## 2 Stochastic Inequality Constrained Optimization Problems
### Difficulty of the Problem  
Solving the SICOPs is more difficult because of (i) time-consuming fitness evaluation, (ii) a large solution space, and (iii) satisfaction of all stochastic inequality constraints. To resolve issues (i) to (iii) simultaneously, the OO method [11] is an attempt to circumvent these difficulties at least for the initial parts of optimization search. OO is adopted to supplement existing search approaches, but is not itself a search method. OO method can greatly reduce the computational complexity by emphasizing order rather than value while acquiring a good enough solution at a very high probability. The OO method has three basic steps. Firstly, a sampling subset is created by using a crude estimate to assess all solutions. A crude estimate provides a computationally fast evaluation of the performance for each solution. OO method specifies that order of solutions can be preserved even when evaluating by a crude estimate [11]. Secondly, a promising subset is selected from the sampling subset. Finally, candidate solutions in the promising subset are assessed using a precise estimate. A precise estimate can provide an accurate evaluation of a solution's performance. The candidate solution with the optimal performance in the promising subset is the required good enough solution. There are already several successful applications of OO method to solve the expensive computing optimization problems, such as flow line production system [14], pull-type production system [15], and assemble-to-order systems [16].  
OO method has emerged as an effective approach to simulation and optimization, but the stochastic inequality constraints dramatically reduce the computational efficiency in the SICOPs. Due to the tight computing budget, optimality in the SICOPs is usually traded off by a superb solution that can be obtained in real-time. In fact, using limited computing time to solve for a superb solution of SICOPs is the goal of this work. The key idea is to narrow the solution space stage by stage or gradually restrict the search range through iterative use of the OO method. The goal softening in the OO method is to relax the optimization goal from searching the best solution to seeking for a superb solution with high probability, which can ease the computational burden of finding the optimum. However, when the solution space is huge, the brute force evaluation using a crude estimate in the conventional OO method is not acceptable. To overcome this drawback, we need to use metamodels to ease the computational burden in the search process. Additionally, to explore more of the large solution space, a moderate population size should be kept in the optimization approach. According to the OO method, a metamodel is effective in separating good solutions from the bad. Therefore, we can use a metamodel-assisted optimization approach to find a small set of significant solutions from a large ranged solution space. In addition, if a precise estimate is applied to evaluate all the candidate solutions, it will be computational time-consuming. Therefore, to enhance efficiency, we can use a refined selection technique to replace the use of precise estimate for evaluating each candidate solution in the promising subset.