Abstract
We consider the problem of sending flow from a source to a destination, where there are flow costs on each arc and fixed costs toward the purchase of capacity. Capacity can be purchased in batches of C units on each arc. We show the problem to be NP-hard in general. If d is the quantity to be shipped from the source to the destination, we give an algorithm that solves the problem in time polynomial in the size of the graph but exponential in \(\left\lfloor {\tfrac{d}{C}} \right\rfloor\). Thus for bounded values of \(\left\lfloor {\tfrac{d}{C}} \right\rfloor\)the problem can be solved in polynomial time. This is useful since a simple heuristic gives a very good approximation of the optimal solution for large values of \(\left\lfloor {\tfrac{d}{C}} \right\rfloor\). We also show a similar result to hold for the case when there are no flow costs but capacity can be purchased either in batches of 1 unit or C units. The results characterizing optimal solutions are used to obtain extended formulations in each of the two cases. The LP-relaxations of the extended formulations are shown to be stronger than the natural formulations considered by earlier authors, even with a family of strong valid inequalities added.
Preview
Unable to display preview. Download preview PDF.
Bibliography
J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, North Holland, Amsterdam (1976).
S. Chopra, D. Bienstock, O. Günlük, C.Y. Tsai,“Minimum cost capacity installation for multicommodity network flows,” Research Report, Northwestern University, January 1995.
M.R. Garey and D.S. Johnson, Computers and Intractability: A guide to the Theory of NP-Completeness, W.H. Freeman and Company, New York (1979). 4. J.M.Y. Leung, T.L. Magnanti and R. Vachani, “Facets and algorithms for capacitated lot sizing,” Mathematical Programming, 45, 331–359.
T.L. Magnanti and P. Mirchandani, “Shortest paths, single origin-destination network design and associated polyhedra,” Networks, Vol. 23, No. 2 (1993) 103–121.
T.L. Magnanti, P. Mirchandani, and R. Vachani, “Modeling and solving the two facility capacitated network loading problem,” Operations Research, Vol. 43, No. 1 (1995) 142–157.
Y. Pochet and L.A. Wolsey, “Lot sizing with constant batches: Formulation and valid inequalities,” Mathematics of Operations Research, 18 (1993) 767–785.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1996 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chopra, S., Gilboa, I., Sastry, S.T. (1996). Algorithms and extended formulations for one and two facility network design. In: Cunningham, W.H., McCormick, S.T., Queyranne, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 1996. Lecture Notes in Computer Science, vol 1084. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61310-2_4
Download citation
DOI: https://doi.org/10.1007/3-540-61310-2_4
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61310-7
Online ISBN: 978-3-540-68453-4
eBook Packages: Springer Book Archive