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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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Literature:

1. Scheduling, Theory, Algorithms, and Systems, Michael Pinedo, Prentice Hall, 1995, or new: Second Addition, 2002Chapters 1 and 2

or

2. Operations Scheduling with Applications in Manufacturingand Services, Michael Pinedo and Xiuli Chao, McGraw Hill, 2000Chapters 1 and 2

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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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1. 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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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

2. 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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3. 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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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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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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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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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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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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P and NP problems

• The efficiency of an algorithm for a given problem is measuredby the maximum (worst-case) number of computational stepsneeded to obtain an optimal solution as a function of the size of theinstance.

• Problems which have a known polynomial algorithm are said to bein class P. These are problems for which an algorithm is knownto exist and it will stop on the correct output while effort isbounded by a polynomial function of the size of the problem.

• For NP (non-deterministic polynomial problems) no simplealgorithm yields optimal solutions in a limited amount ofcomputer time.

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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.