Elsevier

Applied Soft Computing

Volume 13, Issue 1, January 2013, Pages 415-427
Applied Soft Computing

Using objective reduction and interactive procedure to handle many-objective optimization problems

https://doi.org/10.1016/j.asoc.2012.08.030Get rights and content

Abstract

A number of practical optimization problems are posed as many-objective (more than three objectives) problems. Most of the existing evolutionary multi-objective optimization algorithms, which target the entire Pareto-front are not equipped to handle many-objective problems. Though there have been copious efforts to overcome the challenges posed by such problems, there does not exist a generic procedure to effectively handle them. This paper presents a simplify and solve framework for handling many-objective optimization problems. In that, a given problem is simplified by identification and elimination of the redundant objectives, before interactively engaging the decision maker to converge to the most preferred solution on the Pareto-optimal front. The merit of performing objective reduction before interacting with the decision maker is two fold. Firstly, the revelation that certain objectives are redundant, significantly reduces the complexity of the optimization problem, implying lower computational cost and higher search efficiency. Secondly, it is well known that human beings are not efficient in handling several factors (objectives in the current context) at a time. Hence, simplifying the problem a priori addresses the fundamental issue of cognitive overload for the decision maker, which may help avoid inconsistent preferences during the different stages of interactive engagement. The implementation of the proposed framework is first demonstrated on a three-objective problem, followed by its application on two real-world engineering problems.

Highlights

► Simplify and solve strategy to solve many objective optimization problem is proposed. ► Objective reduction and interactive evolutionary optimization procedure is used. ► Five objective storm drainage problem is successfully solved. ► Eleven objective car side impact problem is also successfully handled.

Introduction

Evolutionary multi-objective optimization (EMO) algorithms have been successfully applied to a number of real world problems involving two or three objectives [1]. However, when the number of objectives scale up beyond three, most existing EMO algorithms [2], [3], [4] struggle to simultaneously achieve both convergence and diversity of solutions on the true Pareto-optimal front (POF). As discussed elsewhere [5], [6], EMO algorithms are not particularly applicable for handling many objectives. The corresponding challenges relate to the inability to search for better solutions using the usual Pareto-dominance concept [7], representation of high-dimensional front with a discrete set of points, and difficulty in decision making when the POF cannot be visualized geometrically. In spite of extensive research, these difficulties have been found hard to overcome.

This persuades us to explore a different solution methodology in this paper, which can more effectively handle the challenges posed by many-objective problems. The proposed simplify and solve methodology, outlined in Fig. 1, emanates in pursuit of addressing some fundamental questions including:

  • Given an M-objective problem, are all the given objectives conflicting?.

  • Is attempting to approximate the whole POF, a meaningful goal for many-objective problems, or whether the most preferred solution should be the target?

  • Is the decision maker (DM) cognitively capable of handling any number (many) of objectives?

  • If the DM is engaged, then what is an effective mode of DM's intervention in the solution procedure?

It is often found that during the problem formulation, there is a tendency to pose as many performance-related indicators as possible, as objectives to be optimized simultaneously. However, it may not be obvious to the DM if some of the objectives are non-conflicting, and hence redundant. More so, the analyst solving the optimization problem has little clue whether all the objectives are essential or not. Notably, each additional objective leads to an exponential rise in the complexity of the optimization problem. Under such a scenario, it is pragmatic to investigate if the given M-objective problem could be posed as a problem with fewer number (m) of objectives. Towards it, the simplification step of the proposed framework employs an objective reduction algorithm [8] to identify the essential objectives – the smallest set of conflicting objectives (FT,m=|FT|) which can generate the same POF as the original problem. Subsequently, the non-essential or the redundant objectives are eliminated on the pretext that a simplified problem will lead to less computational effort and a higher search efficiency.

It may be noted that problem solving in the context of EMO algorithms implies approximation of the entire POF, and it is assumed that selecting the best point of interest (among the solutions produced) is a simple task for a well informed DM. While this may be an acceptable approach for two- or three-objective problems, it is far from feasible in the context of many-objective problems. This is because the requirement of population size grows exponentially with an increase in the number of objectives, and it is difficult for the DM to identify the solutions of interest from the innumerable solutions presented. Even if only a small number of solutions are presented, the DM needs to investigate any pair of solutions in the wake of relative preferences for many objectives which is not a straight forward task. It implies that in the context of many-objective problems, the scope of problem solving by EMO algorithms cannot be limited to just presenting a set of solutions to the DM. In fact, the decision making aspects which have largely been ignored by the EMO community become crucial, and these could be even more challenging than the other issues faced by EMO algorithms. Given the above discussion, the solving step of the proposed framework does not endeavor to approximate the entire POF, but employs a multi-criteria decision making (MCDM) approach to engage the DM during the solution procedure to arrive at the solution most preferred by him/her.

The engagement of the DM during the solution procedure could be in different forms. The DM may be allowed to interact only at the beginning of an EMO run, wherein the DM may provide preference – information in the form of one or more reference point(s) [9], [10], one or more reference directions [11], one or more light beam specifics [12], and others [13], following which an EMO algorithm would target convergence of the population near the specific solutions on the POF. However, engaging the DM progressively [14], [15] could be a more effective approach as it would allow the DM to modify his/her preferences as new solutions evolve. In such an approach, the overall process would be more DM-oriented, where the DM would be in-charge and seamlessly involved in the overall optimization-cum-decision-making process. Given this understanding, the solving step of the proposed framework is based on a progressively interactive approach [15] which uses implicitly defined value functions [16] to produce the most preferred solution as the final outcome. The framework requires simple decision making tasks to be performed during the optimization run, which directs the algorithm towards the region of interest. The final solution produced using such an approach provides much higher accuracy with less computational expense when compared to attempting an entire Pareto-front.

While on one hand the benefits of interactively engaging the DM in the solution procedure for many-objective problems are clear, on the other hand the utility of this approach may be impaired due to the cognitive limitations of human beings in handling more than several factors at a time. Research in psychology has shown that individuals have a fixed capacity for processing information regardless of the task in question and irrespective of the processes they employ in solving any given task. A seminal work in Cognitive Science [17], has established through experimental evidence that the span of absolute judgment and the span of immediate memory are limited to holding seven plus or minus two digits of information. In the current context, it implies that the ability of the DM to specify the preferences with consistency for problems comprising of seven or more objectives (i.e., many-objective problems where M > 7) may be questionable. This affirms the promise in the proposed simplify and solve framework, in that, the cognitive load on the DM during the solution phase may be minimized by the preceding simplification.

The remaining paper is organized as follows. A brief survey of the past studies on objective reduction and progressively interactive methods is presented in Section 2. The details of the objective reduction algorithm, and the interactive solution procedure used in this paper are presented in 3 NL-MVU-PCA: nonlinear objective reduction based on Maximum Variance Unfolding and Principal Component Analysis, 4 Progressively interactive evolutionary multi-objective optimization using value function (PI-EMO-VF), respectively. Section 5 details the simplify and solve framework proposed in this paper, whose efficacy is demonstrated in Section 6, on two real world many-objective problems. Finally, the conclusions are drawn in Section 7.

Section snippets

A survey of past studies

In this section, we discuss some of the past studies, which have been performed in the domain of objective reduction and progressively interactive methods, independent of each other.

NL-MVU-PCA: nonlinear objective reduction based on Maximum Variance Unfolding and Principal Component Analysis

This section briefly presents the NL-MVU-PCA algorithm for nonlinear objective reduction. Given an initial objective set F0={f1, …, fM}, this algorithm aims to identify an essential objective set (FT). To achieve this aim, NL-MVU-PCA: (i) treats the objective vectors of the non-dominated solution set obtained from an EMO algorithm, as the input data, (ii) identifies the directions of significant variance (principal components) in the data, (iii) composes a set of objectives that are conflicting

Progressively interactive evolutionary multi-objective optimization using value function (PI-EMO-VF)

The progressively interactive EMO using value function (PI-EMO-VF) [15] is a generic procedure which can be integrated with any EMO algorithm. It equips an EMO to accept preference information from the DM during the intermediate generations of the algorithm, such that the final output is no longer the whole POF, rather the DM's most preferred point on the POF. In essence, the decision making and search work in tandem to make a progress towards a desired solution.

In PI-EMO-VF, the preference

The simplify and solve framework: integration of objective reduction and interactive procedures

The simplify and solve framework proposed in this paper, relies on exploiting the synergy between the state-of-the-art approaches of: (i) problem simplification using objective reduction, and (ii) interactive solution procedure through engagement of the DM. In that, for a given M-objective problem (original objective set being F0), first the nonlinear objective reduction algorithm, namely NL-MVU-PCA is executed, and the dimension (m) and composition of the essential objective set FT (FTF0) is

Demonstration of the proposed framework on real-world problems

In this section, we present the experimental results obtained with the proposed simplify and solve framework on two real-world many-objective problems, comprising of five and 11 objectives, respectively. The parameters for the NSGA-II run are kept the same as in the previous section (for solving the test problem, namely, modified ZDT1). The considered problems: (i) illustrate the earlier argument that in real-world scenarios, it may not always be obvious/intuitive to the DM that not all

Conclusions

In this paper, a simplify and solve framework for handling many-objective problems has been proposed. This framework relies on the integration of: (i) problem simplification guided by a machine learning based objective reduction algorithm, namely NL-MVU-PCA, and (ii) a solution procedure based on interactive engagement of the DM, namely PI-EMO-VF. The proposed framework holds the promise of effectively addressing some of the major challenges posed by many objective problems. In that, the

Acknowledgements

K. Deb and A. Sinha acknowledge the support provided by the Academy of Finland project under Grant No. 133387. A. Sinha also acknowledges the support provided by the Wallenberg Foundation.

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