A method for determining cost-efficient inspection strategies in multistage production systems

This is a summary of the author’s PhD thesis. The thesis, written in English, was defended on 6 March 2006, and was supervised by Rik Van Landeghem and Claude Van Mechelen. A copy is available from the author upon request. This PhD thesis focuses on (cost-)efficient quality control for multistage production processes, a major issue to manufacturers. Subjecting a larger product fraction to inspection, or tightening the acceptance limits, will normally lead to a higher product quality, but will also result in higher costs of inspection, scrap and rework. Quantifying this trade-off and thus establishing ways of finding an economic optimum is at the heart of this research. Thereto, a metaheuristic solution approach, consisting of an evolutionary algorithm combined with simulation, is presented and validated.      

A non-parametric method for fitting ROC curves for evaluation of industrial inspection performance

The use of signal detection theory (SDT) in evaluation of the performance of an inspector on certain tasks requires the identification of the receiver operating characteristic (ROC) curve of the inspector. In this paper a functional form, Y = f(X), for the ROC equation is proposed, and two alternative approaches are developed for estimation of its parameters. This method does not require prior assumptions on the statistical distribution of the noise and signal-plus-noise. The application of the method is illustrated by the use of an example data set.     

Determining the optimum process mean based on quadratic quality loss function and rectifying inspection plan

In 1996, Pulak and Al-Sultan presented a rectifying inspection plan for determining the optimum process mean. However, they did not consider the quality cost for the product within the specification limits and did not point out whether the non-conforming items in the sample of accepted lot is replaced or eliminated from the lot. In this paper, we propose a modified Pulak and Al-Sultan’s model with quadratic quality loss function of product within the specification limits. Assume that the non-conforming items in the sample of accepted lot are replaced by conforming ones. Finally, the numerical results and sensitivity analysis of parameters of modified model and those of Pulak and Al-Sultan are provided for illustration.     

On allocation of redundant components for systems with dependent components

In this paper we consider the problem of optimal allocation of a redundant component for series, parallel and k-out-of-n systems of more than two components, when all the components are dependent. We show that for this problem is naturally to consider multivariate extensions of the joint bivariates stochastic orders. However, these extensions have not been defined or explicitly studied in the literature, except the joint likelihood ratio order, which was introduced by Shanthikumar and Yao (1991). Therefore we provide first multivariate extensions of the joint stochastic, hazard rate, reversed hazard rate order and next we provide sufficient conditions based on these multivariate extensions to select which component performs the redundancy.     

质量控制中完全检验的一种最优设计

本文提出质量控制中完全检验的一种最优设计。所谓完全检验 ,也就是 1 0 0 %检验 ,是指生产出来的每一个产品都要经过检验。生产过程中有两种生产状态 :控制状态和失控状态。当一个产品的质量特征值超过预定的控制界限 ,就立即对生产过程进行检查 ,一旦发现生产失控 ,马上采取调整措施使生产恢复正常。检验后的产品按其质量特征值接受或者重新制作。根据本文所建立的质量成本模型 ,我们可以得到最优的产品质量的规格界限和控制界限 ,对实际企业的生产有积极的指导作用。     

A particle swarm algorithm for inspection optimization in serial multi-stage processes

Implementing efficient inspection policies is much important for the organizations to reduce quality related costs. In this paper, a particle swarm optimization (PSO) algorithm is proposed to determine the optimal inspection policy in serial multi-stage processes. The policy consists of three decision parameters to be optimized; i.e. the stages in which inspection occurs, tolerance of inspection, and size of sample to inspect. Total inspection cost is adopted as the performance measure of the algorithm. A numerical example is investigated in two phases, i.e. fixed sample size and sample size as a decision parameter, to ensure the practicality and validity of the proposed PSO algorithm. It is shown that PSO gives better results in comparison with two other algorithms proposed by earlier works.     

Economic optimization of off-line inspection in a process that also produces non-conforming units when in control and conforming units when out of control

A finite batch of units is produced by a process subject to random failures. The process starts from the in-control state and may shift, while producing a unit, to the out of control state. We consider the case that inspection is conducted after all the units of the batch have been produced, when the production order of the units is preserved. Most research has assumed that while in the in-control state, the process produces only conforming units and in the out of control state, it produces only non-conforming ones. In our research we relax this assumption, i.e. we consider that in the in-control state the process may also produce non-conforming units and in the out of control state, it may also produce conforming ones. Using dynamic programming, we develop an optimal inspection/disposition policy that finds which units to inspect and how to dispose of uninspected units in order to minimize the expected cost, which includes inspection and penalty costs due to classification errors. In addition to the optimal policy, we develop several heuristic policies since the computational complexity of the dynamic programming calculations is O(3^N)$\mathrm{O}(3^N)$. We then perform computational studies to check the behavior of the optimal and heuristic policies and also to compare the latter policies.     

A note on multiple life premiums for dependent lifetimes

We study the properties of multiple life annuity and insurance premiums for general symmetric and survival statuses in the case when the joint distribution of future lifetimes has a dependence structure belonging to some nonparametric neighbourhood of independence. The size of the neighbourhood is controlled by a single parameter, which enables us to model really weak as well as stronger dependencies. We provide bounds on the difference of multiple life premiums for vectors of dependent and independent future lifetimes with the same univariate marginal distributions. Each such upper bound can be treated as a premium loading related to the strength of lifetimes’ dependence.     

An efficient inspection scheme for variables based on Taguchi capability index

Acceptance sampling has been one of practical tools for quality assurance applications, which provide a general rule to the producer and the consumer for product acceptance determination. It has been shown that variables sampling plans requires less sampling compared with attributes sampling plans. Thus, variables sampling plans become more attractive and desirable especially when the required quality level is very high or the allowable fraction non-conforming is very small. This paper attempts to develop an efficient and economic sampling scheme, variables repetitive group sampling plan, by incorporating the concept of Taguchi loss function. The OC curve of the proposed plan is derived based on the exact sampling distribution and the plan parameters are determined by minimizing the average sample number with two constraints specified by the producer and the consumer. The efficiency of the proposed variables RGS is examined and also compared with the existing variables single sampling plan in terms of the sample size required for inspection. In addition, tables of the plan parameters for various combinations of entry parameters are provided and an example is presented for illustration.     

Variables sampling inspection scheme for resubmitted lots based on the process capability index Cpk

This paper attempts to develop a sampling inspection scheme by variables based on process performance index for product acceptance determination, which examines the situation where resampling is permitted on lots not accepted on original inspection. The equations for plan parameters, the required sample size and the corresponding critical value, are derived based on the exact sampling distribution rather than an approximation approach hence the decisions made are more accurate and reliable. Moreover, the efficiency of the proposed variables resubmitted sampling plan is evaluated and compared with the existing variables single sampling plan. For illustrative purpose, an example is presented to demonstrate the use of the derived results for making a decision on product acceptance determination.     

线性对流占优扩散方程的后验误差估计

对于线性对流占优扩散方程,采用特征线有限元方法离散时间导数项和对流项,用分片线性有限元离散空间扩散项,并给出了一致的后验误差估计,其中估计常数不依赖与扩散项系数。     

A fourth order method for y″ = ƒ(x,y, y′)

A fourth order numerical method for solving the general second order differential equation $\text{y″ = ?(x, y, y′)}$ is suggested. This method is self starting, P-stable (when applied to $\text{y″ = ?(x, y)}$), and it involves only two function evaluations per step. Easily calculable and simple truncation error estimates are given, which can be used for automatic error control in computations. The method is illustrated on numerical examples.     

A posteriori error analysis of nonconforming finite volume elements for general second‐order elliptic PDEs

In this article, we study the a posteriori H¹ and L² error estimates for Crouzeix‐Raviart nonconforming finite volume element discretization of general second‐order elliptic problems in ?². The error estimators yield global upper and local lower bounds. Finally, numerical experiments are performed to illustrate the theoretical findings. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A Priori Error Estimates for Finite Element Methods for H (2,1)-Elliptic Equations

The convergence of finite element methods for linear elliptic boundary value problems of second and forth order is well understood. In this article, we introduce finite element approximations of some linear semi-elliptic boundary value problem of mixed order on a two-dimensional rectangular domain Q. The equation is of second order in one direction and forth order in the other and appears in the optimal control of parabolic partial differential equations if one eliminates the control and the state (or the adjoint state) in the first order optimality conditions. We establish a regularity result and estimate for the finite element error of conforming approximations of this equation. The finite elements in use have a tensor product structure, in one dimension we use linear, quadratic or cubic Lagrange elements in the other dimension cubic Hermite elements. For these elements, we prove the error bound O(h ² + τ ^k ) in the energy norm and O((h ² + τ ^k )(h ² + τ)) in the L ²(Q)-norm.     

A Posteriori Verification of Optimality Conditions for Control Problems with Finite-Dimensional Control Space

In this article, we investigate non-convex optimal control problems. We are concerned with a posteriori verification of sufficient optimality conditions. If the proposed verification method confirms the fulfillment of the sufficient condition then a posteriori error estimates can be computed. A special ingredient of our method is an error analysis for the Hessian of the underlying optimization problem. We derive conditions under which positive definiteness of the Hessian of the discrete problem implies positive definiteness of the Hessian of the continuous problem. The article is complemented with numerical experiments.     

A proof of a general maximum principle for optimal controls via a multiplier rule on metric space

A unified proof is given of the maximum principle for optimal control with various kinds of constraints by using a multiplier rule on metric spaces.     

对流扩散方程特征线三角元法的一致估计

利用三角形线性元的积分恒等式,给出了二维非定常对流占优扩散方程的特征线有限元解和真解的一致最优估计,并利用插值后处理算子,得到了有限元解梯度的一致超收敛估计,即只与初值和右端项有关,而与ε无关.     

A Priori and A Posteriori Error Estimates of Streamline Diffusion Finite Element Method for Optimal Control Problem Governed by Convection Dominated Diffusion Equation

In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existence and uniqueness of the discretized scheme.Then a priori and a posteriori error estimates are derived for the state,the co-state and the control.Three numerical examples are presented to illustrate our theoretical results.