# The Minimum Concave Cost Network Flow Problem with fixed numbers of sources and nonlinear arc costs

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## 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 in*r, n, m*, where*r* is the number of nodes,*n* is the number of arcs and*m* the number of sinks in the network.

### Key words

Minimum concave cost network flow fixed number of sources and nonlinear arc costs strongly polynomial algorithm, parametric method## Preview

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### References

- 1.
- 2.Du, D.-Z. and P.M. Pardalos (eds.) (1993),
*Network Optimization Problems*, World Scientific.Google Scholar - 3.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.Google Scholar - 4.Florian, M. and P. Robillard (1971), An implicit enumeration algorithm for the concave cost network flow problem,
*Management Science***18**, 184–193.Google Scholar - 5.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.Google Scholar - 6.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.Google Scholar - 7.Gallo, G. and C. Sodini (1979), Adjacent extreme flows and applications to min concave-cost flow problems,
*Networks***9**, 95–121.Google Scholar - 8.Guisewite, G. and P.M. Pardalos (1990), Minimum concave-cost network flow problems: Applications, complexity and algorithms,
*Annals of Operations Research***25**, 75–100.Google Scholar - 9.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.Google Scholar - 10.Guisewite, G. and P.M. Pardalos (1992), A polynomial time solvable concave network flow problem,
*Network***23**, 143–147.Google Scholar - 11.Guisewite, G. and P.M. Pardalos (1993), Complexity issues in nonconvex network flow problems, in
*Complexity in Numerical Optimization*, ed. P.M. Pardalos, World Scientific, 163–179.Google Scholar - 12.Holmberg, K. and H. Tuy (1993), A production-transportation problem with stochastic demands and concave production costs, Preprint, Department of Mathematics, Linköping University. Submitted.Google Scholar
- 13.Klinz, B. and H. Tuy (1993), Minimum concave-cost network flow problems with a single nonlinear arc cost, in
*Network Optimization Problems*, eds. P.M. pardalos and D.-Z. Du, World Scientific, 125–143.Google Scholar - 14.Lenstra, H.W. Jr. (1983), Integer programming with a fixed number of variables,
*Mathematics of Operations Research***8**, 538–548.Google Scholar - 15.Minoux, M. (1989), Network synthesis and optimum network design problems: models, solution methods and applications,
*Networks***19**, 313–360.Google Scholar - 16.Nemhauser, G.L. and L.A. Wolsey (1988),
*Integer and Combinatorial Optimization*, John Wiley & Sons, New York.Google Scholar - 17.Pardalos, P.M. and S.A. Vavasis (1992), Open questions in complexity theory for nonlinear optimization,
*Math. Prog.***57**, 337–339.Google Scholar - 18.Tardos, E. (1985), A strongly polynomial minimum cost circulation algorithm,
*Combinatorika***5**, 247–255.Google Scholar - 19.Thach, P.T. (1987), A decomposition method for the min concave-cost flow problem with a special structure, Preprint, Institute of Mathematics, Hanoi.Google Scholar
- 20.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.Google Scholar
- 21.Tuy, H. (1992), The complementary convex structure in global optimization,
*Journal of Global Optimization***2**, 21–40.Google Scholar - 22.Tuy, H. and B.T. Tam (1992), An efficient solution method for rank two quasiconcave minimization problems,
*Optimization***24**, 43–56.Google Scholar - 23.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.Google Scholar - 24.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.Google Scholar - 25.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)*.Google Scholar - 26.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)*.Google Scholar - 27.Veinott, A.F. (1969), Minimum concave-cost solution of Leontiev substitution models of multifacility inventory systems,
*Operations Research***17**, 262–291.Google Scholar - 28.Wagner, H.M., and T.M. Whitin (1959), Dynamic version of the economic lot size model,
*Management Science***5**, 89–96.Google Scholar - 29.Wallace, S.W. (1986), Decomposition of the requirement space of a transportation problem into polyhedral cones,
*Mathematical Programming***28**, 29–47.Google Scholar - 30.Zangwill, W.I. (1968), Minimum concave cost flows in certain networks,
*Management Science***14**, 429–450.Google Scholar - 31.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.Google Scholar

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