Pareto memetic algorithm with path relinking for bi-objective traveling salesperson problem

The paper presents an effective version of the Pareto memetic algorithm with path relinking and efficient local search for multiple objective traveling salesperson problem. In multiple objective Traveling salesperson problem (TSP), multiple costs are associated with each arc (link). The multiple costs may for example correspond to the financial cost of travel along a link, time of travel, or risk in the case of hazardous materials. The algorithm searches for new good solutions along paths in the decision space linking two other good solutions selected for recombination. Instead of a simple local search it uses short runs of tabu search based on the steepest version of the Lin–Kernighan algorithm. The efficiency of local search is further improved by the techniques of candidate moves and locked arcs. In the final step of the algorithm the neighborhood of each potentially Pareto-optimal solution is searched for new solutions that could be added to this set. The algorithm is compared experimentally to the state-of-the-art algorithms for multiple objective TSP.     

Heuristics for a coupled-operation scheduling problem

In this paper, we study a strongly NP-hard single machine scheduling problem in which each job consists of two operations that are separated by a time delay which lies within a specified range. The objective is to minimize the makespan. Determining the feasibility and, if applicable, makespan of any proposed permutation of the operations is non-trivial, requiring a longest path algorithm with O(n²) complexity for each permutation. Several heuristic algorithms are proposed: a deterministic and randomized construction algorithm, three descent algorithms and two reactive tabu search algorithms. The local search algorithms use a first improvement neighbourhood and mainly visit only feasible solutions within the search space. Results of extensive computational tests are reported, showing that the heavy computational burden of testing potential solutions renders the local search algorithms uncompetitive in comparison to the construction algorithms. The iterated descent algorithm performs least well.     

Finite-Time Performance Analysis of Static Simulated Annealing Algorithms

Generalized hill climbing (GHC) algorithms provide a framework for modeling local search algorithms for addressing intractable discrete optimization problems. Current theoretical results are based on the assumption that the goal when addressing such problems is to find a globally optimal solution. However, from a practical point of view, solutions that are close enough to a globally optimal solution (where close enough is measured in terms of the objective function value) for a discrete optimization problem may be acceptable. This paper introduces -acceptable solutions, where is a value greater than or equal to the globally optimal objective function value. Moreover, measures for assessing the finite-time performance of GHC algorithms, in terms of identifying -acceptable solutions, are defined. A variation of simulated annealing (SA), termed static simulated annealing (S²A), is analyzed using these measures. S²A uses a fixed cooling schedule during the algorithm's execution. Though S²A is provably nonconvergent, its finite-time performance can be assessed using the finite-time performance measures defined in terms of identifying -acceptable solutions. Computational results with a randomly generated instance of the traveling salesman problem are reported to illustrate the results presented. These results show that upper and lower estimates for the number of iterations to reach a -acceptable solution within a specified number of iterations can be obtained, and that these estimates are most accurate for moderate and high fixed temperature values for the S²A algorithm.     

Hybrid Metaheuristics for the Vehicle Routing Problem with Stochastic Demands

This article analyzes the performance of metaheuristics on the vehicle routing problem with stochastic demands (VRPSD). The problem is known to have a computationally demanding objective function, which could turn to be infeasible when large instances are considered. Fast approximations of the objective function are therefore appealing because they would allow for an extended exploration of the search space. We explore the hybridization of the metaheuristic by means of two objective functions which are surrogate measures of the exact solution quality. Particularly helpful for some metaheuristics is the objective function derived from the traveling salesman problem (TSP), a closely related problem. In the light of this observation, we analyze possible extensions of the metaheuristics which take the hybridized solution approach VRPSD-TSP even further and report about experimental results on different types of instances. We show that, for the instances tested, two hybridized versions of iterated local search and evolutionary algorithm attain better solutions than state-of-the-art algorithms.     

Enhanced speciation in particle swarm optimization for multi-modal problems

In this paper, we present a novel multi-modal optimization algorithm for finding multiple local optima in objective function surfaces. We build from Species-based particle swarm optimization (SPSO) by using deterministic sampling to generate new particles during the optimization process, by implementing proximity-based speciation coupled with speciation of isolated particles, and by including “turbulence regions” around already found solutions to prevent unnecessary function evaluations. Instead of using error threshold values, the new algorithm uses the particle’s experience, geometric mean, and “exclusion factor” to detect local optima and stop the algorithm. The performance of each extension is assessed with leave-it-out tests, and the results are discussed. We use the new algorithm called Isolated-Speciation-based particle swarm optimization (ISPSO) and a benchmark algorithm called Niche particle swarm optimization (NichePSO) to solve a six-dimensional rainfall characterization problem for 192 rain gages across the United States. We show why it is important to find multiple local optima for solving this real-world complex problem by discussing its high multi-modality. Solutions found by both algorithms are compared, and we conclude that ISPSO is more reliable than NichePSO at finding optima with a significantly lower objective function value.     

A Transportation model with multiple criteria and multiple constraint levels

In this paper, a transportation model with multiple criteria and multiple constraint levels (MC²) is formulated by using the framework of MC² linear programming. An algorithm is developed to solve such MC² transportation problems. In this algorithm, the traditional northwest corner rule is adopted to find an initial basic feasible solution for a given MC² transportation problem. Then the MC²-simplex method is applied to locate the set of all potential solutions over possible changes of the objective coefficient parameter and the supply and demand parameter for the MC² transportation problem. A numerical example is illustrated to demonstrate the applicability of the algorithm in solving the MC² transportation problems.     

Primal and dual algorithms for optimization over the efficient set

ABSTRACT

Optimization over the efficient set of a multi-objective optimization problem is a mathematical model for the problem of selecting a most preferred solution that arises in multiple criteria decision-making to account for trade-offs between objectives within the set of efficient solutions. In this paper, we consider a particular case of this problem, namely that of optimizing a linear function over the image of the efficient set in objective space of a convex multi-objective optimization problem. We present both primal and dual algorithms for this task. The algorithms are based on recent algorithms for solving convex multi-objective optimization problems in objective space with suitable modifications to exploit specific properties of the problem of optimization over the efficient set. We first present the algorithms for the case that the underlying problem is a multi-objective linear programme. We then extend them to be able to solve problems with an underlying convex multi-objective optimization problem. We compare the new algorithms with several state of the art algorithms from the literature on a set of randomly generated instances to demonstrate that they are considerably faster than the competitors.     

A heuristic algorithm based on multi-assignment procedures for nurse scheduling

This paper tackles a Nurse Scheduling Problem which consists of generating work schedules for a set of nurses while considering their shift preferences and other requirements. The objective is to maximize the satisfaction of nurses’ preferences and minimize the violation of soft constraints. This paper presents a new deterministic heuristic algorithm, called MAPA (multi-assignment problem-based algorithm), which is based on successive resolutions of the assignment problem. The algorithm has two phases: a constructive phase and an improvement phase. The constructive phase builds a full schedule by solving successive assignment problems, one for each day in the planning period. The improvement phase uses a couple of procedures that re-solve assignment problems to produce a better schedule. Given the deterministic nature of this algorithm, the same schedule is obtained each time that the algorithm is applied to the same problem instance. The performance of MAPA is benchmarked against published results for almost 250,000 instances from the NSPLib dataset. In most cases, particularly on large instances of the problem, the results produced by MAPA are better when compared to best-known solutions from the literature. The experiments reported here also show that the MAPA algorithm finds more feasible solutions compared with other algorithms in the literature, which suggest that this proposed approach is effective and robust.     

A hybrid genetic algorithm for an identical parallel-machine problem with maintenance activity

The scheduling of maintenance activities has been extensively studied, with most studies focusing on single-machine problems. In real-world applications, however, multiple machines or assembly lines process numerous jobs simultaneously. In this paper, we study a parallel-machine scheduling problem in which the objective is to minimize the total tardiness given that there is a maintenance activity on each machine. We develop a branch-and-bound algorithm to solve the problem with a small problem size. In addition, we propose a hybrid genetic algorithm to obtain the approximate solutions when the number of jobs is large. The performance of the proposed algorithms is evaluated based mainly on computational results.     

An improved algorithm for cyclic flowshop scheduling in a robotic cell

This paper addresses a cyclic robot scheduling problem in an automated manufacturing line in which a single robot is used to move parts from one workstation to another. The objective is to minimize the cycle length. Previously known algorithms are either heuristic or at best polynomial of the fifth degree in the number of machines, m. We derive an exact scheduling algorithm solving the problem in O(m³ log m) time.     

An ensemble of discrete differential evolution algorithms for solving the generalized traveling salesman problem

In this paper, an ensemble of discrete differential evolution algorithms with parallel populations is presented. In a single populated discrete differential evolution (DDE) algorithm, the destruction and construction (DC) procedure is employed to generate the mutant population whereas the trial population is obtained through a crossover operator. The performance of the DDE algorithm is substantially affected by the parameters of DC procedure as well as the choice of crossover operator. In order to enable the DDE algorithm to make use of different parameter values and crossover operators simultaneously, we propose an ensemble of DDE (eDDE) algorithms where each parameter set and crossover operator is assigned to one of the parallel populations. Each parallel parent population does not only compete with offspring population generated by its own population but also the offspring populations generated by all other parallel populations which use different parameter settings and crossover operators. As an application area, the well-known generalized traveling salesman problem (GTSP) is chosen, where the set of nodes is divided into clusters so that the objective is to find a tour with minimum cost passing through exactly one node from each cluster. The experimental results show that none of the single populated variants was effective in solving all the GTSP instances whereas the eDDE performed substantially better than the single populated variants on a set of problem instances. Furthermore, through the experimental analysis of results, the performance of the eDDE algorithm is also compared against the best performing algorithms from the literature. Ultimately, all of the best known averaged solutions for larger instances are further improved by the eDDE algorithm.     

A framework for analyzing sub-optimal performance of local search algorithms

This paper presents a framework for analyzing and comparing sub-optimal performance of local search algorithms for hard discrete optimization problems. The β-acceptable solution probability is introduced that captures how effectively an algorithm has performed to date and how effectively an algorithm can be expected to perform in the future. Using this probability, the necessary conditions for a local search algorithm to converge in probability to β-acceptable solutions are derived. To evaluate and compare the effectiveness of local search algorithms, two estimators for the expected number of iterations to visit a β-acceptable solution are obtained. Computational experiments are reported with simulated annealing and tabu search applied to four small traveling salesman problem instances, and the Lin-Kernighan-Helsgaun algorithm applied to eight medium to large traveling salesman problem instances (all with known optimal solutions), to illustrate the application of these estimators.     

Estimation of an optimal solution of a LP problem with unknown objective function

We consider a linear programming problem with unknown objective function. Random observations related to the unknown objective function are sequentially available. We define a stochastic algorithm, based on the simplex method, that estimates an optimal solution of the linear programming problem. It is shown that this algorithm converges with probability one to the set of optimal solutions and that its failure probability is of order inversely proportional to the sample size. We also introduce stopping criteria for the algorithm. The asymptotic normality of some suitably defined residuals is also analyzed. The proposed estimation algorithm is motivated by the stochastic approximation algorithms but it introduces a generalization of these techniques when the linear programming problem has several optimal solutions. The proposed algorithm is also close to the stochastic quasi-gradient procedures, though their usual assumptions are weakened.Mathematics Subject Classification (2000): 90C05, 62L20, 90C15Acknowledgments. I would like to thank two unknown referees for their fruitful suggestions that have helped to improve the paper.     

Minimizing makespan in a two-machine flowshop scheduling with batching and release time

This paper studies two-machine flowshop scheduling with batching and release time, whose objective is to minimize the makespan. We formulate the scheduling problem as a mixed integer programming model and show that it is a strongly NP-hard problem. We derive a lower bound and develop dynamic programming-based heuristic algorithms to solve the scheduling problem. Computational experiments are carried out to evaluate the performance of the heuristic algorithms. The numerical results show that some of the heuristic algorithms can indeed find effective solutions for the scheduling problem.     

带有资源柔性约束的max-NPV分布式多项目调度问题

现有的分布式资源约束多项目调度问题研究中,假定全局资源限量在多项目工期内不可突破且多以工期为优化目标。针对此问题,考虑全局资源可从外部获取,以净现值为目标,构建带有全局资源柔性约束的分布式多项目调度问题的整数规划模型并设计有效的求解算法。首先,界定问题并确定项目现金流的计算方法;然后,针对求解问题的NP-hard属性,设计了遗传-模拟退火混合算法(GA_SA)求解此模型。最后,通过多组数值实验,设计不同算法与GA_SA算法进行比较,并分析了关键参数对多项目净现值的影响。结果表明,GA_SA算法具有较好的求解效果;与传统的全局资源刚性约束条件相比,全局资源柔性使用状态可以显著改善分布式多项目的收益绩效。     

A modified variable neighborhood search for the discrete ordered median problem

This paper presents a modified Variable Neighborhood Search (VNS) heuristic algorithm for solving the Discrete Ordered Median Problem (DOMP). This heuristic is based on new neighborhoods’ structures that allow an efficient encoding of the solutions of the DOMP avoiding sorting in the evaluation of the objective function at each considered solution. The algorithm is based on a data structure, computed in preprocessing, that organizes the minimal necessary information to update and evaluate solutions in linear time without sorting. In order to investigate the performance, the new algorithm is compared with other heuristic algorithms previously available in the literature for solving DOMP. We report on some computational experiments based on the well-known N-median instances of the ORLIB with up to 900 nodes. The obtained results are comparable or superior to existing algorithms in the literature, both in running times and number of best solutions found.     

The transportation problem with exclusionary side constraints and two branch-and-bound algorithms

The transportation problem with exclusionary side constraints, a practical distribution and logistics problem, is formulated as a 0–1 mixed integer programming model. Two branch-and-bound (B&B) algorithms are developed and implemented in this study to solve this problem. Both algorithms use the Driebeek penalties to strengthen the lower bounds so as to fathom some of the subproblems, to peg variables, and to guide the selection of separation variables. One algorithm also strongly exploits the problem structure in selecting separation variables in order to find feasible solutions sooner. To take advantage of the underlying network structure of the problem, the algorithms employ the primal network simplex method to solve network relaxations of the problem. A computational experiment was conducted to test the performance of the algorithms and to characterize the problem difficulty. The commercial mixed integer programming software CPLEX and an existing special purpose algorithm specifically designed for this problem were used as benchmarks to measure the performance of the algorithms. Computational results show that the new algorithms completely dominate the existing special purpose algorithm and run from two to three orders of magnitude faster than CPLEX.     

Multiple objective minimum cost flow problems: A review

In this paper, theory and algorithms for solving the multiple objective minimum cost flow problem are reviewed. For both the continuous and integer case exact and approximation algorithms are presented. In addition, a section on compromise solutions summarizes corresponding results. The reference list consists of all papers known to the authors which deal with the multiple objective minimum cost flow problem.     

Analysis of Static Simulated Annealing Algorithms

Generalized hill climbing (GHC) algorithms provide a framework for modeling local search algorithms to address intractable discrete optimization problems. This paper introduces a measure for determining the expected number of iterations to visit a predetermined objective function level, given that an inferior objective function level has been reached in a finite number of iterations. A variation of simulated annealing (SA), termed static simulated annealing (S²A), is analyzed using this measure. S²A uses a fixed cooling schedule during the algorithm execution. Though S²A is probably nonconvergent, its finite-time performance can be assessed using the finite-time performance measure defined in this paper.     

单体型装配问题及其算法

单核苷酸多态性(SNP)单体型装配问题就是从给定的来自某人染色体的SNP片段中去除错误,重构出尽可能与原来片段一致的单体型.这个问题有几个不同的模型最少片段去除(MFR)问题,最少SNP去除(MSR)问题以及最少错误纠正(MEC)问题.前两个问题的复杂性与算法已有一些学者研究过.第三个问题已被证明是NP完全问题,但这个问题的实际算法还没有.该文对MEC问题给出了一个分支定界算法,这个算法能得到问题的全局最优解.通过这个算法对实际数据的计算说明了MEC模型的合理性,即在一定条件下,通过修正最少的错误重构出的单体型确实是真实的单体型.由于分支定界算法对这样一个NP完全问题不能在可接受的时间内解规模较大的问题,文中又给出了求解MEC问题的两个基于动态聚类的算法,以便对规模较大的问题在可接受的时间内得到近似最优解.数值实际表明这两个算法很快,很有效.这两个算法总能得到与分支定界找到的全局最优解很接近的近似最优解.鉴于MEC问题是NP完全的,这两个算法是有效的、实际的算法.