Production variability in manufacturing systems: Bernoulli reliability case

The problem of production variability in serial manufacturing lines with unreliable machines is addressed. Bernoulli statistics of machine reliability are assumed. Three problems are considered: the problem of production variance, the problem of constant demand satisfaction, and the problem of random demand satisfaction generated by another (unreliable) production line. For all three problems, bounds on the respective variability measures are derived. These bounds show that long lines smooth out the production and reduce the variability. More precisely, these bounds state that the production variability of a line with many machines is smaller than that of a single machine system with production volume and reliability characteristics similar to those of the longer line. Since all the variability measures for a single machine line can be calculated relatively easily, these bounds provide analytical tools for analysis and design of serial production lines from the point of view of the customer demand satisfaction.     

Representation and analysis of transfer lines with machines that have different processing rates

A transfer line is a tandem production system, i.e. a series of machines separated by buffers. Material flows from outside the system to the first machine, then to the first buffer, then to the second machine, the second buffer, and so forth. In some earlier models, buffers are finite, machines are unreliable, and the times that parts spend being processed at machines are equal at all machines. In this paper, a method is provided to extend a decomposition method to large systems in which machines are allowed to take different lengths of time performing operations on parts. Numerical and simulation results are provided.     

一类可靠性模型的适定性

研究由一个可靠机器，一个不可靠机器和一个缓冲库构成的生产线．首先对应于此系统的数学模型化为抽象Cauchy问题，然后运用C0-半群理论证明此模型存在唯一的非负解。     

Properties of Continous Models of Transfer Lines with Unreliable Machines and Finite Buffers

We consider a transfer line consisting of a series of machinesseparated by buffers of finite capacity. The processing of apart on a machine requires a fixed amount of time. In thesesystems, blocking and starvation, which occur as a consequenceof machine failures, are important phenomena. Except for transferlines without buffer storage, no exact models of such systemshave been reported in the literature, even in the case of two-machinelines. Two approximate models have been proposed: the discrete-timemodel and the continuous-flow model. Exact solutions of theemodels have been reported in the case of two-machine Lines,and approximations have been proposed for longer lines. Thepurpose of this paper is to provide properties of the continuous-flowmodel. The main result of the paper is to show that lower andupper bounds on the exact production rate of the transfer linecan be obtained using the continuous flow model. These boundsare obtained by making an appropriate choice of the buffer capacities     

Exact analysis of a two-workstation one-buffer flow line with parallel unreliable machines

This paper examines a model of a serial flow line with two workstations and an intermediate buffer. Each workstation consists of multiple unreliable parallel machines which are not necessarily identical, viz., the processing times, failure times and repair times of the parallel machines at each workstation are assumed to be exponentially distributed with non-identical mean rates. The system under consideration is solved via exact Markovian analysis. More specifically, a recursive algorithm that generates the transition matrix for any value of the intermediate buffer capacity is developed and all possible transition equations are derived and solved analytically. Once the transition equations are solved the performance measures of the model under consideration can be easily evaluated. This model may be used as a decomposition block for solving larger flow lines with parallel unreliable machines at each workstation.     

Unreliable Transfer Lines: Decomposition/Aggregation and Optimization

Complexity has been a long standing obstacle to efficient buffer assignment in transfer lines. For fixed buffer sizes, an approximate transfer line decomposition/aggregation algorithm is developed and its ability to predict line performance is demonstrated via Monte-Carlo simulations. It equates the line with a collection of isolated, unreliable multi-state machines with recursively related statistical parameters. For scalability enhancement, state merging is used to reduce the number of machine states from up to 6 down to 2. An efficient dynamic programming based buffering optimization algorithm which minimizes a combined measure of storage and backlog costs in the transfer line is then presented. Numerical results and comparisons with alternative algorithms are reported.     

带缓冲串行生产系统预防性维护建模及运行参数优化研究

为了保证串行生产系统的产能和提高系统可靠性,提出了带缓冲区的串行生产系统预防性维护决策模型。首先,分析了生产线各执行单元可靠性和运行参数之间的关系,建立了考虑执行单元运行参数和缓冲库存的维护模型。在此基础上,结合串行生产线的特点,建立综合考虑维护成本、有效运行速度和缓冲库存的多目标优化函数。最后,构建启发式算法求解目标函数,并以串行包装生产线为例进行仿真实验分析,结果表明本文所建模型是有效且实用的。     

Sensitivity analysis and decomposition of unreliable production lines with blocking

The analysis of finitebuffered, unreliable production lines is often based on the method of decomposition, where the original system is decomposed into a series of twostage subsystems that can be modeled as quasi birthdeath processes. In this paper, we present methods for computing the gradients of the equilibrium distribution vector for such processes. Then we consider a specific production line with finite buffers and machine breakdowns and develop an algorithm that incorporates gradient estimation into the framework of Gershwin's approximate decomposition. The algorithm is applied to the problem of workforce allocation to the machines of a production line to maximize throughput. It is shown that this problem is equivalent to a convex mathematical programming problem and, therefore, a globally optimal solution can be obtained.     

An analytic formula for the mean throughput of K-station production lines with no intermediate buffers

Using the holding time model (HTM) method, an approximate analytic formula is derived for calculating the average throughput of a K-station production line with exponential service times, manufacturing blocking and no intermediate buffers between adjacent stations. The usefulness of the proposed analytical formula relies on the fact that it can handle the (general) case of workstations with different mean processing times — this being the contribution of this work compared against that of Alkaff and Muth — provided a good estimation of some coefficients involved is being made. By doing this for the balanced lines case, a simple formula is proposed with very good numerical results.     

Modeling and analysis of multi‐stage transfer lines with unreliable machines and finite buffers

This paper models and analyzes multistage transfer lines with unreliable machines and finite buffers. The machines have exponential operation, failure, and repair processes. First, a mixed vector–scalar Markov process model is presented based on some notations of mixed vector–scalar operations. Then, several steadystate system properties are deduced from this model. These include the reversibility and duality of transfer lines, conservation of flow, and the flow rate–idle time relationship. Finally, a fourstage transfer line case is used to compare and evaluate the accuracy of some approximation methods presented in the literature with the exact numerical solutions this model can provide. The properties and their proofs in this paper lay the theoretic foundation for some widely held assumptions in decomposition techniques of long transfer lines in the area of manufacturing systems engineering.     

A polynomial time algorithm for solving a quality control station configuration problem

We study unreliable serial production lines with known failure probabilities for each operation. Such a production line consists of a series of stations; existing machines and optional quality control stations (QCS). Our aim is to simultaneously decide where and if to install the QCSs along the line and to determine the production rate, so as to maximize the steady state expected net profit per time unit from the system.We use dynamic programming to solve the cost minimization auxiliary problem where the aim is to minimize the time unit production cost for a given production rate. Using the above developed O(N²) dynamic programming algorithm as a subroutine, where N stands for the number of machines in the line, we present an O(N⁴) algorithm to solve the Profit Maximization QCS Configuration Problem.     

Multi-state throughput analysis of a two-stage manufacturing system with parallel unreliable machines and a finite buffer

This paper models and analyzes the throughput of a two-stage manufacturing system with multiple independent unreliable machines at each stage and one finite-sized buffer between the stages. The machines follow exponential operation, failure, and repair processes. Most of the literature uses binary random variables to model unreliable machines in transfer lines and other production lines. This paper first illustrates the importance of using more than two states to model parallel unreliable machines because of their independent and asynchronous operations in the parallel system. The system balance equations are then formulated based on a set of new notations of vector manipulations, and are transformed into a matrix form fitting the properties of the Quasi-Birth–Death (QBD) process. The Matrix-Analytic (MA) method for solving the generic QBD processes is used to calculate the system state probability and throughput. Numerical cases demonstrate that solution method is fast and accurate in analyzing parallel manufacturing systems, and thus prove the applicability of the new model and the effectiveness of the MA-based method. Such multi-state models and their solution techniques can be used as a building block for analyzing larger, more complex manufacturing systems.     

Asymptotic variance rate of the output of a transfer line with no buffer storage and cycle-dependent failures

In this study the variability properties of the output of transfer lines are investigated. The asymptotic variance rate of the output of an N-station synchronous transfer line with no interstation buffers and cycle-dependent failures is analytically determined. Unlike the other studies, the analytical method presented in this study yields a closed-form expression for the asymptotic variance rate of the output. The method is based on a general result derived for irreducible recurrent Markov chains. Namely, the limiting variance of the number of visits to a state of an irreducible recurrent Markov chain is obtained from the n-step transition probability function. Thus, the same method can be used in other applications where the limiting variance of the number of visits to a state of an irreducible recurrent Markov chain is of interest. Numerical results show that the asymptotic variance rate of the output does not monotonically increase as the number of stations in the transfer line increases. The asymptotic variance rate of the output may first increase and then decrease depending on the station parameters. This property of the production rate is investigated through numerical experiments and the results are presented.     

Efficiency and production rate of a transfer line with two machines and a finite storage buffer

A markov model for a transfer line with two unreliable machines separated by a finite storage size buffer is introduced. Service time distribution for the two machines is Erlang whereas failure and repair times are assumed to be exponential random variables. The paper presents an efficient method to solve analytically the steady state probabilities of the system. This method is independent of the buffer size. We also include in the paper a study of the behavior of some systems performance measures such as the efficiency of the two machines and the production rate of the system.     

Machine utilizations achieved using balanced FMS production ratios in a simulated setting

Stecke [21] has developed mathematical programming approaches for determining, from a set of part type requirements, the production ratios (part types to be produced next, and their proportions) which maximize overall machine utilizations by balancing machine workloads in a flexible manufacturing system (FMS). These mathematical programming (MP) approaches are aggregate in the sense that they do not take into account such things as contention for transportation resources, travel time for work-in-process, contention for machines, finite buffer space, and dispatching rules. In the current study, the sensitivity of machine utilizations to these aggregations is investigated through simulation modeling. For the situation examined, it is found that achieved machine utilizations are a strong function of some of the factors ignored in the MP methodology, ranging from 9.1% to 22.9% less than those theoretically attainable under the mathematical programming assumptions. The 9.1% degradation results from modeling with nonzero work-in-process travel times (i.e. 2 minutes per transfer) and using only central work-in-process buffers. Resource levels (e.g. the number of automated guided vehicles; the amount of work-in-process; the number of slack buffers) needed to limit the degradation to 9.1% correspond to FMS operating conditions which are feasible in practice.     

Hierarchical production policies in stochastic two-machine flowshops with finite buffers

This paper concerns production planning in manufacturing systems with two unreliable machines in tandem. The problem is formulated as a stochastic control problem in which the objective is to minimize the expected total cost of production, inventories, and backlogs. Since the sizes of the internal and external buffers are finite, the problem is one with state constraints. As the optimal solutions to this problem are extremely difficult to obtain due to the uncertainty in machine capacities as well as the presence of state constraints, a deterministic limting problem in which the stochastic machine capacities are replaced by their mean capacities is considered instead. The weak Lipschitz property of the value functions for the original and limiting problems is introduced and proved; a constraint domain approximation approach is developed to show that the value function of the original problem converges to that of the limiting problem as the rate of change in machine states approaches infinity. Asymptotic optimal production policies for the orginal problem are constructed explicity from the near-optimal policies of the limiting problem, and the error estimate for the policies constructed is obtained. Algorithms for constructing these policies are presented.This work was partly supported by CUHK Direct Grant 220500660, RGC Earmarked Grant CUHK 249/94E, and RGC Earmarked Grant CUHK 489/95E.     

A Decomposition Method for Transfer Line Life Cycle Cost Optimisation

A new method to search best parameters of a transfer line so that the cost of each manufactured part will be minimised. The synchronised transfer lines with parallel machining are considered. Such lines are widely used in mass and large-scale mechanical production. The objective is to minimise the line life cycle cost per part under the given productivity and technological constraints. The design decisions to be optimised are: number of spindles and workstations. This will be accomplished by defining subsets of tasks which are performed by one spindle head and cutting conditions for each spindle. The paper focuses on a mathematical model of the problem and methods used to solve it. This model is formulated in terms of mixed (discrete and non-linear) programming and graph theory. A special decomposition scheme based on the parametric decomposition technique is proposed. For solving the sub-problems obtained after decomposition, a Branch-and-Bound algorithm as well as a shortest path technique are used.     

A Genetic Algorithm for the Allocation of Buffer Storage Capacities in a Production Line with Unreliable Machines

In this paper, we consider a flow-line manufacturing system organized as a series of workstations separated by finite buffers. The failure and repair times of machines are supposed to be exponentially distributed. The production rate of each machine is deterministic, and different machines may have different production rates. The buffer allocation problem consists in determining the buffer capacities with respect to a given optimality criterion, which depends on the average production rate of the line, the buffer acquisition and installation cost, and the inventory cost. For this problem we propose a genetic algorithm where the tentative solutions are evaluated with an approximate method based on the Markov-model aggregation approach.     

Mixed integer programming for scheduling flexible flow lines with limited intermediate buffers

This paper presents new mixed integer programming formulations for scheduling of a flexible flow line with blocking. The flexible flow line consists of several processing stages in series, separated by finite intermediate buffers, where each stage has one or more identical parallel machines. The line produces several different product types and each product must be processed by, at most, one machine in each stage. A product which has completed processing on a machine may remain there and block the machine until a downstream machine becomes available for processing. The objective is to determine a production schedule for all products so as to complete the products in a minimum time. The basic mixed integer programming formulations have been enhanced to model blocking scheduling with alternative processing routes where for each product a set of routes is available for processing. A reentrant flow line where a product visits a set of stages more than once is also considered. Numerical examples are presented to illustrate applications of the various models proposed.     

Modelling and scheduling of an asynchronous cyclic production line with multiple parts

In this study, we model and analyse a production line with asynchronous part transfers, processing time variability, and cyclic scheduling in the same framework. We consider a production line with multiple parts and finite interstation buffers. The line produces a batch of n jobs repetitively using the same order of jobs in every batch. The processing time of a job on a station is a random variable and is assumed to have a phase-type distribution. Parts are transferred between the stations in an asynchronous manner. We first present a continuous time Markov chain model to analyse the performance of this system for a given sequence. A state-space representation of the model and the associated rate matrix are generated automatically. The steady state probabilities of the Markov chain are determined by using a recursive method that exploits the special structure of the rate matrix. The cycle time, the production rate, and the expected Work-In-Progress (WIP) inventory are used as the main performance measures. We then present an approximate procedure to determine the cyclic sequence that minimises the cycle time. We then investigate the effects of operating decisions, system structure, processing time variability, and their interaction in the same framework. Numerical results for the performance evaluation and scheduling of cyclic production lines are also presented.