MIE376 Lec 2 - Simplex

MIE376 Lec 2 - Simplex - Solving the LP Problem LP Software...

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MIE376 - Mathematical Programmingt 1 Solving the LP Problem LP Software Foundations of Simplex Method Simplex by Substitution and = Constraints Variables Unrestricted in Sign (URS)
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MIE376 - Mathematical Programmingt 2 Solving the LP Problem LP Software Foundations of Simplex Method Simplex by Substitution and = Constraints Variables Unrestricted in Sign (URS)
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MIE376 - Mathematical Programmingt 3 LP Software ± INFORMS regularly surveys the marketplace for LP software systems ² For the latest 2009 survey see http://www.lionhrtpub.com/orms/surveys/LP/LP- survey.html ± ECF labs includes the following LP software: ² Excel with Solver – for introductory courses ² OPL/CPLEX - the real McKoy – used in OR courses ² AMPL/Gurobi – the real McKoy – used in OR courses ² GAMS – another real McKoy ² Lindo/Lingo – another introductory package
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MIE376 - Mathematical Programmingt 4 LP System Components ± Front-end GUI/DB application ² Customized software ± Back-End Database System ² Access, Oracle, DB2, MS-SQL ± Back-End Modeling System ² OPL, GAMS ² Usually separate model from data (.mod and .dat) ± Back-End Solver ² CPLEX, MINOS ² Also see: http://www.gams.com/solvers/index.htm
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MIE376 - Mathematical Programmingt 5 Typical LP System Diagram GUI/ DB App Modeling system (.mod, .dat) Solver Database
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MS Excel / Frontline Solver MIE376 - Mathematical Programmingt 6 GUI/ DB App Modeling system Database MS Excel Solver Frontline Solver Add-In
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MIE376 - Mathematical Programmingt 7 AMPL/Gurobi GUI/ DB App Modeling system Database AMPL Solver Gurobi
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MIE376 - Mathematical Programmingt 8 Solving the LP Problem LP Software Foundations of Simplex Method Simplex by Substitution and = Constraints Variables Unrestricted in Sign (URS)
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MIE376 - Mathematical Programmingt 9 Reconsider the Lathe and Drill problem ± Let x 1 be the number of chairs to be manufactured ± Let x 2 be the number of tables to be manufactured ± Let z be the objective function, i.e. profit ± LP: ² Maximize z = x 1 +2*x 2 ² subject to the “constraints” ² x 1 + 3*x 2 16 - lathe capacity ² x 1 + x 2 7 - drill capacity ² x 1 0 - chair variable non-negative ² x 2 0 - table variable non-negative
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MIE376 - Mathematical Programmingt 10 Graphically x 2 x 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 7 6 5 4 3 2 1 Feasible region for x 1 , x 2
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MIE376 - Mathematical Programmingt 11 Key Insight optimal solution is at a “corner point” of the feasible region, technically referred to as basic feasible solutions (bfs) x 2 x 1
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MIE376 - Mathematical Programmingt 12 Simplex method skeleton 1. Find an initial basic feasible solution – usually the origin 2. Move to the most promising adjacent basic feasible solution 3. Repeat step 2 until there is no better adjacent basic feasible solution
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MIE376 - Mathematical Programmingt 13 What if we have thousands of variables?
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This note was uploaded on 04/20/2011 for the course MIE 376 taught by Professor Daniel during the Spring '11 term at University of Toronto.