Robust minimum cost flow problem under consistent flow constraints

The robust minimum cost flow problem under consistent flow constraints (RobMCF≡\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{do...

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Vydané v:	Annals of operations research Ročník 312; číslo 2; s. 691 - 722
Hlavní autori:	Büsing, Christina, Koster, Arie M. C. A, Schmitz, Sabrina
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York, NY Springer US 01.05.2022 Springer Springer Nature B.V
Predmet:	Algorithms Approximation Business and Management Combinatorics Demand Dynamic programming Equal flow problem Graph theory Logistics Mathematical optimization Mathematical research Minimum cost Minimum cost flow problem Operations research Operations Research/Decision Theory Original Research Polynomials Robust flows Robustness Series-parallel digraphs Statistical decision Subcontractors Theory of Computation Germany Equal flow problem Dynamic programming Series-parallel digraphs Minimum cost flow problem Robust flows
ISSN:	1572-9338, 0254-5330, 1572-9338
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Shrnutí:	The robust minimum cost flow problem under consistent flow constraints (RobMCF≡\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\equiv $$\end{document}) is a new extension of the minimum cost flow (MCF) problem. In the RobMCF≡\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\equiv $$\end{document} problem, we consider demand and supply that are subject to uncertainty. For all demand realizations, however, we require that the flow value on an arc needs to be equal if it is included in the predetermined arc set given. The objective is to find feasible flows that satisfy the equal flow requirements while minimizing the maximum occurring cost among all demand realizations. In the case of a finite discrete set of scenarios, we derive structural results which point out the differences with the polynomial time solvable MCF problem in networks with integral demands, supplies, and capacities. In particular, the Integral Flow Theorem of Dantzig and Fulkerson does not hold. For this reason, we require integral flows in the entire paper. We show that the RobMCF≡\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\equiv $$\end{document} problem is strongly NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {NP}$$\end{document}-hard on acyclic digraphs by a reduction from the (3, B2)-Sat problem. Further, we demonstrate that the RobMCF≡\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\equiv $$\end{document} problem is weakly NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {NP}$$\end{document}-hard on series-parallel digraphs by providing a reduction from Partition. If in addition the number of scenarios is constant, we propose a pseudo-polynomial algorithm based on dynamic programming. Finally, we present a special case on series-parallel digraphs for which we can solve the RobMCF≡\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\equiv $$\end{document} problem in polynomial time.
Bibliografia:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	1572-9338 0254-5330 1572-9338
DOI:	10.1007/s10479-021-04426-0