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

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

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

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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?

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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.

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

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Canonical Form

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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.

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

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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 !!!

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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.

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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

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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.

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