on cyclic plans for scheduling of a smart card personalisation system tim nieberg universiteit...
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On Cyclic Plans for Scheduling of a Smart Card
Personalisation System
Tim NiebergUniversiteit Twente, EWI/TWDWMP-Group
Overview / Objectives
Give abstract model of scheduleDefine (L,f)-cyclic schedule
Bounds on Cycle-Time Special Schedules
Tight LoadingSingle-Mode
Optimal Plans for Case Study
Model of Personalisation System
n Smart Cards k Pers. Stations
Loading/Unloading Personalisation
m Graphical Machines Processing Time
Conveyor Belt with n+k+2 slots underneath
J1,…,Jn
S1,…,Sk
Pin,Pout
Ppers
M1,…,Mm
pj
pmax:= max pj
n>1’000 k=4,8,16,32
½ 10-50
k=5 pPR=3
pFO=3/2
pL=4
Assumptions w.r.t. Case Study
For now, we assumeNo time needed for placing cards onto beltNo gap b/w personalisation and graphical
treatmentFlip-Over machines use single slot
Equivalent to real case
No faulty cards
Characterization of Schedules
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:BeltConveyor
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Cyclic SchedulesLnlnL ,...,1each for s.t. 0 and an exists thereSuppose
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L
(L,f)-cyclic Schedules
Definition:A cyclic schedule that involves placing L smart
cards onto the conveyor belt, and that uses f free slots, is called (L,f)-cyclic schedule.
Maximizing Throughput
<=>
Minimizing Cycle Time
Lower Bounds on Cycle Time-> Personalisation
Consider personalisation part of system
Claim 1:Any cyclic schedule has cycle time of at least
Pin+Pout+Ppers+1.
This is the minimal time to personalise a smart card in one of the personalisation stations.
Lower Bounds on Cycle Time-> Graphical Treatment
Consider feasible, (L,f)-cyclic schedule Belt has to advance L+f times + Lpmax for bottleneck machine
+ other f free slots under bottleneck machine Some machine(s) have to process maxF denotes Fth largest processing time in case that F free slots are
arbitrarily presented to graphical machines M1,…,Mm
i.e.
Fm
pp mF },...,{min:max 1
Lower Bounds on Cycle Time-> Graphical Treatment
Claim 2: An (L,f)-cyclic schedule has a cycle time of at least
)max()()( maxFfpLfL
Advancement of Belt
Processing Bottleneck Machine
LB on Processing Non-Bottleneck
Special Schedules:Tight Loading
(k,0)-cyclic schedule All k personalisation stations loaded and unloaded at once
Special Schedules:Tight Loading
(k,0)-cyclic schedule All k personalisation stations loaded and unloaded at once
Special Schedules:Tight Loading
(k,0)-cyclic schedule All k personalisation stations loaded and unloaded at once
Tight Loading: Properties
TL dominates any (L,0)-cyclic scheduleL>k: easy (split into subschedules)L<k: yields equal or worse throughput
For the personalisation stations:Loading is done directly after advancement of beltUnloading occurs just before next advancement
Theorem 1:Any (k,0)-cyclic schedule only loads and unloads
from and to the same slot on the belt. Idea of proof:
Any other schedule results in infeasibility after insertion of at most k new cards.
Corollary:Any other schedule uses at least one free slot per
k smart cards.
Uniqueness of Tight Loading
(Super) Single Mode
At beginning of cycle, a free slot is inserted into system 1.) Personalisation Station unloads if free slot is advanced
underneath 2.) Belt advances 3.) New card is now loaded into Pers. Station
Single Mode is event-driven Advance belt as soon as all task have been completed
Single Mode respects order of smart cards Simple inductive arguement
(Super) Single Mode
SM defines (k,1)-cyclic schedule When personalisation is bottleneck, i.e.
Ppers+Pin+Pout > k + k pmax + max1,then SM is optimalPf: Claim 1 => each Pers. Station is optimally
utilized.
Optimal Schedules for Case Study
Overview of bounds obtained thus far (for (k,f)-cyclic schedules):
From Claim 1:
Optimal Schedules for Case Study
(k,0)-cyclic schedule does not meet bound for any Ppers > 10 in case study
By Theorem 1:Improvement, if exist must use at least one free
slot per k smart cards=> Single Mode
Optimal Plans for Case Study
Note that inserting even more free slots must result in plans with strictly greater cycle time
Notes on the Assumptions
Some assumptions made can be “revoked”Loading/Unloading of conveyor belt always takes
less time than bottleneck task of graphical treatment
Gap b/w Personalisation Stations and Graphical Treatment does not affect arguements presented
Conclusions
A simple characterization of cyclic schedules by the number of free slots they use has been presentedThis characterization was used to show that there
exists only one (k,0)-cyclic schedule (Tight Loading)
Lower bounds on the cycle time of (L,f)-cyclic schedules were given
Using destructive bounding methods, the instances of the CYBERNETIX case study were solved at optimality