In This Article

  1. Introduction
  2. Components of an Optimization Model
  3. Formulation Steps
  4. Worked Example
  5. Formulation Tips
  6. Conclusion

Introduction

Optimization is about finding the best solution from a set of feasible alternatives. Formulating a problem correctly is often the hardest part—once properly formulated, many optimization problems can be solved efficiently using standard algorithms.

A well-formulated optimization problem clearly defines what decisions need to be made, what objective is being optimized, and what constraints limit the solution space.

Components of an Optimization Model

1. Decision Variables

What you're trying to determine—the unknowns in your problem.

  • Should be clearly defined and measurable
  • Can be continuous, integer, or binary
  • Examples: units to produce, dollars to invest, routes to take

2. Objective Function

What you're trying to optimize—maximize or minimize.

  • Express as a mathematical function of decision variables
  • Common objectives: profit, cost, time, distance, utilization
  • Must be quantifiable

3. Constraints

Limitations or requirements that restrict feasible solutions.

  • Resource constraints: Limited capacity, budget, time
  • Requirement constraints: Minimum levels that must be met
  • Logical constraints: Relationships between variables
  • Non-negativity: Variables cannot be negative
ComponentQuestionMathematical Form
Decision VariablesWhat can we control?x₁, x₂, ..., xₙ
ObjectiveWhat do we want to achieve?Max/Min Z = f(x)
ConstraintsWhat are the limitations?g(x) ≤, =, or ≥ b

Formulation Steps

  1. Understand the problem: What decision needs to be made? What's the goal?
  2. Define decision variables: What are the unknowns? What units?
  3. Formulate objective: Write the goal as a function of variables
  4. Identify constraints: What limitations exist? What requirements must be met?
  5. Express constraints mathematically: Translate to equations/inequalities
  6. Add non-negativity: Most variables can't be negative
  7. Verify: Does the model capture the real problem?

Worked Example: Production Planning

Problem Statement

A company makes two products (A and B). Product A generates ₹30 profit, Product B generates ₹20 profit. Each product requires time on two machines:

  • Product A: 2 hours on Machine 1, 1 hour on Machine 2
  • Product B: 1 hour on Machine 1, 2 hours on Machine 2
  • Machine 1 has 100 hours available; Machine 2 has 80 hours available

How many of each product should be made to maximize profit?

Formulation

Decision Variables:

x₁ = number of units of Product A to produce

x₂ = number of units of Product B to produce

Objective Function:

Maximize Z = 30x₁ + 20x₂ (total profit)

Constraints:

2x₁ + x₂ ≤ 100 (Machine 1 capacity)

x₁ + 2x₂ ≤ 80 (Machine 2 capacity)

x₁, x₂ ≥ 0 (non-negativity)

Formulation Tips

Common Mistakes to Avoid

  • Unclear variable definitions: Always specify units and meaning
  • Missing constraints: Think about all limitations
  • Wrong inequality direction: ≤ for limits, ≥ for requirements
  • Inconsistent units: Ensure all terms use same units
  • Forgetting non-negativity: Most real-world variables can't be negative

Best Practices

  • Start with a verbal description before writing equations
  • Use meaningful variable names (not just x, y)
  • Check that constraint coefficients are correct
  • Verify with a simple example—does a known feasible solution satisfy all constraints?

Conclusion

Key Takeaways

  • Optimization models have three components: variables, objective, constraints
  • Decision variables are what you're trying to determine
  • Objective function expresses what to maximize or minimize
  • Constraints define feasible solutions
  • Follow systematic formulation steps
  • Always include non-negativity constraints
  • Verify your model captures the real problem