A heuristic for solving large bin packing problems in two and three dimensions

The more-dimensional bin packing problem (BPP) considered here requires packing a set of rectangular-shaped items into a minimum number of identical rectangular-shaped bins. All items may be rotated and the guillotine cut constraint has to be respected. A straightforward heuristic is presented that is based on a method for the container loading problem following a wall-building approach and on a method for the one-dimensional BPP. 1,800 new benchmark instances are introduced for the two-dimensional and three-dimensional BPP. The instances include more than 1,500 items on average. Applied to these very large instances, the heuristic generates solutions of acceptable quality in short computation times. Moreover, the influence of different instance parameters on the solution quality is investigated by an extended computational study.     

Optimizing the packing of cylinders into a rectangular container: A nonlinear approach

The container loading problem has important industrial and commercial applications. An increase in the number of items in a container leads to a decrease in cost. For this reason the related optimization problem is of economic importance. In this work, a procedure based on a nonlinear decision problem to solve the cylinder packing problem with identical diameters is presented. This formulation is based on the fact that the centers of the cylinders have to be inside the rectangular box defined by the base of the container (a radius far from the frontier) and far from each other at least one diameter. With this basic premise the procedure tries to find the maximum number of cylinder centers that satisfy these restrictions. The continuous nature of the problem is one of the reasons that motivated this study. A comparative study with other methods of the literature is presented and better results are achieved.     

Yard crane scheduling in port container terminals

Yard cranes are the most popular container handling equipment for loading containers onto or unloading containers from trucks in container yards of land scarce port container terminals. However, such equipment is bulky, and very often generates bottlenecks in the container flow in a terminal because of their slow operations. Hence, it is essential to develop good yard crane work schedules to ensure a high terminal throughput. This paper studies the problem of scheduling a yard crane to perform a given set of loading/unloading jobs with different ready times. The objective is to minimize the sum of job waiting times. A branch and bound algorithm is proposed to solve the scheduling problem optimally. Efficient and effective algorithms are proposed to find lower bounds and upper bounds. The performance of the proposed branch and bound algorithm is evaluated by a set of test problems generated based on real life data. The results show that the algorithm can find the optimal sequence for most problems of realistic sizes.     

A pegging approach to the precedence-constrained knapsack problem

The knapsack problem (KP) is generalized to the case where items are partially ordered through a set of precedence relations. As in ordinary KPs, each item is associated with profit and weight, the knapsack has a fixed capacity, and the problem is to determine the set of items to be packed in the knapsack. However, each item can be accepted only when all the preceding items have been included in the knapsack. The knapsack problem with these additional constraints is referred to as the precedence-constrained knapsack problem (PCKP). To solve PCKP exactly, we present a pegging approach, where the size of the original problem is reduced by applying the Lagrangian relaxation followed by a pegging test. Through this approach, we are able to solve PCKPs with thousands of items within a few minutes on an ordinary workstation.     

Heuristic and Exact Algorithms for the Precedence-Constrained Knapsack Problem

The knapsack problem (KP) is generalized taking into account a precedence relation between items. Such a relation can be represented by means of a directed acyclic graph, where nodes correspond to items in a one-to-one way. As in ordinary KPs, each item is associated with profit and weight, the knapsack has a fixed capacity, and the problem is to determine the set of items to be included in the knapsack. However, each item can be adopted only when all of its predecessors have been included in the knapsack. The knapsack problem with such an additional set of constraints is referred to as the precedence-constrained knapsack problem (PCKP). We present some dynamic programming algorithms that can solve small PCKPs to optimality, as well as a preprocessing method to reduce the size of the problem. Combining these, we are able to solve PCKPs with up to 2000 items in less than a few minutes of CPU time.     

Tree search for the stacking problem

The stacking problem is a hard combinatorial optimization problem with high practical interest in, for example, steel storage or container port operations. In this problem, a set of items is stored in a warehouse for a period of time, and a crane is used to place them in a limited number of stacks. Since the entrance and exit of items occurs in an arbitrary order, items may have to be relocated in order to reach and deliver other items below them. The objective of the problem is to find a feasible sequence of movements that delivers all items, while minimizing the total number of movements. We study the scalability of an exact approach to this problem, and propose two heuristic methods to solve it approximately. The two heuristic approaches are a multiple simulation algorithm using semi-greedy construction heuristics, and a stochastic best-first tree search algorithm. The two methods are compared in a set of challenging instances, revealing a superior performance of the tree search approach in most cases.     

A tree search algorithm for solving the multi-dimensional strip packing problem with guillotine cutting constraint

The article presents a tree search algorithm (TRSA) for the strip packing problem in two and three dimensions with guillotine cutting constraint. In the 3D-SPP a set of rectangular items (boxes) and a container with fixed width and height but variable length are given. An arrangement of all boxes within the container has to be determined so that the required length is minimised. The 2D-SPP is analogously defined. The proposed TRSA is based on a tree search algorithm for the container loading problem by Fanslau and Bortfeldt (INFORMS J. Comput. 22:222?C235, 2010). The TRSA generates guillotine packing patterns throughout. In a comparison with all recently proposed 3D-SPP methods the TRSA performs very competitive. Fine results are also achieved for the 2D-SPP.     

A tree search method for the container loading problem with shipment priority

This paper addresses a special kind of container loading problem with shipment priority. We present a tree search method, which is based on a greedy heuristic. In the greedy heuristic, blocks made up of identical items with the same orientation are selected for packing into a container. Five evaluation functions are proposed for block selection, and the different blocks selected by each evaluation function constitute the branches of the search tree. A method of space splitting and merging is also embedded in the algorithm to facilitate efficient use of the container space. In addition, the proposed algorithm covers an important constraint called shipment priority to solve practical problems. The validity of the proposed algorithm is examined by comparing the present results with those of published algorithms using the same data.     

A caving degree approach for the single container loading problem

Inspired by an old adage “Gold corner, silver side and strawy void”, and improved by a new observation “Maximum value in diamond cave”, a new heuristic approach is proposed for solving the three-dimensional single container loading problem. Differing from several previous approaches, its key issue is to pack the outside item into a corner or even a cave in the container such that the item is as compactly and closely as possible with other packed items. Experiments are on two groups of public and difficult benchmarks. For the 47 without-orientation-constraint instances from the OR-Library, experiments indicate an average packing utilization of 94.9%, which improves current best result reported in the literature by 3.9%. For the 800 strongly heterogeneous instances among 1500 representative benchmarks proposed by Bischoff et al., (100 instances in a set), experiments show an average packing utilization of 87.97%, which improves current best record reported in the literature by 0.28%. Besides, new best records are achieved on the latter five sets among the eight sets of strongly heterogeneous benchmarks.     

Mixed-integer programming models for nesting problems

Several industrial problems involve placing objects into a container without overlap, with the goal of minimizing a certain objective function. These problems arise in many industrial fields such as apparel manufacturing, sheet metal layout, shoe manufacturing, VLSI layout, furniture layout, etc., and are known by a variety of names: layout, packing, nesting, loading, placement, marker making, etc. When the 2-dimensional objects to be packed are non-rectangular the problem is known as the nesting problem. The nesting problem is strongly NP-hard. Furthermore, the geometrical aspects of this problem make it really hard to solve in practice. In this paper we describe a Mixed-Integer Programming (MIP) model for the nesting problem based on an earlier proposal of Daniels, Li and Milenkovic, and analyze it computationally. We also introduce a new MIP model for a subproblem arising in the construction of nesting solutions, called the multiple containment problem, and show its potentials in finding improved solutions.     

The stochastic generalized bin packing problem

The Generalized Bin Packing Problem (GBPP) is a recently introduced packing problem where, given a set of bins characterized by volume and cost and a set of items characterized by volume and profit (which also depends on bins), we want to select a subset of items to be loaded into a subset of bins which maximizes the total net profit, while satisfying the volume and bin availability constraints. The total net profit is given by the difference between the total profit of the loaded items and the total cost of the used bins. In this paper, we consider the stochastic version of the GBPP (S-GBPP), where the item profits are random variables to take into account the profit oscillations due to the handling operations for bin loading. The probability distribution of these random variables is assumed to be unknown. By using the asymptotic theory of extreme values a deterministic approximation for the S-GBPP is derived.     

Multiple Container Packing: A Case Study of Pipe Packing

While the problem of packing single containers and pallets has been thoroughly investigated very little attention has been given to the efficient packing of multiple container loads. Normally in practice a multiple container load is packed by a single container algorithm used in a greedy fashion. This paper introduces the issues involved in multiple container loading. It lays out three different strategies for solving the problem: sequential packing using a single container heuristic, pre-allocating items to the containers and choosing container loads using simultaneous packing models. The principal simultaneous models are pattern selection IP models. We present an application of packing pipes in shipping containers using two pattern selection IP models, a pattern selection heuristic, a sequential greedy algorithm and a pre-allocation method. The experimental results use randomly generated data sets. We discuss several useful insights into the methods and show that for this application the pattern selection methods perform best.     

Solving container loading problems by block arrangement

In order to solve heterogeneous single and multiple container loading problems, an algorithm is presented that builds homogeneous blocks of identically orientated items. First a greedy heuristic is presented that generates the desired block arrangements. Second the solutions provided by the greedy heuristic are improved by a tree search. Additional aspects such as load stability and weight distribution within the container are also taken into account. The test cases of Bischoff and Ratcliff are used for benchmarking purposes.     

A PTAS for the chance-constrained knapsack problem with random item sizes

We consider a stochastic knapsack problem where each item has a known profit but a random size that is normally distributed independent of other items. The goal is to select a profit maximizing set of items such that the probability of the total size exceeding the knapsack bound is at most a given threshold. We present a Polynomial Time Approximation Scheme (PTAS) for the problem via a parametric LP reformulation that efficiently computes a solution satisfying the chance constraint strictly and achieving near-optimal profit.     

An iterative three-component heuristic for the team orienteering problem with time windows

This paper studies the team orienteering problem with time windows, the aim of which is to maximize the total profit collected by visiting a set of customers with a limited number of vehicles. Each customer has a profit, a service time and a time window. A service provided to any customer must begin in his or her time window. We propose an iterative framework incorporating three components to solve this problem. The first two components are a local search procedure and a simulated annealing procedure. They explore the solution space and discover a set of routes. The third component recombines the routes to identify high quality solutions. Our computational results indicate that this heuristic outperforms the existing approaches in the literature in average performance by at least 0.41%. In addition, 35 new best solutions are found.     

A branch-and-price approach to the feeder network design problem

In this paper we consider the problem of designing a container liner shipping feeder network. The designer has to choose which port to serve during many rotations that start and end at a central hub. Many operational characteristics are considered, such as variable leg-by-leg speeds and cargo transit times. Realistic instances are generated from the LinerLib benchmark suite. The problem is solved with a branch-and-price algorithm, which can solve most instances to optimality within one hour. The results also provide insights on the cost structure and desirable features of optimal routes. These insights were obtained by means of an analysis where scenarios are generated varying internal and external conditions, such as fuel costs and port demands.     

考虑路径冲突的AGV配置与调度优化

合理调度有限的码头资源以满足船舶的装卸时间要求是自动化集装箱码头的重要目标之一。针对自动化集装箱码头自动导引车(automated guided vehicle,AGV)配置与调度问题,考虑船舶装卸时间要求和AGV运输过程中的路径冲突,提出分阶段调度策略。将船舶装卸作业分为卸船阶段、装卸同步阶段、装船阶段三个阶段,在每个阶段中,建立以最小化最大完工时间和最小化AGV空载和等待时间为双目标的调度优化模型,并设计基于NSGA-Ⅱ的启发式算法求解。根据本阶段的实际完工时间,从最优解集中选择下一阶段AGV的配置与调度方案。最后对比其他调度方案表明本文调度方案能够满足集装箱船的装卸时间要求,且提高了AGV的利用率,更符合码头实际作业要求。     

The Multi-Handler Knapsack Problem under Uncertainty

The Multi-Handler Knapsack Problem under Uncertainty is a new stochastic knapsack problem where, given a set of items, characterized by volume and random profit, and a set of potential handlers, we want to find a subset of items which maximizes the expected total profit. The item profit is given by the sum of a deterministic profit plus a stochastic profit due to the random handling costs of the handlers. On the contrary of other stochastic problems in the literature, the probability distribution of the stochastic profit is unknown. By using the asymptotic theory of extreme values, a deterministic approximation for the stochastic problem is derived. The accuracy of such a deterministic approximation is tested against the two-stage with fixed recourse formulation of the problem. Very promising results are obtained on a large set of instances in negligible computing time.     

Weight distribution considerations in container loading

A major drawback of the current literature on container loading is the lack of consideration of many practical issues. The weight distribution of the cargo is one such aspect which has been largely ignored. The paper considers post-processing approaches to tackle this problem. A new container loading heuristic is put forward in this context and is evaluated against several existing approaches. It is demonstrated that the procedure proposed is capable of producing loading arrangements which combine high space utilisation with an even weight distribution of the cargo.     

Pricing and lot sizing for an EPQ inventory model with rework and multiple shipments

This paper deals with an economic production quantity (EPQ) inventory model with reworkable defective items when a given multi-shipment policy is used. In this work, it is assumed that in each cycle, the rework process of all defective items starts when the regular production process finishes. After the rework process, a portion of reworked items fails. This portion becomes scrap and only the perfect finished items can be delivered to customers at the end of rework process. A profit function is derived to model the inventory problem and it is shown that the profit function is concave. Due to the complexity of the optimization problem, an algorithm is developed to determine the optimal values of manufacturing lot size and price such that the long-run average profit function is maximized. Furthermore, two special cases are identified and explained. Finally, a numerical example is given to illustrate the applicability of the proposed inventory model.