IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 7, NO. 3, JUNE 2003 253 Dynamic Multiobjective Evolutionary Algorithm: Adaptive Cell-Based Rank and Density Estimation Gary G. Yen, Senior Member, IEEE, and Haiming Lu, Member, IEEE Abstract—This paper proposes a new evolutionary approach to multiobjective optimization problems—the dynamic multiob- jective evolutionary algorithm (DMOEA). In DMOEA, a novel cell-based rank and density estimation strategy is proposed to efficiently compute dominance and diversity information when the population size varies dynamically. In addition, a population growing and declining strategies are designed to determine if an individual will survive or be eliminated based on some qualitative indicators. Meanwhile, an objective space compression strategy is devised to continuously refine the quality of the resulting Pareto front. By examining the selected performance metrics on three recently designed benchmark functions, DMOEA is found to be competitive with or even superior to five state-of-the-art MOEAs in terms of maintaining the diversity of the individuals along the tradeoff surface, tending to extend the Pareto front to new areas, and finding a well-approximated Pareto optimal front. Moreover, DMOEA is evaluated by using different parameter settings on the chosen test functions to verify its robustness of converging to an optimal population size, if it exists. Simulations show that DMOEA has the potential of autonomously determining the optimal population size, which is found insensitive to the initial population size chosen. Index Terms—Dynamic population size, multiobjective evolu- tionary algorithm (MOEA), multiobjective optimization, rank/ density estimation approach. I. INTRODUCTION D URING THE past decade, several multiobjective evo- lutionary algorithms (MOEAs) have been proposed and applied in multiobjective optimization problems (MOPs) [1]. These algorithms share the common purpose—searching for a uniformly distributed, near-optimal, and well-extended Pareto front for a given MOP. However, this ultimate goal is far from being accomplished by the existing MOEAs as documented in the literature, e.g., [1]. In one respect, most of the MOPs are very complicated and require computational resources to be homogenously distributed in a high-dimensional search space. On the other hand, those fitter individuals generally have strong tendencies to restrict searching efforts within local areas because of the “genetic drift” phenomenon [2], which results in the loss of diversity. Additionally, most of the existing MOEAs adopt a fixed population size to initiate the evolutionary process. As pointed out in [3], evolutionary algorithms may Manuscript received March 25, 2002; revised August 30, 2002 and January 13, 2003. This work was supported in part by the U.S. Air Force Office of Sci- entific Research under Grant F49620-98-1-0049 and in part by the National Sci- ence Foundation under I/UCRC Measurement and Control Engineering Center. G. G. Yen is with the School of Electrical and Computer Engineering, Okla- homa State University, Stillwater, OK 74078 USA (e-mail: gyen@okstate.edu). H. Lu is with Prediction Corporation, Santa Fe, NM 87505 USA. Digital Object Identifier 10.1109/TEVC.2003.810068 suffer from premature convergence if the population size is too small, whereas an overestimated population size will result in a heavy burden on computation and a long waiting time for fitness improvement. In the cases of single objective (SO) optimization, several methods of determining an optimal population size from dif- ferent perspectives have been proposed [3]–[5]. Since the pur- pose of solving a SO problem is to search for a single optimal solution at the final generation, the distribution characteristics of the final population is not an issue to be concerned. However, in order to solve MOPs, an MOEA needs to approximate a set of nondominated solutions that produces a uniformly distributed Pareto front. In general, the size of the final Pareto set yielded by most MOEAs is bounded by the size of the initial population chosen ad hoc. As indicated in [6], the exact tradeoff surface of an MOP is often unknown a priori. It is difficult to estimate an optimal number of individuals necessary for effective explo- ration of the solution space, compared with a good representa- tion of the tradeoff surface. This difficulty implies that an “esti- mated” size of the initial population may not be appropriate in a real-world application. Therefore, a dynamic population size adjusted autonomously by the online characteristics of popula- tion tradeoff and density distribution information will be more efficient and effective than a constant population size in terms of avoiding premature convergence and unnecessary computa- tional complexity. As highlighted in [6], the issue of dynamic population in MOEAs has not been well attended yet. Although in some elitism-based MOEAs, the main population and elitist archive are separated and updated by exchanging elitists between them, the size of the main population or the sum of the main population and the archive remains fixed [7], [8]. Therefore, either an estimated size of the initial population is needed in some of these algorithms, or a maximum size of the archive is predetermined [9]. Tan et al. [6] proposed an incrementing multiobjective evolutionary algorithm (IMOEA), which de- vises a fuzzy boundary local perturbation technique and a dynamic local fine-tuning method in order to achieve broader neighborhood explorations and eliminate gaps and disconti- nuities along the Pareto front. However, this algorithm adopts a heuristic method to estimate the desired population size, , at generation according to the approximated tradeoff hyperareas of the current generation, but not to the domi- nance and density information of the entire objective space. Therefore, the computation workload may be determined wrongly if the approximation of value is inaccurate, which may force IMOEA to adjust the grid density to reach the incorrect “optimal” population size. Moreover, IMOEA 1089-778X/03$17.00 © 2003 IEEE