Optimal inspection sublots for deteriorating batch production

This paper analyzes a problem of optimal inspection intervals for a failure-prone batch production facility. The model makes a seemingly minor modification to similar models that have appeared recently, but which results in a significant change in the form of the optimal policy. Optimal solutions are developed for both continuous and integer valued sublot sizes. The paper also examines the effects of inspection sublots in an economic lot sizing context and shows that a simple, exact modification of the Economic Production Quantity formula is often possible in the continuous relaxation of the problem.     

Optimal and heuristic inspection schedules for multistage production systems

Consider a single inspection facility that can be quickly switched among multiple inspection tasks. It can be used (for example) for detecting malfunction (or down state) production stages in a multistage production system. We assume that a properly working (or up state) production stage moves to a down state in any period with fixed probability. The stage then stays down until it is inspected and immediately restored back to an up state. Our purpose is to schedule inspections among the different production stages so as to maximize the fraction of good items produced. An optimal inspection schedule for a two stage production system is provided. For the general case of more than two stages, four heuristics are compared. We conclude that the proposed dynamic schedule is easy to derive, always feasible, and outperforms the static schedules.     

Optimal periodic replacement for a deteriorating production system with inspection and minimal repair

Here we discuss an inspection policy model for a deteriorating production system with minimal repair. A minimal repair is resorted to as and when the system is found to be in a failed state during an inspection unless it is apre-set overhaul/replacement time in which case the system is overhauled or replaced. Using a dynamic programming formulation, and assuming that the cost of minimal repair is a non-decreasing function of age, we arrive at the optimal inspection time that maximizes the profit per unit time for a given overhaul/replacement time. The procedure is then extended to determine the optimal periodic overhaul/replacement time and the corresponding optimal number of inspections and their schedule.     

Optimal manufacturing and delivery schedules in a supply chain system of deteriorating items

The market is changing from minute to minute nowadays. Increasing cooperation and pursuing of the optimal interest of the integrated supply chain system become more effective than acting alone in the face of competition. In this research, an integrated inventory policy between a single producer and a multi-buyer is developed. The model extends the research of Lin and Lin (2007 Lin, C and Lin, YS. 2007. A cooperative inventory policy with deteriorating items for a two-echelon model. European Journal of Operational Research, 178(1): 92–111. [Crossref], [Web of Science ®] , [Google Scholar], A cooperative inventory policy with deteriorating items for a two-echelon model. European Journal of Operational Research, 178 (1), 92–111) by changing the single-buyer system to the multi-buyer system. This problem is solved under the assumptions of equal replenishments and production cycles. The producer and buyers collaboratively intend to decrease their joint total cost of operation. We consider the deterioration of items at all levels of producer, buyer and in-transport, and the backorders at buyer's level only. The algorithm to find the optimal solution is given, and Matlab and Maple software are used. A numerical example is also presented.     

Managing a deteriorating process in a batch production environment

When manufacturing processes have a tendency to deteriorate over time and begin producing defective products, there is always a trade-off between the cost and/or rework time that is associated with such defectives and the amount of time that is spent inspecting and restoring the process. Although a considerable amount of attention has been devoted to this trade-off as it is found in high-volume, repetitive manufacturing, we study it in a job shop environment. In particular, we study the way in which inter-job setups affect the inspection policy. After modeling the problem as one of maximizing the throughput of defective-free jobs, we describe a simple algorithm which identifies optimal inspection intervals.     

Optimal control of a deteriorating process with a quadratic loss function

This paper considers the optimal determination of the length of the production run and the initial setting of a process that exhibits a linear drift that can start at a random point in time. Quadratic off‐target costs and time‐based costs of maintenance and salvage value are included in the formulation. The model includes other models proposed in the literature as particular cases. Numerical examples are provided to illustrate the application of the proposed model. Copyright © 2001 John Wiley & Sons, Ltd.     

The performance of rolling production schedules in a process industry

In this paper we study lot-sizing and changeover decisions in production schedules that are implemented on a rolling-horizon basis. The study is carried out by comparing production schedules for the packaging operations of a pharmaceutical company that produces various products on several capacitated production lines. The schedules are obtained by solving a mixed-integer programming formulation by using a heuristic procedure. We examine the effects of the number of periods in the planning horizon, the starting inventory and the demand fluctuation on the schedules. We show empirically that the saving in annual production cost diminishes rapidly as more demand periods are added to the planning horizon. Computational results are reported.     

Determination of the optimal production run and the most profitable process mean for a production process

This paper deals with the problem of joint determination of the optimal process mean and the production run for a process. It extends the Quality Selection problem to production systems in which the process mean (target) shifts to an out-of-control state due to an assignable cause. The problem has been validated using various examples. Moreover, results indicate that an inverse relationship exists for the length of production run and the difference expected profit between in-control state and out-of-control state.     

Optimal designs of the multivariate synthetic chart for monitoring the process mean vector based on median run length

The average run length (ARL) is usually used as a sole measure of performance of a multivariate control chart. The Hotelling's T², multivariate exponentially weighted moving average (MEWMA) and multivariate cumulative sum (MCUSUM) charts are commonly optimally designed based on the ARL. Similar to the case of univariate quality control, in multivariate quality control, the shape of the run length distribution changes in accordance to the magnitude of the shift in the mean vector, from highly skewed when the process is in‐control to nearly symmetric for large shifts. Because the shape of the run length distribution changes with the magnitude of the shift in the mean vector, the median run length (MRL) provides additional and more meaningful information about the in‐control and out‐of‐control performances of multivariate charts, not given by the ARL. This paper provides a procedure for optimal designs of the multivariate synthetic T² chart for the process mean, based on MRL, for both the zero and steady‐state modes. Two Mathematica programs, each for the zero state and steady‐state modes are given for a quick computation of the optimal parameters of the synthetic T² chart, designed based on MRL. These optimal parameters are provided in the paper, for the bivariate case with sample sizes, nin{4, 7, 10}. The MRL performances of the synthetic T², MEWMA and Hotelling's T² charts are also compared. Copyright © 2011 John Wiley & Sons, Ltd.     

Imperfect inspection of a multi‐attribute deteriorating production system—a continuous time model

The reliability of a multi‐attribute deteriorating production system is controlled using versatile identical inspection facilities. An attribute state is dichotomous (up designates proper function versus down). A product item is conforming if all the system attributes are up when it is produced. When a system attribute is detected as down it is restored back to an up state. Inspection of an attribute can rely on observations of the system, recently produced items, or both. Inspection policy determines the inspection capacity, frequency of inspecting each attribute and inspection schedule. These decisions involve a tradeoff between the cost of inspectors and the loss associated with the roportion of non‐conforming items due to lack of adequate inspection. Three models are introduced, analyzed and solved. In the first model, inspection and restoration are perfect, product attribute is up (down) when the system attribute is up (down), and restoration is immediate. The assumptions of perfect inspection and restoration are relaxed in the second model. The third model relaxes in addition the assumption of immediate restoration. An efficient heuristic solution scheme is provided for solving these models. Sensitivity of the solution to system parameters is studied. Numerical experiments provide some insights regarding the combined effect of imperfect production, inspection and restoration, in various conditions of inspection and restoration durations. Copyright © 2001 John Wiley & Sons Ltd.     

Integrated maintenance and production control of a deteriorating production system

We consider a make-to-stock production/inventory system consisting of a single deteriorating machine which produces a single item. We formulate the integrated decisions of maintenance and production using a Markov Decision Process. The optimal dynamic policy is shown to have a rather complex structure which leads us to consider more implementable policies. We present a double-threshold policy and derive exact and approximate methods for evaluating the performance of this policy and computing its optimal parameters. A detailed numerical study demonstrates that the proposed policy and our approximate method for computing its parameters perform extremely well. Finally, we show that policies which do not address maintenance and production control decisions in an integrated manner can perform rather badly.     

A sensitivity analysis of an integrated model for joint determination of economic design of $\overline{x}$‐control charts,economic production quantity and production run length for a deteriorating production system

This study presents an integrated model for the joint economic design of ‐control charts and maintenance schedules and, simultaneously, determines the economic production quantity and production run length for a deteriorating production system. The operating state of the production process is classified as either in control or out of control. In the latter state, the process produces some defective items. An ‐control chart is used to monitor the process mean. Both uniform and non‐uniform inspection schemes are adopted. Inspection and maintenance are performed simultaneously. Replacement cost is assumed to be very high. The process failure mechanism is assumed to follow a general probability distribution with an increasing failure rate. The concept of a truncated production cycle is introduced. The production cycle begins when a new component is installed and ends with a repair after the detection of a failure or after a specified number of inspection intervals, , whichever occurs first. The effects of preventive maintenance on quality control are discussed. Numerical examples are provided to evaluate the performance of the model. Sensitivity analyses are conducted to study the effects of various model parameters. Copyright © 2002 John Wiley & Sons, Ltd.     

Optimal design of the inspection buffer for an unreliable process

Placing an inspection buffer immediately after an unreliable process subject to failure protects the process customers from receiving non-conforming products. Such protection is achieved by screening the buffer contents whenever a shift in the process is detected. Previous studies have suggested that, if properly designed, such buffers can reduce overall costs. In this paper we make two types of contributions to the study of inspection buffers. First, we refine the model proposed by Klastorin et al. (1993), by incorporating a more efficient way of sampling, a more realistic objective function and more accurate cost expressions. Second, we provide analytical results to assist in the calculation of the economically optimal buffer size. Specifically, these include conditions for determining whether an inspection buffer is at all justified and a theorem for bounding the search for the optimal buffer size. The performance of the search bound and the sensitivity of the model are examined computationally.     

An inspection policy for a deteriorating single-unit system characterized by a delay-time model

This paper develops a method for determining the discrete time points of inspection for a deteriorating system which is under condition-based maintenance. The system is regarded as a single component unit, and it has the following states: a normal state, a symptom state and a failed state. A delay-time model is utilized in order to describe the transition of the states. The transition time from a normal state to a symptom state and the transition time from a symptom state to a failed state are assumed to obey independent and identically distributed probability density functions respectively. Two types of probabilities with respect to inspections are considered: failed-dangerous (FD) probability or type I error that a system is wrongly diagnosed as in a normal state when the system is actually in a symptom state and failed-safe (FS) probability or type II error that a system is wrongly diagnosed as in a symptom state when the system is actually in a normal state. The determination method is formulated to minimize the long-run average cost per unit time with a constraint on inspection time. The nature and sensitivity of the proposed method is investigated under varying FD and FS probabilities and coefficients of variation.     

Optimal inspection of a finite population

This article presents a method for determining an optimal quality control (QC) inspection frequency for a manufacturing process where a specified number of items will be produced and where, if the process goes out of control and generates a defective item, it stays out of control. In addition, there is a QC inspection procedure than can detect a defective item. The frequency of inspection is based on minimizing the total cost. The total cost includes the cost of QC inspections plus the cost of manufacturing defective items. In this application the cost of manufacturing defective items is, after identifying a defective item, the cost of checking previously manufactured items until it is determined when the process went out of control, i.e., until a good item is found.     

Optimal inspection policies in a serial production system including scrap rework and repair: An MILP approach

The allocation of inspection effort problem for aerial systems is formulated as a 0-1 mixed integer linear programming problem. This formulation permits any combination of scrap, rework, or repair at each station and allows the problem to be solved using standard MILP software packages. Moreover electronic spread-sheets may be used to easily calculate the relevant coefficients. An additional advantage of this approach when compared with the traditional dynamic programming approach is the ease with which the basic model may be modified. For example, it is shown how the model may easily be modified to include both a material and a production constraint and to select between various material suppliers. Sensitivity analysis is also easily performed with this approach. This model is then used to show that the optimal inspection policy is dependent on whether a production or a material requirement is used.     

Optimal restarting distribution after repair for a Markov deteriorating system

We consider a repairable system such that different completeness degrees are possible for the repair (or corrective maintenance) that go from a ‘minimal’ up to a ‘complete’ repair. Our question is: to what extent must the system be repaired in case of failure for the long-run availability to be optimal? The system evolves in time according to a Markov process as long as it is running, whereas the duration of repairs follows general distributions. After repair, the system starts again in the up-state i with probability d(i). We observe from numerical examples that the optimal restarting distribution d^opt (such that the long-run availability is optimal) is generally random and does not correspond to a new start in a fixed up-state. Sufficient conditions under which the optimal restarting distribution is non-random are given. Also, the optimal restarting distribution is provided for two classical structures in reliability.     

A generalized economic model for joint determination of production run,inspection schedule and control chart design

This paper presents a generalized model for a continuous production process for simultaneous determination of production quantity, inspection schedule and control chart design, with a non-zero inspection time for false alarms. Traditionally, the quality control problem and the inventory control problem have been viewed as two separate problems. Rahim (1994) developed an economic model for joint determination of production quantity, inspection schedule, and control chart design for a typical production process which is subject to a non-Markovian random shock. The model consists of the following cost components: (1) the production setup cost, (2) the inventory holding cost, and (3) the cost of maintaining the quality of the product under the surveillance of an x-chart. The optimal production quanity, the optimal inspection schedules, and the economic design parameters of the control charts were determined by striking a balance among these costs. For mathematical simplicity, it was assumed that production ceases only if the process was found to be out-of-control. However, in reality, this assumption may be inapplicable in many industrial situations. In many production processes the machine must be shut down when a search for the assignable cause is being carried out, even though occasionally the alarm turns out to be false. The purpose of this paper is to generalize the above model to cases where production ceases not only for a true alarm but also for a fixed amount of time whenever there is a false alarm. Examples of Weibull shock models are used to illustrate the proposed generalized model.     

Conditional median run length performance of the synthetic X¯ chart with unknown process parameters

Synthetic-type charts are efficient tools for process monitoring. They are easy to design and implement in practice. The properties of these charts are usually evaluated under the assumption of known process parameters. This assumption is sometimes violated in practice, and process parameters have to be estimated from different phase I data sets collected by different practitioners. This fact causes the between-practitioners variability among the properties of the synthetic-type charts designed for each practitioner. In fact, the shape of the run length distribution of the synthetic-type charts changes with the mean shift size. As a good alternative, the median run length $(MRL)$ metric is argued to evaluate the properties of different control charts. In this paper, the $MRL$ is used as a measure of the synthetic $\overline{X}$ chart's performance, and the conditional $MRL$ properties of the synthetic $\overline{X}$ chart with unknown process parameters are investigated. Both the average $MRL$ ( $AMRL$) and the standard deviation of $MRL$ ( $\text{◂⋅▸}SDMRL$) are used together to investigate the chart's properties when the process parameters are unknown. If the available number of phase I samples is not large enough to reduce the variability of the in-control $MRL$ values to an acceptable level, a bootstrap-type approach is suggested to adjust the control limits of the synthetic $\overline{X}$ chart and to further prevent many unwanted lower in-control $MRL$ values.