Optimization: basic concepts and potential benefits

Optimization is yet another important field in mathematics and engineering, with many useful real-life applications in planning and other areas. In essence, it is a mathematical approach to solving problems that have as a general goal finding the best outcome possible.

Optimization defined

While this may sound somewhat non-specific, optimization has a clear meaning to those in the know, and stands for a series of techniques that include linear and non-linear programming, integer programming, quadratic programming, stochastic programming, and dynamic programming, among others. Programming here means formulating and solving systems of equations, not software coding.

Optimization usually involves casting the problem in the form of a goal, an objective function, and constraints expressed as inequalities, so a solution can be found — something that is easier said than done and requires experience in being able to visualize situations and problems in these terms. The goal then becomes to find the maximum (benefit) or the minimum (cost) of a mathematical function, with the constraints being the limits imposed on the range of solutions. Mathematically, these solutions can lie in a “feasible region” of a 2-D plane or of a 3-D surface. The unfeasible region can be informally thought of as the “don’t go there” region, and reflects what we can’t or won’t do in terms of violating constraints. Although in some simple cases a solution can be found manually, the best solution is typically found iteratively with specialized software on the computer.

Optimization has many real-world applications of interest: in a healthcare context, where planning can be difficult, staff scheduling and resource allocation come to mind immediately. Think of a nurse manager striving to keep a fluctuating census of patients well cared for while planning an allocation of staff to shifts and days off that is fair to everyone, and being told by hospital administrators to do more to reduce the potential for harm on her floor yet not to incur overtime.

One good thing about optimization models is that, once built, a variety of what-if scenarios can be explored relatively easily, either to learn more about a specific situation or because operating conditions change and so does the range of applicability of a given solution.  Not recognizing the latter as a further constraint, however, will doom those involved to achieve little yet waste a great deal of time doing it.

Optimization models are quite powerful, and there is really no better alternative when juggling many simultaneous constraints while striving for one or more objectives.

 

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