For most complex systems, the decision maker needs to make the optimal decision solution in real time with the changing exogenous information in order to minimize cost or maximize profit. While the exogenous information, commonly called covariates or contexts, will influence the performance of feasible solutions, which means that the optimal solution varies as covariates. In addition, with the inherent complexity and uncertainty pervasive in them, these complex systems are rendered intractable to analytical modeling and analysis. However, as the exogenous environment evolves quickly, a large amount of information is unknown. In our research, we adopt correlation among solutions and correlation among covariates, which help share sampling information. Following the Bayesian approach, we develop an efficient sampling policy, integrated knowledge gradient with correlation (CIKG) policy, to search the alternative-covariates pairs.We use multivariate output Gaussian processes with nonseparable covariates structures, linear model of coregionalization (LMC), as a prior belief on unknown performance of alternatives. Finally, a bound is given for the suboptimality. And we prove that, as the sampling budgets grow to infnity, the uncertainties of all underlying true performances vanish, and the optimal solution can be found almost surely for any covariates.