
its dual. Also write the optimal solution of the dual from the optimal table of given
problem:
Maximize :????????????????????????????????????? =?? ?? ?? -?? ?? ?
+?? ?? ??
subject to
?? ?? +?? ?? ?? -?? ?? ?? ?=?? -?? ?? +?? ?? ?? +?? ?? ?? ?=?? ?? ?? ,?? ?? ,?? ?? ?=??
(2015 : 20 Marks)
Solution:
Let ?? 1
,?? 2
be dual variables and ?? be the objective function for dual of the given problem.
Dual of the problem is
Min. ?? ?=2?? 1
+?? 2
?????????????? ????? ???????????????????????????????????????? 1
-?? 2
?=2
4?? 1
+2?? 2
?=-4
-2?? 1
+3?? 2
?=5
?? 1
,?? 2
,?? 3
?=0
Let Min. (?? )?=Max.(-?? '
)
??????????????????????????????????????????????????????????????????? =-2?? 1
-?? 2
Writing given problem in standard form, we get
Max. ?? =-2?? 1
-?? 2
+0?? 1
+0?? 2
+0?? 3
-?? ?? 1
-?? ?? 2
subject to
?? 1
-?? 2
-?? 1
+?? 1
=2
-4?? 1
-2?? 2
+?? 2
=4
-2?? 1
+3?? 2
-?? 3
+?? 2
=5
where ?? 1
,?? 3
are surplus variables, ?? 2
is a slack variable. ?? 1
and ?? 2
are artificial variables
and ?? is a very large quantity. We follow Big-M method for finding solution.
The simplex table is :
?? ?? -2 -1 0 0 0 -?? -??
?? ?? Basis ?? 1
?? 2
?? 1
?? 2
?? 3
?? 1
?? 2
?? 8
-?? ?? 1
1 -1 -1 0 0 1 0 2 -
0 ?? 2
-4 -2 0 1 0 0 0 4 -
-?? ?? 2
-2 3 0 0 -1 0 1 5
5
3
?? ?? =S?? ?? ?? ??
?? -2?? ?? 0 ?? -?? -??
?? ?? =?? ?? -?? ??
-2
-??
-1
+2??
-?? 0 -?? 0 0
Page 3
Edurev123
5. Duality
5.1 Construct the dual of the problem: Maximize ?? =?? ?? ?? +?? ?? +?? ?? subject to the
constraints ?? ?? +?? ?? +?? ?? =?? ,?? ?? ?? -?? ?? ?? +?? ?? ?? =?? ,-?? ?? ?? +?? ?? ?? -?? ?? ?? =?? and
?? ?? ,?? ?? ,?? ?? =?? .
(2010 : 12 Marks)
Solution:
The equation is Max. ?? =2?? 1
+?? 2
+?? 3
such that
?????????????????????????? 1
+?? 2
+?? 3
=6 or -?? 1
-?? 2
-?? 3
=-6????????????????????????????????????????????????(1)
?????????????+3?? 1
-2?? 2
+3?? 3
=3, which can be written as
????????????????3?? 1
-2?? 2
+3?? 3
=3?????????????????????????????????????????????????????????????????????????????????????????????????(2)
and ???????3?? 1
-2?? 2
+3?? 3
=3 or -3?? 1
+2?? 2
-3?? 3
=-3?????????????????????????????????????????(3)
and ?-4?? 1
+3?? 2
-6?? 3
=1 which can be written as
????????????-4?? 1
+3?? 2
-6?? 3
=1??????????????????????????????????????????????????????????????????????????????????????????????????(4)
????????????-4?? 1
+3?? 2
-6?? 3
=1 or 4?? 1
-3?? 2
+6?? 3
=-1????????????????????????????????????????????????(5)
Let ?? 1
,?? 2
,?? 3
,?? 4
and ?? 5
be dual variables.
? from (1), (2), (3), (4) and (5)
Dual of the given primal can be written as
Min. ?? =-6?? 1
+3?? 2
-3?? 3
+?? 4
-?? 5
Subject to
?-?? 1
+3?? 2
-3?? 3
-4?? 4
+4?? 5
=2
?-?? 1
-2?? 2
+2?? 3
+3?? 4
-3?? 5
=1
?-?? 1
+3?? 2
-3?? 3
-6?? 4
+6?? 5
=1
5.2 Solve the following linear programming problem by the simplex method. Write
its dual. Also write the optimal solution of the dual from the optimal table of given
problem:
Maximize :????????????????????????????????????? =?? ?? ?? -?? ?? ?
+?? ?? ??
subject to
?? ?? +?? ?? ?? -?? ?? ?? ?=?? -?? ?? +?? ?? ?? +?? ?? ?? ?=?? ?? ?? ,?? ?? ,?? ?? ?=??
(2015 : 20 Marks)
Solution:
Let ?? 1
,?? 2
be dual variables and ?? be the objective function for dual of the given problem.
Dual of the problem is
Min. ?? ?=2?? 1
+?? 2
?????????????? ????? ???????????????????????????????????????? 1
-?? 2
?=2
4?? 1
+2?? 2
?=-4
-2?? 1
+3?? 2
?=5
?? 1
,?? 2
,?? 3
?=0
Let Min. (?? )?=Max.(-?? '
)
??????????????????????????????????????????????????????????????????? =-2?? 1
-?? 2
Writing given problem in standard form, we get
Max. ?? =-2?? 1
-?? 2
+0?? 1
+0?? 2
+0?? 3
-?? ?? 1
-?? ?? 2
subject to
?? 1
-?? 2
-?? 1
+?? 1
=2
-4?? 1
-2?? 2
+?? 2
=4
-2?? 1
+3?? 2
-?? 3
+?? 2
=5
where ?? 1
,?? 3
are surplus variables, ?? 2
is a slack variable. ?? 1
and ?? 2
are artificial variables
and ?? is a very large quantity. We follow Big-M method for finding solution.
The simplex table is :
?? ?? -2 -1 0 0 0 -?? -??
?? ?? Basis ?? 1
?? 2
?? 1
?? 2
?? 3
?? 1
?? 2
?? 8
-?? ?? 1
1 -1 -1 0 0 1 0 2 -
0 ?? 2
-4 -2 0 1 0 0 0 4 -
-?? ?? 2
-2 3 0 0 -1 0 1 5
5
3
?? ?? =S?? ?? ?? ??
?? -2?? ?? 0 ?? -?? -??
?? ?? =?? ?? -?? ??
-2
-??
-1
+2??
-?? 0 -?? 0 0
??? 2
is outgoing variable and ?? 2
is incoming variable.
?? ?? -2 -1 0 0 0 -??
?? ?? Basis ?? 1
?? 2
?? 1
?? 2
?? 3
?? 1
?? ??
-?? ?? 1
1
3
0 -1 0 -
1
3
1
11
3
11
0 ?? 2
-
16
3
0 0 1 -
2
3
0
22
3
-
-1
1
2
-
2
3
1 0 0 -
1
3
0
5
3
-
?? ?? =S?? ?? ?? ????
-
?? 3
+
2
3
-1 ?? 0
?? +1
3
-??
?? ?? =?? ?? -?? ??
-
8
3
+
?? 3
0 -?? 0
-
(?? +1)
3
0
?
??? 1
is incoming variable and ?? 1
is outgoing variable.
?? ?? -2 -1 0 0 0
?? ?? Basis ?? 1
?? 2
?? 1
?? 2
?? 3
?? ??
-2 ?? 1
1 0 -3 0 -1 11
0 ?? 2
0 0 -16 1 -6 66
-1 ?? 2
0 1 -2 0 -1 9
?? ?? =S?? ?? ?? ????
-2 -1 8 0 3
?? ?? =?? ?? -?? ?? 0 0 -8 0 -3
? All ?? ?? '
?? =0? this is the optimal feasibie solution.
Page 4
Edurev123
5. Duality
5.1 Construct the dual of the problem: Maximize ?? =?? ?? ?? +?? ?? +?? ?? subject to the
constraints ?? ?? +?? ?? +?? ?? =?? ,?? ?? ?? -?? ?? ?? +?? ?? ?? =?? ,-?? ?? ?? +?? ?? ?? -?? ?? ?? =?? and
?? ?? ,?? ?? ,?? ?? =?? .
(2010 : 12 Marks)
Solution:
The equation is Max. ?? =2?? 1
+?? 2
+?? 3
such that
?????????????????????????? 1
+?? 2
+?? 3
=6 or -?? 1
-?? 2
-?? 3
=-6????????????????????????????????????????????????(1)
?????????????+3?? 1
-2?? 2
+3?? 3
=3, which can be written as
????????????????3?? 1
-2?? 2
+3?? 3
=3?????????????????????????????????????????????????????????????????????????????????????????????????(2)
and ???????3?? 1
-2?? 2
+3?? 3
=3 or -3?? 1
+2?? 2
-3?? 3
=-3?????????????????????????????????????????(3)
and ?-4?? 1
+3?? 2
-6?? 3
=1 which can be written as
????????????-4?? 1
+3?? 2
-6?? 3
=1??????????????????????????????????????????????????????????????????????????????????????????????????(4)
????????????-4?? 1
+3?? 2
-6?? 3
=1 or 4?? 1
-3?? 2
+6?? 3
=-1????????????????????????????????????????????????(5)
Let ?? 1
,?? 2
,?? 3
,?? 4
and ?? 5
be dual variables.
? from (1), (2), (3), (4) and (5)
Dual of the given primal can be written as
Min. ?? =-6?? 1
+3?? 2
-3?? 3
+?? 4
-?? 5
Subject to
?-?? 1
+3?? 2
-3?? 3
-4?? 4
+4?? 5
=2
?-?? 1
-2?? 2
+2?? 3
+3?? 4
-3?? 5
=1
?-?? 1
+3?? 2
-3?? 3
-6?? 4
+6?? 5
=1
5.2 Solve the following linear programming problem by the simplex method. Write
its dual. Also write the optimal solution of the dual from the optimal table of given
problem:
Maximize :????????????????????????????????????? =?? ?? ?? -?? ?? ?
+?? ?? ??
subject to
?? ?? +?? ?? ?? -?? ?? ?? ?=?? -?? ?? +?? ?? ?? +?? ?? ?? ?=?? ?? ?? ,?? ?? ,?? ?? ?=??
(2015 : 20 Marks)
Solution:
Let ?? 1
,?? 2
be dual variables and ?? be the objective function for dual of the given problem.
Dual of the problem is
Min. ?? ?=2?? 1
+?? 2
?????????????? ????? ???????????????????????????????????????? 1
-?? 2
?=2
4?? 1
+2?? 2
?=-4
-2?? 1
+3?? 2
?=5
?? 1
,?? 2
,?? 3
?=0
Let Min. (?? )?=Max.(-?? '
)
??????????????????????????????????????????????????????????????????? =-2?? 1
-?? 2
Writing given problem in standard form, we get
Max. ?? =-2?? 1
-?? 2
+0?? 1
+0?? 2
+0?? 3
-?? ?? 1
-?? ?? 2
subject to
?? 1
-?? 2
-?? 1
+?? 1
=2
-4?? 1
-2?? 2
+?? 2
=4
-2?? 1
+3?? 2
-?? 3
+?? 2
=5
where ?? 1
,?? 3
are surplus variables, ?? 2
is a slack variable. ?? 1
and ?? 2
are artificial variables
and ?? is a very large quantity. We follow Big-M method for finding solution.
The simplex table is :
?? ?? -2 -1 0 0 0 -?? -??
?? ?? Basis ?? 1
?? 2
?? 1
?? 2
?? 3
?? 1
?? 2
?? 8
-?? ?? 1
1 -1 -1 0 0 1 0 2 -
0 ?? 2
-4 -2 0 1 0 0 0 4 -
-?? ?? 2
-2 3 0 0 -1 0 1 5
5
3
?? ?? =S?? ?? ?? ??
?? -2?? ?? 0 ?? -?? -??
?? ?? =?? ?? -?? ??
-2
-??
-1
+2??
-?? 0 -?? 0 0
??? 2
is outgoing variable and ?? 2
is incoming variable.
?? ?? -2 -1 0 0 0 -??
?? ?? Basis ?? 1
?? 2
?? 1
?? 2
?? 3
?? 1
?? ??
-?? ?? 1
1
3
0 -1 0 -
1
3
1
11
3
11
0 ?? 2
-
16
3
0 0 1 -
2
3
0
22
3
-
-1
1
2
-
2
3
1 0 0 -
1
3
0
5
3
-
?? ?? =S?? ?? ?? ????
-
?? 3
+
2
3
-1 ?? 0
?? +1
3
-??
?? ?? =?? ?? -?? ??
-
8
3
+
?? 3
0 -?? 0
-
(?? +1)
3
0
?
??? 1
is incoming variable and ?? 1
is outgoing variable.
?? ?? -2 -1 0 0 0
?? ?? Basis ?? 1
?? 2
?? 1
?? 2
?? 3
?? ??
-2 ?? 1
1 0 -3 0 -1 11
0 ?? 2
0 0 -16 1 -6 66
-1 ?? 2
0 1 -2 0 -1 9
?? ?? =S?? ?? ?? ????
-2 -1 8 0 3
?? ?? =?? ?? -?? ?? 0 0 -8 0 -3
? All ?? ?? '
?? =0? this is the optimal feasibie solution.
??????????????????????????????????????????????????????? 1
=11
?????????????????????????????????????????????????????????? ?? =9
???? ,?????????????????????????????????????????????? max
=-2×11-1-9=-31
??????????????????????????????????????????????????????? min?
=31
? Maximum value of ?? =?? min
=31.
5.3 Convert the following LPP into dual LPP :
Minimize????????????????????????????????????????? =?? ?? -?? ?? ?? -?? ?? ??
subject to
?? ?? ?? -?? ?? +?? ?? ?? ?=?? ?? ?? ?? -?? ?? ?? ?=????
-?? ?? ?? +?? ?? ?? +?? ?? ?? ?=????
where ?? ?? ,?? ?? =?? and ?? ?? is unrestricted in sign.
(2021: 15 marks)
Solution:
Given that:
Minimize
?? ?=?? 1
-3?? 2
-2?? 3
subject to
??????????????????????????????????????????????????3?? 1
-?? 2
+2?? 3
?=7???????????????????????????????????????????????????????????????(1)
2?? 1
-4?? 2
?=12?????????????????????????????????????????????????????????????(2)
-4?? 1
+3?? 2
+8?? 3
?=10?????????????????????????????????????????????????????????????(3)
where ?? 1
,?? 2
=0 and ?? 3
is unrestricted in sign.
Since, the constraints (1) in ( = ) type and the problem is of minimization, all the
constraints should be of (=) type.
We multiply constraints (1) by ( -1 ).
So that,
-3?? 1
+?? 2
-2?? 3
=-7
and the constraints with equality sign can be written as
???????????????????????????????????????????????????????????
-4?? 1
+3?? 2
+8?? 3
?=10
4?? 1
-3?? 2
-8?? 3
?=-10
-4?? 1
+3?? 2
+8?? 3
?=10
Page 5
Edurev123
5. Duality
5.1 Construct the dual of the problem: Maximize ?? =?? ?? ?? +?? ?? +?? ?? subject to the
constraints ?? ?? +?? ?? +?? ?? =?? ,?? ?? ?? -?? ?? ?? +?? ?? ?? =?? ,-?? ?? ?? +?? ?? ?? -?? ?? ?? =?? and
?? ?? ,?? ?? ,?? ?? =?? .
(2010 : 12 Marks)
Solution:
The equation is Max. ?? =2?? 1
+?? 2
+?? 3
such that
?????????????????????????? 1
+?? 2
+?? 3
=6 or -?? 1
-?? 2
-?? 3
=-6????????????????????????????????????????????????(1)
?????????????+3?? 1
-2?? 2
+3?? 3
=3, which can be written as
????????????????3?? 1
-2?? 2
+3?? 3
=3?????????????????????????????????????????????????????????????????????????????????????????????????(2)
and ???????3?? 1
-2?? 2
+3?? 3
=3 or -3?? 1
+2?? 2
-3?? 3
=-3?????????????????????????????????????????(3)
and ?-4?? 1
+3?? 2
-6?? 3
=1 which can be written as
????????????-4?? 1
+3?? 2
-6?? 3
=1??????????????????????????????????????????????????????????????????????????????????????????????????(4)
????????????-4?? 1
+3?? 2
-6?? 3
=1 or 4?? 1
-3?? 2
+6?? 3
=-1????????????????????????????????????????????????(5)
Let ?? 1
,?? 2
,?? 3
,?? 4
and ?? 5
be dual variables.
? from (1), (2), (3), (4) and (5)
Dual of the given primal can be written as
Min. ?? =-6?? 1
+3?? 2
-3?? 3
+?? 4
-?? 5
Subject to
?-?? 1
+3?? 2
-3?? 3
-4?? 4
+4?? 5
=2
?-?? 1
-2?? 2
+2?? 3
+3?? 4
-3?? 5
=1
?-?? 1
+3?? 2
-3?? 3
-6?? 4
+6?? 5
=1
5.2 Solve the following linear programming problem by the simplex method. Write
its dual. Also write the optimal solution of the dual from the optimal table of given
problem:
Maximize :????????????????????????????????????? =?? ?? ?? -?? ?? ?
+?? ?? ??
subject to
?? ?? +?? ?? ?? -?? ?? ?? ?=?? -?? ?? +?? ?? ?? +?? ?? ?? ?=?? ?? ?? ,?? ?? ,?? ?? ?=??
(2015 : 20 Marks)
Solution:
Let ?? 1
,?? 2
be dual variables and ?? be the objective function for dual of the given problem.
Dual of the problem is
Min. ?? ?=2?? 1
+?? 2
?????????????? ????? ???????????????????????????????????????? 1
-?? 2
?=2
4?? 1
+2?? 2
?=-4
-2?? 1
+3?? 2
?=5
?? 1
,?? 2
,?? 3
?=0
Let Min. (?? )?=Max.(-?? '
)
??????????????????????????????????????????????????????????????????? =-2?? 1
-?? 2
Writing given problem in standard form, we get
Max. ?? =-2?? 1
-?? 2
+0?? 1
+0?? 2
+0?? 3
-?? ?? 1
-?? ?? 2
subject to
?? 1
-?? 2
-?? 1
+?? 1
=2
-4?? 1
-2?? 2
+?? 2
=4
-2?? 1
+3?? 2
-?? 3
+?? 2
=5
where ?? 1
,?? 3
are surplus variables, ?? 2
is a slack variable. ?? 1
and ?? 2
are artificial variables
and ?? is a very large quantity. We follow Big-M method for finding solution.
The simplex table is :
?? ?? -2 -1 0 0 0 -?? -??
?? ?? Basis ?? 1
?? 2
?? 1
?? 2
?? 3
?? 1
?? 2
?? 8
-?? ?? 1
1 -1 -1 0 0 1 0 2 -
0 ?? 2
-4 -2 0 1 0 0 0 4 -
-?? ?? 2
-2 3 0 0 -1 0 1 5
5
3
?? ?? =S?? ?? ?? ??
?? -2?? ?? 0 ?? -?? -??
?? ?? =?? ?? -?? ??
-2
-??
-1
+2??
-?? 0 -?? 0 0
??? 2
is outgoing variable and ?? 2
is incoming variable.
?? ?? -2 -1 0 0 0 -??
?? ?? Basis ?? 1
?? 2
?? 1
?? 2
?? 3
?? 1
?? ??
-?? ?? 1
1
3
0 -1 0 -
1
3
1
11
3
11
0 ?? 2
-
16
3
0 0 1 -
2
3
0
22
3
-
-1
1
2
-
2
3
1 0 0 -
1
3
0
5
3
-
?? ?? =S?? ?? ?? ????
-
?? 3
+
2
3
-1 ?? 0
?? +1
3
-??
?? ?? =?? ?? -?? ??
-
8
3
+
?? 3
0 -?? 0
-
(?? +1)
3
0
?
??? 1
is incoming variable and ?? 1
is outgoing variable.
?? ?? -2 -1 0 0 0
?? ?? Basis ?? 1
?? 2
?? 1
?? 2
?? 3
?? ??
-2 ?? 1
1 0 -3 0 -1 11
0 ?? 2
0 0 -16 1 -6 66
-1 ?? 2
0 1 -2 0 -1 9
?? ?? =S?? ?? ?? ????
-2 -1 8 0 3
?? ?? =?? ?? -?? ?? 0 0 -8 0 -3
? All ?? ?? '
?? =0? this is the optimal feasibie solution.
??????????????????????????????????????????????????????? 1
=11
?????????????????????????????????????????????????????????? ?? =9
???? ,?????????????????????????????????????????????? max
=-2×11-1-9=-31
??????????????????????????????????????????????????????? min?
=31
? Maximum value of ?? =?? min
=31.
5.3 Convert the following LPP into dual LPP :
Minimize????????????????????????????????????????? =?? ?? -?? ?? ?? -?? ?? ??
subject to
?? ?? ?? -?? ?? +?? ?? ?? ?=?? ?? ?? ?? -?? ?? ?? ?=????
-?? ?? ?? +?? ?? ?? +?? ?? ?? ?=????
where ?? ?? ,?? ?? =?? and ?? ?? is unrestricted in sign.
(2021: 15 marks)
Solution:
Given that:
Minimize
?? ?=?? 1
-3?? 2
-2?? 3
subject to
??????????????????????????????????????????????????3?? 1
-?? 2
+2?? 3
?=7???????????????????????????????????????????????????????????????(1)
2?? 1
-4?? 2
?=12?????????????????????????????????????????????????????????????(2)
-4?? 1
+3?? 2
+8?? 3
?=10?????????????????????????????????????????????????????????????(3)
where ?? 1
,?? 2
=0 and ?? 3
is unrestricted in sign.
Since, the constraints (1) in ( = ) type and the problem is of minimization, all the
constraints should be of (=) type.
We multiply constraints (1) by ( -1 ).
So that,
-3?? 1
+?? 2
-2?? 3
=-7
and the constraints with equality sign can be written as
???????????????????????????????????????????????????????????
-4?? 1
+3?? 2
+8?? 3
?=10
4?? 1
-3?? 2
-8?? 3
?=-10
-4?? 1
+3?? 2
+8?? 3
?=10
Put ?? 3
=?? 3
'
-?? 3
''
so that ?? 3
'
,?? 3
''
=0
and the primal can be written as
Min??? =?? 1
-3?? 2
-2?? 3
'
+2?? 3
''
subject to
-3?? 1
+?? 2
-2?? 3
'
+2?? 3
''
?=-7
2?? 1
-4?? 2
?=12
4?? 1
-3?? 2
-8?? 3
'
+8?? 3
''
?=-10
-4?? 1
+3?? 2
+8?? 3
'
-8?? 3
''
?=10
?? 1
,?? 2
,?? 3
'
,?? 3
''
?=0
? Dual to this LPP is :
Maximize ?? =-7?? 1
+12?? 2
-10?? 3
+10?? 4
subject to
-3?? 1
+2?? 2
+4?? 3
-4?? 4
?=1
?? 1
-4?? 2
-3?? 3
+3?? 4
?=-3
-2?? 1
+0?? 2
-8?? 3
+8?? 4
?=-2
2?? 1
+0?? 2
+8?? 3
-8?? 4
?=2
?? 1
,?? 2
,?? 3
,?? 4
?=0
This can also be written as :
Max??? =-7?? 1
+12?? 2
-10(?? 3
-?? 4
)
subject to
-3?? 1
+2?? 2
+4(?? 3
-?? 4
)?=1
?? 1
-4?? 2
-3(?? 3
-?? 4
)?=-3
-2?? 1
-8(?? 3
-?? 4
)?=-2
2?? 1
+8(?? 3
-?? 4
)?=2
?? 1
,?? 2
,?? 3
,?? 4
?=0
The term (?? 3
-?? 4
) in both objective function and constraints of the dual.
This is always whenever there is equality constraints in the primal. Then, the new
variable (?? 3
-?? 4
)=?? 1
becomes unrestricted in sign, being the difference of two non-
negative variable.
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FAQs on Duality - Mathematics Optional Notes for UPSC
| 1. What is the concept of duality in the context of UPSC exam preparation? |  |
Ans. Duality in the context of UPSC exam preparation refers to the importance of balancing both the theoretical knowledge and practical application of concepts while studying for the exam. It involves understanding the concepts deeply as well as practicing mock tests and previous year question papers to enhance problem-solving skills.
| 2. How can one maintain duality while preparing for the UPSC exam? | |
Ans. One can maintain duality while preparing for the UPSC exam by allocating time for both studying the theoretical aspects of the syllabus and practicing questions to improve problem-solving abilities. It is essential to strike a balance between understanding the concepts and applying them in practice.
| 3. Why is duality important in UPSC exam preparation? | |
Ans. Duality is important in UPSC exam preparation as it helps candidates develop a comprehensive understanding of the concepts and enhances their problem-solving skills. By balancing theoretical knowledge with practical application, candidates can effectively tackle the diverse questions asked in the exam.
| 4. How can one improve duality in UPSC exam preparation? | |
Ans. To improve duality in UPSC exam preparation, candidates can focus on strengthening their conceptual understanding through regular study sessions and revision. Additionally, solving mock tests and previous year question papers can help in enhancing problem-solving abilities and applying theoretical knowledge in practice.
| 5. What are the benefits of maintaining duality in UPSC exam preparation? | |
Ans. Maintaining duality in UPSC exam preparation can lead to a comprehensive understanding of the syllabus, improved retention of concepts, enhanced problem-solving skills, and better performance in the exam. It allows candidates to approach questions from different perspectives and effectively apply their knowledge during the exam.
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