The functions g(X) and h(X) may involve any number of the decision variables as well as other quantities and numerical constants. In some classes of problem some or all of the design variables are not permitted to take negative values. This is ensured by including in the statement of the model the required number of nonnegativity constraints of the form:
Classification of mathematical models Although the mathematical models for optimisation may generally be described in the format defined above, it is useful to classify certain categories of models in order to better understand the applicability of the various techniques available for their solution. The categories, which are frequently encountered, can be described as follows (Smith et al., 1983; Onwubiko, 2000):
* Category 1: Constraint If the model is stated with some constraints, we have a constrained optimisation. If, however, the model is stated without constraints, then we have an unconstrained optimisation;
* Category 2: Linearity If the objective function and all of the constraint functions are linear in terms of the decision variables, then the model is said to be linear. If any of the constraints or any part of the objective function contains a non-linearity the model is said to be non-linear;
* Category 3: Data This category depends on the nature of the data available. A mathematical model is termed deterministic if all data are assumed to be known with certainty, and probabilistic or stochastic if it involves quantities that are stochastic;
* Category 4: Variable If the objective function is a function of one variable, we have single-variable optimisation. On the other hand, if the objective function consists of two or more variables, the model is known as multivariable optimisation. If anyone of its decision variables is discrete, implying that the variable is limited to a fixed or countable set of values the mathematical model is termed a discrete programming. If all variables are integers, the model is a pure integer programming; otherwise, it is a mixed-integer programming;
* Category 5: Time Models, which involve time-dependent interactions, are said to be dynamic, otherwise they are static;
* Category 6: Objective A mathematical model with a single design objective is known as single-criterion optimisation. But in engineering, it is often a problem to formulate a design in which there are several criteria or design objectives. This model is known as multicriteria or multi-objective optimisation. If the objectives are opposing, then the problem becomes finding the best possible design, which still satisfies the opposing objectives. An optimum problem must then be solved, with multiple objectives and constraints to be taken into consideration.