2-opt Algorithm Research Articles

Parallel heuristic local search algorithm on OTIS hyper hexa-cell and OTIS mesh of trees optoelectronic architectures

Heuristic local search algorithms have achieved good results in tackling combinatorial optimization problems, such as Travelling Salesman Problem (TSP). One of the well-known local search algorithms is the 2-opt algorithm. As a local search algorithm, 2-opt has achieved approximate optimal solutions for TSP within a reasonable time, especially for small data instances. However, solving large data instances of TSP using 2-opt requires extensive computation and considerable CPU time. Therefore, this paper presents a parallel version of the 2-opt algorithm, exploiting the features of Optical Transpose Interconnection System (OTIS) in solving the TSP. In this paper, we present the Parallel Repetitive 2-Opt (PRTO) algorithm for solving symmetric TSP on OTIS Hyper Hexa-Cell (OTIS-HHC) and OTIS Mesh of Trees (OTIS-MOT) optoelectronic architectures. We assess the performance of our algorithm analytically in terms of parallel time complexity, speedup, efficiency, cost, and communication cost on both optoelectronic architectures. Furthermore, a set of simulation experiments is conducted on various instances from the standard TSP library. The simulation results confirm that our algorithm is efficient regarding speedup and efficiency. For instance, the PRTO algorithm achieves a speedup of 32.9 for 6880 cities over OTIS-HHC with 36 processors. Moreover, the superiority of PRTO algorithm is shown through solving the TSP on OTIS-HHC and OTIS-MOT; its performance has been compared with the performance of the Parallel Repetitive Nearest Neighbor (PRNN) algorithm in terms of speedup, efficiency, and solution quality. For example, as a best case, the PRTO algorithm has shown 34 times improved speedup over the PRNN algorithm.

A Novel Hybrid Ant Colony Optimization Algorithm for Emergency Transportation Problems During Post-Disaster Scenarios

The increasing impacts of natural disasters have led to concerns regarding predisaster plans and post-disaster responses. During post-disaster responses, emergency transportation is the most important part of disaster relief supply chain operations, and its optimal planning differs from traditional transportation problems in the objective function and complex constraints. In disaster scenarios, fairness and effectiveness are two important aspects. This paper investigates emergency transportation in real-life disasters scenarios and formulates the problem as an integer linear programming model (called cum-MDVRP), which combines cumulative vehicle routing problem and multidepot vehicle routing problem. The cum-MDVRP is NP-hard. To solve it, a novel hybrid ant colony optimization-based algorithm is proposed by combining both saving algorithms and a simple two-step 2-opt algorithm. The proposed algorithm allows ants to go in and out the depots for multiple rounds, so we abbreviate it as ACOMR. Moreover, we present a smart design of the ants’ tabus, which helps to simplify the solution constructing process. The ACOMR could yield good solutions quickly, then the decision makers for emergency responses could do expert planning at the earliest time. Computational results on standard benchmarking data sets show that the proposed cum-MDVRP model performs well, and the ACOMR algorithm is more effective and stable than the existing algorithms.

The modification of hybrid method of ant colony optimization, particle swarm optimization and 3-OPT algorithm in traveling salesman problem

The traveling salesman problem (TSP) is a famous problem in finding the shortest tour to visit every vertex exactly once, except the first vertex, given a set of vertices. This paper discusses three modification methods to solve TSP by combining Ant Colony Optimization (ACO), Particle Swarm Optimization (PSO) and 3-Opt Algorithm. The ACO is used to find the solution of TSP, in which the PSO is implemented to find the best value of parameters α and β that are used in ACO.In order to reduce the total of tour length from the feasible solution obtained by ACO, then the 3-Opt will be used. In the first modification, the 3-Opt is used to reduce the total tour length from the feasible solutions obtained at each iteration, meanwhile, as the second modification, 3-Opt is used to reduce the total tour length from the entire solution obtained at every iteration. In the third modification, 3-Opt is used to reduce the total tour length from different solutions obtained at each iteration. Results are tested using 6 benchmark problems taken from TSPLIB by calculating the relative error to the best known solution as well as the running time. Among those modifications, only the second and third modification give satisfactory results except the second one needs more execution time compare to the third modifications.

Analysis of a High-Performance TSP Solver on the GPU

Graphical Processing Units have been applied to solve NP-hard problems with no known polynomial time solutions. An example of such a problem is the Traveling Salesman Problem (TSP). The TSP is one of the most commonly studied combinatorial optimization problems and has multiple applications in the areas of engineering, transportation, and logistics. This article presents an improved algorithm for approximating the TSP on fully connected, symmetric graphs by utilizing the GPU. Our approach improves an existing 2-opt hill-climbing algorithm with random restarts by considering multiple updates to the current path found in parallel, and it allows k number of updates per iteration, called k-swap . With our k-swap modification, we show a speed-up over the existing algorithm of 4.5× to 22.9× on data sets ranging from 1,400 to 33,810 nodes, respectively.

Search for Prioritized Test Cases in Multi-Objective Environment During Web Application Testing

Regression testing is an expensive procedure that is implemented during maintenance phase of the Software Development Life Cycle of evolving software. During this process, test case prioritization is one of the strategies followed in which test cases are organized in a fashion so as to enhance efficiency in achieving some performance goal. During the process, there could be several aspects to be kept in mind due to resources constraints such as fault severity detected per unit of test cost, severity detection per test case execution, and execution time of test cases to detect all the faults. Keeping all such constraints in mind, the test case prioritization problem becomes a multi-objective problem where some of the objectives have to be maximized and the remaining ones minimized. In this study, experiments were performed on different versions of five web applications. The problem instance was found to vary from 5 $$\times $$ 5 test cases versus fault matrix, to 125 $$\times $$ 125 matrix. Random approach, 2-opt algorithm, improved 2-opt algorithm, greedy approach, additional greedy approach, Weighted Genetic Algorithm and Non-dominated Sorting Genetic Algorithm-II (NSGA-II) were applied to a generate prioritized test sequence which maximizes the Cost Cognizant Average Percentage of Fault Detection value, severity detection and minimizes test case execution cost to expose all the faults. The performances of these algorithms are compared, keeping these parameters in mind, and it is concluded that the performance of NSGA-II algorithm is better than that of all the other tested algorithms throughout all the experiments.

A hybrid algorithm combining glowworm swarm optimization and complete 2-opt algorithm for spherical travelling salesman problems

The Travelling Salesman Problem (TSP) is one of the most well-known combinatorial optimization problems and has attracted a lot of interests from researchers. Many studies have proposed various methods for solving the two-dimensional TSP. In this study, we extend the two-dimensional TSP to the three-dimensional TSP, namely the spherical TSP in which all points (cities) and paths (solutions) are on the surface of a sphere. A hybrid algorithm based on the glowworm swarm optimization (GSO) and the complete 2-opt algorithm is proposed, in which the carriers of the luciferin are transformed from glowworms to edges between cities, and the probabilistic formula and the luciferin updating formula are modified. In addition, the complete 2-opt algorithm is performed to optimize the selected optimal routes every few iterations. Numerical experimental results show that the proposed algorithm has a better performance than the basic GSO in solving the spherical TSP. Meanwhile, the complete 2-opt algorithm can speed up the convergence rate.

Modified artificial bee colony algorithm for scheduling optimization for printed circuit board production

The production scheduling of placement machines strongly influences the efficiency of printed circuit board (PCB) production. This study focuses on the scheduling optimization for PCB production, that is, on maximizing throughput and minimizing total assembly time and head moving distance. Three problems—the automatic nozzle changer (ANC) assignment problem, the nozzle assignment problem, and the component pick-and-place sequence problem—are investigated using a modified artificial bee colony algorithm (MABCA) to enhance the production efficiency of placement machines.The proposed scheduling optimization algorithm comprises three phases. First, the proportional distribution method is applied to optimize ANC assignment. Second, considering the component height as well as the pick, simultaneous pickup, and component shape restrictions, the MABCA generates the nozzle assignment. Third, under the constraints of place restriction and the predetermined nozzle assignment, the MABCA and the 2-opt algorithm generates the place sequence with the minimal PCB assembly time. Finally, the proposed algorithm is experimentally demonstrated. The average number of pickups in the solutions generated by the proposed method, the onsite engineer, and two related studies are 159.3, 167.2, 208.6, and 210.6, respectively. The proposed approach yields the least PCB assembly time. The average total assembly time (Ttotal) improved between on-site engineer and the proposed MABCA is 4.08%.

Cross-layer selective routing for cost and delay minimization in IEEE 802.11ac wireless mesh network

A Wireless Internet-access Mesh NETwork (WIMNET) provides scalable and reliable internet access through the deployment of multiple access points (APs) and gateways (GWs). In this work, we propose a selective routing algorithm aiming at a hierarchical minimization of the operational cost and the maximal end-to-end delay. In particular, by deploying redundant APs/GWs in the network field, the WIMNET becomes robust to the link or AP/GW failure. However, these redundant APs/GWs increase the operational cost like the power consumption. By using Dijkstra algorithm and 2-opt algorithm, the proposed algorithm iteratively deactivates the deployed APs/GWs and performs the routing that reduces the maximal end-to-end delay based on the APs/GWs remaining active. The generated route meets the real-world constraints like fairness criterion. We further propose a cross-layer design to enhance the routing performance by exploiting the MAC-layer frame aggregation technique. The selective routing algorithm is then implemented in the WIMNET simulator proposed by our group. The numerical experiments demonstrate that in both indoor and open space environments, the proposed selective routing greatly reduces the operational cost, i.e., up to \(80\%\) APs/GWs can be deactivated.

A parallel cooperative hybrid method based on ant colony optimization and 3-Opt algorithm for solving traveling salesman problem

This article presented a parallel cooperative hybrid algorithm for solving traveling salesman problem. Although heuristic approaches and hybrid methods obtain good results in solving the TSP, they cannot successfully avoid getting stuck to local optima. Furthermore, their processing duration unluckily takes a long time. To overcome these deficiencies, we propose the parallel cooperative hybrid algorithm (PACO-3Opt) based on ant colony optimization. This method uses the 3-Opt algorithm to avoid local minima. PACO-3Opt has multiple colonies and a master---slave paradigm. Each colony runs ACO to generate the solutions. After a predefined number of iterations, each colony primarily runs 3-Opt to improve the solutions and then shares the best tour with other colonies. This process continues until the termination criterion meets. Thus, it can reach the global optimum. PACO-3Opt was compared with previous algorithms in the literature. The experimental results show that PACO-3Opt is more efficient and reliable than the other algorithms.

A Framework for Automated Content Based Medical Image Queries in Grid

AbstractIn the current work, the effect of implementing a medical image data-set query application on the grid is studied. Medical imaging is extremely data intensive, because of the size of medical image scans. Grids offer immense processing power as well as a possibility for great levels of coarse grain parallelism adequate for tackling queries on medical image datasets in a comparatively shorter time period. Apart from the security issues, which are common in the domain, the possible parallelism of grids is challenging to make use of. In the current study, the max–min method, Genetic Algorithm (GA), wherein genetic material is substituted by strings of bits while natural selection is substituted by fitness functions, Particle Swarm Optimization (PSO), wherein all particles utilize their own memories and optimum solutions are discovered on the basis of the knowledge obtained by the swarm as a whole as well as a modified PSO (PSO with 2-opt algorithms) are suggested. The outcomes of experiments proved th...

Smoothed Analysis of the 2-Opt Algorithm for the General TSP

2-Opt is a simple local search heuristic for the traveling salesperson problem that performs very well in experiments with respect to both running time and solution quality. In contrast to this, there are instances on which 2-Opt may need an exponential number of steps to reach a local optimum. To understand why 2-Opt usually finds local optima quickly in experiments, we study its expected running time in the model of smoothed analysis, which can be considered as a less-pessimistic variant of worst-case analysis in which the adversarial input is subject to a small amount of random noise. In our probabilistic input model, an adversary chooses an arbitrary graph G and a probability density function for each edge according to which its length is chosen. We prove that in this model the expected number of local improvements is O (mnϕ ċ 16 √ln m )= m 1+ o (1) nϕ , where n and m denote the number of vertices and edges of G , respectively, and ϕ denotes an upper bound on the density functions.

Using 2-Opt based evolution strategy for travelling salesman problem

Harmony search algorithm that matches the (µ+1) evolution strategy, is a heuristic method simulated by the process of music improvisation. In this paper, a harmony search algorithm is directly used for the travelling salesman problem. Instead of conventional selection operators such as roulette wheel, the transformation of real number values of harmony search algorithm to order index of vertex representation and improvement of solutions are obtained by using the 2-Opt local search algorithm. Then, the obtained algorithm is tested on two different parameter groups of TSPLIB. The proposed method is compared with classical 2-Opt which randomly started at each step and best known solutions of test instances from TSPLIB. It is seen that the proposed algorithm offers valuable solutions.

A Fractional-Order Weighted and Self-Adaptive Max-Min Ant System with 3-Opt Algorithm for Traveling Salesman Problem

In this paper, a Fractional-order weighted and Self-adaptive Max-Min Ant System (FS-MMAS) is proposed to make full use of spatial information of the traveling salesman problem. Furthermore, it is realized so that ants can select next city according to the complex topography. Most advanced algorithms based on Ant Colony Optimization can't take advantage of spatial information during the traverse tour, which easily leads to local minimums. Through the multi-scale and self-adaptive search, ants can take advantage of information which contributes to finding the global optimization. Finally, the 3-Opt algorithm is used to improve local solutions. The performance of proposed method was investigated on eight different benchmark problems taken from a literature and proved to be better than other well-known methods in terms of solution quality and robustness.

High-precision and ultrafast UV laser system for next-generation flexible PCB drilling

Since conventional mechanical punching technology for Flexible Printed Circuit Board (FPCB) drilling has restricted via-hole size and depth control for multi-layer circuit boards, CO2 and Ultra Violet (UV) laser drilling technologies have been developed. However, the FPCBs for mobile phones and Personal Digital Assistants (PDAs) require smaller via-hole diameters, since the development of thinner and higher circuit density devices is demanded. Currently, UV laser systems are widely used for FPCB drilling of 75–105μm diameter via-holes and inspectors performs quality test manually using microscopes. We developed a high-precision UV laser microfabrication system for next-generation FPCB drilling of 15μm diameter via-holes. The degrees of the precision of the microfabricated via-holes of 15, 35, 50 and 85μm were mean absolute error rate of 4.4, 2.2, 2.3, and 2.2 which was fully satisfied with industrial inspection specification ±10%. The drilling speed of the system of 2800 via-holes per second at stationary state was achieved. In addition, we applied modified Greedy 2-opt algorithm to find out optimal drilling path which reduced the total time of via-hole fabrication. We successfully reduced the production time by 25% compared with the result obtained in the normal Greedy 2-opt algorithm. Moreover, we designed very accurate inspection method using Canny edge detection and geometric pattern matching algorithms and successfully applied it to the Automated Optical Inspection (AOI) module for the inspections of 15μm diameter via-hole which was required for the fabrication of high density FPCB.

A new hybrid method based on Particle Swarm Optimization, Ant Colony Optimization and 3-Opt algorithms for Traveling Salesman Problem

The Traveling Salesman Problem (TSP) is one of the standard test problems used in performance analysis of discrete optimization algorithms. The Ant Colony Optimization (ACO) algorithm appears among heuristic algorithms used for solving discrete optimization problems. In this study, a new hybrid method is proposed to optimize parameters that affect performance of the ACO algorithm using Particle Swarm Optimization (PSO). In addition, 3-Opt heuristic method is added to proposed method in order to improve local solutions. The PSO algorithm is used for detecting optimum values of parameters α and β which are used for city selection operations in the ACO algorithm and determines significance of inter-city pheromone and distances. The 3-Opt algorithm is used for the purpose of improving city selection operations, which could not be improved due to falling in local minimums by the ACO algorithm. The performance of proposed hybrid method is investigated on ten different benchmark problems taken from literature and it is compared to the performance of some well-known algorithms. Experimental results show that the performance of proposed method by using fewer ants than the number of cities for the TSPs is better than the performance of compared methods in most cases in terms of solution quality and robustness.

Motion planning algorithms for the Dubins Travelling Salesperson Problem

Two motion planning algorithms for the so-called Dubins Travelling Salesperson Problem are presented, and compared via simulations with existing algorithms from the literature. The first algorithm–dubbed “k-step look-ahead algorithm”–stems naturally from the formulation of the Dubins Travelling Salesperson Problem as a minimum-time control problem and is suitable for obtaining short tours when the number of cities is relatively small. The second algorithm is an adaptation of the classic 2-Opt algorithm for the Travelling Salesperson Problem and can be applied to larger instances of the Dubins Travelling Salesperson Problem. In this sense, the two algorithms complement each other in terms of their range of applicability. Instead of being decoupled, the combinatorial and motion planning aspects of the Dubins Travelling Salesperson Problem are treated in an integrated manner by both algorithms and no assumptions are made on the magnitude of the intercity distances.

An Extended Self-Organizing Map based on 2-opt algorithm for solving symmetrical Traveling Salesperson Problem

Self-organizing map (SOM) is a powerful variant of neural network for solving optimization problems. Many researchers have reported SOM for Traveling Salesperson Problem; however, problems still exist due to the trapping of the optimization techniques at the local optimal position. In this work, we propose an Extended Self-Organizing Map based on 2-opt algorithm with one-dimensional neighborhood to approach the Symmetrical Traveling Salesperson Problem (STSP). We elaborate our approach for STSP where weights of neurons represent nodes that are placed in the polygonal domain. The selection of winner neuron of SOM has been extended to overcome the problem of trapping of SOM at local optima. The results of SOM are improved through 2-opt local optimization algorithm. We briefly discuss self-organization in neural networks, 2-opt algorithm, and extension applied to SOM. Finally, the algorithm is compared with Kohonen Self-Organizing Map and Evolutionary Algorithm. The results show that our approach performs better as compared to other techniques.

Vehicle Routing to Multiple Warehouses Using a Memetic Algorithm

This paper presents a solution procedure for solving the vehicle routing problem with pick-up and delivery with multiple warehouses (satellites) (Multi warehouse Vehicle Routing Problem with Pickup and Delivery-MDVRPPD) based on a hybrid metaheuristic. The solution strategy includes a memetic algorithm consisting of an evolutionary metaheuristic (genetic algorithm) and a heuristic local search (2-opt algorithm).

Optimization on NoC Mapping Based on Improved Ant Colony Algorithm

Traditional ant colony mapping algorithm not only has big power consumption, but also is easy to be trapped into local optimization on NoC mapping, for which the paper proposes an optimization scheme based on improved ant colony algorithm. Firstly, the parameters are for initialization operation. Secondly, tabu list is used to solve them, and the solutions are for local optimization of optimal solutions by using 2-opt algorithm. Lastly, pheromone rules are updated. Simulation experiment indicates that compared with traditional ant colony mapping algorithm, NoC mapping optimization scheme based on improved ant colony algorithm not only has better performance on mapping power consumption, but also is not easy to be trapped into local optimization.

Application of Imperialist Competitive Algorithm on Solving the Traveling Salesman Problem

The imperialist competitive algorithm (ICA) is a new heuristic algorithm proposed for continuous optimization problems. The research about its application on solving the traveling salesman problem (TSP) is still very limited. Aiming to explore its ability on solving TSP, we present a discrete imperialist competitive algorithm in this paper. The proposed algorithm modifies the original rules of the assimilation and introduces the 2-opt algorithm into the revolution process. To examine its performance, we tested the proposed algorithm on 10 small-scale and 2 large-scale standard benchmark instances from the TSPLIB and compared the experimental results with that obtained by two other ICA-based algorithms and six other existing algorithms. The proposed algorithm shows excellent performance in the experiments and comparisons.