The Stochastic Generalized Assignment Problem with Coordination Constraints

This work introduces the stochastic generalized assignment problem with coordination (SGAPC). The SGAP-C is a novel and challenging variant of stochastic generalized assignment problems that models many-to-many assignments with varying levels of complexity, and studies the benefits of collaboration and coordination. The work is motivated by a disaster relief setting where relief organizations can coordinate with each other to increase the quantity and speed of relief goods delivered to points of distribution. To solve the stochastic programming model, we use the L-shaped decomposition method and study the computational performance of several enhancements to the L-shaped method. We find that using trust regions to regularize the Lshaped method and initializing the L-shaped method with solutions from small-sample instances of the problem reduces mean optimality gaps by approximately 8% over the standard L-shaped method, and reduces the maximum of optimality gaps on instance solved from 65% to 25%. We also find that the multicut L-shaped method performs similarly well on average, but has poorer performance on a few of the most difficult instances solved. We use the L-shaped method with trust regions and warm starts to solve a set of instances of the SGAP-C with varying levels of complexity of coordination to study how increased coordination increases performance of the SGAP-C. We find that compared to independent agents with no coordination, a minimal level of coordination improves the objective function by an average of 55%. On average, this minimal level of coordination captures more than half of the increase in the objective function possible with the maximum level of performance, while incurring only a fraction of the maximum complexity of coordination.