This paper develops a finite-horizon dcnumcrable state-space dynamic programming formulation of the inspector location problem in convergent production lines. The effects of the inspection p i n t location decisions on pn~duction plimning are clarified, and the resulting complexity of the determination of the costs of not detecting defective units is demonstrnted under the assumption that all detected defective units are eliminated from the production line. A sample ~mhlern is solved by a bncknanl induction algorithm to obtain the minimumcost screening program. IntroductionDetermination of a n appropriate inspection policy, involving both sampling plans Lo monitor the quality of a product and the optirnal location of inspection points within a multi-stage production system has been a time-honoured research topic in production planning and control.Consider a defective unit leaving some work station. The defect may or may not be immediately detected, depending on whether there exists a n inspection station Ibllowing t h a t work station or not, a n d depending on t l~c sampling plan used in such a n inspection station. I f a defective unit is detected at a n inspection point, then the unit can either he scrapped, or reworked in a station devoted t o only repair activities, or rerouted back t o downstream work stations t o be reworked based on some preemptive processing role, incurring the associated relevant costs. On the other hand, if adefcctive unit is not detected and passed on t o the upstream stations, then certain wasted manufacturing'times and costs are incurred.The complexity of the determination of a n appropriate inspection policy arises d u e t o the following two characteristics of multi-stage production lines: first, the proportion of defective units entering each station depends on tire inspection policy used in all downstream stations. In particular, in a non-serial production line framework, if mnltiple assembly lines converge t o provide the inputs for a particular station, the proportion of defective units entering t h a t station depends on the inspect.ion policies used in all downstream (parallel) production lines feeding that. station. Secondly, the desired production rate in each work station is affected by the inspection policy used in all downstream stations in the production line, a s well as the demand for the end-items. In particular, since any rework cannot be instantaneous, the inspection policy a t a station affects not only the number of units t h a t will bc processed in all upstream stations, h u t also the number of units t h a t will he processed in all downstream and parallel work stations. T h e existence of these interdependencies renders the identification of the relevant costs of inspecting and not inspw:t.ing any specified prolmrtior of Lhc o~~l , l n~l . s of 11 stat,ion a h a d t.ask.
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