Abstract
We prove that the Minimum Concave Cost Network Flow Problem with fixed numbers of sources and nonlinear arc costs can be solved by an algorithm requiring a number of elementary operations and a number of evaluations of the nonlinear cost functions which are both bounded by polynomials inr, n, m, wherer is the number of nodes,n is the number of arcs andm the number of sinks in the network.
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References
Bellman, R.E. (1958), On a routing problem,Quart. Appl. Math. 16, 87–90.
Du, D.-Z. and P.M. Pardalos (eds.) (1993),Network Optimization Problems, World Scientific.
Ericksson, R.E., C.L. Monma, and A.F. Veinott (1987), Send- and-split method for minimum concave-cost network flows,Mathematics for Operations Research 12, 634–664.
Florian, M. and P. Robillard (1971), An implicit enumeration algorithm for the concave cost network flow problem,Management Science 18, 184–193.
Fredman, M.L. and R.E. Tarjan (1984), Fibonaccy heaps and their uses in improved network optimization algorithms,Proc. 25th IEEE Sympos. Foundations Computer Sci, 338–346.
Gallo, G., C. Sandi, and C, Sodini (1980), An algorithm for the min concave cost flow problem,European Journal of Operations Research 4, 248–259.
Gallo, G. and C. Sodini (1979), Adjacent extreme flows and applications to min concave-cost flow problems,Networks 9, 95–121.
Guisewite, G. and P.M. Pardalos (1990), Minimum concave-cost network flow problems: Applications, complexity and algorithms,Annals of Operations Research 25, 75–100.
Guisewite, G. and P.M. Pardalos (1991), Algorithms for the single source uncapacitated minimum concave-cost network flow problem,Journal of Global Optimization 1, 245–265.
Guisewite, G. and P.M. Pardalos (1992), A polynomial time solvable concave network flow problem,Network 23, 143–147.
Guisewite, G. and P.M. Pardalos (1993), Complexity issues in nonconvex network flow problems, inComplexity in Numerical Optimization, ed. P.M. Pardalos, World Scientific, 163–179.
Holmberg, K. and H. Tuy (1993), A production-transportation problem with stochastic demands and concave production costs, Preprint, Department of Mathematics, Linköping University. Submitted.
Klinz, B. and H. Tuy (1993), Minimum concave-cost network flow problems with a single nonlinear arc cost, inNetwork Optimization Problems, eds. P.M. pardalos and D.-Z. Du, World Scientific, 125–143.
Lenstra, H.W. Jr. (1983), Integer programming with a fixed number of variables,Mathematics of Operations Research 8, 538–548.
Minoux, M. (1989), Network synthesis and optimum network design problems: models, solution methods and applications,Networks 19, 313–360.
Nemhauser, G.L. and L.A. Wolsey (1988),Integer and Combinatorial Optimization, John Wiley & Sons, New York.
Pardalos, P.M. and S.A. Vavasis (1992), Open questions in complexity theory for nonlinear optimization,Math. Prog. 57, 337–339.
Tardos, E. (1985), A strongly polynomial minimum cost circulation algorithm,Combinatorika 5, 247–255.
Thach, P.T. (1987), A decomposition method for the min concave-cost flow problem with a special structure, Preprint, Institute of Mathematics, Hanoi.
Thach, P.T. (1991), A dynamic programming method for min concave-cost flow problems on circuitless single source uncapacitated networks, Preprint, Institute of Mathematics, Hanoi.
Tuy, H. (1992), The complementary convex structure in global optimization,Journal of Global Optimization 2, 21–40.
Tuy, H. and B.T. Tam (1992), An efficient solution method for rank two quasiconcave minimization problems,Optimization 24, 43–56.
Tuy, H., N.D. Dan, and S. Ghannadan (1993), Strongly polynomial time algorithm for certain concave minimization problems on networks,Operations Research Letters 14, 99–109.
Tuy, H., S. Ghannadan, A. Migdalas, and P. Värbrand (1993), Strongly polynomial algorithm for a production-transportation problem with concave production cost,Optimization 27, 205–228.
Tuy, H., S. Ghannadan, A. Migdalas, and P. Värbrand (1993), Strongly polynomial algorithms for two special minimum concave-cost network flow problems,Optimization (to appear).
Tuy, H., S. Ghannadan, A. Migdalas, and P. Värbrand (1993), Strongly polynomial algorithm for a production-transportation problem with a fixed number of nonlinear variables,Mathematical Programming (to appear).
Veinott, A.F. (1969), Minimum concave-cost solution of Leontiev substitution models of multifacility inventory systems,Operations Research 17, 262–291.
Wagner, H.M., and T.M. Whitin (1959), Dynamic version of the economic lot size model,Management Science 5, 89–96.
Wallace, S.W. (1986), Decomposition of the requirement space of a transportation problem into polyhedral cones,Mathematical Programming 28, 29–47.
Zangwill, W.I. (1968), Minimum concave cost flows in certain networks,Management Science 14, 429–450.
Zangwill, W.I. (1969) A backlogging model and a multi-echelon model on a dynamic lot size production system—a network approach,Management Science 15, 509–527.
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On leave from Institute of Mathematics, P.O. Box 631, Bo Ho, Hanoi, Vietnam.
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Tuy, H., Ghannadan, S., Migdalas, A. et al. The Minimum Concave Cost Network Flow Problem with fixed numbers of sources and nonlinear arc costs. J Glob Optim 6, 135–151 (1995). https://doi.org/10.1007/BF01096764
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DOI: https://doi.org/10.1007/BF01096764