ت optimisation: كيف take the best decisions in math and life

هل تعلم أن قرارك في اختيار أفضل طريق للgoing to work هو حل لمشكلة ت optimisation؟

كل يوم، نtake decisions. من اختيار أفضل طريق إلى work إلى تحديد كميات الفواكه التي نشتري من السوق، نحن ن optimise without knowing it. ت optimisation هي فنة take best decisions when we have multiple options and constraints. في هذا المقال، سنستكشف theory behind هذه الفنة، مع examples من الحياة daily.

foundations: المفاهيم basic

Before we dive in, let's define some key concepts.

• Objective function: هي function ن want to maximize or minimize. For example, maximizing profit or minimizing cost.
• Constraints: هي restrictions limit possible solutions. For example, budget constraints or resource limits.
• Feasible region: هي set of solutions that satisfy all constraints.

deep dive 1: ت optimisation linear

Let's start with linear optimisation, which deals with linear objective functions and constraints. Imagine you own a small shop selling two products: apples and oranges. You want to maximize your profit, but you have constraints like limited space and budget.

Let's say:

• Profit per apple: 2 JDs
• Profit per orange: 3 JDs
• Space constraint: 100 apples + 200 oranges ≤ 1000 units
• Budget constraint: 5 JDs per apple + 4 JDs per orange ≤ 200 JDs

We can represent this as:

$$ \text{Maximize } Z = 2x + 3y $$ $$ \text{Subject to:} $$ $$ x + 2y \leq 1000 $$ $$ 5x + 4y \leq 200 $$ $$ x, y \geq 0 $$

deep dive 2: ت optimisation quadratic

Now, let's look at quadratic optimisation, where the objective function is quadratic. Imagine you want to minimize the cost of producing two products, where the cost depends on the square of the quantities.

For example: $$ \text{Minimize } C = x^2 + 2y^2 $$ $$ \text{Subject to:} $$ $$ x + y \geq 10 $$ $$ x, y \geq 0 $$

In this case, the cost function is a paraboloid, and the minimum is at the vertex if there are no constraints. But with constraints, the minimum might be on the boundary.

common mistakes: forget constraints

One of the most common mistakes is to ignore constraints. Remember that the best solution is only valid if it satisfies all constraints.

practice: planning a trip

Let's say you want to plan a trip to three cities with a budget of 300 JDs. The costs to travel between cities are:

• A to B: 50 JDs
• A to C: 80 JDs
• B to C: 30 JDs

What's the cheapest route that visits all three cities?

Possible routes:

1. A → B → C: 50 + 30 = 80 JDs
2. A → C → B: 80 + 30 = 110 JDs
3. A → B → A → C: 50 + 50 + 80 = 180 JDs (more expensive)

The best solution is route 1, with cost 80 JDs.

summary: takeaways key

Let's recap what we've learned:

1. ت optimisation is about find best solution within constraints.
2. Objective function is what we want to maximize or minimize.
3. Constraints define feasible region.
4. Always check that your solution satisfies all constraints.