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Information Sciences

Volume 268, 1 June 2014, Pages 202-219
Information Sciences

INSPM: An interactive evolutionary multi-objective algorithm with preference model

https://doi.org/10.1016/j.ins.2013.12.045Get rights and content

Abstract

In this paper an interactive method for modeling the preferences of a Decision-Maker (DM) is employed to guide a modified version of the NSGA-II algorithm: the Interactive Non-dominated Sorting algorithm with Preference Model (INSPM). The INSPM’s task is to find a non-uniform sampling of the Pareto-optimal front with a detailed sampling of the DM’s preferred regions and a coarse sampling of the non-preferred regions. In the proposed technique, a Radial Basis Function (RBF) network is employed to construct a function which represents the DM’s utility function using ordinal information only, extracted from queries to the DM. The INSPM algorithm calls the DM’s preference model via a Dynamic Crowding Distance (DCD) density control method which provides the mechanism for increasing the sampling in the preferred regions and for decreasing it in non-preferred regions which allows a fine-tunning control of the Pareto-optimal front sampling density.

Introduction

The development of multi-objective approaches for the design of an increasing number of real-world systems is a current trend. Although there are available, at this moment, several Evolutionary Multi-objective Optimization (EMO) techniques that aim to provide representative samplings of the Pareto-sets in multi-objective optimization problems [4], [8], [1], [15], their application to the actual design of real systems still requires a further step in which, given a set of possible solution alternatives, a specific alternative should be chosen to be implemented. This step is usually recognized as a task that is attributed to a decision-maker (DM).

Most of the current techniques of Evolutionary Multi-objective Optimization (EMO) assume an a posteriori preference articulation scheme in which an entire sampling of the Pareto-optimal front is preprocessed before being presented to the DM. Such a detailed sampling represents an inefficient allocation of computational effort. On the one hand, the knowledge about the solutions in the regions which are not preferred has the only role of informing the DM about the value of the trade-offs that are aggregated in the preferred regions. For this purpose, a rough sampling of the non-preferred regions would be enough. On the other hand, it would be desirable a fine sampling of the preferred regions allowing the choice of a well-tunned solution. This fine sampling could become compromised if the computational budget was spent on the generation of a uniform sampling that covers all the Pareto-front including the non-preferred regions. This issue is even more concerning if the number of objectives is more than three. In this case, due to the exponential growth of the Pareto-front size related to the number of objectives, it may become even computationally impossible to produce a fine sampling of it. Therefore, procedures in which the DM progressively states her/his preferences in an interactive environment, which steers the multi-objective optimization algorithm in the search for Pareto-optimal solutions, may become relevant [10], [2].

This work proposes an EMO methodology which combines two features: (i) it interacts with the DM along the execution of the optimization task, such that the result of the optimization is guided by that interaction in a progressive preference articulation and (ii) it ends the optimization procedure with a model for the DM’s preferences, which becomes available for re-utilization in other instances of the same problem.

This paper is structured as follows. In Section 2 the motivation for the development of a methodology with those features is discussed. In Section 3 the definitions of multi-objective optimization and multi-criteria decision-making problems are presented. In Section 4 the NN-DM method, which is the technique employed to construct a model for the DM’s preferences, is explained. In Section 5 the NN-DM model is combined with NSGA-II allowing a fine-tunned steering of the optimization process to the preferred regions of the Pareto-optimal front. In Section 6 the complete INSPM algorithm is presented. In Section 7 some tests are executed to validate the proposed methodology. In Section 8 the conclusions and some directions for future work are presented.

Section snippets

Progressive preference articulation in EMO

Some recent works have developed EMO algorithms with progressive preference articulation. Deb and Sundar [3] combined a preference-based strategy with an EMO methodology. Using an iteration of the proposed Reference-point-based NSGA-II (R-NSGA-II) procedure, in which the DM supplies one or more reference points, the algorithm is able to find a preferred set of solutions near the reference points. Those authors comment that with a number of trade-off solutions in the region of interest the DM

Multi-objective optimization problem

A multi-objective optimization problem can be written asminf(X)=(f1(X),f2(X),,fM(X))s.t.gi(X)0,i=1,2,,khi(X)=0,i=1,2,,rin which fi,i=1,2,,M are the objective functions, gi are the inequality constraint functions, hi are the equality constraint functions, and X=(x1,x2,,xN) is the vector of decision variables. The feasible set, denoted by F, is composed of the vectors X that satisfy all constraints. The solution set of this problem is defined by the concept of dominance. A vector XF is

NN-DM method

The methodology for the construction of a function which models the DM’s preferences is compatible with the Multi-Attribute Utility Theory (MAUT) [6]. The MAUT theory assumes that there is a utility function U which represents the DM’s preferences. In this paper, the NN-DM method is responsible for providing an approximation U^ of the utility function U. This approximation – also called NN-DM model – is built from a partial ranking method based on the ordinal information provided by the DM and

NN-DM method and NSGA-II

This section presents the adaptation performed in the NSGA-II algorithm to allow it to indicate the preferred regions according to a DM’s model. Section 5.1 reviews the Dynamic Crowding Distance (DCD) – an improvement in the original crowding distance (CD) which allows the achievement of a Pareto-optimal front with well-distributed solutions. Section 5.2 introduces the Neural Network Dynamic Crowding Distance (NN-DCD) – a crowding distance weighted by the NN-DM model which allows the INSPM

INSPM algorithm

The next sections explain the implementation details related to the adapted NN-DM method employed to construct the NN-DM model within the INSPM algorithm. The process starts with the initialization of the genetic algorithm and the construction of an initial NN-DM model. While the INSPM algorithm evolves new estimates of the NN-DM model are constructed to maintain the model updated according to the current Pareto-optimal front estimate. After a pre-established number of generations the final

Results

This section presents the results obtained by INSPM in the following situations: (i) guided by the DM’s utility function U (Section 7.1), (ii) guided by the NN-DM model U^ (Section 7.2) and (iii) compared with iTDEA (Section 7.3). The DM’s utility function U is simulated here considering the function given by Eq. (10):U(x)=exp(-x·M·xt)in which x represents an alternative. Three different M matrices are chosen to simulate different kinds of preferences: a balanced preference between the two

Conclusions

This paper proposed an algorithm for interactive multi-objective optimization – the INSPM – based on the NSGA-II algorithm with the replacement of the usual crowding distance operator for a dynamic crowding distance weighted by a DM’s preference model (the NN-DM model). The NN-DM model is constructed using ordinal information provided by the DM about her/his preferences acquired along the algorithm interactions with the DM. The INSPM requires a number of queries from the DM compatible with

Acknowledgements

This work was supported by the Brazilian agencies CAPES, CNPq and FAPEMIG, and by a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme.

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