Showing papers on "Concave function published in 1969"
TL;DR: A broad class of problems can be formulated as minimizing a concave function over the solution set of a Leontief substitution system, which includes deterministic single- and multi-facility economic lot size, lot-size smoothing, warehousing, product-assortment, batch-queuing, capacity-expansion, investment consumption, and reservoir-control problems with concave cost functions.
Abstract: The paper shows that a broad class of problems can be formulated as minimizing a concave function over the solution set of a Leontief substitution system. The class includes deterministic single- and multi-facility economic lot size, lot-size smoothing, warehousing, product-assortment, batch-queuing, capacity-expansion, investment consumption, and reservoir-control problems with concave cost functions. Because in such problems an optimum occurs at an extreme point of the solution set, we can utilize the characterization of the extreme points given in a companion paper to obtain most existing qualitative characterizations of optimal policies for inventory models with concave costs in a unified manner. Dynamic programming recursions for searching the extreme points to find an optimal point are given in a number of cases. We only give algorithms whose computational effort increases algebraically (instead of exponentially) with the size of the problem.
235 citations
44 citations
01 Sep 1969
TL;DR: In this article, the Fibonacci search technique for maximizing a unimodal function of one real variable is generalized to the case of a given first evaluation, which is then employed to determine the optimal sequential search for the maximization of a concave function.
Abstract: : The Fibonacci search technique for maximizing a unimodal function of one real variable is generalized to the case of a given first evaluation. This technique is then employed to determine the optimal sequential search technique for the maximization of a concave function.
5 citations
TL;DR: In this article, a new approach to mathematical programming problems based on a penalty function method is presented, where the given problem is replaced by a second auxiliary problem which can be solved by standard methods since it involves the maximization of a concave function of a single variable over an interval.
Abstract: This paper contains the mathematical validation of a new approach to mathematical programming problems based on a penalty function method. The given problem is replaced by a second „auxiliary“ problem which, in many cases may be solved by standard methods since it involves the maximization of a concave function of a single variable over an interval. The auxiliary problem is defined implicitly in therms of the constituents of the original problem. Examples are presented in order to illustrate the theoretical results.
2 citations