Optimization Interactive

Linear Programming

Optimize resource allocation under constraints. Find the best budget split, pricing mix, or exposure limits to maximize profit.

📐 LP Framework

Decision Variables

What you control (allocations, prices, quantities)

x₁ = social budget, x₂ = search budget...

Objective Function

What to maximize/minimize

max: 1.5x₁ + 2.0x₂ + 1.8x₃

Constraints

Limits and requirements

x₁ + x₂ + x₃ ≤ 100, x₃ ≥ 10

Non-negativity

Variables must be ≥ 0

x₁, x₂, x₃ ≥ 0

Budget & Returns

50 500

Social

ROI: 0.30

ROI: 0.20

Display

ROI: 0.23

Constraints

10 80

5 40

🎯 Optimal Allocation

Social

$40K

40%

$0K

Display

$60K

60%

📊 Performance Comparison

Optimal Allocation

25.5

Total Return Units

Equal Allocation

24.2

Total Return Units

Optimization gain: +5.5%

🏀 Sports Betting Applications

Liability Management

Maximize expected profit while limiting exposure per game, sport, or outcome.

max profit s.t. exposure_i ≤ limit_i

Promo Budget Allocation

Allocate bonus budget across user segments to maximize LTV.

max Σ ROI_i × spend_i s.t. Σ spend ≤ B

Market Mix

Optimize bet offering across markets (spreads, totals, props) for margin.

max margin s.t. diversity constraints

📝 Standard Form

LP Standard Form

Maximize:   c'x
Subject to: Ax ≤ b
            x ≥ 0

• c: objective coefficients (returns)
• x: decision variables (allocations)
• A: constraint matrix
• b: constraint bounds

Solving Methods

→ Simplex: Classic algorithm, walks vertices
→ Interior Point: Faster for large problems
→ Tools: R (lpSolve), Python (scipy, cvxpy)

R Code Equivalent

# Linear Programming with lpSolve
library(lpSolve)

# Objective: maximize returns
# Decision vars: [social, search, display]
obj_coef <- c(0.30, 
              0.20, 
              0.23)

# Constraints matrix
# 1. social + search + display ≤ budget
# 2. social ≤ max_social%
# 3. search ≤ max_search%
# 4. display ≥ min_display%
const_mat <- matrix(c(
  1, 1, 1,          # budget constraint
  1, 0, 0,          # social max
  0, 1, 0,          # search max
  0, 0, -1          # display min (negated for ≤ form)
), nrow = 4, byrow = TRUE)

const_rhs <- c(100, 
               40, 
               50, 
               -10)

const_dir <- c("<=", "<=", "<=", "<=")

# Solve
solution <- lp("max", obj_coef, const_mat, const_dir, const_rhs)
print(solution$solution)  # Optimal allocation
print(solution$objval)    # Maximum return

✅ Key Takeaways

• LP: optimize linear objective with linear constraints
• Simplex/Interior Point algorithms solve efficiently
• Use for budget allocation, liability limits

• Always check constraint feasibility
• Shadow prices show constraint value
• Extend to integer/quadratic for more complex cases