In the business world and various daily situations, decisions are made to minimise costs and maximise profits based on existing constraints. Through linear programming, real-world problems can be translated into mathematical models involving systems of linear inequalities. By graphing these inequalities, a feasible region is formed to locate the optimum values for specific objective functions.
Follow the step-by-step procedure from interpreting word problems to locating optimum solutions on a feasible region.
Translate verbal constraints into inequalities, graph them, and optimise the objective function.
Interpret the problem statement and determine the decision variables \(x\) and \(y\).
Express all given constraints as a system of linear inequalities, including non-negativity constraints such as \(x \geqslant 0\) and \(y \geqslant 0\).
Draw the boundary lines corresponding to the linear equations. Identify and shade the common region that satisfies all constraints simultaneously. Any point inside or on the boundary of this common region is a feasible solution, while any point outside is an infeasible value.
Write the linear equation representing the quantity to be maximised or minimised, typically in the form \(k = ax + by\).
Maximum or optimum points are located at the vertices (corner points) of the feasible region. Alternatively, select a suitable constant value for \(k\), draw the straight line \(ax + by = k\), and move it parallel across the feasible region to locate the optimal vertex.
"at most" / "does not exceed" \(\rightarrow \leqslant\)"at least" / "not less than" \(\rightarrow \geqslant\)"exceeds ... by at least" \(\rightarrow \text{Variable}_1 - \text{Variable}_2 \geqslant \text{value}\)
Identify the existing constraints and decision variables \(x\) and \(y\).
Define a mathematical model as a system of linear inequalities.
Draw graphs and determine the feasible region \(R\).
Determine the objective function \(k = ax + by\).
Determine the optimal solution at the vertices of \(R\).
Objective function
Maximise or minimise \(k\) subject to the constraints.
Feasible region
Optimum values occur at the vertices of \(R\).
Answer in your mind, then press "Check Answer".
Write a linear inequality for the statement: "The number of table \(Q\) (\(y\)) is at most twice that of table \(P\) (\(x\))".
What is the term for a point or set of values that satisfies every constraint in a linear programming problem?
In a linear programming model, where do the maximum or optimum points of an objective function occur on a graph?
Any point along an objective line \(ax + by = k\) is always feasible.
Only points on the line that lie within or on the boundaries of the feasible region \(R\) are feasible solutions.
The optimum value can be found at any random point inside the shaded region.
The optimum (maximum or minimum) value of a linear objective function always occurs at the boundary vertices of the feasible region.
Non-negativity constraints like \(x \geqslant 0\) and \(y \geqslant 0\) are optional.
Real-world physical quantities (such as number of items, area, or time) cannot be negative, so non-negativity constraints are essential to restrict solutions to the first quadrant.
Objective: Test your understanding of linear programming constraints, feasible regions and optimum points.
1 Which inequality represents the constraint "the total surface area of table \(P\) (\(1\text{ m}^2\)) and table \(Q\) (\(2\text{ m}^2\)) is not less than \(30\text{ m}^2\)"?
2 In a graphical linear programming solution, where are the maximum or optimum points that give the optimum value of the objective function located?
3 A value that satisfies all the constraints in a linear programming problem is known as a ________ value.
4 For the objective function \(k = ax + by\), maximum and minimum values occur at the ________ of the feasible region.
Drag each term to the correct meaning. On a phone, tap a term first, then tap the matching box.
There is something wrong with this question.
