Optimization

Optimization or Possibilities

Optimization should be viewed as a tool to aid in decision making. Its purpose is to aid in the selection of better values for the decisions that can be made by a person in solving a problem. To formulate an optimization problem, one must resolve three issues.

First, one must have a representation of the artifact that can be used to determine how the artifact performs in response to the decisions one makes. This representation may be a mathematical model or the artifact itself.

Second, one must have a way to evaluate the performance—an objective function—which is used to compare alternative solutions.

Third, one must have a method to search for the improvement. This section concentrates on the third issue, the methods one might use. The first two items are difficult ones, but discussing them at length is outside the scope of this section.

Example optimization problems are:

determining the optimal thickness of pipe insulation;
finding the best equipment sizes and operating schedules for the design of a new batch process to make a given slate of products;
choosing the best set of operating conditions for a set of experiments to determine the constants in a kinetic model for a given reaction;
finding the amounts of a given set of ingredients one should use for making a carbon rod to be used as an electrode in an arc welder.

For the first problem, one will usually write a mathematical model also often use geometry mathematic of how insulation of varying thicknesses restricts the loss of heat from a pipe. Evaluation requires that one develop a cost model for the insulation (a capital cost in dollars) and the heat that is lost (an operating cost in dollars/year). Some method is required to permit these two costs to be compared, such as a present worth analysis.

Finally, if the model is simple enough, the method one can use is to set the derivative of the evaluation function to zero with respect to wall thickness to find candidate points for its optimal thickness. For the second problem, selecting a best operating schedule involves discrete decisions, which will generally require models that have integer variables.

It may not be possible to develop a mathematical model for the fourth problem if not enough is known to characterize the performance of a rod versus the amounts of the various ingredients used in its manufacture. The rods may have to be manufactured and judged by ranking the rods relative to each other, perhaps based partially or totally on opinions. Pattern search methods have been devised to attack problems in this class.

In this section assume a mathematical model is possible for the problem to be solved. The model may be encoded in a subroutine and be known only implicitly, or the equations may be known explicitly. A general form for such an optimization problem is min F = F(z), such that h(z) = 0 and g(z) £ 0 where F represents a specified objective function that is to be minimized.

Functions h and g represent equality and inequality constraints that must be satisfied at the final problem solution.

Variables z are used to model such things as flows, mole fractions, physical properties, temperatures, and sizes. The objective function F is generally assumed to be a scalar function, one which represents such things as cost, net present value, safety, or flexibility.

Sometimes several objective functions are specified (e.g., minimizing cost while maximizing reliability); these are commonly combined into one function, or else one is selected for the optimization while the others are specified as constraints.

Equations h(z) = 0 are typically algebraic equations, linear or nonlinear, when modeling steady-state processes, or algebraic coupled with ordinary and/or partial differential equations when optimizing time-varying processes. Inequalities g(z) £ 0 put limits on the values variables can take, such as a minimum and maximum temperature, or they restrict one pressure to be greater than another.

An important issue is how to solve large problems that occur in distributed systems. The optimization of distributed systems is discussed in Refs. 52, 120, 244, and 285. For further reading on optimization, readers are directed to Refs. 120 and 244 as well as introductory texts on optimization applied to chemical engineering (Refs. 99 and 225).

The material in this section is part of a more advanced treatment (Ref. 295). Packages There are a number of packages available for optimization, some of which are listed here.

1. Frameworks

GAMS. This framework is commercially available. It provides a uniform language to access several different optimization packages, many of them listed below. It will convert the model as expressed in “GAMS” into the form needed to run the package chosen.
AMPL. This framework is by Fourier and coworkers at Northwestern University. It is well suited for constructing complex models.
ASCEND. This framework is by Westerberg and coworkers at Carnegie-Mellon University. It features an object oriented modeling language and is well suited for constructing complex models.

2. Algebraic optimization with equality and inequality constraints

SQP. A package by Biegler at Carnegie-Mellon University.
MINOS5.4. A package available from Stanford Research Institute (affiliated with Stanford University). This package is the state of the art for mildly nonlinear programming problems.
GRG. A package from Lasdon at the University of Texas, Dept. of Management Science.

3. Linear programming

Most current commercial codes for linear programming extend the Simplex algorithm, and they can typically handle problems with up to 15,000 constraints.

MPSX. From IBM
SCICONIC. From the company of that name
MINOS5.4
Cplex. A package by R. Bixby at Rice University and Cplx, Inc.