Recently, Network Function Virtualization (NFV) has drawn attentions of many network researchers with great deal of flexibilities, and various network service chains can be used in an SDN/NFV environment. With the flexibility of virtual middlebox placement, how to place virtual middleboxes in order to optimize the performance of service chains becomes essential. Some past studies focused on placement problem of consolidated middleboxes which combine multiple functions into a virtual middlebox. However, when a virtual middlebox providing only a single function is considered, the placement problem becomes much more complex. In this paper, we propose a new heuristic method, the gradual switch clustering based virtual middlebox placement method, in order to improve the performance of service chains, with the constraints of end-to-end delay, bandwidth, and operation cost of deploying a virtual middlebox on a switch. The proposed method gradually finds candidate places for each type of virtual middlebox along with the sequential order of service chains, by clustering candidate switches which satisfy the constraints. Finally, among candidate places for each type of virtual middlebox, the best places are selected in order to minimize the end-to-end delays of service chains. The evaluation results, which are obtained through Mininet based extensive emulations, show that the proposed method outperforms than other methods, and specifically it achieves around 25% less end-to-end delay than other methods.