Abstract
The problem considered is as follows:m resources are to be allocated ton activities, with resourcei contributing linearly to the potential for activityj according to the coefficientE(i,j). The objective is to minimize some nonlinear function of the potentials. If the objective function is sufficiently well behaved, the problem can be solved in finitely many steps using the method described in this paper.
Similar content being viewed by others
References
Koopman, B.,Theory of Search, Part 2, Operations Research, Vol. 4, pp. 519–521, 1956.
Washburn, A.,Note on Constrained Maximization of a Sum, Operations Research, Vol. 29, pp. 411–414, 1981.
Clasen, R., Graves, G., andLu, J.,Sortie Allocation by a Nonlinear Programming Model for Determining a Munitions Mix, RAND Report R-1411-DDPAE, 1974.
Bausch, D., andBrown, G.,NDP Fortran and Phar Lap Tools, Interfaces, Vol. 15, pp. 20–25, 1988.
James, R., andShaw, J.,Iline:An Iterative Algorithm for the Weapon Assignment Problem, 28th ORSA/TIMS Meeting, New York, New York, 1989.
Bazaraa, M., andShetty, C.,Nonlinear Programming: Theory and Algorithms, Wiley, New York, New York, 1975.
Zangwill, W.,Nonlinear Programming: A Unified Approach, Prentice-Hall, Englewood Cliffs, New Jersey, Chapter 8, 1970.
Simmons, D.,Nonlinear Programming for Operations Research, Prentice-Hall, Englewood Cliffs, New Jersey, Chapter 8, 1975.
Von Hohenbalken, B.,Simplicial Decomposition in Nonlinear Programming Problems, Mathematical Programming, Vol. 13, pp. 49–68, 1977.
Hearn, D., Lawphongpanich, S., andVentura, J.,Restricted Simplicial Decomposition: Computation and Extensions, Mathematical Programming Study, Vol. 31, pp. 99–118, 1987.
Ahlfeld, D., Dembo, R., Mulvey, J., andZenios, S.,Nonlinear Programming on Generalized Networks, ACM Transactions on Mathematical Software, Vol. 13, pp. 350–367, 1987.
Rockafellar, R.,Network Flows and Monotropic Optimization, Wiley, New York, New York, Chapter 11, 1984.
Ibaraki, T., andKatoh, N.,Resource Allocation Problems, MIT Press, Cambridge, Massachusetts, 1988.
Charnes, A., andCooper, A.,The Theory of Search: Optimum Distribution of Search Effort, Management Science, Vol. 5, pp. 44–50, 1958.
Bakhtin, I., Krasnosel'skii, M., andLevin, A.,Finding the Extremum of a Function on a Polyhedron, USSR Computational Mathematics and Mathematical Physics, Vol. 3, pp. 533–546, 1963.
Lebedev, S.,A Finite Method for Solving Nonlinear Transportation Problems, Ekonomika i Matematica Metody, Vol. 1, pp. 71–82, 1965.
Danskin, J.,The Theory of Max-Min, Springer Verlag, New York, New York, pp. 85–100, 1967.
Einbu, J.,Optimal Allocations of Continuous Resources to Several Activities with a Concave Return Function: Some Theoretical Results, Mathematics of Operations Research, Vol. 3, pp. 82–88, 1978.
Berge, C.,The Theory of Graphs, Wiley, New York, New York, p. 152 1962.
Bazaraa, M., andJarvis, J.,Linear Programming and Network Flows, Wiley, New York, New York, 1977.
Bradley, G., Brown, G., andGraves, G.,Design and Implementation of Large-Scale Primal Transshipment Problems, Management Science, Vol. 24, pp. 1–34, 1977.
Washburn, A.,Search for a Moving Target: The FAB Algorithm, Operations Research, Vol. 31, pp. 739–751, 1983.
Brooke, A., Kendrick, D., andMeeraus, A.,Gams: A User's Guide, Scientific Press, Redwood City, California, 1988.
Dantzig, G., Folkman, J., andShapiro, N.,On the Continuity of the Minimum Set of a Continuous Function, Journal of Mathematical Analysis and Applications, Vol. 17, pp. 519–548, 1967.
Author information
Authors and Affiliations
Additional information
Communicated by D. G. Luenberger
The author thanks S. Toi Lawphongpanich for several helpful references and suggestions.
Rights and permissions
About this article
Cite this article
Washburn, A.R. Finite method for a nonlinear allocation problem. J Optim Theory Appl 85, 705–726 (1995). https://doi.org/10.1007/BF02193063
Issue Date:
DOI: https://doi.org/10.1007/BF02193063