Skip to main content
SpringerLink
Log in

Hybrid approach for Pareto front expansion in heuristics

  • Special Issue Paper
  • Published:
Journal of the Operational Research Society

Abstract

Heuristic search can be an effective multi-objective optimization tool; however, the required frequent function evaluations can exhaust computational sources. This paper explores using a hybrid approach with statistical interpolation methods to expand optimal solutions obtained by multiple criteria heuristic search. The goal is to significantly increase the number of Pareto optimal solutions while limiting computational effort. The interpolation approaches studied are kriging and general regression neural networks. This paper develops a hybrid methodology combining an interpolator with a heuristic, and examines performance on several non-linear bi-objective example problems. Computational experience shows this approach successfully expands and enriches the Pareto fronts of multi-objective optimization problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1
Figure 2

References

  • Agrawal S, Dashora Y, Tiwari MK and Young-Jun S (2008). Interactive particle swarm: A Pareto-adaptive metaheuristic to multiobjective optimization. IEEE Trans Syst Man Cybern Hum 38: 258–277.

    Article  Google Scholar 

  • Casey K, Lim A and Dozier GV (2006). Evolving general regression neural networks for tsunami detection and response. Proceedings of the IEEE Congress on Evolutionary Computation (CEC2006), IEEE Computer Society, Washington, DC, USA, pp 2451–2458.

  • Coello Coello CA, Lamont GB and Van Veldhuizen DA (2007). Evolutionary Algorithms for Solving Multi-objective Problems. 2nd edn. Springer: New York.

    Google Scholar 

  • Coello Coello CA and Lechuga MS (2002). MOPSO: A proposal for multiple objective particle swarm optimization. Proceedings of the IEEE Congress on Evolutionary Computation (CEC2002), IEEE Computer Society, Washington, DC, USA, pp 1051–1056.

  • Coppersmith D and Winograd S (1990). Matrix multiplication via arithmetic progressions. J Symb Comput 9: 251–280.

    Article  Google Scholar 

  • Cressie NAC (1993). Statistics for Spatial Data. John Wiley & Sons: New York.

    Google Scholar 

  • Deb K (2001). Multi Objective Optimization Using Evolutionary Algorithms. John Wiley & Sons: New York.

    Google Scholar 

  • Fieldsend JE and Singh S (2002). A multi-objective algorithm based upon particle swarm optimisation, an efficient data structure and turbulence. Proceedings of the 2002 U.K. Workshop on Computational Intelligence, Birmingham, UK, pp 37–44, http://empslocal.ex.ac.uk/people/staff/jefields/JF_07.pdf.

  • Garrett A, Dozier GV and Deb K (2007). NEMO: Neural enhancement for multiobjective optimization. Proceedings of the IEEE Congress on Evolutionary Computation (CEC2007), IEEE Computer Society, Washington, DC, USA, pp 3108–3113.

  • Goldberg DE, Korb B and Deb K (1989). Messy genetic algorithms: Motivation, analysis and first results. Complex Syst 4: 415–444.

    Google Scholar 

  • Kennedy J and Eberhart RC (1995). Particle swarm optimization. Proceedings of the IEEE International Conference on Neural Networks, IEEE Service Center, Piscataway, NJ, pp 1942–1948.

  • Kleijnen JPC and van Beers WCM (2005). Robustness of kriging when interpolating in random simulation with heterogeneous variances: Some experiments. Eur J Opl Res 165: 826–834.

    Article  Google Scholar 

  • Krige DG (1951). A statistical approach to some basic mine valuation problems on the Witwatersrand. J Chem Metall Min Soc S Af 52: 119–139.

    Google Scholar 

  • Matheron G (1973). The intrinsic random functions and their applications. Adv Appl Probab 5: 439–468.

    Article  Google Scholar 

  • Miettinen KM (1999). Nonlinear Multiobjective Optimization. Kluwer Academic Publishers: Boston.

    Google Scholar 

  • Mitchell TM (1997). Machine Learning. McGraw Hill: Boston.

    Google Scholar 

  • Olea RA (1974). Optimal contour mapping using universal kriging. J Geophys Res 79: 695–702.

    Article  Google Scholar 

  • Polat O and Yıldırım T (2008). Genetic optimization of GRNN for pattern recognition without feature extraction. Expert Syst Appl 34: 2444–2448.

    Article  Google Scholar 

  • Reyes-Sierra M and Coello Coello CA (2006). Multi-objective particle swarm optimizers: A survey of the state-of-the-art. Int J Comput Intell Res 2: 287–308.

    Google Scholar 

  • Sacks J, Welch WJ, Mitchell TJ and Wynn HP (1989). Design and analysis of computer experiments. Stat Sci 4: 409–435.

    Article  Google Scholar 

  • Schaffer JD (1984). Some experiments in machine learning using vector evaluated genetic algorithms. PhD thesis, Vanderbilt University.

  • Simpson TW, Mauery TM, Korte JJ and Mistree F (2001). Kriging models for global approximation in simulation-based multidisciplinary design optimization. AIAA J 39: 2233–2241.

    Article  Google Scholar 

  • Skriver AJV and Andersen KA (2003). The bi-criterion semi-obnoxious location (BSL) problem solved by an ɛ approximation. Eur J Opl Res 146: 517–528.

    Article  Google Scholar 

  • Specht DF (1991). A general regression neural network. IEEE T Neural Networ 2: 568–576.

    Article  Google Scholar 

  • Strassen V (1969). Gaussian elimination is not optimal. Numer Math 13: 354–356.

    Article  Google Scholar 

  • Trochu F (1993). A contouring program based on dual kriging interpolation. Eng Comput 9: 160–177.

    Article  Google Scholar 

  • Werner H and Obach M (2001). New neural network types estimating the accuracy of response for ecological modeling. Ecol Model 146: 289–298.

    Article  Google Scholar 

  • Wu J-D and Ye S-H (2009). Driver identification based on voice signal using continuous wavelet transform and artificial neural network techniques. Expert Syst Appl 36: 1061–1069.

    Article  Google Scholar 

  • Yapicioglu H, Dozier GV and Smith AE (2006). Neural network enhancement of multi-objective evolutionary search. Proceedings of the IEEE Congress on Evolutionary Computation (CEC2006), IEEE Computer Society, Washington, DC, USA, pp 1909–1915.

  • Yapicioglu H, Smith AE and Dozier GV (2007). Solving the semi-desirable facility location problem using bi-objective particle swarm. Eur J Opl Res 177: 733–749.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Anadolu University, Eskisehir, Turkey

    H Yapicioglu

  2. Auburn University, Auburn, USA

    H Liu & A E Smith

  3. North Carolina A&T University, Greensboro, USA

    G Dozier

Authors
  1. H Yapicioglu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A E Smith
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. G Dozier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H Yapicioglu.

Appendices

Appendix A

Test functions

Test Problem 1

Where d(·) is the Euclidean distance between the fixed point a i and point X.

Test Problems 2 and 3

where

Where d(·) is the distance (Euclidean for test Problem 2 and rectilinear for test Problem 3) between fixed point a i and a facility located at point X. M=200, m=1, d 1 =10 and d 2 =30.

Test Problems 4 and 5

Where d(·) is the distance that is Euclidean for test Problem 4 and rectilinear for test Problem 5 between the fixed point a i and a facility located at point X.

Test Problem 6

Where d(·) is the Euclidean distance between the fixed point a i and a facility located at point X and b=−2.

The data for the first five test problems are given in Table A1.

Table A1 Data for the first five test problems
Full size table

Appendix B

Pareto set and Pareto front figures

Figures B1, B2, B3, B4, B5, B6, B7, B8, B9, B10, B11, B12, B13, B14, B15, B16, B17, B18.

Figure B1
figure 3

Pareto front obtained by PSO (Problem 1).

Full size image
Figure B2
figure 4

Merged Pareto front obtained by kriging and PSO (Problem 1).

Full size image
Figure B3
figure 5

Merged Pareto front obtained by GRNN and PSO (Problem 1).

Full size image
Figure B4
figure 6

Pareto front obtained by PSO (Problem 2).

Full size image
Figure B5
figure 7

Merged Pareto front obtained by kriging and PSO (Problem 2).

Full size image
Figure B6
figure 8

Merged Pareto front obtained by GRNN and PSO (Problem 2).

Full size image
Figure B7
figure 9

Pareto front obtained by PSO (Problem 3).

Full size image
Figure B8
figure 10

Merged Pareto front obtained by kriging and PSO (Problem 3).

Full size image
Figure B9
figure 11

Merged Pareto front obtained by GRNN and PSO (Problem 3).

Full size image
Figure B10
figure 12

Pareto front obtained by PSO (Problem 4).

Full size image
Figure B11
figure 13

Merged Pareto front obtained by kriging and PSO (Problem 4).

Full size image
Figure B12
figure 14

Merged Pareto front obtained by GRNN and PSO (Problem 4).

Full size image
Figure B13
figure 15

Pareto front obtained by PSO (Problem 5).

Full size image
Figure B14
figure 16

Merged Pareto front obtained by kriging and PSO (Problem 5).

Full size image
Figure B15
figure 17

Merged Pareto front obtained by GRNN and PSO (Problem 5).

Full size image
Figure B16
figure 18

Pareto front obtained by PSO (Problem 6).

Full size image
Figure B17
figure 19

Merged Pareto front obtained by kriging and PSO (Problem 6).

Full size image
Figure B18
figure 20

Merged Pareto front obtained by GRNN and PSO (Problem 6).

Full size image

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yapicioglu, H., Liu, H., Smith, A. et al. Hybrid approach for Pareto front expansion in heuristics. J Oper Res Soc 62, 348–359 (2011). https://doi.org/10.1057/jors.2010.151

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1057/jors.2010.151

Keywords

Search

Navigation