See references below





D-metric, Δ-metric, ∇-metric, Contribution and Coverage metrics


[1] D. Joshua, D. Knowles, L. Thiele, and E. Zitzler, "A tutorial on the performance assessment of stochastic multiobjective optimizers", TIK-Report No. 214, Computer Engineering and Networks Laboratory, ETH Zurich, February 2006.

[2] K. Deb, "A fast and elitist multiobjective genetic algorithm: NSGA-II", IEEE Trans. Evol. Comp., Vol. 6, No. 2, Apr. 2002.

[3] Y. Colette, P. Siarry, "Three new metrics to measure the convergence of meta-heuristics towards the Pareto frontier and the aesthetic of of a set of solutions in biobjective optimization", available online at

[4] C. Erbas, S. Cerav-Erbas, and D. Pimentel, "Multiobjective optimization and evolutionary algorithms for the application mapping problem in multiprocessor system-on- chip design", IEEE Trans. Evol. Comp., Vol. 10, No. 3, pp. 358-374, Jun. 2006.

[5] H. Meunier, E. G. Talbi, and P. Reininger, "A multiobjective genetic algorithm for radio network optimization", CEC, Vol. 1, pp. 317-324, Piscataway, New Jersey, Jul 2000, IEEE Service Center.

[6] E. Zitzler, "Evolutionary algorithms for multiobjective optimization: methods and applications", Master’s thesis, Swiss federal Institute of technology (ETH), Zurich, Switzerland, Nov. 1999


Description of the metrics

In principle, to assess the performance of two multi-objective optimizers, two basic approaches exist in the literature; the attainment function approach, and the indicator approach. To assess just one of criteria such as i) convergence to the Pareto-optimal front, ii) having an uniform distribution of the Pareto front and iii) having a better coverage of the objective space, many quality indicators (metrics) have been proposed.

  1. D-metric is used to assess the convergence to the Pareto-optimal front. Then,this metric needs a reference set.
  2. is approach used to assess the performance of multi-objective algprithm in term of the uniformity criteria.
  3. ∇-metric is used to compare multiobjective optimizers in term of extent criteria.
  4. The contribution metric evaluate the proportion of Pareto solutions given by each front.
  5. The coverage metric evaluate the dominated area>given by each front