5 Scheduling. Operations Planning and Control

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "5 Scheduling. Operations Planning and Control"

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

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 information

Chapter 8. Operations Scheduling

Chapter 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 information

Operations Management

Operations Management 15-1 Scheduling Operations Management William J. Stevenson 8 th edition 15-2 Scheduling CHAPTER 15 Scheduling McGraw-Hill/Irwin Operations Management, Eighth Edition, by William J. Stevenson Copyright

More information

Copyright 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved.

Copyright 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved. Chapter 16 McGraw-Hill/Irwin Copyright 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved. : Establishing the timing of the use of equipment, facilities and human activities in an organization

More information

1 st year / 2014-2015/ Principles of Industrial Eng. Chapter -3 -/ Dr. May G. Kassir. Chapter Three

1 st year / 2014-2015/ Principles of Industrial Eng. Chapter -3 -/ Dr. May G. Kassir. Chapter Three Chapter Three Scheduling, Sequencing and Dispatching 3-1- SCHEDULING Scheduling can be defined as prescribing of when and where each operation necessary to manufacture the product is to be performed. It

More information

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

Operations Management

Operations Management Operations Management Short-Term Scheduling Chapter 15 15-1 Outline GLOAL COMPANY PROFILE: DELTA AIRLINES THE STRATEGIC IMPORTANCE OF SHORT- TERM SCHEDULING SCHEDULING ISSUES Forward and ackward Scheduling

More information

CHAPTER 1. Basic Concepts on Planning and Scheduling

CHAPTER 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 information

PRODUCTION PLANNING AND SCHEDULING Part 1

PRODUCTION PLANNING AND SCHEDULING Part 1 PRODUCTION PLANNING AND SCHEDULING Part Andrew Kusiak 9 Seamans Center Iowa City, Iowa - 7 Tel: 9-9 Fax: 9-669 andrew-kusiak@uiowa.edu http://www.icaen.uiowa.edu/~ankusiak Forecasting Planning Hierarchy

More information

Scheduling Single Machine Scheduling. Tim Nieberg

Scheduling Single Machine Scheduling. Tim Nieberg Scheduling Single Machine Scheduling Tim Nieberg Single machine models Observation: for non-preemptive problems and regular objectives, a sequence in which the jobs are processed is sufficient to describe

More information

Classification - Examples

Classification - 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 information

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

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

More information

Short-Term Scheduling. Outline

Short-Term Scheduling. Outline Short-Term Scheduling 15 Outline Global Company Profile: Delta Air Lines The Importance of Short-Term Scheduling Scheduling Issues Scheduling Process-Focused Facilities 1 Outline - Continued Loading s

More information

2.3 Scheduling jobs on identical parallel machines

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

More information

Classification - Examples -1- 1 r j C max given: n jobs with processing times p 1,..., p n and release dates

Classification - 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 -1-1 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

More information

Short-Term Scheduling

Short-Term 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 information

PLANNING AND SCHEDULING

PLANNING AND SCHEDULING PLANNING AND SCHEDULING Andrew Kusiak 2139 Seamans Center Iowa City, Iowa 52242-1527 Tel: 319-335 5934 Fax: 319-335 5669 andrew-kusiak@uiowa.edu http://www.icaen.uiowa.edu/~ankusiak Forecasting Balancing

More information

Simultaneous Scheduling of Machines and Material Handling System in an FMS

Simultaneous 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, Warangal-5 5 (A.P) INDIA. **Department of Mech. Engg.,

More information

LOGISTIQUE ET PRODUCTION SUPPLY CHAIN & OPERATIONS MANAGEMENT

LOGISTIQUE 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 information

Lecture Notes 12: Scheduling - Cont.

Lecture 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 information

Introduction 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 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 information

Optimal Scheduling for Dependent Details Processing Using MS Excel Solver

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

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

Single machine models: Maximum Lateness -12- Approximation ratio for EDD for problem 1 r j,d j < 0 L max. structure of a schedule Q...

Single 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 information

Integrated 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 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 information

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 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 information

2006 Prentice Hall, Inc. 15 2. 2006 Prentice Hall, Inc. 15 3. 2006 Prentice Hall, Inc. 15 4

2006 Prentice Hall, Inc. 15 2. 2006 Prentice Hall, Inc. 15 3. 2006 Prentice Hall, Inc. 15 4 Operations Management hapter 5 Short-Term Scheduling Outline The Strategic Importance Of Short- Term Scheduling Scheduling Issues Forward and ackward Scheduling Scheduling riteria PowerPoint presentation

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

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

IEOR 4404 Homework #2 Intro OR: Deterministic Models February 14, 2011 Prof. Jay Sethuraman Page 1 of 5. Homework #2 IEOR 4404 Homework # Intro OR: Deterministic Models February 14, 011 Prof. Jay Sethuraman Page 1 of 5 Homework #.1 (a) What is the optimal solution of this problem? Let us consider that x 1, x and x 3

More information

This supplement focuses on operations

This supplement focuses on operations OPERATIONS SCHEDULING SUPPLEMENT J J-1 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 information

A SIMULATION STUDY FOR DYNAMIC FLEXIBLE JOB SHOP SCHEDULING WITH SEQUENCE-DEPENDENT SETUP TIMES

A SIMULATION STUDY FOR DYNAMIC FLEXIBLE JOB SHOP SCHEDULING WITH SEQUENCE-DEPENDENT SETUP TIMES A SIMULATION STUDY FOR DYNAMIC FLEXIBLE JOB SHOP SCHEDULING WITH SEQUENCE-DEPENDENT SETUP TIMES by Zakaria Yahia Abdelrasol Abdelgawad A Thesis Submitted to the Faculty of Engineering at Cairo University

More information

Flexible Manufacturing System

Flexible 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 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

! 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 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

More information

Applied Algorithm Design Lecture 5

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

More information

WORST-CASE PERFORMANCE ANALYSIS OF SOME APPROXIMATION ALGORITHMS FOR MINIMIZING MAKESPAN AND FLOWTIME

WORST-CASE PERFORMANCE ANALYSIS OF SOME APPROXIMATION ALGORITHMS FOR MINIMIZING MAKESPAN AND FLOWTIME WORST-CASE 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 information

Real-Time Scheduling (Part 1) (Working Draft) Real-Time System Example

Real-Time Scheduling (Part 1) (Working Draft) Real-Time System Example Real-Time 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 information

CS188 Spring 2011 Section 3: Game Trees

CS188 Spring 2011 Section 3: Game Trees CS188 Spring 2011 Section 3: Game Trees 1 Warm-Up: Column-Row 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 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

Completion Time Scheduling and the WSRPT Algorithm

Completion 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 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

Job Scheduling Techniques for Distributed Systems with Heterogeneous Processor Cardinality

Job Scheduling Techniques for Distributed Systems with Heterogeneous Processor Cardinality Job Scheduling Techniques for Distributed Systems with Heterogeneous Processor Cardinality Hung-Jui Chang Jan-Jan Wu Department of Computer Science and Information Engineering Institute of Information

More information

GESTION DE LA PRODUCTION ET DES OPERATIONS PICASSO EXERCICE INTEGRE

GESTION DE LA PRODUCTION ET DES OPERATIONS PICASSO EXERCICE INTEGRE ECAP 21 / PROD2100 GESTION DE LA PRODUCTION ET DES OPERATIONS PICASSO EXERCICE INTEGRE 2004-2005 Prof : Pierre Semal : semal@poms.ucl.ac.be Assistants : Eléonore de le Court : delecourt@poms.ucl.ac.be

More information

Real Time Scheduling Basic Concepts. Radek Pelánek

Real 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 information

COMPLEX EMBEDDED SYSTEMS

COMPLEX EMBEDDED SYSTEMS COMPLEX EMBEDDED SYSTEMS Real-Time Scheduling Summer Semester 2012 System and Software Engineering Prof. Dr.-Ing. Armin Zimmermann Contents Introduction Scheduling in Interactive Systems Real-Time Scheduling

More information

F E M M Faculty of Economics and Management Magdeburg

F E M M Faculty of Economics and Management Magdeburg OTTO-VON-GUERICKE-UNIVERSITY MAGDEBURG FACULTY OF ECONOMICS AND MANAGEMENT Algorithms for On-line Order Batching in an Order-Picking Warehouse Sebastian Henn FEMM Working Paper No. 34, October 2009 F E

More information

The Trip Scheduling Problem

The 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 information

Seradex White Paper A newsletter for manufacturing organizations April, 2004

Seradex 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 information

4.6 Linear Programming duality

4.6 Linear Programming duality 4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal

More information

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

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

More information

Batch Scheduling for Identical Multi-Tasks Jobs on Heterogeneous Platforms

Batch Scheduling for Identical Multi-Tasks Jobs on Heterogeneous Platforms atch Scheduling for Identical Multi-Tasks Jobs on Heterogeneous Platforms Jean-Marc Nicod (Jean-Marc.Nicod@lifc.univ-fcomte.fr) Sékou iakité, Laurent Philippe - 16/05/2008 Laboratoire d Informatique de

More information

An improved on-line algorithm for scheduling on two unrestrictive parallel batch processing machines

An improved on-line algorithm for scheduling on two unrestrictive parallel batch processing machines This is the Pre-Published Version. An improved on-line 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 information

Operating Systems. Scheduling. Lecture 8 Michael O Boyle

Operating 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 information

ECEN 5682 Theory and Practice of Error Control Codes

ECEN 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 information

Chapter 5: CPU Scheduling. Operating System Concepts 8 th Edition,

Chapter 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 Multiple-Processor Scheduling Linux Example

More information

max 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

max 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 information

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

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

More information

Chapter 5: CPU Scheduling!

Chapter 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 information

Scheduling Resources and Costs

Scheduling Resources and Costs Student Version CHAPTER EIGHT Scheduling Resources and Costs McGraw-Hill/Irwin Copyright 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Gannt Chart Developed by Henry Gannt in 1916 is used

More information

Introduction to Scheduling Theory

Introduction 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 information

An Efficient Combination of Dispatch Rules for Job-shop Scheduling Problem

An Efficient Combination of Dispatch Rules for Job-shop Scheduling Problem An Efficient Combination of Dispatch Rules for Job-shop Scheduling Problem Tatsunobu Kawai, Yasutaka Fujimoto Department of Electrical and Computer Engineering, Yokohama National University, Yokohama 240-8501

More information

ANALYTIC HIERARCHY PROCESS (AHP) TUTORIAL

ANALYTIC 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 information

Standard Form of a Linear Programming Problem

Standard 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 information

Informed search algorithms

Informed 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 information

Approximability of Two-Machine No-Wait Flowshop Scheduling with Availability Constraints

Approximability of Two-Machine No-Wait Flowshop Scheduling with Availability Constraints Approximability of Two-Machine No-Wait 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 information

Load Balancing. Load Balancing 1 / 24

Load Balancing. Load Balancing 1 / 24 Load Balancing Backtracking, branch & bound and alpha-beta pruning: how to assign work to idle processes without much communication? Additionally for alpha-beta pruning: implementing the young-brothers-wait

More information

Scheduling Parallel Machine Scheduling. Tim Nieberg

Scheduling 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 information

Project 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 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 information

Material Requirements Planning. Managing Inventories of Items With Dependent Demand

Material 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 information

CPU Scheduling. Prof. Sirer (dr. Willem de Bruijn) CS 4410 Cornell University

CPU 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 information

Research Article Batch Scheduling on Two-Machine Flowshop with Machine-Dependent Setup Times

Research Article Batch Scheduling on Two-Machine Flowshop with Machine-Dependent 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 Two-Machine Flowshop with Machine-Dependent

More information

Reinforcement Learning

Reinforcement Learning Reinforcement Learning LU 2 - Markov Decision Problems and Dynamic Programming Dr. Martin Lauer AG Maschinelles Lernen und Natürlichsprachliche Systeme Albert-Ludwigs-Universität Freiburg martin.lauer@kit.edu

More information

56:171 Operations Research Midterm Exam Solutions Fall 2001

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

More information

Chapter 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 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 Multiple-Processor Scheduling

More information

AN ANALYSIS OF THE IMPORTANCE OF APPROPRIATE TIE BREAKING RULES IN DISPATCH HEURISTICS

AN ANALYSIS OF THE IMPORTANCE OF APPROPRIATE TIE BREAKING RULES IN DISPATCH HEURISTICS versão impressa ISSN 0101-7438 / versão online ISSN 1678-5142 AN ANALYSIS OF THE IMPORTANCE OF APPROPRIATE TIE BREAKING RULES IN DISPATCH HEURISTICS Jorge M. S. Valente Faculdade de Economia Universidade

More information

Automated Scheduling, School of Computer Science and IT, University of Nottingham 1. Job Shop Scheduling. Disjunctive Graph.

Automated 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 information

MIP-Based Approaches for Solving Scheduling Problems with Batch Processing Machines

MIP-Based 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 MIP-Based Approaches for Solving

More information

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

More information

Factoring Algorithms

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

More information

Graphical method. plane. (for max) and down (for min) until it touches the set of feasible solutions. Graphical method

Graphical 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 information

Models in Transportation. Tim Nieberg

Models 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 information

Ecient 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. 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 information

Solutions to Exercises 8

Solutions 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 information

Small Maximal Independent Sets and Faster Exact Graph Coloring

Small 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 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

7 Gaussian Elimination and LU Factorization

7 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 information

Project Scheduling: PERT/CPM

Project 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 information

Offline sorting buffers on Line

Offline 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 information

A Linear Programming Based Method for Job Shop Scheduling

A 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, Orhanlı-Tuzla, 34956 Istanbul, Turkey bulbul@sabanciuniv.edu

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

Lecture 3: Linear Programming Relaxations and Rounding

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

More information

vii TABLE OF CONTENTS CHAPTER TITLE PAGE DECLARATION DEDICATION ACKNOWLEDGEMENT ABSTRACT ABSTRAK

vii 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 information

15 Markov Chains: Limiting Probabilities

15 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 n-step transition probabilities are given

More information

A Genetic Algorithm Approach for Solving a Flexible Job Shop Scheduling Problem

A 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 information

Chapter 6: CPU Scheduling

Chapter 6: CPU Scheduling Chapter 6: CPU Scheduling Basic Concepts Scheduling Criteria Scheduling Algorithms Multiple-Processor Scheduling Real-Time Scheduling Algorithm Evaluation Oct-03 1 Basic Concepts Maximum CPU utilization

More information

Linear Inequalities and Linear Programming. Systems of Linear Inequalities in Two Variables

Linear 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 information

A Study of Crossover Operators for Genetic Algorithm and Proposal of a New Crossover Operator to Solve Open Shop Scheduling Problem

A 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, 774-789 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 information

Chap 4 The Simplex Method

Chap 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 information

Priori ty ... ... ...

Priori 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 information

VENDOR MANAGED INVENTORY

VENDOR 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