QUALITY AND RELIABILITY ENGINEERING INTERNATIONAL Qual. Reliab. Engng. Int. 2001; 17: 407–418 (DOI: 10.1002/qre.422) IMPERFECT INSPECTION OF A MULTI-ATTRIBUTE DETERIORATING PRODUCTION SYSTEM—A CONTINUOUS TIME MODEL GAD RABINOWITZ ∗ AND ORLY YAHALOM Department of Industrial Engineering and Management, Ben Gurion University, Beer-Sheva, Israel 84105 SUMMARY The reliability of a multi-attribute deteriorating production system is controlled using versatile identical inspection facilities. An attribute state is dichotomous (up designates proper function versus down). A product item is conforming if all the system attributes are up when it is produced. When a system attribute is detected as down it is restored back to an up state. Inspection of an attribute can rely on observations of the system, recently produced items, or both. Inspection policy determines the inspection capacity, frequency of inspecting each attribute and inspection schedule. These decisions involve a tradeoff between the cost of inspectors and the loss associated with the proportion of non-conforming items due to lack of adequate inspection. Three models are introduced, analyzed and solved. In the first model, inspection and restoration are perfect, product attribute is up (down) when the system attribute is up (down), and restoration is immediate. The assumptions of perfect inspection and restoration are relaxed in the second model. The third model relaxes in addition the assumption of immediate restoration. An efficient heuristic solution scheme is provided for solving these models. Sensitivity of the solution to system parameters is studied. Numerical experiments provide some insights regarding the combined effect of imperfect production, inspection and restoration, in various conditions of inspection and restoration durations. Copyright  2001 John Wiley & Sons, Ltd. KEY WORDS: inspection; multi-stage; continuous-time; heuristic 1. INTRODUCTION Advanced inspection and communication technolo- gies enable and give advantage to sharing inspection efforts among a multiple of inspection tasks. These tasks could be: the state of a quality attribute in several production stages, different attributes in a single stage, or both. Inline inspection for quality control of multi- attribute production systems requires observation of processors’ attributes, items’ attributes, or both. The model we propose enables any such situation. Several authors (e.g. [1–3]) consider multi-attribute stochastic yield models with product flow decisions (e.g. accept, reprocess, scrap). This is in contrast to our inspection allocation decisions. Similar to our approach, they assume that an inspection is made for each determined quality attribute. A number of models for economical allocation of inspection effort in multistage production systems were reviewed by [4–6]. Recent models in this class were proposed by [7–10]. According to these papers, three general ∗ Correspondence to: G. Rabinowitz, Department of Industrial Engineering and Management, Ben Gurion University, Beer-Sheva, Israel 84105. assumptions were made by most of the authors: each inspection facility can be allocated to only one attribute; the rate of defects generation in each attribute is fixed over time; and the loss from shipping non-conforming items is proportional to their fraction. The last assumption is in contrast with practical experience (see [11]) that suggests a convex dependency due to loss of goodwill and market share. These assumptions explain the fixed inspection allocation policy and the irrelevancy of inspection schedule and process quality control by the works surveyed by the above-mentioned reviews. On [12, p. 161], possible inspection and test strategy problems in flow lines are discussed. They develop a variety of stochastic models for manufacturing systems, some of which involve inspection. Yet, conforming to the preceding literature, they do not mention or model the use and schedule of inspection for detecting malfunction production attributes. The models discussed in [13–16] permit a more realistic formulation under a discrete time setting. Specifically, they allow for: imperfect inspection; imperfect restoration of an attribute when it is detected Received 15 February 2001 Copyright  2001 John Wiley & Sons, Ltd. Revised 1 July 2001