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

Model of Personalisation System

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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processing starts time

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advancesbelt when the times }|{

:BeltConveyor

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Cyclic SchedulesLnlnL ,...,1each for s.t. 0 and an exists thereSuppose

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cyclic. schedule resulting thecall then wehold,

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

Tight Loading has Cycle Time

Compare with LB

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

Cycle Times for Single Mode

Optimal Plans for Case Study

Cycle Times for Single Mode

Optimal Plans for Case Study

Cycle Times for 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

Thank you for your attention…

T.Nieberg@utwente.nl