1. Chapter 5 The Simplex Method
The most popular method for solving Linear Programming Problems
We shall present it as an
Algorithm

2. General Structure of Algorithms
Initialise
Check
for
desired
results
Yes
Stop
Iterate No
Perform a sequence
of repetitive steps

3. to a better extreme point
Construct a feasible
extreme point
Is this
point
optimal ?
Yes
Stop
Iterate No
Move along an edge
to a better extreme point

4. 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. 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.

6. Standard Form
opt=max
~ 
bi ≥ 0 , for all i.

7. Canonical Form

8. Observation
The i-th slack variable measure the “distance” of the point x=(x1,...,xn) 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= (x1,...,xn) 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.

9. Example
x3,x4,x5 are slack variables

10. Why do we do this?
If we use the slack variables as a basis, we obtain a feasible extreme point !!!

11. 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.

12. Example x3,x4,x5 are slack variables
Initial basic feasible solution: x =(0,0,40,30,15), namely
x1 = 0 x2 = 0 x3 = 40 x4 = 30 x5 =15

13. 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.