A GRASP×ELS for the vehicle routing problem with basic three-dimensional loading constraints

This paper addresses an extension of the capacitated vehicle routing problem where the client demand consists of three-dimensional weighted items (3L-CVRP). The objective is to design a set of trips for a homogeneous fleet of vehicles based at a depot node which minimizes the total transportation cost. Items in each vehicle trip must satisfy the three-dimensional orthogonal packing constraints. This problem is strongly connected to real-life transportation systems where the packing of items to be delivered by each vehicle can have a significant impact on the routes. We propose a new way to solve the packing sub-problem. It consists of a two-step procedure in which the z-constraints are first relaxed to get a (x,y) positioning of the items. Then, a compatible z-coordinate is computed to get a packing solution. Items can be rotated but additional constraints such as item fragility, support and LIFO are not considered. This method is included in a GRASP×ELS hybrid algorithm dedicated to the computation of VRP routes. The route optimization alternates between two search spaces: the space of VRP routes and the space of giant trips. The projection from one to the other is done by dedicated procedures (namely the Split and the concatenation algorithms). Moreover, a Local Search is defined on each search space. Furthermore, hash tables are used to store the result of the packing checks and thus save a substantial amount of CPU time. The effectiveness of our approach is illustrated by computational experiments on 3L-CVRP instances from the literature. A new set of realistic instances based on the 96 French districts are also proposed. They range from 19 nodes for the small instances to 255 nodes for the large instances and they can be stated as realistic since they are based on true travel distances in kilometers between French cities. The impact of the hash tables is illustrated as well.     

Exact algorithms for the two-dimensional guillotine knapsack

The two-dimensional knapsack problem requires to pack a maximum profit subset of “small” rectangular items into a unique “large” rectangular sheet. Packing must be orthogonal without rotation, i.e., all the rectangle heights must be parallel in the packing, and parallel to the height of the sheet. In addition, we require that each item can be unloaded from the sheet in stages, i.e., by unloading simultaneously all items packed at the same either y or x coordinate. This corresponds to use guillotine cuts in the associated cutting problem.In this paper we present a recursive exact procedure that, given a set of items and a unique sheet, constructs the set of associated guillotine packings. Such a procedure is then embedded into two exact algorithms for solving the guillotine two-dimensional knapsack problem. The algorithms are computationally evaluated on well-known benchmark instances from the literature.The C++ source code of the recursive procedure is available upon request from the authors.     

Local search techniques for a routing-packing problem

We propose a complex real-world problem in logistics that integrates routing and packing aspects. It can be seen as an extension of the Three-Dimensional Loading Capacitated Vehicle Routing Problem (3L-CVRP) introduced by Gendreau, Iori, Laporte, and Martello (2006). The 3L-CVRP consists in finding a set of routes that satisfies the demand of all customers, minimizes the total routing cost, and guarantees a packing of items that is feasible according to loading constraints. Our problem formulation includes additional constraints in relation to the stability of the cargo, to the fragility of items, and to the loading and unloading policy. In addition, it considers the possibility of split deliveries, so that each customer can be visited more than once. We propose a local search approach that considers the overall problem in a single stage. It is based on a composite strategy that interleaves simulated annealing with large-neighborhood search. We test our solver on 13 real-world instances provided by our industrial partner, which are very diverse in size and features. In addition, we compare our solver on benchmarks from the literature of the 3L-CVRP showing that our solver performs well compared to other approaches proposed in the literature.     

A Computational Study of a Cutting Plane Algorithm for University Course Timetabling

In this paper, we describe a case-study where a Branch-and-Cut algorithm yields the “optimal” solution of a real-world timetabling problem of University courses (University Course Timetabling problem). The problem is formulated as a Set Packing problem with side constraints. To tighten the initial formulation, we utilize well-known valid inequalities of the Set Packing polytope, namely Clique and Lifted Odd-Hole inequalities. We also analyze the combinatorial properties of the problem to introduce new families of cutting planes that are not valid for the Set Packing polytope, and their separation algorithms. These cutting planes turned out to be very effective to yield the optimal solution of a set of real-world instances with up to 69 courses, 59 teachers, and 15 rooms.     

On Lazy Bin Covering and Packing problems

In this paper, we study two interesting variants of the classical bin packing problem, called Lazy Bin Covering (LBC) and Cardinality Constrained Maximum Resource Bin Packing (CCMRBP) problems. For the offline LBC problem, we first prove the approximation ratio of the First-Fit-Decreasing and First-Fit-Increasing algorithms, then present an APTAS. For the online LBC problem, we give a competitive analysis for the algorithms of Next-Fit, Worst-Fit, First-Fit, and a modified HARMONIC_M algorithm. The CCMRBP problem is a generalization of the Maximum Resource Bin Packing (MRBP) problem Boyar et al. (2006) [1]. For this problem, we prove that its offline version is no harder to approximate than the offline MRBP problem.     

A multi-start evolutionary local search for the two-dimensional loading capacitated vehicle routing problem

This paper addresses an extension of the capacitated vehicle routing problem where customer demand is composed of two-dimensional weighted items (2L-CVRP). The objective consists in designing a set of trips minimizing the total transportation cost with a homogenous fleet of vehicles based on a depot node. Items in each vehicle trip must satisfy the two-dimensional orthogonal packing constraints. A GRASP×ELS algorithm is proposed to compute solutions of a simpler problem in which the loading constraints are transformed into resource constrained project scheduling problem (RCPSP) constraints. We denote this relaxed problem RCPSP-CVRP. The optimization framework deals with RCPSP-CVRP and lastly RCPSP-CVRP solutions are transformed into 2L-CVRP solutions by solving a dedicated packing problem. The effectiveness of our approach is demonstrated through computational experiments including both classical CVRP and 2L-CVRP instances. Numerical experiments show that the GRASP×ELS approach outperforms all previously published methods.     

Fair versus Unrestricted Bin Packing

Abstract. We consider the on-line Dual Bin Packing problem where we have n unit size bins and a sequence of items. The goal is to maximize the number of items that are packed in the bins by an on-line algorithm. We investigate unrestricted algorithms that have the power of performing admission control on the items, i.e., rejecting items while there is enough space to pack them, versus fair algorithms that reject an item only when there is not enough space to pack it. We show that by performing admission control on the items, we get better performance compared with the performance achieved on the fair version of the problem. Our main result shows that with an unfair variant of First-Fit, we can pack approximately two-thirds of the items for sequences for which an optimal off-line algorithm can pack all the items. This is in contrast to standard First-Fit where we show an asymptotically tight hardness result: if the number of bins can be chosen arbitrarily large, the fraction of the items packed by First-Fit comes arbitrarily close to five-eighths.     

New bin packing fast lower bounds

In this paper, we address the issue of computing fast lower bounds for the Bin Packing problem, i.e., bounds that have a computational complexity dominated by the complexity of ordering the items by non-increasing values of their volume. We introduce new classes of fast lower bounds with improved asymptotic worst-case performance compared to well-known results for similar computational effort. Experimental results on a large set of problem instances indicate that the proposed bounds reduce both the deviation from the optimum and the computational effort.     

New heuristics for one-dimensional bin-packing

Several new heuristics for solving the one-dimensional bin packing problem are presented. Some of these are based on the minimal bin slack (MBS) heuristic of Gupta and Ho. A different algorithm is one based on the variable neighbourhood search metaheuristic. The most effective algorithm turned out to be one based on running one of the former to provide an initial solution for the latter. When tested on 1370 benchmark test problem instances from two sources, this last hybrid algorithm proved capable of achieving the optimal solution for 1329, and could find for 4 instances solutions better than the best known. This is remarkable performance when set against other methods, both heuristic and optimum seeking.Scope and purposePacking items into boxes or bins is a task that occurs frequently in distribution and production. A large variety of different packing problems can be distinguished, depending on the size and shape of the items, as well as on the form and capacity of the bins (H. Dyckhoff and U. Finke, Cutting and Packing in Production and Distribution: a Typology and Bibliography, Springer, Berlin, 1992). Similar problems occur in minimising material wastage while cutting pieces into particular smaller ones and in the scheduling of identical processors in order to minimise total completion time. This work addresses the basic packing problem, known as the one-dimensional bin packing problem, where it is required to pack a number of items into the smallest possible number of bins of pre-specified equal capacity. Even though this problem is simple to state, it is NP hard, i.e., it is unlikely that there exists an algorithm that could solve every instance of it in polynomial time. Solution of more general realistic packing problems is probably contingent upon the availability of effective and computationally efficient solution procedures for the basic problem. In this work we present several heuristics capable of doing that. Extensive computational testing attests to the power of these heuristics, as well as to their computational efficiency.     

Solving large FPT problems on coarse-grained parallel machines

Fixed-parameter tractability (FPT) techniques have recently been successful in solving NP-complete problem instances of practical importance which were too large to be solved with previous methods. In this paper, we show how to enhance this approach through the addition of parallelism, thereby allowing even larger problem instances to be solved in practice. More precisely, we demonstrate the potential of parallelism when applied to the bounded-tree search phase of FPT algorithms. We apply our methodology to the k-Vertex Cover problem which has important applications in, for example, the analysis of multiple sequence alignments for computational biochemistry. We have implemented our parallel FPT method for the k-Vertex Cover problem using C and the MPI communication library, and tested it on a 32-node Beowulf cluster. This is the first experimental examination of parallel FPT techniques. As part of our experiments, we solved larger instances of k-Vertex Cover than in any previously reported implementations. For example, our code can solve problem instances with k?400 in less than .     

A two-stage tabu search algorithm with enhanced packing heuristics for the 3L-CVRP and M3L-CVRP

The Three-Dimensional Loading Capacitated Vehicle Routing Problem (3L-CVRP) addresses practical constraints frequently encountered in the freight transportation industry. In this problem, the task is to serve all customers using a homogeneous fleet of vehicles at minimum traveling cost. The constraints imposed by the three-dimensional shape of the goods, the unloading order, item fragility, and the stability of the loading plan of each vehicle are explicitly considered. We improved two well-known packing heuristics, namely the Deepest-Bottom-Left-Fill heuristic and the Maximum Touching Area heuristic, for the three-dimensional loading sub-problem and provided efficient implementations. Based on these two new heuristics, an effective tabu search algorithm is given to address the overall problem. Computational experiments on publicly available test instances show our new approach outperforms the current best algorithms for 20 out of 27 instances. Our approach is also superior to the existing algorithm on benchmark data for the closely related problem variant M3L-CVRP (which uses a slightly different unloading order constraint compared to 3L-CVRP).     

Characterizing reinforcement learning methods through parameterized learning problems

The field of reinforcement learning (RL) has been energized in the past few decades by elegant theoretical results indicating under what conditions, and how quickly, certain algorithms are guaranteed to converge to optimal policies. However, in practical problems, these conditions are seldom met. When we cannot achieve optimality, the performance of RL algorithms must be measured empirically. Consequently, in order to meaningfully differentiate learning methods, it becomes necessary to characterize their performance on different problems, taking into account factors such as state estimation, exploration, function approximation, and constraints on computation and memory. To this end, we propose parameterized learning problems, in which such factors can be controlled systematically and their effects on learning methods characterized through targeted studies. Apart from providing very precise control of the parameters that affect learning, our parameterized learning problems enable benchmarking against optimal behavior; their relatively small sizes facilitate extensive experimentation. Based on a survey of existing RL applications, in this article, we focus our attention on two predominant, ??first order?? factors: partial observability and function approximation. We design an appropriate parameterized learning problem, through which we compare two qualitatively distinct classes of algorithms: on-line value function-based methods and policy search methods. Empirical comparisons among various methods within each of these classes project Sarsa(??) and Q-learning(??) as winners among the former, and CMA-ES as the winner in the latter. Comparing Sarsa(??) and CMA-ES further on relevant problem instances, our study highlights regions of the problem space favoring their contrasting approaches. Short run-times for our experiments allow for an extensive search procedure that provides additional insights on relationships between method-specific parameters??such as eligibility traces, initial weights, and population sizes??and problem instances.     

An improved skyline based heuristic for the 2D strip packing problem and its efficient implementation

The best-fit heuristic by Burke et al. (2004) is a simple but effective approach for the 2D Strip Packing (2DSP) problem. In this paper, we propose an improved best-fit heuristic for the 2DSP. Instead of selecting the rectangle with the largest width, we use the fitness number to select the best rectangle fitting into the gap. An efficient implementation pattern with a time complexity of O(n log n) (n is the number of rectangles) is provided for the improved best-fit heuristic. A simple random local search is used to improve the results by trying different sequences. The experiment on the benchmark test sets shows that the final approach is both effective and efficient.     

An improved kernelization algorithm for r-Set Packing

We present a reduction procedure that takes an arbitrary instance of the r-Set Packing problem and produces an equivalent instance whose number of elements is in O(kr−1), where k is the input parameter. Such parameterized reductions are known as kernelization algorithms, and a reduced instance is called a problem kernel. Our result improves on previously known kernelizations by a factor of k. In particular, the number of elements in a 3-Set Packing kernel is improved from a cubic function of the parameter to a quadratic one.     

An incomplete m-exchange algorithm for solving the large-scale multi-scenario knapsack problem

This paper introduces a fast heuristic based algorithm for the max-min multi-scenario knapsack problem. The problem is a variation of the standard 0-1 knapsack problem, in which the profits of the items vary under different scenarios, though the capacity of the knapsack is fixed. The objective of the problem is to find the optimal packing of a set of items so that the minimum total profits of the items in the knapsack over all different scenarios is maximized. For some large-scaled instances, traditional branch-and-bound techniques cannot find an optimal solution within reasonable time, thus we propose a collection of incomplete m-exchange algorithms which are able to produce high quality solutions in just a few minutes of cpu time. Various computational results are also given.     

Solution approaches for the cutting stock problem with setup cost

In this paper, we study the Cutting Stock Problem with Setup Cost (CSP-S) which is a more general case of the well-known Cutting Stock Problem (CSP). In the classical CSP, one wants to minimize the number of stock items used while satisfying the demand for smaller-sized items. However, the number of patterns/setups to be performed on the cutting machine is ignored. In most cases, one has to find the trade-off between the material usage and the number of setups in order to come up with better production plans. In CSP-S, we have different cost factors for the material and the number of setups, and the objective is to minimize total production cost including both material and setup costs. We develop a mixed integer linear program and analyze a special case of the problem. Motivated by this special case, we propose two local search algorithms and a column generation based heuristic algorithm. We demonstrate the effectiveness of the proposed algorithms on the instances from the literature.     

Algorithms for 3D guillotine cutting problems: Unbounded knapsack, cutting stock and strip packing

We present algorithms for the following three-dimensional (3D) guillotine cutting problems: unbounded knapsack, cutting stock and strip packing. We consider the case where the items have fixed orientation and the case where orthogonal rotations around all axes are allowed. For the unbounded 3D knapsack problem, we extend the recurrence formula proposed by [1] for the rectangular knapsack problem and present a dynamic programming algorithm that uses reduced raster points. We also consider a variant of the unbounded knapsack problem in which the cuts must be staged. For the 3D cutting stock problem and its variants in which the bins have different sizes (and the cuts must be staged), we present column generation-based algorithms. Modified versions of the algorithms for the 3D cutting stock problems with stages are then used to build algorithms for the 3D strip packing problem and its variants. The computational tests performed with the algorithms described in this paper indicate that they are useful to solve instances of moderate size.     

A comparative runtime analysis of heuristic algorithms for satisfiability problems

The satisfiability problem is a basic core NP-complete problem. In recent years, a lot of heuristic algorithms have been developed to solve this problem, and many experiments have evaluated and compared the performance of different heuristic algorithms. However, rigorous theoretical analysis and comparison are rare. This paper analyzes and compares the expected runtime of three basic heuristic algorithms: RandomWalk, (1+1) EA, and hybrid algorithm. The runtime analysis of these heuristic algorithms on two 2-SAT instances shows that the expected runtime of these heuristic algorithms can be exponential time or polynomial time. Furthermore, these heuristic algorithms have their own advantages and disadvantages in solving different SAT instances. It also demonstrates that the expected runtime upper bound of RandomWalk on arbitrary k-SAT (k?3) is O(n(k−1)), and presents a k-SAT instance that has Θ(n(k−1)) expected runtime bound.     

A unified approach to tracking performance analysis of the selective partial update adaptive filter algorithms in nonstationary environment

In this paper, a unified approach to mean-square performance analysis of the family of selective partial update (SPU) adaptive filter algorithms in nonstationary environment is presented. Using this analysis, the tracking performance of Max normalized least mean squares (Max-NLMS), N-Max NLMS, the various types of SPU-NLMS algorithms, SPU transform domain LMS (SPU-TD-LMS), the family of SPU affine projection algorithms (SPU-APA), the family of selective regressor APA (SR-APA), the dynamic selection of APA (DS-APA), the family of SPU-SR-APA, the family of SPU-DS-APA, SPU subband adaptive filters (SPU-SAF), and the periodic, sequential, and stochastic partial update LMS, NLMS, and APA as well as classical adaptive filter algorithms can be analyzed with a unified approach. Two theoretical expressions are introduced to study the performance. The analysis is based on energy conservation arguments and does not need to assume a Gaussian or white distribution for the regressors. We demonstrate through simulations that the derived expressions are useful in predicting the performance of this family of adaptive filters in nonstationary environment.     

A grouping genetic algorithm with controlled gene transmission for the bin packing problem

In this study, the one-dimensional Bin Packing Problem (BPP) is approached. The BPP is a classical optimization problem that is known for its applicability and complexity. We propose a method that is referred to as the Grouping Genetic Algorithm with Controlled Gene Transmission (GGA-CGT) for Bin Packing. The proposed algorithm promotes the transmission of the best genes in the chromosomes without losing the balance between the selective pressure and population diversity. The transmission of the best genes is accomplished by means of a new set of grouping genetic operators, while the evolution is balanced with a new reproduction technique that controls the exploration of the search space and prevents premature convergence of the algorithm. The results obtained from an extensive computational study confirm that (1) promoting the transmission of the best genes improves the performance of each grouping genetic operator; (2) adding intelligence to the packing and rearrangement heuristics enhances the performance of a GGA; (3) controlling selective pressure and population diversity tends to lead to higher effectiveness; and (4) GGA-CGT is comparable to the best state-of-the-art algorithms, outperforming the published results for the class of instances Hard28, which appears to have the greatest degree of difficulty for BPP algorithms.