Constrained Optimization

Business Calculus · Axiom Academy

LESSON Constrained Optimization Optimizing when there are limits or requirements 1 What is Constrained Optimization? In real business problems, we rarely get to optimize freely. There are always constraints - limits on resources, requirements to meet, or budgets to stay within. "Maximize profit with a 50,000 budget" Find max P(x) subject to C(x) ≤ 50,000 Equality constraints: Must use exactly 50,000 Inequality constraints: Cannot exceed 50,000 Multiple constraints: Budget AND warehouse capacity For problems with one equality constraint, we can use substitution to reduce the problem to one variable. Identify the constraint equation: g(x, y) = c Solve for one variable: y = h(x) Substitute into objective: f(x, h(x)) Optimize the single-variable function Objective: Maximize area A = xy Constraint: 200 feet of fencing: 2x + 2y = 200 Result: x = 50, y = 50. A square maximizes area! 3 Business Application: Budget Allocation A company wants to maximize sales by spending money on advertising (x) and sales staff (y). Their sales function and budget constraint are: Budget Constraint: 100,000 total Step 1: Solve constraint for y Step 2: Substitute into sales function Step 3: Take derivative and set to zero 25,000 on advertising, 75,000 on sales staff This generates maximum sales of 625,000

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