Shizhe Zhao, Yancheng Wu, Zhongqiang Ren

Bi-Objective Search for the Traveling Salesman Problem with Time Windows and Vacant Penalties

Introduction

Bi-objective search for the traveling salesman problem with time windows and vacant penalties. Solves the Bi-Objective Traveling Salesman Problem with Time Windows & Vacant Penalties (TSP-TW-VP). Presents S-LAP algorithm for Pareto-optimal solutions, achieving 2-8x speedup.

Abstract

This paper investigates a Traveling Salesman Problem with Time Windows and Vacant Penalties (TSP-TW-VP), which plans a path to service a set of machines at different locations within their respective time windows while minimizing two objective functions: the finish time and penalty for machine vacancy. There is often no single solution that optimizes both objectives simultaneously, and the problem thus seeks the Pareto-optimal solutions. TSP-TW-VP generalizes TSP-TW and is therefore NP-hard. To solve the problem, this paper develops an algorithm called Search with Look-Ahead Pruning (S-LAP) that is guaranteed to find all Pareto-optimal solutions for TSP-TW-VP. S-LAP gains computational efficiency by introducing a novel look-ahead pruning rule, and a fast dominance checking method based on both the objective functions and path history. Experimental results show that the proposed look-ahead pruning and fast dominance can speed up the search for 2-8 times over 4 different datasets.

Review

The paper "Bi-Objective Search for the Traveling Salesman Problem with Time Windows and Vacant Penalties" investigates a significant extension to the classical Traveling Salesman Problem with Time Windows (TSP-TW). By introducing the objective of minimizing "vacant penalties" alongside the finish time, the authors address a more realistic and complex scheduling challenge where both timely completion and resource utilization are critical. The bi-objective nature of the TSP-TW-VP, which is confirmed to be NP-hard due to its generalization of TSP-TW, correctly necessitates the pursuit of Pareto-optimal solutions, acknowledging the inherent trade-offs between competing objectives. This problem formulation is highly relevant for practical applications requiring efficient routing and scheduling under stringent time constraints and cost considerations for idleness. To solve this challenging problem, the authors propose an algorithm called Search with Look-Ahead Pruning (S-LAP). A key strength of S-LAP is its guarantee to find *all* Pareto-optimal solutions for the TSP-TW-VP, which is a robust theoretical claim for an NP-hard problem. The computational efficiency of S-LAP is primarily attributed to two innovative technical contributions: a novel look-ahead pruning rule and a fast dominance checking method. The abstract further details that this dominance checking is not merely based on objective values but also incorporates "path history," suggesting a sophisticated approach to avoid redundant computations and ensure optimality while navigating the search space effectively. The experimental evaluation provides compelling evidence of S-LAP's practical utility. The results demonstrate that the proposed look-ahead pruning and fast dominance checking collectively yield a substantial speedup, ranging from 2 to 8 times, across four distinct datasets. This significant enhancement in computational efficiency, coupled with the guarantee of completeness, marks S-LAP as a powerful and valuable contribution. The paper thus offers a robust and theoretically sound solution to a complex bi-objective optimization problem, providing both academic rigor and practical applicability for researchers and practitioners in fields such as operations research, logistics, and artificial intelligence.

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