Abstract
Formulating the minimum concave cost capacitated network flow problem as an integer concave minimization problem, we establish finite branch and bound algorithms, in which the branching operation is the so-called integral rectangular partition and the bounding procedure is performed by the classical minimum linear cost flow problem on subnetworks. For the special case that the flow cost function is concave on a fixed number of arcs and linear on the others, an upper bound of the running time is given.
Zusammenfassung
Es wird das Problem eines kostenminimalen Flusses auf Netzwerken mit konkaven Kosten diskutiert. Mit Hilfe einer äquivalenten Formulierung dieses Problems als eine ganzzahlige konkave Minimumsaufgabe wird ein Branch-and-Bound Algorithmus entwickelt, in dem die ganzzahlige Quaderaufteilung und die klassischen Methoden des linearen kostenminimalen Flußproblems angewendet werden. Für den Fall, daß die Kostenfunktion auf einer festen Anzahl der Kanten konkav ist, wird eine Abschätzung der Laufzeit des Algorithmus angegeben. Numerische Ergebnisse zeigen die Brauchbarkeit der Methode.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahuja RK, Magnanti TL, Orlin, JB (1993) Network flows. Prentice-Hall, New Jersey
Bell G, Lamar B, (1996) Solution methods for nonconvex network flow problem. In: Pardalos PM, Hearn D, Hager W (eds) Network optimization. Lecture Notes in Economics and Mathematical Sciences. Springer, Berlin Heidelberg New York
Benson HP, Erengue SS, Horst R (1990) A Note on adapting methods for continuous global optimization to the discrete case. Annals of Operations Research 25:243–252
Falk IE, Soland RM (1969) An algorithm for separable nonconvex programming problems. Management Science 15:550–569
Guisewite GM (1994) Network Problems. In: Horst R, Pardalos PM (eds) Handbook of global optimization, pp 609–648. Kluwer Academic Publishers, Dordrecht
Guisewite GM, Pardalos PM (1990) Minimum concave-cost network flow problems: Applications, complexity, and algorithms. Annals of Operations Research 25:75–100
Guisewite GM, Pardalos PM (1991) Algorithms for the single-source uncapacitated minimum concave-cost network flow problem. Journal of Global Optimization 3:245–266
Guisewite GM, Pardalos PM (1993) A polynomial time solvable concave network flow problem. Network 23:143–147
Horst R, Tuy H(1996) Global optimization: Deterministic approaches, 3rd edn. Springer, Berlin Heidelberg New York
Horst R, Thoai NV (1996) A decomposition approach for the global minimization of biconcave functions over polytopes. Journal of Optimization Theory and Applications 88:561–583
Klinz B, Tuy H (1993) Minimum concave cost network flow problem with a single nonlinear arc cost. In: Du DZ, Pardalos PM (eds) Network optimization problems. World Scientific, Singapore
Lamar BW (1993) A method for solving network flow problems with general nonlinear arc costs. In: Du DZ, Pardalos PM (eds) Network optimization problems. World Scientific, Singapore
Soland RM (1974) Optimal facility location with concave costs. Operations Research 22:373–382
Thoai, N.V (1994) Global optimization techniques for solving the general quadratic integer programming problem. Research Report, Nr. 94-09, University of Trier; forthcoming in: Computational Optimization and Applications
Author information
Authors and Affiliations
Additional information
Research supported by the “Deutsche Forschungsgemeinschaft” through the project DECOMP.
Rights and permissions
About this article
Cite this article
Horst, R., Thoai, N.V. An integer concave minimization approach for the minimum concave cost capacitated flow problem on networks. OR Spektrum 20, 47–53 (1998). https://doi.org/10.1007/BF01545530
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01545530