• Industrial Engineering and Management Systems
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• v.7 no.2
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• pp.126-132
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• 2008

The bin packing problem (BPP) is an NP-Complete Problem. The problem can be described as there are $N=\{1,2,{\cdots},n\}$ which is a set of item indices and $L=\{s1,s2,{\cdots},sn\}$ be a set of item sizes sj, where $0<sj{\leq}1$, ${\forall}j{\in}N$. The objective is to minimize the number of bins used for packing items in N into a bin such that the total size of items in a bin does not exceed the bin capacity. Assume that the bins have capacity equal to one. In the past, many researchers put on effort to find the heuristic algorithms instead of solving the problem to optimality. Then, the quality of solution may be measured by the asymptotic worst-case ratio or the average-case ratio. The First Fit Decreasing (FFD) is one of the algorithms that its asymptotic worst-case ratio equals to 11/9. Many researchers prove the asymptotic worst-case ratio by using the weighting function and the proof is in a lengthy format. In this study, we found an easier way to prove that the asymptotic worst-case ratio of the First Fit Decreasing (FFD) is not more than 11/9. The proof comes from two ideas which are the occupied space in a bin is more than the size of the item and the occupied space in the optimal solution is less than occupied space in the FFD solution. The occupied space is later called the weighting function. The objective is to determine the maximum occupied space of the heuristics by using integer programming. The maximum value is the key to the asymptotic worst-case ratio.

• Proceedings of the Korean Operations and Management Science Society Conference
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• 2004.05a
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• pp.311-314
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• 2004

The 2 DBPP(Two-Dimensional Bin Packing Problem) is a problem of packing each item into a bin so that no two items overlap and the number of required bins is minimized under the set of rectangular items which may not be rotated and an unlimited number of identical rectangular bins. The 2 DBPP is strongly NP-hard and finds many practical applications in industry. In this paper we discuss a tabu search approach which includes tabu list, intensifying and diversification strategies. The HNFDH(Hybrid Next Fit Decreasing Height) algorithm is used as an internal algorithm. We find that use of the proper parameter and function such as maximum number of tabu list and space utilization function yields a good solution in a reduced time. We present a tabu search algorithm and its performance through extensive computational experiments.

• Communications of the Korean Mathematical Society
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• v.34 no.1
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• pp.213-230
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• 2019

We construct new metric outer measures (multifractal analogues of the Hewitt-Stromberg measure) $H^{q,t}_{\mu}$ and $P^{q,t}_{\mu}$ lying between the multifractal Hausdorff measure ${\mathcal{H}}^{q,t}_{\mu}$ and the multifractal packing measure ${\mathcal{P}}^{q,t}_{\mu}$. We set up a necessary and sufficient condition for which multifractal Hausdorff and packing measures are equivalent to the new ones. Also, we focus our study on some regularities for these given measures. In particular, we try to formulate a new version of Olsen's density theorem when ${\mu}$ satisfies the doubling condition. As an application, we extend the density theorem given in [3].

• Communications of the Korean Mathematical Society
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• v.22 no.4
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• pp.547-552
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• 2007

The unit interval is not homeomorphic to a self-similar Cantor set in which we studied the dimensions of distribution subsets. However we show that similar results regarding dimensions of the distribution subsets also hold for the unit interval since the distribution subsets have similar structures with those in a self-similar Cantor set.

• Polymer(Korea)
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• v.36 no.6
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• pp.685-692
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• 2012

Injection molding operation consists of phases of filling, packing, and cooling. The highest cavity pressure is involved in the packing phase among the operation phases. Thus the cavity pressure largely depends upon velocity to pressure (v/p) switchover timing and magnitude of packing pressure. Developed cavity pressure is directly related to stress concentration in the cavity of mold and it may cause a crack in the mold. Consequently control of cavity pressure is considered very important. In this study, cavity pressure was analyzed in terms of v/p switchover timing and packing pressure through computer simulation and experiment. Cavity pressure was increased as the v/p switchover timing was delayed. Residual pressure after cooling phase was observed when the v/p switchover timing was late, which was due to increased pressurizing time for long filling phase. Cavity pressure was increased proportionally with the packing pressure. Residual pressure after cooling phase was also observed, and it was increased with increasing packing pressure. High cavity pressure and residual pressure have been observed at late v/p switchover and high packing pressure. Compared with simulation and experimental results, the profiles of pressures were very similar however simulation could not predict residual pressure. Packing condition was important for the control of cavity pressure and the optimum condition could be set up using CAE analysis.

• Korean Chemical Engineering Research
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• v.50 no.1
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• pp.66-71
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• 2012

We deal with a problem to determine experimentally as well as theoretically permeability of incompressible viscous flow through packed bed of bidisperse hard spheres in size. For the size ratios of large to small spheres ${\lambda}$=1.25 and 2, we set up bidisperse packing and measured porosity and permeability at various volumetric ratios of small to large spheres ${\gamma}$. Bidisperse packing shows lower porosity and permeability than monodisperse packing does. Variation of porosity as a function of ${\gamma}$ does not match with that of permeability. A theoretical expression for predicting permeability of a viscous flow for packed bed of bidisperse packing is derived based on calculation of drag force acting on each sphere and its predictions are compared with the experimental data and those from some relations previously suggested. It is found that our theory shows better agreement with experimental results than the previous studies and is proved to be quite simple and accurate in estimating the permeability.

• Communications of the Korean Mathematical Society
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• v.23 no.4
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• pp.549-554
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• 2008

We study the multifractal spectrum of two dimensionally indexed classes whose members are distribution sets of a self-similar attractor in the unit interval.

• Communications of the Korean Mathematical Society
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• v.19 no.3
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• pp.469-475
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• 2004

We generalize S. Ikeda's results for perturbed cantor sets showing how we get the dimensions for deformed self-similar sets.

• Journal of the Korean Mathematical Society
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• v.31 no.4
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• pp.539-544
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• 1994

We [1] investigated the Hausdorff dimension and the packing dimension of a certain perturbed Cantor set whose ratios are unformly bounded. In this paper, we consider a specific Cantor set whose ratios are not necessarily uniformly bounded but satisfy some other conditions. In fact, in the hypothesis, only the condition of the unform boundedness of ratios on the set is substituted by a "*-condition". We use energy theory related to Hausdorff dimension in this study while we [1] used Hausdorff density theorem to find the Hausdorff dimension of some perturbed Cantor set. In the end, we given an example which explains aformentioned facts.ned facts.

• Journal of the Korean Mathematical Society
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• v.45 no.3
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• pp.871-881
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• 2008

In this paper, we study the core stability of the dominating set game which has arisen from the cost allocation problem related to domination problem on graphs. Let G be a graph whose neighborhood matrix is balanced. Applying duality theory of linear programming and graph theory, we prove that the dominating set game corresponding to G has the stable core if and only if every vertex belongs to a maximum 2-packing in G. We also show that for dominating set games corresponding to G, the core is stable if it is large, the game is extendable, or the game is exact. In fact, the core being large, the game being extendable and the game being exact are shown to be equivalent.