Generalized Bin Packing Problems

Packing problems make up a fundamental topic of combinatorial optimization. Their importance is confirmed both by their wide range of scientific and technological applications they are able to address and by their theoretical implications. In fact, they are exploited in many fields such as computer science and technologies, industrial applications, transportation and logistics, and telecommunications. From a theoretical perspective, packing problems often appear as sub-problems in order to iteratively solve bigger problems. Although packing problems play a fundamental role in all these settings, there is a gap in terms of comprehensive study in the literature. In fact, the joint presence of both compulsory and non-compulsory items has not been considered yet. This particular setting arises in many real-life applications, not yet addressed or only partially addressed by the current state-of-the-art packing problems. Furthermore, little has been done in terms of unified methodologies, and different techniques have been used in order to solve packing problems with different objective functions. In particular, none of these techniques is able to address the presence of compulsory and non-compulsory items at the same time. In order to overcome a noteworthy portion of this gap, we formulated a new packing problem, named the Generalized Bin Packing Problem (GBPP), characterized by both compulsory and non-compulsory items, and multiple item and bin attributes. Packing problems have also been studied within stochastic settings where the items are affected by uncertainty. In these settings, there are fundamentally two kinds of stochasticity concerning the items: 1) stochasticity of the item attributes, where one attribute is affected by uncertainty and modeled as a random variable or 2) stochasticity of the item availability, i.e., the items are not known a priori but they arrive on-line in an unpredictable way to a decision maker. Although packing problems have been studied according to these stochastic variants, the GBPP with uncertainty on the items is still an open problem. Therefore, we have also studied two stochastic variants of the GBPP, named the Stochastic Generalized Bin Packing Problem (S-GBPP) and the On-line Generalized Bin Packing Problem (OGBPP). Our main results concern the development of models and unified methodologies of these new packing problems, making up, as done for the Vehicle Routing Problem (VRP) with the definition of the so called Rich Vehicle Routing Problems, a new family of advanced packing problems named Generalized Bin Packing Problem

On the generalized bin packing problem

The generalized bin packing problem (GBPP) is a novel packing problem arising in many transportation and logistic settings, characterized by multiple items and bins attributes and the presence of both compulsory and non-compulsory items. In this paper, we study the computational complexity and the approximability of the GBPP. We prove that the GBPP cannot be approximated by any constant, unless P = NP. We also study the particular case of a single bin type and show that when an unlimited number of bins is available, the GBPP can be reduced to the bin packing with rejection (BPR) problem, which is approximable. We also prove that the GBPP satisfies Bellman’s optimality principle and, exploiting this result, we develop a dynamic programming solution approach. Finally, we study the behavior of standard and widespread heuristics such as the first fit, best fit, first fit decreasing, and best fit decreasing.We show that while they successfully approximate previous versions of bin packing problems, they fail to approximate the GBPP

Intermodalism in the Transportation Network of the Roman Empire

In this paper we are proposing a preliminary work for evidencing some features of intermodalism in the transportation network of the Roman Empire. We will show that intermodal terminals for oversea and overland transportations existed such as containers and shipping documents. For them, the Empire that had some specific logistic systems too

Japanese Temple Geometry: A Digital Sangaku About a Regular Pentagon and the Golden Ratio

Sangaku are wooden tablets depicting geometric or mathematical problems. They are objects typical of the Edo period. Since these tables were exposed in the temples, the related geometry is known as the Japanese Temple Geometry. Here we illustrate a digital approach to Sangaku. The specific problem we are discussing in the construction of a regular pentagon

Job order assignment at optimal costs in railway maintenance

Tamping is an important part of railway maintenance. Well tamped ballast reduces track irregularities and increases travel safety and comfort. But if the ballast is in a bad condition, the train speed must be restricted, which leads to delays and penalty costs for the operator. In this paper a novel model for the tamping scheduling problem in a short-term planning horizon is presented. In contrast to other railway maintenance scheduling problems the penalty costs caused by deferring tamping activities are considered in the scheduling process beside the travel costs. Three greedy heuristics are presented and compared in different benchmarks. An outlook discusses issues of interest for further research

Asymptotic results for the Generalized Bin Packing Problem

We present a worst case analysis for the Generalized Bin Packing Problem, a novel packing problem arising in many Transportation and Logistics settings and characterized by multiple item and bin attributes and by the joint presence of both compulsory and non-compulsory items. As a preliminary worst case analysis has recently been proposed in the literature, we extend this study by proposing semi-online and offline algorithms, extending the well known First Fit Decreasing and Best Fit Decreasing heuristics for the Bin Packing Problem. In particular, we show that knowing part of the instance or the whole instance is not enough for computing worst case ratio bounds

A Generalized Bin Packing Problem for parcel delivery in last-mile logistics

Abstract In this paper, we present a new problem arising at a tactical level of setting a last-mile parcel delivery service in a city by considering different Transportation Companies (TC), which differ in cost and service quality. The courier must decide which TCs to select for the service in order to minimize the total cost and maximize the total service quality. We show that the problem can be modeled as a new packing problem, the Generalized Bin Packing Problem with bin-dependent item profits (GBPPI), where the items are the parcels to deliver and the bins are the TCs. The aim of the GBPPI is to select the appropriate fleet from TCs and determine the optimal assignment of parcels to vehicles such that the overall net cost is minimized. This cost takes into account both transportation costs and service quality. We provide a Mixed Integer Programming formulation of the problem, which is the starting point for the development of efficient heuristics that can address the GBPPI for instances involving up to 1000 items. Extensive computational tests show the accuracy of the proposed methods. Finally, we present a last-mile logistics case study of an international courier which addresses this problem

Worst-case analysis for new online bin packing problems

We consider two new online bin packing problems, the online Variable Cost and Size Bin Packing Problem (o-VCSBPP) and the online Generalized Bin Packing Problem (o-GBPP). We take two well-known bin packing algorithms to address them, the First Fit (FF) and the Best Fit (BF). We show that both algorithms have an asymptotic worst-case ratio bound equal to 2 for the o-VCSBPP and this bound is tight. When there are enough bins of a particular type to load all items, FF and BF also have an absolute worst-case ratio bound equal to 2 for the o-VCSBPP, and this bound is also tight. In addition, we prove that no worst-case ratio bound of FF and BF can be computed for the o-GBPP. Therefore, we consider a natural evolution of these algorithms, the First Fit with Rejection and the Best Fit with Rejection, able to reject inconvenient bins at the end of the process. Similarly, we prove that no worst-case ratio of these algorithms can be computed for the o-GBPP. Finally, we give sucient conditions under which algorithms do not admit any performance ratio, and conclude that the worst-case results obtained for the o-VCSBPP and the o-GBPP also hold for the oine variant of these two problem

Branch-and-price and beam search algorithms for the Generalized Bin Packing Problem

Tech. Rep. CIRRELT-2012-01, CIRRELT, Montreal, Canad