An application of the self-organizing map in the non-Euclidean Traveling Salesman Problem

An application of the self-organizing map (SOM) to the Traveling Salesman Problem (TSP) has been reported by many researchers, however these approaches are mainly focused on the Euclidean TSP variant. We consider the TSP as a problem formulation for the multi-goal path planning problem in which paths among obstacles have to be found. We apply a simple approximation of the shortest path that seems to be suitable for the SOM adaptation procedure. The approximation is based on a geometrical interpretation of SOM, where weights of neurons represent nodes that are placed in the polygonal domain. The approximation is verified in a set of real problems and experimental results show feasibility of the proposed approach for the SOM based solution of the non-Euclidean TSP.     

基于泛化竞争和局部渗透机制的自组织网TSP问题求解方法

旅行商问题(TSP)是组合优化中最典型的NP完全问题之一,具有很强的工程背景和应用价值.文章在分析了标准SOM(Self-Organizing Map)算法在求解TSP问题的不足和在寻求总体最优解的潜力的基础上,引入泛化竞争和局部渗透这两个新的学习机制,提出了一种新的SOM算法---渗透的SOM(Infiltrative SOM,ISOM)算法.通过泛化竞争和局部渗透策略的协同作用:总体竞争和局部渗透并举、先倾向总体竞争后倾向局部渗透、在总体竞争基础上的局部渗透,实现了在总体路径寻优指导下的局部路径优化,从而使所得路径尽可能接近最优解.通过对TSPLIB中14组TSP实例的测试结果及与KNIES、SETSP、Budinich和ESOM等类SOM算法的比较,表明该算法既简单又能使解的质量得到很大提高,同时还保持了解的良好的稳健特性.     

An efficient self-organizing map designed by genetic algorithms for the traveling salesman problem

As a typical combinatorial optimization problem, the traveling salesman problem (TSP) has attracted extensive research interest. In this paper, we develop a self-organizing map (SOM) with a novel learning rule. It is called the integrated SOM (ISOM) since its learning rule integrates the three learning mechanisms in the SOM literature. Within a single learning step, the excited neuron is first dragged toward the input city, then pushed to the convex hull of the TSP, and finally drawn toward the middle point of its two neighboring neurons. A genetic algorithm is successfully specified to determine the elaborate coordination among the three learning mechanisms as well as the suitable parameter setting. The evolved ISOM (eISOM) is examined on three sets of TSP to demonstrate its power and efficiency. The computation complexity of the eISOM is quadratic, which is comparable to other SOM-like neural networks. Moreover, the eISOM can generate more accurate solutions than several typical approaches for TSP including the SOM developed by Budinich, the expanding SOM, the convex elastic net, and the FLEXMAP algorithm. Though its solution accuracy is not yet comparable to some sophisticated heuristics, the eISOM is one of the most accurate neural networks for the TSP.     

双获胜节点SOM及其在TSP中的应用

在Kohonen提出的SOM(self-organization map)神经网络的基础上,通过拓广SOM网络的获胜节点数量,引入惩罚修正因子,改进邻域和连接权函数等方法提出一种新的SOM即SOMDW(SOM with double-winner)模型.为了验证该模型的有效性,以旅行商问题(traveling salesman problem,TSP)为例对该模型进行检验,得到了满意的结果.另外为了增强SOMDW网络的动态聚类性能,提高解的精确性,还采用禁忌搜索的搜索方法.     

求解不确定TSP问题的蚂蚁算法

提出了不确定旅行商问题模型,该模型将路径长度看作动态可变的。从实际应用来说,该模型考虑了交通运行中的不确定情况,比经典旅行商问题更具有灵活性及实用价值,利用该模型得到的结果将更适于指导车辆对运行路线的选择。同时提出了一种基于蚂蚁算法的混合方法求解不确定旅行商问题,并给出了解的评价标准。实验结果显示,该方法能够加速蚂蚁算法的收敛性,可以有效求解不确定旅行商问题。     

On the performance of self-organizing maps for the non-Euclidean Traveling Salesman Problem in the polygonal domain

In this paper, two state-of-the-art algorithms for the Traveling Salesman Problem (TSP) are examined in the multi-goal path planning problem motivated by inspection planning in the polygonal domain W. Both algorithms are based on the self-organizing map (SOM) for which an application in W is not typical. The first is Somhom’s algorithm, and the second is the Co-adaptive net. These algorithms are augmented by a simple approximation of the shortest path among obstacles in W. Moreover, the competitive and cooperative rules are modified by recent adaptation rules for the Euclidean TSP, and by proposed enhancements to improve the algorithms’ performance in the non-Euclidean TSP. Based on the modifications, two new variants of the algorithms are proposed that reduce the required computational time of their predecessors by an order of magnitude, therefore making SOM more competitive with combinatorial heuristics. The results show how SOM approaches can be used in the polygonal domain so they can provide additional features over the classical combinatorial approaches based on the complete visibility graph.     

Nonlinear resolving functions for the travelling salesman problem

We propose two approaches to finding lower bounds in the traveling salesman problem (TSP). The first approach, based on a linear specification of the resolving function φ(t, y), uses a two-index TSP model in its solution. This model has many applications. The second approach, based on a nonlinear specification of the resolving function φ(t, y), uses a single-index TSP model. This model is original and lets us significantly reduce the branching procedure in the branch-and-bound method for exact TSP solution. One cannot use the two-index TSP model here due to the nonlinear specification of the resolving function φ(t, y).     

A two-dimensional mapping for the traveling salesman problem

The traveling salesman problem (TSP) is a classic problem in combinatorial optimization. An N-city asymmetric TSP has (N − 1)! tours. We develop a conceptual mapping of an N-city problem to a two-dimensional space and computer code for the defining function and its inverse. The transformation is implemented using color graphics and allows one to better understand the topography of the solution space and study the effects of different search techniques visually, leading to improvements in solution algorithms.     

Self-organizing maps by difference of convex functions optimization

We offer an efficient approach based on difference of convex functions (DC) optimization for self-organizing maps (SOM). We consider SOM as an optimization problem with a nonsmooth, nonconvex energy function and investigated DC programming and DC algorithm (DCA), an innovative approach in nonconvex optimization framework to effectively solve this problem. Furthermore an appropriate training version of this algorithm is proposed. The numerical results on many real-world datasets show the efficiency of the proposed DCA based algorithms on both quality of solutions and topographic maps.     

Comparison of visualization of optimal clustering using self-organizing map and growing hierarchical self-organizing map in cellular manufacturing system

The present research deals with the cell formation problem (CFP) of cellular manufacturing system which is a NP-hard problem thus, the development of optimum machine-part cell formation algorithms has always been the primary attraction in the design of cellular manufacturing system. In this proposed work, the self-organizing map (SOM) approach has been used which is able to project data from a high-dimensional space to a low-dimensional space so it is considered a visualized approach for explaining a complicated CFP data set. However, for a large data set with a high dimensionality, a traditional flat SOM seems difficult to further explain the concepts inside the clusters. We propose one such possible solution for a large CFP data set by using the SOM in a hierarchical manner known as growing hierarchical self-organizing map (GHSOM). In the present work, the two novel contributions using GHSOM are: the choice of optimum architecture through the minimum pattern units extracted at layer 1 for the respective threshold values and selection. Furthermore, the experimental results clearly indicated that the machine-part visual clustering using GHSOM can be successfully applied in identifying a cohesive set of part family that is processed by a machine group. Computational experience specifically with the proposed GHSOM algorithm, on a set of 15 CFP problems from the literature, has shown that it performs remarkably well. The GHSOM algorithm obtained solutions that are at least as good as the ones found the literature. For 75% of the cell formation problems, the GHSOM algorithm improved the goodness of cell formation through GTE performance measure using SOM as well as best one from the literature, in some cases by as much as more than 12.81% (GTE). Thus, comparing the results of the experiment in this paper with the SOM and GHSOM using the paired t-test it has been revealed that the GHSOM approach performed better than the SOM approach so far the group technology efficiency (GTE) measures of performance of the goodness of cell formation is concerned.     

A novel constructive-optimizer neural network for the traveling salesman problem.

In this paper, a novel constructive-optimizer neural network (CONN) is proposed for the traveling salesman problem (TSP). CONN uses a feedback structure similar to Hopfield-type neural networks and a competitive training algorithm similar to the Kohonen-type self-organizing maps (K-SOMs). Consequently, CONN is composed of a constructive part, which grows the tour and an optimizer part to optimize it. In the training algorithm, an initial tour is created first and introduced to CONN. Then, it is trained in the constructive phase for adding a number of cities to the tour. Next, the training algorithm switches to the optimizer phase for optimizing the current tour by displacing the tour cities. After convergence in this phase, the training algorithm switches to the constructive phase anew and is continued until all cities are added to the tour. Furthermore, we investigate a relationship between the number of TSP cities and the number of cities to be added in each constructive phase. CONN was tested on nine sets of benchmark TSPs from TSPLIB to demonstrate its performance and efficiency. It performed better than several typical Neural networks (NNs), including KNIES_TSP_Local, KNIES_TSP_Global, Budinich's SOM, Co-Adaptive Net, and multivalued Hopfield network as wall as computationally comparable variants of the simulated annealing algorithm, in terms of both CPU time and accuracy. Furthermore, CONN converged considerably faster than expanding SOM and evolved integrated SOM and generated shorter tours compared to KNIES_DECOMPOSE. Although CONN is not yet comparable in terms of accuracy with some sophisticated computationally intensive algorithms, it converges significantly faster than they do. Generally speaking, CONN provides the best compromise between CPU time and accuracy among currently reported NNs for TSP.     

Solving TSP by using Lotka–Volterra neural networks

This paper proposes a new approach to solve traveling salesman problem (TSP) by using a class of Lotka–Volterra neural networks (LVNN) with global inhibition. Some stability criteria that ensure the convergence of valid solutions are obtained. It is proved that an equilibrium state is stable if and only if it corresponds to a valid solution of the TSP. Thus, a valid solution can always be obtained whenever the network convergence to a stable state. A set of analytical conditions for optimal settings of LVNN is derived. Simulation results illustrate the theoretical analysis.     

Discrete Spider Monkey Optimization for Travelling Salesman Problem

Meta-heuristic algorithms inspired by biological species have become very popular in recent years. Collective intelligence of various social insects such as ants, bees, wasps, termites, birds, fish, has been investigated to develop a number of meta-heuristic algorithms in the general domain of swarm intelligence (SI). The developed SI algorithms are found effective in solving different optimization tasks. Travelling Salesman Problem (TSP) is the combinatorial optimization problem where a salesman starting from a home city travels all the other cities and returns to home city in the shortest possible path. TSP is a popular problem due to the fact that the instances of TSP can be applied to solve real-world problems, implication of which turns TSP into a standard test bench for performance evaluation of new algorithms. Spider Monkey Optimization (SMO) is a recent addition to SI algorithms based on the social behaviour of spider monkeys. SMO implicitly adopts grouping and regrouping for the interactions to improve solutions; such multi-population approach is the motivation of this study to develop an effective method for TSP. This paper presents an effective variant of SMO to solve TSP called discrete SMO (DSMO). In DSMO, every spider monkey represents a TSP solution where Swap Sequence (SS) and Swap Operator (SO) based operations are employed, which enables interaction among monkeys in obtaining the optimal TSP solution. The SOs are generated using the experience of a specific spider monkey as well as the experience of other members (local leader, global leader, or a randomly selected spider monkey) of the group. The performance and effectiveness of the proposed method have been verified on a large set of TSP instances and the outcomes are compared to other well-known methods. Experimental results demonstrate the effectiveness of the proposed DSMO for solving TSP.     

Formulations and exact algorithms for the vehicle routing problem with time windows

In this paper we review the exact algorithms proposed in the last three decades for the solution of the vehicle routing problem with time windows (VRPTW). The exact algorithms for the VRPTW are in many aspects inherited from work on the traveling salesman problem (TSP). In recognition of this fact this paper is structured relative to four seminal papers concerning the formulation and exact solution of the TSP, i.e. the arc formulation, the arc-node formulation, the spanning tree formulation, and the path formulation. We give a detailed analysis of the formulations of the VRPTW and a review of the literature related to the different formulations. There are two main lines of development in relation to the exact algorithms for the VRPTW. One is concerned with the general decomposition approach and the solution to certain dual problems associated with the VRPTW. Another more recent direction is concerned with the analysis of the polyhedral structure of the VRPTW. We conclude by examining possible future lines of research in the area of the VRPTW.     

Solution of the neutron diffusion equation through multigrid methods implemented on a memory-coupled 25-processor system

For the complex task to attain maximum use and maximum security maintenance of nuclear reactors, we simulate the physical behaviour by way of mathematical models. The stationary solution of the Neutron Diffusion Equation as an eigenvalue problem, presented in this paper, results in a large non-linear partial differential equation system (PDE) whose solution is parallelized and implemented on the DIRMU multiprocessor system. We demonstrate that modern algorithms (e.g. multigrid methods) combined with innovative hardware (e.g. vector-multiprocessor systems) can lead to powerful tools which are needed for real-time simulation of physical events. This approach also allows to solve problems which arise in fluid dynamics, aerodynamics, weather forecast, etc. in a reasonable amount of time.     

Coverage of a Planar Point Set With Multiple Robots Subject to Geometric Constraints

This paper focuses on the assignment of N discrete points among K geometrically constrained robots and determination of the order in which the points should be processed by the robots. This path planning problem is directly motivated by an industrial laser drilling system with two robots that are constrained to translate along a common line while satisfying collision avoidance constraints. The points lie on a planar base plate that translates normal to the axis of motion of the robots. The geometric constraints on the motions of the robots lead to constraints on points that can be processed simultaneously.We use a two step approach to solve the path planning problem: (1) Splitting Problem: Assign the points to the K robots, subject to geometric constraints, to maximize parallel processing of the points. (2) Ordering Problem: Find an order of processing the split points by formulating and solving a multidimensional Traveling Salesman Problem (TSP) in the if-tuple space with an appropriately defined metric to minimize the total travel cost. For K = 2, we solve the splitting problem optimally in O(N³) time by converting it to a maximum cardinality matching problem. Since this is too slow for large datasets, we also provide a greedy O(N log N) algorithm. We provide computational results showing that the greedy algorithm solution is very close to the optimal solution for large datasets. For the ordering problem we present local search based heuristics to improve the multidimensional TSP tour. We give computational results for the ordering problem and for the overall performance gain obtained (over a single robot system) by using our algorithm. Finally, we extend our approach to a K-robot system and give computational results for K = 4.     

Embedded local search approaches for routing optimization

This paper presents a new class of heuristics which embed an exact algorithm within the framework of a local search heuristic. This approach was inspired by related heuristics which we developed for a practical problem arising in electronics manufacture. The basic idea of this heuristic is to break the original problem into small subproblems having similar properties to the original problem. These subproblems are then solved using time intensive heuristic approaches or exact algorithms and the solution is re-embedded into the original problem. The electronics manufacturing problem where we originally used the embedded local search approach, contains the Travelling Salesman Problem (TSP) as a major subproblem. In this paper we further develop our embedded search heuristic, HyperOpt, and investigate its performance for the TSP in comparison to other local search based approaches. We introduce an interesting hybrid of HyperOpt and 3-opt for asymmetric TSPs which proves more efficient than HyperOpt or 3-opt alone. Since pure local search seldom yields solutions of high quality we also investigate the performance of the approaches in an iterated local search framework. We examine iterated approaches of Large-Step Markov Chain and Variable Neighbourhood Search type and investigate their performance when used in combination with HyperOpt. We report extensive computational results to investigate the performance of our heuristic approaches for asymmetric and Euclidean Travelling Salesman Problems. While for the symmetric TSP our approaches yield solutions of comparable quality to 2-opt heuristic, the hybrid methods proposed for asymmetric problems seem capable of compensating for the time intensive embedded heuristic by finding tours of better average quality than iterated 3-opt in many less iterations and providing the best heuristic solutions known, for some instance classes.     

Application of visual clustering properties of self organizing map in machine-part cell formation

Cellular manufacturing (CM) is an approach that includes both flexibility of job shops and high production rate of flow lines. Although CM provides many benefits in reducing throughput times, setup times, work-in-process inventories but the design of CM is complex and NP complete problem. The cell formation problem based on operation sequence (ordinal data) is rarely reported in the literature. The objective of the present paper is to propose a visual clustering approach for machine-part cell formation using self organizing map (SOM) algorithm an unsupervised neural network to achieve better group technology efficiency measure of cell formation as well as measure of SOM quality. The work also has established the criteria of choosing an optimum SOM size based on results of quantization error, topography error, and average distortion measure during SOM training which have generated the best clustering and preservation of topology. To evaluate the performance of the proposed algorithm, we tested the several benchmark problems available in the literature. The results show that the proposed approach not only generates the best and accurate solution as any of the results reported, so far, in literature but also, in some instances the results produced are even better than the previously reported results. The effectiveness of the proposed approach is also statistically verified.     

Tsp-based scheduling in a batch-wise hybrid flow-shop

In the past few years, flexible production systems have allowed an extensive exploitation of new technologies, but have also led to new difficulties in production planning management science. The model presented in this paper extends the traditional HFS (hybrid flow-shop) scheduling problem to the case in which jobs are due to follow strict precedence constraints and batch assignment constraints and the parallel machines at a stage are served by a bottleneck machine. A variant of the well-known TSP problem is used to develop an efficient heuristic solution for the problem. The effectiveness of the proposed approach is validated through a comparison with different heuristics traditionally used in HFS scheduling problems. Furthermore, a simple insertion heuristic based on the TSP model of the problem is tested. Finally, a MIP-based approach is also explored to provide the optimum solutions within much larger times for comparison.     

Scheduling jobs with ready times and precedence constraints on parallel batch machines using metaheuristics

In this paper, we discuss a scheduling problem for parallel batch machines where the jobs have ready times. Problems of this type can be found in semiconductor wafer fabrication facilities (wafer fabs). In addition, we consider precedence constraints among the jobs. Such constraints arise, for example, in scheduling subproblems of the shifting bottleneck heuristic when complex job shop scheduling problems are tackled. We use the total weighted tardiness as the performance measure to be optimized. Hence, the problem is NP-hard and we have to rely on heuristic solution approaches. We consider a variable neighborhood search (VNS) scheme and a greedy randomized adaptive search procedure (GRASP) to compute efficient solutions. We assess the performance of the two metaheuristics based on a large set of randomly generated problem instances and based on instances from the literature. The obtained computational results demonstrate that VNS is a very fast heuristic that quickly leads to high-quality solutions, whereas the GRASP is slightly outperformed by the VNS approach. However, the GRASP approach has the advantage that it can be parallelized in a more natural manner compared to VNS.