The Orienteering Problem with Time Windows

The orienteering problem with time windows, denoted by OPTW, belongs to a class of routeing and scheduling problems that arise in physical distribution. It may be modelled as a problem on a graph. It considers a set of nodes (customers), each with an associated profit and service duration (time window), and a set of arcs, each with an associated travel time. The objective of the problem is to construct an acyclic path beginning at a specified origin and ending at a specified destination that maximizes the total profit while observing time window constraints on all nodes and not exceeding a designated time limit. The problem is classified as NP-hard and, thus, an exact algorithm that executes in reasonable computational time is unlikely to exist. Since the problem is highly-constrained, we were able to develop a heuristic (referred to as the ‘tree’ heuristic) based upon an exhaustive search of the feasible solution space. The tree heuristic systematically generates a list of feasible paths and then selects the most profitable path from the list. In comparison with an insertion heuristic, the tree heuristic was found to produce improved values of total profit for heavily-constrained, modest-sized problems with reasonable computational effort.