An Introduction to Max-plus Algebra

Hiroyuki GotoDepartment of Industrial & Systems Engineering

Hosei University, Japan

Outline of Today's TutorialPart I. IntroductionWhy max-plus algebra?What is max-plus algebra?How to use?

Part II. Relevant TopicsWhere is max-plus algebra?Relevance with close fields• Control theory, graph theory, discrete mathematics

Part III. Miscellaneous Topics & Recent AdvancesExtensions to wider classesExtension stochastic systems

2

Part I. Introduction

3

Why Max-Plus Algebra?Simple Scheduling Problem based on PERT

• PERT: Performance Evaluation and Review Technique

Project with four activities

4

Activity

Days

Predecessors

A dA --

B dB A

C dC A

D dD B, C

AOA (Activity on Arrow) Earliest node times

AON (Activity on Node)

1

2

4

3

AdA C

dC

B

dB DdD

,23 Cdxx ,1 ux ,12 Adxx ),,max( 324 xdxx B ,45 Ddxx

5xy

1 2 4

3

5

C dC

A

dA

B

dB

D

dDu y

Start Finishevent/state

Earliest Node TimesDerive an explicit form

5

,1 ux ,12 Adxx ,23 Cdxx

,45 Ddxx ),,max( 324 xdxx B

5xy

DCBA

CBA

CA

A

dddd

ddd

dd

d

u

u

u

u

u

x

x

x

x

x

),max(

),max(

0

5

4

3

2

1

DCBA dddduxy ),max(5

1 2 4

3

5

C dC

A

dA

B

dB

D

dDu y

Eliminate xi on the right hand-side

Lapse of time: '+' operationSynchronization: 'max' operation

Completion (output) time

Latest Node TimesCalculate the times from downstream to upstream

Slack time (margin)

Critical path 6

1 2 4

3

5

C dC

A

dA

B

dB

D

dDu y

,54 Ddxx

,043 xx

),,min( 432 BC dxdxx ,21 Adxx

yx 5

Eliminate xi using

0

),max(

),max(

5

4

3

2

1

D

D

CBD

ACBD

d

d

ddd

dddd

y

y

y

y

y

x

x

x

x

x

),max(),min( baba

0

)0,max(

)0,max(

0

45

23

24

12

CB

BC

D

C

B

A

D

C

B

A

dd

dd

dxx

dxx

dxx

dxx

f

f

f

fWijW dxxf

A, D, and (B or C)

Features of PERTTraditional PERT can describePrecedence relationships between activitiesDuration time of each activity

Limitation: cannot describe other practical constraints such asA single worker (resource) is assigned to multiple activities

The facilities (resources) process the same job repeatedly

Resource conflict may occur, etc.

7

Modeling and analysis method using max-plus algebra is an useful alternative approach

What is max-plus algebra?Basic operations

8

}{max RR),max( yxyx

yxyx Addition:

Multiplication: max, Ryx

Priorities of operators: '*' > '+' (same as conventional algebra)

Subtraction ‘-’ and division '/' : not defined directly

Zero element: xxx xxeex

)( )0( eUnit element:

0

1

•Examples5)3,5max(35

5),5max(5 )(44

3303 e83535

81795

Correspond to)0log( )1log( e

‘O-plus’

‘O-times’

Behavior of a Production System

Production system with 3 machines

9

Non-concurrency = each machine cannot process multiple materials at the same time

ix

iu

: Job number

: Material feeding time in input iy : Finish time

Earliest processing start times / output time)}1(),(max{)( 1111 kxkudkx

)}1(),(max{)( 2222 kxkudkx

)}1(),(),(max{)( 32133 kxkxkxdkx

)()( 3 kxky

Lapse of time: '+' operationSync., non-concurrency: 'max'

kM1

M2

M3

1x

1d

2x

2d

1u

2u

y3x

3d

: Completion time in machine i

Matrix OperationsSame rules as conventional algebra

10

ijijij ][][][ YXYX

}][]{[max][][][1

ljill

ljil

n

lij ZXZXZX

Addition:

Multiplication:

nm max, RYX

Zero matrix:εeUnit matrix:

pn maxRZ

All elements are e

Diagonal elements: e, off-diagonal elements: e

n

lljilij

1

][][][ ZXZXijijij ][][][ YXYX

•Examples

143

111

21131213

1111

1

111

23

eee

23

11

213

111

1

111

23

eee

Matrix Representation (1)Earliest node times of the four-activity project

11

u

e

d

ed

d

d

D

B

C

A

xx

bxAx

Initial stateDummy task (synchronization)

Linear form in max-plus algebra:

Precedence relations & elapsed times

⇒ MPL (Max-Plus Linear) form

,1 ux ,12 Adxx ,23 Cdxx

,45 Ddxx ),,max( 324 xdxx B

1 2 4

3

5

C dC

A

dA

B

dB

D

dDu y

5xy xy e

Matrix Representation (2)Earliest schedule of the three-machine production system

12

})(,)1(max{)( 11111 dkudkxkx })(,)1(max{)( 22222 dkudkxkx

})1(,)(,)(max{)( 3332313 dkxdkxdkxkx

)()( 3 kxky

)()1()()( 2

1

3

2

1

33

kd

d

k

d

d

d

k

dd

k uxxx

External inputs

)()1()()( kkkk uBxPxFx MPL form:

Non-concurrencyPredecessors

)()( kek xy

1

2

3

1x

1d2x

2d

1u

2u

y3x

3d

How to Solve the Equation?Substitute iteratively

13

bxAx bxAx bAx 1)( I

bbxAA )(

bAAexA )( 23

bAexA )(2

bAAAexA )( 12 ss

bAbAAAe *12 )( s

If the precedence relationships are represented by a DAG (Directed Acyclic Graph),

Cf. In conventional algebra,

)1(, 1 nsss εAεA (nilpotent)

Kleene Star (Closure)

Interpretation of the Representation

MatrixSolution of the recursive linear equation

14

1 2 4

3

5

C dC

A

dA

B

dB

D

dDu y

bxAx

bAx *

A: precedence relationsb: start time

j: source node

: earliest arrival times between two nodes = longest paths

eddddddddd

eeddddd

eddd

ed

e

DDDCBDCBA

CBCBA

CCA

A

)()(

)(

*A

i: destination node

Output timexCy eC : final node

u

e

d

ed

d

d

D

B

C

A

bA ,

Earliest Times of the Production System

Earliest times of the system with 3 machines

15

)()1()()( kkkk uBxPxFx

2

1

3

2

1

33

,, d

d

d

d

d

dd

BPF

)]()1([)( * kkk uBxPFx

3231

2

1*

33231

2

1* ,

dddd

d

d

ddddd

d

d

BFPF

1

2

3

1x

1d2x

2d

1u

2u

y3x

3d

)()( kk xCy

bxAx

bAx *

: earliest processing times between two nodes

: earliest processing times from the external inputs

Focusing on the Latest TimeLatest node times of the four-activity project

16

,54 Ddxx ,043 xx

),,min( 432 BC dxdxx ,21 Adxx

yx 5'min' and '-' operators appear

)][][(min][ ljill

ij YXYX

y

ed

d

e

dd

d

D

D

BC

A

xx

e

d

ed

d

d

D

B

C

A

CA ,

yCxA TT

)][,]min([][][ ijijijij YXYX Min.: Pseudo division:

Solution of the Recursive Equation

Solve (a kind of) recursive linear equationLatest times

17

dxAx T bxAx

bAx *dAx T*

yCxAx TT

ZXYZYX TTT )()(

Cf. Earliest times

e

d

d

ddd

dddd

y

D

D

DCB

DCBA

TTT

)(

)(

)()( ** yACyCA

1 2 4

3

5

C dC

A

dA

B

dB

D

dDu y

: the same as p.5

Part II. Relevant Topics

18

Relevant FieldsModern control theoryState monitoring and control of systems• Control input: start time of a job • Control output: end time of a job• State variable: event occurrence time• System parameter: duration time

Petri netRepresentation of the behavior of event-driven systems• Structure: synchronization, parallel processing, etc.• Place: conditions for event occurrence (non-concurrency,

capacity)• Transition: event occurrence, start/completion of an event • Arc: precedence constraint, sequence of events• Marking: system’s state 19

Relevance with Modern Control Theory

Earliest schedule for the 3-machine production system

20

)()()( ttt BuAxx )()( tt Cxy )()( kk xCy

Cf.)()1()( kkk uBxAx

Generalized representation

Same form as the state-space representationSome methods in modern control theory can be applied• Internal model control, model predictive control, adaptive

control, etc.

)]()1([)( * kkk uBxPFx )()( kk xCy

Relevance with Petri netBehavior of TEGs: expressed by an MPL formTEG: Timed Event GraphAll places have one input and one output transitions

21

M1

M2

M3

1x

1d2x

2d

1u

2u

y3x

3d

y

1x

2x

1u

2u

3x

1d

2d

3dCapacity of placesInternal: 1External: +Inf

Capacity=1

Capacity=+Inf

Ultra-discretization (1)Ramp function

22

)0(0

)0()(

x

xxxR

u

uxxS

)]0exp()ln[exp()(

x

y

O

)(xRy 0)0,max( xx

)(

)(

u

xR

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.60

0.5

u=1u=5u=10R(x)

x

y

max operation is related to exp and log functions

Ultra-discretization (2)Variable transformation

Addition

Multiplication

Zero & unit elements• Zero element:• Unit element:

23

)exp(),exp(),exp( uYzuYyuXx

and let (ultra-discretization)u

yxz )exp()exp()exp( uYuXuZ

u

eeZ

uYuX

u

)ln(lim

yxz

YXYX ),max(

)exp()exp()exp( uYuXuZ

u

eeZ

uYuX

u

)ln(lim

YXYX

Xx ,00,1 Xx

Semiring & DioidSemiringCommutative law:Associative laws:Distributive laws:

Three axioms for e and e:

Dioid (idempotent semiring)+Idempotency

24

),,( D

),()( zyxzyx

xyyx

)()( zyxzyx

)()()( zyzxzyx ),()()( zxyxzyx

,xxx ,xxeex ex

Dzyx ,,

xxx (does not hold in usual algebras)

),,( D is a semiring

is a Dioid

Some Classes of Dioid

Max-plus algebra:Max-times algebra:Min-max algebra:Min-plus algebra:Boolean algebra:

25

),(

)1,0(

),max,],1,0([

),( e),max,},{( R )0,(

),,( D

)1,0(

max),min,},{( R

)0,(min),},{( R

min),max,},1,0({

xxx

Note:These are widely referred to as *** algebra,but are not an algebra in a strict sense because of

Communication GraphState transition graph of a representation matrixNode: state (of jobs, facilities)Weight of an edge: transition time

Example

26

,

3

57

4

A

1 3

2

7

4 5

43

810

92A

,

12

3

A

4A: Reachable with k steps from j -> i maximum cumulative weight

ijk ][ A

Eigenvalue problem

Num. eigenvalues of square matrices: n or smaller

27

ee1

2

1

ee

2

2

1

e

eeee

1

1

111

eeee

xxA

1 2

1 2

1 2

0 1

0,

3

2,

2

1,

1

e

These are all eigenvectors-> indeterminacy for constant offsets(Set e for the minimum non-e element)

Eigenvalue of a reduced matrixOnly one eigenvalueMaximum average cumulative weight among all cyclesAll elements of eigenvectors are non-e

28

ee

5.25.4

5.2

42

73

ee

14

1

42

53

ee

13

1

22

33

l

w

ACp p

p

||

||max

)(

p

wp ||

lp ||

: Path of the cycle: Weight of the path: Length of the path

1 2

3 42

7

1 2

3 42

5

1 2

3 2

3

2

)(AC : Cycle in Graph A

Kleene Star In Max-Plus AlgebraAlso referred to as the Kleene ClosureCollection of symbols of generated by arbitrary repetitions of an operation

In Max-plus algebra

longest paths for all node pairs

29

l

l

AAAeA0

2*

ij][ *A ≠ e : maximum cumulative weights from node j -> i= e : not reachable from node j -> i

e

e

e

e

3812

59

4*

A

1 3

2

7

4 5

43

3

57

4A

(p.25)

Kleene Star In Some Classes

Directed Acyclic Graph (DAG)Today's main target • There is no path with s steps or greater

Connected graph with non-positive maximum circuit weight• Any non-positive circuit cannot be the longest path

30

ls

l

A

0

l

l

AAAeA0

2*

(finite number of terms)

εA l ),( slns

),( slns ls

l

l

AA0

Part III. Miscellaneous Topics & Recent

Advances

31

Tetris-Like ScheduleEarliest schedule of 3 blocks

32

: Block (job) numberk

ix: Upper-end position of resource i

}2)1(,1)1(max{)( 322 kxkxkx

}3)1(,2)1(max{)( 323 kxkxkx

)1()(),1()(),1()( 554411 kxkxkxkxkxkx

}),()1({max)( jijj

i lukxkx

ii lu , : relative upper-/lower- end positions of resource i

)1()( kxkx ii : Resource i is not used

Lapse of time: '+'Non-concurrency: 'max'

k-1

Resource 1 2 3 4 5

k

Block = relative times are fixed

Pre- and post- processing tasks

Facility interference

Matrix RepresentationEarliest times of the Tetris type schedule

33

)1(32

21

)(

k

e

e

e

k xx

)1()( kk xMxMPL form:

}2)1(,1)1(max{)( 322 kxkxkx

}3)1(,2)1(max{)( 323 kxkxkx

)1()( 11 kxkx

)1()( 44 kxkx

1 2 3 4 5

)1()( 55 kxkx

Non-diagonal: completion time in i – start time in j

Diagonal: block depth

k

1k

Duality & Dual SystemEarliest timesState equation

• gives the earliest completion timesOutput equation

• gives the earliest output times

Latest times

• Latest start times

• Latest input times

34

)]()1([)( 0 kkk uBxAx

)()( 0 kk xCy

)]()1([)( 0 kkk TT uCxAx

)()( 0 kk T xBy

PFPA *0 )(

0B

0C

0F

: Input matrix e/e

: Output matrix e/e

: Adjacency matrix e/e

Pls. refer to Ref. [1] for details

)](diag[ kdP

Connected = eNot connected = e

Transition matrix

Consideration of Capacity Constraints

Assumptions so far• Number of jobs that can be processed simultaneously in a

facility = 1• Number of maximum jobs that can exist between two facilities =

+Inf

Consideration of maximum capacity• Specify maximum capacity between two arbitrary nodes

Representation of lag times

35

)]()1([)( 0 kkk uBxAx

)()( 0 kk xCy

PFPA *0 )(

1 2 3

5

)()()( 0)(

01

* khkk hQ

hk uBxHFx

)(

)()(

k

kk

x

xx

*

0*

0

0*

0*

012

1

*

)()(

)()(

PFPPF

FPFPFFF l

k

s

lk

)()( 0 kk xCy Lag time

Ref. [2]

Application to Model Predictive Control

Substitute iteratively to the state equation

Output prediction equation

36

)()1()( kkk BuAxx

)1()()1()1( 2 kkkk BuABuxAx

)1()()1()1( 1 NkkkNk NN BuuBAxAx

)()1()( kkk ΔUΦxY

)1(

)1(

)(

)(

Nk

k

k

k

y

y

y

y

)1(

)1(

)(

)(

Nk

k

k

k

u

u

u

U

1

2

NCA

CA

CA

Φ

CABBCABCA

ε

CABBCA

εεCAB

Δ

21

2

NN

Case of production systems:Problems to determine proper material feeding times by giving due dates

Ref. [3]

)()( kk xCy

Efficient Calculation of the Kleene Star

Time complexity with a naïve method: Efficient Algorithms:1. Topological sort• Based on Depth First Search (DFS) • If the precedence relations are given by an adjacency matrix:• If given by a list:

2. Iterative update of the longest paths• Starting from a unit matrix e , procedures similar to the

elementary transformation are performed

37

)( 4nO Refs. [4], [5]

))(( mnnO n: Num. nodesm: Num. edges

)( 2nO

)( mnO

1 2 3

5 6

4

7

1 2 35 64 7

)( mnO

Extension to Stochastic Systems

Tandem structureDistribution function of summation

• Asymptotically -> central limit theorem

Fork structure (synchronization)Distribution of max.

• Only Weibull distr. (incl. exponential distr.) and Gumbell distr. (incl. double-exponential) are simple, while others are complex

Hard to handle analytically for general cases!• Numerical computations for only small-sized systems are

achieved38

1 2

12

t

dftftf0 2112 )()()(

)(1 tf )(2 tf

t t

)(12 tf

t

)]()([)( 2112 tFtFdt

dtf

tdttftF

0)()(

Another Approach to Stochastic

SystemsUtilize the framework of the Critical Chain Project Management (CCPM) Method• High uncertainty in the execution time of tasks • Detailed probability distribution is not considered

Affinity with max-plus algebra because of the same formulation framework as project scheduling problem

39

Outline of CCPMCurtail (cut) the margin time of each task

Redistribute (paste) the curtailed times to critical points• Insert time buffers

t

)(tf

HP(Highly Possible)90%

ABP(Aggressive But Possible)

50%

Ref. [6]

Insertion of Time BuffersPre-processingCurtail the margin timesIdentify critical or non-critical

Feeding bufferInsert just on the eve of critical -> non-critical points • 1/2 of the cumulative weights

of the non-critical chain

40

Project bufferInsert just before the output• 1/2 of the cumulative weights of the critical chain

Effective for both reducing the lead time and avoid delay for the due date

1 2 3

4

1 2 3

4

P

F

Thank you for listening!

41

Reference BooksMax-plus algebraF. Baccelli, G. Cohen, G.J. Olsder, and J.P. Quadrat, Synchronization and Linearity, John Wiley & Sons, New York, 1992.• Now out of print: can be downloaded via: http://maxplus.org

B. Heidergott, G.J. Olsder, and L. Woude, Max Plus at Work: Modeling and Analysis of Synchronized Systems, Princeton University Press, New Jersey, 2006.

Critical Chain Project ManagementP.L. Leach, Critical Chain Project Management, Second Edition, Artech House, London, 2005.

42

Reference Articles[1] H. Goto, ''Dual Representation of Event-Varying Max-Plus Linear Systems'',

International Journal of Computational Science, vol. 1, no.3, pp.225-242, 2007.

[2] H. Goto, ''Dual Representation and Its Online Scheduling Method for Event-Varying DESs with Capacity Constraints," International Journal of Control, vol.81, no.4, pp.651-660, 2008.

[3] H. Goto, "A Lightweight Model Predictive Controller for Repetitive Discrete Event Systems", Asian Journal of Control, vol. 15, no.4, pp.1081-1090, 2013.

[4] H. Goto, "A Fast Computation for the State Vector in a Max-Plus Algebraic System with an Adjacency Matrix of a Directed Acyclic Graph," Journal of Control, Measurement, and System Integration, vol.4, no.5, pp.361-364, 2011.

[5] H. Goto and H. Takahashi, "Fast Computation Methods for the Kleene Star in Max-Plus Linear Systems with a DAG Structure," IEICE Transactions on Fundamentals, vol.E92-A, no.11, pp.2794-2799, 2009.

[6] H. Goto, N. T. N. TRUC, and H. Takahashi, "Simple Representation of the Critical Chain Project Management Framework in a Max-Plus Linear Form", Journal of Control, Measurement, and System Integration, vol.6, no.5, pp.341-344, 2013.

43