---
title: "Break-even cost comparison of three sites"
subject: "Industrial Engineering and Management"
chapter: 2
key: "semester_6/industrial_engineering_and_management:break-even-analysis-plant-location"
source: https://www.bctengineer.com/q/semester_6/industrial_engineering_and_management/break-even-analysis-plant-location
site: BCT Engineering
---

# Break-even cost comparison of three sites

Industrial Engineering and Management · Chapter 2: Design of Production Systems

- Asked in: year not given, 6 marks
- Video: [Break Even Analysis, Break Even Point, CVP Analysis, Margin Safety, BEP, break-even analysis,  Cost](https://www.youtube.com/watch?v=swRHc2JqaVE) — DWIVEDI GUIDANCE
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/industrial_engineering_and_management/break-even-analysis-plant-location
- Whole chapter: https://www.bctengineer.com/md/semester_6/industrial_engineering_and_management/chapter-2

## Question

Three possible locations X, Y and Z are compared for a factory. The annual fixed cost and variable cost per unit are: X: Rs 600,000 and Rs 40; Y: Rs 900,000 and Rs 30; Z: Rs 1,500,000 and Rs 20. (a) Find the volume ranges for which each site is cheapest. (b) Which site should be chosen for an expected output of 50,000 units a year? Show the total cost of each.

## Short answer

Find break-even volumes by equating total cost equations (TC = FC + vQ). X–Y break-even at 30,000 units, Y–Z at 60,000 units. For 50,000 units, site Y has the lowest total cost (Rs 2,400,000).

## Answer

#### Method
Total annual cost $TC = FC + v\,Q$. Two sites cost the same at the volume where their lines cross; the cheapest site changes at these break-even volumes.

#### (a) Break-even points
X and Y:

$$
\begin{aligned} 600000 + 40Q &= 900000 + 30Q \\ 10Q &= 300000 \Rightarrow Q = 30000 \text{ units} \end{aligned}
$$

Y and Z:

$$
\begin{aligned} 900000 + 30Q &= 1500000 + 20Q \\ 10Q &= 600000 \Rightarrow Q = 60000 \text{ units} \end{aligned}
$$

X and Z cross at $Q = 900000/20 = 45000$, but at that volume Y is cheaper than both (Rs 2,250,000 against Rs 2,400,000), so this crossing is not used.

| Annual volume | Cheapest site |
|---|---|
| 0 to 30,000 | X (lowest fixed cost) |
| 30,000 to 60,000 | Y |
| above 60,000 | Z (lowest variable cost) |

#### (b) Output of 50,000 units
- X: $600000 + 40(50000) = $ Rs 2,600,000
- Y: $900000 + 30(50000) = $ Rs 2,400,000
- Z: $1500000 + 20(50000) = $ Rs 2,500,000

Y is cheapest, which agrees with the range 30,000 to 60,000.

**Answer: Break-evens at 30,000 and 60,000 units; choose site Y (cost Rs 2,400,000 per year) for 50,000 units.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the total annual cost for a site in the break-even comparison?
   A. Variable cost only
   B. TC = FC + v x Q
   C. TC = FC x v x Q
   D. TC = FC / Q
   Answer: B. Total cost = fixed cost + variable cost per unit x volume. (Key point: Total cost model)

2. At what volume do sites X and Y cost the same?
   A. 20,000 units
   B. 45,000 units
   C. 30,000 units
   D. 60,000 units
   Answer: C. 600000 + 40Q = 900000 + 30Q gives 10Q = 300000, so Q = 30,000. (Key point: Break-even X and Y)

3. Why is the X and Z crossing at 45,000 units not used?
   A. Because it is above 60,000
   B. Because Y is cheaper than both at that volume
   C. Because X has zero fixed cost
   D. Because Z has the highest variable cost
   Answer: B. At 45,000 units Y costs Rs 2,250,000 against Rs 2,400,000 for X and Z, so Y is the cheapest and the crossing is irrelevant. (Key point: Why X-Z crossing ignored)

4. Which site is chosen for 50,000 units a year and what is its cost?
   A. Y at Rs 2,400,000
   B. X at Rs 2,600,000
   C. Z at Rs 2,500,000
   D. Z at Rs 2,400,000
   Answer: A. Y: 900000 + 30 x 50000 = Rs 2,400,000, lower than Z (2,500,000) and X (2,600,000). (Key point: Choice at 50,000)

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