Optimization is an integral part of everyday life, often working seamlessly in the background of various familiar scenarios. It guides route planning in navigation apps like Google Maps to find the quickest path, shapes our online shopping experiences by comparing prices across platforms, and manages energy efficiency in homes through smart thermostats. Airlines optimize complex schedules for flights and crews, while personal fitness trackers tailor workout and diet plans to individual goals. Streaming services like Netflix adjust video quality based on internet speed for a smoother experience, social media platforms like Facebook curate personalized feeds, and manufacturing companies streamline production and inventory to match demand efficiently.

In a Formal Way

An optimization problem is typically expressed in the form of a function that needs to be maximized or minimized. This function is known as the objective function. The formal structure of an optimization problem can be outlined as:

1. Objective Function: This is the function we want to optimize. It is denoted as $f(x)$, where $x$ represents the decision variables of the function which can be adjusted to reach the goal i.e., to find the optimal value of $x$ that either maximizes or minimizes $f(x)$: $\max_x \quad f(x)$ or $\min_x \quad f(x)$.

   For instance, consider $f(x) = (x-3)^2$.

   • The minimum value of the function $f(x)$ is 0, i.e., $\min_x \, f(x) = 0$.
   • The value of $x$ at which this minimum occurs is 3, i.e., $\arg\min_x \, f(x) = 3$.

2. Constraints: These are the limitations or conditions that the solution must satisfy. They are often represented as equations or inequalities involving the variables of the function. Constraints define the feasible region within which the optimal solution must subject to: $\text{s.t.} \quad g_i(x)≤0,h_j(x)=0$. Here, $g_i(x)$ and $h_j(x)$ represent the inequality and equality constraints, respectively.

   For instance, consider $f(x) = (x-3)^2$ and $x \leq 2$:

   • The $x$ at which this minimum occurs is 2, i.e., $\arg\min_x \, f(x) = 2 \quad \text{s.t.} \quad x\leq2$.
   • The minimum value of the function $f(x)$ is 1, i.e., $\min_x \, f(x) = 1 \quad \text{s.t.} \quad x\leq2$.

   

Optimization Modeling Example in Production

A small bakery wants to determine the optimal number of loaves of bread to produce each day in order to maximize its daily profit. The bakery produces two types of bread: white bread and whole wheat bread.

• Each loaf of white bread sells for €3, and
• Each loaf of whole wheat bread sells for €4.

However, the bakery has limited resources, including flour and oven space.

• It takes 2 hours in to bake a white bread, and it takes 3 hours to bake a whole wheat bread.
• It takes 0.5 kilograms of flour to bake a white bread, and it takes 0.4 kilograms of flour to bake a whole wheat bread.
• The bakery has 16 hours of oven time available each day and 20 kilograms of flour.

The objective is to maximize the profit $G$. We can model such objective and the limited resources using the notations below:

Decision Variables: Decision variables represent the aspects of the problem we can control or adjust in order to influence the outcome – in this case, the bakery's profit $G$. In our example, the key decision variables are:

• $x$: The number of white bread produced each day.
• $y$: The number of whole wheat bread produced each day.

Constraints: In our bakery example, the constraints are related to resource limitations, specifically flour availability and oven capacity:

1. Flour Constraint: $0.5x + 0.4y \leq 20$ (it reflects the limitation in the amount of flour available. The bakery has a total of 20 kilograms of flour each day)
2. Oven Time Constraint: $2x+3y\leq16$ (it relects the limited amount of oven time each day)