An improved decomposition method for evaluating the performance of transfer lines with unreliable machines and finite buffers

In this paper, we consider transfer lines consisting of a series of machines separated by finite buffers. The processing time of each machine is deterministic and may be not identical. All machines are prone to operation-dependent failures, and the times between failures and the times to repair are assumed to be exponentially distributed. Many analytical methods have been developed to evaluate the performance of such lines. In general, these methods provide fairly accurate results. However, in some real cases where the orders of magnitude of machines’ reliability parameters (mean times between failures and mean times to repair) are not at the same level, the accuracy of these existing methods may not be good enough. The purpose of this paper is to propose an improved decomposition method that performs well even in the situation above. We use generalised exponential distributions instead of exponential distributions to approximate the repair-time distributions of the fictitious machines, and a new ADDX algorithm is developed to calculate the performance parameters such as the production rate and the average buffer levels. Numerical results indicate that the improved decomposition method provides more accurate results and converges in most cases. It is feasible and valid to evaluate the performance of transfer lines with unreliable machines and finite buffers.     

An efficient analytical method for performance evaluation of transfer lines with unreliable machines and finite transfer-delay buffers

Buffers are widely adopted in transfer lines to reduce the fluctuations caused by the imbalances of systems or machine failures. This paper presents an efficient analytical method to evaluate the performance of transfer lines with unreliable machines and finite transfer-delay buffers. Firstly, the buffers with transfer delays are transformed equivalently into a series of perfect machines and buffers without transfer delays. Correspondingly, the initial transfer line is replaced by an equivalent transfer line with more machines and zero-transfer-delay buffers. Since in the equivalent transfer line the orders of magnitude of machines’ reliability parameters (mean times between failures and mean times to repair) are not at the same level, an advanced decomposition method is introduced to analyse the equivalent transfer line, using the general-exponential distributions instead of the exponential distributions to approximate the repair time distributions of the fictitious machines. Finally, extensive simulation and numerical cases are carried out to verify the performance of the developed method.     

Performance evaluation of flow lines with non-identical and unreliable parallel machines and finite buffers

This paper examines serial production lines with unreliable non-identical parallel machines at each workstation and intermediate buffers with finite capacities. All machines are assumed to have exponential service times, times to failure and repair times. An efficient decomposition technique is introduced for the performance evaluation of such lines. Rather than replacing each parallel-machine workstation with an equivalent single-server workstation, the main contribution of this paper is the presentation of a direct approach to derive and apply decomposition equations directly for every parallel machine at each workstation. Experimental results indicate that such a method can provide a computationally efficient algorithm to analyse large serial unreliable multi-server production lines with a good accuracy compared against simulation and other available methods.     

Production Rate of Transfer Lines Without Buffer Storage

We consider transfer lines without buffer storage in between the machines and with synchronous transfer. The processing time of each machine is a constant but machines are unreliable. Our analysis is based on the overall completion time of the transfer line, which is the time between successive transfers of parts within the line. Two techniques are presented. The first one provides lower and upper bounds on the production rate. These bounds can be made as tight as desired. The second technique provides an approximation of the production rate. It is based on the approximation of some distributions by simpler distributions having the same mean and coefficient of variation. We first consider the case of transfer lines with identical processing times and exponential times to failure and times to repair. Extensions to different processing times and general repair times are also discussed.     

A Mean Value Analysis of the Travelling Repairman Problem

We consider the machine repairman problem where the machines are spatially distributed and repaired by a travelling serviceman. We assume identical machines with exponential up times and general repair times. We also assume that the travel times between pairs of locations have identical, but arbitrary distributions. Assuming a work-conserving, non-preemptive service discipline, a semi-Markov model is developed and various performance measures are evaluated. Some applications of the model are discussed. The model developed here, is used elsewhere to develop approximations for a more realistic situation where machines are non-identical and travel time distributions between pairs of locations are also non-identical.     

An Efficient Algorithm for Analysis of Transfer Lines With Unreliable Machines and Finite Buffers

In a recent paper [3], Gershwin proposed a decomposition method for the approximate analysis of transfer lines with unreliable machines and finite buffers. The method is based on a decomposition of the line into a set of two-machine lines. It leads to a set of equations which are solved using an iterative algorithm. Experimental results have shown that this technique is very accurate. However, it may happen that the algorithm fails to converge. In this paper, we propose to replace the original set of equations by an equivalent one, which is again solved using an iterative procedure. This new algorithm is simpler than the previous one, and its computational complexity is lower. Moreover, on all examples we tested, the algorithm always converged.     

An approximation method for performance analysis of assembly/disassembly systems with parallel-machine stations

This note presents an efficient approximation method to evaluate the performance of tree-structured Assembly/Disassembly (AD) systems in which each station consists of multiple identical machines. It is assumed that the times to failure, times to repair, and processing times are exponentially distributed and the capacities of the buffers are finite. The method transforms an AD system with parallel machines in each station to an (approximately) equivalent AD system with a single machine in each station. After the transformation, a decomposition algorithm is used to analyze the performance of the transformed system. The results of computational experiments show that the suggested method gives good estimates in a short time.     

Modeling and Analysis of Assembly Systems with Unreliable Machines and Finite Buffers

An assembly line is a tree-structured manufacturing system in which some machines perform assembly operations. In this paper, we consider assembly lines with the following features: every operation is performed in a fixed amount of time, machines are unreliable, and buffers have finite capacity. Usually, the times to failures of machines are much larger than the processing times. This allows us to approximate the behavior of these systems by a continuous flow model. The behavior of this model is then analyzed using a decomposition technique which is an extension of an earlier technique proposed in the case of transfer lines. An efficient algorithm for calculating performance measures such as production rate and average buffer levels is derived. Experimental results are provided showing mat this approximate method is quite accurate.     

CONWIP ASSEMBLY WITH DETERMINISTIC PROCESSING AND RANDOM OUTAGES

We develop structural results and an approximation for the throughput of an assembly system fed by multi-station fabrication lines where releases are governed by the CONWIP protocol and all machines have deterministic processing times but are subject to random outages. This formulation is motivated by a printed circuit board manufacturing process.

We demonstrate that while throughput of such systems is nondecreasing in machine speed, there are cases where throughput declines when mean time between failures (MTBF) increases or mean time to repair (MTTR) decreases. Using the concept of "deterministic steady state," which describes the behavior of the system in die absence of failures, we derive a simple, closed-form approximation for throughput. Comparisons with simulation show that this approximation is robust over a wide range of conditions. Finally, we observe that throughput tends to be higher when the bottleneck is located in fabrication rather than assembly.     

Estimating the throughput of a cyclic assembly system

We consider a production system consisting of several fabrication lines feeding an assembly station. The machines in the fabrication lines and at assembly are assumed to have general processing time distributions. Releases to the system are governed by the CONWIP protocol. We model this system as an assembly-like queue and develop approximations for the throughput of the system. Comparisons with simulations show that this approximation is robust over a wide range of conditions. Finally, we observe that throughput tends to be higher when machines with higher mean processing times and/or higher variances are in fabrication rather than assembly.     

A station model for continuous materials flow production systems

This study develops a station model for continuous flow production systems. The most prominent use of the model is as a building block for a general and flexible decomposition method to analyse and design continuous materials flow production systems. Station breakdown and a finite capacity buffer are considered. Station inference caused by the blocking and starving phenomena is included in the station model. We assume that the time to station breakdown and station repair are exponentially distributed while the buffer is neither empty nor full. No restrictive assumptions are made about the distributions of the station breakdown and repair times when the station is blocked or starved, that is, while the buffer remains empty or remains full. The production rate and the expected level of the buffer are given in closed form. Numerical results that show the effects of the input parameters on the production rate along with an overview of the decomposition methods are presented.     

Closed-Loop Conveyor Systems with Breakdown and Repair of Unloading Stations

We consider a closed-loop conveyor system having a single loading and multiple unloading stations. The belt of the conveyor moves continuously with a constant speed. Units to be transported on the conveyor system are homogeneous. Interarrival times of units at the loading station are independent and identically distributed (i.i.d.) random variables having exponential distributions. The unloading station(s) are subject to breakdown. When an unloading station fails, 'thesystem operates with the remaining unloading station(s) while the failed station is being repaired. When multiple unloading stations fail, each station is equally likely to be selected for repair. The service times of unloading stations, the time between failures, and the repair times of unloading stations are i.i.d, random variables having exponential distributions. A matrix-geometric solution is obtained which provides an approximation of the steady-state probabilities of the system being in different operating states. In particular, we study the 2-server conveyor systems under different repair policies where we present some numerical results and recommend repair policies for different system parameters.     

Analysis of long flow lines with quality and operational failures

This paper presents approximation methods for the performance analysis of long manufacturing lines, i.e. lines with more than two machines and one buffer, that have both quality and operational failures. We describe three different versions of long flow lines that differ in the locations of the inspection stations and in the sets of machines that each inspection station monitors. We explain a transformation method that approximates long manufacturing lines that have quality and operational failures with long lines that only have operational failures. Such lines can be evaluated by decomposition methods. We introduce other approximations to quantify the effects of the separation of inspections from operations. Comparison with simulation shows that the solution methods provide reliable performance estimates.     

A closed form solution for the G/M/r machine interference model

This paper solves the machine interference problem in which N different machines are looked after by a team of r operatives. The run time of each machine is assumed to have a general distribution, different for each machine and the repair times are assumed to have a negative exponential distribution with different means for the different machines. An explicit expression for the probability that a particular group of machines is found running in the steady state is derived. From this other useful measures for the system can be obtained. It is shown that these depend on the run time distributions only through the means of those distributions.     

Analysis of approximately balanced production lines

This paper develops two analytical formulas for estimating the throughput of a reliable production line with exponential service times and finite intermediate buffers. The formulas apply in the case of an approximately balanced line with identical buffers or near optimal buffer allocations, where the processing times of the machines are close to each other but not necessarily the same. The derivation of the formulas is based on the standard decomposition method. Moreover, it is proved that, in general cases, both formulas provide upper bounds for the throughput obtained by the decomposition method. Numerical experiments show that the proposed formulas achieve good accuracy for approximately balanced production lines. Finally, the formulas are applied to the buffer allocation problem, and two closed-form expressions are obtained for estimating the smallest buffer capacity which is necessary to achieve the desired throughput.     

Repair policies for additively degrading machines

This paper describes a method for determining repair policies for machines whose output degrades additively over time. The novelty of the models is that they consider both the state of the machines as well as the state of the repair facility when making repair decisions. The objective in the models is to maximize long-run production. In one model, we approximate the queue waiting time by a geometric random variable, while in the second model we approximate the waiting time by a sequence of geometric random variables (with different means). We show that as the average repair queue increases, the decision to repair must be made earlier. In addition, we show empirically that the simple geometric waiting time approximation becomes less accurate as the queue length increases and that the approximation understates the expected long-run output of the machine. A plastic moulding facility is used to motivate the problem. Computational results using industry supplied data are presented. The results indicate that substantial (10-20%) productivity improvement can be realized using the derived repair policies instead of policies that do not consider the repair queue.     

Monitoring reliability for a three-parameter Weibull distribution

Control charts are widely used to monitor production processes in the manufacturing industry and are also useful for monitoring reliability. A method to monitor reliability has recently been proposed when the distributions of inter-failure times are exponential and Weibull with known parameters. This method has also been extended to monitor the cumulative time elapsed between a fixed number of failures for the exponential distribution. In this paper, we consider a three-parameter Weibull distribution to model inter-failure times, use a robust estimation technique to estimate the unknown parameters, and extend the proposed method to monitor the cumulative time elapsed between r failures using the three-parameter Weibull distribution. Since the distribution of the sum of independent Weibull random variates is not known (except in specific cases with known parameters), we give two useful moment approximations to be able to apply their scheme. We show how effective the approximations are and the usefulness of the method in detecting a possible instability during production.     

Exact analysis of a discrete material three-station one-buffer merge system with unreliable machines

In this paper, a model of a discrete material flow line consisting of three unreliable machines and one buffer of limited capacity is analysed. A similar system, but with continuous flow of material was examined by Helber and Mehrtens (2001) and Tan (2001). In our system it is assumed that the buffer has two immediate preceding machines, performing the same operations and one immediate succeeding machine that receives material from the buffer. For the case where the buffer reaches its own capacity, one of the two preceding machines has priority over the other to dispose its processed part into the buffer. Processing times are assumed to be deterministic and identical for all machines and are taken as the time unit. Geometrically distributed operation dependent failures at the machines are assumed. All possible transition equations for the examined model are derived and a recursive algorithm that generates the transition matrix for any value N of the storage level is developed. Once the transition matrix is known the performance measures of the model under consideration can be easily evaluated. This model may be used as a building block in a decomposition method to evaluate large production systems with split/merge operations (for example, flow lines with quality inspections and rework loops).     

Availability and throughput of unreliable,unbuffered production lines with non-homogeneous deterministic processing times

This paper studies production lines composed of several serial machines which are subject to random operation-dependent failures. The production lines have no intermediate buffers between adjacent machines. Machines have different deterministic processing times. The purpose of this paper is to propose analytical models to assess steady-state availability and throughput of such lines. Thousands of production line configurations have been experimented to compare the performance of the proposed approach to approximate existing techniques. A general simulation model was developed and statistical tests carried out to prove that the proposed approach is exact and robust to model unreliable, unbuffered, and non-homogeneous production lines.     

Joint optimization of maintenance,buffers and machines in manufacturing lines

This article considers a series manufacturing line composed of several machines separated by intermediate buffers of finite capacity. The goal is to find the optimal number of preventive maintenance actions performed on each machine, the optimal selection of machines and the optimal buffer allocation plan that minimize the total system cost, while providing the desired system throughput level. The mean times between failures of all machines are assumed to increase when applying periodic preventive maintenance. To estimate the production line throughput, a decomposition method is used. The decision variables in the formulated optimal design problem are buffer levels, types of machines and times between preventive maintenance actions. Three heuristic approaches are developed to solve the formulated combinatorial optimization problem. The first heuristic consists of a genetic algorithm, the second is based on the nonlinear threshold accepting metaheuristic and the third is an ant colony system. The proposed heuristics are compared and their efficiency is shown through several numerical examples. It is found that the nonlinear threshold accepting algorithm outperforms the genetic algorithm and ant colony system, while the genetic algorithm provides better results than the ant colony system for longer manufacturing lines.