SRE logo Lambda Notes
VOLUME 32, ISSUE NO. 1, December 1999
A Publication of the Society of Reliability Engineers

TABLE OF CONTENTS

President's Message
From the Editor
Secretary's Report
SRE News and Information
History of SRE Part I
Washington Chapter History
Huntsville Chapter
Upcoming Huntsville Chapter Meetings

Presentations:

Reliability Impact for Scheduled Maintenance Intervals
Reliability of Means
Software Reliability and the Path to Improvement
Utilizing Simulations for Performing Degradation Trends Analyses
Stan Ofsthun Award 1999 Ofthsun Award Winner: "Simplifying the Solution of Redundanccy Allocation Problems"

SRE Officers
Membership Application



President's Message
Duane Cook

Each year I attend the RAM Symposium I come away impressed with the quality of work being done in the Reliability field. From Academia, to Government, to Corporate America, Reliability Engineers are developing new techniques, and more importanily, putting principle to work. I was intrigued by Richard Cassidy's SRE Award winning presentation. He, along with his students, developed an easy way to solve for the optimum solution to a Non-Linear Reliability unction, simply by performing repetitive calculations and sorting the results. The technique is straightforward, and readily applied to many different situations. I was pleased to present our Stan Ofsthun Award to Richard for this paper.

We had a pretty good turnout at our Board of Director's meeting. Our Secretary, George Chernowitz, has sent the minutes out to all of the chapters. One idea adopted was the Corporate Sponsorship program. As George finalizes the documentation, we will be sending out a package outlining it to the chapters. This is a good opportunity to get the companies in your area involved with our organization. Another area of interest is the revision of the By Laws, Constitution, and Operations Manual. Vice President Pat Larter is heading up this effort, but he will need the help of every chapter to make this revision happen.

Finally, please take a look at the new feature in Lambda Notes on the history of SRE. This organization has had a rich past. Judging from the caliber of members we have, I know it will have a wonderful future.



From The Editor
Richard Youngk

I would like to commend the RAMS organizations for the professional effort in coordinating the '99 RAMS in Washington D.C. It was my first and I found it a rewarding experience. The RAMS symposium also proved to be a great forum for distributing the LN. The RAMS attendees used nearly 200 copies distributed at the SRE table in the exhibit hall. I recenfly completed another very rewarding experience; the Reliability Engineering Masters Program at the University of Maryland. My congratulations and thanks Dr.'s Marvin Roush and Ah Mosleh for their leadership and expertise in running this excellent program.

This issue of Lambda Notes (LN) includes improved "Chapter News and Information" and has a robust history of the beginnings of SRE and some of its chapters.

Next issue, we will feature the Ottawa Chapter. Other chapters should begin to prepare for following issues by preparing technical articles which address reliability topics, chapter histories, list of upcoming events, current club member information and by maintaining club meeting summaries.

Thanks to Dave Dyrk at the Buffalo Chapter, several printing errors (missing pages) were identified early in the distribution of the LN January issue. As a result 100% inspection was conducted on the rest of the printing and with an approximate 5% reject rate.

We are coming along toward the regular publication of Lambda Notes. Hope you enjoy this "Double Issue".

Until the next issue



Secretary's Report
George Chernowitz

1. During this year, our objectives were to increase the "value" of our Society to our members and to the Society as a whole. They were as follows:

a. To stand-up a viable "At-Large Chapter".

b. See Lambda Notes as a regular publication. This is absolutely essential to a serious technical society; with a minimum of two issues in any one year, with an objective of four issues/year.

c. Establish a Corporate Associates Program which Industry and Government can join to support our efforts and demonstrate their commitment to high reliability products.

d. Establish a series of rewards for individuals who would build our organization.

e. Develop a reliable database for current and future members.

f. Recognize contributions to "growing SRE" by Chapter members.

2. The "At-Large Chapter" membership has gone extremely well. As of the end of November, we are truly "worldwide", from China to Australia to Taiwan, Russia and on. We have our first "Paper" (an excellent survey of reliability simulation software by Reid Willis) on chapter distribution, and there has been lively ~mail discussion among our members.

In this "Virtual Chapter" we are exercising the widely touted idea of "untethered" organizations, i.e., organizations which are not located in a specific geographical location, but constitute a "floating world" comprised of its participants. Our first year's experience is that there is obviously something lost by the lack of person-to-person and human contact; but at the same time, people who otherwise have no chance whatsoever to participate in a group activity now have potential of fruitful interchange with colleagues in our discipline.

Like any other Chapter, it is heavily dependent on individuals who are willing to make the time and devote the effort to keep things going1 and to collaborate with other members in a proactive rather than reactive way.

3. A second major objective was to see Lambda Notes "up and running" as an essential adjunct to a serious technical society. Requests were made to the Huntsville and Ottawa Chapters each to develop the technical content for a complete issue and they have done that handsomely. Additionally, effort was made to sell advertisements as a means of funding the publication. This, too, has moved, and as of November, we now have advertisements from two book publishers. Publication is now going forward.

4. Our Corporate Membership effort was initiated earlier this year with a direct approach to a major manufacturing organization that has a real interest in reliability.

A unique feature of the SRE Corporate Associates Program is the sharing of Corporate Membership dues with the Chapters with which the Corporate member affiliates. By providing resources to the Chapters, it forwards the SRE concept of "Chapter Orientation". As of the beginning of November, a brochure, and operating manual have been prepared and distributed to the various Chapters.

The concept of Corporate Membership as a means of supporting technical societies is well established. Indeed, many organizations maintain several levels of Corporate Memberships, ranging from sustaining members to "regular" members, often with a substantial level of dues. Our thought for SRE was to have extremely low Corporate Membership dues requirements, since as a technical society1 it is our interest to see an emphasis on reliability, not only for the continued growth of our national standard of living, but also to enhance recognition and support of the reliability discipline.

The jury is still out on the prospects for this effort, and whether or not it "goes" will depend almost entirely on efforts by the respective Chapters and their members. There is high payoff to all concerned, and it is worth our best efforts.

5. The SRE member database has been further developed and kept reasonably up to date, primarily through the addition of At-Large members and inputs from the active chapters.

6. To recognize the contributions made by individuals to the growth and success of our all our chapters, we announced a prize consisting of sets of RAMS Proceedings and a Certificate of Appreciation. We have received nominations from our Chapters and the prizes are on their way.

7. Plans for next year involve the following:

Building the organization through the At-Large Chapter, and the Corporate Associates and Lambda Notes programs. It is our objective that every member of our Society get "full value received" from his/her membership. As important, to encourage the role of SRE in speaking for our profession where professional, specifications and industry practices are involved.

Our Society will grow with membership participation and has a bright future in a world which increasingly depends for its survival on advanced technology and people who can create and make it work.



SRE International Chapter News and Information

Albuquerque Chapter - Ed Masterson - email: masterej@juno.com

The Albuquerque Chapter meets at Kirkland Airforce Base. Members are involved in and interested in defense aerospace issues.

Belvoir Chapter - Woody Rabon (703) 806-7827

Formed in 1995, Belvoir Chapter is represented primarily by U.S. Army CECOM personnel. Areas of interest include defense personnel equipment & product quality include such items as night vision scopes, mine countermeasures and decoys. Meetings are held a the Fort Belvoir facility which is on the Potomac River, south of Washington D.C.

Buffalo Chapter - President, Dave Dyrk (716) 439-3177

The Buffalo Chapter is the founding chapter of SRE. Buffalo Chapter meets in the Buffalo, NY area and is represented by many engineering and manufacturing organizations. The Buffalo chapter coordinates their meetings with other reliability organizations and with the Technical Societies Council of the Niagra Frontier. See the feature "Chapter Profile" in this issue for more detail.

Huntsville Chapter - President, Henry Cook (256) 876-9566

The Huntsville Chapter formed around 1969 during NASA Marshall Space Flight Center's support of the Apollo program. The chapter conducts monthly meetings in the Huntsville area. Membership includes personnel from U.S. Army activities, NASA and many of their support contractors.

India Chapter - P.H. Bhave - (022) 8300363

Montreal Chapter - Ulrich Hass (514) 855-5001 ext 59338

The Montreal Chapter is centered around the Canadian Airforce industry which is faced with reliability issues with the aging F-18 fleet. Bombardier Aerospace is the primary member representation.

Orlando Chapter

The Orlando Chapter is in the process of rebuilding.

Ottawa Chapter - President, Hans Reiche (613) 745-5034

The Ottawa chapter is the second oldest chapter in SRE, formed in 1970. The founding members were Hans Reiche and Chester Soucie from the Canadian Department of National Defense. This chapter has a strong membership and is represented by NORTEL, Nowbridge, DY4, National Research Council, Lockheed Martin, Fleet, Marconi, Computing Devices and Atomic Energy.

Philadelphia Chapter - Dr. Ajay Agarwala (610) 591-4788

The Philadelphia Chapter has met at the Boeing complex near the international airport and is one of the early SRE chapters. The Philadelphia Chapter was founded by Mr. Ken Eagle. The chapter has been supported by Boeing and Lockheed-Martin employees and is in the process of increasing membership. Speakers are in demand for aerospace topics, specifically software and network reliability analysis.

Rockford Chapter - Kevin Walsh (815) 639-6227

The Rockford Chapter formed in the early 1990's and was founded by Andrea Jang from Sunstrand Corp. The chapter meets in the Rockford, Illinois area and is considering ways to include Chicago members in regular meetings. Meetings have included 15 to 20 people and include a variety of topics. Industries represented include Woodward, Barber Coleman, Sunstrand and Microswitch.

Rocky Mountain Chapter - Pat Larter (719) 556-2571

The Rocky Mountain was founded in 1995 and conducts monthly noontime meetings and semi-annual banquets in the Colorado Springs area. Dr. Pat Larter is the founding member of the Chapter and the current president is Sondra Bundgaard. Meeting attendance is strong and is represented by the local defense Airforce agencies, support contractors and by Colorado Technical University. Dr. Ron Sega, former U.S. astronaut and Dean of College of Engineering at University of Colorado at Colorado Springs, is a well known speaker for the Chapter.

S.E. Michigan Chapter - Charles Slattery (810) 986-5848

Meetings are held quarterly in the Detroit area with a strong membership 60 plus. The membership is represented by Ford, GM, Daimler-Chrysler, many automobile suppliers, and General Dynamics.

St. Louis Chapter - Charles Schornak (314) 234-5241

The St Louis Chapter has met in the St Louis area with members from the major aircraft manufacturers and support companies.

Tucson Chapter - Dr. Dimitri Kececioglu - (602) 621-2495

The Tucson Chapter meets at the University of Arizona and is represented by the students of the Reliability Engineering Program at University of Arizona.

Washington Chapter - Richard Youngk - (703) 602-3900 ext. 657

The Washington Chapter was organized in the early 1980's and is supported by reliability professionals in the Washington D.C., northern Virginia, and suburban Maryland areas. The Department of Defense, other government organizations and various private corporations are represented. Several founding members of the Chapter continue to be active and provide academic and professional expertise and experinece in support of the chapter. The following founding members are recognized for their contributions over the past two decades: Dr. Pat Hartman, Bill Lohmar, Palmer Luetjen, Dave Mandel, and Reid Willis.

The At-Large Chapter - George Chernowitz - (201) 945-8203

This year, SRE activated its At Large Chapter, which serves our members who are not physically located close to one of our chapters. It is our first "virtual" chapter connected entirely by email, but otherwise receiving all the same services as a "regular" chapter, and receiving all the benefits of our Society. It's rapidly growing, with members from Beavercreek, Ohio to Freeport, Maine to Taipei, Taiwan. At Large members will have our own area on the SRE Web. Postings are planned of articles of interest, a place for "Q&A" and much else.



The History of SRE

Editor's Notes:
This is the first of a series of articles tracing events in the history of the SRE. To understand where we are going, we should begin with looking at where we have come from. The "headwaters" of SRE lie in upstate New York during the 1960's and show continued growth through the last 4 decades. These articles are appropriately divided in three phases below. Lambda Notes will publish each phase beginning with Phase I in this issue.

Phase I The Foundation
Phase II The Formation of International SRE
Phase III The Growth of SRE

During the first several years, the SRE was isolated to the Buffalo, N.Y. area. These years are included in the "Phase I - The Foundation". Much of the content of these articles is taken verbatim from the early documents of the founding chapter, and thus takes on a Buffalo Chapter perspective. The prime information sources are minutes of meetings, news-letters, and consultations with some of those involved at the time. Phase II begins the expansion of SRE to other chapters in neighboring cities, including Ottawa Canada, Huntsville Alabama, and Philadelphia Pennsylvania. Phase Ill includes the addition of many of the currents chapters and includes the present membership I'd like to especially acknowledge the assistance of Gerry Cohen for his recollections, Bob Nowacki for having the foresight to compile the documents more than 30 years ago, and to Jack Baker for storing the documents over all this time and making them available to me.

David A. Dyrck
SRE Buffalo Chapter
June, 1999



A Short History of the SRE
Part 1 - The Foundation

Background:

It was mid 1960's. The United States had been challenged by former President Kennedy to land a man on the moon and safely return him to Earth. The conflict in Southeast Asia was escalating.

There was a growing interest in improving the reliability of systems and components. The reliability of the vacuum tube was of concern. Applications of the transistor and consumer electronics were in their infancy. Igor Bazovsky had published the text, "Reliability Theory and Practice", in 1961. Reliability Engineering was becoming a recognized discipline.

In Western New York, the major industries were aerospace, electronics, automotive, and medical. Some of the major private employers were Bell Aerosystems, Cornell Aeronautical Labs, General Motors, Houdaille, Linde, Mennen Greatbatch, Moog, Scott Aviation, Sierra Research, Sylvania, Taylor, Westinghouse, Worthington, and Wurlitzer.

The following is a chronology of events in the history of the SRE.

29 April 1966, Buffalo, New York:
Organizational meeting held at Sylvania, now GTE. Initial objectives drafted:

1. To promote principles of reliability.
2. Establish a forum for interchange of reliability experience.
3. To promote reliability education.
4. The interchange of "state of the art" techniques in reliability.
Attendees agree to contact local companies to solicit additional membership.

19 May 1966:
Objectives refined, and have remained essentially unchanged since that time. Gerry Cohen, then Supervisor of Reliability-Maintainability, Sylvania Electronic Systems Division, elected Chairman. Secretary and committee heads appointed. Work started on constitution and by-laws.

29 June 1966:
Operating ground rules established until a constitution is drafted.

First Board of Directors formed:
Chairman G. Cohen, Sylvania
Secretary D. Goodman, Sylvania
Treasurer P. Mangan, Sierra
Membership R. Nowacki, Bell
Programs E. Ehrman/S. Zobel, Bell/Cornell
Publicity S. Antos, Bell
Education N. Jacobs, Westinghouse
Constitution J. Medford, Bell
Newsletter P. Balbich, Firewal

Plans made for meetings, publicity releases, brochures, educational program, and newsletter. Membership dues set at two dollars per year. Advantages of membership to include:

1. Becoming a charter member in a rapidly growing organization.
2. Eligibility to serve on committees and as elected officers.
3. Receive newsletters and other literature.
4. Eligibility for discount prices on reliability textbooks.
5. Be on distribution list for latest specifications and other pertinent reliability information.
6. Eligibility to attend SRE educational programs free of charge.

August 1966:
First quarterly newsletter of the SRE published. (7 pages type written, single spaced!) General agreement reached to pursue an independent course as SRE rather than affiliating with other societies. Members urged to contact other societies concerning associating with SRE to mutual benefit of both yet allow SRE to retain its identity.

3 October 1966:
Paid membership totals 19. 1966 SRE Education Program established with the following content:

Criteria for reliable design, and design reviews
Analytic reliability studies - how they work and what can they do
Malfunction reporting and test data handling
Human factors and maintainability
Reliability in an industrial competitive market

Enrollment is 24.

A package containing a letter of introduction, an SRE brochure, an SRE application form and the educational program was sent to about 140 people.

Technical programs in 1966:

"Organization Of A Professional Group For Reliability", by F. Medford, Bell Aerosystems.
"How DCASO Analyzes a Contractor's Reliability Requirements", by J. Murphy, DCASO
"Reliability and Quality Control", by C. A. Bicking, Carborundum
"Finite Markov Chains", by N. Jagodzinski
"Interaction of Injury Factors in Automobile Accidents", by J. Kihiberg

A social evening featuring lobster tail, prime rib, or filet mignon cost $11.50/couple.

15 November 1966:
SRE Constitution approved unanimously at general meeting to become the official operating document of the SRE.

January 1967:
Membership totals 70 individuals from the following organizations:

American Optical
Bell Aerosystems
Canadian Westinghouse
Carborundum
Cornell Aeronautical Labs
DCASO
The Firewel Company
General Dynamics
IBM
Kamen Aircraft
Lapp Insulator Company
Moog, Inc.
Scott Aviation
Sierra Research
Sylvania Electronic Systems
Taber Instrument Corp.
Westinghouse Electric
Xerox

Second (spring) education program offered the following content:

"Total Reliability - Mathematics or Engineering?" by N. Lewis, Canadian Westinghouse
"Reliability - Its Role in Management", panel discussion
"System Effectiveness - A Case Study" by P. Heggs, Cornell Labs
"Computer Techniques - User vs. Supplier" by R. E. Carroll, Bell and S. B. Bliss, IBM
"Maintainability Design and Prediction" (in two parts) by R. Tillotson, Bell

Technical programs in 1967 included:

"Life-time Distributions" by M. Rosenshine of Cornell Labs
"Reliability and Medical Electronics" by H. Mennen, Mennen-Greatbatch
"Reliability Experiences at Moog" by D. Elmer, Moog Valve
"Space Reliability and Consumer Products" by C. Hock, NASA
"Bayesian Reliability Demonstration Plans" by A. J. Bonis, Bell

February 1967:
As a result of distributing the SRE brochure at Reliability Symposia, membership inquiries were received from Pratt & Whitney, Hartford, CT, Lockheed, Plainfield, NJ, Federal Electric Service, Huntsville AL, and Northern Electric, Ottawa, Ontario, Canada. Board consensus at the time was that the SRE remain localized until such time as territorial expansion is warranted.

April 1967:
SRE newsletter includes first message from the Chairman, Gerry Cohen:

"It is nearly a year since approximately ten men accepted my invitation to come to Sylvania and discuss the organization for reliability engineers.

While those in attendance were enthusiastic and immediately laid the foundation for the present Society of Reliability Engineers, There were some who did not attend who were very skeptical. They indicated there were already enough societies and that there was no need for another. It would appear that there was a need for an SRE since history indicates the following:

1. We now have 98 paid members.
2. They represent more than 25 different organizations.
3. Our members come from 7 different states and 2 companies in Canada.
4. We have conducted two excellent training sessions which were well attended and ably instructed.
5. Our newsletter is second to none and is probably the motivation for our out-of-state members.
6. We have had no difficulty obtaining speakers and our meetings are generally well attended.
7. Area companies have been most cooperative in providing both facilities for meetings, refreshments, and postage and stenographic assistance.
8. Originally, our decision to go independent...was strictly one of monetary consideration...

Apparently we proved there is a need for SRE.

Any message from the Chairman at the close of our current year would be incomplete without acknowledging the work of the Board of Directors. They have been untiring in their efforts and it is they who must be given the credit for all of the above accomplishments. However I might add that if you talk to any one of them you would find that they and their committees drew more benefits by their participation than those that only sat home and read the Newsletter.

Enough of the past - let's look to the future. We already have several speakers lined up for next year and many new ideas for bigger and better programs...

Thank you for your time, your cooperation, and your efforts in making all products more reliable. This goal is certainly worthy of our greatest efforts."

July 1967:
SRE announces a contest to design an insignia that most appropriately symbolizes the interests and goals of the SRE. The prize of $10.00 was later awarded to Stan Antos in September.

Plans being made for one-day tutorial seminar sponsored jointly with ASQC. The seminar would become an annual joint event for more than the next 20 years.

A committee is appointed to make a study of SRE future status, i. e. - remain local, become national, or affiliate with other professional organizations.

Elected officers for 1967-68 were:
Chairman R. Nowacki, Bell
Vice Chair N. Jacobs, Westinghouse
Treasurer J. Rainbolt, Bell
Secretary W. Atkins, DCASO

January 1968:
The sixth SRE newsletter published, continuing with such features as membership information, abstracts and reviews of technical meetings and technical articles, educational opportunities, news briefs, and technical articles.

March 1968:
The SRE tours the Iroquois Brewery - "no notes were taken... hic!" Membership stood at 141 (24 from Canada)

4 May 1968:
First joint SRE/ASQC Reliability - Quality Control Seminar held at Erie County Technical Institute, Buffalo, NY. The theme was "Reliability and Quality: Teamwork for Product Effectiveness". The seminar featured the following subjects: Reliability Management, Statistical Quality Control, Life Cycle Costs, Data Retrieval, Corporate Reliability, Environmental Testing, Configuration Management, Reliability Math, R/QC in Education, and Airline Maintenance. Speakers were from Westinghouse, Rochester Institute of Technology, Carborundum, Sylvania, Dunlop Tire & Rubber, Manufacturers & Traders Trust Company, General Dynamics, Bell Aerosystems, United Airlines, and the Rome Air Development Center. 125 people attend.

June 1968:
SRE receives New York State tax exempt status as a non- rofit professional organization. SRE applies for registration of the SRE insignia to the State of New York. Membership stands at 169.

The elected officers for 1968-69 are:
Chairman N. Jacobs, Westinghouse
Vice Chair W. Atkins, DCASO
Secretary S. Antos, Bell
Treasurer M. Nusbaum, Sylvania

August 1968:
Bob Nowacki writes, in his Chairman's report: "...At this time, the state of health of the organization is excellent. The treasury is solvent and the expansion of activities and chapters is solid evidence of the SRE's vitality. Under the leadership of our newly elected officers and committee leaders, I am confident that the SRE will have continued growth and success...

To be continued:



THE WASHINGTON CHAPTER - A SHORT HISTORY

This story really began in San Francisco. Two Navy reliability analysts, David Mandel and Palmer Luetjen, were attending the 1980 RAMS symposium when they first heard of the Society of Reliability Engineers. It seemed like such a good idea that when they returned they persuaded their Division chief, Mort Buckberg, to help them organize a Washington chapter. Two of the founding members were Harry Feingold and Harry Ascher, co-authors of Repairable Systems Reliability. Harry Feingold was the first chapter president.

Over the years chapter membership has varied between about 25 and 40. The members meet monthly to conduct as little business as possible (5 minutes max if absolutely unavoidable), about an hour of technical presentations, and peripheral socializing. The chapter has heard from visiting notables including Dr. Kocieoglu of the University of Arizona, Nozir Singpurwalla of George Washington University, Dinesh Verma of Lockheed-Martin, and Marvin Roush of the University of Maryland. One memorable event was a dinner meeting featuring a spirited debate between Bayesian and classical statisticians; other discussions have centered on papers presented by members and visitors. Off-site members in Colorado, Louisiana, New Jersey and Maine attend the meetings whenever they are in Washington on business.

The Washingtonians have encouraged the founding of other chapters at Fort Belvoir, Virginia and at Maryland University's Center for Reliability Engineering, and created the TIGER Users Group as a special interest group of the SRE.

The Washington location provides an excellent base for liaison with other reliability organizations, including joint meetings with IEEE and SoLE. Several members are active in other reliability organizations and have presented papers at RAMS, ASME, SoLE, Electric Power Industry, and other symposia. You know how that works: Get a paper accepted for a meeting at some neat place like Las Vegas or Montreal, and the boss has to send you.

Two Washington chapter members represent the SRE at the Partnership for RMS Standards, an oversight group headed by Dr. Vacante of the Defense Management School, that attempts to keep track of the migration of RMS standards and specifications from government to commercial and international publishers. Through this access they have submitted SRE comments on R&M program, FMECA, RCM and other proposed standards. All SRE members are invited to participate in reviewing draft documents. The SRE Standards Committee is looking for help in this important task. The address is reidwillis@juno.com, copy to mandel_david@hq.navsea.navy.mil.



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CHAPTER PROFILE
Huntsville, Alabama

In 1949 Dr. Wernher Von Braun and a team of German rocket scientists arrived in Huntsville to organize the U.S. military's Ordnance Guided Missile Center. Their accomplishments became the core of the Apollo program's success. The Center has since expanded to include the U.S. Army Ballistic Missile Agency, which contains the U.S. Army Aviation and Missile Command, the U.S. Army Space and Strategic Defense Command, and NASA's George C. Marshall Space Flight Center.

In the years since the military chose Huntsville as a center for vital scientific development and research, the area has earned a reputation for having one of the strongest and most diverse high-tech communities in the nation. Wharton Econometric Forecasting Associates' 1997 study ranked Huntsville as the second-highest concentration of technology employment in the United States -- second only to California's Silicon Valley.

Although the city now has numerous high-tech firms, the current make-up of the Huntsville members still consists primarily of professionals from the NASA and Army community. We conduct monthly meetings with diverse presentation topics concerning the reliability professional.

The presentations are provided during lunchtime on the third Tuesday of each month. As part of the attraction, the presentations change location from month-to-month in order to allow us to visit various members' work sites. We also boast an annual symposium (where "symposium" follows the original Latin translation - a party, usually following dinner, for drinking and conversation), that is held at a local picnic area every June.

By Eric E. Hunt, Secretary, Huntsville Chapter

HUNTSVILLE CHAPTER PRESIDENT'S MESSAGE

The Huntsville Chapter of the Society of Reliability Engineers looks forward to the twenty-first century. We have maintained a steady membership throughout the last several years. We are still primarily an organization made up of engineers involved in aerospace and defense. As the surrounding community is diversifying into other high tech areas, our chapter is continually looking for new members from these businesses. As the only active chapter in the southeast, we also continue to attract members from the surrounding states as well. They enjoy our entertaining and informative newsletter, even if they may be unable to attend our monthly presentations. As the President of the Huntsville Chapter, I would like to wish all chapters continued growth and a strengthening of the National Organization.

Mr. Henry B. Cook, Chapter President MEMBER INTERVIEW

There are so many interesting members of the Huntsville Chapter that it is almost insulting to select just one of them to interview. There is Mr. Tom Turner, who became an ASQ Fellow in 1996. He started with the Army's Ballistic Missile Agency in 1959 and, of course, transitioned with it into the Apollo program. He was responsible to ensure the quality, safety and reliability of the Saturn V Instrument Unit (probably had less memory than a Commodore 64 computer!). Then there is Dr. William Wessels who started his reliability career with a mining company in Australia. He's worked on everything from unmanned vehicles to medical equipment. Consider the amount of knowledge and experience - over 60 years worth in these two individuals alone.

However, after I put out an unrelated request out to all the members for a history of the Huntsville Chapter of the SRE, I received the following write-up from Dr. Ron Giuntini. The history he wrote is both entertaining, and a self-interview. I submit it here for your enjoyment and appreciation of a valuable career dedicated to Reliability Engineering.

HUNTSVILLE - A PERSONAL MEMOIR
Dr. Ron Giuntini

It all seems like a blur to me now and in many respects, a nightmare. It is still hard to believe that exactly 30 years ago I was working the John F. Kennedy Space Center in Florida. At the time I was a young guy with an exciting job as the chief scientist of a 300-person reliability and quality engineering operation for ITT. Most of my time was in managing as well as doing reliability, maintainability and availability predictions for the ground support equipment that supported the launches of the Saturn V moon rocket That was a tough job. This is a story in itself but I feel blessed to have been part of this wonderful segment in American History.

Needless to say, right after the successful landing on the moon, NASA made drastic cuts in the entire program and reliability and quality at ITT lost all (every single one of the engineers) because it could not lay off any of the technicians - the Union, you know. I was fortunate in that the ITT Director at KSC thought enough of me and the work I had done for ITT, to make a place for me at the Huntsville office. When I arrived at the local office for an interview, I was startled at both Huntsville and the office. Huntsville was a bad looking place that appeared smaller than it was. Housing was scarce; shopping was worse and, in general, the fall climate stunk. You just can't move from the beach areas of Florida to Huntsville without psychological help. It was a bad experience on my wife, child and me. The local ITT organization was the NASA Marshall Space Flight Center's reliability, maintainability, and quality company. Back then we did all of the RAM and quality work on all MSFC programs.

It's not that way now; each program has it own reliability engineering. We did all of the reliability work. That meant everything from R and M allocations, predictions, assessments, testing, FMECAs, Fault Tree Analysis and many amazing reliability research programs. We were the "reliability people" and we knew if a so-called reliability engineer did not work for us they were only "talkers" of reliability engineering and we did not consider these people as being "real". Today, we would call them "virtual" reliability engineers. There are still plenty of them around today.

We had the best at the time when reliability engineering was in its heyday and at its pinnacle. When I first arrived at the new job in Huntsville, I was informed that all engineers at the company were members of the SRE and that all of the officers were within the company and no people outside the company were allowed to be members. I thought that was odd and asked "why?" I was told that there were a couple of reasons and the first was that "we started the chapter in Huntsville", and second, "the speakers discuss the various reliability research projects we are working on and don't want any outsiders involved." The first project I became involved in was the very first "software reliability" methodology and we coined the term "software reliability". The second, which I did independently was a compatible "human reliability and maintainability process/methodology" for space systems.

This was 27 years before the paper that Bill Wessels and I recently collaborated on, and it went deeper. As I said at one of the SRE meetings, human reliability has a mean-time-between - arrival of about 27 years. If I'm alive in 2025, I'll do it again. The software reliability since that embryonic stage has grown into a growth industry that is bigger, in many respects, from the parent discipline of reliability engineering. I can recall that one of the SRE members left the company to become the director of product assurance at a small manufacturing company known as Space Craft, Inc, later to become SCI. Everyone pondered what to do about it Should he be forced to resign or what? It was decided that he might make a fuss, so after much deliberation and against our collective better judgement, he was allowed to remain a member.

That decision significantly changed the face of the Huntsville Chapter (then known as the Tennessee Valley Chapter) of the SRE. This individual broke the stranglehold we had and before we knew it there were all kinds of people who knew nothing about reliability (albeit one or two of them may have worked on a FMEA) that were allowed to join. Meaningful topics degenerated to nothingness to accommodate these guys who just wanted a night out to drink beer. I remember one disgraceful joint meeting that the SRE had with an unnamed 'quality society' where there was no speaker but instead there were about a dozen belly dancers. This is not a joke.

I would rather think of the olden times as the Bright Years. They were bright, at least, for "reliability". Many wonderful things in the field of reliability were accomplished. All of this coincided with the NASA goal of "putting a man on the moon before the end of this decade", which was 1969, of course. As far as what I was doing, all I can say was that it was very exciting and very few people ever get to experience the true satisfaction that I did. I didn't look at what I was doing as "a job". It was much more. I was inventing reliability at a time when it was needed and appreciated.

It is a different world today. The "quick and dirty" is king. Excellence is out. If you strive for excellence you are considered to be a "dope". It is a sign of the times, and believe me, we as a nation are going to pay dearly for this attitude that permeates our entire social fabric. I could tell you many stories about "reliability" in which I was involved. I teach reliability engineering to the NASA Kennedy Space Center reliability engineers as well as risk engineering and have done that since 1992. I just got back from North Carolina State University where I taught a seminar in design reliability. When you talk three or four days for eight hours a day, the audience occasionally needs a little diversion from the hard material with something that is entertaining. I usually sprinkle in a few stories about the problems in reliability "back then".

Some of the things in which I was involved were; the Skylab problem, the Space Shuffle computer, the SAFEGUARD ABM System (the U.S. Army, of course), Space Shuttle payloads, the simulation modeling of the Space Station resources - and how my method drove out the design envelope for the Space Station resources from power to vacuum to water to crew time to more than 60 consumables, and so on. All of these things are intertwined with reliability, maintainability, availability and all of the spares, tools, test equipment, crew utilization for payload operations, maintenance and general Station "housekeeping".

I think that the Huntsville SRE Chapter today is held together by a "few good men" and hope that somehow we can turn this around and add new members who have the dedication I think some of the old members had. However, long ago, the SRE dropped the ball by not having a certification program and as a result, the ASQ has filled that void. All of us who have the CRE had to go through the ASQ. I also think the Chapter is in good hands under the present officer-ship. I have said numerous times that had it not been for the U.S. Army MICOM {now AMCOM}, the Chapter would likely have evaporated. So, I think that AMCOM is an impetus that can enable the Huntsville Chapter of the SRE grow and flourish in the future. This is the reason, I feel that the AMCOM engineers should have a strong leadership role in the organization. Further, AMCOM and NASA Marshall Space Flight Center are the seats of strong reliability technical staffs and their contractors are at work on some of the nation's most advanced technology. Huntsville is an excellent location and we should be able to make the SRE grow.

RAM engineering and maybe the "Government" suffered a mighty blow a few years back when it went away from imposing the use of various methods, say, FMECA Mil-Std- 1629A, or Mil-Hdbk-217 and a host of others, to just saying these classics are for reference only. I understand that we are all looking for COTS and other commercial, less costly ways of doing business but the U.S. DOD has historically been "brain trust" of technology that force the development of other supporting technologies such as reliability engineering. The shrinkage of our military is symptomatic of the not-so-clearly- hinking public who forget that if it were not for our military we would not be enjoying the lives we have been privileged to have. This is a roundabout way of saying that to improve our outlook in reliability, and all technology, we need a robust military and more civil technology programs such as NASA.

By Dr. Ronald E. Giuntini

HUNTSVILLE CHAPTER NEWSLETTER
CHAPTER SPEAKERS AWARDS

We have had a couple of Huntsville Chapter members win awards this year. This issue of Lambda Notes includes a bio and copy of Mr. C. Richard Cassady's RAMS Symposium presentation that was awarded the Stan Ofsthun award. Mr. Cassady is a Huntsville Chapter member out of Mississippi. Another member, Mr. Woody Bombara was selected as one of the Huntsville Association of Technical Society's "Professionals of the Year". Mr. Bombara is a Washington D.C. native with a BS in Mathematics and an MS in Statistics both from Virginia Tech. Most of his career has been in government programs. He was a statistician for the Army Rocket and Guided Missile Agency until 1961, then transferred to MSFC working as a statistician for the Apollo main engine program until 1974. He then became an Electrical Engineer for the High Energy Astronomy Observatory until 1979. He finally found himself as a Reliability Engineer in the MSFC QA laboratory until 1986. He last worked as a Sr. Reliability Engineer for Calspan. He provides our Chapter with some of the most creative and interesting monthly presentations.



Upcoming Huntsville Chapter Meetings

The Huntsville Chapter takes a summer break from July to August. Monthly meetings will resume in September and continue until June. The initial presentation schedule will be drawn up during the board meeting in August. Topics that vary across the spectrum of reliability engineering are selected to cover all interests. Separate presentations are intentionally identified to cover software, simulations, new technology, and mathematical methods. At least one significant tour of a local activity is identified, and the hosts/locations for each month are assigned. This schedule will be available upon request from the Secretary of the Huntsville Chapter.

SRE
Huntsville Chapter
Board of Directors

President: Mr. Henry Cook, 876-2258,
COOK-HB@redstone.army.mil
Vice President: Dr. William Wessels, 971-4878,
wwessels@bellsouth.net /TD>
Treasurer: Woody Bombara, 650-0200.
Secretary and Editor: Mr. Eric E Hunt, 876-9566,
HUNT-EE@redstone.army.mil, or
RAMEngr@aol.com, or SREMon@aol.com



CHAPTER TECHNICAL PROGRAMS
RECENT HUNTSVILLE CHAPTER TECHNICAL MEETINGS

RELIABILITY IMPACT FROM SCHEDULED MAINTENANCE INTERVALS

This paper provided an approach unique from the standard taught in most reliability engineering courses. The presentation starts with the basic definitions of scheduled maintenance, and the associated reliability assumptions. The time interval between scheduled maintenance was defined as 'T'. For any time, t, the integer number of scheduled maintenance actions that have occurred is given by:

k = Integer { t / T } The probability that the system will survive 'k' maintenance intervals: Pr ( k ) = ( Rsys ( T ) ) k The remaining time to 't', is: t = t - kT, Therefore, the probability that the system will survive to time unit 't' is: Rsys ( t ) = ( Rsys ( T ) ) k ( Rsys ( t ) ) This approach provides the reliability graph shown in Figure 1, which is unique in that it does not assume that the reliability returns to its' original state after each SMI.

Figure 1. An example of the reliability curve created utilizing Dr. Wessels' approach.

Future opportunities for the development of this approach were identified:

  • LP ALGORITHM TO OPTIMIZE SMI WITH CONSTRAINTS
  • IMPLEMENTATION STRATEGY
  • FEEDBACK PERFORMANCE CRITERIA FOR CONTINUOUS IMPROVEMENT
More information on this paper can be obtained from Dr. William R. Wessels at (256) 772-3531. wwessels@bellsouth.net



THE RELIABILITY OF MEANS

In this paper, Space Shuttle Main Engine component data, for which only one failure mode is considered, is used to compare median and mean time between failures. Specifically arithmetic mean, arithmetic median, Weibull mean and Weibull mean time between failures are compared. It is concluded in the presentation that:

  • Weibull median is the most conservative measure of time between failures, with low variance, and the arithmetic mean is the least conservative, with high variance. Therefore, the Weibull median is the preferred measure of time between failures from the customer's viewpoint.
  • At least four failures are recommended for calculating mean or median time between failures, and at least ten failures for the AMSAA Reliability Growth Model.
  • Trending by model or groups of year models using Weibull medians is preferred over trending with the AMSAA model (or other trend by operating time) when a design/process change is made for a particular year model or manufacturing lot.
More information on this paper can be obtained from Mr. Woody Bombara, (256) 650~0200.



SOFTWARE RELIABILITY AND THE PATH TO IMPROVEMENT

It seems that Software Reliability was an "attractive" topic. We had approximately 40 attendees who came to hear our guest speaker Mr. Darryl Davis of Davis Systems speak on the topic. Mr. Davis did an excellent job addressing a collection of concerns in this area that would have taken most people two weeks to cover.

Mr. Davis summarized the problem:

  • Software size, complexity, and importance are rapidly growing. The average late-model car today has more on- board computing power than did the Apollo spacecraft that carried man to the moon.
  • In general, quality, cost, and schedule targets are usually badly missed.
  • Most software organizations are reacting to problems rather than preventing them.
  • Plans break down under stress.
  • Success depends largely on heroic efforts by outstanding individuals.
  • Sound software engineering technologies are not being effectively utilized.

Mr. Davis' proposed approach to the problem involves:

  • Recognize that there is "no silver bullet."
    There is "no silver bullet" with which to kill the monster - past attempts to focus on specific point solutions (such as a specific method or technology) have failed. What is needed is an overall solution. That solution should address the whole system -- the people, methods, and technology - and focus on the process that ties them all together.
  • Focus on the process:

  • Recognize that improvement takes time.
Mr. Davis concludes, "This is no different from the continuous process improvement approach taken in applying Total Quality Management and continuous improvement to other areas."



UTILIZING SIMULATIONS FOR PERFORMING DEGRADATION TREND ANALYSES

Missile systems are originally procured with a predicted shelf life. After fielding, system data is analyzed to assess the original shelf life prediction and perform shelf life extensions whenever possible and necessary. This Army program is referred to as the Stockpile Reliability Program (SRP). Historically, Army missile systems have been procured with an average seven-year shelf life, however, through the SRP the shelf life has been extended to an average of 18 years. SRP utilizes a variety of test methodologies and analysis techniques customized to each missile system. In general, data is collected over the life of the missile system from surveillance (field storage and operations), flight testing, and disassembly / component testing. This data is then analyzed for trends associated with age, manufacturing strata, and exposure environments. Undesirable trends may result in suspension, restriction, or risk acceptance. When the system continues to perform reliably and safely, and still meets a tactical need (i.e., is not obsolete, or replaced by a new system), then a shelf life extension will be coordinated with the Army missile community. For obvious reasons this program attains high visibility. For example, extending a system's shelf life may have a significant impact on the decision to procure a replacement system. Shelf life extensions must be taken seriously, with the greatest possible statistical confidence imposed. Unfortunately, due to significant operating budget shortfalls, the Army SRP is extremely constrained. Accordingly, several new initiatives have been undertaken in SRP to improve the confidence of the analyses at equivalent or reduced costs. One of these initiatives is to extend the use of simulation models created during system development for assessing and predicting missile system shelf life.

In this paper, the process of modifying a simulations model for supporting stockpile reliability program (SRP) analyses of a tactical missile system is described. Basic application of the modified simulation to SRP analyses is discussed. A methodology for supporting degradation trend analyses by predicting future performance is developed. Examples from an ongoing analysis are provided. It is shown that this is a valuable tool with significant growth potential. The model programming and system design considerations that should be incorporated early in program development in order to optimize future applications of the methodology are discussed.

More information on this paper can be obtained from Mr. Eric E. Hunt, (256) 876-9566, HUNT-EE@redstone.army.mil.



STAN OFSTHUN AWARD

The late Stan Ofsthun of Sierra Research, was the first prestident of the SRE in 1971 and 1972. As a member of the Buffalo Chapter from its earliest beginnings, he was responsible for initially leading SRE beyond its beginnings in Buffalo to the present organization with more than twenty chapters worldwide.

This award recognizes the best technical paper by an SRE member accepted for presentation at the Annual RAMS.

The award consists of a certificate of recognition and $1,000 CASH. The "Call for Papers" appears early each year. Abstracts must be submitted to RAMS and to the SRE by April. The criteria for the Stan Ofsthun Award are (1) being an active member of SRE, (2) having the paper approved for presentation at the annual Reliability and Maintainability Symposium (RAMS), and (3) being selected by the SRE Paper Awards Committee. SRE members can submit their papers destined to RAMS, through their local chapter president, to the SRE paper Awards Committee. The Chairman is:

SRE Paper Awards Committee
C/O Dr. J. A. Nachlas
250 New Engineering Bldg.
Virginia Tech
Blacksburg, Va. 24061-0118

Lambda Notes will reprint award winning papers in future issues with the permission of RAMS and IEEE.



Simplifying the Solution of Redundancy Allocation Problems

Wanda F. Rice · Mississippi State University · Starkville
C. Richard Cassady · Mississippi State University · Starkville
Tracy R. Wise · Mississippi State University · Starkville

Key Words: Redundancy, Allocation, Reliability

SUMMARY & CONCLUSIONS

Redundancy allocation is a useful technique in designing systems that have high levels of reliability while satisfying limitations on cost, weight, volume, etc. Performing redundancy allocation typically involves formulating and solving an appropriate mathematical programming problem. In the literature, a wide variety of these problems have been formulated and a large number of solution techniques have been proposed. As a result, redundancy allocation problems are typically perceived to be quite difficult to solve. However, very little analysis has been performed for the purpose of quantifying the true difficulty of these problems. In this paper, four basic redundancy allocation problems are addressed. Strategies are defined for solving these four problems to optimality by either partial or total enumeration. In all four cases, the results indicate that these redundancy allocation problems are very simple to solve. These enumeration strategies should provide insight into developing simpler strategies for solving more complex redundancy allocation problems.

1. INTRODUCTION

In many instances, the initial design phase for a system yields a structure with inadequate reliability. For example, suppose the initial design phase for a system results in a design that consists of three components connected in series. Suppose the reliabilities of the three components are 0.95, 0.90 and 0.92 respectively. The initial design therefore yields a system reliability of 0.7866. Suppose that this level of system reliability performance is inadequate.

In these types of scenarios, system reliability can be improved through the use of component-level redundancy. For example, if a single redundant copy of each component is added to the system described above, the system reliability will increase to 0.9591. If a second redundant copy of each component is added to the system, the resulting system reliability is 0.9984. The difficulty in adding redundant components to a system is two-fold. First, system design may be constrained by limiting factors such as cost, weight and volume. At some point, one or more of these constraints may prevent adding further redundancy to the system design. Second, some components may be in "more need" of redundancy than others. Thus, adding a third level of redundancy to one component may be preferable to adding a second level of redundancy to some other component.

Redundancy allocation refers to the process of identifying the optimal addition of redundancy to a system subject to constraints placed upon the system design. Redundancy allocation is typically performed through the formulation and solution of mathematical programming problems. Most redundancy allocation problems are nonlinear integer programming problems which can be quite difficult to solve.

In this paper, enumeration procedures are defined for solving series system redundancy allocation problems. Four formulations of this type of redundancy allocation problem are addressed. Since the number of potential solutions to these problems may be very large or even infinite, a structured set of guidelines is defined for pruning the solution space. The solution procedures are demonstrated using example problems. Implementation of these guidelines greatly simplifies the solution of these redundancy allocation problems. In addition, the numerical results obtained from implementing these solution procedures provide insight into the size of the feasible region of these problems.

2. NOTATION

m number of components in series
ri reliability of component i
ci cost of component i
wi weight of component i
vi volume of component i
ni number of copies of component i in parallel
R system reliability
C total system cost
W total system weight
V total system volume
R0 minimum system reliability
C0 maximum total system cost
W0 maximum total system weight
V0 maximum total system volume
3. REDUNDANCY ALLOCATION MODELS

Consider a system consisting of m independent components connected in series. Suppose that the reliability of this system, R, is inadequate, and active redundancy (as opposed to standby redundancy) is going to be used to increase the reliability of the system. Note that redundant components of type i (i = 1, 2, ... , m) are assumed to be independent of and identical to the original component i. Let ri denote the reliability of a component of type i, and let ni denote the number of copies of component i placed in parallel. The system reliability is thus given by
. (1)
In addition to reliability considerations, the design of the system may also be constrained by cost, weight and volume. Let ci denote the cost of a copy of component i, let wi denote the weight of a copy of component i, and let vi denote the volume of a copy of component i. The total system cost, total system weight, and total system volume are thus given by
, (2)
, (3)
and
, (4)
respectively.

Four formulations of the redundancy allocation problem for a series system are considered. The first formulation, P1, is designed to achieve a maximum system reliability subject to C0, an upper limit on the total system cost. Therefore, this problem is formulated as

P1:maxR
s.t.C£C0
ni³1, integer i = 1, 2, , m.
The second formulation of the redundancy allocation problem, P2, is designed to achieve a maximum system reliability subject to C0, W0, an upper limit on the total system weight, and V0, an upper limit on the total system volume. Therefore, this problem is formulated as
P2:maxR
s.t.C£C0
W£W0
V£V0
ni³1, integer i = 1, 2, , m.
The third formulation of the redundancy allocation problem, P3, is designed to achieve a minimum total system cost subject to R0, a lower limit on system reliability. Therefore, this problem is formulated as
P3:minC
s.t.R³R0
ni³1, integer i = 1, 2, , m.
The fourth and final formulation, P4, is designed to achieve a minimum total system cost subject to R0, W0 and V0. Therefore, this problem is formulated as
P4:minC
s.t.R³R0
W£W0
V£V0
ni³1, integer i = 1, 2, , m.
Note that each of these formulations is a nonlinear integer programming problem. Chern (Ref. 3) proved that all four of these problems are NP-hard. Therefore, the four problems are perceived to be quite difficult to solve.

4. SOLUTION OF THE REDUNDANCY ALLOCATION PROBLEMS

Solving redundancy allocation problems has been a subject of great interest in the literature for many years. Tillman, Hwang and Kuo (Ref. 16) identify 143 references on redundancy (or reliability) allocation published prior to 1977. These references consider a wide variety of solution procedures for several formulations of the redundancy allocation problem. In addition, these allocation problems are applied to many system configurations. In the time since this survey paper, the focus of redundancy allocation research has been on the development of solution procedures for either more complex formulations of the allocation problem or allocation problems for more complex systems (Refs. 2,4,5,9,11,12,13,14,15). Recently, the use of stochastic optimization techniques and neural networks in solving redundancy allocation problems has received a significant amount of attention (Refs. 1,5,6,7,8,9,10,13,17). These research efforts tend to support the assertion that redundancy allocation problems are quite difficult to solve. However, in this paper a set of strategies are defined for solving the four defined redundancy allocation problems by total or partial enumeration. Although the problems considered are the simplest of the redundancy allocation problems, the results obtained here may prove useful in defining more efficient solution procedures for more complex redundancy allocation problems.

Consider a system comprised of three (m = 3) components connected in series. This system is used to demonstrate the enumeration procedure for each of the four defined problems. Equivalent procedures for series systems having m > 3 can easily be defined using the logic that follows. Suppose for this system:

r1 = 0.9 r2 = 0.7 r3 = 0.8
c1 = 2 c2 = 1 c3 = 3.
Suppose it is of interest to solve P1 for this system with a total system cost limit of 17. A quick inspection of the model parameters reveals that n1 £ 8, n2 £ 17 and n3 £ 5. This implies a potential solution set of 8(17)(5) = 680 solutions. However, a more efficient strategy can be defined which permits the investigation of only the feasible solutions. First, an upper bound on n1 can be defined.
(5)
Since n1 must be integer-valued, this upper bound can be simplified to
. (6)
Given the value of n1, an upper bound on n2 can be defined.
. (7)
Given the values of n1 and n2, an upper bound on n3 can be defined.
. (8)
These upper bounds can then be used to define the limits for a computer program which enumerates all the feasible solutions for any specific realization of P1. Mathematica code for solving P1 (m = 3) and counting the number of feasible solutions is given in the Appendix. For the defined example, there are only 83 feasible solutions and the optimal redundancy allocation is n1 = 2, n2 = 4 and n3 = 3. The resulting system reliability is 0.9741.

For P2, the additional limitations on total system weight and total system volume result in three restrictions on each of the decision variables. For n1, these restrictions are
. (9)
The resulting upper bound on n1 can thus be defined by
. (10)
Following the same logic as used for P1, the following upper bounds on n2 and n3 can be defined.
(11)
(12)
These upper bounds can then be used to define the limits for a computer program which enumerates all the feasible solutions for any specific realization of P2. Mathematica code for solving P2 (m = 3) and counting the number of feasible solutions is given in the Appendix. Consider the following parameter values for a realization of P2.

C0 = 25 c1 = 4 c2 = 1 c3 = 3
W0 = 19 w1 = 1 w2 = 2 w3 = 2.5
V0 = 24 v1 = 3 v2 = 2.3 v3 = 1
r1 = 0.92 r2 = 0.88 r3 = 0.95
For this problem, there are 72 feasible solutions and the optimal redundancy allocation is n1 = 3, n2 = 4 and n3 = 3. The resulting system reliability is 0.9992.

For P3, the bounds on n1, n2 and n3 are lower bounds rather than upper bounds. Specifically, a minimum number of components of each type must be included in the system to achieve the required reliability level, R0. These minimum values serve as the starting point for the enumeration strategy used to solve P3. In order for the system reliability to meet or exceed R0, the reliability of each set of components must be at least R0. Therefore,
. (13)
This relationship reduces to
. (14)
Applying the integer restriction on n1 yields the lower bound
. (15)
Given the value of n1, the lower bound on n2 can be derived as follows.
(16)
(17)
(18)
Likewise, given the values of n1 and n2 the corresponding expression serves as the lower bound for n3.
(19)
Although these lower bounds do restrict the values of n1, n2 and n3, the lack of an upper bound implies that each realization of P3 has an infinite number of feasible solutions. Therefore, a partial enumeration strategy must be defined for solving P3.

The partial enumeration strategy used in this paper limits the exploration of the feasible region of P3 in a way that facilitates the quick identification of the optimal solution. First, the lower bound on n1 is identified. Then, the corresponding lower bound on n2 is computed, followed by the computation of the appropriate lower bound on n3. The total system cost for these values of n1, n2 and n3 is computed and this solution is initialized as the "current optimal solution." Given n1 and n2, there is no reason to consider a value of n3 larger than its lower bound. Doing so would only result in a feasible solution with a larger total system cost. Second, n1 is held constant and n2 is incremented by 1. The corresponding value of n3 is computed and the new solution is compared to the current optimal solution. If necessary, the current optimal solution is updated. This second stage is repeated until the values of n1 and n2 automatically imply a sub-optimal solution. Third, n1 is incremented by 1. The corresponding values of n2 and n3 are computed and the new solution is compared to the current optimal solution. If necessary, the current optimal solution is updated. Then, the search continues with the second stage described above. This third stage is repeated until the value of n1 automatically implies a sub-optimal solution. At that point, the search is complete and the optimal solution has been identified.

Mathematica code for solving P3 (m = 3) and counting the number of feasible solutions evaluated is given in the Appendix. Consider the following parameter values for a realization of P3.

R0 = 0.995 r1 = 0.9 r2 = 0.875 r3 = 0.91
c1 = 4 c2 = 1 c3 = 3.
For this problem, 15 feasible solutions are explored and the optimal redundancy allocation is n1 = 3, n2 = 3 and n3 = 3. The resulting total system cost is 24.

For P4, the strategies used in solving P2 and P3 are combined. The lower bounds used in solving P3 are used as the lower bounds on n1, n2 and n3. The upper bounds used in solving P2 are modified by removing the cost term in each upper bound. These modified expressions are used as the upper bounds on n1, n2 and n3. These upper and lower bounds are then used to perform a total enumeration of the feasible solutions of P4. Mathematica code for solving P4 (m = 3) and counting the number of feasible solutions is given in the Appendix. Consider the following parameter values for a realization of P4.

R0 = 0.95 r1 = 0.7 r2 = 0.8 r3 = 0.93
W0 = 40 w1 = 3 w2 = 6 w3 = 4
V0 = 50 v1 = 2 v2 = 3 v3 = 7
c1 = 5 c2 = 4 c3 = 3
For this problem, there are only 8 feasible solutions and the optimal redundancy allocation is n1 = 3, n2 = 3 and n3 = 2. The resulting total system cost is 33.

5. IMPLICATIONS

The examples presented in this paper, as well as examples solved by the authors in the research leading to this paper, imply that for problems P1, P2 and P4 the number of feasible solutions is relatively small. In addition, these examples imply that P3 can be solved by considering a small number of feasible solutions. Therefore, the authors have concluded that the mathematical programming problems corresponding to the four basic series system redundancy allocation problems are not very difficult to solve. Even when the number of solutions that must be enumerated is large, modern computing power still provides an almost instantaneous optimal solution. The authors anticipate that the strategies similar to those defined in this paper could be used to solve some other redundancy allocation problems. In addition, the insights provided in this paper may be used to assist in the development of more efficient strategies for solving more complex redundancy allocation problems.

APPENDIX

Mathematica Code for P1

C0=17;
c1=2;c2=1;c3=3;
r1=0.9;r2=0.7;r3=0.8;
Rel[n1_,n2_,n3_]:=(1-(1-r1)^n1)(1-(1-r2)^n2)
                  (1-(1-r3)^n3);
numsol=0;
rmax=0;
Do[numsol++;
   If[Rel[n1,n2,n3]>rmax,rmax=Rel[n1,n2,n3];
                n1max=n1;n2max=n2;n3max=n3];,
{n1,1,Floor[C0/c1]},
{n2,1,Floor[(C0-c1*n1)/c2]},
{n3,1,Floor[(C0-c1*n1-c2*n2)/c3]}];
Print["The number of feasible solutions is "    
      ,numsol,"."];
Print["The maximum system reliability is " 
      ,Rel[n1max,n2max,n3max],"."];
Print["n1 = ",n1max," n2 = ",n2max," n3 = " 
      ,n3max];
Mathematica Code for P2
C0=50;W0=38;V0=48;
c1=4;c2=1;c3=3;
w1=1;w2=2;w3=2.5;
v1=3;v2=2.3;v3=1;
r1=0.92;r2=0.88;r3=0.95;
Rel[n1_,n2_,n3_]:=(1-(1-r1)^n1)(1-(1-r2)^n2)
                  (1-(1-r3)^n3);
numsol=0;
rmax=0;
Do[numsol++;
   If[Rel[n1,n2,n3]>rmax,rmax=Rel[n1,n2,n3];
                n1max=n1;n2max=n2;n3max=n3];,
{n1,1,Min[Floor[C0/c1],Floor[W0/w1],
          Floor[V0/v1]]},    
{n2,1,Min[Floor[(C0-c1*n1)/c2],
          Floor[(W0-w1*n1)/w2],
          Floor[(V0-v1*n1)/v2]]},
{n3,1,Min[Floor[(C0-c1*n1-c2*n2)/c3],
          Floor[(W0-w1*n1-w2*n2)/w3],
          Floor[(V0-v1*n1-v2*n2)/v3]]}];
Print["The number of feasible solutions is " 
      ,numsol,"."];
Print["The maximum system reliability is " 
      ,Rel[n1max,n2max,n3max],"."];
Print["n1 = ",n1max," n2 = ",n2max," n3 = " 
      ,n3max];
Mathematica Code for P3
c1=4;c2=1;c3=3;
r1=0.9;r2=0.875;r3=0.91;
R0=0.995;

Cost[n1_,n2_,n3_]:=c1*n1+c2*n2+c3*n3;
Rel[n1_,n2_,n3_]:=(1-(1-r1)^n1)(1-(1-r2)^n2)
                  (1-(1-r3)^n3);
Rel1[n1_]:=1-(1-r1)^n1
Rel2[n2_]:=1-(1-r2)^n2
numsol=0;
mincost=10000000.0;
n1=Ceiling[Log[1-R0]/Log[1-r1]];
Label[compn2];
n2=Ceiling[Log[1-(R0/Rel1[n1])]/Log[1-r2]];
Label[compn3];
n3=Ceiling[Log[1-(R0/(Rel1[n1]*Rel2[n2]))]
           /Log[1-r3]];
numsol++;
If[Cost[n1,n2,n3]<=mincost,
   mincost=Cost[n1,n2,n3];
   n1min=n1;n2min=n2;n3min=n3];
n2++;
If[Cost[n1,n2,0]]< mincost,Goto[compn3]];
n1++;
If[Cost[n1,0,0]]< mincost,Goto[compn2]];
Print["The number of feasible solutions 
      evaluated was ",numsol,"."];
Print["The minimum total system cost is ",
      Cost[n1min,n2min,n3min],"."];
Print["n1 = ",n1min," n2 = ",n2min," n3 = " 
      ,n3min];
Mathematica Code for P4
R0=0.95;W0=40;V0=50;
c1=5;c2=4;c3=3;
w1=3;w2=6;w3=4;
v1=2;v2=3;v3=7;
r1=0.7;r2=0.9;r3=0.8;

Cost[n1_,n2_,n3_]:=c1*n1+c2*n2+c3*n3;
Rel[n1_,n2_,n3_]:=(1-(1-r1)^n1)(1-(1-r2)^n2)
                  (1-(1-r3)^n3);
Rel1[n1_]:=1-(1-r1)^n1
Rel2[n2_]:=1-(1-r2)^n2
numsol=0;
mincost=100000000.0;
Do[numsol++;
   If[Cost[n1,n2,n3]<=mincost,
      mincost=Cost[n1,n2,n3];
      n1min=n1;n2min=n2;n3min=n3],
{n1,Ceiling[Log[1-R0]/Log[1-r1]],
    Min[Floor[W0/w1],Floor[V0/v1]]},
{n2,Ceiling[Log[1-(R0/Rel1[n1])]/Log[1-r2]],
    Min[Floor[(W0-w1*n1)/w2],
        Floor[(V0-v1*n1)/v2]]},
{n3,Ceiling[Log[1-(R0/(Rel1[n1]*Rel2[n2]))]/
            Log[1-r3]],
    Min[Floor[(W0-w1*n1-w2*n2)/w3],
        Floor[(V0-v1*n1-v2*n2)/v3]]}];
Print["The number of feasible solutions is " 
      ,numsol,"."];
Print["The minimum total system cost is " 
      ,Cost[n1min,n2min,n3min],"."];
Print["n1 = ",n1min," n2 = ",n2min," n3 = " 
      ,n3min];

REFERENCES:

1. J.E. Angus, K. Ames, "Simulated annealing algorithm for system cost minimization subject to reliability constraints", Communications in Statistics, Part B: Simulation and Computation, vol 26, 1997 no 2, pp 783-790.

2. W-T. K. Chien, W. Kuo, "Optimization of the burn-in times through redundancy allocation", Proceedings of the 2nd Industrial Engineering Research Conference, 1993, pp 579-583.

3. M-S. Chern, "On the computational complexity of reliability redundancy allocation in a series system", Operations Research Letters, vol 11, 1992 no 5, pp 309-315.

4. D.W. Coit, "System reliability optimization considering variability of empirical component reliability estimates", Proceedings of the 6th Industrial Engineering Research Conference, 1997, pp 42-47.

5. D.W. Coit, A.E. Smith, "Considering risk profiles in design optimization for series-parallel systems", Proceedings of the Annual Reliability & Maintainability Symposium, 1997, pp. 271-277.

6. D.W. Coit, A.E. Smith, "Penalty guided genetic search for reliability design optimization", Computers & Industrial Engineering, vol 30, 1996 no 4, pp 895-904.

7. D.W. Coit, A.E. Smith, "Solving the redundancy allocation problem using a combined neural network/genetic algorithm approach", Computers & Operations Research, vol 36, 1996 no 6, pp 515-526.

8. D.W. Coit, A.E. Smith, "Reliability optimization of series-parallel systems using a genetic algorithm", IEEE Transactions on Reliability, vol 45, 1996 no 2, pp 254-260.

9. D.W. Coit, A.E. Smith, "Stochastic formulations of the redundancy allocation problem", Proceedings of the 5th Industrial Engineering Research Conference, 1996, pp 459-463.

10. D.W. Coit, A.E. Smith, "Optimization approaches to the redundancy allocation problem for series-parallel systems", Proceedings of the 4th Industrial Engineering Research Conference, 1995, pp 342-349.

11. D.L. Fugate, "A reliability allocation methods for combination series-parallel systems", Proceedings of the Annual Reliability & Maintainability Symposium, 1992, pp 432-435.

12. J.B. Ghosh, C.E. Wells, "Determining optimal redundancy for systems with random lifetimes", IEEE Transactions on Reliability, vol 39, 1990 no 3, pp 309-313.

13. S.R.V. Majety, S. Venkatasubramanian, A.E. Smith, "Optimal reliability allocation in series-parallel systems from components' discrete cost-reliability data sets: a nested simulated annealing approach", Proceedings of the 5th Industrial Engineering Research Conference, 1996, pp 435-440.

14. K.B. Misra, U. Sharma, "Multicriteria optimization for combined reliability and redundancy allocation in systems employing mixed redundancies", Microelectronics and Reliability, vol 31, 1991 no 2, pp 323- 335.

15. D. Nowicki, "Reliability allocation with partial redundancy", Proceedings of the Annual Reliability & Maintainability Symposium, 1991, pp 400-404.

16. F.A. Tillman, C-L. Hwang, W. Kuo, "Optimization techniques for system reliability with redundancy a review", IEEE Transactions on Reliability, vol 26, 1977 no 3, pp 148-155.

17. V.V. Vinod, S. Ghose, "Neural network optimization for redundancy allocation", Microelectronics and Reliability, vol 34, 1994 no 1, pp 115-123.

BIOGRAPHY

Wanda F. Rice
Mississippi State University
PO Box 9542
Mississippi State, Mississippi 39762 USA
wfr1@ra.msstate.edu

Wanda Rice is a masters student in the Department of Industrial Engineering at Mississippi State University. She received her B.S. in mathematics from Mississippi State University.

C. Richard Cassady, Ph.D.
Mississippi State University
PO Box 9542
Mississippi State, Mississippi 39762 USA
cassady@engr.msstate.edu

Richard Cassady is an assistant professor in the Department of Industrial Engineering at Mississippi State University. He received his B.S., M.S. and Ph.D. all in industrial and systems engineering from Virginia Polytechnic Institute and State University. His research interests are in reliability and maintainability engineering, statistical quality control, and operations research modeling. His research has been sponsored by the National Science Foundation. He is a senior member of IIE, and a member of ASQ, SRE and INFORMS.

Tracy R. Wise
Mississippi State University
PO Box 9542
Mississippi State, Mississippi 39762 USA

Tracy Wise is a senior in the Department of Industrial Engineering at Mississippi State University.



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