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On Cyclic Plans for Scheduling of a Smart Card Personalisation System Tim Nieberg Universiteit Twente, EWI/TW DWMP-Group

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Overview / Objectives Give abstract model of schedule Define (L,f)-cyclic schedule Bounds on Cycle-Time Special Schedules Tight Loading Single-Mode Optimal Plans for Case Study

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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 J 1,…,J n S 1,…,S k P in,P out P pers M 1,…,M m p j p max := max p j n>1’000 k=4,8,16,32 ½ 10-50 k=5 p PR =3 p FO =3/2 p L =4

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Model of Personalisation System

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Assumptions w.r.t. Case Study For now, we assume No time needed for placing cards onto belt No gap b/w personalisation and graphical treatment Flip-Over machines use single slot Equivalent to real case No faulty cards

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Characterization of Schedules

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Cyclic Schedules

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(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

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Lower Bounds on Cycle Time -> Personalisation Consider personalisation part of system Claim 1: Any cyclic schedule has cycle time of at least P in +P out +P pers +1. This is the minimal time to personalise a smart card in one of the personalisation stations.

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Lower Bounds on Cycle Time -> Graphical Treatment Consider feasible, (L,f)-cyclic schedule Belt has to advance L+f times + Lp max for bottleneck machine + other f free slots under bottleneck machine Some machine(s) have to process max F denotes F th largest processing time in case that F free slots are arbitrarily presented to graphical machines M 1,…,M m i.e.

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Lower Bounds on Cycle Time -> Graphical Treatment Claim 2: An (L,f)-cyclic schedule has a cycle time of at least Advancement of Belt Processing Bottleneck Machine LB on Processing Non-Bottleneck

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Special Schedules: Tight Loading (k,0)-cyclic schedule All k personalisation stations loaded and unloaded at once

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Special Schedules: Tight Loading (k,0)-cyclic schedule All k personalisation stations loaded and unloaded at once

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Special Schedules: Tight Loading (k,0)-cyclic schedule All k personalisation stations loaded and unloaded at once

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Tight Loading: Properties TL dominates any (L,0)-cyclic schedule L>k: easy (split into subschedules) L

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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

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(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

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(Super) Single Mode SM defines (k,1)-cyclic schedule When personalisation is bottleneck, i.e. P pers +P in +P out > k + k p max + max 1, then SM is optimal Pf: Claim 1 => each Pers. Station is optimally utilized.

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Optimal Schedules for Case Study Overview of bounds obtained thus far (for (k,f)-cyclic schedules): From Claim 1:

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Optimal Schedules for Case Study Tight Loading has Cycle Time Compare with LB

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Optimal Schedules for Case Study (k,0)-cyclic schedule does not meet bound for any P pers > 10 in case study By Theorem 1: Improvement, if exist must use at least one free slot per k smart cards => Single Mode

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Optimal Plans for Case Study Cycle Times for Single Mode

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Optimal Plans for Case Study Cycle Times for Single Mode

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Optimal Plans for Case Study Cycle Times for Single Mode

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Optimal Plans for Case Study Note that inserting even more free slots must result in plans with strictly greater cycle time

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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

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Conclusions A simple characterization of cyclic schedules by the number of free slots they use has been presented This 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

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Thank you for your attention… T.Nieberg@utwente.nl

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