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Algorithm for turnpike policies in the dynamic lot size model

100%
Bylka S.
Applicationes Mathematicae
|
1996-1997
|
tom 24
|
nr 1
57-75
EN
This article considers optimization problems in a capacitated lot sizing model with limited backlogging. Nothing is assumed about the cost function in the case of finite restrictions of the size on the stock and backlogs. The holding and backlogging costs are functions assumed to be stationary or nearly stationary in time. In both cases, it is shown that there exists an optimal infinite inverse policy and a periodical turnpike policy. Some forward and backward procedures are adopted that determine an optimal infinite inverse policy and a strong turnpike policy relative to the class of standard or batch ordering type policies. Some remarks on the existence of planning and forecast horizons are also given.
2
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Forecast horizon and planning horizon paths in time-indexed network

100%
Bylka S.
Banach Center Publications
|
2006
|
tom 71
|
nr 1
63-82
EN
The problem of existence of a forecast (or planning) horizon has been considered in many special models, more or less precisely. We specify and investigate this problem for families of cheapest paths in networks with weakly ordered nodes. In a discrete network, the standard forward algorithm finds the subnetwork generated by optimal paths. The proposed forward procedure reduces subnetworks such that the forecast horizon remains unchanged. Based on the final subnetwork, we have an answer to the forecast horizon questions. In particular, we show that many questions about rationality of initial decisions become NP-hard. To improve the performance of heuristics, we introduce the notion of potentially rational initial decisions.
3
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Arithmetically maximal independent sets in infinite graphs

100%
Bylka S.
Discussiones Mathematicae Graph Theory
|
2005
|
tom 25
|
nr 1-2
167-182
EN
Families of all sets of independent vertices in graphs are investigated. The problem how to characterize those infinite graphs which have arithmetically maximal independent sets is posed. A positive answer is given to the following classes of infinite graphs: bipartite graphs, line graphs and graphs having locally infinite clique-cover of vertices. Some counter examples are presented.
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The representation of multi-hypergraphs by set intersections

63%
Bylka S., Komar J.
Discussiones Mathematicae Graph Theory
|
2007
|
tom 27
|
nr 3
565-582
EN
This paper deals with weighted set systems (V,𝓔,q), where V is a set of indices, $𝓔 ⊂ 2^V$ and the weight q is a nonnegative integer function on 𝓔. The basic idea of the paper is to apply weighted set systems to formulate restrictions on intersections. It is of interest to know whether a weighted set system can be represented by set intersections. An intersection representation of (V,𝓔,q) is defined to be an indexed family $R = (R_v)_{v∈ V}$ of subsets of a set S such that $|⋂_{v∈ E} R_v| = q(E)$ for each E ∈ 𝓔. A necessary condition for the existence of such representation is the monotonicity of q on 𝓔 i.e., if F ⊂ 𝓔 then q(F) ≥ q(𝓔). Some sufficient conditions for weighted set systems representable by set intersections are given. Appropriate existence theorems are proved by construction of the solutions. The notion of intersection multigraphs to intersection multi- hypergraphs - hypergraphs with multiple edges, is generalized. Some conditions for intersection multi-hypergraphs are formulated.
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