Public Policy Modeling Simplex Method Tuesday, October 13, 2015 Hun Myoung Park, Ph.D. Public Management & Policy Analysis Program Graduate School of International.

Slides:



Advertisements
Similar presentations
Hillier and Lieberman Chapter 4.
Advertisements

Operation Research Chapter 3 Simplex Method.
SIMPLEX METHOD FOR LP LP Model.
Chapter 6 Linear Programming: The Simplex Method
The Simplex Method The geometric method of solving linear programming problems presented before. The graphical method is useful only for problems involving.
Dr. Sana’a Wafa Al-Sayegh
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Computational Methods for Management and Economics Carla Gomes Module 6a Introduction to Simplex (Textbook – Hillier and Lieberman)
Computational Methods for Management and Economics Carla Gomes Module 8b The transportation simplex method.
Chapter 6 Linear Programming: The Simplex Method Section 3 The Dual Problem: Minimization with Problem Constraints of the Form ≥
Linear Inequalities and Linear Programming Chapter 5
Computational Methods for Management and Economics Carla Gomes Module 6b Simplex Pitfalls (Textbook – Hillier and Lieberman)
The Simplex Method: Standard Maximization Problems
The Simplex Algorithm An Algorithm for solving Linear Programming Problems.
Introduction to the Simplex Algorithm Active Learning – Module 3
Operation Research Chapter 3 Simplex Method.
Solving Linear Programs: The Simplex Method
Optimization Mechanics of the Simplex Method
Linear Programming (LP)
The Simplex Method.
5.6 Maximization and Minimization with Mixed Problem Constraints
MIT and James Orlin © Chapter 3. The simplex algorithm Putting Linear Programs into standard form Introduction to Simplex Algorithm.
LINEAR PROGRAMMING SIMPLEX METHOD.
Linear Programming - Standard Form
Learning Objectives for Section 6.2
Chapter 3 Linear Programming Methods 高等作業研究 高等作業研究 ( 一 ) Chapter 3 Linear Programming Methods (II)
Chapter 6 Linear Programming: The Simplex Method
The Two-Phase Simplex Method LI Xiao-lei. Preview When a basic feasible solution is not readily available, the two-phase simplex method may be used as.
8. Linear Programming (Simplex Method) Objectives: 1.Simplex Method- Standard Maximum problem 2. (i) Greedy Rule (ii) Ratio Test (iii) Pivot Operation.
Simplex Algorithm.Big M Method
Duality Theory LI Xiaolei.
Chapter 6 Linear Programming: The Simplex Method Section 2 The Simplex Method: Maximization with Problem Constraints of the Form ≤
Topic III The Simplex Method Setting up the Method Tabular Form Chapter(s): 4.
Barnett/Ziegler/Byleen Finite Mathematics 11e1 Learning Objectives for Section 6.4 The student will be able to set up and solve linear programming problems.
Kerimcan OzcanMNGT 379 Operations Research1 Linear Programming: The Simplex Method Chapter 5.
1 1 © 2003 Thomson  /South-Western Slide Slides Prepared by JOHN S. LOUCKS St. Edward’s University.
1 1 © 2003 Thomson  /South-Western Slide Slides Prepared by JOHN S. LOUCKS St. Edward’s University.
Duality Theory  Every LP problem (called the ‘Primal’) has associated with another problem called the ‘Dual’.  The ‘Dual’ problem is an LP defined directly.
1© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ Linear Programming: The Simplex Method.
Simplex Method Adapting to Other Forms.  Until now, we have dealt with the standard form of the Simplex method  What if the model has a non-standard.
Solving Linear Programming Problems: The Simplex Method
Water Resources Development and Management Optimization (Linear Programming) CVEN 5393 Mar 4, 2011.
4  The Simplex Method: Standard Maximization Problems  The Simplex Method: Standard Minimization Problems  The Simplex Method: Nonstandard Problems.
Mechanical Engineering Department 1 سورة النحل (78)
Linear Programming Erasmus Mobility Program (24Apr2012) Pollack Mihály Engineering Faculty (PMMK) University of Pécs João Miranda
1 1 Slide © 2005 Thomson/South-Western Linear Programming: The Simplex Method n An Overview of the Simplex Method n Standard Form n Tableau Form n Setting.
Chapter 4 Linear Programming: The Simplex Method
Chapter 6 Linear Programming: The Simplex Method Section 4 Maximization and Minimization with Problem Constraints.
University of Colorado at Boulder Yicheng Wang, Phone: , Optimization Techniques for Civil and Environmental Engineering.
Simplex Method for solving LP problems with two variables.
An-Najah N. University Faculty of Engineering and Information Technology Department of Management Information systems Operations Research and Applications.
Simplex Method Simplex: a linear-programming algorithm that can solve problems having more than two decision variables. The simplex technique involves.
Part 3. Linear Programming 3.2 Algorithm. General Formulation Convex function Convex region.
 LP graphical solution is always associated with a corner point of the solution space.  The transition from the geometric corner point solution to the.
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Linear Programming: An Algebraic Approach 4 The Simplex Method with Standard Maximization.
1 Simplex algorithm. 2 The Aim of Linear Programming A Linear Programming model seeks to maximize or minimize a linear function, subject to a set of linear.
Decision Support Systems INF421 & IS Simplex: a linear-programming algorithm that can solve problems having more than two decision variables.
The Simplex Method. and Maximize Subject to From a geometric viewpoint : CPF solutions (Corner-Point Feasible) : Corner-point infeasible solutions 0.
Solving Linear Program by Simplex Method The Concept
The Simplex Method The geometric method of solving linear programming problems presented before. The graphical method is useful only for problems involving.
Linear programming Simplex method.
10CS661 OPERATION RESEARCH Engineered for Tomorrow.
The Simplex Method: Standard Minimization Problems
ENGM 631 Optimization Ch. 4: Solving Linear Programs: The Simplex Method.
Solving Linear Programming Problems: Asst. Prof. Dr. Nergiz Kasımbeyli
Well, just how many basic
The Simplex Method The geometric method of solving linear programming problems presented before. The graphical method is useful only for problems involving.
Linear programming Simplex method.
Presentation transcript:

Public Policy Modeling Simplex Method Tuesday, October 13, 2015 Hun Myoung Park, Ph.D. Public Management & Policy Analysis Program Graduate School of International Relations

2 Simplex Method Algorithm to solving LP problems. Developed by George Dantzig in 1947 Search for optimal solution by examining corner point feasible solutions (corner solutions) Iteration algorithm keeps searching until finding the optimal solution. Efficient algorithm to reach the optimal solution by focusing on corner solutions Informative to provide valuable information such as shadow price (sensitivity analysis)

3 Augmented Form 1 Convert the functional inequality constraints to equivalent equality constraints by introducing slack variables Original formAugmented form Max Z=3x 1 +5x 2, Subject to x 1 <= 4 2x 2 <= 12 3x 1 + 2x 2 <= 18 Z-3x 1 -5x 2 =0 Subject to x 1 + s 1 = 4 2x 2 + s 2 = 12 3x 1 + 2x 2 + s 3 = 18

4 Augmented Form 2 The original form has been augmented by some supplementary variables (slack variables) Augmented solution is a solution for the original variables that has been augmented by the corresponding values of the slack variables. Basic solution is an augmented corner-point solution and Basic feasible (BF) solution is an augmented corner point feasible (CPF) solution

5 Simplex Method 1 Tabular form is often used Use augmented form with equality constraints Construct a simplex tableau A simplex tableau summarizes coefficients in the middle and result values on the right side Objective function Z always appears on the top. The Z value appear on the right side column Right side contains the (initial) basic feasible solution. See Excel Worksheet for an example.

6 Simplex Method 2 An example of an Initial Simplex Tableau BasicCoefficientsRHS Zx1x2s1s2s3 Z S S S

7 Simplex Method 4 Initialization  Optimality Test  Iteration  Optimality test  … Initialization: construct the initial simplex tableau –Decision variables remain nonbasic variables (set equal to zero) –Slack variables become basic variables that appear on the left. Optimality test: if and only if every coefficient in row for Z is nonnegative (>=0) current BF solution is optimal. Otherwise, go to an iteration.

8 Simplex Method 5 Iteration 1: determine the entering basic variable (to be basic variables in the next iteration) with the negative coefficient having the largest absolute value. The column for such variable is called pivot column. Iteration 2: determine the leaving basic variable (to be nonbasic variables in the next iteration) using minimum ratio test. See pp of Hillier & Lieberman (2010)

9 Simplex Method 6 Minimum ratio test: –Pick out each positive (>0) coefficient in the pivot column –Divide right side values by positive coefficients –Identify the row having the smallest ratio –The row (basic variable) is leaving basic variable and is called pivot row. –The value in pivot column and row is pivot number. Minimum ratio test determine which basic variable drops to zero first as the entering basic variable is increased.

10 Simplex Method 7 Iteration 3: solve for the new basic feasible solution by Gauss-Jordan method of elimination –Divide the pivot row by the pivot number –Identify a row other than pivot row that has nonzero coefficient in the pivot column –Multiply the pivot row by the coefficient and then subtract the pivot row from the row of interest. –Repeat it to other row having nonzero coefficient in the pivot column Optimality Test to see if every coefficients in row Z are nonnegative.

11 Further Directions Duality Theory: primal versus dual problem Transportation and Assignment Integer (Binary) Programming Goal Programming