4120_lecture-12-F10 - Computational Methods for Management...

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Unformatted text preview: Computational Methods for Management and Economics Carla Gomes Lecture 12 Reading: 4.1-4.6 of textbook Outline Recap Simplex How to handle simplexs pitfalls Simplex Method Iterative procedure involving the following steps: 1. Initialization find initial CPF solution Whenever possible pick (0,0) as initial solution 1. Optimality test (check value of Z of adjacent CPF solutions) 1. Iteration find a better CPF solution; go to 2. 1. Consider edges that emanate from current CPF solution and pick the one that increases Z at a faster rate 2. Stop at the first new constraint boundary Wyndor Glass 2 4 6 8 8 6 4 2 Production rate for windows Production rate for doors Feasible region (2, 6) Optimal solution 10 W D P = 3600 = 300D + 500W P = 3000 = 300D + 500W P = 1500 = 300D + 500W CPF Edge of Feasible region 1 Z=0 Z=30 Let D = the number of doors to produce W = the number of windows to produce Maximize P = 3 D + 5 W 2 Z=36 Z=27 3 Simplex Method - Algebraic / Tabular Form Tabular form more compact form it records only the essential information namely: 1. Coefficients of variables 2. The constants on the right hand sides 3. Basic variable in each equation Note: only x j vars are basic and non-basic we can think of Z as the basic var. of objective function. Simplex Method in Tabular Form Assumption: Standard form max; only <= functional constraints; all vars have non-negativity constraints; rhs are positive Initialization Introduce slack variables Decision variables non-basic variables (set to 0) Slack variables basic variables (set to corresponding rhs) Transform the objective function and the constraints into equality constraints Optimality Test Current solution is optimal iff all the coefficients of objective function are non-negative Simplex Method in Tabular Form (cont.) Iteration: Move to a better BFS Step1 Entering Variable non-basic variable with the most negative coefficient in the objective function. Mark that column as the pivot column . Step2 Leaving basic variable apply the minimum ratio test : Consider in the pivot column only the coeffcients that are strictly positive Divide each of theses coefficients into the rhs entry for the same row Identify the row with the smallest of these ratios The basic variable for that row is the leaving variable; mark that row as the pivot row; The number in the intersection of the pivot row with the pivot column is the pivot number Simplex Method in Tabular Form (cont.) Iteration: Move to a better BFS Step3 Solve for the new BFS by using elementary row operations to construct a new simplex tableau in proper form Divide pivot row by the pivot number for each row (including objective function) that has a negative coefficient in the pivot column, add to this row the product of the absolute value of this coefficient and the new pivot row....
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This note was uploaded on 02/25/2011 for the course AEM 4120 taught by Professor Gomes,c. during the Fall '08 term at Cornell University (Engineering School).

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4120_lecture-12-F10 - Computational Methods for Management...

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