BSOP209 The Simplex Method of Linear Programming

BSOP209 The Simplex Method of Linear Programming - CD...

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CD Tutorial 3 The Simplex Method of Linear Programming Tutorial Outline CONVERTING THE CONSTRAINTS TO EQUATIONS SETTING UP THE FIRST SIMPLEX TABLEAU SIMPLEX SOLUTION PROCEDURES SUMMARY OF SIMPLEX STEPS FOR MAXIMIZATION PROBLEMS ARTIFICIAL AND SURPLUS VARIABLES SOLVING MINIMIZATION PROBLEMS S UMMARY K EY T ERMS S OLVED P ROBLEM D ISCUSSION Q UESTIONS P ROBLEMS
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T3-2 CD T UTORIAL 3T HE S IMPLEX M ETHOD OF L INEAR P ROGRAMMING Most real-world linear programming problems have more than two variables and thus are too com- plex for graphical solution. A procedure called the simplex method may be used to find the optimal solution to multivariable problems. The simplex method is actually an algorithm (or a set of instruc- tions) with which we examine corner points in a methodical fashion until we arrive at the best solu- tion—highest profit or lowest cost. Computer programs and spreadsheets are available to handle the simplex calculations for you. But you need to know what is involved behind the scenes in order to best understand their valuable outputs. CONVERTING THE CONSTRAINTS TO EQUATIONS The first step of the simplex method requires that we convert each inequality constraint in an LP for- mulation into an equation. Less-than-or-equal-to constraints ( ) can be converted to equations by adding slack variables , which represent the amount of an unused resource. We formulate the Shader Electronics Company’s product mix problem as follows, using linear programming: Maximize profit = $7 X 1 + $5 X 2 subject to LP constraints: where X 1 equals the number of Walkmans produced and X 2 equals the number of Watch-TVs produced. To convert these inequality constraints to equalities, we add slack variables S 1 and S 2 to the left side of the inequality. The first constraint becomes 2 X 1 + 1 X 2 + S 1 = 100 and the second becomes 4 X 1 + 3 X 2 + S 2 = 240 To include all variables in each equation (a requirement of the next simplex step), we add slack vari- ables not appearing in each equation with a coefficient of zero. The equations then appear as Because slack variables represent unused resources (such as time on a machine or labor-hours avail- able), they yield no profit, but we must add them to the objective function with zero profit coeffi- cients. Thus, the objective function becomes Maximize profit = $7 X 1 + $5 X 2 + $0 S 1 + $0 S 2 SETTING UP THE FIRST SIMPLEX TABLEAU To simplify handling the equations and objective function in an LP problem, we place all of the coefficients into a tabular form. We can express the preceding two constraint equations as S OLUTION M IX X 1 X 2 S 1 S 2 Q UANTITY (RHS) S 1 2 1 1 0 100 S 2 4 3 0 1 240 The numbers (2, 1, 1, 0) and (4, 3, 0, 1) represent the coefficients of the first equation and second equation, respectively.
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This note was uploaded on 08/10/2010 for the course MGMT 400 taught by Professor Frankenstein during the Spring '10 term at DeVry Chicago O'Hare.

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BSOP209 The Simplex Method of Linear Programming - CD...

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