An interactive system for the loading of cargo aircraft

The paper describes an interactice, computer based procedure for solving the variant of the vehicle loading problem encountered when loading containers and pallets into an aircraft. The procedure is heuristic and practical oriented; it is presently being implemented by Scandinavian Airlines.     

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.     

An effective recursive partitioning approach for the packing of identical rectangles in a rectangle

In this work, we deal with the problem of packing (orthogonally and without overlapping) identical rectangles in a rectangle. This problem appears in different logistics settings, such as the loading of boxes on pallets, the arrangements of pallets in trucks and the stowing of cargo in ships. We present a recursive partitioning approach combining improved versions of a recursive five-block heuristic and an L-approach for packing rectangles into larger rectangles and L-shaped pieces. The combined approach is able to rapidly find the optimal solutions of all instances of the pallet loading problem sets Cover I and II (more than 50?000 instances). It is also effective for solving the instances of problem set Cover III (almost 100?000 instances) and practical examples of a woodpulp stowage problem, if compared to other methods from the literature. Some theoretical results are also discussed and, based on them, efficient computer implementations are introduced. The computer implementation and the data sets are available for benchmarking purposes.     

Complexity of flow time minimization in a crossdock truck scheduling problem with asymmetric handover relations

We address a novel truck scheduling problem arising in crossdocking logistics, in which inbound trucks carry items (pallets) which must be sorted and loaded onto outbound trucks. We minimize the utilisation of the warehouse by focusing on the synchronisation between the different related trucks. The problem is to assign the trucks to the doors of the warehouse and sequence them, in order to minimize the total time spent in the system by the pallets. We discuss the complexity of the problem, showing that even with a single door the problem is NP-hard in general, and discuss some special cases.     

A simple and effective recursive procedure for the manufacturer's pallet loading problem

In this paper we present a simple and effective heuristic to solve the problem of packing the maximum number of rectangles of sizes (l,w) and (w,l) into a larger rectangle (L,W) without overlapping. This problem appears in the loading of identical boxes on pallets, namely the manufacturer's pallet loading (MPL), as well as in package design and truck or rail car loading. Although apparently easy to be optimally solved, the MPL is claimed to be NP-complete and several authors have proposed approximate methods to deal with it. The procedure described in the present paper can be seen as a refinement of Bischoff and Dowsland's heuristic and can easily be implemented on a microcomputer. Using moderate computational resources, the procedure was able to find the optimal solution of 99.9% of more than 20?000 examples analysed.     

Scheduling of a machining center

A machining center is an advanced NC (Numerical Control) machine that has the capability to perform a variety of operations on a part by automatically changing the cutting tools. Because of its versatile processing capabilities, a machining center is often a production bottleneck, and effective scheduling can result in significant improvement of system performance. The problem, however, is very difficult since many factors such as machine setups, pallets, tool magazine, and possible tool overlapping among different part types, etc., have to be considered. This paper presents an optimization-based approach for the scheduling of a machining center with two pallets. A novel “separable” problem formulation that considers the above mentioned factors is presented. Lagrangian relaxation is applied to decompose the problem into simple subproblems, which are efficiently solved without encountering complexity difficulties. The subgradient method is then used to update the multipliers. Testing results indicate that the approach is effective, and the algorithm provides a valuable tool for solving stand-alone machining center problems. The approach also points out a direction on how to consider machining centers within a job shop environment.     

An Application of the Micro to Product Design and Distribution

This article illustrates an application of the micro-computer to some common problems in product design and distribution. One problem which receives particular attention is that of pallet loading. Existing algorithms for determining efficient layouts on rectangular pallets are shown to suffer from a number of shortcomings, and an improved method is presented. Extensions of this method to more complex pallet designs are discribed. Initially a program for mainframe computer operation was produced, and this was subsequently transferred to a micro-computer. The problems of such a transfer are discussed, and the flexibility and power of the micro-computer as a management tool are illustrated.     

Automatic aircraft cargo load planning

The goal of this paper is the development of a new mixed integer linear program designed for optimally loading a set of containers and pallets into a compartmentalised cargo aircraft. It is based on real-world problems submitted by a professional partner. This model takes into account strict technical and safety constraints. In addition to the standard goal of optimally positioning the centre of gravity, we also propose a new approach based on the moment of inertia. This double goal implies an increase in aircraft efficiency and a decrease in fuel consumption. Cargo loading generally remains a manual, or at best a computer-assisted, and time-consuming task. A fully automatic software was developed to quickly compute optimal solutions. Experimental results show that our approach achieves better solutions than manual planning, within only a few seconds.     

Storage controlled pile-up systems,theoretical foundations

This paper presents the theoretical foundations for controlling pile-up systems. A pile-up system consists of one or more stacker cranes picking up bins from a conveyor and placing them onto pallets with respect to costumer orders. The bins usually arrive at a conveyor from an orderpicking system.We give a mathematical definition of the pile-up problem, define a data structure for control algorithms, introduce polynomial time algorithms for deciding whether a system can be blocked by making bad decisions, and show that the pile-up problem is in general NP-complete. For pile-up systems with a restricted storage capacity or with a fixed number of pile-up places the pile-up problem is proved to be solvable very efficiently.     

Scheduling the truckload operations in automatic warehouses

This work presents a scheduling problem that arises in an automatic storage/retrieval warehouse system AS/RS involving the scheduling of the truck load operations. The truck loading operations are modelled as job shop problem with recirculation. The loads are considered as jobs, the pallets of a load are seen as the job’s operations. The forklifts are the machines. The minimization of the makespan allows minimizing the idle time of the warehouse’s equipments.     

Manufacturer’s mixed pallet design problem

We study a problem faced by a major beverage producer. The company produces and distributes several brands to various customers from its regional distributors. For some of these brands, most customers do not have enough demand to justify full pallet shipments. Therefore, the company decided to design a number of mixed or “rainbow” pallets so that its customers can order these unpopular brands without deviating too much from what they initially need. We formally state the company’s problem as determining the contents of a pre-determined number of mixed pallets so as to minimize the total inventory holding and backlogging costs of its customers over a finite horizon. We first show that the problem is NP-hard. We then formulate the problem as a mixed integer linear program, and incorporate valid inequalities to strengthen the formulation. Finally, we use company data to conduct a computational study to investigate the efficiency of the formulation and the impact of mixed pallets on customers’ total costs.     

Stochastic single vehicle routing problem with delivery and pick up and a predefined customer sequence

In this paper we study the routing of a single vehicle that delivers products and picks up items with stochastic demand. The vehicle follows a predefined customer sequence and is allowed to return to the depot for loading/unloading as needed. A suitable dynamic programming algorithm is proposed to determine the minimum expected routing cost. Furthermore, the optimal routing policy to be followed by the vehicle’s driver is derived by proposing an appropriate theorem. The efficiency of the algorithm is studied by solving large problem sets.     

Three-stage heuristic algorithm for three-dimensional irregular packing problem

This paper studies a new practical problem which can be decomposed into three three-dimensional packing problems: three-dimensional irregular packing with variable-size cartons problem, three-dimensional variable-size bin packing problem, and the single container loading problem. Since the three sub-problems are NP-hard, searching a good solution becomes more difficult. In this paper, mathematical models of each sub-problem are developed and three-stage heuristic algorithms are proposed to solve this new problem. Experiments are conducted with random instances generated by real-life case. Computational results indicate that the proposed algorithm is efficient and can yield satisfactory results.     

A MIP-based slicing heuristic for three-dimensional bin packing

The mixed-case palletization problem is a common problem in warehousing and logistics where boxes of rectangular shapes are stacked on top of each other to form pallets. The problem shares common features with three-dimensional bin packing but requires boxes to be adequately supported. We propose a mixed integer programming formulation that maximizes the density of the bottom layers and the compactness of the pallet to ensure stability for top layers. We use a relative-position formulation with slicing that minimizes height, maximizes the fill rate of slices, and pushes boxes towards the vertical axis in order to consolidate fragmented space. Apart from common non-overlap and dimension-related constraints, we explicitly model the fill rates and force lower slices to have an equal or higher density than upper slices. As expected, the formulation could only handle small instances. To tackle larger instances, we embedded the formulation in an iterative approach that packs subsets of boxes sequentially. The approach was found to provide stable pallets and to outperform the branch-and-bound approach of Martello et al. (Oper Res 48(2):256–267, 2000).     

The convex hull of two core capacitated network design problems

The network loading problem (NLP) is a specialized capacitated network design problem in which prescribed point-to-point demand between various pairs of nodes of a network must be met by installing (loading) a capacitated facility. We can load any number of units of the facility on each of the arcs at a specified arc dependent cost. The problem is to determine the number of facilities to be loaded on the arcs that will satisfy the given demand at minimum cost.This paper studies two core subproblems of the NLP. The first problem, motivated by a Lagrangian relaxation approach for solving the problem, considers a multiple commodity, single arc capacitated network design problem. The second problem is a three node network; this specialized network arises in larger networks if we aggregate nodes. In both cases, we develop families of facets and completely characterize the convex hull of feasible solutions to the integer programming formulation of the problems. These results in turn strengthen the formulation of the NLP.Research of this author was supported in part by a Faculty Grant from the Katz Graduate School of Business, University of Pittsburgh.     

Diversification strategies in local search for a nonbifurcated network loading problem

This paper examines a variant of the network loading problem, a network design problem found in the telecommunications industry. In this problem, facilities of fixed capacity must be installed on the edges of an undirected network to carry the flow from a central vertex to a set of demand vertices. The objective is to minimize the total installation costs. In this work, the nonbifurcated version of the problem is considered, where the demand at any given vertex must be satisfied through a single path. The proposed heuristics alternate between a construction phase and a local search phase. Each new construction phase, except the first one, is part of a diversification strategy aimed at providing a new starting point for the following local search phase. Different diversification strategies are tested and compared on large-scale instances with up to 500 vertices.     

成品油配送多车舱车辆指派及路径优化问题研究

针对成品油配送中多车型、多车舱的车辆优化调度难题,综合考虑多车型车辆指派、多车舱车辆装载及路径安排等决策,以派车成本与油耗成本之和的总成本最小为目标,建立了多车型多车舱的车辆优化调度模型。为降低模型求解的复杂性,本文提出一种基于C-W节约算法的“需求拆分→合并装载”的车辆装载策略,并综合利用Relocate和Exchange算子进行并行邻域搜索改进,获得优化的成品油配送方案。最后,通过算例验证了本文提出的模型与算法用于求解大规模成品油配送问题的有效性。并通过数据实验揭示了以下规律:1)多车舱车辆相对于单车舱车辆在运营成本上具有优越性;2)大型车辆适合远距离配送,小型车辆适合近距离配送;3)多车型车辆混合配送相对于单车型车辆配送在运营成本上具有优越性。这些规律可为成品油配送公司的车辆配置提供决策参考。     

Heuristics with a new block strategy for the single and multiple containers loading problems

A heuristic algorithm using new block strategy for the heterogeneous single and multiple containers loading problem (CLP) is proposed in this paper. In order to solve the single CLP, this algorithm fills unused spaces with the homogeneous load-blocks of identically oriented boxes and splits residual space into three child-spaces starting with an empty container. An initial container pattern is first built applying this approach recursively until all boxes are stowed or no unused spaces are left. And then, alternative container patterns are generated after replacing the load-blocks of the pattern-determining spaces in the initial container pattern with the alternative-blocks previously stored. Finally, an improvement procedure compares these alternatives with the initial container pattern to identify an improved container pattern. An algorithm for the multiple CLP uses the single CLP algorithm to generate an initial solution and uses improvement procedures to improve the initial solution. Numerical experiments with 715 test cases for the single CLP and 47 test cases for the multiple the CLP revealed the excellent performance of this algorithm.     

An efficient method to determine the optimal configuration of a flexible manufacturing system

A frequently encountered design issue for a flexible manufacturing system (FMS) is to find the lowest cost configuration, i.e. the number of resources of each type (machines, pallets, ...), which achieves a given production rate. In this paper, an efficient method to determine this optimal configuration is presented. The FMS is modelled as a closed queueing network. The proposed procedure first derives a heuristic solution and then the optimal solution. The computational complexity for finding the optimal solution is very reasonable even for large systems, except in some extreme cases. Moreover, the heuristic solution can always be determined and is very close (and often equal) to the optimal solution. A comparison with the previous method of Vinod and Solberg shows that our method performs very well.     

A hybrid genetic algorithm for the container loading problem

This paper presents a hybrid genetic algorithm (GA) for the container loading problem with boxes of different sizes and a single container for loading. Generated stowage plans include several vertical layers each containing several boxes. Within the procedure, stowage plans are represented by complex data structures closely related to the problem. To generate offspring, specific genetic operators are used that are based on an integrated greedy heuristic. The process takes several practical constraints into account. Extensive test calculations including procedures from other authors vouch for the good performance of the GA, above all for problems with strongly heterogeneous boxes.