Objective Function and Constraints Explained Simply

Every optimisation problem has two mathematical ingredients: a goal and the rules that limit it. The objective function is the goal as a formula, e.g. profit Z = 60x + 30y for two products x and y. The constraints are the rules, written as inequalities like 2x + y ≤ 100 (machine hours) or x ≥ 0 (no negative production). Together they form a linear programming problem.

These two ingredients make linear programming work in the real world. Writing the objective defines success in numbers; listing every constraint captures the true limits. The feasible region holds the answer, and the best answer always sits at one of its corner points.

Key Concepts

• Objective function: Z = c₁x + c₂y, to be maximised or minimised; c₁ and c₂ are per-unit profit or cost rates.
• Decision variables: x and y, the quantities being chosen.
• Constraints: linear inequalities or equations that the variables must satisfy, such as 3x + 4y ≤ 120.
• Feasible region: the set of all (x, y) points satisfying every constraint simultaneously; only these points are allowed.
• Binding constraint: a constraint exactly met (equality holds) at the optimum; others have slack left over.

How It Works

• Step 1 — Build the objective: multiply each variable by its per-unit rate and add: Z = (profit per table)x + (profit per chair)y.
• Step 2 — State the direction: write "maximise Z" for profit or output, "minimise Z" for cost or time.
• Step 3 — Convert each limit: resource caps become ≤, minimum requirements become ≥, exact balances become equations.
• Step 4 — Add x ≥ 0, y ≥ 0: negative production is meaningless in almost every real problem.
• Step 5 — Verify consistency: units must match across each inequality (hours with hours, rupees with rupees).

Worked Example

A bakery sells cakes (x) and pastries (y). Each cake gives ₹40 profit and needs 3 units of flour; each pastry gives ₹55 profit and needs 4 units of flour. Only 120 units of flour are available. Write the objective function and constraints, then check whether making 20 cakes and 15 pastries is feasible.

1. Objective function: maximise Z = 40x + 55y (total profit in rupees).
2. Flour constraint: 3x + 4y ≤ 120.
3. Non-negativity: x ≥ 0, y ≥ 0.
4. Test (20, 15): flour used = 3(20) + 4(15) = 60 + 60 = 120 ≤ 120, and both variables are non-negative, so the plan is feasible — it exactly uses up the flour.
5. Profit: Z = 40(20) + 55(15) = ₹1625. Testing (30, 10) gives flour = 130 > 120 — infeasible, proving constraints genuinely restrict choices.

Common Misconceptions

• Treating the objective function as a constraint: it is the goal to optimise, not a limit; it never restricts the feasible region.
• Ignoring the direction of optimisation: maximising cost or minimising profit are classic sign errors.
• Assuming every constraint binds: some constraints have slack (unused resource) at the optimum; only binding ones shape the answer.

Real-World Applications

• Manufacturing: maximising profit under material, labour and machine limits.
• Logistics: minimising transport cost subject to supply and demand constraints.
• Agriculture: maximising crop yield under land, water and fertiliser limits.
• Finance: maximising portfolio return under budget and risk constraints.

The objective function says what you want; the constraints say what is allowed. Write both in the same variables, keep units consistent, add non-negativity, and every plan can be tested by substitution.