Strategic Cost Management · Linear Programming

Introduction to Linear Programming and Formulation

Updated 11 October 2026 · Fact-checked

Linear programming (LP) is a method to find the best value of a linear objective, such as maximum profit or minimum cost, subject to linear limits on scarce resources. To formulate a problem, define decision variables, write the objective function, list the constraints, and add non-negativity conditions.

Understand Introduction to Linear Programming and Formulation

Every business has limited resources: machine hours, labour hours, materials, cash, storage space. You must decide how to use them to earn the most profit or incur the least cost. Linear programming (LP) is a mathematical technique that does this when all relationships are straight-line (linear).

An LP model has four components. Decision variables are the quantities you control, such as units of product A and product B. The objective function is the linear expression you want to maximise (profit, contribution) or minimise (cost). Constraints are linear inequalities or equations that show the resource limits and other requirements. Non-negativity conditions say that variables cannot be negative, because you cannot produce minus 10 units.

LP rests on some assumptions. Linearity: the objective and each constraint are linear, so doubling output doubles resource use and profit. Proportionality and additivity: each unit uses the same resources, and totals are the sum of the parts, with no interaction between products. Divisibility: variables may take fractional values. Certainty: all coefficients (profit per unit, resource use, availability) are known and constant. Single objective: one goal is optimised at a time. Finiteness: a finite number of variables and constraints exist.

These assumptions are also the limitations. Real costs may not be linear because of discounts and learning effects. Fractional units may be meaningless, as with machines or vehicles; then integer programming is needed. Data is rarely certain, and managers often have several goals. Even so, LP gives a clear, defensible allocation of scarce resources.

Formulation is the skill that earns marks. Once the model is written correctly, the graphical or simplex method solves it. Most errors in LP answers start in formulation, not in arithmetic.

Key rules to remember

Objective function (maximisation): Maximise Z = c₁x₁ + c₂x₂ + … + cₙxₙ; cⱼ is the contribution or profit per unit of xⱼ. Use contribution per unit, not profit after fixed costs, when fixed costs do not change with the decision.
Objective function (minimisation): Minimise Z = c₁x₁ + c₂x₂ + … + cₙxₙ; cⱼ is the cost per unit or per use of xⱼ.
Resource constraint: a₁₁x₁ + a₁₂x₂ + … + a₁ₙxₙ ≤ b₁; aᵢⱼ is the use of resource i per unit of xⱼ; bᵢ is the availability. Use ≤ for limited resources.
Requirement constraint: a₂₁x₁ + a₂₂x₂ + … + a₂ₙxₙ ≥ b₂; Use ≥ for minimum requirements such as minimum nutrients, minimum output or contracted supply.
Non-negativity: x₁, x₂, …, xₙ ≥ 0; Always state it. Leaving it out loses marks.

How to solve Introduction to Linear Programming and Formulation questions

Use this sequence for any formulation question. Read the whole problem once before writing anything.

1Identify what the manager must decide and define the decision variables with units, for example x = units of product A per month.
2Identify the goal: maximise contribution or minimise cost. Check the per-unit figure given; if selling price and variable cost are both given, compute contribution.
3Write the objective function using the per-unit figures for each variable.
4List every limited resource or requirement (machine hours, labour, material, demand limit, minimum quantity) and write one constraint for each, choosing ≤, ≥ or = from the wording.
5Check that units match in each constraint, such as hours with hours and kg with kg.
6Add the non-negativity conditions.
7Present the complete model neatly: objective, subject to constraints, non-negativity. If asked, move on to solve it.

Quickest way: Table-first formulation

When to use it: Use when the question gives several products and resources and time is short.

Draw a small grid with products as columns and resources as rows, filling in per-unit use and the availability figure.
Put contribution or cost per unit in the top row.
Read the objective function across the top row.
Read each constraint across its resource row, with availability on the right.
Mark each row ≤ or ≥ from the words 'maximum', 'at most', 'at least', 'minimum'.
Write non-negativity last and re-check units.

Common mistakes in Introduction to Linear Programming and Formulation

Using selling price instead of contribution per unit in the objective function.
Students see the price first and use it directly.
Fix: Compute selling price minus variable cost per unit. Use it unless the question states profit per unit.
Choosing the wrong inequality sign.
The wording 'at least' or 'not more than' is skimmed.
Fix: Resource limits and maximum demand take ≤. Minimum requirements and minimum supply take ≥.
Mixing units in a constraint, such as minutes per unit against hours available.
Data is given in different units across the question.
Fix: Convert everything to one unit before writing the constraint.
Omitting non-negativity conditions.
Students treat it as obvious.
Fix: Write x₁, x₂ ≥ 0 as the last line of every model.
Including fixed costs in the objective function.
All costs given in the question seem relevant.
Fix: Fixed costs that do not change with the decision do not affect the optimum. Leave them out and deduct them only when asked for total profit.
Defining variables vaguely, for example 'let x be product A'.
Students rush to the equations.
Fix: State the quantity and period, such as 'x = units of A produced per week'.

Worked examples

Example 1

Sundaram Industries makes two products, P and Q. Contribution is ₹40 per unit of P and ₹30 per unit of Q. Each unit of P needs 2 machine hours and 3 labour hours. Each unit of Q needs 4 machine hours and 2 labour hours. Available per week: 80 machine hours and 60 labour hours. Market demand for Q is at most 12 units a week. Formulate the LP model to maximise contribution.

Show the solution

Decision variables: x₁ = units of P produced per week; x₂ = units of Q produced per week.
Objective: contribution is ₹40 per unit of P and ₹30 per unit of Q, so Maximise Z = 40x₁ + 30x₂.
Machine hours: 2x₁ + 4x₂ ≤ 80.
Labour hours: 3x₁ + 2x₂ ≤ 60.
Demand limit for Q: x₂ ≤ 12.
Non-negativity: x₁, x₂ ≥ 0.

Answer: Maximise Z = 40x₁ + 30x₂, subject to 2x₁ + 4x₂ ≤ 80; 3x₁ + 2x₂ ≤ 60; x₂ ≤ 12; x₁, x₂ ≥ 0.

Example 2

Kaveri Feeds mixes two ingredients, A and B, to make a cattle feed. Ingredient A costs ₹20 per kg and B costs ₹30 per kg. Each kg of A has 3 units of protein and 1 unit of fibre. Each kg of B has 2 units of protein and 4 units of fibre. Each batch must contain at least 24 units of protein and at least 16 units of fibre. Also, ingredient B must not exceed 5 kg per batch. Formulate the LP model to minimise cost per batch.

Show the solution

Decision variables: x₁ = kg of A per batch; x₂ = kg of B per batch.
Objective: cost is ₹20 per kg of A and ₹30 per kg of B, so Minimise Z = 20x₁ + 30x₂.
Protein requirement (at least): 3x₁ + 2x₂ ≥ 24.
Fibre requirement (at least): x₁ + 4x₂ ≥ 16.
Limit on B (not more than): x₂ ≤ 5.
Non-negativity: x₁, x₂ ≥ 0.

Answer: Minimise Z = 20x₁ + 30x₂, subject to 3x₁ + 2x₂ ≥ 24; x₁ + 4x₂ ≥ 16; x₂ ≤ 5; x₁, x₂ ≥ 0.

Exam tips

In Section A, expect MCQs on LP assumptions, the meaning of terms like objective function and feasible region, and which sign suits a given condition. Learn the assumptions as a short list.
In descriptive questions, always define variables first. Marks are usually given for variables, objective, each constraint and non-negativity separately.
Check whether the question gives contribution, profit or only price and cost. Compute contribution if needed and show the working.
If the question asks to formulate and solve, formulate cleanly first, because an error in formulation carries through every later step.
For 'state limitations of LP' questions, link each limitation to an assumption: linearity, certainty, divisibility and single objective.

Practice questions from Linear Programming

Introduction to Linear Programming and Formulation in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Introduction to Linear Programming and Formulation: frequently asked questions

What are the main assumptions of linear programming?

The main assumptions are linearity, proportionality and additivity, divisibility, certainty of data, a single objective, and non-negative variables. If one fails in a real problem, the LP solution may not be reliable.

How do I decide between maximisation and minimisation?

If the question talks about profit, contribution or revenue, you maximise. If it talks about cost, wastage or time, you minimise. The objective function always follows the goal stated in the question.

Should fixed costs be included in the objective function?

Not when they stay the same whatever the product mix, because they do not change the optimal solution. Use contribution per unit and deduct fixed costs at the end only if the question asks for net profit.

What is the difference between a constraint and the objective function?

The objective function is what you want to optimise, such as maximum contribution. Constraints are the limits on resources or requirements that the solution must respect. Both are linear expressions in the decision variables.