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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?

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An Introduction to Max-plus Algebra

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  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.

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