Optimization Problems in Operations Research
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LECTURE 2
Operations Research
Example 1 (1.1 – Bronson and Naadimuthu)
The Village Butcher Shop traditionally makes its
meat loaf from a combination of lean ground beef
and ground pork. The ground beef contains 80
percent meat and 20 percent fat, and costs 80 cents
per pound; the ground pork contains 68 percent
meat and 32 percent fat, and costs 60 cents per
pound. How much of each kind of meat should the
shop use in each pound of meat loaf if it wants to
minimize its costs and to keep the fat content of the
meat loaf to no more than 25 percent?
Example 1 (1.1 – Bronson and Naadimuthu)
Formulation:
x = amount of ground beef, y = amount of ground pork
Minimize: z = 80x + 60y
Subject To: 0.2x + 0.32y <= 0.25
x + y = 1
Hidden constraint: all variables nonnegative (x >= 0, y >= 0
)
Solve graphically!
Example 2 (1.8 – Bronson and Naadimuthu)
A hiker plans to go on a camping trip. There are five items the
hiker wishes to take with her, but together they exceed the 60-
lb weight limit she feels she can carry. To assist herself in the
selection process she has assigned a value to each item in
ascending order of importance:
Which items should she take to maximize the total value?
Formulate this problem.
Example 2 (1.8 – Bronson and Naadimuthu)
Maximize: z = 100v + 60w + 70x + 15y + 15z
Subject To: 52v + 23w + 35x + 15y + 7z <= 60
v, w, x, y, z <= 1
all values nonnegative and integral
Solution?
Integer Programming
Now You Do It.
- 1.13 Bronson & Naadimuthu -
A 400-meter medley relay involves 4 different swimmers, who
successively swim 100 meters of the backstroke, breaststroke, butterfly,
and freestyle. A couch has six very fast swimmers whose expected times
in the individual events are below. What should be the relay assignment
be (formulate the problem first!)?
Now You Do It.
- Hillier & Lieberman 3.1.8 -
The WorldLight Company produces two light fixtures (products
1 & 2) that require both metal frame parts and electrical
components. Management wants to determine how many
units of each product to produce so as to maximize profit. For
each unit of product 1, 1 unit of frame parts and 2 units of
electrical components are required. For each unit of product
2, 3 units of frame parts and 2 units of electrical components
are required. The company has 200 units of frame parts, and
300 units of electrical components. Each unit of product 1
gives a profit of $1, and each unit of product 2 gives a profit of
$2. Any excess over 60 units of product 2 brings no profit, so
such an excess has been ruled out. Formulate and solve via
graphical method.
This collection of examples from Operations Research covers topics such as minimizing costs in meat loaf production, maximizing value in item selection for a camping trip, and solving relay assignment problems in a 400-meter medley. It also includes a profit maximization scenario for a company producing two types of light fixtures. The content provides formulations, constraints, and solutions to these optimization problems.
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Presentation Transcript
Operations Research LECTURE 2
Example 1 (1.1 Bronson and Naadimuthu) The Village Butcher Shop traditionally makes its meat loaf from a combination of lean ground beef and ground pork. The ground beef contains 80 percent meat and 20 percent fat, and costs 80 cents per pound; the ground pork contains 68 percent meat and 32 percent fat, and costs 60 cents per pound. How much of each kind of meat should the shop use in each pound of meat loaf if it wants to minimize its costs and to keep the fat content of the meat loaf to no more than 25 percent?
Example 1 (1.1 Bronson and Naadimuthu) Formulation: x = amount of ground beef, y = amount of ground pork Minimize: z = 80x + 60y Subject To: 0.2x + 0.32y <= 0.25 x + y = 1 Hidden constraint: all variables nonnegative (x >= 0, y >= 0) Solve graphically!
Example 2 (1.8 Bronson and Naadimuthu) A hiker plans to go on a camping trip. There are five items the hiker wishes to take with her, but together they exceed the 60- lb weight limit she feels she can carry. To assist herself in the selection process she has assigned a value to each item in ascending order of importance: Item 1 2 3 4 5 Weight 52 23 35 15 7 Value 100 60 70 15 15 Which items should she take to maximize the total value? Formulate this problem.
Example 2 (1.8 Bronson and Naadimuthu) Maximize: z = 100v + 60w + 70x + 15y + 15z Subject To: 52v + 23w + 35x + 15y + 7z <= 60 v, w, x, y, z <= 1 all values nonnegative and integral Solution? Integer Programming
Now You Do It. - 1.13 Bronson & Naadimuthu - A 400-meter medley relay involves 4 different swimmers, who successively swim 100 meters of the backstroke, breaststroke, butterfly, and freestyle. A couch has six very fast swimmers whose expected times in the individual events are below. What should be the relay assignment be (formulate the problem first!)? Backstroke 65 67 68 67 71 69 Breaststroke Butterfly 73 70 72 75 69 71 Freestyle 57 58 55 59 57 59 A B C D E F 63 65 69 70 75 66
Now You Do It. - Hillier & Lieberman 3.1.8 - The WorldLight Company produces two light fixtures (products 1 & 2) that require both metal frame parts and electrical components. Management wants to determine how many units of each product to produce so as to maximize profit. For each unit of product 1, 1 unit of frame parts and 2 units of electrical components are required. For each unit of product 2, 3 units of frame parts and 2 units of electrical components are required. The company has 200 units of frame parts, and 300 units of electrical components. Each unit of product 1 gives a profit of $1, and each unit of product 2 gives a profit of $2. Any excess over 60 units of product 2 brings no profit, so such an excess has been ruled out. Formulate and solve via graphical method.
