In this chapter, we proposed a two stage sequential framework for the optimization of stochastic simulations with heterogeneous variances. The proposed two-stage framework is based on the Kriging model and iteratively incorporates the optimal computing budget allocation techniques and the modified expected improvement function to drive and improve the estimation of the global optimum. We first illustrate our approach with several numerical examples. The empirical results indicate that the proposed approach is effective in obtaining the optimal solutions and require less computer time than other Kriging based optimization techniques
Table 5.11: Optimal results for 15 different legs.
Port legs Distance(NM) Optimal bunker
inventory safety level
Hakata - Kwangyang 171 1.2%
Kwangyang - Pusan 86 1.1%
Pusan - Shanghai 475 2.3%
Shanghai - Kaoshiung 639 2.4%
Kaoshiung - Hong Kong 350 2.3%
Hong Kong - Yantian 45 0.9%
Yantian - Singapore 1500 3.2%
Singapore -Rotterdam 8339 7.4%
Rotterdam - Hamburg 223 1.6%
Hamburg - Thamesport 325 1.9%
Thamesport - Colombo 6727 7.1%
Colombo - Singapore 1567 3.2%
Singapore - Hong Kong 1460 3.1%
Hong Kong - Kaoshiung 350 2.0%
Kaoshiung - Hakata 904 2.7%
proposed to address stochastic simulations. We also applied the approach to a real complex ocean liner simulation model to determine the optimal bunker inventory safety levels for a fuel management problem. The results from this problem provided invaluable insights on the inventory levels on a service route, and clearly illustrated the interrelatedness of the bunker fuel management and safety levels.
0 0.5 1
Figure 5.7: One dimensional example with proposed algorithm.(Higher variance)
Figure 5.8: Contour plot of the two dimension test function.
Rotterdam Hamburg
Thamesport
Colombo Singapore
Hong Kong Kaohsiung
Hakata Kwangyang
Pusan
Shanghai
Yantian
Singapore
Kaohsiung Hong Kong
6XH]&DQDO
Figure 5.9: AEX service route(distances in nautical miles).
BAYESIAN METAMODELING AND DESIGN APPROACH
FOR STOCHASTIC SIMULATIONS
6.1 Introduction
Based on the discussion in the previous chapters, the kriging model is one of the more promising metamodels for the applications in the computer simulation as it is more adaptable than the regression based models and not as complicated and time consuming as artificial intelligence techniques. However, as indicated in Chapter 3 and Chapter 4, the performance of kriging model highly deteriorates when the noise varies, also seeYin et al.(2008),Li et al.(2010a). Yin et al.(2008) and Ankenman et al. (2010) propose the modified nugget effect model and the stochastic kriging model respectively to address the more general heteroscedastic case. Although these models enable much better modeling of stochastic systems, several issues in parameter estimation arise when the noise levels are high, as mentioned in Chapter 4.
we provide a simple quadratic test function example. Consider the test func-tion y = x2+ ε(x), where the noise function ε(x) is normally distributed with a zero mean and constant variance σε2. Suppose the response surface is unknown and
observations are obtained from 11 evenly distributed locations from -5 to 5. Two different scenarios are considered: a low variance scenario where σε2 = σlow2 = 1, and a high variance scenario where σε2 = σ2high = 10. 50 replicates are taken at each observed location and the simulation response is estimated with a MNEK model of the form Y (x) = Z(x) + "(x) where Z is a Gaussian random process that models the unknown response surface (x2 in this case) and "(x) describes the random noise inherent in the stochastic simulation. To model the unknown response surface, Z is assumed to have a constant mean β0 and the covariance between outputs at two design points, xi and xj, is described by a Gaussian covariance function σZ2R(Z(xi), Z(xj)) = σZ2 exp(−φ(xi − xj)2), where φ is the sensitivity parameter that controls how fast the correlation decays with the dis-tance. With the observed data, the parameters of the model are estimated using the Maximum Likelihood Estimation (MLE) method.
Table 6.1: Empirical mean and standard deviation and quantiles of parameter estimates and the predictor’s output.
estimator scenarios est. mean est. std dev est. 25th est. 75th quantile quantile
βˆ0 low variance 70.35 43.1 62.60 73.80
high variance 69.87 44.6 55.30 81.70
ˆ
σz2 low variance 2722.0 4.8425 × 103 2364.0 3139.0 high variance 2878.0 1.2087 × 104 1628.0 3922.0
φˆZ low variance 0.64 1.1 0.54 0.71
high variance 0.72 1.4 0.50 0.81
Y (xˆ 0) low variance -0.28 0.1 -0.38 -0.19
high variance -0.30 0.9 -0.41 -0.14
Table 6.1 shows the empirical mean, standard deviation and quantiles of the estimated parameters and the model predictor’s output at point x0 = 0 in both the low variance and the high variance scenarios based on 1000 macroreplica-tions. The variabilities of the estimated parameters in the high variance scenario
is significantly larger than in the low variance scenario. As the designs for both scenarios are fixed, this suggests that the random noise affects the parameter estimations. Furthermore, the increase in variance and variability in the param-eters result in an increase of the overall prediction variability. This increase in the overall prediction variability as the simulation noise increases is also noted in Ankenman et al. (2010).
In Chapter 4, we considered only the sensitivity parameter φZas unknown and estimated, and decomposed the overall prediction error into three error compo-nents, the model misspecification error, the random noise error and the parameter estimation error for stochastic simulations. As observed in Yin et al.(2009), the last two components increase as the variability of the random noise increases.
This parameter estimation error is usually ignored when applying the traditional plug-in estimator of the predictor and the prediction error. This can lead to over-confident confidence/prediction intervals which can translate to design decisions that result in the actual system performing very differently from the expected per-formance in the design phase. Figure6.1 plots the plug-in average Mean Squared Prediction Error (MSPE) of the MNEK model and the observed MSPE in the high variance scenario over 1000 macroreplications. As seen, the plug-in MSPE of the MNEK model underestimates the actual observed prediction error. This phenomenon is also highlighted and studied in den Hertog et al. (2005) using a bootstrapping approach for deterministic kriging.
In this chapter, we propose a Bayesian approach for kriging metamodeling for stochastic simulations. This general approach overcomes some of the problems identified in Chapter 4 by appropriately accounting for all parameter uncertain-ties in the model and its predictor. We discuss this metamodeling approach for certain special cases and also for the general case to fully account for all parameter uncertainties and noise levels. This modeling form can also lead to more effective ways of planning simulation experiments by balancing the need for exploring new points or regions in the design space and placing more points at observation areas to reduce the uncertainties in the metamodel form, its parameters and measure-ment errors. This chapter is organized as follows. In Section 2, we first describe our kriging metamodel formulation, then derive the predictive distributions of the
−5 0 5 0.02
0.04 0.06 0.08 0.1 0.12 0.14
x
Prediction Error
Plug−in MSPE for MNEK Observed MSPE
Figure 6.1: Average plug-in MSPE of MNEK and the observed MSPE for the high variance scenario.
model for specific special cases. Then, we propose a Markov Chain Monte Carlo (MCMC) approach to derive the predictive distribution for the general case.