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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
4. Example-3
(Previous example)		6. Example-5
(Next example)

5. Example-4


Find dual from primal conversion
MIN z = x1 - 3x2 - 2x3
subject to
3x1 - x2 + 2x3 <= 7
2x1 - 4x2 >= 12
-4x1 + 3x2 + 8x3 = 10
and x1,x2 >= 0 and x3 unrestricted in sign

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

MIN `z_x``=``````x_1`` - ``3``x_2`` - ``2``x_3`
subject to
```3``x_1`` - ````x_2`` + ``2``x_3`≤`7`
```2``x_1`` - ``4``x_2`≥`12`
` - ``4``x_1`` + ``3``x_2`` + ``8``x_3`=`10`
and `x_1,x_2 >= 0; ``x_3` unrestricted in sign


Since objective function is minimizing, all `<=` constraints (1) can be converted to `>=` type by multipling both sides by -1

MIN `z_x``=``````x_1`` - ``3``x_2`` - ``2``x_3`
subject to
` - ``3``x_1`` + ````x_2`` - ``2``x_3`≥`-7`
```2``x_1`` - ``4``x_2`≥`12`
` - ``4``x_1`` + ``3``x_2`` + ``8``x_3`=`10`
and `x_1,x_2 >= 0; ``x_3` unrestricted in sign


In primal, There are `3` variables and `3` constraints, so in dual there must be `3` constraints and `3` variables

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

In primal, The right hand side constants `b_1=-7,b_2=12,b_3=10` becomes coefficient of objective function in dual

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

The `x_3` variable in the primal is unrestricted in sign, therefore the `3^(rd)` constraint in the dual shall be equality.

Let `y1,y2,y3` be the dual variables

Since `3^(rd)` constraint in the primal is equality, the corresponding dual variable `y_3` will be unrestricted in sign.

Dual is (Solution steps of Dual by Simplex method)

MAX `z_y``=`` - ``7``y_1`` + ``12``y_2`` + ``10``y_3`
subject to
` - ``3``y_1`` + ``2``y_2`` - ``4``y_3`≤`1`
`````y_1`` - ``4``y_2`` + ``3``y_3`≤`-3`
` - ``2``y_1`` + ``8``y_3`=`-2`
and `y_1,y_2 >= 0; ``y_3` unrestricted in sign


This material is intended as a summary. Use your textbook for detail explanation.
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4. Example-3
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