Model and Solution Robustness

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 

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In general, the coefficients of variation decrease with smaller values of 02. This is definitely desirable since it indicates that for higher expected profits there is diminishing uncertainty in the model, thus signifying model and solution robustness. It is also observed that for values of 02 approximately greater than or equal to 2, the coefficient of variation remain at a static value of 0.5237, thus indicating overall stability and a minimal degree of uncertainty in the model. 

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. 

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. 

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. 

Fig. 26.2 The graph of obtained Pareto-optimal solutions for the deterministic model and the robust model with F = 5...

Marketers and other analysts have been addressing price in this context for some time, see for example Rao [126] for a review. However, we are particularly interested in the situation when the pricing decisions are incorporated with inventory decisions. In this context, there are a number of researchers to consider pricing and inventory problems that are specific to retail industries, where production decisions are usually not incorporated. Although these problems are contained within the scope of a manufacturing price and inventory problem, focusing on the characteristics of the retail industry can lead to more robust models and solutions specific to the situation. Therefore in this section we address research specific to the area of retail. ... 

The conductor-like screening model (COSMO) is a continuum method designed to be fast and robust. This method uses a simpler, more approximate equation for the electrostatic interaction between the solvent and solute. Line the SMx methods, it is based on a solvent accessible surface. Because of this, COSMO calculations require less CPU time than PCM calculations and are less likely to fail to converge. COSMO can be used with a variety of semiempirical, ah initio, and DFT methods. There is also some loss of accuracy as a result of this approximation. 

Rawlings etal. (1992) analysed the stability of a eontinuous erystallizer based on the linearization of population and solute balanee. Their model did not depend on a lumped approximation of partial differenee equations and sueeess-fully predieted the oeeurrenee of sustained oseillations. They demonstrated that simple proportional feedbaek eontrol using moments of CSD as measurements ean stabilize the proeess. It was eoneluded that the relatively high levels of error in these measurements require robust design for effeetive eontrol. 

One of the reasons why the pair of decreasing values of 0, with a fixed value of 03 leads to increasing profit is due to the decrease in production shortfalls and, at the same time, increase in production surpluses. Typically, the fixed penalty cost for shortfalls is lower than surpluses. A good start would be to select a lower operating value of 0 j to achieve both high model feasibility as well as increased profit. Moreover, lower values of 03 correspond to decreasing variation in the recourse penalty costs, which implies solution robustness. 

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... 

The surface tensions themselves in the GB/SA and MST-ST models were developed by taking collections of experimental data for the free energy of solvation in a specific solvent, removing the electrostatic component as calculated by the GB or MST model, and fitting the surface tensions to best reproduce the residual free energy given the known SASA of the solute atoms. Such a multilinear regression procedure requires a reasonably sized collection of data to be statistically robust, and limitations in data have thus restricted these models to water, carbon tetrachloride, chloroform, and octanol as solvents. 

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