MAT016: Optimization
|
|
- Georgia Hall
- 5 years ago
- Views:
Transcription
1 MAT016: Optimization M.El Ghami URL: melghami/ March 29, 2011
2 Outline for today The Simplex method in matrix notation Managing a production facility The linear programming problem (LP) or (LO) Examples The Simplex method Matrix notation The primal Simplex method Examples The dual simplex method 1
3 Managing a production facility A company produces three products. The per-unit profit, labor usage, and pollution produced per unit are given in the following table Profit Labor Usage Pollution Product 1 6 NOK 4 hours lb Product 2 4 NOK 3 hours lb Product 3 3 NOK 2 hours lb At most labor hours can be used to produce the three products, and government regulations require that the company produce at most 2lb of pollution. 2
4 Linear programming formulation Let x i =units produced of product i. maximize z = 6x 1 + 4x 2 + 3x 3 subject to 4x 1 + 3x 2 + 2x x x x 3 2 x 1, x 2, x
5 Application of LO Business Logistics Economics Engineering 4
6 The linear programming problem Decision variables (variables) are written as: x j, j = 1, 2,..., n. Objective function: maximize or minimize linear function of decision variables, ζ = c 1 x 1 + c 2 x c n x n. Equivalent: maximize ζ or minimize ζ. Constraints: equality or inequality associated with some linear combination of the decision variables a 1 x 1 + a 2 x a n x n {, =, } b. 5
7 Convert of the constraints Convert of inequality to an other inequality: a 1 x 1 + a 2 x a n x n b, can be written a 1 x 1 a 2 x 2... a n x n b. Convert of inequality to equality: a 1 x 1 + a 2 x a n x n b, can be written a 1 x 1 + a 2 x a n x n + w = b, where w 0, which we call slack variable. 6
8 Convert of the constraints (count) Convert of equality to inequalities: a 1 x 1 + a 2 x a n x n = b, can be written in form of two inequality constraints a 1 x 1 + a 2 x a n x n b, a 1 x 1 + a 2 x a n x n b. 7
9 Formulation of LP Standard form of LP. maximize c 1 x 1 + c 2 x c n x n subject to a 11 x 1 + a 12 x a 1n x n b 1 a 21 x 1 + a 22 x a 2n x n b 2... a m1 x 1 + a m2 x a mn x n b m x 1, x 2,..., x n 0. n: number of variables. m: number of constraints. (x 1, x 2,..., x n ) is feasible if it satisfies all constraints. It is optimal if it attains the desired maximum. 8
10 Example of infeasible problem maximize 5x 1 + 4x 2 subject to x 1 + x 2 2 2x 1 2x 2 9 x 1, x 2 0. Note that 2x 1 2x 2 9 x 1 + x 2 4.5, which contradict with x 1 + x 2 2. This problem is infeasible. 9
11 Example of unbounded problem maximize x 1 4x 2 subject to 2x 1 + x 2 1 x 1 2x 2 2 x 1, x 2 0. Let x 2 = 0 and x 1 is arbitrarily large. As long as x 1 is greater than 2 the solution will be feasible, and as it gets large the objective function does too. This problem is called unbounded. Definition 1 A problem is unbounded if it has feasible solutions with arbitrarily large objective values. 10
12 Exercises Exercise 1 Jim and Jane have a small furniture workshop in their garage, where they assemble from purchased parts and finish the furniture in preparation for sale. they are presently limiting their production to table and chairs. Each chair requires 4 h to assemble and 2 h to finish, but each table require 2 h to assemble and 4 h to finish. Jim works only on the assembly operation. Jane works only on the finishing operation, and each puts in no more than an 8-h day. It is known that each chair can be sold for 3000 NOK and each table for 2000 NOK. If Jim and Jane wants to maximize the total income. What is a mathematical model for this problem? 11
13 Next How to solve a linear programming? Simplex method. 12
14 Simplex method: Example 1 How the Simplex method works? Example 1: maximize 3x 1 + 2x 2 subject to 2x 1 + 4x 2 8 (I) 4x 1 + 2x 2 8 x 1, x
15 Simplex method: Example (cont) Introduce the slack variables: maximize 3x 1 + 2x 2 = ζ subject to 2x 1 + 4x 2 + w 1 = 8 (II) 4x 1 + 2x 2 + w 2 = 8 x 1, x 2, w 1, w 2, 0. Equivalent to maximize 3x 1 + 2x 2 = ζ subject to 8 2x 1 4x 2 = w 1 (II) 8 4x 1 2x 2 = w 2 x 1, x 2, w 1, w 2, 0. 14
16 Example (cont) Simplex method is an iterative process in which we start with a solution x 1, x 2, w 1, w 2 and then look for new solution x 1, x 2, w 1, w 2 which is better in the following sense 3x 1 + 2x 2 3 x x 2 we continue until we arrive at the solution that can t improve. Initial feasible solution (starting point) of this example is x 1 = 0, x 2 = 0, w 1 = 8, w 2 = 8, the objective value is ζ = 0. 15
17 Example (cont) This solution can be improved? If yes. How? The coefficients of x 1 and x 2 are positive (Coefficient of x 1 is large than coefficient of x 2 ). Increase the value of x 1 and then the value of ζ increase. Make sure that all slack variables stay nonnegative. Since x 2 = 0. Then we have to make sure that { 8 x 1 2 = 4, 8 } 4 = 2 = x 1 2. New solution is and ζ = 6. (x 1 = 2, x 2 = 0, w 1 = 4, w 2 = 0). 16
18 Example (cont) Next step Rewrite the equations of our system in such a way that x 1, w 1, and ζ. are expressed as functions of w 2, and x 2. By using the second constraint we have x 1 = x w 2. Performing the appropriate row operations, we can eliminate x 1 from other equations. Then we have x w 2 = ζ x w 2 = x 1 (III) 4 3x w 2 = w 1. 17
19 Example (cont) w 2, and x 2 are called independent variables and x 1, and w 1 are called dependent variables. Increase x 2, the value of ζ increase also (coefficient of x 2 is positive). Make sure that x 1, and w 1 remain nonnegative. { x 2 4, 4 }, 3 then x New solution is ( x 1 = 4 3, x 2 = 4 ) 3, w 1 = 0, w 2 = 0. and ζ =
20 Example (cont) We will write our equations: ζ, x 1, and x 2 as functions of w 1, and w 2. Solving the last equation in (III) for x 2, we get x 2 = w w 2. Performing the appropriate row operations, we eliminate x 2 from other equations. We have w w 2 = ζ w w 2 = x 1 (IV) w w 2 = x 2. Note that all coefficients of the variables presented in the objective function (w 1 and w 2 ) are negative. This means that we can t improve the objective function. 19
21 Example (cont) Definition 2 The systems of equations (II), (III), and (IV ) are called dictionaries. The dependent variables are called basic variables and B denote the set of indices corresponding to the basic variables. The independent variables are called nonbasic variables and N denote the set of indices corresponding to the nonbasic variables. The solutions we have obtained by setting the nonbasic variables to zero are called basic feasible solutions. 20
22 Simplex method We consider a standard (LP). maximize subject to nj=1 c j x j nj=1 a ij x j b i, i = 1, 2,..., m x j 0, j = 1, 2,..., n. 21
23 Simplex method (cont) Introduce slack variables and a name for the objective function. maximize subject to nj=1 c j x j = ζ nj=1 a ij x j + w i = b i, i = 1, 2,..., m x j 0, j = 1, 2,..., n. w i 0, i = 1, 2,..., m. For simplicity, we denote (x 1,..., x n, w 1,..., w m ) = ( x 1,..., x n, x n+1,..., x n+m ). 22
24 Simplex method (cont) Our problem become maximize nj=1 c j x j = ζ subject to b i n j=1 a ij x j = x n+i, i = 1, 2,..., m We have N = {1, 2,..., n} and B = {n + 1, n + 2,..., n + m}. But this will change later (just after one iteration). maximize j N c j x j + ζ = ζ subject to b i j N ā ij x j = x n+i, i B. We put bars over the coefficients to indicate that they change as the algorithm progresses. After one iteration: One variable goes from basic to nonbasic (leaving variable). One goes from nonbasic to basic (entering variable). 23
25 Simplex method (cont) Entering variable: x k such that k N + = { j N : c j > 0 }. If N + = then the current solution is optimal. If N + contains more than one element, we will pick an index k having the largest coefficient c k. Leaving variable: chosen to preserve nonnegativity of the current basic variables. If x k is the entering variable, its value will be increased from zero to a positive value. The basic variables will change: We must ensure that x i = b i ā ik x k, i B. x k = x i 0 b i ā ik x k 0, i B. min i B:ā ik >0 b i ā ik x k = Thus Leaving variable: pick l from Pivot: Step from one dictionary to the next. ( max i B ) 1 ā ik. b i { i B : āik b i is maximal }. 24
26 Unboundedness Let x k be the leaving variable, and let l { i B : āik b i is maximal If ālk 0 i B, āik 0. Then our problem is unbounded. (the b l b i entering variable can be increased indefinitely to produce large objective value). We consider the following dictionary: }. ζ = 5 + x 3 x 1 x 2 = 5 + 2x 3 3x 1 x 4 = 7 4x 1 x 5 = x 1. The entering variable is x 3 and the ratios are: 2 5, 0 7, 0 0. This problem is unbounded. Because all these ratios are nonpositive. 25
27 Geometry: Example 1 We consider the following linear problem maximize 3x 1 + 2x 2 subject to 4x 1 + 2x 2 8 2x 1 + 4x 2 8 x 1, x 2 0. See Blackboard 26
28 We consider a LP in standard form Matrix notation maximize subject to nj=1 c j x j nj=1 a ij x j b i, i = 1, 2,..., m x j 0, j = 1, 2,..., n. Introduce slack variables x n+i = b i Problem in matrix form n j=1 a ij x j, i = 1, 2,..., m. maximize c T x subject to Ax = b x 0, 27
29 Matrix notation (cont) Where A = a 11 a a 1n 1 a 21 a a 2n a m1 a m2... a mn 1 b = b 1. b m, c = c 1. c n 0. 0, and x = x 1. x n x n+1. x n+m 28
30 Matrix notation (cont) In Component notation, the ith component of Ax can be broken up into a basic part and a nonbasic part: n+m j=1 We can writ the Matrix A a ij x j = A = j B [ a ij x j + B N ]. j N a ij x j. Separations of x and c into basic and nonbasic parts, then Constraints: Objective function: x = x B x N c = c B c N. Ax = b Bx B + Nx N = b. ζ = c T x c T B x B + c T N x N. Matrix B is m m and invertible! Why? 29
31 Matrix notation (cont) x B and ζ in terms of x N x B = B 1 b B 1 Nx N ζ = c T B x B + c T N x N = c T B B 1 b ( (B 1 N ) T cb c N ) T xn. Dictionary ζ = c T B B 1 b ( (B 1 N ) T cb c N ) T xn x B = B 1 b B 1 Nx N. 30
32 Dual variables (z 1,..., z n, y 1,..., y m ) ( ) z 1,..., z n, z n+1,..., z n+m The dual dictionary corresponding to the primal dictionary is ζ = c T B B 1 b ( B 1 b ) T zb z N = ( B 1 N ) T cb c N + ( B 1 N ) T zb. Dual solution associated to this dictionary is: z B = 0 z N = ( B 1 N ) T cb c N. Solution: ζ = c T B B 1 b 31
33 Dictionary Primal dictionary: ζ = ζ z N T x N x B = x B B 1 Nx N. Dual dictionary ζ = ζ z N T x N z N = z N + B 1 Nz B. 32
34 Example We consider a LP. maximize 5x 1 + 4x 2 + 3x 3 subject to 2x 1 + 3x 2 + x 3 5 4x 1 + x 2 + 2x 3 11 (I) 3x 1 + 4x 2 + 2x 3 8 x 1, x 2, x 3 0. First we introduce the slack variables. Our problem is equivalent to maximize 5x 1 + 4x 2 + 3x 3 = ζ subject to 2x 1 + 3x 2 + x 3 + x 4 = 5 4x 1 + x 2 + 2x 3 + x 5 = 11 (II) 3x 1 + 4x 2 + 2x 3 + x 6 = 8 x 1, x 2, x 3, x 4, x 5, x
35 Matrix notation for this example Constraints: x 1 x 2 x 3 x 4 x 5 x 6 5 = Objective function: [ ] x 1 x 2 x 3 x 4 x 5. x 6 34
36 Example (cont) Basic variables: x 1, x 5, x 6. Nonbasic variables: x 2, x 3, x 4. Ax = = = 2x 1 + 3x 2 + x 3 + x 4 4x 1 + x 2 +2x 3 +x 5 3x 1 + 4x 2 + 2x 3 +x 6 2x 1 +3x 2 + x 3 + x 4 4x 1 + x 5 +x 2 +2x 3 3x 1 + x 6 +4x 2 + 2x = Bx B + Nx N. x 1 x 5 + x x 2 x 3 x 4 35
37 Example (cont) B = = B 1 = B 1 b = B 1 N = =
38 Example (cont) ( B 1 N ) T cb c N = = c T B B 1 b = [ ] = 12.5 Associated Primal Solution: x N = 0 x B = B 1 b =
CHAPTER 2. The Simplex Method
CHAPTER 2 The Simplex Method In this chapter we present the simplex method as it applies to linear programming problems in standard form. 1. An Example We first illustrate how the simplex method works
More informationThe use of shadow price is an example of sensitivity analysis. Duality theory can be applied to do other kind of sensitivity analysis:
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
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 informationORF 307: Lecture 2. Linear Programming: Chapter 2 Simplex Methods
ORF 307: Lecture 2 Linear Programming: Chapter 2 Simplex Methods Robert Vanderbei February 8, 2018 Slides last edited on February 8, 2018 http://www.princeton.edu/ rvdb Simplex Method for LP An Example.
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 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 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 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 information9.1 Linear Programs in canonical form
9.1 Linear Programs in canonical form LP in standard form: max (LP) s.t. where b i R, i = 1,..., m z = j c jx j j a ijx j b i i = 1,..., m x j 0 j = 1,..., n But the Simplex method works only on systems
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 informationTIM 206 Lecture 3: The Simplex Method
TIM 206 Lecture 3: The Simplex Method Kevin Ross. Scribe: Shane Brennan (2006) September 29, 2011 1 Basic Feasible Solutions Have equation Ax = b contain more columns (variables) than rows (constraints),
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 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 informationOptimization (168) Lecture 7-8-9
Optimization (168) Lecture 7-8-9 Jesús De Loera UC Davis, Mathematics Wednesday, April 2, 2012 1 DEGENERACY IN THE SIMPLEX METHOD 2 DEGENERACY z =2x 1 x 2 + 8x 3 x 4 =1 2x 3 x 5 =3 2x 1 + 4x 2 6x 3 x 6
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 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 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 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 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 informationEND3033 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 informationThe Simplex Algorithm
8.433 Combinatorial Optimization The Simplex Algorithm October 6, 8 Lecturer: Santosh Vempala We proved the following: Lemma (Farkas). Let A R m n, b R m. Exactly one of the following conditions is true:.
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 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 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 informationMetode Kuantitatif Bisnis. Week 4 Linear Programming Simplex Method - Minimize
Metode Kuantitatif Bisnis Week 4 Linear Programming Simplex Method - Minimize Outlines Solve Linear Programming Model Using Graphic Solution Solve Linear Programming Model Using Simplex Method (Maximize)
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 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 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 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 informationPart 1. The Review of Linear Programming
In the name of God Part 1. The Review of Linear Programming 1.2. Spring 2010 Instructor: Dr. Masoud Yaghini Outline Introduction Basic Feasible Solutions Key to the Algebra of the The Simplex Algorithm
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 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 informationExample Problem. Linear Program (standard form) CSCI5654 (Linear Programming, Fall 2013) Lecture-7. Duality
CSCI5654 (Linear Programming, Fall 013) Lecture-7 Duality Lecture 7 Slide# 1 Lecture 7 Slide# Linear Program (standard form) Example Problem maximize c 1 x 1 + + c n x n s.t. a j1 x 1 + + a jn x n b j
More informationIntroduction to linear programming using LEGO.
Introduction to linear programming using LEGO. 1 The manufacturing problem. A manufacturer produces two pieces of furniture, tables and chairs. The production of the furniture requires the use of two different
More informationFundamental Theorems of Optimization
Fundamental Theorems of Optimization 1 Fundamental Theorems of Math Prog. Maximizing a concave function over a convex set. Maximizing a convex function over a closed bounded convex set. 2 Maximizing Concave
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 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 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 informationCSCI5654 (Linear Programming, Fall 2013) Lecture-8. Lecture 8 Slide# 1
CSCI5654 (Linear Programming, Fall 2013) Lecture-8 Lecture 8 Slide# 1 Today s Lecture 1. Recap of dual variables and strong duality. 2. Complementary Slackness Theorem. 3. Interpretation of dual variables.
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 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 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 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 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 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 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 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 informationChapter 2: Linear Programming Basics. (Bertsimas & Tsitsiklis, Chapter 1)
Chapter 2: Linear Programming Basics (Bertsimas & Tsitsiklis, Chapter 1) 33 Example of a Linear Program Remarks. minimize 2x 1 x 2 + 4x 3 subject to x 1 + x 2 + x 4 2 3x 2 x 3 = 5 x 3 + x 4 3 x 1 0 x 3
More informationLecture 2: The Simplex method
Lecture 2 1 Linear and Combinatorial Optimization Lecture 2: The Simplex method Basic solution. The Simplex method (standardform, b>0). 1. Repetition of basic solution. 2. One step in the Simplex algorithm.
More informationMATH 4211/6211 Optimization Linear Programming
MATH 4211/6211 Optimization Linear Programming Xiaojing Ye Department of Mathematics & Statistics Georgia State University Xiaojing Ye, Math & Stat, Georgia State University 0 The standard form of a Linear
More information3. THE SIMPLEX ALGORITHM
Optimization. THE SIMPLEX ALGORITHM DPK Easter Term. Introduction We know that, if a linear programming problem has a finite optimal solution, it has an optimal solution at a basic feasible solution (b.f.s.).
More informationSystems Analysis in Construction
Systems Analysis in Construction CB312 Construction & Building Engineering Department- AASTMT by A h m e d E l h a k e e m & M o h a m e d S a i e d 3. Linear Programming Optimization Simplex Method 135
More informationAM 121: Intro to Optimization Models and Methods
AM 121: Intro to Optimization Models and Methods Fall 2017 Lecture 2: Intro to LP, Linear algebra review. Yiling Chen SEAS Lecture 2: Lesson Plan What is an LP? Graphical and algebraic correspondence Problems
More informationMATH2070 Optimisation
MATH2070 Optimisation Linear Programming Semester 2, 2012 Lecturer: I.W. Guo Lecture slides courtesy of J.R. Wishart Review The standard Linear Programming (LP) Problem Graphical method of solving LP problem
More informationAM 121 Introduction to Optimization: Models and Methods Example Questions for Midterm 1
AM 121 Introduction to Optimization: Models and Methods Example Questions for Midterm 1 Prof. Yiling Chen Fall 2018 Here are some practice questions to help to prepare for the midterm. The midterm will
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 informationLinear programming: algebra
: algebra CE 377K March 26, 2015 ANNOUNCEMENTS Groups and project topics due soon Announcements Groups and project topics due soon Did everyone get my test email? Announcements REVIEW geometry Review geometry
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 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 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 informationIntroduction to linear programming
Chapter 2 Introduction to linear programming 2.1 Single-objective optimization problem We study problems of the following form: Given a set S and a function f : S R, find, if possible, an element x S that
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 informationCSCI 1951-G Optimization Methods in Finance Part 01: Linear Programming
CSCI 1951-G Optimization Methods in Finance Part 01: Linear Programming January 26, 2018 1 / 38 Liability/asset cash-flow matching problem Recall the formulation of the problem: max w c 1 + p 1 e 1 = 150
More informationDr. Maddah ENMG 500 Engineering Management I 10/21/07
Dr. Maddah ENMG 500 Engineering Management I 10/21/07 Computational Procedure of the Simplex Method The optimal solution of a general LP problem is obtained in the following steps: Step 1. Express the
More informationLinear programs, convex polyhedra, extreme points
MVE165/MMG631 Extreme points of convex polyhedra; reformulations; basic feasible solutions; the simplex method Ann-Brith Strömberg 2015 03 27 Linear programs, convex polyhedra, extreme points A linear
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 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 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 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 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 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 informationOperations Research. Duality in linear programming.
Operations Research Duality in linear programming Duality in linear programming As we have seen in past lessons, linear programming are either maximization or minimization type, containing m conditions
More informationIntroduction to optimization
Introduction to optimization Geir Dahl CMA, Dept. of Mathematics and Dept. of Informatics University of Oslo 1 / 24 The plan 1. The basic concepts 2. Some useful tools (linear programming = linear optimization)
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 informationDecember 2014 MATH 340 Name Page 2 of 10 pages
December 2014 MATH 340 Name Page 2 of 10 pages Marks [8] 1. Find the value of Alice announces a pure strategy and Betty announces a pure strategy for the matrix game [ ] 1 4 A =. 5 2 Find the value of
More informationA Parametric Simplex Algorithm for Linear Vector Optimization Problems
A Parametric Simplex Algorithm for Linear Vector Optimization Problems Birgit Rudloff Firdevs Ulus Robert Vanderbei July 9, 2015 Abstract In this paper, a parametric simplex algorithm for solving linear
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 informationIE 400: Principles of Engineering Management. Simplex Method Continued
IE 400: Principles of Engineering Management Simplex Method Continued 1 Agenda Simplex for min problems Alternative optimal solutions Unboundedness Degeneracy Big M method Two phase method 2 Simplex for
More informationFebruary 22, Introduction to the Simplex Algorithm
15.53 February 22, 27 Introduction to the Simplex Algorithm 1 Quotes for today Give a man a fish and you feed him for a day. Teach him how to fish and you feed him for a lifetime. -- Lao Tzu Give a man
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 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 informationLinear Programming. Linear Programming I. Lecture 1. Linear Programming. Linear Programming
Linear Programming Linear Programming Lecture Linear programming. Optimize a linear function subject to linear inequalities. (P) max " c j x j n j= n s. t. " a ij x j = b i # i # m j= x j 0 # j # n (P)
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 informationIP Cut Homework from J and B Chapter 9: 14, 15, 16, 23, 24, You wish to solve the IP below with a cutting plane technique.
IP Cut Homework from J and B Chapter 9: 14, 15, 16, 23, 24, 31 14. You wish to solve the IP below with a cutting plane technique. Maximize 4x 1 + 2x 2 + x 3 subject to 14x 1 + 10x 2 + 11x 3 32 10x 1 +
More informationCO350 Linear Programming Chapter 6: The Simplex Method
CO50 Linear Programming Chapter 6: The Simplex Method rd June 2005 Chapter 6: The Simplex Method 1 Recap Suppose A is an m-by-n matrix with rank m. max. c T x (P ) s.t. Ax = b x 0 On Wednesday, we learned
More informationSimplex Method for LP (II)
Simplex Method for LP (II) Xiaoxi Li Wuhan University Sept. 27, 2017 (week 4) Operations Research (Li, X.) Simplex Method for LP (II) Sept. 27, 2017 (week 4) 1 / 31 Organization of this lecture Contents:
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 informationIntroduction to Mathematical Programming
Introduction to Mathematical Programming Ming Zhong Lecture 22 October 22, 2018 Ming Zhong (JHU) AMS Fall 2018 1 / 16 Table of Contents 1 The Simplex Method, Part II Ming Zhong (JHU) AMS Fall 2018 2 /
More informationThe simplex algorithm
The simplex algorithm The simplex algorithm is the classical method for solving linear programs. Its running time is not polynomial in the worst case. It does yield insight into linear programs, however,
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 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 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 informationIntroduction to Linear Programming
Nanjing University October 27, 2011 What is LP The Linear Programming Problem Definition Decision variables Objective Function x j, j = 1, 2,..., n ζ = n c i x i i=1 We will primarily discuss maxizming
More information3. Vector spaces 3.1 Linear dependence and independence 3.2 Basis and dimension. 5. Extreme points and basic feasible solutions
A. LINEAR ALGEBRA. CONVEX SETS 1. Matrices and vectors 1.1 Matrix operations 1.2 The rank of a matrix 2. Systems of linear equations 2.1 Basic solutions 3. Vector spaces 3.1 Linear dependence and independence
More informationLinear programming. Saad Mneimneh. maximize x 1 + x 2 subject to 4x 1 x 2 8 2x 1 + x x 1 2x 2 2
Linear programming Saad Mneimneh 1 Introduction Consider the following problem: x 1 + x x 1 x 8 x 1 + x 10 5x 1 x x 1, x 0 The feasible solution is a point (x 1, x ) that lies within the region defined
More informationLecture 11 Linear programming : The Revised Simplex Method
Lecture 11 Linear programming : The Revised Simplex Method 11.1 The Revised Simplex Method While solving linear programming problem on a digital computer by regular simplex method, it requires storing
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 informationCSC Design and Analysis of Algorithms. LP Shader Electronics Example
CSC 80- Design and Analysis of Algorithms Lecture (LP) LP Shader Electronics Example The Shader Electronics Company produces two products:.eclipse, a portable touchscreen digital player; it takes hours
More informationChapter 7 Network Flow Problems, I
Chapter 7 Network Flow Problems, I Network flow problems are the most frequently solved linear programming problems. They include as special cases, the assignment, transportation, maximum flow, and shortest
More informationLinear Programming Inverse Projection Theory Chapter 3
1 Linear Programming Inverse Projection Theory Chapter 3 University of Chicago Booth School of Business Kipp Martin September 26, 2017 2 Where We Are Headed We want to solve problems with special structure!
More information