A Parallel Grasp for the Steiner Tree Problem in Graphs Using a Hybrid Local Search Strategy

In this paper, we present a parallel greedy randomized adaptive search procedure (GRASP) for the Steiner problem in graphs. GRASP is a two-phase metaheuristic. In the first phase, solutions are constructed using a greedy randomized procedure. Local search is applied in the second phase, leading to a local minimum with respect to a specified neighborhood. In the Steiner problem in graphs, feasible solutions can be characterized by their non-terminal nodes (Steiner nodes) or by their key-paths. According to this characterization, two GRASP procedures are described using different local search strategies. Both use an identical construction procedure. The first uses a node-based neighborhood for local search, while the second uses a path-based neighborhood. Computational results comparing the two procedures show that while the node-based variant produces better quality solutions, the path-based variant is about twice as fast. A hybrid GRASP procedure combining the two neighborhood search strategies is then proposed. Computational experiments with a parallel implementation of the hybrid procedure are reported, showing that the algorithm found optimal solutions for 45 out of 60 benchmark instances and was never off by more than 4% of the optimal solution value. The average speedup results observed for the test problems show that increasing the number of processors reduces elapsed times with increasing speedups. Moreover, the main contribution of the parallel algorithm concerns the fact that larger speedups of the same order of the number of processors are obtained exactly for the most difficult problems.     

Greedy Randomized Adaptive Search Procedures

Today, a variety of heuristic approaches are available to the operations research practitioner. One methodology that has a strong intuitive appeal, a prominent empirical track record, and is trivial to efficiently implement on parallel processors is GRASP (Greedy Randomized Adaptive Search Procedures). GRASP is an iterative randomized sampling technique in which each iteration provides a solution to the problem at hand. The incumbent solution over all GRASP iterations is kept as the final result. There are two phases within each GRASP iteration: the first intelligently constructs an initial solution via an adaptive randomized greedy function; the second applies a local search procedure to the constructed solution in hope of finding an improvement. In this paper, we define the various components comprising a GRASP and demonstrate, step by step, how to develop such heuristics for combinatorial optimization problems. Intuitive justifications for the observed empirical behavior of the methodology are discussed. The paper concludes with a brief literature review of GRASP implementations and mentions two industrial applications.     

Local search with a generalized neighborhood in the optimization problem for pseudo-Boolean functions

In the optimization problem for pseudo-Boolean functions we consider a local search algorithm with a generalized neighborhood. This neighborhood is constructed for a locally optimal solution and includes nearby locally optimal solutions. We present some results of simulations for pseudo-Boolean functions whose optimization is equivalent to the problems of facility location, set covering, and competitive facility location. The goal of these experiments is to obtain a comparative estimate for the locally optimal solutions found by the standard local search algorithm and the local search algorithm using a generalized neighborhood.     

Applying the pilot method to improve VNS and GRASP metaheuristics for the design of SDH/WDM networks

Variable neighborhood search (VNS) and Greedy randomized adaptive search procedure (GRASP) are among the well studied local search based metaheuristics providing good results for many combinatorial optimization problems throughout the last decade. While they are usually explored in different environments one may encounter quite obvious commonalities. Based on previous successful applications of these two types of metaheuristics on various network design problems in telecommunications, we further enhance these approaches by incorporating ideas from the pilot method. The different heuristics are compared among each other as well as against objective function values obtained from a mathematical programming formulation based on a commercial solver. The problem instances cover a large variety of networks and demand patterns.     

A filter-and-fan approach to the job shop scheduling problem

The job shop scheduling problem (JSSP) is a notoriously difficult problem in combinatorial optimization. Extensive investigation has been devoted to developing efficient algorithms to find optimal or near-optimal solutions. This paper proposes a new heuristic algorithm for the JSSP that effectively combines the classical shifting bottleneck procedure (SBP) with a dynamic and adaptive neighborhood search procedure. Our new search method, based on a filter-and-fan (F&F) procedure, uses the SBP as a subroutine to generate a starting solution and to enhance the best schedules produced. The F&F approach is a local search procedure that generates compound moves by a strategically abbreviated form of tree search. Computational results carried out on a standard set of 43 benchmark problems show that our F&F algorithm performs more robustly and effectively than a number of leading metaheuristic algorithms and rivals the best of these algorithms.     

A dual local search framework for combinatorial optimization problems with TSP application

In practice, solving realistically sized combinatorial optimization problems to optimality is often too time-consuming to be affordable; therefore, heuristics are typically implemented within most applications software. A specific category of heuristics has attracted considerable attention, namely local search methods. Most local search methods are primal in nature; that is, they start the search with a feasible solution and explore the feasible space for better feasible solutions. In this research, we propose a dual local search method and customize it to solve the traveling salesman problem (TSP); that is, a search method that starts with an infeasible solution, explores the dual space—each time reducing infeasibility, and lands in the primal space to deliver a feasible solution. The proposed design aims to replicate the designs of optimal solution methodologies in a heuristic way. To be more specific, we solve a combinatorial relaxation of a TSP formulation, design a neighborhood structure to repair such an infeasible starting solution, and improve components of intermediate dual solutions locally. Sample-based evidence along with statistically significant t-tests support the superiority of this dual design compared to its primal design counterpart.     

Computing Approximate Solutions of the Maximum Covering Problem with GRASP

We consider the maximum covering problem, a combinatorial optimization problem that arises in many facility location problems. In this problem, a potential facility site covers a set of demand points. With each demand point, we associate a nonnegative weight. The task is to select a subset of p > 0 sites from the set of potential facility sites, such that the sum of weights of the covered demand points is maximized. We describe a greedy randomized adaptive search procedure (GRASP) for the maximum covering problem that finds good, though not necessarily optimum, placement configurations. We describe a well-known upper bound on the maximum coverage which can be computed by solving a linear program and show that on large instances, the GRASP can produce facility placements that are nearly optimal.     

Restart strategies for GRASP with path-relinking heuristics

GRASP with path-relinking is a hybrid metaheuristic, or stochastic local search (Monte Carlo) method, for combinatorial optimization. A restart strategy in GRASP with path-relinking heuristics is a set of iterations {i ₁, i ₂, …} on which the heuristic is restarted from scratch using a new seed for the random number generator. Restart strategies have been shown to speed up stochastic local search algorithms. In this paper, we propose a new restart strategy for GRASP with path-relinking heuristics. We illustrate the speedup obtained with our restart strategy on GRASP with path-relinking heuristics for the maximum cut problem, the maximum weighted satisfiability problem, and the private virtual circuit routing problem.     

Speeding up continuous GRASP

Continuous GRASP (C-GRASP) is a stochastic local search metaheuristic for finding cost-efficient solutions to continuous global optimization problems subject to box constraints (Hirsch et al., 2007). Like a greedy randomized adaptive search procedure (GRASP), a C-GRASP is a multi-start procedure where a starting solution for local improvement is constructed in a greedy randomized fashion. In this paper, we describe several improvements that speed up the original C-GRASP and make it more robust. We compare the new C-GRASP with the original version as well as with other algorithms from the recent literature on a set of benchmark multimodal test functions whose global minima are known. Hart’s sequential stopping rule (1998) is implemented and C-GRASP is shown to converge on all test problems.     

GRASP-based heuristic algorithm for the multi-product multi-vehicle inventory routing problem

In this paper, we introduce an improved Greedy Randomized Adaptive Search Procedure (GRASP) based heuristic for the multi-product multi-vehicle inventory routing problem (MMIRP). The inventory routing problem, which combines the vehicle-routing problem and the inventory control decisions, is one of the most important problems in combinatorial optimization field. To deal with the MMIRP, we develop a GRASP-based heuristic (GBH). Each GBH iteration consists of two sequential phases; the first phase is a Greedy Randomized Procedure, in which, the best tradeoff between the inventory holding cost and routing cost is looked. Then, in the second phase, as local search for the GRASP, we use the Tabu search (TS) meta-heuristic to improve the solution found in the first phase. The GBH two phases are repeated until some stopped criterion is met. Our proposed method is evaluated on two benchmark data sets, and successfully compared with two state-of-the-art algorithms.     

Multiobjective GRASP with Path Relinking

In this paper we review and propose different adaptations of the GRASP metaheuristic to solve multiobjective combinatorial optimization problems. In particular, we describe several alternatives to specialize the construction and improvement components of GRASP when two or more objectives are considered. GRASP has been successfully coupled with Path Relinking for single-objective optimization. Moreover, we propose different hybridizations of GRASP and Path Relinking for multiobjective optimization. We apply the proposed GRASP with Path Relinking variants to two combinatorial optimization problems, the biobjective orienteering problem and the biobjective path dissimilarity problem. We report on empirical tests with 70 instances and 30 algorithms, that show that the proposed heuristics are competitive with the state-of-the-art methods for these problems.     

A hybrid metaheuristic for the vehicle routing problem with stochastic demand and duration constraints

The vehicle routing problem with stochastic demands (VRPSD) consists in designing optimal routes to serve a set of customers with random demands following known probability distributions. Because of demand uncertainty, a vehicle may arrive at a customer without enough capacity to satisfy its demand and may need to apply a recourse to recover the route’s feasibility. Although travel times are assumed to be deterministic, because of eventual recourses the total duration of a route is a random variable. We present two strategies to deal with route-duration constraints in the VRPSD. In the first, the duration constraints are handled as chance constraints, meaning that for each route, the probability of exceeding the maximum duration must be lower than a given threshold. In the second, violations to the duration constraint are penalized in the objective function. To solve the resulting problem, we propose a greedy randomized adaptive search procedure (GRASP) enhanced with heuristic concentration (HC). The GRASP component uses a set of randomized route-first, cluster-second heuristics to generate starting solutions and a variable-neighborhood descent procedure for the local search phase. The HC component assembles the final solution from the set of all routes found in the local optima reached by the GRASP. For each strategy, we discuss extensive computational experiments that analyze the impact of route-duration constraints on the VRPSD. In addition, we report state-of-the-art solutions for a established set of benchmarks for the classical VRPSD.     

EB-GLS: an improved guided local search based on the big valley structure

Local search is a basic building block in memetic algorithms. Guided local search (GLS) can improve the efficiency of local search. By changing the guide function, GLS guides a local search to escape from locally optimal solutions and find better solutions. The key component of GLS is its penalizing mechanism which determines which feature is selected to penalize when the search is trapped in a locally optimal solution. The original GLS penalizing mechanism only makes use of the cost and the current penalty value of each feature. It is well known that many combinatorial optimization problems have a big valley structure, i.e., the better a solution is, the more the chance it is closer to a globally optimal solution. This paper proposes to use big valley structure assumption to improve the GLS penalizing mechanism. An improved GLS algorithm called elite biased GLS (EB-GLS) is proposed. EB-GLS records and maintains an elite solution as an estimate of the globally optimal solutions, and reduces the chance of penalizing the features in this solution. We have systematically tested the proposed algorithm on the symmetric traveling salesman problem. Experimental results show that EB-GLS is significantly better than GLS.     

A variable neighborhood search for graph coloring

Descent methods for combinatorial optimization proceed by performing a sequence of local changes on an initial solution which improve each time the value of an objective function until a local optimum is found. Several metaheuristics have been proposed which extend in various ways this scheme and avoid being trapped in local optima. For example, Hansen and Mladenovic have recently proposed the variable neighborhood search method which has not yet been applied to many combinatorial optimization problems. The aim of this paper is to propose an adaptation of this new method to the graph coloring problem.     

A two-phase method for selecting IMRT treatment beam angles: Branch-and-Prune and local neighborhood search

This paper presents a new two-phase solution approach to the beam angle and fluence map optimization problem in Intensity Modulated Radiation Therapy (IMRT) planning. We introduce Branch-and-Prune (B&P) to generate a robust feasible solution in the first phase. A local neighborhood search algorithm is developed to find a local optimal solution from the Phase I starting point in the second phase. The goal of the first phase is to generate a clinically acceptable feasible solution in a fast manner based on a Branch-and-Bound tree. In this approach, a substantially reduced search tree is iteratively constructed. In each iteration, a merit score based branching rule is used to select a pool of promising child nodes. Then pruning rules are applied to select one child node as the branching node for the next iteration. The algorithm terminates when we obtain a desired number of angles in the current node. Although Phase I generates quality feasible solutions, it does not guarantee optimality. Therefore, the second phase is designed to converge Phase I starting solutions to local optimality. Our methods are tested on two sets of real patient data. Results show that not only can B&P alone generate clinically acceptable solutions, but the two-phase method consistently generates local optimal solutions, some of which are shown to be globally optimal.     

Vehicle routing problems with time windows and multiple service workers: a systematic comparison between ACO and GRASP

This paper systematically compares an ant colony optimization (ACO) and a greedy randomized adaptive search procedure (GRASP) metaheuristic. Both are used to solve the vehicle routing problem with time windows and multiple service workers. In order to keep the results comparable, the same route construction heuristic and local search procedures are used. It is shown that ACO clearly outperforms GRASP in the problem under study. Additionally, new globally best results for the used benchmark problems are presented.     

Expanding Neighborhood GRASP for the Traveling Salesman Problem

In this paper, we present the application of a modified version of the well known Greedy Randomized Adaptive Search Procedure (GRASP) to the TSP. The proposed GRASP algorithm has two phases: In the first phase the algorithm finds an initial solution of the problem and in the second phase a local search procedure is utilized for the improvement of the initial solution. The local search procedure employs two different local search strategies based on 2-opt and 3-opt methods. The algorithm was tested on numerous benchmark problems from TSPLIB. The results were very satisfactory and for the majority of the instances the results were equal to the best known solution. The algorithm is also compared to the algorithms presented and tested in the DIMACS Implementation Challenge that was organized by David Johnson.     

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.     

A hybrid algorithm for the manufacturing cell formation problem

In this paper we compare different heuristic methods for the manufacturing cell formation problem considering part process sequence: a GRASP algorithm, a reactive GRASP algorithm and a hybrid algorithm which combines reactive GRASP and tabu search. All algorithms are tested with a set of instances from the literature. The results from the GRASP algorithm are compared to those of the reactive GRASP in order to evaluate the advantages of automatically adjusting the parameter value within the randomized greedy procedure. Also the reactive GRASP results are compared to those of the hybrid algorithm to evaluate the contribution to solution quality of replacing the local search phase of the GRASP algorithm with tabu search.     

A Combinatorial Approach to Piecewise Linear Time Series Analysis

Over recent years, several nonlinear time series models have been proposed in the literature. One model that has found a large number of successful applications is the threshold autoregressive model (TAR). The TAR model is a piecewise linear process whose central idea is to change the parameters of a linear autoregressive model according to the value of an observable variable, called the threshold variable. If this variable is a lagged value of the time series, the model is called a self-exciting threshold autoregressive (SETAR) model. In this article, we propose a heuristic to estimate a more general SETAR model, where the thresholds are multivariate. We formulate the task of finding multivariate thresholds as a combinatorial optimization problem. We develop an algorithm based on a greedy randomized adaptive search procedure (GRASP) to solve the problem. GRASP is an iterative randomized sampling technique that has been shown to quickly produce good quality solutions for a wide variety of optimization problems. The proposed model performs well on both simulated and real data.