1 / 43

An Introduction to Max-plus Algebra

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?

rian
Download Presentation

An Introduction to Max-plus Algebra

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. An Introduction to Max-plus Algebra Hiroyuki Goto Department of Industrial & Systems Engineering Hosei University, Japan

  2. 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

  3. Part I. Introduction

  4. Why Max-Plus Algebra? • Simple Scheduling Problem based on PERT • PERT: Performance Evaluation and Review Technique • Project with four activities AON (Activity on Node) B dB B dB D dD 2 A dA 1 2 4 5 AOA (Activity on Arrow) Earliest node times D dD A dA 1 4 C dC C dC 3 3 Finish Start event/state

  5. Earliest Node Times • Derive an explicit form Eliminate xi on the right hand-side Completion (output) time Lapse of time: '+' operation Synchronization: 'max' operation B dB D dD A dA 1 2 4 5 C dC 3

  6. Latest Node Times • Calculate the times from downstream to upstream • Slack time (margin) • Critical path Eliminate xi using B dB A dA D dD 1 2 4 5 C dC 3 A, D, and (B or C)

  7. 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. Modeling and analysis method using max-plus algebrais an useful alternative approach

  8. What is max-plus algebra? • Basic operations Addition: ‘O-plus’ ‘O-times’ Multiplication: • Priorities of operators: '*' > '+' (same as conventional algebra) • Subtraction ‘-’ and division '/' : not defined directly Correspond to Zero element: 0 Unit element: 1 • Examples

  9. Behavior of a Production System • Production system with 3 machines Non-concurrency = each machine cannot process multiple materials at the same time M1 k : Job number : Completion time in machine i M3 : Material feeding time in input i : Finish time M2 Lapse of time: '+' operation Sync., non-concurrency: 'max' Earliest processing start times / output time

  10. Matrix Operations • Same rules as conventional algebra Addition: Multiplication: Zero matrix: All elements are e Unit matrix: Diagonal elements: e, off-diagonal elements: e • Examples

  11. Matrix Representation (1) • Earliest node times of the four-activity project Dummy task (synchronization) Initial state Precedence relations & elapsed times B dB D dD A dA 1 2 4 5 Linear form in max-plus algebra: C dC 3 ⇒ MPL (Max-Plus Linear) form

  12. Matrix Representation (2) • Earliest schedule of the three-machine production system 1 3 2 Non-concurrency Predecessors External inputs MPL form:

  13. How to Solve the Equation? • Substitute iteratively Cf. In conventional algebra, • If the precedence relationships are represented by a DAG (Directed Acyclic Graph), Kleene Star (Closure) (nilpotent)

  14. Interpretation of the Representation Matrix • Solution of the recursive linear equation A: precedence relations b: start time j: source node B dB D dD A dA i: destination node 1 2 4 5 : earliest arrival times between two nodes = longest paths C dC 3 • Output time : final node

  15. Earliest Times of the Production System • Earliest times of the system with 3 machines 1 3 2 : earliest processing times between two nodes : earliest processing times from the external inputs

  16. Focusing on the Latest Time • Latest node times of the four-activity project Min.: Pseudo division: 'min' and '-' operators appear

  17. Solution of the Recursive Equation • Solve (a kind of) recursive linear equation • Latest times Cf. Earliest times B dB A dA D dD 1 2 4 5 : the same as p.5 C dC 3

  18. Part II. Relevant Topics

  19. 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

  20. Relevance with Modern Control Theory • Earliest schedule for the 3-machine production system Generalized representation Cf. • 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.

  21. 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 M1 Capacity=1 M3 Capacity=+Inf M2 Capacity of places Internal: 1 External: +Inf

  22. Ultra-discretization (1) • Ramp function • max operation is related to exp and log functions

  23. Ultra-discretization (2) • Variable transformation • Addition • Multiplication • Zero & unit elements • Zero element: • Unit element: and let (ultra-discretization)

  24. Semiring & Dioid • Semiring • Commutative law: • Associative laws: • Distributive laws: • Three axioms for e and e: • Dioid (idempotent semiring) • +Idempotency is a semiring (does not hold in usual algebras) is a Dioid

  25. Some Classes of Dioid • Max-plus algebra: • Max-times algebra: • Min-max algebra: • Min-plus algebra: • Boolean algebra: Note: These are widely referred to as *** algebra, but are not an algebra in a strict sense because of

  26. Communication Graph • State transition graph of a representation matrix • Node: state (of jobs, facilities) • Weight of an edge: transition time • Example 2 5 4 1 3 4 3 7 : Reachable with ksteps from j -> imaximum cumulative weight

  27. Eigenvalue problem • Num. eigenvalues of square matrices: n or smaller 1 2 1 2 0 1 1 2 0 These are all eigenvectors -> indeterminacy for constant offsets (Set e for the minimum non-e element)

  28. Eigenvalue of a reduced matrix • Only one eigenvalue • Maximum average cumulative weight among all cycles • All elements of eigenvectors are non-e 4 3 2 : Cycle in Graph A 1 2 : Path of the cycle 7 : Weight of the path 4 3 2 : Length of the path 1 2 5 2 3 2 1 2 3

  29. Kleene Star In Max-Plus Algebra • Also referred to as the KleeneClosure • Collection of symbols of generated by arbitrary repetitions of an operation • In Max-plus algebra • longest paths for all node pairs (p.25) 2 5 4 1 3 4 3 7 ≠ e : maximum cumulative weights from node j -> i =e :not reachable from node j -> i

  30. Kleene Star In Some Classes • Directed Acyclic Graph (DAG) • Today's main target • There is no path with ssteps or greater • Connected graph with non-positive maximum circuit weight • Any non-positive circuit cannot be the longest path (finite number of terms)

  31. Part III. Miscellaneous Topics & Recent Advances

  32. Tetris-Like Schedule • Earliest schedule of 3 blocks • Block = relative times are fixed • Pre- and post- processing tasks • Facility interference k k : Block (job) number : Upper-end position of resource i k-1 Lapse of time: '+' Non-concurrency: 'max' Resource 1 2 3 4 5 : relative upper-/lower- end positions of resource i : Resource i is not used

  33. Matrix Representation • Earliest times of the Tetris type schedule Diagonal: block depth 1 2 3 4 5 Non-diagonal: completion time in i – start time in j MPL form:

  34. 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 Pls. refer to Ref. [1] for details Transition matrix : Input matrix e/e : Output matrix e/e : Adjacency matrixe/e Connected = e Not connected = e

  35. Consideration of Capacity Constraints Ref. [2] • 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 Lag time 5 1 2 3

  36. Application to Model Predictive Control • Substitute iteratively to the state equation • Output prediction equation Ref. [3] Case of production systems: Problems to determine proper material feeding times by giving due dates

  37. 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 Refs. [4], [5] n: Num. nodes m: Num. edges 1 2 3 4 1 5 2 3 4 6 7 5 6 7

  38. 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 t t 1 2 12 t

  39. 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 Ref. [6] • Outline of CCPM • Curtail (cut) the margin time of each task • Redistribute (paste) the curtailed times to critical points • Insert time buffers t ABP (Aggressive But Possible) 50% HP (Highly Possible) 90%

  40. 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 1 2 3 4 1 2 3 P 4 F • 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

  41. Thank you for listening!

  42. 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.

  43. 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.

More Related