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194 Niels J. Olieman and Eligius M.T. HendrixRobustness bounds can be defined, which are proportional to the radius of shapes enclosedby the Happy set and shapes enclosing the happy set. Finding the radius of the largest enclosedand smallest enclosing shapes, are global optimisation problems, where it is essentialto find global optima to guarantee a lower bound or upper bound respectively.From an end-user perspective, the extreme values of robustness bounds, can be useful fordecision support. From a technical perspective, robustness bounding methods are useful inthe context of feasible domain reduction and efficient conditional sampling techniques.References[1] Aharon Ben-Tal and Arkadi Nemirovski. Robust optimization − methodology and applications. MathematicalProgramming series B, 92:453–480, 2002.[2] P. Bjerager. Probability integration by directional simulation. ASCE Journal of Engineering Mechanics,8(114):1285–1302, 1988.[3] X. Du and W. Chen. Towards a better understanding of modeling feasibility robustness in engineering design.ASME Journal of Mechanical Design, 122(4):385–394, 2000.[4] Saeed Ghahramani. Fundamentals of Probability Theory. Prentice Hall, Upper Saddle River, New Jersey 07458,2000.[5] G.R. Grimmett and D.R. Stirzaker. Probability and Random Processes 3ed. Oxford University Press, Great ClarendonStreet, Oxford ox2 6dp, 2001.[6] H.M. Markowitz. Portfolio selection. Jounal of Finance, 7:77–91, 1952.[7] Jinsuo Nie and Bruce R. Ellingwood. Directional methods for structural reliability analysis. Structural Safety,22(3):233–249, 2000.[8] G. Taguchi. Introduction to Quality Engineering: Designing Quality into Products and Processes. Asian ProductivityOrganization, Tokyo, 1986.