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Proceedings of GO 2005, pp. 29 – 34.Analysis of a nonlinear optimization problem related tosemi-on-line bin packing problemsJános Balogh, 1 József Békési, 1 Gábor Galambos, 1 and Mihály Csaba Markót 21 University of Szeged, Faculty of Juhász Gyula Teacher Training College, Department of Computer Science, 6701 Szeged,POB. 396. Hungary{balogh, bekesi, galambos}@jgytf.u-szeged.hu2 Advanced Concepts Team, European Space Agency, ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The NetherlandsMihaly.Csaba.Markot@esa.intAbstractKeywords:In the paper we deal with lower bounds constructed for the asymptotic performance ratio of semion-linebin packing and batched bin packing algorithms. We determine the bounds as the solutionsof a related nonlinear optimization problem using theoretical analysis and a reliable numericalglobal optimization method. The results improves the lower bounds given in [9] for some specialcases of the batched bin packing problem (fixed, finite number of different elements in two batches),answering a question raised in [9] regarding the optimal bounds.semi-on-line bin packing problems, nonlinear optimization, branch–and–bound, interval arithmetic1. IntroductionBin packing is a well-known combinatorial optimization problem. In the classical one–dimensionalcase we are given a set of items represented by a list L = {x 1 , x 2 , . . . , x n } of real numbersin (0, 1], and an infinite list of unit capacity bins. Each item x i has to be assigned to a uniquebin such that the sum of the elements in each bin does not exceed 1. Our aim is to minimizethe number of used bins. It is well-known that finding an optimal packing is NP-hard [7].Consequently, large number of papers have been published which look for polynomial timealgorithms with an acceptable approximative behavior. The on-line bin packing algorithms putitems into a bin as they appear without knowing anything about the subsequent elements(neither the sizes nor the number of the elements). Off-line algorithms can use more information:most of them examine the entire list before they apply their strategy to pack the items.The so called semi-on-line algorithms [2] are between the on-line and off-line ones. For such algorithmsat least one of the following operations is allowed: repacking of some items [4–6,11],lookahead of the next several elements [8], or some kind of preordering.The efficiency of different algorithms is generally measured by two different methods: theinvestigation of the worst-case behavior, or – assuming some probability distribution of theelements – a probability analysis. In this paper we will concentrate on the asymptotic worstcaseratio which can be defined as follows: denote A(L) the number of bins used by the (eitherrepacking or non-repacking) algorithm A to pack the elements of a list L, and let L ∗ the numberof bins in an optimal packing. If{ }A(L)R A (k) := maxk|L∗ = k(1)denotes the maximum ratio of A(L)/L ∗ for any list L with L ∗ = k, then the asymptotic performance