Foundation of the Simplex
Method


Constraints Boundary Equations
2 Dimensional
Space
Line
3 Dimensional
Space
Plane
n Dimensional
Space
HyperPlane


Graphical approach is very limited based on number of variables. The simplex method overcomes this obstacle
Optimal solutions are on the boundaries of the feasible region.





Corner-Point Feasible (CPF) solution is a feasible solution that does not lie on any line segment connection to other feasible solution.
For any linear programming problem with n decision variables, each CPF solution lies at the intersection of n constraint boundaries.
CPF solution is the simultaneous solution of a system if n constraint boundaries equations.
We call these constraint equations Defining Equations.



 n -decision variables ( n non-negativity constraints) m functional constraint
Total of n + m constraints
CPF solutions

 n m



Set of equations
Solve simultaneously
Corner-point non feasible solutions
No solution


Simplex method moves form the Current CPF solution to an
Adjacent CPF solution !


What is the path followed in the process?
What does the adjacent CPF solution mean?

x
-x
1 x
1
1
+ x
2
≤ 4
≤ 6
+ 2 x
3
≤ 4 x
3
≤ 4 x
1
≥ 0 x
2 x
3
≥ 0
≥ 0







A CPF solution lies at the intersection of n constraint boundaries
CPF solution satisfies the other constraint as well
An edge is a line segment that lies at the intersection of n -1 constraint boundaries
2 CPF solutions are adjacent if the line segment connecting them is an edge of the feasible region
Emanating from each CPF are n edges which lead us to n adjacent CPF solutions
In any iteration of simplex method we are moving from current CPF solution to an adjacent one along with on of the edges .


Property 1:


When there is only one Optimal Solution it should be a CPF solution
When there is multiple Optimal solutions at least two must be adjacent CPF solutions.

It suggests:
 we just need to search the CPF solutions to find the optimal solution.


Property 2:

There are only a finite number of CPF solutions.


 n number of decision variables m number of functional constraints number of different sets of defining equations
 
 n n m




Property 3:

If no adjacent CPF solution is better than the current CPF solution, then the Optimal
Solution is found

Type of constraint
Non-Negativity
Functional (≤)
Functional (=)
Functional (≥)
Form of
Constraint x
∑a ij
∑a ij
∑a ij j
≥ 0 x x x j j j
≤ b
= b
≥ b i i i
Constraint in augmented form
∑a ij
∑a ij
∑a ij x j x j x j x j
≥ 0
+ x
+ x
+ x n+i n+i n+i
x sj
≤ b i
= b i
≥ b i
Indicating variable x j x n+I x n+I x n+I

1) Deleting one non basic variable, entering basic variable


 the variable was an indicating variable in current solution it was used to define one of the constraints as defining constraint deleting it from non-basics removes that constraint form the defining constraints

2) Moving away from this current solution by increasing this one variable, while keeping the rest ( n – 1) non basic variables at 0


 other non basic variables are indicating variables.
we keep them at 0 which means, we are keeping n -1 other defining constraint as defining constraint at this stage

3) Stopping when the first of the basic variables (leaving basic variable) reaches 0

 when a basic variable reaches 0 it will become an indicating variable.
so it defines a new constraint as the defining constraint.

I n each iteration we are changing one of our defining constraints, which means that we are moving from one CPF solution to an adjacent one.