A new iterative-doubling Greedy–Lookahead algorithm for the single container loading problem

The aim of the Single Container Loading Problem (SCLP) is to pack three-dimensional boxes into a three-dimensional container so as to maximize the volume utilization of the container. We propose a new block building approach that constructs packings by placing one block (of boxes) at a time until no more boxes can be loaded. The key to obtaining high quality solutions is to select the right block to place into the right free space cuboid (or residual space) in the container. We propose a new heuristic for evaluating the fitness of residual spaces, and use a tree search to decide the best residual space-block pair at each step. The resultant algorithm outperforms the best known algorithms based on the 1600 commonly used benchmark instances even when given fewer computational resources. We also adapted our approach to address the full support constraint. The computational results for the full support support variant on the 1600 instances similarly show a significant improvement over existing techniques even when given substantially fewer computational resources.     

A reference length approach for the 3D strip packing problem

In the three-dimensional strip packing problem (3DSP), we are given a container with an open dimension and a set of rectangular cuboids (boxes) and the task is to orthogonally pack all the boxes into the container such that the magnitude of the open dimension is minimized. We propose a block building heuristic based on extreme points for this problem that uses a reference length to guide its solution. Our 3DSP approach employs this heuristic in a one-step lookahead tree search algorithm using an iterative construction strategy. We tested our approach on standard 3DSP benchmark test data; the results show that our approach produces better solutions on average than all other approaches in literature for the majority of these data sets using comparable computation time.     

Fast local search for the maximum independent set problem

Given a graph G=(V,E), the independent set problem is that of finding a maximum-cardinality subset S of V such that no two vertices in S are adjacent. We introduce two fast local search routines for this problem. The first can determine in linear time whether a maximal solution can be improved by replacing a single vertex with two others. The second routine can determine in O(mΔ) time (where Δ is the highest degree in the graph) whether there are two solution vertices than can be replaced by a set of three. We also present a more elaborate heuristic that successfully applies local search to find near-optimum solutions to a wide variety of instances. We test our algorithms on instances from the literature as well as on new ones proposed in this paper.     

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.     

General variable neighborhood search for the uncapacitated single allocation <Emphasis Type="Italic">p</Emphasis>-hub center problem

In this paper we propose a general variable neighborhood search heuristic for solving the uncapacitated single allocation p-hub center problem (USApHCP). For the local search step we develop a nested variable neighborhood descent strategy. The proposed approach is tested on benchmark instances from the literature and found to outperform the state-of-the-art heuristic based on ant colony optimization. We also test our heuristic on large scale instances that were not previously considered as test instances for the USApHCP. Moreover, exact solutions were reached by our GVNS for all instances where optimal solutions are known.     

Scheduling just-in-time part supply for mixed-model assembly lines

With increasing cost competition and product variety, providing an efficient just-in-time (JIT) supply has become one of the greatest challenges in the use of mixed-model assembly line production systems. In the present paper, therefore, we propose a new approach for scheduling JIT part supply from a central storage center. Usually, materials are stored in boxes that are allotted to the consumptive stations of the line by a forklift. For such a real-world problem, a new model, a complexity proof as well as different exact and heuristic solution procedures are provided. Furthermore, a direct comparison with a simple two-bin kanban system is provided. Such a system is currently applied in the real-world industrial process that motivates our research. It becomes obvious that this policy is considerably outperformed according to the resulting inventory- and α-service levels. Moreover, at the interface between logistics and assembly operations, strategic management implications are obtained. Specifically, based on the new approach, it is the first time a statistical analysis is being made as to whether widespread Level Scheduling policies, which are well-known from the Toyota Production System, indeed facilitate material supply. Note that in the literature it is frequently claimed that this causality exists.     

Variable neighborhood search for harmonic means clustering

Harmonic means clustering is a variant of minimum sum of squares clustering (which is sometimes called K-means clustering), designed to alleviate the dependance of the results on the choice of the initial solution. In the harmonic means clustering problem, the sum of harmonic averages of the distances from the data points to all cluster centroids is minimized. In this paper, we propose a variable neighborhood search heuristic for solving it. This heuristic has been tested on numerous datasets from the literature. It appears that our results compare favorably with recent ones from tabu search and simulated annealing heuristics.     

A heuristic for the circle packing problem with a variety of containers

In this paper we present a heuristic algorithm based on the formulation space search method to solve the circle packing problem. The circle packing problem is the problem of finding the maximum radius of a specified number of identical circles that can be fitted, without overlaps, into a two-dimensional container of fixed size. In this paper we consider a variety of containers: the unit circle, unit square, rectangle, isosceles right-angled triangle and semicircle. The problem is formulated as a nonlinear optimization problem involving both Cartesian and polar coordinate systems.Formulation space search consists of switching between different formulations of the same problem, each formulation potentially having different properties in terms of nonlinear optimization. As a component of our heuristic we solve a nonlinear optimization problem using the solver SNOPT.Our heuristic improves on previous results based on formulation space search presented in the literature. For a number of the containers we improve on the best result previously known. Our heuristic is also a computationally effective approach (when balancing quality of result obtained against computation time required) when compared with other work presented in the literature.     

A hybrid GRASP/VND algorithm for two- and three-dimensional bin packing

The three-dimensional bin packing problem consists of packing a set of boxes into the minimum number of bins. In this paper we propose a new GRASP algorithm for solving three-dimensional bin packing problems which can also be directly applied to the two-dimensional case. The constructive phase is based on a maximal-space heuristic developed for the container loading problem. In the improvement phase, several new moves are designed and combined in a VND structure. The resulting hybrid GRASP/VND algorithm is simple and quite fast and the extensive computational results on test instances from the literature show that the quality of the solutions is equal to or better than that obtained by the best existing heuristic procedures.     

An effective greedy heuristic for the Social Golfer Problem

The Social Golfer Problem (SGP) is a combinatorial optimization problem that exhibits a lot of symmetry and has recently attracted significant attention. In this paper, we present a new greedy heuristic for the SGP, based on the intuitive concept of freedom among players. We use this heuristic in a complete backtracking search, and match the best current results of constraint solvers for several SGP instances with a much simpler method. We then use the main idea of the heuristic to construct initial configurations for a metaheuristic approach, and show that this significantly improves results obtained by local search alone. In particular, our method is the first metaheuristic technique that can solve the original problem instance optimally. We show that our approach is also highly competitive with other metaheuristic and constraint-based methods on many other benchmark instances from the literature.     

MIP-based approaches for the container loading problem with multi-drop constraints

In this paper, we present approaches based on a mixed integer linear programming model (MIP) for the problem of packing rectangular boxes into a container or truck, considering multi-drop constraints. We assume that the delivery route of the container is already known in advance and that the volume of the cargo is less than or equal to the container volume. Considering the sequence that the boxes should be unloaded, the aim is to avoid additional handling when each drop-off point of the route is reached, as well as ensuring that the boxes do not overlap each other and the cargo loading is stable. Computational tests with the proposed model and the approaches were performed with randomly generated instances and instances from the literature using an optimization solver embedded into a modeling language. The results validate the model and the approaches, but indicate that they are able to handle only problems of a moderate size. However, the model and the approaches can be useful to motivate future research to solve larger problems, as well as to solve more general problems considering integrated vehicle routing and container loading problems.     

An efficient heuristic algorithm for the capacitated p-\! median problem

The capacitated $p$ -median problem (CPMP) is one of the well-known facility-location problems. The objective of the problem is to minimize total cost of locating a set of capacitated service points and allocating a set of demand points to the located service points, while the total allocated demand for each service point is not be greater than its capacity limit. This paper presents an efficient heuristic algorithm based on the local branching and relaxation induced neighborhood search methods for the CPMP. The proposed algorithm is a heuristic technique that utilizes a general mixed integer programming solver to explore neighborhoods. The parameters of the proposed algorithm are tuned by design of experiments. The proposed method is tested on a large set of benchmark instances. The results show that the method outperforms the best method found in the literature.     

Heuristics for the container loading problem

The knapsack container loading problem is the problem of loading a subset of rectangular boxes into a rectangular container of fixed dimensions such that the volume of the packed boxes is maximized. A new heuristic based on the wall-building approach is proposed, which decomposes the problem into a number of layers which again are split into a number of strips. The packing of a strip may be formulated and solved optimally as a Knapsack Problem with capacity equal to the width or height of the container. The depth of a layer as well as the thickness of each strip is decided through a branch-and-bound approach where at each node only a subset of branches is explored.Several ranking rules for the selection of the most promising layer depths and strip widths are presented and the performance of the corresponding algorithms is experimentally compared for homogeneous and heterogeneous instances. The best ranking rule is then used in a comprehensive computational study involving large-sized instances. These computational results show that instances with a total box volume up to 90% easily may be solved to optimality, and that average fillings of the container volume exceeding 95% may be obtained for large-sized instances.     

A mathematical formulation and complexity considerations for the blocks relocation problem

The blocks relocation problem (BRP) may be defined as follows: given a set of homogeneous blocks stored in a two-dimensional stock, which relocations are necessary to retrieve the blocks from the stock in a predefined order while minimizing the number of those relocations? In this paper, we first prove NP-hardness of the BRP as well as a special case, closing open research questions. Moreover, we propose different solution approaches. First, a mathematical model is presented that provides optimal solutions to the general BRP in cases where instances are small. To overcome such limitation, some realistic assumption taken from the literature is introduced, leading to the definition of a binary linear programming model. In terms of computational time, this approach is reasonably fast to be used to solve medium-sized instances. In addition, we propose a simple heuristic based upon a set of relocation rules. This heuristic is used to generate “good” quality solutions for larger instances in very short computational time, and, consequently, is proposed for tackling problem instances where solutions are required (almost) immediately. Solution quality of the heuristic is measured against optimal solutions obtained using a state-of-the-art commercial solver and both of them are compared with reference results from literature.     

A goal-driven prototype column generation strategy for the multiple container loading cost minimization problem

In the multiple container loading cost minimization problem (MCLCMP), rectangular boxes of various dimensions are loaded into rectangular containers of various sizes so as to minimize the total shipping cost. The MCLCMP can be naturally modeled as a set cover problem. We generalize the set cover formulation by introducing a new parameter to model the gross volume utilization of containers in a solution. The state-of-the-art algorithm tackles the MCLCMP using the prototype column generation (PCG) technique. PCG is an effective technique for speeding up the column generation technique for extremely hard optimization problems where their corresponding pricing subproblems are NP-hard. We propose a new approach to the MCLCMP that combines the PCG technique with a goal-driven search. Our goal-driven prototype column generation (GD-PCG) algorithm improves the original PCG approach in three respects. Computational experiments suggest that all three enhancements are effective. Our GD-PCG algorithm produces significantly better solutions for the 350 existing benchmark instances than all other approaches in the literature using less computation time. We also generate two new set instances based on industrial data and the classical single container loading instances.     

A local search heuristic for the Multi-Commodity k-splittable Maximum Flow Problem

The Multi-Commodity $k$ -splittable Maximum Flow Problem consists of maximizing the amount of flow routed through a network such that each commodity uses at most $k$ paths and such that edge capacities are satisfied. The problem is $\mathcal NP $ -hard and has application in a.o. telecommunications. In this paper, a local search heuristic for solving the problem is proposed. The heuristic is an iterative shortest path procedure on a reduced graph combined with a local search procedure to modify certain path flows and prioritize the different commodities. The heuristic is tested on benchmark instances from the literature and solves 83 % of the instances to optimality. For the remaining instances, the heuristic finds good solution values which on average are 1.04 % from the optimal. The heuristic solves all instances in less than a second. Compared to other heuristics, the proposed heuristic again shows superior performance with respect to solution quality.     

A fast meta-heuristic approach for the $$(\alpha ,\beta )-k$$-feature set problem

The feature selection problem aims to choose a subset of a given set of features that best represents the whole in a particular aspect, preserving the original semantics of the variables on the given samples and classes. In 2004, a new approach to perform feature selection was proposed. It was based on a NP-complete combinatorial optimisation problem called (\(\alpha ,\beta \))-k-feature set problem. Although effective for many practical cases, which made the approach an important feature selection tool, the only existing solution method, proposed on the original paper, was found not to work well for several instances. Our work aims to cover this gap found on the literature, quickly obtaining high quality solutions for the instances that existing approach can not solve. This work proposes a heuristic based on the greedy randomised adaptive search procedure and tabu search to address this problem; and benchmark instances to evaluate its performance. The computational results show that our method can obtain high quality solutions for both real and the proposed artificial instances and requires only a fraction of the computational resources required by the state of the art exact and heuristic approaches which use mixed integer programming models.     

Planning Activities in a Network of Logistic Platforms with Shared Resources

This paper has been motivated by the study of a real application, the transshipment container terminal of Gioia Tauro in Italy. The activities in a container terminal concern with the movement of containers from/to mother vessels and feeders and with the handling and storage of containers in the yard. For such type of applications both operational (e.g., scheduling) and tactical (e.g., planning) models, currently available in the literature, are not useful in terms of operations management and resources optimization. Indeed, the former models are too detailed for the complexity of the systems, while the latter are not able to capture the operational constraints in representing those activities which limit the nominal capacity. Herein, the container terminal, or more in general a service or production system, is represented as a network of complex substructures or platforms. The idea is to formalize the concept of platform capacity, which is used to represent the operational aspects of the container terminal in a mathematical model for the tactical planning. The problem, which consists in finding an allocation of resources in each platform in order to minimize the total delay on the overall network and on the time horizon, is modelled by a mathematical programming formulation for which we carry out a computational analysis using CPLEX-MIP solver. Moreover, we present a dynamic programming based heuristic to solve larger instances in short computational time. On all but one of the smaller instances, the heuristic solutions are also optimal. On the larger instances, the maximum gap, i.e. the percentage deviation, between the heuristic solutions and the best solutions computed by CPLEX-MIP within the time limit of 3600 s, has been 6.3%.     

A heuristic procedure for the Capacitated m-Ring-Star problem

In this paper we propose a heuristic method to solve the Capacitated m-Ring-Star Problem which has many practical applications in communication networks. The problem consists of finding m rings (simple cycles) visiting a central depot, a subset of customers and a subset of potential (Steiner) nodes, while customers not belonging to any ring must be “allocated” to a visited (customer or Steiner) node. Moreover, the rings must be node-disjoint and the number of customers allocated or visited in a ring cannot be greater than the capacity Q given as an input parameter. The objective is to minimize the total visiting and allocation costs. The problem is a generalization of the Traveling Salesman Problem, hence it is NP-hard. In the proposed heuristic, after the construction phase, a series of different local search procedures are applied iteratively. This method incorporates some random aspects by perturbing the current solution through a “shaking” procedure which is applied whenever the algorithm remains in a local optimum for a given number of iterations. Computational experiments on the benchmark instances of the literature show that the proposed heuristic is able to obtain, within a short computing time, most of the optimal solutions and can improve some of the best known results.     

A “reduce and solve” approach for the multiple-choice multidimensional knapsack problem

The multiple-choice multidimensional knapsack problem (MMKP) is a well-known NP-hard combinatorial optimization problem with a number of important applications. In this paper, we present a “reduce and solve” heuristic approach which combines problem reduction techniques with an Integer Linear Programming (ILP) solver (CPLEX). The key ingredient of the proposed approach is a set of group fixing and variable fixing rules. These fixing rules rely mainly on information from the linear relaxation of the given problem and aim to generate reduced critical subproblem to be solved by the ILP solver. Additional strategies are used to explore the space of the reduced problems. Extensive experimental studies over two sets of 37 MMKP benchmark instances in the literature show that our approach competes favorably with the most recent state-of-the-art algorithms. In particular, for the set of 27 conventional benchmarks, the proposed approach finds an improved best lower bound for 11 instances and as a by-product improves all the previous best upper bounds. For the 10 additional instances with irregular structures, the method improves 7 best known results.