This paper proposes an exact model for the planning and design problem of fog networks. More precisely, a mathematical model is proposed to simultaneously determine the optimal location, the capacity and the number of fog node(s) as well as the interconnection between the installed fog nodes and the cloud. The goal of the model is to minimize the delay in the network and the amount of traffic sent to the cloud data center. To address this multi-objective optimization problem, three optimization techniques are used: the weighted sum, the hierarchical and the trade-off methods. The weighted sum method aggregates all the lone objective functions into a single objective by applying a weighted vector. The hierarchical method takes a sequential approach by tightly constraining the more important objective function. The trade-off method solves a single objective function and translates all other objective functions into constraints. These methods are then compared in terms of average delay, amount of traffic sent to the cloud and amount of CPU time required to find optimal solution(s). Since we are dealing with a multi-objective optimization problem and that multiple optimal solutions can be found, the fuzzy-based mechanism and the hypervolume indicator have been used. Computational results show that as the problem size increases, the delay and the traffic also increase in a linear form; whereas, the solution time increases in non-polynomial time. The weighted sum method was able to achieve the best trade-off results for the delay and the traffic, whereas the hierarchical method was able to return minimum delay but with worse traffic going to the cloud. As the model considers realistic edge device traffic parameters, constraints, and various topology aspects, it can be helpful for the planning and deployment of fog networks and how they operate within a cloud infrastructure.