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32 János Balogh, József Békési, Gábor Galambos, and Mihály Csaba MarkótConcerning the choice of the y values, the most obvious method is to consider the y 1 , . . . , y ksystem as k equidistant points in the (0.25, 0.5] interval, i.e. y j := 0.5 − 0.25(j−1)k, j = 1, . . . , k.For this y-system, solving the received LP (if we fix y-s, (9) is an LP), we get the lower boundsfor (9). The results are displayed in the third column of our Table 1. In this way all values ofthe third column of our Table 1 improve the lower bound for the on-line bin packing problemwith restricted repacking and for the same version of fully dynamic bin packing as well.Our construction (without considering repacking) can be used for the two batched bin packingproblem (2 − BBP P , [9]). in cases when p different sizes are allowed. In our constructionthis means p = k + 1 lists, because of the list L 0 .In this way, if we used the aforementioned equidistant points as the y-system of our construction,we exactly get back the results of Gutin at al. for the two batched bin packing problemwhen p number of different elements are allowed. The third column of our Table 1 containsthe values which are corresponding to their Table on page 77 of [9] for the two batchedbin packing problem.Note that the construction in [9] was similar, but the main point is different: they used onlyequidistant y j -values (in our terminology), while we can consider any y-systems (allowing notonly equidistant points). In this way our construction can be considered as a generalization ofthe one of [9]. The construction of [9] placed some elements of size s in the list L 0 , 0 < s < 1 2 ,while L j contained n/j elements, each of size 1 − js (j = 1, . . . , m, where m = ⌊ 1s⌋). In ourconstruction x j = 1 − js , and y j = js, respectively. (Note, that since our first list has an index0, their p value corresponds to k + 1 in our construction; see Table 1, columns 1–2.)The essence of the results of the LP approach (either considering equidistant y-systemsor not) is that the obtained bounds are valid for the two above mentioned semi-on-line binpacking problems, allowing restricted repacking.In the following we answer a question raised in [9], namely, whether their bound givenfor the cases when only p (p ≥ 2) different elements are allowed, can be improved? In ourcontext, the corresponding problem is whether we can produce better lower bounds from ourconstruction, as compared to the ones obtained from the equidistant y-points. This questionleads us to a nonlinear optimization problem which can be described in a short form using theobjective function and constraints of Theorem 1:max b ∗ (y 1 , . . . , y k ),subject to 0.25 ≤ y k < . . . < y 1 ≤ 0.5,(9)where b ∗ (y 1 , . . . , y k ) denotes the optimal solution of the minimization problem (3) subject tothe constraints (4-7) for a fixed system of y 1 , . . . , y k . Recall that the 0.25 ≤ y k restriction isallowed due to (8). The obtained problem (9) is a so called max-min problem. Nevertheless,if we fix the y-s, then we get back the LP (3-7), and again any y-values deliver a lower bound.Now, the question is: which y-system should be considered to maximize (9).3. The analysis and solution of the nonlinear optimizationproblem – asymptotic behavior and special casesProblem (9) seems to be very hard to solve from a global optimization point of view in its presentedform. Following the transformations introduced in details in [1], we obtain an equivalent,but more convenient form (with one single objective and easily manageable bound constraintsfor the variables):max1 − y kf(y) = 1 +y k + 1 y 1+ ∑ k y i−1i=2 y i− k ,subject to 0.25 ≤ y k < . . . < y 1 ≤ 0.5.(10)