Re-entrant Lines Research Articles

On the Strong Approximation for a Simple Reentrant Line in Light Traffic Under First-buffer First-served Service Discipline

On the Strong Approximation for a Simple Reentrant Line in Light Traffic Under First-buffer First-served Service Discipline

The BAR Approach for Multiclass Queueing Networks with SBP Service Policies

The basic adjoint relationship (BAR) approach is an analysis technique based on the stationary equation of a Markov process. This approach was introduced to study heavy-traffic, steady-state convergence of generalized Jackson networks in which each service station has a single job class. We extend it to multiclass queueing networks operating under static-buffer-priority (SBP) service disciplines. Our extension makes a connection with Palm distributions that allows one to attack a difficulty arising from queue-length truncation, which appears to be unavoidable in the multiclass setting. For multiclass queueing networks operating under SBP service disciplines, our BAR approach provides an alternative to the “interchange of limits” approach that has dominated the literature in the last twenty years. The BAR approach can produce sharp results and allows one to establish steady-state convergence under three additional conditions: stability, state space collapse (SSC) and a certain matrix being “tight.” These three conditions do not appear to depend on the interarrival and service-time distributions beyond their means, and their verification can be studied as three separate modules. In particular, they can be studied in a simpler, continuous-time Markov chain setting when all distributions are exponential. As an example, these three conditions are shown to hold in reentrant lines operating under last-buffer-first-serve discipline. In a two-station, five-class reentrant line, under the heavy-traffic condition, the tight-matrix condition implies both the stability condition and the SSC condition. Whether such a relationship holds generally is an open problem.

Instability of LRTF multiclass queueing networks

We provide an example of a strictly subcritical multiclass queueing network which is unstable under the longest remaining time first (LRTF) service discipline. It is a reentrant line with two servers and four customer classes, Poisson arrivals and deterministic service times. Our instability proof can be generalized to a class of light-tailed stochastic model primitives, including independent, identically distributed exponential service times. A similar example of an unstable deterministic, strictly subcritical LRTF network is also given. Our arguments work for any service discipline such that there are no departures from any network station during its busy period.

Instability of SRPT, SERPT and SJF multiclass queueing networks

We provide two examples of strictly subcritical multiclass queueing networks which are unstable under the shortest remaining processing time (SRPT) service protocol. Both of them are reentrant lines with two servers and eight customer classes. The customer service times in our first system are deterministic, yielding an example of an unstable shortest remaining expected processing time (SERPT) network. In the second one, the service times in one customer class are randomized. Both our examples show also system instability under the shortest job first (SJF) discipline. A simulation study of robustness of our results with respect to changes in the customer interarrival and service times is also provided. Our results indicate that size-based service policies may not use the available resources efficiently in a multiserver network setting and in fact cause instability effects. This is in sharp contrast with their satisfactory performance for single-server queues.

Instability of LAS multiclass queueing networks

We provide an example of a strictly subcritical multiclass queueing network which is unstable under the least attained service (LAS) service protocol. It is a reentrant line with two servers and six customer classes. The customer interarrival times in our system are bounded below and have finite exponential moments, while the corresponding service times are deterministic. As a special case, we obtain a deterministic, strictly subcritical unstable LAS network.

Characteristic Curves and Cycle Time Control of Re-Entrant Lines

The characteristic curve (CC) of a production system is a function describing the behavior of cycle time ( CT ) versus throughput ( TP ). In systems with unlimited buffers, this function typically has a knee-type shape. Operating the system below the knee is not efficient, since TP can be increased without an appreciable increase in CT . Operating above the knee is also counterproductive— CT becomes extremely large without a significant increase of TP . Thus, the desirable operating point is at the knee. In order to operate the system at or close to the knee, some knowledge of its CC is necessary. In this paper, an analytical/empirical method for calculating CCs of single-product re-entrant lines is developed using the so-called bottleneck workcenter model of systems at hand. Based on this method, open- and closed-loop raw material release policies, which ensure operating at the desired point of CC, are provided. The development is carried out in terms of the First Buffer First Served (FBFS) and Last Buffer First Served (LBFS) dispatch policies, although CCs for other dispatch policies can be investigated in a similar manner. In addition, it is shown by simulations that the method developed can be used for evaluating CC of a model of a scaled-down wafer fab, which includes multiple workcenters and batching, as long as the severity of the bottleneck workcenter is sufficiently high.

Throughput Maximization of Capacitated Re-Entrant Lines Through Fluid Relaxation

This paper extends the scheduling methodology for complex stochastic networks that is based on the solution of a “fluid” relaxation (FR) at each decision point of the original scheduling problem to stochastic networks with blocking and deadlocking effects. For a clearer and more concrete treatment, the presented results are developed in the operational context of a re-entrant line (RL) with finite buffering capacity at each workstation; these RLs are characterized as “capacitated RLs (CRLs).” From a methodological standpoint, the paper results are enabled by a pre-established ability to control the underlying resource allocation for deadlock freedom and by the further ability to express the corresponding deadlock avoidance policy as a set of linear inequalities on the system state. Also, the employed FR for this new regime differs from the FRs that have been employed in past implementations of the method since it must account for the blocking effects that take place in the considered CRLs. The efficacy of the presented scheduling method is assessed through numerical experimentation that compares, for a set of “benchmark” CRLs, the performance of the scheduling policies obtained through this method to 1) the performance of the corresponding optimal scheduling policies and also to 2) the performance of some other heuristic scheduling policies for these systems that are adapted from the relevant literature. Finally, an additional set of experiments demonstrates and assesses the scalability of the presented method by applying it to some pretty large system configurations. Note to Practitioners —While the real-time management/scheduling of complex resource allocation systems (RASs) is a thriving area in general, the particular problem of scheduling such systems with extensive blocking and deadlocking effects in their operation has received very limited attention. To a large extent, this is due to the fact that the effective scheduling of this particular RAS class requires the initial resolution of an additional problem, of a more combinatorial type, that concerns the establishment of “liveness” for the underlying workflow, i.e., the ability of all the activated jobs to proceed to their completion securing successfully all the required resources for the execution of their various processing stages, and avoiding the formation of any deadlocks or livelocks. On the other hand, this problem of liveness enforcement for the considered RAS has received extensive attention within a certain part of the controls community during the past decades, and the currently available results provide a broad range of methods and policies for supporting the necessary supervision. This paper combines the aforementioned results on the RAS liveness-enforcing supervision with a scheduling methodology that has been very popular in the context of other hard scheduling problems in order to develop a complete scheduling methodology for the considered RAS. Extensive numerical experimentation reported in this paper demonstrates and assesses the efficacy of the presented method. Finally, in an effort to provide more concrete and simpler exposition for the presented developments, the results are presented in the operational context of a “capacitated re-entrant line (CRL),” i.e., a particular RAS class modeling the operation of RLs with finite buffering capacity at its various workstations.

Bi-objective scheduling for re-entrant hybrid flow shop with learning effect and setup times

The production scheduling problem in hybrid flow shops is a complex combinatorial optimization problem observed in many real-world applications. The standard hybrid flow shop problem involves often unrealistic assumptions. In order to address the realistic assumptions, four additional traits were added to the proposed problem. These include re-entrant line, setup times, position-dependent learning effects, and the consideration of maximum completion time together with total tardiness as objective function. Since the proposed problem is non-deterministic polynomial-time (NP)-hard, a meta-heuristic algorithm is proposed as the solution procedure. The solution procedure is categorized as an a priori approach. To show the efficiency and effectiveness of the proposed algorithm, computational experiments were done on various test problems. Computational results show that the proposed algorithm can obtain an effective and appropriate solution quality for our investigated problem

An efficient bi-objective algorithm to solve re-entrant hybrid flow shop scheduling with learning effect and setup times

This paper deals with a bi-objective hybrid flow shop scheduling problem minimizing the maximum completion time (makespan) and total tardiness, in which we consider re-entrant lines, setup times and position-dependent learning effects. The solution method based on genetic algorithm is proposed to solve the problem approximately, which belongs to non-deterministic polynomial-time (NP)-hard class. The solution procedure is categorized through methods where various solutions are found and then, the decision-makers select the most adequate (a posteriori approach). Taguchi method is applied to set the parameters of proposed algorithm. To demonstrate the validation of proposed algorithm, the full enumeration algorithm is used to find the Pareto-optimal front for special small problems. To show the efficiency and effectiveness of the proposed algorithm in comparison with other efficient algorithm in the literature (namely MLPGA) on our problem, the experiments were conducted on three dimensions of problems: small, medium and large. Computational results are expressed in terms of standard multi-objective metrics. The results show that the proposed algorithm is able to obtain more diversified and competitive Pareto sets than the MLPGA.

Move blocking strategy applied to re-entrant manufacturing line scheduling

Model predictive control (MPC)-based approach to fab-wide scheduling has been suggested to solve constraint-aware production optimization and in-process inventory level control simultaneously at each scheduling instance. However, application of this approach to real fab suffers from computational difficulties brought by the need to solve a huge optimization problem on-line as real fab scheduling problems are characterized by long cycle times, multiple product types, hundreds of machines/processing steps and re-entrant product flows. This study explores the use of an offset-blocking strategy combined with a modified recursive least square (RLS) estimation in the fab-wide scheduler, in order to alleviate the difficulty. The strategy is tested on a modified version of published case study called Intel Mini-Fab (IMF) problem. Despite its simplicity, the blocking strategy showed excellent performance in the face of realistic demand changes and plant/model mismatch.

A Method for “Sweet Point” Operation of Re-entrant Lines

Production systems with infinite buffers typically have the lead time (IT) vs. throughput (TP) characteristic curve (CC) with a knee-type shape. Operating the system below the knee is not efficient, since TP can be increased without an appreciable increase in IT. Operating the system above the knee is also counterproductive, since IT becomes extremely large without a significant increase of TP. Thus, the desirable operating point is at the knee, which is why it is often referred to in practice as the “sweet point”. In this paper, using an empirical/analytical approach, CCs of re-entrant lines described by the bottleneck workcenter- based model with exponential machines are analyzed, the positions of the sweet point are quantified, and an open- and closed-loop job release strategies that ensure the operation at the knee are developed. The development is carried out in terms of the First Buffer First Served and Last Buffer First Served dispatch policies, although CCs for other policies can be investigated in a similar manner.

Design and comparative performance analysis of a heuristic-based parameterised Banker’s algorithm using the CRL scheduling problems

In this paper, we suggest an efficient sub-optimal deadlock avoidance policy (DAP), called a heuristic-based parameterised Banker’s algorithm (H-pBA). It has the main three steps such as (i) for a state , explicitly identifying -job instances and feasible -processing steps for them to be proceeded, (ii) generating a new state by applying the -choice to the state and (iii) determining the safety of the state by checking if all the resources required by all the job instances in the state up to completion are available. The algorithm iteratively applies these steps to a state by changing the values of and , up to their upper bounds in the worst case, until it makes a decision about the safety. For performance evaluation, we perform a comparative performance analysis by conducting numerical experiments using a set of prototypical examples for capacitated re-entrant line (CRL) configurations, while comparing with the most common polynomial-kernel DAPs reported in the literature. By suggesting two performance indices representing the operational flexibility and effectiveness, we show that the performance of the suggested H-pBA is consistently superior to that of other PK-DAPs, and furthermore, it is beneficial to integrate it with the first-buffer first-serve policy for system throughput in the CRL.

Cycle Time Design and Simulation for a Certain Kind of Automatic Production Line

This paper proposes a cycle time design rule for a certain kind of time-constraint and re-entrant automatic production line. A computation model of the productive time and productivity is presented. Then, the cycle time under different schemes according to a real technology process is given. The design rule of this kind of line is deduced by a lot of simulations at last.

Stability of multi-class queueing networks with infinite virtual queues

We generalize the standard multi-class queueing network model by allowing both standard queues and infinite virtual queues which have an infinite supply of work. We pose the general problem of finding policies which allow some of the nodes of the network to work with full utilization, and yet keep all the standard queues in the system stable. Toward this end we show that re-entrant lines, systems of two re-entrant lines through two service stations, and rings of service stations can be stabilized with priority policies under certain parameter restrictions. The analysis throughout the paper depends on model and policy and illustrates the difficulty in solving the general problem.

Study of Re-Entrant Lines Modeling Based on Object-Oriented Petri Net

This article puts forward object-oriented Petri net modeling method, which possesses good encapsulation and modularity compared with current ordinary modeling method. On the macro level, it divides the re-entrant lines into different object modules according to the technology, so that the complexity of models is largely reduced through message delivery between objects. In the micro level, it explains objects' internal operational mechanism, in another word, each object's internal operation cannot be affected by other objects and environment. At last, it makes modeling and dynamic analysis by taking LED chips' processing flow for example, showing that re-entrant lines model based on object-oriented Petri net possesses good modeling ability.

Equilibria, Stability, and Transients in Re-Entrant Lines Under FBFS and LBFS Dispatch and Constant Release

A model of a re-entrant line, consisting of the bottleneck workcenter and time delays representing other workcenters, is considered. Its mathematical description is provided and performance metrics are introduced. The steady states of this model and their stability properties are investigated under two dispatch policies-first buffer first served (FBFS) and last buffer first served (LBFS)-and under constant release rate. The transients due to machine downtime are also analyzed. It is shown that, although LBFS may be viewed as having superior steady-state characteristics, it induces longer and more volatile transients than FBFS and, in some cases, periodic and chaotic regimes.

Computationally efficient neuro-dynamic programming approximation method for the capacitated re-entrant line scheduling problem

This paper presents a computationally efficient neuro-dynamic programming approximation method for the capacitated re-entrant line scheduling problem by reducing the number of feature functions. The method is based on a statistical assessment of the significance of the various feature functions. This assessment can be made by combining the weighted principal components with a thresholding algorithm. The efficacy of the new feature functions selected is tested by numerical experiments. The results indicate that the feature selection method presented here can extract a small number of significant features with the potential capability of providing a compact representation of the target value function in a neuro-dynamic programming framework. Moreover, the linear parametric architecture considered holds considerable promise as a way to provide effective and computationally efficient approximations for an optimal scheduling policy that consistently outperforms the heuristics typically employed.

Re-entrant lines with unreliable asynchronous machines and finite buffers: performance approximation and bottleneck identification

In this paper we study a re-entrant line with unreliable asynchronous exponential machines and finite buffers. First, an approximation method is presented to estimate the throughput of the re-entrant line. The idea of the approximation is to transform an M-machine re-entrant line into a 2M-machine serial line. Then, a system approach to identify a c-bottleneck based on blockage and starvation information is proposed, where a c-bottleneck machine is the machine whose improvement in machine capacity leads to the largest improvement in system throughput compared with improving all other machines. It is shown that the approximation method results in acceptable accuracy, and the bottleneck identification method can correctly detect the bottlenecks in most cases.

A two-layer dynamic scheduling method for minimising the earliness and tardiness of a re-entrant production line

This study focuses on the dynamic scheduling problem of a re-entrant production line, in which all of the parts are assumed to have the same processing routes and be processed on every machine. A two-layer dynamic scheduling method is proposed for the dynamic scheduling of the re-entrant line with the objective of minimising total earliness and tardiness. This method consists of two layers. The top layer is to select the appropriate scheduling policy, and the bottom layer is to generate the scheduling by using the policy selected in the top layer. In the top layer, three different rules are constructed for selecting scheduling policies, namely the lateness comparison rule, the lateness variation comparison rule, and the equal interval switching rule. In the bottom layer, three different scheduling policies are proposed to generate the real-time scheduling for manufacturing, namely the FCFS (first come first service) scheduling policy, the PI (proportional-integral) scheduling policy, and the fuzzy PI scheduling policy. Considering that the real-time status of manufacturing changes constantly, it is necessary to switch among different scheduling policies to adapt to this change. Numerical experiments are performed in the situations with and without urgent jobs. The results show that the proposed two-layer dynamic scheduling method outperforms any single scheduling policy (e.g., the FCFS policy, the PI policy and the fuzzy PI policy) for the dynamic scheduling of a re-entrant production line.

Diffusion approximation of a multitype re-entrant line under smaller-buffer-first-served policy

This paper studies a multitype re-entrant line under smaller-buffer-first-served policy, which is an extension of first-buffer-first-served re-entrant line. We prove a heavy traffic limit theorem. The key to the proof is to prove the uniform convergence of the corresponding critical fluid model.