1 set #1 dr. lee heung wing joseph email : [email protected] phone: 2766 6951 office : 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
)()(
)()(
)()()()(