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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 ???