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Penn - ESE 204 - Class Notes - Week 1

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Penn - ESE 204 - Class Notes - Week 1

School: University of Pennsylvania
Department: Electrical Engineering
Course: Decision Models
Professor: Rakesh Vohra
Term: Fall 2020
Tags: math analysis
Name: ESE 204, Week 1 Linear Programming Primer
Description: These notes cover the basics of linear programming which is essential in the first half of the course.
Uploaded: 01/09/2019
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background image Reading assignment: Introduction to Linear Programming September 3, 2014 Surprising as it may seem, simple algebraic inequalities play a crucial (often cen- tral) role in modeling. Examples include: 1. Online media organizations often guarantee their sponsors a minimum number of product advertisements aired (or links clicked) per episode viewed by retail
audiences.
2. The number of United States dollars that US banks may lend to customers is capped at a percentage (usually much greater than %100) decided by federal
regulators.
3. The elapsed time of a digital signal traveling from Chicago to New York is no less than 3.75 milliseconds, the intercity distance divided by the speed of light. 4. The number of available machine hours available in most production lines is limited in part by a weighted sum of the number of available operators, floor
managers, and maintenance staff.
5. The amount of energy needed by a 10 ton helicopter to lift itself x meters off the ground is at least 10 × 907.185 × x joules. 6. The number of distinct “comma-free” k-letter words one can spell with an al- phabet of n distinct letters is less than or equal to 1
k
d |k µ (d)n k /d , where µ(d) is called the Mo¨obis function and summation is taken over all divisors d of k. This
was the basis for a 1957 paper by Crick, Griffith, and Orgel claiming that there
could not exist 20 distinct amino acid words.
7. The number of animals eaten by great white sharks annually is no less than 0.4 times the number attacked. Compare this with the number eaten by lions, which
comprise less that 0.3 times the number attacked, and with insects eaten by drag-
onflies, which make up 95% of those targeted.
In operations management, and nearly everywhere else, creating accurate mod- els of large systems with only small number of inequalities pays big dividends. The
“toy” examples shown below are rather simplified, but illustrate several of the key ideas
1
background image underlying Linear Programming as a powerful tool for constructing realistic, compli-
cated, and interesting models in science, business, and engineering.
Morpheus Problem The Morpheus 1 company makes two kinds of liquid soporifics: white soma and red soma. 2 Each gallon of white soma can be sold for $1, white each gallon of red soma can be sold for $2. The production capacity of the company limits them to
producing a total of 2,000 gallons of soma. Each gallon of white soma requires 1 hour
of labor to process and package. Each gallon of red soma requires 8/3 hours of labor
to process and package. The company has a total of 4,000 hours of labor available.
Government regulation rations the production of white soma. Morpheus has a license
that permits it to produce up to 1,500 gallons of white soma. What mix of white and
red soma should be produced to maximize the revenue of the Morpheus company?
The model The meaning of the word “model” depends a great deal on context. For the
next several weeks, it will simply mean a collection of (in)equalities. For the example
at hand, let x be the number of gallons of white soma produced, and y the number of
red soma produced. Constraints on production can be summarized by
(a) x + y ≤ 2, 000 due to limited production capacity. (b) x + (8/3)y ≤ 4, 000 due to limited availability of labor. (c) x ≤ 1500 due to license restrictions. (d) x + 2y is equal to total profit. (e) x ≥ 0 and y ≥ 0. We do not entertain scenarios, however plausible, in which negative quantities of either drug are produced. Listing inequalities in a bulleted format (as above) is cumbersome and unnecessary in most circumstances. We’ll write ours in tableau format, shown below. maximize 1x +2y subject to 1x +1y 2, 000 1x + 8 3 y 4, 000 1x +0y 1, 500 x , y ≥ 0 Figure 1 shows regions in the x-y plane representing production levels satisfying the various constraints. 1 The Roman god of sleep or dreams. 2 Soma is a narcotic distributed in Aldous Huxley’s Brave New Worldthat induces euphoria and halluci- nations. 2
background image Figure 1: Each point (x, y) in the chart shown above represents a pair of production
quantities for White and Red Soma. Production quantities which would require more
hours of labor than are available are colored light blue; production quantities which
exceed the legal limit on White Soma manufacturing are colored green, etc. Thus, the
feasible production quantities are those lying in the white portion of the chart.
Pro Shop Problem The Pro Shop makes golf hats and visors. Golf hats require 4 minutes on
the cutting machine and 3 minutes on the stitching machine. Visors require 3 minutes
on the cutting machine and 1 minute on the stitching machine. The cutting machine is
available for at most 2 hours each day and the stitching machine for at most 1 hour each
day. The company must make at least 6 golf hats each day to fulfill standing orders. If
the profit on a golf hat is
$1.10 and on a visor is $0.90, how many of each should be made each day to maximize profit? Model If x is the number of hats produced and y is the number of visors, then the salient
features of this problem are
(a) 4x + 3y ≤ 120 due to limited availability of the cutting machine. (b) 3x + y ≤ 60 due to limited availability of the stitching machine. (c) x ≥ 6 due to prior commitments with long term customers. (d) Total profit is 1.10x + 0.90y. (e) Both x and y are nonnegative. We don’t entertain scenarios in which it makes sense to produce negative quantities of either product. 3

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School: University of Pennsylvania
Department: Electrical Engineering
Course: Decision Models
Professor: Rakesh Vohra
Term: Fall 2020
Tags: math analysis
Name: ESE 204, Week 1 Linear Programming Primer
Description: These notes cover the basics of linear programming which is essential in the first half of the course.
Uploaded: 01/09/2019
11 Pages 74 Views 59 Unlocks
  • Better Grades Guarantee
  • 24/7 Homework help
  • Notes, Study Guides, Flashcards + More!
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