Chapter 8 Network Models Part 1. Prof. Dr. Arslan M. ÖRNEK

Size: px
Start display at page:

Download "Chapter 8 Network Models Part 1. Prof. Dr. Arslan M. ÖRNEK"

Transcription

1 Chapter 8 Network Models Part 1 Prof. Dr. Arslan M. ÖRNEK

2 Network Optimization Models Networks arise in numerous settings: Transportation Electrical and communication networks Production planning problems Distribution problems Project management Facilities location Financial planning etc. Recent developments in OR: Efficient network algorithms 2

3 Network Optimization Models Many network models: Special type of LP Minimum cost network flow problem (MCNFP) In this chapter: Shortest path problem Minimum spanning tree problem Maximum flow problem Project scheduling 3

4 8.1. Basic Definitions A graph or network is defined by two sets of symbols: Nodes: A set of points or vertices (call it V) are called nodes (or vertices) of a graph or network. Arcs: An arc consists of an ordered pair of vertices and represents a possible direction of motion (flow) that may occur between vertices. Node 1 2 Arc (1,2) Node (vertex)

5 8.1. Basic Definitions Edges (links, branches, undirected arcs) : Lines connecting certain pairs of nodes in a network. The flow through an edge is allowed in both directions. 5

6 8.1. Basic Definitions 6

7 8.1. Basic Definitions On an edge, the flow will be in only one direction at a time, a simultaneous flow to both directions requires two directed arcs in both directions. A network with only directed arcs: A directed network A network with only undirected arcs: An undirected network You can convert a network with both directed and undirected arcs to a directed network. (How?) 7

8 8.1. Basic Definitions V = {1, 2, 3, 4} and A = {(1, 2), (2, 3), (3, 4), (4, 3), (4, 1)} For any arc (j, k), node j is the initial node, and node k is the terminal node. The arc ( j, k) is said to go from node j to node k.

9 8.1. Basic Definitions A sequence of arcs such that every arc has exactly one vertex in common with the previous arc is called a chain (or undirected path). A path is a directed chain in which the terminal node of each arc is identical to the initial node of the next arc.

10 8.1. Basic Definitions A path from node i to node j has a sequence of arcs whose directions are towards node j. A chain from node i to node j has a sequence of arcs whose directions are either towards or away from node j. 10

11 8.1. Basic Definitions Nodes A and B are factories, Nodes D and E are warehouses, and Node C is a distribution center, arcs represent shipping channels. A B C E : path from A to E (We can send a flow from A to E on this path) B C A D : undirected path (chain) from B to D (cannot send flow) 11

12 8.2. Shortest Path Problems Assume that each arc in the network has a length associated with it. Suppose we start at a particular node (say, node 1). The problem of finding the shortest path (path of minimum length) from node 1 to any other node in the network is called a shortest path problem.

13 8.2. Shortest Path Problems Example 1: Suppose that when power is sent from plant 1 (node 1) to city 1 (node 6), it must pass through substations (nodes 2 to 5). For any pair of nodes between which power can be transported, distances (in miles) between the nodes are shown. Powerco wants the power sent from plant 1 to city 1 to travel the minimum possible distance.

14 8.2. Solution: Dijkstra s Algorithm Find the shortest path from a node (say, node 1) to all other nodes. To begin, we label node 1 with a permanent label of 0. Then we label each node i that is connected to node 1 by a single arc with a temporary label equal to the length of the arc joining node 1 to node i. Each other node (except, of course, for node 1) will have a temporary label of. Choose the node with the smallest temporary label and make this label permanent. See the rest on the example:

15 8.2. Shortest Path Problems Dijkstra s Algorithm

16 8.2. Shortest Path Problems Dijkstra s Algorithm

17 8.2. Shortest Path Problems Dijkstra s Algorithm

18 8.2. Shortest Path Problems Dijkstra s Algorithm

19 The Shortest Path Problem Example 2 Problem: Determine the shortest routes from the origin to all destinations. The arc lengths are times of travel between cities. 19

20 The Shortest Path Problem Example Network Representation 20

21 The Shortest Path Problem Example [0*, 16, 9, 35, inf., inf., inf.] Label 3 as permanent: [0*, 16, 9*, 35, inf., inf., inf.] Update labels for connected nodes: [0*, 16, 9*, 24, inf., 31, inf.] Label 2 as permanent: [0*, 16*, 9*, 24, inf., 31, inf.] Update labels for connected nodes: [0*, 16*, 9*, 24, 41, 31, inf.] Label 4 as permanent: 21

22 The Shortest Path Problem Example [0*, 16*, 9*, 24*, 41, 31, inf.] Update labels for connected nodes: [0*, 16*, 9*, 24*, 38, 31, 43] Label 6 as permanent: [0*, 16*, 9*, 24*, 38, 31*, 43] Update labels for connected nodes: [0*, 16*, 9*, 24*, 38, 31*, 43] Label 5 as permanent: [0*, 16*, 9*, 24*, 38*, 31*, 43] Update labels for connected nodes: 22

23 The Shortest Path Problem Example [0*, 16*, 9*, 24*, 38*, 31*, 43*]. Work backwards: = 8 NO, = 5 NO, = 19 YES Go back to node 4: 24-9 = 15 YES Go back to node 3: 9-0 = 9 YES Shortest path from node 1 to node 7 : with length

24 The Shortest Path Problem Example Network with Optimal Routes 24

25 The Shortest Path Problem Example Shortest Travel Time from Origin to Each Destination 25

26 The Shortest Path Problem Example Mathematical Model Formulation as a 0-1 integer programming problem: x ij = 1 if branch (i,j) is selected as part of the shortest route, and 0 if it is not selected. Minimize Z = 16x x x x x x x x x x x x 67 subject to: x 12 + x 13 + x 14 = 1 x 12 - x 24 - x 25 = 0 x 13 - x 34 - x 36 = 0 x 14 + x 24 + x 34 - x 45 - x 46 - x 47 = 0 x 25 + x 45 - x 57 = 0 x 36 + x 46 - x 67 = 0 x 47 + x 57 + x 67 = 1 x ij = 0 or 1 26

27 Example 3: 8.2. Shortest Path Problems I have just purchased (at time 0) a new car for $12,000. To avoid the high maintenance costs associated with an older car, I may trade in my car and purchase a new car. At any time, it costs $12,000 to purchase a new car. My goal is to minimize the net cost (purchasing costs + maintenance costs - money received in trade-ins) incurred during the next five years. Formulate this problem as a shortest-path problem. Age of Car (Years) Annual Maintenance cost Age of Car (Years) Trade-in Price 0 $2,000 1 $7,000 1 $4,000 2 $6,000 2 $5,000 3 $2,000 3 $9,000 4 $1,000 4 $12,000 5 $0

28 Example 3 solution: 8.2. Shortest Path Problems Our goal is to minimize the net cost incurred during the next five years. Our network will have six nodes. Node i is the beginning of year i and for i<j, an arc (i,j) corresponds to purchasing a new car at the beginning of year i and keeping it until the beginning of year j. The length of arc (i,j) (call it c ij ) is the total net cost incurred from year i to j. etc.

29 8.2. Shortest Path Problems Example 3 solution: We now see that the length of any path from node 1 to node 6 is the net cost incurred during the next five years corresponding to a particular trade-in strategy. For example, suppose I trade in the car at the beginning of year 3 and next trade in the car at the end of year 5 (the beginning of year 6). This strategy corresponds to the path in the graph. Exercise: Solve this shortest path problem using Dijkstra s!

30 8.3. Maximum Flow Problems Many situations can be modeled by a network in which the arcs may be thought of as having a capacity that limits the quantity of a product that may be shipped through that arc. In these situations, it is often desired to transport the maximum amount of flow from a starting point (called the source) to a terminal point (called the sink). Such problems are called maximum flow problems. We will see the LP model and Ford-Fulkerson/Augmented Path method for solving the maximum flow problems.

31 8.3. Maximum Flow Problems Example: Sunco Oil wants to ship the maximum possible amount of oil (per hour) via pipeline from node s o to node s i. The various arcs represent pipelines of different diameters. The maximum number of barrels of oil (millions of barrels per hour) that can be pumped through each arc is shown. Formulate an LP that can be used to determine the maximum number of barrels of oil per hour.

32 8.3. Maximum Flow Problems Example (cont.):

33 8.3. Maximum Flow Problems Example (cont.): Artificial arc

34 8.3. Maximum Flow Problems Example (cont.):

35 8.3. Maximum Flow Problems Example: Fly-by-Night Airlines must determine how many connecting flights can be arranged between Juneau, Alaska, and Dallas, Texas daily. Connecting flights must stop in Seattle and then stop in Los Angeles or Denver. Because of limited landing space, Fly-by-Night is limited to making the number of daily flights between pairs of cities shown in Table 9. Set up a maximum-flow problem whose solution will tell the airline how to maximize the number of connecting flights daily from Juneau to Dallas.

36 8.3. Maximum Flow Problems Network for the problem: Exercise: Write the LP for the model.

37 8.3. Maximum Flow Problems Example: Five men and five women are at a dance. The goal of the matchmaker is to match each woman with a man in a way that maximizes the number of people who are matched with compatible mates. Table 10 describes the compatibility of the couples. Draw a network that makes it possible to represent the problem of maximizing the number of compatible pairings as a maximum-flow problem.

38 Network: 8.3. Maximum Flow Problems

39 8.3. Maximum Flow Problems The Ford Fulkerson Method Assume a feasible flow has been found. Given a feasible flow, how can we tell if it is an optimal flow (max x 0 ), if it is nonoptimal, how can we modify it to increase the flow? 39

40 8.3. Maximum Flow Problems The Ford Fulkerson Method

41 8.3. Maximum Flow Problems The Ford Fulkerson Method If the labeling process results in the sink being labeled, then there will be a chain of arcs (call it C) leading from the source to the sink. By adjusting the flow of the arcs in C, we can increase the total flow from source to sink. Note: Labeling a node means establishing a chain from the source to the node.

42 8.3. Maximum Flow Problems The Ford Fulkerson Method

43 8.3. Maximum Flow Problems The Ford Fulkerson Method

44 8.3. Maximum Flow Problems The Ford Fulkerson Method

45 8.3. Maximum Flow Problems The Ford Fulkerson Method

46 8.3. Maximum Flow Problems The Ford Fulkerson Method

47 8.3. Maximum Flow Problems The Ford Fulkerson Method

48 8.3. Maximum Flow Problems Example: Sunco Oil - Try to label the sink: Label the source, and then arc (so, 1) and node 1; then label arc (1, 2) and node 2; finally, label arc (2, si) and node si. - Thus, C = (so, 1) (1, 2) (2, si). Each arc in C is a forward arc, so increase the flow through each arc in C by min (2, 3, 2) = 2 units.

49 8.3. Maximum Flow Problems Example: Sunco Oil Label arc (so, 2) and node 2 (thus (so, 2) is a forward arc). Then label arc (1, 2) and node 1. Arc (1, 2) is a backward arc, because node 1 was unlabeled before we labeled arc (1, 2), and arc (1, 2) is in R. Nodes 1 and 2 are labeled, so label arc (1, 3) and node 3. Arc (1, 3) is a forward arc, because node 3 has not yet been labeled. Finally label arc (3, si) and node si. Arc (3, si) is a forward arc, because node si has not yet been labeled.

50 8.3. Maximum Flow Problems Example: Sunco Oil We have now labeled the sink via the chain C = (so, 2) - (1, 2) - (1, 3) - (3, si). Except arc (1, 2), all arcs in the chain are forward arcs. Because i(so, 2) = 3; i(1, 3) = 4; i(3, si) = 1; and r(1, 2) = 2, increase the flow on all forward arcs in C by 1 and decrease the flow in all backward arcs by 1.

51 8.3. Maximum Flow Problems Example: Sunco Oil

52 Example: 8.3. Maximum Flow Problems

53 Example: 8.3. Maximum Flow Problems The minimum cut (with capacity 7) corresponds to nodes 1, 3, and si and consists of arcs (so, 1), (so, 3) and (2, 3).

54 8.3. Maximum Flow Problems Problem: How many tram trips can we make from location O to location T if the number of tram trips between any two stations is limited?

55 Augmenting Path Algorithm An augmenting path: Directed path from source to sink in the residual network such that every arc on it has positive residual capacity. Residual capacity of the augmenting path: The minimum residual (remaining) arc capacity on the augmenting path. This is the maximum amount that we can send through that path from source to the sink.

56 Augmenting Path Algorithm 56

57 Iteration 0 Example 1 Iteration 1 O-B-E-T Iteration 3 O-A-B-D-T Iteration 4 O-B-D-T Iteration 2 O-A-D-T

58 Iteration 5 O-C-E-D-T Iteration 6 O-C-E-T Iteration 7 O-C-E-B-D-T

59 Example 2 Network of a railway system: Maximize the number of trains from Omaha to St. Louis.

60 Example 2 Augmenting path: , Flow = 4

61 Example 2 Augmenting path: 1-4-6, Flow = =8

62 Example 2 Augmenting path: 1-3-6, Flow = = 14

63 Example 2 Augmenting path: , Flow = = 15

64 Example 2 Optimal Solution: Maximum Flow = 15

65 Finding an Augmented Path - Find all nodes directly connected to the source with arcs having positive residual capacities. - From each of these nodes, find new ones that are directly connected to them with such arcs. - Continue to fan-out in this manner, if an augmenting path exists, you will find it!

66 Max-flow-min-cut Theorem Applied to Seervada Park Example Maximum Flow = Minimum Cut = 14

Scheduling Shop Scheduling. Tim Nieberg

Scheduling Shop Scheduling. Tim Nieberg Scheduling Shop Scheduling Tim Nieberg Shop models: General Introduction Remark: Consider non preemptive problems with regular objectives Notation Shop Problems: m machines, n jobs 1,..., n operations

More information

Chapter 10: Network Flow Programming

Chapter 10: Network Flow Programming Chapter 10: Network Flow Programming Linear programming, that amazingly useful technique, is about to resurface: many network problems are actually just special forms of linear programs! This includes,

More information

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

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

More information

Minimum cost maximum flow, Minimum cost circulation, Cost/Capacity scaling

Minimum cost maximum flow, Minimum cost circulation, Cost/Capacity scaling 6.854 Advanced Algorithms Lecture 16: 10/11/2006 Lecturer: David Karger Scribe: Kermin Fleming and Chris Crutchfield, based on notes by Wendy Chu and Tudor Leu Minimum cost maximum flow, Minimum cost circulation,

More information

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

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

More information

5.1 Bipartite Matching

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

More information

Social Media Mining. Graph Essentials

Social Media Mining. Graph Essentials Graph Essentials Graph Basics Measures Graph and Essentials Metrics 2 2 Nodes and Edges A network is a graph nodes, actors, or vertices (plural of vertex) Connections, edges or ties Edge Node Measures

More information

Chapter 6: Graph Theory

Chapter 6: Graph Theory Chapter 6: Graph Theory Graph theory deals with routing and network problems and if it is possible to find a best route, whether that means the least expensive, least amount of time or the least distance.

More information

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

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

More information

Max Flow, Min Cut, and Matchings (Solution)

Max Flow, Min Cut, and Matchings (Solution) Max Flow, Min Cut, and Matchings (Solution) 1. The figure below shows a flow network on which an s-t flow is shown. The capacity of each edge appears as a label next to the edge, and the numbers in boxes

More information

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.

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

More information

Approximation Algorithms

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

More information

Network optimization is a special type of linear programming model. Network models have three

Network optimization is a special type of linear programming model. Network models have three Chapter 11 Network Optimization 11.1 Introduction Network optimization is a special type of linear programming model. Network models have three main advantages over linear programming: 1. They can be solved

More information

Data Structures and Algorithms Written Examination

Data Structures and Algorithms Written Examination Data Structures and Algorithms Written Examination 22 February 2013 FIRST NAME STUDENT NUMBER LAST NAME SIGNATURE Instructions for students: Write First Name, Last Name, Student Number and Signature where

More information

2 SYSTEM DESCRIPTION TECHNIQUES

2 SYSTEM DESCRIPTION TECHNIQUES 2 SYSTEM DESCRIPTION TECHNIQUES 2.1 INTRODUCTION Graphical representation of any process is always better and more meaningful than its representation in words. Moreover, it is very difficult to arrange

More information

Problem Set 7 Solutions

Problem Set 7 Solutions 8 8 Introduction to Algorithms May 7, 2004 Massachusetts Institute of Technology 6.046J/18.410J Professors Erik Demaine and Shafi Goldwasser Handout 25 Problem Set 7 Solutions This problem set is due in

More information

Week_11: Network Models

Week_11: Network Models Week_11: Network Models 1 1.Introduction The network models include the traditional applications of finding the most efficient way to link a number of locations directly or indirectly, finding the shortest

More information

2. (a) Explain the strassen s matrix multiplication. (b) Write deletion algorithm, of Binary search tree. [8+8]

2. (a) Explain the strassen s matrix multiplication. (b) Write deletion algorithm, of Binary search tree. [8+8] Code No: R05220502 Set No. 1 1. (a) Describe the performance analysis in detail. (b) Show that f 1 (n)+f 2 (n) = 0(max(g 1 (n), g 2 (n)) where f 1 (n) = 0(g 1 (n)) and f 2 (n) = 0(g 2 (n)). [8+8] 2. (a)

More information

GRAPH THEORY LECTURE 4: TREES

GRAPH THEORY LECTURE 4: TREES GRAPH THEORY LECTURE 4: TREES Abstract. 3.1 presents some standard characterizations and properties of trees. 3.2 presents several different types of trees. 3.7 develops a counting method based on a bijection

More information

Cpt S 223. School of EECS, WSU

Cpt S 223. School of EECS, WSU The Shortest Path Problem 1 Shortest-Path Algorithms Find the shortest path from point A to point B Shortest in time, distance, cost, Numerous applications Map navigation Flight itineraries Circuit wiring

More information

A Review And Evaluations Of Shortest Path Algorithms

A Review And Evaluations Of Shortest Path Algorithms A Review And Evaluations Of Shortest Path Algorithms Kairanbay Magzhan, Hajar Mat Jani Abstract: Nowadays, in computer networks, the routing is based on the shortest path problem. This will help in minimizing

More information

Midterm Practice Problems

Midterm Practice Problems 6.042/8.062J Mathematics for Computer Science October 2, 200 Tom Leighton, Marten van Dijk, and Brooke Cowan Midterm Practice Problems Problem. [0 points] In problem set you showed that the nand operator

More information

The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

More information

1. Write the number of the left-hand item next to the item on the right that corresponds to it.

1. Write the number of the left-hand item next to the item on the right that corresponds to it. 1. Write the number of the left-hand item next to the item on the right that corresponds to it. 1. Stanford prison experiment 2. Friendster 3. neuron 4. router 5. tipping 6. small worlds 7. job-hunting

More information

Network Flow I. Lecture 16. 16.1 Overview. 16.2 The Network Flow Problem

Network Flow I. Lecture 16. 16.1 Overview. 16.2 The Network Flow Problem Lecture 6 Network Flow I 6. Overview In these next two lectures we are going to talk about an important algorithmic problem called the Network Flow Problem. Network flow is important because it can be

More information

Linear Programming. March 14, 2014

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

More information

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.

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

More information

IE 680 Special Topics in Production Systems: Networks, Routing and Logistics*

IE 680 Special Topics in Production Systems: Networks, Routing and Logistics* IE 680 Special Topics in Production Systems: Networks, Routing and Logistics* Rakesh Nagi Department of Industrial Engineering University at Buffalo (SUNY) *Lecture notes from Network Flows by Ahuja, Magnanti

More information

at which branching takes place, a "middleman," if you will. See the transship model panel. ABSTRACT

at which branching takes place, a middleman, if you will. See the transship model panel. ABSTRACT Optimal Solution of Discrete Resource Allocation Problems with SAS/OR Software by LTC Doug McAllaster, US Army Logistics Management College, Fort Lee, VA ABSTRACT This paper is a tutorial on how to use

More information

COUNTING SUBSETS OF A SET: COMBINATIONS

COUNTING SUBSETS OF A SET: COMBINATIONS COUNTING SUBSETS OF A SET: COMBINATIONS DEFINITION 1: Let n, r be nonnegative integers with r n. An r-combination of a set of n elements is a subset of r of the n elements. EXAMPLE 1: Let S {a, b, c, d}.

More information

Dynamic Programming 11.1 AN ELEMENTARY EXAMPLE

Dynamic Programming 11.1 AN ELEMENTARY EXAMPLE Dynamic Programming Dynamic programming is an optimization approach that transforms a complex problem into a sequence of simpler problems; its essential characteristic is the multistage nature of the optimization

More information

Linear Programming. Solving LP Models Using MS Excel, 18

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

More information

4 UNIT FOUR: Transportation and Assignment problems

4 UNIT FOUR: Transportation and Assignment problems 4 UNIT FOUR: Transportation and Assignment problems 4.1 Objectives By the end of this unit you will be able to: formulate special linear programming problems using the transportation model. define a balanced

More information

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

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

More information

Dynamic programming. Doctoral course Optimization on graphs - Lecture 4.1. Giovanni Righini. January 17 th, 2013

Dynamic programming. Doctoral course Optimization on graphs - Lecture 4.1. Giovanni Righini. January 17 th, 2013 Dynamic programming Doctoral course Optimization on graphs - Lecture.1 Giovanni Righini January 1 th, 201 Implicit enumeration Combinatorial optimization problems are in general NP-hard and we usually

More information

Practical Guide to the Simplex Method of Linear Programming

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

More information

1 Solving LPs: The Simplex Algorithm of George Dantzig

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.

More information

CMPSCI611: Approximating MAX-CUT Lecture 20

CMPSCI611: Approximating MAX-CUT Lecture 20 CMPSCI611: Approximating MAX-CUT Lecture 20 For the next two lectures we ll be seeing examples of approximation algorithms for interesting NP-hard problems. Today we consider MAX-CUT, which we proved to

More information

A Labeling Algorithm for the Maximum-Flow Network Problem

A Labeling Algorithm for the Maximum-Flow Network Problem A Labeling Algorithm for the Maximum-Flow Network Problem Appendix C Network-flow problems can be solved by several methods. In Chapter 8 we introduced this topic by exploring the special structure of

More information

Computational Approach for Assessment of Critical Infrastructure in Network Systems

Computational Approach for Assessment of Critical Infrastructure in Network Systems Computational Approach for Assessment of Critical Infrastructure in Network Systems EMIL KELEVEDJIEV 1 Institute of Mathematics and Informatics Bulgarian Academy of Sciences ABSTRACT. Methods of computational

More information

EdExcel Decision Mathematics 1

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

More information

Lecture 1: Course overview, circuits, and formulas

Lecture 1: Course overview, circuits, and formulas Lecture 1: Course overview, circuits, and formulas Topics in Complexity Theory and Pseudorandomness (Spring 2013) Rutgers University Swastik Kopparty Scribes: John Kim, Ben Lund 1 Course Information Swastik

More information

Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs

Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs MCS-236: Graph Theory Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010 Chapter 7: Digraphs Strong Digraphs Definitions. A digraph is an ordered pair (V, E), where V is the set

More information

Linear Programming Notes V Problem Transformations

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

More information

The Assignment Problem and the Hungarian Method

The Assignment Problem and the Hungarian Method The Assignment Problem and the Hungarian Method 1 Example 1: You work as a sales manager for a toy manufacturer, and you currently have three salespeople on the road meeting buyers. Your salespeople are

More information

SYSM 6304: Risk and Decision Analysis Lecture 5: Methods of Risk Analysis

SYSM 6304: Risk and Decision Analysis Lecture 5: Methods of Risk Analysis SYSM 6304: Risk and Decision Analysis Lecture 5: Methods of Risk Analysis M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu October 17, 2015 Outline

More information

Special cases in Transportation Problems

Special cases in Transportation Problems Unit 1 Lecture 18 Special cases in Transportation Problems Learning Objectives: Special cases in Transportation Problems Multiple Optimum Solution Unbalanced Transportation Problem Degeneracy in the Transportation

More information

Scheduling Home Health Care with Separating Benders Cuts in Decision Diagrams

Scheduling Home Health Care with Separating Benders Cuts in Decision Diagrams Scheduling Home Health Care with Separating Benders Cuts in Decision Diagrams André Ciré University of Toronto John Hooker Carnegie Mellon University INFORMS 2014 Home Health Care Home health care delivery

More information

Internet Control Message Protocol (ICMP)

Internet Control Message Protocol (ICMP) SFWR 4C03: Computer Networks & Computer Security Jan 31-Feb 4, 2005 Lecturer: Kartik Krishnan Lecture 13-16 Internet Control Message Protocol (ICMP) The operation of the Internet is closely monitored by

More information

CS311H. Prof: Peter Stone. Department of Computer Science The University of Texas at Austin

CS311H. Prof: Peter Stone. Department of Computer Science The University of Texas at Austin CS311H Prof: Department of Computer Science The University of Texas at Austin Good Morning, Colleagues Good Morning, Colleagues Are there any questions? Logistics Class survey Logistics Class survey Homework

More information

MEASURES OF VARIATION

MEASURES OF VARIATION NORMAL DISTRIBTIONS MEASURES OF VARIATION In statistics, it is important to measure the spread of data. A simple way to measure spread is to find the range. But statisticians want to know if the data are

More information

Decision Mathematics 1 TUESDAY 22 JANUARY 2008

Decision Mathematics 1 TUESDAY 22 JANUARY 2008 ADVANCED SUBSIDIARY GCE 4736/01 MATHEMATICS Decision Mathematics 1 TUESDAY 22 JANUARY 2008 Additional materials: Answer Booklet (8 pages) Graph paper Insert for Questions 3 and 4 List of Formulae (MF1)

More information

6.02 Practice Problems: Routing

6.02 Practice Problems: Routing 1 of 9 6.02 Practice Problems: Routing IMPORTANT: IN ADDITION TO THESE PROBLEMS, PLEASE SOLVE THE PROBLEMS AT THE END OF CHAPTERS 17 AND 18. Problem 1. Consider the following networks: network I (containing

More information

Networks and Paths. The study of networks in mathematics began in the middle 1700 s with a famous puzzle called the Seven Bridges of Konigsburg.

Networks and Paths. The study of networks in mathematics began in the middle 1700 s with a famous puzzle called the Seven Bridges of Konigsburg. ame: Day: etworks and Paths Try This: For each figure,, and, draw a path that traces every line and curve exactly once, without lifting your pencil.... Figures,, and above are examples of ETWORKS. network

More information

. 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9

. 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9 Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a

More information

Decision Mathematics D1 Advanced/Advanced Subsidiary. Tuesday 5 June 2007 Afternoon Time: 1 hour 30 minutes

Decision Mathematics D1 Advanced/Advanced Subsidiary. Tuesday 5 June 2007 Afternoon Time: 1 hour 30 minutes Paper Reference(s) 6689/01 Edexcel GCE Decision Mathematics D1 Advanced/Advanced Subsidiary Tuesday 5 June 2007 Afternoon Time: 1 hour 30 minutes Materials required for examination Nil Items included with

More information

Load balancing Static Load Balancing

Load balancing Static Load Balancing Chapter 7 Load Balancing and Termination Detection Load balancing used to distribute computations fairly across processors in order to obtain the highest possible execution speed. Termination detection

More information

Chinese postman problem

Chinese postman problem PTR hinese postman problem Learning objectives fter studying this chapter, you should be able to: understand the hinese postman problem apply an algorithm to solve the problem understand the importance

More information

PROTOTYPE TO DESIGN A LEASED LINE TELEPHONE NETWORK CONNECTING LOCATIONS TO MINIMIZE THE INSTALLATION COST

PROTOTYPE TO DESIGN A LEASED LINE TELEPHONE NETWORK CONNECTING LOCATIONS TO MINIMIZE THE INSTALLATION COST PROTOTYPE TO DESIGN A LEASED LINE TELEPHONE NETWORK CONNECTING LOCATIONS TO MINIMIZE THE INSTALLATION COST Abstract ARCHANA SHARMA Research Scholar, Computer Science and Enginering Department,, NIMS University,

More information

Chapter 15: Dynamic Programming

Chapter 15: Dynamic Programming Chapter 15: Dynamic Programming Dynamic programming is a general approach to making a sequence of interrelated decisions in an optimum way. While we can describe the general characteristics, the details

More information

Solutions to Homework 6

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

More information

Foundations of Operations Research

Foundations of Operations Research Foundations of Operations Research Master of Science in Computer Engineering Roberto Cordone roberto.cordone@unimi.it Tuesday 13.15-15.15 Thursday 10.15-13.15 http://homes.di.unimi.it/~cordone/courses/2013-for/2013-for.html

More information

Algorithm Design and Analysis

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;

More information

Operations and Supply Chain Management Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras

Operations and Supply Chain Management Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Operations and Supply Chain Management Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Lecture - 36 Location Problems In this lecture, we continue the discussion

More information

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

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) +...

More information

Load Balancing and Termination Detection

Load Balancing and Termination Detection Chapter 7 Load Balancing and Termination Detection 1 Load balancing used to distribute computations fairly across processors in order to obtain the highest possible execution speed. Termination detection

More information

Max Flow. Lecture 4. Optimization on graphs. C25 Optimization Hilary 2013 A. Zisserman. Max-flow & min-cut. The augmented path algorithm

Max Flow. Lecture 4. Optimization on graphs. C25 Optimization Hilary 2013 A. Zisserman. Max-flow & min-cut. The augmented path algorithm Lecture 4 C5 Optimization Hilary 03 A. Zisserman Optimization on graphs Max-flow & min-cut The augmented path algorithm Optimization for binary image graphs Applications Max Flow Given: a weighted directed

More information

CAB TRAVEL TIME PREDICTI - BASED ON HISTORICAL TRIP OBSERVATION

CAB TRAVEL TIME PREDICTI - BASED ON HISTORICAL TRIP OBSERVATION CAB TRAVEL TIME PREDICTI - BASED ON HISTORICAL TRIP OBSERVATION N PROBLEM DEFINITION Opportunity New Booking - Time of Arrival Shortest Route (Distance/Time) Taxi-Passenger Demand Distribution Value Accurate

More information

Seminar. Path planning using Voronoi diagrams and B-Splines. Stefano Martina stefano.martina@stud.unifi.it

Seminar. Path planning using Voronoi diagrams and B-Splines. Stefano Martina stefano.martina@stud.unifi.it Seminar Path planning using Voronoi diagrams and B-Splines Stefano Martina stefano.martina@stud.unifi.it 23 may 2016 This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International

More information

Network (Tree) Topology Inference Based on Prüfer Sequence

Network (Tree) Topology Inference Based on Prüfer Sequence Network (Tree) Topology Inference Based on Prüfer Sequence C. Vanniarajan and Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology Madras Chennai 600036 vanniarajanc@hcl.in,

More information

Graph Theory Problems and Solutions

Graph Theory Problems and Solutions raph Theory Problems and Solutions Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles November, 005 Problems. Prove that the sum of the degrees of the vertices of any finite graph is

More information

Optimization with Big Data: Network Flows

Optimization with Big Data: Network Flows Optimization with Big Data: Network Flows Sertac Karaman Assistant Professor of Aeronautics and Astronautics Laboratory for Information and Decision Systems Institute for Data, Systems, and Society Massachusetts

More information

Chapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling

Chapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling 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

More information

Graph Theory Origin and Seven Bridges of Königsberg -Rhishikesh

Graph Theory Origin and Seven Bridges of Königsberg -Rhishikesh Graph Theory Origin and Seven Bridges of Königsberg -Rhishikesh Graph Theory: Graph theory can be defined as the study of graphs; Graphs are mathematical structures used to model pair-wise relations between

More information

V. Adamchik 1. Graph Theory. Victor Adamchik. Fall of 2005

V. Adamchik 1. Graph Theory. Victor Adamchik. Fall of 2005 V. Adamchik 1 Graph Theory Victor Adamchik Fall of 2005 Plan 1. Basic Vocabulary 2. Regular graph 3. Connectivity 4. Representing Graphs Introduction A.Aho and J.Ulman acknowledge that Fundamentally, computer

More information

Warshall s Algorithm: Transitive Closure

Warshall s Algorithm: Transitive Closure CS 0 Theory of Algorithms / CS 68 Algorithms in Bioinformaticsi Dynamic Programming Part II. Warshall s Algorithm: Transitive Closure Computes the transitive closure of a relation (Alternatively: all paths

More information

A simpler and better derandomization of an approximation algorithm for Single Source Rent-or-Buy

A simpler and better derandomization of an approximation algorithm for Single Source Rent-or-Buy A simpler and better derandomization of an approximation algorithm for Single Source Rent-or-Buy David P. Williamson Anke van Zuylen School of Operations Research and Industrial Engineering, Cornell University,

More information

6.852: Distributed Algorithms Fall, 2009. Class 2

6.852: Distributed Algorithms Fall, 2009. Class 2 .8: Distributed Algorithms Fall, 009 Class Today s plan Leader election in a synchronous ring: Lower bound for comparison-based algorithms. Basic computation in general synchronous networks: Leader election

More information

INTRODUCTION TO MATHEMATICAL MODELLING

INTRODUCTION TO MATHEMATICAL MODELLING 306 MATHEMATICS APPENDIX 2 INTRODUCTION TO MATHEMATICAL MODELLING A2.1 Introduction Right from your earlier classes, you have been solving problems related to the real-world around you. For example, you

More information

WAN Wide Area Networks. Packet Switch Operation. Packet Switches. COMP476 Networked Computer Systems. WANs are made of store and forward switches.

WAN Wide Area Networks. Packet Switch Operation. Packet Switches. COMP476 Networked Computer Systems. WANs are made of store and forward switches. Routing WAN Wide Area Networks WANs are made of store and forward switches. To there and back again COMP476 Networked Computer Systems A packet switch with two types of I/O connectors: one type is used

More information

Level 2 Routing: LAN Bridges and Switches

Level 2 Routing: LAN Bridges and Switches Level 2 Routing: LAN Bridges and Switches Norman Matloff University of California at Davis c 2001, N. Matloff September 6, 2001 1 Overview In a large LAN with consistently heavy traffic, it may make sense

More information

Greatest Common Factor and Least Common Multiple

Greatest Common Factor and Least Common Multiple Greatest Common Factor and Least Common Multiple Intro In order to understand the concepts of Greatest Common Factor (GCF) and Least Common Multiple (LCM), we need to define two key terms: Multiple: Multiples

More information

A new Branch-and-Price Algorithm for the Traveling Tournament Problem (TTP) Column Generation 2008, Aussois, France

A new Branch-and-Price Algorithm for the Traveling Tournament Problem (TTP) Column Generation 2008, Aussois, France A new Branch-and-Price Algorithm for the Traveling Tournament Problem (TTP) Column Generation 2008, Aussois, France Stefan Irnich 1 sirnich@or.rwth-aachen.de RWTH Aachen University Deutsche Post Endowed

More information

Dimitris Chatzidimitriou. Corelab NTUA. June 28, 2013

Dimitris Chatzidimitriou. Corelab NTUA. June 28, 2013 AN EFFICIENT ALGORITHM FOR THE OPTIMAL STABLE MARRIAGE PROBLEM Dimitris Chatzidimitriou Corelab NTUA June 28, 2013 OUTLINE 1 BASIC CONCEPTS 2 DEFINITIONS 3 ROTATIONS 4 THE ALGORITHM DIMITRIS CHATZIDIMITRIOU

More information

Phonics. High Frequency Words P.008. Objective The student will read high frequency words.

Phonics. High Frequency Words P.008. Objective The student will read high frequency words. P.008 Jumping Words Objective The student will read high frequency words. Materials High frequency words (P.HFW.005 - P.HFW.064) Choose target words. Checkerboard and checkers (Activity Master P.008.AM1a

More information

Linear Programming. April 12, 2005

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

More information

Session 7 Bivariate Data and Analysis

Session 7 Bivariate Data and Analysis Session 7 Bivariate Data and Analysis Key Terms for This Session Previously Introduced mean standard deviation New in This Session association bivariate analysis contingency table co-variation least squares

More information

Lecture 16 : Relations and Functions DRAFT

Lecture 16 : Relations and Functions DRAFT CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence

More information

Session 6 Number Theory

Session 6 Number Theory Key Terms in This Session Session 6 Number Theory Previously Introduced counting numbers factor factor tree prime number New in This Session composite number greatest common factor least common multiple

More information

Lecture 10 Scheduling 1

Lecture 10 Scheduling 1 Lecture 10 Scheduling 1 Transportation Models -1- large variety of models due to the many modes of transportation roads railroad shipping airlines as a consequence different type of equipment and resources

More information

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

! 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

More information

Bicolored Shortest Paths in Graphs with Applications to Network Overlay Design

Bicolored Shortest Paths in Graphs with Applications to Network Overlay Design Bicolored Shortest Paths in Graphs with Applications to Network Overlay Design Hongsik Choi and Hyeong-Ah Choi Department of Electrical Engineering and Computer Science George Washington University Washington,

More information

Full and Complete Binary Trees

Full and Complete Binary Trees Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full

More information

Common Multiples. List the multiples of 3. The multiples of 3 are 3 1, 3 2, 3 3, 3 4,...

Common Multiples. List the multiples of 3. The multiples of 3 are 3 1, 3 2, 3 3, 3 4,... .2 Common Multiples.2 OBJECTIVES 1. Find the least common multiple (LCM) of two numbers 2. Find the least common multiple (LCM) of a group of numbers. Compare the size of two fractions In this chapter,

More information

3.1 Solving Systems Using Tables and Graphs

3.1 Solving Systems Using Tables and Graphs Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system

More information

Introduction to LAN/WAN. Network Layer

Introduction to LAN/WAN. Network Layer Introduction to LAN/WAN Network Layer Topics Introduction (5-5.1) Routing (5.2) (The core) Internetworking (5.5) Congestion Control (5.3) Network Layer Design Isues Store-and-Forward Packet Switching Services

More information

Prentice Hall Mathematics: Course 1 2008 Correlated to: Arizona Academic Standards for Mathematics (Grades 6)

Prentice Hall Mathematics: Course 1 2008 Correlated to: Arizona Academic Standards for Mathematics (Grades 6) PO 1. Express fractions as ratios, comparing two whole numbers (e.g., ¾ is equivalent to 3:4 and 3 to 4). Strand 1: Number Sense and Operations Every student should understand and use all concepts and

More information

QoS optimization for an. on-demand transportation system via a fractional linear objective function

QoS optimization for an. on-demand transportation system via a fractional linear objective function QoS optimization for an Load charge ratio on-demand transportation system via a fractional linear objective function Thierry Garaix, University of Avignon (France) Column Generation 2008 QoS optimization

More information

VEHICLE ROUTING PROBLEM

VEHICLE ROUTING PROBLEM VEHICLE ROUTING PROBLEM Readings: E&M 0 Topics: versus TSP Solution methods Decision support systems for Relationship between TSP and Vehicle routing problem () is similar to the Traveling salesman problem

More information

Operation Research. Module 1. Module 2. Unit 1. Unit 2. Unit 3. Unit 1

Operation Research. Module 1. Module 2. Unit 1. Unit 2. Unit 3. Unit 1 Operation Research Module 1 Unit 1 1.1 Origin of Operations Research 1.2 Concept and Definition of OR 1.3 Characteristics of OR 1.4 Applications of OR 1.5 Phases of OR Unit 2 2.1 Introduction to Linear

More information