Appropriate Combination of Crossover Operator and Mutation Operator in Genetic Algorithms for the Travelling Salesman Problem

Genetic algorithms(GAs)are very good metaheuristic algorithms that are suitable for solving NP-hard combinatorial optimization problems.AsimpleGAbeginswith a set of solutions represented by a population of chromosomes and then uses the idea of survival of the fittest in the selection process to select some fitter chromosomes.It uses a crossover operator to create better offspring chromosomes and thus,converges the population.Also,it uses a mutation operator to explore the unexplored areas by the crossover operator,and thus,diversifies the GA search space.A combination of crossover and mutation operators makes the GA search strong enough to reach the optimal solution.However,appropriate selection and combination of crossover operator and mutation operator can lead to a very good GA for solving an optimization problem.In this present paper,we aim to study the benchmark traveling salesman problem(TSP).We developed several genetic algorithms using seven crossover operators and six mutation operators for the TSP and then compared them to some benchmark TSPLIB instances.The experimental studies show the effectiveness of the combination of a comprehensive sequential constructive crossover operator and insertion mutation operator for the problem.The GA using the comprehensive sequential constructive crossover with insertion mutation could find average solutions whose average percentage of excesses from the best-known solutions are between 0.22 and 14.94 for our experimented problem instances.

A Novel Insertion Solution for the Travelling Salesman Problem

The studypresents theHalfMax InsertionHeuristic (HMIH) as a novel approach to solving theTravelling SalesmanProblem (TSP). The goal is to outperform existing techniques such as the Farthest Insertion Heuristic (FIH) andNearest Neighbour Heuristic (NNH). The paper discusses the limitations of current construction tour heuristics,focusing particularly on the significant margin of error in FIH. It then proposes HMIH as an alternative thatminimizes the increase in tour distance and includes more nodes. HMIH improves tour quality by starting withan initial tour consisting of a ‘minimum’ polygon and iteratively adding nodes using our novel Half Max routine.The paper thoroughly examines and compares HMIH with FIH and NNH via rigorous testing on standard TSPbenchmarks. The results indicate that HMIH consistently delivers superior performance, particularly with respectto tour cost and computational efficiency. HMIH’s tours were sometimes 16% shorter than those generated by FIHand NNH, showcasing its potential and value as a novel benchmark for TSP solutions. The study used statisticalmethods, including Friedman’s Non-parametric Test, to validate the performance of HMIH over FIH and NNH.This guarantees that the identified advantages are statistically significant and consistent in various situations. Thiscomprehensive analysis emphasizes the reliability and efficiency of the heuristic, making a compelling case for itsuse in solving TSP issues. The research shows that, in general, HMIH fared better than FIH in all cases studied,except for a few instances (pr439, eil51, and eil101) where FIH either performed equally or slightly better thanHMIH. HMIH’s efficiency is shown by its improvements in error percentage (δ) and goodness values (g) comparedto FIH and NNH. In the att48 instance, HMIH had an error rate of 6.3%, whereas FIH had 14.6% and NNH had20.9%, indicating that HMIH was closer to the optimal solution. HMIH consistently showed superior performanceacross many benchmarks, with lower percentage error and higher goodness values, suggesting a closer match tothe optimal tour costs. This study substantially contributes to combinatorial optimization by enhancing currentinsertion algorithms and presenting a more efficient solution for the Travelling Salesman Problem. It also createsnew possibilities for progress in heuristic design and optimization methodologies.

Propagation and Pinning of Travelling Wave for Nagumo Type Equation

In this paper, we study the propagation and its failure to propagate (pinning) of a travelling wave in a Nagumo type equation, an equation that describes impulse propagation in nerve axons that also models population growth with Allee effect. An analytical solution is derived for the traveling wave and the work is extended to a discrete formulation with a piecewise linear reaction function. We propose an operator splitting numerical scheme to solve the equation and demonstrate that the wave either propagates or gets pinned based on how the spatial mesh is chosen.

Effect of travelling magnetic field on interface morphology in directionally solidified Sn-Cd alloy

The diversity of interface morphologies is observed for directionally solidified Sn-0.65%Cd alloy under a travelling magnetic field （TMF） in the 4 mm-diameter sample. Under an upward TMF, planar and cellular interface morphologies transform alternately with increasing magnetic flux density （B≤10.3 mT）. The interface morphology transforms from shallow cellular to deep cellular morphology under a weak downward TMF （B=3.2 mT）. When the magnetic flux density increases further, both sides of the interface morphology appear to be slightly inconsistent, but they roughly tend to be planar under a strong downward TMF （BS10.3 mT）. The interface instability may be attributed to the flow driven by the TMF. Moreover, the shape of interface appears to be almost flat under an upward TMF, but deflective under a downward TMF.

THE EQUIVALENT CIRCUIT AND CONTROL SYSTEM OF TRAVELLING WAVE TYPE ULTRASONIC MOTOR

The ultrasonic motor is a sort of new type of micromotor with special structure. By use of piezoelectric converse effect of ceramics, the electrical energy is transformed into mechanical energy. Its operating principle is quite different from that of the traditional motors. In this paper the equivalent circuits of the ring stator and even the whole motor are proposed after studying the equivalent circuit of piezoelectric vibrator. Then the paper makes detailed analyses of each part of the control system, which has been simplified by the equivalent circuit. The theory in this paper has been proved through experiments.

On the Value of "Wild Play Grounds" in Promoting Local "Carefree Travelling"

'Wild Play Grounds' are the wild places which can attract people to visit.And the 'Wild Play Grounds' are generally the naturally existing places with unique charm,rather than the established tourist attractions or scenic spots.There are no formal organizations for the daily management.Both tickets and clear geographical borders can not be found here.In this research,'Carefree Travelling' is an integrated project of tourism resources in Shandong Province promoted in 2007.Because of their particular characteristics,the 'Wild Play Grounds' have a certain relationship with 'Carefree Travelling',and they are the natural resources of 'Carefree Travelling'.The development of 'Wild Play Grounds' is the further exploitation of the 'Carefree Travelling' resources as well as a better expanding of the tourist market.Relying on the 'Wild Play Grounds',it also has many advantages to develop 'Carefree Travelling',and it will bring good economic and social benefits.Local government should play an important role in developing the 'Wild Play Grounds'.

Speed Control of Travelling Wave Type Ultrasonic Motors Using Artificial Neural Network

Ultrasonic motor (USM) is a newly developed motor, and it has some excellent performances and useful features, therefore, it has been expected to be of practical use. However, the driving principle of USM is different from that of other electromagnetic type motors, and the mathematical model is complex to apply to motor control. Furthermore, the speed characteristics of the motor have heavy nonlinearity and vary with driving conditions. Hence, the precise speed control of USM is generally difficult. This paper proposes a new speed control scheme for USM using an artificial neural network. An accurate tracking response can be obtained by random initialization of the weights of the network owing to the powerful on line learning capability. Two prototype ultrasonic motors of travelling wave type were fabricated, both having 100 mm outer diameters of stator and piezoelectric ceramic. The usefulness and validity of the proposed control scheme are examined in experiments.

ASYMPTOTIC METHOD OF TRAVELLING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR REACTION DIFFUSION EQUATION

In this article the travelling wave solution for a class of nonlinear reaction diffusion problems are considered. Using the homotopic method and the theory of travelling wave transform, the approximate solution for the corresponding problem is obtained.

BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS FOR THE GENERALIZED DODD-BULLOUGH-MIKHAILOV EQUATION

In this paper, the generalized Dodd-Bullough-Mikhailov equation is studied. The existence of periodic wave and unbounded wave solutions is proved by using the method of bifurcation theory of dynamical systems. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.Some exact explicit parametric representations of the above travelling solutions are obtained.

The classification of travelling wave solutions and superposition of multi-solutions to Camassa-Holm equation with dispersion

Under the travelling wave transformation, the Camassa-Holm equation with dispersion is reduced to an integrable ordinary differential equation （ODE）, whose general solution can be obtained using the trick of one-parameter group. Furthermore, by using a complete discrimination system for polynomial, the classification of all single travelling wave solutions to the Camassa-Holm equation with dispersion is obtained. In particular, an affine subspace structure in the set of the solutions of the reduced ODE is obtained. More generally, an implicit linear structure in the Camassa-Holm equation with dispersion is found. According to the linear structure, we obtain the superposition of multi-solutions to Camassa-Holm equation with dispersion.

Exact travelling wave solutions for （1＋ 1）-dimensional dispersive long wave equation

A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a （1＋1）-dimensional dispersive long wave equation with general coefficients to an elementary integral form and obtained its all possible exact travelling wave solutions including rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions double periodic solutions. This method can be also applied to many other similar problems.

TRAVELLING WAVE SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS BY USING SYMBOLIC COMPUTATION

A Riccati equation involving a parameter and symbolic computation are used to uniformly construct the different forms of travelling wave solutions for nonlinear evolution equations.It is shown that the sign of the parameter can be applied in judging the existence of various forms of travelling wave solutions.An efficiency of this method is demonstrated on some equations,which include Burgers Huxley equation,Caudrey Dodd Gibbon Kawada equation,generalized Benjamin Bona Mahony equation and generalized Fisher equation.

Solution of Travelling Wave for Nonlinear Disturbed Long-Wave System

The approximate expressions of the travelling wave solutions for a class of nonlinear disturbed long-wave system are constructed using the generalized variational iteration method.

Separation of inclusions from liquid metal contained in a triangle/square pipe by travelling magnetic field

By using plug flow and trajectory model, the elimination efficiency of the inclusions from liquid metals purified by travelling magnetic field (TMF) in either a triangle or a square pipe was analyzed theoretically. The ways to improve the elimination efficiency were suggested. The results using different kinds of pipes were reciprocally compared. It is determined that by means of TMF to eliminate inclusions the efficiency is affected by the diameter of the inclusions, in which the inclusions can be removed most efficiently, is optimized.

A new auxiliary equation method for finding travelling wave solutions to KdV equation

In this paper, a new auxiliary equation method is used to find exact travelling wave solutions to the （1＋1）-dimensional KdV equation. Some exact travelling wave solu- tions with parameters have been obtained, which cover the existing solutions. Compared to other methods, the presented method is more direct, more concise, more effective, and easier for calculations. In addition, it can be used to solve other nonlinear evolution equations in mathematical physics.

CONVERGENCE RATES TO TRAVELLING WAVES FOR A RELAXATION MODEL WITH LARGE INITIAL DISTURBANCE

This paper is concerned with the convergence rates to travelling waves for a relaxation model with general flux functions. Compared with former results in this direction, the main novelty in this paper lies in the fact that the initial disturbance can be chosen large in suitable norm. Our analysis is based on the L^1-stability results obtained by C. Mascia and R. Natalini in [12].

THE NONLINEAR STABILITY OF TRAVELLING WAVE SOLUTIONS FOR A REACTING FLOW MODEL WITH SOURCE TERM

In this paper, author considers a 3 x 3 system for a reacting flow model propesed by [9]. Since this model has source term, it can be considered as a relaxation approximation to 2 x 2 systems of conservation laws, which include the well-known p-system. From this viewpoint, the author establishes the global existence and the nonlinear stability of travelling wave solutions by L-2 energy method.

NEW EXACT TRAVELLING WAVE SOLUTIONS TO THREE NONLINEAR EVOLUTION EQUATIONS

Based on the computerized symbolic computation, some new exact travelling wave solutions to three nonlinear evolution equations are explicitly obtained by replacing the tanhξ in tanh-function method with the solutions of a new auxiliary ordinary differential equation.

Study on Atmospheric Travelling Wave Solutions and Review of Its Present Developments

The scientific achievements of travelling waves in a barotropic atmosphere are introduced, including i) the existence conditions of periodic solutions (wavetrain solutions) and solitary wave solutions (pulse solutions), together with the solution finding methods and a series of related problems, ii) seeking solutions of monotonous wave (wave front) and of nonmonotonous travelling wave (oscillatory wave) by using phase plane shooting technique and hi) progress in the study of travelling wave solution at home and abroad. The investigation of travelling wave solutions in recent years has been found in mathematics, physics, chemistry, biology and other sciences. Over the past decade the problem has been the subject of much interest and become an important area of research. So it is no doubt of great significance to investigate the travelling wave solutions and thereby explain phenomena of weather.

Classification of All Single Travelling Wave Solutions to Calogero-Degasperis-Focas Equation

Under the travelling wave transformation, Calogero-Degasperis-Focas equation is reduced to an ordinary differential equation. Using a symmetry group of one parameter, this ODE is reduced to a second-order linear inhomogeneous ODE. Furthermore, we apply the change of the variable and complete discrimination system for polynomial to solve the corresponding integrals and obtained the classification of all single travelling wave solutions to Calogero- Degasperis-Focas equation.