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The objective of coalition formation is to partition the agent set that gives the highest utility to the system. Over the past three decades, the process of coalition formation has been applied to various real-life applications where agents need to form efficient groups to accomplish a task. This article presents a study of the state-of-the-art approaches on the applications of coalition formation. In particular, it surveys the algorithmic approaches for optimizing the system's welfare. The algorithms are then analyzed based on a framework that consists of two dimensions: (i) the features of the problem environment, which gives an overview of the complexity level of the environment, and (ii) the features of the problem solver, which gives an overview of the solution quality. Our study analyses the approaches in terms of the framework mentioned above, justifies the use of the approaches in a particular problem setting, presents guidance to choose the right algorithmic approach for a problem at hand, and classifies the state-of-the-art approaches according to their basic working principles. This article also presents possible future directions of work to the research community. This study shows that theoretical models need more research before they can be deployed in the real world.