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How is optimization used to allocate limited resources among competing jobs?
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optimization
Constraint set bounded by the five lines x1 = 0, x2 = 0, x1 = 8, x2 = 5, and x1 + x2 = 10. These enclose an infinite number of points that represent feasible solutions.
In mathematics, optimization is used to allocate limited resources among competing jobs by formulating the problem with three fundamental elements: an objective function, decision variables, and constraints.1
- Objective Function: This is a single numerical quantity to be maximized or minimized, such as total profit, production costs, or the number of tasks completed.1
- Decision Variables: These represent the quantities to be manipulated, such as the specific amount of resources allocated to each individual job or activity.1
- Constraints: These are the restrictions on the variables, such as the total availability of labor, raw materials, or time. A process cannot consume more resources than are available, nor can it use a negative amount.1
A primary technique for this is linear programming, which models these relationships as linear equations and inequalities. By defining the system as a set of activities that require inputs and produce outputs, mathematicians can use methods like the simplex method to find the optimal distribution of resources that satisfies all constraints while achieving the desired goal.123