The effect of testing errors on a repetitive testing process

Repetitive testing is a fairly common practice in the final testing stage of a chip manufacturing. Decisions on setting initial lot size and the number of testing repetitions are crucial to the effectiveness of the testing process. The task of setting optimal parameters for a testing process is often difficult in practical situations due to uncertainties in both incoming product yield and testing equipment condition during the testing process. In this paper, we investigate a repetitive testing process where the testing equipment may shift randomly from an in-control state to an inferior state during the testing process which, correspondingly, results in different testing errors. We develop a quantitative model that helps us to find optimal test parameters that maximizes system performance. Based on the model, we performed extensive numerical experiments to test the effects of incoming product defective rate, testing equipment shift rate, especially, type II testing errors on decision and system performance. We find that test equipment condition may significantly affect the optimal decisions on the number of test repetitive and initial testing batch size. Further, we find that, while a small type II testing error may have negligible negative effect of system performance, the effect increases as the error or the incoming product yield increases. The results of this research may potentially provide practitioners with insights and a quantitative tool for designing an efficient repetitive testing process.     

Multisource Bayesian sequential binary hypothesis testing problem

We consider the problem of testing two simple hypotheses about unknown local characteristics of several independent Brownian motions and compound Poisson processes. All of the processes may be observed simultaneously as long as desired before a final choice between hypotheses is made. The objective is to find a decision rule that identifies the correct hypothesis and strikes the optimal balance between the expected costs of sampling and choosing the wrong hypothesis. Previous work on Bayesian sequential hypothesis testing in continuous time provides a solution when the characteristics of these processes are tested separately. However, the decision of an observer can improve greatly if multiple information sources are available both in the form of continuously changing signals (Brownian motions) and marked count data (compound Poisson processes). In this paper, we combine and extend those previous efforts by considering the problem in its multisource setting. We identify a Bayes optimal rule by solving an optimal stopping problem for the likelihood-ratio process. Here, the likelihood-ratio process is a jump-diffusion, and the solution of the optimal stopping problem admits a two-sided stopping region. Therefore, instead of using the variational arguments (and smooth-fit principles) directly, we solve the problem by patching the solutions of a sequence of optimal stopping problems for the pure diffusion part of the likelihood-ratio process. We also provide a numerical algorithm and illustrate it on several examples.     

Optimal testing strategies in overlapped design process

Testing is an important activity in product development. Past studies, which are developed to determine the optimal scheduling of tests, often focused on single-stage testing of sequential design process. This paper presents an analytical model for the scheduling of tests in overlapped design process, where a downstream stage starts before the completion of upstream testing. We derive optimal stopping rules for upstream and downstream stages’ testing, together with the optimal time elapsed between beginning the upstream tests and beginning the downstream development. We find that the cost function is first convex then concave increasing with respect to upstream testing duration. A one-dimensional search algorithm is then proposed for finding the unique optimum that minimizes the overall cost. Moreover, the impact of different model parameters, such as the problem-solving capacity and opportunity cost, on the optimal solution is discussed. Finally, we compare the testing strategies in overlapped process with those in sequential process, and get some additional results. The methodology is illustrated with a case study at a handset design company.     

Effects of regulatory delays and uncertainty on pricing decisions

In this paper we introduce a simple decision rule that a single product firm may use for filing for a price change to offset variations of the marginal cost. We consider a regulatory body whose response to the price change request involves a time delay with an exponential distribution. Two possibilities regarding the response of the regulatory body are considered. In one case it is assumed to be a binary approval process in which the rate adjustment is either approved in its entirety or rejected. In the second case we consider a partial approval process with a more general distribution. Decision rules for each case are developed. Finally we derive a multi-stage decision rule in which filing decisions are continuously updated based on temporal variations of the cost function. The multi-stage pricing decision model assumes that marginal cost escalation satisfies a Markovian jump process.This work was completed while the authors were with Bell Laboratories, USA.     

技术不确定环境下新产品策略的设计应用

技术的动态发展和企业间的竞争对企业新产品策略有很大影响,直接决定新产品的引进周期。本文在产业技术动态变化的随机环境下构建随机动态规划模型,关注产业技术进步、投资成本和产品市场竞争等影响因素,探讨企业进行新产品引进的周期选择,对新产品引进的周期和质量决策进行方法设计和应用举例。利用随机动态规划模型得出新产品引进的最优时间周期,用算例分析技术进步和产品研发成本对企业引进周期策略的影响,采取策略迭代的方法进行求解,发现技术进步较快时企业的新产品引进步伐也较快,研发成本的提高使企业的新产品引入步伐降低。     

The impact of inspection errors,imperfect maintenance and minimal repairs on an imperfect production system

This paper develops an integrated model of production lot-sizing, maintenance and quality for considering the possibilities of inspection errors, preventive maintenance (PM) errors and minimal repairs for an imperfect production system with increasing hazard rates. In this study, a PM activity is imperfect in that a production system cannot be recovered as good as new and might cause the production system to shift to the out-of-control state with a certain probability. Numerical analyses are used to simulate the effect of changes in various parameters on the optimal solution for which the time that the process remains in the in-control state is assumed to follow a Weibull distribution. In addition, we investigate the effects of inspection errors and PM errors on the minimum total cost of the optimal inspection interval, inspection frequency and production quantity.     

Optimal systems for equipment maintenance and replacement under Markovian deterioration

Deterioration of equipment is modeled as a multistate discrete time controlled Markov process. The states are classified according to the degree of deterioration. The problem of design of optimal systems for equipment maintenance and replacement is considered when the decision-maker may take, in each stage, one of many available maintenance actions, classified according to their “stochastic effectiveness”; no action and replacement are included as alternatives. It is assumed that the transition probabilities satisfy two conditions which effectively describe a trend for monotonically increasing expected deterioration and rate of deterioration. Under these assumptions it is proved in the paper that the optimal (cost minimizing) decision system in an infinite horizon is of the control limit rule type, rapidly obtained by policy improvement algorithms. A numerical example is presented for a specific practical application; detailed data are available from the authors on request.     

Economic Production Lot Size Model with Variable Production Rate and Imperfect Quality

The classical economic production lot size (EPL) model assumes a constant production rate that is predetermined and inflexible, and perfect quality. Recent models have removed the assumption of perfect quality while maintaining the inflexible production rate assumption. Production rates in many cases, such as orders filled by a machine, can be changed. Moreover, unit production cost and process quality depend on the production rate. In this paper, we extend the EPL model to cases where the production rate is a decision variable. Unit production cost becomes a function of the production rate. Also, the quality of the production process deteriorates with increased production rate. We solve the proposed model for special cost and quality functions and illustrate the results with a numerical example. The results show that, for cases where increases in the production rate lead to a significant deterioration in quality, the optimal production rate may be smaller than the rate that minimizes unit production cost. For cases where quality is largely independent of the production rate, the optimal production rate may be larger than the rate that minimizes unit production cost.     

Optimal preventive maintenance for equipment with two quality states and general failure time distributions

We present an economic model for the optimization of preventive maintenance in a production process with two quality states. The equipment starts its operation in the in-control state but it may shift to the out-of-control state before failure or scheduled preventive maintenance. The time of shift and the time of failure are generally distributed random variables. The two states are characterized by different failure rates and revenues. We first derive the structure of the optimal maintenance policy, which is defined by two critical values of the equipment age that determine when to perform preventive maintenance depending on the actual (observable) state of the process. We then provide properties of the optimal solution and show how to determine the optimal values of the two critical maintenance times accurately and efficiently. The proposed model and, in particular, the behavior of the optimal solution as the model parameters and the shift and failure time distributions change are illustrated through numerical examples.     

Optimum life testing plans in presence of hybrid censoring: A cost function approach

Hybrid censoring scheme is a combination of Type‐I and Type‐II censoring schemes. Determination of optimum hybrid censoring scheme is an important practical issue in designing life testing experiments to enhance the information on reliability of the product. In this work, we consider determination of optimum life testing plans under hybrid censoring scheme by minimizing the total cost associated with the experiment. It is shown that the proposed cost function is scale invariant for some selected distributions. Optimum solution cannot be obtained analytically. We propose a method for obtaining the optimum solution and consider Weibull distribution for illustration. We also studied the sensitivity of the optimal solution to the misspecification of parameter values and cost components through a well‐designed sensitivity analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

Reverse programming the optimal process mean problem to identify a factor space profile

For the manufacturer that intends to reduce the processing costs without sacrificing product quality, the identification of the optimal process mean is a problem frequently revisited. The traditional method to solving this problem involves making assumptions on the process parameter values and then determining the ideal location of the mean based upon various constraints such as cost or the degree of quality loss when a product characteristic deviates from its desired target value. The optimal process mean, however, is affected not only by these settings but also by any shift in the variability of a process, thus making it extremely difficult to predict with any accuracy. In contrast, this paper proposes the use of a reverse programming scheme to determine the relationship between the optimal process mean and the settings within an experimental factor space. By doing so, one may gain increased awareness of the sensitivity and robustness of a process, as well as greater predictive capability in the setting of the optimal process mean. Non-linear optimization programming routines are used from both a univariate and multivariate perspective in order to illustrate the proposed methodology.     

Approximation of Markov decision processes with general state space

We deal with a discrete-time finite horizon Markov decision process with locally compact Borel state and action spaces, and possibly unbounded cost function. Based on Lipschitz continuity of the elements of the control model, we propose a state and action discretization procedure for approximating the optimal value function and an optimal policy of the original control model. We provide explicit bounds on the approximation errors. Our results are illustrated by a numerical application to a fisheries management problem.     

Sequential variable sampling plan for normal distribution

In this paper, a sequential variable sampling plan is studied. Suppose that the quality of an item in a batch is measured by a normally distributed random variable with a known variance, but the mean is unknown with a normal prior distribution. Then by using Bayesian approach and considering a Markov decision process, the optimality equations for the minimum total expected cost are formulated. We show that an optimal decision rule will have a control limit structure. An algorithm for a sequence of ϵ-optimal decisions is introduced. Then, the statistical procedure for conducting the sequential sampling plan is presented.     

A periodic review production and maintenance model with random demand,deteriorating equipment,and binomial yield

In many environments, product yield is heavily influenced by equipment condition. Despite this fact, previous research has either focused on the issue of maintenance, ignoring the effect of equipment condition on yield, or has focused on the issue of production, omitting the possibility of actively changing the machine state. We formulate a Markov decision process model of a single-stage production system in which demand is random. The product yield has a binomial distribution that depends on the equipment condition, which deteriorates over time. The objective is to choose simultaneously the equipment maintenance schedule as well as the quantity to produce in a way that minimizes the sum of expected production, backorder, and holding costs. After proving some results about the structural properties of the optimal policy, numerical problems are used to compare this method to the typical approach of solving the maintenance and production problems sequentially. The results show that the simultaneous solution provides substantial gains over the sequential approach. In the cases studied, the proposed method resulted in an average cost savings of approximately 18%.     

Economic design of a complete inspection plan with interactive quality improvement

Consider the monitoring of an ongoing production process subject to complete inspection. Inferior quality products are either reworked or scrapped to assure good outgoing quality. Whenever the quality characteristic of a product exceeds a predetermined action limit, remedial action is taken to restore the process to an in-control state. In addition, the decision maker has a learning opportunity to improve the process by investing in resources to identify and eliminate the causes of deviation from the target quality. Taking a learning action would improve the mean value of the quality characteristic of items produced in an out-of-control state. This paper proposes a cost model to determine the optimal number of learning actions to be taken and the optimal action limit. The model considers the trade-off between quality cost and process improvement cost.     

Dynamic repositioning strategy in a bike-sharing system; how to prioritize and how to rebalance a bike station

Bike-sharing systems are becoming increasingly popular in large cities. The natural imbalance and the stochasticity of bike’s arrivals and departures lead operators to develop redistribution strategies in order to ensure a sufficiently high quality of service for users. Using a Markov decision process approach, we develop an implementable decision-support tool which may help the operator to decide at any point of time (i) which station should be prioritized, and (ii) which number of bikes should be added or removed at each station. Our objective is to minimize the rate of arrival of unsatisfied users who find their station empty or full. The existence of an optimal inventory level at each station is proven. It may vary over time but does not depend on the capacity of the truck which operates the repositioning. Next, we compute the relative value function of the system, together with the average cost and the optimal state. These results are used to derive a policy for station’s prioritization using a one-step policy improvement method. We evaluate our policy in comparison with the optimal one and with other intuitive ones in an extended version of our model. From our numerical experiments, we show that only a little intervention of the operator can significantly enhance the quality of service, and that the rule of thumb for bike repositioning is to prioritize the closer, the more active, the closer to be full or empty, and the more imbalanced stations if no reversing in the imbalance is anticipated.     

Heuristic procedures for a stochastic batch service problem

This paper considers a multi-class batch service problem that involves a class-dependent waiting cost and a service cost in determining customer batch sizes. Unlike a fixed service cost used widely in standard models, the service cost considered in this work is incurred only if the total service time is over the capacity. We formulate this problem as an infinite horizon Markov decision process, and exploit its structural properties to establish theoretical results, including bounds on the optimal action space. We use the results to improve the value iteration procedure. Furthermore, we design heuristic algorithms for large problems. The numerical experiments demonstrate that the class-dependent waiting cost has a considerable influence on the optimal customer batch size. Finally, we evaluate the efficiency of the proposed value iteration procedure and the quality of the heuristic solutions.     

Decision-making in testing process performance with fuzzy data

Over the years, numerous process capability indices (PCIs) have been proposed to the manufacturing industry to provide numerical measures of process performance. Most research efforts have focused on developing and investigating PCIs that assess process capability by precise measurements of output quality. However, real observations of continuous quantities are not precise numbers; in practice, they are more or less imprecise. Since observations of continuous random variables are imprecise the values of related test statistics become imprecise. Therefore, decision rules for statistical tests have to be adapted to this situation. This article presents a set of confidence intervals that produces triangular fuzzy numbers for the estimation of C_pk index using Buckley’s approach with some modification. Additionally, a three-decision testing rule and step-by-step procedure are developed to assess process performance based on fuzzy critical values and fuzzy p-values. This concept is also illustrated with an example for testing process performance.     

An EMQ model with inspections and random machine failures

In the classical Economic Manufacturing Quantity (EMQ) model, it is assumed that all items produced are of perfect quality and the production facility never breaks down. However, in real production, the product quality is usually a function of the state of the production process which may deteriorate over time and the production facility may fail randomly. In this paper, we study the effect of machine failures on the optimal lot size and on the optimal number of inspections in a production cycle. The formula for the long-run expected average cost per unit time is obtained for a generally distributed time to failure. An optimal production/inspection policy is found by minimising the expected average cost.