Production Schemes
Student Guide
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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Student Guide (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.
b) Write the piecewise function for the cost per unit for production of units in the Wichita factory.
That is, state the function 𝐶(𝑥), where 𝑥 is the number of units produced.
c) What is the cost per unit for 35,000 units?
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Student Guide (continued)
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?
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.
___
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, 𝐶(𝑥)is the
cost function, where 𝑥 is the number of units produced. The plant’s maximum capacity is 42,000 units per
day.
0.35 𝑖𝑓 𝑥 ≤ 8,000
0.75 𝑖𝑓 8,000 < 𝑥 ≤ 20,000
𝐶(𝑥) = {
𝑥
0.83 − (
) 𝑖𝑓 20,000 < 𝑥 ≤ 42,000
200,000
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Student Guide (continued)
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.
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.
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Student Guide (continued)
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.
a) At what rate is the cost per unit decreasing for production levels above 12,000?
b) State the function for the domain over [12000, 38000].
c) What is the cost per unit at the production level of 19,000?
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.
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Student Guide (continued)
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
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.
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.

First, consider giving the orders to separate factories. Use these values and the
following formula to calculate the average weighted cost per unit.
(𝑐𝑜𝑠𝑡 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝐵)(#𝑜𝑓 𝑢𝑛𝑖𝑡𝑠) + (𝑐𝑜𝑠𝑡 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝐶)(#𝑜𝑓 𝑢𝑛𝑖𝑡𝑠)
𝑇𝑜𝑡𝑎𝑙 # 𝑜𝑓 𝑢𝑛𝑖𝑡𝑠

Second, consider sending both orders to a single factory that has the lowest cost
per unit to produce 37,000 units.
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Student Guide (continued)
c) Compare the two options. Decide how you will send the orders out, and document your
decision by completing the daily production report below.
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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