Robust optimization in simulation: Taguchi and response surface methodology

Optimization of simulated systems is tackled by many methods, but most methods assume known environments. This article, however, develops a `robust' methodology for uncertain environments. This methodology uses Taguchi's view of the uncertain world, but replaces his statistical techniques by Response Surface Methodology (RSM). George Box originated RSM, and Douglas Montgomery recently extended RSM to robust optimization of real (non-simulated) systems. We combine Taguchi's view with RSM for simulated systems. We illustrate the resulting methodology through classic Economic Order Quantity (EOQ) inventory models, which demonstrate that robust optimization may require order quantities that differ from the classic EOQ

Metamodel variability analysis combining bootstrapping and validation techniques

Research on metamodel-based optimization has received considerably increasing interest in recent years, and has found successful applications in solving computationally expensive problems. The joint use of computer simulation experiments and metamodels introduces a source of uncertainty that we refer to as metamodel variability. To analyze and quantify this variability, we apply bootstrapping to residuals derived as prediction errors computed from cross-validation. The proposed method can be used with different types of metamodels, especially when limited knowledge on parameters’ distribution is available or when a limited computational budget is allowed. Our preliminary experiments based on the robust version of the EOQ model show encouraging results

Statistical Testing of Optimality Conditions in Multiresponse Simulation-based Optimization (Revision of 2005-81)

This paper studies simulation-based optimization with multiple outputs. It assumes that the simulation model has one random objective function and must satisfy given constraints on the other random outputs. It presents a statistical procedure for test- ing whether a specific input combination (proposed by some optimization heuristic) satisfies the Karush-Kuhn-Tucker (KKT) first-order optimality conditions. The pa- per focuses on "expensive" simulations, which have small sample sizes. The paper applies the classic t test to check whether the specific input combination is feasi- ble, and whether any constraints are binding; it applies bootstrapping (resampling) to test the estimated gradients in the KKT conditions. The new methodology is applied to three examples, which gives encouraging empirical results.Stopping rule;metaheuristics;response surface methodology;design of experiments

Bootstrapping the economy -- a non-parametric method of generating consistent future scenarios

The fortune and the risk of a business venture depends on the future course of the economy. There is a strong demand for economic forecasts and scenarios that can be applied to planning and modeling. While there is an ongoing debate on modeling economic scenarios, the bootstrapping (or resampling) approach presented here has several advantages. As a non-parametric method, it directly relies on past market behaviors rather than debatable assumptions on models and parameters. Simultaneous dependencies between economic variables are automatically captured. Some aspects of the bootstrapping method require additional modeling: choice and ransformation of the economic variables, arbitrage-free consistency, heavy tails of distributions, serial dependence, trends and mean reversion. Results of a complete economic scenario generator are presented, tested and discussed.economic scenario generator (ESG); asset-liability management (ALM); bootstrapping; resampling; simulation; Monte-Carlo simulation; non-parametric model; yield curve model

Expected Improvement in Efficient Global Optimization Through Bootstrapped Kriging - Replaces CentER DP 2010-62

This article uses a sequentialized experimental design to select simulation input com- binations for global optimization, based on Kriging (also called Gaussian process or spatial correlation modeling); this Kriging is used to analyze the input/output data of the simulation model (computer code). This design and analysis adapt the clas- sic "expected improvement" (EI) in "efficient global optimization" (EGO) through the introduction of an unbiased estimator of the Kriging predictor variance; this estimator uses parametric bootstrapping. Classic EI and bootstrapped EI are com- pared through various test functions, including the six-hump camel-back and several Hartmann functions. These empirical results demonstrate that in some applications bootstrapped EI finds the global optimum faster than classic EI does; in general, however, the classic EI may be considered to be a robust global optimizer.Simulation;Optimization;Kriging;Bootstrap

Simulation-Optimization via Kriging and Bootstrapping:A Survey (Revision of CentER DP 2011-064)

Abstract: This article surveys optimization of simulated systems. The simulation may be either deterministic or random. The survey reflects the author’s extensive experience with simulation-optimization through Kriging (or Gaussian process) metamodels. The analysis of these metamodels may use parametric bootstrapping for deterministic simulation or distribution-free bootstrapping (or resampling) for random simulation. The survey covers: (1) Simulation-optimization through "efficient global optimization" (EGO) using "expected improvement" (EI); this EI uses the Kriging predictor variance, which can be estimated through parametric bootstrapping accounting for estimation of the Kriging parameters. (2) Optimization with constraints for multiple random simulation outputs and deterministic inputs through mathematical programming applied to Kriging metamodels validated through distribution-free bootstrapping. (3) Taguchian robust optimization for uncertain environments, using mathematical programming— applied to Kriging metamodels— and distribution- free bootstrapping to estimate the variability of the Kriging metamodels and the resulting robust solution. (4) Bootstrapping for improving convexity or preserving monotonicity of the Kriging metamodel.

Design of Experiments: An Overview

Design Of Experiments (DOE) is needed for experiments with real-life systems, and with either deterministic or random simulation models. This contribution discusses the different types of DOE for these three domains, but focusses on random simulation. DOE may have two goals: sensitivity analysis including factor screening and optimization. This contribution starts with classic DOE including 2k-p and Central Composite designs. Next, it discusses factor screening through Sequential Bifurcation. Then it discusses Kriging including Latin Hyper cube Sampling and sequential designs. It ends with optimization through Generalized Response Surface Methodology and Kriging combined with Mathematical Programming, including Taguchian robust optimization.simulation;sensitivity analysis;optimization;factor screening;Kriging;RSM;Taguchi

Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations

We consider stochastic programs where the distribution of the uncertain parameters is only observable through a finite training dataset. Using the Wasserstein metric, we construct a ball in the space of (multivariate and non-discrete) probability distributions centered at the uniform distribution on the training samples, and we seek decisions that perform best in view of the worst-case distribution within this Wasserstein ball. The state-of-the-art methods for solving the resulting distributionally robust optimization problems rely on global optimization techniques, which quickly become computationally excruciating. In this paper we demonstrate that, under mild assumptions, the distributionally robust optimization problems over Wasserstein balls can in fact be reformulated as finite convex programs---in many interesting cases even as tractable linear programs. Leveraging recent measure concentration results, we also show that their solutions enjoy powerful finite-sample performance guarantees. Our theoretical results are exemplified in mean-risk portfolio optimization as well as uncertainty quantification.Comment: 42 pages, 10 figure

Transit Costs and Cost Efficiency: Bootstrapping Nonparametric Frontiers.

This paper explores a selection of recently proposed bootstrapping techniques to estimate non-parametric convex (DEA) cost frontiers and efficiency scores for transit firms. Using a sample of Norwegian bus operators, the key results can be summarised as follows: (i) the bias implied by uncorrected cost efficiency measures is numerically important (close to 25%), (ii) the bootstrapped-based test rejects the constant returns to scale hypothesis (iii) explaining patterns of efficiency scores using a two-stage bootstrapping approach detects only one significant covariate, in contrast to earlier results highlighting, e.g., the positive impact of high-powered contract types. Finally, comparing the average inefficiency obtained for the Norwegian data set with an analogous estimate for a smaller French sample illustrates how the estimated differences in average efficiency almost disappear once sample size differences are accounted for.