13. A Probabilistic Characterization of Allocation Performance in a Worker-Constrained Job Shop

We analyze a dual resource constrained (DRC) job shop in which both machines and workers are limited, and we seek to minimize L _max, the maximum job lateness. An allocation of workers to machine groups is required to generate a schedule, and determining a schedule that minimizes L _max is NP-hard. This chapter details a probabilistic method for evaluating the quality of a specific (but arbitrary) allocation in a given DRC job shop scheduling problem based on two recent articles by Lobo et al. (2013a) The first article Lobo et al. (2013b) describes a lower bound on L _max given an allocation, and an algorithm to find an allocation yielding the smallest such lower bound, while the second article Lobo et al. (2013b) establishes criteria for verifying the optimality of an allocation. For situations where the optimality criteria are not satisfied, Lobo et al. (2013c) presents HSP, a heuristic search procedure to find allocations enabling the Virtual Factory (a heuristic scheduler developed by Hodgson et al. in 1998) to generate schedules with smaller L _max than can be achieved with allocations yielding article 1’s final lower bound. From simulation replications of the given DRC job shop scheduling problem, we estimate the distribution of the difference between (a) the “best” (smallest) L _max value achievable with a Virtual Factory–generated schedule, taken over all feasible allocations; and (b) the final lower bound of Lobo et al. (2013b). To evaluate the quality of a specific allocation for the given problem, we compute the difference between L _max for the Virtual Factory–generated schedule based on the specific allocation, and the associated lower bound (b) for the problem; then we refer this difference to the estimated distribution to judge the likelihood that the specific allocation yields the Virtual Factory’s “best” schedule (a) for the given problem. We present several examples illustrating the usefulness of our approach, and summarize the lessons learned in this work.