How a Bakery Turns a Profit Question Into a Few Inequalities

A bakery with limited flour, sugar and oven time wants to know how many cakes and biscuits to make for the greatest profit. Problems like this — getting the best result from limited resources — are everywhere, and linear programming turns them into precise mathematics.

This lesson covers the language of linear programming, its advantages and limitations, and formulating word problems.

**A linear programming problem seeks the maximum or minimum value — the optimisation — of a linear objective function such as $Z=ax+by$, subject to linear inequalities called constraints, together with the non-negativity restrictions $x\ge 0$ and $y\ge 0$.

Terminology:

- Decision variables — the unknown quantities, such as x cakes and y tins of biscuits
- Objective function** — the linear expression to be maximised or minimised, such as profit $Z=50x+30y$
- Constraints — linear inequalities from limited resources, such as $2x+y\le 40$
- Non-negativity restrictions — $x\ge 0$ and $y\ge 0$, since quantities cannot be negative
- Optimisation — finding the values of the variables that give the best value of Z

Worked example. Each cake uses 2 kg of flour and each tin of biscuits 1 kg, with 40 kg available, so the flour constraint is $2x+y\le 40$. Making 15 cakes and 10 tins uses $30+10=40$ kg, exactly the limit, and earns $Z=50(15)+30(10)=1050$ rupees.

An everyday example. A shopkeeper deciding how many sacks of rice and wheat to stock in a storeroom of fixed size is solving a linear programming problem.

The substance. Every expression must be linear — terms such as $x^2$ or $xy$ turn it into a different and harder kind of problem.

Linear programming finds the best use of limited resources with a clear, checkable method, but it assumes that relationships are linear, quantities can be divided and data are known exactly, and it is used in manufacturing, transport, diet planning and agriculture.

Advantages:

- Finds the best allocation of scarce resources
- Makes decisions systematic and easy to explain
- Shows which constraints actually limit the result

Limitations:

- Real relationships are not always linear — bulk discounts break linearity
- Solutions may be fractional, such as 12.5 tables, when only whole numbers make sense
- It handles a single objective, while real decisions often balance several
- It assumes that costs, profits and resources are known with certainty

Application areas:

- Manufacturing — the product mix for greatest profit
- Transport — the cheapest way to deliver goods
- Diet planning — the least-cost meals that meet nutritional needs
- Agriculture — dividing land between crops

Worked example. A diet must provide at least 8 units of vitamin A. If food P gives 2 units per packet and food Q gives 1 unit, the constraint is $2x+y\ge 8$; 3 packets of P and 2 of Q give $6+2=8$ units, just meeting the requirement.

An everyday example. A school canteen planning the cheapest weekly menu that still meets nutrition guidelines faces exactly this kind of diet problem.

The substance. A linear model is only as good as its assumptions — if profit per item falls as more are sold, a linear objective function misleads.

To formulate a linear programming problem, identify the decision variables, write the objective function to maximise or minimise, turn each limited resource or requirement into a linear inequality, and add the non-negativity restrictions.

Steps:

- Define the variables, with units
- Write Z for the profit or cost
- Write one constraint for each resource or requirement
- Add $x\ge 0$ and $y\ge 0$

Worked example. A carpenter makes chairs and tables. A chair needs 2 hours of carpentry and 1 hour of polishing and earns ₹300; a table needs 3 hours of carpentry and 2 hours of polishing and earns ₹500. There are 60 carpentry hours and 36 polishing hours a week, and at least 5 tables must be made.

- Variables: x chairs and y tables per week
- Maximise $Z=300x+500y$
- Carpentry: $2x+3y\le 60$
- Polishing: $x+2y\le 36$
- Orders: $y\ge 5$
- $x\ge 0$ and $y\ge 0$

Checking a plan. Making 12 chairs and 12 tables uses $24+36=60$ carpentry hours and $12+24=36$ polishing hours, meets the order constraint, and earns $Z=3600+6000=9600$ rupees.

An everyday example. A farmer dividing a fixed area of land between wheat and mustard, with limited water and labour, writes exactly this kind of formulation before deciding.

The substance. **Read 'at least' as $\ge$ and 'at most' as $\le$** — reversing a single inequality changes the whole set of possible plans.