Transition Firing Sequence Research Articles

A Novel Control-Theory-Based Approach to Scheduling of High-Throughput Screening System for Enzymatic Assay

Nowadays, high-throughput screening (HTS) systems are widely used in pharmaceutical industries and laboratories for the discovery of new drugs and biomedical substances. It is important to efficiently schedule them so as to reduce the cost. In the operation of an HTS system, a microplate may visit some resources more than once and complex time window constraints are imposed on some activities and activity sequences. Moreover, with the consistency requirement, a one-microplate cyclic schedule is necessary. Thus, its scheduling problem is very challenging. This article studies the scheduling problem of an HTS system for an enzymatic assay, a typical application of HTSs, from the perspective of control theory. The system is modeled by resource-oriented Petri nets (ROPNs). With the model, necessary and sufficient conditions under which a feasible cyclic schedule exists are established. Then, we determine how many microplates should be in the system for concurrent processing and the transition firing sequence to obtain the activity sequence for an optimal and feasible schedule. In this way, a feasible and optimal cyclic schedule can be found by very simple computations, which shows that polynomial algorithms for an optimal schedule exist, while the existing studies apply mathematical programming with exponential computation complexity. Also, its efficient implementation is given.

Petri-Net-Based Scheduling of Flexible Manufacturing Systems Using an Estimate Function

In this paper, a novel admissible estimate function is proposed to schedule flexible manufacturing systems (FMSs) by using heuristic search. The FMSs to be scheduled are modeled by P-timed Petri nets. The problem is to make the system evolve from the initial marking to a given final marking by firing a sequence of transitions. The structure of jobs in an FMS is always symmetrical to utilize the shared resources, but the processing time of each job is asymmetrical to reduce the global process time. By utilizing the structural symmetry of a Petri net model of an FMS, a partial reachability graph is generated such that the notorious state explosion problem is mitigated. For each generated marking, the proposed estimate function is used to provide an estimated cost for firing the transition sequence. Then, we can select the marking with the smallest cost from the generated markings and compute its successors. This process is continued until the system reaches the final marking. With the proposed method, the performance is evaluated in terms of the cost of the obtained transition firing sequence and the number of the expanded markings. The cost provided by the proposed estimate function is closer to the optimal cost than the previous work, i.e., the proposed method can find a transition firing sequence with less expanded markings and minimal process time from a marking to the final marking. Experimental results are used to demonstrate and evaluate the proposed approach.

On liveness analysis in mono-T-semiflow Petri nets based on choice-join pairs

This work introduces novel structural conditions to guarantee the liveness of Mono T-Semiflow (MTS) systems for any valid initial marking. In particular it shows that in MTS the existence of problematic choice-join pairs, a pure structural objects, is equivalent to the existence of transition firing sequences leading to deadlock markings. Moreover, such sequences always include an output transition of a choice place pc, capable of removing all the tokens from pc, but incapable of marking all the input places of a join transition tj In this situation tj will be dead, and since the net is MTS, then the net will reach a deadlock marking.

Estimation of Least-Cost Transition Firing Sequences in Labeled Petri Nets by Using Basis Reachability Graph

This paper develops an approach for solving the problem of least-cost transition firing sequences estimation in labeled Petri nets with unobservable transitions. Each transition in the net is labeled as either observable or unobservable, and has a nonnegative cost. Additionally, we assume that the net system is bounded, the unobservable subnet is acyclic, and the cost of each unobservable transition is strictly positive. We propose the method mainly on the basis of the notions of basis marking and basis reachability graph (BRG), which is a compact representation of the reachability graph of the net system. By sacrificing extra storage space to keep the BRG and other necessary information in memory, some time-consuming computations are moved off-line. This makes our proposed approach more feasible for on-line applications. Finally, we present a brief survey and comparison of some representative methods that use labeled Petri nets as a formalism in the literature.

FMS Scheduling under Availability Constraint with Supervisor Based on Timed Petri Nets

This paper proposes an optimal solution to large-scale Flexible Manufacturing System (FMS) scheduling problems under availability constraints based on Timed Petri Nets (TPNs). First a decomposition method of TPNs is proposed, then a mathematical model is derived based on their properties. The mathematical model is built to determine the optimal firing sequence of TPN transitions to minimize the total manufacturing time. The resulting firing sequence of TPN transitions is used to generate the manufacturing system supervisor operated by TPN and digital controllers. Several numerical examples and comparative studies are provided in this paper in order to prove the new approach’s efficiency.

Petri net representation and reachability analysis of 0–1 integer linear programming problems

In this paper, we investigate a general algorithm for converting the 0–1 integer linear programming problem (0-1IP) into an optimal transition firing sequence problem (OFSP) of a Petri net (PN). The general 0-1IP can be visualized graphically and then analyzed using the PN theory after application of our proposed conversion algorithm. The proposed algorithm is applied to a traveling salesman problem, a vehicle routing problem and an automated guided vehicles (AGV) routing problem. A PN reduction technique is employed to reduce the size of the PN. Valid inequalities are derived using reachability analysis of the converted PN model. These inequalities are imposed on the original 0-1IP. Computational results show that the total computational time for solving an AGVRP with the valid inequalities derived using the reachability analysis is significantly reduced.

Propositional syntax and semantics induced knowledge re-structuring in a fuzzy logic network for ad hoc reasoning

Traditional approaches to fuzzy reasoning usually employ Generalized Modus Ponens, Generalized Modus Tollens and Generalized Hypothetical Syllogisms to derive fuzzy inferences from a given set of fuzzy observations/facts and fuzzy if-then rules. However, the above approaches occasionally fail to generate inferences, although semantically plausible fuzzy inferences follow as a natural consequence of the given set of rules and fuzzy facts/observations. This paper serves to fill this void. It extends propositional syntax and semantics in the context of fuzzy reasoning to transform/re-structure individual rules to a suitable form (by negating and transposing propositions from antecedent to consequent and vice-versa), so as to get the propositions present in the antecedent (consequent) instantiated (refuted) by positive (negative) facts to enable the rules for forward (backward) firing. Next fuzzy compositional rule of inference is used to perform forward/backward reasoning. Consequently, when the rules are embedded in a fuzzy Petri net like structure, the rules apparently fire randomly in forward or backward direction without maintaining any topological order in transition-firing. Such random firing of rules in distributed parts of the network is referred to as ad hoc reasoning. Here, we transform such randomly ordered forward/backward firing sequence of transitions in the network into a fixed topological order of transition-firing in forward direction only by replacing backward firing transitions into equivalent forward firing transitions. The proposed method of automated reasoning has successfully been applied in diagnostic application of an electronic rectifier circuit, and the results are appealing.

Schedulability Analysis for Noncyclic Operation of Time-Constrained Cluster Tools With Time Variation

We consider a scheduling problem of a robotized cluster tool for semiconductor manufacturing, which should control wafer delays within a chamber so as not to exceed a specified limit under time variation. The prior research has proposed a ${\rm p}^{+}$ -time event graph, an extension of Petri nets, for modeling the scheduling problem and developed a method of verifying whether a cyclic ${\rm p}^{+}$ -time event graph or a cluster tool, which repeats identical work cycles, can satisfy the time constraints under time variation. In this paper, we extend and simplify the schedulability analysis method in the prior research for a noncyclic event graph or a cluster tool which performs start-up and close-down operation for a lot or lot switching, and obtain specialized results. We assume that a robot task sequence or firing sequence of transitions is given. Based on the schedulability analysis, we also propose a way of modifying a not always schedulable noncyclic ${\rm p}^{+}$ -time event graph with some qualifications to be always schedulable, that is, we prolong token holding times at some places or equivalently delay firings of some transitions so as to be always feasible.

Optimal robotic cell scheduling with controllers using mathematically based timed Petri nets

This paper presents an optimal solution to the robotic cell scheduling problem for robot movement controllers using timed Petri nets (TPNs). The suggested TPN approach is used to generate a mathematical transition model, based on a From/To transition matrix and the properties of the TPN, that considers all possible movements of robots between cell stations. The mathematical model thus obtained is solved to identify the optimal firing sequence of TPN transitions for the considered robotic cell problem to minimize the time elapsed before the firing of the last transition (the cycle time). Finally, the optimal sequence of transitions is used to generate robotic cell controllers and construct the final TPN model. A numerical example is used to demonstrate the proposed approach.

Comparative trace semantics of time Petri nets

In the paper, a family of trace equivalencies in the interleaving, step, partial order, and nondeterministic semantics is defined and studied in the context of time Petri nets. The equivalences studied are based both on the classical concept of transition firing sequences and on time processes, i.e., time extensions of net processes with global transition firing moments. Relationships between the equivalences are established, and a hierarchy of classes of equivalent time Petri nets is constructed.

Universal petri net

A universal inhibitor Petri net executing an arbitrary given inhibitor Petri net is constructed. An inhibitor Petri net graph, its marking, and transition firing sequence are encoded as 10 scalar nonnegative integer variables and are represented by the corresponding places of the universal net. An algorithm using only these scalar variables and executing an arbitrary inhibitor net is developed based on the state equation and is encoded by the universal inhibitor Petri net. Subnets that implement arithmetic, comparison, and copy operations are employed.

An Algorithm for Seeking the Minimal Cost Maximal Flow Based on Petri Net

In this paper, Petri net is adopted for seeking the minimal cost maximal flow of a network. The cost Petri net is proposed and the enabled rules of transitions are given. In addition, an algorithm for obtaining the minimal cost maximal flow based on cost Petri net is developed. The proposed algorithm, which is different from the others, is to find the firing sequence of transitions by labeling places and increase flows in this sequence. Finally, an example is given to demonstrate the application of the algorithm. The results show that the given method can find the minimal cost maximal flow of a network efficiently.

Least-Cost Transition Firing Sequence Estimation in Labeled Petri Nets With Unobservable Transitions

This paper proposes an approach for estimating the least-cost transition firing sequence(s) that matches (match) the observation of a sequence of labels produced by transition activity in a given labeled Petri net. Each transition in the labeled net is associated with a (possibly empty) label and also with a nonnegative cost which captures its likelihood (e.g., in terms of the amount of workload or power required to execute the transition). Given full knowledge of the structure of the labeled Petri net and the observation of a sequence of labels, we aim at finding the transition firing sequence(s) that is (are) consistent with both the observed label sequence and the Petri net, and also has (have) the least total cost (i.e., the least sum of individual transition costs). The existence of unobservable transitions makes this task extremely challenging since the number of firing sequences that might be consistent can potentially be infinite. Under the assumption that the unobservable transitions in the net form an acyclic subnet and have strictly positive costs, we develop a recursive algorithm that is able to find the least-cost firing sequence(s) by reconstructing only a finite number of firing sequences. In particular, if the unobservable transitions in the net are contact-free, the proposed recursive algorithm finds the least-cost transition firing sequences with complexity that is polynomial in the length of the observed sequence of labels.

Petri Net Decomposition Approach to Optimization of Route Planning Problems for AGV Systems

In this paper, we propose a Petri Net (PN) decomposition approach to the optimization of route planning problems for automated guided vehicles (AGVs) in semiconductor fabrication bays. An augmented PN is developed to model the concurrent dynamics for multiple AGVs. The route planning problem to minimize the total transportation time is formulated as an optimal transition firing sequence problem for the PN. The PN is decomposed into several subnets such that the subnets are made independent by removing the original shared places and creating its own set of resource places for each subnet with the appropriate connections. The partial solution derived at each subnet is not usually making a feasible solution for the entire PN. The penalty function algorithm is used to integrate the solutions derived at the decomposed subnets. The optimal solution for each subnet is repeatedly generated by using the shortest-path algorithm in polynomial time with a penalty function embedded in the objective function. The effectiveness of the proposed method is demonstrated for a practical-sized route planning problem in semiconductor fabrication bay from computational experiments.

Least-cost planning sequence estimation in labelled Petri nets

This paper develops a recursive algorithm for estimating the least-cost planning sequence in a manufacturing system that is modelled by a labelled Petri net. We consider a setting where we are given a sequence of labels that represents a sequence of tasks that need to be executed during a manufacturing process, and we assume that each label (task) can potentially be accomplished by a number of different transitions, which represent alternative ways of accomplishing a specific task. The processes via which individual tasks can be accomplished and the interactions among these processes in the given manufacturing system are captured by the structure of the labelled Petri net. Moreover, each transition in this net is associated with a non-negative cost that captures its execution cost (eg, in terms of the amount of workload or power required to execute the transition). Given the sequence of labels (ie, the sequence of tasks that has to be accomplished), we need to identify the transition firing sequence(s) (ie, the sequence(s) of activities) that has (have) the least total cost and accomplishes (accomplish) the desired sequence of tasks while, of course, obeying the constraints imposed by the manufacturing system (ie, the dynamics and structure of the Petri net). We develop a recursive algorithm that finds the least-cost transition firing sequence(s) with complexity that is polynomial in the length of the given sequence of labels (tasks). An example of two parallel working machines is also provided to illustrate how the algorithm can be used to estimate least-cost planning sequences.

Decomposition of Petri nets and Lagrangian relaxation for solving routing problems for AGVs

In this paper, we propose a Lagrangian relaxation method for solving routing problems for multiple AGVs by decomposition of timed Petri nets. The AGV routing problem is represented by an optimal transition firing sequence problem for timed Petri nets. The timed Petri net is decomposed into several subnets in which the subproblem for each subnet can be easily solved by Dijkstra's algorithm. We show that each subproblem generated by each subnet is polynomially solvable. The optimality of the solution can be evaluated by the duality gap derived by the Lagrangian relaxation method. The performance of the proposed method is compared with a conventional optimisation algorithm with the penalty function method. The effectiveness of the proposed method is demonstrated.

분해우선작업을 가지는 페트리 넷 기반의 분해순서계획모델

This paper presents a Petri Net (PN) based disassembly sequence planning model with precedence operations. All feasible disassembly sequences are generated by a disassembly tree and a disassembly sequence is determined using the disassembly precedence and disassembly value matrix, The precedence of disassembly operations is determined through a disassembly tree and the value of disassembly is induced by economic analysis in the end-of-life phase. To solve the disassembly sequence planning model with precedence operations, a heuristic algorithm based on PNs is developed. The developed algorithm generates and searches a partial reachability graph to arrive at an optimal or near-optimal disassembly sequence based on the firing sequence of transitions of the PN model. A refrigerator is shown as an example to demonstrate the effectiveness of proposed model.

Lagrangian Relaxation Technique for Solving Scheduling Problems by Decomposition of Timed Petri Nets

In this paper, we propose Lagrangian relaxation technique for solving scheduling problems by decomposition of timed Petri nets. The scheduling problem is represented by the transition firing sequence problem to minimize a given objective function. The timed Petri net is decomposed into several subnets so that the subproblem for each subnet can be easily solved by a shortest path algorithm. The optimality of solution can be evaluated by the duality gap derived by Lagrangian relaxation method. The performance of the proposed method is compared with the conventional optimization algorithm with penalty function method. The results show that the duality gap within 1.5% can be derived for AGVs routing problems. The effectiveness of the proposed method is demonstrated by comparing the performance between the conventional method.

Petri Net Modeling and Decomposition Method for Solving Production Scheduling Problems

Considering the need to develop general scheduling problem solver, the recent integration of Petri Nets as modeling tools into effective optimization methods for scheduling problems is very promising. The paper addresses a Petri Net modeling and decomposition method for solving a wide variety of scheduling problems. The scheduling problems are represented as the optimal transition firing sequence problems for timed Petri Nets. The Petri Net is decomposed into several subnets in which each subproblem can be easily solved by Dijkstra' algorithm. The approach is applied to a flowshop scheduling problem. The performance of the proposed algorithm is compared with that of a simulated annealing method.

Sequence analysis for time petri nets

In this paper is proposed a new methodology to analyse Time Petri nets. This new approach is based on the class graph and it allows temporal analysis of any valid sequence of transition firings. For that, it is proposed a procedure that find valid dates for all events over a transition firing sequence present in the graph. These valid dates are presented by means of an inequalities system, that is constructed following the sequence path in the class graph. Then, it is possible to calculate, for instance, the maximal time interval between two events in the sequence, as well as the latter date of a given event.