# Dual Simplex Method

## Dual Simplex Method:

Steps involved in Dual Simplex Method are as follows:

- Write the given linear programming problem in its standard basic feasible solution by adding appropriate slack variables.
- If the existing basic solution is feasible ,then use simplex method (using slack variables) to obtain optimum solution.
- If the existing solution is infeasible, then the values of basic variables less than or equal to zero, go to next step.

- check the optimality of the solution:
- If the solution is not optimum then add an artificial constant so that the optimality condition is satisfied.
- If the solution is optimum then go to next step.

- take the most negative value as the leaving variable & its corresponding row is the key row.
- Obtain the ratio of net evaluation to the corresponding coefficient in the key row. leave those with positive & zero denominators.the entering has the smallest value of ratio column corresponding to entering vector is the key column.
- Reduce the leading element to unity and all other entriesto zero.
- Repeat the process until the optimum solution.

Let us consider one example of simplex method to understand the topic.

## Example:

Minimize: z=3x_{1}+x_{2}

Subject to: x_{1}+x_{2}=1

2x_{1}+3x_{2} = 2 & x_{1,}x_{2} = 0

* Solution: *Converting the minimizing problem into maximizing LPP

Minimize: -z=-3x_{1-}x_{2}

Subject to: -x_{1}– x_{2}=-1

-2x_{1}-3x_{2} = -2 & x_{1,}x_{2} = 0

Now introducing slack variables s_{1}, s_{2}, s_{3} & the artificial variables the given LPP takes the form:

Maximize: -z=-3x_{1}-x_{2}+0s_{1}+0s_{2}

Subject to: -x_{1}-x_{2}+s_{1}=-1

-2x_{1}-3x_{2}+s_{2} = -2 &

where, s_{1}, s_{2}, x_{1}, x_{2} =0

Now, we form initial simplex table:

(1) Initial Simplex Table:

(2) First Simplex Table:

(3) Second Simplex Table:

Here, all C_{j}-Z_{j} =0. Hence, the optimality reached.

We get the solution,

x_{1}=0, x_{2}=1

z=3(0)=(1)

=1

Maximized Value of Z is 1.

Get detailed knowledge of the remaining methods of operations research … through our posts Big M method, simplex method & Graphical Method…