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 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.     

A variable neighborhood-based heuristic for the heterogeneous fleet vehicle routing problem

The heterogeneous fleet vehicle routing problem is investigated using some adaptations of the variable neighborhood search (VNS). The initial solution is obtained by Dijkstra’s algorithm based on a cost network constructed by the sweep algorithm and the 2-opt. Our VNS algorithm uses several neighborhoods which are adapted for this problem. In addition, a number of local search methods together with a diversification procedure are used. Two VNS variants, which differ in the order the diversification and Dijkstra’s algorithm are used, are implemented. Both variants appear to be competitive and produce new best results when tested on the data sets from the literature. We also constructed larger data sets for which benchmarking results are provided for future comparison.     

A nonmonotone GRASP

A greedy randomized adaptive search procedure (GRASP) is an iterative multistart metaheuristic for difficult combinatorial optimization problems. Each GRASP iteration consists of two phases: a construction phase, in which a feasible solution is produced, and a local search phase, in which a local optimum in the neighborhood of the constructed solution is sought. Repeated applications of the construction procedure yields different starting solutions for the local search and the best overall solution is kept as the result. The GRASP local search applies iterative improvement until a locally optimal solution is found. During this phase, starting from the current solution an improving neighbor solution is accepted and considered as the new current solution. In this paper, we propose a variant of the GRASP framework that uses a new “nonmonotone” strategy to explore the neighborhood of the current solution. We formally state the convergence of the nonmonotone local search to a locally optimal solution and illustrate the effectiveness of the resulting Nonmonotone GRASP on three classical hard combinatorial optimization problems: the maximum cut problem (MAX-CUT), the weighted maximum satisfiability problem (MAX-SAT), and the quadratic assignment problem (QAP).     

Hybridized evolutionary local search algorithm for the team orienteering problem with time windows

The orienteering problem (OP) consists in finding an elementary path over a subset of vertices. Each vertex is associated with a profit that is collected on the visitor’s first visit. The objective is to maximize the collected profit with respect to a limit on the path’s length. The team orienteering problem (TOP) is an extension of the OP where a fixed number m of paths must be determined. This paper presents an effective hybrid metaheuristic to solve both the OP and the TOP with time windows. The method combines the greedy randomized adaptive search procedure (GRASP) with the evolutionary local search (ELS). ELS generates multiple distinct child solutions using a mutation mechanism. Each child solution is further improved by a local search procedure. GRASP provides multiple starting solutions to the ELS. The method is able to improve several best known results on available benchmark instances.     

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.     

A GRASP algorithm for the multi-criteria minimum spanning tree problem

This paper proposes a GRASP (Greedy Randomized Adaptive Search Procedure) algorithm for the multi-criteria minimum spanning tree problem, which is NP-hard. In this problem a vector of costs is defined for each edge of the graph and the problem is to find all Pareto optimal or efficient spanning trees (solutions). The algorithm is based on the optimization of different weighted utility functions. In each iteration, a weight vector is defined and a solution is built using a greedy randomized constructive procedure. The found solution is submitted to a local search trying to improve the value of the weighted utility function. We use a drop-and-add neighborhood where the spanning trees are represented by Prufer numbers. In order to find a variety of efficient solutions, we use different weight vectors, which are distributed uniformly on the Pareto frontier. The proposed algorithm is tested on problems with r=2 and 3 criteria. For non-complete graphs with n=10, 20 and 30 nodes, the performance of the algorithm is tested against a complete enumeration. For complete graphs with n=20, 30 and 50 nodes the performance of the algorithm is tested using two types of weighted utility functions. The algorithm is also compared with the multi-criteria version of the Kruskal’s algorithm, which generates supported efficient solutions. This work was funded by the Municipal Town Hall of Campos dos Goytacazes city. The used computer was acquired with resource of CNPq.     

Discrete particle swarm optimization for the team orienteering problem

Orienteering problem is a well researched routing problem which is a generalization of the traveling salesman problem. Team orienteering problem (TOP) is the extended version of the orienteering problem with more than one member in the team. In this paper the first known discrete particle swarm optimization (DPSO) algorithm has been developed for 2, 3 and 4-member TOP. In the DPSO meta-heuristic novel methods have been introduced for the initial particle generation process. Reduced variable neighborhood search and 2-opt were applied as the local search tools. The efficacy of the algorithm was tested using seven commonly used benchmark problem sets ranging in size from 21 to 102 nodes. The results of the DPSO algorithm were compared against seven other heuristic algorithms that have been developed for TOP. It was concluded that the developed DPSO algorithm for the TOP is competitive and robust across the benchmark problem sets.     

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.     

A GRASP for Coloring Sparse Graphs

We first present a literature review of heuristics and metaheuristics developed for the problem of coloring graphs. We then present a Greedy Randomized Adaptive Search Procedure (GRASP) for coloring sparse graphs. The procedure is tested on graphs of known chromatic number, as well as random graphs with edge probability 0.1 having from 50 to 500 vertices. Empirical results indicate that the proposed GRASP implementation compares favorably to classical heuristics and implementations of simulated annealing and tabu search. GRASP is also found to be competitive with a genetic algorithm that is considered one of the best currently available for graph coloring.     

A reactive GRASP with path relinking for capacitated clustering

This paper presents a greedy randomized adaptive search procedure (GRASP) coupled with path relinking (PR) to solve the problem of clustering n nodes in a graph into p clusters. The objective is to maximize the sum of the edge weights within each cluster such that the sum of the corresponding node weights does not exceed a fixed capacity. In phase I, both a heaviest weight edge (HWE) algorithm and a constrained minimum cut algorithm are used to select seeds for initializing the p clusters. Feasible solutions are obtained with the help of a self-adjusting restricted candidate list that sequentially guides the assignment of the remaining nodes. At each major GRASP iteration, the list length is randomly set based on a probability density function that is updated dynamically to reflect the solution quality realized in past iterations. In phase II, three neighborhoods, each defined by common edge and node swaps, are explored to attain local optimality. The following exploration strategies are investigated: cyclic neighborhood search, variable neighborhood descent, and randomized variable neighborhood descent (RVND). The best solutions found are stored in an elite pool.     

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.     

Effective Heuristic Procedures for a Field Technician Scheduling Problem

This paper addresses a field technician scheduling problem faced by many service providers in telecommunication industry. The problem is to assign a set of jobs, at different locations with time windows, to a group of field technicians with different job skills. Such a problem can be viewed as a generalization of the well-known vehicle routing problem with time windows since technician skills need to be matched with job types. We designed and tested several heuristic procedures for solving the problem, namely a greedy heuristic, a local search algorithm, and a greedy randomized adaptive search procedure (GRASP). Our computational results indicate that GRASP is the most effective among them but requires more CPU time. However, the unique structure of GRASP allows us to exploit parallelism to achieve linear speed-up with respect to the number of machines used.     

Diversified local search strategy under scatter search framework for the probabilistic traveling salesman problem

This paper focuses on introducing a concept of diversified local search strategy under the scatter search framework for the probabilistic traveling salesman problem (PTSP). Different combinations of three commonly used local search methods in the PTSP, i.e., 1-shift, 2-opt, and 3-opt, were used to investigate its effects. A set of numerical experiments were conducted to test the validity of the proposed strategy based on randomly generated test instances. The numerical results and the permutation test showed that the diversified local search strategy, especially by combining 1-shift and 2-opt algorithms, can most effectively solve the homogeneous and heterogeneous PTSP in most of the tested instances in comparison with the single local search strategy under scatter search framework.     

A Reactive Local Search-Based Algorithm for the Multiple-Choice Multi-Dimensional Knapsack Problem

In this paper, we approximately solve the multiple-choice multi-dimensional knapsack problem. We propose an algorithm which is based upon reactive local search and where an explicit check for the repetition of configurations is added to the local search. The algorithm starts by an initial solution and improved by using a fast iterative procedure. Later, both deblocking and degrading procedures are introduced in order (i) to escape to local optima and, (ii) to introduce diversification in the search space. Finally, a memory list is applied in order to forbid the repetition of configurations. The performance of the proposed approaches has been evaluated on several problem instances. Encouraging results have been obtained.     

Heuristics for the flow line problem with setup costs

This paper presents two new heuristics for the flowshop scheduling problem with sequence-dependent setup times (SDSTs) and makespan minimization objective. The first is an extension of a procedure that has been very successful for the general flowshop scheduling problem. The other is a greedy randomized adaptive search procedure (GRASP) which is a technique that has achieved good results on a variety of combinatorial optimization problems. Both heuristics are compared to a previously proposed algorithm based on the traveling salesman problem (TSP). In addition, local search procedures are developed and adapted to each of the heuristics. A two-phase lower bounding scheme is presented as well. The first phase finds a lower bound based on the assignment relaxation for the asymmetric TSP. In phase two, attempts are made to improve the bound by inserting idle time. All procedures are compared for two different classes of randomly generated instances. In the first case where setup times are an order of magnitude smaller than the processing times, the new approaches prove superior to the TSP-based heuristic; for the case where both processing and setup times are identically distributed, the TSP-based heuristic outperforms the proposed procedures.     

Scatter search for the single source capacitated facility location problem

This paper considers the Single Source Capacitated Facility Location Problem (SSCFLP). We propose a Scatter Search approach to provide upper bounds for the optimal solution of the problem. The proposed approach uses GRASP to initialize the Reference Set. Solutions of the Reference Set are combined using a procedure that consists of two phases: (1) the initialization phase and (2) the improvement phase. During the initialization phase each client is assigned to an open facility to obtain a solution that is then improved with the improvement phase. Also, a tabu search algorithm is applied. In order to evaluate the proposed approach we use different sets of test problems. According to the results obtained we observe that the method provides good quality solutions with reasonable computational effort.     

Avoiding local optima in thep-hub location problem using tabu search and GRASP

In the discretep-hub location problem, various nodes interact with each other by sending and receiving given levels of traffic (such as telecommunications traffic, data transmissions, airline passengers, packages, etc.). It is necessary to choosep of the given nodes to act as hubs, which are fully interconnected; it is also necessary to connect each other node to one of these hubs so that traffic can be sent between any pair of nodes by using the hubs as switching points. The objective is to minimize the sum of the costs for sending traffic along the links connecting the various nodes. Like many combinatorial problems, thep-hub location problem has many local optima. Heuristics, such as exchange methods, can terminate once such a local optimum is encountered. In this paper, we describe new heuristics for thep-hub location problem, based on tabu search and on a greedy randomized adaptive search procedure (GRASP). These recently developed approaches to combinatorial optimization are capable of examining several local optima, so that, overall, superior solutions are found. Computational experience is reported in which both tabu search and GRASP found optimal hub locations (subject to the assumption that nodes must be assigned to the nearest hub) in over 90% of test problems. For problems for which such optima are not known, tabu search and GRASP generated new best-known solutions.     

Algorithms for single machine total tardiness scheduling with sequence dependent setups

We consider the problem of scheduling a single machine to minimize total tardiness with sequence dependent setup times. We present two algorithms, a problem space-based local search heuristic and a Greedy Randomized Adaptive Search Procedure (GRASP) for this problem. With respect to GRASP, our main contributions are—a new cost function in the construction phase, a new variation of Variable Neighborhood Search in the improvement phase, and Path Relinking using three different search neighborhoods. The problem space-based local search heuristic incorporates local search with respect to both the problem space and the solution space. We compare our algorithms with Simulated Annealing, Genetic Search, Pairwise Interchange, Branch and Bound and Ant Colony Search on a set of test problems from literature, showing that the algorithms perform very competitively.