Optimizing Drone Deliveries: A Linear Algebra Problem in Real Life

Maths
1

Optimizing Drone Deliveries: A Linear Algebra Problem in Real Life

:rocket: Problem Statement:

In the age of automation and rapid delivery, drones are becoming a vital part of last-mile logistics. Suppose we have a delivery drone that can:

  • Carry up to 5 kilograms of packages per trip, and
  • Complete 3 deliveries before it needs to return for charging.

Each delivery involves exactly one package, and packages are of two types:

  • Type A: 1.5 kg
  • Type B: 2.5 kg

The challenge:

What is the maximum number of 2.5 kg packages the drone can deliver in one trip while still making all 3 deliveries without exceeding the 5 kg weight limit?

:writing_hand: Step-by-Step Analysis

We are asked to maximize the number of heavier packages (2.5 kg) under the given constraints:

  • The drone can carry 3 packages in total.
  • The combined weight must not exceed 5 kg.

Let’s define variables:

  • Let x be the number of 2.5 kg packages (Type B).
  • Let y be the number of 1.5 kg packages (Type A).

:bullseye: Constraints:

  1. x + y = 3 → Because the drone must carry exactly 3 packages.
  2. 2.5x + 1.5y ≤ 5 → The total weight must be ≤ 5 kg.
  3. x, y ≥ 0 and must be integers (can’t carry half a package).

:abacus: Solving the Equations

Substitute the first constraint into the second:

Since:

y = 3 – x
Plug into the weight equation:
2.5x + 1.5(3 – x) ≤ 5
2.5x + 4.5 – 1.5x ≤ 5
(2.5 – 1.5)x + 4.5 ≤ 5
x + 4.5 ≤ 5
x ≤ 0.5

But x must be a whole number, so the maximum integer value satisfying this is:

x = 0

:white_check_mark: Final Answer:

  • Maximum number of 2.5 kg packages = 0

So the drone cannot deliver even one 2.5 kg package if it must make all 3 deliveries and stay under the 5 kg limit.

Let’s check:

  • 3 packages × 1.5 kg = 4.5 kg :white_check_mark:
  • If we try 1 package of 2.5 kg + 2 × 1.5 kg = 5.5 kg :cross_mark:

:light_bulb: Insight & Real-Life Implication

This problem illustrates a real-world constraint-based optimization, often encountered in logistics, supply chain, and robotics. It combines:

  • Algebra (equation solving),
  • Inequalities (weight limits), and
  • Integer programming (discrete decision variables).

It also highlights:

  • How heavier items reduce total quantity, and
  • Why companies must consider package weight classes for route planning.

:brain: Extension: What If…

Q: What if the drone could make only 2 deliveries instead of 3?

New constraint: x + y = 2
Repeat the same process:

Substitute y = 2 – x in the weight constraint:

2.5x + 1.5(2 – x) ≤ 5
2.5x + 3 – 1.5x ≤ 5
x + 3 ≤ 5 → x ≤ 2
:white_check_mark: x = 2 is valid!

Total weight: 2 × 2.5 = 5 kg → Exactly fits.

Answer: It could carry 2 heavy packages in that case.

:memo: Conclusion

The Delivery Drone Optimization problem is a compact example of how math governs modern-day technology. By modeling constraints using algebra, we can make better, data-driven decisions in logistics.