A Convex Hull Cheapest Insertion Heuristic for Precedence Constrained Traveling Salesperson Problems or Sequential Ordering Problems
Mithun Goutham, Stephanie Stockar
TL;DR
This work addresses a precedence-constrained variant of the TSP, including SOP, by extending the Convex Hull Cheapest Insertion heuristic to respect precedence—yielding the Adapted CHCI (ACHCI). ACHCI builds an initial subtour from the convex hull of the depot and parent nodes and incrementally inserts remaining nodes within feasible segments that honor the precedence relations, performing insertions in both possible directions and selecting the cheaper tour. Empirical results on TSPLIB instances augmented with precedence constraints show ACHCI significantly outperforms the Nearest Neighbor baseline, achieving up to a 97% cost reduction when delivery nodes cluster near the centroid; performance degrades under different spatial configurations, with a worst-case typical complexity of $O(n^3)$. The method offers a practical heuristic for pickup-delivery logistics and manufacturing settings, with potential parallelization and avenues for future extensions to multi-commodity and dial-a-ride problems.
Abstract
The convex hull cheapest insertion heuristic is a well-known method that efficiently generates good solutions to the Traveling Salesperson Problem. However, this heuristic has not been adapted to account for precedence constraints that restrict the order in which locations can be visited. Such constraints result in the precedence constrained traveling salesperson problem or the sequential ordering problem, which are commonly encountered in applications where items have to be picked up before they are delivered. In this paper, we present an adapted version of this heuristic that accounts for precedence constraints in the problem definition. This algorithm is compared with the widely used Nearest Neighbor heuristic on the TSPLIB benchmark data with added precedence constraints. It is seen that the proposed algorithm is particularly well suited to cases where delivery nodes are centrally positioned, with pickup nodes located in the periphery, outperforming the Nearest Neighbor algorithm in 97\% of the examined instances.