(273b) A Novel Objective Reduction Method: Application to the Comparison of Risk Metrics

A novel objective reduction method: application to the
comparison of risk metrics

Daniel
Vázquez, Rubén Ruiz-Femenia, José A. Caballero

Keywords

Multi-objective Optimization, Inherent Safety, stochastic
optimization, sample average approximation

Typically, stochastic programming
models optimize the expected value of an indicator that
asses the performance of the system, providing a design solution that
behaves the best when considering all the scenarios simultaneously. However,
this approach may provoke that the value of this indicator worsens after the
uncertainty is revealed in a particular unfavorable
scenario. As decision makers may have different attitude towards the risk
associated with a design under uncertainty, it would be advisable to incorporate
risk management techniques to identify the trade-off between risk and
performance measures. This approach increases the complexity of the problem to
be solved because not only the problem is transformed into a multi-objective
optimization (MOO) problem –in
which the expected performance measure and its associated risk must be
simultaneously optimized–
but also the simultaneous incorporation of uncertainty
leads to a stochastic multi-objective optimization (SMOO) problem, which is inherently
high-dimensional because it captures multiple realizations of random data¹.

Multi-objective
optimization provides a set of Pareto optimal alternatives that feature the
property that it is not possible to find another feasible solution which is
better in one of the objectives without worsening at least one of the others. The
main limitation of multi-objective optimization is that its computational
burden exponentially grows with the number of objectives. Consequently, in many
applications, it is impractical to compute the entire Pareto set. One possible
way to overcome this limitation consists of omitting some of them, thereby
reducing the associated complexity. However this dimensionality reduction
should be applied trying to maintain the Pareto dominance structure, as Brockhoff and Zitzler ² formally
stated.

Following the
dimensionality reduction approach, in this work we propose a modification of
the objective reduction models 2 and 3 of the work by Vazquez et al.³, in which a mixed-integer linear
programming formulation, inspired in the Set Covering Problem, is presented.
First, we apply the model 1 of the previous work in its original form to a case
study of a petrochemical supply chain⁴.
This model provides the minimum subset of objectives that preserve the dominance
structure completely unaltered. Keeping intact the dominance structure of the
Pareto frontier is often too strict for practical applications, especially if
the minimum number of non-redundant objectives is still too large. For that
reason, in models 2 and 3 the dominance structure is allowed
to be modified when some objectives are removed, and its degree of the
degeneration is quantified by an error (denoted as d-error). By relaxing
this constraint of immutability, model 2 offers the minimum objective subset
while keeping the d-error below some given threshold (this model belongs
to the type of problems known as the d-Minimum Objective Subset, d-MOOS
problems). Likewise, in the third model we fix the maximum number k of objectives to preserve, and the
model finds the subset of at most k
objectives with the minimum d-error (this model is also known as the
Minimum Objective Subset of Size k
with Minimum Error, k-EMOSS problem).
In the original work, the d-error in models 2 and 3 are computed using the
infinity norm, while in this work we propose the use of the norm 1 in order to increase the sensitivity of these models. In
addition, we introduce a fourth model based on new metric denominated the
non-redundancy ratio. This new concept is defined as the percentage of the
Pareto solutions needed to maintain completely the dominance structure of the
non-redundant subset of objectives of a given size.

On the other
hand, in this work we consider the following metrics to control and manage
risk: Downside Risk (DR_Ω)⁵, Financial Risk (FR_Ω) ⁶, Worst Case, Value-at-Risk (VaR) ⁷, Opportunity
Value (OV) ⁸ and Conditional
Value-at-Risk (CVaR)⁹. Figure 1 shows a geometric interpretation of these risk
metrics. DR_Ω
and FR_Ω
require the definition of a target (denoted by Ω),
whereas VaR, OV and CVaR
are calculated for a probability target in the cumulative probability
distribution function (usually 0.05). To deal with the high dimensionality of
the SMOO problem formulated, we solved it using a decomposition strategy based
on the Sample Average Approximation algorithm¹⁰ . Under this approach, the stochastic problem (second
stage) contains only one objective (the economic) and all the risk metrics are
computed after solving this problem. For that reason, a Pareto filter is used
to ensure that the solutions belong to the Pareto frontier.

The results
highlight how the different risk metrics affect the dominance structure of the
problem, and trade-off among them, with the additional key feature of measuring
the error incurred after removing some objectives.

Figure 1. Geometric
interpretation of the risk metrics.

References

(1)          Cao, Y.; Fuentes-Cortes, L. F.; Chen,
S.; Zavala, V. M., Scalable modeling and solution of stochastic multiobjective
optimization problems. Comput Chem Eng 2017, 99, 185-197.

(2)          Brockhoff, D.; Zitzler, E., Are All
Objectives Necessary? On Dimensionality Reduction in Evolutionary
Multiobjective Optimization. In Parallel
Problem Solving from Nature - PPSN IX, Springer Berlin / Heidelberg: 2006;
Vol. 4193/2006, pp 533-542.

(3)          Vázquez, D.; Fernández-Torres, M. J.;
Ruiz-Femenia, R.; Jiménez, L.; Caballero, J. A., MILP method for objective
reduction in multi-objective optimization. Comput
Chem Eng 2018, 108, 382-394.

(4)          Guillén-Gosálbez, G.; Grossmann, I.,
Optimal design and planning of sustainable chemical supply chains under
uncertainty. AIChE J. 2009, 55, (1), 99-121.

(5)          Eppen, G. D.; Martin, R. K.; Schrage,
L., A Scenario Approach to Capacity Planning. Operations Research 1989,
37, (4), 517-527.

(6)          Barbaro, A.; Bagajewicz, M. J.,
Managing financial risk in planning under uncertainty. AIChE J. 2004, 50, (5),
963-989.

(7)          Jorion, P., Value at risk : the new benchmark for managing financial risk. 3rd
ed.; McGraw-Hill: New York, 2007; p xvii, 602 p.

(8)          Aseeri, A.; Bagajewicz, M. J., New
measures and procedures to manage financial risk with applications to the
planning of gas commercialization in Asia. Comput.
Chem. Eng. 2004, 28, (12),
2791-2821.

(9)          Rockafellar, R. T.; Uryasev, S.,
Optimization of Conditional Value-at-Risk. Journal
of Risk 2000, 2, 21-41.

(10)        Medina-González,
S.; Pozo, C.; Corsano, G.; Guillén-Gósalbez, G.; Espuña, A., Using Pareto
filters to support risk management in optimization under uncertainty:
Application to the strategic planning of chemical supply chains. Comput Chem Eng 2017, 98, 236-255.