23rd International Conference on Multiple Criteria Decision Making MCDM 2015 - Bridging Disciplines

August 2nd–7th, 2015, Hamburg, Germany

Polysemy of Robustness in Multiobjective Optimization

Margaret M. Wiecek

Department of Mathematical Sciences
Clemson University
Clemson, SC 29634
USA

Robust multiobjective optimization addressing decision making under multiple criteria and uncertainty has become a research field of vital interest in the last decade starting with concepts of robustness in engineering design (Deb & Gupta, 2004; Azarm & Li, 2005). Earlier efforts had used multiobjective optimization methods to resolve single-objective decision-making problems under uncertainty (Kouvelis & Yu, 1997).
In multiobjective optimization problems (MOPs), uncertainty may be assumed in the elements needed to define and solve MOPs such as objective and constraint functions, parameters converting MOPs into single-objective problems, and preference information provided by decision makers (DMs). Additionally, uncertainty can be modeled with continuous or discrete sets depending on the real-life context. In effect, uncertain MOPs may take on different forms leading to multiple concepts of robust counterpart problems (RCs) and to various notions of robust efficient solutions that convey multiple meanings to DMs. In view of this diversity, in the talk we interweave highlights of studies on robust multiobjective optimization with our research to obtain a comprehensive overview of this field.
We examine the column-wise (Soyster, 1973) and row-wise (Ben-Tal & Nemirovski, 1998) uncertainty in the constraints of the MOP. For each model we show that the efficient solutions of an RC can be found as the efficient solutions of a related deterministic problem. We demonstrate the findings on an Internet network requiring multiobjective routing under polyhedral traffic uncertainty.
We also apply the row-wise model to six scalarization formulations of the multiobjective linear program (MOLP) in which the scalarizing parameters remain uncertain, a situation being common for many DMs. The min-max solution to the MOLP emerges as robust (weakly) efficient to five out of the six formulations and, in this way, clearly resolves the challenge of choosing a scalarization formulation.
MOPs with uncertainty in the objective functions may lead to RCs making use of the (traditional) point domination or a (new) set domination. Again, in each case the RC is reduced into a computationally tractable deterministic MOP and the relationship between their efficient sets is examined. The objective-wise uncertainty helps to obtain stronger results. However, not only the coefficients in the objective functions but the number of objective functions may be uncertain. We analyze MOPs with an uncertain, possibly infinite, number of objective functions using two approaches to constructing a RC: all-in-one (AiO) and all-at-once (AAO). In the AiO approach, the RC assumes the form of a single MOP with an infinite number of vector-valued objective functions. In the AAO approach, the RC involves an infinite number of MOPs, each having a finite number of vector-valued objective functions. Under some conditions, these RCs can be reduced respectively to an MOP with a finite number of objective functions and to a finite number of MOPs, while the efficient set remains unchanged.
In conclusion we argue that, in comparison to single-objective optimization, multiobjective optimization offers many more research opportunities to exploit the concept robustness which, in turn, strikes with multiple meanings and interpretations.