# The TSP is a special case of VRP, which means VRP is NP-hard.

Save this PDF as:

Size: px
Start display at page:

Download "The TSP is a special case of VRP, which means VRP is NP-hard."

## Transcription

1 49 6. ROUTING PROBLEMS 6.1. VEHICLE ROUTING PROBLEMS Vehicle Routing Problem, VRP: Customers i=1,...,n with demands of a product must be served using a fleet of vehicles for the deliveries. The vehicles, with given maximum capacities, are situated at a central depot (or several depots) to which they must return. Determine a routing schedule that minimizes the total cost of the deliveries. The TSP is a special case of VRP, which means VRP is NP-hard. Graph formulation for a single depot problem: Input: Graph G=(V,E), with costs (or distances) c ij on the edges and demands d i on the vertices, V =n, m vehicles with capacities K, j=1,...,m. A central depot is at vertex 0. j Problem: Determine cycles R,..., R for the vehicles that start from vertex 0 and service all 1 m vertices such that the load of vehicle j doesn't exceed its capacity and the total cost of the cycles is minimized. A route means a sequence of customers for each vehicle. Variations and modifications: - Service demand for each edge instead of vertices ÿ multiple postmen problem. - Number of vehicles may be a decision variable, instead of a fixed value. Initially there is an unlimited number of vehicles available. Number of vehicles or total cost to be minimzed. - The graph may be directed or undirected. - Pick-up problem: goods collected from the customers. Combination: customers may have either demand (import) or take away (export) needs. - Multiple products, specialized vehicles for some products. - Due dates / deadlines for deliveries. - Travel time / cost of a vehicle may be constrained. Applications: - distribution plan for a wholesale dealer - garbage disposal - mail delivery, mailbox collection - security company's rounds - elevator maintenance - school bus routing - airline schedules - snow plows Also variable costs may be included in the cost function. These cost are proportional to the amount of load. These probelms belong to logistics and transportation, not considered here. EXACT METHODS Because the problem is "harder" than the TSP, exact methods are suitable for small instances only (number of customers < 100). Versions of Branch and Bound, Branch and Cut and Dynamic Programming algorithm have been tried. The Integer Programming formulation can be based on - partition of routes, or - construction of routes for vehicles (cf. TSP models), or - product flow.

2 50 HEURISTICS Many heuristics use similar principles as the TSP heuristics, keeping an eye on the capacity constraints. Most methods can be classified into the following categories: Constructive methods: tours are built up by adding nodes to partial tours or combining subtours, regarding capacities and costs. Two-phase methods: consist of 1) clustering of vertices and 2) route construction. The order of these phases may be "clustering first, route later" or "route first, clustering later". We introduce two methods: the Clarke and Wright method is an example of the first, the Sweeping heuristic of the second category. Assume an complete, undirected graph with symmetric costs and an unlimited number of identical vehicles with capacity K. METHOD OF CLARKE AND WRIGHT 1. Starting solution: each of the n vehicles serves one customer. 2. For all pairs of nodes i,j, i j, calculate the savings for joining the cycles using egde [i,j]: s ij = c 0i + c 0j - c ij. 3. Sort the savings in decreasing order. 4. Take edge [i,j] from the top of the savings list. Join two separate cycles with edge [i,j], if (i) the nodes belong to separate cycles (ii) the maximum capacity of the vehicle is not exceeded (iii) i and j are first or last customer on their cycles. 5. Repeat 4 until the savings list is handled or the capacities don't allow more merging. The cycles if i and j are NOT united in sep 4, if the nodes belong to the same cycle OR the capacity is exceeded OR either node is an interior node of the cycle. Improvement: Optimize the tour of each vehicle with a TSP heuristic Example 6.1. Given a complete graph with a central depot 0 and 9 customers. (The costs are not directly proportional to the euclidean distances of the diagram.)

3 51 Symmetric costs: c ij Demands: i d i Capacity of a vehicle: K = 40 Solution with Clarke & Wright: Savings s ij = c 0i + c 0j - c ij Symmetry: s ij = sji s ij Ordered savings: [4,5], [1,2], [1,3], [1,4], [2,6], [5,7], [4,7], [1,5], [3,4], [2,3], [7,9], [3,5], [8,9], [1,6], [5,9], [6,8], [2,4],... Initial solution: cycles 0-1-0, 0-2-0,...,

4 Edge [4,5]: Join cycles and 0-5-0: result , load d + d = 20 < K. 4 5 Edge [1,2]: Join and 0-2-0: result , load d + d = 25 < K. 1 2 Edge [1,3]: Capacity limit: d +d +d = 43 > K Edge [1,4]: Capacity limit: d +d +d = 42 > K Edge [2,6]: Join cycles and 0-6-0: result , load d+d+d = 30 < K Edge [5,7]: Join cycles and 0-7-0: result , load d+d+d = 29 < K Edge [4,7]: Condition 4(i) doesn't hold: nodes belong to same cycle. Edge [1,5]: Condition 4(iii) doesn't hold: node 5 is an interior node of its route. Edge [3,4]: Capacity limit: d +d +d +d = 47 > K Edge [2,3]: Condition 4(iii) doesn't hold: node 2 is an interior node. Edge [7,9]: Join cycles and 0-9-0: result , load d +d +d +d = 35 < K Edge [3,5]: Condition 4(iii) doesn't hold: node 5 is an interior node. Edge [8,9]: Join cycles and 0-8-0: result , load d +d +d +d +d = 39 < K Edge [1,6]: Condition 4(i) doesn't hold: nodes belong to same cycle. Edge [5,9]: Condition 4(i) doesn't hold: nodes belong to same cycle. Edge [6,8]: Capacity limit: (d +d +d )+(d +d +d +d +d ) = 69 > K Result: three routes that cannot be united because of capacity limit. Solution: Route: Load: Cost: Total cost f = SWEEP HEURISTIC This heuristic is of the type "clustering first, route later". Assume the customers are points in a plane with euclidean distances as costs. The distance between (x, y ) and (x,y ) is i i j j. 1. Compute the polar coordinates of each customer with respect to the depot. Sort the customers by increasing polar angle. 2. Add loads to the first vehicle from the top of the list as long as the capacity allows. Continue with the next vehicle until all customers are included. Now the customers have been clustered by vehicles. 3. For each vehicle, optimize its route by a suitable TSP method. Graphically, in step 1 we rotate a ray centered at the depot.the starting angle can be chosen arbitrarily. Source: Comment: Also for the TSP, clustering techniques can be used for splitting large scale TSP instances to smaller parts.

5 THE POSTMAN PROBLEM In a weighted graph, find a minimum weight closed walk that includes every edge at least once. A graph is an Eulerian graph if it has an Eulerian cycle (= Eulerian circuit). Graph G has an Eulerian cycle if and only if every vertex has an even degree. An Eulerian cycle, when it exists, is an optimum postman's tour, because every edge is traversed exactly once. An Eulerian cycle can be found with the following method: EULERIAN CYCLE 1. Choose a starting node i:=i 0 with a degree \$2. 2. If an untraversed edge [i,j] incident to i exists, go from i to j (edge [i,j] is added to the path). Set i:=j and repeat step 2. If all edges from i are traversed, i=i 0 and a cycle is formed. 3. Remove edges of the previous cycle. If untraversed edges remain, go to Connect the cycles at shared nodes. The resulting walk is an Eulerian cycle. For instance, a depth-first search can be used when choosing the next vertex. Different choices may produce different solutions because there are more than one Eulerian cycles in a almost all graphs. Example 6.2. Find an Eulerian cycle in the following graph. An Eulerian cycle exists, because every vertex has an even degree. Start with node 1. First cycle: Choose starting node for the next cycle: 2 Second cycle: Shared nodes 2 and 4. Choose connection at node 2. Insert the second cycle between 2-3: is an Eulerian cycle. Assume a connected, undirected graph G=(V,E) with nonnegative weights c on the edges. The ij following algorithm uses as a subroutine a shortest path algorithm and a minimum weight perfect matching algorithm. Both of these, and the Eulerian cycle algorithm, are polynomial and the resulting postman's algorithm is also polynomial. POSTMAN ALGORITHM 1. Identify odd-degree vertices, denote this set by V. If none exists, go to 5. O 2. Find a shortest path between all pairs of odd degree vertices i,j 0 V O 3. Form a complete graph K for V Owith weights w ij= length of the shortest path determined in step Find a minimum weigth perfect matching in K. Duplicate the implied edges of the matching (the shortest paths) in G. This augmented multigraph, call it G, is an Eulerian graph. E 5. Form an Eulerian cycle in G. This is the optimum postman's tour. E

6 54 In the augmented graph all edges have an even degree and an Eulerian cycle can be formed. This algorithm finds a minimum length duplication of edges that makes an Eulerian cycle possible. Example 6.3. Find a postman's route in the following graph G. Vertices of odd degree: V = {1,2,4,5}. O Lengths of shortest paths shown in graph K. Minimum weight perfect matching: {[1,5], [2,4]} Add edges of the shortest paths and to G. Augmented graph G : E Construct an Eulerian cycle in G e.g. with following choices: E First cycle: Second cycle: Connect the cycles at node 1 to form an Eulerian cycle: This is the optimum postman's tour, with cost 93. There are alternative optimum tours: in all of these the edges 1-6, 6-5, 2-3, 3-4 are are traversed twice and the cost is 93. The method can be modified for directed graphs. Unfortunately, there is no polynomial time exact algorithm for mixed graphs, because that problem is NP-hard.

### 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

### Research Paper Business Analytics. Applications for the Vehicle Routing Problem. Jelmer Blok

Research Paper Business Analytics Applications for the Vehicle Routing Problem Jelmer Blok Applications for the Vehicle Routing Problem Jelmer Blok Research Paper Vrije Universiteit Amsterdam Faculteit

### 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

### 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

### Outline. NP-completeness. When is a problem easy? When is a problem hard? Today. Euler Circuits

Outline NP-completeness Examples of Easy vs. Hard problems Euler circuit vs. Hamiltonian circuit Shortest Path vs. Longest Path 2-pairs sum vs. general Subset Sum Reducing one problem to another Clique

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

### 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

### ! 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

### Computer Algorithms. NP-Complete Problems. CISC 4080 Yanjun Li

Computer Algorithms NP-Complete Problems NP-completeness The quest for efficient algorithms is about finding clever ways to bypass the process of exhaustive search, using clues from the input in order

### 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)

### 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

### Complexity Theory. IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar

Complexity Theory IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar Outline Goals Computation of Problems Concepts and Definitions Complexity Classes and Problems Polynomial Time Reductions Examples

### 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

### ! 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

### Vehicle Routing and Scheduling. Martin Savelsbergh The Logistics Institute Georgia Institute of Technology

Vehicle Routing and Scheduling Martin Savelsbergh The Logistics Institute Georgia Institute of Technology Vehicle Routing and Scheduling Part I: Basic Models and Algorithms Introduction Freight routing

### 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

### 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

### Homework Exam 1, Geometric Algorithms, 2016

Homework Exam 1, Geometric Algorithms, 2016 1. (3 points) Let P be a convex polyhedron in 3-dimensional space. The boundary of P is represented as a DCEL, storing the incidence relationships between the

### OFTECHNOLO..1_GY: X; 0 t;0~~~~~~~~~~~~~~ : ;. i : wokn. f t,0l'' f:fad 02 'L'S0 00';0000f ::: 000';0' :-:::

... -. -. _,.,...,..-. - - - - soft,, LI,,, '-'J''

### 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.

### Fleet Management Optimisation

Fleet Management Optimisation Sindre Soltun Master of Science in Communication Technology Submission date: January 2007 Supervisor: Steinar Andresen, ITEM Co-supervisor: Per Stein, Nordisk Mobiltelefon

### Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph

### Freight Sequencing to Improve Hub Operations in the Less-Than-Truckload Freight Transportation Industry

Freight Sequencing to Improve Hub Operations in the Less-Than-Truckload Freight Transportation Industry Xiangshang Tong, Ph.D. Menlo Worldwide Logistics Kimberly Ellis, Ph.D. Virginia Tech Amy Brown Greer

### 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

### NP-Completeness. CptS 223 Advanced Data Structures. Larry Holder School of Electrical Engineering and Computer Science Washington State University

NP-Completeness CptS 223 Advanced Data Structures Larry Holder School of Electrical Engineering and Computer Science Washington State University 1 Hard Graph Problems Hard means no known solutions with

### Approximation Algorithms. Scheduling. Approximation algorithms. Scheduling jobs on a single machine

Approximation algorithms Approximation Algorithms Fast. Cheap. Reliable. Choose two. NP-hard problems: choose 2 of optimal polynomial time all instances Approximation algorithms. Trade-off between time

### 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

### Charles Fleurent Director - Optimization algorithms

Software Tools for Transit Scheduling and Routing at GIRO Charles Fleurent Director - Optimization algorithms Objectives Provide an overview of software tools and optimization algorithms offered by GIRO

### THE PROBLEM WORMS (1) WORMS (2) THE PROBLEM OF WORM PROPAGATION/PREVENTION THE MINIMUM VERTEX COVER PROBLEM

1 THE PROBLEM OF WORM PROPAGATION/PREVENTION I.E. THE MINIMUM VERTEX COVER PROBLEM Prof. Tiziana Calamoneri Network Algorithms A.y. 2014/15 2 THE PROBLEM WORMS (1)! A computer worm is a standalone malware

### Routing Problems. Viswanath Nagarajan R. Ravi. Abstract. We study the distance constrained vehicle routing problem (DVRP) [20, 21]: given a set

Approximation Algorithms for Distance Constrained Vehicle Routing Problems Viswanath Nagarajan R. Ravi Abstract We study the distance constrained vehicle routing problem (DVRP) [20, 21]: given a set of

### 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

### Strategic Planning and Vehicle Routing Algorithm for Newspaper Delivery Problem: Case study of Morning Newspaper, Bangkok, Thailand

Strategic Planning and Vehicle Routing Algorithm for Newspaper Delivery Problem: Case study of Morning Newspaper, Bangkok, Thailand Arunya Boonkleaw, Nanthi Sutharnnarunai, PhD., Rawinkhan Srinon, PhD.

### 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

### Two objective functions for a real life Split Delivery Vehicle Routing Problem

International Conference on Industrial Engineering and Systems Management IESM 2011 May 25 - May 27 METZ - FRANCE Two objective functions for a real life Split Delivery Vehicle Routing Problem Marc Uldry

### 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

### 11. APPROXIMATION ALGORITHMS

11. APPROXIMATION ALGORITHMS load balancing center selection pricing method: vertex cover LP rounding: vertex cover generalized load balancing knapsack problem Lecture slides by Kevin Wayne Copyright 2005

### MATHEMATICAL ENGINEERING TECHNICAL REPORTS. An Improved Approximation Algorithm for the Traveling Tournament Problem

MATHEMATICAL ENGINEERING TECHNICAL REPORTS An Improved Approximation Algorithm for the Traveling Tournament Problem Daisuke YAMAGUCHI, Shinji IMAHORI, Ryuhei MIYASHIRO, Tomomi MATSUI METR 2009 42 September

### Chapter 1. Introduction

Chapter 1 Introduction Intermodal freight transportation describes the movement of goods in standardized loading units (e.g., containers) by at least two transportation modes (rail, maritime, and road)

### Transportation Logistics

Transportation Logistics Lecture held by Gilbert Laporte Winter Term 2004 (University of Vienna) Summarized by Guenther Fuellerer Contents 1 Travelling Sales Person Problem (TSP) 5 1.1 Applications of

### Euler Paths and Euler Circuits

Euler Paths and Euler Circuits An Euler path is a path that uses every edge of a graph exactly once. An Euler circuit is a circuit that uses every edge of a graph exactly once. An Euler path starts and

### 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

### Cost Models for Vehicle Routing Problems. 8850 Stanford Boulevard, Suite 260 R. H. Smith School of Business

0-7695-1435-9/02 \$17.00 (c) 2002 IEEE 1 Cost Models for Vehicle Routing Problems John Sniezek Lawerence Bodin RouteSmart Technologies Decision and Information Technologies 8850 Stanford Boulevard, Suite

### 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

### Graph Theory and Complex Networks: An Introduction. Chapter 06: Network analysis

Graph Theory and Complex Networks: An Introduction Maarten van Steen VU Amsterdam, Dept. Computer Science Room R4.0, steen@cs.vu.nl Chapter 06: Network analysis Version: April 8, 04 / 3 Contents Chapter

### Graph theory and network analysis. Devika Subramanian Comp 140 Fall 2008

Graph theory and network analysis Devika Subramanian Comp 140 Fall 2008 1 The bridges of Konigsburg Source: Wikipedia The city of Königsberg in Prussia was set on both sides of the Pregel River, and included

### Scheduling. Open shop, job shop, flow shop scheduling. Related problems. Open shop, job shop, flow shop scheduling

Scheduling Basic scheduling problems: open shop, job shop, flow job The disjunctive graph representation Algorithms for solving the job shop problem Computational complexity of the job shop problem Open

### 4. How many integers between 2004 and 4002 are perfect squares?

5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started

### 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

### Chapter 6 TRAVELLING SALESMAN PROBLEM

98 hapter 6 TRVLLING SLSMN PROLM 6.1 Introduction The Traveling Salesman Problem (TSP) is a problem in combinatorial optimization studied in operations research and theoretical computer science. Given

### A Scatter Search Algorithm for the Split Delivery Vehicle Routing Problem

A Scatter Search Algorithm for the Split Delivery Vehicle Routing Problem Campos,V., Corberán, A., Mota, E. Dep. Estadística i Investigació Operativa. Universitat de València. Spain Corresponding author:

### Divide-and-Conquer. Three main steps : 1. divide; 2. conquer; 3. merge.

Divide-and-Conquer Three main steps : 1. divide; 2. conquer; 3. merge. 1 Let I denote the (sub)problem instance and S be its solution. The divide-and-conquer strategy can be described as follows. Procedure

### Solving the Travelling Salesman Problem Using the Ant Colony Optimization

Ivan Brezina Jr. Zuzana Čičková Solving the Travelling Salesman Problem Using the Ant Colony Optimization Article Info:, Vol. 6 (2011), No. 4, pp. 010-014 Received 12 July 2010 Accepted 23 September 2011

### 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

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

### 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

### Graph Theory and Complex Networks: An Introduction. Chapter 06: Network analysis. Contents. Introduction. Maarten van Steen. Version: April 28, 2014

Graph Theory and Complex Networks: An Introduction Maarten van Steen VU Amsterdam, Dept. Computer Science Room R.0, steen@cs.vu.nl Chapter 0: Version: April 8, 0 / Contents Chapter Description 0: Introduction

### P13 Route Plan. E216 Distribution &Transportation

P13 Route Plan Vehicle Routing Problem (VRP) Principles of Good Routing Technologies to enhance Vehicle Routing Real-Life Application of Vehicle Routing E216 Distribution &Transportation Vehicle Routing

### CAD Algorithms. P and NP

CAD Algorithms The Classes P and NP Mohammad Tehranipoor ECE Department 6 September 2010 1 P and NP P and NP are two families of problems. P is a class which contains all of the problems we solve using

### JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004

Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February

### 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

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

### A Sublinear Bipartiteness Tester for Bounded Degree Graphs

A Sublinear Bipartiteness Tester for Bounded Degree Graphs Oded Goldreich Dana Ron February 5, 1998 Abstract We present a sublinear-time algorithm for testing whether a bounded degree graph is bipartite

### 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

### 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;

### Class One: Degree Sequences

Class One: Degree Sequences For our purposes a graph is a just a bunch of points, called vertices, together with lines or curves, called edges, joining certain pairs of vertices. Three small examples of

### Sport Timetabling. Outline DM87 SCHEDULING, TIMETABLING AND ROUTING. Outline. Lecture 15. 1. Problem Definitions

Outline DM87 SCHEDULING, TIMETABLING AND ROUTING Lecture 15 Sport Timetabling 1. Problem Definitions Marco Chiarandini DM87 Scheduling, Timetabling and Routing 2 Problems we treat: single and double round-robin

### The Clar Structure of Fullerenes

The Clar Structure of Fullerenes Liz Hartung Massachusetts College of Liberal Arts June 12, 2013 Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 1 / 25

### New Exact Solution Approaches for the Split Delivery Vehicle Routing Problem

New Exact Solution Approaches for the Split Delivery Vehicle Routing Problem Gizem Ozbaygin, Oya Karasan and Hande Yaman Department of Industrial Engineering, Bilkent University, Ankara, Turkey ozbaygin,

### Reductions & NP-completeness as part of Foundations of Computer Science undergraduate course

Reductions & NP-completeness as part of Foundations of Computer Science undergraduate course Alex Angelopoulos, NTUA January 22, 2015 Outline Alex Angelopoulos (NTUA) FoCS: Reductions & NP-completeness-

### Answers to some of the exercises.

Answers to some of the exercises. Chapter 2. Ex.2.1 (a) There are several ways to do this. Here is one possibility. The idea is to apply the k-center algorithm first to D and then for each center in D

### Asking Hard Graph Questions. Paul Burkhardt. February 3, 2014

Beyond Watson: Predictive Analytics and Big Data U.S. National Security Agency Research Directorate - R6 Technical Report February 3, 2014 300 years before Watson there was Euler! The first (Jeopardy!)

### VEHICLE ROUTING AND SCHEDULING PROBLEMS: A CASE STUDY OF FOOD DISTRIBUTION IN GREATER BANGKOK. Kuladej Panapinun and Peerayuth Charnsethikul.

1 VEHICLE ROUTING AND SCHEDULING PROBLEMS: A CASE STUDY OF FOOD DISTRIBUTION IN GREATER BANGKOK By Kuladej Panapinun and Peerayuth Charnsethikul Abstract Vehicle routing problem (VRP) and its extension

### Exploring the use of hyper-heuristics to solve a dynamic bike-sharing problem

Exploring the use of hyper-heuristics to solve a dynamic bike-sharing problem Emily Myers, Ghofran Shaker July 21, 2015 Abstract In a self-service bike-share system we have a finite number of stations

### 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

### Home Page. Data Structures. Title Page. Page 1 of 24. Go Back. Full Screen. Close. Quit

Data Structures Page 1 of 24 A.1. Arrays (Vectors) n-element vector start address + ielementsize 0 +1 +2 +3 +4... +n-1 start address continuous memory block static, if size is known at compile time dynamic,

### Near Optimal Solutions

Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NP-Complete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.

### A CENTROID-BASED HEURISTIC ALGORITHM FOR THE CAPACITATED VEHICLE ROUTING PROBLEM

Computing and Informatics, Vol. 30, 2011, 721 732 A CENTROID-BASED HEURISTIC ALGORITHM FOR THE CAPACITATED VEHICLE ROUTING PROBLEM Kwangcheol Shin Department of Computer Science Korea Advanced Institute

### 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

### 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

### Compact Representations and Approximations for Compuation in Games

Compact Representations and Approximations for Compuation in Games Kevin Swersky April 23, 2008 Abstract Compact representations have recently been developed as a way of both encoding the strategic interactions

### 12 Abstract Data Types

12 Abstract Data Types 12.1 Source: Foundations of Computer Science Cengage Learning Objectives After studying this chapter, the student should be able to: Define the concept of an abstract data type (ADT).

### 8.1 Min Degree Spanning Tree

CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree

### Part 2: Community Detection

Chapter 8: Graph Data Part 2: Community Detection Based on Leskovec, Rajaraman, Ullman 2014: Mining of Massive Datasets Big Data Management and Analytics Outline Community Detection - Social networks -

### 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

### Coordination in vehicle routing

Coordination in vehicle routing Catherine Rivers Mathematics Massey University New Zealand 00.@compuserve.com Abstract A coordination point is a place that exists in space and time for the transfer of

### Distributed Computing over Communication Networks: Maximal Independent Set

Distributed Computing over Communication Networks: Maximal Independent Set What is a MIS? MIS An independent set (IS) of an undirected graph is a subset U of nodes such that no two nodes in U are adjacent.

### Extremal Wiener Index of Trees with All Degrees Odd

MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 70 (2013) 287-292 ISSN 0340-6253 Extremal Wiener Index of Trees with All Degrees Odd Hong Lin School of

### 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,

### 2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

### Graphical degree sequences and realizations

swap Graphical and realizations Péter L. Erdös Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences MAPCON 12 MPIPKS - Dresden, May 15, 2012 swap Graphical and realizations Péter L. Erdös

### 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

### Solving the Vehicle Routing Problem with Multiple Trips by Adaptive Memory Programming

Solving the Vehicle Routing Problem with Multiple Trips by Adaptive Memory Programming Alfredo Olivera and Omar Viera Universidad de la República Montevideo, Uruguay ICIL 05, Montevideo, Uruguay, February

### 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

### Constraint Satisfaction Problems. Constraint Satisfaction Problems. Greedy Local Search. Example. 1 Greedy algorithm. 2 Stochastic Greedy Local Search

Constraint Satisfaction Problems June 19, 2007 Greedy Local Search Constraint Satisfaction Problems Greedy Local Search Bernhard Nebel, Julien Hué, and Stefan Wölfl Albert-Ludwigs-Universität Freiburg

### 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,

### Residential waste management in South Africa: Optimisation of vehicle fleet size and composition

Residential waste management in South Africa: Optimisation of vehicle fleet size and composition by Wouter H Bothma 26001447 Submitted in partial fulfillment of the requirements for the degree of Bachelors

### A cluster-based optimization approach for the multi-depot heterogeneous fleet vehicle routing problem with time windows

European Journal of Operational Research 176 (2007) 1478 1507 Discrete Optimization A cluster-based optimization approach for the multi-depot heterogeneous fleet vehicle routing problem with time windows

### MATHEMATICS Unit Decision 1

General Certificate of Education January 2007 Advanced Subsidiary Examination MATHEMATICS Unit Decision 1 MD01 Tuesday 16 January 2007 9.00 am to 10.30 am For this paper you must have: an 8-page answer