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independent variables, subject to the applicable constraints, that yield the optimum
value of the objective function. In many circumstances there may be several solutions
to an optimization problem.
There are many algorithms for solving linear and nonlinear optimization problems.
Good summaries may be found in Beveridge and Schechter (1970) and in Edgar and
Himmelblau (2001). Aspen Plus provides the following proprietary methods:
￿ SQP: successive quadratic programming
￿ Complex: a black-box pattern search
Each is described in Aspen Plus's Help.
13.1 OPTIMIZATION EXAMPLE
A simple example is the trade-off between the number of theoretical stages and the
reflux ratio in a distillation column required to obtain a target product composition.
For this case the independent variables are the number of theoretical stages, a discrete
variable that takes on integer values, and the reflux ratio, a continuous variable that is
constrained between a minimum value which corresponds to a practical value near an
infinite number of theoretical stages, and a maximum which corresponds to a minimum
number of stages. The objective function has contributions from the following factors.
￿ Number of theoretical stages, N
￿ Installed cost of the column, where each stage adds an increment of height to the
column, I$ (dollars/theoretical stage)
￿ Expected life of the column, L (years)
￿ Annual maintenance of the column, M$ (dollars/year)
￿ Annual reboiler load, R (Btu)
￿ Annual condenser load, C (Btu)
￿ Cost of heating, H$ (dollars/Btu)
￿ Cost of cooling, C$ (dollars/Btu)
￿ Operating labor, O$ (dollars/year)
A simplified objective function, , which represents the total cost of the column
over its life is defined by
= N I$ + M$ L + R H$ L + C C$ L + O$ L
(13.4)
The implementation of the objective function is carried out using the entry Model
Analysis, Optimization. The initial input form on which the objective function is defined
is given in Figure 13.1. In this example, variables defined in the figure are taken from
the process during each iteration, and are used in calculation of the objective function.
The various elements that are required for optimization are given in the tabs across
the top of the figure. Figure 13.2 provides for the formal definition of the objective
function and its constraints. Note that it is possible to code the objective function in the
objective function frame in Fortran. Check boxes to select to maximize or minimize
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