This paper studies a robust variant of the team orienteering problem with decreasing profits (TOP-DP), where a fleet of vehicles are dispatched to serve customers with decreasing profits in a limited time horizon. The service times at customers are assumed to be uncertain, which are characterized by a budgeted uncertainty set. Our goal is to determine the set of customers to be served and the routes for the vehicles such that the collected profit is maximized; meanwhile, all the planned routes remain feasible for any realization of service times within the uncertainty set. We propose a two-index robust formulation for the problem, which is defined using constraints based on dynamic programming recursive equations and can be directly solved by a general-purpose optimization solver. We also present a route-based formulation for the problem, which is solved by a tailored branch-and-price (B\&P) algorithm. To tackle large-size instances efficiently, we further implement a tabu search (TS) algorithm. Numerical tests show that our B&P algorithm can solve most instances with 100 customers to optimality within 30 minutes and that the TS algorithm can find high-quality solutions within a few seconds. Moreover, we find that in most cases, the robust solutions can significantly reduce the probability of deadline violations in simulation tests with only a slight compromise of profit, compared to the solutions generated by the deterministic model.
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