VIII. RISK-BASED DESIGN OPTIMIZATION METHOD UTILIZING M-E DESIGN
VIII.3 Surrogate Model Construction: Adaptive Training Point Selection Process
To efficiently construct a surrogate model that accurately predicts pavement performance across the entire domain space, an adaptive selection technique is presented. The location of training points for the surrogate model is determined through an optimization routine, combining both an exploration and exploitation optimization process. The exploration process improves the predictive accuracy of the GP across the entire domain space and guarantees that the GP is accurate within a specified tolerance across the space. The exploitation routine refines the GP model around a local optimum to provide greater accuracy in a specific area of interest.
VIII.3.1 Quantity of Training Points (NTP)
A Latin Hypercube (LHC) sampling routine is utilized to generate potential training points. The development of the potential points is computationally inexpensive. Training values (outputs) for the surrogate model are required only when selection of a training point has been made and are found utilizing the MEPDG design software, therefore selection of the training points does not require evaluation of the MEPDG functions.
The surrogate model is initialized with randomly chosen points, a sub-set selected from the full set of potential training points, and exploration and exploitation routines are performed (in parallel) until convergence criteria is reached for both methods. The training points not selected in the initialization routine are considered as candidate points which can become training points through the exploration and exploitation routines. This process of pre-selecting candidate points by the LHC sampling method is not required.
New training points could be selected as any feasible solution in the domain space. The LHC process was utilized here to reduce the computational cost associated with the exploration routine.
VIII.3.1.a Exploration Routine
The Exploration routine explores the design domain and selects additional training points that will most significantly improve the accuracy of the model predictions across the entire design space. Improvement in accuracy is defined in this routine as a reduction in the GP variance. This algorithm selects a new training point in a region of the domain
where the GP variance is a maximum, otherwise stated as the maximum distance from all other training points.
This exploration routine chooses the next potential training point based on the average GP variance for that candidate point across all distress models. The improvement of the GP through the exploration process is quantified by the variance of a set of verification points randomly chosen across the domain space. The verification points are not utilized as training points or candidate points, therefore maintaining consistency throughout the construction process. Additional training points, selected from a pre- defined candidate pool, are added to the surrogate model at each iteration of the exploration routine and are chosen as the points that minimize the average variance for the candidate points across all MEPDG distress modes.
VIII.3.1.b Exploitation Routine
The exploitation routine chooses additional training points for the surrogate model utilizing a construction cost function. This process provides model refinement in the region of the design space where a local minimum, and potentially a global minimum, exists.
The selection of the next training point for the surrogate can be performed with a cost function which includes an initial construction cost and an additive maintenance cost, similar to that shown in Equation VIII.6.
Minimize CostPerLaneMile =$20,000*HMAthick +$7,500*GBthick +$125,000pf Ave,
(VIII.6) Equation VIII.6 defines the initial construction cost per lane mile or road as a function of two significant material properties: asphalt and granular base layer thicknesses. The maintenance cost is treated as a function of the average probability of failure across all distress modes.
VIII.3.1.c Stopping Criteria
The minimum required quantity of training points (NTP) for the surrogate model is determined by the stopping criteria for the exploration and exploitation routines. The exploration routine stopping criteria is best defined when the addition of a new training point does not significantly improve the accuracy of the surrogate model across the domain. The selection routine from the pool of candidate points will not always reduce the average GP variance for the remaining candidate points. Although the point of greatest GP variance is removed from the candidate points, the mean of the GP variance is impacted by the change in quantity. Further, the GP model is retrained at each iteration, so the GP variance for each candidate point is likely to change based on the updated GP parameters. Therefore, improvement is defined as a significant reduction in GP variance for the set of verification points which remains constant through the construction process.
The verification points will quantify the performance of the model across the domain, independent of the location and quantity of the training points.
The stopping criterion for the exploitation routine is dependent on the cost function utilized. The cost function utilized in this dissertation is a function of two design variables and the probability of failure for the design. Additional constraints could limit the feasible solutions and provide a stopping criterion for this routine. Stopping criteria could include a budgetary constraint, which for this formulation, would also require a minimum reliability level. The unconstrained problem in the exploitation routine does not restrict the probability of failure for a pavement, which may not be acceptable to some agencies. However, the increased use of warranty contracts for pavement construction can use this routine as a financial decision-making process.
For the analysis here, the exploitation routine is left unconstrained, allowing for a better investigation into the performance of the exploration routine and impact on accuracy in predictions by the GP. The constraints on the exploitation routine will always reduce the number of training points, as a function of feasible cost and performance requirements, which is an important aspect to the purpose of RBDO.