Optimization of one-dimensional Bin Packing Problem with island parallel grouping genetic algorithms

Highlights

• Novel hybrid bin-oriented heuristics with polynomial running times are proposed.

• Parallel processing, grouping genetic algorithms, and novel hybrid bin oriented heuristics are combined.

• A set of robust Island Parallel Genetic Algorithms are proposed.

Abstract

The well-known one-dimensional Bin Packing Problem (BPP) of whose variants arise in many real life situations is a challenging NP-Hard combinatorial optimization problem. Metaheuristics are widely used optimization tools to find (near-) optimal solutions for solving large problem instances of BPP in reasonable running times. With this study, we propose a set of robust and scalable hybrid parallel algorithms that take advantage of parallel computation techniques, evolutionary grouping genetic metaheuristics, and bin-oriented heuristics to obtain solutions for large scale one-dimensional BPP instances. A total number of 1318 benchmark problems are examined with the proposed algorithms and it is shown that optimal solutions for 88.5% of these instances can be obtained with practical optimization times while solving the rest of the problems with no more than one extra bin. When the results are compared with the existing state-of-the-art heuristics, the developed parallel hybrid grouping genetic algorithms can be considered as one of the best one-dimensional BPP algorithms in terms of computation time and solution quality.

Introduction

Many practical optimization problems require efficient grouping (clustering) of a set of items into a collection of disjoint subsets according to some specific constraints. The well-known one-dimensional Bin Packing Problem (BPP) is such a NP-hard combinatorial grouping problem that often occurs in real life including engineering, logistics, and manufacturing (Garey and Johnson, 1979, Johnson et al., 1974). The BPP appears as the main problem or a significant subproblem in a large number of industrial applications (Camacho et al., 2013, Fleszar and Charalambous, 2011, Fleszar, 2012). Two/three dimensional versions of BPP are frequently faced during manufacturing processes. For instance, in clothing, construction, glass, plastic, or metal industries, components need to be cut from the fewest sheets of material (Dahmani, Clautiaux, Krichen, & Talbi, 2013). Similarly, when designing the layout of the pages of a newspaper, the editor needs to arrange the articles on pages of fixed dimensions. In the shipping and transportation industries, packages of identical heights have to be loaded in the minimum number of rectangular bins.

Informally, the BPP is the process of packing n items into a number of same size and shape (for ease of planning and transportation) bins with a capacity of c where the objective is to minimize the number of bins required to contain n items (Gupta and Ho, 1999, Martello and Toth, 1989). The number of bins is assumed to be unlimited, each with capacity c $>$ 0. A set of n items is to be packed into bins and the size of item i $\in \{1\text{,}\dots \text{,}n\}$ is $s_i$ $>$ 0, as shown in Eq. (1):

$$\forall k:{\displaystyle \sum _{i\in \mathit{bin}(k)}}{s}_i⩽c$$

The goal of one-dimensional BPP is to find the minimum number of bins, M, required for packing all n items (Eq. (2)):

$$M⩾⌈\left({\displaystyle \sum _{i=1}^n}{s}_i\right)/c⌉$$

Modern parallel computation environments such as grid and high performance computing have received intensive attention from industry and academia in recent years because of their ability to solve hard problems more efficiently and accurately. In order to overcome the processing power limits, multi-core processors integrate many cores into one chip and deliver higher total computing power. By developing parallel algorithms that can efficiently use all of the available processors, the processing power supplied by multi-core processors can increase the solution quality of one-dimensional BPP (Fernandez, Gil, Banos, & Montoya, 2013).

In addition to these benefits of multi-core processors, genetic algorithms (GAs) continue to be promising tools for numerous NP-Hard optimization problems where exact solution methods tend to fail because of the exponential search spaces (Cantu-Paz, 2000, Holland, 1975, Luque and Alba, 2011, Mitchell, 1996, Zitzler and Thiele, 1999). The study of Rohlfshagen and Bullinaria (2007) that we have parallelized is such a recent technique for solving the one-dimensional BPP that views individuals as (eukaryotic) genes instead of (prokaryotic) genomes in order to improve the traditional design of GAs.

In this study, our main motivation was to solve the one dimensional BPP with island parallel grouping genetic algorithms (GGAs) that take advantage of state-of-the-art computation tools. Island parallel GAs are very efficient tools for the solution of many NP-Hard problems (Cantu-Paz, 2000, Gordon and Whitley, 1993, Lim et al., 2000, Luque and Alba, 2011). They bring many of the state-of-the-art computational tools together for better results.

With the proposed novel algorithms, we have combined state-of-the-art computation tools; parallel processing, GGAs, and bin oriented heuristics to efficiently solve the intractable one-dimensional BPP. The majority (88.5%) of the 1318 benchmark problem instances are solved optimally while the rest of the solutions produce only one extra bin. Novel hybrid bin-oriented heuristics with polynomial running times are proposed for GGAs and used to reinsert the remaining items into bins after the process of crossover and mutation. The solution quality of sequential GGAs is improved by the use of subpopulations on multi-core processors and it is shown that there is a potential to solve the BPP even more accurately if additional processors can be supplied.

In Section 2, we briefly review the relevant and the best performing sequential and parallel recent methods in chronological order. Section 3 explains the GGAs, canonical GA crossovers, Falkenauer’s crossover, molecular genetics exon shuffling crossover, mutation, and inversion operator. Section 4 gives brief information about the use of BPP heuristics in the developed algorithms. Section 5 defines the proposed parallel hybrid GGAs and the parameter settings. Experimental comparisons of the developed algorithms are given in Section 6. Finally, conclusions and further research directions are discussed in Section 7.