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 where FastQ Company would be indifferent between acquiring the small-volume model laser copier, 1024S, and the medium-volume model laser copier, 1024M, we need to compare the total costs of each copier at different volume levels.
Let's denote the volume level in copies by V.
For the small-volume copier 1024S:
Total cost = Annual machine rental + (direct material and direct labor) * V + (variable overhead costs) * V
Total cost for 1024S = 8,000 + 0.02 * V + 0.12 * V
For the medium-volume copier 1024M:
Total cost = Annual machine rental + (direct material and direct labor) * V + (variable overhead costs) * V
Total cost for 1024M = 11,000 + 0.02 * V + 0.07 * V
We need to find the volume level where these costs are equal:
8,000 + 0.02 * V + 0.12 * V = 11,000 + 0.02 * V + 0.07 * V
Simplifying the equation:
0.12 * V - 0.07 * V = 11,000 - 8,000
0.05 * V = 3,000
V = 3,000 / 0.05
V = 60,000
Therefore, FastQ Company would be indifferent between acquiring the small-volume model laser copier, 1024S, and the medium-volume model laser copier, 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 following decision rule:
1. If the estimated number of copies to be sold at an establishment is less than 60,000 copies, choose the small-volume model laser copier, 1024S.
2. If the estimated number of copies to be sold at an establishment is between 60,000 and 350,000 copies, choose the medium-volume model laser copier, 1024M.
3. If the estimated number of copies to be sold at an establishment is greater than 350,000 copies, choose the large-volume model laser copier, 1024G.
This decision rule is based on the volume levels where FastQ Company would be indifferent between each pair of copiers. By applying this decision rule consistently, FastQ Company can select the most profitable machine without having to make a separate cost calculation for each establishment.
a. In order to calculate the volume level at which 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 each copier model.
The annual rental cost for the 1024S copier is $8,000, while the annual rental cost for the 1024M copier is $11,000. The direct material and direct labor costs for both models are $0.02 per copy, while the variable overhead costs for the 1024S copier are $0.12 per copy and for the 1024M copier are $0.07 per copy.
To calculate the volume level of copies at which FastQ Company would be indifferent between the two copier models, we need to find the point at which the total costs of each model are equal.
Let's assume that x represents the number of copies.
For the 1024S copier:
Total cost = Annual rental cost + (Direct material and direct labor cost + Variable overhead cost) * Number of copies produced
Total cost for 1024S = $8,000 + ($0.02 + $0.12) * x = $8,000 + $0.14 * x = $0.14x + $8,000
For the 1024M copier:
Total cost for 1024M = $11,000 + ($0.02 + $0.07) * x = $11,000 + $0.09 * x = $0.09x + $11,000
Setting the total costs equal to each other and solving for x:
$0.14x + $8,000 = $0.09x + $11,000
$0.05x = $3,000
x = $3,000 / $0.05
x = 60,000
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 level reaches 60,000 copies.
b. The decision rule that would enable FastQ Company to select the most profitable machine without having to make a separate cost calculation for each establishment is based on the volume levels at which the company would be indifferent between different copier models.
To specify the decision rule, we need to determine the volume at which FastQ would be indifferent between the small and medium copiers, as well as the volume at which FastQ would be indifferent between the medium and large copiers.
From part a, we found that FastQ would be indifferent between the 1024S and 1024M copiers at a volume level of 60,000 copies. Similarly, we can determine the volume level at which FastQ would be indifferent between the 1024M and 1024G copiers.
Using the same approach as part a, we compare the total costs of the 1024M and 1024G copiers:
Total cost for 1024G = $20,000 + ($0.02 + $0.03) * x = $20,000 + $0.05 * x = $0.05x + $20,000
Setting the total costs equal to each other and solving for x:
$0.09x + $11,000 = $0.05x + $20,000
$0.09x - $0.05x = $20,000 - $11,000
$0.04x = $9,000
x = $9,000 / $0.04
x = 225,000
Therefore, FastQ Company would be indifferent between the medium-volume model laser copier, 1024M, and the large-volume model laser copier, 1024G, at a volume level of 225,000 copies.
The decision rule would be as follows:
- If the estimated number of copies to be sold is less than 60,000, FastQ should acquire the small-volume model, 1024S.
- If the estimated number of copies to be sold is between 60,000 and 225,000, FastQ should acquire the medium-volume model, 1024M.
- If the estimated number of copies to be sold is greater than 225,000, FastQ should acquire the large-volume model, 1024G.
To calculate the volume level 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 operating each machine.
Let's assume x represents the number of copies produced annually.
For the small-volume model, 1024S:
Annual machine rental cost = $8,000
Variable operating cost per copy = $0.02
Total cost for the small-volume model = Annual machine rental cost + (Variable operating cost per copy * Number of copies)
Total cost for 1024S = $8,000 + ($0.02 * x)
For the medium-volume model, 1024M:
Annual machine rental cost = $11,000
Variable operating cost per copy = $0.02
Total cost for the medium-volume model = Annual machine rental cost + (Variable operating cost per copy * Number of copies)
Total cost for 1024M = $11,000 + ($0.02 * x)
Now we need to find the volume level where the two costs are equal:
$8,000 + ($0.02 * x) = $11,000 + ($0.02 * x)
Solving for x, we get:
$8,000 = $11,000
x = 300,000
So, 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 annual copies produced reaches 300,000.
To 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, we can use the same approach for comparing the medium-volume model and the large-volume model.
Using the same method as above, we find that FastQ Company would be indifferent to acquiring either the medium-volume model, 1024M, or the large-volume model, 1024G, when the annual copies produced reaches 775,000.
Therefore, the decision rule for FastQ Company should be as follows:
- If the estimated annual copies produced is less than 300,000, choose the small-volume model, 1024S.
- If the estimated annual copies produced is between 300,000 and 775,000, choose the medium-volume model, 1024M.
- If the estimated annual copies produced is greater than 775,000, choose the large-volume model, 1024G.