A genetic algorithm for two-dimensional bin packing with due dates Julia A. Bennell a,n , Lai Soon Lee b , Chris N. Potts c a School of Management, University of Southampton, Southampton SO17 1BJ, UK b Department of Mathematics, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia c School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK article info Article history: Received 15 February 2012 Accepted 21 April 2013 Keywords: Cutting and packing Two-dimensional bin packing Due date Scheduling Genetic algorithms abstract This paper considers a new variant of the two-dimensional bin packing problem where each rectangle is assigned a due date and each bin has a ﬁxed processing time. Hence the objective is not only to minimize the number of bins, but also to minimize the maximum lateness of the rectangles. This problem is motivated by the cutting of stock sheets and the potential increased efﬁciency that might be gained by drawing on a larger pool of demand pieces by mixing orders, while also aiming to ensure a certain level of customer service. We propose a genetic algorithm for searching the solution space, which uses a new placement heuristic for decoding the gene based on the best ﬁt heuristic designed for the strip packing problems. The genetic algorithm employs an innovative crossover operator that considers several different children from each pair of parents. Further, the dual objective is optimized hierarchically with the primary objective periodically alternating between maximum lateness and number of bins. As a result, the approach produces several non-dominated solutions with different trade-offs. Two further approaches are implemented. One is based on a previous Uniﬁed Tabu Search, suitably modiﬁed to tackle this revised problem. The other is randomized descent and serves as a benchmark for comparing the results. Comprehensive computational results are presented, which show that the Uniﬁed Tabu Search still works well in minimizing the bins, but the genetic algorithm performs slightly better. When also considering maximum lateness, the genetic algorithm is considerably better. & 2013 Elsevier B.V. All rights reserved. 1. Introduction Cutting and packing problems have been the subject of exten- sive research over a number of years motivated by a wide range of real world applications. A typology of the cutting and packing literature can be found in Wäscher et al. (2007). This paper focusses on two-dimensional packing in which rectangles of speciﬁed dimensions are to be cut from identical stock sheets. It is related to the “two-dimensional, rectangular single bin size, bin packing problem”, which is known as 2DRSBSBPP in Wäscher et al. (2007). For brevity, we use the term two-dimensional bin packing problem, or 2DBPP, henceforth. However, rather than focussing only on the packing problem alone, we also examine issues of production planning and scheduling. In a classic 2DBPP, time is not considered an issue, and it is assumed that a rectangle can be allocated to any bin and the bins can be processed in any order. This is acceptable provided all the bins can be processed within a single production period. In this paper, we consider the issue of supplying the rectangles in a timely manner across multiple production periods, assuming that the capacity of the cutting process results in potential delays to the times at which the rectangles become available to the customer. More precisely, we assume that each stock sheet has a cutting time, and that each rectangle has an associated due date that speciﬁes the time by which it should ideally be cut and available to the customer. As a result, the goal in this problem is to minimize the maximum lateness of the rectangles with respect to their due dates (DD) while using as few stock sheets as possible. This problem description captures the trade-off between using an ideal cutting with small waste which may result in some rectangles being delayed past their due dates, and using the due dates to group the rectangles for cutting which may result in more stock sheets being used than necessary. There are few examples in the literature of implementations that tackle both the packing problem and the production planning/ scheduling problem. These papers mainly focus on cutting-stock problems and tend to separate the problem of determining the cutting patterns and the scheduling of the parts. Gramani and França (2006) examine a similar problem, which can be viewed as one of the lot sizing, where pieces may be cut early and incur a holding cost but are not permitted to be late. They use a network shortest path model to generate a solution and deal with the packing element by selecting from available patterns. Nonas and Thorstenson (2000) also tackle the combined cutting-stock and Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/ijpe Int. J. Production Economics 0925-5273/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpe.2013.04.040 n Corresponding author. Tel.: +44 2380595671. E-mail address: J.A.Bennell@soton.ac.uk (J.A. Bennell). Please cite this article as: Bennell, J.A., et al., A genetic algorithm for two-dimensional bin packing with due dates. International Journal of Production Economics (2013), http://dx.doi.org/10.1016/j.ijpe.2013.04.040i Int. J. Production Economics ∎ (∎∎∎∎) ∎∎∎–∎∎∎