SOLVING THE OR MODEL
In OR, we do not have a single general technique to solve all mathematical models that can arise in practice. Instead, the type and complexity of the mathematical model dictate the nature of the solution method. For example, in Section 1.1 the solution of the tickets problem requires simple ranking of alternatives based on the total purchasing price, whereas the solution of the rectangle problem utilizes differential calculus to determine the maximum area.
The most prominent OR technique is linear programming. It is designed for models with linear objective and constraint functions. Other techniques include integer programming (in which the variables assume integer values), dynamic programming (in which the original model can be decomposed into more manageable subproblems), network programming (in which the problem can be modeled as a network), and nonlinear programming (in which functions of the model are nonlinear). These are only a few among many available OR tools.
A peculiarity of most OR techniques is that solutions are not generally obtained in (formula like) closed forms. Instead, they are determined by algorithms. An algorithm provides fixed computational rules that are applied repetitively to the problem, with each repetition (called iteration) moving the solution closer to the optimum. Be-cause the computations associated with each iteration are typically tedious and voluminous, it is imperative that these algorithms be executed on the computer.
Some mathematical models may be so complex that it is impossible to solve them by any of the available optimization algorithms. In such cases, it may be necessary to abandon the search for the optimal solution and simply seek a good solution using heuristics or rules of thumb.