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Problem Solving Skills
(14021601-3 )
Lecture 4
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Modelling:
Let’s think about the problem
a bit more
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Solving real problems is a two step process:
Problem Model Solution
Important observation
Rule #1: Be sure you understand the problem,
and all the basic terms and expressions used to
define it.
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Puzzle: refresh your mind
There is a horseshoe with six holes for nails,
which looks like that:
Using two straight-line cuts, chop the horseshoe into
six separate parts so that each part has exactly one hole.
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Rule #3
Rule #3: Solid calculations and reasoning
are more meaningful when you build a
model of the problem by defining its
variables, constraints, and objectives.
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PuzzleA manufacturing enterprise that produces just two types of
items: chairs and tables:
• the profit per chair is $20,
• the profit per table is $30.
To build a chair, a single unit of wood is required and
three man-hours of labor. To build a table, six units of
wood are required and one man-hour of labor.
The production process has some restrictions: all the
machines can only process 288 units of wood per day and
there are only 99 man-hours of available labor each day.
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Puzzle
Question: How many chairs and tables should
the company build to maximize its profit?
Let’s build a model…
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PuzzleUsing Rule #3, we should construct a model of the
problem by specifying the following:
Variables: There are only two variables, x and y, with each
variable corresponding to the number of items (chairs and
tables, respectively) to be produced.
Constraints: In this puzzle there are only two constraints:
(1) the 288 wood units available for processing, and
(2) the 99 man-hours available for labor.
Objective: In this particular problem/puzzle, the objective
is to maximize the total profit.
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Puzzle
Objective:
maximize $20x + $30y
For example, if we produce 10 chairs (i.e. x = 10)
and 15 tables (i.e. y = 15), the daily profit would be:
$20 × 10 + $30 × 15 = $200 + $450 = $650.
Of course, the larger number of chairs and tables we
produce, the higher the profit. If we produce 20 tables
(instead of 15), the profit would be:
$20 × 10 + $30 × 20 = $200 + $600 = $800.
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Puzzle
Constraints:
• to build a chair, a single unit of wood is required and
three man-hours of labor,
• to build a table, we need six units of wood and one
man-hour of labor.
x + 6y ≤ 288 (wood)
3x + y ≤ 99 (labor)
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Puzzle
The model: the mathematical model for the profit
maximization problem of the manufacturing company can
be formulated as follows:
maximize $20x + $30y
subject to:
x + 6y ≤ 288
3x + y ≤ 99
where x ≥ 0 and y ≥ 0, and where both of the variables
x and y can only take on integer values.
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Puzzle
Solution: It might not be that obvious that the
solution to this profit maximization problem is:
x = 18 and y = 45
which implies a profit of $1,710.
This is the best we can do: it is impossible to achieve
a higher profit by producing a different number of
chairs and tables (while staying within the constraints
of wood units and available man-hours).
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Puzzle
Questions: However, there are some additional
questions we should ask:
• Is the model adequate for the problem?
• Did we include all the relevant information?
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Observation
There are always many possible models one
can construct for a given problem…
Let’s consider an “ideal” map for a city
(map – a model of real world), which can
be used for various routing decisions.
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A map
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A good model
A good model should be precise enough to allow
for a meaningful solution, but on the other hand,
it should not be so complex that it is too difficult
to use. A good model should satisfy two
intuitive requirements:
• It should be general enough, so that irrelevant
details of the problem are hidden.
• It should be specific enough, so that we can
derive a meaningful solution.
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Many issues to consider How precise is the model (in terms of real-
world environment it models)?
How difficult is it to find a solution in the
model?
What is the tradeoff between precision of the
model and quality of the solution?
What is the frequency of use of the model?
How much time do we have to find a
solution?
What is the “cost” of using inferior solution?
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Chairs and tables
Is this model of any good?
maximize $20x + $30y
subject to:
x + 6y ≤ 288
3x + y ≤ 99
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Puzzle
Mrs. Brown celebrated her birthday. One of the
guests asked her about her age. Mrs. Brown
replied that the total of her age and the age of her
husband, Mr. Brown, is 140, and then she added:
“My husband is twice the age I was when my
husband was my age.” How old is Mrs. Brown?
Puzzle
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Puzzle
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Lesson learned
Rule #3: Solid calculations and reasoning
are more meaningful when you build a
model of the problem by defining its
variables, constraints, and objectives.
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Money and percentages
John inherited from his uncle 25% more
money than his sister, Jane. He wants to give
her part of his money so both of them get the
same inheritance.
What % of his money should John give to
Jane?
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Money and percentages
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Importance of models in daily lives
There is a common perception that car
accidents is a relatively minor problem of
local travel, whereas being killed by
terrorists is a major risk when going
oversees.
Simple model (statistics from 1985, USA):
• 45,000 killed in car accidents,
• 17 killed by terrorists.
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Importance of models in daily lives
A simple probabilistic model will put
perceptions in a perspective. The chances
are:
• 1 : 1,600,000 – to be killed by terrorists,
• 1 : 68,000 – to choke to death,
• 1 : 75,000 – to die in bicycle crash,
• 1 : 20,000 – to die by drowning,
• 1 : 5,300 – to die in car accident.
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Importance of models in daily livesMany people are excited by discoveries
of “unusual” relationships:
• Christopher Columbus discovered New
World in 1492 and his fellow Italian Enrico
Fermi discovered the new world of atom in
1942.
• President Kennedy’s secretary was named
Lincoln, while President Lincoln’s secretary
was named Kennedy.
• Each word in the name Ronald Wilson
Reagan (former US President) has 6 letters
(thus a connection with “666”).
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Importance of models in daily lives
Simple models may protect us from a
tendency to drastically underestimate the
frequency of coincidences…
How many of you have “aunt Molly” (or
some other family member) who had a clear
dream one night that “uncle Jack” had a
serious car accident just hours before his
Holden Commodore hit a tree?
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Importance of models in daily lives
Let’s build a simple model… Assume the
probability that a particular dream
matches (in some details) an event in real
life to be
1 : 10,000.
It means, that such occurrence is quite
unlikely: the chances of non-predictive
dream are 9,999 out of 10,000…
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Importance of models in daily lives
Further, assume that sequences of dreams
are independent in a sense that whether or
not a dream matches events of the following
day is independent of whether or not other
dream matches events of the other day.
These assumptions are important
components of our model. Now we can
proceed…
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Importance of models in daily livesProbability of having a non-predictive dream
is:
0.9999.
Probability of having two non-predictive
dreams is:
0.9999 × 0.9999.
Probability of having three non-predictive
dreams is:
0.9999 × 0.9999 × 0.9999.
(multiplication principle)
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Importance of models in daily lives
Probability of having n non-predictive
dreams is: 0.9999n.
If a person dreams every night, than the
probability of having n = 365 non-predictive
dreams (full year of non-predictive dreams)
is 0.9999365 ≈ 0.964.
Conclusion: About 96.4% of the people
who dream every night will have only non-
predictive dreams during a one-year span.
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Importance of models in daily livesConclusion: About 3.6% of the people who
dream every night will have a predictive
dream…
Note that 3.6% translates into millions of
people…
Note also, that even if we change the
probability that a particular dream matches (in
some details) an event in real life to be
1 : 1,000,000, the number of predictive
dreams is still quite significant...
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Importance of models in daily livesCoincidences are much more common than
most people realize.
The key issue: ability to distinguish between
occurrences of general and specific events:
• What is the probability that there are two
people with the same birthday in a group?
• What is the probability that there is another
person in the group with birthday of January
24th?
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Importance of models in daily lives
The key issue: ability to distinguish between
occurrences of general and specific events:
If we have a spinner with 26 letters and we
spun it 100 times and we record the letters in
a sequence, the probability we get the words
“JAMES” or “DOG” are pretty slim. On the
other hand, the probability of getting some
word is pretty high…
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Importance of models in daily lives
Examples:
• sequence of first letters of the names of the
months:
JFMAMJJASOND
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Importance of models in daily lives
Examples:
• sequence of first letters of the names of the
months:
JFMAMJJASOND
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Importance of models in daily lives
Examples:
• sequence of the first letters of the names of
the months:
JFMAMJJASOND
• sequence of the first letters of the names of
the planets:
MVEMJSUNP
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Importance of models in daily lives
Examples:
• sequence of the first letters of the names of
the months:
JFMAMJJASOND
• sequence of the first letters of the names of
the planets:
MVEMJSUNP
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Lesson learned
Rule #3: Solid calculations and reasoning
are more meaningful when you build a
model of the problem by defining its
variables, constraints, and objectives.
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Some experiments
• Puzzle-Based Learning web-site:
www.PuzzleBasedLearning.edu.au
Experiment with Puzzle 3.1 – set different number of
items to produce; set different constraints. Try to find the
optimal solution for these cases…
Any Question ???