Genetic Algorithms For Solving Multi-objective Optimization Problems With Interval Parameters

Multi-objective optimization problems (MOPs) with interval parameters are verypopular and important in real-world applications. However, there has been very feweffective method of solving these problems up to date because of their intervalparameters and multiple objectives. According to different actual demands, threekinds of genetic algorithms (GAs) were proposed to solve the above problems in thisdissertation.First, a GA for obtaining a Pareto set approximation was given to face thecommon demand of a MOP. In this method, through defining a lower limit ofpossibility degree of interval dominance, the dominance relation of an intervalmulti-objective optimization problem (IMOP) based on the above lower limit and theproperty of the corresponding Pareto set are given; the proposed dominance relation isemployed to modify the fast non-dominated sorting approach in NSGA-II, and a novelGA for an IMOP is then developed. Further, after theoretically analyzing theperformance of the proposed method, it was applied to six IMOPs and compared withtwo typical optimization methods. The experimental results confirm its advantages.Then, to meet a decision maker (DM)'s demand for a most preferred solution(set), three GAs incorporating a search-cum-decision-making procedure for solvingIMOPs were studied in the context of two kinds of preference presentations. Byconstructing the theory of a preference polyhedron for an optimization problem withinterval parameters, an interactive genetic algorithm (IGA) for an IMOP based onpreference polyhedron was proposed. The algorithm periodically provides a part ofnon-dominated solutions to a DM, and a preference polyhedron, based on whichdifferent evolutionary individuals with the same ranks are distinguished, is createdtaking the worst solution as vertex in the objective space. Further, a preferencedirection is elicited from the above preference polyhedron, and a metric based on thepreference direction, reflecting the approximation performance of an evolutionaryindividual, is designed to rank different individuals with the same ranks andpreferences. Consequently, an IGA for an IMOP based on the preference direction wasdeveloped. Both of the above methods were applied to four interval bi-objectiveoptimization problems, and compared with an a posterior method. The experimentalresults show that both of the methods are superior than the a posterior method, andthe most preferred solution that fits the DM's preferences can be obtained. Additionally, an IGA was presented to obtain a most preferred solution set meetingthe DM's preferences by utilizing the relative importance of objectives. In thisalgorithm, the preference informations are interactively input, and the preferenceregion in the objective space is then obtained; based on this region, differentevolutionary individuals with the same ranks are further discriminated, to guide thepopulation to evolve towards the DM's really preference regions. The proposedmethod was applied to four IMOPs, and compared with an a priori method and an aposterior method. The experimental results confirm its advantages of the proposedalgorithm, and more optimal solutions which fit the DM's preferences can be derived.Finally, to satisfy the special request of uncertain optimization problems, aset-based GA was proposed to effectively solve many-objective optimizationproblems with interval parameters. In this method, the original optimization problemis transformed into a bi-objective one with deterministic parameters by takinghyper-volume and imprecision as two new objectives; a set-based Pareto dominancerelation is defined to modify the fast non-dominated sorting approach in NSGA-II;moreover, set-based evolutionary schemes are suggested. The proposed method wasapplied to four benchmark many-objective optimization problems with intervalparameters and compared with a state-of-the-art method. The experimental resultsindicate that a trade-off Pareto set approximation between convergence anduncertainty can be found by the proposed method.Three kinds of GAs proposed in this dissertation not only offer several practicalways to solve IMOPs, but also enrich the researches of interval mathematics.