A Cooperative Coordination Solver for Travelling Thief Problems

In the travelling thief problem (TTP), a thief undertakes a cyclic tour through a set of cities, and according to a picking plan, picks a subset of available items into a rented knapsack with limited capacity. The overall aim is to maximise profit while minimising renting cost. TTP combines two interdependent components: the travelling salesman problem (TSP) and the knapsack problem (KP). Existing TTP approaches typically solve the TSP and KP components in an interleaved fashion: the solution of one component is fixed while the solution of the other is changed. This indicates poor coordination between solving the two components, which may lead to poor quality TTP solutions. The 2-OPT heuristic is often used for solving the TSP component, which reverses a segment in the tour. Within the TTP context, 2-OPT does not take into account the picking plan, which can result in a lower objective value. This in turn can result in the tour modification to be rejected by a solver. To address this, we propose an extended form of 2-OPT in order to change the picking plan in coordination with modifying the tour. Items deemed as less profitable and picked in cities earlier in the reversed segment are replaced by items that tend to be equally or more profitable and not picked in cities later in the reversed segment. The picking plan is further changed through a modified form of the bit-flip search, where changes in the picking state are only permitted for boundary items, which are defined as lowest profitable picked items or highest profitable unpicked items. This restriction reduces the amount of time spent on the KP component, allowing more tours to be evaluated by the TSP component within a given time budget. The two modified heuristics form the basis of a new cooperative coordination solver, which is shown to outperform several state-of-the-art TTP solvers on a broad range of benchmark TTP instances.