Abstract
In this paper, we propose a primal-dual algorithm for solving a class ofproduction-transportation problems. Among m(≥ 2) sources two factoriesexist, which produce given goods at some concave cost and supply them to nterminals. We show that one can globally minimize the total cost ofproduction and transportation by solving a Hitchcock transportation problemwith m sources and n terminals and a minimum linear-cost flow problem withm+n nodes. The number of arithmetic operations required by the algorithm ispseudo-polynomial in the problem input length.
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References
Ahuja, R.K., T.L. Magnanti and J.B. Orlin, Network flows: Theory, Algorithms and Applications, Prentice Hall, N.J., (1993).
Erickson, R.E., C.L. Monma and A.F. Veinott, “Send-and-split method for minimum-concave-cost network flows”, Mathematics of Operations Research 12(1987), 634–664.
Ford, L.R. and D.R. Fulkerson, Flows in Networks, Princeton University Press, N.J., (1962).
Fredman, M.L. and R.E. Tarjan, “Fibonacci heaps and their uses in improved network optimization algorithms”, Journal of ACM 34(1987), 596–615.
Gal, T., “Linear parametric programming —a brief survey”, Mathematical Programming Study 21(1984), 43–68.
Garey, M.S. and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman, N.Y., (1979).
Guisewite, G.M., “Network problems”, in R. Horst and P.M. Pardalos (eds.), Handbook of Global Optimization, Kluwer Academic Publishers (Dordrecht, 1995).
Guisewite, G.M. and P.M. Pardalos, “Minimum concave-cost network flow problems: applications, complexity and algorithms”, Annals of Operations Research 25(1990), 75–100.
Guisewite, G.M. and P.M. Pardalos, “A polynomial time solvable concave network flow problem”, Networks 23(1993), 143–149.
Horst, R. and H. Tuy, Global Optimization: Deterministic Approaches, Springer-Verlag, Berlin, (1990).
Iri, M., “A new method of solving transportation network problems”, Journal of the Operations Research Society of Japan 3(1960), 27–87.
Klinz, B. and H. Tuy, “Minimum concave-cost network flow problems with a single nonlinear arc cost”, Dungzhu Du and P.M. Pardalos (eds.), Network Optimization Problems, World Scientific, Singapore, (1993), 125–143.
Kuno, T. and T. Utsunomiya, “A decomposition algorithm for solving certain classes of production-transportation problems with concave production cost”, Journal of Global optimization 8(1996), 67–80.
Kuno, T. and T. Utsunomiya, “Minimizing a linear multiplicative-type function under network flow constraints”, Technical Report ISE-TR-95-124, Institute of Information Sciences and Electronics, University of Tsukuba, Ibaraki, (1995), to appear in Operations Research Letters.
Mangasarian, O.L., Nonlinear Programming, McGraw-Hill (N.Y., 1969).
Orlin, J.B., “A faster strongly polynomial minimum cost flow algorithm”, 20th ACM Symposium on Theory of Computing(1988), 377–387.
Pardalos, P.M. and S.A. Vavasis, “Quadratic programming with one negative eigenvalue is NP-hard”, Journal of Global Optimization 1(1991), 15–22.
Tuy, H., “The complementary convex structure in global optimization”, Journal of Global Optimization 2(1992), 21–40.
Tuy, H., N.D. Dan and S. Ghannadan, “Strongly polynomial time algorithms for certain concave minimization problems on networks”, Operations Research Letters 14(1993), 99–109.
Tuy, H., S. Ghannadan, A. Migdalas and P. Väbrand, “Strongly polynomial algorithm for a production-transportation problem with concave production cost”, Optimization 27(1993), 205–227.
Tuy, H., S. Ghannadan, A. Migdalas and P. Värbrand, “Strongly polynomial algorithm for a production-transportation problem with a fixed number of nonlinear variables”, Preprint, Department of Mathematics, Linköping, (1993) to appear in Mathematical Programming.
Tuy, H., S. Ghannadan, A. Migdalas and P. Värbrand, “The minimum concave cost network flow problems with fixed number of sources and nonlinear arc costs”, Journal of Global Optimization 6(1995), 135–151.
Tuy, H. and B.T. Tam, “An efficient solution method for rank two quasiconcave minimization problems”, Optimization 24(1992), 43–56.
Zangwill, W.L., “A deterministic multi-product, multi-facility production and inventory system”, Operations Research 14(1966), 486–508.
Zangwill, W.L., “A backlogging model and multi-echelon model of a dynamic economic lot size production system — a network approach”, Management Science 15(1969), 506–527.
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Kuno, T., Utsunomiya, T. A Pseudo-Polynomial Primal-Dual Algorithm for Globally Solving a Production-Transportation Problem. Journal of Global Optimization 11, 163–180 (1997). https://doi.org/10.1023/A:1008278625289
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DOI: https://doi.org/10.1023/A:1008278625289