BOWEN UNIVERSITY, IWO, OSUN STATEMATHEMATICS AND STATISTICS PROGRAMMESTA 322(OPERATIONS RESEARCH I) LECTURE NOTE

DUAL SIMPLEX METHODOmoyajowo A.C

adefunke.omoyajowo@bowen.edu.ng

WE SHALL BE GUIDED BY THE FOLLOWING:

• Recap of the Steps in Simplex Method

• Iterative steps of Dual Simplex Method(DSM)

• Simplex method versus Dual Simplex Method

• Difference between Dual-Primal and Dual Simplex Method

• Solving an LPP using Dual Simplex Method.

RECAP OF THE STEPS IN SIMPLEX METHOD

• Step 0: Using the standard form, determine a starting feasible solution by setting n-m appropriate (non-basic variables) at zero level.

• Step 1: Select an entering variable from among the current (zero) non-basic variables which when increased above zero can improve the value of the objective function. If non exist, stop, the current basic solution is optimal. Otherwise, go to step 2.

• Step 2: select a leaving variable from among the current basic variables that must be set to zero (becomes non basic variable) when the entering variables becomes basic.

• Step 3: Determine the new basic solution by making the entering variable basic and the leaving variable Non –basic. Go to step 1. The process above is called iterative process.

STEPS OR ITERATIVE PROCESS IN DUAL SIMPLEX METHOD(DSM)

STEPS OR ITERATIVE PROCESS IN DUAL SIMPLEX METHOD(DSM) CONT’D

STEPS OR ITERATIVE PROCESS IN DUAL SIMPLEX METHOD(DSM) CONT’D

SIMPLEX METHOD VERSUS DUAL SIMPLEX METHOD

• Simplex method starts with a non-optimal but feasible solution whereas dual simplex method starts with an optimal but infeasible solution.

• Simplex method maintains the feasibility during successive iterations while dual simplex method maintains the optimality.

DIFFERENCE BETWEEN DUAL-PRIMAL AND DUAL SIMPLEX METHOD

Dual-primal Dual Simplex Method

1. The change in the inequality sign in the constraints depend on the optimal objective (if it is maximization or minimization).

The change in the inequality sign in the constraint changes to ≤ for both optimalobjective, be it maximization or minimization.

2. The dual is the complete opposite of the original equation called primal.

Dual simplex method is a complete opposite of the conditions in the normal simplex method.

3. The dual or primal can be solved using anymethod that is suitable( simplex method, two phase method, etc.).

Dual Simplex method is a method on its own and so no other method is acceptable.

SOLVING AN LPP USING DUAL SIMPLEX METHOD

Example

Use the dual simplex method to solve the following LPP.

��� � = 2�� + ��+3��

Subject to

�� − 2��+��≥ 4

2�� + ��+��≤ 8

�� − ��≥ 0

��, ��, ��≥0

SOLVING AN LPP USING DUAL SIMPLEX METHOD

Remember the first step: Every inequality in the constraint must be in ≤ form.

So, the equation becomes this form:

��� � = 2�� + ��+3��

Subject to

−�� + 2�� − ��≤ -4

2�� + �� +��≤ 8

−�� + ��≤ 0

��, ��, ��≥0

SOLVING AN LPP USING DUAL SIMPLEX METHOD CONT’D

We write in standard form by adding the slack variables to the constraint to make it become an equation

��� � − 2�� − �� −3�� = 0

Subject to

−�� + 2�� − ��+�� = -4

2�� + ��+�� + �� = 8

−�� + �� +��= 0

��, ��, ��,��,��,��≥0

SOLVING AN LPP USING DUAL SIMPLEX METHOD

We arrange into the simplex tableau and solve following the steps above.

Note that The values in yellow are the pivot column, those in red are the pivot row while the single value in blue color is the pivot element.

Iteration Basic

Variables

Non-Basic variables

�� �� �� �� �� ��

solution

0

�� Leaves

�� enters

Z -2 -1 -3 0 0 0 0

�� -1 2 -1 1 0 0 -4

�� 2 1 1 0 1 0 8

�� -1 0 1 0 0 1 0

1

Feasibility

condition

satisfied

Z 0 -5 -1 -2 0 0 8

�� 1 -2 1 -1 0 0 4

�� 0 5 -1 2 1 0 0

�� 0 -2 2 -1 0 1 4

Solution is given as Z=8, ��=4, ��=0, ��=0

ON A FINAL NOTE

•It should be noted that if the next iteration contains a negative coefficient in the solution, then the feasibility condition has not been satisfied. Therefore, we repeat the steps all over again until there is feasibility.

THANK YOU FOR READING.