Showing papers on "Concave function published in 1968"

Minimum Concave Cost Flows in Certain Networks

Abstract: The literature is replete with analyses of minimum cost flows in networks for which the cost of shipping from node to node is a linear function. However, the linear cost assumption is often not realistic. Situations in which there is a set-up charge, discounting, or efficiencies of scale give rise to concave functions. Although concave functions can be minimized by an exhaustive search of all the extreme points of the convex feasible region, such an approach is impractical for all but the simplest of problems. In this paper some theorems are developed which explicitly characterize the extreme points for certain single commodity networks. By exploiting this characterization algorithms are developed that determine the minimum concave cost solution for networks with a single source and a single destination, for acyclic single source multiple destination networks, and for acyclic single destination multiple source networks. An interesting theorem then establishes that for either single source or single destination networks the multi-commodity case can be reduced to the single commodity case. Applications to the concave warehouse problem, a single product production and inventory model, a multi-product production and inventory model, and a plant location problem are included.

A minimax control problem for sampled linear systems

Abstract: A linear differential system is subject to a bounded control and a bounded disturbance. The controller receives the value of the state at a finite number of fixed sampling times. The cost is a convex or concave function of the state at a fixed final time. Given any control law, there is a maximum cost over all perturbations, the guaranteed performance for this control law. It is desired to find the minimum of this number over all control laws. Fenchel's theory of conjugate convex functions is used to set up an algorithm dual to the dynamic programming approach. This algorithm deals with the support functions of reachable sets, more easily determined than other descriptions of these sets. A discrete minimax principle is generally incorrect for problems of this class.

Optimal inventory policy with multiple set-up costs

Abstract: We consider a deterministic, single product, discrete review, finite time horizon inventory problem, called the multiple set-up cost problem. The holding cost in each period is a nondecreasing and sometimes concave function. The distinguishing feature of our model is the ordering cost function which is neither concave nor convex. The ordering cost is such that a natural interpretation of it consists in assuming that the order in period i is delivered in trucks with capacity Mi and that the cost of delivery for each truck is a nondecreasing concave function of the amount delivered by that truck. We establish the existence of an optimal production schedule such that for each period 1 there are no partially filled trucks in period i if the inventory entering period i is positive and 2 the inventory at the end of period i is less than Mi. Exploiting this information, an efficient algorithm is developed. In part II, we study the stationary, infinite horizon version of the multiple set-up cost problem. We single out a countable set S of schedules, each of which possess a periodic property in addition to properties 1 and 2 above, and we show that S contains a schedule with minimal cost per unit time. Moreover, if the ratio of demand per period to Mi is rational, then S contains a schedule with minimal discounted cost.

Programming problems with convex fractional functions

Abstract: The main result proved in this paper is that the ratio of the square of a nonnegative convex function to a strictly positive concave function is convex over a convex domain. Some particular cases of this result and a few applications to mathematical programming are also considered.

Investment Behaviour with Utility a Concave Function of Wealth

Abstract: In its simplest terms this paper is concerned with the following problem. An individual is confronted with two assets: (1) a bond for which the payoff at time t + 1 per dollar invested at time t is certain and equal to or greater than one; and (2) a share of stock for which the payoff at t + 1 per dollar invested at t is a random variable with a known distribution and an expected value greater than the bond’s payoff. The individual’s wealth less his investment in the share is invested in the bond. Investment in the bond may be negative, that is, he may go in debt. How will the individual allocate his wealth between the two assets if his utility is a concave function of his wealth and his objective is the maximization of the expected value of his utility one period hence?