To find the crossover point between the current process-focused shop and the proposed repetitive manufacturing process, we need to determine the production volume at which the total costs of both processes are equal. Given data:
- Current process (Process A):
- Fixed costs = $9,000 per year
- Variable costs = $0 per unit
- New process (Process B):
- Fixed costs = $90,000 per year
- Variable costs = unknown (let's denote it as vvv per unit)
The crossover point is the production quantity xxx where total costs are equal:
Total cost Process A=Total cost Process B\text{Total cost Process A}=\text{Total cost Process B}Total cost Process A=Total cost Process B
9,000+0×x=90,000+v×x9,000+0\times x=90,000+v\times x9,000+0×x=90,000+v×x
Simplifying:
9,000=90,000+vx9,000=90,000+vx9,000=90,000+vx
vx=9,000−90,000=−81,000vx=9,000-90,000=-81,000vx=9,000−90,000=−81,000
Since variable costs cannot be negative, this suggests that without knowing the exact variable cost vvv for the new process, we cannot calculate the crossover point numerically. If the variable cost vvv for the new process is given, the crossover point xxx can be calculated by:
x=9,000−90,000v=−81,000vx=\frac{9,000-90,000}{v}=\frac{-81,000}{v}x=v9,000−90,000=v−81,000
Since production volume xxx cannot be negative, this implies the new process must have a positive variable cost less than zero for crossover to exist, which is impossible. Therefore, the crossover point formula is:
Crossover point=Fixed cost differenceVariable cost difference=90,000−9,0000−v=81,000v\text{Crossover point}=\frac{\text{Fixed cost difference}}{\text{Variable cost difference}}=\frac{90,000-9,000}{0-v}=\frac{81,000}{v}Crossover point=Variable cost differenceFixed cost difference=0−v90,000−9,000=v81,000
Here, the variable cost difference is from process A to B: since process A has zero variable cost, the difference is 0−v=−v0-v=-v0−v=−v. To get a positive crossover point, we consider absolute values:
x=81,000vx=\frac{81,000}{v}x=v81,000
So, once the variable cost per unit vvv of the new process is known, the crossover point is calculated by dividing the fixed cost difference by the variable cost per unit of the new process. Summary:
- Current process total cost = $9,000 (fixed) + $0 (variable) = $9,000
- New process total cost = $90,000 (fixed) + v×xv\times xv×x (variable)
- Crossover point x=90,000−9,000v=81,000vx=\frac{90,000-9,000}{v}=\frac{81,000}{v}x=v90,000−9,000=v81,000
This is the production volume where both processes cost the same
. If you provide the variable cost per unit for the new process, I can calculate the exact crossover point.
