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 Check for desired results Yes Stop Iterate No Perform a sequence of repetitive steps

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

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.

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

Canonical Form

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.

Example x3,x4,x5 are slack variables

Why do we do this? 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 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.

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

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.