A Summary of Inspection Policies of One Shot Systems

Inspection due to its relatively simple implementation is increasingly used in industry to identify the system state, to reveal system fault, and to make maintenance decisions. Researchers and engineers have almost always investigated optimal inspection policies for standby systems under an assumption that the servicing systems fail following exponential distributions. In many cases, however, the exponential distribution assumption can restrict the applicability of inspection policies due to its memory-less nature. In this paper, we consider the optimal inspection problem of a stochastically failing warm-standby system that will be needed at a random time when a failure of the main servicing system occurs. If the warm-standby system is not operational at that time, the system incurs a large penalty, which we want to avoid through inspections. To address this issue, we investigate an optimal inspection policy where the failure of the servicing system follows Weibull distribution. A model to schedule limited optimal inspection times is presented so that the total cost consisting of the expected penalty and inspection costs is minimized. The structural properties of the optimal policy are discussed in detail, and a solution algorithm is presented to find the optimal inspection times, which can guarantee the existence and uniqueness of the optimal policy. Thus, the commonly used inspection policies limited to the exponential cases turn out to be our special cases. Numerical simulations are provided to demonstrate the proposed approach.

We treat optimal equidistant and optimal non-equidistant inspection times for interval censoring of exponential distributions. We provide in particular a new approach for determining the optimal non-equidistant inspection times. The resulting recursive formula is related to a formula for optimal spacing of quantiles for asymptotically best linear estimates based on order statistics and to a formula for optimal cutpoints by the discretisation of continuous random variables. Moreover, we show that by the censoring with the optimal non-equidistant inspection times as well as with optimal equidistant inspection times, there is no loss of information if the number of inspections is converging to infinity. Since optimal equidistant inspection times are easier to calculate and easier to handle in practice, we study the efficiency of optimal equidistant inspection times with respect to optimal non-equidistant inspection times. Moreover, since the optimal inspection times are only locally optimal, we also provide some results concerning maximin efficient designs.

Operational Risk Management: Optimal Inspection Policy

This study considers an integrated model of economic production quantity and preventive maintenance to incorporate possibilities of minimal repair, inspection time and preventive maintenance error. We determine the optimal inspection interval, inspection frequency and economic production quantity. It assumes the process will shift to the out-ofcontrol state with a general deterioration distribution. If the process is in the in-control state, preventive maintenance is performed. The correctly preventive maintenance will decrease the failure rate. But, incorrectly preventive maintenance will cause the process to the out- of-control state. If the process is in the out-of-control state, there are two types. One is a relatively mild production process shift, where a minimal repair can remove the out-ofcontrol state to the in-control state. The other is a more serious shift, the process needs to replacement. When the process is in the out-of-control state, an amount of defective goods will be produced. In this study, numerical analysis is used to analyze the influence of minimal repair, inspection time and preventive maintenance error on cost.

The main role in the occurrence of accidental destruction of metal structures is played by the operational defects, the most dangerous among them are fatigue cracks, which are the cause of brittle fracture. One of the efficient methods for increasing the reliability of defect detection is the use of several types of non-destructive testing with different physical phenomena underlying them. To obtain expert estimates of the detectability of fatigue cracks in the metal structures, a survey of specialists with subsequent statistical processing of its results was conducted. Three samples were subjected to statistical analysis: the detectability of a defect (32 combinations of types of non-destructive testing), the cost of inspection by each combination of types, and the time of inspection. The formation of samples of the cost and time of the inspection was carried out according to the current price lists considering the expert opinions on the complexity of the operations performed. Statistical analysis showed no correlation between the probability of detecting a defect, the cost and time of inspection of a metal structure by various combinations of types of non-destructive testing. To determine the optimal combinations of types of non-destructive testing, it is proposed to use Pareto method related to the multicriteria optimization methods and representing the process of simultaneous optimization of several conflicting objective functions in a given domain of definition. Further development of the proposed methodology involves the transition from point estimates of defect detection to more accurate probabilistic ones, the expansion of the statistical base and a reasonable increase in the number of considered factors.

Inspection scheduling for imperfect production processes under free repair warranty contract

PurposeThe purpose of the paper is to show that modern markets are characterized by rapidly changing environments and numerous external forces. Under these conditions, product lifetimes are rapidly reducing. Therefore, products require optimum inspection policies to maintain high quality and reduce costs in the competitive market. This study aims to establish optimal inspection policies of reliability analysis for quantal‐response product with Weibull lifetime components.Design/methodology/approachThis study considers a product consisting of m different components in series with lifetimes that follow Weibull distributions, and applies a competing failure model to examine the proposed series system for quantal‐response products. The maximum likelihood estimators of parameters of the Weibull distribution are derived based on the quantal‐response data in the proposed series system. The statistical features of the model are illustrated through a numerical example of two‐component series products, and the properties of the maximum likelihood estimators were studied via Monte Carlo simulation under a two‐stage inspected scheme for various sampling sizes and inspection time conditions.FindingsSimulation results demonstrate not only that the optimum inspection condition is the inspection times at T1=0.2 and T2=0.5 for the two‐stage inspected scheme, but also that the economical sampling size is 150 for both cases.Originality/valueThis research results can be applied to the analysis of one‐shot products, e.g. firework, ammunition, airbag, injector, with Weibull components lifetime distribution or the stockpile storage test.

Purpose – The purpose of this paper is to determine a nearly optimal inspection sequence for a series system consisting of two components subject to gradual deterioration and whose failures are not self-announcing and can be detected only through inspection. Design/methodology/approach – The problem is tackled in the context of condition-based maintenance (CBM) with a maintenance model in the class of the control-limit policies for which the maintenance decision is made following inspection by comparison of the deterioration level to critical thresholds. A mathematical model is developed to express the total expected cost per time unit as a function of the inspection instants. Findings – For any given series system composed of two components with known critical deterioration threshold levels, and for any given set of costs related to inspection, inactivity due to failure, and preventive and corrective replacements of each component, a nearly optimal inspection sequence of the system is derived such as the total expected cost is reduced. Research limitations/implications – Due to the complexity of the cost model with the inspection instants (×1, ×2, ×3, …) being the decision variables, it has not been possible to derive the optimal solution. A quasi-optimal sequence of inspection times is derived along with the corresponding total average cost per time unit. Practical implications – In many practical situations in which CBM is implemented, a tradeoff between inspection costs and inactivity and replacement costs has to be balanced when determining the intervals between successive inspections at which the degradation level of the components should be assessed and compared to predetermined critical threshold levels. Inspecting too often would increase inspection costs but in the same time it would also increase the probability to avoid a failure and end up with a preventive replacement, whereas not inspecting often enough would increase the probability to end up with a failure increasing replacement and inactivity costs. Originality/value – While the inspection problem has been largely treated for single component systems, inspection policies become much more complex when considering multi-component systems. A two-component series system is considered in this paper.

To ensure the safety and reliability of ship structures during their service lifetime, inspections are essential to evaluate corrosion or fatigue damage and schedule maintenance or repair. This paper aims to propose a cost-benefit model of risk based inspection and repair planning for ship structures subjected to corrosion deterioration. The models of expected costs and expected benefit are formulated. The benefit-cost ratio is adopted to select the optimal inspection and repair planning. The planning problem is formulated as an optimization problem where the total benefit-cost ratio in the expected lifetime is maximized with a constraint on the minimum acceptable reliability index. A component subjected to corrosion deterioration is presented to illustrate the proposed approach. In addition, sensitivity studies are presented to account for the effects of uncertainties in basic parameters on the benefit-cost ratio and the optimal inspection and repair planning. The results indicate that the proposed approach can effectively integrate the economy with reliability of the inspection and repair planning. Furthermore, a balance can be achieved between the risk cost and total expected inspection and repair costs, which is very effective in selecting the optimal inspection and repair planning. Parametric studies show that the benefit-cost ratio is sensitive to the time of inspection and repair, the reference period, the repair criterion and the failure criterion.

Efficiency analysis of optimal inspection management for reinforced concrete structures under carbonation-induced corrosion risk

: A model is presented of a Markov process whose state is unknown except when an inspection is performed. The evolution of the process is governed by fixed transition probability matrices P and Q under non-inspection and inspection, respectively. The costs of inspection and non-inspection depend on the current state. The objective is to characterize inspection policies which minimize expected total discounted cost. The following specific models are presented. Model I is a process which starts in state 0 and is terminated when, on inspection, the state is found to exceed some fixed value M. In Model II the process is repaired (reverts to state 0) when the state at inspection exceeds M. Simple conditions are given which imply that the optimal inspection interval is a non-increasing function of the last observed state. Model III is an inventory process with uncertain supply as well as demand. Given order size n, the number received is Binomial (n;p). Costs of ordering, storage, and shortage are incorporated. In the single period case, conditions are given which imply that the optimal order size is non-increasing in current inventory. This result extends to the undiscounted multiperiod case provided the holding cost is zero. A counterexample is given for a two period case with linear, non-zero holding, shortage, and ordering costs. (Author)

PurposeThe purpose of this paper is to derive the optimal inspection policy for a power supply system which consists of two components: a capacitor bank and a transformer. The capacitor bank has two states: healthy and hidden failure. Its failure is detected only by inspection, and the delay in detection increases the cost of electricity consumption. The transformer has three states: healthy, defective and obvious failure. It continues to work after being defective, but if the defect is not detected and repaired in time, it becomes an obvious failure. The defect is detected only by inspection, but the obvious failure is self-announcing.Design/methodology/approachThe expected total cost of the system over a limited time horizon for a given inspection schedule is first modeled mathematically, and then an efficient genetic algorithm is designed to find the nearly optimal non-periodic inspection policy such that the expected total cost of the system is minimized.FindingsThe results show that, firstly, the higher the failure rate of the system components, the non-periodic inspection policy is more economical than the periodic one. Secondly, the proposed genetic algorithm reaches the nearly optimal solution in a shorter time with less computational effort compared to the A-star search algorithm.Originality/valueUsing the proposed approach, the nearly optimal non-periodic inspection schedule over a finite time horizon can be easily obtained for the power supply system. As a result of the implementation of this policy, the inspection and maintenance costs have been reduced, and it can have a significant impact on energy price.

This paper derives an inspection policy for an economic production quantity (EPQ) model under the assumption that a process may produce non-conforming (NC) items. In various stages of a production process, a department receiving an order uses a single sampling inspection policy to detect NC items. Under such a policy, a lot is accepted if the number of NC items in the inspected sample is equal to or less than the acceptance number. The proposed model considers both EPQ- and quality-related costs. Moreover, economic production order quantity, sample size and acceptance number are considered decision variables. A numerical example is presented, and a set of sensitivity analysis are provided to highlight the effectiveness of the proposed model. The results reveal that when the inspection cost is high, the classical EPQ model achieves a lower expected total cost for the production system compared with the EPQ model with the inspection. In contrast, when the NC cost is high, the EPQ model with the inspection policy outperforms the classical EPQ model, which can significantly decrease the expected total cost.

An EMQ model with a production process subject to random deterioration is considered. The process can be monitored through inspections, and both the lot size and the inspection schedule are subject to control. The “in-control” periods are assumed to be generally distributed and the inspections are imperfect, i.e., the true state of the process is not necessarily revealed through an inspection. The objective is the joint determination of the lot size and the inspection schedule, minimizing the long-run expected average cost per unit time. Both discrete and continuous cases are examined. A dynamic programming formulation is considered in the case where the inspections can be performed only at discrete times, which is typical for the parts industry. In the continuous case, an optimum inspection schedule is obtained for a given production time and given number of inspections by solving a nonlinear programming problem. A two-dimensional search procedure can be used to find the optimal policy. In the exponential case, the structure of the optimal inspection policy is established using Lagrange's method, and it is shown that the optimal inspection times can be found by solving a nonlinear equation. Numerical studies indicate that the optimal policy performs much better than the optimal policy with periodic inspections considered previously in the literature. The case of perfect inspections is discussed, and an extension of the results obtained previously in the literature is presented. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 165–186, 1998

Keywords: quarantine pest, plant health policy, optimization, import phytosanitary inspection, 'reduced checks', optimal allocation of resources, multinomial logistic regression, the Netherlands World trade is a major vector of spread of quarantine plant pests. Border phytosanitary inspection is a major barrier against introductions of quarantine pests through imported commodities, although the inspection resources are limited. This thesis provides conceptual and empirical insights that may help optimise import inspection under limited resources. The developed conceptual models analysed inspection policies under capacity constraints and in the absence of capacity constraints. The empirical models focused on finding the optimal inspection policies of propagating materials imported in the Netherlands. The results indicate that inspection effort should focus on commodities, whose inspection yields ceteris paribus greater marginal reduction in the expected costs of pest introduction. The results show that under binding capacity constraints, inspection of chrysanthemum cuttings in the Netherlands has a high marginal benefit, ranging from 8 to 49 euros for every marginal euro of inspection cost. The results further indicate that in the presence of fixed inspection costs, attaining the unconstrained allocation of inspection effort from the current, capacity constrained levels, is relatively inexpensive and greatly reduces costs to society. To expand current inspection capacities, the costs and likelihoods of pest introduction should be carefully estimated. Using the developed models for optimal allocation of inspection resources, the efficacy of the 'reduced checks' import inspection system in the EU was analysed. The results indicate that the expected costs of pest introduction in the EU under reduced checks could further be reduced if the economic impacts of pest introduction through various commodities are accounted for when calculating the frequencies of reduced checks. Finally, a multinomial logistic model was developed to analyse factors that determine the likelihoods of rejecting imported commodities due to phytosanitary and non-phytosanitary reasons. The results suggest that the geographical position of the exporting country, the characteristics of the importing company, the size of imported shipments, and the intended use of the commodity, among others, are significant factors based on which shipments of plant commodities can be targeted for inspection. Inspecting agencies can considerably facilitate the design of optimal inspection frameworks for the management of import phytosanitary risks by sound data-recording procedures that enable scientific analysis and provide a solid basis for reliable and applicable results.