Chapter 5_Notes - Linear Programming: The Simplex Method...

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Kerimcan Ozcan MNGT 379 Operations Research 1 Linear Programming: The Simplex Method Chapter 5
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Kerimcan Ozcan MNGT 379 Operations Research 2 An Overview of the Simplex Method Standard Form Tableau Form Setting Up the Initial Simplex Tableau Improving the Solution Calculating the Next Tableau Solving a Minimization Problem Special Cases Steps Leading to the Simplex Method Formulate Formulate Problem  Problem  as LP as LP Put In Put In Standard  Standard  Form Form Put In Put In Tableau  Tableau  Form Form Execute Execute Simplex  Simplex  Method Method
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Kerimcan Ozcan MNGT 379 Operations Research 3 Standard Form A Minimization Problem MIN 2 x 1 - 3 x 2 - 4 x 3 s. t. x 1 + x 2 + x 3 < 30 2 x 1 + x 2 + 3 x 3 > 60 x 1 - x 2 + 2 x 3 = 20 x 1 , x 2 , x 3 > 0 An LP is in standard form when: All variables are non-negative All constraints are equalities Putting an LP formulation into standard form involves: Adding slack variables to “< “ constraints Subtracting surplus variables from “> constraints. Problem in Standard Form MIN 2 x 1 - 3 x 2 - 4 x 3 s. t. x 1 + x 2 + x 3 + s 1 = 30 2 x 1 + x 2 + 3 x 3 - s 2 = 60 x 1 - x 2 + 2 x 3 = 20 x 1 , x 2 , x 3 , s 1 , s 2 > 0
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Kerimcan Ozcan MNGT 379 Operations Research 4 Tableau Form A set of equations is in tableau form if for each equation: its right hand side (RHS) is non-negative, and there is a basic variable. (A basic variable for an equation is a variable whose coefficient in the equation is +1 and whose coefficient in all other equations of the problem is 0.) To generate an initial tableau form: An artificial variable must be added to each constraint that does not have a basic variable. Problem in Tableau Form MIN 2 x 1 - 3 x 2 - 4 x 3 + 0 s 1 - 0 s 2 + Ma 2 + Ma 3 s. t. x 1 + x 2 + x 3 + s 1 = 30 2 x 1 + x 2 + 3 x 3 - s 2 + a 2 = 60 x 1 - x 2 + 2 x 3 + a 3 = 20 x 1 , x 2 , x 3 , s 1 , s 2 , a 2 , a 3 > 0
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Kerimcan Ozcan MNGT 379 Operations Research 5 Setting Up Initial Simplex Tableau The simplex tableau is a convenient means for performing the calculations required by the simplex method. Step 1: If the problem is a minimization problem, multiply the objective function by -1. Step 2: If the problem formulation contains any constraints with negative right-hand sides, multiply each constraint by -1. Step 3: Add a slack variable to each < constraint. Step 4: Subtract a surplus variable and add an artificial variable to each > constraint. Step 5:
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Chapter 5_Notes - Linear Programming: The Simplex Method...

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