Robustness, model/solution

Up to this point, it is assumed that prices are deterministic, which is true for contract demand and procurement but is not necessarily true for spot demand and procurement prices. Therefore, an important value chain planning requirement is the consideration of uncertain prices and price scenarios. Now, uncertain spot demand prices are under consideration and it is illustrated how price uncertainty can be integrated into the model in order to reach robust planning solutions. 

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. 

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. 

The solutions of (3.68)-(3.69) for positive parameters and generic initial conditions are oscillations, of amplitude fixed by the initial conditions, around the fixed point Z — a /a2, P = b2/b. The equation of this family of closed trajectories is ai In Z + b2 In P — biP — a2Z = constant. The oscillations are suggestive of the population oscillations observed in some real predator-prey systems, but they suffer from an important drawback the existence of the continuous family of oscillating trajectories is structurally unstable systems similar to (3.68)-(3.69) but with small additional terms either lack the oscillations, or a single limit cycle is selected out of the continuum. Thus, the model (3.68)-(3.69) can not be considered a robust model of biological interactions, which are never known with enough... 

From the sensitivities and model solution, we can then calculate the gradient of the objective function and the Gauss-Newton approximation of the Hessian matrix. Reliable and robust numerical optimization programs are available to find the optimal values of the parameters. These programs are generally more efficient if we provide the gradient in addition to the objective function. The Hessian is normally needed only to calculate the confidence intervals after the optimal parameters are determined. If we define e to be the residual vector... 

In this section, we describe problems of thermo-elasticity and focus on their robust numerical solution. The considered problems are important for the investigation of nuclear waste repositories, as is illustrated by the simple model problem shown in Figure 1. 

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 best robust solution for the decision-making (with the studied model parameter range and the depreciation years m =10) was found by using an -metric method, where the robust model parameters are obtained nearest to the ideal point. Figure 3. 

The scaling factor, /), is a function of a probability of the worst-case scenario, i.e. Pw = 0.25. The result was also calculated without any scaling (p , = 1). The selected robust optimal solution is closer to the stochastic model solution E(A=0) for smaller and closer to the worst case analysis solution W(A=7Vp) for larger p ,. 

A solution in this case is to use robust models and robust statistics (such as median as measure of location and median of absolute deviation around the median as measure of spread) in estimating PCA parameters. The aim is to construct models and estimates clearly describing the majority of the data. Moreover, construction of robust models allows a proper identification of outlying observations. A review illustrating the basis of robust techniques in data analysis and chemometrics can be found in reference [91]. 

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