Robust model

Filters are designed to remove unwanted information, but do not address the fact that processes involve few events monitored by many measurements. Many chemical processes are well instrumented and are capable of producing many process measurements. However, there are far fewer independent physical phenomena occurring than there are measured variables. This means that many of the process variables must be highly correlated because they are reflections of a limited number of physical events. Eliminating this redundancy in the measured variables decreases the contribution of noise and reduces the dimensionality of the data. Model robustness and predictive performance also require that the dimensionality of the data be reduced. 

SuLEA, T., Oprea, T.I., Muresan, S., and Chan, S.L. Different method for steric field evaluation in CoMEA improves model robustness./. Chem. Inf. Comput. Sci. 1997, 37, 1162-1170. 

A widely used approach to establish model robustness is the randomization of response [25] (i.e., in our case of activities). It consists of repeating the calculation procedure with randomized activities and subsequent probability assessments of the resultant statistics. Frequently, it is used along with the cross validation. Sometimes, models based on the randomized data have high q values, which can be explained by a chance correlation or structural redundancy [26]. If all QSAR models obtained in the Y-randomization test have relatively high values for both and LOO (f, it implies that an acceptable QSAR model cannot be obtained for the given dataset by the current modeling method. 

It is desirable to demonstrate that the proposed stochastic formulations provide robust results. According to Mulvey, Vanderbei, and Zenios (1995), a robust solution remains close to optimality for all scenarios of the input data while a robust model remains almost feasible for all the data of the scenarios. In refinery planning, model robustness or model feasibility is as essential as solution optimality. For example, in mitigating demand uncertainty, model feasibility is represented by an optimal solution that has almost no shortfalls or surpluses in production. A trade-off exists... 

In Chapter 3 of this book we discussed the problem of multisite refinery integration under deterministic conditions. In this chapter, we extend the analysis to account for different parameter uncertainty. Robustness is quantified based on both model robustness and solution robustness, where each measure is assigned a scaling factor to analyze the sensitivity of the refinery plan and integration network due to variations. We make use of the sample average approximation (SAA) method with statistical bounding techniques to generate different scenarios. 

Risk is modeled in terms of variance in both prices of imported cmde oil CrCosta and petroleum products Pry/, represented by first stage variables, and forecasted demand DRef, yr, represented by the recourse variables. The variability in the prices represents the solution robustness in which the model solution will remain close to optimal for all scenarios. On the other hand, variability ofthe recourse term represents the model robustness in which the model solution will almost be feasible for all scenarios. This technique gives rise to a multiobjective optimization problem in which... 

This chapter addresses the planning, design and optimization of a network of petrochemical processes under uncertainty and robust considerations. Similar to the previous chapter, robustness is analyzed based on both model robustness and solution robustness. Parameter uncertainty includes process yield, raw material and product prices, and lower product market demand. The expected value of perfect information (EVPI) and the value of the stochastic solution (VSS) are also investigated to illustrate numerically the value of including the randomness of the different model parameters. 

Remark 2 (Algorithmic approach) The MINLP model for the illustrative example can be solved using the v2-GBD by projecting on the binary variables, z, z2, z ,, zuzb. Kokossis and Floudas (1990) proposed an effective initialization procedure that makes the solution of the MINLP model robust. This initialization strategy consists of the following steps ... 

Mathematical modeling of physical processes in fuel cells inevitably involves some assumptions that may or may not be valid under all circumstances. Furthermore approximations have to be introduced to make the computational models robust and tractable. These approximations in the mathematical models lead to the so called modeling errors . That is if the equations posed are solved exactly, the difference between this exact solution and the corresponding true but usually unknown physical reality is known as the modeling error. However, it is rarely the situation that the solution to the mathematical models is exact due to the inherent numerical errors such as round off errors, iteration convergence and discretization errors, among oth-... 

In practice decision makers typically are risk averse and the expected value approach does not take into account the variability of the solutions obtained under the probability distributions or scenarios considered for the uncertain parameters. Rosenhead et al. (1972) introduced the aspect of robustness as a criterion for strategic planning to address this issue. Building on the notion of robustness, Mulvey et al. (1995) developed the concept of robust optimization distinguishing between two different types of robust models. A model is solution robust if the solution obtained remains close to optimality for any realization of the uncertain parameters. The model itself is robust if it remains (almost) feasible for any realization of the uncertain parameters (model robust).36 Here, only solution robustness is of interest as the most important elements of uncertainty in production network design, namely demand volumes, costs, prices and exchange rates, should not lead to infeasibility problems under different scenarios considered. 

Assuming that it is often not possible to obtain a feasible solution under all possible realizations of the uncertain parameters, Mulvey et al. use a multicriteria objective function that penalizes infeasibilities to trade off model robustness and solution robustness. 

Cook, R. D. and Nachtsheim, C. J. (1982). Model robust, linear-optimal design A review. Technometrics, 24, 49-54. 

A useful measure of the estimation capability of a design over a class of models is estimation capacity, which was introduced by Sun (1993) and then used by Li and Nachtsheim (2000) in a criterion for the construction of model-robust factorial designs. Estimation capacity (EC) is defined as... 

Let ei (d) denote the efficiency of design d when the true model fT = /, where /, is the ith model in some model space. A design d is model robust for F if it has maximum average weighted efficiency. We write... 

Table 2. Design indices for 16-run EC-optimal orthogonal designs over four model spaces of Li and Nachtsheim (2000) (for model-robust orthogonal designs), Bingham...

Model-robust orthogonal designs Model-robust parameter designs Addelman A Sun h ... 

Table 3. Summary of properties of 20-run model-robust orthogonal designs pq denotes the percentage of designs for which ECq = 100% ave( C ) is the average ECq over all designs with / factors ...

Cheng, C. S., Steinberg, D. M., and Sun, D. X. (1999). Minimum aberration and model robustness for two-level fractional factorial designs. Journal of the Royal Statistical Society B, 61, 85-93. 

Jones, B., Li, W., Nachtsheim, C. J., and Ye, K. Q. (2005). Model discrimination—Another perspective of model-robust designs. Journal of Statistical Planning and Inferences. In press. 

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