Computer Solutions of the Traveling Salesman Problem

Heuristics and Learning Models for Dubins MinMax Traveling Salesman Problem

This paper addresses a MinMax variant of the Dubins multiple traveling salesman problem (mTSP). This routing problem arises naturally in mission planning applications involving fixed-wing unmanned vehicles and ground robots. We first formulate the routing problem, referred to as the one-in-a-set Dubins mTSP problem (MD-GmTSP), as a mixed-integer linear program (MILP). We then develop heuristic-based search methods for the MD-GmTSP using tour construction algorithms to generate initial feasible solutions relatively fast and then improve on these solutions using variants of the variable neighborhood search (VNS) metaheuristic. Finally, we also explore a graph neural network to implicitly learn policies for the MD-GmTSP using a learning-based approach; specifically, we employ an S-sample batch reinforcement learning method on a shared graph neural network architecture and distributed policy networks to solve the MD-GMTSP. All the proposed algorithms are implemented on modified TSPLIB instances, and the performance of all the proposed algorithms is corroborated. The results show that learning based approaches work well for smaller sized instances, while the VNS based heuristics find the best solutions for larger instances.

Combined DFT, SCAPS-1D, and wxAMPS frameworks for design optimization of efficient Cs2BiAgI6-based perovskite solar cells with different charge transport layers

In this study, combined DFT, SCAPS-1D, and wxAMPS frameworks are used to investigate the optimized designs of Cs2BiAgI6 double perovskite-based solar cells. First-principles calculations are employed to investigate the structural stability, optical responses, and electronic contribution of the constituent elements in Cs2BiAgI6 absorber material, where SCAPS-1D and wxAMPS simulators are used to scrutinize different configurations of Cs2BiAgI6 solar cells. Here, PCBM, ZnO, TiO2, C60, IGZO, SnO2, WS2, and CeO2 are used as ETL, and Cu2O, CuSCN, CuSbS2, NiO, P3HT, PEDOT:PSS, spiro-MeOTAD, CuI, CuO, V2O5, CBTS, CFTS are used as HTL, and Au is used as a back contact. About ninety-six combinations of Cs2BiAgI6-based solar cell structures are investigated, in which eight sets of solar cell structures are identified as the most efficient structures. Besides, holistic investigation on the effect of different factors such as the thickness of different layers, series and shunt resistances, temperature, capacitance, Mott-Schottky and generation-recombination rates, and J-V (current-voltage density) and QE (quantum efficiency) characteristics is performed. The results show CBTS as the best HTL for Cs2BiAgI6 with all eight ETLs used in this work, resulting in a power conversion efficiency (PCE) of 19.99%, 21.55%, 21.59%, 17.47%, 20.42%, 21.52%, 14.44%, 21.43% with PCBM, TiO2, ZnO, C60, IGZO, SnO2, CeO2, WS2, respectively. The proposed strategy may pave the way for further design optimization of lead-free double perovskite solar cells.

Precedence-Constrained Colored Traveling Salesman Problem: An Augmented Variable Neighborhood Search Approach

A colored traveling salesman problem (CTSP) as a generalization of the well-known multiple traveling salesman problem utilizes colors to distinguish the accessibility of individual cities to salesmen. This work formulates a precedence-constrained CTSP (PCTSP) over hypergraphs with asymmetric city distances. It is capable of modeling the problems with operations or activities constrained to precedence relationships in many applications. Two types of precedence constraints are taken into account, i.e., 1) among individual cities and 2) among city clusters. An augmented variable neighborhood search (VNS) called POPMUSIC-based VNS (PVNS) is proposed as a main framework for solving PCTSP. It harnesses a partial optimization metaheuristic under special intensification conditions to prepare candidate sets. Moreover, a topological sort-based greedy algorithm is developed to obtain a feasible solution at the initialization phase. Next, mutation and multi-insertion of constraint-preserving exchanges are combined to produce different neighborhoods of the current solution. Two kinds of constraint-preserving k -exchange are adopted to serve as a strong local search means. Extensive experiments are conducted on 34 cases. For the sake of comparison, Lin-Kernighan heuristic, two genetic algorithms and three VNS methods are adapted to PCTSP and fine-tuned by using an automatic algorithm configurator-irace package. The experimental results show that PVNS outperforms them in terms of both search ability and convergence rate. In addition, the study of four PVNS variants each lacking an important operator reveals that all operators play significant roles in PVNS.

Traveling Transporter Problem: Arranging a New Circular Route in a Public Transportation System Based on Heterogeneous Non-Monotonic Urban Data

Hybrid computational intelligent systems that synergize learning-based inference models and route planning strategies have thrived in recent years. In this article, we focus on the non-monotonicity originated from heterogeneous urban data, as well as heuristics based on neural networks, and thereafter formulate the traveling transporter problem (TTP). TTP is a multi-criteria optimization problem and may be applied to the circular route deployment in public transportation. In particular, TTP aims to find an optimized route that maximizes passenger flow according to a neural-network-based inference model and minimizes the length of the route given several constraints, including must-visit stations and the requirement for additional ones. As a variation of the traveling salesman problem (TSP), we propose a framework that first recommends new stations’ location while considering the herding effect between stations, and thereafter combines state-of-the-art TSP solvers and a metaheuristic named Trembling Hand , which is inspired by self-efficacy for solving TTP. Precisely, the proposed Trembling Hand enhances the spatial exploration considering the structural patterns, previous actions, and aging factors. Evaluation conducted on two real-world mass transit systems, Tainan and Chicago, shows that the proposed framework can outperform other state-of-the-art methods by securing the Pareto-optimal toward the objectives of TTP among comparative methods under various constrained settings.

Path planning of patrol robot based on improved discrete electrostatic discharge algorithm

Safety is the premise of the stable and sustainable development of the chemical industry, safety accidents will not only cause casualties and economic losses, but also cause panic among workers and nearby residents. Robot safety inspection based on the fire risk level in a chemical industrial park can effectively reduce process accident losses and can even prevent accidents. The optimal inspection path is an important support for patrol efficiency, therefore, in this study, the fire risk level of each location to be inspected, which is obtained by the electrostatic discharge algorithm (ESDA)–nonparallel support vector machine evaluation model, is combined with the optimisation of the inspection path; that is, the fire risk level is used to guide the inspection path planning. The inspection path planning problem is a typical travelling salesman problem (TSP). The discrete ESDA (DESDA), based on the ESDA, is proposed. In view of the shortcomings of the long convergence time and ease of falling into the local optimum of the DESDA, further improvements are proposed in the form of the IDESDA, in which the greedy algorithm is used for the initial population, the 2-opt algorithm is applied to generate new solutions, and the elite set is joined to provide the best segment for jumping out of the local optimum. In the experiments, 11 public calculation examples were used to verify the algorithm performance. The IDESDA exhibited higher accuracy and better stability when solving the TSP. Its application to chemical industrial parks can effectively solve the path optimisation problem of patrol robots.

Homotopic Convex Transformation: A New Landscape Smoothing Method for the Traveling Salesman Problem

This article proposes a novel landscape smoothing method for the symmetric traveling salesman problem (TSP). We first define the homotopic convex (HC) transformation of a TSP as a convex combination of a well-constructed simple TSP and the original TSP. The simple TSP, called the convex-hull TSP, is constructed by transforming a known local or global optimum. We observe that controlled by the coefficient of the convex combination, with local or global optimum: 1) the landscape of the HC transformed TSP is smoothed in terms that its number of local optima is reduced compared to the original TSP and 2) the fitness distance correlation of the HC transformed TSP is increased. Furthermore, we observe that the smoothing effect of the HC transformation depends highly on the quality of the used optimum. A high-quality optimum leads to a better smoothing effect than a low-quality optimum. We then propose an iterative algorithmic framework in which the proposed HC transformation is combined within a heuristic TSP solver. It works as an escaping scheme from local optima aiming to improve the global searchability of the combined heuristic. Case studies using the 3-Opt and the Lin-Kernighan local search as the heuristic solver show that the resultant algorithms significantly outperform their counterparts and two other smoothing-based TSP heuristic solvers on most of the test instances with up to 20 000 cities.

A Two-Echelon Electric Vehicle Routing Problem with Time Windows and Battery Swapping Stations

Driven by the new laws and regulations concerning the emission of greenhouse gases, it is becoming more and more popular for enterprises to adopt cleaner energy. This research proposes a novel two-echelon vehicle routing problem consisting of mixed vehicles considering battery swapping stations, which includes one depot, multiple satellites with unilateral time windows, and customers with given demands. The fossil fuel-based internal combustion vehicles are employed in the first echelon, while the electric vehicles are used in the second echelon. A mixed integer programming model for this proposed problem is established in which the total cost, including transportation cost, handling cost, fixed cost of two kinds of vehicles, and recharging cost, is minimized. Moreover, based on the variable neighborhood search, a metaheuristic procedure is developed to solve the problem. To validate its effectiveness, extensive numerical experiments are conducted over the randomly generated instances of different sizes. The computational results show that the proposed metaheuristic can produce a good logistics scheme with high efficiency.

Optimization and benchmarking of the thermal cycling algorithm

Optimization plays a significant role in many areas of science and technology. Most of the industrial optimization problems have inordinately complex structures that render finding their global minima a daunting task. Therefore, designing heuristics that can efficiently solve such problems is of utmost importance. In this paper we benchmark and improve the thermal cycling algorithm [Phys. Rev. Lett. 79, 4297 (1997)PRLTAO0031-900710.1103/PhysRevLett.79.4297] that is designed to overcome energy barriers in nonconvex optimization problems by temperature cycling of a pool of candidate solutions. We perform a comprehensive parameter tuning of the algorithm and demonstrate that it competes closely with other state-of-the-art algorithms such as parallel tempering with isoenergetic cluster moves, while overwhelmingly outperforming more simplistic heuristics such as simulated annealing.

Discrete Social Spider Algorithm for Solving Traveling Salesman Problem

The Social Spider Algorithm (SSA) was introduced based on the information-sharing foraging strategy of spiders to solve the continuous optimization problems. SSA was shown to have better performance than the other state-of-the-art meta-heuristic algorithms in terms of best-achieved fitness values, scalability, reliability, and convergence speed. By preserving all strengths and outstanding performance of SSA, we propose a novel algorithm named Discrete Social Spider Algorithm (DSSA), for solving discrete optimization problems by making some modifications to the calculation of distance function, construction of follow position, the movement method, and the fitness function of the original SSA. DSSA is employed to solve the symmetric and asymmetric traveling salesman problems. To prove the effectiveness of DSSA, TSPLIB benchmarks are used, and the results have been compared to the results obtained by six different optimization methods: discrete bat algorithm (IBA), genetic algorithm (GA), an island-based distributed genetic algorithm (IDGA), evolutionary simulated annealing (ESA), discrete imperialist competitive algorithm (DICA) and a discrete firefly algorithm (DFA). The simulation results demonstrate that DSSA outperforms the other techniques. The experimental results show that our method is better than other evolutionary algorithms for solving the TSP problems. DSSA can also be used for any other discrete optimization problem, such as routing problems.

Infochemistry and the Future of Chemical Information Processing

Nowadays, information processing is based on semiconductor (e.g., silicon) devices. Unfortunately, the performance of such devices has natural limitations owing to the physics of semiconductors. Therefore, the problem of finding new strategies for storing and processing an ever-increasing amount of diverse data is very urgent. To solve this problem, scientists have found inspiration in nature, because living organisms have developed uniquely productive and efficient mechanisms for processing and storing information. We address several biological aspects of information and artificial models mimicking corresponding bioprocesses. For instance, we review the formation of synchronization patterns and the emergence of order out of chaos in model chemical systems. We also consider molecular logic and ion fluxes as information carriers. Finally, we consider recent progress in infochemistry, a new direction at the interface of chemistry, biology, and computer science, considering unconventional methods of information processing.

Travelling Santa Problem: Optimization of a Million-Households Tour Within One Hour

Finding the shortest tour visiting all given points at least ones belongs to the most famous optimization problems until today [travelling salesman problem (TSP)]. Optimal solutions exist for many problems up to several ten thousand points. The major difficulty in solving larger problems is the required computational complexity. This shifts the research from finding the optimum with no time limitation to approaches that find good but sub-optimal solutions in pre-defined limited time. This paper proposes a new approach for two-dimensional symmetric problems with more than a million coordinates that is able to create good initial tours within few minutes. It is based on a hierarchical clustering strategy and supports parallel processing. In addition, a method is proposed that can correct unfavorable paths with moderate computational complexity. The new approach is superior to state-of-the-art methods when applied to TSP instances with non-uniformly distributed coordinates.

MAPSOFT: A Multi-Agent based Particle Swarm Optimization Framework for Travelling Salesman Problem

This paper proposes a Multi-Agent based Particle Swarm Optimization (PSO) Framework for the Traveling salesman problem (MAPSOFT). The framework is a deployment of the recently proposed intelligent multi-agent based PSO model by the authors. MAPSOFT is made up of groups of agents that interact with one another in a coordinated search effort within their environment and the solution space. A discrete version of the original multi-agent model is presented and applied to the Travelling Salesman Problem. Based on the simulation results obtained, it was observed that agents retrospectively decide on their next moves based on consistent better fitness values obtained from present and prospective neighborhoods, and by reflecting back to previous behaviors and sticking to historically better results. These overall attributes help enhance the conventional PSO by providing more intelligence and autonomy within the swarm and thus contributed to the emergence of good results for the studied problem.

An Improved Whale Optimization Algorithm for the Traveling Salesman Problem

The whale optimization algorithm is a new type of swarm intelligence bionic optimization algorithm, which has achieved good optimization results in solving continuous optimization problems. However, it has less application in discrete optimization problems. A variable neighborhood discrete whale optimization algorithm for the traveling salesman problem (TSP) is studied in this paper. The discrete code is designed first, and then the adaptive weight, Gaussian disturbance, and variable neighborhood search strategy are introduced, so that the population diversity and the global search ability of the algorithm are improved. The proposed algorithm is tested by 12 classic problems of the Traveling Salesman Problem Library (TSPLIB). Experiment results show that the proposed algorithm has better optimization performance and higher efficiency compared with other popular algorithms and relevant literature.

DMRVR: Dynamic Milk-Run Vehicle Routing Solution Using Fog-Based Vehicular Ad Hoc Networks

Milk-run tours with time windows are an essential strategy to collect goods to minimize production and transportation costs. Due to unexpected events at the supplier production or traffic congestion, delays can occur during the vehicle route execution, causing non-compliance between the logistics operator and the company. This paper describes the DMRVR (Dynamic Milk-Run Vehicle Routing) solution that uses a dynamic routing algorithm along with fog-based vehicular ad hoc networks for implementing the collection of goods in milk-run operations that respect the company’s time window. When a production delay occurs, the supplier sends a message through the vehicular network to alert the pickup vehicle, forcing it to make dynamic route changes to satisfy the constraints. We have implemented a queue with a timeout and retransmission features to improve the vehicular network’s message delivery. To assess the DMRVR solution, we analyzed the efficiency of the dynamic vehicle routing and the vehicular network impacts. In the experiments, we used an event-based network simulator OMNeT++ bidirectionally coupled with SUMO (Simulation of Urban Mobility), aiming to make the most realistic simulations. Simulation results show the average route time was lower than the time limit imposed by the company with the DMRVR solution. In dense vehicular network scenarios, the message delivery success rate is higher. Conversely, when the vehicular network scenario is sparse, it is necessary to balance network coverage and distribute more RSUs in specific places.

Modeling the Municipal Waste Collection Using Genetic Algorithms

Calculating adequate vehicle routes for collecting municipal waste is still an unsolved issue, even though many solutions for this process can be found in the literature. A gap still exists between academics and practitioners in the field. One of the apparent reasons why this rift exists is that academic tools often are not easy to handle and maintain by actual users. In this work, the problem of municipal waste collection is modeled using a simple but efficient and especially easy to maintain solution. Real data have been used, and it has been solved using a Genetic Algorithm (GA). Computations have been done in two different ways: using a complete random initial population, and including a seed in this initial population. In order to guarantee that the solution is efficient, the performance of the genetic algorithm has been compared with another well-performing algorithm, the Variable Neighborhood Search (VNS). Three problems of different sizes have been solved and, in all cases, a significant improvement has been obtained. A total reduction of 40% of itineraries is attained with the subsequent reduction of emissions and costs.

A Hybrid Tabu Search and 2-opt Path Programming for Mission Route Planning of Multiple Robots under Range Limitations

The application of an unmanned vehicle system allows for accelerating the performance of various tasks. Due to limited capacities, such as battery power, it is almost impossible for a single unmanned vehicle to complete a large-scale mission area. An unmanned vehicle swarm has the potential to distribute tasks and coordinate the operations of many robots/drones with very little operator intervention. Therefore, multiple unmanned vehicles are required to execute a set of well-planned mission routes, in order to minimize time and energy consumption. A two-phase heuristic algorithm was used to pursue this goal. In the first phase, a tabu search and the 2-opt node exchange method were used to generate a single optimal path for all target nodes; the solution was then split into multiple clusters according to vehicle numbers as an initial solution for each. In the second phase, a tabu algorithm combined with a 2-opt path exchange was used to further improve the in-route and cross-route solutions for each route. This diversification strategy allowed for approaching the global optimal solution, rather than a regional one with less CPU time. After these algorithms were coded, a group of three robot cars was used to validate this hybrid path programming algorithm.

COEBA: A Coevolutionary Bat Algorithm for Discrete Evolutionary Multitasking

Multitasking optimization is an emerging research field which has attracted lot of attention in the scientific community. The main purpose of this paradigm is how to solve multiple optimization problems or tasks simultaneously by conducting a single search process. The main catalyst for reaching this objective is to exploit possible synergies and complementarities among the tasks to be optimized, helping each other by virtue of the transfer of knowledge among them (thereby being referred to as Transfer Optimization). In this context, Evolutionary Multitasking addresses Transfer Optimization problems by resorting to concepts from Evolutionary Computation for simultaneous solving the tasks at hand. This work contributes to this trend by proposing a novel algorithmic scheme for dealing with multitasking environments. The proposed approach, coined as Coevolutionary Bat Algorithm, finds its inspiration in concepts from both co-evolutionary strategies and the metaheuristic Bat Algorithm. We compare the performance of our proposed method with that of its Multifactorial Evolutionary Algorithm counterpart over 15 different multitasking setups, composed by eight reference instances of the discrete Traveling Salesman Problem. The experimentation and results stemming therefrom support the main hypothesis of this study: the proposed Coevolutionary Bat Algorithm is a promising meta-heuristic for solving Evolutionary Multitasking scenarios.

Failure-Robot Path Complementation for Robot Swarm Mission Planning

Currently, unmanned vehicles are widely used in different fields of exploration. Due to limited capacities, such as limited power supply, it is almost impossible for one unmanned vehicle to visit multiple wide areas. Multiple unmanned vehicles with well-planned routes are required to minimize an unnecessary consumption of time, distance, and energy waste. The aim of the present study was to develop a multiple-vehicle system that can automatically compile a set of optimum vehicle paths, complement failed assignments, and avoid passing through no-travel zones. A heuristic algorithm was used to obtain an approximate solution within a reasonable timeline. The A* Search algorithm was adopted to determine an alternative path that does not cross the no-travel zone when the distance array was set, and an improved two-phased Tabu search was applied to converge any initial solutions into a feasible solution. A diversification strategy helped identify a global optimal solution rather than a regional one. The final experiments successfully demonstrated a group of three robot cars that were simultaneously dispatched to each of their planned routes; when any car failed during the test, its path was immediately reprogrammed by the monitoring station and passed to the other cars to continue the task until each target point had been visited.

Replica symmetry and replica symmetry breaking for the traveling salesperson problem

We study the energy landscape of the traveling salesperson problem (TSP) using exact ground states and a novel linear programming approach to generate excited states with closely defined properties. We look at four different ensembles, notably the classic finite dimensional Euclidean TSP and the mean-field-like (1,2)-TSP, which has its origin directly in the mapping of the Hamiltonian circuit problem on the TSP. Our data supports previous conjectures that the Euclidean TSP does not show signatures of replica symmetry breaking neither in two nor in higher dimension. On the other hand the (1,2)-TSP exhibits some signature which does not exclude broken replica symmetry, making it a candidate for further studies in the future.

A Branch-and-Bound-Based Crossover Operator for the Traveling Salesman Problem

In this article, the new crossover operator BBX for Evolutionary Algorithms (EAs) for traveling salesman problems (TSPs) is introduced. It uses branch-and-bound to find the optimal combination of the (directed) edges present in the parent solutions. The offspring solutions created are at least as good as their parents and are only composed of parental building blocks. The operator is closer to the ideal concept of crossover in EAs than existing operators. This article provides the most extensive study on crossover operators on the TSP, comparing BBX to ten other operators on the 110 instances of the TSPLib benchmark set in EAs with four different population sizes. BBX, with its better ability to reuse and combine building blocks, surprisingly does not generally outperform the other operators. However, it performs well in certain scenarios. Besides presenting a novel approach to crossover on the TSP, the study significantly extends and refines the body of knowledge on the field with new conclusions and comparison results.

Principal component analysis of the excluded area of two-dimensional hard particles

The excluded area between a pair of two-dimensional hard particles with given relative orientation is the region in which one particle cannot be located due to the presence of the other particle. The magnitude of the excluded area as a function of the relative particle orientation plays a major role in the determination of the bulk phase behavior of hard particles. We use principal component analysis (PCA) to identify the different types of excluded areas corresponding to randomly generated two-dimensional hard particles modeled as non-self-intersecting polygons and star lines (line segments radiating from a common origin). Only three principal components are required to have an excellent representation of the value of the excluded area as a function of the relative particle orientation for sufficiently anisotropic particles. Independent of the particle shape, the minimum value of the excluded area is always achieved when the particles are antiparallel to each other. The property that affects the value of the excluded area most strongly is the elongation of the particle shape. PCA identifies four limiting cases of excluded areas with one to four global minima at equispaced relative orientations. We study selected particle shapes using Monte Carlo simulations.

Dynamic Flying Ant Colony Optimization (DFACO) for Solving the Traveling Salesman Problem

This paper presents an adaptation of the flying ant colony optimization (FACO) algorithm to solve the traveling salesman problem (TSP). This new modification is called dynamic flying ant colony optimization (DFACO). FACO was originally proposed to solve the quality of service (QoS)-aware web service selection problem. Many researchers have addressed the TSP, but most solutions could not avoid the stagnation problem. In FACO, a flying ant deposits a pheromone by injecting it from a distance; therefore, not only the nodes on the path but also the neighboring nodes receive the pheromone. The amount of pheromone a neighboring node receives is inversely proportional to the distance between it and the node on the path. In this work, we modified the FACO algorithm to make it suitable for TSP in several ways. For example, the number of neighboring nodes that received pheromones varied depending on the quality of the solution compared to the rest of the solutions. This helped to balance the exploration and exploitation strategies. We also embedded the 3-Opt algorithm to improve the solution by mitigating the effect of the stagnation problem. Moreover, the colony contained a combination of regular and flying ants. These modifications aim to help the DFACO algorithm obtain better solutions in less processing time and avoid getting stuck in local minima. This work compared DFACO with (1) ACO and five different methods using 24 TSP datasets and (2) parallel ACO (PACO)-3Opt using 22 TSP datasets. The empirical results showed that DFACO achieved the best results compared with ACO and the five different methods for most of the datasets (23 out of 24) in terms of the quality of the solutions. Further, it achieved better results compared with PACO-3Opt for most of the datasets (20 out of 21) in terms of solution quality and execution time.

Collision-Free Path Planning and Delivery Sequence Optimization in Noncoplanar Radiation Therapy

Radiation therapy is among the top three cancer treatments in current medical services. The novel noncoplanar radiation therapy which claimed the best characteristics in almost all dosimetric properties encountered the challenges of the potential collision and the long time delivering. In this paper, we proposed a brand new scheme which uses a combined method of the collision avoidance path planning based on an improved probability roadmap method (PRM) and the delivery sequence optimization based on a modified genetic algorithm (GA) to solve the problems in noncoplanar radiation therapy. A uniform sampling strategy, an improved connection strategy, and an efficient local planner are introduced to optimize the roadmap result and accelerate the roadmap construction. The GA is improved by the elitist selection, the local search strategy, and the similar substitution strategy to achieve a better performance both in convergence rate and optimal solution. Experiments are carried out on the simulation platform with typical therapy system models. The results show that our proposed methods work well with the radiation therapy system in a compact working area. Collision is avoided and time consumption is reduced. We believe that our proposed algorithms could solve the problems in current radiation therapy and promote their clinic applications.

Centralized bundle generation in auction-based collaborative transportation

In horizontal collaborations, carriers form coalitions in order to perform parts of their logistics operations jointly. By exchanging transportation requests among each other, they can operate more efficiently and in a more sustainable way. This exchange of requests can be organized through combinatorial auctions, where collaborators submit requests for exchange to a common pool. The requests in the pool are grouped into bundles, and these are offered to participating carriers. From a practical point of view, offering all possible bundles is not manageable, since the number of bundles grows exponentially with the number of traded requests. We show how the complete set of bundles can be efficiently reduced to a subset of attractive ones. For this we define the Bundle Generation Problem (BuGP). The aim is to provide a reduced set of offered bundles that maximizes the total coalition profit, while a feasible assignment of bundles to carriers is guaranteed. The objective function, however, could only be evaluated whether carriers reveal sensitive information, which would be unrealistic. Thus, we develop a proxy for the objective function for assessing the attractiveness of bundles under incomplete information. This is used in a genetic algorithms-based framework that aims at producing attractive and feasible bundles, such that all requirements of the BuGP are met. We achieve very good solution quality, while reducing the computational time for the auction procedure significantly. This is an important step towards running combinatorial auctions of real-world size, which were previously intractable due to their computational complexity. The strengths but also the limitations of the proposed approach are discussed.

A Heuristic Initialized Stochastic Memetic Algorithm for MDPVRP With Interdependent Depot Operations

The vehicle routing problem (VRP) is a widely studied combinatorial optimization problem. We introduce a variant of the multidepot and periodic VRP (MDPVRP) and propose a heuristic initialized stochastic memetic algorithm to solve it. The main challenge in designing such an algorithm for a large combinatorial optimization problem is to avoid premature convergence by maintaining a balance between exploration and exploitation of the search space. We employ intelligent initialization and stochastic learning to address this challenge. The intelligent initialization technique constructs a population by a mix of random and heuristic generated solutions. The stochastic learning enhances the solutions' quality selectively using simulated annealing with a set of random and heuristic operators. The hybridization of randomness and greediness in the initialization and learning process helps to maintain the balance between exploration and exploitation. Our proposed algorithm has been tested extensively on the existing benchmark problems and outperformed the baseline algorithms by a large margin. We further compared our results with that of the state-of-the-art algorithms working under MDPVRP formulation and found a significant improvement over their results.