The Generalized Bin Packing Problem under Uncertainty ROBERTO TADEI, GUIDO PERBOLI Department of Control and Computer Engineering, Politecnico di Torino Corso Duca degli Abruzzi, 24, I-10129, Turin, ITALY {roberto.tadei, guido.perboli}@polito.it Abstract: In this paper we introduce the Generalized Bin Packing Problem under Uncertainty, a new packing problem where, given a set of items characterized by volume and stochastic revenue and a set of bins characterized by volume and cost, we want to select a subset of items to be loaded into a subset of bins which maximizes the total profit, given by the difference between the expected total revenue of the loaded items and the total cost of the used bins, by satisfying the volume and bin availability constraints. The item revenues are random variables with unknown probability distribution. By using the asymptotic theory of extreme values a nonlinear integer determin- istic model is derived. Key-Words: packing, stochastic revenue, asymptotic approximation, nonlinear integer deterministic model. 1 Introduction Given a set of items characterized by volume and stochastic revenue and a set of bins characterized by volume and cost, the Generalized Bin Packing Problem under Uncertainty (GBPPu) chooses a subset of items to be loaded into a subset of bins which maximizes the total profit, given by the difference between the expected total revenue of the loaded items and the total cost of the used bins, by satisfying the volume and bin availability constraints. The item revenues are random variables with unknown probability distribution. They are composed by a deterministic revenue plus a random term, which represents the revenue oscillations due to the handling operations of the bins which are necessary for item loading and dispatching. The GBPPu frequently arises in real-life applications, in particular in logistics, where the freight consolidation is essential to optimize the delivery process. In this case, a series of handling operations for the bins must be performed at the logistic platforms and these operations could highly affect the final total revenue of the loading [8]. In this paper we introduce a stochastic model for the GBPPu. In most papers dealing with uncertainty, the probability distribution of the random variables is given and their expected value can be calculated. This is not the case of the GBPPu, where the probability distribution of the stochastic item revenue is unknown, because it is unmeasurable in practice and any strong assumption on its shape would be arbitrary. We show that, by using some results of the asymptotic theory of extreme values, the probability distribution of the maximum item revenue becomes a Gumbel (or double exponential) distribution and the expected total revenue of the loaded items can be easily calculated. 2 Literature review The GBPPu is the stochastic version of the Generalized Bin Packing Problem (GBPP), introduced by Baldi et al. [1] as a generalization of the Variable Cost and Size Bin Packing Problem [2]. The GBPPu literature does not exist yet, being this paper the first one on this problem. Also the literature on the GBBP is quite limited, due to its recent introduction. For this reason, in the following we will also consider some relevant literature on a similar problem, which is a special case of the GBPPu, i.e. the Stochastic Bin Packing Problem. Recent Researches in Applied and Computational Mathematics ISBN: 978-1-61804-002-2 163