The deployment of deep neural network (DNN) models in software applications is increasing rapidly with the exponential growth of artificial intelligence. Currently, such models are deployed manually by developers in the cloud considering several user requirements, while the decision of model selection and user assignment is difficult to take. With the rise of edge computing paradigm, companies tend to deploy applications as close as possible to the user. Considering this system, the problem of DNN model selection and the inference serving becomes harder due to the introduction of communication latency between nodes. We present an automatic method for DNN placement and inference in edge computing; a mathematical formulation to the DNN Model Variant Selection and Placement (MVSP) problem is presented, it considers the inference latency of different model-variants, communication latency between nodes, and utilization cost of edge computing nodes. Furthermore, we propose a general heuristic algorithm to solve the MVSP problem. We provide an analysis of the effects of hardware sharing on inference latency, on an example of GPU edge computing nodes shared between different DNN model-variants. We evaluate our model numerically, and show the potentials of GPU sharing, with decreased average latency by 33% of millisecond-scale per request for low load, and by 21% for high load. We study the tradeoff between latency and cost and show the pareto optimal curves. Finally, we compare the optimal solution with the proposed heuristic and showed that the average latency per request increased by more than 60%. This can be improved using more efficient placement algorithms.