The multi-dimensional orthogonal packing problem (OPP) is a well studied optimization problem [3,9]. Given a set of items with rectangular shapes, the problem is to decide whether there is a non-overlapping packing of these items in a rectangular bin. Rotation of items is not allowed. Fekete and Schepers introduced a tuple of interval graphs as data structures to store a feasible packing, and g...

The bin packing problem has been extensively studied and numerous variants have been considered. The k-item bin packing problem is one of the variants introduced by Krause et al. in Journal of the ACM 22(4). In addition to the formulation of the classical bin packing problem, this problem imposes a cardinality constraint that the number of items packed into each bin must be at most k. For the o...

Online Bin Stretching is a semi-online variant of bin packing in which the algorithm has to use the same number of bins as an optimal packing, but is allowed to slightly overpack the bins. The goal is to minimize the amount of overpacking, i.e., the maximum size packed into any bin. We give an algorithm for Online Bin Stretching with a stretching factor of 1.5 for any number of bins. We build o...

We consider a generalized one-dimensional bin-packing model where the cost of a bin is a nondecreasing concave function of the utilization of the bin. Four popular heuristics from the literature of the classical bin-packing problem are studied: First Fit (FF), Best Fit (BF), First Fit Decreasing (FFD), and Best Fit Decreasing (BFD). We analyze their worst-case performances when they are applied...

In this paper, we present a branch-and-cut algorithm for the Vehicle Routing Problem with Two-dimensional Loading Constraints (2L-CVRP). The 2L-CVRP is a combination of the Capacitated Vehicle Routing Problem (CVRP) and the Two-dimensional Bin Packing Problem. We propose a strategy based on emphasizing cut generation using known families of valid inequalities for the CVRP. Checking packing feas...

The Two-Dimensional Finite Bin Packing Problem (2BP) consists of determining the minimum number of large identical rectangles, bins, that are required for allocatingwithout overlapping a given set of 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. In this...

The two-dimensional non-oriented bin packing problem with due dates packs a set of rectangular items, which may be rotated by 90 degrees, into identical rectangular bins. The bins have equal processing times. An item’s lateness is the difference between its due date and the completion time of its bin. The problem packs all items without overlap as to minimize maximum lateness Lmax. The paper pr...

In this paper we address the problem of assigning seats in a train for a group of people traveling together. We consider two variants of the problem. One is a variant of two-dimensional knapsack where we consider the train as having fixed size and the objective is to maximize the utilization of the seats in the train. The other is a variant of two-dimensional bin packing where all requests must...

Packing problems arise in a wide variety of application areas. The basic problem is that of determining an efficient arrangement of different objects in a region without any overlap and with minimal wasted gap between shapes. This paper presents a novel evolutionary approach based on three new evolutionary operators for optimizing the arrangement of irregular shapes. In this approach, each chro...