Simplex Method - Introduction to Operations Research - Lecture Slides, Slides of Operational Research

Download Simplex Method - Introduction to Operations Research - Lecture Slides and more Slides Operational Research in PDF only on Docsity!

The Simplex Method

The most popular method for solving

Linear Programming Problems

We shall present it as an

Algorithm

General Structure of Algorithms

Initialise

Perform a sequence of repetitive steps

Check for desired results

Stop

No

Yes

Iterate

Missing Details

 Initialisation :

• How do we represent a feasible extreme point algebraically?

 Optimality Test :

• How do we determine whether a given extreme point is optimal?

 Iteration :

• How do we move a long an edge to a better adjacent extreme point?

5.1 initialisation

Transform the LP problem given in a standard

form into a canonical form.

This involves the introduction of slack

variables , one for each functional constraint.

Thus if we start with n variables and m

functional constraints, we end up with n+m variables and m functional equality constraints.

Canonical Form

a 11 x 1 + a 12 x 2 + ... + a 1 n xn + x (^) n + 1 = b 1

a 21 x 1 + a 22 x 2 + ... + a 2 n x (^) n + + xn + 2 = b 2

..... ..... ... .... .... ..... ..... ... .... ....

a (^) m 1 x 1 + am 2 x 2 + ... + amn xn + + x (^) n + m = b (^) m

x (^) j ≥ 0 , j = 1,..., n + m

max z = c (^) j j = 1

n ∑ x^ j

Observation

The i-th slack variable measure the “distance” of the point x=(x 1 ,...,x (^) n ) from the hyperplane defining the i-th constraint (This is not a Euclidean distance).

Thus, if the i-th slack variable is equal to zero the point x= (x 1 ,...,x (^) n ) is on the i-th hyperplane. Otherwise it is not.

The original variables “measure” the distance to the hyperplanes defining the respective non-negativity constraints.

Why do we do this?

If we use the slack variables as a basis , we

obtain a feasible extreme point !!!

a 11 x 1 + a 12 x 2 + ... + a 1 n xn + x (^) n + 1 = b 1

a 21 x 1 + a 22 x 2 + ... + a 2 n x (^) n + + xn + 2 = b 2

..... ..... ... .... .... ..... ..... ... .... ....

a (^) m 1 x 1 + am 2 x 2 + ... + amn xn + + x (^) n + m = b (^) m

x (^) j ≥ 0 , j = 1,..., n + m

5.5.1 Definition

A basic feasible solution is a basic solution

that satisfies the non-negativity constraint.

Observation :

A basic feasible solution is an extreme point

of the feasible region.

Thus:

Initialisation involves constructing a basic

feasible solution using the slack varaibles.

Summary of the Initialisation Step

Select the slack variables as basic

 Comments :

• Simple
• Not necessarily good selection: the first basic feasible solution can be (very) far from the optimal solution.