Combinatorial optimization problems are often considered NP-hard problems in the field of decision science and the industrial revolution. As a successful transformation to tackle complex dimensional problems, metaheuristic algorithms have been implemented in a wide area of combinatorial optimization problems. Metaheuristic algorithms have been evolved and modified with respect to the problem nature since it was recommended for the first time. As there is a growing interest in incorporating necessary methods to develop metaheuristics, there is a need to rediscover the recent advancement of metaheuristics in combinatorial optimization. From the authors’ point of view, there is still a lack of comprehensive surveys on current research directions. Therefore, a substantial part of this paper is devoted to analyzing and discussing the modern age metaheuristic algorithms that gained popular use in mostly cited combinatorial optimization problems such as vehicle routing problems, traveling salesman problems, and supply chain network design problems. A survey of seven different metaheuristic algorithms (which are proposed after 2000) for combinatorial optimization problems is carried out in this study, apart from conventional metaheuristics like simulated annealing, particle swarm optimization, and tabu search. These metaheuristics have been filtered through some key factors like easy parameter handling, the scope of hybridization as well as performance efficiency. In this study, a concise description of the framework of the selected algorithm is included. Finally, a technical analysis of the recent trends of implementation is discussed, along with the impacts of algorithm modification on performance, constraint handling strategy, the handling of multi-objective situations using hybridization, and future research opportunities.

Air pollution is a worldwide problem faced by most countries across the world. Prediction of air pollution is crucial in air quality research since it is related to public health effects. The symmetry concept of fuzzy data transformation from a single point (crisp) to a fuzzy number is essential for the forecasting model. Fuzzy time series (FTS) is applied for predicting air pollution; however, it has a limitation caused by utilizing an arbitrary number of intervals. This study involves predicting the daily air pollution index using the FTS Markov chain (FTSMC) model based on a grid method with an optimal number of partitions, which can greatly develop the model accuracy for air pollution. The air pollution index (API) data, which was collected from Klang, Malaysia, is considered in the analysis. The model has been validated using three statistical criteria, which are the root mean (RMSE), the mean absolute percentage error (MAPE), and the Thiels’ U statistic. Also, the model’s validation has been investigated by comparison with some of the famous statistical models. The results of the proposed model demonstrated outperformed the other models. Thus, the proposed model could be a better option in air pollution forecasting that can be useful for managing air quality.

A Markov chain is commonly used in stock market analysis, manpower planning, and in many other areas because of its efficiency in predicting long run behavior. However, the Air Quality Index (AQI) suffers from not using a Markov chain in its forecasting approach. Therefore, this paper proposes a simple forecasting tool to predict the future air quality with a Markov chain model. The proposed method introduces the Markov chain as an operator to evaluate the distribution of the pollution level in the long term. Initial state vector and state transition probability were used in forecasting the behavior of Air Pollution Index (API) that has been obtained from the observed frequency for one state shift to another. The study explores that regardless of the present status of API, in the long run, the index shows a probability of 0.9231 for a good state, and a moderate and unhealthy state with a probability of 0.0722 and 0.0037, while for very unhealthy and hazardous states a probability of 0.0001 and 0.0009. The outcome of this study reveals that the model development could be used as a forecasting method that able to help government to project a prevention action plan during hazy weather.