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Deadlock avoidance in mixed capacity flexible manufacturing systems

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Title:
Deadlock avoidance in mixed capacity flexible manufacturing systems
Physical Description:
Book
Language:
English
Creator:
Mohan, Sridhar
Publisher:
University of South Florida
Place of Publication:
Tampa, Fla.
Publication Date:

Subjects

Subjects / Keywords:
Colored Petri Nets
polynomial complexity
real time control
Dissertations, Academic -- Industrial Engineering -- Masters -- USF   ( lcsh )
Genre:
government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
ABSTRACT: This research addressed the design and implementation of a polynomial-complexity deadlock avoidance controller for a flexible manufacturing cell modeled using Colored Petri Nets. The cell model is robust to changes in the part types to be manufactured in the system and is automatically generated using the interaction of the resources in the cell and the technological capabilities of the machines. The model also captures dynamic routing flexibility options. The framework introduced separates the cell model from the control logic allowing the system designer to implement and test various control algorithms using the same cell model. The controller adopts the neighborhood deadlock avoidance policy to resolve deadlocks and control the resource allocation decisions within the system. The evaluation of the performance of systems controlled by not maximally permissive algorithms is important in determining the applicability of the control algorithms. There are many polynomial time deadlock avoidance algorithms proposed for the control of general resource allocation systems. However, the permissiveness of these algorithms is not quantified and the applicability of these algorithms in terms of effective resource utilization remains unanswered. The performance of automated manufacturing cells controlled using the neighborhood deadlock avoidance policy is benchmarked by comparing its performance with other control policies.
Thesis:
Thesis (M.S.I.E.)--University of South Florida, 2004.
Bibliography:
Includes bibliographical references.
System Details:
System requirements: World Wide Web browser and PDF reader.
System Details:
Mode of access: World Wide Web.
Statement of Responsibility:
by Sridhar Mohan.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 68 pages.

Record Information

Source Institution:
University of South Florida Library
Holding Location:
University of South Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 001478784
oclc - 56563322
notis - AJS2474
usfldc doi - E14-SFE0000428
usfldc handle - e14.428
System ID:
SFS0025120:00001


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Deadlock avoidance in mixed capacity flexible manufacturing systems
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ABSTRACT: This research addressed the design and implementation of a polynomial-complexity deadlock avoidance controller for a flexible manufacturing cell modeled using Colored Petri Nets. The cell model is robust to changes in the part types to be manufactured in the system and is automatically generated using the interaction of the resources in the cell and the technological capabilities of the machines. The model also captures dynamic routing flexibility options. The framework introduced separates the cell model from the control logic allowing the system designer to implement and test various control algorithms using the same cell model. The controller adopts the neighborhood deadlock avoidance policy to resolve deadlocks and control the resource allocation decisions within the system. The evaluation of the performance of systems controlled by not maximally permissive algorithms is important in determining the applicability of the control algorithms. There are many polynomial time deadlock avoidance algorithms proposed for the control of general resource allocation systems. However, the permissiveness of these algorithms is not quantified and the applicability of these algorithms in terms of effective resource utilization remains unanswered. The performance of automated manufacturing cells controlled using the neighborhood deadlock avoidance policy is benchmarked by comparing its performance with other control policies.
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PAGE 1

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PAGE 4

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PAGE 5

r ' n n r r ', n r ' # nn r ', n ,$ n n n # r ',' n rn + r ',', n n + r ',' n r ',' n rn n .# ) ) .. n .. ', r ',' rn / r ',', rn rn ',' n # r ',' / n . ## r n #n r n #n rr n # # ) r n nn ) r r nn

PAGE 6

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PAGE 7

n n ', n n n n rn #nn #nn n #n rn # n n ', ,n .' nn / n .', n r r n .' / nn n r rn n rnnnn n r nn ., n r ', n rn n n n ., n r n rn n n #n r n ', nn ) r n

PAGE 8

n ) n n nn n!n n n n n #n ) n n n n n n nn n n nnn n' ) n n nnn n' ) / n n /n n n n n n ) n n n n nn /nn ) n n nn n n n n nn nnn n' ) n n n n n /& nn n n n nnn n n n nn n n /' ) n n n n n n n n /n nn

PAGE 9

n n rn r n n n n & /n n' r n n n n n / n ) n n r n n n r n' r & n n n n n n n' n n n n & /n n n n n n n n n n /n n& n n n' r n n n n /n n nn n& ' n / n r n n n n' n n n n' r n n n' r r n n n /n

PAGE 10

n n n rn r nn n n nn n / r n #n r & +& & & & .& n & r & -& ,& .,& & nn ,. & #n /n n & & n & r & n #n nn& n n n /n n n n/ n n ,' #n & & & #n n /n nn n' n #n n n /n & /n n n n n' n n #n n n n n / n n nn n n nn /n n n n n nn& / n n & /n n n n nn& n n n' n / n r r, / n n & / n n n n n r n r r n r & n n n n ) n n n /n & / n /nn n "& /n n n n "' ) n n n / / n n n /n / n n n nn $ r n' ) n n'

PAGE 11

,' n n' n n n' #n' # n n .' n /n' n n& / /n n n & n n / n n ',' ) r n n n / n nn n ) n n n nn' Machine M1Machine M2 Parts from M1 requires M2 Parts from M2 requires M1n ' Machine 1Machine 3 Machine 4 Machine 2n ',' n n n ) /n n nn'

PAGE 12

' #n' n n n r ,' n ) n / /n n n ' n' ) n n n n nn n n n n& n nnn n n/ & ' r & / n n & n& n / nn #n n & n / n / n n n nn ) n n / /n n & n n n #! n /n /n n n n n r & n nn n& ) nn n n r n ) nn n nn n ) nn n n n n' nn n n n r & n n n n n n / n r n & / n

PAGE 13

) n n nn n n nn n n n n' ) r / /n n n n n n n' n / & n n' n n / & r ) n n nn n nn / n nn / n n n n n n ) n n / n n ' n n / nn n n n nn' rn nn nn n n nn n n nn n nn n n n n & /n n nn n n "n' & nn nn& n & n n nn n n n n n' ) n n n n n n nn n n n n& /n n n nn n n' ) n n n n n n n n #n n n n nn / & nn n n n n / n r r

PAGE 14

A Non maximally permissiveallows a subset of the safe states Safe states: Maximally Permissive Controller allows for all safe states to be visited Unsafe states: Deadlocked or leading to inevitable deadlock n ' rn #nn #nn n n n n nn /n n nn ) n n n /& n n n/ & /n n nn / n n' n "n #n n n n #n' n / n! n r /n n n n #n # ) n n n n n r /n n /'

PAGE 15

) n n n n n n n n n #n n r n n # /n n & ' + ) #n n n& /n n & n n n' /& & r & #n n' n / & ' + & n n / & /n n #n n n n n n n ) n & nnn n n n& n n n n& /n ) n n n n n n n& ' (n& ' r & n n n + & nn rn #n r# n n n n ) n n n n n n & ' + (n& ' r & / n n n & & n

PAGE 16

n ) n nn & n n n / n / n' ) n (n& r & / #n & n n n (n& r n! nnn n n n nn #n n n n n n ) #n n & ' + & #n n & n n / n nn n n n n nn n n n nn& nn n #n n n & / n n n n (n .' n n & ' & & (n .& nn n & ' ) n n n n n n n n ' / n& /n n nn n n n& '' ) /n n n nn n n n ) nn n n n / n n n' nn nn n n n /& nn nn' ) nn nn n n nn n n +

PAGE 17

n n' n nn n& n n n& ' ) n n / n #n & ' + n& ' & & / n n nn & n n n r r n n / n n #n ) n / n n n / nnn n' n n n & ' & n n /n n n n / & ' r & r n n n n' ) n / n n n ) n n & ' + & n n& ' ) r / n' n n/ & ' r & / n n n& n n n' & n / / n & / / nn & n n n % n n #n & /n n nn nn n' ) n #n n

PAGE 18

r #n n n n n n #n n #n & n #n # & /n n n n n& & / n n /' ) # nnn / / nn /n / n n n / nn& / # / n n' ) n n n n nn' r& n n & /& n / #n /' ) n n n n nn /n & n #n n n n /& nn& n + n n n n' n n! n /n nn ) # n' nn' ) n n n nnn' ) n n nnn n n nn' ) n n n n & n n n n n n n! "n n "n "n ) nnn n n n ) n n n /n n nn n ) n n n n n n n

PAGE 19

n n' nn n ,& n n n n +& ,.& /n n n n n n n n r / n n r #n & ,+& & n r n #n & n n #n nn nnn n n r ) n n #' n nn& # n n ) n n n n n n n & /n # n ) n # n n n /' ) n n n /n n n n n n nn' n n n / n n n n nn n n nn' ) / n n n n n n & n n (n .n #n n n /& ) "n n / n # n ./n n n n'

PAGE 20

) n n # n (n .' ) $ ) n n n n /nn ) n n /n n nn n n n n n n n /n n nn n ,' ) n n n n n nn n ' ) #n n n .n n .' ) n ) n n # /n n nn n' ) n n /& n ', n # n n r n .& n n n n n n nn n n n nn n n #n ) n #n #n n n n n

PAGE 21

n nn #n n (n .n n n n n' ) #n n /n n n n n& n / n n #n n #n n / n n ' I1 pi_M1 pi_M3 pi_M2 pi_M4 RM1 M1R M3R RM2 M2R O1 RM4 M4R pi_R RM3 rc_R rc_M1 rc_M4 rc_M2 rc_M3n ' #n rn # n #n n n nn ) n #n nn / / ) / & nn nn n r & /n nn nn n n r n r ) n / n n n / n' ) nn n n #n & n n& n

PAGE 22

/ n n n nn' ) nn r / / rn r & r / ) n n / nn n #n ) n n #n n n & #& #& n /& n & n n nn & n / n n n n n #& # n n #n n nnn n n n #n n nn n n n ) /n nn nn nn n n n r r ) n nn n n n nn n & / n n n n n /n n& ----& --& --&'''& n n n n n n ) n nn n n n n /n n n n n /n' r r n n n n n

PAGE 23

n n n n / n ) ' n r --n r n r. -nn n .' n n nn n n n' ) n n / nn n / / n n nn n / n' & n r n ---n n n r. n n& n n -,. n -,. & n n / n n n n nnn r. / --, ) nn n n n n n / n nn n n n / n' ) ' # n n r --n r -n r -n r. -) nn n ) & n nnn n n n n n n& n ) n nn / n n ) n n n n & / n n n ) n n nn n n n' ) nnn n n & n r r

PAGE 24

n & & nnn ) & / ' n *& * n r r & r ,& r & r. !, ,! .! ) n / /n n n (n .& n , & n n n & n nn n n / & nn n n ,' ) nnn & /n nnn & n /n n r --' ) n nn n n n n ) * n ) n / nnn /n nnn ,& r

PAGE 25

r r n r -' ) n& n n n n n ) ' ) #n n n #& #& n & & /& n n n n n n n ) n n n ) nn n #n n n / & r r n n n r ) nn /n ---/n n n n ) & r /& n n r r n n r n nn n n n /n /n & n n n n' ) n n nn n n n n' ) n nn n n n / n /n n n' r r n n r r r

PAGE 26

n nn n nn n n n n /n & n n n /n n' n n & n n & n nn n ) n n /' n & --n /n n n n n n nn r /n & ) & / rn r n #n #n r & /n /n r n n #n r n & nnn n n n nn' ) / n n rn n / / & n n n r n nn n n r n n n n n n n n r r n n n n r ) rn n n n /n n & ! ----! --+

PAGE 27

n n ! --rn n n & / nn / n' / & --' & -' n n n nn n n n n nn nn n n nn n n ) n n n n ', n nn n n n nn n n n n nn # nn /& n n n !n ) n nn n n n n' ) n n n n n n n #n n & & & +& ,& ) n #n n n n n n n n ',' n n& / n n # n & & + n # n n n n ',' & n n n n /& n # n&

PAGE 28

n n /n n n nn n n / n n n # ) n # n & n n n n n ',' n & n n n n n n ',' s s n ',' n #& n n n n n' ) n n n n n nn n n /n ) n n n n n n & nn n n ) n n n nn n n ) n n n n n n ) nn n n n n n& r r & / r n rn & & r n ) n n n n !n n n! ,-

PAGE 29

' ) n n n n # n n' ) n n ) n n n n ) n rn n n /n n $ rrr n n n n & / n n n /n n n n n ', / n n nn n n ) ,' n nn n n n /& n nn n / n n n & n / n nn n ) n n n / & n n n ) n n nn n n& r n r n n r

PAGE 30

/ r n / n n ) n n n / n n n /n nn n n n ,' ,' n /n "& & r &r & r ,& r & r. & / n n n nn & & & .&,& nnn n #n n ' ) n n n nn n -n n n ) n n& & n r. r ,' ) nn ) n n & n n r n r.& . ) n r n n nn ) r n n r & n r n n ) r n& r ) n rn' ) / n n r n /n' ) n n n n /n n / n ) ',' ,,

PAGE 31

) ',' ) n rn n n n n n n n ) n n n n' ) n n n & n n n n n n n n n n n & n # n& n n n ) & / n n n n & n / n n / n n n n' ) n n n nn nn nn n ) & n n & n & n' '& rn ) n n n /&

PAGE 32

, n n #& n rn n / / & , ) & /n n n n n n n n n n ) / n n & n / n n n n n nn n n n& n n n rn ) n rn n n rn n ,.

PAGE 33

n & n n #n' n / n n n & nn n n' n / n # & n n n n n' n n .& n & n n n n n n n n n n' ) n n / n n n' n / n ,' ) n n / / & , ) nn n rn n n n / n n .' r

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, , / & nn # n n n / n n ,' n n #n n n n n ', n n' n n # n # n' ) n n n n /n n n n nn n n n n n ' ) nn r & n n n n n

PAGE 35

" n n nn n n n n' n nn n / n nn n nn n / n nn n "' "& n "!n n & n n n /n n & /n n n n n n n /n n " n n' ) nn "! n n n n n / /n n n /n n& & n& n& nn n nn .& n & n n n n /' ) n n n nn / n' ) r n "' n n n ) n n n n n nn n nnn' ) n n n n & n nn n

PAGE 36

) n n n n n n n n n /n n n n n& n n' ) n n / n n n n' ) # n n n ) n n n& n n' ) n nn /n n' ) n n n / n/ n n & n n n n n ) n n n$ # & ,' rn r& .' ) n n n n& & n n nn /n ) n n n n n & n n / n/ & n / n/' n ) n nn n n n & n n n n n nn' ) n n n n! n n n n n nn n n n n n n n n n' ,+

PAGE 37

) n n / n n .', nn / / n n .' n n n n / & PART 1 MACHINE 1 HANDLER MATERIAL PART 2PART N MACHINE 2MACHINE M PART CLASSMACHINE CLASS CELL MODEL CLASS CONTROLLER CLASS MANUFACTURING SYSTEM SUPERVISORY CONTROL allow/disallow allow/disallown .' nn / # ) ) n n & /n n n' ) n& n n n n & n & n n' rn ) n n& n n n n nn& n n n n' ) " n n nn n nnn n n n n n n n $ ) n& & n

PAGE 38

Find deadlock(); generate color(); Number_of_machinesNumber_of_partsCreate machines();Create parts();Create A_Matrix(); Attributes: neighborhood matrixstate vectorcapacity matrix Update Color() part (); Methods: colorpart typepart number ; Attributes: parts_present Capacity Part ClassCell Class Controller Class Find enabled transition(); Attributes: S Matrix D Matrix Methods: Machine Class Attributes:Methods: VectMachine(); Methods: Create neighborhood matrix();Create State Vector();n .',' n r r n n n r ) n /n n n n & ) n nn / n& n ) n nn n / n n & n nn n #n $ ) n n n n n n& ) n r & n # ) n n n n n n

PAGE 39

n n n n ) n n n /n n n n nn n n' n n ) n nn n & n n nnn' ) / & n n n n nn n n n n n & /n n n n! n' ) n nn / & n n / n ) n n n n' ) nn & n & n n n n n n n' n ) nn n n n n "!n n n nn !n nn n "!n ) n n n ) nn / n/ ) / n .' n

PAGE 40

nnnn n& # n n n nnn' ) n n n n n # n nn n n n # ) n n nn' nn /n n n & nn nn & n n n / n/ nn' nn n /& n n n # n nn n n/& n n n n / n / nn' & nn n' ) nn n nn n n n n n ) nn n n & /n n nn nn' ) nn n n n n ) n n / n n nn n n r ) n n n n n nn n ) n nn r n n n ', /n # n n n n

PAGE 41

Calculate the cellCPN model Input Machines and operations Calculate the Neighborhood Matrix Calculate the Initial Marking Check for an enabledtransition in the CPN model Allow/Disallow using Control Policy Update State Vector and CPN Marking Check for events in queue transitions in pending Input the Part Typesto be processedInput the Parts to be processed ExitSimulation Add transition tothe pending events queue Allow NO YES YES NO Disallown .' / nn

PAGE 42

) n # nn n nn n' ) # n n nn n n n' ) n n nn n # n n nn& n n n nn n n n! nn n nn ) n n n / n n ) n # n n n' n n / n n n nn ) n / n n n n n n' n n n n n n + & # n n n n n n n n ', & /n nnn n # n / n nn # /n n n' ) "n # n n n n n n n n n n n n / n' & n n /

PAGE 43

n n n n nnn n / /n n n nn & # ) & # ) "n n # /n n n n n n n r / n nnn n n n ,' n r n nnn n r ', n n n nn & n / n n n nn n n n n n n' n r r n n n' n r r r n / n n' n n n n & n /n n& n /n n' ) n n n ) n n n n n n n n r & nnn n' n n & n n n #' n!!n! n n n n n' nn #& / n / ) n n n n & / n n n nn nn #' n # n n n n / n n nn' / r

PAGE 44

n n & n n n / n n n n n n n n ) r ' # # n n n nr n n nn n n n ) n n n n # n / n ) r ' ) nn n n n n' ) n n / /nn n ) r / n n rn #n n n n r & /n n' nn nn n n / n n n / n n / n n /n n n& nn n /n n nn n n n n n n n

PAGE 45

n n n nn n /n nn r# n' ) / n ) r ',' n n & / nn nn n n # nn n / ) n nn n n / n n nn nn n ) r ',' n # nn n nr n nr n n n nn n nn nn n n n n n ) "n n n n n n n n nn' ) # n & /n n /n n nn n # /n n n nn n n /n nn n nnn n nnn n /n / n n'

PAGE 46

n n n n n / n n n & /n n n n n n n r ',' ) n n n n / n n n r ',& rn r r n nn r, rn r n nn r n /n r n n n n ) n n / n ) r ' n nn n n& /n r, r r r r n& n n / n # #, n r r #, #. n r r r nn n n' ) r ' # # ) # n + & n n n # n n n / n' ) n n n n n r /n n' ) n n& # n r -n n /n n --! n n n n n n # n! nn n / n n n n' n n ) / nn / & n r r n ) r ' n n +

PAGE 47

& ,nn n ,n r ) n n / n ) r .' n& n / r n r .--n n n n r -, r -n n' n ) ) r / r n n! n n n #' n # n -' n n n # n n & /n n / n / n n' ) r n n n n n n' ) # /n / / n nn /n n n n' ) n n n n n n n & /n n /n n n ) n& n & n n n n& n n n n n n n n ) n n n # / / n / nn / n n n n n' ) n n n n n n' ) r .' n nn n nn

PAGE 48

n n n n n n & / / nn n n nn n n /n n /' ) n n n / n n r ' ) n n n & n n n n ) # n /n n n n n' ) n n / n ) r r n # n / n n /n / nn& # / n n n' ) n n /n n n' n n # n / /n n n n /n n n n' ) r r n rn n nn n nn n n n n n n n n n nn # n n ) n nn & n nn' ) n # nn nn n n nn n n nn / nn n& /n nn nn n n n /n n' ) n n & n nn n& nn n n / #' n & n n n # / n ) n n n .-

PAGE 49

! n n n /n n n' ) n n n # / n n n ) n n n nn & n n /n n' ,' ) n n n / /nn n ' ) n /n n nn .' n n n & # n' r nn n& nnn n n / nn n& / # n

PAGE 50

Minimize M s.t M-xiK 0 i I (5.1) xikjtikj xik-1l i I xprjxqsj+H(1-yprqsj) tprj j J, (p,q) I xqsjxprj+H(yprqsj) tqsj j J, (p,q) I xp(r-1)lxqsj+H(1-yprqsj) E j J, (p,q) I xq(s-1)lxprj+H(yprqsj) E j J, (p,q) I where xiK completion time of last operation K of job i xijk completion time of job i operation k on machine j tijk processing time of job i operation k on machine j yprqsj =1 if job p operation r follows job q operation s on machine j. 0 otherwise H large positive number E small positive number I set of all jobs J set of all machines M makespan (5.2) (5.3) (5.4) n r ' rn n rnnnn n r nn M1 M2 M6M5 M3M4 I/O TurningPT:30 TurningPT:30 DrillingPT:20 PT:10 Punching DrillingPT:20 MillingPT:40n r ',' n rn n n n .,

PAGE 51

Capcity:1 Capcity:1 Capcity:2 Capcity:2 Capcity:2 M4 M3 M2 PT:10 PT:30 PT:40 I/O Milling PT:20 Drilling Punching Turning M1 n r ' n rn n

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n n ) r / / n (n .n n n n nnn n& ) n n .n n / n nn n r n n n' nn n nn n n n n r n n n' ) n n n n n r n n n n n n ) #n n .n n ) n n n n #' ) n n n n n n & / n nnn n n nn n / #' ) / n #' ) n n n / # n nn n' ) nn # /n n n n n ..

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n nn n' / # n n n n n n n n nn n ) n nn nn ) n n n # n n n & / n nn n # n / n n n n nnn ) n n n n #n (n .n n n /n n n n n n n n r nn' ) n / nn n n' n nn n n n' ) n nn / n n / & /n n nn n n' n n / nn n nn n n nn #n n n n' n n n n n n ) #n n n n ', / n n n n n n' n n n r n n n nn' r

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n n ) nn n n nn n n' ) n n n!n n n' r n n n n n ,, n ) #n n n ', n n ) / n n n "n "n n "n n n n ) n n n / n n n n n n n n / n' ) n n n n n

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& ' r& ' n n n r $ #n r r n n n n r n & $ -. r & +' & ' %& ' n n n n /n n /' r r n r r & $ ,. .& -' n& % & n r r r n r & r & . ,.& r & r r n n n r n r "n! n #nn '& '& / n& & r & '& %& ) % n & ' !n n n n n r r n r r & $ ., & r & n& n #n n n n n' r r n r r & $ +-.& & *' & *' r n n #n r r r n r r & $ & + & *'& & *' r rn & *' #n n n n n r r n r r & $ +.& n .' n& r #' ) n& n #nn n n r! n r #n r r n n n r r r & ,& & n& r #'& ) n& rn& n n n n /n n n $, ,. & n& r #'& ) n& rn& n ) n n n n #n r n r n r n & r &

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, n& r #'& ) n& rn& ! n n n n r r n r r & $ & n& r #'& ) n& rn& n n #n n n r r r n r & r $ + +& ,---' & ( '& *& r '& n& & #n #n n rn n n' r r n r n r r & & r r & ,-' r & ( '& *& r '& n& & #n #n #n n' r r n n n r & $,+ r & ,-' *& % ' nn n #n r r r & +$, ,& %& '& & & # n n n /n n r $,+ ,+ & + r& n n n #n& #' nn' r & $+ ++ & r r& ) #n $#n& n nn' r r r r n & $ .. r & + ,n ' n n r r nr r n r # n& nn& -' #& *' r r n n n r n nn r r r r # n& n n ) & ,---' ,, #& *'& nn& '& & r nn& ' nn& n n n rn r r n n r r r n r & r & & ,--,' /& ' *n& ' r! n n n' r r n r r & $ .--& ,. nn& n& #' r n nn n r r n r r & $+. r + r & r nn& ' n& #' r n nn n n $ "n r r n r n r r & & r r +& .+

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, nn& ' #& *' n n n n n n /n r r r r n & $,+ r & nn& ' #& *' n!n n n n n /n n nnn n n' n r n & & , & ,---' ,+ nn& ' #& *' n n nn "! n n "n n r r n r n r r & & & ,-' nn& ' #& *' n n n n /n n nnn n n' r r r r n & $ r r + & ,-' nn& & /& r n& #' r n nn n /n nn r n n r r r & & ., .+& nn& & /& r n& #' r n n n n n n n r r n & ,& --+ ,& nn& & /& r n& #' r #n!n n #nn n n r r r r n & ., $ .. r & nn& & /& r n& #' r n n n n r r n r r & r $ +& +' & *'& & r '& & n '& & #n& n r r n n n n #n! & '& n& *& r n/ & '& n& ( *& ) ' n n n n n n #n r r n r r & $ & -' n & n nn n n n n n #n r r n r & , $ ,-.& r n & '& (& ' *n& n n n n r r n r r & $+ r + r & + n & '& (& ' *n& n n n r! n $n n r r & $ ,+ +& .'

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(n& rr r r n r nn r r nr # n& n nn& nn& ,---' .(n& & ) ' n n n /n nn n' r r n r r & r $ -& (n& & ) ' n n rn n nn r r n r r & $ .,. ., & ,---' ., (n & '& %n & *' n & ' n n n n n n!n n r r n n n r & r $,. r ,. r & & r '& r& % #& #' #n n n n rn r r n r & , $ r r & r r

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r

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n #n n # nn ) & n /n nn n ) nn n nn /n n!' n nn n n! nn n nn' nn n n n n & n n n n n n nn' n n nn n nn n /n nn n' ) nnn #n n n nnn!n' ) #n n nnn n #n n n n nn& nnn n #n #n n n & r n /& ) nn n nnn n #n ) n nn n #n n n nn n n & n / n nn n n nn n n n n & n n #n / n n n' '& nn n ) n n #n & / n n & n' '& nn n n n n #n n r

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n n n #n n n' #n n n & / n n / n nn n nn' n #n n #n n & n /n nnn n / n n n ) n n n & nn ) n / n n' & n nn n / n ',' n #n n T1T2 T3 P1 P2P3n ' #n r

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n n n (1,0,0)(0,1,1)(1,1,0) Mo Mo T1T2T3 (1,0,1) T3 T2n ',' nn ) r

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n n n / /n n nn n n #n ) #n & nn n ) n & n n n n ) n nn n #! n n n n n n n n / !n n / n n ) nn & n nn& n n & n! nn ) n r n / n n nn & n r & n r nn n r & n' ) r n r r n nnn #n #! # & n & r r /& n n & nn # n n & nnn ) n n nn& n n n & & n n& n # ) ) #& n n n # n & rr

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n n n nn n& nnn n n n& n n nnnn n n' r

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n n ) ' ) r r ) ',' ) ) ' ) r

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n n n ) .' ) ) r ) r ) ' ) ) ' ) ) '+' ) + ) ' ) ) -' ) r +

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n n n n ) n "!n nn /& n & n rn & /n n n ) n n n n /n r n ) r n "' ) / n /n nnn n n n n n nn n & n nn' / n ) /n / n nn $ n /n n n ) / n /n & # ) ,' rn ' .' r ' ' rn +' #n ' -' r n r r r r r n r n r n r r r r n n ) n n n ) / n & r

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n n n n n n n n n ' n n n & /n n n nn' ) n nn / & n n n n n n' ) nn & n n & n n n n n' r r r r r ) nn / n n ) & n n ) n n n n ' ) n & n' r r r n r ) nn n& n n' ) n nn n n ' ) nn n n / nn & ' r r r & '& n n r r ) n n n & '' ) n n nn nn n n& '& ,' /n ) n nn ,& n nn