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Solution of a Large-Scale Traveling Salesman Problem

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Abstract

It is shown that a certain tour of 49 cities, one in each of the 48 states and Washington, D.C., has the shortest road distance. Operations Research, ISSN 0030-364X, was published as Journal of the Operations Research Society of America from 1952 to 1955 under ISSN 0096-3984.

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... One such challenge is the Nurse Rostering Problem (NRP), where the allocation of nursing staff is vital for both patient care and operational efficiency. In oncology departments, efficient nurse scheduling is essential to ensure continuous, high-quality care [1][2][3][4]. ...
... Early efforts to address NRP primarily involved heuristic methods and basic mathematical programming [2,16]. These methods, although foundational, were eventually replaced by more sophisticated optimisation techniques, particularly Mixed Integer Linear Programming (MILP). ...
... LP is a methodological cornerstone in OR that focuses on maximising or minimising a linear objective function under a set of linear constraints [2]. LP's foundational role is pivotal in addressing a wide array of optimisation challenges across diverse fields, from resource allocation to financial planning, providing a structured approach to decisionmaking [18]. ...
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Background/Objectives: This study addresses the complex challenge of Nurse Rostering (NR) in oncology departments, a critical component of healthcare management affecting operational efficiency and patient care quality. Given the intricate dynamics of healthcare settings, particularly in oncology clinics, where patient needs are acute and unpredictable, optimizing nurse schedules is paramount for enhancing care delivery and staff satisfaction. Methods: Employing advanced Integer Programming (IP) techniques, this research develops a comprehensive model to optimise NR. The methodology integrates a variety of constraints, including legal work hours, staff qualifications, and personal preferences, to generate equitable and efficient schedules. Through a case study approach, the model’s implementation is explored within a clinical setting, demonstrating its practical application and adaptability to real-world challenges. Results: The implementation of the IP model in a clinical setting revealed significant improvements in scheduling efficiency and staff satisfaction. The model successfully balanced workload distribution among nurses, accommodated individual preferences to a high degree, and ensured compliance with work-hour regulations, leading to optimised shift schedules that support both staff well-being and patient care standards. Conclusions: The findings underscore the effectiveness of IP in addressing the complexities of NR in oncology clinics. By facilitating a strategic allocation of nursing resources, the proposed model contributes to operational excellence in healthcare settings, underscoring the potential of Operations Research in enhancing healthcare delivery and management practices.
... Letting L denote the value of the lower bound and U the length of a known TSP tour, then ≡ U −L ≥ 0 is called the integrality gap with respect to the relaxation and tour. An important observation of Dantzig et al. [6] is that if a variable x e has reduced cost in the LP dual solution greater than , then x e can be eliminated from the problem, since the corresponding edge e cannot be included in any optimal TSP tour. Similarly, if the reduced cost of x e is less than − , then the value of x e can be fixed to 1, since edge e must be present in every optimal tour. ...
... Similarly, if the reduced cost of x e is less than − , then the value of x e can be fixed to 1, since edge e must be present in every optimal tour. Concerning this, Dantzig et al. [6] write the following, where E is used in place of . ...
... A long-term goal of this line of research is to build edge elimination into a tool that can enhance the TSP cutting-plane method, following general ideas outlined in Dantzig et al. [6], [7]. The target is to extend the range of current exact solvers, such as the Concorde code. ...
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Hougardy and Schroeder (WG 2014) proposed a combinatorial technique for pruning the search space in the traveling salesman problem, establishing that, for a given instance, certain edges cannot be present in any optimal tour. We describe an implementation of their technique, employing an exact TSP solver to locate k-opt moves in the elimination process. In our computational study, we combine LP reduced-cost elimination together with the new combinatorial algorithm. We report results on a set of geometric instances, with the number of points n ranging from 3038 up to 115,475. The test set includes all TSPLIB instances having at least 3000 points, together with 250 randomly generated instances, each with 10,000 points, and three currently unsolved instances having 100,000 or more points. In all but two of the test instances, the complete-graph edge sets were reduced to under 3n edges. For the three large unsolved instances, repeated runs of the elimination process reduced the graphs to under 2.5n edges.
... The solution is to find a route for them, such that each location is visited exactly once, and the overall cost of travel (or distance) is minimized. Using Dantzig-Fulkerson-Johnson formulation [11], we have: ...
... Then, the TSP can be modeled with the following integer linear programming problem. [11] The problem can also be modeled with a graph ( , , ) with V being the set of customer locations, and E being roads connecting any 2 locations, with ( , ) being the cost to travel between locations i and j. ...
... This heuristic is more efficient than the previous ones because it significantly reduces the search space and thus speeds up the performance. [11] This table shows the experimental results of the NN, C-S, 2-Opt, 3-Opt and the Lin-Kernighan heuristics, measured in average % improvements from the Held-Karp theoretical lower bound of the TSP problem solution. We can see that the NN method performs the worst on average. ...
Thesis
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Route optimization has been historically an important topic for logistics and supply chain design. Because routing problems are usually NP-hard, numerous route optimization heuristics, approximation algorithms and metaheuristics are designed to find near-optimal solutions efficiently. Each kind of solution is associated with its own constraints and challenges, and the academia has also been looking for ML-based methods to improve on these optimization solvers. In this paper, I will present a study of route optimization problems, specifically the Traveling Salesperson and the Vehicle Routing Problem, with their variations, the algorithms and heuristics used to solve the problem throughout history, and analyses of the methods. Next, I will analyze the metaheuristics used to find better solutions for the routing problems, and present a study of how current Machine Learning models can improve the design of metaheuristic solvers to optimization problems. Lastly, I will also present a case study analysis of the route planning problem and optimization model used by the supply chain department of AmeriGas, the largest propane distributor in the United States. Using the knowledge from the previous parts, I will try to analyze AmeriGas' business model, staffing and route optimization model. I will identify challenges faced by AmeriGas and raise some suggestions on improving its staffing and route planning model.
... The Traveling Salesman Problem (TSP) is one of the most well-known combinatorial problems in optimization, where the objective is to find the shortest path that visits all the cities in a set exactly once and returns to the origin city [4]. Formally, the TSP can be represented by a complete graph G = (V, E), where V is the set of vertices (cities) and E is the set of edges with associated weights (distances or costs between cities). ...
... Algorithm 1: PSO for the Traveling Salesman Problem Data: Graph G = (V, E), number of particles n, maximum iterations maxIter Result: Best solution found for the TSP 1 Initialize n particles with random city sequences; 2 Calculate the cost of each particle using the objective function; 3 Set the initial position as pBest for each particle; 4 Identify the global best solution gBest; 5 for t ← 1 to maxIter do 6 for each particle i in the swarm do 7 Update the particle's velocity: ...
Preprint
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The Traveling Salesman Problem (TSP) is a well-known combinatorial optimization problem that aims to find the shortest possible route that visits each city exactly once and returns to the starting point. This paper explores the application of Particle Swarm Optimization (PSO), a population-based optimization algorithm, to solve TSP. Although PSO was originally designed for continuous optimization problems, this work adapts PSO for the discrete nature of TSP by treating the order of cities as a permutation. A local search strategy, including 2-opt and 3-opt techniques, is applied to improve the solution after updating the particle positions. The performance of the proposed PSO algorithm is evaluated using benchmark TSP instances and compared to other popular optimization algorithms, such as Genetic Algorithms (GA) and Simulated Annealing (SA). Results show that PSO performs well for small to medium-sized problems, though its performance diminishes for larger instances due to difficulties in escaping local optima. This paper concludes that PSO is a promising approach for solving TSP, with potential for further improvement through hybridization with other optimization techniques.
... The VRP was first studied by Dantzig et al. in 1954, marking a significant milestone in the field of operational research [34]. Subsequently, Clarke and Wright [4] expanded on Dantzig and Ramser's method by introducing the classical saving method, which became widely adopted in logistics optimization. ...
... The VRP was first studied by Dantzig et al. in 1954, marking a significant milestone in the field of operational research [34]. Subsequently, Clarke and Wright [4] expanded on Dantzig and Ramser's method by introducing the classical saving method, which became widely adopted in logistics optimization. ...
Article
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The vehicle routing problem (VRP) is a crucial group of transportation problems, and traditional capacitated VRP (CVRP) directly handles external logistics with a homogeneous vehicle fleet. This paper examines the CVRP from an in-plant logistic view with a heterogeneous fleet in an automotive factory. The homogeneous and heterogeneous vehicle fleets are compared to address the factory's actual in-plant logistics issues. Firstly, simulated annealing (SA), tabu search (TS) algorithms and mathematical modeling were utilized. Then, a hybrid approach was proposed, and all suggested meta-heuristic algorithms were evaluated for homogeneous and heterogeneous vehicle fleets. According to the results, the reduction rates using CVRP with the heterogeneous fleet are 17% and 36% for fleet area and distribution costs, respectively. Also, to examine the effect of the hybrid algorithm parameters on the result, traveling distance was calculated for different scenarios, and multiple regression analyses were applied. According to the multiple regression analysis, the hybrid algorithm's most affected parameter was the cooling coefficient.
... This was recently improved to 3/2 − 10 −34 by Gurvits, Klein and Leake [GKL24]. Like many algorithms for Metric-TSP, their method exploits an optimal solution to the following linear program (LP) relaxation, also known as the Subtour Elimination LP [DFJ54]: ...
... Importance of solving the Subtour Elimination LP Apart from being a crucial component in the breakthrough result of [KKOG21], the importance of solving (1) has been recognized since the dawn of mathematical programming. It inspired the cutting plane method, introduced by Dantzig, Fulkerson and Johnson [DFJ54] to solve TSP exactly. Applegate, Bixby, Chvátal and Cook [ABCC03] implemented their method into the Concorde solver, which is capable of solving very large real-world instances. ...
Preprint
We present a nearly linear work parallel algorithm for approximating the Held-Karp bound for the Metric TSP problem. Given an edge-weighted undirected graph G=(V,E) on m edges and ϵ>0\epsilon>0, it returns a (1+ϵ)(1+\epsilon)-approximation to the Held-Karp bound with high probability, in O~(m/ϵ4)\tilde{O}(m/\epsilon^4) work and O~(1/ϵ4)\tilde{O}(1/\epsilon^4) depth. While a nearly linear time sequential algorithm was known for almost a decade (Chekuri and Quanrud'17), it was not known how to simultaneously achieve nearly linear work alongside polylogarithmic depth. Using a reduction by Chalermsook et al.'22, we also give a parallel algorithm for computing a (1+ϵ)(1+\epsilon)-approximate fractional solution to the k-edge-connected spanning subgraph (kECSS) problem, with the same complexity. To obtain these results, we introduce a notion of core-sequences for the parallel Multiplicative Weights Update (MWU) framework (Luby-Nisan'93, Young'01). For the Metric TSP and kECSS problems, core-sequences enable us to exploit the structure of approximate minimum cuts to reduce the cost per iteration and/or the number of iterations. The acceleration technique via core-sequences is generic and of independent interest. In particular, it improves the best-known iteration complexity of MWU algorithms for packing/covering LPs from poly(lognnz(A))poly(\log nnz(A)) to polylogarithmic in the product of cardinalities of the core-sequence sets where A is the constraint matrix of the LP. For certain implicitly defined LPs such as the kECSS LP, this yields an exponential improvement in depth.
... The methods used to solve TSP can be examined in three groups as exact, heuristic, and metaheuristic methods. Examples of exact methods contain dynamic programming [1], branch-andbound [2], branch-and-cut [3], cutting plane [4], and LP relaxation [5]. Exact methods are used to resolve smallsized TSPs and produce solutions within a reasonable amount of time. ...
... Each node represents a city, while each edge represents the path between the two cities. The mathematical model of TSP can be presented as follows [5]: ...
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The traveling salesman problem (TSP) is a frequently studied problem by researchers today and belongs to the class of combinatorial optimization problems. It can be used to solve many current world problems such as scheduling, circuit design, layout design of plants in factories, route planning and printed circuit design. Therefore, researchers working in the field of optimization methods use it as a realistic test environment and evaluate the performance of new algorithms on it. In this study, a discrete version of the orca predation algorithm (OPA) was developed, called DOPA. DOPA is a novel discrete and permutation-coded optimization algorithm. As in OPA, it consists of two phases: chasing and attacking. However, these phases in DOPA were arranged to be able to solve a combinatorial optimization problem such as TSP. In the chasing phase, the distances between the orcas were calculated with the Hamming distance, and the speed values were obtained by using these distances. The positions of the orcas were updated using the speed values and 2-opt algorithm. In the attacking phase, the positions of the orcas were calculated by order crossover (OX1) operator. In the position adjustment procedure, swap local search operator was used. Thus, the convergence of the algorithm was accelerated. DOPA has much less parameters than OPA. Therefore, it is a simple and effective algorithm with few parameters. The parameters of DOPA were optimized by the Taguchi statistical method. It was tested on 67 well-known TSP instances. The results of different performance measures and tests showed that DOPA is a highly competitive and alternative method compared to other methods.
... Dantzig-Fulkerson-Johnson formulation. The classical formulation (Dantzig et al., 1954) used in most OR textbooks implies binary variables x ij , i < j, for all edges {i, j} ∈ E, indicating that an edge is traversed by the tour: ...
... Note that Equation (9) for subsets S of size 2 is equivalent to the upper bound of the variables. In the original formulation by Dantzig et al. (1954) given above, we get redundant constraints if N is even, because for every set S with |S| = N 2 we have a constraint corresponding to S and a constraint corresponding to V \ S. If we remove the redundant constraints, e.g., by only taking subsets of size N 2 containing node 1, we are left with 2 N −1 − 1 − N − N (N −1) 2 subtour elimination constraints. In order to create an equivalent QUBO formulation, one needs to incorporate constraints into the objective function as penalty terms, involving binary variables only. ...
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Quantum optimization has emerged as a promising frontier of quantum computing, providing novel numerical approaches to mathematical optimization problems. The main goal of this paper is to facilitate interdisciplinary research between the Operations Research (OR) and Quantum Computing communities by providing an OR scientist's perspective on selected quantum-powered methods for discrete optimization. To this end, we consider three quantum-powered optimization approaches that make use of different types of quantum hardware available on the market. To illustrate these approaches, we solve three classical optimization problems: the Traveling Salesperson Problem, Weighted Maximum Cut, and Maximum Independent Set. With a general OR audience in mind, we attempt to provide an intuition behind each approach along with key references, describe the corresponding high-level workflow, and highlight crucial practical considerations. In particular, we emphasize the importance of problem formulations and device-specific configurations, and their impact on the amount of resources required for computation (where we focus on the number of qubits). These points are illustrated with a series of experiments on three types of quantum computers: a neutral atom machine from QuEra, a quantum annealer from D-Wave, and a gate-based device from IBM.
... In existing literature, exact and metaheuristic algorithms are two approaches to tackle TSP as shown in table 1. In case of exact algorithms, following are major exact algorithms in literature introduced to encounter with TSP; Dantzig et al. [19] introduced a methodology to solve the large-scale TSP. Later, Petberg [20] proposed a branch and cut method to get the optimal solution of symmetric TSP. ...
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In this study, Genetic Algorithm (GA), a sort of randomized direct, iterative search methodology built around natural selection, is employ in computers to discover approximations of solutions to optimisation and search issues. GA employs operators including selection, crossover, and mutation to tackle. In case of NP-hard issues, particularly for travelling salesman problem (TSP), the GAs is beneficial. To reduce the overall distance, we propose a novel crossover operator with its python code for the TSP. Along with the Python pseudo coding, we additionally introduced a mutation operator to enhance the consummation of GA in determining the shortest distance in the TSP. To emphasize the proposed crossover and mutation operator, we also illustrate different steps using examples. We integrated path representation with our developed crossover and mutation operator as it is apparent method to represent a tour. Cite this article as: Sharma MK, Chaudhary S, Rathour L, Mishra VN. Modified genetic algorithm with novel crossover and mutation operator for travelling salesman problem. Sigma
... On a different note [6], focused on the exact solution of the STSWSN-PC. This was achieved through the introduction of an integer linear programming (ILP) formulation in which the feasible region is modeled using the generalized version of the subtour elimination constraints (GSECs), proposed by [11], that are exponential in number. This prevents the possibility to explicitly defining the formulation and solving it directly through the MILP solver at hand, using it as a black-box optimization tool. ...
Article
Given an undirected graph whose edge weights change over time slots, the sub-tree scheduling for wireless sensor networks with partial coverage asks to partition the vertices of in non-empty trees such that the total weight of the trees is minimized. In this article, we show that the problem is NP-hard in both the cases where () is part of the input and () is a fixed instance parameter. In both our proofs we reduce the cardinality of the Steiner tree problem. Then, in order to provide easy-to-implement and effective computational tools to benchmark heuristics, we introduce new polynomial-size integer linear programming formulations for the problem. Being defined by a polynomial number of variables and constraints, the formulations can be used to address instances of the problem by means of off-the-shelf mixed-integer linear programming solvers with minimum implementation efforts. All proposed formulations share the property that the integrality requirement can be relaxed on subsets of variables. This enables several algorithmic choices to solve the models, including Benders' decomposition approaches and branch-and-bound with specific branching priorities. We experimentally identify the best resolution method for each formulation. Moreover, the experimental results obtained on benchmark instances of the problem, compared against those reported in the literature, support the effectiveness arising from using the new proposed formulations.
... In the second case, we add a standard subtour-elimination constraint, e.g., the one by Dantzig, Fulkerson, and Johnson [23] and rerun the algorithm (if the tour contains a cycle on vertices, we add a constraint stating that at most − 1 edges between these vertices can be selected). This strategy can be understood as simultaneously applying column and row generation [24] -new variables (e.g., columns) to split intervals, and new constraints (e.g., rows) to eliminate subtours. ...
Conference Paper
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We formalize a Keplerian Travelling Salesperson Problem (ktsp) as a special version of tsp, where the "cities" correspond to objects on Keplerian orbits. In this context, the salesperson is modeled as a "spacecraft" tasked with achieving rendezvous (matching position and velocity) with each orbital object while minimizing the required Δ from the propulsion system, subject to operational constraints such as time-of-flight limits. Unlike the classical tsp, which has been extensively studied, the Keplerian version introduces unique complexities that are relevant to real-world scenarios. These complexities arise from the dynamic movement of orbital objects and the resulting changes in transfer costs between them. The Keplerian Vehicle Routing Problem (kvrp) and Keplerian low-thrust Travelling Salesperson Problem (ktsp-lt) are also introduced, extending this framework further to scenarios involving multiple spacecraft that must collaboratively visit designated targets as well as spacecraft equipped with a low-thrust propulsion system. The specific problems modelled are related to design tasks that have gained prominence in the past decade as advanced trajectory optimization challenges, particularly highlighted in the Global Trajectory Optimization Competition series. These problems have been instrumental in automating the design of multi-rendezvous trajectories in asteroid belt missions, addressing challenges such as mission target selection. They have also been applied to plan active debris removal missions, design future asteroid mining strategies as well as ivestigate on-orbit servicing concepts. In this work, we explore different encodings of these problems into well-known combinatorial optimization tasks, focusing on Integer Linear Programming (ilp) formulations. We provide and study a umber of initial instances of various complexity. A key aspect of our work is the detailed process of unrolling time, which is critical for translating dynamic problems like ktsp into discrete optimization tasks. We also propose a possible strategy to mitigate the complexity involved in the resulting time-indexed formulations of ILPs: Interval-based Dynamic Discretization Discovery. We conclude with a preliminary experimental evaluation, comparing the performance of various ilp solvers to that of the beam search approach, commonly used to incrementally build mission plans in these cases.
... We employ the L1 distance metric = | − | to determine the ground distance between two histograms and . Following the EMD calculation, the highest similarity dataset order (HSDO) is determined by solving a traveling salesman problem (TSP) [128]. The objective is to minimize the total EMD by identifying the order of datasets that exhibit the highest similarity starting from N1-1. ...
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Manufacturing wound rotor synchronous machines (WRSMs) for electric vehicle traction systems necessitates rigorous quality inspection to ensure optimal product performance and efficiency. This paper presents a novel visual anomaly detection method for monitoring the needle winding process of WRSMs, utilizing unsupervised learning with convolutional autoencoders (CAEs) and the structural similarity index measure (SSIM). The method identifies deviations from the desired orthocyclic winding pattern during each stage of the winding process, enabling early detection of winding errors and preventing resource wastage and potential damage to the product or winding machinery. Trajectory-synchronized frame extraction aligns the visual inspection system with the winding trajectory, ensuring precise monitoring traceable to a specific point in the winding process. We present the comprehensive Winding Anomaly Dataset (WAD), which comprises images of WRSM rotor prototypes with and without winding faults recorded in different lighting conditions. The proposed reconstruction-based anomaly detection technique is trained on fault-free data only and utilizes the introduced masked mean structural dissimilarity index measure (MMSDIM) to focus on the relevant sections of the winding during inference. Comprehensive comparative analysis reveals that the CAE with unregularized latent space and the maximum mean discrepancy Wasserstein autoencoder (MMD-WAE) outperform the beta variational autoencoder (beta-VAE) in terms of anomaly detection performance, with the CAE and WAE delivering comparable results. Extensive testing confirms the approach’s effectiveness, achieving 95.6 % recall at 100 % precision, an AUROC of 99.9 %, and an average precision of 99.1 % on the challenging WAD, considerably outperforming state-of-the-art visual anomaly detection models. This work thus offers a robust solution for WRSM production quality monitoring and promotes the incorporation of visual inspection in electric drive manufacturing systems.
... Here's a literature review table that in Figure 1, summarizes key studies in stochastic transportation optimization from 1965 to 2024, highlighting the use of different distributions, methodologies, and contributions in this field [11][12][13][14][15][16][17][18][19][20][21][22][23]. ...
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We propose a novel solution approach combining stochastic programming techniques with Pareto distribution characteristics. This approach involves reformulating the problem into a tractable optimization model using probabilistic constraints and employing advanced algorithms to solve the resulting mixed-integer programming problem. Numerical experiments illustrate the effectiveness of the proposed method and highlight its practical implications for transportation network design and management under uncertain conditions. The results demonstrate that incorporating Pareto-distributed uncertainties into the transportation problem provides a more realistic and adaptable framework for decision-making. The proposed solution approach offers valuable insights for managing complex transportation systems where both stochastic and deterministic factors play a crucial role.
... The Traveling Salesman Problem (TSP) has served as a fundamental challenge in combinatorial optimization, driving the development of various solving techniques. Notably, seminal work in integer programming by Dantzig et al. (1954) [14] introduced cutting planes, laying the groundwork for advanced branch and cut methods (Grötschel and Holland, 1991[23]; Padberg and Rinaldi, 1991[42]; Applegate et al., 1998[5] ; 2003a [3]). Polyhedral approaches (Grötschel and Padberg, 1985 [24, 25]) and branch and bound algorithms (Little et al., 1963 [38]; Held and Karp, 1971[28]) also stemmed from TSP research. ...
Thesis
The dissertation provides a detailed account of our research on developing a quantum algorithm for the Traveling Salesperson Problem (TSP). TSP involves finding the shortest route through a city network, a known NP-hard problem. We introduce a quantum polynomial time algorithm addressing the existence of a Hamiltonian cycle in a graph and the more complex task of determining the shortest Hamiltonian cycle in the context of TSP. Preliminary results suggest our quantum algorithm offers exponential speedup over classical counterparts, albeit with a small probability.
... There are integer or mixed-integer linear programs that require exponentially large numbers of linear constraints. A well-known example is the traveling salesman problem, which has exponentially many constraints to eliminate subtours [4]. A general group of these problems are mixed integer linear programs to be solved by Benders decomposition. ...
Article
A crucial problem in combinatorial optimization is the submodular function maximization (SFM), and in many cases it involves graphs on which the maximization is specified. The problem is well-studied and hence there are several proposed algorithms in the literature. The greedy strategy quickly finds a feasible solution that guarantees an approximation of (1-1/e). However, there are many applications that expect an optimal result within a reasonable computational time. One popular method for finding the global optimum is the constraint generation (CG) algorithm. Traditionally, the initial feasible solution of CG is given by the greedy algorithm. It turns out that choosing different starting point than the greedy solution might lead to better performance in terms of running time. In this paper we introduce such a strategy which is beneficial on non-complete bipartite graphs. Benchmarking results on different versions of the solution methods are shown to demonstrate the efficiency of the proposed methods.
... In (Dantzig et al., 1954), the author proposed an early deterministic solution to TSPTW by using linear programming (LP) relaxation. This approach involves solving the integer formulation and continuously adding a well-selected linear inequality to the list of constraints. ...
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This work aims to compare two distinct approaches for solving a Travelling Salesman Problem with time window constraints. Given an environment with a fixed number of cities (points of interest), a robot must determine a route such that each city is visited in an imposed time interval. Both of the examined techniques have the objective of identifying the path with the lowest cost in terms of the distance traveled. The initial approach employs an exact method by defining the requirements as a mixed integer linear programming (MILP) optimization problem. The second method involves a meta-heuristic approach, using an ant colony procedure to solve the optimization problem. Besides qualitative information, the performed quantitative comparison relies on multiple numerical simulations performed in a MATLAB environment. We thus highlight the advantages and disadvantages of both methods, by taking into consideration criteria as the simulation time and the relative difference between the obtained costs versus the number of cities.
... Ensuring connectedness is a similar problem to the traveling salesperson problem (TSP), where a satisfactory solution describes a single tour passing through all the cities that need to be visited. A known issue in the Dantzig-Fulkerson-Johnson formulation [49] of the TSP is that solutions often contain sub-tours, i.e., tours through subsets of cities. An approach for dealing with these disconnected sub-tours is the "finger-inthe-dike" method [50], where the problem is initially solved without banning sub-tours, and ad-hoc constraints are introduced iteratively to the original formulation to ban them as they appear in the sampled solutions. ...
Preprint
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Graph partitioning has many applications in powersystems from decentralized state estimation to parallel simulation. Focusing on parallel simulation, optimal grid partitioning minimizes the idle time caused by different simulation times for the sub-networks and their components and reduces the overhead required to simulate the cuts. Partitioning a graph into two parts such that, for example, the cut is minimal and the subgraphs have equal size is an NP-hard problem. In this paper we show how optimal partitioning of a graph can be obtained using quantum annealing (QA). We show how to map the requirements for optimal splitting to a quadratic unconstrained binary optimization (QUBO) formulation and test the proposed formulation using a current D-Wave QPU. We show that the necessity to find an embedding of the QUBO on current D-Wave QPUs limits the problem size to under 200 buses and notably affects the time-to-solution. We finally discuss the implications on near-term implementation of QA in combination to traditional CPU or GPU based simulation.
... Consequently, route planning has become a priority to reduce travel times and distances in product delivery. One of the first problems posed to address this situation was the Traveling Salesman Problem (TSP), which seeks to find the shortest route for a traveling salesman who must visit a specified group of cities and return to the original starting point [1]. ...
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In the everchanging landscape of human mobility and commerce, efficient route planning has become paramount. This paper addresses the open vehicle routing problem (OVRP), a major logistical challenge in route optimization for a fleet of vehicles serving geographically dispersed customers. Using a heuristic approach, we explore the complexities of OVRP, comparing the results with advanced optimization methods. This study not only highlights the effectiveness of mathematical modeling, but also explores the practicality of heuristic algorithms such as Greedy, Nearest Neighbor and 2-opt to provide quality solutions. The findings highlight the nuanced interplay between solution quality and computational efficiency, providing valuable insights for addressing real-world logistics challenges. Recommendations delve into optimization opportunities and the integration of emerging technologies, ensuring adaptable solutions to the intricate the problem of open vehicle routing.
... The problem arises from the typical problem of planning a tour: It is about planning a round trip to different cities so that the traveling salesman or the truck arrives back at the starting point (the home town or the depot) at the end and has covered a minimum distance. While in 1954 only one instance with 49 cities could be solved optimally [6], in 2006 an instance with 1.9 million places in the world was provably solved almost optimally (to within 0.01%). ...
Chapter
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Optimization is a fundamental topic in mathematics that deals with finding the best solution to a problem from a set of possible solutions. This chapter provides an overview of mathematical optimization, its main objectives, and the methods used to solve optimization problems. It also introduces basic problems such as the modeling of binary decision trees, the pooling problem, the clique problem, and flow models. It concludes with an outlook on online optimization and learning optimization methods, which represent promising areas of research in the field. Overall, this chapter serves as a useful introduction to mathematical optimization, its basic problems, and their practical applications. The chapter is based on a white paper from Bärmann et al. [2]
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The Traveling Salesman Problem (TSP) is a well-known problem in combinatorial optimization, in which a salesperson who wishes to sell goods in a specific set of cities is seeking for the shortest possible course that would take him or her to all the cities before returning to the original point where he or she is based. Looking at what we have come up with, it becomes evident that we want to find the shortest distance possible. Even though Algorithm TSP is a well-researched problem, the fact that it is NP-resistant has encouraged a lot of work in various solutions. As such, the aim of this paper is to present a brief discussion of TSP, its history, and various approaches developed throughout the years. Thus, when considering the brute force approach implemented in Python we examine its flaws in terms of the methodological approach and possible computational performance. Using the notion of visualization, we also offer a means for an intuitive understanding of the solution space with particular focus placed on the interdependence between data, computations, and graphics.
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In 2018, a fort of a graph was introduced as a non-empty subset of vertices in which no vertex outside of the set has exactly one neighbor in the set. Since then, forts have been used to characterize zero forcing sets, model the zero forcing number as an integer program, and generate lower bounds on the zero forcing number of a Cartesian product. In this article, we investigate the number of minimal forts of a graph, where a fort is minimal if every proper subset is not a fort. In particular, we show that the number of minimal forts of a graph of order at least six is strictly less than Sperner’s bound, a famous bound due to Emanuel Sperner (1928) on the size of a collection of subsets where no subset contains another. Then, we derive an explicit formula for the number of minimal forts for several families of graphs, including the path, cycle, and spider graphs. Moreover, we show that the asymptotic growth rate of the number of minimal forts of the spider graph is bounded above by that of the path graph. We conjecture that the asymptotic growth rate of the path graph is extremal over all trees. Finally, we develop methods for constructing minimal forts of graph products using the minimal forts of the original graphs. In the process, we derive explicit formulas and lower bounds on the number of minimal forts for additional families of graphs, such as the wheel, sunlet, and windmill graphs. Most notably, we show that a family of windmill graphs has an exponential number of minimal forts with a maximum asymptotic growth rate of cube root of three, which is the largest asymptotic growth rate we have observed. We conjecture that there exist families of graphs with a larger asymptotic growth rate.
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I translated "Linus Schrage: Optimization Modeling with LINGO" into three Japanese books, sold three books on Kindle, and canceled the contract for distributing them on RG. The other two books, including IP and portfolio analysis, can be obtained from LINDO Japan (LINDO.JP), and the original book can be obtained from LINDO.COM.
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