Mathematical Experiences in Business, Industry and Government

Mathematical Experiences in Business, Industry and Government

Applications of mathematics to projects in business, industry, and government (BIG) offer a wealth of exciting problems for mathematicians. A wonderful sampling of BIG topics was presented at the MAA Contributed Paper session entitled ?Mathematical Experiences in Business, Industry, and Government,? during the Joint MAA?AMS meetings in Atlanta. This article discusses highlights of the BIG projects presented at the session. The paper session was sponsored by the Business, Industry and Government Special Interest Group of the MAA (BIG SIGMAA), and was organized by Phil Gustafson of Mesa State College and Michael Monticino of the University of North Texas.

The United States Military Entrance Processing Command (MEPCOM) recently conducted an A-Z Business Practice Review to improve the efficiency of the organization. Tyge Rugenstein and Darrall Henderson of the US Military Academy have worked with the Military Entrance Processing Station (MEPS) Location Optimization Study (MLOS). This study attempts to find the optimal distribution of MEPS to support the needs of the thousands of recruiters for the four military services, Coast Guard, and the FBI. At the US Military Academy, math and operations research majors addressed a reduced version of this problem with a simple but robust 0-1 integer programming model. Since the students completed the project, MEPCOM has prepared the model for full implementation by obtaining the endorsement of all the services, confirming the minimizing function and constraints. The partnership between MEPCOM and USMA continues to grow and provide an excellent venue for both faculty and students to study and solve relevant problems.

Supply chain groups are often faced with questions such as how much inventory should be stocked, what happens to manufacturing capacity if actual sales exceed predictions, or how much product x is expected to sell next month to customer y. The mathematics that is used to get these answers is embedded in a variety of systems running on laptops, desktops, and servers. Charlie Stevens of General Electric discussed some of the specific mathematical tools used daily within the GE Plastics Materials Management group that addresses several of the above questions. Three specific tools discussed were: (1) Manugistics ? a tool used to forecast sales demand using Fourier and Lewandowski methods, (2) SAILS ? an optimization engine system that allows us to determine the minimum distribution costs that can be achieved in a supply chain distribution network of plants, warehouses and customer locations and (3) Crystal Ball ? a Monte Carlo simulation tool that allows us to study the impact to customer service based on a variety of probability distributions for demand and manufacturing capabilities.

Greg Coxson of Lockheed Martin Maritime Systems and Sensors discussed Costas arrays, which originated in sonar waveform studies. Costas arrays are permutation matrices with the property that any nontrivial translation in rows and columns, with no wrap-around, results in no more than one 1 on top of another 1. There appear to be two types of arrays ? those formed by a set of generators, and ?spurious? ones requiring an exhaustive search to find. Greg was part of a small industry-university team consisting of NJIT Mount Laurel students and Lockheed Martin engineers who sought to advance the array orders for which the full set is known. When the work began, full sets of Costas Arrays were known up to order 23. The team achieved its initial goal of finding the full list for order 24, and then continued on to order 25; it is well on the way to compiling the full list for order 26. The trend in number of Costas arrays as a function of order, based on exhaustive search, indicates a steep drop to zero by order 30, yet arrays can be generated in large numbers for arbitrarily high orders using number-theoretic generators. The team exploited array symmetries, and the use of sophisticated computing methods, including the formation of a ?virtual cluster? by networking individual computers and the optimization of low-level computations.

Marcus Pendergrass of Convergent Corporation presented a channel model for short-range ultrawideband (UWB) radio channels. Development of UWB devices for consumer electronics, tracking, positioning, automobile collision avoidance, and a host of other applications, is expected to lead to products that will appear on the market in 2005. The model Marcus discussed was developed through an industrial/academic collaboration under the auspices of the IEEE 802.15.3a standards task group, for the purpose of aiding in the development of a standard for UWB wireless personal area networks. The model is based upon the well-known Saleh-Valenzuela model, but with fading statistics modified to more accurately represent the phenomenology associated with ultrawideband signals. Marcus gave an overview of UWB technology and the associated standards, and then presented the mathematical channel model. He also discussed the process of tuning the channel model to fit several sets of measurement data that were collected as part of the model development process.

Gregory Battle of Morehouse College presented a Markov model to structure the trafficking of tasks in a given governmental office with three components. The Fleet Support Office of the Naval Personnel Research and Development Center was used as a typical case. The probabilities of predicted office states are represented from the matrix equation Q(k) = Q(0)Pk for a given period k using an empirically constructed transition matrix P. A long range prediction of various office states will be computed using a mean passage of time matrix M. Once the cost of these long-range predicted states (given as a probability of occurrence) are calculated per unit time (e.g. daily, or weekly), then a cost equation can be formulated. The model allows managers to assign tasks to a particular office component that predicts functional office states at acceptable productivity rates with more efficient use of manpower resources resulting in an optimization of costs per time interval of budgeting.

Abbe Herzig of the University of Albany, State University of New York, discussed some of the statistical challenges she faced as a statistician in the product-testing division of Consumer Reports. Consumer Reports is one of the largest-circulation magazines in the U.S. It reports on consumer products and services, based on extensive research from a staff of engineers, scientists, survey researchers, statisticians, and journalists. Each product category tested for Consumer Reports presents a unique set of statistical requirements, involving a broad range of statistical design, data analysis, and presentation tools. Herzig presented some specific examples of product tests, including nutritional quality of breakfast cereals (cluster analysis) and food sensory tests (incomplete block designs and linear models). One popular food sensory example presented was a test of beer quality.

Jim Pomfret of Bloomsburg University described a recent project he and Youmin Lu conducted for an aluminum products manufacturer. This project estimated the effect on output of modifying the process flow in a chemical bathing and anodizing line. Jim described the input data provided by the company, the breakthrough in writing a Java simulation, and the analysis of results. Follow-up data from the company showed that actual operations agreed well with predictions.

Katharine Gurski of George Washington University spoke on ?The Effect of the Rayleigh Instability on Anisotropic Crystalline Rods,? a project she conducted along with G.B. McFadden at the National Institute of Standards and Technology and M.J. Miksis of Northwestern University. Properties of strength, toughness, ductility, and hardness of materials are greatly influenced by the conditions of solidification. Katharine studied the effect of anisotropic surface energy on the stability of a solidifying crystalline rod, both free-standing and on a surface. The study was partially motivated by the apparent stability of elongated nanowires in a bridge configuration or grown epitaxially on a heterogeneous surface. The rod was assumed to be smooth with a uniform cross-section given by a two-dimensional equilibrium shape. The investigators applied the analysis to examples with uniaxial or cubic anisotropy and showed that the anisotropic surface energy played a significant role in establishing the stability of the free standing rod or rod on a surface.

Nick Coult of Augsburg College discussed how the seismic exploration industry has need for sophisticated mathematical methods. As is often the case in industrial settings, the problems are not always clearly understood or explained at the outset of a project, and often the requirements and goals are fluid as the project proceeds. Nick described a particular project, travel-time data compression, in which some significant aspects of the problem were not known at the start. In particular, though travel-time data is relatively smooth and thus highly compressible using wavelet algorithms, it also often contains ?holes? or regions of missing data, which present a major obstacle for compression. Thus, the project evolved to include a significant new aspect ? developing a robust, efficient algorithm for smoothly filling in missing data to increase its compressibility using wavelet methods.

Tracy Bibelnieks of Augsburg College spoke on applications of linear, non-linear and integer programming, combined with statistical approaches, to analysis of customer behavior in direct mail and retail markets. Tracy explained how Fingerhut Corporation used basic linear programming methods to optimize the allocation of marketing/advertising dollars across segments of the customer database as well as integer programming to make catalog mailing decisions on a individual customer basis. This approach was a significant component of a larger project, which ultimately changed the way the company viewed and practiced campaign management for direct mail marketing. The method is now being used with other direct mail industries and in applications within the financial and e-commerce sectors. Tracy credits the IBM and Fingerhut teams for allowing the presentation of the applications.

In this article we have seen many exciting applications of mathematics to projects in business, industry, and government. In a variety of settings, mathematics is a key component to many important projects in the world around us. Who uses math? The answer includes many of the mathematicians, scientists, and engineers whose projects and products help improve the quality of our everyday lives.

Acknowledgement: The author gratefully appreciates the input provided by the speakers for the content appearing in this article, and for their participation in the paper session.

Phil Gustafson is Associate Professor of Mathematics at Mesa State College in Grand Junction, CO, and is Vice Chair for Programs for BIG SIGMAA.

Tyge Rugenstein: Tyge.Rugenstein@usma.edu
Charlie Stevens: charles.stevens@gepex.ge.com
Greg Coxson: gregory.e.coxson@lmco.com
Marcus Pendergrass: Marcus.Pendergrass@ConvergentAnalysis.com
Gregory Battle: KarmaNupe88@msn.com
Abbe Herzig: AHerzig@uamail.albany.edu
Jim Pomfret: Pomfret@bloomu.edu
Katharine Gurski: kgurski@gwu.edu
Nick Coult: coult@augsburg.edu
Tracy Bibelnieks: bibelnie@augsburg.edu