Optimisation: problem objective function

The problem is to find values for the variables v to vn that optimise the objective function that give the maximum or minimum value, within the constraints. 

Optimisation may be used, for example, to minimise the cost of reactor operation or to maximise conversion. Having set up a mathematical model of a reactor system, it is only necessary to define a cost or profit function and then to minimise or maximise this by variation of the operational parameters, such as temperature, feed flow rate or coolant flow rate. The extremum can then be found either manually by trial and error or by the use of numerical optimisation algorithms. The first method is easily applied with MADONNA, or with any other simulation software, if only one operational parameter is allowed to vary at any one time. If two or more parameters are to be optimised this method becomes extremely cumbersome. To handle such problems, MADONNA has a built-in optimisation algorithm for the minimisation of a user-defined objective function. This can be activated by the OPTIMIZE command from the Parameter menu. In MADONNA the use of parametric plots for a single variable optimisation is easy and straight-forward. It often suffices to identify optimal conditions, as shown in Case A below. 

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. 

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. 

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. 

Apart from N (an integer) the optimisation problem is a standard Nonlinear Programming (NLP) problem, with the inner optimisation problem providing the values of the outer objective function and constraints. In the outer problem Mujtaba... 

Thus the multiperiod optimisation problem is formulated as a sequence of two independent dynamic optimisation problems (PI and P2), with the total time minimised by a proper choice of the off cut variables in an outer problem (PO) and the quasi-steady state conditions appearing as a constraint in P2. The formulation is very similar to those presented by Mujtaba and Macchietto (1993) discussed in Chapter 5. For each iteration of PO, a complete solution of PI and P2 is required. Thus, even for an intermediate sub-optimal off cut recycle, a feasible quasi-steady state solution is calculated. The gradients of the objective function with respect to each decision variable (Rl or xRl) in problem PO were evaluated by a finite difference scheme (described in previous chapters) which again requires a complete solution of problem PI and P2 for each gradient evaluation (Mujtaba, 1989). 

In addition to the above-mentioned problem, numerical difficulties may arise. The system (model equations) describing the multicomponent off-cut recycle operation needs to be reinitialised at the end of each main-cut and off-cut to accommodate the next off-cut to the reboiler. To optimise these initial conditions (new mixed reboiler charge and its composition) it is essential to obtain the objective function gradients with respect to these initial conditions. 

Solution of optimisation problems using rigorous mathematical methods have received considerable attention in the past (Chapter 5). It is worth mentioning here that these techniques require the repetitive solution of the model equations (to evaluate the objective function and the constraints and their gradients with respect to the optimisation variables) and therefore computationally can be very expensive. 

A NLP optimisation problem can be viewed as the minimisation of an objective function F(d,x) in a design space d subject to equality c(x)a.n inequality g(x) constraints. Mathematically, the formulation is ... 

Figure 3.41 illustrates the optimisation of a flowsheeting problem in a two dimensional space. Contours of the objective function F are plotted, where the two variables x, and Xj are bounded by upper and lower values. The overall heat and... 

After this example, the reader is encouraged to revisit the Example 3.4 and reformulate it as an optimisation problem. We suggest as objective function the sale value of the total LNG and LPG produced. Optimisation variables could be the outlet hot gas temperature after the cold box, as well as the pressure after expander. 

The Equation (7.6) indicates that as design variables we may choose x-,. (conversion), yp (fraction of hydrogen in purge) and Mk (molar ratio). Unfortunately, there are no design heuristics to guide their selection. This issue is in fact a constraint optimisation problem, the objective function being the Economic Potential at the level of recycle stricture Level 3 (see later in this chapter). 

For integration of precedence-constrained production sequencing and scheduling, the authors [19-21] proposed a GA with variability of chromosome size to optimise the class of problems for one as well as two objective functions in one as well as multiple production line environments. 

More generally, the process optimisation problem is solved using an optimisation solver, which interacts with the process simulator to minimise the objective function. The optimisation solver can be based on any one of many optimisation techniques. One group of possible optimisation techniques are the gradient-search methods. These methods rely on analytic or semi-analytic expressions for the objective function and objective function... 

As a precursor to the optimisation, an optimum mesh density for SimLCM needs to be found for the particular problem (geometry, materials, etc.) under study. This involves finding a mesh which is fine enough to ensure an adequately converged objective function for a wide range of sets of the design variables P, v, but coarse enough not to increase computa-... 

As explained in section 3.1.2 an interval FE analysis can be formulated as a numerical optimisation problem, with all uncertain parameters as design variables and the desired output quantity as the objective function. The optimisation procedures then has to be repeated for each individual output quantity of interest. This section presents an efficient approximate approach to reduce the calculation time, while keeping acceptable accuracy. 

The proposed optimisation algorithm enables one to find local optimum of a general objective function. The evaluation of objective functions requires only a modest amount of computation (in the essence the multiplication of matrices Z, 2 and Z, 4b by vectors of variances of measured quantities). Even problems of realistic dimensionality can be solved efficiently on personal computers. 

Flowsheets can be built up from generic units and then evaluated using the operations for an extended type, T, which provides the information necessary for the global optimisation algorithm. The variable type being used is intervals. Each unit calculates the output streams (where the input and output streams are usually a vector) and the cost associated with the unit as intervals. The summation of the costs provides the objective function. The design constraints are added outside the module to the optimisation problem. 

Interval gradient types have been used, and the global optimisation algorithm uses the NE and MV underestimation scheme (Bogle Byrne, 1999, p.1341-1350) to construct a linear relaxation of the objective function and constraints in terms of the optimisation variables. The solution of this linear relaxation provides the necessary bounds on the optimisation problem. 

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