Cost volume profit analysis - TONTUF Money

1. Identifying Limiting Factors in a Scarce Resource Situation and Selecting an Appropriate Technique

Limiting Factors:

• Definition: Limiting factors, also known as constraints, are resources that are in short supply and restrict the ability to achieve desired levels of output or performance.
• Examples: Machine hours, labor hours, raw materials, financial capital.

Techniques for Managing Limiting Factors:

• Contribution Margin Analysis: Focus on maximizing the contribution margin (sales revenue minus variable costs) per unit of the limiting resource.
• Linear Programming: Use linear programming to optimize production schedules and resource allocation when dealing with multiple constraints.
• Theory of Constraints: Identify and manage the bottleneck resource that limits overall system performance and maximize its efficiency.

2. Determining the Optimal Production Plan with a Single Limiting Factor

Single Limiting Factor:

• Definition: A situation where only one resource or constraint restricts production or decision-making.

Approach:

1. Identify the Limiting Factor: Determine which resource is in short supply.
2. Calculate Contribution Margin per Unit of Limiting Factor: Compute the contribution margin (profit per unit) for each product or activity divided by the amount of the limiting factor required.
3. Prioritize Products/Activities: Rank products or activities based on the highest contribution margin per unit of the limiting factor.
4. Allocate Resources: Produce or perform the activities in order of priority until the limiting factor is fully utilized.

Example in Make-or-Buy Decisions:

• If the limiting factor is machine hours, compare the contribution margin per machine hour for in-house production versus buying from an external supplier. Opt for the option that provides the highest contribution margin per machine hour.

3. Formulating and Solving Multiple Scarce Resource Problems

Linear Programming:

• Definition: A mathematical technique for optimizing a linear objective function subject to linear constraints.
• Graphical Method: Use when there are two variables. Plot the constraints and objective function on a graph to find the feasible region and optimal solution.
• Simultaneous Equations: Use when dealing with more than two variables or when linear programming graphs are impractical. Set up equations based on constraints and solve for the optimal values.

Example:

• Graphical Method: Plot the constraints on a graph, identify the feasible region, and find the point that maximizes the objective function within that region.
• Simultaneous Equations: Solve a system of linear equations derived from the constraints to find the optimal solution.

4. Shadow Prices (Dual Prices) and Their Implications

Shadow Prices:

• Definition: The value of one additional unit of a scarce resource in terms of its impact on the objective function (e.g., profit). It indicates how much the objective function would improve if the constraint were relaxed by one unit.
• Calculation: Typically obtained from linear programming solutions. It represents the change in the optimal value of the objective function with a one-unit increase in the constraint's availability.

Implications:

• Decision-Making: Helps prioritize where to allocate resources or make investments. If the shadow price is high, it suggests a significant benefit from increasing the resource.
• Performance Management: Indicates areas where performance could be improved by managing or acquiring more of the scarce resource.

5. Calculating Slack and Implications

Slack:

• Definition: The amount of unused capacity in a constraint. It represents the difference between the available and required resources for a constraint in a linear programming problem.
• Calculation: For a constraint in the form ( a_1x_1 + a_2x_2 \leq b ), slack is ( b - (a_1x_1 + a_2x_2) ).

Implications:

• Decision-Making: Indicates whether a constraint is fully utilized or has room for additional activity. High slack suggests that the constraint could handle more production or tasks.
• Performance Management: Helps identify underused resources and opportunities for increasing production or efficiency.

Summary of Techniques

• Limiting Factors: Use contribution margin analysis, linear programming, or Theory of Constraints.
• Optimal Production Plan: Prioritize based on contribution margin per unit of limiting factor.
• Multiple Scarce Resource Problems: Use linear programming (graphical or simultaneous equations).
• Shadow Prices: Assess the value of additional resources and guide investment decisions.
• Slack: Evaluate unused resource capacity and potential for increased output.

These techniques and concepts help in managing scarce resources efficiently and making informed decisions to optimize production and profitability.