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Bin Packing Problem with Uncertain Volumes and Capacities Jin Peng · Bo Zhang
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Abstract Bin packing problem is a traditional problem in combinatorial optimization and applied widely in modern society. This paper presents bin packing problem with uncertain volumes and capacities. First, two types of uncertain programming models are mathematically constructed for the uncertain bin packing problem, including expected value model and chance-constrained programming model. Next, the proposed models can be transformed into the corresponding deterministic forms by taking advantage of uncertainty theory. Finally, a numerical experiment is presented to show the performances of the proposed models and algorithms. Keywords Bin Packing Problem · Uncertainty Theory · Uncertain Variable · Uncertain Programming Model
1 Introduction Bin packing problem is a historically interesting problem, in which objects of different volumes must be packed into a finite number of bins of fixed capacity in a way that minimizes the number of bins used. The bin packing problem has been widely studied due to its various applications in transportation and logistics industry. There is an important connection between bin packing and another very important collection of operations research questions. The bin packing problem can also be viewed as a special case of the cutting stock problem(de Carvalho, 2002). When the number of bins is restricted to 1 and each item is characterized Jin Peng Institute of Uncertain Systems, Huanggang Normal University, Hubei 438000, China Tel.: +86.713.8621602 Fax: +86.713.8621601 E-mail:
[email protected] Bo Zhang School of Mathematics and Statistics, Huazhong Normal University, Hubei 430079, China
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by both a volume and a value, the problem of maximizing the value of items that can fit in the bin is known as the knapsack problem. There are many variations of the bin packing problem, such as 2D packing (Clautiaux et al., 2007; Lodi et al., 2002; Pisinger and Sigurd, 2005), 3D packing (Martello et al., 2000), multiobjective packing (Liu et al., 2008), online packing (Babel et al., 2004), dynamic bin packing (Coffman et al., 1983; Epstein and Levy, 2010), multi-stage bin packing(Puchinger and Raidl, 2007), variable sized packing (Bang-Jensen and Larsen, 2012; Boyar and Favrholdt, 2012; Kang and Park, 2003), and so on. In computational complexity theory, the bin packing problem is a type of combinatorial NP-hard problem. Despite the fact that it is NP-hard, optimal solutions to very large instances can be produced with sophisticated algorithms. In many cases, approximation optimal solutions may be obtained by using efficient heuristic packing algorithms (Johnson, 1974). For example, the first fit algorithm provides a fast but often non-optimal solution, involving placing each item into the first bin in which it will fit. The algorithm can be made much more effective by first sorting the list of elements into decreasing order (sometimes known as the first-fit decreasing algorithm), although this still does not guarantee an optimal solution, and for longer lists may increase the running time of the algorithm. It is known, however, that there always exists at least one ordering of items that allows first-fit to produce an optimal solution. Many evolutionary heuristic algorithms (Coffman et al., 1983), including genetic algorithms (Chan et al., 2007) and particle swarm algorithms (Liu et al., 2008), have been developed to find the optimal solutions for various bin packing problem. Despite the fact that some bin packing problems are NPcomplete, they are amenable to dynamic programming solutions (Han et al., 2010) or to approximately optimal heuristic solutions (Gendreau et al., 2004; Stawowy, 2008). In the above mentioned research, the bin packing problems were investigated in a deterministic environment, in which the volumes (weights) of the items, the capacities (costs) of the bins, etc., are positive crisp values. However, some indeterminacy factors might occur in the problems. In practice, due to the lack of adequate sample data, or detail sample data are not easy to get because of economic reasons or technical difficulties, the volumes (weights) of the items and the capacities (costs) of the bins are not sharply known in advance. As a result, they are described by some empirical data such as “about 2 cubic feet” or “approximately 3 cubic meters”. In this situations, it is not suitable to employ the classical models and algorithms to study the bin packing problem directly. Probabilistic approaches are borrowed in the analysis of bin packing algorithm (Foster and Vohra, 1989; Coffman et al., 2008). Moreover, some researchers employed probability theory to investigate the stochastic bin packing problem (Coffman et al., 1980; Rhee, 1985; Perboli et al., 2012) and considered the stochastic version of the generalized bin packing problem. Lessmore was
Bin Packing Problem with Uncertain Volumes and Capacities
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founded in current literature on fuzzy bin packing problem (Kim et al., 2001; Nasibov, 2004). The indeterminacy phenomena can be divided into distinct types. One is stochastic phenomena, and another is uncertain phenomena. When the sample size is too small or even no-sample to estimate a probability distribution, we have to invite some domain experts to evaluate their belief degree that each event will occur. Since human beings usually overweight unlikely events, the belief degree may have much larger variance than the real frequency. Probability theory is inappropriate in this case because it may lead to counterintuitive results. In order to distinguish from randomness, this phenomenon was named uncertainty (Liu, 2012). In order to deal with the uncertain phenomena, Liu (2007) founded an uncertainty theory, which has become a branch of axiomatic mathematics for modeling human uncertainty. In theoretical aspect, uncertain process (Liu, 2008), uncertain differential equation (Chen and Liu, 2010), have been established. From a practical aspect, uncertain programming (Zhang and Peng, 2012), uncertain network(Zhang and Peng, 2012), uncertain finance (Peng and Yao, 2011), etc., have also been developed. As an illustration of researching uncertain combinatorial optimization problems, we initially consider uncertain bin packing (UBP) problem, in detail, the bin packing problem with uncertain volumes and uncertain capacities. The conventional bin packing model are extended to uncertain programming models in which the uncertainty distributions of volumes of items and capacities of bins are assumed. The paper is organized as follows: Section 2 presents some necessary preliminary concepts and results selected from uncertainty theory. In Section 3, the UBP problem is introduced as the extension of classical bin packing problem. In Section 4, two types of uncertain programming models for UBP are presented, which includes expected value model and chance-constrained programming model. Section 5 illustrates an example. The last section contains a summary.
2 Preliminaries As an efficient tool of modeling the behavior of uncertain phenomena, uncertainty theory is employed to deal with uncertain bin packing problem. Uncertainty theory is a branch of mathematics based on normality, duality, subadditivity, and product axioms. In this section, we present some related preliminaries from uncertainty theory. The first fundamental concept in uncertainty theory is uncertain measure that is interpreted as the personal belief degree (not frequency) of an uncertain event that may occur. Let Γ be a nonempty set, and L a σ-algebra over Γ . For any Λ ∈ L, Liu (2007) presented an axiomatic uncertain measure M{Λ} to express the chance that uncertain event Λ occurs. The set function M{·} satisfies the following three axioms:
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(i) (Normality) M{Γ } = 1; (ii) (Duality) M{Λ} + M{Λc } = 1 for any Λ ∈ L; (iii) (Subadditivity) For every countable sequence of events {Λi }, we have P M {∪i Λi } ≤ i M{Λi }. The triplet (Γ, L, M) is called an uncertainty space. Liu (2009) defined product uncertain measure by way of the fourth axiom of uncertainty theory, which makes much differences in operations between uncertainty theory and probability theory. Let (Γk , Lk , Mk ) be uncertainty spaces for k = 1, 2, · · · Write Γ = Γ1 × Γ2 × · · · , L = L1 × L2 × · · · Then the product uncertain measure M on the product σ-algebra L is defined by the following axiom: (iv) (Product Axiom) (∞ ) ∞ Y ^ M Λk = Mk {Λk } k=1
k=1
where Λk are arbitrarily chosen events from Lk for k = 1, 2, · · ·, respectively. An uncertain variable is defined as a measurable function from an uncertainty space to the set of real numbers (Liu,2007). An uncertain variable ξ can be characterized by its uncertainty distribution Φ : < → [0, 1], which is defined by Liu (2007) as follows Φ(x) = M γ ∈ Γ ξ(γ) ≤ x . Let ξ be an uncertain variable with uncertainty distribution Φ. Then the inverse function Φ−1 is called the inverse uncertainty distribution of ξ. The expected value of uncertain variable ξ is defined by Liu (2007) as Z +∞ Z 0 E[ξ] = M{ξ ≥ r}dr − M{ξ ≤ r}dr −∞
0
provided that at least one of the two integrals is finite. As a useful representation of expected value, it has been proved by Liu (2007) that Z 1 E[ξ] = Φ−1 (α)dα 0 −1
where Φ is the inverse uncertainty distribution of uncertain variable ξ. Liu (2007) introduced the independence concept of uncertain variables. The uncertain variables ξ1 , ξ2 , · · · , ξm are independent if and only if (m ) \ M {ξi ∈ Bi } = min M {ξi ∈ Bi } i=1
for any Borel sets B1 , B2 , · · · , Bm of
