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.

An Improved Farmland Fertility Algorithm with Hyper-Heuristic Approach for Solving Travelling Salesman Problem

Travelling Salesman Problem(TSP)is a discrete hybrid optimization problem considered NP-hard.TSP aims to discover the shortest Hamilton route that visits each city precisely once and then returns to the starting point,making it the shortest route feasible.This paper employed a Farmland Fertility Algorithm(FFA)inspired by agricultural land fertility and a hyper-heuristic technique based on the Modified Choice Function(MCF).The neighborhood search operator can use this strategy to automatically select the best heuristic method formaking the best decision.Lin-Kernighan(LK)local search has been incorporated to increase the efficiency and performance of this suggested approach.71 TSPLIB datasets have been compared with different algorithms to prove the proposed algorithm’s performance and efficiency.Simulation results indicated that the proposed algorithm outperforms comparable methods of average mean computation time,average percentage deviation(PDav),and tour length.

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.

Use the Power of a Genetic Algorithm to Maximize and Minimize Cases to Solve Capacity Supplying Optimization and Travelling Salesman in Nested Problems

Using Genetic Algorithms (GAs) is a powerful tool to get solution to large scale design optimization problems. This paper used GA to solve complicated design optimization problems in two different applications. The aims are to implement the genetic algorithm to solve these two different (nested) problems, and to get the best or optimization solutions.

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.

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.

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.

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.

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'.

THE SMOOTH AND NONSMOOTH TRAVELLING WAVESOLUTIONS IN A NONLINEAR WAVE EQUATION

The travelling wave solutions (TWS) in a class of P. D. E. is studied. The travelling wave equation of this P. D. E. is a planar cubic polynomial system in three-parameter space. The study for M became the topological classifications of bifurcations of phase portraits defined by the planar system. B using the theory of planar dynamical systems to do qualitative analysis, all topological classifications of the cubic polynomial system can be obtained. Returning the results of the phase plane analysis to TWS, u(xi), and considering discontinuity of the right side of the equation of TWS when xi = x - ct is varied along a phase orbit and passing through a singular curve, all conditions of existence of smooth and nonsmooth travelling waves are given.

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.

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.

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.

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.

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.

Finite Travelling Wave Solutions for a Semilinear System of Parabolic Type

Aict f Finjte rmvedrig wave (M) so1uhons fOr the fOllowhg sechear syttem (I){u_t-u_(xx)+u^mv^p=0 u_t-v_(xx)+u^q=0 -∞<x<+∞,t>0,p,q>0,m≥0 are studied. SolutiOns to (I) of the fOrm u (x, t)=lt(ct--x), v(x, t)=v (cl--X) are called W soIutiOns if there exjstS a fwite ', such that u({)=v(j)=0 for t<{,':=ct--x. It is proVed that if Pq+nl<l, fOr any ed c thele erktS an FTW that is inhque up to phase transIahons and Is unbOunded, whena no rm ekist if pq+m> l. The asmpptohc weve profileS near the front as well as far from it are also determined. If I)q^m = l. the exjstence of travebe wave soluhons to (I) is proved. The plnof in Esqniruis's paper(1990) for the one m=0 co be sdriplified by using the methOd develOped in thjs paper.

Solution of ODE u″+p(u)(u′)2+q(u)=0 and Applications to Classifications of All Single Travelling Wave Solutions to Some Nonlinear Mathematical Physics Equations

Under the travelling wave transformation, some nonlinear partial differential equations such as Camassa-Holm equation, High-order KdV equation, etc., are reduced to an integrable ODE expressed by u＂ ＋p（u）（u＇）^2 ＋ q（u） = 0 whose generai solution can be given. Furthermore, combining complete discrimination system for polynomiai, the classifications of all single travelling wave solutions to these equations are obtained. The equation u＂＋p（u）（u＇）^2＋q（u） = 0 includes the equation （u＇）^2 = f（u） as a special case, so the proposed method can be also applied to a large number of nonlinear equations. These complete results cannot be obtained by any indirect method.

Eliminating shrinkage defects and improving mechanical performance of large thin-walled ZL205A alloy castings by coupling travelling magnetic fields with sequential solidification

ZL205 A alloys with large thin-walled shape were continuously processed by coupling travelling magnetic fields(TMF)with sequential solidification,to eliminate the shrinkage defects and optimize the mechanical performance.Through experiments and simulations,the parameter optimization of TMF and the influence on feeding behavior,microstructure and properties were systematically studied.The results indicate that the magnetic force maximizes at the excitation current of 20 A and frequency of 200 Hz under the experimental conditions of this study,and increases from center to side-walls,which is more convenient to process thin-walled castings.TMF can break secondary dendritic arm and dendrites overlaps,widen feeding channels,prolong the feeding time,optimize the feeding paths,eliminate shrinkage defects and improve properties.Specifically,for as-cast state,TMF with excitation current of 20 A increases ultimate tensile strength,elongation and micro-hardness from 186 MPa,7.3%and 82.1 kg/mm^(2) to 221 MPa,11.7%and 100.5 kg/mm^(2),decreases porosity from 1.71%to 0.22%,and alters brittle fracture to ductile fracture.