EASYLOCAL++: an object‐oriented framework for the flexible design of local‐search algorithms

Local search is a paradigm for search and optimization problems, which has recently evidenced to be very effective for a large number of combinatorial problems. Despite the increasing interest of the research community in this subject, there is still a lack of a widely‐accepted software tools for local search. We propose EASY LOCAL , an object‐oriented framework for the design and the analysis of local‐search algorithms. The abstract classes that compose the framework specify and implement the invariant part of the algorithm and are meant to be specialized by concrete classes that supply the problem‐dependent part. The framework provides the full control structures of the algorithms, and the user has only to write the problem‐specific code. Furthermore, the framework comes with some tools that simplify the analysis of the algorithms. The architecture of EASY LOCAL provides a principled modularization for the solution of combinatorial problems by local search and helps the user by deriving a neat conceptual scheme of the application. It also supports the design of combinations of basic techniques and/or neighborhood structures. The framework has been tested in some applicative domains and has proved to be flexible enough in the implementation of algorithms for the solution of various scheduling problems. Copyright © 2003 John Wiley & Sons, Ltd.     

An extended Akers graphical method with a biased random‐key genetic algorithm for job‐shop scheduling

This paper presents a local search, based on a new neighborhood for the job‐shop scheduling problem, and its application within a biased random‐key genetic algorithm. Schedules are constructed by decoding the chromosome supplied by the genetic algorithm with a procedure that generates active schedules. After an initial schedule is obtained, a local search heuristic, based on an extension of the 1956 graphical method of Akers, is applied to improve the solution. The new heuristic is tested on a set of 205 standard instances taken from the job‐shop scheduling literature and compared with results obtained by other approaches. The new algorithm improved the best‐known solution values for 57 instances.     

Enhanced genetic algorithms for a bi‐objective bus driver rostering problem: a computational study

In this work, the bus driver rostering problem is considered in the context of a noncyclic rostering, with two objectives representing either the company or the drivers’ interests. A network model and a proof of the NP‐hardness of the problem are presented, along with a bi‐objective memetic algorithm that combines a specific decoder with a utopian/lexicographic elitism, a strength Pareto fitness evaluation, and a local search procedure. By taking real and benchmark instances the computational behavior of the memetic algorithm is compared with simpler versions to assess the effects of the embedded components. The developed algorithm is a valuable tool for bus companies’ planning departments insofar as it yields at low computing times a pool of good quality rosters that reconcile contradictory objectives. This study shows that simple enhancements in standard bi‐objective genetic algorithms may improve the results for this difficult combinatorial problem.     

Design and analysis of stochastic local search for the multiobjective traveling salesman problem

Stochastic local search (SLS) algorithms are typically composed of a number of different components, each of which should contribute significantly to the final algorithm's performance. If the goal is to design and engineer effective SLS algorithms, the algorithm developer requires some insight into the importance and the behavior of possible algorithmic components. In this paper, we analyze algorithmic components of SLS algorithms for the multiobjective travelling salesman problem. The analysis is done using a careful experimental design for a generic class of SLS algorithms for multiobjective combinatorial optimization. Based on the insights gained, we engineer SLS algorithms for this problem. Experimental results show that these SLS algorithms, despite their conceptual simplicity, outperform a well-known memetic algorithm for a range of benchmark instances with two and three objectives.     

ADAPTIVE RANKS CLONE AND k‐NEAREST NEIGHBOR LIST–BASED IMMUNE MULTI‐OBJECTIVE OPTIMIZATION

Artificial immune systems (AIS) are computational systems inspired by the principles and processes of the vertebrate immune system. The AIS‐based algorithms typically exploit the immune system's characteristics of learning and adaptability to solve some complicated problems. Although, several AIS‐based algorithms have proposed to solve multi‐objective optimization problems (MOPs), little focus have been placed on the issues that adaptively use the online discovered solutions. Here, we proposed an adaptive selection scheme and an adaptive ranks clone scheme by the online discovered solutions in different ranks. Accordingly, the dynamic information of the online antibody population is efficiently exploited, which is beneficial to the search process. Furthermore, it has been widely approved that one‐off deletion could not obtain excellent diversity in the final population; therefore, a k‐nearest neighbor list (where k is the number of objectives) is established and maintained to eliminate the solutions in the archive population. The k‐nearest neighbors of each antibody are founded and stored in a list memory. Once an antibody with minimal product of k‐nearest neighbors is deleted, the neighborhood relations of the remaining antibodies in the list memory are updated. Finally, the proposed algorithm is tested on 10 well‐known and frequently used multi‐objective problems and two many‐objective problems with 4, 6, and 8 objectives. Compared with five other state‐of‐the‐art multi‐objective algorithms, namely NSGA‐II, SPEA2, IBEA, HYPE, and NNIA, our method achieves comparable results in terms of convergence, diversity metrics, and computational time.     

Optimizing a bi‐objective vehicle routing problem that appears in industrial enterprises

In this paper, a new solution method is implemented to solve a bi‐objective variant of the vehicle routing problem that appears in industry and environmental enterprises. The solution involves designing a set of routes for each day in a period, in which the service frequency is a decision variable. The proposed algorithm, a muti‐start multi‐objective local search algorithm (MSMLS), minimizes total emissions produced by all vehicles and maximizes the service quality measured as the number of times that a customer is visited by a vehicle in order to be served. The MSMLS is a neighbourhood‐based metaheuristic that obtains high‐quality solutions and that is capable of achieving better performance than other competitive algorithms. Furthermore, the proposed algorithm is able to perform rapid movements thanks to the easy representation of the solutions.     

A novel local search algorithm with configuration checking and scoring mechanism for the set k‐covering problem

The set k‐covering problem, an extension of the classical set covering problem, is an important NP‐hard combinatorial optimization problem with extensive applications, including computational biology and wireless network. The aim of this paper is to design a new local search algorithm to solve this problem. First, to overcome the cycling problem in local search, the set k‐covering configuration checking (SKCC) strategy is proposed. Second, we use the cost scheme of elements to define the scoring mechanism so that our algorithm can find different possible good‐quality solutions. Having combined the SKCC strategy with the scoring mechanism, a subset selection strategy is designed to decide which subset should be selected as a candidate solution component. After that, a novel local search framework, as we call DLL_ccsm (diversion local search based on configuration checking and scoring mechanism), is proposed. DLL_ccsm is evaluated against two state‐of‐the‐art algorithms. The experimental results show that DLL_ccsm performs better than its competitors in terms of solution quality in most classical instances.     

Hybrid algorithms for the twin–screw extrusion configuration problem

The twin-screw configuration problem (TSCP) arises in the context of polymer processing, where twin-screw extruders are used to prepare polymer blends, compounds or composites. The goal of the TSCP is to define the configuration of a screw from a given set of screw elements. The TSCP can be seen as a sequencing problem as the order of the screw elements on the screw axis has to be defined. It is also inherently a multi-objective problem since processing has to optimize various conflicting parameters related to the degree of mixing, shear rate, or mechanical energy input among others. In this article, we develop hybrid algorithms to tackle the bi-objective TSCP. The hybrid algorithms combine different local search procedures, including Pareto local search and two phase local search algorithms, with two different population-based algorithms, namely a multi-objective evolutionary algorithm and a multi-objective ant colony optimization algorithm. The experimental evaluation of these approaches shows that the best hybrid designs, combining Pareto local search with a multi-objective ant colony optimization approach, outperform the best algorithms that have been previously proposed for the TSCP.     

Minimum common string partition: on solving large‐scale problem instances

Minimum common string partition is an NP‐hard combinatorial optimization problem from the bioinformatics field. The current state‐of‐the‐art algorithm is a hybrid technique known as construct, merge, solve, and adapt (CMSA). This algorithm combines two main algorithmic components: generating solutions in a probabilistic way and solving reduced subinstances obtained from the tackled problem instances, if possible, to optimality. However, the CMSA algorithm was not intended for application to very large problem instances. Therefore, in this paper we present a technique that makes CMSA, and other available algorithms for this problem, applicable to problem instances that are about one order of magnitude larger than the largest problem instances considered so far. Moreover, a reduced variable neighborhood search (RVNS) for solving the tackled problem, based on integer programming, is introduced. The experimental results show that the modified CMSA algorithm is very strong for problem instances based on rather small alphabets. With growing alphabet size, it turns out that RVNS has a growing advantage over CMSA.     

双邻域查找组播聚合算法

光传输网络中聚合组播问题是一个完全NP 难问题,提出了一种解决聚合组播问题的双邻域查找算法.该算法使得生成的聚合树数量在满足波长约束的前提下,带宽浪费比率尽可能地小.基于贪婪策略定义了一种优先聚合规则以生成初始解;定义了两种邻域结构,使邻域查找具有效率;提出了跳坑策略以跳出局部最优解并且将查找引向有希望的方向.模拟实验结果表明:该算法可以有效地进行组播树的聚合,当轻载时,组播组阻塞比率始终为0;当重载时,与其他算法相比,平均带宽浪费比率降低25%以上.因此,对不同的网络状况都能获得较好的性能.     

混合算法的邻域结构变更研究及在排样问题上的应用

本文研究了全局搜索算法和局部搜索算法的混合机制,设计了基于邻域搜索和遗传算法的混合搜索算法。该算法结合了遗传算法的全局搜索特性和邻域局部贪婪搜索特性;在分析排样问题碰靠过程特征的基础上,构建了排样问题邻域假设,当邻域假设满足时,遗传算法+邻域搜索能很好发挥作用;当不能判断邻域结构是否满足邻域假设时,提出了建立遗传算法+匹配变邻域的搜索算法,该算法兼顾了组合优化中邻域搜索的局部搜索无效的情况,实现了匹配的变邻域混合算法在排样优化问题中的应用。实例结果标明,排样图形不一样,其求解难度不一样,该算法均搜索到了更好的排样模式,验证了算法的有效性。     

A two‐phase harmony search algorithm for continuous optimization

Harmony search (HS) algorithm is inspired by the music improvisation process in which a musician searches for the best harmony and continues to polish the harmony to improve its aesthetics. The efficiency of evolutionary algorithms depends on the extent of balance between diversification and intensification during the course of the search. An ideal evolutionary algorithm must have efficient exploration in the beginning and enhanced exploitation toward the end. In this paper, a two‐phase harmony search (TPHS) algorithm is proposed that attempts to strike a balance between exploration and exploitation by concentrating on diversification in the first phase using catastrophic mutation and then switches to intensification using local search in the second phase. The performance of TPHS is analyzed and compared with 4 state‐of‐the‐art HS variants on all the 30 IEEE CEC 2014 benchmark functions. The numerical results demonstrate the superiority of the proposed TPHS algorithm in terms of accuracy, particularly on multimodal functions when compared with other state‐of‐the‐art HS variants; further comparison with state‐of‐the‐art evolutionary algorithms reveals excellent performance of TPHS on composition functions. Composition functions are combined, rotated, shifted, and biased version of other unimodal and multimodal test functions and mimic the difficulties of real search spaces by providing a massive number of local optima and different shapes for different regions of the search space. The performance of the TPHS algorithm is also evaluated on a real‐life problem from the field of computer vision called camera calibration problem, ie, a 12‐dimensional highly nonlinear optimization problem with several local optima.     

An MO‐GVNS algorithm for solving a multiobjective hybrid flow shop scheduling problem

This paper addresses the multiobjective hybrid flow shop (MOHFS) scheduling problem. In the MOHFS problem considered here, we have a set of jobs that must be performed in a set of stages. At each stage, we have a set of unrelated parallel machines. Some jobs may skip stages. The evaluation criteria are the minimizations of makespan, the weighted sum of the tardiness, and the weighted sum of the earliness. For solving it, an algorithm based on the multiobjective general variable neighborhood search (MO‐GVNS) metaheuristic, named adapted MO‐GVNS, is proposed. This work also presents and compares the results obtained by the adapted MO‐GVNS with those of four other algorithms: multiobjective reduced variable neighborhood search, nondominated sorting genetic algorithm II (NSGA‐II), and NSGA‐III, and another MO‐GVNS from the literature. The results were evaluated based on the Hypervolume, Epsilon, and Spacing metrics, and statistically validated by the Levene test and confidence interval charts. The results showed the efficiency of the proposed algorithm for solving the MOHFS problem.     

LOCAL++: A C++ framework for local search algorithms

Local search is an emerging paradigm for combinatorial search which has recently been shown to be very effective for a large number of combinatorial problems. It is based on the idea of navigating the search space by iteratively stepping from one solution to one of its neighbors, which are obtained by applying a simple local change to it. In this paper we present LOCAL++, an object‐oriented framework to be used as a general tool for the development and implementation of local search algorithms in C++. The framework comprises a hierarchy of abstract template classes, one for each local search technique taken into account (i.e. hill‐climbing, simulated annealing and tabu search). Each class specifies and implements the invariant part of the algorithm built according to the technique, and is supposed to be specialized by a concrete class once a given search problem is considered, so as to implement the problem‐dependent part of the algorithm. LOCAL++ comprises also a set of abstract classes for creating new techniques by combining different search techniques and different neighborhood relations. The architecture of LOCAL++ provides a principled modularization for the solution of combinatorial search problems, and helps the designer deriving a neat conceptual scheme of the application, thus facilitating the development and debugging phases. LOCAL++ proved to be flexible enough for the implementation of the algorithms solving various scheduling problems. Copyright © 2000 John Wiley & Sons, Ltd.     

Two‐node‐connected star problem

In this study, a two‐node‐connected star problem (2NCSP) is introduced. We are given a simple graph and internal and external costs for each link of the graph. The goal is to find the minimum‐cost spanning subgraph, where the core is two‐node‐connected and the remaining external nodes are connected to the core. First, we show that the 2NCSP belongs to the class of NP‐hard computational problems. Therefore, a greedy randomized adaptive search procedure (GRASP) heuristic is developed, enriched with a variable neighborhood descent (VND). The neighborhood structures include exact integer linear programming models to find the best paths and two‐node‐connected replacements, as well as a shaking operation in order to prevent being trapped in a local minima. The ring star problem (RSP) represents a relevant model in network optimization, where the core is a ring instead of an arbitrary two‐node‐connected graph. We contrast our GRASP/VND methodology with a previous reference work on the RSP in order to highlight the effectiveness of our heuristic. The heuristic is competitive, and the best results produced for several instances so far are under study. In this study, a discussion of the results and trends for future work are provided.     

Evolutionary algorithms for the multi‐objective test data generation problem

Automatic test data generation is a very popular domain in the field of search‐based software engineering. Traditionally, the main goal has been to maximize coverage. However, other objectives can be defined, such as the oracle cost, which is the cost of executing the entire test suite and the cost of checking the system behavior. Indeed, in very large software systems, the cost spent to test the system can be an issue, and then it makes sense by considering two conflicting objectives: maximizing the coverage and minimizing the oracle cost. This is what we did in this paper. We mainly compared two approaches to deal with the multi‐objective test data generation problem: a direct multi‐objective approach and a combination of a mono‐objective algorithm together with multi‐objective test case selection optimization. Concretely, in this work, we used four state‐of‐the‐art multi‐objective algorithms and two mono‐objective evolutionary algorithms followed by a multi‐objective test case selection based on Pareto efficiency. The experimental analysis compares these techniques on two different benchmarks. The first one is composed of 800 Java programs created through a program generator. The second benchmark is composed of 13 real programs extracted from the literature. In the direct multi‐objective approach, the results indicate that the oracle cost can be properly optimized; however, the full branch coverage of the system poses a great challenge. Regarding the mono‐objective algorithms, although they need a second phase of test case selection for reducing the oracle cost, they are very effective in maximizing the branch coverage. Copyright © 2011 John Wiley & Sons, Ltd.     

Multiobjective pseudo‐variable neighborhood descent for a bicriteria parallel machine scheduling problem with setup time

A multiobjective variable neighborhood descent (VND) based heuristic is developed to solve a bicriteria parallel machine scheduling problem. The problem considers two objectives, one related to the makespan and the other to the flow time, where the setup time depends on the sequence, and the machines are identical. The heuristic has a set of neighborhood structures based on swap, remove, and insertion moves. We propose changing the local search inside the VND to a sequential search through the neighborhoods to obtain nondominated points for the Pareto‐front quickly. In the numerical tests, we consider a single‐objective version of the heuristic, comparing the results on 510 benchmark instances to show that it is quite effective. Moreover, new instances are generated in accordance with the literature for the bicriteria problem, showing the ability of the proposed heuristic to return an efficient set of nondominate solutions compared with the well‐known nondominated sorting genetic algorithm II.     

求解柔性作业车间调度问题的新型改进Jaya算法

为有效解决复杂的柔性作业车间调度问题,以最小化最大完成时间为目标,提出了一种结合了变邻域搜索算法的新型改进Jaya算法来求解。为不断挖掘和优化探索最优解,提高算法求解的结果质量,通过Jaya算法的原理重新提出一种解的更新机制,此外在Jaya算法原理的基础上嵌入一种变邻域搜索策略,并在传统邻域结构的基础上重新设计了两种新型邻域结构,扩大了邻域搜索范围,增强了Jaya算法的局部搜索能力,避免算法因失去解的多样性从而陷入局部最优。运用基准算例对该算法的求解性能进行了验证,并与其他算法的仿真结果进行对比,结果表明该改进算法的求解效率更高。     

混合流水车间调度的变邻域禁忌搜索算法

针对混合流水车间系统的最小化Makespan调度问题,提出一种基于关键路径理论的变邻域禁忌搜索算法,讨论其关键技术。在该算法中,提出基于关键路径的毗邻域概念,防止搜索算法陷入局部最优解,采用变邻域搜索策略,在无法改进解时,实现对移动毗邻域的搜索。仿真结果表明,该算法获得的调度结果优于简化禁忌搜索和启发式算法。     

Rough‐fuzzy quadratic minimum spanning tree problem

A quadratic minimum spanning tree problem determines a minimum spanning tree of a network whose edges are associated with linear and quadratic weights. Linear weights represent the edge costs whereas the quadratic weights are the interaction costs between a pair of edges of the graph. In this study, a bi‐objective rough‐fuzzy quadratic minimum spanning tree problem has been proposed for a connected graph, where the linear and the quadratic weights are represented as rough‐fuzzy variables. The proposed model is formulated by using rough‐fuzzy chance‐constrained programming technique. Subsequently, three related theorems are also proposed for the crisp transformation of the proposed model. The crisp equivalent models are solved with a classical multi‐objective solution technique, the epsilon‐constraint method and two multi‐objective evolutionary algorithms: (a) nondominated sorting genetic algorithm II (NSGA‐II) and (b) multi‐objective cross‐generational elitist selection, heterogeneous recombination, and cataclysmic mutation (MOCHC) algorithm. A numerical example is provided to illustrate the proposed model when solved with different methodologies. A sensitivity analysis of the example is also performed at different confidence levels. The performance of NSGA‐II and MOCHC are analysed on five randomly generated instances of the proposed model. Finally, a numerical illustration of an application of the proposed model is also presented in this study.