Analysis of the suitability of using blind crossover operators in genetic algorithms for solving routing problems

Two new crossover operators in genetic algorithms for solving some combinatorial problems with ordering are presented. One is enhanced order crossover (EOX). The other, GREE, is a heuristic crossover for a class of combinatorial optimization problems, such as traveling salesman problems (TSPs). Genetic algorithms using GREE as unique crossover run very fast and get good solutions. Combining GREE with EOX, genetic algorithms can find optimal or very near optimal solutions in a rather short time. >

We review the recent work done by our group on theoretically analyzing the behavior of genetic algorithms(GAs) and searching for new ideas and strategies to design more efficient GAs. By introducing the concept of degree of population diversity and using the tools from finite Markov chain, we presented fairly clear characterizations of premature convergence in GAs and proved some results on the relationships between premature convergence and GA’s control parameters such as the population size, the mutation and crossover probabilities, and some population statistics. In the mean time, we have also obtained a result that depicts the search ability of the crossover operators in GAs. An upper bound on the convergence rate for the GAs has been established. Based on the theoretical analyses, we have proposed a novel GA called Genetic Algorithm with No Genetic Operators (GANGO), which needs neither to maintain a population nor to use the conventional genetic operators, and yet has the same search mechanisms as the classical GAs. As another development in improving the performance of GAs, a new concept called Dual Fitness Function was proposed which, when incorporated into the classical GAs will enable the algorithms to judge whether the current population is in the local or global attractive basins. And in this way, the control parameters of GAs can be adjusted adaptively. Some applications of our results to fields such as fuzzy systems, multi-destination routing problem (MDR) are briefly

Vehicle routing problems (VRPs) are a fundamental class of combinatorial optimization problems (COPs) in computer science and operations research, with diverse real-world applications in logistics, transportation, and manufacturing. The intrinsic NP-hard nature makes VRPs exponentially expensive to be solved by exact solvers. As an alternative, heuristic solvers deliver suboptimal solutions within reasonable time but require extensive hand-crafted rules and domain expertise tailored to each specific problem. Recently, neural combinatorial optimization (NCO), which leverages machine learning to learn heuristics in a data-driven manner, has gained significant attention. These neural solvers demonstrate strong performance while reducing computational overhead and reliance on domain expertise compared to traditional solvers. However, they face significant generalization challenges. For example, their performance may degrade substantially when applied to instances with different data distributions, problem scales, or constraints than those encountered during training. This generalization issue severely limits their practical applicability. This thesis aims to systematically address these challenges and advance the development of generalizable neural solvers for VRPs. The thesis begins by enhancing the cross-distribution generalization of neural VRP solvers through an adversarial training approach. We introduce an ensemble-based Collaborative Neural Framework (CNF) that adversarially trains multiple models in a collaborative manner, promoting robustness against adversarial attacks while also boosting performance on clean instances. In this framework, an attacker generates hard instances by perturbing the data distribution, challenging the models to adapt and thereby strengthening cross-distribution generalization. Additionally, a neural router is designed to efficiently distribute training instances among the models, thereby improving load balancing and collaborative efficacy. Extensive experiments on two classic routing problems -- the travelling salesman problem (TSP) and the capacitated vehicle routing problem (CVRP) -- demonstrate the effectiveness and versatility of CNF in enhancing the cross-distribution generalization. Next, the thesis further expands neural solvers to simultaneously address both cross-size and cross-distribution generalization challenges. We propose Omni-VRP, a generic meta-learning framework that enables effective training of an initial model with the capacity for rapid adaptation to new tasks during inference, where each task corresponds to a set of instances with the same problem scale and data distribution. Additionally, we introduce a simple yet efficient approximation method to reduce the training overhead associated with second-order derivative calculations. Extensive experiments on both synthetic and benchmark instances of TSP and CVRP validate the effectiveness of our approach. Finally, the thesis pioneers the study of cross-problem generalization, pushing the boundaries of neural VRP solvers in learning generalizable representations across diverse constraints. We propose MVMoE, a unified neural solver capable of solving 16 VRP variants simultaneously in a zero-shot manner through attribute composition. To efficiently enhance model capacity, we employ a mixture-of-experts architecture with a hierarchical gating mechanism, striking a balance between empirical performance and computational complexity. Experimentally, our method significantly promotes zero-shot generalization on 10 unseen VRP variants, and showcases decent results on the few-shot setting and real-world benchmark instances. Overall, this thesis represents a pioneering exploration and substantial advancement in developing generalizable neural solvers for VRPs. It investigates innovative designs across various components, including model architectures, training algorithms, and problem settings, to enhance the generalization of neural solvers across diverse contexts such as varying data distributions, problem scales, and constraints. These contributions encourage neural solvers to learn more robust and generalizable representations, paving the way for the first generation of foundation models capable of addressing a wide range of COPs. Ultimately, the insights gained here not only propel the field of NCO forward but also enrich the broader landscape of learning-based optimization methods.

Genetic Algorithm(GA) is a guided random search and the guiding direction always aims at the family whose ancestors have schemata with high fitness.Based on the results,the crossover operation in GA was redesigned by using the principle of random uniform design sampling and combining the locale search strategy.Then a new GA called Genetic Algorithm based on Uniform Design Sampling(UDS) was presented.The new GA was applied to solve the 2-way graph partitioning problem.Compared to simple GA and good point GA for solving this problem,the simulation results show that the new GA has superiority in terms of speed and accuracy and overcomes premature convergence.

Based on Latin Hypercube Sampling method, the crossover operation in GA is redesigned; combined with immune mechanism, chromosome concentration is defined and clonal selection strategy is designed, thus an Immune Genetic Algorithm is given based on Latin Hypercube Sampling for solving the Maximum Clique Problem in this paper. The examples shows the new algorithm in solution quality, convergence speed, and other indicators is better than the classical genetic algorithm and good point set genetic algorithm. On the other hand, the new algorithm is not inferior to such classical algorithms as dynamic local search, and it got better solutions to some examples.KeywordsMaximum Clique Problem(MCP)Genetic Algorithm (GA)Latin Hypercube Sampling(LHS)Artificial Immune Systems(AIS)Immune Genetic Algorithm based on Latin Hypercube Sampling(LIGA)

A hybrid algorithm (HA) integrated genetic algorithm (GA) with ant colony optimization (ACO) for solving Traveling Salesman Problems(TSP) was studied to get better optimization performance than each single algorithm, and complement merits each other and avoid each own demerits. The hybrid algorithm runs GA first and then ACO. A new strategy called GSA was proposed aiming at the key link in the HA that converts genetic solution from GA into information pheromone to distribute in ACO. GSA takes new matrix which is formed by the combination of the former 90% of individual from genetic solution and 10% of individual by random generation as the basis of transformation of pheromone value. The best combination of genetic operators in GA was also discussed. Several TSP were used as simulation tests to test genetic operators matching and optimization performance of HA. The results show that PMX crossover matched with IVM mutation in the GA is the best combination of genetic operators which is able to make GA improve the precision of optimal solution, and HA using the best combination operators and GSA strategy is successful and available to search for optimal solution in high efficiency and has good convergence.

The probabilistic travelling salesman problem (PTSP) is an extension of the well-known travelling salesman problem (TSP), which has been extensively studied in the field of combinatorial optimization. The goal of the TSP is to find the minimum length of a tour to all customers, given the distances between all pairs of customers whereas the objective of the PTSP is to minimize the expected length of the a priori tour where each customer requires a visit only with a given probability (Bertsimas, 1988; Bertsimas et al., 1990; Jaillet, 1985). The main difference between the PTSP and the TSP is that in the PTSP the probability of each node being visited is between 0.0 and 1.0 while in TSP the probability of each node being visited is 1.0. Due to the fact that the element of uncertainty not only exists, but also significantly affects the system performance in many real-world transportation and logistics applications, the results from the PTSP can provide insights into research in other probabilistic combinatorial optimization problems. Moreover, the PTSP can also be used to model many real-world applications in logistical and transportation planning, such as daily pickup-delivery services with stochastic demand, job sequencing involving changeover cost, design of retrieval sequences in a warehouse or in a cargo terminal operations, meals on wheels in senior citizen services, trip-chaining activities, vehicle routing problem with stochastic demand, and home delivery service under e-commerce (Bartholdi et al., 1983; Bertsimas et al., 1995; Campbell, 2006; Jaillet, 1988; Tang & Miller-Hooks, 2004). Early PTSP computational studies, dating from 1985, adopted heuristic approaches that were modified from the TSP (e.g., nearest neighbor, savings approach, spacefilling curve, radial sorting, 1-shift, and 2-opt exchanges) (Bartholdi & Platzman, 1988; Bertsimas, 1988; Bertsimas & Howell, 1993; Jaillet, 1985, 1987; Rossi & Gavioli, 1987). With its less than satisfactory performance in yielding solution quality, researchers in the recent years switch to metaheuristic methods, such as ant colony optimization (Bianchi, 2006; Branke & Guntsch, 2004), evolutionary algorithm (Liu et al., 2007), simulated annealing (Bowler et al., 2003), threshold accepting (Tang & Miller-Hooks, 2004) and scatter search (Liu, 2006, 2007, 2008). Because the genetic algorithm (GA), a conceptual framework of the population-based metaheuristic method, has been shown to yield promising outcomes for solving various complicated optimization problems in the past three decades (Back et al., 1997; Davis, 1991; O pe n A cc es s D at ab as e w w w .ite ch on lin e. co m

Recent works using deep learning to solve routing problems such as the traveling salesman problem (TSP) have focused on learning construction heuristics. Such approaches find good quality solutions but require additional procedures such as beam search and sampling to improve solutions and achieve state-of-the-art performance. However, few studies have focused on improvement heuristics, where a given solution is improved until reaching a near-optimal one. In this work, we propose to learn a local search heuristic based on 2-opt operators via deep reinforcement learning. We propose a policy gradient algorithm to learn a stochastic policy that selects 2-opt operations given a current solution. Moreover, we introduce a policy neural network that leverages a pointing attention mechanism, which can be easily extended to more general k-opt moves. Our results show that the learned policies can improve even over random initial solutions and approach near-optimal solutions faster than previous state-of-the-art deep learning methods for the TSP. We also show we can adapt the proposed method to two extensions of the TSP: the multiple TSP and the Vehicle Routing Problem, achieving results on par with classical heuristics and learned methods.

In the aftermath of a large disaster, the routing of vehicles carrying critical supplies can greatly impact the arrival times to those in need. Because it is critical that the deliveries are both fast and fair to those being served, it is not clear that the classic cost-minimizing routing problems properly reflect the relevant priorities in disaster relief. In this paper, we take the first steps toward developing new methodologies for these problems. We focus specifically on two alternative objective functions for the traveling salesman problem (TSP) and the vehicle routing problem (VRP): one that minimizes the maximum arrival time (minmax) and one that minimizes the average arrival time (minavg). To demonstrate the potential impact of using these new objective functions, we bound the worst-case performance of optimal TSP solutions with respect to these new variants and extend these bounds to include multiple vehicles and vehicle capacity. Similarly, we examine the potential increase in routing costs that results from using these alternate objectives. We present solution approaches for these two variants of the TSP and VRP, which are based on well-known insertion and local search techniques. These are used in a series of computational experiments to help identify the types of instances in which TSP and VRP solutions can be significantly different from optimal minmax and minavg solutions.

The effects of two new crossover operators on genetic algorithm performance

Crisscross optimization algorithm and its application

To solve the Logistics Distribution Vehicle Routing Problem(VRP), the Modified Shuffled Frog leaping Algorithm(MSFLA) is advanced. The Greedy Method is applied to solve the VRP with step-by-step answers, which turns VRP into Traveling Salesman Problem(TSP)seeking for route K. Then SFLA was used to offer solution to TSP. But the traditional model of SFLA is not fit for discrete combinatorial optimization problems. So the traditional one is be improved, and the memetic algorithm is brought in. The computational results demonetrate that compared to ant colony optimization, local search algorithm, genetic algorithm and the hybrid genetic algorithm, SFLA is faster. It has a stronger ability to achieve the most optimistic result, and is easier to be implemented.

Combinatorial optimization problems are broadly studied in the literature. On the one hand, their challenging characteristics, such as the constraints and number of potential solutions, inspires their use to test new solution techniques. On the other hand, the practical application of these problems provides support of daily tasks of people and companies. Vehicle routing problems constitute a well-known class of combinatorial optimization problems, from which the Traveling Salesman Problem (TSP) is one of the most elementary ones. TSP corresponds to finding the shortest route that visits all cities within a path returning to the start city. Despite its simplicity, the difficulty in finding its exact solution and its direct application in practical problems in multiple areas make it one of the most studied problems in the literature. Algorithms inspired by biological phenomena are being successfully applied to solve optimization tasks, mainly combinatorial optimization problems. Those inspired by the collective behavior of insects produce good results for solving such problems. This article proposes the VRoptBees, a framework inspired by honeybee behavior to tackle vehicle routing problems. The framework provides a flexible and modular tool to easily build solutions to vehicle routing problems. Together with the framework, two examples of implementation are described, one to solve the TSP and the other to solve the Capacitated Vehicle Routing Problem (CVRP). Tests were conducted with benchmark instances from the literature, showing competitive results.

The wide usage of tensor computation in deep neural networks (DNNs) has boosted the high demand for high-performance and flexible library implementation on different hardware platforms, which is time-cost and inefficient. Deep learning compilers (DLCs) are therefore proposed, such as Ansor, to search the optimization computation combinations automatically. Ansor can generate high-performance tensor programs by employing Genetic Algorithm (GA) in its auto-tuning process. However, the structure information of an individual is easily destroyed by the uniform crossover operator, which leads to low search efficiency. In this paper, we propose a two-point crossover operator applied in Ansor called Ansor-TPC, which can optimize tensor computation with higher efficiency. The tensor expression can be computed with random schedules regarded as individuals. When performing the crossover operator, Ansor-TPC exchanges parent genes at two points instead of every point, which can preserve the structure information of programs and find the optimal schedule combination in the large search space. A high-performance program is generated for targeted hardware based on the optimized schedule configuration. Ansor-TPC is compared with the benchmarks at different levels. In terms of average performance, Ansor-TPC achieves 1.07--23.2× performance speedup. In terms of the best performance, Ansor-TPC outperforms by up to 1.06--23.0×.

Ant colony optimization algorithm with mutation mechanism and its applications