Discrete Event Dynamic Systems Research Articles

A Stochastic Petri Net Approach for Inventory Rationing in Multi-Echelon Supply Chains

Manufacturing supply chains are considered as discrete event dynamical systems (DEDS) where coordination of material and information flows is essential to satisfy customer orders and to improve the bottomline of the constituent organizations. A critical problem that is often faced by distribution centres that hold finished good inventory is that of inventory rationing. Inventory rationing is a useful strategy to tackle the problem of conflicting objectives i.e., minimizing inventory costs (holding and backorder) on the one hand and achieving the desired customer service levels (CSLs) on the other. The focus of this paper is to formulate Generalized Stochastic Petri net models to address the inventory rationing problem in the context of multi-echelon make-to-stock distribution chains, where the goods flow through multiple echelons, typically from product manufacturers all the way up-to the retail outlets. The statistical inventory control (SIC) policies modeled by the GSPN are (R, s, S) and a variant that we propose, (R?, s, S). We compare the performance of the model under two rationing settings. The first setting considers a case without cooperation, where the individual local stockpoints maximize their own performance. The second setting considers a case with cooperation, where the local stockpoints cooperate with each other to maximize the overall system performance. We provide a methodology to approximately determine the optimal rational fractions with different weights assigned to expected backorder and holding cost components (b/h). We present some interesting results obtained after rigorous numerical experimentation on the model.

Petri Net Controller for Network Connectivity

Petri Net (PN) is a commonly used versatile tool for analysis of Discrete Event Dynamic systems (DEDS). In this paper PN is applied to demonstrate network connectivity under various connectivity options. The network components are susceptible to failure and get repaired in due course. A PN based complete failure repair dynamics of the network connectivity for various optional routings are presented. This provides a means for evaluating the network connectivity for reliability.

Optimal control of processing times in single-stage discrete event dynamic systems with blocking

This note concerns an optimal control problem in single-stage discrete event dynamic systems with finite buffers and blocking. The system is modeled as a deterministic queue, slated to process a finite sequence of jobs. Each job is characterized by its arrival time, service time, and due date, and has associated with it a cost function that penalizes short service times, buffer times, and lateness of completion times with respect to the due date. The sequencing (order) of the jobs at the server is given, and the variable parameter consists of the jobs' service times. Even though the cost function associated with each job is assumed to be convex, the aggregated cost functional is not convex. Therefore, much of the analysis focuses on a decomposition of the problem into a finite sequence of reduced-order convex programming problems which can be solved one at a time. This approach has been investigated in the past, but the present note provides an analysis under the most general and realistic assumptions considered to date.

Eigenvectors of interval matrices over max–plus algebra

The behaviour of a discrete-event dynamic system is often conveniently described using a matrix algebra with operations max and plus. Such a system moves forward in regular steps of length equal to the eigenvalue of the system matrix, if it is set to operate at time instants corresponding to one of its eigenvectors. However, due to imprecise measurements, it is often unappropriate to use exact matrices. One possibility to model imprecision is to use interval matrices. We show that the problem to decide whether a given vector is an eigenvector of one of the matrices in the given matrix interval is polynomial, while the complexity of the existence problem of a universal eigenvector remains open. As an aside, we propose a combinatorial method for solving two-sided systems of linear equations over the max–plus algebra.

ISSUES IN OPTIMAL CONTROL OF DYNAMIC DISCRETE-EVENT SYSTEMS

We define the notion of Dynamic Discrete-Event Systems, a class of time-varying systems, and present a simple approach to optimal control of such systems. More specifically, we use limited-lookahead online control and an algorithm which tries to maximize the benefit of the executed sequences of events, while at the same time ensuring that unwanted (illegal) sequences are avoided. We use examples to illustrate the different types of problems that can arise if such control is used, for example, overspecialization and failure to take advantage of available resources. These issues are used to formulate a list of desirable properties for a new algorithm that optimizes the control of dynamic systems.

The Potential Structure of Sample Paths and Performance Sensitivities of Markov Systems

We study the structure of sample paths of Markov systems by using performance potentials as the fundamental units. With a sample path-based approach, we show that performance sensitivity formulas (performance gradients and performance differences) of Markov systems can be constructed intuitively, by first principles, with performance potentials (or equivalently, perturbation realization factors) as building blocks. In particular, we derive sensitivity formulas for two Markov chains with possibly different state spaces. The proposed approach can be used to obtain flexibly the sensitivity formulas for a wide range of problems, including those with partial information. These formulas are the basis for performance optimization of discrete event dynamic systems, including perturbation analysis, Markov decision processes, and reinforcement learning. The approach thus provides insight on on-line learning and performance optimization and opens up new research directions. Sample path based algorithms can be developed.

Development of a new dioid algebraic model for manufacturing with the scheduling decision making capability

The dioid algebraic model, which is developed in this paper, captures the discontinuous nature of Discrete-Event Dynamic Systems and has widespread applicational capabilities. As a further improvement over existing dioid algebraic models, decision making capability is introduced to the developed dioid algebraic model. With this new capability, sequencing decisions at each machine can be represented in the model. Furthermore, the developed algebraic model is capable of representing job and resource unavailability, technological constraints, alternative process plans, and is easily modifiable to include new jobs and exclude finished jobs. In a possible application, the model can be used in representing dynamic scheduling problems and can be used as an important part of a real-time control and scheduling scheme of a manufacturing system.

The balance problem of min–max systems is co-NP hard

Min–max systems are timed discrete event dynamic systems containing min, max and plus operations. A min–max system is balanced if it admits a global cycle time (generalized eigenvalue) for every possible assignment of parameters. This paper proves that the testing of the balance property for min–max systems is co-NP hard by showing that the test of inseparability is co-NP hard. The latter result is established on a new complexity result, namely, that the testing of existence of non-trivial fixed points for monotone Boolean functions is NP-complete.

A Remark on Inseparability of Min–Max Systems

Min-max systems, or min-max-plus systems are algebraic models of discrete-event dynamic systems. They are nonlinear extensions of the well known linear max-plus system models. The structural property "inseparability" of min-max systems was proposed by us in a recent paper. It turns out that inseparability is equivalent to the property of "irreducibility" proposed recently by van der Woude and Subiono, although the two properties appear in very different forms. We will prove this fact here.

Optimal Legal Firing Sequence of Petri Nets Using Linear Programming

Petri nets (PNs) are a reliable graphical and mathematical modeling tool for the formal modeling and validation of systems (W. Reisig, A Primer in Petri Net Design, Springer-Verlag: Berlin, Heidelberg, 1992). Applications of PNs include discrete event dynamic systems (DEDS) that are recognized as being concurrent, asynchronous, distributed, parallel, and/or nondeterministic. It is also a powerful formal method for the analysis of concurrent, embedded, and distributed finite state systems (K. Varpaaniemi, Series A: Research Reports, No. 26, Helsinki University of Technology, Digital Systems Laboratory, Oct. 1993). The reachability analysis of PNs is strategically significant as it captures the dynamic behavior of the system as well as providing efficient verification of the correctness of the model. Few linear programming (LP)-based methods can be found that address the reachability problem, and some of these are suitable for optimal control problems. However, due to an inherent state explosion they are difficult to implement; other methods run easily into deadlock as they lack appropriate mechanisms to avoid the firing of critical transitions (T. Matsumoto and A. Tarek, in Proceedings of the 35th IEEE Conference on Decision and Control, Kobe, Japan, 1996-12, pp. 4459–4468). In this paper an improved and easy to implement method is proposed that combines the Optimality Principle and Linear Programming (OP + LP) techniques to find an Optimal Legal Firing Sequence (OLFS) in PNs. This method can be applied to ordinary PNs with self-loops, avoids deadlocks, and can also be used for general PNs having cycles.

Fault Diagnosis Based on Limit Measurements of Process Variables

Abstract Industry uses mostly information on measurements passing limits as inputs to diagnostic systems. Asking the question on what can be achieved by such systems, we aim at an optimal design of diagnostic systems. The approach is, in contrast to the currently available techniques, based on continuous models that are mapped into discrete-event dynamic systems in the form of nondeterministic automata, with faults being constraint to have event-dynamic, that is, they occur and persist. We compute domains in which faults can be isolated or detected and discuss guidelines on how to design an application-optimal fault detection and isolation mechanisms.

On the polynomial dynamic system approach to software development

In this paper the development of reactive software is transformed into a control problem, and the supervisory control theory for discrete event dynamic systems is suggested to solve this control problem. The operating environment under consideration is viewed as a controlled plant, the software under development as the corresponding controller, and the software requirements as the corresponding control objective. This idea leads to a constructive approach of software design, which ensures properties required a priori of the software under development. In this way the validation of the software under development is reduced to the validation of properties independent of implementation process. We reveal the inconsistence in using the concept of reachability to specify software requirements and clarify six different definitions of reachability. Two different definitions of invariance for specifying software requirements are also clarified. We then show how to synthesize the required controller or obtain software design solutions if the underlying software requirements are specified by several new combinations of reachability and invariance. The topic of this paper falls into the scope of software cybernetics that explores the interplay between software and control.

A new Petri net modeling technique for the performance analysis of discrete event dynamic systems

An interesting modeling problem is the need to model one or more of the system modules without exposition to the other system modules. This modeling problem arises due to our interest in these modules or incomplete knowledge, or inherent complexity, of the rest of the system modules. Whenever the performance measures (one or more) of the desired modules are available through previous performance studies, data sheets, or previous experimental works, the required performance measures for the whole system can be predicted from our proposed modeling technique. The incomplete knowledge problem of the dynamic behavior of some system modules has been studied by control theory. In the control area, such systems are known as partially observed discrete event dynamic systems, or POS systems. To the best of our knowledge, the performance evaluation of the POS system has not been addressed by the Petri net theory yet. Therefore, in this paper, we propose a new modeling technique for solving this kind of problem based on using the Petri net theory (i.e. Stochastic Reward Nets (SRNs)) in conjunction with the optimal control theory. In this technique, we develop an SRN Equivalent Model (EM) for the modeled system. The SRN EM-model consists of two main nets and their interface nets. One of the main nets represents the part(s) of interest or the known part(s) of the overall POS system that allows us to model its dynamic behavior and evaluate its performance measures. The other main net represents the remaining part(s) of the overall POS system that feeds the part(s) of interest. The well-known maximum principles have been used to develop an algorithm for determining the unknown transition rates of the proposed model. Numerical simulations are given to show that the proposed approach is more effective than the conventional modeling techniques, especially when dealing with systems having a large number of states.

Selecting the best stochastic system for large scale problems in DEDS

We consider the problem of selecting the stochastic system with the best expected performance measure, when the number of alternative systems is large. We consider the case of discrete event dynamic systems (DEDS) where the standard clock simulation technique can be used for simulating multiple systems using only one sample path. In this paper, we use a two-phase procedure that uses the standard clock simulation technique. In the first phase, we screen out non-competent alternatives and construct a confidence set that contains the best alternative with a pre-specified large probability. In the second phase, we use the indifference-zone ranking and selection procedure to select the best expected alternative among the survivals of the first phase. We implement this algorithm for solving a practical simulation optimization problem.

Automated Solving the Control Synthesis Problem for the Extended Class Of D.E.D.S.

A generalization of the author's methods published recently is presented in this paper. It extends their validity to the wider class of DEDS (discrete event dynamic systems) to be controlled. Both the previous methods and the innovated one are suitable for the kind of DEDS described by ordinary Petri nets (OPN). However, while the previous methods can be successfully used only in case of DEDS described by the special class of OPN - so called state machines (SM) where each transition has only the single input place and the single output place - the innovated method can be used in case of DEDS described by the wider class of OPN - the bounded OPN (having a limited number of marks in their places) without any restriction on their structure. Thus, the validity of the methods is extended because the class of bounded OPN is undoubtedly wider than the class of OPN represented by SM. Both the original approach and the proposed one can alternatively utilize either ordinary directed graphs (ODG) or bipartite directed graphs (BDG).

Supervisory Control of Extended Timed Event Graphs

This paper describes the dynamic behavior of extended timed event graphs (ETEG) related to place delay in the dioid framework. By Cofer and Garg's supervisory control theory (1996), we address control problems of extended timed event graphs. Supervisory control of extended timed event graphs (a class of discrete event dynamic systems) is studied in the dioid framework. A necessary and sufficient condition for the ideals of the set of firing time sequences of transitions to be controllable is presented. We prove all the strongly controllable subsets can form a complete lattice.

Semi-markov decision problems and performance sensitivity analysis

Recent research indicates that Markov decision processes (MDPs) can be viewed from a sensitivity point of view; and the perturbation analysis (PA), MDPs, and reinforcement learning (RL) are three closely related areas in optimization of discrete-event dynamic systems that can be modeled as Markov processes. The goal of this paper is two-fold. First, we develop the PA theory for semi-Markov processes (SMPs); and then we extend the aforementioned results about the relation among PA, MDP, and RL to SMPs. In particular, we show that performance sensitivity formulas and policy iteration algorithms of semi-Markov decision processes can be derived based on the performance potential and realization matrix. Both the long-run average and discounted-cost problems are considered. This approach provides a unified framework for both problems, and the long-run average problem corresponds to the discounted factor being zero. The results indicate that performance sensitivities and optimization depend only on first-order statistics. Single sample path-based implementations are discussed.

Optimization of tandem queue systems with finite buffers

We consider the problem of optimizing the mean processing rate in a tandem queue system (TQS), where items enter in a fixed order. Stations in the TQS are selected and optimized so as to reduce the mean processing time of an item in the system. The concept of critical path is introduced, with which a PA method is presented. In order to reduce the number of items in a simulation, regenerative properties are discussed for the TQS so that the steady performance measure can be obtained in a regenerative cycle. Furthermore, through a simulation in a regenerative cycle, the stations that should be optimized are identified, and the optimal decision to improve these stations is determined by introducing slack degree. Scope and purpose The study of discrete event dynamic systems (DEDS) was proposed in the early 1980s, to model many modern complex stochastic systems, especially, many complex artificial systems. The methodology of perturbation analysis (PA) has been developed to improve the efficiency of DEDS simulation, while Critical Path has been defined and adopted to simplify the PA method, which avoids the tedious analysis in perturbation generation and propagation. This paper aims to study a kind of tandem queue systems with finite buffers, to optimize its processing rate by minimizing the mean processing time of an item in the system under some conditions. In the process of simulation, a number of methodologies, including PA, critical path, and regeneration will be discussed and ordinal optimization will be adopted to improve the efficiency of simulation.

Sizing of an industrial plant under tight time constraints using two complementary approaches: (max,+) algebra and computer simulation

In this article (max,+) spectral theory results are applied in order to solve the problem of sizing in a real-time constrained plant. The process to control is a discrete event dynamic system without conflict. Therefore, it can be modeled by a timed event graph, a class of Petri net, whose behavior can be described with linear equations in the (max,+) algebra. First the sizing of the process without constraint is solved. Then we propose to design a simulation model of the plant to validate the sizing of the process.

Max-algebra: the linear algebra of combinatorics?

Let a⊕ b=max( a, b), a⊗ b= a+ b for a,b∈ R := R∪{−∞} . By max-algebra we understand the analogue of linear algebra developed for the pair of operations (⊕,⊗) extended to matrices and vectors. Max-algebra, which has been studied for more than 40 years, is an attractive way of describing a class of nonlinear problems appearing for instance in machine-scheduling, information technology and discrete-event dynamic systems. This paper focuses on presenting a number of links between basic max-algebraic problems like systems of linear equations, eigenvalue–eigenvector problem, linear independence, regularity and characteristic polynomial on one hand and combinatorial or combinatorial optimisation problems on the other hand. This indicates that max-algebra may be regarded as a linear-algebraic encoding of a class of combinatorial problems. The paper is intended for wider readership including researchers not familiar with max-algebra.