Optimisation: problem

A Classic Optimisation Problem Predicting Crystal Structures... 

At times, it is possible to build an empirical mathematical model of a process in the form of equations involving all the key variables that enter into the optimisation problem. Such an empirical model may be made from operating plant data or from the case study results of a simulator, in which case the resultant model would be a model of a model. Practically all of the optimisation techniques described can then be appHed to this empirical model. 

In general, dynamic programming is an algorithmic scheme for solving discrete optimisation problems that have overlapping subproblems. In a dynamic... 

A common problem in pre-formulation of the cosmetic product including lipstick is the optimisation of the mixture composition aimed to obtain a product with the required characteristic. Mixture design represents an efficient approach for solving such optimisation problem [10]. It has been proved to be an effective tool to select the best lipstick formulation [11]. 

The formulae developed in this section are presented as an illustration of a simple optimisation problem in design, and to provide an estimate of economic pipe diameter that is based on UK costs and in SI units. The method used is essentially that first published by Genereaux (1937). 

Since in this mixture design problem we have to identify a mixture whose constituents perform different functions, i.e., the solvent needs to have high solubility for the solute while the anti-solvent needs to reduce the solubility, we have to solve two different single compound design problems (involving subproblem 1M, 2m and 3M) to identify the candidate solvents and anti-solvents. The mutually miscible pairs are identified in sub-problem 4M and the final optimisation problem is solved in sub-problem 5M. 

Jackson, R. Trans. Inst. Chem. Eng. 45 (1967) T160. An approach to the numerical solution of time-dependent optimisation problems in two-phase contacting devices. 

In addition to the above technical improvement and new technologies to emerge, some UGS owners operating several UGS facilities (like Gaz de France operating 13 UGS facilities in France, 6 in Germany and involve in 3 others UGS facilities in Slovak Republic) have done R D works in order to solve the technical and economical optimisation problem due to the multiplicity of available sites [18] [19] [20] ... 

The performance criteria of a batch distillation column can be measured in terms of maximum profit, maximum product or minimum time (Mujtaba, 1999). In distillation, whether batch, continuous or extractive, purity of the main products must be specified as it is driven by the customer demand and product prices. The amount of product and the operation time can be dictated by economics (maximum profit) or one of them can be fixed and the other is obtained (minimum time with fixed amount of product or maximum distillate with fixed operation time). The calculation of each of these will require formulation and solution of optimisation problems. A brief description of these optimisation problems is presented below. Further details will be provided in Chapter 5. 

In general, the optimisation problem to optimise the operation of conventional batch distillation can be stated as follows ... 

A summary of several example cases illustrated in Mujtaba and Macchietto (1998) is presented below. Instead of carrying out the investigation in a pilot-plant batch distillation column, a rigorous mathematical model (Chapter 4) for a conventional column was developed and incorporated into the minimum time optimisation problem which was numerically solved. Further details on optimisation techniques are presented in later chapters. 

Essential Features of Optimisation Problems Every optimisation problem will have ... 

A set of variables that satisfy the items 2 and 3 precisely will provide feasible solution of the optimisation problem. 

Reklaitis et al. (1983), Edgar and Himmelblau (1988) have discussed several solution methods for solving linear and nonlinear optimisation problems. Here, some of the optimisation techniques used in batch distillation will be discussed. 

Batch distillation is inherently a dynamic process and thus results to optimal control or dynamic optimisation problems (unless batch distillation task is carried out in a continuous distillation column). 

The optimal operation of a batch column depends of course on the objectives one wishes to achieve at the end of the process. Depending on the objective function and any associated constraints, a variety of dynamic optimisation problems were defined in the past for conventional batch distillation column. Brief formulations of these optimisation problems are presented in the following subsections. Situations in which each formulation can be applied are discussed. 

Converse and Gross (1963), Converse and Huber (1965), Murty et al. (1980), Diwekar et al. (1987), Mujtaba (1989), Logsdon (1990) and Logsdon et al. (1990) considered an optimisation problem which maximises the amount of distillate... 

All the optimisation problem formulations presented above were aimed to achieve optimal operation policies for a variety of objective functions but for a single period operation (i.e. single distillation task). In single period operation only one product cut is made from both binary and multicomponent mixtures and optimal operation policy is restricted only to that period. 

Reference Column Type Model Type Mixture Operation Optimisation Problem... 

Summary of the Past Work on the Solution of Optimisation Problems... 

Depending on the numerical techniques available for solving optimal control or optimisation problems the model reformulation or development of simplified version of the original model was always the first step. In the Sixties and Seventies simplified models represented by a set of Ordinary Differential Equations (ODEs) were developed. The explicit Euler or Runge-Kutta methods (Huckaba and Danly, 1960 Domenech and Enjalbert, 1981) were used to integrate the model equations and the Pontryagin s Maximum Principle was used to obtain optimal operation policies (Coward, 1967 Robinson, 1969, 1970 etc.). 

Diwekar (1992, 1995) has extensively used Pontryagin s Maximum Principle for solving all types of optimisation problems (section 5.2) using the short-cut model presented in Chapter 4. Refer to the original references for example problems. 

In this approach, the process variables are partitioned into dependent variables and independent variables (optimisation variables). For each choice of the optimisation variables (sometimes referred to as decision variables in the literature) the simulator (model solver) is used to converge the process model equations (described by a set of ODEs or DAEs). Therefore, the method includes two levels. The first level performs the simulation to converge all the equality constraints and to satisfy the inequality constraints and the second level performs the optimisation. The resulting optimisation problem is thus an unconstrained nonlinear optimisation problem or a constrained optimisation problem with simple bounds for the associated optimisation variables plus any interior or terminal point constraints (e.g. the amount and purity of the product at the end of a cut). Figure 5.2 describes the solution strategy using the feasible path approach. 

In this approach, the ODE or DAE process models are discretised into a set of algebraic equations (AEs) using collocation or other suitable methods and are solved simultaneously with the optimisation problem. Application of the collocation techniques to ODEs or DAEs results in a large system of algebraic equations which appear as constraints in the optimisation problem. This approach results in a large sparse optimisation problem. 

Nonlinear Programming (NLP) Based Dynamic Optimisation Problem-Feasible Path Approach... 

The optimisation problem presented in the previous section can be written more formally as Find the set of decision variables, y (given by Equation 5.7) which will... 

This constrained nonlinear optimisation problem can be solved using a Successive Quadratic Programming (SQP) algorithm. In the SQP, at each iteration of optimisation a quadratic program (QP) is formed by using a local quadratic approximation to the objective function and a linear approximation to the nonlinear constraints. The resulting QP problem is solved to determine the search direction and with this direction, the next step length of the decision variable is specified. See Chen (1988) for further details. 

In the NLP optimisation problem presented in section 5.7.2, the model equations (equality constraints) can now be replaced by a set of discrete AEs. Once... 

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