4.10 Break-Even Analysis: A Fundamental Tool for Capacity Evaluation
Break-even analysis is a fundamental tool used to evaluate capacity alternatives by determining the point at which total revenue equals total costs. Since capacity utilization typically involves fixed costs, the objective is to identify the output quantity that generates sufficient revenue to cover both fixed and variable costs.
The break-even point (BEP) or break-even quantity (QBEP) represents the volume of output at which total revenue (TR) equals total cost (TC). At this point, the organization neither realizes a profit nor incurs a loss. Mathematically, at QBEP, the following equation holds:
[latex]TC\;=\;FC\;+\;VC[/latex]
TC = total cost
FC = total fixed cost
VC = total variable cost
Fixed costs are incurred regardless of the output quantity and remain constant within the relevant range of production. Examples of fixed costs include rental expenses, property taxes, equipment costs, heating and cooling expenses, and certain administrative costs.
To calculate the break-even point, the following notation and relationships are used:
[latex]TC=FC+VC[/latex]
[latex]VC=Q\times v[/latex] (Variable Cost = Quantity × Variable Cost per Unit)
[latex]TR=Q\times r[/latex] (Total Revenue = Quantity × Revenue per Unit)
[latex]P=TR-TC=Q\times r-(FC+Q\times v)[/latex] (Profit = Total Revenue – Total Cost)
[latex]Q_{BEP}=FC\div(r-v)[/latex] (Break-Even Quantity = Fixed Cost ÷ (Revenue per Unit – Variable Cost per Unit))
By rearranging the equations and substituting the appropriate values, organizations can determine the break-even quantity (QBEP) for a given capacity alternative. This quantity represents the minimum output level required to recover the fixed costs associated with that alternative.
Break-even analysis provides valuable insights into the viability of capacity alternatives by quantifying the relationship between costs, revenue, and output levels. It serves as a crucial decision-making tool, enabling organizations to assess the potential risks and returns associated with each alternative and make informed choices aligned with their financial objectives and operational capabilities.
Example
The management of a pizza place would like to add a new line of small pizza, which will require leasing new equipment for a monthly payment of $4,000. Variable costs would be $4 per pizza, and pizzas would retail for $9 each.
- How many pizzas must be sold per month in order to break even?
- What would the profit (loss) be if 1200 pizzas are made and sold in a month?
- How many pizzas must be sold to realize a profit of $10,000 per month?
- If demand is expected to be 700 pizzas per month, will this be a profitable investment?
Solution
- [latex]Q_{BEP}=FC\div(r–v)=4000\div(9–4)=800[/latex] pizzas month
- [latex]\text{total revenue}-\text{total cost}=1200\times9-1200\times4-4000=\$2000[/latex] (i.e. a profit)
- [latex]P=\$10000=Q(r–v)–FC[/latex];
Solving for Q will give us: [latex]Q=(10000+4000)\div(9–4)=2800[/latex] - Producing less than 800 (i.e. QBEP) pizzas will bring in a loss. Since 700 < 800 (QBEP), it is not a profitable investment.
Break-Even Analysis for “Make” or “Buy” Decisions
Organizations often face the strategic decision of whether to produce a product or service in-house (“make”) or outsource it to an external supplier (“buy”). Break-even analysis can provide valuable insights to guide this decision by identifying the quantity at which the total cost of making a product in-house equals the total cost of buying it from a supplier.
The key question to be addressed is: “For what quantities would buying the product be preferred to making it in-house, and vice versa?”
Let’s define the following variables:
vm = per unit variable cost of “make”
vb = per unit variable cost of “buy”
FC = Fixed cost associated with in-house production
To determine the break-even quantity (Q), we set the total cost of “make” equal to the total cost of “buy”:
Total Cost of “Make” = Total Cost of “Buy”
[latex]Q\times v_m+FC=Q\times v_b[/latex]
Rearranging the equation, we get:
[latex]FC=Q\times (v_b-v_m)[/latex]
[latex]Q=FC\div(v_b-v_m)[/latex]
The break-even quantity (Q) represents the output level at which the total cost of making the product in-house is equal to the total cost of buying it from a supplier.
If the required quantity is greater than the break-even quantity (Q > FC ÷ (vb – vm)), it would be more cost-effective to make the product in-house. Conversely, if the required quantity is less than the break-even quantity (Q < FC ÷ (vb – vm)), it would be more economical to buy the product from a supplier.
By calculating the break-even quantity and comparing it to the anticipated demand, organizations can make informed decisions regarding the “make” or “buy” strategy that minimizes costs and maximizes profitability. This analysis considers the trade-off between the fixed costs associated with in-house production and the variable costs of outsourcing.
It is important to note that while break-even analysis provides valuable insights, it should be complemented with other factors, such as quality considerations, supply chain risks, strategic alignment, and long-term implications, to arrive at a comprehensive decision that aligns with the organization’s overall objectives.
Example
The ABX Company has developed a new product and is wondering if they should make this product in-house or have a capable supplier make the product for them. The costs associated with each option are provided in the following table:
| Fixed Cost (annual) | Variable Cost | |
|---|---|---|
| Make in-house | $160,000 | $100 |
| Buy | $150 |
- What is the break-even quantity at which the company will be indifferent between the two options?
- If the annual demand for the new product is estimated at 1000 units, should the company make or buy the product?
- For what range of demand volume will it be better to make the product in-house?
Solution
- [latex]Q_{BEP}=FC\div(v_b–v_m)=160,000\div(150–100)=3200[/latex]
- [latex]\text{Total cost of "make"}=1000\times100+160,000=\$260,000[/latex]; [latex]\text{Total cost of "buy"}=1000\times150=\$150,000[/latex]
Thus, it will be better to buy since it will be less costly in total. - It will always be better to use the option with the lower variable cost for quantities greater than the break-even quantity.
This can also be proven as follows:
We want “make” to be better than “buy” in this part of the question. Thus, for any quantity Q, we need to have:
Total cost of “make” < Total cost of “buy”
160,000 + 100Q < 150Q
160,000 < 50Q
3200 <Q
Break-Even Analysis for Capacity Choice: Machine A vs. Machine B
When faced with multiple options for producing a product, organizations must evaluate the trade-offs between fixed and variable costs to determine the most cost-effective solution. Break-even analysis can provide valuable insights to guide this decision by identifying the output quantity at which the total cost of using one machine equals the total cost of using another.
The key question to be addressed is: “For what quantities would Machine A be preferred over Machine B, and vice versa?”
Let’s define the following variables:
FCA = Fixed cost associated with Machine A
vA = Variable cost per unit of using Machine A
FCB = Fixed cost associated with Machine B
vB = Variable cost per unit of using Machine B
Q = Output quantity
To determine the break-even quantity (Q), we set the total cost of using Machine A equal to the total cost of using Machine B:
Total Cost of A = Total Cost of B
[latex]Q\times V_A+FC_A=Q\times V_B+FC_B[/latex]
Rearranging the equation, we get:
[latex]FC_A-FC_B=Q\times (v_B-v_A)[/latex]
[latex]Q=(FC_A-FC_B)\div(v_B-v_A)[/latex]
The break-even quantity (Q) represents the output level at which the total cost of using Machine A is equal to the total cost of using Machine B.
If the required quantity is greater than the break-even quantity (Q > (FCA – FCB) ÷ (vB – vA)), it would be more cost-effective to use Machine A. Conversely, if the required quantity is less than the break-even quantity (Q < (FCA – FCB) ÷ (vB – vA)), it would be more economical to use Machine B.
It is recommended that the total cost expressions for each option be written down and simplified to ensure that no additional cost factors are overlooked. The total cost expressions can provide a comprehensive view of all cost components, including fixed costs, variable costs, and other relevant factors specific to the production process or machines under consideration.
By calculating the break-even quantity and comparing it to the anticipated demand, organizations can make informed decisions regarding the most cost-effective machine or production option, minimizing costs and maximizing profitability while aligning with their operational requirements and strategic objectives.
Example
The ABX Company has developed a new product and is going to make this product in-house. To be able to do this, they need to get a new equipment to be able to do the special type of processing required by the new product design. They have found two suppliers that sell such equipment. They are wondering which supplier they go ahead with. The costs associated with each option are provide in the following table:
| Fixed Cost (annual) | Variable Cost | |
|---|---|---|
| Supplier A | $160,000 | $150 |
| Supplier B | $200,000 | $100 |
- What is the break-even quantity at which the company will be indifferent between the two options?
- If the annual demand for the new product is estimated at 1000 units, which supplier should the company use?
- For what range of demand volume each supplier will be better?
Solution
- [latex]Q_{BEP}=(FC_B–FC_A)\div(V_A–V_B)=(200,000–160,000)\div(150–100)=40,000\div50=800[/latex]
- [latex]\text{Total cost of Supplier A}=1000\times150+160,000=\$310,000[/latex]; [latex]\text{Total cost of Supplier B}=1000\times100+200,000=\$300,000[/latex]
Thus, it will be better to go with Supplier B, since it will be less costly in total. - It will always be better to use the option with the lower variable cost for quantities greater than the break-even quantity.
This can also be proven as follows:
Let’s see for what quantities Supplier B will be better than Supplier A. In that case, for the quantity Q, we need to have:
Total cost of Supplier B < Total cost of Supplier A
200,000 + 100Q < 160,000 + 150Q
40,000 < 50Q
800 < Q
This means that for quantities above 800 units, Supplier B will be cheaper in total. Thus, for quantities less than 800, Supplier A will be cheaper in total.
“7 Strategic Capacity Planning” from Introduction to Operations Management Copyright © by Hamid Faramarzi and Mary Drane is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.—Modifications: used section Break Even Analysis, some paragraphs rewritten; added additional explanations.
