Fitness landscape analysis and memetic algorithms for the quadratic assignment problem

In this paper, a fitness landscape analysis for several instances of the quadratic assignment problem (QAP) is performed, and the results are used to classify problem instances according to their hardness for local search heuristics and meta-heuristics based on local search. The local properties of the fitness landscape are studied by performing an autocorrelation analysis, while the global structure is investigated by employing a fitness distance correlation analysis. It is shown that epistasis, as expressed by the dominance of the flow and distance matrices of a QAP instance, the landscape ruggedness in terms of the correlation length of a landscape, and the correlation between fitness and distance of local optima in the landscape together are useful for predicting the performance of memetic algorithms-evolutionary algorithms incorporating local search (to a certain extent). Thus, based on these properties, a favorable choice of recombination and/or mutation operators can be found. Experiments comparing three different evolutionary operators for a memetic algorithm are presented.     

Bag of local landscape features for fitness landscape analysis

Evolutionary computation is a research field dealing with black-box and complex optimization problems whose fitness landscapes are usually unknown in advance. It is difficult to select an appropriate evolutionary algorithm and parameters for a given problem due to the black-box setting although many evolutionary algorithms have been developed. In this context, several landscape features have been proposed and their usefulness examined for understanding the problem. In this paper, we propose a novel feature vector by focusing on the local landscape in order to characterize the fitness landscape. The proposed landscape features are a vector form and composed of a histogram of quantized local landscape features. We introduce two implementation methods of this concept, called the bag of local landscape patterns (BoLLP) and the bag of evolvability (BoEvo). The BoLLP uses the fitness pattern of the neighbors of a certain candidate solution, and the BoEvo uses the number of better candidate solutions in the neighbors as the local landscape features. Furthermore, the hierarchical versions of the BoLLP and the BoEvo, concatenated feature vectors with different sample sizes, are considered to capture the landscape characteristic with various resolutions. We extract the proposed landscape feature vectors from well-known continuous optimization benchmark functions and the BBOB benchmark function set to investigate their properties; the visualization of the proposed landscape features, clustering and running time prediction experiments are conducted. Then the effectiveness of the proposed landscape features for the fitness landscape analysis is discussed based on the experimental results.     

Iterated local search using an add and delete hyper-heuristic for university course timetabling

Hyper-heuristics are (meta-)heuristics that operate at a higher level to choose or generate a set of low-level (meta-)heuristics in an attempt of solve difficult optimization problems. Iterated local search (ILS) is a well-known approach for discrete optimization, combining perturbation and hill-climbing within an iterative framework. In this study, we introduce an ILS approach, strengthened by a hyper-heuristic which generates heuristics based on a fixed number of add and delete operations. The performance of the proposed hyper-heuristic is tested across two different problem domains using real world benchmark of course timetabling instances from the second International Timetabling Competition Tracks 2 and 3. The results show that mixing add and delete operations within an ILS framework yields an effective hyper-heuristic approach.     

Scatter search for mixed blocking flowshop scheduling

Empty or limited storage capacities between machines introduce various types of blocking constraint in the industries with flowshop environment. While large applications demand flowshop scheduling with a mix of different types of blocking, research in this area mainly focuses on using only one kind of blocking in a given problem instance. In this paper, using makespan as a criterion, we study permutation flowshops with zero capacity buffers operating under mixed blocking conditions. We present a very effective scatter search (SS) algorithm for this. At the initialisation phase of SS, we use a modified version of the well-known Nawaz, Enscore and Ham (NEH) heuristic. For the improvement method in SS, we use an Iterated Local Search (ILS) algorithm that adopts a greedy job selection and a powerful NEH-based perturbation procedure. Moreover, in the reference set update phase of SS, with small probabilities, we accept worse solutions so as to increase the search diversity. On standard benchmark problems of varying sizes, our algorithm very significantly outperforms well-known existing algorithms in terms of both the solution quality and the computing time. Moreover, our algorithm has found new upper bounds for 314 out of 360 benchmark problem instances.     

Exploring the fitness landscape and the run-time behaviour of an iterated local search algorithm for cost-based abduction

Cost-based abduction (CBA) is an important problem in reasoning under uncertainty, and can be considered a generalization of belief revision. CBA is known to be NP-hard and has been a subject of considerable research over the past decade. In this paper, we investigate the fitness landscape for CBA, by looking at fitness–distance correlation for local minima and at landscape ruggedness. Our results indicate that stochastic local search techniques would be promising on this problem. We go on to present an iterated local search algorithm based on hill-climbing, tabu search, and simulated annealing. We compare the performance of our algorithm to simulated annealing, and to Santos' integer linear programming method for CBA.     

A multi-restart iterated local search algorithm for the permutation flow shop problem minimizing total flow time

A variety of metaheuristics have been developed to solve the permutation flow shop problem minimizing total flow time. Iterated local search (ILS) is a simple but powerful metaheuristic used to solve this problem. Fundamentally, ILS is a procedure that needs to be restarted from another solution when it is trapped in a local optimum. A new solution is often generated by only slightly perturbing the best known solution, narrowing the search space and leading to a stagnant state. In this paper, a strategy is proposed to allow the restart solution to be generated from a group of solutions drawn from local optima. This allows an extension of the search space, while maintaining the quality of the restart solution. A multi-restart ILS (MRSILS) is proposed, with the performance evaluated on a set of benchmark instances and compared with six state of the art metaheuristics. The results show that the easily implementable MRSILS is significantly better than five of the other metaheuristics and comparable to or slightly better than the remaining one.     

Heuristic approaches for the family traveling salesman problem

In this article, we address the family traveling salesman problem (FTSP), an NP‐hard problem that may be seen as a generalization of the traveling salesman problem. In the FTSP, the set of cities is partitioned into several families and one wants to find the minimum cost route that visits a given number of cities in each family. We propose two metaheuristics, a population‐based method and a local search method, and a hybrid algorithm, which combines a branch‐and‐cut algorithm with a local search procedure. All the proposed methods improve the best known upper bounds from the literature. The local search method and the hybrid algorithm improve the best known upper bounds for all the benchmark instances with unknown optimal value, while the population‐based method improves the best known upper bounds for all instances, except two. Furthermore, we developed an instance generator to create additional test instances with different visit patterns. These newly generated instances were considered in the computational experiment and, thus, we provide optimal values or upper bounds for each instance. Additionally, we created a website dedicated to the FTSP, where this information is made available to the scientific community ( http://familytsp.rd.ciencias.ulisboa.pt/ ).     

Multi-start iterated local search metaheuristic for the multi-mode resource-constrained project scheduling problem

In this paper a multi-start iterated local search (MS-ILS) algorithm is presented as a new and effective approach to solve the multi-mode resource-constrained project scheduling problem (MRCPSP). The MRCPSP is a well-known project scheduling NP-Hard optimization problem, in which there is a trade-off between the duration of each project activity and the amount of resources they require to be completed. The proposed algorithm generates an initial solution, performs a local search to obtain a local optimum, subsequently, for a certain number of iterations, makes a perturbation to that local optimum and performs a new local search on the perturbed solution. This whole process then restarts with a different initial solution for a certain number of restarts. The algorithm was tested on benchmark instances of projects with 30, 50 and 100 activities from well-known libraries. The obtained results were compared to recent benchmark results from the literature. The proposed algorithm outperforms other solution methods found in related literature for the largest tested instances (100 activities), while for smaller instances it shows to be quite competitive, in terms of the average deviation against known lower bounds.     

Metaheuristics based on decision hierarchies for the traveling purchaser problem

In this paper we address the traveling purchaser problem, an NP‐hard problem that generalizes the traveling salesman problem. We present several metaheuristics that combine genetic algorithms and local search. The genetic algorithms are induced by different hierarchic orderings of the decision making regarding the route and the acquisition of the items. Computational experiments were carried out with benchmark instances and the results show that the proposed metaheuristics are a suitable tool to solve high‐dimensioned instances for which the exact methods do not provide solutions within a reasonable CPU time. For several instances, best new upper bounds for the optimum value of the objective function were obtained.     

Multidimensional knapsack problem: a fitness landscape analysis.

Fitness landscape analysis techniques are used to better understand the influence of genetic representations and associated variation operators when solving a combinatorial optimization problem. Five representations are investigated for the multidimensional knapsack problem. Common mutation operators, such as bit-flip mutation, are employed to generate fitness landscapes. Measures such as fitness distance correlation and autocorrelation are applied to examine the landscapes associated with the tested genetic encodings. Furthermore, additional experiments are made to observe the effects of adding heuristics and local optimization to the representations. Encodings with a strong heuristic bias are more efficient, and the addition of local optimization techniques further enhances their performance.     

求解车辆路径问题的改进扰动机制的ILS算法

针对车辆路径问题,提出一种改进的迭代局部搜索(ILS)算法。该算法基于破坏再重建(Ruin and Recreate)的思想,设计了一种新的扰动机制。扰动过程包含破坏和重建两个阶段,即先使用一种兼顾随机性和相关性的破坏方法对解进行破坏,并引入扰动因子控制解的破坏强度,然后再随机选择基本贪婪插入和改进贪婪插入算法完成解的修复。利用国际标准测试案例对常规扰动机制和改进后的扰动机制进行了测试,并与量子进化算法、蜂群算法进行了比较,实验结果表明改进后的ILS算法更加有效。     

Iterated Local Search for single-machine scheduling with sequence-dependent setup times to minimize total weighted tardiness

We present an Iterated Local Search (ILS) algorithm for solving the single-machine scheduling problem with sequence-dependent setup times to minimize the total weighted tardiness. The proposed ILS algorithm exhibits several distinguishing features, including a new neighborhood structure called Block Move and a fast incremental evaluation technique, for evaluating neighborhood solutions. Applying the proposed algorithm to solve 120 public benchmark instances widely used in the literature, we achieve highly competitive results compared with a recently proposed exact algorithm and five sets of best solutions of state-of-the-art metaheuristic algorithms in the literature. Specifically, ILS obtains the optimal solutions for 113 instances within a reasonable time, and it outperforms the previous best-known results obtained by metaheuristic algorithms for 34 instances and matches the best results for 82 instances. In addition, ILS is able to obtain the optimal solutions for the remaining seven instances under a relaxed time limit, and its computational efficiency is comparable with the state-of-the-art exact algorithm by Tanaka and Araki (Comput Oper Res 40:344–352, 2013). Finally, on analyzing some important features that affect the performance of ILS, we ascertain the significance of the proposed Block Move neighborhood and the fast incremental evaluation technique.     

A three-phase algorithm for flowshop scheduling with blocking to minimize makespan

This paper proposes a three-phase algorithm (TPA) for the flowshop scheduling problem with blocking (BFSP) to minimize makespan. In the first phase, the blocking nature of BFSP is exploited to develop a priority rule that creates a sequence of jobs. Using this as the initial sequence and a variant of the NEH-insert procedure, the second phase generates an approximate solution to the problem. Then, utilizing a modified simulated annealing algorithm incorporated with a local search procedure, the schedule generated in the second phase is improved in the third phase. A pruning procedure that helps evaluate most solutions without calculating their complete makespan values is introduced in the local search to further reduce the computational time needed to solve the problem. Results of the computational experiments with Taillard's benchmark problem instances show that the proposed TPA algorithm is relatively more effective and efficient in minimizing makespan for the BFSP than the state-of-the-art procedures. Utilizing these results, 53 out of 60 new tighter upper bounds have been found for large-sized Taillard's benchmark problem instances.     

MaxSAT-based large neighborhood search for high school timetabling

High School Timetabling (HSTT) is a well known and wide spread problem. The problem consists of coordinating resources (e.g. teachers, rooms), times, and events (e.g. lectures) with respect to various constraints. Unfortunately, HSTT is hard to solve and just finding a feasible solution for simple variants of HSTT have been proven to be NP-complete. We propose a new algorithm for HSTT which combines local search with a novel maxSAT-based large neighborhood search. A local search algorithm is used to drive an initial solution into a local optimum and then more powerful large neighborhood search (LNS) techniques based on maxSAT are used to further improve the solution. We prove the effectiveness of our approach with experimental results on instances taken from the Third International Timetabling Competition 2011 and the XHSTT-2014 benchmark archive. We were able to model 27 out of 39 instances. The remaining 12 instances were not modeled because the currently used maxSAT formulation for XHSTT does not support resource assignments in general. For the instances which could be modeled, our algorithm shows good performance when compared to other XHSTT state-of-the-art solvers and for several instances new best known upper bounds have been computed.     

校车路径问题的改进迭代局部搜索算法

针对考虑站点服务时间、学生最大乘车时间约束的校车路径问题(SBRP),提出一种改进迭代局部搜索(ILS)算法以提升求解质量。该算法使用大规模邻域搜索(LNS)算法作为扰动算子;在解的破坏过程中,设计一组解的破坏因子并赋以一定的选择概率,每隔若干次迭代后根据解的质量自适应更改破坏因子的选择概率,进而调整解的破坏程度。为提升ILS解的多样性,算法采用了基于偏差系数的邻域解接受准则。在国际基准测试案例上进行了测试,测试结果表明在ILS算法中使用自适应调整破坏程度的LNS扰动比常规扰动和其他破坏扰动的求解质量有大幅提升;与蚁群算法的比较结果进一步验证了改进算法的有效性。     

A hybrid variable neighborhood search algorithm for solving the limited-buffer permutation flow shop scheduling problem with the makespan criterion

This paper investigates the limited-buffer permutation flow shop scheduling problem (LBPFSP) with the makespan criterion. A hybrid variable neighborhood search (HVNS) algorithm hybridized with the simulated annealing algorithm is used to solve the problem. A method is also developed to decrease the computational effort needed to implement different types of local search approaches used in the HVNS algorithm. Computational results show the higher efficiency of the HVNS algorithm as compared with the state-of-the-art algorithms. In addition, the HVNS algorithm is competitive with the algorithms proposed in the literature for solving the blocking flow shop scheduling problem (i.e., LBPFSP with zero-capacity buffers), and finds 54 new upper bounds for the Taillard's benchmark instances.     

Fitness landscapes, memetic algorithms, and greedy operators for graph bipartitioning

The fitness landscape of the graph bipartitioning problem is investigated by performing a search space analysis for several types of graphs. The analysis shows that the structure of the search space is significantly different for the types of instances studied. Moreover, with increasing epistasis, the amount of gene interactions in the representation of a solution in an evolutionary algorithm, the number of local minima for one type of instance decreases and, thus, the search becomes easier. We suggest that other characteristics besides high epistasis might have greater influence on the hardness of a problem. To understand these characteristics, the notion of a dependency graph describing gene interactions is introduced. In particular, the local structure and the regularity of the dependency graph seems to be important for the performance of an algorithm, and in fact, algorithms that exploit these properties perform significantly better than others which do not. It will be shown that a simple hybrid multi-start local search exploiting locality in the structure of the graphs is able to find optimum or near optimum solutions very quickly. However, if the problem size increases or the graphs become unstructured, a memetic algorithm (a genetic algorithm incorporating local search) is shown to be much more effective.     

GA performance distributions and randomly generated binary constraint satisfaction problems

We investigate the variable performance of a genetic algorithm (GA) on randomly generated binary constraint satisfaction problem instances which occur near the phase transition from soluble to non-soluble. We first carry out a conventional landscape analysis and observe, next to a number of common features related to the interaction structure, important differences between the instances, such as the number of solutions, the quality of the paths to the solutions, and the lengths and extent of the neutral paths for mutation. We then split the dynamics of the GA into two phases: the ascent towards the high fitness region, and from this high fitness region to a solution. To gain further information about the GA's behavior in the first phase, we construct two models based on the much simpler fully separable functions, and try to match instances which show a similar performance distribution. Although far from exact, this technique of comparing with analog search problems gives a hint about the landscape elements that are responsible for the GA's slow ascent.     

Combining local branching and descent method for solving the multiple-choice knapsack problem with setups

In this paper, the multiple-choice knapsack problem with setups is tackled with an iterative method, where both local branching and descent method cooperate. First, an iterative procedure is designed for solving a series of mixed integer programming problems combined with a special reduced subproblem; that is, a combined model built by injecting some valid cardinality constraints. Second, the local branching-based learning strategy is embedded into an iterative search to mimic the variable neighborhood descent method, such that the local branching strategy drives the search process for enhancing the quality of the solutions. Third, the proposed method is experimentally analyzed on benchmark instances extracted from the literature, where its provided (lower) bounds are compared to those reached by methods published in the literature and the Cplex solver. Finally, its performance is evaluated by providing a statistical analysis.     

Breakout Local Search for maximum clique problems

The maximum clique problem (MCP) is one of the most popular combinatorial optimization problems with various practical applications. An important generalization of MCP is the maximum weight clique problem (MWCP) where a positive weight is associate to each vertex. In this paper, we present Breakout Local Search (BLS) which can be applied to both MC and MWC problems without any particular adaptation. BLS explores the search space by a joint use of local search and adaptive perturbation strategies. Extensive experimental evaluations using the DIMACS and BOSHLIB benchmarks show that the proposed approach competes favourably with the current state-of-art heuristic methods for MCP. Moreover, it is able to provide some new improved results for a number of MWCP instances. This paper also reports for the first time a detailed landscape analysis, which has been missing in the literature. This analysis not only explains the difficulty of several benchmark instances, but also justifies to some extent the behaviour of the proposed approach and the used parameter settings.