A hybrid adaptive large neighborhood search heuristic for lot-sizing with setup times

This paper presents a hybrid of a general heuristic framework and a general purpose mixed-integer programming (MIP) solver. The framework is based on local search and an adaptive procedure which chooses between a set of large neighborhoods to be searched. A mixed integer programming solver and its built-in feasibility heuristics is used to search a neighborhood for improving solutions. The general reoptimization approach used for repairing solutions is specifically suited for combinatorial problems where it may be hard to otherwise design suitable repair neighborhoods. The hybrid heuristic framework is applied to the multi-item capacitated lot sizing problem with setup times, where experiments have been conducted on a series of instances from the literature and a newly generated extension of these. On average the presented heuristic outperforms the best heuristics from the literature, and the upper bounds found by the commercial MIP solver ILOG CPLEX using state-of-the-art MIP formulations. Furthermore, we improve the best known solutions on 60 out of 100 and improve the lower bound on all 100 instances from the literature.     

Cooperative Strategies for Solving the Bicriteria Sparse Multiple Knapsack Problem

For hard optimization problems, it is difficult to design heuristic algorithms which exhibit uniformly superior performance for all problem instances. As a result it becomes necessary to tailor the algorithms based on the problem instance. In this paper, we introduce the use of a cooperative problem solving team of heuristics that evolves algorithms for a given problem instance. The efficacy of this method is examined by solving six difficult instances of a bicriteria sparse multiple knapsack problem. Results indicate that such tailored algorithms uniformly improve solutions as compared to using predesigned heuristic algorithms.     

Local search intensified: Very large-scale variable neighborhood search for the multi-resource generalized assignment problem

We introduce a heuristic for the Multi-Resource Generalized Assignment Problem (MRGAP) based on the concepts of Very Large-Scale Neighborhood Search and Variable Neighborhood Search. The heuristic is a simplified version of the Very Large-Scale Variable Neighborhood Search for the Generalized Assignment Problem. Our algorithm can be viewed as a k-exchange heuristic; but unlike traditional k-exchange algorithms, we choose larger values of k resulting in neighborhoods of very large size with high probability. Searching this large neighborhood (approximately) amounts to solving a sequence of smaller MRGAPs either by exact algorithms or by heuristics. Computational results on benchmark test problems are presented. We obtained improved solutions for many instances compared to some of the best known heuristics for the MRGAP within reasonable running time. The central idea of our heuristic can be used to develop efficient heuristics for other hard combinatorial optimization problems as well.     

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.     

A linear programming approach for linear programs with probabilistic constraints

We study a class of mixed-integer programs for solving linear programs with joint probabilistic constraints from random right-hand side vectors with finite distributions. We present greedy and dual heuristic algorithms that construct and solve a sequence of linear programs. We provide optimality gaps for our heuristic solutions via the linear programming relaxation of the extended mixed-integer formulation of Luedtke et al. (2010) [13] as well as via lower bounds produced by their cutting plane method. While we demonstrate through an extensive computational study the effectiveness and scalability of our heuristics, we also prove that the theoretical worst-case solution quality for these algorithms is arbitrarily far from optimal. Our computational study compares our heuristics against both the extended mixed-integer programming formulation and the cutting plane method of Luedtke et al. (2010) [13]. Our heuristics efficiently and consistently produce solutions with small optimality gaps, while for larger instances the extended formulation becomes intractable and the optimality gaps from the cutting plane method increase to over 5%.     

Scatter search for project scheduling with resource availability cost

This paper considers a project scheduling problem with the objective of minimizing resource availability costs, taking into account a deadline for the project and precedence relations among the activities. Exact methods have been proposed for solving this problem, but we are not aware of existing heuristic methods. Scatter search is used to tackle this problem, and our implementation incorporates advanced strategies such as dynamic updating of the reference set, the use of frequency-based memory within the diversification generator, and a combination method based on path relinking. We also analyze the merit of employing a subset of different types when combining solutions. Extensive computational experiments involving more than 2400 instances are performed. For small instances, the performance of the proposed procedure is compared to optimal solutions generated by an exact cutting plane algorithm and upper and lower bounds from the literature. For medium and larger size instances, we developed simple multi-start heuristics and comparative results are reported.     

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%.     

The crane scheduling problem: models and solution approaches

In this paper, we study the crane scheduling problem for a vessel after the vessel is moored on a terminal and develop both exact and heuristic solution approaches for the problem. For small-sized instances, we develop a time-space network flow formulation with non-crossing constraints for the problem and apply an exact solution approach to obtain an optimal solution. For medium-sized instances, we develop a Lagrangian relaxation approach that allows us to obtain tight lower bounds and near-optimal solutions. For large-sized instances, we develop two heuristics and show that the error bounds of our heuristics are no more than 100%. Finally, we perform computational studies to show the effectiveness of our proposed solution approaches.     

Improved state space relaxation for constrained two-dimensional guillotine cutting problems

Christofides and Hadjiconstantinou (1995) introduced a dynamic programming state space relaxation for obtaining upper bounds for the Constrained Two-dimensional Guillotine Cutting Problem. The quality of those bounds depend on the chosen item weights, they are adjusted using a subgradient-like algorithm. This paper proposes Algorithm X, a new weight adjusting algorithm based on integer programming that provably obtains the optimal weights. In order to obtain even better upper bounds, that algorithm is generalized into Algorithm X2 for obtaining optimal two-dimensional item weights. We also present a full hybrid method, called Algorithm X2D, that computes those strong upper bounds but also provides feasible solutions obtained by: (1) exploring the suboptimal solutions hidden in the dynamic programming matrices; (2) performing a number of iterations of a GRASP based primal heuristic; and (3) executing X2H, an adaptation of Algorithm X2 to transform it into a primal heuristic. Extensive experiments with instances from the literature and on newly proposed instances, for both variants with and without item rotation, show that X2D can consistently deliver high-quality solutions and sharp upper bounds. In many cases the provided solutions are certified to be optimal.     

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.     

An algorithm and upper bounds for the weighted maximal planar graph problem

In this paper, we investigate the weighted maximal planar graph (WMPG) problem. Given a complete, edge-weighted, simple graph, the WMPG problem involves finding a subgraph with the highest sum of edge weights that is maximal planar, namely, it can be embedded in the plane without any of its edges intersecting, and no additional edge can be added to the subgraph without violating its planarity. We present a new integer linear programming (ILP) model for this problem. We then develop a cutting-plane algorithm to solve the WMPG problem based on the proposed ILP model. This algorithm enables the problem to be solved more efficiently than previously reported algorithms. New upper bounds are also provided, which are useful in evaluating the quality of heuristic solutions or in generating initial solutions for meta-heuristics. Computational results are reported for a set of 417 test instances of size varying from 6 to 100 nodes including 105 instances from the literature and 312 randomly generated instances. The computational results indicate that instances with up to 24 nodes can be solved optimally in reasonable computational time and the new upper bounds for larger instances significantly improve existing upper bounds.     

Algorithms for common due-date assignment and sequencing on a single machine with sequence-dependent setup times

This paper focuses on the single machine sequencing and common due-date assignment problem for the objective of minimizing the sum of the penalties associated with earliness, tardiness and due-date assignment. Unlike the previous research articles on this class of scheduling problem, we consider sequence-dependent setup times that make the problem much more difficult. To solve the problem, a branch and bound algorithm, which incorporates the method to obtain lower and upper bounds as well as a dominance property to reduce the search space, is suggested that gives the optimal solutions for small-sized instances. Heuristic algorithms are suggested to obtain solutions for large-sized problems within a reasonable computation time. The performances of both the optimal and heuristic algorithms, in computational experiments on randomly generated test instances, are reported.     

Application of the out-of-kilter algorithm to the asymmetric traveling salesman problem

This paper presents a heuristic method that finds optimum or near-optimum solutions to the asymmetric traveling salesman problem. The method uses the out-of-kilter algorithm to search for a neighbourhood. When subtours are produced by a flow-augmenting path of the out-of-kilter algorithm, it patches them into a Hamiltonian cycle. It extends the neighbourhood space by exchanging an even number of arcs, and it also exchanges arcs by a non-sequential primary change. Instances from real applications were used to test the algorithm, along with randomly generated problems. The new heuristic algorithm produced optimum solutions for 16 out of 28 real-world instances from TSPLIB and other sources. Also, compared with four efficient heuristics, it produced the best solutions for all except six instances. It also produced relatively good solutions in reasonable times for 216 randomly generated instances from nine instance generators.     

Sensitivity analysis of a greedy heuristic for knapsack problems

In this paper, we carry out parametric analysis as well as a tolerance limit based sensitivity analysis of a greedy heuristic for two knapsack problems—the 0–1 knapsack problem and the subset sum problem. We carry out the parametric analysis based on all problem parameters. In the tolerance limit based approach, we use a definition of the sensitivity analysis problem that is polynomial for polynomial heuristics. One of the interesting and counterintuitive results described in this paper is that the tolerance limits obtained from the heuristic sensitivity analysis cannot be used as bounds for the tolerance limits of the sensitivity analysis of optimal solutions in most cases.     

A hybrid multistart heuristic for the uncapacitated facility location problem

We present a multistart heuristic for the uncapacitated facility location problem, based on a very successful method we originally developed for the p-median problem. We show extensive empirical evidence to the effectiveness of our algorithm in practice. For most benchmarks instances in the literature, we obtain solutions that are either optimal or a fraction of a percentage point away from it. Even for pathological instances (created with the sole purpose of being hard to tackle), our algorithm can get very close to optimality if given enough time. It consistently outperforms other heuristics in the literature.     

A Hybrid Heuristic for the p-Median Problem

Given n customers and a set F of m potential facilities, the p-median problem consists in finding a subset of F with p facilities such that the cost of serving all customers is minimized. This is a well-known NP-complete problem with important applications in location science and classification (clustering). We present a multistart hybrid heuristic that combines elements of several traditional metaheuristics to find near-optimal solutions to this problem. Empirical results on instances from the literature attest the robustness of the algorithm, which performs at least as well as other methods, and often better in terms of both running time and solution quality. In all cases the solutions obtained by our method were within 0.1% of the best known upper bounds.     

Experimental analysis of heuristics for the bottleneck traveling salesman problem

In this paper we develop efficient heuristic algorithms to solve the bottleneck traveling salesman problem (BTSP). Results of extensive computational experiments are reported. Our heuristics produced optimal solutions for all the test problems considered from TSPLIB, JM-instances, National TSP instances, and VLSI TSP instances in very reasonable running time. We also conducted experiments with specially constructed ‘hard’ instances of the BTSP that produced optimal solutions for all but seven problems. Some fast construction heuristics are also discussed. Our algorithms could easily be modified to solve related problems such as the maximum scatter TSP and testing hamiltonicity of a graph.     

Beam search heuristics for quadratic earliness and tardiness scheduling

In this paper, we present beam search heuristics for the single machine scheduling problem with quadratic earliness and tardiness costs, and no machine idle time. These heuristics include classic beam search procedures, as well as filtered and recovering algorithms. We consider three dispatching heuristics as evaluation functions, in order to analyse the effect of different rules on the performance of the beam search procedures. The computational results show that using better dispatching heuristics improves the effectiveness of the beam search algorithms. The performance of the several heuristics is similar for instances with low variability. For high variability instances, however, the detailed, filtered and recovering beam search (RBS) procedures clearly outperform the best existing heuristic. The detailed beam search algorithm performs quite well, and is recommended for small- to medium-sized instances. For larger instances, however, this procedure requires excessive computation times, and the RBS algorithm then becomes the heuristic of choice.     

The ring-star problem: A new integer programming formulation and a branch-and-cut algorithm

A ring star in a graph is a subgraph that can be decomposed into a cycle (or ring) and a set of edges with exactly one vertex in the cycle. In the minimum ring-star problem (mrsp) the cost of a ring star is given by the sum of the costs of its edges, which vary, depending on whether the edge is part of the ring or not. The goal is to find a ring-star spanning subgraph minimizing the sum of all ring and assignment costs. In this paper we show that the mrsp can be reduced to a minimum (constrained) Steiner arborescence problem on a layered graph. This reduction is used to introduce a new integer programming formulation for the mrsp. We prove that the dual bound generated by the linear relaxation of this formulation always dominates the one provided by an early model from the literature. Based on our new formulation, we developed a branch-and-cut algorithm for the mrsp. On the primal side, we devised a grasp heuristic to generate good upper bounds for the problem. Computational tests with these algorithms were conducted on a benchmark of public domain. In these experiments both our exact and heuristics algorithms had excellent performances, noticeably in dealing with instances whose optimal solution has few vertices in the ring. In addition, we also investigate the minimum spanning caterpillar problem (mscp) which has the same input as the mrsp and admits feasible solutions that can be viewed as ring stars with paths in the place of rings. We present an easy reduction of the mscp to the mrsp, which makes it possible to solve to optimality instances of the former problem too. Experiments carried out with the mscp revealed that our branch-and-cut algorithm is capable to solve to optimality instances with up to 200 vertices in reasonable time.     

Minimising the number of gap-zeros in binary matrices

We study a problem of minimising the total number of zeros in the gaps between blocks of consecutive ones in the columns of a binary matrix by permuting its rows. The problem is referred to as the Consecutive Ones Matrix Augmentation Problem, and is known to be NP-hard. An analysis of the structure of an optimal solution allows us to focus on a restricted solution space, and to use an implicit representation for searching the space. We develop an exact solution algorithm, which is linear-time in the number of rows if the number of columns is constant, and two constructive heuristics to tackle instances with an arbitrary number of columns. The heuristics use a novel solution representation based upon row sequencing. In our computational study, all heuristic solutions are either optimal or close to an optimum. One of the heuristics is particularly effective, especially for problems with a large number of rows.