| Home | Need | Issues | Impediments | Principles | Pitfalls | Skills | Cases | Work | Pedagogy | Sites | People |

# Case Study: High Performance Manufacturing

A working draft of resources and reports from an NSF-sponsored project intended to strengthen the role of mathematics in Advanced Technological Education (ATE) programs. Intended as a resource for ATE faculty and members of the mathematical community. Comments are welcome by e-mail to the project directors: Susan L. Forman or Lynn A. Steen.

Today's internationally competitive manufacturing requires both high performance and rapid responsiveness. Customers expect products to be free of flaws, and marketplace innovation makes new products rapidly obselete. This means that production processes must be highly efficient yet thoroughly revamped every year or two.

Mathematical and statistical methods aid this enterprise in many ways:

• Statistical analysis transforms data gathered from manufacturing processes to forms meaningful for process and quality control.

• Mathematical modeling translates manufacturing problems into quantitative forms suitable for attack by computer methods.

Such methods assist in the design of materials (e.g., high-strength ceramics, poloymeric systems), in manufacturing processes (e.g., crystal growth, molding, joining, curing, and coating), and in the evolution of manufacuring methods (e.g., solid modeling, rapid prototyping, and molecular manufacturing). Mathematical methods are also used widely in management decision-making in manufacturing.

### Underlying Analysis

"Six Sigma," developed more than a decade ago at Motorola, is a statistical quality control method that some say "combines the art of the efficiency expert with the science of the computer geek." Technically, six sigma refers to the infinitesimally tiny number of errors (one in a billion) found six standard deviations way from the average in a normal error curve. Of course, no company can really achieve that kind of perfection. But like a religion, six sigma offers an ideal state toward which adherents continually strive. Big no-nonsense companies like Allied Signal, Motorola, and General Electric swear by it. "Six sigma has galvanized our company with an intensity the likes of which I have never seen in my forty years at G.E.", says G.E. chairman John F. Welch.

To achieve six sigma reliability the company breaks a customer's requirements into individual steps and then, based on how the systems interact, sets optimum specifications for each step to achieve the desired result. For example, if a customer wants to be billed on the same day each month, each step in the process (e.g. information transmission to billing, delays in mailing room) is analyzed to set performance specifications and provide backup for inevitable contingencies (illness, breakdown, etc.). Managers like six sigma since it promotes teamwork. "Six sigma gets people from all over the organization working together to improve the end product, not just their individual piece of it."

Techniques like six sigma are among the new manufacturing strategies promoted by the Advanced Integrated Manufacturing (AIM) Center Center in Dayton, Ohio. A partnership between Sinclair Community College and the University of Dayton, AIM is a "customer-driven" developer and provider of professional and educational services whose mission is to assist companies in implementing advanced manufacturing technologies, processes, and techniques. The AIM Center is supported by the ATE program to serve as a National Center for the purpose of (a) developing interdisciplinary curriculum materials for an associates degree in manufacturing engineering technology and (b) providing substantial faculty development opportunities. Two examples of their recent work illustrate the relevance of mathematical thinking to manufacturing processes:

• Whiteford Foods in Ohio produces over one million frozen beef patties per day for fast food outlets. The logisitics of managing the packaging and shipment of these beef patties had become a nightmare, so with the assistance of the nearby Advanced Integrated Manufacturing (AIM) Center, Whiteford developed a just-in-time inventory system for all the dry goods (spacing paper, plastic bags. shipping cartons) that are used in the manufacturing process. The Center then worked with Whiteford to help develop a strategic plan for placing equipment and channeling product flow in a more productive fashion. The result was a 50% decrease in the time required to move products along the various stages of the production process.

• McCauley Propeller Systems, a manufacturer of airplane propellers, design a new production facility that more than doubled productivity and reduced waste by over 60 per cent. They did this through a coordinated program involving computer simulations of various assembly line layouts and options followed by a full-scale mock-up that they used to test the simulations in a realistic work environment. A one-week workshop organized by the AIM Center introduced McCauley production workers to modern manufacturing concepts and let them work in the mock assembly room to suggest improvements.

### Curriculum Issues

One activity of the AIM Center is to develop an Associate of Science curriculum in manufacturing engineering technology that can, if carefully designed, fulfill the objectives of the various national standards in occupational skills (from NSSB), science (from NRC and AAAS), workforce preparation (from SCANS), mathematics (from NCTM and AMATYC), manufacturing, and engineering (from ABET). Mathematics is naturally embedded in applications that permeate such courses as Design; Quality Management; Systems, Controls, and Automation; Physical Sciences; and Production and Operations Management. This A.S. degree program presumes that students have completed the full "college-intending" recommendations of NCTM, including algebra, geometry, trigonometry, and data analysis. Thus the program can build on this foundation with courses in pre-calculus, calculus, probability, and statistics.

The curriculum for this degree is based on an understanding of learning that recognizes three distinct but related components required to support the contemporary view of high performance manufacturing:

• Authentic learning tasks provide particular examples, in depth, of tasks or activities with broad importance. These focused tasks, embedded in or based on real situations, provide integrated, hands-on experience in one specific context.
• Transfer of learning requires significant engagement with at least two different instances of the same idea or set of principles. Generally, two or more authentic tasks based on the same underlying concept (e.g., percentages, scheduling, standard deviation) are necessary to enable an individual to transfer to new domains ideas first learned in a different but specific context.
• Abstraction and generalization, the centerpiece of "academic" study, arise indirectly as consequences of repeated involvement with authentic learning tasks and collateral tasks designed to promote transfer of learning. Very few students learn to abstract or generalize by studying absract and general ideas. Most create generalizaions out of repeated contextualized tasks united by a common theme.
Thus, in this curriculum, mathematics, science, and technology are taught primarily through hands-on experiences. Some instructors worry that too much hands-on work enables students to delay making the transition in physics from pre-Newtonian intuition to more formal methods, or in mathematics from the specific to the general, from the concrete to the abstract. These issues are important not only for students who may transfer to four-year degree programs, but also for everyone who must now be prepared to change jobs, careers, and even their profession.

Most production workers in high performance manufacturing industries are now expected to have AA/AS degrees. Because of the rapid pace of change, industries can no longer afford to hire people to do only one thing. Early success with an associate's degree helps create the capacity for life long learning. Moreover, it establishes the potential for mobility into management. Moreover, in today's interconnected world, workers are often called on to explain production processes to outsiders (clients, customers, visitors, politicians). So understanding and communication skills are critically important for manufacturing employees who now occupy the front line in public relations for many companies.

### Supporting Mathematics

Some of the mathematical know-how used to improve manufacturing processes pertains to production techniques--measurement, calculation, alignment, tolerances, etc. Other aspects pertain to quality control--sampling, quality control charts, error estimates. Still others pertain to management of the production process itself--scheduling, inventory control, etc. The underlying mathematics involves topics in algebra (simultaneous equations), geometry (indirect measurement), finite mathematics (linear programming), calculus (optimization).

### References

Claudia H. Deutsch. "Six Sigma Enlightenment: Managers Seek Corporate Nirvana Through Qualilty Control." New York Times, Dec. 7, 1998, p C1.

| Top | Home | Case Studies | Geographic Information Systems (GIS) | Precision Agriculture |
| Global Positioning Systems (GPS) | High-Performance Manufacturing | Image Processing |

Supported by the Advanced Technological Educaiton (ATE) program at the National Science Foundation. Opinions and information on this site are those of the authors and do not represent the views of either the ATE program or the National Science Foundation.

Copyright © 1999.   Last Updated: October 12, 1999.   Comments to: Susan L. Forman or Lynn A. Steen.