The problem of determining the critical power spectral density (PSD) function of a partially specified stationary Gaussian load process which maximizes the response of a linear system has been considered. The partial specification of the load is given only in terms of its total average energy. The critical input PSD turns out to be highly narrow banded which fails to capture the erratic nature of the excitation. Consequently, the trade-off curve between the maximum linear system response and the disorder in the input process, quantified in terms of its entropy rate, has been generated. The Pareto optimization theory is used to tackle the conflicting objectives of simultaneous maximization of the system response and the input entropy rate. Consequently, the non-linear multi-objective optimization has been carried out using a Multi-criteria Genetic Algorithm scheme. An illustrative example of determining the critical input of an axially vibrating rod excited by a partially specified stationary Gaussian load process has been considered.