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Set #1

Dr. LEE Heung Wing Joseph

Email : majlee@polyu.edu.hkPhone: 2766 6951Office : HJ639

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Subject Code: AMA522Location and Time: Wednesday, 18:30, at DE303

Recommended background knowledge: Calculus, Linear Algebra, Probability,Fortran or C Programming, Combinatorics.

Assessment:2 Assignments, 10%Project, 15%Test, 15%Final Examination 60%

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Aim

To provide a sound understanding of the fundamental techniques and algorithms for scheduling problems from a range ofcommercial and service sectors.

Objectives

To give an understanding of the methods and techniques thatare available for building scheduling systems.

To introduce modern approaches for dealing with schedulingproblems including treating uncertainties.

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Adopted Text

1. Scheduling, Theory, Algorithms, and Systems, Michael Pinedo, Prentice Hall, 1995.

NEW: Second edition, 2002

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Lecture Notes will be available for download online at:

http://www.acad.polyu.edu.hk/~majlee/ama522.html

All announcements for the subject will be made in lecturesand put on the web site

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INTRODUCTION TO SCHEDULING

Contents

1. Definition of Scheduling

2. Examples

3. Terminology

4. Classification of Scheduling Problems

5. P and NP problems

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Gantt Chart

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Definition of Scheduling

Scheduling concerns optimal allocation or assignment of resources, over time, to a set of tasks or activities.

machines Mi, i=1,...,m (ith machine)jobs Jj, j=1,...,n (jth job)

• Schedule may be represented by Gantt charts.

J3 J2

J2

J3

J1

J1J1

J3

J4

M3

M2

M1

Machine oriented Gantt chart

M1 M2

M2

M1

M3

M3 M2

J1

J2

J3

Job oriented Gantt chart

M1

M1J4t

t

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Bicycle Assembly• 3 workers within a team• each task has its own duration• precedence constraints• no preemption

T7

T2

T1

T5

T4

T6

T9

T8 T10

T3

Examples

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T1

T6T4 T8

T2

T10

T7

T3

7

T9T5

14 21 39

2 4 6 14

2 5 6 14 21

Task assignment

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T1

T6

T4

T8

T2

T10

T3

7

T9

T5

14 16 34

2 4 6 9

2 9 17 25

Improved task assignment

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T7

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An optimal task assignment

T1

T6T4 T8

T2

T10T3

7

T9T5

14 32

2 5 7 14 322416

T7

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Scheduling Systems

• Enterprise Resource Planning (ERP)– Common for larger businesses

• Materials Requirement Planning (MRP)– Very common for manufacturing companies

• Advanced Planning and Scheduling (APS)– Most recent trend– Considered “advanced feature” of ERP

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Scheduling Problem

• Allocate scarce resources to tasks

• Combinatorial optimization problem

Maximize profit

Subject to constraints

• Mathematical techniques and heuristics

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Seminar A B C D E F G H I J K L M NPeriods 2 8 4,

51,2

3,4,5

6, 7,8

2,3

1,2

5 6,7

3,4

8 2,3,4

Period 1 2 3 4 5 6 7 8Room1 D D C C F BRoom2 I I E E E G GRoom3 H H J K KRoom4 N N N MRoom5 A L L

Classroom Assignment

• one day seminar• 14 seminars• 5 rooms• 8:00 - 5:00pm• no seminars during the lunch hour 12:00 - 1:00pm

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Soft Drink Bottling

• single machine• 4 flavours• each flavour has its own filling time• cleaning and changeover time between the bottling of

successive flavoursaim: to minimise cycle time, sufficient: to minimise the total changeover time

6550

238

436

50702

4

3

2

1

4321

f

f

f

f

ffff f1 - f2 - f3 - f4 - f1

2+3+2+50 = 57

f2 - f3 - f4 - f1 - f2

3+2+50+2 = 57

f3 - f4 - f2 - f1 - f3

2+5+6+70 = 83

f4 - f2 - f3 - f1 - f4

4+3+8+50 = 66

f1 - f2 - f4 - f3 - f1

2+4+6+8 = 20optimal:

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Terminology

• Scheduling is the allocation, subject to constraints, of resources to objects being placed in space-time, so that the total cost is minimised. Schedule includes the spacial and temporal information.

• Sequencing is the construction, subject to constraints, of an order inwhich activities are to be carried out.Sequence is an order in which activities are carried out.

• Timetabling is the allocation, subject to constraints, of resources to objects being placed in space-time, so that the set of objectives are satisfied as much as possible. Timetable shows when particular events are to take place.

• Rostering is the placing, subject to constraints, of resources into slots in a pattern. Roster is a list of people's names that shows which jobs they are to doand when.

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Example 1.1.1 A Paper Bag Factory

1. Printing of the logo

2. gluing of the side of the bag

3. sewing of one end of the bag

• Colours, size may affect processing speed.

• Late delivery implies loss of goodwill.

• Sequence dependent setup time.

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Example 1.1.2 Gate Assignments at an Airport

• Dozens of gates and hundreds of airplanes arriving and departing according to a schedule each day.

• Gates as well as the airplanes are not all identical.• Some gates are in locations where it is difficult to

bring in the planes• Certain planes may have to be towed to their

gates.• Weather, or events at other airports may cause

randomness in the schedule.

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Example 1.1.3 Scheduling Tasks in CPU

• Multi-tasking computer.• Exact processing time are not known in

advance, only the distributions may be known.

• Priority level.• The operation system slices the tasks into

little pieces.• Preemption allowed.

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Classification of Scheduling Problems

machines j=1,…,mjobs i =1,…,n(i, j) processing step, or operation, of job j on machine i

Job data

Processing time pij - processing time of job j on machine i

Release date rj - earliest time at which job j can start its processing

Due date dj - committed shipping or completion date of job j

Weight wj - importance of job j relative to the other jobs in the system

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Scheduling problem:

| | machine environment

job characteristics

optimality criteria

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Machine characteristics

Single machine 1

Identical machines in parallel Pm • m machines in parallel• Job j requires a single operation and may be processed

on any of the m machines• If job j may be processed on any one machine belonging to a

given subset Mj

Pm | Mj | ...

• Machines in parallel with different speeds Qm • Unrelated machines in parallel Rm

machines have different speeds for different jobs

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Machine Configurations

Single-Machine Parallel-Machine

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Flow shop Fm

• m machines in series• all jobs have the same routing• each job has to be processed on each one of the m machines

(permutation)first in first out (FIFO) Fm | prmu | ...

Flexible flow shop FFs

• s stages in series with a number of machines in parallel• at each stage job j requires only one machine• FIFO discipline is usually between stages

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Machine Configurations

Flow Shop

Job Shop

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Open shop Om

• m machines• each job has to be processed on each of the m machines • scheduler determines the route for each job

Job shop Jm

• m machines• each job has its own route• job may visit a machine more then once (recirculation)

Fm | recrc | ...

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Job characteristics

Release date rj - earliest time at which job j can start its processing

Sequence dependent setup times sjk - setup time between jobs j and k sijk - setup time between jobs j and k depends on the machine

Preemptions prmp - jobs can be interrupted during processing

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Precedence constraints prec - one or more jobs may have to be completed before another job is allowed to start its processing

may be represented by an acyclic directed graph G=(V,A)V={1,…,n} corresponds to the jobs(j, k) A iff jth job must be completed before kth

chains each job has at most one predecessor and one successor

outree each job has at mostone predecessor

intree each job has at mostone successor

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Breakdowns brkdwn - machines are not continuously available

Machine eligibility restrictions Mj - Mj denotes the set of machinesthat can process job j

Permutation prmu - in the flow shop environment the queues in frontof each machine operates according to the FIFO discipline

Blocking block - in the flow shop there is a limited buffer in betweentwo successive machines, when the buffer is full the upstream machineis not allowed to release a completed job.

No wait no-wait- jobs are not allowed to wait between twosuccessive machines

Recirculation recrc - in the job shop a job may visit a machinemore than once

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Constraints

blocking

Machine Eligibility

Completion time

Start time Buffer Space

Jobs Machines

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Optimality criteria

We define for each job j:

Cij completion time of the operation of job j on machine i

Cj time when job j exits the system

Lj = Cj - dj lateness of job j

Tj = max(Cj - dj , 0) tardiness of job j

otherwise0

if1 jjj

dCU unit penalty of job j

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Due Date Penalties

jdjC

jL

jdjC

jT

jdjC

jU1

jdjC

In practice

Tardiness

Unit Penalty (Late or Not)

Lateness

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Possible objective functions to be minimised:

Makespan Cmax - max (C1,...,Cn)

Maximum lateness Lmax - max (L1,...,Ln)

Total weighted completion time wjCj - weighted flow time

Total weighted tardiness wjTj

Weighted number of tardy jobs wjUj

Examples

Bicycle assembling: precedence constrained parallel machinesP3 | prec | Cmax

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Summary

Scheduling is a decision making process with the goal ofoptimising one or more objectives

Production scheduling problems are classified based onmachine environment, job characteristics, andoptimality criteria.

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Classes of Schedules

• Nondelay Schedule: A feasible schedule is called nondelay if no machine is kept idle while on operation is waiting for processing.

• Active Schedule: A feasible schedule is called active if it is not possible to construct another schedule by changing the order of processing on the machines and having at least one operation finishing earlier and no operation finishing later.

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• Semi-Active Schedule: A feasible schedule is called semi-active if no operation can be completed earlier without changing the order of processing on any one of the machines.

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Complexity Hierarchies

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Classic Scheduling Theory

• Look at a specific machine environment with a specific objective

• Analyze to prove an optimal policy or to show that no simple optimal policy exists

• Thousands of problems have been studied in detail with mathematical proofs!

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Example: single machine

• Lets say we have– Single machine (1), where– the total weighted completion time should be

minimized (wjCj)

• We denote this problem as

jjCw||1

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Optimal Solution

• Theorem: Weighted Shortest Processing time first - called the WSPT rule - is optimal for

• In decreasing order of wj /pj .

• Note: The SPT rule starts with the job that has the shortest processing time, moves on the job with the second shortest processing time, etc.

jjCw||1

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Proof (by contradiction)• Suppose it is not true and schedule S is optimal• Then there are two adjacent jobs, say job j followed

by job k such that

• Do a pairwise interchange to get schedule S ’

k

k

j

j

p

w

p

w

j k

k j

kj ppt

kj ppt

t

t

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Proof (continued)

The weighted completion time of the two jobs under S is

The weighted completion time of the two jobs under S ‘ is

Now:

Contradicting that S is optimal.

jkjkk wpptwpt )()(

kkjjj wpptwpt )()(

jjkkk

kkjkjj

kkkjjjkkjjj

wpptwpt

wptwpwpt

wptwpwptwpptwpt

)()(

)()(

)()()()(