A Cost-Minimization Model for Multicharacteristic Component Inspection

In this paper a mathematical model is developed for determining the optimal number of inspections for multicharacteristic components where failure can be catastrophic. The model optimizes the expected total cost per accepted component resulting from (1) type I errors, (2) type II errors, (3) cost of added inspection, and (4) ordering of quality characteristics for inspection. The model considers components with several characteristics to be inspected. Failure to meet the quality requirements of any one characteristic results in the rejection of the component. Taking into consideration all three costs referred to above, a mathematical expression for expected total cost per accepted component is obtained. Also, an expression is developed for finding the optimal sequence of characteristic inspection. Finally, a computational procedure is outlined to determine the optimal sequence of characteristic inspection and the optimal number of inspections using the two expressions stated above.     

A complete inspection plan for dependent multicharacteristic critical components

Here a complete inspection plan for critical components with several characteristics is designed for the case where the characteristics’ defective rates are statistically dependent. The plan minimizes total expected cost resulting from type-I and II errors and the cost of inspection. A mathematical model and an algorithm for obtaining the optimal number of repeat inspections are developed and demonstrated by an example. The output of the model is an optimal inspection plan to control the quality of critical components.     

Repeat inspection planning using dynamic programming

In this paper we propose a dynamic programming approach to the problem of determination of the inspection sequence of multi-characteristic critical components, and the number of repeat inspection for each characteristic. The model presented here considers the case of several classification of a product by an inspector. An inspector could classify a product as non-defective, to be reworked, or to be scrapped, with respect to a certain characteristic. The model accounts as well for possible misclassification by the inspector. The dynamic programming algorithm searches for a solution that minimizes the total cost of inspection per accepted component. The total cost includes the cost of false rejection of good items, the cost due to false acceptance of an item which is either reworkable or to be scrapped, the cost of inspection, and the cost of rework.     

Economic design of offline inspections for a batch production process

We consider a batch production process that can be either stable or unstable, in which inspection is performed offline after production of the batch is completed. The quality of a batch can be estimated with a desired level of certainty by inspecting only a sample of its units. In order to minimise the expected total cost per batch, which includes the costs of inspection, of false acceptance and of false rejection, we propose an economic inspection plan in which only a fraction of the batches, rather than each batch, is inspected. We prove that the expected total cost is a strictly quasiconvex function of the inspection interval. We establish necessary and sufficient conditions for the optimal inspection interval to be finite and propose an efficient algorithm to obtain its value. We demonstrate for the case of a single-sampling plan where the proposed economic approach outperforms the common procedure of inspecting every batch, and the proposed algorithm is very efficient.     

Carryover effects in multi-attribute inspection

Consider a product that has several quality attributes for inspection where the inspection of some attributes may have carryover effect(s) on subsequent inspections of other attributes. In this case, not all the attributes can be inspected on the same unit and therefore more than one unit may be required to complete one inspection cycle of all the attributes. Furthermore, if the inspection time of each attribute is considered, the number of attributes and their inspection sequence on each inspected unit may affect the time-span of an inspection cycle. In this paper, heuristic procedures are developed to find the inspection sequences of all the attributes that minimize the total number of inspected units and to balance the inspection time of each inspected unit such that the time-span of an inspection cycle is minimized. An integrated circuit card inspection example is given to illustrate the implementation of these procedures.     

An Effective Double Sampling Scheme for the c Control Chart

In the present paper is developed a statistical process control inspection procedure based on a new simple‐to‐implement and effective double sampling scheme for the c control chart, aimed at the minimization of the number of inspected observation units and warranting fixed levels for the type I and II error risks. In particular, the formulations of the false alarm risk α, the power P of the chart, and the expected number of inspected observation units for the developed inspection procedure are given, whereas a macro of Microsoft Excel is adopted to solve the tackled problem. In order to illustrate the application of the developed approach and to investigate on the influence of several operating parameters, numerical examples are carried out and the related considerations are given. Finally, by comparing the performance of the developed inspection procedure with that of the related classic c chart scheme, meaningful reduction of the number of the inspected observation units can be achieved by adopting the proposed approach. Copyright © 2013 John Wiley & Sons, Ltd.     

Use of weighted least squares in relating component outage rates to age of vehicle

A field survey was conducted to measure the outage rates of various vehicle safety components. The population of vehicles sampled had been subject to a random checklane vehicle inspection program which annually inspected about 6% of the vehicles in the state. A recently developed unified approach to the analysis of multivariate categorical data was utilized to relate the various component outage rates to the age of the vehicle. A method is presented for utilizing commonly available least squares regression programs to accomplish the weighed least squares computations. The results relate component outage rates to the age of the vehicle and have implications for vehicle inspection programs.     

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.     

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.     

A joint spare part and maintenance inspection optimisation model using the Delay-Time concept

Spare parts and maintenance are closely related logistics activities where maintenance generates the need for spare parts. When preventive maintenance is present, it may need more spare parts at one time because of the planned preventive maintenance activities. This paper considers the joint optimisation of three decision variables, e.g., the ordering quantity, ordering interval and inspection interval. The model is constructed using the well-known Delay-Time concept where the failure process is divided into a two-stage process. The objective function is the long run expected cost per unit time in terms of the three decision variables to be optimised. Here we use a block-based inspection policy where all components are inspected at the same time regardless of the ages of the components. This creates a situation that the time to failure since the immediate previous inspection is random and has to be modelled by a distribution. This time is called the forward time and a limiting but closed form of such distribution is obtained. We develop an algorithm for the optimal solution of the decision process using a combination of analytical and enumeration approaches. The model is demonstrated by a numerical example.     

The optimal frequency and sequencing of tests in the inspection of multicharacteristic components

IIE Transactions - A product (component) often requires inspection on its characteristics, of which nonconformance of one would result in the rejection of the component. However, the need for...     

A process monitoring procedure based on a surrogate variable for dichotomous performance variable

A monitoring procedure using a surrogate variable is proposed for a process with a binary performance variable. Using the procedure, every item is inspected on a surrogate variable. When an item is rejected, the latest number of consecutively accepted items up to this point is compared with a predetermined number, r, to decide whether to stop the process for an investigation. The conditional distribution of the surrogate variable given the performance variable is assumed to be normal. A cost model is developed, which incorporates costs of inspection, misclassification, investigation, opportunity, and correction. We develop methods of finding the optimum r value and screening limit and analyze the effects of cost parameters on the optimal solution.     

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.     

On the optimal scheduling of periodic tests and maintenance for reliable redundant components

Periodically, some m of the n redundant components of a dependable system may have to be taken out of service for inspection, testing or preventive maintenance. The system is then constrained to operate with lower (n−m) redundancy and thus with less reliability during these periods. However, more frequent periodic inspections decrease the probability that a component fail undetected in the time interval between successive inspections. An optimal time schedule of periodic preventive operations arises from these two conflicting factors, balancing the loss of redundancy during inspections against the reliability benefits of more frequent inspections.Considering no other factor than this decreased redundancy at inspection time, this paper demonstrates the existence of an optimal interval between inspections, which maximizes the mean time between system failures. By suitable transformations and variable identifications, an analytic closed form expression of the optimum is obtained for the general (m, n) case. The optimum is shown to be unique within the ranges of parameter values valid in practice; its expression is easy to evaluate and shown to be useful to analyze and understand the influence of these parameters.Inspections are assumed to be perfect, i.e. they cause no component failure by themselves and leave no failure undetected. In this sense, the optimum determines a lowest bound for the system failure rate that can be achieved by a system of n-redundant components, m of which require for inspection or maintenance recurrent periods of unavailability of length t.The model and its general closed form solution are believed to be new [2] and [5]. Previous work [1], [4] and [10] had computed optimal values for an estimation of a time average of system unavailability, but by numerical procedures only and with different numerical approximations, other objectives and model assumptions (one component only inspected at a time), and taking into account failures caused by testing itself, repair and demands (see in particular [6], [7] and [9]).System properties and practical implications are derived from the closed form analytical expression. Possible extensions of the model are discussed. The model has been applied to the scheduling of the periodic tests of nuclear reactor protection systems.     

A fuzzy knowledge based method for maintenance planning in a power system

The inspection planning in electric power industry is used to assess the safety and reliability of system components and to increase the ability of failure situation identification before it actually occurs. It reflects the implications of the available information on the operational and maintenance history of the system. The output is a ranked list of components, with the most critical ones at the top, which indicates the selection of the components to be inspected.In this paper, we demonstrate the use of a fuzzy relational database model for manipulating the data required for the criticality component ranking in thermal power systems inspection planning, incorporating criteria concerning aspects of safety and reliability, economy, variable operational conditions and environmental impacts. Often, qualitative thresholds and linguistic terms are used for the component criticality analysis. Fuzzy linguistic terms for criteria definitions along with fuzzy inference mechanisms allow the exploitation of the operators' expertise.The proposed database model ensures the representation and handling of the aforementioned fuzzy information and additionally offers to the user the functionality for specifying the precision degree by which the conditions involved in a query are satisfied.In order to illustrate the behavior of the model, a case study is given using real inspection data.     

Availability and cost functions for periodically inspected preventively maintained units

Unavailability and cost rate functions are developed for components whose failures can occur randomly but they are detected only by periodic testing or inspections. If a failure occurs between consecutive inspections, the unit remains failed until the next inspection. Components are renewed by preventive maintenance periodically, or by repair or replacement after a failure, whichever occurs first (age-replacement). The model takes into account finite repair and maintenance durations as well as costs due to testing, repair, maintenance and lost production or accidents. For normally operating units the time-related penalty is loss of production. For standby safety equipment it is the expected cost of an accident that can happen when the component is down due to a dormant failure, repair or maintenance. The objective of maintenance optimization is to minimize the total cost rate by proper selection of two intervals, one for inspections and one for replacements. General conditions and techniques are developed for solving optimal test and maintenance intervals, with and without constraints on the production loss or accident rate. Insights are gained into how the optimal intervals depend on various cost parameters and reliability characteristics.     

Multiattribute Bayesian Acceptance Sampling Plans for Screening and Scrapping Rejected Lots

A general model for multiattribute Bayesian acceptance sampling plans is developed which incorporates the multiattribute utility function of a decision maker in its design. The model accommodates various dispositions of rejected lots such as screening and scrapping. The disposition of rejected lots is shown to have a substantial impact on the solution approach used and on the ease of incorporation of multiattribute utility functions in terms of their measurement complexity, functional form, and parameter estimation. For example, if all attributes are screenable upon rejection, and the prior distributions of lot quality on each attribute are independent, then an optimal multiattribute sampling plan can be obtained simply by solving for an optimal single sampling plan on each attribute independently. A discrete search algorithm, based on pattern search, is also developed and shown to be very effective in obtaining an optimal multiattribute inspection plan when such separability cannot be accomplished.     

Optimal inspection decisions for the block mats of the Eastern-Scheldt barrier

To prevent the southwest of The Netherlands from flooding, the Eastern-Scheldt storm-surge barrier was constructed, has to be inspected and, when necessary, repaired. Therefore, one is interested in obtaining optimal rates of inspection for which the expected maintenance cost is minimal and the barrier is safe. For optimisation purposes, a maintenance model was developed for part of the sea-bed protection of the Eastern-Scheldt barrier, namely the block mats. This model enables optimal inspection decisions to be determined on the basis of the uncertainties in the process of occurrence of scour holes and, given that a scour hole has occurred, of the process of current-induced scour erosion. The stochastic processes of scour-hole initiation and scour-hole development was regarded as a Poisson process and a gamma process, respectively. Engineering knowledge was used to estimate their parameters.     

Improved method of estimating the product quality after multiple inspections

A complex product is often inspected more than once in a sequential manner to ensure the product’s quality. Based on the number of defects discovered during each round of inspection process, we can estimate the number of defects still remaining in the product. For each defect, the probability that the defect will be detected during each inspection cycle is usually assumed to be a known ‘constant’. However, in many practical situations, some defects are easily detected, while others are much more difficult to identify. In this paper, we propose a ‘beta-geometric’ inspection model in which the heterogeneity in detection probability is described by a beta distribution. In a numerical study, we show that our more realistic inspection model clearly outperforms traditional estimation methods that are based on the assumption of a constant detection probability.