A constraint programming–shapley framework for resource valuation in multi-mode resource-constrained project scheduling

Abstract

This study proposes a framework based on Constraint Programming with Boolean Satisfiability (CP-SAT) that links project scheduling to project management decisions by quantifying how individual activities and resource types influence project outcomes through Shapley values. A cooperative game is defined on the chosen mode project network, where the payoff represents cumulative resource exposure along precedence-consistent paths, after first solving the multi-mode resource constrained project scheduling problem (MRCPSP) with a CP-SAT formulation that enforces mode selection and precedence/resource feasibility. A precedence-based approximation is then used to construct a resource–precedence Shapley matrix, whose entries measure each resource type's marginal contribution under interaction effects and extend traditional utilization-based indicators. Aggregating this matrix across resource types yields an aggregated Shapley priority score for each activity, providing interpretable rankings of resource criticality suitable for staffing, procurement, and contracting decisions. Experiments on 3,287 PSPLIB multi-mode instances (J10–J20) show that CP-SAT reproduces benchmark average makespans while remaining computationally efficient (mean runtime of approximately 0.11–0.28 s per instance family). Nonparametric analyses (Shapiro–Wilk, Kruskal–Wallis, Dunn tests) reveal statistically distinct Shapley distributions across instance classes, identifying stable and transitional regions in resource importance as network complexity and size grow. Complementary Spearman analyses show that Shapley values are only weakly associated with makespan but exhibit consistent moderate negative correlations with resource strength, indicating that the proposed valuation captures structural resource scarcity effects rather than merely mirroring schedule length. Overall, the CP-SAT–Shapley framework offers a reproducible resource valuation and decision-support pipeline that transforms optimized multi-mode schedules into quantitative, explainable resource criticality evidence, addressing the gap between powerful MRCPSP solvers, which optimize schedules, and practical project management, which requires interpretable guidance on which resources to prioritize under budget, staffing, or procurement constraints.