Estimation of f-COCOMO Model
Parameters Using Optimization
Techniques
University of Alabama at Birmingham
Birmingham, Alabama, USA
Leonard J. Jowers
Dr. James J. Buckley
Dr. Kevin D. Reilly
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Introduction

This presentation is concerned with promoting
use of optimization methods to estimate
COCOMO model parameters.

We briefly describe fuzzy COCOMO, showing
how fuzzy arithmetic is applied to the model.

We describe the issue which may be
addressed using this technique.

We provide an example and note future work.
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University of Alabama at Birmingham
Department of Computer and Information Sciences
LEONARD J. JOWERS
Bachelor of Sciences, 1969, University of Alabama.
Master of Arts, 1972, University of Alabama.
PhD candidate, University of Alabama at Birmingham.
2003-pres.
2002-pres.
2001-2002
1982-2001
1974-1982
1972-1974
1970-1972
1967-1970
1965-1968
Doctorial student. UAB
President. AuditSoft, Inc. Birmingham, Alabama, USA.
Executive Vice-President. Imaging Business Machines, LLC.
President. Computer Utilization Services Corporation.
VP of Operations. AOM Corporation, Birmingham, Alabama, USA.
Unit Supervisor & systems analyst, LTV Aerospace, Langley AFB, Virginia, USA.
Head of Software Department & systems analyst, Applied Computer Data Services, Tuscaloosa, AL
Programmer, UNIVAC, Bluebell, Pennsylvania, USA.
Student programmer, University Of Alabama Computing Center, Tuscaloosa, Alabama, USA
Publications on simulating fuzzy systems, numerical computing. Latest book Simulating Continuous Fuzzy
Systems (Springer), jointly with Dr. Buckley.
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University of Alabama at Birmingham
Department of Mathematics
JAMES J.
BUCKLEY, Ph.D.
Mathematics;
Georgia Tech in
1970
1970-1976 Mathematics Department at University
of South Carolina.
1976-pres. Mathematics Department at U. of
Alabama at Birmingham.
Numerous publications in fuzzy sets/fuzzy logic
and 9 books.
Department of Computer and Information Sciences
KEVIN D.
REILLY, Ph.D.
Math Biol. (Th.
Biol. & Biophy.);
U. of Chicago in
1966.
1966-1970 Information Scientist,
Institute of Library Research, UCLA
1968-1970 Lecturer, Computer Science,
School of Engineering, UCLA
1969-1970 Senior Lecturer, School of Business,
University of Southern California
1970-Pres. Professorial staff, Computer &
Information Sciences, UAB
Numerous publications on simulating fuzzy
systems.
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Origins of Fuzzy Logic

Lotfi Zadeh founded fuzzy logic in 1965.

A basic principle of fuzzy logic is, “Everything is a
matter of degree”.

Whereas Boolean logic postulates the concept of
truth as a function from a linguistic expression onto
the set {0, 1}, fuzzy logic postulates the concept of
truth as a function from a linguistic expression onto
the interval [0,1].
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Fuzzy Numbers
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Fuzzy Arithmetic
Using a-cuts, create membership functions for
C = A + B: C[a]=[cL,cR] = [aL,aR] + [bL,bR] = [aL+bL, aR+bR].
C = A - B: C[a]=[cL,cR] = [aL,aR] - [bL,bR] = [aL-bR, aR-bL].
C = A * B: C[a]=[cL,cR] = [aL,aR] * [bL,bR] = [bL, bR].
Where bL = min{aL bL, aL bR, aR bL, aR bR}.
bR = max{aL bL, aL bR, aR bL, aR bR}.
and if 0 is not in the support of B,
C = A / B: C[a]=[cL,cR] =[aL,aR]/[bL,bR] = [aL, aR] * [(1/bR),(1/bL].
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f-COnstructive COst MOdel

Start with classical COCOMO.

Making one or more parameters into a fuzzy
variable causes the result to be a fuzzy variable.

Use definitions and tables for COCOMO, but
allow specific uncertainty in one or more
parameters.
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Motivation

At the start of a project, some schedule and budget
requirements are known.

Values for f-COCOMO project parameters may be
generated from expert experience or data.
However, project resources and methods may not be
fixed.



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One or more fuzzy parameters may be optimized to
meet the schedule or budget.
By improving the estimation of parameters,
management may make adjustments to improve the
project.
Optimization to Solve Inverse Problems

An inverse problem is one for which an answer
is known but the question is not.

Such problems are sometimes difficult to solve
by analytical methods.

Optimization techniques such as Monte Carlo
methods or Genetic Algorithms are available for
such problems.
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Method – a Simple Example


Consider a ‘Nominal’ project of Size 5K for which a
budget of 16 PM has been decreed (but PM=17.26!).

PM = A x SizeE < 16.

E=B+0.01 x SSFj
Allow a couple of Scale Factors more freedom.



Use Fuzzy Monte Carlo to optimize TEAM and PMAT for


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TEAM – nominal to Very High
PMAT – nominal to Very High
ln(PM) = ln(A x SizeE)< ln(16).
ln(PM) = 0.0161 x TEAM + 0.0161 x PMAT < 0.0526.
Method – a Simple Example (Cont’d.)

It is determined that it is possible to meet the decreed
budget by raising TEAM and PMAT (lowering their
nominal values) to less than Very High.



Also



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TEAM – 1.43/1.51/2.62
PMAT – 1.62/1.76/2.99
Other constraints may be put into the optimization; such as, limits
on costs of improving a Scale Factor.
Effort Multipliers can be handled as fuzzy variables also.
With specification of a defuzzification of compounded results,
additional understanding may be possible.
Summary

Starting with crisp COCOMO, one may represent
linguistic parameters as fuzzy variables to create a
fuzzy COCOMO.

Fuzzy arithmetic is such that operational using fuzzy
variables tends to increase fuzziness in results.

Using decreed limitations on person-months, one may
use optimization techniques to determine fuzzy values
for parameters.
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Future Work

Computer sources will be available on request from
jowersl@cis.uab.edu after publication of Monte Carlo
Studies with Fuzzy Random Numbers, to appear
Sringer-Verlag, 2007.

On-going research into extending this method to
multi-objective f-COCOMO.
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Major References
Barry W. Boehm, Chris Abts, A. Windsor Brown, Sunita Chulani, Bradford K. Clark, Ellis
Horowitz, Ray Madachy, Donald Reifer, and Bert Steece, Software Cost Estimation with
COCOMO II, Prentice Hall PTR, Upper Saddle River, NJ, 2000.
J.J. Buckley and L.J. Jowers, Simulating continuous fuzzy systems, Springer-Verlag,
Heidelberg, Germany, 2005.
George J. Klir and Bo Yuan, Fuzzy sets and fuzzy logic: Theory and applications,
Prentice-Hall, Inc., Upper Saddle River, NJ, USA, 1995.
Petr Musilek, Witold Pedrycz, Giancarlo Succi, and Marek Reformat, Software cost
estimation with fuzzy models, SIGAPP Appl. Comput. Rev. 8 (2000), no. 2, 24-29.
L. Zadeh, Fuzzy sets, Inf. Control 8 (1965), 338-353.
Lotfi A. Zadeh, Computing with Words and Its Application to Information Processing,
Decision and Control, The 2003 IEEE International Conference on Information
Reuseand Integration (2003), Keynote speech.
Toward a Generalized Theory of Uncertainty (GTU)An Outline, January 20, 2005, To
appear in Information Sciences,
http://www.bisc.cs.berkeley.edu/BISCSE2005/Zadeh2005.pdf.
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Questions?
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