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Break-even cost comparison of three sites

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).

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.

Method

Total annual cost TC=FC+v QTC = 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:

600000+40Q=900000+30Q10Q=300000⇒Q=30000 units\begin{aligned} 600000 + 40Q &= 900000 + 30Q \\ 10Q &= 300000 \Rightarrow Q = 30000 \text{ units} \end{aligned}

Y and Z:

900000+30Q=1500000+20Q10Q=600000⇒Q=60000 units\begin{aligned} 900000 + 30Q &= 1500000 + 20Q \\ 10Q &= 600000 \Rightarrow Q = 60000 \text{ units} \end{aligned}

X and Z cross at Q=900000/20=45000Q = 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 volumeCheapest site
0 to 30,000X (lowest fixed cost)
30,000 to 60,000Y
above 60,000Z (lowest variable cost)

(b) Output of 50,000 units

  • X: 600000+40(50000)=600000 + 40(50000) = Rs 2,600,000
  • Y: 900000+30(50000)=900000 + 30(50000) = Rs 2,400,000
  • Z: 1500000+20(50000)=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.

✓ Answer checked by Claude Opus 5.5 (high effort)

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