Statistical Testing of Optimality Conditions in Multiresponse Simulation-Based Optimization (Replaced by Discussion Paper 2007-45)

This paper derives a novel procedure for testing the Karush-Kuhn-Tucker (KKT) first-order optimality conditions in models with multiple random responses.Such models arise in simulation-based optimization with multivariate outputs.This paper focuses on expensive simulations, which have small sample sizes.The paper estimates the gradients (in the KKT conditions) through low-order polynomials, fitted locally.These polynomials are estimated using Ordinary Least Squares (OLS), which also enables estimation of the variability of the estimated gradients.Using these OLS results, the paper applies the bootstrap (resampling) method to test the KKT conditions.Furthermore, it applies the classic Student t test to check whether the simulation outputs are feasible, and whether any constraints are binding.The paper applies the new procedure to both a synthetic example and an inventory simulation; the empirical results are encouraging.stopping rule;metaheuristics;RSM;design of experiments

Statistical Testing of Optimality Conditions in Multiresponse Simulation-Based Optimization (Replaced by Discussion Paper 2007-45)

This paper derives a novel procedure for testing the Karush-Kuhn-Tucker (KKT) first-order optimality conditions in models with multiple random responses.Such models arise in simulation-based optimization with multivariate outputs.This paper focuses on expensive simulations, which have small sample sizes.The paper estimates the gradients (in the KKT conditions) through low-order polynomials, fitted locally.These polynomials are estimated using Ordinary Least Squares (OLS), which also enables estimation of the variability of the estimated gradients.Using these OLS results, the paper applies the bootstrap (resampling) method to test the KKT conditions.Furthermore, it applies the classic Student t test to check whether the simulation outputs are feasible, and whether any constraints are binding.The paper applies the new procedure to both a synthetic example and an inventory simulation; the empirical results are encouraging.

A systematic investigation into the flowback cleanup of hydraulic-fractured wells in unconventional gas plays

This paper conducts an extensive investigation into fracture cleanup efficiency by considering several pertinent parameters instantaneously over a wide practical range. Injection, shut-in and production stages of the fracturing operation were simulated for 32 sets consisting of 113,072 runs. To perform such a large number of simulation runs, a computer code was utilised to routinely read input data, implement the simulation runs and produce output data. In each set (which consists of 4096 runs), instantaneous impacts of twelve different parameters (i.e., fracture and matrix permeability, Brooks matrix capillary pressure (Pc) parameters, and Brooks-Corey relative permeability parameters) were investigated. To sample the domain of variables, full factorial experimental design (two-level FFS) was employed. The linear surface methodology was used to map the simulation output, which is the loss in gas production (GPL), compared to the clean case (i.e., 100% clean-up) after three production periods of 10, 30 and 365 days. The impact of various combinations of fracture fluid injection volume, fracture length, shut-in soaking time, matrix permeability variation range and drawdown on GPL were studied in different sets. Additionally, more simulation sets were performed to capture the impact of hysteresis, layering and mobile formation water on the clean-up efficiency. Results indicated that in line with some literature data, factors that controlled the mobility of FF inside the fracture had the most significant impact on cleanup efficiency. It was also noted that injecting high volumes of FF, into very tight formations significantly delayed clean-up and impaired gas production. The effect of varying other parameters such as extending soaking time or increasing pressure down in such a case delivered negligible GPL improvement. Introducing hysteresis made clean-up slightly faster in all production periods. The impact of the gravity segregation was discussed in this study. Considering the layered systems, it was indicated that in the top layer, the fracture mobility coefficients were more important than the ones in the bottom layer whist capillary pressure seems to become more important in deeper layers compared to the top layers. Additionally, a slower clean-up was observed for sets with larger initial water saturation compared to those cases with immobile water saturation due to the detrimental effect of mobile water on gas production. In some cases, with significantly high values of water saturation, using chemicals (which IFT reducing agents) to reduce Pc could reduce GPL and improve cleanup efficiency. These findings contribute to the further understanding of the fracture fluid cleanup process and provide practical guidelines to achieve economically successful hydraulic fracturing operations, which are popular but expensive for tight and ultra-tight reservoirs

Determination of Cap Model Parameters using Numerical Optimization Method for Powder Compaction

Many advantages are inherent to the successful powder metallurgy (P/M) process especially in high volume manufacturing. The strength/density distribution of the compacted product is crucial to overall success. The finite element analysis (FEA) method has become an effective way to numerically simulate strength/density distribution in a P/M compact. The modified Drucker-Prager cap (DPC) model has been shown to be a suitable constitutive relationship for metal powder compaction simulation. The calibration of the modified DPC model involves a procedure known as a triaxial compression test. Equipment for completing a triaxial compression test on metal powders is neither readily available nor standardized in the P/M industry. A robust calibration procedure that requires only simple experimental tests would increase the usability of the simulation procedure. This research created a universal cost/time-effective calibration method to accurately determine all parameters of a modified DPC model by using a combination of numerical simulation methods, numerical optimization methods and common material testing techniques. The use of the triaxial compression test is eliminated and the new method relies only upon conventional compaction equipment, standard geometry and readily available metallographic techniques. The DPC parameters were determined by applying the proposed method on ferrous powders. The predicted DPC parameters were verified on a compressed product with complex geometry

Pattern Search Ranking and Selection Algorithms for Mixed-Variable Optimization of Stochastic Systems

A new class of algorithms is introduced and analyzed for bound and linearly constrained optimization problems with stochastic objective functions and a mixture of design variable types. The generalized pattern search (GPS) class of algorithms is extended to a new problem setting in which objective function evaluations require sampling from a model of a stochastic system. The approach combines GPS with ranking and selection (R&S) statistical procedures to select new iterates. The derivative-free algorithms require only black-box simulation responses and are applicable over domains with mixed variables (continuous, discrete numeric, and discrete categorical) to include bound and linear constraints on the continuous variables. A convergence analysis for the general class of algorithms establishes almost sure convergence of an iteration subsequence to stationary points appropriately defined in the mixed-variable domain. Additionally, specific algorithm instances are implemented that provide computational enhancements to the basic algorithm. Implementation alternatives include the use modern R&S procedures designed to provide efficient sampling strategies and the use of surrogate functions that augment the search by approximating the unknown objective function with nonparametric response surfaces. In a computational evaluation, six variants of the algorithm are tested along with four competing methods on 26 standardized test problems. The numerical results validate the use of advanced implementations as a means to improve algorithm performance