What is Linear Programming? — A Scientific Modelling Tool

Linear programming is a powerful mathematical modelling technique used to find the best possible outcome in a situation where there are limitations or restrictions. The word linear tells us that all the relationships involved are straight lines when drawn on a graph, and programming here means planning or optimising — not computer coding!

In MYP Sciences, we study models as tools that help us represent, understand, and make predictions about real systems. Linear programming is one type of mathematical model — it translates a real-world situation into a set of equations and inequalities, then uses those to make optimal decisions.

Linear Programming

A mathematical modelling technique used to find the maximum or minimum value of a quantity (called the objective function) subject to a set of restrictions called constraints, all expressed as linear inequalities.

Note

Where does LP fit in MYP Sciences? In the Thinking with Models unit, we explore how scientists build mathematical models to represent natural systems, test predictions, and make decisions. Linear programming is an example of an optimisation model — a model specifically designed to answer the question: given these real constraints, what is the best possible outcome?

Linear programming is used extensively in science and the real world to solve problems like:

An ecologist deciding how to allocate conservation funding across habitats to maximise biodiversity protection
A pharmacologist designing drug dosing schedules to maximise therapeutic effect while staying within safe toxicity limits
A factory deciding how many of two products to make in order to maximise profit
A dietitian planning meals to minimise cost while meeting nutritional needs
A farmer deciding how much of each crop to plant to maximise yield while protecting soil health

Analogy

Imagine you are running a small bakery. You can bake cakes or cookies, but you only have a limited amount of flour, butter, and time. Linear programming helps you figure out exactly how many of each to bake so that you make the most money possible — without breaking those resource limits. The mathematical model represents your bakery system, and the optimal solution is your prediction of the best strategy.

In MYP Sciences, linear programming connects to the Thinking with Models unit because the graphical model we build (a shaded feasible region on a coordinate plane) represents all the possible solutions to a real system, and we then identify the optimal one — just as a scientist uses a model to identify the best outcome from all possible scenarios.

Note

A key scientific question to keep in mind: Every model has limitations. As you work through linear programming, ask yourself — what assumptions are we making? When might this model break down? We will return to this at the end of the unit.

Reviewing Inequalities

Before tackling linear programming, let's do a quick review of inequalities — you have seen these before in Mathematics, and they are the building blocks of every constraint in an LP model.

Inequality

A mathematical statement that shows the relationship between two expressions that are not necessarily equal, using the symbols $<$ , $>$ , $\leq$ , or $\geq$ .

The four inequality symbols and their meanings:

Symbol	Meaning	Example
$<$	strictly less than	$x < 5$
$>$	strictly greater than	$y > 2$
$\leq$	less than or equal to	$x \leq 10$
$\geq$	greater than or equal to	$y \geq 0$

Note

In most real-life linear programming problems, we use $\leq$ or $\geq$ rather than strict inequalities, because a constraint like "you have at most 100 kg of flour" includes using exactly 100 kg as a valid option. Strict inequalities ( $<$ , $>$ ) produce dashed boundary lines on graphs and mean the boundary itself is not included.

Solving a simple inequality works almost exactly like solving an equation, with one critical rule:

Warning

Flip the inequality sign whenever you multiply or divide both sides by a negative number.

For example: $- 2 x \leq 8$ → divide both sides by $- 2$ → $x \geq - 4$ (sign flips!)

Example

Solve: $3 x + 4 \leq 19$

Step 1: Subtract 4 from both sides: $3 x \leq 15$

Step 2: Divide both sides by 3: $x \leq 5$

Interpretation: Any value of $x$ that is 5 or less satisfies this inequality.

For LP problems, we will be working with inequalities in two variables ( $x$ and $y$ ). These describe entire regions on a graph rather than just a range of values on a number line — which is what we explore in the next section.

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