Chapter 5 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

Construct a feasible extreme point Move along an edge to a better extreme point Is this point optimal ? 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 form into a form.  Transform the LP problem given in a standard form into a canonical form.  This involves the introduction of, one for each functional constraint.  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 and m functional constraints.  Thus if we start with n variables and m functional constraints, we end up with n+m variables and m functional equality constraints.

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

Canonical Form

ObservationObservation  The i-th measure the “distance” of the point x=(x 1,...,x n ) from the defining the i-th constraint (This is not a Euclidean distance).  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 the point x= (x 1,...,x n ) is. Otherwise it is not.  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 “measure” the distance to the hyperplanes defining the respective constraints.  The original variables “measure” the distance to the hyperplanes defining the respective non-negativity constraints.

ExampleExample x 3,x 4,x 5 are slack variables

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

5.5.1 Definition A basic feasible solution is a basic solution that satisfies the constraint. A basic feasible solution is a basic solution that satisfies the non-negativity constraint. : Observation: A basic feasible solution is an of the feasible region. A basic feasible solution is an extreme point of the feasible region.Thus: involves constructing a using the. Initialisation involves constructing a basic feasible solution using the slack varaibles.

Example basic feasible solution: x =(0,0,40,30,15), namely Initial basic feasible solution: x =(0,0,40,30,15), namely x 1 = 0 x 2 = 0 x 1 = 0 x 2 = 0 x 3 = 40 x 4 = 30 x 5 =15 x 3,x 4,x 5 are slack variables

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.