In linear programming problems, the requirement for variables is that they must be non-negative. This means that the values of the decision variables cannot be less than zero. The rationale behind this requirement often relates to the context of many real-world problems where negative values would not make sense. For example, in a problem involving production, negative quantities of products do not exist as one cannot produce a negative amount of an item.
While there are instances in optimization involving variables that can take negative values, the standard formulation of linear programming typically constrains variables to be non-negative unless explicitly stated otherwise. This constraint helps in defining a feasible region in which solutions can be found.
The other potential characteristics of variables, like being integers only or continuous, depend on the specific type of linear programming problem being addressed. "Must be integers only" refers to integer programming, a specific subset of linear programming that applies when decisions must be made in whole units. Continuous variables, on the other hand, can take any value within a given range, which may not necessarily fulfill the non-negativity requirement without additional constraints. Thus, the focus on non-negativity ensures the applicability of linear programming to a broad range of practical scenarios.