Number of bins and maximum lateness minimization in two-dimensional bin packing (original) (raw)

An Analysis of Solutions to the 2D Bin Packing Problem and Additional Complexities

2021

The Bin Packing problem in 2 space is an NP-Hard combinatorial problem in optimization of packing and arrangement of objects in a given space. It has a wide variety of applications ranging from logistics in retail industries to resource allocation in cloud computing. In this paper, we discuss the mathematical formulation of this problem. Furthermore, we analyse its time complexity, its NP-Hardness and some of its stochastic solutions with their efficiencies. We then propose additional complexities that would make the problem more fit for industrial use and discuss in depth the domains in which it might prove to be useful. We conclude while suggesting areas of improvement in operations research on this subject.

Maximum lateness minimization in one-dimensional bin packing

Omega, 2016

In the One-dimensional Bin Packing problem (1-BP) items of different lengths must be assigned to a minimum number of bins of unit length. Regarding each item as a job that requires unit time and some resource amount, and each bin as the total (discrete) resource available per time unit, the 1-BP objective is the minimization of the makespan C max ¼ max j fC j g. We here generalize the problem to the case in which each item j is due by some date d j : our objective is to minimize a convex combination of C max and L max ¼ max j fC j À d j g. For this problem we propose a time-indexed Mixed Integer Linear Programming formulation. The formulation can be decomposed and solved by column generation relegating single-bin packing to a pricing problem to be solved dynamically. We use bounds to (individual terms of) the objective function to address the oddity of activation constraints. In this way, we get very good gaps for instances that are considered difficult for the 1-BP.

On-line bin packing ? A restricted survey

ZOR Zeitschrift f�r Operations Research Methods and Models of Operations Research, 1995

In the classical bin packing problem, one is asked to pack items of various sizes into the minimum number of equal-sized bins. In the on-line version of this problem, the packer is given the items one by one and must immediately and irrevocably assign every item to its bin, without knowing the future items. Beginning with the first results in the early 1970's, we survey-from the worst case point of view-the approximation results obtained for on-line bin packing, higher dimensional versions of the problem, lower bounds on worst ratios and related results.

Models and algorithms for three-stage two-dimensional bin packing

European Journal of Operational Research, 2007

We consider the three-stage two-dimensional bin packing problem (2BP) which occurs in real-world applications such as glass, paper, or steel cutting. We present new integer linear programming formulations: models for a restricted version and the original version of the problem are developed. Both only involve polynomial numbers of variables and constraints and effectively avoid symmetries. Those models are solved using CPLEX. Furthermore, a branch-and-price (B&P) algorithm is presented for a set covering formulation of the unrestricted problem, which corresponds to a Dantzig-Wolfe decomposition of the polynomially-sized model. We consider column generation stabilization in the B&P algorithm using dual-optimal inequalities. Fast column generation is performed by applying a hierarchy of four methods: (a) a fast greedy heuristic, (b) an evolutionary algorithm, (c) solving a restricted form of the pricing problem using CPLEX, and finally (d) solving the complete pricing problem using CPLEX. Computational experiments on standard benchmark instances document the benefits of the new approaches: The restricted version of the integer linear programming model can be used to quickly obtain nearoptimal solutions. The unrestricted version is computationally more expensive. Column generation provides a strong lower bound for 3-stage 2BP. The combination of all four pricing algorithms and column generation stabilization in the proposed B&P framework yields the best results in terms of the average objective value, the average run-time, and the number of instances solved to proven optimality.

The cutting-stock approach to bin packing: Theory and experiments

2002

We report on results of an experimental study of the Gilmore-Gomory cutting-stock heuristic GG61, GG63] and related LP-based approaches to bin packing, as applied to instances generated according to discrete distributions (previously studied theoretically in such papers as CCG + 91, CCG + 00, CJSW93, CJSW97, KRS98]). We examine the questions of how best to solve the knapsack problems used to generate columns in the Gilmore-Gomory approach, how the various algorithms' running times and solution qualities scale with key instance parameters, and how the algorithms compare to more traditional bin packing heuristics.