Like many surrogate-based approaches, the solution process of PWAS unfolds in two phases: the initial sampling phase and the active-learning phase (see Fig. ). Due to the nature of our target problems, where experiments and simulations are expensive to run, we assume to operate under a fixed computational budget, i.e., only N max experiments/simulations are allowed. During the initial sampling phase, PWAS draws N init samples from the feasible domain. Different strategies for initial sampling are applied based on the specifics of the problem . Since there is limited or no information about the current problem during the initial sampling stage, static DoE approaches, i.e., space-filling approaches, can be a good starting point, especially when no complex constraint is involved. However, we also stress that using static DoE approaches initially does not restrict the exploration capability of PWAS in the active-learning stage, which will be the case for normal DoE approaches. In general, when only box constraints are present, the Latin Hypercube Sampling (LHS) method is employed . In cases where linear equality and/or inequality constraints exist alongside integer and/or categorical variables, the algorithm initially employs LHS and then discards any infeasible samples. Should the number of feasible samples generated be insufficient, two strategies are used: (1) if there exist only continuous variable, polytope sampler, specifically, the Double Description Method 80 is implemented; (2) if there exist integer/categorical variables, a method involving solving mixedinteger linear programming (MILP) problems sequentially is implemented. These MILP problems utilize the exploration functions discussed later in this section as their objective functions, ensuring adherence to given constraints, and thereby obtaining feasible samples. This strategy is particularly useful when constraints are hard to fulfill by random sampling.