Relaxation heuristics for a generalized assignment problem

We propose relaxation heuristics for the problem of maximum profit assignment of n tasks to m agents (n > m), such that each task is assigned to only one agent subject to capacity constraints on the agents. Using Lagrangian or surrogate relaxation, the heuristics perform a subgradient search obtaining feasible solutions. Relaxation considers a vector of multipliers for the capacity constraints. The resolution of the Lagrangian is then immediate. For the surrogate, the resulting problem is a multiple choice knapsack that is again relaxed for continuous values of the variables, and solved in polynomial time. Relaxation multipliers are used with an improved heuristic of Martello and Toth or a new constructive heuristic to find good feasible solutions. Six heuristics are tested with problems of the literature and random generated problems. Best results are less than 0.5% from the optimal, with reasonable computational times for an AT/386 computer. It seems promising even for problems with correlated coefficients.     

GRASP heuristic with path-relinking for the multi-plant capacitated lot sizing problem

This paper addresses the independent multi-plant, multi-period, and multi-item capacitated lot sizing problem where transfers between the plants are allowed. This is an NP-hard combinatorial optimization problem and few solution methods have been proposed to solve it. We develop a GRASP (Greedy Randomized Adaptive Search Procedure) heuristic as well as a path-relinking intensification procedure to find cost-effective solutions for this problem. In addition, the proposed heuristics is used to solve some instances of the capacitated lot sizing problem with parallel machines. The results of the computational tests show that the proposed heuristics outperform other heuristics previously described in the literature. The results are confirmed by statistical tests.     

A new Lagrangean approach to the pooling problem

We present a new Lagrangean approach for the pooling problem. The relaxation targets all nonlinear constraints, and results in a Lagrangean subproblem with a nonlinear objective function and linear constraints, that is reformulated as a linear mixed integer program. Besides being used to generate lower bounds, the subproblem solutions are exploited within Lagrangean heuristics to find feasible solutions. Valid cuts, derived from bilinear terms, are added to the subproblem to strengthen the Lagrangean bound and improve the quality of feasible solutions. The procedure is tested on a benchmark set of fifteen problems from the literature. The proposed bounds are found to outperform or equal earlier bounds from the literature on 14 out of 15 tested problems. Similarly, the Lagrangean heuristics outperform the VNS and MALT heuristics on 4 instances. Furthermore, the Lagrangean lower bound is equal to the global optimum for nine problems, and on average is 2.1% from the optimum. The Lagrangean heuristics, on the other hand, find the global solution for ten problems and on average are 0.043% from the optimum.     

New heuristics for the maximum diversity problem

The maximum diversity problem (MDP) consists of identifying, in a population, a subset of elements, characterized by a set of attributes, that present the most diverse characteristics among the elements of the subset. The identification of such solution is an NP-hard problem. Some heuristics are available to obtain approximate solutions for this problem. In this paper, we propose different GRASP heuristics for the MDP, using distinct construction procedures and including a path-relinking technique. Performance comparison among related work and the proposed heuristics is provided. Experimental results show that the new GRASP heuristics are quite robust and are able to find high-quality solutions in reasonable computational times. G.C. Silva’s work sponsored by CAPES MSc scholarship. L.S. Ochi’s work sponsored by CNPq research grants 304103/2003-9 and 550059/2005-9. S.L. Martins’s work sponsored by CNPq research grant 475124/03-0. A. Plastino’s work sponsored by CNPq research grants 300879/00-8 and 475124/03-0.     

An Almost Optimal Heuristic for Preemptive C max Scheduling of Dependent Tasks on Parallel Identical Machines

We consider the problem of scheduling preemptable, dependent tasks on parallel, identical machines to minimize the makespan. The computational complexity of this problem remains open if the number of machines is fixed and larger than 2. The aim of this paper is to compare two heuristic algorithms on a basis of a computational experiment. The solutions generated by the heuristics are compared with optimal solutions obtained by a branch-and-bound algorithm. Computational results show that the heuristic based on node ordering finds optimal schedules for 99.9% of instances with the maximum relative deviation from optimum of 4.8%.     

Greedy Randomized Adaptive Search and Variable Neighbourhood Search for the minimum labelling spanning tree problem

This paper studies heuristics for the minimum labelling spanning tree (MLST) problem. The purpose is to find a spanning tree using edges that are as similar as possible. Given an undirected labelled connected graph, the minimum labelling spanning tree problem seeks a spanning tree whose edges have the smallest number of distinct labels. This problem has been shown to be NP-hard. A Greedy Randomized Adaptive Search Procedure (GRASP) and a Variable Neighbourhood Search (VNS) are proposed in this paper. They are compared with other algorithms recommended in the literature: the Modified Genetic Algorithm and the Pilot Method. Nonparametric statistical tests show that the heuristics based on GRASP and VNS outperform the other algorithms tested. Furthermore, a comparison with the results provided by an exact approach shows that we may quickly obtain optimal or near-optimal solutions with the proposed heuristics.     

Average-weight-controlled bin-oriented heuristics for the one-dimensional bin-packing problem

Bin-oriented heuristics for one-dimensional bin-packing problem construct solutions by packing one bin at a time. Several such heuristics consider two or more subsets for each bin and pack the one with the largest total weight. These heuristics sometimes generate poor solutions, due to a tendency to use many small items early in the process. To address this problem, we propose a method of controlling the average weight of items packed by bin-oriented heuristics. Constructive heuristics and an improvement heuristic based on this approach are introduced. Additionally, reduction methods for bin-oriented heuristics are presented. The results of an extensive computational study show that: (1) controlling average weight significantly improves solutions and reduces computation time of bin-oriented heuristics; (2) reduction methods improve solutions and processing times of some bin-oriented heuristics; and (3) the new improvement heuristic outperforms all other known complex heuristics, in terms of both average solution quality and computation time.     

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.     

DVRP: a hard dynamic combinatorial optimisation problem tackled by an evolutionary hyper-heuristic

In this paper we propose and evaluate an evolutionary-based hyper-heuristic approach, called EH-DVRP, for solving hard instances of the dynamic vehicle routing problem. A hyper-heuristic is a high-level algorithm, which generates or chooses a set of low-level heuristics in a common framework, to solve the problem at hand. In our collaborative framework, we have included three different types of low-level heuristics: constructive, perturbative, and noise heuristics. Basically, the hyper-heuristic manages and evolves a sophisticated sequence of combinations of these low-level heuristics, which are sequentially applied in order to construct and improve partial solutions, i.e., partial routes. In presenting some design considerations, we have taken into account the allowance of a proper cooperation and communication among low-level heuristics, and as a result, find the most promising sequence to tackle partial states of the (dynamic) problem. Our approach has been evaluated using the Kilby’s benchmarks, which comprise a large number of instances with different topologies and degrees of dynamism, and we have compared it with some well-known methods proposed in the literature. The experimental results have shown that, due to the dynamic nature of the hyper-heuristic, our proposed approach is able to adapt to dynamic scenarios more naturally than low-level heuristics. Furthermore, the hyper-heuristic can obtain high-quality solutions when compared with other (meta) heuristic-based methods. Therefore, the findings of this contribution justify the employment of hyper-heuristic techniques in such changing environments, and we believe that further contributions could be successfully proposed in related dynamic problems.     

Automated antenna positioning algorithms for wireless fixed-access networks

This article addresses a real-life problem - obtaining communication links between multiple base station sites, by positioning a minimal set of fixed-access relay antenna sites on a given terrain. Reducing the number of relay antenna sites is considered critical due to substantial installation and maintenance costs. Despite the significant cost saved by eliminating even a single antenna site, an inefficient manual approach is employed due to the computational complexity of the problem. From the theoretical point of view we show that this problem is not only NP hard, but also does not have a constant approximation. In this paper we suggest several alternative automated heuristics, relying on terrain preprocessing to find educated potential points for positioning relay stations. A large-scale computer-based experiment consisting of approximately 7,000 different scenarios was conducted. The quality of alternative solutions was compared by isolating and displaying factors that were found to affect the standard deviation of the solutions supplied by the tested heuristics. The results of the simulation based experiments show that the saving potential increases when more base stations are needed to be interconnected. The designs of a human expert were compared to the automatically generated solutions for a small subset of the experiment scenarios. Our studies indicate that for small networks (e.g., connecting up to ten base stations), the results obtained by human experts are adequate although they rarely exceed the quality of automated alternatives. However, the process of obtaining these results in comparison to automated heuristics is longer. In addition, when more base station sites need to be interconnected, the human approach is easily outperformed by our heuristics, both in terms of better results (fewer antennas) and in significant shorter calculation times.     

Heuristic Solution Methods for the Multilevel Generalized Assignment Problem

The multilevel generalized assignment problem is a problem of assigning agents to tasks where the agents can perform tasks at more than one efficiency level. A profit is associated with each assignment and the objective of the problem is profit maximization. Two heuristic solution methods are presented for the problem. The heuristics are developed from solution methods for the generalized assignment problem. One method uses a regret minimization approach whilst the other method uses a repair approach on a relaxation of the problem. The heuristics are able to solve moderately large instances of the problem rapidly and effectively. Procedures for deriving an upper bound on the solution of the problem are also described. On larger and harder instances of the problem one heuristic is particularly effective.     

Scatter search for a real-life heterogeneous fleet vehicle routing problem with time windows and split deliveries in Brazil

In this paper, we consider a real-life heterogeneous fleet vehicle routing problem with time windows and split deliveries that occurs in a major Brazilian retail group. A single depot attends 519 stores of the group distributed in 11 Brazilian states. To find good solutions to this problem, we propose heuristics as initial solutions and a scatter search (SS) approach. Next, the produced solutions are compared with the routes actually covered by the company. Our results show that the total distribution cost can be reduced significantly when such methods are used. Experimental testing with benchmark instances is used to assess the merit of our proposed procedure.     

Greedy and Local Search Heuristics for Unconstrained Binary Quadratic Programming

In this paper, a greedy heuristic and two local search algorithms, 1-opt local search and k-opt local search, are proposed for the unconstrained binary quadratic programming problem (BQP). These heuristics are well suited for the incorporation into meta-heuristics such as evolutionary algorithms. Their performance is compared for 115 problem instances. All methods are capable of producing high quality solutions in short time. In particular, the greedy heuristic is able to find near optimum solutions a few percent below the best-known solutions, and the local search procedures are sufficient to find the best-known solutions of all problem instances with n 100. The k-opt local searches even find the best-known solutions for all problems of size n 250 and for 11 out of 15 instances of size n = 500 in all runs. For larger problems (n = 500, 1000, 2500), the heuristics appear to be capable of finding near optimum solutions quickly. Therefore, the proposed heuristics—especially the k-opt local search—offer a great potential for the incorporation in more sophisticated meta-heuristics.     

Resource leveling in a machine environment

We address resource leveling problems in a machine environment. Given a set of m machines, one or more renewable resources, and a set of n tasks, each assigned to exactly one of the machines. Each task has a processing time, an earliest start time, a deadline, and resource requirements. There are no precedence relations between the tasks. The tasks have to be sequenced on the machines while minimizing a function of the level of resource utilization from each resource over time. We provide various complexity results including a polynomial time algorithm for a one machine special case. We also propose an exact method using various techniques to find optimal or close-to-optimal solutions. The computational experiments show that our exact method significantly outperforms heuristics and a commercial MIP solver.     

Finding preferred subsets of Pareto optimal solutions

Multi-objective optimization algorithms can generate large sets of Pareto optimal (non-dominated) solutions. Identifying the best solutions across a very large number of Pareto optimal solutions can be a challenge. Therefore it is useful for the decision-maker to be able to obtain a small set of preferred Pareto optimal solutions. This paper analyzes a discrete optimization problem introduced to obtain optimal subsets of solutions from large sets of Pareto optimal solutions. This discrete optimization problem is proven to be NP-hard. Two exact algorithms and five heuristics are presented to address this problem. Five test problems are used to compare the performances of these algorithms and heuristics. The results suggest that preferred subset of Pareto optimal solutions can be efficiently obtained using the heuristics, while for smaller problems, exact algorithms can be applied.     

Lagrangian heuristics for instructor scheduling in executive development programmes

In this paper we present the problem of scheduling instructors in a university management development programme. Problems of similar structure arise in a number of scheduling applications like assigning officials to athletic competitions, inspectors to sites and maintenance crews to jobs. The problem is formulated as a zero-one linear integer programme but is difficult to solve in real life situations because of problem size. The bounds on total assignments for different nested time periods give sub-problems that can be solved as network flow problems. Four Lagrangian relaxation heuristics are developed using different relaxations of the problem. Computational results are reported on 1350 random problems. In over 85% of these problems, the heuristics find solutions within 1% of the optimal. Heuristic performance is also analyzed in terms of average percent deviation from optimal, percent of times optimal solution is found and the cpu time. Computational results on two significantly larger real problems indicate that the heuristics are capable of solving real sized problems with tolerable deviations of around 4% from the optimal. An integrated strategy utilizing the strengths of the optimal and heuristic approaches is described for schedule generation and updating.     

On the Significance of the Initial Solution in Travelling Salesman Heuristics

The quality requirements set by edge exchange heuristics on their initial solutions are evaluated in connection with the travelling salesman problem. The performance of the heuristics is measured using the expected value of the best solution achievable in a certain computing time. The computational results show that the use of initial solutions generated by applying a construction heuristic, instead of random initial solutions, typically improves the performance of edge exchange heuristics. The improvement, however, is dependent on the edge exchange heuristic to be used, the properties of the problem, and the computing time available.     

Efficient and effective heuristics for the coordinated capacitated lot-size problem

Coordinating procurement decisions for a family of products that share a constrained resource, such as an ocean shipping container, is an important managerial problem. However due to the problem’s difficult mathematical properties, efficient and effective solution procedures for the problem have eluded researchers. This paper proposes two heuristics, for the capacitated, coordinated dynamic demand lot-size problem with deterministic but time-varying demand. In addition to inventory holding costs, the problem assumes a joint setup cost each time any member of the product family is replenished and an individual item setup cost for each item type replenished. The objective is to meet all customer demand without backorders at minimum total cost. We propose a six-phase heuristic (SPH) and a simulated annealing meta-heuristic (SAM). The SPH begins by assuming that each customer demand is met by a unique replenishment and then it seeks to iteratively maximize the net savings associated with order consolidation. Using SPH to find a starting solution, the SAM orchestrates escaping local solutions and exploring other areas of the solution state space that are randomly generated in an annealing search process. The results of extensive computational experiments document the effectiveness and efficiency of the heuristics. Over a wide range of problem parameter values, the SPH and SAM find solutions with an average optimality gap of 1.53% and 0.47% in an average time of 0.023 CPU seconds and 0.32 CPU seconds, respectively. The heuristics are strong candidates for application as stand alone solvers or as an upper bounding procedure within an optimization based algorithm. The procedures are currently being tested as a solver in the procurement software suite of a nationally recognized procurement software provider.     

Heuristics for multiple facility location in a network using the variance criterion

In cyclic networks the p-variance location problem is NP-hard, and therefore it is suitable to use heuristic methods to find approximate solutions to the problem. To this end, two strategies are explored, the first using combinatorial search procedures over the vertex set, whereas the second searches for the solution over the entire network. The initial vertex set solutions are generated by using tabu search, variable neighbourhood search and interchange procedures. The heuristics have been tested on instances of up to 30 vertices and 70 edges, and their performances compared.     

The mixed and multi model line balancing problem: a comparison

The balancing problem deals with the assignment of tasks to work stations. We can distinguish two approaches in the literature on the mixed model line balancing problem, that both transform this problem into a single model line balancing problem. These approaches use combined precedence diagrams and adjusted task processing times, respectively.An experiment was carried out to compare several heuristics based on the combined precedence diagram. A new optimisation method has been developed. The results indicate that the position of common tasks in the precedence diagram of the different models has a significant effect on both the CPU time and the unequal distribution of the total work content of single models among work stations. Moreover, good solutions with respect to the number of required stations go together with long CPU times. For several instances, we decreased the CPU times considerably without deteriorating the performance of the methods, by using a reversed combined precedence diagram.