Solving the pallet loading problem

This paper presents new bounds, heuristics, and an exact algorithm for the Pallet Loading Problem (PLP). PLP maximizes the number of boxes placed on a rectangular pallet. All boxes have identical rectangular dimensions and, when placed, must be located completely within the pallet. Boxes may be rotated 90° so long as they are placed with edges parallel to the pallet’s edges. The set of all PLP instances with an area ratio (pallet area divided by box area) less than 101 boxes can be represented by 3,080,730 equivalent classes. Our G5-heuristic finds optimal solutions to 3,073,724 of these 3,080,730 classes and in the remaining 7006 classes only differs from the best known bound by one box. We develop three other heuristics that solve another 54 instances. Finally, we solve the 6952 remaining classes with our exact HVZ algorithm. Only a subset of these classes has been solved previously.     

The minimum size instance of a Pallet Loading Problem equivalence class

The Pallet Loading Problem (PLP) maximizes the number of identical rectangular boxes placed within a rectangular pallet. Boxes may be rotated 90° so long as they are packed with edges parallel to the pallet’s edges, i.e., in an orthogonal packing. This paper defines the Minimum Size Instance (MSI) of an equivalence class of PLP, and shows that every class has one and only one MSI. We develop bounds on the dimensions of box and pallet for the MSI of any class. Applying our new bounds on MSI dimensions, we present an algorithm for MSI generation and use it to enumerate all 3,080,730 equivalence classes with an area ratio (pallet area divided by box area) smaller than 101 boxes. Previous work only provides bounds on the ratio of box dimensions and only considers a subset of all classes presented here.     

A fast algorithm for identifying minimum size instances of the equivalence classes of the Pallet Loading Problem

In this paper, a novel and fast algorithm for identifying the Minimum Size Instance (MSI) of the equivalence class of the Pallet Loading Problem (PLP) is presented. The new algorithm is based on the fact that the PLP instances of the same equivalence class have the property that the aspect ratios of their items belong to an open interval of real numbers. This interval characterises the PLP equivalence classes and is referred to as the Equivalence Ratio Interval (ERI) by authors of this paper. The time complexity of the new algorithm is two polynomial orders lower than that of the best known algorithm. The authors of this paper also suggest that the concept of MSI and its identifying algorithm can be used to transform the non-integer PLP into its equivalent integer MSI.     

A simulated annealing based hyperheuristic for determining shipper sizes for storage and transportation

The current drive to reduce packaging waste has led many companies to consider the use of multi-trip containers or shippers in which to transport their products in order to reduce packaging waste. The efficiency of such systems obviously depends on selecting shipper dimensions in such a way as to ensure high volumetric utilisation. As is the case with many practical problems the efficiency/solution quality can be improved if problem specific information is used to enhance the operation of a meta-heuristic solution approach. The problem can be modelled as a p-median problem but is too large to be solved in reasonable time without further modification. Four such modifications, all based on properties of the physical problem, are introduced and incorporated into a hyperheuristic driven simulated annealing solution approach.     

货物尺寸相同的2维装箱问题的等价类(英文)

在生产与储运领域,把小长方体货物（盒子）装入大长方体箱子是一项重要的工作.本文涉及的问题是:把相同尺寸（a×b×c）的盒子装到一个箱子X×Y×Z中,使所装入箱子的盒子数量为最大.由于某些条件的限止,有时要求货物只能按一个重力方向进行装箱,从而使装箱问题变为把尺寸相同的2维盒子（a×b）填装到一个2维箱子X×Y中.本文讨论当盒子尺寸（a×b包括 b×a）给定,箱子尺寸充分大时,在本文所给的等价意义下,共有多少种互不等价的箱子X×Y.     

Heuristics for container loading of furniture

We consider a container loading problem that occurs at a typical furniture manufacturer. Each furniture item has an associated profit. Given container dimensions and a set of furniture items, the problem is to determine a subset of items with maximal profit sum that is loadable in the container. In the studied company, the problem arises hundreds of times daily during transport planning. Instances may contain more than one hundred different items with irregular shapes. To solve this complex problem we apply a set of heuristics successively that each solve one part of the problem. Large items are combined in specific structures to ensure proper protection of the items during transportation and to simplify the problem. The solutions generated by the heuristic has an average loading utilization of 91.3% for the most general instances with average running times around 100 seconds.     

Computing the asymptotic worst-case of bin packing lower bounds

This paper addresses the issue of computing the asymptotic worst-case of lower bounds for the Bin Packing Problem. We introduce a general result that allows to bound the asymptotic worst-case performance of any lower bound for the problem and to derive for the first time the asymptotic worst-case of the well-known bound L₃ by Martello and Toth. We also show that the general result allows to easily derive the asymptotic worst-case of several lower bounds proposed in the literature.     

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.     

Designing vehicle routes for a mix of different request types,under time windows and loading constraints

This article introduces and solves a new rich routing problem integrated with practical operational constraints. The problem examined calls for the determination of the optimal routes for a vehicle fleet to satisfy a mix of two different request types. Firstly, vehicles must transport three-dimensional, rectangular and stackable boxes from a depot to a set of predetermined customers. In addition, vehicles must also transfer products between pairs of pick-up and delivery locations. Service of both request types is subject to hard time window constraints. In addition, feasible palletization patterns must be identified for the transported products. A practical application of the problem arises in the transportation systems of chain stores, where vehicles replenish the retail points by delivering products stored at a central depot, while they are also responsible for transferring stock between pairs of the retailer network. To solve this very complex combinatorial optimization problem, our major objective was to develop an efficient methodology whose required computational effort is kept within reasonable limits. To this end, we propose a local search-based framework for optimizing vehicle routes, in which feasible loading arrangements are identified via a simple-structured packing heuristic. The algorithmic framework is enhanced with various memory components which store and retrieve useful information gathered through the search process, in order to avoid any duplicate unnecessary calculations. The proposed solution approach is assessed on newly introduced benchmark instances.     

Optimizing Supply Shortage Decisions in Base Stock Distribution Operations

This paper addresses policies and agreements between suppliers and customers for handling supply shortages in base-stock systems under uncertain demand. We investigate the impacts that backlogging and expediting decisions have on inventory and transportation costs in these systems and develop a model for deciding whether a supplier should completely backlog, completely expedite, or employ some combination of backlogging and expediting shortages. Our results indicate that practical cases exist where some combination of both expediting and backlogging supply shortages outperforms either completely expediting or backlogging all shortages. Including transportation costs in our model provides incentive to employ `hybrid' policies that partially expedite and partially backlog excess demands within a given period. Our model demonstrates how inventory policy decisions directly impact transportation costs and provides a heuristic approach for jointly minimizing expected inventory and transportation costs.     

A prototype column generation strategy for the multiple container loading problem

The multiple container loading cost minimization problem (MCLCMP) is a practical and useful problem in the transportation industry, where products of various dimensions are to be loaded into containers of various sizes so as to minimize the total shipping cost. The MCLCMP can be naturally formulated as a set cover problem and solved using column generation techniques, which is a popular method for handling huge numbers of variables. However, the direct application of column generation is not effective because feasible solutions to the pricing subproblem is required, which for the MCLCMP is NP-hard. We show that efficiency can be greatly improved by generating prototypes that approximate feasible solutions to the pricing problem rather than actual columns. For many hard combinatorial problems, the subproblem in column generation based algorithms is NP-hard; if suitable prototypes can be quickly generated that approximate feasible solutions, then our strategy can also be applied to speed up these algorithms.     

A heuristic for single-warehouse multiretailer supply chains with all-unit transportation cost discounts

In supply chain management research, transportation costs, if explicitly considered at all, are frequently assumed to be linear. These costs often have a more complex form, such as an all-unit discount structure – this piecewise cost function adds significant complexity when included in supply chain management problems and is therefore often ignored due to solution time or tractability concerns. We present and evaluate a new heuristic procedure which provides good solutions to problems involving all-unit discount cost functions while significantly reducing solution times. The general nature of this procedure does not require assumptions about the supply chain structure or policies, and is therefore applicable in a wide range of settings.     

A solution approach to the inventory routing problem in a three-level distribution system

We consider the infinite horizon inventory routing problem in a three-level distribution system with a vendor, a warehouse and multiple geographically dispersed retailers. In this problem, each retailer faces a demand at a deterministic, retailer-specific rate for a single product. The demand of each retailer is replenished either from the vendor through the warehouse or directly from the vendor. Inventories are kept at both the retailers and the warehouse. The objective is to determine a combined transportation (routing) and inventory strategy minimizing a long-run average system-wide cost while meeting the demand of each retailer without shortage. We present a decomposition solution approach based on a fixed partition policy where the retailers are partitioned into disjoint and collectively exhaustive sets and each set of retailers is served on a separate route. Given a fixed partition, the original problem is decomposed into three sub-problems. Efficient algorithms are developed for the sub-problems by exploring important properties of their optimal solutions. A genetic algorithm is proposed to find a near-optimal fixed partition for the problem. Computational results show the performance of the solution approach.     

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.     

Computing the nearest stable matrix pairs

In this paper, we study the nearest stable matrix pair problem: given a square matrix pair (E,A), minimize the Frobenius norm of (Δ_E,Δ_A) such that (E+Δ_E,A+Δ_A) is a stable matrix pair. We propose a reformulation of the problem with a simpler feasible set by introducing dissipative Hamiltonian matrix pairs: A matrix pair (E,A) is dissipative Hamiltonian if A=(J−R)Q with skew‐symmetric J, positive semidefinite R, and an invertible Q such that Q^TE is positive semidefinite. This reformulation has a convex feasible domain onto which it is easy to project. This allows us to employ a fast gradient method to obtain a nearby stable approximation of a given matrix pair.     

The impact of distribution system characteristics on computational tractability

Mathematical programming models that seek optimal design and operational plans for distribution systems can be computationally intractable. This paper examines the extent to which distribution configuration and demand characteristics affect the ease of obtaining an optimal solution. Problem characteristics, which are reflected in a model by the parameter values, can render the model for one distribution scenario to be computationally intractable and that for another to yield an optimal solution easily. We introduce echelon-flow-based valid inequalities and use them to explicate the extent to which problem characteristics impact computational tractability.     

On pallets for Fox colorings of spatial graphs

We introduce the notion of pallets of quandles and define coloring invariants for spatial graphs which give a generalization of Fox colorings studied in Ishii and Yasuhara (1997) [4]. All pallets for dihedral quandles are obtained from the quotient sets of the universal pallets under a certain equivalence relation. We study the quotient sets and classify their elements.     

Relation between spectral classes of a self-similar Cantor set

A self-similar Cantor set is completely decomposed as a class of the lower (upper) distribution sets. We give a relationship between the distribution sets in the distribution class and the subsets in a spectral class generated by the lower (upper) local dimensions of a self-similar measure. In particular, we show that each subset of a spectral class is exactly a distribution set having full measure of a self-similar measure related to the distribution set using the strong law of large numbers. This gives essential information of its Hausdorff and packing dimensions. In fact, the spectral class by the lower (upper) local dimensions of every self-similar measure, except for a singular one, is characterized by the lower or upper distribution class. Finally, we compare our results with those of other authors.     

Computing Chernoff's Distribution

A distribution that arises in problems of estimation of monotone functions is that of the location of the maximum of two-sided Brownian motion minus a parabola. Using results from the first author's earlier work, we present algorithms and programs for computation of this distribution and its quantiles. We also present some comparisons with earlier computations and simulations.     

Performance evaluation of distribution strategies for the inventory routing problem

Direct shipping strategy is an easy-to-implement distribution strategy frequently used in industrial distribution systems. In this paper, an analytic method is developed for performance evaluation of the strategy for the infinite horizon inventory routing problem with delivery frequency constraint. With the method, the effectiveness of direct shipping strategy can be represented as a function of some system parameters. We demonstrate that the effectiveness of direct shipping is at least the square root of the smallest utilization ratio of vehicle capacity. This implies that the effectiveness of the strategy can reach 100% (respectively, 94.86%) whenever the demand rate of each retailer is 100% (respectively, 90%) of the vehicle capacity multiplied by the upper bound of the delivery frequency. This insight can help a firm answer questions such as: under what conditions direct shipping strategy is effective and why, and how effective the strategy is under a specific condition? In case direct shipping strategy is proven ineffective, a more general Fixed Partition Policy (FPP) that combines direct shipping strategy and multiple-stop shipping strategy must be used. An analytic method is also developed for performance evaluation of general FPPs. We demonstrate that the effectiveness of an FPP depends on the total demand rate of the retailers in each partition (each retailer set) and their closeness level. This insight provides a useful guideline to the design of effective FPPs. The analytic methods make the performance improvement of a distribution system possible through adjusting its system parameters.