Foundation of the Simplex Method

 Constraints Boundary Equations  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. 2 Dimensional Space 3 Dimensional Space n Dimensional Space LinePlaneHyperPlane

 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 Set of equations Solve simultaneously CPF solutions 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 1 + x 2 ≤ 6 -x 1 + 2x 3 ≤ 4 x 1 ≤ 4 x 3 ≤ 4 x 1 ≥ 0 x 2 ≥ 0 x 3 ≥ 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

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

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