An Introduction to Max-plus Algebra

Hiroyuki Goto Department of Industrial & Systems Engineering

Hosei University, Japan

Outline of Today's Tutorial

Part I. Introduction

Why max-plus algebra?

What is max-plus algebra?

How to use?

Part II. Relevant Topics

Where is max-plus algebra?

Relevance with close fields

• Control theory, graph theory, discrete mathematics

Part III. Miscellaneous Topics & Recent Advances

Extensions to wider classes

Extension stochastic systems

1

Part I. Introduction

2

Why Max-Plus Algebra?

Simple Scheduling Problem based on PERT • PERT: Performance Evaluation and Review Technique

Project with four activities

3

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)

A

dA C

dC

B

dB D

dD

,23 Cdxx ,1 ux ,12 Adxx

),,max( 324 xdxx B ,45 Ddxx

5xy C dC

A

dA

B

dB

D

dD

u y

Start Finish event/state

Earliest Node Times

Derive an explicit form

4

,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

C dC

A

dA

B

dB

D

dD u y

Eliminate xi on

the right hand-side

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

Completion (output) time

Latest Node Times

Calculate the times from downstream to upstream

Slack time (margin)

Critical path 5

C dC

A

dA

B

dB

D

dD u 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

f

WijW dxxf

A, D, and (B or C)

Features of PERT

Traditional PERT can describe

Precedence relationships between activities

Duration time of each activity

Limitation: cannot describe other practical constraints such as

A single worker (resource) is assigned to multiple activities

The facilities (resources) process the same job repeatedly

Resource conflict may occur, etc.

6

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

What is max-plus algebra?

Basic operations

7

}{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

•Examples

5)3,5max(35

5),5max(5

)(44

3303 e

83535

81795

Correspond to

)0log(

)1log( e

‘O-plus’ ‘O-times’

Behavior of a Production System

Production system with 3 machines

8

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

ix

iu

: Job number

: Material feeding time in input i

y : 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: '+' operation Sync., non-concurrency: 'max'

k M1

M2

M3

1x

1d

2x

2d

1u

2u

y3x

3d

: Completion time in machine i

Matrix Operations

Same rules as conventional algebra

9

ijijij ][][][ YXYX

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

ljill

ljil

n

l

ij ZXZXZX

Addition:

Multiplication:

nm max, RYX

Zero matrix: ε

eUnit matrix:

pn maxRZ

All elements are

Diagonal elements: e, off-diagonal elements:

n

l

ljilij

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

10

u

e

d

ed

d

d

D

B

C

A

xx

bxAx

Initial state Dummy 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

C dC

A

dA

B

dB

D

dD

u y

5xy

xy e

Matrix Representation (2)

Earliest schedule of the three-machine production system

11

})(,)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-concurrency Predecessors

)()( kek xy

1

2

3

1x

1d

2x

2d

1u

2u

y3x

3d

How to Solve the Equation?

Substitute iteratively

12

bxAx bxAx

bAx1)( 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 Matrix

Solution of the recursive linear equation

13

C dC

A

dA

B

dB

D

dD u y

bxAx

bAx *

A: precedence relations b: 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 time

xCy 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

14

)()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

1d

2x

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 Time

Latest node times of the four-activity project

15

,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 equation

Latest times

16

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

C dC

A

dA

B

dB

D

dD u y

: the same as p.5

Part II. Relevant Topics

17

Relevant Fields

Modern control theory

State 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 net

Representation 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

18

Relevance with Modern Control Theory

Earliest schedule for the 3-machine production system

19

)()()( ttt BuAxx

)()( tt Cxy )()( kk xCy Cf.

)()1()( kkk uBxAx

Generalized representation

Same form as the state-space representation

Some 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 net

Behavior of TEGs: expressed by an MPL form

TEG: Timed Event Graph

All places have one input and one output transitions

20

M1

M2

M3

1x

1d

2x

2d

1u

2u

y3x

3d

y

1x

2x

1u

2u

3x

1d

2d

3dCapacity of places Internal: 1 External: +Inf

Capacity=1

Capacity=+Inf

Ultra-discretization (1)

Ramp function

21

)0(0

)0()(

x

xxxR

u

uxxS

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

x

y

O

)(xRy 0)0,max( xx

)(

)(

u

xR

0

0.5

-1 -0.5 0 0.5

y

x

u=1

u=5

u=10

R(x)

max operation is related to exp and log functions

Ultra-discretization (2)

Variable transformation

Addition

Multiplication

Zero & unit elements • Zero element:

• Unit element: 22

)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 ,0

0,1 Xx

Semiring & Dioid

Semiring

Commutative law:

Associative laws:

Distributive laws:

Three axioms for e and :

Dioid (idempotent semiring)

+Idempotency

23

),,( 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:

24

),(

)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 Graph

State transition graph of a representation matrix

Node: state (of jobs, facilities)

Weight of an edge: transition time

Example

25

,

3

57

4

A

7

4 5

3

810

9

2A

,

12

3

A

4A

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

ij

k ][ A

Eigenvalue problem

Num. eigenvalues of square matrices: n or smaller

26

ee1

2

1

ee

2

2

1

ee

eee

1

11

11

eeee

xxA

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- element)

Eigenvalue of a reduced matrix

Only one eigenvalue

Maximum average cumulative weight among all cycles

All elements of eigenvectors are non-

27

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

3 4 2

7

3 4 2

5

3 2

3

2

)(AC : Cycle in Graph A

Kleene Star In Max-Plus Algebra

Also referred to as the Kleene Closure

Collection of symbols of generated by arbitrary repetitions of an operation

In Max-plus algebra

longest paths for all node pairs

28

l

l

AAAeA0

2*

ij][ *A ≠ : maximum cumulative

weights from node j -> i

= : not reachable from node j -> i

e

e

e

e

3812

59

4*

A

7

4 5

3

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

29

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

30

Tetris-Like Schedule

Earliest schedule of 3 blocks

31

: Block (job) number k

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 Representation

Earliest times of the Tetris type schedule

32

)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 System

Earliest times

State equation

• gives the earliest completion times

Output equation

• gives the earliest output times

Latest times

• Latest start times

• Latest input times

33

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

)()( 0 kk xCy

)]()1([)( 0 kkk TTuCxAx

)()( 0 kk TxBy

PFPA *

0 )(

0B

0C

0F

: Input matrix e/

: Output matrix e/

: Adjacency matrix e/

Pls. refer to Ref. [1] for details

)](diag[ kdP

Connected = e

Not connected =

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

34

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

)()( 0 kk xCy

PFPA *

0 )(

1 2 3

5

)()()( 0

)(

0

1

* khkk hQ

h

k uBxHFx

)(

)()(

k

kk

x

xx

*

0

*

0

0

*

0

*

012

1

*

)()(

)()(

PFPPF

FPFPFFF

l

k

s

l

k

)()( 0 kk xCy Lag time

Ref. [2]

Application to Model Predictive Control

Substitute iteratively to the state equation

Output prediction equation

35

)()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

36

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

))(( mnnO n: Num. nodes

m: Num. edges

)( 2nO

)( mnO

)( mnO

Extension to Stochastic Systems

Tandem structure

Distribution 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 achieved

37

1 2

12

t

dftftf0

2112 )()()(

)(1 tf )(2 tf

t t

)(12 tf

t

)]()([)( 2112 tFtFdt

dtf

t

dttftF0

)()(

Another Approach to Stochastic Systems

Utilize 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

38

Outline of CCPM

Curtail (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 Buffers

Pre-processing

Curtail the margin times

Identify critical or non-critical

Feeding buffer

Insert just on the eve of critical -> non-critical points

• 1/2 of the cumulative weights of the non-critical chain

39

Project buffer

Insert 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!

40

Reference Books

Max-plus algebra

F. 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 Management

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

41

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. 42