Determining Process Capacity: Intractability and Efficient Special Cases

Most operations management textbooks use the following simple approximation to illustrate the computation of the capacity of a process: the capacity of each resource is first calculated by examining that resource in isolation; process capacity is then defined as the smallest among the capacities of the resources, that is, bottleneck capacity. In a recent paper, Gurvich and Van Mieghem [Gurvich I, Van Mieghem JA (2015) Collaboration and multitasking in networks: Architectures, bottlenecks, and capacity. Manufacturing Service Oper. Management 17(1):16–33.] show that, in the presence of collaboration and multitasking, this “bottleneck formula” can be significantly inaccurate, and they obtain a necessary and sufficient condition under which it correctly determines process capacity. We provide further clarity on determining process capacity by showing that it is hard to compute process capacity exactly and also to approximate it to within a reasonable factor. These results are based on a novel characterization, which we establish, of process capacity that relates it to the fractional chromatic number of the associated “collaboration graph.” An important implication is that it is unlikely that we can replace the bottleneck formula with a simple but close approximation of process capacity. On the positive side, we show that capacity can be efficiently computed for processes for which the collaboration graph is a perfect graph. From a practical viewpoint, our analysis for general processes results in a natural hierarchy of subclasses of policies that require an increasing amount of sophistication in implementation and management: while process capacity is the maximum long-term process rate achievable over all feasible policies, we provide a precise expression for the maximum process rate over policies in each subclass of this hierarchy, thus highlighting the trade-off between operational difficulty and the achievable process rate. The online appendix is available at https://doi.org/10.1287/msom.2017.0689 .

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