Abstract
The stochastic transportation problem can be formulated as a convex transportation problem with nonlinear objective function and linear constraints. We compare several different methods based on decomposition techniques and linearization techniques for this problem, trying to find the most efficient method or combination of methods. We discuss and test a separable programming approach, the Frank-Wolfe method with and without modifications, the new technique of mean value cross decomposition and the more well known Lagrangean relaxation with subgradient optimization, as well as combinations of these approaches. Computational tests are presented, indicating that some new combination methods are quite efficient for large scale problems.
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M. S. Bazaraa and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, John Wiley & Sons: New York, 1979.
J.F. Benders, “Partitioning Procedures for Solving Mixed-Variables Programming Problems,” Numerische Matematik, Vol. 4, pp. 238–252, 1962.
D. P. Bertsekas and P. Tseng, “Fortran Codes for Network Optimization: The Relax Codes for Linear Minimum Cost Network Flow Problems,” Annals of Operations Research, Vol. 13, pp. 125–190, 1988.
P.M. Camerini, L. Fratta, and F. Maffioli, “On Improving Relaxation Methods by Modified Gradient Techniques,” Mathematical Programming Study, Vol. 3, pp. 26–34, 1975.
L. Cooper and L.J. LeBlanc, “Stochastic Transportation Problems and Other Network Related Convex Problems,” Naval Research Logistics Quarterly, Vol. 24, pp. 327–336, 1977.
H. Crowder, “Computational improvements for subgradient optimization,” in Symposia Mathematica, Vol. XIX, pp. 357–372, Academic Press: London, 1976.
G.B. Dantzig and P. Wolfe, “Decomposition Principle for Linear Programs,” Operations Research, Vol. 8, pp. 101–111, 1960.
S.E. Elmaghraby, “Allocation Under Uncertainty when the Demand has Continuous D.F.,” Management Science, Vol. 6, pp. 270–294, 1960.
M. Frank and P. Wolfe, “An Algorithm for Quadratic Programming,” Naval Research Logistics Quarterly, Vol. 3, pp. 95–110, 1956.
M. Held, P. Wolfe, and H.P. Crowder, “Validation of Subgradient Optimization,” Mathematical Programming, Vol. 6, pp. 62–88, 1974.
K. Holmberg, “Separable programming applied to the stochastic transportation problem,” Research Report LiTH-MAT-R-1984-15, Department of Mathematics, Linköping Institute of Technology, Sweden, 1984.
K. Holmberg, “Linear Mean Value Cross Decomposition: A Generalization of the Kornai-Liptak Method,” European Journal of Operational Research, Vol. 62, pp. 55–73, 1992.
K. Holmberg, “Computational tests of decomposition and linearization methods for the stochastic transportation problem,” Working Paper LiTH-MAT/OPT-WP-1993-01, Optimization, Department of Mathematics, Linköping Institute of Technology, Sweden, 1993.
K. Holmberg, “A Convergence Proof for Linear Mean Value Cross Decomposition”, Zeitschrift für Operations Research, Vol. 39, No. 2, 1994.
K. Holmberg and K. Jörnsten, “Cross Decomposition Applied to the Stochastic Transportation Problem,” European Journal of Operational Research, Vol. 17, pp. 361–368, 1984.
J. Kornai and T. Liptak, “Two-Level Planning,” Econometrica, Vol. 33, pp. 141–169, 1965.
T. Larsson and A. Migdalas, “An Algorithm for Nonlinear Programs over Cartesian Product Sets,” Optimization, Vol. 21, pp. 535–542, 1990.
L.J. LeBlanc, R.V. Helgason, and D.E. Boyce, “Improved efficiency of the Frank-Wolfe algorithm for convex network programs,” Transportation Science, Vol. 19, pp. 445–462, 1985.
M. Patriksson, “Partial linearization in nonlinear programming,” Research Report LiTH-MAT-R-1991-11, Department of Mathematics, Linköping Institute of Technology, Sweden, 1991. Accepted for publication in JOTA.
B.T. Poljak, “A General Method of Solving Extremum Problems,” Soviet Mathematics Doklady, Vol. 8, pp. 593–397, 1967.
B.T. Poljak, “Minimization of Unsmooth Functionals,” USSR Computational Mathematics and Mathematical Physics, Vol. 9, pp. 14–29, 1969.
L. Qi, “Forest Iteration Method for Stochastic Transportation Problem,” Mathematical Programming Study, Vol. 25, pp. 142–163, 1985.
J. Sun, K.-H. Tsai, and L. Qi, “A simplex method for network programs with convex separable piecewise linear costs and its application to stochastic transshipment problems,” in D.-Z. Du and P. Pardalos (Eds.), Network Optimization Problems, pp. 283–300, World Scientific Publishing Co.: Singapore, 1993.
T.J. Van Roy, “Cross Decomposition for Mixed Integer Programming,” Mathematical Programming, Vol. 25, pp. 46–63, 1983.
K. Vlahos, “Convergence proof for mean value cross decomposition applied to convex problems,” Research paper, Decision Science Group, London Business School, England, 1991.
A. Weintraub, C. Ortiz, and J. Gonzáles, “Accelerating Convergence of the Frank-Wolfe Algorithm,” Transportation Research, Vol. 19, pp. 113–122, 1985.
A.C. Williams, “A Stochastic Transportation Problem,” Operations Research, Vol. 11, pp. 759–770, 1963.
D. Wilson, “An a Priori Bounded Model for Transportation Problems with Stochastic Demand and Integer Solutions,” AIIE Transactions, Vol. 4, pp. 186–193, 1972.
D. Wilson, “Tighter Bounds for Stochastic Transportation Models,” AIIE Transactions, Vol. 5, pp. 180–185, 1973.
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Holmberg, K. Efficient decomposition and linearization methods for the stochastic transportation problem. Comput Optim Applic 4, 293–316 (1995). https://doi.org/10.1007/BF01300860
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DOI: https://doi.org/10.1007/BF01300860