Most, if not all, real-world applications require finding the best solutions to meet a set of objectives. Hence, solving various kinds of optimization problems has been a focal point in scientific and computational study. There are significant results available in the literature on computational optimization.

Different from many other optimization problems, this paper tackles a challenging multi-objective optimization problem in which both quantitative and qualitative objectives need to be optimized. The search domain may not be continuous but discrete within a subspace in an n-dimensional Euclidean space. Furthermore, the objectives can be implicit with uncertainty. The authors call such an optimization problem a hybrid multi-objective optimization problem. According to them, there are not many available computational algorithms available to solve this kind of challenging optimization problem.

To address this challenge, the authors reported several intelligent, practical approaches. Among them, the most interesting one involves using interval-valued parameters to handle uncertainty and to model implicit categorical objectives. Through clustering potential solutions during a machine learning process, the proposed genetic computing algorithm uses a similarity-based strategy, which estimates interval valued objectives to enlarge the searching space efficiently and effectively. Using a precision method, users can specify their interval preferences easily in the course of approaching optimal solutions. By taking an evolutionary computing approach, instead of an analytic approach, theoretically, computers can learn and approach acceptable solutions with limited human intervention. With a sorting scheme according to the user’s preferences, the algorithm guides the search toward the user’s preferred region.

As a case study, the authors applied the algorithm on an interior layout design optimization problem with parameter discretization within intervals. In the case study, there are two objective functions. One of them is numeric. The other is the user’s aesthetic requirements, which of course are uncertain and implicit. As reported in this paper, the algorithm successfully solved the problem with somewhat better performance, though not very significant, on almost all indicators comparing against those obtained with four other methods.

This paper should benefit readers working on evolutionary computing and optimization. However, people working in other areas of computing such as data sciences, machine learning, and others may also benefit. This is because using intervals to model uncertainty and categorical objectives can be applied to a variety of applications. For instance, in data science and machine learning, mixed quantitative and qualitative features and data uncertainty are not easy to handle. Intervals have been successfully applied to encounter the kinds of objectives in optimization that are reported in this paper. This approach should be applicable to other areas of computing as well.