Optimisation linear programming

Linear programming is an optimisation technique that can be used when the objective function and constraints can be expressed as a linear function of the variables see Driebeek (1969), Williams (1967) and Dano (1965). 

Production costs per tonne of base oil are calculated by dividing the total annual costs by the total annual production of base oils. Net feedstock cost can be calculated in several ways, but it will not necessarily be identical to the cost of crude oil. As the base oil plant in a sense competes with fuel production units for feedstock, the basic feedstock cost to the lubricant base oil complex should be determined by the alternative value of that feedstock if it were used to make mainstream fuels products. The by-products of base oil manufacture also have values for blending into fuel streams or in some cases for direct sale as speciality products, such as waxes and bitumen. Credit must be given for these products so that the net value of the hydrocarbon content of the base oil can be calculated. Refineries use sophisticated linear programming computer models to optimise refinery operations based on different crude oil input, process yields, market prices, production targets, etc. 

More efficient is a generic tree representation of the separations based on tasks. The sequencing can be formulated as a structural optimisation problem where standard techniques based on Mixed Integer Linear Programming (MILP) apply. The tasks consist of simple distillation columns, as well as of hybrids for complex column arrangements, modelled by appropriate shortcut or semi-rigorous methods. Details can be found in Doherty and Malone (2001). 

Kocis, G.R. and Grossmann, I.E. (1988) Global optimisation of nonconvex mixed-integer non linear programming (MINLP) problems in process synthesis. Industrial Engineering Chemistry Research, 27 (8), 1407-1421. 

The gasoline blend problem is formulated as a Mixed Integer Non-Linear Programming (MINLP) problem, where the fuel composition is to be optimised, subject to product attributes (target properties) and process specifications. Considering the multiple types of eonstraint equations, the general gasoline blend problem is formulated as ... 

An efficient and robust optimisation algorithm is primordial for this solution strategy. Rao Sawyer (1995) applied Powell s method to tackle the optimisation. Koyliioglu et al. (1995) defined a linear programming solution for this purpose. The input interval vector defines the number of constraints and, therefore, strongly influences the performance of the procedure. Also, because of the required execution of the deterministic FE analysis in each goal function evaluation, the optimisation approach is numerically expensive. Therefore, this approach is best suited for rather small FE models with a limited number of input uncertainties, unless approximate methods can be used that avoid the expensive iterative calculation of the entire FE system of equations. 

Non-linear programming technique (NLP) is used to solve the problems resulting from syntheses optimisation. This NLP approach involves transforming the general optimal control problem, which is of infinite dimension (the control variables are time-dependant), into a finite dimensional NLP problem by the means of control vector parameterisation. According to this parameterisation technique, the control variables are restricted to a predefined form of temporal variation which is often referred to as a basis function Lagrange polynoms (piecewise constant, piecewise linear) or exponential based function. A successive quadratic programming method is then applied to solve the resultant NLP. 

Figure 12.120 shows the feasible operating area. Most MVC include a linear program (LP) for optimisation. This technology can only find the most profitable node, where constraints cross. There may be a more profitable operating point within the feasible operating area. A nonlinear optimisation technique would be required to identify this point this is covered later in this section. 

An adaptive control system can automatically modify its behaviour according to the changes in the system dynamics and disturbances. They are applied especially to systems with non-linear and unsteady characteristics. There are a number of actual adaptive control systems. Programmed or scheduled adaptive control uses an auxiliary measured variable to identify different process phases for which the control parameters can be either programmed or scheduled. The "best" values of these parameters for each process state must be known a priori. Sometimes adaptive controllers are used to optimise two or more process outputs, by measuring the outputs and fitting the data with empirical functions. 

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. 

MOGP is based on the more traditional optimisation method genetic programming (GP), which is a type of GA [53,54]. The main difference between GP and a GA is in the chromosome representation in a GA an individual is usually represented by a fixed-length linear string, whereas in GP individuals are represented by treelike structures hence, they can vary in shape and size as the population undergoes evolution. The internal nodes of the tree, typically represent mathematical operators, and the terminal nodes, typically represent variables and constant values thus, the chromosome can represent a mathematical expression as shown in Fig. 4. 

---