Supply chain simulation tools and techniques: a survey <s> Robustness analysis <s> Two types of sampling plans are examined as alternatives to simple random sampling in Monte Carlo studies. These plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function. <s> BIB001 </s> Supply chain simulation tools and techniques: a survey <s> Robustness analysis <s> The application of computer simulation have been proposed and implemented to solve the problems of variation in integrated manufacturing systems. However, a simulation model only acts as a tool in examining performance. It is essentially a trial and error methodology, and does not directly provide explanations for observed system behaviours. Therefore, in this paper the use of Taguchi's Experiment Design in simulation is studied to solve decision-making problems in integrated manufacturing systems. In particular, we use the case of a steel soakingpit/rolling-mill plant to demonstrate how the approach works. The results reveal that this approach can take into account the evaluation and optimisation of operating conditions in complex systems simultaneously. <s> BIB002
Taguchi is the Japanese engineer who designed cars (for Toyota) that operate satisfactorily in many environments; see  . His method is applied to simulation by  and BIB002 . Next,  use Taguchi's view Figure 2 The simulation model of the current supply chain structure but not his statistical methods, because simulation experiments enable the exploration of many more factors and scenarios than are possible in real-life experiments.  try to minimise expected cost (as in classic optimisation), but also consider cost variance due to environmental disturbances (as Taguchi proposes). For illustration, I focus on the current model. After the SB screening (Section 3.3), only three important controllable factors remain (and six important environmental factors). The challenge is to 'optimise' these controllable factors, denoted by (say) x j ( j = 1, …, k ). A second-order polynomial approximation of the I/O behaviour of the simulation model is: with the overall mean β 0 , the first-order effects β j , the interactions (cross-products) β j; j' and the error terme i -which represents noise caused by both the PRN and the lack of fit of the approximation in Equation (4) in scenario i. To estimate the coefficients of the second-order polynomial in Equation (4),  augment the 2 k full factorial design with a one-factor-at-a-time design, so that each factor is observed at more than two values. For the environmental factors, robustness analysis is not interested in a functional relationship like Equation (4). Following Taguchi,  treat these factors as noise. Unlike Taguchi, they sample environmental scenarios through LHS; BIB001 . Further, as in a Taguchian design, the design ('inner array') for the controllable factors is crossed (combined) with the design ('outer array') for the environmental factors. This crossed design is not replicated (unlike SB; see Section 3.3): the standard error in each design point due to pure replication (using different PRN) turns out to be much smaller than the standard error across the environmental scenarios. The optimisation of Equation (4) for different environmental scenarios should account for the box constraints on the inputs: only changes of 5% and 25% are allowed. Mathematically, these constraints are incorporated through Lagrangian multipliers, which quantify the shadow prices of the constraints. All controllable factors turn out to minimise the mean costs when they are set at their lower boundary values. (Because these bounds are specified rather arbitrarily, I raise the question whether these boundary values should not be revised, and followed-up, by a new analysis?) Next, the response variable y in Equation (4) is replaced by the output variance. One factor again has its optimal value at its lower boundary. The other factors, however, have conflicting optimal values when considering both outputs. Yet, these optimal input values are only estimates, so maybe the truly optimal inputs are the same for both outputs (mean and variance)? To answer this question, a confidence region is needed for the optimal values. Standard confidence regions do not hold, because the estimated optimal simulation inputs are nonlinear functions of the simulation outputs and the corresponding regression estimates. Yet, a confidence region can be computed through bootstrapping. Assuming that the estimated regression parameters corresponding with Equation (4) are normally distributed, the bootstrap resamples from a multivariate normal distribution with a vector of means equal to the original estimates, and with a covariance matrix equal to the estimated matrix that is a standard output of regression software. The resulting bootstrapped estimated regression parameters (say) give estimated optimal inputs . Bootstrapping repeats this sampling (say) 1,000 times, to get a confidence region for the values that minimise the mean and the variance of the output respectively. Using this region, management may select a robust solution that satisfies their preferences. Note: Bootstrapping is computationally inexpensive, compared with the computer effort required to generate the simulation output. Finally, comparing the optimum solution for the controllable factors accounting for many environmental scenarios that are generated through LHS, and the solution accounting for a single scenario, namely the base scenario, suggests that risk considerations do make a difference; see  .