Urmila M. Diwekar

Optimization Under Uncertainty

In previous chapters, we looked at various optimization problems. Depending on the decision variables, objectives, and constraints, the problems were classified as LP, NLP, IP, MILP, or MINLP. However, as stated above, the future cannot be perfectly forcasted but instead should be considered random or uncertain. Optimization under uncertainty refers to this branch of optimization where there are uncertainties involved in the data or the model, and is popularly known as Stochastic Programming or stochastic optimization problems. In this terminology, stochastic refers to the randomness, and programming refers to the mathematical programming techniques like LP, NLP, IP, MILP, and MINLP. In the discrete optimization chapter, we came across probabilistic techniques like Simulated Annealing and Genetic Algorithms; these techniques are sometimes referred to as the stochastic optimization techniques because of the probabilistic nature of the method. In general, however, Stochastic Programming and stochastic optimization involves optimal decision making under uncertainty. For example, consider the LP example stated in Chapter 2 where, instead of having a fixed maximum supply of chemical X2, the supply can be uncertain, as shown in the following Stochastic Programming (optimization) problem.

An Efficient Sampling Approach to Multiobjective Optimization

This paper presents a new approach to multiobjective optimization based on the principles of probabilistic uncertainty analysis. At the core of this approach is an efficient nonlinear multiobjective optimization algorithm, Minimizing Number of Single Objective Optimization Problems (MINSOOP), to generate a true representation of the whole Pareto surface. Results show that the computational savings of this new algorithm versus the traditional constraint method increase dramatically when the number of objectives increases. A real world case study of multiobjective optimal design of a best available control technology for Nitrogen Oxides (NOx) and Sulfur Oxides (SOx) reduction illustrates the usefulness of this approach.

Better Optimization of Nonlinear Uncertain Systems (BONUS): A New Algorithm for Stochastic Programming Using Reweighting through Kernel Density Estimation

A new nonlinear programming algorithm is proposed for stochastic programming problems. This method relies on sampling to estimate the probabilistic objective function and constraints. The computational burden of excessive model calculations for determining the search direction is bypassed through a reweighting method using Kernel Density Estimation. The improvements accomplished by this algorithm called Better Optimization of Nonlinear Uncertain Systems (BONUS) are presented through two real world case studies involving parameter design for off-line quality control of a chemical reactor, and optimal capacity expansion for electric utilities in uncertain markets.

Efficient ant colony optimization for computer aided molecular design: Case study solvent selection problem

Abstract In this paper, we propose a novel computer-aided molecular design (CAMD) methodology for the design of optimal solvents based on an efficient ant colony optimization (EACO) algorithm. The molecular design problem is formulated as a mixed integer nonlinear programming (MINLP) model in which a solvent performance measure is maximized (solute distribution coefficient) subject to structural feasibility, property, and process constraints. In developing the EACO algorithm, the better uniformity property of Hammersley sequence sampling (HSS) is exploited. The capabilities of the proposed methodology are illustrated using a real world case study for the design of an optimal solvent for extraction of acetic acid from waste process stream using liquid–liquid extraction. The UNIFAC model based on the infinite dilution activity coefficient is used to estimate the mixture properties. New solvents with better targeted properties are proposed.

Water networks security: A two-stage mixed-integer stochastic program for sensor placement under uncertainty

This work describes a stochastic approach for the optimal placement of sensors in municipal water networks to detect maliciously injected contaminants. The model minimizes the expected fraction of the population at risk and the cost of the sensors. Our work explicitly includes uncertainties in the attack risk and population density, so that the resulting problem involves optimization under uncertainty. In our formulation, we include the location of a number of sensors as first stage decision variables of a two-stage mixed-integer stochastic linear problem; the second stage evaluates the population at risk for the scenario obtained in the first stage and that information is then used to modify the first stage decisions for the next iteration. Since the model is integer in the first stage, a generalized framework based on the stochastic decomposition algorithm allows us to solve the problem in a reasonable computational time. The paper describes the mixed-integer stochastic model and the algorithmic framework, and compares the deterministic and stochastic optimal solutions. The network used as our case study has been derived through the water network simulator EPANET 1.0; four acyclic water flow patterns are considered. Results show a significant effect of uncertainty in sensor placement and total cost.

Stochastic maximum principle for optimal control under uncertainty

Abstract Optimal control problems involve the difficult task of determining time-varying profiles through dynamic optimization. Such problems become even more complex in practical situations where handling time dependent uncertainties becomes an important issue. Approaches to stochastic optimal control problems have been reported in the finance literature and are based on real option theory, combining Ito’s Lemma and the dynamic programming formulation. This paper describes a new approach to stochastic optimal control problems in which the stochastic dynamic programming formulation is converted into a stochastic maximum principle formulation. An application of such method has been reported by Rico-Ramirez et al. ( Computers and Chemical Engineering, 2003, 27, 1867 ) but no details of the derivation were provided. The main significance of this approach is that the solution to the partial differential equations involved in the dynamic programming formulation is avoided. The classical isoperimetric problem illustrates this approach.

Thermodynamic uncertainties in batch processing and optimal control

Abstract Batch distillation is an important separation process for small-scale production especially in pharmaceutical, specialty chemical and biochemical industries. Although batch distillation units require lower capital cost than continuous units, the unsteady state nature of the process, results in higher operating costs. Optimal control in batch distillation is a mode of operation which allows us to optimize the column operating policy by selecting a trajectory for reflux ratio. Due to the uncertainties in thermodynamic models the reflux ratio profile obtained is often suboptimal. Recently a new method was proposed by Rico-Ramirez et al. [Comput. Chem. Eng. 27 (2003) 1867] to include time-dependent uncertainties in current formulations of batch distillation optimal control for ideal systems. In this paper, a general approach is proposed to handle both dynamic and static uncertainties in thermodynamics for more complex non-ideal systems. The static uncertainties result from the inaccuracies associated with predicting vapor–liquid equilibrium using group contribution methods such as UNIFAC. The unsteady state nature of batch distillation translates these static uncertainties into time-dependent uncertainties. A new Ito process representation is proposed for the dynamic behavior of relative volatility for non-ideal mixtures. Numerical case studies are presented to demonstrate the usefulness of this approach for batch as well as bio-processing.

Real option theory from finance to batch distillation

Batch distillation processes have gained renewed interest because of the recent development in small-scale industries producing high-value-added, low-volume specialty chemicals. The flexibility and unsteady state nature of batch distillation constitute a challenge for the designer. A particularly difficult problem is the optimal control problem involving open loop solution for the reflux ratio profile. This is because of the complexity of the formulation and the large computational effort associated to its solution. The mathematical and numerical complexities of the optimal control problem get worse when uncertainty is present in the formulation. In this work, by applying the optimality conditions from the real option theory based on the Ito’s Lemma [Investment under uncertainity (1994); Memoirs Am. Math. Soc. 4 (1951) 1; Appl. Math. Opt. 4 (1974) 374], the mathematical tools needed to solve optimal control problems in batch distillation columns when uncertainties in the state variables are present have been developed. Furthermore, the coupled maximum principle and NLP approach developed by Diwekar [Am. Inst. Chem. Eng. J. 38 (1992) 1551] has been extended for solving the optimal control problem in the uncertain case. This new algorithm has been implemented in the MultiBatchDS batch distillation process simulator. Finally, a numerical case-study is presented to show the scope and application of the proposed approach. # 2003 Elsevier Ltd. All rights reserved.

Uncertainties in parameter estimation and optimal control in batch distillation

Abstract Optimal control problems in batch distillation involve finding a trajectory for the reflux ratio so as to maximize a performance index. Then the controller is asked to follow this trajectory in an open loop fashion. It is important to minimize the effect of uncertainties in thermodynamic models on the optimal control profiles to achieve a better operating performance. The non-linear parameter estimation problem in vapor–liquid equilibrium modeling involves determining the values of model parameters, which provide the best fit to experimental data. It was shown previously by Gau et al. [Fluid Phase Equilibria, 168, 1–8, (2000)] that, using a global optimization procedure based on interval-Newton technique combined with interval-branch-and-bound can significantly reduce the error between the predicted and experimental data. Using this method, it was also shown that for some of the data sets published in DECHEMA, the parameters estimated correspond to local minima. The effect of locally and globally optimal parameter estimates on batch distillation optimal control profiles is demonstrated in this work. Since batch distillation is a dynamic process, the static (parametric) uncertainties are translated into time-dependent uncertainties. The time-dependent changes in relative volatility for the two cases are analyzed and represented by Ito processes. Next, the optimal control problem is solved using the maximum principle and NLP approach. Numerical case studies show that using globally optimal parameter estimates versus locally optimal parameter estimates results in a better product yield and the minimum error between the specified purity and the purity that is achieved.

Model-based approach to study the impact of biofuels on the sustainability of an ecological system

The importance and complexity of sustainability have been well recognized and a formal study of sustainability based on system theory approaches is imperative as many of the relationships between various components of the ecosystem could be nonlinear, intertwined and non-intuitive. A mathematical model capable of yielding qualitative inferences can serve as an important tool for policy makers as it can be simulated under various important scenarios and also help in evaluating different strategies and technologies. In this article, we consider a simplified ecological food web which comprises a macro-economic system, an industrial production sector, an energy generation sector, and elements of a human society along with a rudimentary legal system. The energy sector is designed to supply energy to the other components of the ecosystem either by using a finite, non-renewable energy source or by a combination of non-renewable source and biomass. Many of the components of the ecosystem depend directly or indirectly on the biomass used for energy production. Subsequently, this model is used to study the impact of using biomass for the production of energy on the sustainability of other components of ecosystem. We have also simulated the model under two commonly foreseen scenarios of population explosion and consumption increase to understand the effect of using biomass for the production of energy on the sustainability of the various components of the system.