5 Scheduling. Operations Planning and Control


 Corey Walker
 2 years ago
 Views:
Transcription
1 5 Scheduling Operations Planning and Control
2 Some Background Machines (resources) are Machines process jobs (molding machine, x ray machine, server in a restaurant, computer ) Machine Environment Single Machine Parallel Machines (identical vs. different) Flow Shops: different machines (e.g. assembly lines) Each job must be processed by each machine exactly once All jobs have the same routing A job cannot begin processing on the second machine until it has completed processing on the first Job Shops Each job may have its own routing Open Shops (e.g. car repair shop) Jobs have no specific routing 2
3 Scheduling Algorithms Instance: particular set of data for the model Exact algorithm: Optimum solution for every instance why Heuristic algorithm: a good solution, we hope, optimal or close to optimal for every instance 3
4 Measures Completion time of job i, Ci Flow time of Job i: Fi=Ci ri, ri as release date Lateness of job i: Li=Ci di, di as due date Tardiness of job i: Ti = max {0;Li}, Li>0 Earliness of job i: Ei=max{0, Li} Number of tardy jobs: Ni 4
5 Some Application Gantt Charts Developed by Henry Gantt (1911) Consider the following four job, three machine jobshop scheduling problem Assume the following sequences: on M on M on M3 5
6 Some Application Gantt Charts (cont.): Example The Gantt Chart oriented towards machines is: Last operation of job 1 is on machine 3 and is completed at time 14. So, completion time, Ci : C 1 = 14 (on machine 3) Also: C 2 = 11 (on machine 3); C 3 = 13 (on machine 1) and C 4 = 10 (on machine 1) 6
7 Some Background Gantt Charts (cont.): Example The makespan is: Since F i = C i r i but r i = 0 in this example for all i, then flowtime and completion time are the same. Total flowtime is: Min 7
8 Some Background Gantt Charts (cont.): Example The lateness and tardiness of a job The total lateness is Number of tardy jobs is (δ i = 1) The total tardiness is The maximum tardiness is 8
9 Common Sequencing Rules FCFS. First Come First Served. Jobs processed in the order they come to the shop SPT. Shortest Processing Time. Jobs with the shortest processing time are scheduled first EDD. Earliest Due Date. Jobs are sequenced according to their due dates CR. Critical Ratio. Schedule the next job with the shortest CR value. Compute the ratio of remaining time until the due date and processing time of the job
10 Common Sequencing Rules A machining center in a job shop for a local fabrication company has 5 unprocessed jobs (1 to 5). Given the processing times and due dates, apply the sequencing rules to determine scheduling options Mean Flow time, Average Tardiness and # tardy jobs Job Number Processing Time Due Date
11 Single Machine Scheduling Introduction Applicability Single Machine Aggregated Machines Bottleneck Machines Methods Simple Methods Target on performance measures Optimization procedures (heuristic, optimal) Sequence dependent setup times Static vs. Dynamic scheduling 11
12 Single Machine Scheduling Minimizing Flow time What if your in process inventory costs dominate? Minimize total flow time tends to minimize total holding costs Example Proposed sequence: Total flowtime (F)=? F = p 1 + (p 1 +p 2 ) + (p 1 +p 2 +p 3 )+...+(p 1 +p p n ) F = n p 1 + (n 1) p p n For this problem F = High value P and delivers asap
13 Single Machine Scheduling Minimizing Flow time Shortest Processing Time (SPT) Sequence of jobs ordered from the smallest to largest processing times Is this optimal? Theorem. SPT sequencing minimizes total flowtime on a single machine with zero release times. Proof. We assume an optimal schedule is not an SPT sequence. S is optimal sequence (assumed) with iand then j S is a schedule where j comes before I The set of jobs B comes before i/j or j/i and A comes after p i > p j TF(S) = TF(B) + (t+p i ) + (t+p i +p j ) + TF(A) TF(S ) = TF(B) + (t+ p j ) + (t+ p j +p i ) + TF(A) TF(S) TF(S )= p i p j > 0» t is the completion of the last job in B, TF(A) and TF(B) are total flowtimes of jobs in A and B 13
14 Single Machine Scheduling Minimizing Flow time Example (cont.) Optimal Schedule is Completion times C 1 = 11, C 2 = 2, C 3 = 7, C 4 = 4 and C 5 =15 Total Flow time = Total Completion Time = 39 Remarks on the SPT Rule Minimizes total time jobs spend in the system (because all release times are 0) Minimizes the mean number of jobs waiting to be processed (mean work in progress) Also minimizes Total Lateness (Li=Ci di). Why? 14 Professor penalty Parkinson Law
15 Single Machine Scheduling Maximal Tardiness and Maximal Lateness Due date oriented measure Earliest due date sequence (EDD) Minimizes the Maximal Tardiness (T max ), T max =max{0;li} Minimizes the Maximal Lateness (L max ), Li=Ci di Example EDD sequence is Tardiness of the jobs is (0, 0, 2, 1, 0) 15 Customer satisfaction Minmax
16 Single Machine Scheduling Number of Tardy Jobs EDD may have several jobs somewhat tardy If the fixed cost component of jobs being tardy dominates we wish to have the most of them on time Moore s Algorithm Step1. Compute the tardiness for each job in the EDD sequence. Set N T =0, and let k be the first position containing a tardy job. If no job is tardy go to step 4. Step 2. Find the job with the largest processing time in positions 1 to k. Step 3. Remove job j* from the sequence, set N T =N T+1, and repeat Step1. Step 4. Place the removed N T jobs in any order at the end of the sequence. This sequence minimizes the number of tardy jobs 16
17 Single Machine Scheduling Number of Tardy Jobs Example EDD sequence Step 1: The tardiness is (0, 0, 2, 1, 0) Job 4 in the third position is the first tardy job; Step 2: The processing times for jobs 5, 3 and 4 are 4, 3, 2, respectively; largest processing time for job 5 Step 3: Remove job 5, go to step 1 Step 1: EDD sequence is ; completion times (3, 5, 7, 11) and tardiness (0, 0, 0, 0) Go to step 4. Step 4: schedule that minimizes the number of tardy jobs is / and has 1 tardy job: Jobs 5 17
18 Single Machine Scheduling Precedence Constraints: Lawler s Algorithm Objective function g i is a non decreasing function of the flow time F i Examples = minimizing maximum lateness ;0 minimizing maximum tardiness 18
19 Single Machine Scheduling Precedence Constraints: Lawler s Algorithm Concept Back scheduling At each step determine the set of jobs V not required to precede any other Among V choose job k that satisfies and corresponds to the processing time of the current sequence Job k is scheduled last Determine again V and is reduced by 19
20 Lawler s Algorithm Example An automotive painting and repair has 6 cars waiting to repairs. Three (1,2,3) from car rental and he agreed to finish these cars based on the due dates. Cars 4,5,6 from a retailer dealer, he agreed that car 4 be completed first (customer is waiting). The processing times and due dates are available for each job. How should be the schedule to minimize the maximum tardiness? Job Pi Due Date
21 Single Machine Scheduling Minimizing Set up Times Sequence dependent set up times The time to change from one product to another may be significant and may depend on the previous part produced p ij = time to process job j if it immediately follows job i Examples: Electronics industry Paint shops Injection molding Minimizes makespan, since it also considers set up Problem is equivalent to the traveling salesman problem (TSP) Plastic Tops 21
22 Single Machine Scheduling Minimizing Set up Times SST = Shortest Set up Time Heuristic A metal products manufacturer has contracted to ship metal braces each day for four customers. Each brace requires a different set up on the rolling mill: *Job C cannot follow job D, because of quality problems SST heuristic:» Step 1 starting arbitrarily by choosing one Job: A» Step 2 B has the smallest set up time following A; A B» Step 3 C has the smallest set up time of all the remaining jobs following B; A B C» Step 4 D is the last remaining job; A B C D A with a makespan of =14 Starting point 22
23 Single Machine Scheduling Minimizing Set up Times A regret based algorithm A regret is a penalty for a decision that was not made Each job must be included once: at least one element from each row Pick the smallest element in each row and their sum is the lower bound on makespan Reduced matrix Row reduction Column reduction Sum of reduced coef = lower bound Find the reduced matrix! Has a 0 in each column and each row 23
24 Single Machine Scheduling Minimizing Set up Times A regret based algorithm (cont.) The reduced matrix is 1 2 If job B does not follow (come after) job A, some other job must follow A» C adds no set up time (is already in the lower bound) Some job must precede B: D has 0 set up» Thus, we have 0 regret not to chose B to follow A A B, A?;? B; 24
25 Single Machine Scheduling Minimizing Set up Times A regret based algorithm (cont.) The reduced matrix is (cont.) 1 2 Zero cell C D» If C does not precede D then we must select other job to precede D (? D)» A can precede D : A D with a regret of 1 time unit» C? : C A with zero regret time unit» Select the jobs pair based on the highest regret value 25
26 Single Machine Scheduling Minimizing Set up Times A regret based algorithm (cont.) Find the cycle sequence that minimizes the set up time Data p ij element of the set up time matrix (even if reduced) R ij regret for element ij, where p ij = 0 C max makespan of the partial sequence L iteration n jobs 26
27 Single Machine Scheduling Minimizing Set up Times A regret based algorithm (cont.) Step 0: C(max) = 0 and L = 1 Step 1: Reduce the Matrix 3 27
28 Single Machine Scheduling Minimizing Set up Times A regret based algorithm (cont.) Step 2: Calculate the regret 3 Step 3 Choose the largest regret : 17 Step 4 Assign a job pair: Job 2 immediately follows job 5 (5 2) L = 1+1; We prohibit
29 Single Machine Scheduling Minimizing Set up Times A regret based algorithm (cont.) New Matrix Step 1: reduce the matrix C(max) = = 24 29
30 Single Machine Scheduling Minimizing Set up Times A regret based algorithm (cont.) Step 2 Calculate the regret Step 3 Chose the largest regret: 9 Step 4 Assign a job pair: 3 1 Prohibit 1 3 Step 1 Reduce the matrix: not possible 30
31 Single Machine Scheduling Minimizing Set up Times A regret based algorithm (cont.) Step 2 Calculate regret Step 3 Choose the largest regret: 3 Step 4 Assign job pair : 1 4; Partial sequence: 5 2, Prohibit 4 1 and 4 3 (to keep from being chosen) Final Matrix Choose 2 3 and 4 5 > sequence The total set up time is 24 31
32 Parallel Machines Introduction A job can be processed in any of the machines The time to process a job is the same on any machine A job consists of a single operation Decision Which machine processes the job? In what order? 32
33 Parallel Machines Introduction List Schedule Sequence of all jobs Assign the next job on the list to the machine with the smallest amount of work assigned Procedure Step 0. Let H i =0, i=1,2,...,m be the assigned workload on machine i, L=([1],[2],...,[n]) the ordered list sequence, C j =0, j=1,2,...,n, and k=1 Step 1. Let j*= L k and H i* =mini =1,m {Hi}; Assign job j* to be processed on machine i*, C j* =H i* +p j*,hi*=h i* +p j* Step 2. Set k=k+1, if k>n, stop. Otherwise go to step 1. 33
34 Parallel Machines Flow Time Consider a facility with 3 identical machines and 15 jobs that need to be done as soon as possible; Processing times(after SPT): 34
35 Parallel Machines Makespan Use the longest processing time list (LPT) Truck Assign the next job on the list to the machine with the least total processing time assigned (heuristic) 35
36 Flow Shops Introduction Jobs processed sequentially on multiple machines All jobs processed in the same order Makespan on a Two Machine Flow Shop Johnson s Algorithm Example Machines Jobs Total Time
37 Flow Shops Makespan on a Two Machine Flow Shop Johnson s Algorithm Example (cont.) Natural schedule for example has a total makespan of 22 h Sequence : makespan of 17» Is this optimal? The makespan must be as large as the sum of the processing times on either machine Makespan must account for unavoidable idle times» For each machine, add the minimum processing time of a job in the other machine» Example: the bound becomes 17, so is optimal 37
38 Flow Shops Makespan on a Two Machine Flow Shop Johnson s Algorithm Step 1» Select the job with the lowest processing time on each machine from the schedulable job list» If the list is empty, the procedure is finished» If the processing time of the job selected is from machine 1, go to step 2, otherwise, go to step 3 Step 2» Schedule the job in the earliest position of the sequence and remove it from the schedulable job list» Return to step 1 Step 3» Schedule the job in the latest position of the sequence and remove it from the schedulable job list» Return to step 1 Johnson s algorithm provides the optimal solution 38
39 Flow Shops Heuristics CDS Heuristic Convert a m machine problem into a two machine problem. How? Procedure Start with: k=1 and l=m; then k=2 and l=m 1; until: k=m 1 and l=2 m 1 schedules are generated Use the best of these m 1 schedules 39
40 Flow Shops Heuristics CDS Heuristic Data Use the CDS to solve the problem First use the Johnson s algorithm for machines 1 and 4 40
41 Flow Shops Heuristics CDS Heuristic Second combine M1 with M2 to pseudomachine 1 and M3 with M4 to pseudomachine 2 Finally combine M1+M2+M3 into pseudomachine 1 and M2+M3+M4 into pseudomachine 2 41
42 Flow Shops Heuristics CDS Heuristic Gantt Chart for CDS schedule 42
43 Job Shops Introduction Different routings for different jobs Difficult to schedule precedence constraints (n!) m possible schedules 43
44 Job Shops Two Machine Job Shops Jackson (1956) adapted the Johnson s algorithm to minimize makespan Job Sets with Machines A and B Machine A: {AB}, {A}, {BA} Machine B: {BA}, {B}, {AB} Why in this order? The order of jobs within the set is to be determined Procedure Machine A: {AB} jobs ordered by Johnson s Algorithm, then {A} in any Shortest Processing Time {BA} jobs in reverse Johnsons order Machine B: {BA} jobs reverse Johnsons order, then {B} in SPT {AB} jobs in ordered by Johnson s Algorithm 44
45 Job Shops Two Machine Job Shops Jackson (1956) adapted the Johnson s algorithm to minimize makespan Example 45
46 Job Shops Two Machine Job Shops Jackson (1956) adapted the Johnson s algorithm to minimize makespan Example 46
47 Stochastic Scheduling: Static Case Single machine case. Suppose that processing times are random variables. If the objective is to minimize average weighted flow time, jobs are sequenced according to expected weighted SPT. That is, if job times are t 1, t 2,..., and the respective weights are u 1, u 2,... then job i precedes job i+1 if E(t i )/u i < E(t i+1 )/u i+1.
48 Stochastic Scheduling: Static Case (continued) Multiple Machines. Requires the assumption that the distribution of job times is exponential, (memoryless property). Assume two parallel machines processing n jobs. Then the optimal sequence is to schedule the jobs according to LEPT (longest expected processing time first). Johnsons algorithm for scheduling n jobs on two machines (flow shop) in the deterministic case has a natural extension to the stochastic case as long as the job times are exponentially distributed.
Factors to Describe Job Shop Scheduling Problem
Job Shop Scheduling Job Shop A work location in which a number of general purpose work stations exist and are used to perform a variety of jobs Example: Car repair each operator (mechanic) evaluates plus
More informationChapter 8. Operations Scheduling
Chapter 8 Operations Scheduling Buffer Soldering Visual Inspection Special Stations Buffer workforce Production Management 161 Scheduling is the process of organizing, choosing and timing resource usage
More informationOperations Management
151 Scheduling Operations Management William J. Stevenson 8 th edition 152 Scheduling CHAPTER 15 Scheduling McGrawHill/Irwin Operations Management, Eighth Edition, by William J. Stevenson Copyright
More informationCopyright 2009 by The McGrawHill Companies, Inc. All Rights Reserved.
Chapter 16 McGrawHill/Irwin Copyright 2009 by The McGrawHill Companies, Inc. All Rights Reserved. : Establishing the timing of the use of equipment, facilities and human activities in an organization
More information1 st year / 20142015/ Principles of Industrial Eng. Chapter 3 / Dr. May G. Kassir. Chapter Three
Chapter Three Scheduling, Sequencing and Dispatching 31 SCHEDULING Scheduling can be defined as prescribing of when and where each operation necessary to manufacture the product is to be performed. It
More informationScheduling 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 informationOperations Management
Operations Management ShortTerm Scheduling Chapter 15 151 Outline GLOAL COMPANY PROFILE: DELTA AIRLINES THE STRATEGIC IMPORTANCE OF SHORT TERM SCHEDULING SCHEDULING ISSUES Forward and ackward Scheduling
More informationCHAPTER 1. Basic Concepts on Planning and Scheduling
CHAPTER 1 Basic Concepts on Planning and Scheduling Scheduling, FEUP/PRODEI /MIEIC 1 Planning and Scheduling: Processes of Decision Making regarding the selection and ordering of activities as well as
More informationPRODUCTION PLANNING AND SCHEDULING Part 1
PRODUCTION PLANNING AND SCHEDULING Part Andrew Kusiak 9 Seamans Center Iowa City, Iowa  7 Tel: 99 Fax: 9669 andrewkusiak@uiowa.edu http://www.icaen.uiowa.edu/~ankusiak Forecasting Planning Hierarchy
More informationScheduling Single Machine Scheduling. Tim Nieberg
Scheduling Single Machine Scheduling Tim Nieberg Single machine models Observation: for nonpreemptive problems and regular objectives, a sequence in which the jobs are processed is sufficient to describe
More informationClassification  Examples
Lecture 2 Scheduling 1 Classification  Examples 1 r j C max given: n jobs with processing times p 1,...,p n and release dates r 1,...,r n jobs have to be scheduled without preemption on one machine taking
More informationApproximation Algorithms. Scheduling. Approximation algorithms. Scheduling jobs on a single machine
Approximation algorithms Approximation Algorithms Fast. Cheap. Reliable. Choose two. NPhard problems: choose 2 of optimal polynomial time all instances Approximation algorithms. Tradeoff between time
More informationShortTerm Scheduling. Outline
ShortTerm Scheduling 15 Outline Global Company Profile: Delta Air Lines The Importance of ShortTerm Scheduling Scheduling Issues Scheduling ProcessFocused Facilities 1 Outline  Continued Loading s
More information2.3 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
More informationClassification  Examples 1 1 r j C max given: n jobs with processing times p 1,..., p n and release dates
Lecture 2 Scheduling 1 Classification  Examples 11 r j C max given: n jobs with processing times p 1,..., p n and release dates r 1,..., r n jobs have to be scheduled without preemption on one machine
More informationShortTerm Scheduling
Strategic Implications of By scheduling effectively, companies use assets more effectively and create greater capacity per dollar invested, which, in turn, lowers cost This added capacity and related flexibility
More informationPLANNING AND SCHEDULING
PLANNING AND SCHEDULING Andrew Kusiak 2139 Seamans Center Iowa City, Iowa 522421527 Tel: 319335 5934 Fax: 319335 5669 andrewkusiak@uiowa.edu http://www.icaen.uiowa.edu/~ankusiak Forecasting Balancing
More informationSimultaneous Scheduling of Machines and Material Handling System in an FMS
Simultaneous Scheduling of Machines and Material Handling System in an FMS B. Siva Prasad Reddy* and C.S.P. Rao** *Department of Mech. Engg., KITS, Warangal5 5 (A.P) INDIA. **Department of Mech. Engg.,
More informationLOGISTIQUE ET PRODUCTION SUPPLY CHAIN & OPERATIONS MANAGEMENT
LOGISTIQUE ET PRODUCTION SUPPLY CHAIN & OPERATIONS MANAGEMENT CURSUS CONTENTS 1) Introduction 2) Human resources functions 3) A new factory 4) Products 5) Services management 6) Methods 7) Planification
More informationLecture Notes 12: Scheduling  Cont.
Online Algorithms 18.1.2012 Professor: Yossi Azar Lecture Notes 12: Scheduling  Cont. Scribe:Inna Kalp 1 Introduction In this Lecture we discuss 2 scheduling models. We review the scheduling over time
More informationIntroduction to production scheduling. Industrial Management Group School of Engineering University of Seville
Introduction to production scheduling Industrial Management Group School of Engineering University of Seville 1 Introduction to production scheduling Scheduling Production scheduling Gantt Chart Scheduling
More informationOptimal 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
More informationVEHICLE 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 informationSingle machine models: Maximum Lateness 12 Approximation ratio for EDD for problem 1 r j,d j < 0 L max. structure of a schedule Q...
Lecture 4 Scheduling 1 Single machine models: Maximum Lateness 12 Approximation ratio for EDD for problem 1 r j,d j < 0 L max structure of a schedule 0 Q 1100 11 00 11 000 111 0 0 1 1 00 11 00 11 00
More informationIntegrated support system for planning and scheduling... 2003/4/24 page 75 #101. Chapter 5 Sequencing and assignment Strategies
Integrated support system for planning and scheduling... 2003/4/24 page 75 #101 Chapter 5 Sequencing and assignment Strategies 5.1 Overview This chapter is dedicated to the methodologies used in this work
More information5 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
More information2006 Prentice Hall, Inc. 15 2. 2006 Prentice Hall, Inc. 15 3. 2006 Prentice Hall, Inc. 15 4
Operations Management hapter 5 ShortTerm Scheduling Outline The Strategic Importance Of Short Term Scheduling Scheduling Issues Forward and ackward Scheduling Scheduling riteria PowerPoint presentation
More informationApproximation Algorithms
Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NPCompleteness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms
More informationIEOR 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
More informationThis supplement focuses on operations
OPERATIONS SCHEDULING SUPPLEMENT J J1 J OPERATIONS SCHEDULING LEARNING GOALS After reading this supplement, you should be able to: 1. Define new performance measures (beyond flow time and past due) for
More informationA SIMULATION STUDY FOR DYNAMIC FLEXIBLE JOB SHOP SCHEDULING WITH SEQUENCEDEPENDENT SETUP TIMES
A SIMULATION STUDY FOR DYNAMIC FLEXIBLE JOB SHOP SCHEDULING WITH SEQUENCEDEPENDENT SETUP TIMES by Zakaria Yahia Abdelrasol Abdelgawad A Thesis Submitted to the Faculty of Engineering at Cairo University
More informationFlexible Manufacturing System
Flexible Manufacturing System Introduction to FMS Features of FMS Operational problems in FMS Layout considerations Sequencing of Robot Moves FMS Scheduling and control Examples Deadlocking Flow system
More informationChapter 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 NPhard problem. What should I do? A. Theory says you're unlikely to find a polytime algorithm. Must sacrifice one
More information! Solve problem to optimality. ! Solve problem in polytime. ! Solve arbitrary instances of the problem. #approximation algorithm.
Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NPhard problem What should I do? A Theory says you're unlikely to find a polytime algorithm Must sacrifice one of three
More informationApplied 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
More informationWORSTCASE PERFORMANCE ANALYSIS OF SOME APPROXIMATION ALGORITHMS FOR MINIMIZING MAKESPAN AND FLOWTIME
WORSTCASE PERFORMANCE ANALYSIS OF SOME APPROXIMATION ALGORITHMS FOR MINIMIZING MAKESPAN AND FLOWTIME PERUVEMBA SUNDARAM RAVI, LEVENT TUNÇEL, MICHAEL HUANG Abstract. In 1976, Coffman and Sethi conjectured
More informationRealTime Scheduling (Part 1) (Working Draft) RealTime System Example
RealTime Scheduling (Part 1) (Working Draft) Insup Lee Department of Computer and Information Science School of Engineering and Applied Science University of Pennsylvania www.cis.upenn.edu/~lee/ CIS 41,
More informationCS188 Spring 2011 Section 3: Game Trees
CS188 Spring 2011 Section 3: Game Trees 1 WarmUp: ColumnRow You have a 3x3 matrix of values like the one below. In a somewhat boring game, player A first selects a row, and then player B selects a column.
More information! Solve problem to optimality. ! Solve problem in polytime. ! Solve arbitrary instances of the problem. !approximation algorithm.
Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NPhard problem What should I do? A Theory says you're unlikely to find a polytime algorithm Must sacrifice one of
More informationCompletion Time Scheduling and the WSRPT Algorithm
Completion Time Scheduling and the WSRPT Algorithm Bo Xiong, Christine Chung Department of Computer Science, Connecticut College, New London, CT {bxiong,cchung}@conncoll.edu Abstract. We consider the online
More informationAlgorithm 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 informationJob Scheduling Techniques for Distributed Systems with Heterogeneous Processor Cardinality
Job Scheduling Techniques for Distributed Systems with Heterogeneous Processor Cardinality HungJui Chang JanJan Wu Department of Computer Science and Information Engineering Institute of Information
More informationGESTION DE LA PRODUCTION ET DES OPERATIONS PICASSO EXERCICE INTEGRE
ECAP 21 / PROD2100 GESTION DE LA PRODUCTION ET DES OPERATIONS PICASSO EXERCICE INTEGRE 20042005 Prof : Pierre Semal : semal@poms.ucl.ac.be Assistants : Eléonore de le Court : delecourt@poms.ucl.ac.be
More informationReal Time Scheduling Basic Concepts. Radek Pelánek
Real Time Scheduling Basic Concepts Radek Pelánek Basic Elements Model of RT System abstraction focus only on timing constraints idealization (e.g., zero switching time) Basic Elements Basic Notions task
More informationCOMPLEX EMBEDDED SYSTEMS
COMPLEX EMBEDDED SYSTEMS RealTime Scheduling Summer Semester 2012 System and Software Engineering Prof. Dr.Ing. Armin Zimmermann Contents Introduction Scheduling in Interactive Systems RealTime Scheduling
More informationF E M M Faculty of Economics and Management Magdeburg
OTTOVONGUERICKEUNIVERSITY MAGDEBURG FACULTY OF ECONOMICS AND MANAGEMENT Algorithms for Online Order Batching in an OrderPicking Warehouse Sebastian Henn FEMM Working Paper No. 34, October 2009 F E
More informationThe Trip Scheduling Problem
The Trip Scheduling Problem Claudia Archetti Department of Quantitative Methods, University of Brescia Contrada Santa Chiara 50, 25122 Brescia, Italy Martin Savelsbergh School of Industrial and Systems
More informationSeradex White Paper A newsletter for manufacturing organizations April, 2004
Seradex White Paper A newsletter for manufacturing organizations April, 2004 Using Project Management Software for Production Scheduling Frequently, we encounter organizations considering the use of project
More information4.6 Linear Programming duality
4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal
More informationBranchandPrice Approach to the Vehicle Routing Problem with Time Windows
TECHNISCHE UNIVERSITEIT EINDHOVEN BranchandPrice 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
More informationBatch Scheduling for Identical MultiTasks Jobs on Heterogeneous Platforms
atch Scheduling for Identical MultiTasks Jobs on Heterogeneous Platforms JeanMarc Nicod (JeanMarc.Nicod@lifc.univfcomte.fr) Sékou iakité, Laurent Philippe  16/05/2008 Laboratoire d Informatique de
More informationAn improved online algorithm for scheduling on two unrestrictive parallel batch processing machines
This is the PrePublished Version. An improved online algorithm for scheduling on two unrestrictive parallel batch processing machines Q.Q. Nong, T.C.E. Cheng, C.T. Ng Department of Mathematics, Ocean
More informationOperating Systems. Scheduling. Lecture 8 Michael O Boyle
Operating Systems Scheduling Lecture 8 Michael O Boyle 1 Scheduling We have talked about context switching an interrupt occurs (device completion, timer interrupt) a thread causes a trap or exception may
More informationECEN 5682 Theory and Practice of Error Control Codes
ECEN 5682 Theory and Practice of Error Control Codes Convolutional Codes University of Colorado Spring 2007 Linear (n, k) block codes take k data symbols at a time and encode them into n code symbols.
More informationChapter 5: CPU Scheduling. Operating System Concepts 8 th Edition,
Chapter 5: CPU Scheduling, Silberschatz, Galvin and Gagne 2009 Chapter 5: CPU Scheduling Basic Concepts Scheduling Criteria Scheduling Algorithms Thread Scheduling MultipleProcessor Scheduling Linux Example
More informationmax cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x
Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study
More informationComplexity 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
More informationChapter 5: CPU Scheduling!
Chapter 5: CPU Scheduling Operating System Concepts 8 th Edition, Silberschatz, Galvin and Gagne 2009 Chapter 5: CPU Scheduling Basic Concepts Scheduling Criteria Scheduling Algorithms Thread Scheduling
More informationScheduling Resources and Costs
Student Version CHAPTER EIGHT Scheduling Resources and Costs McGrawHill/Irwin Copyright 2011 by The McGrawHill Companies, Inc. All rights reserved. Gannt Chart Developed by Henry Gannt in 1916 is used
More informationIntroduction to Scheduling Theory
Introduction to Scheduling Theory Arnaud Legrand Laboratoire Informatique et Distribution IMAG CNRS, France arnaud.legrand@imag.fr November 8, 2004 1/ 26 Outline 1 Task graphs from outer space 2 Scheduling
More informationAn Efficient Combination of Dispatch Rules for Jobshop Scheduling Problem
An Efficient Combination of Dispatch Rules for Jobshop Scheduling Problem Tatsunobu Kawai, Yasutaka Fujimoto Department of Electrical and Computer Engineering, Yokohama National University, Yokohama 2408501
More informationANALYTIC HIERARCHY PROCESS (AHP) TUTORIAL
Kardi Teknomo ANALYTIC HIERARCHY PROCESS (AHP) TUTORIAL Revoledu.com Table of Contents Analytic Hierarchy Process (AHP) Tutorial... 1 Multi Criteria Decision Making... 1 Cross Tabulation... 2 Evaluation
More informationStandard Form of a Linear Programming Problem
494 CHAPTER 9 LINEAR PROGRAMMING 9. THE SIMPLEX METHOD: MAXIMIZATION For linear programming problems involving two variables, the graphical solution method introduced in Section 9. is convenient. However,
More informationInformed search algorithms
CmpE 540 Principles of Artificial Intelligence Informed search algorithms Pınar Yolum pinar.yolum@boun.edu.tr Department of Computer Engineering Boğaziçi University Chapter 4 (Sections 1 3) (Based mostly
More informationApproximability of TwoMachine NoWait Flowshop Scheduling with Availability Constraints
Approximability of TwoMachine NoWait Flowshop Scheduling with Availability Constraints T.C. Edwin Cheng 1, and Zhaohui Liu 1,2 1 Department of Management, The Hong Kong Polytechnic University Kowloon,
More informationLoad Balancing. Load Balancing 1 / 24
Load Balancing Backtracking, branch & bound and alphabeta pruning: how to assign work to idle processes without much communication? Additionally for alphabeta pruning: implementing the youngbrotherswait
More informationScheduling Parallel Machine Scheduling. Tim Nieberg
Scheduling Parallel Machine Scheduling Tim Nieberg Problem P C max : m machines n jobs with processing times p 1,..., p n Problem P C max : m machines n jobs with processing times p 1,..., p { n 1 if job
More informationProject and Production Management Prof. Arun Kanda Department of Mechanical Engineering Indian Institute of Technology, Delhi
Project and Production Management Prof. Arun Kanda Department of Mechanical Engineering Indian Institute of Technology, Delhi Lecture  15 Limited Resource Allocation Today we are going to be talking about
More informationMaterial Requirements Planning. Managing Inventories of Items With Dependent Demand
Material Requirements Planning Managing Inventories of Items With Dependent Demand 1 Demand Types Dependent Demand: Demand for items that are component parts to be used in the production of finished goods
More informationCPU Scheduling. Prof. Sirer (dr. Willem de Bruijn) CS 4410 Cornell University
CPU Scheduling Prof. Sirer (dr. Willem de Bruijn) CS 4410 Cornell University Problem You are the cook at the state st. diner customers continually enter and place their orders Dishes take varying amounts
More informationResearch Article Batch Scheduling on TwoMachine Flowshop with MachineDependent Setup Times
Hindawi Publishing Corporation Advances in Operations Research Volume 2009, Article ID 153910, 10 pages doi:10.1155/2009/153910 Research Article Batch Scheduling on TwoMachine Flowshop with MachineDependent
More informationReinforcement Learning
Reinforcement Learning LU 2  Markov Decision Problems and Dynamic Programming Dr. Martin Lauer AG Maschinelles Lernen und Natürlichsprachliche Systeme AlbertLudwigsUniversität Freiburg martin.lauer@kit.edu
More information56:171 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
More informationChapter 5: CPU Scheduling. Operating System Concepts 7 th Edition, Jan 14, 2005
Chapter 5: CPU Scheduling Operating System Concepts 7 th Edition, Jan 14, 2005 Silberschatz, Galvin and Gagne 2005 Outline Basic Concepts Scheduling Criteria Scheduling Algorithms MultipleProcessor Scheduling
More informationAN ANALYSIS OF THE IMPORTANCE OF APPROPRIATE TIE BREAKING RULES IN DISPATCH HEURISTICS
versão impressa ISSN 01017438 / versão online ISSN 16785142 AN ANALYSIS OF THE IMPORTANCE OF APPROPRIATE TIE BREAKING RULES IN DISPATCH HEURISTICS Jorge M. S. Valente Faculdade de Economia Universidade
More informationAutomated Scheduling, School of Computer Science and IT, University of Nottingham 1. Job Shop Scheduling. Disjunctive Graph.
Job hop cheduling Contents 1. Problem tatement 2. Disjunctive Graph. he hifting Bottleneck Heuristic and the Makespan Literature: 1. cheduling, heory, Algorithms, and ystems, Michael Pinedo, Prentice Hall,
More informationMIPBased Approaches for Solving Scheduling Problems with Batch Processing Machines
The Eighth International Symposium on Operations Research and Its Applications (ISORA 09) Zhangjiajie, China, September 20 22, 2009 Copyright 2009 ORSC & APORC, pp. 132 139 MIPBased Approaches for Solving
More informationVehicle 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
More informationFactoring 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
More informationGraphical method. plane. (for max) and down (for min) until it touches the set of feasible solutions. Graphical method
The graphical method of solving linear programming problems can be applied to models with two decision variables. This method consists of two steps (see also the first lecture): 1 Draw the set of feasible
More informationModels in Transportation. Tim Nieberg
Models in Transportation Tim Nieberg Transportation Models large variety of models due to the many modes of transportation roads railroad shipping airlines as a consequence different type of equipment
More informationEcient approximation algorithm for minimizing makespan. on uniformly related machines. Chandra Chekuri. November 25, 1997.
Ecient approximation algorithm for minimizing makespan on uniformly related machines Chandra Chekuri November 25, 1997 Abstract We obtain a new ecient approximation algorithm for scheduling precedence
More informationSolutions to Exercises 8
Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 8 (1) Suppose that G is a graph in which every vertex has degree at least k, where k 1, and in which every cycle contains at least 4 vertices.
More informationSmall Maximal Independent Sets and Faster Exact Graph Coloring
Small Maximal Independent Sets and Faster Exact Graph Coloring David Eppstein Univ. of California, Irvine Dept. of Information and Computer Science The Exact Graph Coloring Problem: Given an undirected
More informationOperation 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 information7 Gaussian Elimination and LU Factorization
7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method
More informationProject Scheduling: PERT/CPM
Project Scheduling: PERT/CPM CHAPTER 8 LEARNING OBJECTIVES After completing this chapter, you should be able to: 1. Describe the role and application of PERT/CPM for project scheduling. 2. Define a project
More informationOffline sorting buffers on Line
Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: rkhandekar@gmail.com 2 IBM India Research Lab, New Delhi. email: pvinayak@in.ibm.com
More informationA Linear Programming Based Method for Job Shop Scheduling
A Linear Programming Based Method for Job Shop Scheduling Kerem Bülbül Sabancı University, Manufacturing Systems and Industrial Engineering, OrhanlıTuzla, 34956 Istanbul, Turkey bulbul@sabanciuniv.edu
More informationLecture 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 informationLecture 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
More informationvii TABLE OF CONTENTS CHAPTER TITLE PAGE DECLARATION DEDICATION ACKNOWLEDGEMENT ABSTRACT ABSTRAK
vii TABLE OF CONTENTS CHAPTER TITLE PAGE DECLARATION DEDICATION ACKNOWLEDGEMENT ABSTRACT ABSTRAK TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF ABBREVIATIONS LIST OF SYMBOLS LIST OF APPENDICES
More information15 Markov Chains: Limiting Probabilities
MARKOV CHAINS: LIMITING PROBABILITIES 67 Markov Chains: Limiting Probabilities Example Assume that the transition matrix is given by 7 2 P = 6 Recall that the nstep transition probabilities are given
More informationA Genetic Algorithm Approach for Solving a Flexible Job Shop Scheduling Problem
A Genetic Algorithm Approach for Solving a Flexible Job Shop Scheduling Problem Sayedmohammadreza Vaghefinezhad 1, Kuan Yew Wong 2 1 Department of Manufacturing & Industrial Engineering, Faculty of Mechanical
More informationChapter 6: CPU Scheduling
Chapter 6: CPU Scheduling Basic Concepts Scheduling Criteria Scheduling Algorithms MultipleProcessor Scheduling RealTime Scheduling Algorithm Evaluation Oct03 1 Basic Concepts Maximum CPU utilization
More informationLinear Inequalities and Linear Programming. Systems of Linear Inequalities in Two Variables
Linear Inequalities and Linear Programming 5.1 Systems of Linear Inequalities 5.2 Linear Programming Geometric Approach 5.3 Geometric Introduction to Simplex Method 5.4 Maximization with constraints 5.5
More informationA Study of Crossover Operators for Genetic Algorithm and Proposal of a New Crossover Operator to Solve Open Shop Scheduling Problem
American Journal of Industrial and Business Management, 2016, 6, 774789 Published Online June 2016 in SciRes. http://www.scirp.org/journal/ajibm http://dx.doi.org/10.4236/ajibm.2016.66071 A Study of Crossover
More informationChap 4 The Simplex Method
The Essence of the Simplex Method Recall the Wyndor problem Max Z = 3x 1 + 5x 2 S.T. x 1 4 2x 2 12 3x 1 + 2x 2 18 x 1, x 2 0 Chap 4 The Simplex Method 8 corner point solutions. 5 out of them are CPF solutions.
More informationPriori ty ... ... ...
.Maintenance Scheduling Maintenance scheduling is the process by which jobs are matched with resources (crafts) and sequenced to be executed at certain points in time. The maintenance schedule can be prepared
More informationVENDOR MANAGED INVENTORY
VENDOR MANAGED INVENTORY Martin Savelsbergh School of Industrial and Systems Engineering Georgia Institute of Technology Joint work with Ann Campbell, Anton Kleywegt, and Vijay Nori Distribution Systems:
More information