Economic control and inspection policies for high-speed unreliable production systems

In this paper, we consider a high-speed production process, which produces defects at a known rate while in control. When the process goes out of control, it produces defects at a higher rate. In this study, we revisit the role of the distribution of the process in-control time when managing such systems. Specifically, we focus on two management schemes, a control policy and an inspection policy. In the control policy, when the number of defects produced reaches a threshold, the process is stopped and inspected. In contrast, in the inspection policy, the process is stopped and inspected periodically. We derive the operating characteristics of the system and devise schemes for finding the optimal policy parameters for each policy. We also investigate the behavior of the optimal policy parameters, compare the performances of the control and inspection policies and identify the environments in which each of these policies out performs the other one using a numerical experiment.     

Maintenance scheduling of a manufacturing system subject to deterioration

This paper presents an integrated model for the joint determination of both optimal inspection strategy and optimal repair policy for a manufacturing system whose resulting output is subject to system state. An appropriate maintenance strategy is essential to optimize revenue from a manufacturing system which is in continuous operation and subject to deterioration. The optimum policy balances the amount of maintenance required to increase availability against the loss of revenue arising from the down time: insufficient maintenance leads to an increase in the number of defective items, low profit and low maintenance cost; excessive maintenance results in a reduction in the proportion of defective items, high profit and high maintenance cost. In this paper, an intensity control model adapted to partial information provides an optimal inspection intensity and repair degree of the system as an optimal control process to yield maximum revenue. The solution is obtained through formulating an equivalent deterministic Hamilton-Jacobi equation. A numerical example is provided to illustrate the behavior of the optimal control process. The optimal control process determines a solution to both optimum inspection frequency and optimal replacement policy which results in an optimal production run length of the system.     

Optimal Design of VSI ―X Control Charts for Monitoring Correlated Samples

This paper develops an economic design of variable sampling interval (VSI)―X control charts in which the next sample is taken sooner than usual if there is an indication that the process is off‐target. When designing VSI―X control charts, the underlying assumption is that the measurements within a sample are independent. However, there are many practical situations that violate this hypothesis. Accordingly, a cost model combining the multivariate normal distribution model given by Yang and Hancock with Bai and Lee's cost model is proposed to develop the design of VSI charts for correlated data. An evolutionary search method to find the optimal design parameters for this model is presented. Also, we compare VSI and traditional ―X charts with respect to expected cost per unit time, utilizing hypothetical cost and process parameters as well as various correlation coefficients. The results indicate that VSI control charts outperform the traditional control charts for larger mean shift when correlation is present. In addition, there is a difference between the design parameters of VSI charts when correlation is present or absent. Copyright © 2005 John Wiley & Sons, Ltd.     

Economic Screening of a Continuously Manufactured Product

In this paper, the problem concerning the economical screening of a continuously manufactured product is considered. Here, the cost of inspection and cost of passing defective items are assumed to be given. Our objective is to find a screening policy that minimizes the long run expected total cost per item. Similar to Klein's approach [7], we will use a Markovian sequential decision model to formulate the problem, and show how to find an optimal solution via linear programming. We will also discuss its implementation in practice and look at an illustrative example.     

Optimization for condition-based maintenance with semi-Markov decision process

The semi-Markov decision model is a powerful tool in analyzing sequential decision processes with random decision epochs. In this paper, we have built the semi-Markov decision process (SMDP) for the maintenance policy optimization of condition-based preventive maintenance problems, and have presented the approach for joint optimization of inspection rate and maintenance policy. Through numerical examples, the improvement of this method is compared with the scheme, which optimizes only over the inspection rate. We also find that under a special case when the deterioration rate at each failure stage is the same, the optimal policy obtained by SMDP algorithm is a dynamic threshold-type scheme with threshold value depending on the inspection rate.     

Economic optimization of off-line inspection

In this paper we solve the problem of determining the optimal inspection/disposition policy for a finite batch of items produced by a machine that is subject to random breakdowns. In particular, we identify which units should be inspected and in which order so as to minimize costs. The operational implications of the optimal policy are analyzed with a selected set of numerical results. We place special emphasis on three different policies: the cost minimizing policy; the policy of perfect information, i.e., we insist on determining the quality of each unit; and the policy of zero-defects,i.e., we insist that all accepted units are known to conform to specifications, allowing the rejection of units of unknown quality. We also show how the optimal inspection/disposition policy is incorporated into the optimization of the batch size.     

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.     

Univariate X¯ control charts for individual characteristics in a multinormal model

The early work on multivariate statistical process control was built upon Hotelling's T ² control chart which was developed to simultaneously monitor the means of correlated quality variables. This chart, however, has a drawback, namely, the problem of identifying the responsible variable(s) when an out-of-control signal occurs. One alternative is to use a separate X¯ control chart for each individual characteristic with equal risks, based on Bonferroni inequality. In this study, we show that, from an economic perspective, it may be desirable to have unequal type I risks for the individual charts, because of different inspection and restoration costs associated with each variable. We obtain their risk ratios, which are measures of relative importance of the variables monitored. Then, based on these risk ratios, we develop computer algorithms for finding the exact control limits for individual variables from a multinormal distribution, in the sense that the overall type I risk of the charts is equal to the desired value. Numerical studies show that the proposed methods give optimal or near-optimal results from an economic as well as statistical point of view.     

Optimisation design of attribute control charts for multi-station manufacturing system subjected to quality shifts

The qualities of products are a major concern in any production system; thus implementing efficient inspection policies is of great importance to reduce quality-related costs. This article addresses the problem of finding optimal inspection policies for the multi-station manufacturing system (MMS) subjected to quality shifts to minimise total quality-related cost. Each station of the MMS may stay at either in-control condition or out-of-control condition, which may lead to different nonconforming product rates. Markov chain method is used to calculate the steady-state probability distribution (SSPD). Based on the SSPD, the cost structure of this MMS is analysed. The economical optimisation model of attribute control charts (ACCs) is then established, in which the decision variables are the control chart parameters: sampling interval, sample size and control limit. The ACCs optimisation model is resolved by the proposed integrated algorithm combining heuristic rule and tabu search. This approach is verified through an application case taken from a mobile phone shell production company. The results of comparative analysis show that the proposed model is much more economical than both the current outgoing inspection strategy and the regular np control chart. The sensitivity analysis of four input parameters is also conducted.     

An Annotated Bibliography on Methods for Non-linear least Squares Computations Including Test Problems

An optimal inspection policy will inspect either every item produced or no item when (a) product characteristics are well modeled as iid and (b) overall inspection cost is a sum of individually and identically determined costs for each of the items encountered. This result is widlsly known for special cases such as iid Bernoulli product characteristics with single-sample lot acceptance-sampling plans. We show that the result holds true much more generally and over a much wider class of inspection plans, even when independent inspection errors are possible. We examine the assumptions that lead to all-or-none optimality and discuss the practil:al meaning of all-or-none results to practitioners. Examples are given to demonstrate that both “other” cost structures and “informative” inspections (i.e., lack of independence) can lead to optimal policies that are not of the all-or-none type.     

The Economic Design of Sampling Control Strategies for a Class of Industrial Process

Many authors have proposed models for the optimal design of control charts for the control of the defect rate for industrial processes. These models have limited the alternatives available subsequent to sampling to either rebuilding the process, or continuing to operate the process in its current state. Although the cost of rebuilding the process is treated as an adjustable parameter in these models, it can not be a negligible amount. If it were, the obvious optimal strategy would be to rebuild the process, possibly without samplings, after each batch or unit produced. This paper considers the determination of the optimal sample size and sample interval for the case where the process does not have to be rebuilt, but rather may be simply adjusted at a nominal cost after each sample. A model, which assumes a continuum of process fraction defectives, is developed and then used to explore the effect of fixed and variable sampling cost, specification limits and measurement errors on the optimal sampling policy and costs.     

The Study of Cost Rate for Multiple Cold Standby System Replacement Policies

We consider a replacement policy for a multiple-component cold-standby system,and atfer we analyze this plicy,we want to get the mean total cost rate and the preventive policy to make it the lowest.In this system,the failure rate of the component in poeration is constant,and the inspection will control all the processes of the operation.The system is inspected at random points over time to determine whether it is to be replaced.During the process,the replacement decision is based on the number of failed components at the time of inspection.     

Sequential degradation-based burn-in test with multiple periodic inspections

Burn-in has been proven effective in identifying and removing defective products before they are delivered to customers. Most existing burn-in models adopt a one-shot scheme, which may not be sufficient enough for identification. Borrowing the idea from sequential inspections for remaining useful life prediction and accelerated lifetime test, this study proposes a sequential degradation-based burn-in model with multiple periodic inspections. At each inspection epoch, the posterior probability that a product belongs to a normal one is updated with the inspected degradation level. Based on the degradation level and the updated posterior probability, a product can be disposed, put into field use, or kept in the test till the next inspection epoch. We cast the problem into a partially observed Markov decision process to minimize the expected total burn-in cost of a product, and derive some interesting structures of the optimal policy. Then, algorithms are provided to find the joint optimal inspection period and number of inspections in steps. A numerical study is also provided to illustrate the effectiveness of our proposed model.     

Joint optimisation of tracking capability and price in a supply chain with endogenous pricing

Tracking systems have been widely used to resolve the issues of product recall and food safety. Thus far, few researches have been done on designing the tracking capability from the perspective of supply chain. In this paper, using the traceable unit size at the manufacturer level to measure the tracking capability, we propose a non-convex non-linear programming to jointly optimise the tracking capability and price considering the tracking cost and recall cost in a supply chain with endogenous pricing. Results show that, in both centralised and decentralised supply chains, there is a unique tracking capability and retailing/wholesale price with closed-form solutions to optimise the supply chain profit. When the cost ratio (unit tracking cost/unit recall cost) is sufficiently large and small, the optimal tracking strategy is barcode tracking and unit tracking, respectively, and otherwise, the optimal tracking strategy is batch tracking with an economic traceable unit size which depends on the cost ratio, quality inspection threshold, supply defection rate and the supplier’s tracking capability. Furthermore, in the context of large and small cost ratio, we find that improving tracking capability will enlarge and mitigate the effect of double marginalisation, respectively. In particular, we find that the strict tracking regulation policy is more robust than the subsidy policy to improve the supply chain tracking capability.     

Using economically designed Shewhart and adaptive ―X charts for monitoring the quality of tiles

This paper presents the economic design of ―X control charts for monitoring a critical stage of the main production process at a tile manufacturer in Greece. Two types of ―X charts were developed: a Shewhart‐type chart with fixed parameters and adaptive charts with variable sampling intervals and/or sample size. Our prime motivation was to improve the statistical control scheme employed for monitoring an important quality characteristic of the process with the objective of minimizing the relevant costs. At the same time we tested and confirmed the applicability of the theoretical models supporting the economic design of control charts with fixed and variable parameters in a practical situation. We also evaluated the economic benefits of moving from the broadly used static charts to the application of the more flexible and effective adaptive control charts. The main result of our study is that, by redesigning the currently employed Shewhart chart using economic criteria, the quality‐related cost is expected to decrease by approximately 50% without increasing the implementation complexity. Monitoring the process by means of an adaptive ―X chart with variable sampling intervals will increase the expected cost savings by about 10% compared with the economically designed Shewhart chart at the expense of some implementation difficulty. Copyright © 2006 John Wiley & Sons, Ltd.     

Optimal inspection policies with predictive and preventive maintenance

A model is proposed to study the inspection and maintenance policy of systems whose failures can be detected only by periodic tests or inspections. Using predictive techniques, the time of the system failure can be predicted for some failure modes. If the system is found failed in an inspection, a corrective maintenance action is carried out. If the system is in a good condition but the predictive test diagnoses a failure in the period until the next inspection, then the system is replaced. The cost rate function is obtained for general distribution function of the signal time of a future failure and for one specific distribution function recently proposed. An algorithm is presented to find the optimal time between inspections and predictive tests and the optimal system replacement times for an age replacement policy. Numerical experiments illustrate the model.     

THE EFFECTS OF INSPECTION ERRORS TO THE IMPERFECT EMQ MODEL

The presence of Type I and Type II inspection errors may seriously affect the product quality. In this paper we incorporate them into the EMQ (Economic Manufacturing Quantity) model under the imperfect production system. We derive the expected total cost when the shift of the production process follows a general distribution and the inspection interval is arbitrary. The objective is to determine the optimal production cycle length and optimal inspection number while minimizing the total cost. Results for special shift distributions such as exponential and Weibull are established. Sensitivity analysis of the optimal production cycle length and analysis of the cost surplus for incorporating inspection errors under exponential shift distribution are also provided.     

Economic Design of CSP-1 Plan Under the Dependent Production Process and Linear Inspection Cost

Chen and Chou presented the economic design of type I continuous sampling plan (CSP-1 plan) under the linear inspection cost. However, it can be argued that the production process is seldom independent. In this paper, we further propose the problem concerning the economic design of CSP-1 plan under the dependent production process and linear inspection cost. A solution procedure is developed to find the unique combination (i*, f*) that will meet the average outgoing quality limit (AOQL) requirement while also minimizing the total expected cost per unit produced during one inspection cycle.     

Joint production and maintenance planning with machine deterioration and random yield

Production, yield and maintenance are three key components for sustaining the competitiveness of a manufacturing firm. In this paper, we investigate a joint production and maintenance planning problem in a periodic review environment subject to stochastic demand and random yields. The manufacturing system deteriorates from period to period according to a discrete-time Markov chain. The objective is to find an integrated lot sizing and maintenance policy for the system such that the aggregate cost associated with production, holding, backlogging and maintenance is minimised. We formulate this integrated planning problem as a Markov decision process and analyse the structural properties of the optimal policies. We prove that the optimal production and the maintenance policies both exhibit a control limit structure and show that the solution to the finite-horizon problems converges to that of the infinite-horizon problem.     

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.