Keywords: multi-objective optimization, structural reliability, hypo-ellipse, life cycle cost, bi-level optimization, first/second order reliability method (FORM/SORM), Pareto front, reliability index, response surface.

In this paper, the problem of the optimizing life cycle cost using multi-objective optimization is addressed. An approximation of the Pareto optimal front using a hypo-ellipse approach shows an effective tool to cut down the number of iterations required to form the front. An underlying, but often inexplicit, objective of structural design is to maximize the utility of a system being designed while minimizing its life cycle cost, which is simply defined as the aggregation of all cost components. Construction cost, annual maintenance and operation costs, repair costs and expected damage and failure costs (economic losses, deaths and injuries) are examples for different life cycle objectives. Multi-objective optimization (MOOP) allows trade-off between the several conflicting cost functions [2,3,4]. Analysis of life cycle cost optimization (LCCO) using MOOP gives the decision maker/designer more liberty in determining the most appropriate design alternative. Here, a special case of MOOP, bi-objective optimization (BOOP), is considered. The two cost objectives are the initial (construction) cost and the expected failure cost. The potential of using LCCO methods in design is promising but still in their infancy due to the difficulties and the lengthy analyses required to achieve the optimal solution.

One problem is addressed in this paper is the generation of the Pareto optimal front. Classical methods of MOOP such as weighted-sum or -constraint methods can be used to form the optimal front. However, classical methods suffer from the need of iterations and their lack of ability to produce diverse solutions along the optimal front [1,4]. Evolutionary methods such as genetic algorithms show excellent ability to produce the Pareto optimal front and to cover the disadvantages of classical methods [1]. In cases where the asymptotic methods (first order/second order reliability methods) are used to evaluate the failure probability included in the expected failure cost, the LCCO turns out to be a bi-level optimization problem [5,2]. Forming a Pareto optimal front for this type of problem is a complicated and unstable numerical process. Therefore, an approximate yet acceptable methodology is presented here based on a hypo-ellipse approach.

The bi-level optimization inherent in the life cycle cost optimization is tackled using a surrogate model. Different approaches have been proposed to solve the bi-level problem such as uni-level and decoupling approaches. The first approach suffers from hard constraint on the limit state function in the transformed domain (equal to zero) which may lead to numerical instability. The second approach considers the hard constraint as an inequality constraint, which is a more robust way of handling the constraint on the reliability index. However, this approach employed semi-infinite optimization algorithms which solve the inner optimization problem approximately. The proposed surrogate model is based on a low order polynomial response surface for the probability of failure constraint. The proposed surrogate model along with the proposed hypo-ellipse approach saves a huge number of analysis iterations and reduces the computational expenses.

The proposed method is applied to a structural design problem is discussed in order to show the acceptability and the strength of the technique. It is concluded that the trade-off front can be generated through three simple optimization processes. The approximation approach shows an acceptable trend to mimic the most sophisticated MOOP/BOOP techniques. Objective functions requiring classical bi-level optimizations are thus no longer an obstacle to produce an optimal front within an acceptable cost of analysis. Extended applications of the proposed methodology show its potential to handle 3-D problems and different optimization schemes (max-max, min-min problems). The proposed method is suitable for large scale structural problems.

1

Abdelatif H., Sh.S., Maes, M.: "Multi-objective Life Cycle Utility Optimization". Proceedings of 11th IFIO 7.5 Reliability and Optimization of Structural Systems, Banff, Canada. (2003) 353-360.

2

Abdelatif H., Sh.S., Maes, M.: "A Comparison of Algorithms for Structural Life Cycle Utility Optmization". Proceedings of International Conference on Structural and Geotechnical Engnieering and Construction Technology (IC-SGECT'04), Mansoura, Egypt. (2004) 108-120

3

Coello Coello, C.A.., Christiansen, A.D.: "MOSES A Multiobjective Optimization Tool for Engineering Design". Engineering Optimization. (1999) 31(3) 337-368. doi:10.1080/03052159908941377

4

Coello Coello, C.A.., Veldhuizen, D.A, Lamont, G.B.: "Evolutionary Algorithms for Solving Multi-Objective Problems". Kluwer Academic Publishers. (2002).

5

Rackwitz, R.: "Optimization-the Basis of Code Making and Reliability". Structural Safety. (2000) 22 27-60. doi:10.1016/S0167-4730(99)00037-5

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