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Home > Operation Research calculators > Primal to dual conversion example
4. Primal to dual conversion example ( Enter your problem )
( Enter your problem )
  1. Formulation and Rules
  2. Example-1
  3. Example-2
  4. Example-3
  5. Example-4
  6. Example-5
  7. Example-6
 		Other related methods
  1. Formulate linear programming model
  2. Graphical method
  3. Simplex method (BigM method)
  4. Two-Phase method
  5. Primal to dual conversion
  6. Dual simplex method
  7. Integer simplex method
  8. Branch and Bound method
  9. 0-1 Integer programming problem
  10. Revised Simplex method
5. Example-4
(Previous example)		7. Example-6
(Next example)

6. Example-5


Find dual from primal conversion
MAX z = 3x1 + x2 + 2x3 - x4
subject to
2x1 - x2 + 3x3 + x4 = 1
x1 + x2 - x3 + x4 = 3
and x1,x2 >= 0 and x3,x4 unrestricted in sign

Solution:
Primal is (Solution steps of Primal by Simplex method)

MAX `z_x``=````3``x_1`` + ````x_2`` + ``2``x_3`` - ````x_4`
subject to
```2``x_1`` - ````x_2`` + ``3``x_3`` + ````x_4`=`1`
`````x_1`` + ````x_2`` - ````x_3`` + ````x_4`=`3`
and `x_1,x_2 >= 0; ``x_3,x_4` unrestricted in sign


In primal, There are `4` variables and `2` constraints, so in dual there must be `4` constraints and `2` variables

In primal, The coefficient of objective function `c_1=3,c_2=1,c_3=2,c_4=-1` becomes right hand side constants in dual

In primal, The right hand side constants `b_1=1,b_2=3` becomes coefficient of objective function in dual

In primal, objective function is maximizing, so in dual objective function must be minimizing

The `x_3,x_4` variables in the primal are unrestricted in sign, therefore the `3^(rd),4^(th)` constraints in the dual shall be equalities.

Let `y1,y2` be the dual variables

Since `1^(st),2^(nd)` constraints in the primal are equalities, the corresponding dual variables `y_1,y_2` will be unrestricted in sign.

Dual is (Solution steps of Dual by Simplex method)

MIN `z_y``=``````y_1`` + ``3``y_2`
subject to
```2``y_1`` + ````y_2`≥`3`
` - ````y_1`` + ````y_2`≥`1`
```3``y_1`` - ````y_2`=`2`
`````y_1`` + ````y_2`=`-1`
and ```y_1,y_2` unrestricted in sign


This material is intended as a summary. Use your textbook for detail explanation.
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5. Example-4
(Previous example)		7. Example-6
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