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Solution Possibilities - Introduction to Operations Research - Lecture Slides, Slides of Operational Research

Central University of Jammu and KashmirOperational Research

These are the important key points of lecture slides of Introduction to Operations Research are:Solution Possibilities, Multiple Optimal Solutions, Nonbasic Variable, Unbounded Solutions, Increased Indefinitely, Greedy Rule, Ratio Test, Non Positive, Non Feasible Solution, Standard Form

Typology: Slides

2012/2013

Uploaded on 01/09/2013

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Solution Possibilities
Multiple Optimal Solutions:
A zero reduced cost of a nonbasic variable in the
final simplex tableau indicates the existence of
multiple optimal solutions.
Observation:
If there are two optimal solutions, then there must
be infinitely many optimal solutions [why?]
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Solution Possibilities

 Multiple Optimal Solutions :

A zero reduced cost of a nonbasic variable in the

final simplex tableau indicates the existence of

multiple optimal solutions.

 Observation :

If there are two optimal solutions, then there must

be infinitely many optimal solutions [why?]

Example

There is a nonbasic variable (x 5 ) whose

reduced cost is equal to zero. If we put it in

the basis the resulting solution will also be

optimal (why?)

What is the “geometric” explanation?

BV Eq. # x 1 x 2 x 3 x 4 x 5 RHS

x 2 1 0 1 1 0 - 2 10

x 4 2 0 0 - 1 1 1 5

x 1 3 1 0 0 0 1 15

Z Z 0 0 2 0 0 80

Example

According to the Greedy Rule, x 2 is the new basic

variable.

If we conduct the Ratio Test on the x 2 column we

fail to find a variable to be taken out of the basis.

This means that x 2 can be increased indefinitely.

BV Eq. # x 1 x 2 x 3 x 4 x 5 RHS

x 5 1 0 - 6 0 1 1 25

x 1 2 1 - 2 0 6 0 40

x 3 3 0 0 1 1 0 10
Z Z 0 - 3 0 2 0 80

Recipe

 If every coefficient in the column of the new

basic variable is non-positive, the solution is

unbounded.

Observations :

This is a very important issue

Problems in standard form always have feasible

solutions.

Cycling

Is it possible that the simplex procedure we

described will never stop ??????

The answer is Yes!

Reason : If there is a change in the basis but not in

the value of the objective function, i.e. a basic

variable whose value is zero leaves the basis, we can

cycle indefinitely between the two solutions.

A basis with one or more of the basic variables equal

zero is called degenerate.

 - x 5 1 1/2 - 11/2 - 5/2 - x 6 2 1/2 - 3/2 - 1/2 - x - Z Z -10 - x 1 1 1 11 - BV Eq. # x 1 x 2 x 3 x 4 x 5 x 6 x 7 RHS
  • x 6 2 0 4 2 - 8 -
    • x 7 3 0 11 5 - 18 - - Z Z 0 - 53 -
  • Table 5.5.2 Example
  • Table - x 1 1 1 0 1/2 - 4 - 3/4 11/4^0 BV Eq. # x 1 x 2 x 3 x 4 x 5 x 6 x 7 RH S - x 2 2 0 1 1/2 - 2 - 1/4 1/4^0 - x 7 3 0 0 - 1/2 4 3/4 - 11/4 - Z Z 0 0 - 29/2 98 27/4 53/4 - x 3 1 2 0 1 - 8 - 3/2 11/2^0 BV Eq. # x 1 x 2 x 3 x 4 x 5 x 6 x 7 RH S - x 2 2 - 1 1 0 2 1/2 - 5/2 - x - Z Z 29 0 0 - 18 -
    • Table
  • Table
    • x 3 1 - 2 4 1 0 1/2 - 9/2 BV Eq. # x 1 x 2 x 3 x 4 x 5 x 6 x 7 RH S
    • x 4 2 - 1/2 1/2^0 1 1/4 - 5/4
      • x - Z Z 20 9 0 0 - 21/2 141/2
  • Table - x 5 1 - 4 8 2 0 1 - BV Eq. # x 1 x 2 x 3 x 4 x 5 x 6 x 7 RHS - x 4 2 1/2 - 3/2 - 1/2 - x - Z Z -
  • Table