Research papers Struct Multidisc Optim 22, 188–197  Springer-Verlag2001 Fuzzy multicriterion design using immune network simulation J. Yoo and P. Hajela Abstract Structural optimization problems have been traditionallyformulatedintermsofcrisplydefinedobjec- tive and constraint functions. With a shift in application focustowardsmorepracticalproblems,thereisaneedto incorporate fuzzy or noncrisp information into an opti- mizationproblemstatement.Suchpracticaldesignprob- lems often deal with the allocation of resources to sat- isfymultiple,andfrequentlyconflictingdesignobjectives. The present paper deals with a genetic algorithm based optimization procedure for solving multicriterion design problemswheretheobjectiveorconstraintfunctionsmay notbecrisplydefined.Theapproachusesageneticalgo- rithm based simulation of the biological immune system to solve the multicriterion design problem; fuzzy set the- ory is adopted to incorporate imprecisely defined infor- mationintotheproblemstatement.Anotablestrengthof theproposedapproachisitsabilitytogenerateaPareto- Edgeworthfrontofcompromisesolutionsinasingleexe- cutionoftheGA. Key words multicriterion design, fuzzy logic, immune networks,fuzzydesign,geneticalgorithms 1 Introduction There has been significant recent activity in improving and developing new tools for formal design optimiza- tion. While these efforts have been quite successful, new challenges have emerged, brought about by the desire to apply formal design optimization to problems of in- creasing complexity and scale. In using such methods in Received May 8, 2000 J.YooandP.Hajela Mechanical Engineering, Aeronautical Engineering & Mech- anics,RensselaerPolytechnicInstitute,Troy,NY12180,USA e-mail: yooj3@rpi.edu, hajela@rpi.edu practical design problems, issues that warrant continued attention from developers include problem dimensional- ity as indicated by the number of design variables and constraints, the ability to account for discrete design variables, and to efficiently account for multiple, and often conflicting criteria in the design problem state- ment. Yet another consideration that cannot be ignored in a move to migrate design optimization tools into a practical design environment, is the ability to incor- porate fuzzy or imprecisely defined information into the solution process in a rational manner. The problem of fuzzy multicriterion design is the subject of the present paper. A commonly adopted approach to the multicriterion design problem is to treat one of the multiple criteria as the scalar objective function for the problem, and to for- mulate appropriate design constraints to accommodate the requirements on the other criteria. While the appar- ent simplicity afforded by this approach is quite attrac- tive, an affective argument can be made against its use. At a philosophical level, one can always contend that thereisa“natural”separationofcriteriaandconstraints in any design problem. Additionally, when one formu- latescriteriaasconstraints,theabilitytolearnaboutthe extent of the feasible set is somewhat compromised. In other words, a multiple criterion approach offers a so- lution in which a trade-off pattern emerges, indicating howimprovementinanyonecriterionwouldadverselyaf- fect the other criteria. Multicriterion design has emerged as a subject of special interest in mathematical nonlin- ear programming. One of the earliest efforts in this area may be attributed to an Italian economist Pareto, who in 1896, introduced the concept within the framework of welfare economics (Pareto 1896). Ramifications of this work in optimization theory, operations research, and control theory were recognized only in the late 1960’s. Applications of multicriterion optimization in engineer- ing design have also been studied (Baier 1977; Osyczka 1981;Koski1979).Recently,therehasbeenanincreased interest in extending the scope of multicriterion design to those engineering systems that are described by fuzzy information. In some engineering systems, it is not pos-