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4120_lecture11-F10

4120_lecture11-F10 - Computational Methods for Management...

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Computational Methods for Management and Economics Carla Gomes Lecture 10 Reading: 4.1-4.3 of textbook
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Outline Simplex: algebraic procedure By hand Using IOR Java applet (CD textbook)
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Simplex as an algebraic procedure System of functional constraints n (5) variables (5) and m (3) equations 2 degrees of freedom, (i.e., we can set those two variables to any arbitrary values); they are the nonbasic variables; the other variables are the basic variables; Simplex chooses to set the non-basic variables to ZERO. Simplex solves the simultaneous equations to set the values of the basic variables;
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Properties of Basic Solutions A basic solution is composed of: 1. Basic variables – 1. number of basic variables equals number of functional constraints 2. Their values results from solving the system of functional constraints (non-basic variables set to 0) 3. They form the basis 1. Non-basic variables – 1. number of non-basic variables equals (total number of variables - number of functional constraints) 2. They are set to ZERO Basic Feasible solution – it is a basic solution that satisfies the non- negativity constraints Adjacent basic feasible solutions – all but one of their basic (non-basic) variables are the same moving from one basic feasible solution to an adjacent one involves switching one variable from non-basic to basic and one variable from basic to non-basic (check graph)
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Simplex Procedure Initialization – origin whenever possible (decision variables 0) (okay if standard form with positive RHS’s; basic feasible solution (BFS): each equation has a basic variable with coefficient 1 (slack variable = RHS) and the variable does not appear in any other eq; the decision variables are the non-basic variables set to 0) Optimality test – is current BFS optimal? ( the coefficients of the objective function of the non-basic variables gives the rate of improvement in Z )
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Iteration – move to a better adjacent BFS a)Variable entering the basis Consider non-basic variables (Graphically - Consider edges emanating from current CPF solution) Pick
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