The Firefighter Problem is a graph-based optimization problem in which a threat called ‘fire’ has to be prevented from spreading on a graph by protecting graph vertices. Solutions to this problem are permutations, which are used to determine the order in which vertices of the graph are to be protected. In this paper, a crossover meta-operator Seq-X is proposed which works by applying various crossover operators available for the permutation-based representation. The quality of solutions generated by sequences of different crossover operators is analyzed during a preliminary run of the optimization algorithm on small problem instances and this information is used by Seq-X to decide which operator to select when the crossover operation has to be performed. The experiments described in this paper show that information collected when optimizing solutions for problem instances based on graphs with 1000 vertices can effectively be used for graphs with up to 10000 vertices.

This paper presents a visualization method useful for understanding the behaviour of an evolutionary algorithm tasked with discovering the propagation mechanism driving the spread of an epidemic. In this paper, the explainability in computational intelligence is addressed at two levels: first, a bi-objective Genetic Programming approach is used, which allows constructing a human-readable description of the disease propagation model based on observations or simulation results. Second, a visualization technique is used for understanding the working of the evolutionary algorithm itself with the aim of discovering clusters of similar solutions, which can later be analyzed in detail. Experimental results show that the proposed Genetic Programming approach is able to discover a human-readable representation of the epidemic propagation mechanism effectively. The visual analysis allowed discovering two clusters of solutions, one in which solutions are similar to the exact representation of the propagation mechanism, and second in which solutions are logically equivalent to the propagation mechanism but are much more bloated.

Abstract This paper addresses a problem of systemic risk minimization in which the optimization algorithm has to simultaneously minimize the number of companies affected by a wave of bankruptcies simulated on a graph as well as the level of reserves the companies keep to avoid going bankrupt. A MOEA/D-NN algorithm (where NN stands for a neural network) is proposed, which optimizes parameters of a machine learning model (a neural network) used in turn to determine the level of reserves the companies keep, based on several attributes describing each node in the graph. In the experiments, the proposed MOEA/D-NN algorithm was found to outperform comparison methods: evolutionary algorithms optimizing the level of reserves for all companies and a method based on the training of neural networks on a dataset previously collected by an evolutionary algorithm solving “training” instances of the optimization problem. The neural networks optimized by MOEA/D-NN were also tested on problem instances based on REDS graphs generated using varying values ofR,E, andSparameters and were found to be applicable to these instances for certain ranges of parameters. TheRparameter controlling the possibility of generating long-distance connections was found to have a bigger impact on the performance of the optimized neural networks than the other two parameters.

We consider search spaces associated with neural network channel configuration. Architectures and their accuracy are visualised using low-dimensional Euclidean embedding (LDEE). Optimisation dynamics are captured using local optima networks (LONs). LONs are a compression of a fitness landscape: the nodes are local optima and the edges are search transitions between them. Several neural architecture search algorithms are tested on the search space and we discover that iterated local search (ILS) is a competitive algorithm for neural channel configuration. We additionally implement a landscape-aware ILS which performs well. Observations from the search and landscape space analyses bring visual clarity and insight to the science of neural network channel design: the results indicate that a high number of channels, kept constant throughout the network, is beneficial.

Abstract The Inventory Routing Problem (IRP) is a combinatorial optimization problem that combines routing decisions with inventory management. In this paper, an approach to solving the IRP is studied, which aims at using an external knowledge source (a known good solution or user interaction) to improve the results attained by an evolutionary algorithm solving an IRP instance. The proposed method improves the best solution found by the evolutionary algorithm by modifying schedules for some of the retailers according to those present in the known good solution or to schedules provided by a domain expert. The experiments shown that to improve the optimization results it suffices to perform a few repetitions of the knowledge import procedure. This observation motivates further research on user-interactive optimization algorithms for the IRP, because the number of interactions needed to improve the results can easily be handled by the user.

This paper studies a method of generating hard instances of the Inventory Routing Problem (IRP) using evolutionary algorithms. The difficulty of the generated instances is measured by the time used by the CPLEX solver to solve a given instance. In order to ensure that feasible problem instances are generated, the Infeasibility Driven Evolutionary Algorithm (IDEA) is used along with seeding of the population with perturbed copies of an instance taken from a well-known set of benchmark instances. In the experiments a significant increase in problem solving time was observed with respect to IRP instances proposed in the literature. Generated IRP instances do not show any unusual properties that would make them unrealistic and they are made available as benchmarks for optimization algorithms. In the paper an attempt was made to compare generated problem instances and to draw conclusions regarding the structure of these instances, however, this has proven to be a difficult task.

In this paper feasibility-preserving genetic operators for hybrid algorithms using TSP solvers for the Inventory Routing Problem (IRP) are studied. The IRP is a problem of jointly optimizing delivery schedules and routes for vehicles transporting products from a supplier to a number of retailers. This optimization problem is highly constrained, because limits on inventory levels as well as the vehicle capacity have to be taken into account. Moreover, the IRP is a generalization of the TSP and solving the TSP effectively is an important part of obtaining good solutions to the IRP. In this paper evolutionary algorithms are used for solving the IRP, but finding good routes is delegated to a state-of-the-art TSP solver. Therefore, genetic operators used in this paper focus on constructing the delivery schedule and not on optimizing the routes. In the experimental part of the paper an evolutionary algorithm using feasibility-preserving genetic operators is compared to the Infeasibility Driven Evolutionary Algorithm (IDEA) that uses the selective pressure to obtain feasible solutions. Presented results suggest that designing good feasibility-preserving genetic operators is important, because allowing the optimization algorithm to generate infeasible solutions and handling infeasibility in IDEA using the selective pressure leads to inferior results.

This paper addresses the Multiobjective Travelling Salesman Problem (MoTSP) with the aim to study the effects of including random immigrants in the population of solutions processed by the evolutionary algorithm. Random immigrants are typically used in evolutionary optimization in order to increase the diversity of the population and to allow the algorithm to explore a larger area of the search space. Introducing random immigrants incurs a certain overhead which is especially significant in combinatorial optimization, because local search procedures are usually employed, which, while effective in improving the solutions, are computationally expensive. In this paper several strategies of introducing new specimens are tested with the aim of improving the effectiveness of the optimization process given a limited computation time. In the experiments the proposed approach was tested on kroABnnn instances of the MoTSP. It was found to improve the results of multiobjective optimization in terms of both the hy-pervolume and the IGD indicators. The most effective immigration strategy turned out to be to decrease the number of immigrants with time.

This paper studies a bi-objective optimization problem in which the goal is to optimize connections between entities in the market in order to make the entire system resilient to shocks of varying magnitudes. As observed in the literature concerning contagion on the inter-bank market, no system structure maximizes resilience to shocks of all sizes. For larger shocks more connections between entities make the crisis worse, and for smaller shocks more connected systems turn out to be more resilient than less connected ones.

This paper proposes a method for visualizing combinatorial search spaces named low-dimensional Euclidean embedding (LDEE). The proposed method transforms the original search space, such as a set of permutations or binary vectors, to Rk(with k = 2 or 3 in practice) while aiming to preserve spatial relationships existing in the original space. The LDEE method uses the t-distributed stochastic neighbor embedding (t-SNE) to transform solutions from the original search space to the Euclidean one. In this paper, it is mathematically shown that the assumptions underlying the t-SNE method are valid in the case of permutation spaces with the Mallows distribution. The same is true for other metric spaces provided that the distribution of points is assumed to be normal with respect to the adopted metric. The embedding obtained using t-SNE is further refined to ensure visual separation of individual solutions. The visualization obtained using the LDEE method can be used for analyzing the behavior of the population in a population-based metaheuristic, the working of the genetic operators, etc. Examples of visualizations obtained using this method for the four peaks problem, the firefighter problem, the knapsack problem, the quadratic assignment problem, and the traveling salesman problem are presented in this paper.

This paper concerns using evolutionary algorithms for optimization in epidemics prevention. Spreading of the epidemic is simulated on a graph and the goal of optimization is to determine which vertices in the graph to vaccinate in order to minimize the number of vertices affected by the epidemic. Decisions whether to vaccinate a vertex or not are represented using a binary genotype.

The Firefighter Problem (FFP) is a combinatorial optimization problem in which the goal is to find the best way of protecting nodes in a graph from spreading fire or other threat given limited resources. Because of high computational complexity the FFP is often solved using metaheuristic methods such as evolutionary algorithms (EAs). The problem that arises is that many crossover operators used in EAs were developed with problems such as the TSP or the flowshop problem in mind and are therefore not optimized towards the FFP.

This paper concerns knowledge discovery from ultra-high frequency financial time series of order book data from the London Stock Exchange Rebuild Order Book database. It focuses on extracting frequent patterns from order book shapes, which characterize particular stock market states. Because order book shapes vary considerably in time, it is necessary to convert the original price-volume representation to real vectors of a fixed length. In this paper two methods of representing order books are studied: using Gaussian Mixture Models (GMMs), and Gaussian density-based representation (GD). Both methods produce real vectors of a constant length and frequent patterns are defined as codebooks of clusters of these real vectors. To obtain useful knowledge we would like these patterns to represent different states of the market, be stable in time, and be universal. We define universality as periodical reoccurence, which makes historical knowledge usable for future decision making. Because of these time-related requirements a typical spatial clustering is not sufficient. This paper studies the usage of an evolutionary algorithm to obtain patterns with desirable properties.

Multiobjective optimization of portfolios aims at finding sets of stocks which are expected to provide a possibly high return while retaining a moderate level of risk. The Pareto front of portfolios generated by the optimization algorithm represents attainable trade-offs between returns obtained by the portfolios and the level of risk involved in the investment. This paper studies the relationship between location of portfolios in the Pareto front and future returns of these portfolios. It is observed that the highest future returns can be obtained for the portfolios with the highest return and risk measures observed in the past but also for those with the lowest return and risk in the Pareto front. Neither constantly selecting portfolios with high return on historical data nor, conversely, those with low historical risk (but also low return) yields high future returns. Based on these observations a method is proposed for adaptively selecting the best portfolios for investment from solutions contained in the Pareto front. The proposed method selects high-return (but also high-risk) portfolios or low-risk (but also low-return) portfolios based on the behaviour of the stock marked index in the time period preceding the moment of investment. The proposed method outperforms both the strategy of always selecting the portfolios with the highest return from the past and the risk-averse strategy of selecting portfolios with low risk measure.

Neural networks are widely used in time-series forecasting. One of the issues that arise in neural networks applications is that when a neural network is trained for too long the quality of the predictions tends to deteriorate. To overcome this problem various methods of early stopping are employed. This paper proposes a new approach to early stopping issue in neural network training. In the approach presented the validation series is chosen based on its mean dynamic correlation with forecasted series. The approach is verified by application to macroeconomic data where suitable sets of series are commonly available.