Brian Gluss

Bio: Brian Gluss is an academic researcher from Illinois Institute of Technology. The author has contributed to research in topics: Parametric statistics & Exponential integrator. The author has an hindex of 6, co-authored 11 publications receiving 172 citations.

Further remarks on line segment curve-fitting using dynamic programming

Abstract: In a recent paper, Bellman showed how dynamic programming could be used to determine the solution to a problem previously considered by Stone. The problem comprises the determination, given N, of the N points of subdivision of a given interval (a, b and the corresponding line segments, that give the best least squares fit to a function g(x) in the interval. Bellman confined himself primarily to the analytical derivation, suggesting briefly, however, how the solution of the equation derived for each particular point of subdivision ui could be reduced to a discrete search. In this paper, the computational procedure is considered more fully, and the similarities to some of Stone's equations are indicated. It is further shown that an equation for u2 involving no minimization may be found. In addition, it is shown how Bellman's method may be applied to the curve-fitting problem when the additional constraints are added that the ends of the line segments must be on the curve.

On Various Versions of the Defective Coin Problem.

Abstract: The problem of ascertaining the minimum number of weighings which suffice to determine the defective coin in a set of N coins of the same appearance, given an equal arm balance and the information that there is precisely one defective coin present, is well known. A large number of ingenious solutions exist, some based upon sequential procedures and some not. The problem in the case where there are known to be two or more defective coins is far more complex because we cannot draw any simple definite conclusions at the end of a single test. We shall analyze this in detail in the following paper. The purpose of this paper is to indicate a systematic way in which the theory of dynamic programming can be used to provide a computational solution to the determination of optimal and suboptimal testing policies. We shall illustrate this by means of some numerical results obtained using a digital computer.

An alternative solution to the “lost at sea” problem

Abstract: A problem posed by Bellman and considered by Isbell is as follows: Suppose one is a mile from a straight shore with no means whatsoever of ascertaining its direction. What is the optimum path to follow so as to (a) minimize the maximum distance travelled in reaching shore, (b) minimize the statistical expectation of the distance travelled, or (c) maximize the probability of reaching shore within a given distance travelled? Isbell found the solution to (a); in the present paper a sequence of approximations to the optimal policy for (b) is considered.

Approximately optimal one-dimensional search policies in which search costs vary through time

Abstract: Consider a model in which there are N neighboring cells in one of which there is an object that it is required to find. The a priori probabilities of the object being in cells 1, …, N are p1, …, pN respectively, and the costs of examination of these cells are tl, …, tN respectively; the search policy is considered to be optimal when the statistical expectation of the total cost of the search is minimized. For the case in which the ti†s are constant throughout the search, an optimal policy solution has previously been found by Bellman and by Smith. In the present paper it is assumed that the costs comprise a travel cost dependent upon the distance from the last cell examined, in addition to a fixed examination cost: initially, assuming that the searcher is next to cell 1, ti = i + t, where t is constant; and from then onwards, assuming that the jth cell has just been examined, ti = | i - j | + t. An optimal search strategy is found in the case where the pi† s are all equal, and approximately optimal strategies in the case where pi is proportional to i. The latter case has application to defense situations where complete searches occur at successive intervals of time, and hence the enemy objects are thinned out the nearer they come to the defense base.

The minimax path in a search for a circle in a plane

Abstract: The problem has been considered by Isbell of determining the path that minimizes the maximum distance along the path to a line in the same plane whose distance from the starting point of the search is known, while its direction is unknown. We consider in this article the analogous problem for the search for a circle of known radius, of known distance from a starting point in its plane. It is further shown that Isbell's solution is a limiting case of the problem posed as the radius of the circle tends to infinity. The problem appears to be of some practical significance, since it is equivalent to that of searching for an object a given distance away which will be spotted when we get sufficiently close—that is, within a specific radius.

Searching in the Plane

Abstract: In this paper we initiate a new area of study dealing with the best way to search a possibly unbounded region for an object. The model for our search algorithms is that we must pay costs proportional to the distance of the next probe position relative to our current position. This model is meant to give a realistic cost measure for a robot moving in the plane. We also examine the effect of decreasing the amount of a priori information given to search problems. Problems of this type are very simple analogues of non-trivial problems on searching an unbounded region, processing digitized images, and robot navigation. We show that for some simple search problems, knowing the general direction of the goal is much more informative than knowing the distance to the goal.

Optimum Uniform Piecewise Linear Approximation of Planar Curves

Abstract: Two-dimensional digital curves are often uniformly approximated by polygons or piecewise linear curves. Several algorithms have been proposed in the literature to find such curves. We present an algorithm that finds a piecewise linear curve with the minimal number of segments required to approximate a curve within a uniform error with fixed initial and final points. We compare our optimal algorithm to several suboptimal algorithms with respect to the number of linear segments required in the approximation and the execution time of the algorithm.

The Status of Mathematical Inventory Theory

Abstract: This paper surveys the current status of mathematical inventory theory. The review is limited to studies which seek optimal policies for dynamic inventory models. Models with certain and uncertain demands are discussed. Particular attention is focused on multi-item and/or multi-echelon inventory systems.