We consider the multi-armed bandit problem with penalties for switching that include setup delays and costs, extending the former results of the author for the special case with no switching delays. A priority index for projects with setup delays that characterizes, in part, optimal policies was introduced by Asawa and Teneketzis in 1996, yet without giving a means of computing it. We present a fast two-stage index computing method, which computes the continuation index (which applies when the project has been set up) in a first stage and certain extra quantities with cubic (arithmetic-operation) complexity in the number of project states and then computes the switching index (which applies when the project is not set up), in a second stage, with quadratic complexity. The approach is based on new methodological advances on restless bandit indexation, which are introduced and deployed herein, being motivated by the limitations of previous results, exploiting the fact that the aforementioned index is the Whittle index of the project in its restless reformulation. A numerical study demonstrates substantial runtime speed-ups of the new two-stage index algorithm versus a general one-stage Whittle index algorithm. The study further gives evidence that, in a multi-project setting, the index policy is consistently nearly optimal.
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