Determining the production lot size with a heuristic inspection policy for controlling the quality of input materials and products

This study simultaneously determines the optimal production lot size and an inspection policy for input materials and products, where an unreliable process produces products with a discrete general shift distribution. This work proposes a heuristic inspection policy for materials and products, by first obtaining the inspection range for the input material without considering product inspection, and by further determining the product inspection range based on the obtained range of the input material inspection. The optimal inspection policy shows that common policies of no or full inspection are never optimal. This study includes the optimal production lot size based on the obtained inspection policy. Numerical examples demonstrate the impacts of input quality level, process reliability and unit nonconforming cost on the optimal solution, which adopts a discrete Weibull shift distribution to model the process failure time. Finally, this study addresses the conclusions.     

Optimal lot size and offline inspection policy

In this study, we investigate the effects of using process status at the end of the production lot (PSPL), on determining the optimal policies for products inspection and production lot size. First, we obtain the optimal product inspection policies for different PSPL for a given lot in the in-control or out-of-control state. Properties for the inspection policy are explored. Then, the expected total cost function, which includes setup cost, process maintenance cost and quality-related control cost, is obtained. The optimal production lot size that minimizes the expected total cost per item is determined. Our proposed inspection policy is compared with the three policies of no inspection, full inspection, and disregarding the first s items policy, in which only items from s+1 until the end of the production lot are inspected. Differences in the minimum expected total cost per item between our proposed inspection policy and the other three policies are investigated with a numerical example.     

Optimal inspection and economical production quantity strategy for an imperfect production process

This paper considers a problem relating to the integration of production, maintenance, inspection and inventory. The optimal inspection interval, inspection frequency and production quantity were studied, assuming an imperfect production process and a maintenance policy that permits minimal repair, shortages and inspection errors. When the process is in the in-control state, preventive maintenance is performed. If the process is in the out-of-control state, there are two possibilities. The first is a relatively mild production process shift, where minimal repair can bring the process back into the in-control state. The second possibility is a more serious shift, where minimal repair cannot bring the process back into the in-control state. In this case, it will be necessary to halt production and perform repair or replacement. It is accepted that, when a production process is in the out-of-control state, an amount of defective goods will be produced. Within the context of a deteriorating production process system, the advantages of preventive maintenance were explored, using numerical analysis to analyse the impact of inspection errors, permitted shortage conditions and minimal repair on cost.     

An inspection policy for shredder equipment used in steel production lines considering buffer level and operating time

On the basis of scrap metal recycling, the shredding process transforms a large variety of objects composed predominantly of steel in crunched scrap. The main equipment used in this process is called shredder: a device consisting of several hammers attached in a horizontal rotor. Due to high efforts involved in the transformation process, the maintenance is an essential aspect to ensure the continuity of the shredder operation. A non-efficient maintenance policy can have negative consequences for the shredder performance, whose failure can stop all downstream processes for which it provides the basic input (crunched scrap). In order to improve the performance of the steel production line, we are introducing an opportunistic inspection policy that makes use of the crushed scrap buffer level. The policy guides the execution of an inspection after a time T of continuous operation, when the system fails, when the buffer is full or empty, or when the opportunity criterion (operation time ≥ t AND buffer level ≥ h) is verified, whichever occurs first. At an inspection, the shredder is stopped, the hammers are inspected, and the failed and defective ones are replaced by new units. A discrete-event simulation model was developed and applied to the context of a real steel production line of a company. The optimal opportunistic inspection policy was obtained, and its performance was compared with that of the policy currently adopted by the company. The results showed the possibility of obtaining significant economic savings. A sensitivity analysis showed a cost reduction of around 95 %.     

Simultaneous determination of the optimal production–inventory and product inspection policies for a deteriorating production system

In this paper, we attempt to find a method for the optimization of production–inventory and product inspection policies for deteriorating production systems. Taking advantage of the nature of a deteriorating production system, a strategy would be not to inspect the first s items of the batch. Therefore, an inspection policy which disregards the first s (DTF-s) items of the batch is proposed. Under the DTF-s policy, we do not inspect the first s produced items but inspect only those items from the (s+1)th till the end of the production run. The objective of this study was the joint determination of the production lot size and the inspection policy s, resulting in a minimization of the expected average cost per unit time. Based on this model, the underlying conditions necessary for the existence of an optimal policy are given. Two commonly used inspection strategies, no inspection and full inspection are discussed. Under both inspection strategies, an optimal production–inventory lot is bounded by the traditional economic quantity. The case of full inspection is shown to be an extension of previously reported results. The option of investing in the process of quality improvement is also discussed. Finally, numerical examples are given to illustrate the method and its advantages in the conclusion.Scope and purposeThis paper considers the relationship between production, inventory and inspection in a deteriorating production system which may transit from the “in-control” state to the “out-of-control” state after a period of operation. Once the transition to the “out-of-control” state has occurred, it is assumed that some percentage of the items produced are defective or of substandard quality. However, in many cases, defects in a defective item can only be identified by an inspection process which carries an inspection cost. Those inspected items which are found to be defective are reworked at some cost before being shipped. On the other hand, defective items which are not inspected will be passed to the customer, incurring a much larger warranty cost. In order to operate such a system economically, tradeoffs among production setup, inventory, inspection and defective cost must be analyzed. Deterioration of the production system is an inherent process in all manufacturing industries. An understanding of the relationship among production, inventory and inspection for such systems will help managers to maintain efficient and economic control of operations.     

A mixed inspection policy for CSP-1 and precise inspection under inspection errors and return cost

This paper is to deal with a mixed policy between precise inspection and CSP-1 with inspection errors (Types I and II) and return cost. First, the concept of a renewal reward process is employed to obtain the long-term net profit of the mixed inspection policy. Then, with respect to non-repairable and repairable products, we determine the following four decision variables such that the unit net profit is maximal: (1) the optimal clearance number, (2) the optimal sampling frequency, and (3) the proportions which should be taken precise inspection for the non-inspected items in the procedure of CSP-1 and the non-defectives identified by CSP-1. Overall, the analytical results indicate that depending on seven parameters (Type I error, Type II error, the selling price of an item, the unit repair cost, the unit return cost, the unit cost of precise inspection, and the process defective fraction), there are three possible optimal inspection policies for CSP-1: “Do not inspect”, “Do 100% inspection”, or any setting for (i, s); for both of the non-defectives identified in CSP-1 and the non-inspected items, there are three possible proportions which should be performed a precise inspection: all, none, or any proportion.     

Two EPQ models with imperfect production processes,inspection errors,planned backorders,and sales returns

In practice the items received in a lot may contain defective items, and during the screening process to eliminate the defective items, the inspector may incorrectly classify a non-defective item as defective (a Type I error) or incorrectly classify a defective item as non-defective (a Type II error). In this paper, we develop two economic production quantity models with imperfect production processes, inspection errors, planned backorders, and sales returns. A closed form solution is obtained for the optimal production lot size and the maximum shortage level for both models. We provide two numerical examples, one in which the defective probability and Type I and Type II inspection errors follow uniform distributions, and the second in which we assume they follow beta distributions. Sensitivity analyses are performed to see the impact of the defective probability, the probability of the Type I inspection error, the probability of the Type II inspection error, the holding cost, and the backordering cost on the optimal solutions. We obtain similar results on the sensitivity analyses for both numerical examples. The results show that the time factor of when to sell the defective items has a significant impact on the optimal production lot size and the backorder quantity. The results also show that if customers are willing to wait for the next production when a shortage occurs, it is profitable for the company to have planned backorders although it incurs a penalty cost for the delay.     

Production/inventory systems with a stochastic production rate under a continuous review policy

A single production facility is dedicated to producing one product with completed units going directly into inventory. The unit production time is a random variable. The demand for the product is given by a Poisson process and is supplied directly from inventory when available, or is backordered until it is produced by the production facility. Relevant costs are a linear inventory holding cost, a linear backorder cost, and a fixed setup cost for initiating a production run. The objective is to find a control policy that minimizes the expected cost per time unit.The problem may be modeled as an M/G/1 queueing system, for which the optimal decision policy is a two-critical-number policy. Cost expressions are derived as functions of the policy parameters, and based on convexity properties of these cost expressions, an efficient search procedure is proposed for finding the optimal policy. Computational test results demonstrating the efficiency of the search procedure and the behavior of the optimal policy are presented.     

A generalized age replacement policy with random delivery time and inspection

This paper is concerned with an age replacement policy that is more general because spare part random delivery time, age-dependent minimal repair and inspection are considered. Introducing the cost due to ordering, repairs, shortage, holding and inspection, we derive the expected cost per unit time in the long run as a criterion of optimality and seek the optimum age replacement policy by minimizing that cost. Various special cases are discussed and a numerical example is finally given to illustrate the method.     

An optimal production lot-sizing problem for an imperfect process with imperfect maintenance and inspection time length

This article develops an integrated model in considering the situations of an imperfect process with imperfect maintenance and inspection time for the joint determination of both economic production quantity (EPQ) and preventive maintenance (PM). This imperfect process has a general deterioration distribution with increasing hazard rate. Even with periodic PM, such a production system cannot be recovered as good as new. This means that the system condition depends on how long it runs. Also, the PM level can be distinct due to the maintenance cost. For convenience, it is assumed the age of system is reduced in proportional to the PM level. Further, during a production cycle, we need an inspection to see if the process is in control. This inspection might demand a considerable amount of time. In this article, we take PM level and inspection time into consideration to optimise EPQ with two types of out-of-control states. To see how the method works, we use a Weibull shock model to show the optimal solutions for the least costs.     

Economic optimization for an off-line inspection, disposition and rework model

An economic off-line inspection, disposition, and rework (IDR) model for a batch produced from an unreliable production system was recently proposed by Wang et al. [Economic optimization of off-line inspection with rework consideration, European Journal of Operational Research, 194 (2009) 807–813], where the process is assumed to possess a discrete general shift distribution. Unfortunately, there are some important flaws in the proposed IDR model; in particular, while obtaining optimal IDR policy, they incorrectly assumed that the process shift distribution had the memoryless property. As a result, the purpose of this paper is to reformulate the IDR model, and to develop a solution procedure to find the optimal IDR policy.     

An integrated production–distribution inventory system involving probabilistic defective and errors in quality inspection under variable setup cost

This paper presents a probabilistic defective vendor–buyer integrated inventory model with the consideration of quality inspection errors at the buyer's end and setup cost as function of capital investment. An integrated inventory model is established to find the optimal solutions of lot size, setup cost, and the total number of shipments from the vendor to the buyer in one production run, so that the joint expected total cost incurred has the minimum value. We consider three types of continuous probabilistic defective function to find the associated cost of the system. The expected total cost function is derived for each of these three distributions, its convexity is proved via differential calculus. An efficient iterative algorithm is designed to obtain the optimal solution of the model. The computational effort and time are small for the proposed algorithm and it is simple to implement. Numerical examples and sensitivity analysis are used to demonstrate the application and the performance of the proposed methodology. The computational results indicate that if we make decisions with the capital investment in reducing setup cost, it will help to lower the system cost, and we obtain a significant amount of savings to increase the competitive edge in business.     

Investigating imperfect process and demand effects on inspection scheduling and supply chain replenishment policy

The challenge of supply chain management (SCM) is to have a supervision in managing operations that is critical to coordinate with partners and have high quality product. Global companies are now striving to engage in production improvement and product design efficiency to provide a good support for competition in any parts of the world. Recently, the development and improved capability of production and information technology have changed the ways in which companies and manufacturers operate. With the assist of advanced production philosophy and collaborative design and manufacturing (CDM), the quality of product and manufacturing has been notably enhanced. However, a traditional single-stage inventory model considers a case in which depletion of inventory is merely caused by a constant rare. In practice, there is an inventory loss due to deterioration and imperfect production. The purpose of this paper is to investigate the effects of stock- and warranty-dependent demand, and take the relationship among imperfect, warranty policy and inspection scheduling into consideration. An integrated two-stage production inventory deteriorating model for replenishment policy and inspection plan is developed utilizing time-weighted inventory approach. A numerical example is presented to demonstrate the theory. The results show that fixed selling rate, the holding cost and the unit inspection cost are the critical factors in applying the deteriorating inventory model.     

A new strategy for defect inspection by the virtual inspection in semiconductor wafer fabrication

The defect of process equipments is a major factor that impairs the yields in the mass production of semiconductor wafer fabrication and it is a main supervision means to use high-resolution defect inspection tools to detect and monitor the defect damage. Due to the high investment costs of these inspection tools and the resulting decrease in the throughput, how to improve the sampling rate is an important issue for the associated inspection strategy. This paper proposes a new concept and implementation of virtual inspection (VI) to enhance the detection and monitoring of defect in semiconductor production process. The underlying theory of the VI concept is that the state variables identifications (SVIDs) of process equipments can reflect the process quality effectively and loyally. The approach of VI is to combine the application of the fault detection and classification (FDC), and the defect library and the re-engineering of inspection procedure to reach the full-scope of strategic objective. VI enables the defect monitoring to enter a new era by promoting the monitoring level of defect inspection from the previous lot-sampling basis to the wafer-sampling level, and hence upgrades the sampling strategy from random-sampling to full and right-sampling. In this study, various typical defect cases are utilized to illustrate how to create VI models and verify the reliability of the proposed approach. Furthermore, a feasible architecture of the VI implementation for mass production in semiconductor factory is presented in the paper.     

An optimal batch size for a production system operating under periodic delivery policy

A manufacturing system that procures raw materials from suppliers in a lot and processes them into a finished product is considered in this research. An ordering policy is proposed for raw materials to meet the requirements of a production facility which, in turn, must deliver finished products demanded by outside buyers at fixed points in time. First, a general cost model is formulated considering both raw materials and finished products. Then, using this model, a simple procedure is developed to determine an optimal ordering policy for procurement of raw materials, and the manufacturing batch size, to minimize the total cost of meeting customer demands in time. The dependent relationships between production batch size and rawmaterial purchasing quantity, and various delivery patterns considered in recent literature are critically reviewed. The quality of the solution is evaluated. Numerical examples are provided.     

Investigating the defect detection effectiveness and cost benefit of nominal inspection teams

Inspection is an effective but also expensive quality assurance activity to find defects early during software development. The defect detection process, team size, and staff hours invested can have a considerable impact on the defect detection effectiveness and cost-benefit of an inspection. In this paper, we use empirical data and a probabilistic model to estimate this impact for nominal (noncommunicating) inspection teams in an experiment context. Further, the analysis investigates how cutting off the inspection after a certain time frame would influence inspection performance. Main findings of the investigation are: 1) Using combinations of different reading techniques in a team is considerably more effective than using the best single technique only (regardless of the observed level of effort). 2) For optimizing the inspection performance, determining the optimal process mix in a team is more important than adding an inspector (above a certain team size) in our model. 3) A high level of defect detection effectiveness is much more costly to achieve than a moderate level since the average cost for the defects found by the inspector last added to a team increases more than linearly with growing effort investment. The work provides an initial baseline of inspection performance with regard to process diversity and effort in inspection teams. We encourage further studies on the topic of time usage with defect detection techniques and its effect on inspection effectiveness in a variety of inspection contexts to support inspection planning with limited resources.     

Optimal production run length in deteriorating production processes with fuzzy elapsed time

This paper investigates the optimal production run length in deteriorating production processes, where the elapsed time until the production process shifts is characterized as a fuzzy variable, also the setup cost and the holding cost are characterized as fuzzy variables, respectively. A mathematical formula representing the expected average cost per unit time is derived, and some properties are obtained to establish an efficient solution procedure. Since there is no closed-form expression for the optimal production run length, an approximate solving approach is presented. Finally, two numerical examples are given to illustrate the procedure of searching the optimal solutions.     

Specifying and optimizing robotic motion for visual quality inspection

Installation or even just modification of robot-supported production and quality inspection is a tedious process that usually requires full-time human expert engagement. The resulting parameters, e.g. robot velocities specified by an expert, are often subjective and produce suboptimal results. In this paper, we propose a new approach for specifying visual inspection trajectories based on CAD models of workpieces to be inspected. The expert involvement is required only to select – in a CAD system – the desired points on the inspection path along which the robot should move the camera. The rest of the approach is fully automatic. From the selected path data, the system computes temporal parametrization of the path, which ensures smoothness of the resulting robot trajectory for visual inspection. We then apply a new learning method for the optimization of robot speed along the specified path. The proposed approach combines iterative learning control and reinforcement learning. It takes a numerical estimate of image quality as input and produces the fastest possible motion that does not result in the degradation of image quality as output. In our experiments, the algorithm achieved up to 53% cycle time reduction from an initial, manually specified motion, without degrading the image quality. We show experimentally that the proposed algorithm achieves better results compared to some other policy learning approaches. The described approach is general and can be used with different types of learning and feedback signals.     

Impact of inspection errors on the formulation of a multi-objective optimization process targeting model under inspection sampling plan

The literature on inspection systems has demonstrated that such systems are error prone. Such systems exhibit type I and type II errors. Type I error is classifying a conforming item as non-conforming while type II error is classifying a non-conforming item as conforming. The errors may arise from the measurement system or the inspectors. It is essential to assess the impact of the inspection errors on the optimal parameters and objective functions of process targeting models. The purpose of this paper is to assess the impact of the inspection errors on the optimal parameters and objectives functions values of Duffuaa and El-Ga’aly multi-objectives optimization model recently developed for process targeting (2013a). In order to accomplish the objectives of the paper, Duffuaa and El-Ga’aly multi-objective optimization model is extended by introducing the measurement errors in the inspection system and penalties to mitigate the effect of the errors. The results of the extended model is compared with the previous model and employed to studying the impact of the errors on the values of objective function and the optimal process parameters in a multi-objectives environment. The results indicated that inspection errors have significant impact on the profit and uniformity objectives.     

基于延迟时间理论的n中取k系统检测区间模型

延迟时间理论广泛应用于系统维修领域,能够更为细致和准确地反映系统的运行状态.不同于一般维修理论只用一个阶段描述系统的故障过程,该理论将系统的故障过程分为两个阶段:正常阶段和缺陷运行阶段.系统部件的状态有正常、缺陷和故障3种情况.延迟时间理论主要应用于单部件系统和串联系统的维修和检测,目前尚未应用于n中取k系统.对此,运用延迟时间理论确定n中取k系统的最优维修检测区间,分别针对两种情况进行分析:一种是系统中部件的初始缺陷发生时间服从指数分布的情况,一种是其服从非指数分布的情况.建立单位时间维修费用模型和设计仿真过程,以单位时间维修费用最小为目标进行优化,求解系统的最优维修检测区间.最后通过数值案例表明所提出方法的有效性.