Process modeling using linear programming

The proposed approach discretizes one variable (concentrations used here or flowrates) of the bilinear term generated at the splitting points. As a result, a mixed integer linear programming (MILP) model is generated. The discretized concentrations are now parameters (7)C for the water using units DCRj for regeneration processes). 

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. 

The example used here does not represent any particular process and is simpler than most real life plant cases. It does, how ever, have the elements needed to talk our way through the development of a process design linear program model. All models should have the following ... 

This problem can be cast in linear programming form in which the coefficients are functions of time. In fact, many linear programming problems occurring in applications may be cast in this parametric form. For example, in the petroleum industry it has been found useful to parameterize the outputs as functions of time. In Leontieff models, this dependence of the coefficients on time is an essential part of the problem. Of special interest is the general case where the inputs, the outputs, and the costs all vary with time. When the variation of the coefficients with time is known, it is then desirable to obtain the solution as a function of time, avoiding repetitions for specific values. Here, we give by means of an example, a method of evaluating the extreme value of the parameterized problem based on the simplex process. We show how to set up a correspondence between intervals of parameter values and solutions. In that case the solution, which is a function of time, would apply to the values of the parameter in an interval. For each value in an interval, the solution vector and the extreme value may be evaluated as functions of the parameter. 

In order to make the problem solvable, a linearized process model has been derived. This enables the use of standard Mixed Integer Linear Programming (MILP) techniques, for which robust solvers are commercially available. In order to ensure the validity of the linearization approach, the process model was verified with a significant amount of real data, collected from production databases and production (shift) reports. 

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