Operations Research Assistant Professor Dr. Sana’a Wafa Al-Sayegh 2 nd Semester 2008-2009 ITGD4207 University of Palestine.

Slides:



Advertisements
Similar presentations
Solving Linear Programming Problems
Advertisements

Tuesday, March 5 Duality – The art of obtaining bounds – weak and strong duality Handouts: Lecture Notes.
Geometry and Theory of LP Standard (Inequality) Primal Problem: Dual Problem:
Understanding optimum solution
Chapter 5 Sensitivity Analysis: An Applied Approach
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. 1 Chapter 5 Sensitivity Analysis: An Applied Approach to accompany Introduction to.
Introduction to Sensitivity Analysis Graphical Sensitivity Analysis
1 Lecture 3 MGMT 650 Sensitivity Analysis in LP Chapter 3.
1/53 Slide Linear Programming: Sensitivity Analysis and Interpretation of Solution n Introduction to Sensitivity Analysis n Graphical Sensitivity Analysis.
Chapter 7 Linear Programming Models Part One n Basis of Linear Programming n Linear Program formulati on.
Chapter 3 Linear Programming: Sensitivity Analysis and Interpretation of Solution MT 235.
Sensitivity analysis BSAD 30 Dave Novak
1 Linear Programming Using the software that comes with the book.
1 1 Slide LINEAR PROGRAMMING Introduction to Sensitivity Analysis Professor Ahmadi.
LINEAR PROGRAMMING: THE GRAPHICAL METHOD
Chapter 3 An Introduction to Linear Programming
John Loucks Modifications by A. Asef-Vaziri Slides by St. Edward’s
1 1 Slide LINEAR PROGRAMMING: THE GRAPHICAL METHOD n Linear Programming Problem n Properties of LPs n LP Solutions n Graphical Solution n Introduction.
1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University.
1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University.
Kerimcan OzcanMNGT 379 Operations Research1 LP: Sensitivity Analysis and Interpretation of Solution Chapter 3.
Readings Readings Chapter 3
Presentation: H. Sarper
1 1 Slide © 2005 Thomson/South-Western Slides Prepared by JOHN S. LOUCKS ST. EDWARD’S UNIVERSITY.
1 1 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole.
Operations Research Assistant Professor Dr. Sana’a Wafa Al-Sayegh 2 nd Semester ITGD4207 University of Palestine.
Managerial Decision Making and Problem Solving
1 1 Slide Linear Programming (LP) Problem n A mathematical programming problem is one that seeks to maximize an objective function subject to constraints.
Stevenson and Ozgur First Edition Introduction to Management Science with Spreadsheets McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies,
Chapter 6 Sensitivity Analysis & Duality
Operations Research Assistant Professor Dr. Sana’a Wafa Al-Sayegh 2 nd Semester ITGD4207 University of Palestine.
1 Chapter 10 The primal linear program solution answers the tactical question when it tells us how much to produce. But the dual can have far greater impact.
THE GALAXY INDUSTRY PRODUCTION PROBLEM -
1 LINEAR PROGRAMMING Introduction to Sensitivity Analysis Professor Ahmadi.
Chapter 6 Supplement 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.
5-1 Wyndor (Before What-If Analysis). 5-2 Using the Spreadsheet to do Sensitivity Analysis The profit per door has been revised from $300 to $200. No.
1 1 Slide © 2009 South-Western, a part of Cengage Learning Slides by John Loucks St. Edward’s University.
Linear Programming: Sensitivity Analysis and Interpretation of Solution Pertemuan 5 Matakuliah: K0442-Metode Kuantitatif Tahun: 2009.
Sensitivity Analysis Consider the CrossChek hockey stick production problem:   Management believes that CrossChek might only receive $120 profit from the.
1 1 Slide © 2001 South-Western College Publishing/Thomson Learning Anderson Sweeney Williams Anderson Sweeney Williams Slides Prepared by JOHN LOUCKS QUANTITATIVE.
1 The Dual in Linear Programming In LP the solution for the profit- maximizing combination of outputs automatically determines the input amounts that must.
1 1 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or.
Chapter 2 Introduction to Linear Programming n Linear Programming Problem n Problem Formulation n A Maximization Problem n Graphical Solution Procedure.
Professional software packages such as The WinQSB and LINDO provide the following LP information: Information about the objective function: –its optimal.
Chapter 6 Simplex-Based Sensitivity Analysis and Duality
Advanced Operations Research Models Instructor: Dr. A. Seifi Teaching Assistant: Golbarg Kazemi 1.
 Minimization Problem  First Approach  Introduce the basis variable  To solve minimization problem we simple reverse the rule that is we select the.
Sensitivity Analysis Consider the CrossChek hockey stick production problem: Management believes that CrossChek might only receive $120 profit from the.
1 1 Slide © 2005 Thomson/South-Western Simplex-Based Sensitivity Analysis and Duality n Sensitivity Analysis with the Simplex Tableau n Duality.
1 1 Slide © 2000 South-Western College Publishing/ITP Slides Prepared by JOHN LOUCKS.
Sensitivity analysis continued… BSAD 30 Dave Novak Source: Anderson et al., 2013 Quantitative Methods for Business 12 th edition – some slides are directly.
Kerimcan OzcanMNGT 379 Operations Research1 Linear Programming Chapter 2.
Sensitivity Analysis Introduction to Sensitivity Analysis
1 Lecture 12 Chapter 7 Duality and Sensitivity in Linear Programming 7.1 Objective values usually can be interpreted as either minimize cost or maximize.
P RIMAL -D UAL LPP. T HE R EDDY M IKKS C OMPANY - PROBLEM Reddy Mikks company produces both interior and exterior paints from two raw materials, M 1 and.
1 1 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole.
Linear Programming: Sensitivity Analysis and Duality
St. Edward’s University
Chapter 5 Sensitivity Analysis: An Applied Approach
Chapter 5 Simplex-Based Sensitivity Analysis and Duality
Chap 9. General LP problems: Duality and Infeasibility
Chapter 5. Sensitivity Analysis
Sensitivity Analysis and
St. Edward’s University
Section 3.4 Sensitivity Analysis.
Linear Programming – Sensitivity Analysis
Presentation transcript:

Operations Research Assistant Professor Dr. Sana’a Wafa Al-Sayegh 2 nd Semester ITGD4207 University of Palestine

ITGD4207 Operations Research Chapter 4 Linear Programming Duality and Sensitivity analysis

Linear Programming Duality and Sensitivity analysis Dual Problem of an LPP Definition of the dual problem Examples (1,2 and 3) Sensitivity Analysis Examples (1 and 2)

Given a LPP (called the primal problem), we shall associate another LPP called the dual problem of the original (primal) problem. We shall see that the Optimal values of the primal and dual are the same provided both have finite feasible solutions. This topic is further used to develop another method of solving LPPs and is also used in the sensitivity (or post-optimal) analysis. Dual Problem of an LPP

Definition of the dual problem Given the primal problem (in standard form) Maximize subject to

the dual problem is the LPP Minimize subject to

If the primal problem (in standard form) is Minimize subject to

Then the dual problem is the LPP Maximize subject to

1. In the dual, there are as many (decision) variables as there are constraints in the primal. We usually say y i is the dual variable associated with the ith constraint of the primal. 2. There are as many constraints in the dual as there are variables in the primal. We thus note the following:

3. If the primal is maximization then the dual is minimization and all constraints are  If the primal is minimization then the dual is maximization and all constraints are  4.In the primal, all variables are  0 while in the dual all the variables are unrestricted in sign.

5. The objective function coefficients c j of the primal are the RHS constants of the dual constraints. 6. The RHS constants b i of the primal constraints are the objective function coefficients of the dual. 7. The coefficient matrix of the constraints of the dual is the transpose of the coefficient matrix of the constraints of the primal.

Example1 Write the dual of the LPP subject to Maximize

Thus the primal in the standard form is: subject to Maximize

Hence the dual is: subject to Minimize

Example2 Write the dual of the LPP subject to Minimize

Thus the primal in the standard form is: subject to Minimize

Hence the dual is: Maximize subject to

Example3 Write the dual of the LPP subject to Maximize

subject to Thus the primal in the standard form is:

Minimize subject to Hence the dual is:

Theorem: The dual of the dual is the primal (original problem). Proof. Consider the primal problem (in standard form) Maximize subject to

Sensitivity Analysis Sensitivity analysis is the study of how changes –in the coefficients of a linear program affect the optimal solution or –in the value of right hand sides of the problem affect the optimal solution

Sensitivity Analysis cont’d Using sensitivity analysis we can answer questions such as: 1.How will a change in a coefficient of the objective function affect the optimal solution? –We can define a range of optimality for each objective function coefficient by changing the objective function coefficients, one at a time

Sensitivity Analysis cont’d 2.How will a change in the right-hand-side value for a constraint affect the optimal solution? The feasible region may change when RHSs (one at a time) are changed and perhaps cause a change in the optimal solution to the problem

Sensitivity Analysis cont’d 3.How much value is added/reduced to the objective function if I have a larger/smaller quantity of a scarce resource? Sensitivity analysis is important to the manager who must operate in a dynamic environment with imprecise estimates of the coefficients

Example 1 LP Formulation Max 5x 1 + 7x 2 s.t. x 1 < 6 2x 1 + 3x 2 < 19 x 1 + x 2 < 8 x 1, x 2 > 0

Example 1 Graphical Solution x x 2 < 19 x 2 x 2 x1x1x1x1 x 1 + x 2 < 8 Max 5 x 1 + 7x 2 x 1 < 6 Optimal: x 1 = 5, x 2 = 3, z = 46

Objective Function Coefficients Let us consider how changes in the objective function coefficients might affect the optimal solution. The range of optimality for each coefficient provides the range of values over which the current solution will remain optimal. Managers should focus on those objective coefficients that have a narrow range of optimality and coefficients near the endpoints of the range.

Example 1 Changing Slope of Objective Function x1x1x1x1 FeasibleRegion x 2 x 2

Range of Optimality Graphically, the limits of a range of optimality are found by changing the slope of the objective function line within the limits of the slopes of the binding constraint lines. The slope of an objective function line, Max c 1 x 1 + c 2 x 2, is -c 1 /c 2, and the slope of a constraint, a 1 x 1 + a 2 x 2 = b, is -a 1 /a 2.

Right-Hand Sides Let us consider how a change in the right-hand side for a constraint might affect the feasible region and perhaps cause a change in the optimal solution. The improvement in the value of the optimal solution per unit increase in the right-hand side is called the dual price. The range of feasibility is the range over which the dual price is applicable. As the RHS increases, other constraints will become binding and limit the change in the value of the objective function.

Example 2 Consider the following linear program: Min 6x 1 + 9x 2 ($ cost) s.t. x 1 + 2x 2 < 8 10x x 2 > 30 x 2 > 2 x 1, x 2 > 0

Example 2 Optimal Solution According to the output: x 1 = 1.5 x 2 = 2.0 Objective function value = 27.00

Example 2 Range of Optimality Question Suppose the unit cost of x 1 is decreased to $4. Is the current solution still optimal? What is the value of the objective function when this unit cost is decreased to $4?

Example 2 Range of Optimality Answer The output states that the solution remains optimal as long as the objective function coefficient of x 1 is between 0 and 12. Because 4 is within this range, the optimal solution will not change. However, the optimal total cost will be affected: 6x 1 + 9x 2 = 4(1.5) + 9(2.0) = $24.00.

Example 2 Range of Optimality Question How much can the unit cost of x 2 be decreased without concern for the optimal solution changing?

Example 2 Range of Optimality Answer The output states that the solution remains optimal as long as the objective function coefficient of x 2 does not fall below 4.5.

Example 2 Range of Feasibility Answer A dual price represents the improvement in the objective function value per unit increase in the right- hand side. A negative dual price indicates a negative improvement in the objective, which in this problem means an increase in total cost because we're minimizing. Since the right-hand side remains within the range of feasibility, there is no change in the optimal solution. However, the objective function value increases by $4.50.