Production Schemes Worksheet

Production Schemes
Worksheet
Piecewise Defined Functions, Modeling, Domain/Range
Your company has three factories that are located in Wichita, Kansas, Seattle, Washington, and
Omaha, Nebraska. Each site produces the same consumer product, but each has a different set
of advantages and disadvantages. Therefore, the cost schedule for producing a given number of
units is different for each plant.
All of the unit numbers are for daily production runs. The
company receives orders from its retailers on a just-in-time inventory model, so we will assume
that the entire order must be filled on the day it is ordered.
For each factory, you must first consider the special circumstances and how its cost function is
affected. As the company’s production analyst, your goal is to use your knowledge of the three
plants to decide how each day’s orders should be manufactured so that the company’s overall
cost per unit is minimized.
Directions:
Complete each of the following tasks, reading the directions carefully as you go. Be sure to show all work
where indicated and to insert images of graphs when needed. Make sure that all graphs or screenshots
include appropriate information, such as titles and labeled axes. Use the built-in Equation Editor to type
equations with mathematical symbols that cannot be typed from the keyboard.
You will be graded on the work you show, or on your solution process, in addition to your answers. Make
sure to show all of your work and to answer each question as you complete the task. Type all of your
work into this document so you can submit it to your teacher for a grade. You will be given partial credit
based on the work you show and the completeness and accuracy of your explanations.
Your teacher will give you further directions as to how to submit your work. You may be asked to upload
the document, e-mail it to your teacher, or hand in a hard copy.
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Worksheet (continued)
1. Factory in Wichita
The cost per unit for Wichita’s factory is graphed as a piecewise function over the domain [0, 35000].
a. Write one or two sentences to describe the cost function for the Wichita factory. (5 points)
b. Write the piecewise function for the cost per unit for production of units in the Wichita factory.
That is, state the function ( ), where is the number of units produced. (10 points)
c.
What is the cost per unit for 35,000 units? (5 points)
d. According to the graph, if only one unit was produced, the cost per unit would be $1.00. In reality,
the company would not call in its worker and start up the equipment to produce one unit. What
safeguard do you think that many manufacturers use to avoid this type of problem? (5 points)
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Worksheet (continued)
e. Of the following three statements, choose the best option and write a viable argument showing
how it is reflected by the Cost Function for Wichita’s factory. (10 points)
___ The electricity contract with the utility company is structured so that higher daily energy
usage is charged at a lower rate.
___ The plant’s production processes are performed primarily by robots that are able to work
longer hours, when needed, at no additional cost.
___ Overtime wages were required to produce at levels above 25,000 units.
2. Factory in Seattle
Seattle’s factory is the company’s newest. It was constructed with sustainable building techniques and
materials, so its cost structure has been affected in several ways. The Seattle plant uses a mix of solar
power (not very reliable with so many clouds), thermal power, and wind. It has an array of batteries, so at
the beginning of each production day, there is “free” stored energy that can produce up to 8,000 units. Of
course, energy is just one factor of the cost of production, but it is significant.
The cost function for the Seattle factory is given by the following piecewise function. That is,
cost function, where
( )is the
is the number of units produced. The plant’s maximum capacity is 42,000 units per
day.
( )
{
(
)
a. Sketch a graph to model Seattle’s cost structure over the domain [0, 42000]. Be sure to label the
axes and any endpoints where the graph breaks. (5 points)
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Worksheet (continued)
b. Describe the function over each part of its domain. State whether it is constant, increasing, or
decreasing, and state the slope over each part. (5 points)
3. Factory in Omaha
Omaha’s factory has yet another type of cost structure. Its cost function is provided graphically. Its
maximum capacity is 38,000 units per day.
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Worksheet (continued)
a. At what rate is the cost per unit decreasing for production levels above 12,000?(5 points)
b. State the function for the domain over [12000, 38000]. (10 points)
c.
What is the cost per unit at the production level of 19,000? (5 points)
Omaha’s city council approved a special growth incentive that decreases the company’s tax burden for
production levels above last year’s average. Explain how this is reflected by the Cost Function for
Omaha’s factory.
4. Analysis and Making Production Decisions
You have now provided your company’s management with powerful tools that model the cost of
production in the three factories. When you get an order for a certain number of units, you can be
informed of the cost that would be incurred by each factory if chosen to fulfill the order. In reality, the
decision must also consider many other factors, but in our scenario, we want to fill our orders at the
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Worksheet (continued)
lowest average cost per unit. Consider each of the following questions as a separate, unrelated day of
business.
a. On Monday, you have a single request: Order A for 15,000 units. It must be fulfilled by a single
factory. To which factory do you send the order? Explain your decision. Support your argument
with numbers. (10 points)
b. On Tuesday, you have two orders. You may send each order to a separate factory OR both to
the same factory. If they are both sent to be fulfilled by a single factory, you must use the total of
the two orders to find that factory’s cost per unit for production on this day. Remember that the
goal is to end the day with the lowest cost per unit to produce the company’s products. Order B
is 7,000 units, and Order C is 30,000 units. (15 points, 5 points per part)

First, consider giving the orders to separate factories. Use these values and the following
formula to calculate the average weighted cost per unit.
(
)(
)
(
)(
)

Second, consider sending both orders to a single factory that has the lowest cost per unit to
produce 37,000 units.

Compare the two options. Decide how you will send the orders out, and document your
decision by completing the daily production report below.
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Worksheet (continued)
Production Report for Tuesday
Order
# of Units
Factory
B
7,000
______
C
30,000
______
Total # of units produced for the company today:
Average cost per unit for all production today:
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______
______