Reliability-based design optimization (RBDO) algorithms typically assume a designer's prior knowledge of the objective function along with its explicit mathematical formula and the probability distributions of random design variables. These assumptions may not be valid in many industrial cases where there is limited information on variable variability and the objective function is subjective without mathematical formula. A new methodology is developed in this research to model and solve problems with qualitative objective functions and limited information about random variables. Causal graphs and design structure matrix are used to capture designer's qualitative knowledge of the effects of design variables on the objective. Maximum entropy theory and Monte Carlo simulation are used to model random variables' variability and derive reliability constraint functions. A new optimization problem based on a meta-objective function and transformed deterministic constraints is formulated, which leads close to the optimum of the original mathematical RBDO problem. The developed algorithm is tested and validated with the Golinski speed reducer design case. The results show that the algorithm finds a near-optimal reliable design with less initial information and less computation effort as compared to other RBDO algorithms that assume full knowledge of the problem.