IE 212: Computational Methods for Industrial Engineering
Summer 2015
Homework #2: Defining Output, Input and Processing Tasks
Tuesday, June 30th, 2015
DELIVERABLE
For each of the problems shown below, clearly define what the output, input and processing tasks
should be to meet the requirements you have been given.
PROBLEMS
1) (10 pts) A program calculates the volume of a pyramid. A pyramid is a polyhedron formed by
connecting a polygonal base and a point called the apex. An example of a pyramid with base B and
height h is depicted in Figure 11. The formula to calculate the volume of a pyramid is:
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V = 3 𝑏ℎ
Where:
V: Volume of the pyramid (in cubic units of length)
b: Area of the base (in square units of length)
h: Height from the center of the base to the apex (in
units of length)
Figure 1. A pyramid with
base B and height h
2) (10 pts) A program calculates the maximum number of pieces p (or bounded/unbounded regions)
of a circle that can be created with a given number of straight cuts (a problem known in
mathematics as the lazy caterer’s sequence). A pizza is usually used to describe the situation (see
Figure 22). For example, three cuts across a pizza will produce six individual slices if the cuts all
meet at a common point; however, seven slices will result otherwise.
Figure 2. Illustration of the number of pieces p of
a circle that can be made by straight cuts
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http://en.wikipedia.org/wiki/Pyramid_(geometry)
http://edublognss.wordpress.com/2013/04/16/famous-mathematical-sequences-and-series/
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The maximum number p of pieces that can be created with a given number of cuts n, where n ≥ 0,
is given by the formula:
𝑝=
𝑛2 + 𝑛 + 2
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3) (10 pts) A program calculates TCo, the optimal cost per unit time. The formula to calculate TCo is3:
𝑇𝐶𝑜 = 𝑈𝐶 × 𝐷 + 𝑉𝐶𝑜
Where:
𝑉𝐶𝑜 = 𝐻𝐶 × 𝑄𝑜
𝑄𝑜 = √
VCo:
Qo:
UC:
HC:
RC:
D:
2 × 𝑅𝐶 × 𝐷
𝐻𝐶
Optimal value of variable cost per unit time (in dollars)
Optimal order quantity (in units)
Unit cost (in dollars)
Holding cost (in dollars)
Reorder cost (in dollars)
Demand (in units)
4) (10 pts) A program calculates the overall equipment effectiveness (OEE) based on the total
productive maintenance (TPM) method. TPM is a method to increase the effectiveness of
production environments, especially through methods for increasing the effectiveness of
equipment. OEE is calculated using the following formula4:
𝑂𝐸𝐸 = 𝐴𝑣𝑎𝑖𝑙𝑎𝑏𝑖𝑙𝑖𝑡𝑦 × 𝑃𝑒𝑟𝑓𝑜𝑟𝑚𝑎𝑛𝑐𝑒 × 𝑄𝑢𝑎𝑙𝑖𝑡𝑦
Where:
𝐴𝑣𝑎𝑖𝑙𝑎𝑏𝑖𝑙𝑖𝑡𝑦 =
𝐴𝑐𝑡𝑢𝑎𝑙 𝑂𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔 𝑇𝑖𝑚𝑒
𝑃𝑙𝑎𝑛𝑛𝑒𝑑 𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑇𝑖𝑚𝑒
Actual Operating Time: Number of hours in which plant was actually operating.
Planed Production Time: Plant operating hours – plant shutdown time (in hours).
𝑃𝑒𝑟𝑓𝑜𝑟𝑚𝑎𝑛𝑐𝑒 =
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𝑇𝑜𝑡𝑎𝑙 𝑂𝑢𝑡𝑝𝑢𝑡
𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑂𝑢𝑡𝑝𝑢𝑡 𝑎𝑡 𝑅𝑎𝑡𝑒𝑑 𝑆𝑝𝑒𝑒𝑑
http://www.wiley.com/legacy/wileychi/waters/supp/Equations.pdf
http://www.makigami.info/cms/tools#TOTAL PRODUCTIVE MAINTENANCE
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Total Output: Total quantity of units produced during the running
time including good, bad, reworked, QC samples, and
product during changeover.
Potential Output at Rated Speed: Potential quantity (in units) given the designed speed of
the equipment.
𝑄𝑢𝑎𝑙𝑖𝑡𝑦 =
𝐺𝑜𝑜𝑑 𝑂𝑢𝑡𝑝𝑢𝑡
𝑇𝑜𝑡𝑎𝑙 𝑜𝑢𝑡𝑝𝑢𝑡
Good Output: Total quantity good product (in units) produced through
the process that are available for sale without any type
of rework or re-processing.
TO OBTAIN FULL CREDIT IN THIS HOMEWORK YOU MUST:
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Prepare your solutions electronically, i.e., type your answer in a word processor such as MS
Word or similar.
You must properly cite any references that you consult to solve the problems included in this
assignment.
You must use the most simplified expression (or formula) when reporting the processing steps
for each problem.
Turn in a hard-copy of your solution (i.e., a paper version) at the beginning of the lecture
session on the due date specified at the top of page #1.
The last page of this document must be the front page of your assignment. Make sure that this
page displays:
− Your full name (added electronically in the space provided).
All the pages of your assignment must be stapled together.
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IE 212: Computational Methods for Industrial Engineering
Summer 2015
Homework #2: Defining Output, Input and Processing Tasks
Tuesday, June 30th, 2015
Student Name: