A contribution to the stochastic flow shop scheduling problem

This paper deals with performance evaluation and scheduling problems in m machine stochastic flow shop with unlimited buffers. The processing time of each job on each machine is a random variable exponentially distributed with a known rate. We consider permutation flow shop. The objective is to find a job schedule which minimizes the expected makespan. A classification of works about stochastic flow shop with random processing times is first given. In order to solve the performance evaluation problem, we propose a recursive algorithm based on a Markov chain to compute the expected makespan and a discrete event simulation model to evaluate the expected makespan. The recursive algorithm is a generalization of a method proposed in the literature for the two machine flow shop problem to the m machine flow shop problem with unlimited buffers. In deterministic context, heuristics (like CDS [Management Science 16 (10) (1970) B630] and Rapid Access [Management Science 23 (11) (1977) 1174]) and metaheuristics (like simulated annealing) provide good results. We propose to adapt and to test this kind of methods for the stochastic scheduling problem. Combinations between heuristics or metaheuristics and the performance evaluation models are proposed. One of the objectives of this paper is to compare the methods together. Our methods are tested on problems from the OR-Library and give good results: for the two machine problems, we obtain the optimal solution and for the m machine problems, the methods are mutually validated.     

Multi-armed bandit-based hyper-heuristics for combinatorial optimization problems

There are significant research opportunities in the integration of Machine Learning (ML) methods and Combinatorial Optimization Problems (COPs). In this work, we focus on metaheuristics to solve COPs that have an important learning component. These algorithms must explore a solution space and learn from the information they obtain in order to find high-quality solutions. Among the metaheuristics, we study Hyper-Heuristics (HHs), algorithms that, given a number of low-level heuristics, iteratively select and apply heuristics to a solution. The HH we consider has a Markov model to produce sequences of low-level heuristics, which we combine with a Multi-Armed Bandit Problem (MAB)-based method to learn its parameters. This work proposes several improvements to the HH metaheuristic that yields a better learning for solving problem instances. Specifically, this is the first work in HHs to present Exponential Weights for Exploration and Exploitation (EXP3) as a learning method, an algorithm that is able to deal with adversarial settings. We also present a case study for the Vehicle Routing Problem with Time Windows (VRPTW), for which we include a list of low-level heuristics that have been proposed in the literature. We show that our algorithms can handle a large and diverse list of heuristics, illustrating that they can be easily configured to solve COPs of different nature. The computational results indicate that our algorithms are competitive methods for the VRPTW (2.16% gap on average with respect to the best known solutions), demonstrating the potential of these algorithms to solve COPs. Finally, we show how algorithms can even detect low-level heuristics that do not contribute to finding better solutions to the problem.     

Heuristics and metaheuristics for the maximum diversity problem

This paper presents extensive computational experiments to compare 10 heuristics and 20 metaheuristics for the maximum diversity problem (MDP). This problem consists of selecting a subset of maximum diversity from a given set of elements. It arises in a wide range of real-world settings and we can find a large number of studies, in which heuristic and metaheuristic methods are proposed. However, probably due to the fact that this problem has been referenced under different names, we have only found limited comparisons with a few methods on some sets of instances. This paper reviews all the heuristics and metaheuristics for finding near-optimal solutions for the MDP. We present the new benchmark library MDPLIB, which includes most instances previously used for this problem, as well as new ones, giving a total of 315. We also present an exhaustive computational comparison of the 30 methods on the MDPLIB. Non-parametric statistical tests are reported in our study to draw significant conclusions.     

Revisiting dynamic programming for finding optimal subtrees in trees

In this paper we revisit an existing dynamic programming algorithm for finding optimal subtrees in edge weighted trees. This algorithm was sketched by Maffioli in a technical report in 1991. First, we adapt this algorithm for the application to trees that can have both node and edge weights. Second, we extend the algorithm such that it does not only deliver the values of optimal trees, but also the trees themselves. Finally, we use our extended algorithm for developing heuristics for the k-cardinality tree problem in undirected graphs G with node and edge weights. This NP-hard problem consists of finding in the given graph a tree with exactly k edges such that the sum of the node and the edge weights is minimal. In order to show the usefulness of our heuristics we conduct an extensive computational analysis that concerns most of the existing problem instances. Our results show that with growing problem size the proposed heuristics reach the performance of state-of-the-art metaheuristics. Therefore, this study can be seen as a cautious note on the scaling of metaheuristics.     

Search Heuristics for Box Decomposition Methods

In this paper we study search heuristics for box decomposition methods that solve problems such as global optimization, minimax optimization, or quantified constraint solving. For this we unify these methods under a branch-and-bound framework, and develop a model that is more convenient for studying heuristics for such algorithms than the traditional models from Artificial Intelligence. We use the result to prove various theorems about heuristics and apply the outcome to the box decomposition methods under consideration. We support the findings with timings for the method of quantified constraint solving developed by the author.     

An empirical study of hyperheuristics for managing very large sets of low level heuristics

Hyperheuristics give us the appealing possibility of abstracting the solution method from the problem, since our hyperheuristic, at each decision point, chooses between different low level heuristics rather than different solutions as is usually the case for metaheuristics. By assembling low level heuristics from parameterised components we may create hundreds or thousands of low level heuristics, and there is increasing evidence that this is effective in dealing with every eventuality that may arise when solving different combinatorial optimisation problem instances since at each iteration the solution landscape is amenable to at least one of the low level heuristics. However, the large number of low level heuristics means that the hyperheuristic has to intelligently select the correct low level heuristic to use, to make best use of available CPU time. This paper empirically investigates several hyperheuristics designed for large collections of low level heuristics and adapts other hyperheuristics from the literature to cope with these large sets of low level heuristics on a difficult real-world workforce scheduling problem. In the process we empirically investigate a wide range of approaches for setting tabu tenure in hyperheuristic methods, for a complex real-world problem. The results show that the hyperheuristic methods described provide a good way to trade off CPU time and solution quality.     

Parametric enhancements of the Esau–Williams heuristic for the capacitated minimum spanning tree problem

The Capacitated Minimum Spanning Tree Problem is NP-hard and several heuristic solution methods have been proposed. They can be classified as classical ones and metaheuristics. Recent developments have shown that metaheuristics outperform classical heuristics. However, they require long computation times and there are difficulties in their parameter calibration and coding phases. This explains the popularity of the Esau–Williams (EW) heuristic in practice, and its use in many successful metaheuristics and second-order greedy methods. In this work, we are concerned with the EW heuristic and we propose simple new enhancements. Based on our computational experiments, we can say that they considerably improve its accuracy with minor increase in computation time, and without harming its simplicity.     

Pareto Simulated Annealing for Fuzzy Multi-Objective Combinatorial Optimization

The paper presents a metaheuristic method for solving fuzzy multi-objective combinatorial optimization problems. It extends the Pareto simulated annealing (PSA) method proposed originally for the crisp multi-objective combinatorial (MOCO) problems and is called fuzzy Pareto simulated annealing (FPSA). The method does not transform the original fuzzy MOCO problem to an auxiliary deterministic problem but works in the original fuzzy objective space. Its goal is to find a set of approximately efficient solutions being a good approximation of the whole set of efficient solutions defined in the fuzzy objective space. The extension of PSA to FPSA requires the definition of the dominance in the fuzzy objective space, modification of rules for calculating probability of accepting a new solution and application of a defuzzification operator for updating the average position of a solution in the objective space. The use of the FPSA method is illustrated by its application to an agricultural multi-objective project scheduling problem.     

A comprehensive review and evaluation of permutation flowshop heuristics

In this work we present a review and comparative evaluation of heuristics and metaheuristics for the well-known permutation flowshop problem with the makespan criterion. A number of reviews and evaluations have already been proposed. However, the evaluations do not include the latest heuristics available and there is still no comparison of metaheuristics. Furthermore, since no common benchmarks and computing platforms are used, the results cannot be generalised. We propose a comparison of 25 methods, ranging from the classical Johnson's algorithm or dispatching rules to the most recent metaheuristics, including tabu search, simulated annealing, genetic algorithms, iterated local search and hybrid techniques. For the evaluation we use the standard test of Taillard [Eur. J. Operation. Res. 64 (1993) 278] composed of 120 instances of different sizes. In the evaluations we use the experimental design approach to obtain valid conclusions on the effectiveness and efficiency of the different methods tested.     

A Taxonomy of Hybrid Metaheuristics

Hybrid metaheuristics have received considerable interest these recent years in the field of combinatorial optimization. A wide variety of hybrid approaches have been proposed in the literature. In this paper, a taxonomy of hybrid metaheuristics is presented in an attempt to provide a common terminology and classification mechanisms. The taxonomy, while presented in terms of metaheuristics, is also applicable to most types of heuristics and exact optimization algorithms.As an illustration of the usefulness of the taxonomy an annoted bibliography is given which classifies a large number of hybrid approaches according to the taxonomy.     

A discussion of scalarization techniques for multiple objective integer programming

In this paper we consider solution methods for multiobjective integer programming (MOIP) problems based on scalarization. We define the MOIP, discuss some common scalarizations, and provide a general formulation that encompasses most scalarizations that have been applied in the MOIP context as special cases. We show that these methods suffer some drawbacks by either only being able to find supported efficient solutions or introducing constraints that can make the computational effort to solve the scalarization prohibitive. We show that Lagrangian duality applied to the general scalarization does not remedy the situation. We also introduce a new scalarization technique, the method of elastic constraints, which is shown to be able to find all efficient solutions and overcome the computational burden of the scalarizations that use constraints on objective values. Finally, we present some results from an application in airline crew scheduling as evidence. This research is partially supported by University of Auckland grant 3602178/9275 and by the Deutsche Forschungsgemeinschaft grant Ka 477/27-1.     

Two phase algorithms for the bi-objective assignment problem

In this paper, we present several algorithms for the bi-objective assignment problem. The algorithms are based on the two phase method, which is a general technique to solve multi-objective combinatorial optimisation (MOCO) problems.     

Greedy algorithm for the general multidimensional knapsack problem

In this paper, we propose a new greedy-like heuristic method, which is primarily intended for the general MDKP, but proves itself effective also for the 0-1 MDKP. Our heuristic differs from the existing greedy-like heuristics in two aspects. First, existing heuristics rely on each item’s aggregate consumption of resources to make item selection decisions, whereas our heuristic uses the effective capacity, defined as the maximum number of copies of an item that can be accepted if the entire knapsack were to be used for that item alone, as the criterion to make item selection decisions. Second, other methods increment the value of each decision variable only by one unit, whereas our heuristic adds decision variables to the solution in batches and consequently improves computational efficiency significantly for large-scale problems. We demonstrate that the new heuristic significantly improves computational efficiency of the existing methods and generates robust and near-optimal solutions. The new heuristic proves especially efficient for high dimensional knapsack problems with small-to-moderate numbers of decision variables, usually considered as “hard” MDKP and no computationally efficient heuristic is available to treat such problems. Supported in part by the NSF grant DMI 9812994.     

Case-based reasoning for repetitive combinatorial optimization problems, part I: Framework

This article presents a case-based reasoning approach for the development of learning heuristics for solving repetitive operations research problems. We first define the subset of problems we will consider in this work: repetitive combinatorial optimization problems. We then present several general forms that can be used to select previously solved problems that might aid in the solution of the current problem, and several different techniques for actually using this information to derive a solution for the current problem. We describe both fixed forms and forms that have the ability to change as we solve more problems. A simple example for the 0–1 knapsack problem is presented.     

A tour construction heuristic for the travelling salesman problem

The tour construction heuristic that generates initial tours for the tour improvement heuristics plays an important role in solving the travelling salesman problem (TSP). With the help of an effective tour construction heuristic, the performance of a heuristic can be improved. In this study we present a new tour construction algorithm, the construction priority (CP). We incorporate the performance of the CP into metaheuristics such as tabu search, genetic algorithm, space smoothing, and noising methods. The performance of the CP is empirically compared with the nearest neighbour (NN) approach. Extensive computational comparison shows that the CP is considerably more effective compared to NN.     

Manpower allocation using genetic annealing

In this paper, we focus on a real size manpower allocation problem. It was modeled after a real world problem of distributing the salesmen force over the branches of a company. The problem includes multiple objectives and the number of salesmen at each branch is unspecified. Conventional integer programming approach and conventional metaheuristics seem to have problems with solving the large size version of this problem. The versatility of our proposed heuristics based on a modification of genetic annealing is exemplified through solving the real size manpower allocation problem. For comparison sake, several small sized versions were solved using our method, conventional integer programming approach, and some well known metaheuristics.     

Theoretical insights into the augmented-neural-network approach for combinatorial optimization

The augmented-neural-network (AugNN) approach has been applied lately to some NP-Hard combinatorial problems, such as task scheduling, open-shop scheduling and resource-constraint project scheduling. In this approach the problem of search in the solution-space is transformed to a search in a weight-matrix space, much like in a neural-network approach. Some weight adjustment strategies are then used to converge to a good set of weights for a locally optimal solution. While empirical results have demonstrated the effectiveness of the AugNN approach vis-à-vis a few other metaheuristics, little theoretical insights exist which justify this approach and explain the effectiveness thereof. This paper provides some theoretical insights and justification for the AugNN approach through some basic theorems and also describes the algorithm and the formulation with the help of examples.     

Hybridizing exact methods and metaheuristics: A taxonomy

The interest about hybrid optimization methods has grown for the last few years. Indeed, more and more papers about cooperation between heuristics and exact techniques are published. In this paper, we propose to extend an existing taxonomy for hybrid methods involving heuristic approaches in order to consider cooperative schemes between exact methods and metaheuristics. First, we propose some natural approaches for the different schemes of cooperation encountered, and we analyse, for each model, some examples taken from the literature. Then we recall and complement the proposed grammar and provide an annotated bibliography.     

DC-GRASP: directing the search on continuous-GRASP

Several papers in the scientific literature use metaheuristics to solve continuous global optimization. To perform this task, some metaheuristics originally proposed for solving combinatorial optimization problems, such as Greedy Randomized Adaptive Search Procedure (GRASP), Tabu Search and Simulated Annealing, among others, have been adapted to solve continuous global optimization problems. Proposed by Hirsch et al., the Continuous-GRASP (C-GRASP) is one example of this group of metaheuristics. The C-GRASP is an adaptation of GRASP proposed to solve continuous global optimization problems under box constraints. It is simple to implement, derivative-free and widely applicable method. However, according to Hedar, due to its random construction, C-GRASP may fail to detect promising search directions especially in the vicinity of minima, which may result in a slow convergence. To minimize this problem, in this paper we propose a set of methods to direct the search on C-GRASP, called Directed Continuous-GRASP (DC-GRASP). The proposal is to combine the ability of C-GRASP to diversify the search over the space with some efficient local search strategies to accelerate its convergence. We compare the DC-GRASP with the C-GRASP and other metaheuristics from literature on a set of standard test problems whose global minima are known. Computational results show the effectiveness and efficiency of the proposed methods, as well as their ability to accelerate the convergence of the C-GRASP.