Linear Programming Week 2

(1)

P R E P A R E D B Y : H A N S I K A K H U R A N A

D E P A R T M E N T O F C O M M E R C E G A R G I C O L L E G E

Linear Programming

Week 2

(2)

In this week, we will be studying …

How to break a tie, if the simplex table has the same value for an Entering Variable or for an Exiting

Variable

How to find out if a solution is Unbounded

How to find out if the problem has Multiple Optimal Solutions

Solving some important questions

2

(3)

(A) Breaking the Tie in a Simplex Table

1. 

If there is a tie for the Entering Variable

- 

If two or more non-basic variables are tied for the highest, positive value in the Cj-Zj row, we can

choose either of these variables

- 

The optimum solution will be reached regardless of which variable you choose as the entering variable

Hansika Khurana, Department of Commerce, Gargi College

3

(4)

Breaking the Tie in a Simplex Table

2.  If there is a tie for the Departing/Exiting Variable (This is an important concept)

-  If the same common ratio corresponds to a Basic and a Non-Basic variable, we will choose the Basic Variable as the exiting variable.

-  If the tie is between both Basic Variables, then we can choose either. This is called a case of Degeneracy. This may lead to the same value of Zj (solution) at different stages of the simplex problem.

-  There is a also a possibility in the above case, to reach the same simplex table again and again. This is called

Cycling and happens in Degenerate Solutions.

4

(5)

(B) Identifying Unbounded Solutions

  Remember this – we studied earlier that the Ratio column is arrived at by dividing each value in the Solution/Quantity column with the Key column.

  We also studied that the ratio column should have a positive value.

  Unbounded solutions occur when each entry in the Key Column is either negative or zero.

  In such a case, it becomes impossible to calculate the values in the Ration column

  In unbounded problems, we cannot arrive at a solution for the LPP

Refer Video sent on whatsapp

5

(6)

(C) Identifying Multiple Optimal Solutions

  Multiple Optimal Solutions mean that there is more than one table which gives us the same Solution/Quantity in the Zj row.

  In simplex, the presence of a multiple optimal solution can be determined from the final table itself.

  Remember again – we have studied that for a

maximization case, the final table is obtained when each value in the Cj-Zj row is either 0 or negative.

  If, in the final table, there is a 0 value in the Cj-Zj row for a non-basic variable, this implies that there is more than one optimal basic feasible solution.

6

(7)

We obtain the other optimal solutions by performing additional iterations of the final simplex table.

While we do this, each time we will choose a non- basic variable with a 0 Cj-Zj entry as the entering variable.

Refer Video sent on whatsapp

7

(8)

(D) Some Important Questions

8

(9)

9

(10)

Next Topic will be …

10

Linear Programming Week 2