Problem Solving Skills (14021601-3 ) Lecture 4 Modelling: Let ’ s think about the problem a bit more 2 1
Important observation Solving real problems is a two step process: Model Solution Problem Rule #1 : Be sure you understand the problem, and all the basic terms and expressions used to define it. 3 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. 4 2
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. 5 Puzzle A 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. 6 3
Puzzle Question : How many chairs and tables should the company build to maximize its profit? Let ’ s build a model … 7 Puzzle Using 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. 8 4
Puzzle Objective : maximize $20 x + $30 y 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. 9 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 + 6 y ≤ 288 (wood) 3 x + y ≤ 99 (labor) 10 5
Puzzle The model : the mathematical model for the profit maximization problem of the manufacturing company can be formulated as follows: maximize $20 x + $30 y subject to: x + 6 y ≤ 288 3 x + y ≤ 99 where x ≥ 0 and y ≥ 0, and where both of the variables x and y can only take on integer values. 11 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). 12 6
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? 13 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. 14 7
A map 15 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. 16 8
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? 17 Chairs and tables Is this model of any good? maximize $20 x + $30 y subject to: x + 6 y ≤ 288 3 x + y ≤ 99 18 9
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? 19 Puzzle 10
Puzzle 21 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. 22 11
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? 23 Money and percentages 24 12
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. 25 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. 26 13
Importance of models in daily lives Many 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 ” ). 27 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? 28 14
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 … 29 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 … 30 15
Importance of models in daily lives Probability 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) 31 Importance of models in daily lives Probability of having n non-predictive 0.9999 n . dreams is: If a person dreams every night, than the probability of having n = 365 non-predictive dreams (full year of non-predictive dreams) 0.9999 365 ≈ 0.964. is Conclusion : About 96.4% of the people who dream every night will have only non- predictive dreams during a one-year span. 32 16
Importance of models in daily lives Conclusion : 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... 33 Importance of models in daily lives Coincidences 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 24 th ? 34 17
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 … 35 Importance of models in daily lives Examples : • sequence of first letters of the names of the months: JFMAMJJASOND 36 18
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