LP Study Buddy

0% Complete

ADHD Study Tips

Work in short bursts (10-15 min) with breaks in between!

Topics

Quick Games

Tools

1. Formulating Linear Programming Problems

Turning real-world problems into mathematical equations

Key Concepts

  • Objective Function: What you want to maximize or minimize
  • Constraints: Limitations or requirements
  • Decision Variables: What you can control

ADHD Tips

Make it fun: Use examples from your life (tacos, video games, etc.)

Quick sketch: Jot down variables and constraints without perfection

Example

Taco & Burger Stand:

You have 10 buns and 5 patties. Tacos need 1 bun, burgers need 1 bun + 1 patty. Profit: $2/taco, $3/burger.

Goal Maximize Profit: P = 2x + 3y
Constraints Buns: x + y ≤ 10
Constraints Patties: y ≤ 5
Variables x ≥ 0, y ≥ 0

Practice

Pizza & Wings:

You have 20 slices of pizza and 30 wings. A pizza party needs 4 slices, a wing party needs 10 wings. Profit: $15/pizza party, $20/wing party.

2. Graphical Solutions (2D Problems)

Visualizing and solving LP problems with two variables

Key Concepts

  • Plot constraints: Draw each inequality as a line
  • Feasible region: Where all constraints overlap
  • Optimal solution: At a corner point of the feasible region

ADHD Tips

Use colors: Different colors for each constraint

Interactive tools: Try online graphing calculators

Interactive Graph

x 0 2 4 6 8 10 10 8 6 4 0 y

3. Conversion to Standard Form

Rewriting LP problems for the simplex method

Conversion Rules

≤ Constraints

Add a slack variable:

x + y ≤ 10 → x + y + s = 10

s ≥ 0

≥ Constraints

Add a surplus variable:

x + y ≥ 10 → x + y - s = 10

s ≥ 0

Free Variables

Replace with difference:

x unrestricted → x = x⁺ - x⁻

x⁺ ≥ 0, x⁻ ≥ 0

Practice

Convert to standard form:

Maximize z = 3x₁ + 2x₂

Subject to: x₁ + x₂ ≤ 4

x₁ - x₂ ≥ 1

x₁ ≥ 0, x₂ unrestricted

4. Simplex Method

Step-by-step algorithm for solving LP problems

Steps

1

Set up initial tableau

Include slack/surplus variables and objective function

2

Choose pivot column

Most negative entry in bottom row (for minimization)

3

Choose pivot row

Smallest non-negative ratio (RHS ÷ pivot column)

4

Perform row operations

Make pivot element 1, others in column 0

5

Repeat until optimal

No negatives in bottom row (minimization)

Interactive Tableau

Basic x₁ x₂ s₁ s₂ RHS
s₁ 1 1 1 0 4
s₂ 1 -1 0 -1 1
z -3 -2 0 0 0

5. Duality

Understanding the dual problem and its relationship to the primal

Primal to Dual Rules

Maximization → Minimization

(and vice versa)

Constraints → Variables

Each primal constraint becomes a dual variable

Variables → Constraints

Each primal variable becomes a dual constraint

Coefficients transpose

Constraint matrix flips rows and columns

Example

Primal

Max z = 3x₁ + 2x₂

x₁ + x₂ ≤ 4

x₁ - x₂ ≤ 1

x₁, x₂ ≥ 0

Dual

Min w = 4y₁ + y₂

y₁ + y₂ ≥ 3

y₁ - y₂ ≥ 2

y₁, y₂ ≥ 0

Practice

Find the dual of:

Min z = 2x₁ + 5x₂

3x₁ + x₂ ≥ 6

x₁ + 2x₂ ≥ 4

x₁, x₂ ≥ 0

Made with DeepSite LogoDeepSite - 🧬 Remix