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Research Papers

# Joint Probability Formulation for Multiobjective Optimization Under Uncertainty

[+] Author and Article Information
Sirisha Rangavajhala

Civil and Environmental Engineering,  Vanderbilt University, VU Station B # 351831,2301 Vanderbilt Place,Nashville, TN 37235-1831

Civil and Environmental Engineering,  Vanderbilt University, VU Station B # 351831,2301 Vanderbilt Place,Nashville, TN 37235-1831sankaran.mahadevan@vanderbilt.edu

J. Mech. Des 133(5), 051007 (Jun 08, 2011) (11 pages) doi:10.1115/1.4003540 History: Received September 07, 2010; Revised January 20, 2011; Published June 08, 2011; Online June 08, 2011

## Abstract

This paper presents a new approach to solve multiobjective optimization problems under uncertainty. Unlike the existing state-of-the-art, where means/variances of the objectives are used to ensure optimality, we employ a distributional formulation. The proposed formulations are based on joint probability, i.e., probability that all objectives are simultaneously bound by certain design thresholds under uncertainty. For minimization problems, these thresholds can be viewed as the desired upper bounds on the individual objectives. The tradeoffs are illustrated using the so-called decision surface, which is the surface of maximized joint probabilities for a set of design thresholds. Two optimization formulations to generate the decision surface are proposed, which provide the designer with the distinguishing capability that is not available in the existing literature, namely, decision making under uncertainty, while ensuring joint objective performance: (1) Maximum probability design: Given a set of thresholds (preferences within each objective), find a design that maximizes the joint probability while using a probabilistic aggregation as against an ambiguous weight-based method. (2) Optimum threshold design: Given a designer-specified joint probability, find a set of thresholds that satisfy the joint probability specification while allowing for a specification of preferences among the objectives.

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## Figures

Figure 1

Deterministic Pareto frontier

Figure 2

Proposed MOUU formulation

Figure 3

Two-bar truss example

Figure 4

Two-bar truss: deterministic and RBDO results

Figure 5

Two-bar truss: proposed results

Figure 6

An illustration of the TSTO design concept

Figure 7

Simplified TSTO analysis components

Figure 8

TSTO: deterministic and RBDO Pareto surfaces

Figure 9

TSTO problem: OTD formulation results

Figure 10

Side impact: deterministic and MOUU Pareto

Figure 11

Side impact problem: Proposed results

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