The use of shadow price is an example of sensitivity analysis. Duality theory can be applied to do other kind of sensitivity analysis:
|
|
- Betty Merritt
- 6 years ago
- Views:
Transcription
1 Sensitivity analysis The use of shadow price is an example of sensitivity analysis. Duality theory can be applied to do other kind of sensitivity analysis: Changing the coefficient of a nonbasic variable In the original optimal solution (the current solution), all the nonbasic variables equal to 0. So if the coefficients of a nonbasic variable was changed, all the constraints will still be satisfied. The problem is whether the current solution is still optimal. This is the same question as whether the dual solution is still feasible. Since only one constraint has been changed, this should be easy to check. Example 1 Consider this situation: A furniture company makes desks, tables and chairs. Resource Desk Table Chair Available Lumber (ft 2 ) Finishing hours Carpentry hours Profit Let x 1 be the number of desks produced; x 2 be the number of tables produced; x 3 be the number of chairs produced: Max z =60x 1 +30x 2 +20x 3 x 1 +6x 2 + x 3 4 4x 1 +2x x x x x 3 Last table The dual problem is x x x Min y 0 =4y 1 +20y 2 +y 3 (1) y 1 +4y 2 +2y y 1 +2y y 3 30 y y y 3 20 y 1,y 2,y 3 0 1
2 Suppose that the profit of a new type of table is $40 and it takes 2.5 hours to finish. x 2, the number of table to be produced is a nonbasic variable in our optimal solution. The new problem is Max z =60x x 2 +20x 3 The dual problem changes to x 1 +6x 2 + x 3 4 4x x x x x x 3 Min y 0 =4x 1 +20x 2 +x 3 y 1 +4y 2 +2y y y y 3 40 (2) y y y 3 20 The complementary solution of the current optimal solution is (0, 10, 10, 0, 5, 0). Sowehave y 1 =0,y 2 =10,y 3 =10. They still satisfy (2). The current solution is still optimal by the strong duality theorem. Suppose that the new table makes $45 profit. Then (2) becomes 6y y y 3 45 which is not satisfied by y 1 =0,y 2 =10,y 3 =10. The current solution is not optimal any more. To find the new optimal solution, we apply reoptimization techniques. In this case, we can apply the simplex method starting with the current basic feasible solution. Adding a new variable If a new variable is introduced, i.e. a new activity is being considered, we can first treat it as a nonbasic variable and see if that still gives us the optimal solution by using duality. In the previous example (the original version), suppose that they are also considering making computer tables. The resource requirements and profit are The new primal problem: Resource Computer Table Lumber (ft 2 ) 4 Finishing hours 2 Carpentry hours 1 Profit 25 Max z =60x 1 +30x 2 +20x 3 +25x 4 2
3 The new dual problem: x 1 +6x 2 + x 3 +4x 4 4 4x 1 +2x x 3 +2x x x x 3 + x 4 Min y 0 =4y 1 +20y 2 +y 3 (3) y 1 +4y 2 +2y y 1 +2y y 3 30 y y y y 1 +2y 2 + y 3 25 (4) Compare (3) with (1) we see that the only difference is the new constraint (4). y 1 =0,y 2 = 10,y 3 =10satisfy (4). The current solution is still optimal. If the profit of a computer table is $35, then the current solution is not optimal any more. We can apply the simplex method starting with the current basic feasible solution. The general situation Max z = cx changes to Ax b x 0 Max z = cx Ax b x 0 Our strategy is: Start with those new parameters and do exactly the operations as before that lead to the final table. Then we check the result. We do not have to actually do the operations. All those operations are recorded in the final table: t T = y A c y y b S A S S b If we have started with A, b and c, the last table would be y A c y y b S A S S b Depending on the changes, this table may or may not be in the proper form. 3
4 b b Example 2 Consider our old example: Max z =60x 1 +30x 2 +20x 3 x 1 +6x 2 + x 3 4 4x 1 +2x x x x x 3 We have The final table is We have A = , b = c = x x x y = S = Suppose that b = ThenewlasttablewouldbethesameexcepttheRHSis y b = = 300 S b = =
5 The current solution is still optimal. If b = 4 30 y b = S b = = = The current solution is not optimal (for it is not feasible). We can find the allowable range to stay feasible for each component of b. Suppose that b 1 b = 20 S b = b 1 20 = b For that solution to be feasible (then optimal) we need b If we allow only the second component of b to change, b = 4 b b 2 S b = b 2 = 2b b We need b 2, b 2 16, b 2 24 The range is 16 b Similarly we can calculate the allowable range to stay feasible for the third component of b is 22 b Wecanusethesametechniquetofind the allowable range to stay optimal for c. So c 1 60, c 2 35, c c = c 1 c 2 c 3 y A c = c 1 c 2 c = 60 c 1 35 c 2 20 c 3 5
6 Every thing is changed: The table now becomes S A = A = , b = c = = y A c = = 0 2 S b = y b = = = x x x This table is not proper. First we make the column of x 1 right: x x x There is still a negative number on the RHS. Multiply the last row with 1: x x x
7 x 1 can not be a basic variable. The only one can be a basic variable in this row is x 5. Let x 1 be the leaving basic variable and x 5 the entering basic variable. We change the table into the proper form: That gives us the new optimal solution The dual simplex method x x x x 1 =0,x 2 =0,x 3 =16. Initially, the row 0 are all positive but the right hand side maybe negative. (If both of them are positive, the solution is optimal.) 1. Pick one row that has a negative right hand side (say row i). The corresponding basic variable is the leaving basic variable. 2. To find the entering basic variable: for every negative numbers a ij in row i, calculate the ratio The number in the top row and jth column. a ij The one with the smallest ratio (but not 0) is the entering basic variable. 3. The entry at the row of the leaving basic variable and the column of the entering basic variable is the pivot. Multiply the pivot row by 1. Use the pivot to change the table into the proper form. 4. If there is no more negative number at the right hand side, the current solution is optimal; otherwise continue. In step 2, if there is no negative number, then the row represents the equation a i1 x 1 + a i2 x a in x n = b i Here all a ij 0 but b i < 0. Sincex j 0 for all j, there is no solution. There is no feasible solution for this LP model. In each step, the dual solution is always feasible but suboptimal (while the solution to the primal problem is not feasible but superoptimal). We make the value of the objective function better for the dual problem after every iteration. This is doing the simplex method on the dual problem but using the table of the primal problem. Example 3 (page 265) Consider this problem: Max z = 4y 1 12y 2 1y 3 7
8 y 1 +3y 3 3 2y 2 +2y 3 5 y 1,y 2,y 3 0 Change the functional constraint to the form: The simplex table is y 1 3y 3 3 2y 2 y 3 5 z y 1 y 2 y 3 y 4 y y y z y 1 y 2 y 3 y 4 y y y z y 1 y 2 y 3 y 4 y y y That gives us the optimal solution.
END3033 Operations Research I Sensitivity Analysis & Duality. to accompany Operations Research: Applications and Algorithms Fatih Cavdur
END3033 Operations Research I Sensitivity Analysis & Duality to accompany Operations Research: Applications and Algorithms Fatih Cavdur Introduction Consider the following problem where x 1 and x 2 corresponds
More informationSlack Variable. Max Z= 3x 1 + 4x 2 + 5X 3. Subject to: X 1 + X 2 + X x 1 + 4x 2 + X X 1 + X 2 + 4X 3 10 X 1 0, X 2 0, X 3 0
Simplex Method Slack Variable Max Z= 3x 1 + 4x 2 + 5X 3 Subject to: X 1 + X 2 + X 3 20 3x 1 + 4x 2 + X 3 15 2X 1 + X 2 + 4X 3 10 X 1 0, X 2 0, X 3 0 Standard Form Max Z= 3x 1 +4x 2 +5X 3 + 0S 1 + 0S 2
More informationLecture 11: Post-Optimal Analysis. September 23, 2009
Lecture : Post-Optimal Analysis September 23, 2009 Today Lecture Dual-Simplex Algorithm Post-Optimal Analysis Chapters 4.4 and 4.5. IE 30/GE 330 Lecture Dual Simplex Method The dual simplex method will
More informationc) Place the Coefficients from all Equations into a Simplex Tableau, labeled above with variables indicating their respective columns
BUILDING A SIMPLEX TABLEAU AND PROPER PIVOT SELECTION Maximize : 15x + 25y + 18 z s. t. 2x+ 3y+ 4z 60 4x+ 4y+ 2z 100 8x+ 5y 80 x 0, y 0, z 0 a) Build Equations out of each of the constraints above by introducing
More informationReview Solutions, Exam 2, Operations Research
Review Solutions, Exam 2, Operations Research 1. Prove the weak duality theorem: For any x feasible for the primal and y feasible for the dual, then... HINT: Consider the quantity y T Ax. SOLUTION: To
More informationChap6 Duality Theory and Sensitivity Analysis
Chap6 Duality Theory and Sensitivity Analysis The rationale of duality theory Max 4x 1 + x 2 + 5x 3 + 3x 4 S.T. x 1 x 2 x 3 + 3x 4 1 5x 1 + x 2 + 3x 3 + 8x 4 55 x 1 + 2x 2 + 3x 3 5x 4 3 x 1 ~x 4 0 If we
More informationSensitivity Analysis and Duality
Sensitivity Analysis and Duality Part II Duality Based on Chapter 6 Introduction to Mathematical Programming: Operations Research, Volume 1 4th edition, by Wayne L. Winston and Munirpallam Venkataramanan
More information"SYMMETRIC" PRIMAL-DUAL PAIR
"SYMMETRIC" PRIMAL-DUAL PAIR PRIMAL Minimize cx DUAL Maximize y T b st Ax b st A T y c T x y Here c 1 n, x n 1, b m 1, A m n, y m 1, WITH THE PRIMAL IN STANDARD FORM... Minimize cx Maximize y T b st Ax
More informationPart 1. The Review of Linear Programming
In the name of God Part 1. The Review of Linear Programming 1.5. Spring 2010 Instructor: Dr. Masoud Yaghini Outline Introduction Formulation of the Dual Problem Primal-Dual Relationship Economic Interpretation
More informationWorked Examples for Chapter 5
Worked Examples for Chapter 5 Example for Section 5.2 Construct the primal-dual table and the dual problem for the following linear programming model fitting our standard form. Maximize Z = 5 x 1 + 4 x
More informationChapter 4 The Simplex Algorithm Part I
Chapter 4 The Simplex Algorithm Part I Based on Introduction to Mathematical Programming: Operations Research, Volume 1 4th edition, by Wayne L. Winston and Munirpallam Venkataramanan Lewis Ntaimo 1 Modeling
More information4. Duality and Sensitivity
4. Duality and Sensitivity For every instance of an LP, there is an associated LP known as the dual problem. The original problem is known as the primal problem. There are two de nitions of the dual pair
More informationSpecial cases of linear programming
Special cases of linear programming Infeasible solution Multiple solution (infinitely many solution) Unbounded solution Degenerated solution Notes on the Simplex tableau 1. The intersection of any basic
More informationMVE165/MMG631 Linear and integer optimization with applications Lecture 5 Linear programming duality and sensitivity analysis
MVE165/MMG631 Linear and integer optimization with applications Lecture 5 Linear programming duality and sensitivity analysis Ann-Brith Strömberg 2017 03 29 Lecture 4 Linear and integer optimization with
More informationMAT016: Optimization
MAT016: Optimization M.El Ghami e-mail: melghami@ii.uib.no URL: http://www.ii.uib.no/ melghami/ March 29, 2011 Outline for today The Simplex method in matrix notation Managing a production facility The
More informationIntroduction to Mathematical Programming IE406. Lecture 13. Dr. Ted Ralphs
Introduction to Mathematical Programming IE406 Lecture 13 Dr. Ted Ralphs IE406 Lecture 13 1 Reading for This Lecture Bertsimas Chapter 5 IE406 Lecture 13 2 Sensitivity Analysis In many real-world problems,
More informationMath Models of OR: Sensitivity Analysis
Math Models of OR: Sensitivity Analysis John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 8 USA October 8 Mitchell Sensitivity Analysis / 9 Optimal tableau and pivot matrix Outline Optimal
More information(includes both Phases I & II)
Minimize z=3x 5x 4x 7x 5x 4x subject to 2x x2 x4 3x6 0 x 3x3 x4 3x5 2x6 2 4x2 2x3 3x4 x5 5 and x 0 j, 6 2 3 4 5 6 j ecause of the lack of a slack variable in each constraint, we must use Phase I to find
More informationSensitivity Analysis
Dr. Maddah ENMG 500 /9/07 Sensitivity Analysis Changes in the RHS (b) Consider an optimal LP solution. Suppose that the original RHS (b) is changed from b 0 to b new. In the following, we study the affect
More informationIntroduction. Very efficient solution procedure: simplex method.
LINEAR PROGRAMMING Introduction Development of linear programming was among the most important scientific advances of mid 20th cent. Most common type of applications: allocate limited resources to competing
More informationLinear Programming: Chapter 5 Duality
Linear Programming: Chapter 5 Duality Robert J. Vanderbei September 30, 2010 Slides last edited on October 5, 2010 Operations Research and Financial Engineering Princeton University Princeton, NJ 08544
More information(includes both Phases I & II)
(includes both Phases I & II) Dennis ricker Dept of Mechanical & Industrial Engineering The University of Iowa Revised Simplex Method 09/23/04 page 1 of 22 Minimize z=3x + 5x + 4x + 7x + 5x + 4x subject
More informationSEN301 OPERATIONS RESEARCH I LECTURE NOTES
SEN30 OPERATIONS RESEARCH I LECTURE NOTES SECTION II (208-209) Y. İlker Topcu, Ph.D. & Özgür Kabak, Ph.D. Acknowledgements: We would like to acknowledge Prof. W.L. Winston's "Operations Research: Applications
More information1 Review Session. 1.1 Lecture 2
1 Review Session Note: The following lists give an overview of the material that was covered in the lectures and sections. Your TF will go through these lists. If anything is unclear or you have questions
More informationDual Basic Solutions. Observation 5.7. Consider LP in standard form with A 2 R m n,rank(a) =m, and dual LP:
Dual Basic Solutions Consider LP in standard form with A 2 R m n,rank(a) =m, and dual LP: Observation 5.7. AbasisB yields min c T x max p T b s.t. A x = b s.t. p T A apple c T x 0 aprimalbasicsolutiongivenbyx
More information4.7 Sensitivity analysis in Linear Programming
4.7 Sensitivity analysis in Linear Programming Evaluate the sensitivity of an optimal solution with respect to variations in the data (model parameters). Example: Production planning max n j n a j p j
More informationF 1 F 2 Daily Requirement Cost N N N
Chapter 5 DUALITY 5. The Dual Problems Every linear programming problem has associated with it another linear programming problem and that the two problems have such a close relationship that whenever
More information21. Solve the LP given in Exercise 19 using the big-m method discussed in Exercise 20.
Extra Problems for Chapter 3. Linear Programming Methods 20. (Big-M Method) An alternative to the two-phase method of finding an initial basic feasible solution by minimizing the sum of the artificial
More informationSAMPLE QUESTIONS. b = (30, 20, 40, 10, 50) T, c = (650, 1000, 1350, 1600, 1900) T.
SAMPLE QUESTIONS. (a) We first set up some constant vectors for our constraints. Let b = (30, 0, 40, 0, 0) T, c = (60, 000, 30, 600, 900) T. Then we set up variables x ij, where i, j and i + j 6. By using
More informationSummary of the simplex method
MVE165/MMG630, The simplex method; degeneracy; unbounded solutions; infeasibility; starting solutions; duality; interpretation Ann-Brith Strömberg 2012 03 16 Summary of the simplex method Optimality condition:
More informationOPERATIONS RESEARCH. Michał Kulej. Business Information Systems
OPERATIONS RESEARCH Michał Kulej Business Information Systems The development of the potential and academic programmes of Wrocław University of Technology Project co-financed by European Union within European
More informationUnderstanding the Simplex algorithm. Standard Optimization Problems.
Understanding the Simplex algorithm. Ma 162 Spring 2011 Ma 162 Spring 2011 February 28, 2011 Standard Optimization Problems. A standard maximization problem can be conveniently described in matrix form
More informationSimplex method(s) for solving LPs in standard form
Simplex method: outline I The Simplex Method is a family of algorithms for solving LPs in standard form (and their duals) I Goal: identify an optimal basis, as in Definition 3.3 I Versions we will consider:
More informationMath Models of OR: Some Definitions
Math Models of OR: Some Definitions John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA September 2018 Mitchell Some Definitions 1 / 20 Active constraints Outline 1 Active constraints
More informationSensitivity Analysis and Duality in LP
Sensitivity Analysis and Duality in LP Xiaoxi Li EMS & IAS, Wuhan University Oct. 13th, 2016 (week vi) Operations Research (Li, X.) Sensitivity Analysis and Duality in LP Oct. 13th, 2016 (week vi) 1 /
More informationFarkas Lemma, Dual Simplex and Sensitivity Analysis
Summer 2011 Optimization I Lecture 10 Farkas Lemma, Dual Simplex and Sensitivity Analysis 1 Farkas Lemma Theorem 1. Let A R m n, b R m. Then exactly one of the following two alternatives is true: (i) x
More informationCO350 Linear Programming Chapter 8: Degeneracy and Finite Termination
CO350 Linear Programming Chapter 8: Degeneracy and Finite Termination 27th June 2005 Chapter 8: Finite Termination 1 The perturbation method Recap max c T x (P ) s.t. Ax = b x 0 Assumption: B is a feasible
More information1. Algebraic and geometric treatments Consider an LP problem in the standard form. x 0. Solutions to the system of linear equations
The Simplex Method Most textbooks in mathematical optimization, especially linear programming, deal with the simplex method. In this note we study the simplex method. It requires basically elementary linear
More informationCO350 Linear Programming Chapter 6: The Simplex Method
CO350 Linear Programming Chapter 6: The Simplex Method 8th June 2005 Chapter 6: The Simplex Method 1 Minimization Problem ( 6.5) We can solve minimization problems by transforming it into a maximization
More informationStandard Form An LP is in standard form when: All variables are non-negativenegative All constraints are equalities Putting an LP formulation into sta
Chapter 4 Linear Programming: The Simplex Method An Overview of the Simplex Method Standard Form Tableau Form Setting Up the Initial Simplex Tableau Improving the Solution Calculating the Next Tableau
More informationLecture 10: Linear programming duality and sensitivity 0-0
Lecture 10: Linear programming duality and sensitivity 0-0 The canonical primal dual pair 1 A R m n, b R m, and c R n maximize z = c T x (1) subject to Ax b, x 0 n and minimize w = b T y (2) subject to
More informationThe Strong Duality Theorem 1
1/39 The Strong Duality Theorem 1 Adrian Vetta 1 This presentation is based upon the book Linear Programming by Vasek Chvatal 2/39 Part I Weak Duality 3/39 Primal and Dual Recall we have a primal linear
More informationMarch 13, Duality 3
15.53 March 13, 27 Duality 3 There are concepts much more difficult to grasp than duality in linear programming. -- Jim Orlin The concept [of nonduality], often described in English as "nondualism," is
More informationThe dual simplex method with bounds
The dual simplex method with bounds Linear programming basis. Let a linear programming problem be given by min s.t. c T x Ax = b x R n, (P) where we assume A R m n to be full row rank (we will see in the
More informationNote 3: LP Duality. If the primal problem (P) in the canonical form is min Z = n (1) then the dual problem (D) in the canonical form is max W = m (2)
Note 3: LP Duality If the primal problem (P) in the canonical form is min Z = n j=1 c j x j s.t. nj=1 a ij x j b i i = 1, 2,..., m (1) x j 0 j = 1, 2,..., n, then the dual problem (D) in the canonical
More informationLinear Programming Duality
Summer 2011 Optimization I Lecture 8 1 Duality recap Linear Programming Duality We motivated the dual of a linear program by thinking about the best possible lower bound on the optimal value we can achieve
More information4.6 Linear Programming duality
4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP Different spaces and objective functions but in general same optimal
More informationFoundations of Operations Research
Solved exercises for the course of Foundations of Operations Research Roberto Cordone The dual simplex method Given the following LP problem: maxz = 5x 1 +8x 2 x 1 +x 2 6 5x 1 +9x 2 45 x 1,x 2 0 1. solve
More informationBrief summary of linear programming and duality: Consider the linear program in standard form. (P ) min z = cx. x 0. (D) max yb. z = c B x B + c N x N
Brief summary of linear programming and duality: Consider the linear program in standard form (P ) min z = cx s.t. Ax = b x 0 where A R m n, c R 1 n, x R n 1, b R m 1,and its dual (D) max yb s.t. ya c.
More informationmin 4x 1 5x 2 + 3x 3 s.t. x 1 + 2x 2 + x 3 = 10 x 1 x 2 6 x 1 + 3x 2 + x 3 14
The exam is three hours long and consists of 4 exercises. The exam is graded on a scale 0-25 points, and the points assigned to each question are indicated in parenthesis within the text. If necessary,
More informationThe Simplex Algorithm and Goal Programming
The Simplex Algorithm and Goal Programming In Chapter 3, we saw how to solve two-variable linear programming problems graphically. Unfortunately, most real-life LPs have many variables, so a method is
More informationChapter 1 Linear Programming. Paragraph 5 Duality
Chapter 1 Linear Programming Paragraph 5 Duality What we did so far We developed the 2-Phase Simplex Algorithm: Hop (reasonably) from basic solution (bs) to bs until you find a basic feasible solution
More informationThe Graphical Method & Algebraic Technique for Solving LP s. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 1
The Graphical Method & Algebraic Technique for Solving LP s Métodos Cuantitativos M. En C. Eduardo Bustos Farías The Graphical Method for Solving LP s If LP models have only two variables, they can be
More informationReview Questions, Final Exam
Review Questions, Final Exam A few general questions 1. What does the Representation Theorem say (in linear programming)? 2. What is the Fundamental Theorem of Linear Programming? 3. What is the main idea
More informationLecture 10: Linear programming. duality. and. The dual of the LP in standard form. maximize w = b T y (D) subject to A T y c, minimize z = c T x (P)
Lecture 10: Linear programming duality Michael Patriksson 19 February 2004 0-0 The dual of the LP in standard form minimize z = c T x (P) subject to Ax = b, x 0 n, and maximize w = b T y (D) subject to
More informationLinear programs Optimization Geoff Gordon Ryan Tibshirani
Linear programs 10-725 Optimization Geoff Gordon Ryan Tibshirani Review: LPs LPs: m constraints, n vars A: R m n b: R m c: R n x: R n ineq form [min or max] c T x s.t. Ax b m n std form [min or max] c
More informationSimplex tableau CE 377K. April 2, 2015
CE 377K April 2, 2015 Review Reduced costs Basic and nonbasic variables OUTLINE Review by example: simplex method demonstration Outline Example You own a small firm producing construction materials for
More informationRelation of Pure Minimum Cost Flow Model to Linear Programming
Appendix A Page 1 Relation of Pure Minimum Cost Flow Model to Linear Programming The Network Model The network pure minimum cost flow model has m nodes. The external flows given by the vector b with m
More informationYinyu Ye, MS&E, Stanford MS&E310 Lecture Note #06. The Simplex Method
The Simplex Method Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye (LY, Chapters 2.3-2.5, 3.1-3.4) 1 Geometry of Linear
More informationLecture 5. x 1,x 2,x 3 0 (1)
Computational Intractability Revised 2011/6/6 Lecture 5 Professor: David Avis Scribe:Ma Jiangbo, Atsuki Nagao 1 Duality The purpose of this lecture is to introduce duality, which is an important concept
More information(P ) Minimize 4x 1 + 6x 2 + 5x 3 s.t. 2x 1 3x 3 3 3x 2 2x 3 6
The exam is three hours long and consists of 4 exercises. The exam is graded on a scale 0-25 points, and the points assigned to each question are indicated in parenthesis within the text. Problem 1 Consider
More informationmin3x 1 + 4x 2 + 5x 3 2x 1 + 2x 2 + x 3 6 x 1 + 2x 2 + 3x 3 5 x 1, x 2, x 3 0.
ex-.-. Foundations of Operations Research Prof. E. Amaldi. Dual simplex algorithm Given the linear program minx + x + x x + x + x 6 x + x + x x, x, x. solve it via the dual simplex algorithm. Describe
More informationIntroduction to Mathematical Programming IE406. Lecture 10. Dr. Ted Ralphs
Introduction to Mathematical Programming IE406 Lecture 10 Dr. Ted Ralphs IE406 Lecture 10 1 Reading for This Lecture Bertsimas 4.1-4.3 IE406 Lecture 10 2 Duality Theory: Motivation Consider the following
More informationIE 5531: Engineering Optimization I
IE 5531: Engineering Optimization I Lecture 7: Duality and applications Prof. John Gunnar Carlsson September 29, 2010 Prof. John Gunnar Carlsson IE 5531: Engineering Optimization I September 29, 2010 1
More informationLINEAR PROGRAMMING 2. In many business and policy making situations the following type of problem is encountered:
LINEAR PROGRAMMING 2 In many business and policy making situations the following type of problem is encountered: Maximise an objective subject to (in)equality constraints. Mathematical programming provides
More information3. Duality: What is duality? Why does it matter? Sensitivity through duality.
1 Overview of lecture (10/5/10) 1. Review Simplex Method 2. Sensitivity Analysis: How does solution change as parameters change? How much is the optimal solution effected by changing A, b, or c? How much
More informationThe augmented form of this LP is the following linear system of equations:
1 Consider the following LP given in standard form: max z = 5 x_1 + 2 x_2 Subject to 3 x_1 + 2 x_2 2400 x_2 800 2 x_1 1200 x_1, x_2 >= 0 The augmented form of this LP is the following linear system of
More information(b) For the change in c 1, use the row corresponding to x 1. The new Row 0 is therefore: 5 + 6
Chapter Review Solutions. Write the LP in normal form, and the optimal tableau is given in the text (to the right): x x x rhs y y 8 y 5 x x x s s s rhs / 5/ 7/ 9 / / 5/ / / / (a) For the dual, just go
More informationA Review of Linear Programming
A Review of Linear Programming Instructor: Farid Alizadeh IEOR 4600y Spring 2001 February 14, 2001 1 Overview In this note we review the basic properties of linear programming including the primal simplex
More informationDuality Theory, Optimality Conditions
5.1 Duality Theory, Optimality Conditions Katta G. Murty, IOE 510, LP, U. Of Michigan, Ann Arbor We only consider single objective LPs here. Concept of duality not defined for multiobjective LPs. Every
More informationMATH 445/545 Test 1 Spring 2016
MATH 445/545 Test Spring 06 Note the problems are separated into two sections a set for all students and an additional set for those taking the course at the 545 level. Please read and follow all of these
More informationLecture 4: Algebra, Geometry, and Complexity of the Simplex Method. Reading: Sections 2.6.4, 3.5,
Lecture 4: Algebra, Geometry, and Complexity of the Simplex Method Reading: Sections 2.6.4, 3.5, 10.2 10.5 1 Summary of the Phase I/Phase II Simplex Method We write a typical simplex tableau as z x 1 x
More informationSimplex Algorithm Using Canonical Tableaus
41 Simplex Algorithm Using Canonical Tableaus Consider LP in standard form: Min z = cx + α subject to Ax = b where A m n has rank m and α is a constant In tableau form we record it as below Original Tableau
More informationThe Primal-Dual Algorithm P&S Chapter 5 Last Revised October 30, 2006
The Primal-Dual Algorithm P&S Chapter 5 Last Revised October 30, 2006 1 Simplex solves LP by starting at a Basic Feasible Solution (BFS) and moving from BFS to BFS, always improving the objective function,
More informationDuality in LPP Every LPP called the primal is associated with another LPP called dual. Either of the problems is primal with the other one as dual. The optimal solution of either problem reveals the information
More informationThe Dual Simplex Algorithm
p. 1 The Dual Simplex Algorithm Primal optimal (dual feasible) and primal feasible (dual optimal) bases The dual simplex tableau, dual optimality and the dual pivot rules Classical applications of linear
More informationLINEAR PROGRAMMING. Introduction
LINEAR PROGRAMMING Introduction Development of linear programming was among the most important scientific advances of mid-20th cent. Most common type of applications: allocate limited resources to competing
More informationLinear Programming Duality P&S Chapter 3 Last Revised Nov 1, 2004
Linear Programming Duality P&S Chapter 3 Last Revised Nov 1, 2004 1 In this section we lean about duality, which is another way to approach linear programming. In particular, we will see: How to define
More informationLecture 5 Simplex Method. September 2, 2009
Simplex Method September 2, 2009 Outline: Lecture 5 Re-cap blind search Simplex method in steps Simplex tableau Operations Research Methods 1 Determining an optimal solution by exhaustive search Lecture
More informationMATH 445/545 Homework 2: Due March 3rd, 2016
MATH 445/545 Homework 2: Due March 3rd, 216 Answer the following questions. Please include the question with the solution (write or type them out doing this will help you digest the problem). I do not
More informationLinear and Combinatorial Optimization
Linear and Combinatorial Optimization The dual of an LP-problem. Connections between primal and dual. Duality theorems and complementary slack. Philipp Birken (Ctr. for the Math. Sc.) Lecture 3: Duality
More informationΩ R n is called the constraint set or feasible set. x 1
1 Chapter 5 Linear Programming (LP) General constrained optimization problem: minimize subject to f(x) x Ω Ω R n is called the constraint set or feasible set. any point x Ω is called a feasible point We
More informationChapter 1: Linear Programming
Chapter 1: Linear Programming Math 368 c Copyright 2013 R Clark Robinson May 22, 2013 Chapter 1: Linear Programming 1 Max and Min For f : D R n R, f (D) = {f (x) : x D } is set of attainable values of
More information56:171 Operations Research Fall 1998
56:171 Operations Research Fall 1998 Quiz Solutions D.L.Bricker Dept of Mechanical & Industrial Engineering University of Iowa 56:171 Operations Research Quiz
More informationThe Simplex Method. Formulate Constrained Maximization or Minimization Problem. Convert to Standard Form. Convert to Canonical Form
The Simplex Method 1 The Simplex Method Formulate Constrained Maximization or Minimization Problem Convert to Standard Form Convert to Canonical Form Set Up the Tableau and the Initial Basic Feasible Solution
More informationLinear Programming Redux
Linear Programming Redux Jim Bremer May 12, 2008 The purpose of these notes is to review the basics of linear programming and the simplex method in a clear, concise, and comprehensive way. The book contains
More informationSimplex Method in different guises
Simplex Method in different guises The Furniture problem Max 0x + 0x 2 + 20x, subject to x 0, 8x + x 2 + 2x 48, 4x + 2x 2 +.x 20, 2x +.x 2 +.x 8. Canonical form: slack variables s = (s, s 2, s ) 0. Constraints
More informationThe Simplex Algorithm
The Simplex Algorithm How to Convert an LP to Standard Form Before the simplex algorithm can be used to solve an LP, the LP must be converted into a problem where all the constraints are equations and
More informationLesson 27 Linear Programming; The Simplex Method
Lesson Linear Programming; The Simplex Method Math 0 April 9, 006 Setup A standard linear programming problem is to maximize the quantity c x + c x +... c n x n = c T x subject to constraints a x + a x
More informationM340(921) Solutions Problem Set 6 (c) 2013, Philip D Loewen. g = 35y y y 3.
M340(92) Solutions Problem Set 6 (c) 203, Philip D Loewen. (a) If each pig is fed y kilograms of corn, y 2 kilos of tankage, and y 3 kilos of alfalfa, the cost per pig is g = 35y + 30y 2 + 25y 3. The nutritional
More information1. Introduce slack variables for each inequaility to make them equations and rewrite the objective function in the form ax by cz... + P = 0.
3.4 Simplex Method If a linear programming problem has more than 2 variables, solving graphically is not the way to go. Instead, we ll use a more methodical, numeric process called the Simplex Method.
More informationECE 307- Techniques for Engineering Decisions
ECE 307- Techniques for Engineering Decisions Lecture 4. Dualit Concepts in Linear Programming George Gross Department of Electrical and Computer Engineering Universit of Illinois at Urbana-Champaign DUALITY
More informationAM 121: Intro to Optimization
AM 121: Intro to Optimization Models and Methods Lecture 6: Phase I, degeneracy, smallest subscript rule. Yiling Chen SEAS Lesson Plan Phase 1 (initialization) Degeneracy and cycling Smallest subscript
More informationLecture Notes 3: Duality
Algorithmic Methods 1/11/21 Professor: Yossi Azar Lecture Notes 3: Duality Scribe:Moran Bar-Gat 1 Introduction In this lecture we will present the dual concept, Farkas s Lema and their relation to the
More informationOPTIMISATION 3: NOTES ON THE SIMPLEX ALGORITHM
OPTIMISATION 3: NOTES ON THE SIMPLEX ALGORITHM Abstract These notes give a summary of the essential ideas and results It is not a complete account; see Winston Chapters 4, 5 and 6 The conventions and notation
More informationChapter 5 Linear Programming (LP)
Chapter 5 Linear Programming (LP) General constrained optimization problem: minimize f(x) subject to x R n is called the constraint set or feasible set. any point x is called a feasible point We consider
More informationExample. 1 Rows 1,..., m of the simplex tableau remain lexicographically positive
3.4 Anticycling Lexicographic order In this section we discuss two pivoting rules that are guaranteed to avoid cycling. These are the lexicographic rule and Bland s rule. Definition A vector u R n is lexicographically
More informationThe Simplex Method. Lecture 5 Standard and Canonical Forms and Setting up the Tableau. Lecture 5 Slide 1. FOMGT 353 Introduction to Management Science
The Simplex Method Lecture 5 Standard and Canonical Forms and Setting up the Tableau Lecture 5 Slide 1 The Simplex Method Formulate Constrained Maximization or Minimization Problem Convert to Standard
More informationDEPARTMENT OF STATISTICS AND OPERATIONS RESEARCH OPERATIONS RESEARCH DETERMINISTIC QUALIFYING EXAMINATION. Part I: Short Questions
DEPARTMENT OF STATISTICS AND OPERATIONS RESEARCH OPERATIONS RESEARCH DETERMINISTIC QUALIFYING EXAMINATION Part I: Short Questions August 12, 2008 9:00 am - 12 pm General Instructions This examination is
More informationmaxz = 3x 1 +4x 2 2x 1 +x 2 6 2x 1 +3x 2 9 x 1,x 2
ex-5.-5. Foundations of Operations Research Prof. E. Amaldi 5. Branch-and-Bound Given the integer linear program maxz = x +x x +x 6 x +x 9 x,x integer solve it via the Branch-and-Bound method (solving
More information