You just started a summer intership with the successful management consulting firm of Kirk, Spock, and McCoy. You first day on the job was a busy one, as the following problems were presented to you.
FastQ Company, a specialist in printing, has established 500 convenience-copying centers throughout the country. In order to upgrade its services, the company is considering three new models of laser copying machines for use in producing high-quality copies. These high-quality copies would be added to the growing list of products offered in the FastQshops. The selling price to the customer for each laser copy would be the same, no matter which machine is installed in the shop. The three models of laser copying machines under consideration are 1024S, a small-volume model; 1024M, a medium-volume model; and 1024G, a large-volume mode. The annual rental cost oand the operating cost vary with the size of each machine. The machine capacities and costs are as follows:
Copier Model
1024S 1024M 1024G
Annual capacity (copies) 100,000 350,000 800,000
Costs:
Annual machine rental $ 8,000.00 $11,000.00 $20,000.00
Direct material and direct labor 0.02 0.02 0.02
Variable overhead costs 0.12 0.07 0.03
a. Calculate the volume level in copies where FastQ Company would be indifferent to acquiring either the small-volume model laser copier, 1024S, or the medium-volume model laser copier, 1024M.
b. The management of FastQ Company is able to estimate the number of copies to be sold at each establishment. Present a decision rule that would enable FASTQ Company to select the most profitable machine without having to make a separate cost calculation for each establishment. (Hint: To specify a decision rule, determine the volume at which FastQ would be indifferent between the small and medium copiers. Then determine the volume at which FastQ would be indifferent between the medium and large copiers).
a. To calculate the volume level in copies where FastQ Company would be indifferent to acquiring either the small-volume model laser copier, 1024S, or the medium-volume model laser copier, 1024M, we need to compare the costs of using each machine.
The total cost of using the small-volume model 1024S is the sum of the annual machine rental cost, direct material and direct labor cost, and variable overhead costs.
Total cost of 1024S = Annual machine rental + Direct material and direct labor + Variable overhead costs
= $8,000 + ($0.02 * Volume) + ($0.12 * Volume)
= $8,000 + $0.14 * Volume
Similarly, the total cost of using the medium-volume model 1024M is:
Total cost of 1024M = Annual machine rental + Direct material and direct labor + Variable overhead costs
= $11,000 + ($0.02 * Volume) + ($0.07 * Volume)
= $11,000 + $0.09 * Volume
To find the volume level where FastQ Company would be indifferent to acquiring either 1024S or 1024M, we need to set the total costs of both machines equal and solve for Volume:
$8,000 + $0.14 * Volume = $11,000 + $0.09 * Volume
Simplifying the equation, we get:
$0.14 * Volume - $0.09 * Volume = $11,000 - $8,000
$0.05 * Volume = $3,000
Volume = $3,000 / $0.05
Volume = 60,000 copies
Therefore, FastQ Company would be indifferent to acquiring either the small-volume model laser copier, 1024S, or the medium-volume model laser copier, 1024M, when the volume of copies produced is 60,000.
b. To select the most profitable machine without making a separate cost calculation for each establishment, FastQ Company can use the following decision rule:
1. Determine the volume at which FastQ Company would be indifferent between the small and medium copiers (previously calculated as 60,000 copies).
2. Determine the volume at which FastQ Company would be indifferent between the medium and large copiers.
3. If the estimated number of copies to be sold at an establishment is less than the volume at which FastQ Company is indifferent between the small and medium copiers (60,000), select the small-volume model 1024S.
4. If the estimated number of copies to be sold at an establishment is greater than or equal to the volume at which FastQ Company is indifferent between the small and medium copiers (60,000), but less than the volume at which FastQ Company is indifferent between the medium and large copiers, select the medium-volume model 1024M.
5. If the estimated number of copies to be sold at an establishment is greater than or equal to the volume at which FastQ Company is indifferent between the medium and large copiers, select the large-volume model 1024G.
By following this decision rule, FastQ Company can select the most profitable machine without having to make a separate cost calculation for each establishment.
a. To calculate the volume level in copies where FastQ Company would be indifferent to acquiring either the small-volume model laser copier, 1024S, or the medium-volume model laser copier, 1024M, we need to compare the total costs of each machine.
For the small-volume model, the annual machine rental cost is $8,000, direct material and direct labor cost is $0.02 per copy, and variable overhead costs are $0.12 per copy.
For the medium-volume model, the annual machine rental cost is $11,000, direct material and direct labor cost is $0.02 per copy, and variable overhead costs are $0.07 per copy.
Let's assume the volume level at which the company would be indifferent between the two copier models is V.
The total cost for the small-volume model can be calculated as follows:
Total Cost (1024S) = Annual Machine Rental + (Direct Material and Direct Labor Cost + Variable Overhead Costs) * Volume
= $8,000 + ($0.02 + $0.12) * V
= $8,000 + $0.14V
The total cost for the medium-volume model can be calculated as follows:
Total Cost (1024M) = Annual Machine Rental + (Direct Material and Direct Labor Cost + Variable Overhead Costs) * Volume
= $11,000 + ($0.02 + $0.07) * V
= $11,000 + $0.09V
To find the volume level where FastQ Company would be indifferent between the two models, we need to set the total costs equal to each other and solve for V:
$8,000 + $0.14V = $11,000 + $0.09V
$0.05V = $3,000
V = $3,000 / $0.05
V = 60,000
Therefore, FastQ Company would be indifferent to acquiring either the small-volume model (1024S) or the medium-volume model (1024M) at a volume level of 60,000 copies.
b. To select the most profitable machine without making separate cost calculations for each establishment, FastQ Company can use the decision rule mentioned in the hint.
First, determine the volume level at which FastQ would be indifferent between the small and medium copiers. We have already calculated this value as 60,000 copies.
Next, determine the volume level at which FastQ would be indifferent between the medium and large copiers.
The total cost for the medium-volume model (1024M) is given as $11,000 + ($0.02 + $0.07) * V.
The total cost for the large-volume model (1024G) can be calculated as follows:
Total Cost (1024G) = Annual Machine Rental + (Direct Material and Direct Labor Cost + Variable Overhead Costs) * Volume
= $20,000 + ($0.02 + $0.03) * V
= $20,000 + $0.05V
Setting the total costs equal to each other and solving for V:
$11,000 + $0.09V = $20,000 + $0.05V
$0.04V = $9,000
V = $9,000 / $0.04
V = 225,000
Therefore, FastQ Company would be indifferent to acquiring either the medium-volume model (1024M) or the large-volume model (1024G) at a volume level of 225,000 copies.
In summary, the decision rule for FastQ Company would be as follows:
- For copy volume below 60,000, use the small-volume model (1024S).
- For copy volume between 60,000 and 225,000, use the medium-volume model (1024M).
- For copy volume above 225,000, use the large-volume model (1024G).
a. To calculate the volume level at which FastQ Company would be indifferent between acquiring the small-volume model 1024S and the medium-volume model 1024M, we need to compare their costs.
The costs for each model include the annual machine rental, direct material and direct labor costs, and variable overhead costs.
For the small-volume model 1024S, the annual rental cost is $8,000, the direct material and direct labor cost is $0.02 per copy, and the variable overhead cost is $0.12 per copy.
For the medium-volume model 1024M, the annual rental cost is $11,000, the direct material and direct labor cost is $0.02 per copy, and the variable overhead cost is $0.07 per copy.
To find the volume level where FastQ Company would be indifferent between the two models, we need to compare the total costs for each model at different volume levels. Let's assume the volume is represented by the number of copies produced.
For the small-volume model 1024S, the total cost can be calculated as follows:
Total Cost (1024S) = Annual rental cost + (Direct material and direct labor cost + Variable overhead costs) * Volume
Total Cost (1024S) = $8,000 + ($0.02 + $0.12) * Volume
Total Cost (1024S) = $8,000 + $0.14 * Volume
For the medium-volume model 1024M, the total cost can be calculated as follows:
Total Cost (1024M) = Annual rental cost + (Direct material and direct labor cost + Variable overhead costs) * Volume
Total Cost (1024M) = $11,000 + ($0.02 + $0.07) * Volume
Total Cost (1024M) = $11,000 + $0.09 * Volume
To find the volume level where FastQ Company would be indifferent between the two models, we set the total costs equal to each other and solve for the volume level:
$8,000 + $0.14 * Volume = $11,000 + $0.09 * Volume
Simplifying the equation:
$0.14 * Volume - $0.09 * Volume = $11,000 - $8,000
$0.05 * Volume = $3,000
Dividing both sides of the equation by $0.05:
Volume = $3,000 / $0.05
Volume = 60,000 copies
Therefore, FastQ Company would be indifferent between acquiring the small-volume model 1024S and the medium-volume model 1024M at a volume level of 60,000 copies.
b. To select the most profitable machine without having to make a separate cost calculation for each establishment, FastQ Company can use a decision rule based on the volume levels where they would be indifferent between the different copier models.
Based on the calculation in part a, FastQ Company would be indifferent between the small and medium copier models at a volume level of 60,000 copies. Similarly, we can calculate the volume level where FastQ Company would be indifferent between the medium and large copier models.
Let's assume this volume level is represented by V.
Using the same approach as in part a, setting the total costs for the medium and large copier models equal to each other, we can solve for V:
$11,000 + $0.09 * V = $20,000 + $0.03 * V
$0.09 * V - $0.03 * V = $20,000 - $11,000
$0.06 * V = $9,000
Dividing both sides of the equation by $0.06:
V = $9,000 / $0.06
V = 150,000 copies
Therefore, FastQ Company would be indifferent between acquiring the medium-volume model 1024M and the large-volume model 1024G at a volume level of 150,000 copies.
Based on these volume levels, the decision rule for selecting the most profitable machine without separate cost calculations for each establishment would be:
- If the estimated number of copies to be sold at an establishment is less than 60,000 copies, select the small-volume model 1024S.
- If the estimated number of copies to be sold at an establishment is between 60,000 and 150,000 copies (inclusive), select the medium-volume model 1024M.
- If the estimated number of copies to be sold at an establishment is greater than 150,000 copies, select the large-volume model 1024G.
By using this decision rule, FastQ Company can choose the most profitable machine based on the estimated volume of copies to be sold at each establishment without having to perform separate cost calculations for each establishment.