# A short OPL tutorial

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 A short OPL tutorial Truls Flatberg January 17, Introduction This note is a short tutorial to the modeling language OPL. The tutorial is by no means complete with regard to all features of OPL. More detailed information can be found in the Language User s Manual and the Language Reference Manual provided with the OPL installation. The motivation of this note is to show a few examples to get people quickly started using OPL. OPL is a modeling language for describing (and solving) optimization problems. The motivation for using modeling languages to model optimization problems is primarily due to two reasons: It provides a syntax that is close to the mathematical formulation, thus making it easier to make the transition from the mathematical formulation to something that can be solved by the computer. It enables a clean separation between the model and the accompanying data. The same model can then be solved with different input data with little extra effort. OPL is just one example of a modeling language. Other examples include AMPL, Mosel and GAMS. Modeling languages are typically used for linear and integer optimization problems, but it is also possible to formulate quadratic problems in OPL. Before starting on the examples, we give a short overview of the structure of an OPL model. A model typically includes these four parts: 1. Data declarations. This part declares the data parameters used in the model, typically coefficients and index sets for the decision variables. Individual data parameters can be declared by giving their type and name, e.g. int a = 4; Sets of a specific type are declared by enclosing the type within curly braces 1

2 {int} I = {1, 2,, 4}; The sets can then be used to create arrays with an entry for each element in the set float A[I] = [0, 1, 5, 2]; 2. Decision variables. These are declared with the keyword dvar, e.g. dvar float+ x; declares x to be a non-negative real variable. Similarly to data, it is possible to create arrays of decision variables for a specific set.. Objective function. The objective function is declared with the keyword minimize or maximize depending on what you want to do. A simple example is maximize 4*x; 4. Constraints. The constraints are declared using the keyword subject to. The following declares two linear inequalities involving the decision variables x and y: subject to { 4*x + 2*y <= 1; -2*x + y >= 5; } Note that all statements in the model file are terminated by a semicolon. In addition to keywords used for describing optimization models, OPL provides a scripting language for non-modelling purposes. This scripting language can be used to display information about the solution, to interact with the model and alter it, or do multiple runs on the same model. The scripting language has a slightly different syntax from the modeling part and is encapsulated in an execute block. For example, execute { writeln("x=", x); } will print the value of the variable x. All model and data files discussed in this note can be downloaded from the author s website The easiest way to run the examples is to open the accompanying project file in OPL Studio. 2

3 2 A first model We will start out with a minimal OPL model to solve the following linear optimization problem: minimize 4x 2y subject to x y 1 x 0 The OPL model is as follows: 1 dvar float x ; 2 dvar float y ; 4 minimize 5 4 x 2 y ; 6 7 subject to { 8 x y >= 1 ; 9 x >= 0 ; 10 } Listing 1: simple.mod Note how similar the two formulations are. The constraint in line 9 can be removed and replaced with an explicit declaration of the variable x as type float+. Production planning We will continue with a production planning problem. A company produces two products: doors and windows. It has three production facilities with limited production time available: Factory 1 produces the metal frame Factory 2 produces the wooden frame Factory produces glass and mounts the parts The products are produced in series of 200 items and each series generates a revenue depending on the product. Each series require a given amount of time of each factory s capacity. The problem is to find the number of series of each product to maximize the revenue. This can be modelled as the following linear

4 program: maximize r d x d + r w x w subject to t d,1 x d + t w,1 x w c 1 t d,2 x d + t w,2 x w c 2 t d, x d + t w, x w c x 0, y 0 where r p is the revenue for product p, t p,i is the time needed at factory i to produce a series of product p and c i is the time available at factory i. The series produced is given by x p for each product p. The following is a formulation of the problem in OPL: 1 { string } Products =... ; 2 { string } Factories =... ; Listing 2: prod.mod 4 float r [ Products ] =... ; 5 float t [ Products ] [ Factories ] =... ; 6 float c [ Factories ] =... ; 7 8 dvar float+ x [ Products ] ; 9 10 maximize 11 sum ( p in Products ) r [ p ] x [ p ] ; 12 1 subject to { 14 forall ( i in Factories ) 15 sum ( p in Products ) t [ p ] [ i ] x [ p ] <= c [ i ] ; 16 } Lines 1 and 2 declare two arrays of strings defining the set of products and the set of factories. Note the use of... to indicate that the value will be provided in a separate data file. Lines 4-6 define the input data as real numbers: revenues, time required for each product and the capacity of each factory. Line 8 declares a non-negative decision variable for each product, lines declare the objective and lines 1-16 declare the constraints, one for each factory by using the forall keyword. Suppose now that we want to solve the problem with the following data: Hours/series Hours at disposal Door Window Factory Factory Factory 2 18 Revenue/series

5 This can be translated to the following OPL data file: Listing : prod.dat 1 Products = { Door, Window } ; 2 Factories = { Factory 1, Factory 2, Factory } ; 4 r = [ ] ; 5 t = [ [ 1 0 ] [ 0 2 ] ] ; 6 c = [ ] ; Note that the separation between model and data facilitates easy updating if part of the problem is changed. For instance, adding a new product to the problem, will only affect the data file and the model can be left unchanged. 4 Linear programming This example shows how to solve a general linear programming problem, and illustrates how scripting can be combined with the model to display results and obtain details on the solution. The problem we want to solve can be written in matrix form as max c T x s.t. Ax b, x 0 The OPL model is as follows 1 int m =... ; 2 int n =... ; 4 range Rows = 1.. m ; 5 range Cols = 1.. n ; 6 7 float A [ Rows ] [ Cols ] =... ; 8 float b [ Rows ] =... ; 9 float c [ Cols ] =... ; dvar float+ x [ Cols ] ; 12 1 maximize 14 sum ( j in Cols ) c [ j ] x [ j ] ; subject to 17 { 18 forall ( i in Rows ) 19 ct : Listing 4: lp.mod 5

6 20 sum ( j in Cols ) A [ i ] [ j ] x [ j ] <= b [ i ] ; 21 } In lines 4-5 we use a special kind of set, range, that can be used to represent a sequence of integer values. Note also that we give a name to the constraint in line 19. This name will be used as an identifier when we later want to obtain information about the constraint. The problem instance can be specified in a separate data file. The following is the input for an example with four constraints and four variables. 1 m = 4 ; 2 n = 4 ; 4 A = [ [ ] 5 [ ] 6 [ ] 7 [ ] 8 ] ; 9 10 b = [ ] ; c = [ ] ; Listing 5: lp.dat In addition to the modeling features, OPL has a scripting language that can be used for several purposes. In the following example we show how to print both primal and dual solution information for the LP problem. 22 execute { Listing 6: lp.mod 2 writeln ( Optimal value :, cplex. getobjvalue ( ) ) ; 24 writeln ( Primal solution : ) ; 25 write ( x = [ ) ; 26 for ( var j in Cols ) 27 { 28 write ( x [ j ], ) ; 29 } 0 writeln ( ] ) ; 1 2 writeln ( Dual solution : ) ; write ( y = [ ) ; 4 for ( var i in Rows ) 5 { 6 write ( ct [ i ]. dual, ) ; 7 } 8 writeln ( ] ) ; 9 } 6

7 Note the slightly different syntax used in the scripting language, e.g. variables are declared using the var keyword while iteration over a set is done using the for keyword. If an execute block is placed after the objective and the constraints, it will be performed as a postprocessing step. If placed beforehand, it can also be used as a preprocessing step. 5 Integer programming In this section we show how to model and solve a problem with integer variables, in this case 0-1 variables. Consider the following problem: We want to construct a sequence of n numbers. The i th position should give the number of times the number i occurs in the sequence, where the indexing starts at zero. The following is an example of a legal sequence for n = 4: pos value The problem of constructing a feasible sequence for a given n can be solved as an integer problem. Let x ij be a 0-1 variable that is one if position i has the value j and zero otherwise, and consider the following integer program min 0 subject to (i) n 1 j=0 x ij = 1, for i = 0,..., n 1 n 1 (ii) k=0 x ki j n(1 x ij ), for i, j = 0,..., n 1 n 1 (iii) k=0 x ki j n(x ij 1), for i, j = 0,..., n 1 (iv) x ij {0, 1} for i, j = 0,..., n 1 Note that we do not need an objective function as we are only interested in a feasible solution. Constraint (i) ensures that each position has a value. While constraints (ii) and (iii) ensure that x ij = 1 implies that n 1 k=0 x ki = j. The following is the corresponding OPL model: 1 int n = 1 0 ; 2 range N = 0.. n 1; 4 dvar boolean x [ N ] [ N ] ; 5 6 minimize 0 ; 7 8 subject to 9 { 10 forall ( i in N ) 11 sum ( j in N ) x [ i ] [ j ] == 1 ; Listing 7: ip.mod 7

8 12 1 forall ( i in N, j in N ) 14 sum ( k in N ) x [ k ] [ i] j <= n (1 x [ i ] [ j ] ) ; forall ( i in N, j in N ) 17 sum ( k in N ) x [ k ] [ i] j >= n ( x [ i ] [ j ] 1); 18 } execute { 21 for ( var i in N ) 22 { 2 for ( var j in N ) 24 { 25 if ( x [ i ] [ j]==1) 26 { 27 write ( j, ) ; 28 } 29 } 0 } 1 writeln ( ) ; 2 } Note that 0-1 variables are declared using the keyword boolean, while general integer variables are declared using the int keyword. 6 Minimum spanning tree We end this tutorial with a slightly more complex model. In the minimum spanning tree problem we want to find a spanning tree of minimum weight in an undirected graph G = (V, E) with weight function w : E R. This problem can be solved efficiently as a combinatorial problem using e.g. Kruskal s algorithm. Here we will show how to formulate and solve the problem as an integer problem. The following is a valid formulation of the problem as an integer problem: min e E w ex e s.t. (i) (ii) e E x e = V 1 e δ(u) x e 1, for all U V (iii) x e {0, 1}, for all e E Constraint (ii) is a cut constraint that ensures that we do not get a solution with subcycles. Note that there is an exponential number of these constraints, clearly limiting the size of the problems solvable in OPL. 8

9 This can be implemented as an OPL model using some additional features of OPL: 1 int n =... ; 2 range Nodes = 1.. n ; { int} NodeSet = asset ( Nodes ) ; 4 5 tuple Edge 6 { 7 int u ; 8 int v ; 9 } Listing 8: mst.mod 10 { Edge} Edges with u in Nodes, v in Nodes =... ; 11 {int} Nbs [ i in Nodes ] = { j <i, j> in Edges } ; 12 1 range S = 1.. ftoi ( round (2ˆ n ) ) ; 14 {int} Sub [ s in S ] = {i i in Nodes : 15 ( s div ftoi ( round ( 2 ˆ ( i 1)))) mod 2 == 1 } ; 16 {int} Compl [ s in S ] = NodeSet diff Sub [ s ] ; float Cost [ Edges ] =... ; dvar boolean x [ Edges ] ; constraint ctcut [ S ] ; 2 24 minimize 25 sum ( e in Edges ) Cost [ e ] x [ e ] ; subject to { 28 forall ( s in S : 0 < card ( Subset [ s ] ) < n ) 29 ctcut [ s ] : 0 sum ( i in Sub [ s ], j in Nbs [ i ] inter Compl [ s ] ) x[<i, j>] 1 + sum ( i in Compl [ s ], j in Nbs [ i ] inter Sub [ s ] ) x[<i, j>] 2 >= 1 ; 4 ctall : 5 sum ( e in Edges ) x [ e ] == n 1; 6 } 7 8 {Edge} Used = {e e in Edges : x [ e ] == 1 } ; 9 40 execute 41 { 42 writeln ( Used edges =, Used ) ; 4 } The graph is represented as a list of edges where each edge is an ordered tuple 9

10 of two nodes. These are declared in line For each node we record the set of neighbouring nodes in line 11. Lines 1-16 declares all possible subsets of nodes and the complement of these subsets. Since we have an ordered tuple for each edge we need to check for cutting edges in both directions when implmenting the cut constraints in lines 0-1. The following is data for an example with 12 nodes: 1 n = 1 2 ; 2 Listing 9: mst.mod Edges = { <1,>, <2,>, <,4>, <1,5>, <2,6>, <1,8>, <7,8>, 4 <,5>, <9,10>, <,9>, <10,12>, <11,12>, <,12>, 5 <4,5>, <5,11>, <1,2>, <5,8>, <,11>}; 6 7 Cost = [ ] ; 10

### Discuss the size of the instance for the minimum spanning tree problem.

3.1 Algorithm complexity The algorithms A, B are given. The former has complexity O(n 2 ), the latter O(2 n ), where n is the size of the instance. Let n A 0 be the size of the largest instance that can

### Jeffrey Kantor. August 28, 2006

Mosel August 28, 2006 Announcements Mosel Enrollment and Registration - Last Day for Class Change is August 30th. Homework set 1 was handed out today. A copy is available on the course web page http://jkantor.blogspot.com

### EXCEL SOLVER TUTORIAL

ENGR62/MS&E111 Autumn 2003 2004 Prof. Ben Van Roy October 1, 2003 EXCEL SOLVER TUTORIAL This tutorial will introduce you to some essential features of Excel and its plug-in, Solver, that we will be using

### 2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

### 1 Solving LPs: The Simplex Algorithm of George Dantzig

Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.

### Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can

### Proximal mapping via network optimization

L. Vandenberghe EE236C (Spring 23-4) Proximal mapping via network optimization minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping Introduction this lecture:

### Approximation Algorithms

Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

### 4.6 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

### C&O 370 Deterministic OR Models Winter 2011

C&O 370 Deterministic OR Models Winter 2011 Assignment 1 Due date: Friday Jan. 21, 2011 Assignments are due at the start of class on the due date. Write your name and ID# clearly, and underline your last

### Linear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc.

1. Introduction Linear Programming for Optimization Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1.1 Definition Linear programming is the name of a branch of applied mathematics that

### Linear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued.

Linear Programming Widget Factory Example Learning Goals. Introduce Linear Programming Problems. Widget Example, Graphical Solution. Basic Theory:, Vertices, Existence of Solutions. Equivalent formulations.

### Introduction to AMPL A Tutorial

Introduction to AMPL A Tutorial September 13, 2000 AMPL is a powerful language designed specifically for mathematical programming. AMPL has many features and options; however this tutorial covers a small

### Lecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method

Lecture 3 3B1B Optimization Michaelmas 2015 A. Zisserman Linear Programming Extreme solutions Simplex method Interior point method Integer programming and relaxation The Optimization Tree Linear Programming

### 5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1

5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition

### What is Linear Programming?

Chapter 1 What is Linear Programming? An optimization problem usually has three essential ingredients: a variable vector x consisting of a set of unknowns to be determined, an objective function of x to

### Definition of a Linear Program

Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1

### Mathematical finance and linear programming (optimization)

Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may

### Practical Guide to the Simplex Method of Linear Programming

Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear

### Linear Programming. March 14, 2014

Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1

### Linear Programming I

Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins

### An Expressive Auction Design for Online Display Advertising. AUTHORS: Sébastien Lahaie, David C. Parkes, David M. Pennock

An Expressive Auction Design for Online Display Advertising AUTHORS: Sébastien Lahaie, David C. Parkes, David M. Pennock Li PU & Tong ZHANG Motivation Online advertisement allow advertisers to specify

### Special Situations in the Simplex Algorithm

Special Situations in the Simplex Algorithm Degeneracy Consider the linear program: Maximize 2x 1 +x 2 Subject to: 4x 1 +3x 2 12 (1) 4x 1 +x 2 8 (2) 4x 1 +2x 2 8 (3) x 1, x 2 0. We will first apply the

### ! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. #-approximation algorithm.

Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of three

Online Adwords Allocation Shoshana Neuburger May 6, 2009 1 Overview Many search engines auction the advertising space alongside search results. When Google interviewed Amin Saberi in 2004, their advertisement

### princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of

### Discrete Optimization

Discrete Optimization [Chen, Batson, Dang: Applied integer Programming] Chapter 3 and 4.1-4.3 by Johan Högdahl and Victoria Svedberg Seminar 2, 2015-03-31 Todays presentation Chapter 3 Transforms using

### CHAPTER 9. Integer Programming

CHAPTER 9 Integer Programming An integer linear program (ILP) is, by definition, a linear program with the additional constraint that all variables take integer values: (9.1) max c T x s t Ax b and x integral

### Linear Programming: Chapter 11 Game Theory

Linear Programming: Chapter 11 Game Theory Robert J. Vanderbei October 17, 2007 Operations Research and Financial Engineering Princeton University Princeton, NJ 08544 http://www.princeton.edu/ rvdb Rock-Paper-Scissors

### 5.1 Bipartite Matching

CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the Ford-Fulkerson

### Basic Components of an LP:

1 Linear Programming Optimization is an important and fascinating area of management science and operations research. It helps to do less work, but gain more. Linear programming (LP) is a central topic

### Permutation Betting Markets: Singleton Betting with Extra Information

Permutation Betting Markets: Singleton Betting with Extra Information Mohammad Ghodsi Sharif University of Technology ghodsi@sharif.edu Hamid Mahini Sharif University of Technology mahini@ce.sharif.edu

### Solutions to Homework 6

Solutions to Homework 6 Debasish Das EECS Department, Northwestern University ddas@northwestern.edu 1 Problem 5.24 We want to find light spanning trees with certain special properties. Given is one example

### INTEGER PROGRAMMING. Integer Programming. Prototype example. BIP model. BIP models

Integer Programming INTEGER PROGRAMMING In many problems the decision variables must have integer values. Example: assign people, machines, and vehicles to activities in integer quantities. If this is

Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should I do? A. Theory says you're unlikely to find a poly-time algorithm. Must sacrifice one

### Solving NP Hard problems in practice lessons from Computer Vision and Computational Biology

Solving NP Hard problems in practice lessons from Computer Vision and Computational Biology Yair Weiss School of Computer Science and Engineering The Hebrew University of Jerusalem www.cs.huji.ac.il/ yweiss

### Recovery of primal solutions from dual subgradient methods for mixed binary linear programming; a branch-and-bound approach

MASTER S THESIS Recovery of primal solutions from dual subgradient methods for mixed binary linear programming; a branch-and-bound approach PAULINE ALDENVIK MIRJAM SCHIERSCHER Department of Mathematical

### Why? A central concept in Computer Science. Algorithms are ubiquitous.

Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online

### CSE 326, Data Structures. Sample Final Exam. Problem Max Points Score 1 14 (2x7) 2 18 (3x6) 3 4 4 7 5 9 6 16 7 8 8 4 9 8 10 4 Total 92.

Name: Email ID: CSE 326, Data Structures Section: Sample Final Exam Instructions: The exam is closed book, closed notes. Unless otherwise stated, N denotes the number of elements in the data structure

### 24. The Branch and Bound Method

24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that no

### Introduction to Algorithms Review information for Prelim 1 CS 4820, Spring 2010 Distributed Wednesday, February 24

Introduction to Algorithms Review information for Prelim 1 CS 4820, Spring 2010 Distributed Wednesday, February 24 The final exam will cover seven topics. 1. greedy algorithms 2. divide-and-conquer algorithms

### ! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. !-approximation algorithm.

Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of

### 3 Does the Simplex Algorithm Work?

Does the Simplex Algorithm Work? In this section we carefully examine the simplex algorithm introduced in the previous chapter. Our goal is to either prove that it works, or to determine those circumstances

### Sensitivity Analysis 3.1 AN EXAMPLE FOR ANALYSIS

Sensitivity Analysis 3 We have already been introduced to sensitivity analysis in Chapter via the geometry of a simple example. We saw that the values of the decision variables and those of the slack and

### Modeling and Solving the Capacitated Vehicle Routing Problem on Trees

in The Vehicle Routing Problem: Latest Advances and New Challenges Modeling and Solving the Capacitated Vehicle Routing Problem on Trees Bala Chandran 1 and S. Raghavan 2 1 Department of Industrial Engineering

### DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

### Nonlinear Programming Methods.S2 Quadratic Programming

Nonlinear Programming Methods.S2 Quadratic Programming Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard A linearly constrained optimization problem with a quadratic objective

### Linear Programming. April 12, 2005

Linear Programming April 1, 005 Parts of this were adapted from Chapter 9 of i Introduction to Algorithms (Second Edition) /i by Cormen, Leiserson, Rivest and Stein. 1 What is linear programming? The first

### 1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.

Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S

### Linear Programming Notes VII Sensitivity Analysis

Linear Programming Notes VII Sensitivity Analysis 1 Introduction When you use a mathematical model to describe reality you must make approximations. The world is more complicated than the kinds of optimization

### 56:171 Operations Research Midterm Exam Solutions Fall 2001

56:171 Operations Research Midterm Exam Solutions Fall 2001 True/False: Indicate by "+" or "o" whether each statement is "true" or "false", respectively: o_ 1. If a primal LP constraint is slack at the

### Facility Location: Discrete Models and Local Search Methods

Facility Location: Discrete Models and Local Search Methods Yury KOCHETOV Sobolev Institute of Mathematics, Novosibirsk, Russia Abstract. Discrete location theory is one of the most dynamic areas of operations

### 7.1 Modelling the transportation problem

Chapter Transportation Problems.1 Modelling the transportation problem The transportation problem is concerned with finding the minimum cost of transporting a single commodity from a given number of sources

### Linear Programming Notes V Problem Transformations

Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material

### A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form

Section 1.3 Matrix Products A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form (scalar #1)(quantity #1) + (scalar #2)(quantity #2) +...

### Algorithm Design and Analysis

Algorithm Design and Analysis LECTURE 27 Approximation Algorithms Load Balancing Weighted Vertex Cover Reminder: Fill out SRTEs online Don t forget to click submit Sofya Raskhodnikova 12/6/2011 S. Raskhodnikova;

### Linear Programming. Solving LP Models Using MS Excel, 18

SUPPLEMENT TO CHAPTER SIX Linear Programming SUPPLEMENT OUTLINE Introduction, 2 Linear Programming Models, 2 Model Formulation, 4 Graphical Linear Programming, 5 Outline of Graphical Procedure, 5 Plotting

### Instituto Superior Técnico Masters in Civil Engineering. Lecture 2: Transport networks design and evaluation Xpress presentation - Laboratory work

Instituto Superior Técnico Masters in Civil Engineering REGIÕES E REDES () Lecture 2: Transport networks design and evaluation Xpress presentation - Laboratory work Eng. Luis Martínez OUTLINE Xpress presentation

### IEOR 4404 Homework #2 Intro OR: Deterministic Models February 14, 2011 Prof. Jay Sethuraman Page 1 of 5. Homework #2

IEOR 4404 Homework # Intro OR: Deterministic Models February 14, 011 Prof. Jay Sethuraman Page 1 of 5 Homework #.1 (a) What is the optimal solution of this problem? Let us consider that x 1, x and x 3

### Dynamic Programming. Applies when the following Principle of Optimality

Dynamic Programming Applies when the following Principle of Optimality holds: In an optimal sequence of decisions or choices, each subsequence must be optimal. Translation: There s a recursive solution.

### Applied Algorithm Design Lecture 5

Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design

### Factoring Algorithms

Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors

### How to speed-up hard problem resolution using GLPK?

How to speed-up hard problem resolution using GLPK? Onfroy B. & Cohen N. September 27, 2010 Contents 1 Introduction 2 1.1 What is GLPK?.......................................... 2 1.2 GLPK, the best one?.......................................

### Permutation Betting Markets: Singleton Betting with Extra Information

Permutation Betting Markets: Singleton Betting with Extra Information Mohammad Ghodsi Sharif University of Technology ghodsi@sharif.edu Hamid Mahini Sharif University of Technology mahini@ce.sharif.edu

### Linear Programming in Matrix Form

Linear Programming in Matrix Form Appendix B We first introduce matrix concepts in linear programming by developing a variation of the simplex method called the revised simplex method. This algorithm,

### Can linear programs solve NP-hard problems?

Can linear programs solve NP-hard problems? p. 1/9 Can linear programs solve NP-hard problems? Ronald de Wolf Linear programs Can linear programs solve NP-hard problems? p. 2/9 Can linear programs solve

### 1. Sorting (assuming sorting into ascending order) a) BUBBLE SORT

DECISION 1 Revision Notes 1. Sorting (assuming sorting into ascending order) a) BUBBLE SORT Make sure you show comparisons clearly and label each pass First Pass 8 4 3 6 1 4 8 3 6 1 4 3 8 6 1 4 3 6 8 1

### Parameters and Expressions

7 Parameters and Expressions A large optimization model invariably uses many numerical values. As we have explained before, only a concise symbolic description of these values need appear in an AMPL model,

### Branch-and-Price Approach to the Vehicle Routing Problem with Time Windows

TECHNISCHE UNIVERSITEIT EINDHOVEN Branch-and-Price Approach to the Vehicle Routing Problem with Time Windows Lloyd A. Fasting May 2014 Supervisors: dr. M. Firat dr.ir. M.A.A. Boon J. van Twist MSc. Contents

### UNIT 1 LINEAR PROGRAMMING

OUTLINE Session : Session 2: Session 3: Session 4: Session 5: Session 6: Session 7: Session 8: Session 9: Session 0: Session : Session 2: UNIT LINEAR PROGRAMMING Introduction What is Linear Programming

### EdExcel Decision Mathematics 1

EdExcel Decision Mathematics 1 Linear Programming Section 1: Formulating and solving graphically Notes and Examples These notes contain subsections on: Formulating LP problems Solving LP problems Minimisation

### Using Excel to Find Numerical Solutions to LP's

Using Excel to Find Numerical Solutions to LP's by Anil Arya and Richard A. Young This handout sketches how to solve a linear programming problem using Excel. Suppose you wish to solve the following LP:

### 1.1 Logical Form and Logical Equivalence 1

Contents Chapter I The Logic of Compound Statements 1.1 Logical Form and Logical Equivalence 1 Identifying logical form; Statements; Logical connectives: not, and, and or; Translation to and from symbolic

### 3. Linear Programming and Polyhedral Combinatorics

Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

### Optimal Scheduling for Dependent Details Processing Using MS Excel Solver

BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 8, No 2 Sofia 2008 Optimal Scheduling for Dependent Details Processing Using MS Excel Solver Daniela Borissova Institute of

### Solutions to Linear Algebra Practice Problems 1. form (because the leading 1 in the third row is not to the right of the

Solutions to Linear Algebra Practice Problems. Determine which of the following augmented matrices are in row echelon from, row reduced echelon form or neither. Also determine which variables are free

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### LABEL PROPAGATION ON GRAPHS. SEMI-SUPERVISED LEARNING. ----Changsheng Liu 10-30-2014

LABEL PROPAGATION ON GRAPHS. SEMI-SUPERVISED LEARNING ----Changsheng Liu 10-30-2014 Agenda Semi Supervised Learning Topics in Semi Supervised Learning Label Propagation Local and global consistency Graph

### Lecture 11: 0-1 Quadratic Program and Lower Bounds

Lecture : - Quadratic Program and Lower Bounds (3 units) Outline Problem formulations Reformulation: Linearization & continuous relaxation Branch & Bound Method framework Simple bounds, LP bound and semidefinite

### MATHEMATICS Unit Decision 1

General Certificate of Education January 2008 Advanced Subsidiary Examination MATHEMATICS Unit Decision 1 MD01 Tuesday 15 January 2008 9.00 am to 10.30 am For this paper you must have: an 8-page answer

### Programming Fundamental. Instructor Name: Lecture-2

Programming Fundamental Instructor Name: Lecture-2 Today s Lecture What is Programming? First C++ Program Programming Errors Variables in C++ Primitive Data Types in C++ Operators in C++ Operators Precedence

### Chapter 5. Linear Inequalities and Linear Programming. Linear Programming in Two Dimensions: A Geometric Approach

Chapter 5 Linear Programming in Two Dimensions: A Geometric Approach Linear Inequalities and Linear Programming Section 3 Linear Programming gin Two Dimensions: A Geometric Approach In this section, we

### Linear Programs: Variables, Objectives and Constraints

8 Linear Programs: Variables, Objectives and Constraints The best-known kind of optimization model, which has served for all of our examples so far, is the linear program. The variables of a linear program

### A Constraint Programming based Column Generation Approach to Nurse Rostering Problems

Abstract A Constraint Programming based Column Generation Approach to Nurse Rostering Problems Fang He and Rong Qu The Automated Scheduling, Optimisation and Planning (ASAP) Group School of Computer Science,

### A Note on the Bertsimas & Sim Algorithm for Robust Combinatorial Optimization Problems

myjournal manuscript No. (will be inserted by the editor) A Note on the Bertsimas & Sim Algorithm for Robust Combinatorial Optimization Problems Eduardo Álvarez-Miranda Ivana Ljubić Paolo Toth Received:

### Chapter 6. Linear Programming: The Simplex Method. Introduction to the Big M Method. Section 4 Maximization and Minimization with Problem Constraints

Chapter 6 Linear Programming: The Simplex Method Introduction to the Big M Method In this section, we will present a generalized version of the simplex method that t will solve both maximization i and

### Noncommercial Software for Mixed-Integer Linear Programming

Noncommercial Software for Mixed-Integer Linear Programming J. T. Linderoth T. K. Ralphs December, 2004. Revised: January, 2005. Abstract We present an overview of noncommercial software tools for the

### 56:171. Operations Research -- Sample Homework Assignments Fall 1992 Dennis Bricker Dept. of Industrial Engineering University of Iowa.

56:171 Operations Research -- Sample Homework Assignments Fall 1992 Dennis Bricker Dept. of Industrial Engineering University of Iowa Homework #1 (1.) Linear Programming Model Formulation. SunCo processes

### Duality in General Programs. Ryan Tibshirani Convex Optimization 10-725/36-725

Duality in General Programs Ryan Tibshirani Convex Optimization 10-725/36-725 1 Last time: duality in linear programs Given c R n, A R m n, b R m, G R r n, h R r : min x R n c T x max u R m, v R r b T

### One last point: we started off this book by introducing another famously hard search problem:

S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani 261 Factoring One last point: we started off this book by introducing another famously hard search problem: FACTORING, the task of finding all prime factors

### Scheduling Algorithm with Optimization of Employee Satisfaction

Washington University in St. Louis Scheduling Algorithm with Optimization of Employee Satisfaction by Philip I. Thomas Senior Design Project http : //students.cec.wustl.edu/ pit1/ Advised By Associate

### Kerby Shedden October, 2007. Overview of R

Kerby Shedden October, 2007 Overview of R R R is a programming language for statistical computing, data analysis, and graphics. It is a re-implementation of the S language, which was developed in the 1980

### Transportation Polytopes: a Twenty year Update

Transportation Polytopes: a Twenty year Update Jesús Antonio De Loera University of California, Davis Based on various papers joint with R. Hemmecke, E.Kim, F. Liu, U. Rothblum, F. Santos, S. Onn, R. Yoshida,

### The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 72.

ADVANCED SUBSIDIARY GCE UNIT 4736/01 MATHEMATICS Decision Mathematics 1 THURSDAY 14 JUNE 2007 Afternoon Additional Materials: Answer Booklet (8 pages) List of Formulae (MF1) Time: 1 hour 30 minutes INSTRUCTIONS

### Solving Linear Programs

Solving Linear Programs 2 In this chapter, we present a systematic procedure for solving linear programs. This procedure, called the simplex method, proceeds by moving from one feasible solution to another,

### Quantitative Methods in Regulation

Quantitative Methods in Regulation (DEA) Data envelopment analysis is one of the methods commonly used in assessing efficiency for regulatory purposes, as an alternative to regression. The theoretical

### Recursive Algorithms. Recursion. Motivating Example Factorial Recall the factorial function. { 1 if n = 1 n! = n (n 1)! if n > 1

Recursion Slides by Christopher M Bourke Instructor: Berthe Y Choueiry Fall 007 Computer Science & Engineering 35 Introduction to Discrete Mathematics Sections 71-7 of Rosen cse35@cseunledu Recursive Algorithms

### Lecture 2: August 29. Linear Programming (part I)

10-725: Convex Optimization Fall 2013 Lecture 2: August 29 Lecturer: Barnabás Póczos Scribes: Samrachana Adhikari, Mattia Ciollaro, Fabrizio Lecci Note: LaTeX template courtesy of UC Berkeley EECS dept.