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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
3. Example-2
(Previous example)		5. Example-4
(Next example)

4. Example-3


Find dual from primal conversion
MIN z = x1 + 2x2
subject to
2x1 + 4x2 <= 160
x1 - x2 = 30
x1 >= 10
and x1,x2 >= 0

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

MIN `z_x``=``````x_1`` + ``2``x_2`
subject to
```2``x_1`` + ``4``x_2`≤`160`
`````x_1`` - ````x_2`=`30`
`````x_1`≥`10`
and `x_1,x_2 >= 0; `


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

MIN `z_x``=``````x_1`` + ``2``x_2`
subject to
` - ``2``x_1`` - ``4``x_2`≥`-160`
`````x_1`` - ````x_2`=`30`
`````x_1`≥`10`
and `x_1,x_2 >= 0; `


In primal, There are `2` variables and `3` constraints, so in dual there must be `2` constraints and `3` variables

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

In primal, The right hand side constants `b_1=-160,b_2=30,b_3=10` becomes coefficient of objective function in dual

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

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

Since `2^(nd)` constraint in the primal is equality, the corresponding dual variable `y_2` will be unrestricted in sign.

Dual is (Solution steps of Dual by Simplex method)

MAX `z_y``=`` - ``160``y_1`` + ``30``y_2`` + ``10``y_3`
subject to
` - ``2``y_1`` + ````y_2`` + ````y_3`≤`1`
` - ``4``y_1`` - ````y_2`≤`2`
and `y_1,y_3 >= 0; ``y_2` unrestricted in sign


This material is intended as a summary. Use your textbook for detail explanation.
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3. Example-2
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