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Proactive content caching in a fog radio access network (F-RAN) is an efficient technique used to alleviate delivery delay and traffic congestion. However, the symmetric caching of the content is impractical due to the dissimilarity among the contents popularity. Therefore, in this paper, a multi-objective random caching scheme to balance the successful transmission probability (STP) and delay in wireless backhauled F-RAN is proposed. First, stochastic geometry tools are utilized to derive expressions of the association probability, STP, and average delivery delay. Next, the complexity is reduced by considering the asymptotic STP and delay in the high signal-to-noise ratio (SNR) regime. Then, aiming at maximizing the STP or minimizing the delay, the multi-objective cache placement optimization problem is formulated. A novel projected multi-objective cuckoo search algorithm (PMOCSA) is proposed to obtain the Pareto front of the optimal cache placement. The numerical results show that PMOCSA outperforms the original multi-objective cuckoo search algorithm (MOCSA) in terms of convergence to a feasible Pareto front and its rate. It also shows that the proposed multi-objective caching scheme significantly outperforms the well-known benchmark caching schemes by up to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>40</mn><mo>%</mo></mrow></semantics></math></inline-formula> higher STP and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>85</mn><mo>%</mo></mrow></semantics></math></inline-formula> lower average delay.