Subset Of Pareto-optimal Solutions Research Articles

ON THE CONCEPT OF VECTOR OPTIMIZATION IN THE TASKS OF MULTICRITERIAL CHOICE

Problems of multi-criteria choice with estimates of applicants by a number of local criteria are considered. Known approaches to the solution of such problems use their peculiarities with the allocation of the corresponding classes of problems and a number of assumptions that significantly limit the possibilities of their practical application. This is the use of fixed scales to compare values that do not correspond to real ranges of values, the assumption of linear relationships between quality assessments and decision attributes, the absence of nonlinear relationships between estimates of local criteria and parameter values of solutions from different ranges. The given circumstances can lead to the loss of adequacy of the taken decisions and cause unacceptable consequences. An approach to procedures of multi-criteria optimization based on the content of optimization as a choice of a solution applicant with the best values of the chosen vector criterion within the elements of the admissible solution set is outlined. To find a solution, it is suggested to use a generalized criterion in the form of a scalar product of vectors, one of which contains components of the values of local criteria for the solution pretender, and the other is a virtual vector of the target reference in a given space with the best or predicted values. To compare the applicants, we propose approaches to forming a subset of Pareto optimal solutions as a set of solutions by consistent local criteria, using a series of concessions on the local criteria represented by mathematical models with the allocation of alternatives to change the values of local criterion estimates, allowing the formation of consistent solutions. The results of the analysis allow us to use the proposed approaches to form a subset of effective solutions in the decision support subsystems, providing operational management of dynamic processes of service and production.

Ra-dominance: A new dominance relationship for preference-based evolutionary multiobjective optimization

While traditional Pareto-based evolutionary multi-objective optimization (EMO) algorithms have shown an excellent balance between convergence and diversity on a wide range of practical problems with two or three objectives in real applications, the decision maker (DM) is interested in a unique set of solutions rather than the whole population on Pareto optimal front (POF). In addition, Pareto-based EMO algorithms have some shortcomings in dealing with many-objective problems because of insufficient selection pressure toward trade-off solutions. Due to the above, it is crucial to incorporate DM preference information into EMO and seek a representative subset of Pareto optimal solutions with an increase in the number of objectives. This paper proposes a new dominance relationship, called Ra-dominance, which can improve diversity among the Pareto-equivalent solutions increase the selection pressure in evolutionary process. It has the ability to guide the population toward areas more responsive to the needs of the DM according to a reference point and preference angle. We use the new dominance relationship in the NSGA-II algorithm, and the efficacy and usefulness of the modified procedure are assessed through two- to ten-objective problems. Experimental results show that the algorithm applying this new dominance relationship is highly competitive when compared with four state-of-the-art preference-based EMO methods.

Preference-based cone contraction algorithms for interactive evolutionary multiple objective optimization

We introduce a family of interactive evolutionary algorithms for Multiple Objective Optimization (MOO). In the phase of preference elicitation, a Decision Maker (DM) is asked to compare some pairs of solutions from the current population. Such holistic information is represented by a set of compatible instances of achievement scalarizing or quasi-convex preference functions, which contribute to the construction of preference cones in the objective space. These cones are systematically contracted during the evolutionary search, because an incremental specification of the DM's pairwise comparisons is progressively reducing the DM's most preferred region in the objective space. An inclusion of evolved solutions in this region is used along with the dominance relation to revise an elitism principle of the employed optimization algorithm. In this way, the evolutionary search is focused on a subset of Pareto optimal solutions that are particularly interesting to the DM. We investigate, moreover, how the convergence is influenced by the use of some pre-defined and newly proposed self-adjusting (dynamic) interaction patterns. We also propose a new way for visualizing the progress of an evolutionary search. It supports understanding the differences between effects of a selection pressure imposed by various optimization algorithms. • We propose interactive evolutionary algorithms for multiple objective optimization. • Interactive cone contraction is used to bias the evolutionary search. • The proposed methods outperform selected state-of-the-art algorithms. • We introduce a novel approach for visualizing a progress in the evolutionary search. • Novel interaction patterns are used to control the questioning of a decision maker.

Preference-based evolutionary algorithm for airport surface operations

In addition to time efficiency, minimisation of fuel consumption and related emissions has started to be considered by research on optimisation of airport surface operations as more airports face severe congestion and tightening environmental regulations. Objectives are related to economic cost which can be used as preferences to search for a region of cost efficient and Pareto optimal solutions. A multi-objective evolutionary optimisation framework with preferences is proposed in this paper to solve a complex optimisation problem integrating runway scheduling and airport ground movement problem. The evolutionary search algorithm uses modified crowding distance in the replacement procedure to take into account cost of delay and fuel price. Furthermore, uncertainty inherent in prices is reflected by expressing preferences as an interval. Preference information is used to control the extent of region of interest, which has a beneficial effect on algorithm performance. As a result, the search algorithm can achieve faster convergence and potentially better solutions. A filtering procedure is further proposed to select an evenly distributed subset of Pareto optimal solutions in order to reduce its size and help the decision maker. The computational results with data from major international hub airports show the efficiency of the proposed approach.

Multiobjective evolutionary algorithms based on target region preferences

Abstract Incorporating decision makers' preferences is of great significance in multiobjective optimization. Target region-based multiobjective evolutionary algorithms (TMOEAs), aiming at a well-distributed subset of Pareto optimal solutions within the user-provided region(s), are extensively investigated in this paper. An empirical comparison is performed among three TMOEA instantiations: T-NSGA-II, T-SMS-EMOA and T-R2-EMOA. Experimental results show that T-SMS-EMOA has the best overall performance regarding the hypervolume indicator within the target region, while T-NSGA-II is the fastest algorithm. We also compare TMOEAs with other state-of-the-art preference-based approaches, i.e., DF-SMS-EMOA, RVEA, AS-EMOA and R-NSGA-II to show the advantages of TMOEAs. A case study in the mission planning of earth observation satellite is carried out to verify the capabilities of TMOEAs in the real-world application. Experimental results indicate that preferences can improve the searching ability of MOEAs, and TMOEAs can successfully find nondominated solutions preferred by the decision maker.

Finding preferred subsets of Pareto optimal solutions

Multi-objective optimization algorithms can generate large sets of Pareto optimal (non-dominated) solutions. Identifying the best solutions across a very large number of Pareto optimal solutions can be a challenge. Therefore it is useful for the decision-maker to be able to obtain a small set of preferred Pareto optimal solutions. This paper analyzes a discrete optimization problem introduced to obtain optimal subsets of solutions from large sets of Pareto optimal solutions. This discrete optimization problem is proven to be NP-hard. Two exact algorithms and five heuristics are presented to address this problem. Five test problems are used to compare the performances of these algorithms and heuristics. The results suggest that preferred subset of Pareto optimal solutions can be efficiently obtained using the heuristics, while for smaller problems, exact algorithms can be applied.