Abstract
A partitioning algorithm for solving the general minimum cost multicommodity flow problem for directed graphs is presented in the framework of a network flow method and the dual simplex method. A working basis which is considerably smaller than the number of capacitated arcs in the given network is employed and a set of simple secondary constraints is periodically examined. Some computational aspects and preliminary experimental results are discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M.Bellmore, G.Bennington and S.Lubore, “Further extension of the tanker scheduling problem”, MTP-339. The Mitre Corporation, Washington, D.C., 1969.
S.P.Bradley, “Solution techniques for the traffic assignment problem”, ORC 65-35. Operations Research Center, University of California, Berkeley, 1965.
S.Clarke and J.Surkis, “An operations research approach to racial desegregation of school systems”,Socio-Econ. Plan. Sci. 1 (1968) 259–272.
R.J.Clasen, “RS MSUB-linear programming subroutine”. The RAND Corporation, Santa Monica, California, 1961.
G.B.Dantzig, “Upper bounds, secondary constraints, and block triangularity in linear programming”,Econometrica 23 (1955) 174–183.
G.B.Dantzig and P.Wolfe, “The decomposition algorithm for linear programs”,Econometrica 29 (1961) 767–778.
L.R.Ford and D.R.Fulkerson, “A suggested computation for maximal multi-commodity network flows”,Management Science 5 (1958) 97–101.
L.R.Ford and D.R.Fulkerson,Flows in networks (Princeton University Press, Princeton, 1963).
A.M.Geoffrion, “Elements of large-scale mathematical programming”,Management Science 16 (1970) 652–675.
M.D.Grigoriadis, “A dual generalized upper bounding technique”,Management Science 17 (1971) 269–284.
M.D.Grigoriadis and K.Ritter, “A decomposition method for structured linear and nonlinear programs”,Journal of Computer and System Sciences 3 (1969) 335–360.
M.D.Grigoriadis and W.F.Walker, “A treatment of transportation problems by primal partition programming”,Management Science 14 (1968) 565–599.
M.D. Grigoriadis and W.W. White, “Computational experience with a multicommodity network flow algorithm”, NATO Conference on Applications of Optimization Methods for Large-Scale Resource-Allocation Problems”, Elsinore, Denmark, 1971. To appear inOptimization methods for resource allocation, R. Cottle and J. Krarup, eds. (English Universities Press, 1972). Also, Scientific Center Report No. 320-3011, IBM Corporation, Philadelphia, Pa., 1972.
G.Hadley,Linear programming (Addison-Wesley, Reading, Mass., 1962).
J.K.Hartman and L.S.Lasdon, “A generalized upper bounding algorithm for multicommodity flow networks”, Technical Memorandum No. 193. Operations Research Department, Case Western Reserve University, Cleveland, Ohio, 1970.
T.C.Hu, “Multicommodity network flows”,Operations Research 11 (1963) 344–360.
IBM, “Mathematical programming system/360” (IBM Corporation, White Plains, New York, 1969).
W.S.Jewell, “A primal-dual multi-commodity flow algorithm”, ORC 66-24. Operations Research Center, University of California, Berkeley, 1966.
E.L.Johnson, “Networks and basic solutions”,Operations Research 14 (1966) 619–623.
L.S.Lasdon,Optimization theory for large systems (McMillan, New York, 1970).
C.E.Lemke, “The dual method of solving the linear programming problem”,Naval Research Logistics Quarterly 1 (1954) 36–47.
J.B.Rosen, “Primal partition programming for block diagonal matrices”,Numerische Mathematik 6 (1964) 250–260.
R.Saigal, “Multicommodity flowsin directed networks”, ORC 67-38. Operations Research Center, University of California, Berkeley, 1967.
M.Sakarovitch, “The multi-commodity maximum flow problem”, ORC 66-25. Operations Research Center, University of California, Berkeley, 1966.
M.Simonnard,Programmation lineaire (Dunod, Paris, 1962).
J.A.Tomlin, “Minimum-cost multicommodity network flows”,Operations Research 14 (1966) 45–51.
W.W.White and A.M.Bomberault, “A network algorithm for empty freight car allocation”IBM Systems Journal 8 (1969) 147–169.
W.W.White and E.Wrathall, “A system for railroad traffic scheduling”, IBM Scientific Center Technical Report 320-2993. Philadelphia, Pennsylvania, 1970.
R.D.Wollmer, “The Dantzig-Wolfe decomposition principle and minimum cost multicommodity network flows”, P-4191. The Rand Corp., Santa Monica, California, 1969.
W.I.Zangwill, “Concave cost flows in certain networks”,Management Science 14 (1968) 429–450.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Grigoriadis, M.D., White, W.W. A partitioning algorithm for the multicommodity network flow problem. Mathematical Programming 3, 157–177 (1972). https://doi.org/10.1007/BF01584987
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01584987