Generalized Orienteering Problem with Resource Dependent Rewards

Abstract

We introduce a generalized Orienteering Problem where, as usual, a vehicle is routed from a prescribed start node, through a directed network, to a prescribed destination node, collecting rewards at each node visited, in order to maximize the total reward along the path. In our generalization, transit on arcs in the network and reward collection at nodes both consume a variable amount of the same limited resource. We exploit this resource trade-off through a specialized branch-and-bound algorithm that relies upon partial path relaxation problems which often yield tight bounds and lead to substantial pruning in the enumeration tree. We present the Smuggler Search Problem as an important real-world application of our generalized Orienteering Problem. Numerical results show that our algorithm applied to the Smuggler Search Problem outperforms standard Mixed-Integer Nonlinear Programming solvers for moderate to large problem instances. We demonstrate model enhancements that allow practitioners to represent realistic search planning scenarios by accounting for multiple heterogeneous searchers and complex smuggler motion.

Chapter 10: Vehicle Routing Problems with Profits

“…In this problem each city is assigned a number of scores for different attributes and the overall function to optimize is a function of these attribute scores. A different variant of the OP, where a limited resource is consumed on the traversed arcs and on the visited nodes, was studied by Pietz and Royset [86], who presented a specialized BB algorithm. Two variants of the OP with variable profits were introduced by Erdogǎn and Laporte [42].…”

Time dependent orienteering problem with time windows and service time dependent profits

“…For example, the fishing problem, in which there is a limited allowed time for fishing at each location, having variable amount of fish (Erdoǧan & Laporte 2013). Other examples are the tourist trip planning (Yu et al 2015), the traffic routing problem (Zhu et al 2014), and finding the shortest path problem for target searching (Guitouni &Masri 2014 andPietz &Royset 2013).…”

“…Then, they provided valid inequalities to strengthen the formulation for both concave and convex profit collection functions and developed a unified branch-and-cut algorithm for the two versions. Pietz & Royset (2013) defined the Generalized Orienteering Problem with Resource Dependent Rewards (GOP-RDR) where rewards are collected by visiting each vertex with a concave reward function, and the reward level depends on the amount of limited resources consumed. Arcs in the network are passed while expending the same resources used for reward collection.…”

Minisum and maximin aerial surveillance over disjoint rectangles

“…Interested readers are referred to Vansteenwegen et al (2011) and Chekuri et al (2012) for the orienteering and related problems. Pietz and Royset (2013) generalized the orienteering problem with resource-dependent rewards. Instead of a fixed reward, reward level depends on the resource expended at each visited node.…”