Robert E. D. Woolsey

Aggregating diophantine equations

SummaryMathews [1897] has given a theorem for aggregating two diophantine equations with positive integer coefficients into a single equation that has the same solution set as its parents over the nonnegative integers. Building on this result,Elmaghraby andWig [1970] show how to shrink the inequality constraints of a bounded variable integer program to a single constraint equation. However, such applications are limited, as we show, by a greater than exponential growth in coefficient size as successive constraints are aggregated into one. To mitigate this situation, we give new theorems and implementation procedures that provide exponential order reductions in the coefficient growth attending the aggregation process.ZusammenfassungMathews [1897] hat ein Theorem zur Zusammenfassung zweier diophantischer Gleichungen mit positiven ganzzahligen Koeffizienten zu einer einzigen Gleichung mit derselben Lösungsmenge wie die beiden ursprünglichen Gleichungen entwickelt. Aufbauend auf dieses Ergebnis zeigtenElmaghraby undWig [1970] eine Möglichkeit, die Ungleichungen eines ganzzahligen Optimierungsproblems mit begrenzten Variablen sukzessive auf eine einzige Gleichung zu reduzieren. Die praktische Anwendbarkeit ist jedoch begrenzt. Bei der sukzessiven Zusammenfassung der Nebenbedingungen zu einer einzigen wachsen die Koeffizienten stärker als exponentiell an. Um diesen Nachteil zu mindern, werden hier neue Theoreme und Anwendungsprozeduren entwickelt. Diese gewährleisten, daß das Anwachsen der Koeffizienten im Verlaufe des Aggregationsprozesses um einen Faktor exponentieller Ordnung geringer ist.

OR Practice-Solving Complex Chemical Equilibria Using a Geometric-Programming Based Technique

Determining the composition of a chemical system at equilibrium is an important problem that arises in many fields of science and engineering. For complex equilibria, the use of a digital computer is required. The chemist often finds the use of current computer codes an inefficient and frustrating experience. This paper presents a globally convergent algorithm for the solution of chemical equilibrium problems. This algorithm has been made both efficient and easy to use and is now employed successfully by chemists on important problems. It is based on the application of geometric programming principles to solve systems of nonlinear equations. Although presented in a form that can be quickly understood by the practitioner, the methodology is mathematically rigorous. Computer programs that require no familiarity with the details of the methodology were developed and extensively tested. These programs can be used as general research tools for investigating the solution of nonlinear equations in areas other than chemistry.

The Fifth Column: on Doing Well by Doing Good and An Offer of Free Education

The author discusses his belief that an operations research program at a state university should actively look for project work to do pro bono publico for state, county, and city governments and for private-sector enterprises.

The Fifth Column: An Exercise in Allocating Federal Funds or Intimidating the Media with Integer Programming

Once upon a time, there was a name less county in Colorado that went through an annual exercise to allocate the funds rebated to the county by the federal government. Upon receipt of a large check from the feds, the county commissioners sent around a memo telling everybody how much money was available that year. It was but a short time before every ad ministrator of every section of county gov ernment appeared, wish list in hand, hop ing to soak up as much of the bounty as possible. The county commissioners went into executive session and, in the fullness of time, emerged with the list of that years winners and losers. There were fearful rumors that this process was not only performed in a less than optimum manner but was (gasp) politically moti vated. Every year the editor of the largest newspaper in the county looked forward to this exercise. This was because he took

Being wrong with Clarke & Wright

Abstract I received a letter from the distinguished Editor of this special issue asking me to submit a general article addressing the topic of “OR and the Rude Intrusions of the Real World”. He made this request to me because of my being noted for my “no-nonsense, practical bent”. In the article which follows, we will discover that I have managed to become the sage, nestor-like, statesman of the profession that I presently am by having blown it, big time, in my earlier excursions into reality.

A Novena to St. Jude, or Four Edifying Case Studies in Mathematical Programming

R. E. D. WOOLSEY is principal scientist in The Institute for Opera tions Research, and professor at the Colorado School of Mines. He is co-owner and operator of the Woolsey Ranch in Williamson County, Texas, and a consultant to Sandia Corporation and the Information Systems Division of General Electric Company in the general areas of operations research and mathematical/integer/geometric programming. He has previously held positions with Control Data Corporation and Sandia Corporation, where he authored several major research reports. His basic interest is the study of methods to solve real world problems.

AN ALGORITHM TO MINIMIZE SINGLE VARIABLE POLYNOMIAL, FUNCTIONS FROM ANY STARTING POINT WITH QUADRATIC CONVERGENCE

Search methods often display non-convergence or excessive convergence time on certain classes of nonlinear functions arising in engineering design. The authors will define a new geometric -programming based search method for single variable polynomials that displays quadratic convergence from any starting point Comparison over a group of test problems is made with a version of Newtons method

An analysis of a model of the MPS MX missile system using geometric programming

Abstract Six years ago, Mishins and Sarin defined and solved a cost optimization model of the MPS MX missile system using a computer (Appl. Math. Modelling 1985, 9, 139-142) . We show that this model can be solved virtually by inspection for a general case, using some recently published approaches of geometric programming.

OR Forum—The Keynote Speech

This note records the substance of a keynote speech recently given to a conference of people dealing with modeling problems and building systems to solve them, primarily in the military-industrial sector.

A Note on Kabak's “Some Aspects of Optimal Design”

Abstract Kabak in (1) considers a systems steady-state availability with mean uptime of U and mean downtime of D. He then formulates a model of total system cost of: as an approximation to the actual model which may be written as: The approximation is used because it is assumed that D << U. In the above formulas K 1 U a is the cost of equipment that has a mean uptime of U; and K 2 D −b is the cost of equipment that has a mean downtime of D. The downtime costs are assumed to be proportional to the downtime and are therefore given by the last term in the above equations. Kabak solves the first equation, (the approximation), with geometric programming because, he points out that the exact model “does not lend itself to solution by geometric programming.” However, this author asserts that the “necessary” conditions for the solution of any geometric program can show that the exact models geometric programming solution is tightly bounded and gives considerable information as to the effect of approximating the...