NMJM #6 ✨ Mathematical Optimization ✨

Hello fellow Magicians!🧙‍♂️

Mathematical Optimization offers algorithmic solutions to optimization problems. Yes, you heard that right—solutions. These methods find optimal solutions to the problems you outline. But as we’ll explore, this comes at the cost of the upfront work needed to model the problem precisely. Let’s dive in!

🚀 Methods Under Mathematical Optimization:

• Linear Programming (LP)

• Mixed Integer Programming (MIP)

• Quadratic Programming

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🔍 Linear Programming (LP)

The most basic form of Mathematical Optimization, linear programming, is valid when the problem's constraints and objective can be modeled with linear equations.

Example Problem:

A factory produces two products, A and B, each requiring different amounts of labor and materials:

• Product A: Requires 2 units of labor and 1 unit of material.

• Product B: Requires 1 unit of labor and 2 units of material.

The factory has limited resources:

• Labor constraint: 100 hours available.

• Materials constraint: 80 units available.

The goal is to maximize profit, where:

• Profit for Product A: $40 per unit.

• Profit for Product B: $30 per unit.

Constraints and Objective:

• Variables:
  x = units of Product A
  y = units of Product B

• Objective:
  Maximize profit: Z = 40x + 30y

• Constraints:

  • Labor: 2x + y ≤ 100

  • Materials: x + 2y ≤ 80

  • Non-negativity: x ≥ 0, y ≥ 0

🟦 🟠 Geometric Perspective: The Value of Linearity

Let's dig into why all this linearity in constraints and the objective is valuable. It’s helpful to view this problem geometrically.

Visualizing Constraints:
Each constraint is a line (it's linear, after all!), and we can plot these lines on a graph to understand how they shape the solution space.

More Than Lines—Defining a Feasible Region:
These are not just linear equations—they're linear inequalities. Each constraint has a direction, defining a feasible region where all constraints are satisfied without violation.

The feasible region is crucial because any optimal solution will lie somewhere within it. Here lies the power: the optimal solution, if it exists, will always be at one of the corner points (vertices) of the feasible region.

Objective Function's Role:
By plotting the objective function Z, we can visualize its direction—often“up and out.” The optimal solution is at the last corner point within the feasible region, aligning with the objective’s direction without violating any constraints.

🤔 But That's a Simple Problem…

Linear programming scales well! Even with multiple variables and constraints, the principle remains: optimal solutions exist at corner points, whether it's in 2D or 30D or even thousands of dimentions.

🚀 Expansions on Linear Programming

Mixed-Integer Programming (MIP): Allows for boolean (integer) constraints, introducing flexibility by sacrificing some precision.

Quadratic Programming: Enables quadratic objectives, which means the solution may no longer be at a corner, but clever algorithms can still find it!

✅ Pros and Cons of Mathematical Optimization

Advantages:

• Guaranteed Optimal Solution: Once formulated, It can and will be solved!

• Computational Efficiency: Knowing the solution exists at corner points makes finding solutions fast.

• Wide Applicability: From project scheduling to battery management, to portfolio risk management —mathematical optimization is commonly used to solve valuable problems.

Disadvantages:

• Complexity with Size/Non-Linearity: Large problems with non-linear objectives can become time-consuming to solve.

• Requires Precise Modeling: Every parameter and constraint must be well-understood.

• Brittle in the face of constraint variation: Often, constraints shift slightly moment to moment—any shift in constraint would need to be directly implemented in the constraint logic if it is to be respected by the optimization.

• One Solution Only: While the solution is optimal, it may not always suit user preferences for multiple options.

🛠️A Tool Using Mathematical Optimization

• Motion.AI: Motion is a scheduling tool I use every day to help plan out my days and projects to finish my work. It uses LP in the backend to optimize your schedule based on constraints like:

  • Work Hours

  • Deadlines

  • Completion time estimates

Check it out here if you want to give it a try and support the newsletter!

https://get.usemotion.com/34nmt4f5cg5x

📝 Author's Notes:

I lean towards Mathematical Optimization when the math behind the system is well-defined. For problems like bin packing or scheduling, where constraints are clear, this approach is unbeatable. But if you want multiple options or there's ambiguity in the formulation, other techniques may serve you better.