Linear programming, process modeling

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). 

Scheduling of refinery processes environmental impact. e-constraint method along with a MILP method Scheduling of refinery processes was modeled as a mixed-integer linear programming (MILP) model. Song et al. (2002)... 

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. 

Many process engineers think of linear programming (L.P.) as a sophisticated mathematical tool, which is best applied by a few specialists extremely well grounded in theory. This is certainly true for your company s central linear program. The layman does not write a linear program, he only provides input that will model the process in which he is interested. 

Process Design vs. Accounting Linear Program Models... 

The process design linear program model is best written with flexibility in mind, such as extra matrix rows to provide flexibility in recycling, adding outside streams intermediate in the process, and determining component incremental values at each processing stage. This subject is discussed more fully later in this chapter. 

Figure 1. Process model for illustrating how process design linear programming can be achieved.

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. 

WASP/TOXIWASP/WASTOX. The Water Quality Analysis Simulation Program (WASP, 3)is a generalized finite-difference code designed to accept user-specified kinetic models as subroutines. It can be applied to one, two, and three-dimensional descriptions of water bodies, and process models can be structured to include linear and non-linear kinetics. Two versions of WASP designed specifically for synthetic organic chemicals exist at this time. TOXIWASP (54) was developed at the Athens Environmental Research Laboratory of U.S. E.P.A. WASTOX (55) was developed at HydroQual, with participation from the group responsible for WASP. Both codes include process models for hydrolysis, biolysis, oxidations, volatilization, and photolysis. Both treat sorption/desorption as local equilibria. These codes allow the user to specify either constant or time-variable transport and reaction processes. 

This chapter focuses on a new approach that allows for the comprehensive planning and optimization of multi-stage production processes - the quant-based combinatorial optimization. First, a distinction is drawn between classical approaches such as Linear Programming (LP) and the quant-based combinatorial approach. Before going into the special characteristics and requirements of the process industry the one model approach with quant-based combinatorial optimization is introduced. Then we will give two examples of how this new approach is applied to real life problems. 

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. 

Both the mixing process and the approximation of the product profiles establish nonconvex nonlinearities. The inclusion of these nonlinearities in the model leads to a relatively precise determination of the product profiles but do not affect the feasibility of the production schedules. A linear representation of both equations will decrease the precision of the objective but it will also eliminate the nonlinearities yielding a mixed-integer linear programming model which is expected to be less expensive to solve. 

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