Welcome to Limiting Factors!
In a perfect world, a business would have unlimited materials, endless labor hours, and plenty of machine time to make everything they want. But in the real world, something usually holds us back. Maybe you have plenty of customers, but not enough staff to serve them. Or maybe you have the staff, but you’ve run out of raw materials. Limiting factors (also called "bottlenecks" or "constraints") are simply the things that prevent a business from expanding further.
In this chapter, we will learn how to make the best possible decisions to maximize profit when resources are scarce. Don't worry if this seems tricky at first; we will break it down into simple, logical steps!
1. Single Limiting Factor Analysis
When a business has only one resource in short supply (e.g., just labor hours), we use a simple ranking method. The goal is to get the "biggest bang for our buck"—or more accurately, the most contribution for every unit of the scarce resource we use.
The 5-Step "Golden Rule" for Single Limiting Factors
If you follow these steps, you can solve almost any single limiting factor problem:
Step 1: Calculate the Contribution per unit for each product. \( (\text{Selling Price} - \text{Variable Costs}) \).
Step 2: Identify how much of the limiting factor each unit uses (e.g., 2kg of material or 3 hours of labor).
Step 3: Calculate the Contribution per unit of limiting factor. This is the most important step!
Formula: \( \frac{\text{Contribution per unit}}{\text{Limiting factor required per unit}} \)
Step 4: Rank the products. The product that gives the highest contribution per unit of the scarce resource is Rank 1.
Step 5: Allocate the scarce resource. Fill the demand for Rank 1 first, then use whatever is left for Rank 2, and so on.
An Everyday Analogy
Imagine you are baking for a school fair. You have only 10 eggs (your limiting factor).
• A Cake gives $10 profit but uses 5 eggs. (That’s $2 profit per egg).
• A Batch of Cookies gives $6 profit but uses 2 eggs. (That’s $3 profit per egg).
Even though the cake has a higher total profit, you should bake the cookies first because they use your limited eggs more efficiently!
Quick Review: Key Takeaway
Always rank products based on Contribution per unit of the limiting factor, NOT the highest contribution per unit of the product itself.
2. Multiple Limiting Factors: Linear Programming
What happens if you are short on both labor and materials? We can’t just use a simple ranking because what's best for labor might not be best for materials. For this, we use Linear Programming.
Did you know? Linear programming is used by airlines to schedule flights and by refineries to blend fuel. It's a powerful tool for finding the "optimal" solution.
The Process of Linear Programming
Don't let the math scare you. It’s just a series of logical steps:
1. Define the Variables: Usually, this is \( x \) and \( y \) (e.g., Let \( x = \text{Product A} \) and \( y = \text{Product B} \)).
2. Define the Objective Function: This is what we want to maximize. Usually: \( \text{Maximize Contribution (C)} = (\text{Contrib per unit of } x) \cdot x + (\text{Contrib per unit of } y) \cdot y \).
3. Define the Constraints: These are the "rules" we must follow. For example, if you only have 1,000kg of material: \( (\text{Material for } x) \cdot x + (\text{Material for } y) \cdot y \le 1,000 \).
Note: Always include the "non-negativity" constraint: \( x, y \ge 0 \), because you can't produce a negative amount of a product!
4. Drawing the Graph
To find the best mix of \( x \) and \( y \), we draw the constraints on a graph.
Trick for drawing lines: For each constraint, find where the line hits the axes.
• Assume \( x = 0 \) and find the value of \( y \).
• Assume \( y = 0 \) and find the value of \( x \).
Connect these two points with a straight line.
The Feasible Region: This is the area on the graph where all constraints are satisfied. It is usually the area closest to the corner (0,0) where all the shaded areas overlap.
5. Finding the Optimal Point
The "best" combination of products is almost always at one of the corners of the feasible region. You can find the exact point by using simultaneous equations where two constraint lines cross.
3. Shadow Prices and Slack
Once we have a plan, managers often ask "What if?" questions. This is where shadow prices and slack come in.
What is Slack?
Slack occurs when you have leftover resources. If you have 100 hours of labor but your optimal production plan only uses 90, you have 10 hours of "slack."
• Binding Constraint: A resource that is completely used up (Zero slack).
• Non-binding Constraint: A resource that has some left over (Positive slack).
What is a Shadow Price?
The Shadow Price is the extra contribution you would earn if you had one more unit of a scarce resource.
Example: If the shadow price of labor is $5, and someone offers you one extra hour of labor for $2, you should take it! You will make $5 more in contribution while only paying $2 extra.
Important Note: The shadow price only tells you the extra contribution. It is the maximum premium (over the normal variable cost) a company should be willing to pay for one more unit of the resource.
Quick Review: Shadow Prices
• If a resource has Slack, its shadow price is zero (because having more of something you already have leftovers of won't increase your profit).
• Shadow price is only valid for a certain range of additional resources.
4. Common Mistakes to Avoid
1. Ranking by Profit instead of Contribution: Always use contribution! Fixed costs don't change based on your production mix in the short term, so they are irrelevant for this decision.
2. Forgetting the units: Make sure you check if the limiting factor is in hours, kilograms, or liters. Sometimes the question gives you "minutes" but the capacity in "hours." Always convert them to be the same!
3. Misinterpreting Shadow Price: Remember that the shadow price is the increase in contribution. It already takes into account the normal variable cost of the resource.
Summary Takeaway
Limiting Factor analysis is all about efficiency. When you have one constraint, Rank by contribution per unit of that constraint. When you have multiple constraints, use Linear Programming to find the feasible region and use the corners to find the optimal production plan. Shadow prices help managers decide how much to pay for extra resources to grow the business further.