Optimization: Max/Min Problems, Business Applications, Constraints

TAKEAWAYS:

• Optimization finds the best possible value (max profit, min cost) using derivatives and constraints.
• Unconstrained optimization uses first/second derivative tests to locate extrema (peaks/troughs).
• Constrained optimization (e.g., budget limits) requires substitution or Lagrange multipliers (basic form).
• Business applications include pricing strategies, production planning, and resource allocation.
• Graphical methods (e.g., feasible region) solve linear constraints visually before calculus.
• Exam focus: Solve word problems, interpret economic scenarios, and justify answers with calculus.

1. What is Optimization?

Optimization is the process of finding the best possible value (maximum or minimum) of a function under given conditions. In business, this means:

• Maximizing profit (revenue – cost).
• Minimizing cost (production, logistics, waste).
• Optimizing resource use (labor, materials, time).

Key Idea: A function $f(x)$ has a local maximum at $x=a$ if $f(a)\ge f(x)$ for all $x$ near $a$, and a local minimum if $f(a)\le f(x)$.

2. Unconstrained Optimization (No Restrictions)

When there are no constraints, we use calculus to find extrema.

Step 1: Find the First Derivative

Set $f^{\prime }(x)=0$ to locate critical points (potential maxima/minima).

Step 2: Second Derivative Test

• If $f^{\prime \prime }(x)>0$: Local minimum (concave up).
• If $f^{\prime \prime }(x)<0$: Local maximum (concave down).
• If $f^{\prime \prime }(x)=0$: Test fails; use first derivative or graph.

Example 1: Maximizing Profit (Unconstrained)

Problem: A company’s profit (in ₹) from selling $x$ units is $P(x)=-2x^2+100x-100$. Find the production level for maximum profit.

Solution:

1. Find $P^{\prime }(x)$: $P^{\prime }(x)=-4x+100$.
2. Set $P^{\prime }(x)=0$: $-4x+100=0$ $x=25$.
3. Second derivative test: $P^{\prime \prime }(x)=-4<0$ → Local maximum at $x=25$.
4. Max profit: $P(25)=-2(25{)}^2+100(25)-100=1125$.

Real-World Tie:

• Daraz’s dynamic pricing adjusts prices (like $P(x)$) to maximize sales revenue during festivals (e.g., Dashain sales). The "$x=25$" here represents the optimal price point where demand (and thus profit) peaks before declining.

3. Constrained Optimization (With Restrictions)

Most real problems have constraints (e.g., budget, time, resources). We solve these using:

1. Substitution method (for simple constraints).
2. Lagrange multipliers (advanced; basic form covered here).

Example 2: Minimizing Cost with a Budget Constraint

Problem: A factory produces two products, A and B. Cost functions are:

• $C_A=2x^2+3x$ (for A)
• $C_B=x^2+5x$ (for B) Constraint: Total cost ≤ ₹500. Find the optimal production mix to minimize waste.

Solution:

1. Total cost constraint: $2x^2+3x+x^2+5x\le 500$ $3x^2+8x-500=0$.
2. Solve for $x$: Use quadratic formula: $x=\frac{-8\pm \sqrt{64+6000}}{6}$. Feasible solution: $x\approx 10$ (units of A and B).
3. Verify with second derivative: $C^{\prime }(x)=6x+8$, $C^{\prime \prime }(x)=6>0$ → Minimum at $x=10$.

Real-World Tie:

• NTC’s network optimization uses similar constraints to allocate bandwidth (a "budget") across regions (products A/B) to minimize latency (cost). The quadratic cost functions here mimic real-world diminishing returns on investment.

4. Graphical Optimization (Linear Constraints)

For linear constraints, we use the feasible region method:

1. Plot constraints as inequalities.
2. Identify the corner points (vertices of the feasible region).
3. Evaluate the objective function at these points.

Example 3: Maximizing Profit with Resource Limits

Problem: A company makes two products:

• Product X: 3 labor hours, ₹2 profit.
• Product Y: 2 labor hours, ₹3 profit. Constraints:
• Labor ≤ 30 hours.
• Materials ≤ 20 units (X needs 1, Y needs 2). Maximize profit.

Solution:

1. Define variables: $x$ = units of X, $y$ = units of Y.
2. Constraints:
   • $3x+2y\le 30$ (labor).
   • $x+2y\le 20$ (materials).
   • $x,y\ge 0$.
3. Feasible region: Plot inequalities and find corner points:
   • (0, 0), (10, 0), (0, 15), (4, 9).
4. Evaluate profit $P=2x+3y$:
   • $P(0,0)=0$
   • $P(10,0)=20$
   • $P(0,15)=45$
   • $P(4,9)=35$ Maximum profit = ₹45 at (0, 15).

Real-World Tie:

• Pathao’s driver allocation uses this method to assign drivers (X/Y) to routes (constraints: max trips/day, fuel limits) to maximize earnings (profit). The corner points represent optimal driver-route combinations.

5. Optimization in Business Scenarios

Example 4: Loan Interest Optimization (Bank Scenario)

Problem: A bank offers loans with interest $I=0.05t^2-2t+100$, where $t$ = years. Find the optimal loan term for maximum interest.

Solution:

1. Find $I^{\prime }(t)$: $I^{\prime }(t)=0.1t-2$.
2. Set $I^{\prime }(t)=0$: $0.1t-2=0$ → $t=20$ years.
3. Second derivative: $I^{\prime \prime }(t)=0.1>0$ → Minimum interest at $t=20$. Wait! This seems counterintuitive. Let’s re-examine:
   • The bank wants maximum interest, so we need a maximum of $I(t)$.
   • The parabola $I(t)$ opens upward (since $I^{\prime \prime }(t)>0$), so there is no maximum—interest grows without bound as $t$ increases.
   • Real constraint: Loan terms are typically ≤ 10 years. So, evaluate $I(t)$ at $t=10$: $I(10)=0.05(100)-20+100=95$.
   • Conclusion: Under a 10-year constraint, the bank’s maximum interest occurs at the longest allowed term.

Real-World Tie:

• Nepal’s NMB Bank uses similar models to set loan terms. For example, a ₹1,00,000 loan at 10% interest with a 5-year term will yield different returns than a 10-year term. Optimization helps banks balance risk (constraint) and profit (objective).

6. Common Mistakes to Avoid

1. Ignoring constraints: Solving $f^{\prime }(x)=0$ without checking feasibility.
2. Misapplying second derivative test: Forgetting to check if $f^{\prime \prime }(x)=0$.
3. Graphical errors: Not shading the feasible region correctly.
4. Unit confusion: Mixing up units (e.g., cost in ₹ vs. labor hours).
5. Assuming all extrema are global: Local maxima/minima may not be the best solution.

In the Real World

1. eSewa’s Transaction Fees:

   • Idea: Unconstrained optimization of fee structures.
   • How: eSewa adjusts transaction fees (like $P(x)$) to maximize revenue while keeping users engaged. The "profit function" here is revenue minus user churn costs.

2. Khalti’s Dynamic Pricing:

   • Idea: Constrained optimization for payment processing.
   • How: Khalti sets transaction limits (constraints) to minimize fraud (cost) while maximizing approval rates (profit). The feasible region represents safe transaction volumes.

3. NTC’s Network Routing:

   • Idea: Graphical optimization for bandwidth allocation.
   • How: NTC uses linear constraints (e.g., max data per region) to route traffic through optimal paths, minimizing latency (like minimizing $C(x)$ in Example 2).

4. Daraz’s Warehouse Placement:

   • Idea: Multi-variable optimization for logistics.
   • How: Daraz places warehouses (variables $x,y$) to minimize delivery time (objective) under constraints like road capacity and demand. The solution involves solving for corner points in a 2D feasible region.

5. Nepal Stock Exchange (NEPSE) Investments:

   • Idea: Portfolio optimization (like Example 3).
   • How: Investors allocate funds between stocks (X) and bonds (Y) to maximize returns under risk constraints (e.g., max 30% in volatile stocks). The corner points represent optimal portfolios.

Exam Tip

1. Always check constraints: If a problem has limits (e.g., "no more than 100 units"), solve within that range.
2. Label your answers: State whether you’re finding a maximum or minimum, and justify with calculus (e.g., "since $f^{\prime \prime }(x)>0$, this is a minimum").
3. Graphical problems:
   • Draw the feasible region clearly.
   • Test all corner points for the objective function.
4. Word problems:
   • Define variables explicitly (e.g., "let $x$ = units of Product A").
   • Translate constraints into equations (e.g., "labor ≤ 30 hours" → $3x+2y\le 30$).
5. Second derivative test:
   • If $f^{\prime \prime }(x)=0$, use the first derivative test (check sign changes around critical points).
6. Real-world tie-ins:
   • Exams often ask for business interpretations. For example:
     • If optimizing profit, say: "The company should produce 25 units to maximize profit of ₹1125."
     • If minimizing cost, say: "The optimal production mix is 10 units of A and 10 units of B to minimize waste."

Practice Questions (Exam-Style)

1. A company’s revenue is $R(x)=50x-0.5x^2$. Find the production level for maximum revenue.
2. Minimize $C(x)=x^2+4x+10$ subject to $x\ge 2$.
3. A farmer has 100 meters of fencing for a rectangular field. Find dimensions for maximum area.
4. Solve graphically: Maximize $P=3x+4y$ subject to:
   • $2x+y\le 20$
   • $x+2y\le 20$
   • $x,y\ge 0$.
5. Ncell’s data plan optimization: A plan costs ₹500 with $D$ GB data. Revenue is $R(D)=500D-0.1D^2$. Find the optimal data allocation for max revenue.

Key Formulas to Memorize

Final Visual Summary

mindmap
  root((Optimization in Business))
    Unconstrained
      First Derivative: f'(x) = 0
      Second Derivative: f''(x) test
      Example: Profit maximization
    Constrained
      Substitution method
      Lagrange multipliers (basic)
      Example: Budget limits
    Graphical
      Feasible region
      Corner points
      Example: Production planning
    Real-World
      eSewa: Fee optimization
      NTC: Bandwidth allocation
      Daraz: Warehouse placement
    Exam Tips
      Check constraints
      Label answers
      Draw graphs clearly