Solving the Generalized Traveling Salesman Problem Using Sequential Constructive Crossover Operator in Genetic Algorithm

The generalized travelling salesman problem(GTSP),a generalization of the well-known travelling salesman problem(TSP),is considered for our study.Since the GTSP is NP-hard and very complex,finding exact solutions is highly expensive,we will develop genetic algorithms(GAs)to obtain heuristic solutions to the problem.In GAs,as the crossover is a very important process,the crossovermethods proposed for the traditional TSP could be adapted for the GTSP.The sequential constructive crossover(SCX)and three other operators are adapted to use in GAs to solve the GTSP.The effectiveness of GA using SCX is verified on some GTSP Library(GTSPLIB)instances first and then compared against GAs using the other crossover methods.The computational results show the success of the GA using SCX for this problem.Our proposed GA using SCX,and swap mutation could find average solutions whose average percentage of excesses fromthe best-known solutions is between 0.00 and 14.07 for our investigated instances.

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.

On the “Onion Husk” Algorithm for Approximate Solution of the Traveling Salesman Problem

The paper describes some implementation aspects of an algorithm for approximate solution of the traveling salesman problem based on the construction of convex closed contours on the initial set of points (“cities”) and their subsequent combination into a closed path (the so-called contour algorithm or “onion husk” algorithm). A number of heuristics related to the different stages of the algorithm are considered, and various variants of the algorithm based on these heuristics are analyzed. Sets of randomly generated points of different sizes (from 4 to 90 and from 500 to 10,000) were used to test the algorithms. The numerical results obtained are compared with the results of two well-known combinatorial optimization algorithms, namely the algorithm based on the branch and bound method and the simulated annealing algorithm. .

A Comparison of Death of a Salesman & The Glass Menagerie ——The use of expressionistic devices

Expressionism was an aesthetic movement originated in painting in the early 1920’s. But it has also exerted great influence on other forms of arts, including dramatic works. This paper intends to compare the application of expressionistic devices such as flashbacks, and the use of symbols and music in Death of a Salesman and The Glass Menagerie.

A Free Exchange of Space and Time-An Analysis of Some Creative Techniques in “Death of a Salesman”

"Death of a Salesman"is important not only in its contents, but also in its creative techniques. This paper will focus on Miller's specific methods with which he depicts Willy's inner world. To be exact, he borrowed some techniques from the films to achieve a free exchange of space and time, that is, to make Willy's remembrance and illusion audio-visual, making the play characteristic of the films.

On the Symbolism in Arthur Miller’s Death of a Salesman

Death of a Salesman is the greatest masterpiece of Arthur Miller.The play is about a common man trying to reach his American dream and the disillusionment of his dream.This paper mainly analyzes the symbols that are used in the play to embody the theme.

Improved ant colony optimization algorithm for the traveling salesman problems

Ant colony optimization （ACO） is a new heuristic algo- rithm which has been proven a successful technique and applied to a number of combinatorial optimization problems. The traveling salesman problem （TSP） is among the most important combinato- rial problems. An ACO algorithm based on scout characteristic is proposed for solving the stagnation behavior and premature con- vergence problem of the basic ACO algorithm on TSP. The main idea is to partition artificial ants into two groups： scout ants and common ants. The common ants work according to the search manner of basic ant colony algorithm, but scout ants have some differences from common ants, they calculate each route＇s muta- tion probability of the current optimal solution using path evaluation model and search around the optimal solution according to the mutation probability. Simulation on TSP shows that the improved algorithm has high efficiency and robustness.

“Alone” VS. “Lonely”:Eudora Welty's Idyllic View on Life inDeath of a Traveling Salesman

In Death of a Traveling Salesman,Eudora Welty's idyllic view on life in the countryside,her nostalgic feelings for the loss of the American Southern tradition,and her protest against the industrialization of the city life are demonstrated by compar ing the"alone"of the couple in the countryside and the"lonely"of R. J. Bowman as a traveling salesman in the city.

A Hybrid Genetic Algorithm for the Traveling Salesman Problem with Pickup and Delivery

In this paper, a hybrid genetic algorithm （GA） is proposed for the traveling salesman problem （TSP） with pickup and delivery （TSPPD）. In our algorithm, a novel pheromone-based crossover operator is advanced that utilizes both local and global information to construct offspring. In addition, a local search procedure is integrated into the GA to accelerate convergence. The proposed GA has been tested on benchmark instances, and the computational results show that it gives better convergence than existing heuristics.

Finding the Minimum Ratio Traveling Salesman Tour by Artificial Ants

Ants of artificial colony are able to generate good solutions to the famous traveling salesman problem (TSP). We propose an artificial ants algorithm for solving the minimum ratio TSP, which is more general than the standard TSP in combinatorial optimization area. In the minimum ratio TSP, another criterion concerning each edge is added, that is, the traveling salesman can have a benefit if he travels from one city to another. The objective is to minimize the ratio between total costs or distances and total benefits. The idea of this type of optimization is in some sense quite similar to that of traditional cost-benefit analysis in management science. Computational results substantiate the solution quality and efficiency of the algorithm.

Collision-free Scheduling of Multi-bridge Machining Systems： A Colored Traveling Salesman Problem-based Approach

Multi-bridge machining systems(MBMS) have gained wide applications in industry due to their high production capacity and efficiency. They contain multiple bridge machines working in parallel within their partially overlapping workspaces.Their scheduling problems can be abstracted into a serial-colored travelling salesman problem in which each salesman has some exclusive cities and some cities shared with its neighbor(s). To solve it, we develop a greedy algorithm that selects a neighboring city satisfying proximity. The algorithm allows a salesman to select randomly its shared cities and runs accordingly many times. It can thus be used to solve job scheduling problems for MBMS. Subsequently, a collision-free scheduling method is proposed to address both job scheduling and collision resolution issues of MBMS. It is an extension of the greedy algorithm by introducing time window constraints and a collision resolution mechanism. Thus, the augmented greedy algorithm can try its best to select stepwise a job for an individual machine such that no time overlaps exist between it and the job sequence of the neighboring machine dealt in the corresponding overlapping workspace; and remove such a time overlap only when it is inevitable. Finally, we conduct a case study of a large triplebridge waterjet cutting system by applying the proposed method.

Tabu Search Implementation on Traveling Salesman Problem and Its Variations: A Literature Survey

The Traveling Salesman Problem (TSP) and its allied problems like Vehicle Routing Problem (VRP) are one of the most widely studied problems in combinatorial optimization. It has long been known to be NP-hard and hence research on developing algorithms for the TSP has focused on approximate methods in addition to exact methods. Tabu search is one of the most widely applied metaheuristic for solving the TSP. In this paper, we review the tabu search literature on the TSP and its variations, point out trends in it, and bring out some interesting research gaps in this literature.

Traveling Salesman Problem Using an Enhanced Hybrid Swarm Optimization Algorithm

The traveling salesman problem( TSP) is a well-known combinatorial optimization problem as well as an NP-complete problem. A dynamic multi-swarm particle swarm optimization and ant colony optimization( DMPSO-ACO) was presented for TSP.The DMPSO-ACO combined the exploration capabilities of the dynamic multi-swarm particle swarm optimizer( DMPSO) and the stochastic exploitation of the ant colony optimization( ACO) for solving the traveling salesman problem. In the proposed hybrid algorithm,firstly,the dynamic swarms,rapidity of the PSO was used to obtain a series of sub-optimal solutions through certain iterative times for adjusting the initial allocation of pheromone in ACO. Secondly,the positive feedback and high accuracy of the ACO were employed to solving whole problem. Finally,to verify the effectiveness and efficiency of the proposed hybrid algorithm,various scale benchmark problems were tested to demonstrate the potential of the proposed DMPSO-ACO algorithm. The results show that DMPSO-ACO is better in the search precision,convergence property and has strong ability to escape from the local sub-optima when compared with several other peer algorithms.

ISPO: A New Way to Solve Traveling Salesman Problem

This paper first introduces the concepts of mobile operators and mobile sequence, with which it redefines the rate of particle swarm optimization algorithm and the formula of position updating. Combining this discrete PSO algorithm with neighbors, the paper puts forward Hybrd Particle Swarm Optimization Algorithm, whose effectiveness is verified at the end of this paper.

Fast Algorithm for the Travelling Salesman Problem and the Proof of P = NP

In the theory of computational complexity, the travelling salesman problem is a typical one in the NP class. With the aid of a brand-new approach named “maximum-deleting method”, a fast algorithm is constructed for it with a polynomial time of biquadrate, which greatly reduces the computational complexity. Since this problem is also NP-complete, as a corollary, P = NP is proved to be true. It indicates the crack of the well-known open problem named “P versus NP”.

The Quantum Approximate Algorithm for Solving Traveling Salesman Problem

The Quantum Approximate Optimization Algorithm(QAOA)is an algorithmic framework for finding approximate solutions to combinatorial optimization problems.It consists of interleaved unitary transformations induced by two operators labelled the mixing and problem Hamiltonians.To fit this framework,one needs to transform the original problem into a suitable form and embed it into these two Hamiltonians.In this paper,for the well-known NP-hard Traveling Salesman Problem(TSP),we encode its constraints into the mixing Hamiltonian rather than the conventional approach of adding penalty terms to the problem Hamiltonian.Moreover,we map edges(routes)connecting each pair of cities to qubits,which decreases the search space significantly in comparison to other approaches.As a result,our method can achieve a higher probability for the shortest round-trip route with only half the number of qubits consumed compared to IBM Q’s approach.We argue the formalization approach presented in this paper would lead to a generalized framework for finding,in the context of QAOA,high-quality approximate solutions to NP optimization problems.

Applying the Method for Solving Traveling Salesman Problem Based on Backtracking Algorithm to Order Picking

In the distribution center, the way of order picking personnel to pick goods has two kinds: single picking and batch picking. Based on the way of the single picking and assumed warehouse model, in order to reduce the walking path of order picking, the order picking problem is transformed into the traveling salesman problem in this paper. Based on backtracking algorithm, the order picking path gets optimized. Finally verifing the optimization method under the environment of VC++6.0, order picking path in the warehouse model get optimized, and compared with the traditional order picking walking paths. The results show that in small and medium-sized warehouse, the optimization method proposed in this paper can reduce order picking walking path and improve the work efficiency as well as reduce the time cost.

Death of a Salesman and American Dream

American Dream refers to the belief of the majority of Americans.Through hard work,thrift,and determination,they can surely achieve a better life and live a comfortable life.Such an American Dream has a profound influence on several generations of Americans.American Dream has a close relationship with people' s working life and cultural activities.Death of a Salesman is a typical play about the pursuit and the illusionary of the Dream follower.In Death of a Salesman,the American Dream is the main hero as well as Willy Loman' s spiritual pillar.The pursuit,the realization,and the disillusionment of the American Dream develop with Willy and his family.