Linear Programming for A-Level and IB: Practical Guide to Graphical Methods
This work has been verified by our teacher: day before yesterday at 1:27
Homework type: Essay
Added: 18.01.2026 at 8:35
Summary:
Master Linear Programming for A-Level and IB using graphical methods: modelling, feasible regions, corner points, objective-line approach and exam tips.
Linear Programming: A Comprehensive Guide for A-Level and IB Students
Linear programming (commonly abbreviated as LP) stands as a vital mathematical technique widely used to optimise outcomes—such as maximising profit or minimising cost—subject to a set of linear restrictions. Imagine a small bakery allocating flour, sugar and time between cakes and scones to generate the highest profit. The baker must make decisions within constraints: resource limits, demands, or labour hours. LP translates these real-world situations into manageable mathematical models, enabling decision-makers in industries ranging from agriculture to aviation to reach the best solutions efficiently.
At the level of A-Level or IB study in the United Kingdom, linear programming is typically taught with a focus on problems involving two decision variables, so their graphical solution can be visualised and understood deeply. This essay will guide you through the essential aspects of linear programming as encountered at pre-university level: converting real situations into LP models, graphical methods, interpretation, and tackling special cases, finishing with a worked example and some practical strategies for exam success. Brief references to the broader field—such as higher-dimensional problems, the simplex method, and integer programming—will point the way for further study.
---
Converting Real Situations into Linear Programming Models
The first step in LP is to translate a practical problem into a mathematical form—a model. This process begins by clearly identifying the decisions to be made.Decision Variables
Define your variables precisely, stating what they represent and their units. For example, if a juice company produces orange juice and lemonade, let \( x \) be “litres of orange juice bottled per day” and \( y \) “litres of lemonade bottled per day.” Crisp variable definitions avoid confusion and errors later.Writing Constraints in Plain English
Before hurrying into algebra, jot down each restriction as a normal sentence. For instance: “No more than 60 litres of oranges are available a day.” Expressing constraints clearly helps avoid the classic blunder of reversing an inequality.Converting Constraints to Mathematical Language
Each English statement becomes a linear equation or inequality: - “Total oranges used must not exceed 60 litres” translates to \( x \leq 60 \). - “At least 40 bottles of lemonade must be produced” converts to \( y \geq 40 \). - “10 minutes labour per orange juice, 8 minutes per lemonade, 8 hours total” turns into \( 10x + 8y \leq 8 \times 60 \) (remember to match units).Some constraints are equalities (e.g., “the output must be exactly 1000 units”), but most in practical questions are in the form of inequalities.
Non-Negativity Constraints
Decision variables such as products made, hours worked, or resources used cannot be negative. Formally, this is written as \( x \geq 0 \), \( y \geq 0 \). On the graph, this ensures solutions live in the first quadrant.Integer Requirements
In some cases—number of staff, number of lorries—fractions are nonsensical. You need “\( x, y \) integer.” When these apply, simply rounding the optimal solution from the graph may not work, a point returned to later.Units and Clarity
Finally, keep units consistent and always label axes and variables. If the objective is profit in pounds, ensure every coefficient and variable that forms it is expressed in those terms.---
The Objective Function
The heart of LP is the linear objective: a formula to maximise or minimise. It might seek the highest possible profit, e.g., \( \text{Profit} = 20x + 15y \), or the lowest cost, \( \text{Cost} = 8x + 5y \). Begin your model explicitly with an instruction: Maximise or Minimise! Check the units—students often forget, leading to nonsensical results (e.g., profit in “bottles” instead of pounds).---
Preparing for a Graphical Solution
Graphical methods empower us to “see” how constraints interact. These methods are only practical for two decision variables; any more (e.g., three types of products) requires advanced algebraic algorithms such as the simplex method.Plot-Ready Constraints
Rewrite constraints in the form \( y = mx + c \) or use intercepts. To plot \( x + 2y \leq 40 \), set \( x = 0 \) to get \( y \)-intercept (\( y = 20 \)), then \( y = 0 \) to get \( x \)-intercept (\( x = 40 \)). Draw the straight line joining these points.Drawing Constraints
On squared paper, use a ruler for accuracy. Label each line with its equation and a reference (C1, C2, etc.). Solid lines indicate “\( \leq \)” or “\( \geq \)” (the boundary is included); dashed lines are for strict inequalities, although A-Level and IB questions nearly always use inclusive bounds.---
Determining the Feasible Region
The feasible region is the set of all points—combinations of decision variables—that satisfy every constraint, including non-negativity.Shading Techniques
For each constraint, use a test point (the origin is often easiest) to determine which side of the line to shade. For example, with \( x + y \leq 10 \), try \( (0,0) \): \( 0 + 0 = 0 \leq 10 \), so the region containing the origin is feasible for this constraint. Only the area that meets *all* inequalities is the feasible region, usually a polygon in the first quadrant. Mark its vertices clearly (A, B, C,...).---
Two Graphical Solution Methods
(A) The Corner-Point Method
A vital theorem in LP states the optimum must occur at a vertex (corner point) of the feasible region. The process is: - List all corner points. - Find the coordinates (read from axes or solve where lines intersect). - Plug each into the objective function. - The vertex with the best value (highest for maximisation, lowest for minimisation) is the solution. - If multiple vertices give the same objective value, the whole edge joining them is optimal.(B) The Parallel Objective-Line Method
Draw the objective as a straight line—e.g., “profit \( = £200 \)”—by plotting values when \( x = 0 \) and \( y = 0 \). Then “slide” this line in the direction that increases (or decreases) the objective while keeping it parallel, stopping as it last touches the feasible region. The contact point is the optimal vertex. This method is fast and visually intuitive—excellent for checking work in exams.---
Calculating Corners Algebraically
Intersections of two constraints are found by solving simultaneous linear equations. The elimination or substitution methods are both standard GCSE/AS techniques. Keep calculations exact (as fractions where necessary) until the final answer. Always check points for feasibility by testing them in *all* constraints, not just the two you used to find them.---
Special Cases
Certain patterns can arise:- Infeasibility: No feasible region—constraints are contradictory. For instance, “\( x+y \leq 5 \)” and “\( x+y \geq 8 \)” is impossible. - Unbounded region: The feasible region goes on forever (e.g., is open towards higher profits), and the objective has no maximum or minimum within the given constraints. - Multiple optima: The objective line is parallel to an edge of the feasible region, so every point along that edge is optimal—report the range of solutions. - Degeneracy: More than two constraints meet at a single point, leading to a “pointy” vertex. At A-Level, this has no practical impact.
---
Integer Programming Considerations
While graphical LP delivers continuous solutions, often problems require whole numbers (people, cars, parcels). Graphically, the optimal vertex may not be at integer coordinates. Systematically check nearby integer points within the feasible region; naive rounding can violate constraints or miss a better integer answer. True integer programming uses advanced methods like branch-and-bound, but for small problems, “trial and improvement” suffices.---
Sensitivity and Interpreting the Solution
A “binding” constraint is one that holds as an equality at the optimal point—it “binds” or “pins” the solution. Should you relax (or tighten) it, the solution might change. A “non-binding” constraint has “slack” (unused capacity or surplus) and could be altered a little without affecting the answer. A simple example of sensitivity: if the total labour hours available increase, could the profit be increased further?At more advanced levels, the “shadow price” represents how much the objective would improve per unit increase in a binding resource—handy for deciding where best to allocate investment or scarce resources.
---
Worked Example
Problem: A farm produces barley (\( x \)) and oats (\( y \)), both in tonnes per season. Each tonne of barley earns £120, each tonne of oats £90. The farmer has 100 acres of land. Barley needs 2 acres/tonne, oats 1 acre/tonne. Demand requires at least 10 tonnes of oats and at least 20 tonnes overall. What mix of crops maximises income?Model: - Let \( x \) = tonnes of barley, \( y \) = tonnes of oats (\( x \geq 0, y \geq 0 \)). - Maximise income: \( 120x + 90y \) - Land: \( 2x + y \leq 100 \) - Oats demand: \( y \geq 10 \) - Total: \( x + y \geq 20 \)
Graph: - Draw axes for \( x \) (barley) and \( y \) (oats), marking units. - Plot \( 2x + y = 100 \): \( x=0, y=100 \); \( y=0, x=50 \). - Plot \( y=10 \) (horizontal line). - Plot \( x + y = 20 \): \( x=0, y=20 \); \( y=0, x=20 \). - Non-negativity: only consider \( x, y \geq 0 \).
Feasible region: Marked by intersection of all regions above constraints (including \( x, y \geq 0 \)), shade accordingly.
Vertices: - (0,10): Surplus barley is negative—rule out. - (0,100): Not all constraints satisfied. - Junctions calculated: - \( y=10, x+y=20 \rightarrow x=10, y=10 \) - \( y=10, 2x+y=100 \rightarrow 2x+10=100 \rightarrow x=45, y=10 \) - \( x+y=20, 2x+y=100 \rightarrow \) subtract gives \( x=80 \) (impossible, as then \( y=-60 \)) - \( 2x+y=100, y=0 \rightarrow x=50 \) - \( x=0, y=10 \)
Evaluate objective at feasible vertices: - (10,10): \( 120*10+90*10=£2100 \) - (45,10): \( 120*45+90*10=£6300 \) So, optimal mix is 45 tonnes barley, 10 tonnes oats. Land used is \( 2*45+10=100 \) acres.
---
Common Mistakes and Exam Strategies
Beware the pitfalls: - Missing non-negativity or integer constraints. - Reversing inequalities (“at least” vs “at most”). - Incorrect line/region shading; *always* test a point. - Failing to label axes or lines. - Rounding intermediate answers.Top Exam Tips: - Always define variables and units first. - Double check every shaded region, verify each vertex algebraically. - Highlight your final answer, giving values with units and objective achieved. - If time is short, use the parallel-line method for a quick solution before confirming precisely.
---
Extensions and Further Study
For more complicated or higher-dimensional problems (more than two variables), the graphical approach breaks down. The simplex algorithm is the standard method, and further topics include duality, integer programming, and network flows—all of which underpin much of industrial scheduling, logistics and supply chain management in the real world, including British supermarkets, rail operators, and NHS staffing rotas.---
Conclusion
Linear programming demands care at every step: thoughtful modelling, precise graphing, and systematic checking. Mastering these skills equips students to solve a broad class of practical problems, from production to diet planning, and builds a strong foundation for advanced mathematical and analytical study. Above all, consistent practice with a range of LP problems, using clear methodical steps, is the surest path to exam confidence and future success.---
Quick Checklist
- Define variables, with units. - State “Maximise/Minimise” and objective function. - Write constraints in words, then algebra. - Include non-negativity and integer requirements. - Rearrange constraints for graphing. - Draw and label all lines accurately; plot intercepts. - Mark and annotate feasible region and all vertices. - Evaluate and select optimum, justifying why it is feasible and optimal. - Interpret solution, check for special cases (infeasible, multiple optima, etc). - State final answer with units and brief explanation.---
Further Reading
- Bostock, L. & Chandler, S., *Core Maths for A-Level* (Nelson Thornes) — an excellent and clear exposition. - Paul Longley et al, *Quantitative Methods for Business* (Standard UK textbook). - MEI resources (www.mei.org.uk) — concise LP notes and practice for A-Level Maths and Further Maths. - Cambridge University Mathematical Tripos past papers for more advanced applications and challenges.*End of Essay*
Rate:
Log in to rate the work.
Log in