## Info

Time

Figure 2. Quality of tea as a function of water temperature and processing time.

To make the preceding system more realistic, other variables could be included in the optimization problem: the amount of stirring, the amount of tea in the tea bag, the storage history of the tea leaves (temperature and humidity), and so on. All these factors will have some effect on the quality. Furthermore, it is difficult to measure and define quality, because it depends on the taste and preference of individuals, among other things. Despite the simplicity of the preceding example, it illustrates the basic problem of optimization and the need to formulate the problem, define the system, find a function (or mathematical model) that describes the system, and find the system's optimum.

More explicitly, all optimization problems must consider several important elements (2,3):

Selection of Decision Variables. The most critical element in optimization problems is to select all the appropriate variables that contribute to the decisionmaking process of attaining the optimum. These input, or independent variables affect the performance of the process or system and can be adjusted, changed, and controlled.

Definition of the Performance Function. Another important element of optimization problems is the process of defining which dependent variables require optimization (maximization or minimization). These measured variables, called performance or objective functions, determine and reflect the performance of the system or process.

Identification of Constraints. In most optimization problems, certain constraints may limit the values that dependent or independent variables can take, restricting, therefore, the region of allowable solutions. Such constraints may be imposed by physicochemical or other impossibilities (in the tea example, the temperature of the water cannot be more than 100°C). They may also represent restrictions endogenous to the system (eg, federal or state requirements on the composition of a food product, grading guidelines, and equipment restrictions). Whatever the case might be, such constraints must be taken into account, because they influence the solution of a given problem.

Development of a Mathematical Model to Describe the System. Attempts to solve a problem by trial and error require painstaking physical experimentation and application of statistical theory. Instead of this, a mathematical model is used for most cases of optimization. In some cases, mathematical models may already exist (eg, thermal processing and drying), but in other situations models must be developed based on available scientific and engineering knowledge. It is necessary to be aware of how to develop a model (4,5). Sometimes the complexity of the system might be such that a model cannot be developed or, if developed, cannot be solved. This is common in biochemical and biological systems such as foods, and usually an approximating (or graduating) function is developed, which represents the true nature of the system as closely as possible (6). Such functions are usually polynomials or transformations thereof, and a limited number of experiments are required to find their constant coefficients (7,8). Data collection methods (experimental design) and model fitting (linear, nonlinear, and other regression) techniques are important elements of finding the appropriate approximating function (3,7-11).

Attaining the Optimum. The final element of all optimization problems is to choose an appropriate method to optimize the system. It is imperative to realize that the scope of every optimization method is to adjust and readjust the values of the decision variables to locate the settings that maximize or minimize (optimize) the performance function(s), while ensuring that all variables (dependent or independent) satisfy the constraints of the system.

Figure 3 gives typical relationships between the important elements of an optimization problem and their role in its solution. The following section discusses the available optimization techniques and methods. This information

Figure 3. The important elements of an optimization problem and their interactions.

should be complemented with the more in-depth discussion given in the references cited therein.

### OPTIMIZATION METHODS Basic Theoretical Background

Among the various optimization techniques, each is best suited to operate on a particular type of mathematical model. Consequently, when formulating a problem and developing a specific model, the strong and weak points of the various optimization techniques must be kept in mind.

The basis for all optimization methods is the classic theory of maxima and minima. Mathematically, the theory is concerned with finding the minimum or maximum (extreme points) of a function of n variables, f(xlt x2, ■ ■ ■, xn), where n is any integer greater than zero. To simplify the concept, assume that fis a function of only one variable x (Fig. 4). In this example, point A is located at the boundary and it is a local minimum. Point B, where the first derivative f' is discontinuous, is a global maximum. Point C, where f' = zero, is a local minimum, D (/' = 0) and G are local maxima, and E is the global minimum. Points C, D, E, and F are stationary points, because their first derivative f' is zero. A stationary point is a maximum (local or

Figure 4. A schematic representation of a function fix) and its first derivative f '(x).

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