Lesson Notes By Weeks and Term v4 - SHS 2

Lesson Video

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Performance objectives

• Model real-life scenarios using algebraic expressions.
• Set up objective functions for optimization.
• Formulate constraints based on given conditions.
• Understand how these models apply to business and planning.

Lesson summary

In our daily lives, especially in business and planning, we constantly face situations with limited resources. A farmer has a limited amount of land and fertilizer. A small chop bar has a limited budget for ingredients. A tailor has a limited amount of fabric and time. How do we make the best decisions to get the most out of what we have? This lesson introduces a powerful mathematical tool called Linear Programming. It is a method used to find the best possible outcome (like maximizing profit or minimizing cost) in a situation described by a set of linear relationships. By applying algebra, we can model these real-life challenges and find concrete, optimal solutions.

Lesson notes

What is Linear Programming?

Linear Programming is a mathematical technique for optimizing a linear function, subject to a set of linear inequality or equality constraints. In simple terms, it helps us find the "best" solution from a range of possible options.

The key components of a linear programming problem are: Decision Variables: These are the quantities we need to determine. They are the "unknowns" in the problem, usually represented by variables like `x` and `y`. *Example:* If a carpenter makes tables and chairs, the decision variables would be `x` = number of tables to make, and `y` = number of chairs to make. Objective Function: This is a linear equation that expresses the main goal of the problem. It's the quantity we want to maximize (e.g., profit) or minimize (e.g., cost). It is written in the form `P = ax + by` or `C = ax + by`. *Example:* If the profit on a table is GHC 100 and on a chair is GHC 60, the objective function for profit (P) would be: P = 100x + 60y. Constraints: These are the limitations or restrictions in the problem, expressed as a system of linear inequalities. They represent the limited resources like time, materials, or money. *Example:* If each table takes 4 hours of labour and each chair takes 2 hours, and there are only 80 hours of labour available per week, the constraint would be: 4x + 2y ≤ 80. Non-negativity Constraints: In most real-world problems, the decision variables cannot be negative. We cannot produce -5 tables. So, we almost always include the constraints x ≥ 0 and y ≥ 0. Feasible Region: This is the graphical representation of all possible combinations of the decision variables that satisfy all the constraints. It is the area on the graph where all the shaded regions of the inequalities overlap. Vertices (Corner Points): These are the points where the boundary lines of the feasible region intersect. A fundamental principle of linear programming (the Corner Point Theorem) states that the optimal solution (maximum or minimum) will always occur at one of these vertices. Step-by-Step Method for Solving Linear Programming Problems Identify and Define Variables: Determine what you need to find and assign variables (e.g., `x`, `y`). Formulate the Objective Function: Write the equation for the quantity to be maximized or minimized. Formulate the Constraints: Write all the limitations as a system of linear inequalities. Don't forget `x ≥ 0` and `y ≥ 0`. Graph the Constraints: Draw a Cartesian plane. For each inequality, graph its boundary line and shade the region that satisfies it. The common overlapping area is the feasible region. Find the Vertices: Determine the coordinates of all the corner points of the feasible region. Some will be intercepts on the axes, while others are found by solving the system of equations for the two lines that intersect at that point. Test the Vertices: Substitute the coordinates of each vertex into the objective function. State the Conclusion: Identify the vertex that gives the maximum or minimum value and state the answer clearly in the context of the original problem.

Evaluation guide

• Find the factors and zeros of a polynomial function using conventional and personal strategy.
• Collaborative Learning, Experiential Learning, Problem -based Learning, Project -Based
• understanding
• Graphically, we have: