Increasing product complexity, manufacturing environment complexity and an increased emphasis on product quality are all factors leading to uncertainties in production processes. These uncertainties are in the form of unplanned machine maintenance, varying production yields and rework, among others. In planning for production, an adequate model must incorporate these uncertainties into the representation of the production process. This paper treats the aggregate planning problem for a single product with random demand and random capacity. In the single-period problem, random capacity does not affect the optimal policy but results in a unimodal, nonconvex cost function. In the multiple-period and infinite-horizon settings order-up-to policies that are dependent on the distribution of capacity are shown to be optimal in spite of a nonconvex cost. In the infinite-horizon setting an intuitive description of the situation leads to the notion of a class of extended myopic policies, requiring the consideration of review periods of uncertain length.

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Spatial quantization error and displacement error are inherent in automated visual inspection systems. This paper discusses the effect of spatial quantization errors and displacement errors on the precision dimensional measurements for an edge segment. Probabilistic analysis in terms of the resolution of the image is developed for two-dimensional (2-D) quantization errors. Expressions for the mean and variance of these errors are developed. The probability density function (pdf) of the quantization error is derived. The position and orientation errors of the active head are assumed to be normally distributed. A probabilistic analysis in terms of these errors is developed for the displacement errors. Through integrating the spatial quantization errors and the displacement errors, we can compute the total error in the active vision inspection system. Based on the developed analysis, we investigate whether a given set of sensor setting parameters in an active system is suitable to obtain a desired accuracy for specific dimensional measurements. In addition, based on this approach, one can determine sensor positions and view directions which meet the necessary tolerance and accuracy of inspection.

This article presents an optimized sensor planning system for active visual inspection of three‐dimensional manufacturing computer‐aided design (CAD) models. Quantization errors and displacement errors are inevitable in active visual inspection. To obtain high accuracy for dimensioning the entities of three‐dimensional CAD models, minimization of these errors is essential. Spatial quantization errors result in digitization. The errors are serious when the size of the pixel is significant compared to the allowable tolerance in the object dimension on the image. In placing the active sensor to perform inspection, displacement of the sensors in orientation and location is common. The difference between observed dimensions obtained by the displaced sensor and the actual dimensions is defined as displacement errors. The density functions of quantization errors and displacement errors depend on camera resolution and camera locations and orientations. The sensor constraints, such as resolution, focus, field‐of‐view, and visibility constraints, restrict sensor placement. To obtain a satisfactory view of the targeted entities of the CAD models, these constraints have to be satisfied. In this article, we focus on the edge line segments as the inspected entities. We use a genetic algorithm to minimize the probabilistic magnitude of the errors subject to the sensor constraints. Since the objective functions and constraint functions are both complicated and nonlinear, traditional nonlinear programming may not be efficient and it may trap at a local minimum. Using crossover operations, mutation operations, and the stochastic selection in the genetic algorithm, trapping can be avoided. Experiments are conducted and the performance of the genetic algorithm is presented. Given the CAD model and the entities to be inspected, the active visual inspection planning system obtains the sensor setting that maximizes the probabilities of a required accuracy for each entity. © 2001 John Wiley & Sons, Inc.

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