Abstract
Transportation and assignment models have been widely used in many applications. Their use was motivated, among other reasons, by the existence of efficient solution methods and their occurrence as sub-problems in the solution of combinatorial problems. A previous study [10] observed that, in large-scale Transportation and Assignment Problems, 95 percent of the pivots were zero or degenerate pivots. This study investigates the ratio of zero pivots to the total number of pivots and verifies the above observation under conditions of small rim variability. Rules are introduced that pay special attention to the zero pivot phenomenon, and significantly reduce CPU time in both phase-1 (generating the initial basic feasible solution) and in phase-2 (selecting the variable leaving the base and the variable entering the base). When these rules were applied, they reduced the CPU time substantially: a 500×500 assignment problem was solved in 1.3 seconds.
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References
M.L. Balinski, “On two special classes of transportation polytopes”,Mathematical Programming Study 1 (1974) 43–58.
M. Bellmore and G. Nemhauser, “The travelling salesman problem: a survey”,Operations Research 16 (1968) 538–550.
Bushnell, R. and G. Venkatramn, “The optimum solution to a school bus scheduling problem”, Wayne State University, Detroit, MI, (1973).
A. Charnes, F. Glover, D. Karney, D. Klingman and J. Stutz, “Past, present and future of development, computational efficiency and practical use of large scale transportation and transhipment computer codes”, Research Rep. CS 131, Center for Cybernetic Studies, University of Texas, Austin (1973).
L. Cooper, “The transportation-location problem”,Operations Research (1972) 94–108.
R.D. Davis, “On the delivery problem and some related topics”, Doctoral dissertation, Northwestern University (1968).
G.B. Dantzig,Linear programming and extensions (Princeton University Press, Princeton, NJ 1963).
E.W. Dijkstra, “A note on two problems in connection with graphs”,Numerische Mathematik 1 (1959) 269–271.
M. Florian and M. Klein, “An experimental evaluation of some methods of solving the assignment problem”,Canadian Operating Research 8 (1970) 101–106.
B. Gavish and P. Schweitzer, “An algorithm for combining truck trips”,Transportation Science 8 (1974) 13–23.
A. Geoffrion and G. Graves, “Multicommodity distribution system design by benders decomposition”, Working Paper No. 109, Western Management Science Institute, University of California, Los Angeles, (1973).
F. Glover and D. Klingman, “Locating stepping-stone paths in distribution problems via the predecessor index method”,Transportation Science 4 (1970) 220–225.
F. Glover, D. Karney, D. Klingman and A. Napier, “A computation study on start procedures; basic change criteria and solution algorithms for transportation problems”,Management Science 20 (1974) 793–813.
F. Glover, D. Karney and J. Stutz, “The augmented threaded index method”, Research Rept. CS 144, Center for Cybernetic Studies, The University of Texas, Austin (1973).
F. Glover and D. Klingman, “Real world applications of network related problems and breakthroughs in solving them efficiently”,ACM Transactions on Mathematical Software 1 (1) (1975) 47–55.
R.S. Hatch, “Optimization strategies for large scale assignment and transportation problems”, DSA, Rockville, Maryland (1974).
D. Karney and D. Klingman, “Implementation and computational study on an in-core out-of-core primal network code”, Research Rept. CS 158, Center for Cybernetic Studies, University of Texas, Austin (1973).
S. Lee, “An experimental study of the transportation algorithms”, Master's Thesis, Graduate School of Business, University of California, Los Angeles (1968).
K.G. Murty, “Solving the fixed charge transportation problem by ranking the extreme points”,Operations Research (16) (1968) 268–279.
V. Srinivasan and G.L. Thompson, “Benefit-cost analysis of coding techniques for the primal transportation algorithm”,Journal of the ACM 20 (2) (1973) 194–213.
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Part of this research was done while the author was at IBM Israel Scientific Center.
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Gavish, B., Schweitzer, P. & Shlifer, E. The zero pivot phenomenon in transportation and assignment problems and its computational implications. Mathematical Programming 12, 226–240 (1977). https://doi.org/10.1007/BF01593789
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DOI: https://doi.org/10.1007/BF01593789