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The problem addressed in this study is that of determining how to allocate the workstation processing and buffering capacity in a capacitated re-entrant line to the job instances competing for it, in order to maximize its long-run/steady-state throughput, while maintaining the logical correctness of the underlying material flow, i.e., deadlock-free operations. An approximation scheme for the optimal policy that is based on neuro-dynamic programming theory is proposed, and its performance is assessed through a numerical experiment. The derived results indicate that the proposed method holds considerable promise for providing a viable, computationally efficient approach to the problem and highlight directions for further investigation. Note to Practitioners-Sequencing and scheduling problems arising in the context of contemporary manufacturing environments are known to be extremely hard. For this reason, in most practical situations, these problems have been resolved through the application of a number of heuristics-i.e., "rules of thumb" that are expected to provide reasonable performance. Things are complicated even further in the automated versions of these environments, since the applied sequencing and scheduling logic must guarantee, in addition to good performance, logically correct and smooth operation. Both the logical and the performance-oriented control problem of flexibly automated production systems can be-and have been-addressed through formal systems theory. However, a challenging remaining problem is the approximation of the derived optimal policies in a way that will maintain near optimality, and at the same time, it will be computationally tractable in the context of the "real-world" applications. Our past work has addressed this approximation problem primarily with respect to the optimal logical control policy. The work presented in this paper undertakes the complementary problem of approximating the optimal scheduling policy. To this end, we employ some recently emerged results from a field known as neuro-dynamic programming (which is essentially a systematic approximation framework for dynamic programming). Our results indicate that the proposed approximation framework holds considerable promise toward developing a systematic analytical meth- odology for deriving near-optimal and logically correct scheduling policies for flexibly automated production systems. More specifically, it is shown that, when applied to some prototypical problems concerning the scheduling of re-entrant lines with finite buffering capacity at their workstations, the proposed approximation framework: 1) effectively integrates past results concerning the logical control of these environments and 2) the obtained performance is consistently superior to the performance provided by the typically used heuristics.