A three-dimensional spatial resource-constrained project scheduling problem: Model and heuristic

For a class of complex engineering projects executed in limited construction sites, spatial resources with three dimensions usually become a bottleneck that hampers their smooth implementation. Serious time-space conflicts frequently occur when multiple activities are carried out in parallel during some time periods. We propose a new project scheduling problem with three-dimensional spatial resource constraints (3D-sRCPSP), and a four-dimensional time-space is adopted to build the model for our 3D-sRCPSP. Firstly, in order to express the impacts of constrained sites on scheduling activities, two types of spatial constraints are refined and modeled: non-overlapping among active subspaces and not exceeding the total space, and then an integer programming model for the 3D-sRCPSP is formalized. Secondly, a novel heuristic algorithm is developed to solve the 3D-sRCPSP, which is embedded with 36 priority rules (PRs) and 3 instructive space allocation strategies. Besides 25 PRs for the traditional RCPSP, 11 new forms of PRs are extracted based on the features of 3D spatial resources. Thirdly, extensive numerical experiments are implemented to validate our model and heuristics. The instances are obtained by configuring the specific parameters of 3D spatial resources for benchmarks from PSPLIB library. One distinctive finding is that some existing PRs that perform well in the traditional RCPSP do not act best in the 3D-sRCPSP. On the other hand, the optimal solutions of very smallscale instances can be obtained by Gourbi solver, but our customized heuristic algorithm is more effective than Gourbi for general instances with medium/large sizes. Overall, the designed heuristics can effectively eliminate time-space conflicts in the planning stage of a project. Finally, as extension studies, the decision tree model is constructed to adaptively select the best PRs for each instance according to the indicators of project instances. These results can help project managers schedule activities and allocate spatial resources more accurately when encountering narrow construction sites.