Abstract
In the well-known fixed-charge linear programming problem, it is assumed, for each activity, that the value of the fixed charge incurred when the level of the activity is positive does not depend upon which other activities, if any, are also undertaken at a positive level. However, we have encountered several practical problems where this assumption does not hold. In an earlier paper, we developed a new problem, called the interactive fixed-charge linear programming problem (IFCLP), to model these problems. In this paper, we show how to construct the convex envelopes and other convex underestimating functions for the objective function for problem (IFCLP) over various rectangular subsets of its domain. Using these results, we develop a specialized branch-and-bound algorithm for problem (IFCLP) which finds an exact optimal solution for the problem in a finite number of steps. We also discuss the main properties of this algorithm.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Erenguc, S. S., andBenson, H. P.,The Interactive Fixed Charge Linear Programming Problem, Naval Research Logistics Quarterly, Vol. 33, pp. 157–177, 1986.
Tuy, H.,Concave Programming under Linear Constraints, Soviet Mathematics, Vol. 5, pp. 1437–1440, 1964.
Taha, H. A.,Concave Minimization over a Convex Polyhedron, Naval Research Logistics Quarterly, Vol. 20, pp. 533–548, 1973.
Zwart, P. B.,Global Maximization of a Convex Function with Linear Inequality Constraints, Operations Research, Vol. 22, pp. 602–609, 1974.
Cabot, A. V.,Variations on a Cutting Plane Method for Solving Concave Minimization Problems with Linear Constraints, Naval Research Logistics Quarterly, Vol. 21, pp. 265–274, 1974.
Majthay, A., andWhinston, A.,Quasiconcave Minimization Subject to Linear Constraints, Discrete Mathematics, Vol. 9, pp. 35–59, 1974.
Rosen, J. B.,Global Minimization of a Linearly Constrained Concave Function by Partition of Feasible Domain, Mathematics of Operations Research, Vol. 8, pp. 215–230, 1983.
Benson, H. P.,A Finite Algorithm for Concave Minimization over a Polyhedron, Naval Research Logistics Quarterly, Vol. 32, pp. 165–177, 1985.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
Martos, B.,The Direct Power of Adjacent Vertex Programming Methods, Management Science, Vol. 12, pp. 241–252, 1965.
Falk, J. E., andHoffman, K. R.,A Successive Underestimation Method for Concave Minimization Problems, Mathematics of Operations Research, Vol. 1, pp. 251–259, 1976.
Todd, M. J.,The Computation of Fixed Points and Applications, Springer-Verlag, Berlin, Germany, 1976.
Al-Khayyal, F. A.,Jointly Constrained Bilinear Programming and Related Problems, Georgia Institute of Technology, School of Industrial and Systems Engineering, Report Series No. J-83-3, 1983.
Falk, J. E.,Lagrange Multipliers and Nonconvex Programs, SIAM Journal on Control, Vol. 7, pp. 534–545, 1969.
Benson, H. P., andErenguc, S. S.,Using Convex Envelopes to Solve the Interactive Fixed Charge Linear Programming Problem, University of Florida, Center for Econometrics and Decision Sciences, College of Business Administration, Discussion Paper No. 119, 1985.
Kao, E.,A Multi-Product Lot-Size Model with Individual and Joint Set-Up Costs, Operations Research, Vol. 27, pp. 279–289, 1979.
Author information
Authors and Affiliations
Additional information
Communicated by P. L. Yu
The authors would like to thank an anonymous referee for his helpful suggestions.
Rights and permissions
About this article
Cite this article
Benson, H.P., Erenguc, S.S. Using convex envelopes to solve the interactive fixed-charge linear programming problem. J Optim Theory Appl 59, 223–246 (1988). https://doi.org/10.1007/BF00938310
Issue Date:
DOI: https://doi.org/10.1007/BF00938310