Traveling Salesman Problem Optimization Research Articles

On approximating a new generalization of traveling salesman problem

The current best-known performance guarantees for the extensively studied Traveling Salesman Problem (TSP) of determinate approximation algorithms is 32, achieved by Christofides' algorithm 47 years ago. This paper investigates a new generalization problem of the TSP, termed the Minimum-Cost Bounded Degree Connected Subgraph (MBDCS) problem. In the MBDCS problem, the goal is to identify a minimum-cost connected subgraph containing n=|V| edges from an input graph G=(V,E) with degree upper bounds for particular vertices. We show that for certain special cases of MBDCS, the aim is equivalent to finding a minimum-cost Hamiltonian cycle for the input graph, same as the TSP. To appropriately solve MBDCS, we initially present an integer programming formulation for the problem. Subsequently, we propose an algorithm to approximate the optimal solution by applying the iterative rounding technique to solution of the integer programming relaxation. We demonstrate that the returned subgraph of our proposed algorithm is one of the best guarantees for the MBDCS problem in polynomial time, assuming P≠NP. This study views the optimization of TSP as finding a minimum-cost connected subgraph containing n edges with degree upper bounds for certain vertices, and it may provide new insights into optimizing the TSP in future research.

Traveling Salesman Problem of optimal debris removal sequence using non-population gradient search

A new approach for debris removal sequence optimization with multiple spacecraft is developed and tested. The motion planning of the multiple rendezvous involves complex Traveling Salesman Problem (TSP) combinatorial decision, in which both the continuous parameters of the orbital maneuvers and integer indices of the debris are determined. Expected value and variance of trajectory conditioned on preceding visit, not the state of target itself, are propagated and analyzed. With the metric, likelihood of expected trajectory can be given, and the mixed integer problem is reformulated into continuous parameter estimation with maximum likelihood. A non-population gradient based optimizer is then used to search the target sequence. The method of TSP optimization is compared to the continuous parameter estimator of Extended Kalman Filter. Using this method, the classic static TSP-LIB benchmarks with up to 100 nodes are tested. We then extend the method to the TSP with space dynamics. Effectiveness and efficiency of the proposed method are tested by a sequential visit of near-earth debris considering J2 effects with a single and multiple spacecrafts.

Comparative Study of Variations in Quantum Approximate Optimization Algorithms for the Traveling Salesman Problem.

The traveling salesman problem (TSP) is one of the most often-used NP-hard problems in computer science to study the effectiveness of computing models and hardware platforms. In this regard, it is also heavily used as a vehicle to study the feasibility of the quantum computing paradigm for this class of problems. In this paper, we tackle the TSP using the quantum approximate optimization algorithm (QAOA) approach by formulating it as an optimization problem. By adopting an improved qubit encoding strategy and a layer-wise learning optimization protocol, we present numerical results obtained from the gate-based digital quantum simulator, specifically targeting TSP instances with 3, 4, and 5 cities. We focus on the evaluations of three distinctive QAOA mixer designs, considering their performances in terms of numerical accuracy and optimization cost. Notably, we find that a well-balanced QAOA mixer design exhibits more promising potential for gate-based simulators and realistic quantum devices in the long run, an observation further supported by our noise model simulations. Furthermore, we investigate the sensitivity of the simulations to the TSP graph. Overall, our simulation results show that the digital quantum simulation of problem-inspired ansatz is a successful candidate for finding optimal TSP solutions.

Statistical Modeling of Quartiles, Standard Deviation, and Buffer Time Index of Optimal Tour in Traveling Salesman Problem and Implications for Travel Time Reliability

The traveling salesman problem (TSP) plays an important role in the field of transportation and logistics. While most studies focus on developing algorithms to find the shortest path and explore the average length of the shortest paths, the degree to which the shortest path deviates from its mean has not been studied. The study of deviation is important because it has implications for travel time reliability. Previous studies have used various indicators to measure this deviation, mainly including standard deviation, quantiles, and buffer time index (BTI). Therefore, this study aims to develop an empirical model to estimate the standard deviation, quantiles, and BTI for the optimal TSP tour. Experiments are performed to find the shortest path connecting N customers, which are generated randomly in a specified service area, using a genetic algorithm. The service area is a rectangle with ratio of length to width ranging from 1:1 to 8:1. Two types of lengths are considered: Euclidean and Manhattan. The number of customers considered ranges from 10 to 100 with intervals of 10. In the experimental design, the customers are generated randomly 500 times. The quartiles and standard deviations of the 500 shortest paths are recorded. The BTI is also calculated. Regression models are developed to estimate quartiles, standard deviation, and BTI using number of customers and parameters of service area as predictive variables. The models perform well on the testing data set. The constructed models can be used to estimate the standard deviation and reliability of travel time.

Optimization of Traveling Salesman Problem on Scheduling Tour Packages using Genetic Algorithms

Ciayumajakuning or often referred to as Cirebon, Indramayu, Majalengka and Kuningan, is located on the east side of West Java. In supporting the regional economy, the West Java government program, especially the community, is focused on the tourism sector. the short time of tourists and the number of tourist destinations that want to be visited, make tourists have to schedule their tour as effectively as possible. By using the Traveling Salesman Problem optimization can minimize the distance of tourist visits by considering the availability of time. The solution to this problem is by combining Traveling Salesman Problems and Genetic Algorithms that can represent candidates for scheduling solutions into chromosomes randomly, then evaluate using the fitness function and then do the selection. The selection method used in this study is the roulette wheel selection method, then crossovers and mutations. In each generation, chromosomes are evaluated based on the value of the fitness function. After several generations the genetic algorithm will produce the best chromosome, which is the optimal solution. this study Code A = Starting Point B = Buddhist Dayak Tribe Segandu Earth, Indramayu Code C = Mangrove Forest Sustainable Beach Karangsong Code D = Heritage Museum Kasepuhan Palace Cirebon Code E = Keraton Kacirebonan Kode F = Goa Sunyaragi Cirebon Code G = Starting Point. The Total Distance to be Passed is 17,706 Km

Conceptual Framework for Finding Approximations to Minimum Weight Triangulation and Traveling Salesman Problem of Planar Point Sets

We introduce a novel Conceptual Framework for finding approximations to both Minimum Weight Triangulation (MWT) and optimal Traveling Salesman Problem (TSP) of planar point sets. MWT is a classical problem of Computational Geometry with various applications, whereas TSP is perhaps the most researched problem in Combinatorial Optimization. We provide motivation for our research and introduce the fields of triangulation and polygonization of planar point sets as theoretical bases of our approach, namely, we present the Isoperimetric Inequality principle, measured via Compactness Index, as a key link between our two stated problems. Our experiments show that the proposed framework yields tight approximations for both problems.

Continuous relaxations for the traveling salesman problem

In this work, we aim to explore connections between dynamical systems techniques and combinatorial optimization problems. In particular, we construct heuristic approaches for the traveling salesman problem (TSP) based on embedding the relaxed discrete optimization problem into appropriate manifolds. We explore multiple embedding techniques -- namely, the construction of new dynamical systems on the manifold of orthogonal matrices and associated Procrustes approximations of the TSP cost function. Using these dynamical systems, we analyze the local neighborhood around the optimal TSP solutions (which are equilibria) using computations to approximate the associated \emph{stable manifolds}. We find that these flows frequently converge to undesirable equilibria. However, the solutions of the dynamical systems and the associated Procrustes approximation provide an interesting biasing approach for the popular Lin--Kernighan heuristic which yields fast convergence. The Lin--Kernighan heuristic is typically based on the computation of edges that have a `high probability' of being in the shortest tour, thereby effectively pruning the search space. Our new approach, instead, relies on a natural relaxation of the combinatorial optimization problem to the manifold of orthogonal matrices and the subsequent use of this solution to bias the Lin--Kernighan heuristic. Although the initial cost of computing these edges using the Procrustes solution is higher than existing methods, we find that the Procrustes solution, when coupled with a homotopy computation, contains valuable information regarding the optimal edges. We explore the Procrustes based approach on several TSP instances and find that our approach often requires fewer $k$-opt moves than existing approaches. Broadly, we hope that this work initiates more work in the intersection of dynamical systems theory and combinatorial optimization.

A variable neighborhood search for flying sidekick traveling salesman problem

AbstractThe efficiency and dynamism of unmanned aerial vehicles, or drones, have presented substantial application opportunities in several industries in the last years. Notably, logistic companies have given close attention to these vehicles to reduce delivery time and operational cost. A variant of the traveling salesman problem (TSP), called the flying sidekick traveling salesman problem, was introduced involving drone‐assisted parcel delivery. The drone launches from the truck, proceeds to deliver parcels to a customer, and then is recovered by the truck at a third location. While the drone travels through a trip, the truck delivers parcels to other customers as long as the drone has enough battery to hover waiting for the truck. This work proposes a hybrid heuristic where the initial solution is created from the optimal TSP solution reached by a TSP solver. Next, an implementation of the general variable neighborhood search is employed to obtain the delivery routes of truck and drone. Computational experiments show the potential of the algorithm to improve significantly delivery time. Furthermore, we provide a new set of instances based on the well‐known traveling salesman problem library instances.

PENGARUH INISIALISASI POPULASI RANDOM SEARCH PADA ALGORITMA BEREVOLUSI DALAM OPTIMASI TRAVELLING SALESMAN PROBLEM (TSP)

<p>Traveling Salesman Problem (TSP) is a problem where each initial route of departure and return path between regions remains the same. The problem with TSP is how to get the optimum results to get the shortest path that will be passed, to solve TSP problems, one way can be by using evolved algorithms. Evolution Algorithm (AE) is a method that uses natural selection as the main idea in solving a particular problem. This algorithm is implemented through computer simulations starting from the individual population that will be randomly generated and then evaluated to achieve the best solution. Random search is included in part of stochastic optimization and global optimization. Random search is a direct search method that does not require derivatives to search for continue domains. The purpose of this study was to see the effect of population initialization with random search on algorithms evolving in TSP optimization. Results of initialization strategy implementation Random search has shown more optimal results compared to pure initialization without using random search. This is due to the initialization searching for space for random initialization.</p>

On the min-cost Traveling Salesman Problem with Drone

Over the past few years, unmanned aerial vehicles (UAV), also known as drones, have been adopted as part of a new logistic method in the commercial sector called “last-mile delivery”. In this novel approach, they are deployed alongside trucks to deliver goods to customers to improve the quality of service and reduce the transportation cost. This approach gives rise to a new variant of the traveling salesman problem (TSP), called TSP with drone (TSP-D). A variant of this problem that aims to minimize the time at which truck and drone finish the service (or, in other words, to maximize the quality of service) was studied in the work of Murray and Chu (2015). In contrast, this paper considers a new variant of TSP-D in which the objective is to minimize operational costs including total transportation cost and one created by waste time a vehicle has to wait for the other. The problem is first formulated mathematically. Then, two algorithms are proposed for the solution. The first algorithm (TSP-LS) was adapted from the approach proposed by Murray and Chu (2015), in which an optimal TSP solution is converted to a feasible TSP-D solution by local searches. The second algorithm, a Greedy Randomized Adaptive Search Procedure (GRASP), is based on a new split procedure that optimally splits any TSP tour into a TSP-D solution. After a TSP-D solution has been generated, it is then improved through local search operators. Numerical results obtained on various instances of both objective functions with different sizes and characteristics are presented. The results show that GRASP outperforms TSP-LS in terms of solution quality under an acceptable running time.

Comments on: Continuous approximation models in freight distribution management

The paper presents a review of the use of continuous approximation modeling for freight distribution management. The authors take a unique approach to reviewing continuous approximation models in the freight distribution literature. They anchor their review around an in-depth analysis of two fundamental results upon which much of the continuous approximation literature is built: the asymptotic approximation formula for the optimal traveling salesman problem (TSP) tour length from Beardwood et al. (1959) and the strip strategy to build implementable TSP tours from Daganzo (1984). Whereas other continuous approximation survey papers focus primarily on application areas and the resulting advances in methodology [e.g., Langevin et al. (1996) for the period up to 1996 and Ansari et al. (2017) for the period from 1996 to 2016], this paper structures the review in the context of these two seminal contributions. As such, the authors create a valuable narrative of the evolution of continuous approximation models.

Capacitated Vehicle Routing Problem Solving using Adaptive Sweep and Velocity Tentative PSO

Vehicle Routing Problem (VRP) has become an integral part in logistic operations which determines optimal routes for several vehicles to serve customers. The basic version of VRP is Capacitated VRP (CVRP) which considers equal capacities for all vehicles. The objective of CVRP is to minimize the total traveling distance of all vehicles to serve all the customers. Various methods are used to solve CVRP, among them the most popular way is splitting the task into two different phases: assigning customers under different vehicles and then finding optimal route of each vehicle. Sweep clustering algorithm is well studied for clustering nodes. On the other hand, route optimization is simply a traveling salesman problem (TSP) and a number of TSP optimization methods are applied for this purpose. In Sweep, cluster formation staring angle is identified as an element of CVRP performance. In this study, a heuristic approach is developed to identify appropriate starting angle in Sweep clustering. The proposed heuristic approach considers angle difference of consecutive nodes and distance between the nodes as well as distances from the depot. On the other hand, velocity tentative particle swarm optimization (VTPSO), the most recent TSP method, is considered for route optimization. Finally, proposed adaptive Sweep (i.e., Sweep with proposed heuristic) plus VTPSO is tested on a large number of benchmark CVRP problems and is revealed as an effective CVRP solving method while outcomes compared with other prominent methods.

APPROXIMATING ASYMMETRIC TSP IN EXPONENTIAL TIME

Let G be a complete directed graph with n vertices and integer edge weights in range [0,M]. It is well known that an optimal Traveling Salesman Problem (TSP) in G can be solved in 2n time and space (all bounds are given within a polynomial factor of the input length, i.e., poly(n, log M)) and this is still the fastest known algorithm. If we allow a polynomial space only, then the best known algorithm has running time 4nnlog n. For TSP with bounded weights there is an algorithm with 1.657n · M running time. It is a big challenge to develop an algorithm with 2n time and polynomial space. Also, it is well-known that TSP cannot be approximated within any polynomial time computable function unless P=NP. In this short note we propose a very simple algorithm that, for any 0 < ε < 1, finds (1+ε)-approximation to asymmetric TSP in 2nε−1 time and ε−1 · poly(n, log M) space. Thereby, for any fixed ε, the algorithm needs 2n steps and polynomial space to compute (1 + ε)-approximation.

Improved Ant Colony Algorithm for Optimal Solution TSP

Based on the ant colony algorithm analysis and research, this paper proposed an improved ant colony algorithm. Through updating pheromone and optimal search strategy, then applied to the Traveling Salesman Problem (TSP), effectively improved the searching capability of the algorithm. Finally through the simulation testing and analysis, verified that the improved ant colony algorithm is effective, and has good performance.

On the asymptotic behavior of subtour-patching heuristics in solving the TSP on permuted Monge matrices

We examine the performance of different subtour-patching heuristics for solving the strongly $\mathcal{NP}$ -hard traveling salesman problem (TSP) on permuted Monge matrices. We prove that a well-known heuristic is asymptotically optimal for the TSP on product matrices and k-root cost matrices. We also show that the heuristic is provably asymptotically optimal for general permuted Monge matrices under some mild conditions. Our theoretical results are strongly supported by the findings of a large-scale experimental study on randomly generated numerical examples, which show that the heuristic is not only asymptotically optimal, but also finds optimal TSP tours with high probability that increases with the problem size. Thus the heuristic represents a practical tool to solve large instances of the problem.

Routing for Relief Efforts

In the aftermath of a large disaster, the routing of vehicles carrying critical supplies can greatly impact the arrival times to those in need. Because it is critical that the deliveries are both fast and fair to those being served, it is not clear that the classic cost-minimizing routing problems properly reflect the relevant priorities in disaster relief. In this paper, we take the first steps toward developing new methodologies for these problems. We focus specifically on two alternative objective functions for the traveling salesman problem (TSP) and the vehicle routing problem (VRP): one that minimizes the maximum arrival time (minmax) and one that minimizes the average arrival time (minavg). To demonstrate the potential impact of using these new objective functions, we bound the worst-case performance of optimal TSP solutions with respect to these new variants and extend these bounds to include multiple vehicles and vehicle capacity. Similarly, we examine the potential increase in routing costs that results from using these alternate objectives. We present solution approaches for these two variants of the TSP and VRP, which are based on well-known insertion and local search techniques. These are used in a series of computational experiments to help identify the types of instances in which TSP and VRP solutions can be significantly different from optimal minmax and minavg solutions.