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.     

Online variable-sized bin packing with conflicts

We study a new kind of online bin packing with conflicts, motivated by a problem arising when scheduling jobs on the Grid. In this bin packing problem, the set of items is given at the beginning, together with a set of conflicts on pairs of items. A conflict on a pair of items implies that they cannot be assigned to a common bin. The online scenario is realized as follows. Variable-sized bins arrive one by one, and items need to be assigned to each bin before the next bin arrives. We analyze the online problem as well as semi-online versions of it, which are the variant where the sizes of the arriving bins are monotonically non-increasing as well as the variant where they are monotonically non-decreasing.     

On the sum minimization version of the online bin covering problem

Given a set of m identical bins of size 1, the online input consists of a (potentially infinite) stream of items in (0,1]. Each item is to be assigned to a bin upon arrival. The goal is to cover all bins, that is, to reach a situation where a total size of items of at least 1 is assigned to each bin. The cost of an algorithm is the sum of all used items at the moment when the goal is first fulfilled. We consider three variants of the problem, the online problem, where there is no restriction of the input items, and the two semi-online models, where the items arrive sorted by size, that is, either by non-decreasing size or by non-increasing size. The offline problem is considered as well.     

Relations between capacity utilization,minimal bin size and bin number

We consider the two-dimensional bin packing problem given a set of rectangular items, find the minimal number of rectangular bins needed to pack all items. Rotation of the items is not permitted. We show for any integer \({k} \ge 3\) that at most \({k}-1\) bins are needed to pack all items if every item fits into a bin and if the total area of items does not exceed \({k}/4\) -times the bin area. Moreover, this bound is tight. Furthermore, we show that only two bins are necessary to pack all items if the total area of items is not larger than the bin area, and if the height of each item is not larger than a third of the bin height and the width of every item does not exceed half of the bin width.     

Bin packing with general cost structures

Following the work of Anily et?al., we consider a variant of bin packing called bin packing with general cost structures (GCBP) and design an asymptotic fully polynomial time approximation scheme (AFPTAS) for this problem. In the classic bin packing problem, a set of one-dimensional items is to be assigned to subsets of total size at most 1, that is, to be packed into unit sized bins. However, in GCBP, the cost of a bin is not 1 as in classic bin packing, but it is a non-decreasing and concave function of the number of items packed in it, where the cost of an empty bin is zero. The construction of the AFPTAS requires novel techniques for dealing with small items, which are developed in this work. In addition, we develop a fast approximation algorithm which acts identically for all non-decreasing and concave functions, and has an asymptotic approximation ratio of 1.5 for all functions simultaneously.     

Selfish Square Packing

We consider a game-theoretical bin packing problem. The 1D (one dimensional) case has been treated in the literature as the ʼselfish bin packing problemʼ. We investigate a 2D version, in which the items to be packed are squares and the bins are unit squares. In this game, a set of items is packed into bins. Each player controls exactly one item and is charged with a cost defined as the ratio between the area of the item and the occupied area of the respective bin. One at a time, players selfishly move their items from one bin to another, in order to minimize the costs they are charged. At a Nash equilibrium, no player can reduce the cost he is charged by moving his item to a different bin. In the 2D case, to decide whether an item can be placed in another bin with other items is NP-complete, so we consider that players use a packing algorithm to make this decision. We show that this game converges to a Nash equilibrium, independently of the packing algorithm used. We prove that the price of anarchy is at least 2.27. We also prove that, using the NFDH packing algorithm, the asymptotic price of anarchy is at most 2.6875.     

Random fuzzy EOQ model with imperfect quality items

This paper investigates an economic order quantity (EOQ) problem with imperfect quality items, where the percentage of imperfect quality items in each lot is characterized as a random fuzzy variable while the setup cost per lot, the holding cost of each unit item per day, and the inspection cost of each unit item are characterized as fuzzy variables, respectively. In order to maximize the expected long-run average profit, a random fuzzy EOQ model is constructed. Since it is almost impossible to find an analytic method to solve the proposed model, a particle swarm optimization (PSO) algorithm based on the random fuzzy simulation is designed. Finally, the effectiveness of the designed algorithm is illustrated by a numerical example.     

Efficient algorithms for the offline variable sized bin-packing problem

We addresses a variant of the classical one dimensional bin-packing problem where several types of bins with unequal sizes and costs are presented. Each bin-type includes limited and/or unlimited identical bins. The goal is to minimize the total cost of bins needed to store a given set of items, each item with some space requirements. Four new heuristics to solve this problem are proposed, developed and compared. The experiments results show that higher quality solutions can be obtained using the proposed algorithms.     

A one-dimensional bin packing problem with shelf divisions

Given bins of size B, non-negative values d and Δ, and a list L of items, each item e∈L with size s_e and class c_e, we define a shelf as a subset of items packed inside a bin with total item sizes at most Δ such that all items in this shelf have the same class. Two subsequent shelves must be separated by a shelf division of size d. The size of a shelf is the total size of its items plus the size of the shelf division. The class constrained shelf bin packing problem (CCSBP) is to pack the items of L into the minimum number of bins, such that the items are divided into shelves and the total size of the shelves in a bin is at most B. We present hybrid algorithms based on the First Fit (Decreasing) and Best Fit (Decreasing) algorithms, and an APTAS for the problem CCSBP when the number of different classes is bounded by a constant C.     

An agent-based approach to the two-dimensional guillotine bin packing problem

The two-dimensional guillotine bin packing problem consists of packing, without overlap, small rectangular items into the smallest number of large rectangular bins where items are obtained via guillotine cuts. This problem is solved using a new guillotine bottom left (GBL) constructive heuristic and its agent-based (A–B) implementation. GBL, which is sequential, successively packs items into a bin and creates a new bin every time it can no longer fit any unpacked item into the current one. A–B, which is pseudo-parallel, uses the simplest system of artificial life. This system consists of active agents dynamically interacting in real time to jointly fill the bins while each agent is driven by its own parameters, decision process, and fitness assessment. A–B is particularly fast and yields near-optimal solutions. Its modularity makes it easily adaptable to knapsack related problems.     

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.     

New lower bounds for the three-dimensional finite bin packing problem

The three-dimensional finite bin packing problem (3BP) consists of determining the minimum number of large identical three-dimensional rectangular boxes, bins, that are required for allocating without overlapping a given set of three-dimensional rectangular items. The items are allocated into a bin with their edges always parallel or orthogonal to the bin edges. The problem is strongly NP-hard and finds many practical applications. We propose new lower bounds for the problem where the items have a fixed orientation and then we extend these bounds to the more general problem where for each item the subset of rotations by 90° allowed is specified. The proposed lower bounds have been evaluated on different test problems derived from the literature. Computational results show the effectiveness of the new lower bounds.     

一类带多重核元集在线装箱问题

本文讨论了一类在线变尺寸装箱问题,假定箱子的尺寸可以是不同的.箱子是在线到达的,仅当箱子到达后其尺寸才知道.给定一个带有核元的物品表及其上的核元关系图.我们的目标是要将表中元素装入到达的箱子中,保证任何箱子所装物品不互为核元,即所装物品对应的点所导出的子图是个空图,并使得所用的箱子总长最小.我们证明了该问题是NPHard的,并给出了基于图的点染色、图的团分解和基于背包问题的近似算法,给出了算法的时间复杂度和性能界.     

The static stochastic knapsack problem with normally distributed item sizes

This paper develops exact and heuristic algorithms for a stochastic knapsack problem where items with random sizes may be assigned to a knapsack. An item’s value is given by the realization of the product of a random unit revenue and the random item size. When the realization of the sum of selected item sizes exceeds the knapsack capacity, a penalty cost is incurred for each unit of overflow, while our model allows for a salvage value for each unit of capacity that remains unused. We seek to maximize the expected net profit resulting from the assignment of items to the knapsack. Although the capacity is fixed in our core model, we show that problems with random capacity, as well as problems in which capacity is a decision variable subject to unit costs, fall within this class of problems as well. We focus on the case where item sizes are independent and normally distributed random variables, and provide an exact solution method for a continuous relaxation of the problem. We show that an optimal solution to this relaxation exists containing no more than two fractionally selected items, and develop a customized branch-and-bound algorithm for obtaining an optimal binary solution. In addition, we present an efficient heuristic solution method based on our algorithm for solving the relaxation and empirically show that it provides high-quality solutions.     

A 17/10-approximation algorithm fork-bounded space on-line variable-sized bin packing

A version of thek-bounded space on-line bin packing problem, where a fixed collection of bin sizes is allowed, is considered. By packing large items into appropriate bins and closing appropriate bins, we can derive an algorithm with worst-case performance bound 1.7 fork≥3. This research is supported by the Science Foundation under State Education Committee of China. The earlier version was done in Institute of Applied Mathematics, Academia Sinica.     

New and improved level heuristics for the rectangular strip packing and variable-sized bin packing problems

We consider two types of orthogonal, oriented, rectangular, two-dimensional packing problems. The first is the strip packing problem, for which four new and improved level-packing algorithms are presented. Two of these algorithms guarantee a packing that may be disentangled by guillotine cuts. These are combined with a two-stage heuristic designed to find a solution to the variable-sized bin packing problem, where the aim is to pack all items into bins so as to minimise the packing area. This heuristic packs the levels of a solution to the strip packing problem into large bins and then attempts to repack the items in those bins into smaller bins in order to reduce wasted space. The results of the algorithms are compared to those of seven level-packing heuristics from the literature by means of a large number of strip-packing benchmark instances. It is found that the new algorithms are an improvement over known level-packing heuristics for the strip packing problem. The advancements made by the new and improved algorithms are limited in terms of utilised space when applied to the variable-sized bin packing problem. However, they do provide results faster than many existing algorithms.     

Lower and upper bounding procedures for the bin packing problem with concave loading cost

We address the one-dimensional bin packing problem with concave loading cost (BPPC), which commonly arises in less-than-truckload shipping services. Our contribution is twofold. First, we propose three lower bounds for this problem. The first one is the optimal solution of the continuous relaxation of the problem for which a closed form is proposed. The second one allows the splitting of items but not the fractioning of bins. The third one is based on a large-scale set partitioning formulation of the problem. In order to circumvent the challenges posed by the non-linearity of the objective function coefficients, we considered the inner-approximation of the concave load cost and derived a relaxed formulation that is solved by column generation. In addition, we propose two subset-sum-based heuristics. The first one is a constructive heuristic while the second one is a local search heuristic that iteratively attempts to improve the current solution by selecting pairs of bins and solving the corresponding subset sum-problem. We show that the worst-case performance of any BPPC heuristic and any concave loading cost function is bounded by 2. We present the results of an extensive computational study that was carried out on large set of benchmark instances. This study provides empirical evidence that the column generation-based lower bound and the local search heuristic consistently exhibit remarkable performance.     

Approximation algorithms for the oriented two-dimensional bin packing problem

Given a set of rectangular items which may not be rotated and an unlimited number of identical rectangular bins, we consider the problem of packing each item into a bin so that no two items overlap and the number of required bins is minimized. The problem is strongly NP-hard and finds practical applications in cutting and packing. We discuss a simple deterministic approximation algorithm which is used in the initialization of a tabu search approach. We then present a tabu search algorithm and analyze its average performance through extensive computational experiments.     

Generalized assignment problem: Truthful mechanism design without money

We propose truthful approximation mechanisms for strategic variants of the generalized assignment problem (GAP) in a payment-free environment. In GAP, a set of items has to be optimally assigned to a set of bins without exceeding the capacity of any singular bin. In our strategic variant, bins are held by strategic agents and each agent may hide its willingness to receive some items in order to obtain items of higher values. The model has applications in auctions with budgeted bidders.     

Space defragmentation for packing problems

One of main difficulties of multi-dimensional packing problems is the fragmentation of free space into several unusable small parts after a few items are packed. This study proposes a defragmentation technique to combine the fragmented space into a continuous usable space, which potentially allows the packing of additional items. We illustrate the effectiveness of this technique using the two- and three-dimensional bin packing problem, where the aim is to load all given items (represented by rectangular boxes) into the minimum number of identical bins. Experimental results based on well-known 2D and 3D bin packing data sets show that our defragmentation technique alone is able to produce solutions approaching the quality of considerably more complex meta-heuristic approaches for the problem. In conjunction with a bin shuffling strategy for incremental improvement, our resultant algorithm outperforms all leading meta-heuristic approaches based on the commonly used benchmark data by a significant margin.