Operations Management and Strategic Management · Optimum Allocation of Resources - LPP

Linear Programming Problem Formulation for CMA Intermediate

Updated 10 October 2026 · Fact-checked

A linear programming problem (LPP) finds the best value of a linear objective, such as maximum profit or minimum cost, subject to linear constraints on scarce resources. To formulate it, 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, material, money, space. Linear programming is a method to decide how to use these limited resources so that profit is highest or cost is lowest.

An LPP has four components:

Decision variables: the unknown quantities you control, such as units of product A (x₁) and product B (x₂).
Objective function: a linear expression to maximise (profit, contribution) or minimise (cost, wastage).
Constraints: linear inequalities or equations showing the resource limits, demand limits or minimum requirements.
Non-negativity restriction: the variables cannot be negative, because you cannot make a negative number of units.

Linear programming rests on certain assumptions: linearity (the objective and constraints are straight-line relationships, so doubling output doubles resource use and profit), proportionality, additivity (total use of a resource is the sum of use by each activity), divisibility (variables can take fractional values), certainty (all coefficients are known and constant), finiteness (a limited number of activities and constraints) and non-negativity.

Common applications include product mix, diet or blending problems, cutting-stock and trim problems, production scheduling, transportation, assignment, investment portfolio selection and advertising media selection.

Formulation is the step where most marks are earned. If your model is right, the graphical or simplex solution that follows is only mechanical work.

Key rules to remember

General form of an LPP: Maximise or Minimise Z = c₁x₁ + c₂x₂ + … + cₙxₙ; cⱼ is the profit or cost per unit of variable xⱼ. Z is the objective function.
Constraints: a₁₁x₁ + a₁₂x₂ + … + a₁ₙxₙ (≤, =, ≥) b₁, and so on for each resource; aᵢⱼ is the amount of resource i used by one unit of xⱼ. bᵢ is the availability or requirement.
Non-negativity: x₁, x₂, …, xₙ ≥ 0; Always write this line. It is a required part of the model.
Typical constraint signs: Resource limit: ≤ | Minimum requirement: ≥ | Exact requirement: =; Maximisation problems usually have ≤ constraints. Minimisation problems usually have ≥ constraints.
Profit per unit: Selling price per unit − Variable cost per unit; Use contribution per unit as the objective coefficient when fixed costs are not affected by the decision.

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 you must decide and define the decision variables with units, for example x₁ = number of units of product A produced per week.
2Find the objective: is it to maximise profit or contribution, or to minimise cost? Work out the coefficient for each variable.
3Write the objective function as Z = c₁x₁ + c₂x₂ + …
4List each scarce resource or condition (machine hours, labour hours, material, demand limit, minimum requirement) and form one constraint for each.
5For each constraint, put the per-unit usage as coefficients, choose the correct sign (≤, ≥ or =), and put the availability or requirement on the right side.
6Check that units are consistent on both sides, for example minutes against hours.
7Add the non-negativity condition x₁, x₂ ≥ 0.
8Write the complete model neatly and re-read the problem to confirm no condition is missed.

Quickest way: Table-first formulation

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

Draw a small table with products as columns and resources as rows. Fill in the usage per unit and the availability.
Add one row for profit or cost per unit.
Read the objective function from the last row.
Read each constraint straight across a resource row, as usage × variables with the sign and the availability.
Scan the text for hidden conditions such as a maximum demand, a minimum order or a ratio, and add them as extra constraints.

Common mistakes in Introduction to Linear Programming and Formulation

Using selling price instead of profit or contribution in the objective function.
Students copy the first rupee figure they see in the question.
Fix: Check what the question asks to maximise. If costs are given, subtract variable cost from selling price per unit.
Mixing units, such as machine time in minutes against availability in hours.
The data are given in different units in different sentences.
Fix: Convert everything to one unit before writing the constraint.
Using the wrong inequality sign.
Students do not think about whether the figure is a limit or a minimum.
Fix: 'At most', 'available' or 'maximum' means ≤. 'At least' or 'minimum requirement' means ≥.
Leaving out the non-negativity condition.
It looks obvious, so students skip it.
Fix: Always end the model with x₁, x₂ ≥ 0.
Defining variables vaguely, for example 'x = product'.
Students rush to the equations.
Fix: State what each variable measures and the time period, for example x₁ = units of A per month.
Missing a ratio or demand condition hidden in the text.
Students stop after the resource constraints.
Fix: Re-read the problem after formulating. Convert a statement such as 'A must be at least twice B' to x₁ ≥ 2x₂, or x₁ − 2x₂ ≥ 0.

Worked examples

Example 1

Sundaram Furnishings makes tables (T) and chairs (C). A table needs 4 hours of carpentry and 2 hours of finishing. A chair needs 3 hours of carpentry and 1 hour of finishing. Carpentry has 240 hours and finishing has 100 hours available per week. Profit is ₹700 per table and ₹500 per chair. Formulate the LPP to maximise profit.

Show the solution

Decision variables: x₁ = number of tables per week, x₂ = number of chairs per week.
Objective: maximise profit Z = 700x₁ + 500x₂.
Carpentry constraint: 4x₁ + 3x₂ ≤ 240.
Finishing constraint: 2x₁ + x₂ ≤ 100.
Non-negativity: x₁, x₂ ≥ 0.

Answer: Maximise Z = 700x₁ + 500x₂ subject to 4x₁ + 3x₂ ≤ 240; 2x₁ + x₂ ≤ 100; x₁, x₂ ≥ 0.

Example 2

A cattle feed firm mixes two ingredients, X and Y. Each kg of X costs ₹30 and contains 4 units of protein and 2 units of fibre. Each kg of Y costs ₹20 and contains 2 units of protein and 3 units of fibre. Each bag of feed must have at least 40 units of protein and at least 30 units of fibre. Formulate the LPP to minimise the cost of one bag.

Show the solution

Decision variables: x₁ = kg of X per bag, x₂ = kg of Y per bag.
Objective: minimise cost Z = 30x₁ + 20x₂.
Protein requirement is a minimum, so the sign is ≥: 4x₁ + 2x₂ ≥ 40.
Fibre requirement is a minimum, so the sign is ≥: 2x₁ + 3x₂ ≥ 30.
Non-negativity: x₁, x₂ ≥ 0.

Answer: Minimise Z = 30x₁ + 20x₂ subject to 4x₁ + 2x₂ ≥ 40; 2x₁ + 3x₂ ≥ 30; x₁, x₂ ≥ 0.

Exam tips

In MCQs, check the sign of each constraint and the objective coefficients before looking at the options. Wrong options usually swap a sign or a coefficient.
In written answers, always start with 'Let x₁ = …'. A complete answer defines the variables, the objective function, each constraint and non-negativity.
Draw the usage table in your rough work. It prevents missed constraints and unit errors.
Know the assumptions by name. Theory questions ask for them, often with a short explanation of each.
Do not solve unless asked. If the question says 'formulate', stop after the model and use the saved time on other questions.

Practice questions from Optimum Allocation of Resources - LPP

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, finiteness and non-negativity. In short, the relationships are straight lines, variables can be fractions, the data are known and fixed, and no variable is negative.

How do I decide between ≤ and ≥ in a constraint?

Ask whether the figure is a ceiling or a floor. A ceiling such as available hours or maximum demand takes ≤. A floor such as a minimum nutrient or minimum order takes ≥.

Do I need to solve the LPP in a formulation question?

No, not if the question says 'formulate' or 'form the model'. Write the objective function, constraints and non-negativity. Solving is done by the graphical or simplex method, which are separate topics.

Should fixed costs be included in the objective function?

Not if they stay the same whatever the product mix. Use contribution or profit per unit as the coefficient. Include a cost only if it changes with the decision, or if the question tells you to.