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In this paper, we describe and compare serial, parallel, and distributed solver implementations for large batches of Traveling Salesman Problems using the Lin-Kernighan Heuristic (LKH) and the Concorde exact TSP Solver. Parallel and distributed solver implementations are useful when many medium to large size TSP instances must be solved simultaneou...

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**Context 1**

... values for parallel, distributed-2, and distributed-3 executions are 6, 12, and 18, respectively. Table 4 shows a 4-way ANOVA using Minitab 17.0 statistical software. The differences between the group means of the main effects (TSP size, number of TSPs (batch), algorithm type (parallel, distributed-2, and distributed-3), solver type (LKH or Concorde)) and their interactions (two, three, and four level interactions) are all statistically significant. ...

## Citations

... The Kernighan Lin-based techniques are well-studied in the literature to rectify the leading combinatorial issues, such as graph partitioning, chip layout design, and machine routing, etc., [27][28][29][30][31][32][33][34]. Moreover, these techniques become hugely influential for upgrading the solutions provided by global search techniques, such as simulated annealing algorithm [34], and artificial bee colony algorithm [31] for the mentioned issues. ...

The many-to-many assignment problem (M- MAP), and the CPU/FPGA scheduling problem are two correlated issues in the field of combinatorial optimization. The framework for Kernighan Lin-driven logarithmic barrier approach (KD-LBA) is made to solve the many-to-many assignment problem. KD-LBA is a deterministic technique, which initiates to achieve the globally optimal solutions for MMAP using logarithmic barrier function-based gradient descent technique. Then, the obtained solution is optimized further using the Kernighan Lin-based local search method. Successive Kernighan Lin-driven logarithmic barrier approach (Successive KD-LBA) is also proposed to sort the issue of scheduling in a CPU/FPGA heterogeneous system. It solves the CPU/ FPGA scheduling problem by transforming it into an MMAP. KD-LBA outperforms the state-of-art methods in terms of convergence speed for MMAPs with a group size greater than 40. Successive KD-LBA presents novel scheduling solutions for the CPU/FPGA scheduling problem as compared to the existing works, regarding average makespan and computation time.

... A local search procedure in EAX is used to determine good combinations of building blocks of parent solutions for generating even better offspring solutions from very high-quality parent solutions. More recently, Ozden et al. (2017) described and compared serial, parallel and distributed solver implementations for large batches of traveling salesman problems using the Lin-Kernighan heuristic (LKH) and the Concorde solver [6,7]. Finally, we quote the work of Staněk et al. (2019) who investigated two variants of the Euclidean TSP based on socalled turning angles [8]. ...

... A local search procedure in EAX is used to determine good combinations of building blocks of parent solutions for generating even better offspring solutions from very high-quality parent solutions. More recently, Ozden et al. (2017) described and compared serial, parallel and distributed solver implementations for large batches of traveling salesman problems using the Lin-Kernighan heuristic (LKH) and the Concorde solver [6,7]. Finally, we quote the work of Staněk et al. (2019) who investigated two variants of the Euclidean TSP based on socalled turning angles [8]. ...

... Looking at the above model, we easily identify the "location-allocation" component, Constraints (2)-(3); the "mTSP" component, Constraints (5)-(9); and the linking constraints, Constraints (4)- (6). The hardness of the problem together with many data-driven applications in which upon collecting the data a solution must be found in a very short time justifies the development of a heuristic algorithm for the problem. ...

This work focuses on the optimization of a last-mile delivery system with multiple transportation modes. In this scenario, parcels need to be delivered to each customer point. The major feature of the problem is the combination of a fleet of road vehicles (vans) with a drone. Each van visits a subset of demand nodes to be determined according to the route of the van. The drone serves the customers not served by vans. At the same time, considering the safety, policy and terrain as well as the need to replace the battery, the drone needs to be transported by truck to the identified station along with the parcel. From each such station, the drone serves a subset of customers according to a direct assignment pattern, i.e., every time the drone is launched, it serves one demand node and returns to the station to collect another parcel. Similarly, the truck is used to transport the drone and cargo between stations. This is somewhat different from the research of other scholars. In terms of the joint distribution of the drone and road vehicle, most scholars will choose the combination of two transportation tools, while we use three. The drone and vans are responsible for distribution services, and the trucks are responsible for transporting the goods and drone to the station. The goal is to optimize the total delivery cost which includes the transportation costs for the vans and the delivery cost for the drone. A fixed cost is also considered for each drone parking site corresponding to the cost of positioning the drone and using the drone station. A discrete optimization model is presented for the problem in addition to a two-phase heuristic algorithm. The results of a series of computational tests performed to assess the applicability of the model and the efficiency of the heuristic are reported. The results obtained show that nearly 10% of the cost can be saved by combining the traditional delivery mode with the use of a drone and drone stations.

... To this end, we use the Concorde TSP Solver ( Applegate, Bixby, Chvatal, & Cook, 2006;Cook, 2011 ), code downloaded from http: //www.math.uwaterloo.ca/tsp/concorde/index.html , generally considered to be the state-of-the-art exact method for solving the TSP (recently, e.g., Ozden, Smith, & Gue, 2017 ). Note that Concorde only solves the symmetric TSP. ...

Seeing the huge success of sharing platforms such as Uber, Lyft, and Airbnb, where owners of under-used assets are connected with users willing to pay for the use of these assets, it is not surprising that retailers aim to transfer the basic idea of the sharing economy to their last-mile deliveries. In crowdshipping, the under-used assets are transport capacities of private drivers and the users are the retailers aiming for additional and cost-efficient delivery capacities for their home deliveries. A major drawback of crowdshipping is that retailers can hardly guarantee their promised delivery services when subcontracting individuals. To avoid this problem, different retailers are establishing crowdshipping platforms offering a reward to the employees of their distribution centers for crowdshipping online orders on their way back from work. We investigate the resulting optimization problem for matching crowdshipping supply and demand in this context. We present an efficient exact solution procedure based on Benders decomposition, which maximizes the number of matched shipments while considering the employees’ minimum expected earnings per time unit. This procedure is shown to solve instances of real-world size before a work shift is over and the shipments have to be loaded into the trunks of the employees’ cars. Furthermore, we show the impact of crowdshipping on all main stakeholders and identify critical success factors.

... Such methods include metaheuristics like genetic algorithms [4], tabu search [5], simulated annealing, other bio-inspired optimizations [6]. Other types of approaches are machine learning [7] or distributed algorithms [8], [9]. ...

We propose our solution to a particular practical problem in the domain of vehicle routing and scheduling. The generic task is finding the best allocation of the minimum number of \emph{mobile resources} that can provide periodical services in remote locations. These \emph{mobile resources} are based at a single central location. Specifications have been defined initially for a real-life application that is the starting point of an ongoing project. Particularly, the goal is to mitigate health problems in rural areas around a city in Romania. Medically equipped vans are programmed to start daily routes from county capital, provide a given number of examinations in townships within the county and return to the capital city in the same day. From the health care perspective, each van is equipped with an ultrasound scanner, and they are scheduled to investigate pregnant woman each trimester aiming to diagnose potential problems. The project is motivated by reports currently ranking Romania as the country with the highest infant mortality rate in the European Union. We developed our solution in two phases: modeling of the most relevant parameters and data available for our goal and then design and implement an algorithm that provides an optimized solution. The most important metric of an output scheduling is the number of vans that are necessary to provide a given amount of examination time per township, followed by total travel time or fuel consumption, number of different routes, and others. Our solution implements two probabilistic algorithms out of which we chose the one that performs the best.

... In order to solve a TSP with even a moderate number of cities and in the interest of optimized tour, an extremely huge search space should be investigated and a massive computational time will be required. Although exact algorithms similar to the brute force approach (BFA) that allow for evaluating all possible solutions are guaranteed to find the optimal solution, they can only be applied to small TSPs up to 10 cities [7,8], and thus, for average and big TSPs the direct methods are, in fact, useless. Therefore, the development and application of heuristic algorithms that could find the optimal or near-optimal solution in a limited time frame [7] have been massively studied. ...

... Although exact algorithms similar to the brute force approach (BFA) that allow for evaluating all possible solutions are guaranteed to find the optimal solution, they can only be applied to small TSPs up to 10 cities [7,8], and thus, for average and big TSPs the direct methods are, in fact, useless. Therefore, the development and application of heuristic algorithms that could find the optimal or near-optimal solution in a limited time frame [7] have been massively studied. Algorithms like but not limited to tabu search [9], Lin-Kernighan heuristic [10] have been significantly improved over years as successful methods for obtaining the optimal or near-optimal solutions [11] as well as genetic algorithm (GA) [8,[12][13][14][15][16][17]. ...

The variety of methods used to solve the traveling salesman problem attests to the fact that the problem is still vibrant and of concern to researchers in this area. For problems with a large search space, similar to the traveling salesman problem, evolutionary algorithms such as genetic algorithm are very powerful and can be used to obtain optimized solutions. However, the challenge in applying a genetic algorithm to the traveling salesman problem is the choice of appropriate operators that could produce legal tours. In the literature, additional repair algorithms have been introduced and employed and the offspring produced by these genetic algorithm operators are modified to ensure that the generated chromosomes represent legal tours. Rather than sticking to repair algorithms, a double-chromosome approach is proposed in this article. The proposed method can be employed to optimize problems similar to the traveling salesman problem. The double-chromosome approach has been tested with a variety of traveling salesman problems, and the results indicated that the proposed method has a high rate of convergence toward the shortest tour.

... Branch & X [Mezmaz et al., 2014, Chakroun et al., 2013b, Herrera et al., 2017, Taoka et al., 2008, Ponz-Tienda et al., 2017, Ismail et al., 2014, Paulavicius et al., 2011, Christou and Vassilaras, 2013, McCreesh and Prosser, 2015, Eckstein et al., 2015, Carvajal et al., 2014, Borisenko et al., 2017, Gmys et al., 2017, Liu and Kao, 2013, Bak et al., 2011, Gmys et al., 2016, Silva et al., 2015, Barreto and Bauer, 2010, Vu and Derbel, 2016, Chakroun and Melab, 2015, Paulavičius and Žilinskas, 2009, Posypkin and Sigal, 2008, Chakroun et al., 2013a, Aitzai and Boudhar, 2013, Ozden et al., 2017, Cauley et al., 2011, Xu et al., 2009, Aldasoro et al., 2017, Pages-Bernaus et al., 2015, Lubin et al., 2013, Adel et al., 2016, Borisenko et al., 2011, Boukedjar et al., 2012, Carneiro et al., 2011, Galea and Le Cun, 2011, Herrera et al., 2013, Sanjuan-Estrada et al., 2011] Dynamic programming [Dias et al., 2013, Aldasoro et al., 2015, Maleki et al., 2016, Tan et al., 2009, Stivala et al., 2010, Boyer et al., 2012, Boschetti et al., 2016, Kumar et al., 2011, Rashid et al., 2010, Tran, 2010 Interior point method [Huebner et al., 2017, Hong et al., 2010, Lubin et al., 2012, Lucka et al., 2008 Problem-specific exact algorithms [Li et al., 2015, Rossbory and Reisner, 2013, Kollias et al., 2014, Bozdag et al., 2008 Problem-specific heuristics [Dobrian et al., 2011, Ozden et al., 2017, Ismail et al., 2011, Bożejko, 2009, Lancinskas et al., 2015, Koc and Mehrotra, 2017, Redondo et al., 2016, Hemmelmayr, 2015, Benedicic et al., 2014, Gomes et al., 2008, Baumelt et al., 2016, Luo et al., 2015. Single-solution based metaheuristics: ...

... Branch & X [Mezmaz et al., 2014, Chakroun et al., 2013b, Herrera et al., 2017, Taoka et al., 2008, Ponz-Tienda et al., 2017, Ismail et al., 2014, Paulavicius et al., 2011, Christou and Vassilaras, 2013, McCreesh and Prosser, 2015, Eckstein et al., 2015, Carvajal et al., 2014, Borisenko et al., 2017, Gmys et al., 2017, Liu and Kao, 2013, Bak et al., 2011, Gmys et al., 2016, Silva et al., 2015, Barreto and Bauer, 2010, Vu and Derbel, 2016, Chakroun and Melab, 2015, Paulavičius and Žilinskas, 2009, Posypkin and Sigal, 2008, Chakroun et al., 2013a, Aitzai and Boudhar, 2013, Ozden et al., 2017, Cauley et al., 2011, Xu et al., 2009, Aldasoro et al., 2017, Pages-Bernaus et al., 2015, Lubin et al., 2013, Adel et al., 2016, Borisenko et al., 2011, Boukedjar et al., 2012, Carneiro et al., 2011, Galea and Le Cun, 2011, Herrera et al., 2013, Sanjuan-Estrada et al., 2011] Dynamic programming [Dias et al., 2013, Aldasoro et al., 2015, Maleki et al., 2016, Tan et al., 2009, Stivala et al., 2010, Boyer et al., 2012, Boschetti et al., 2016, Kumar et al., 2011, Rashid et al., 2010, Tran, 2010 Interior point method [Huebner et al., 2017, Hong et al., 2010, Lubin et al., 2012, Lucka et al., 2008 Problem-specific exact algorithms [Li et al., 2015, Rossbory and Reisner, 2013, Kollias et al., 2014, Bozdag et al., 2008 Problem-specific heuristics [Dobrian et al., 2011, Ozden et al., 2017, Ismail et al., 2011, Bożejko, 2009, Lancinskas et al., 2015, Koc and Mehrotra, 2017, Redondo et al., 2016, Hemmelmayr, 2015, Benedicic et al., 2014, Gomes et al., 2008, Baumelt et al., 2016, Luo et al., 2015. Single-solution based metaheuristics: ...

... Problem-specific heuristics, other heuristics, matheuristics, and multi-search algorithms: Problem-specific heuristics have been parallelized for a variety of optimization problems, including a graph theory problem [Dobrian et al., 2011], TSPs [Ozden et al., 2017, Ismail et al., 2011, a FSSP [Bożejko, 2009], a facility location problem [Lancinskas et al., 2015], a mixed integer linear program [Koc and Mehrotra, 2017], and several other problems [Redondo et al., 2016, Hemmelmayr, 2015, Benedicic et al., 2014, Gomes et al., 2008, Baumelt et al., 2016, Luo et al., 2015. We found four studies which parallelize heuristics that differ from all types described above: an agent-based heuristic [Benedicic et al., 2014], an auction-based heuristic [Sathe et al., 2012], a Monte Carlo simulation inside a heuristic-randomization process [Juan et al., 2013], and a random search algorithm [Sancı andİşler, 2011]. ...

Solving optimization problems with parallel algorithms has a long tradition in OR. Its future relevance for solving hard optimization problems in many fields, including finance, logistics, production and design, is leveraged through the increasing availability of powerful computing capabilities. Acknowledging the existence of several literature reviews on parallel optimization, we did not find reviews that cover the most recent literature on the parallelization of both exact and (meta)heuristic methods. However, in the past decade substantial advancements in parallel computing capabilities have been achieved and used by OR scholars so that an overview of modern parallel optimization in OR that accounts for these advancements is beneficial. Another issue from previous reviews results from their adoption of different foci so that concepts used to describe and structure prior literature differ. This heterogeneity is accompanied by a lack of unifying frameworks for parallel optimization across methodologies, application fields and problems, and it has finally led to an overall fragmented picture of what has been achieved and still needs to be done in parallel optimization in OR. This review addresses the aforementioned issues with three contributions: First, we suggest a new integrative framework of parallel computational optimization across optimization problems, algorithms and application domains. The framework integrates the perspectives of algorithmic design and computational implementation of parallel optimization. Second, we apply the framework to synthesize prior research on parallel optimization in OR, focusing on computational studies published in the period 2008-2017. Finally, we suggest research directions for parallel optimization in OR.

... Recently, Ezugwu et al. [25] proposed a hybrid optimization algorithm which fuses the SOS and SA algorithm together. On the other hand, Ozden et al. [26] demonstrated how the parallel computing techniques can be used for TSP and significantly decrease the overall computational time with the increase of CPU utilization. ...

The travelling salesman problem (TSP) is one of the NPC combinatorial optimization problems and still now it remains as an interesting and challenging problem in the field of combinatorial optimization. In this paper, we propose a consecutive route filtering approach to solving the symmetric TSP with the help of probe concept such that the worse routes are filtered out step by step by using a rigorous predesigned step proportion. In this way, it is important to set up a reasonable value of the step proportion which is needed in each step during the filtering process. Actually, our proposed algorithm is implemented on the set of symmetric TSP benchmarks with both small and large numbers of cities from the TSPLIB dataset. It is demonstrated by the experimental results that our proposed algorithm can obtain the best results in some cases and generally get the approximation results close to the best known solutions.

... Such methods include: metaheuristics like genetic algorithms [4], tabu search [5], simulated annealing, other bio-inspired optimizations [6]. Other types of approaches are machine learning [7] or distributed algorithms [8], [9]. ...

... Distributed computation is a promising approach to this kind of problem [20]. For example, large batches of Traveling Salesman Problems [21] was solved with distributed computation. Recent researches on solving heuristic problems adopted distributed computation [22][23][24][25]. ...

Weapon target allocation (WTA) is a classic NP-complete problem in the field of military operations research. In this paper, we addressed the multi-constraint WTA problems in multilayer defense scenario. To solve large-scale WTA problems effectively, a distributed MAX-MIN Ant System (MMAS) algorithm based on distributed computing framework Spark was developed and improved. An experiment environment comprising virtual machines was built for implementing the distributed MMAS. First, a small-scale WTA example, whose theoretical optimal solution can be obtained by existing optimization software, was taken as a benchmark problem to assess the performance of distributed MMAS. The result shows that it can find high-quality and robust approximate solutions. Then a large-scale WTA problem was constructed and used to further evaluate the performance of distributed MMAS in the experiment environment. The result shows that the distributed MMAS can also achieve high-quality approximate solutions with high robustness and computational efficiency even for large scale WTA problems. Our study demonstrates it is a promising approach for solving large-scale iteration-dependent optimization problems like WTA by means of incorporating heuristic optimization algorithms such as Ant Colony Optimization into distributed computing framework.

... Avoiding local stagnation requires adding other disturbing operations into an ACO algorithm, but such an action inevitably prolongs the convergence [8], [9]. Given the multi-agent nature of ACOs, a few recent studies have explored the likelihood of efficiently applying parallel ACO algorithms to solve TSPs [14]- [19]. ...

... As ACOs are prone to local maxima, as-observed in many existing studies [9], [10], [19], we make a small change to the basic ACO for TSPs in Algorithm 1 to deal with such stagnation. In the probability-based stochastic controller (1), instead of directing ant k in city i to next city j according to the highest probability k ij p , we make a pool of three potential cities with the three highest probability k ij p and direct ant k in city i to next city j by a random pick from the pool. ...

In recent years some comparative studies have explored the use of parallel ant colony optimization (ACO) algorithms over the traditionally sequential ACOs to solve the traveling salesman problem (TSP). However, these studies did not take a systematical approach to assess the performance of both algorithms on a comparable ground. In this paper, we aim to make a comparison of both the quality of the solutions and the running time as a result of the application of a sequential ACO and a parallel ACO to Eil51, Eil76 and KroA100 on a normalized and thus, comparable ground. Our study reaffirmed that the parallel algorithm is superior in computing efficiency over the sequential algorithm, particularly for larger TSPs. We also found that such a comparison could be meaningless if the size of the TSPs keeps increasing. We revealed that the worst solution among 10 repeated runs obtained from the parallel ACO was still better than the best solution among 10 repeated runs obtained from the sequential ACO, though both did not reach the global optimal solution within 300 iterations. The proposed parallel ACO has a very high consistency because at least one best solution was found within an error of 0.5% to the global optimal solution in every three repeats for all three cases.