prepared by Imran Ismail

Slides:



Advertisements
Similar presentations
Linear Inequalities in 2 Variables
Advertisements

Linear Programming Problem
LIAL HORNSBY SCHNEIDER
CCMIII U2D4 Warmup This graph of a linear programming model consists of polygon ABCD and its interior. Under these constraints, at which point does the.
1 1 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or.
Linear Programming Models: Graphical Methods
Chapter 2 Linear Programming Models: Graphical and Computer Methods © 2007 Pearson Education.
Chapter 2: Introduction to Linear Programming
Optimization Linear Programming and Simplex Method
LINEAR PROGRAMMING: THE GRAPHICAL METHOD
Chapter 3 An Introduction to Linear Programming
1 1 Slide © 2009 South-Western, a part of Cengage Learning Slides by John Loucks St. Edward’s University.
College Algebra Fifth Edition James Stewart Lothar Redlin Saleem Watson.
Chapter 5 Linear Inequalities and Linear Programming Section 2 Systems of Linear Inequalities in Two Variables.
FORMULATION AND GRAPHIC METHOD
Graphical Solutions Plot all constraints including nonnegativity ones
1 1 Slide LINEAR PROGRAMMING: THE GRAPHICAL METHOD n Linear Programming Problem n Properties of LPs n LP Solutions n Graphical Solution n Introduction.
Stevenson and Ozgur First Edition Introduction to Management Science with Spreadsheets McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies,
Chapter 5 Linear Inequalities and Linear Programming Section R Review.
LINEAR PROGRAMMING SIMPLEX METHOD.
1 1 Slide © 2005 Thomson/South-Western Slides Prepared by JOHN S. LOUCKS ST. EDWARD’S UNIVERSITY.
1 1 Slide Linear Programming (LP) Problem n A mathematical programming problem is one that seeks to maximize an objective function subject to constraints.
L4 Graphical Solution Homework See new Revised Schedule Review Graphical Solution Process Special conditions Summary 1 Read for W for.
MATH 527 Deterministic OR Graphical Solution Method for Linear Programs.
Linear Programming Terminology. Contents 1.What is a Mathematical Model? 2.Illustration of LPP: Maximization Case 3.What.
Chapter 7 Introduction to Linear Programming
1 1 Slide © 2005 Thomson/South-Western Chapter 2 Introduction to Linear Programming n Linear Programming Problem n Problem Formulation n A Maximization.
Systems of Inequalities in Two Variables Sec. 7.5a.
Linear Programming: A Geometric Approach3 Graphing Systems of Linear Inequalities in Two Variables Linear Programming Problems Graphical Solution of Linear.
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 5 Systems and Matrices Copyright © 2013, 2009, 2005 Pearson Education, Inc.
1 1 Slide © 2001 South-Western College Publishing/Thomson Learning Anderson Sweeney Williams Anderson Sweeney Williams Slides Prepared by JOHN LOUCKS QUANTITATIVE.
D Nagesh Kumar, IIScOptimization Methods: M3L2 1 Linear Programming Graphical method.
Linear Programming Problem. Definition A linear programming problem is the problem of optimizing (maximizing or minimizing) a linear function (a function.
Systems Analysis Methods
Chapter 2 Introduction to Linear Programming n Linear Programming Problem n Problem Formulation n A Maximization Problem n Graphical Solution Procedure.
1Barnett/Ziegler/Byleen Finite Mathematics 12e Learning Objectives for Section 5.2  The student will be able to solve systems of linear inequalities graphically.
3.4 Solving Systems of Linear Inequalities ©2001 by R. Villar All Rights Reserved.
Kerimcan OzcanMNGT 379 Operations Research1 Linear Programming Chapter 2.
1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University.
Operations Research By: Saeed Yaghoubi 1 Graphical Analysis 2.
Sullivan Algebra and Trigonometry: Section 12.9 Objectives of this Section Set Up a Linear Programming Problem Solve a Linear Programming Problem.
Linear Programming: A Geometric Approach3 Graphing Systems of Linear Inequalities in Two Variables Linear Programming Problems Graphical Solution of Linear.
1 Optimization Linear Programming and Simplex Method.
1 Introduction to Linear Programming Linear Programming Problem Linear Programming Problem Problem Formulation Problem Formulation A Simple Maximization.
1 1 Slide Graphical solution A Graphical Solution Procedure (LPs with 2 decision variables can be solved/viewed this way.) 1. Plot each constraint as an.
Chapter 9: Systems of Equations and Inequalities; Matrices
Chapter 2 Linear Programming Models: Graphical and Computer Methods
An Introduction to Linear Programming
EMGT 6412/MATH 6665 Mathematical Programming Spring 2016
An Introduction to Linear Programming Pertemuan 4
Special Cases in simplex method applications
Chapter 2 An Introduction to Linear Programming
Chapter 5 Linear Inequalities and Linear Programming
Chapter 5 Linear Inequalities and Linear Programming
Systems of Equations and Inequalities
Limiting factor analysis
Copyright © Cengage Learning. All rights reserved.
Linear Programming in Two Dimensions
Chapter 5 Linear Inequalities and Linear Programming
Copyright © Cengage Learning. All rights reserved.
Graphing Systems of Inequalities
Introduction to Optimization and Linear Programming (LP)
Introduction Finding the solutions to a system of linear equations requires graphing multiple linear inequalities on the same coordinate plane. Most real-world.
Max Z = x1 + x2 2 x1 + 3 x2  6 (1) x2  1.5 (2) x1 - x2  2 (3)
Systems Analysis Methods
Linear Programming Problem
Graphical solution A Graphical Solution Procedure (LPs with 2 decision variables can be solved/viewed this way.) 1. Plot each constraint as an equation.
College Algebra Sixth Edition
Presentation transcript:

prepared by Imran Ismail SMU EMIS 8374 LP Solutions The Graphical Method prepared by Imran Ismail updated 19 January 2006

Consider the following LP Maximize X + 2Y Subject to: X + Y  4 X - 2Y  2 -2X + Y  2 X, Y  0 This problem is in two dimensions and can be solved graphically.

Finding the Feasible Region We begin by graphing the constraints on an XY coordinate system to determine the set of all points that satisfy all the constraints. This set of points is known as the feasible region for the LP. Since both variables must be non-negative, we know that the feasible region must be within the first quadrant.

The set of solutions to the equation X + Y = 4 can be represented by the straight line passing through the points (4,0) and (0,4) in the XY plane. The shaded triangle below represents the set of points satisfying the constraint X + Y  4, X  0, Y  0.

Similarly for constraint X-2Y  2, we get

And for constraint 3, -2X+Y  2: Since X and Y must be nonnegative, the feasible region for the LP is the bounded by the three lines described (and shown) above and the X and Y axes. Any point inside or on the boundary of the region described above is a feasible solution to the LP.

Finding an Optimal Solution The objective function is given as an equation whose value has yet to be determined. The information provided in the LP gives us two clues: The function has to be maximized The function must satisfy all constraints i.e. it must lie in the feasible regions

In such an instance, we can start by assuming an objective function value of 0 (zero). This gives us the following equation X + 2Y = 0 The line above will provide us with a set of points that will have an objective function value of 0. When drawn on the graph, the segment of this line that intersects the feasible region will represent the set of feasible solutions with an objective function value of zero. In this case, the set is just the single point (0,0).

Similarly, the line X + 2Y = 2 describes the set of points that have an objective function value of two. So, the segment of this line that intersects the feasible region represents the set of feasible solutions with an objective function value of two. We can continue this method till we reach an objective function value such that it no longer intersects the feasible region.

By continuing on in this fashion, we can find an optimal solution for the LP by "pushing" the objective function line up until it last touches the feasible region.

This occurs when we graph the line X + 2Y = 22/3 = 7.333 which intersects the feasible region at the point (2/3, 10/3). Since there are no feasible solutions with a greater objective function value than 22/3, we say that X = 2/3, Y = 10/3 is an optimal solution and that 22/3 is the optimal value for the objective function.

It is important to investigate how other objective functions might behave given the same feasible region.

The “Brute Force” Method Using the fact that if an LP has an optimal solution, then it has an extreme point solution, we can use a "brute force" method to find an optimal solution by: testing each extreme point to see if it is feasible and then comparing the objective function values. Recall that an extreme, or corner-point, solution is formed by the intersection of two of the constraints.

In our example we can identify the extreme points by labeling them A, B, C, D and E, as shown below.

Point A is formed by the intersection of: X + Y = 4 and –2X + Y = 2 and similarly: B: X - 2Y = 2 and X + Y = 4 C: X - 2Y = 2 and Y = 0 D: X = 0 and Y = 0 E: -2X + Y = 2 and X = 0 Solving these equations and plugging them into the objective function, we can calculate the objective function value all of the extreme points of the feasible region.

The co-ordinate points of A give us the maximum objective function value. X-co-ord Y co-ord Obj. Val A .66 3.33 7.33 B 4.66 C 2 D E 4

Conclusion The graphical method and the brute force method will always obtain the same result. In our example, it was found that the optimal solution is such that: Xopt=2/3 = 0.66 Yopt=10/3 = 3.33 Zopt = 22/3 = 7.333

Unique Optimal Solution

Infeasibility

Other Special Cases of LP Adding the constraint Y  4 to the original LP makes the problem infeasible. Removing the constraint X + Y  4from the original LP makes the problem unbounded. Changing the objective function of the original LP to gives an LP with multiple optimal solutions.

Unboundedness

Alternative Optima