Solving combinatorial optimization problems with single seekers society algorithm

Contributors

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.

This paper investigates the application of a new class of neighborhood search algorithms—cyclic transfers—to multivehicle routing and scheduling problems. These algorithms exploit the two-faceted decision structure inherent to this problem class: First, assigning demands to vehicles and, second, routing each vehicle through its assigned demand stops. We describe the application of cyclic transfers to vehicle routing and scheduling problems. Then we determine the worst-case performance of these algorithms for several classes of vehicle routing and scheduling problems. Next, we develop computationally efficient methods for finding negative cost cyclic transfers. Finally, we present computational results for three diverse vehicle routing and scheduling problems, which collectively incorporate a variety of constraint and objective function structures. Our results show that cyclic transfer methods are either comparable to or better than the best published heuristic algorithms for several complex and important vehicle routing and scheduling problems. Most importantly, they represent a novel approach to solution improvement which holds promise in many vehicle routing and scheduling problem domains.

The chapter by Teodor Crainic in this book discusses the long haul truck routing problem. Some of the key differences between the long haul truck routing problem and the vehicle routing problems considered in this chapter are as follows: 1. In the long haul truck routing problem, the routes can extend over several days whereas, in the vehicle routing problem described herein, the routes are of one-day duration. 2. In the long haul truck routing problem, the locations can be scattered over a wide region, even the entire United States, whereas in the vehicle routing problems considered herein, the locations requiring service are packed in a small region.

Many combinatorial optimization problems can be reduced to the LP problems using the results from the field called “polyhedral combinatorics”. The main goal of polyhedral combinatorics is to represent the convex envelope of feasible points of the given combinatorial problem in the form of a system of linear equalities and inequalities. Obtained by this way LP problems often have an exponentially growing number of constraints. We know that at each step of ellipsoid method for LP we have to use no more than one constraint from the set of constraints which are not fulfilled at current point. In many cases the problem of finding such a constraint can be formulated in the form of a new combinatorial (in some sense polar to the original) optimization problem (so-called separation problem).

An adaptive memory methodology for the vehicle routing problem with simultaneous pick-ups and deliveries

This paper presents a predictive robust and stable approach for a two-machine flow shop scheduling problem with machine disruption and uncertain job processing time. Indeed, a general approach is proposed that can be used for robustness and stability optimization in an m-machine flow shop or job shop scheduling problem. The robustness measure is the total expected realized completion time. The expected sum of squared aberration between each jobs’ completion time in the realized and initial schedules is the stability measure. We proposed and compared two methods to deal with such an NP-hard problem; a method based on decomposing the problem into sub-problem and solving each sub-problem, and a theorem-based method. The extensive computational results indicated that the second method has a better performance in terms of robustness and stability, especially in large-sized problems. In other words, the second method is preferable because of the better manufacturer responsiveness to the customer and the production staff satisfaction enhancement.

The job shop scheduling problem (JSSP) is a notoriously difficult problem in combinatorial optimization. Extensive investigation has been devoted to developing efficient algorithms to find optimal or near-optimal solutions. This paper proposes an improved immune clonal selection algorithm, called improved clonal selection algorithm for the JSSP. The new algorithm has the advantage of preventing from prematurity and fast convergence speed. Numerous well-studied benchmark examples in job-shop scheduling problems were utilized to evaluate the proposed approach. The computational results show that the proposed algorithm could obtain the high-quality solutions within reasonable computing times, and the results indicate the effectiveness and flexibility of the immune memory clonal selection algorithm.

An effective job shop scheduling (JSS) in the manufacturing industry is helpful to meet the production demand and reduce the production cost, and to improve the ability to compete in the ever increasing volatile market demanding multiple products.In so many combinatorial optimization problems, job shop scheduling problems have earned a reputation for being difficult to solve. Job-shop scheduling is essentially an ordering problem. A new encoding scheme for a classic job-shop scheduling problem is presented. The aim is to find an allocation for each job and to define the sequence of jobs on each machine so that the resulting schedule has a minimal completion time.Genetic algorithm that has demonstrated considerable success in providing efficient solutions to many non polynomial-hard optimization problems is used to solve job-shop scheduling problem. The schedules given by genetic algorithms are constructed using a priority rule and under several constraints. After a schedule is obtained a checking operation is applied to ensure that the solution is feasible. The approach is tested on a set of instances. The results validate the effectiveness of the algorithm.

Niching ethods are useful extension of evolutionary algorithm that can permit population to search many peaks in parallel in multimodal domains. Traditional niching methods that can effectively be used to solve function optimization problems (FOPs) are not suitable for solving combinatorial optimization problems (COPs). We propose a new genetic algorithm (GA) for forming niches that can be efficiently applied to COPs. For this, we propose four requirements that are needed for an ideal method of forming niches for COPs. We then discuss a niching method we designed that satisfies these requirements. The proposed GA was applied to job shop scheduling problems (JSP) to demonstrate its effectiveness.

SYNOPTIC ABSTRACTBranch exchange techniques, such as the well-known 2-opt and 3-opt procedures, are among the most powerful heuristics available for the solution of the classic vehicle routing problem. The imposition of time window constraints on customer delivery time within the vehicle routing problem, however, introduces several complexities that can reduce the power of these techniques. In this paper, two test data sets for the vehicle routing and scheduling problem with time window constraints are studied. Initial solutions are obtained using a variety of heuristics. These solutions are then improved using branch exchange procedures modified to incorporate the time window constraints.

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.

In This Issue

With the development of machine learning in various fields, it can also be applied to combinatorial optimization problems, automatically discovering generic and fast heuristic algorithms based on training data, and requires fewer theoretical and empirical knowledge. Pointer network improves the attention mechanism, instead of allocating different attention to hidden states of encoder to generate context vectors, using attention as a pointer to select an element of the input sequence at every step of decoding, which solves the problem of variable dictionary size of the output sequence. Pointer net (Ptr-Net) applied to three combinatorial optimization problems, convex hull, Delaunay triangulation, and traveling salesman problem (TSP), obtains good approximate solutions. Point matching is also a special kind of combinatorial optimization problems that is to obtain the optimal corresponding references, which can be modeled by Ptr-Net. However, Ptr-Net can’t be used to solve point matching problem because it doesn’t take full advantage of the correspondences between the two point sets. We propose multi-pointer network, which draws the idea from multi-label classification, to address this limitation by pointing out a set of input elements. These applications are all based on supervised learning to approximate expected known solutions. However, high-quality labeled data is often expensive, unreliable, or simply unavailable and may be infeasible for new problem statements, making supervised learning being unpractical. Reinforcement learning, as another research hotspot in the field of machine learning, does not require labeled sample data. It interacts with the environment through trial-and-error mechanism and focuses more on learning problem-solving strategies. We introduce a framework to tackle combinatorial optimization problems using neural networks and reinforcement learning, focusing on the traveling salesman problem. We also introduce a framework, a unique combination of reinforcement learning and graph embedding network, to solve graph optimization problems, focusing on maximum cut (MAXCUT) and minimum vertex cover (MVC) problems.

The vehicle routing problem(VRP) is an important issue in practical use. VRPs are one of combinatorial optimization problems. For solving such combinatorial problems, several evolutionary computation methods have been proposed. In practical situation, many complex constraint conditions and desired computation time are obstacles to use evolutionary computation methods to such problems. In this paper, an evolutionary computation based solver is developed. From some computational experiments for real tanker truck scheduling problem, effectiveness of the proposed method is shown.