Chapter 3 Problem Solving Methods
Chapter 3Problem Solving
Methods
Engineers Solve Problems Problem solving is a powerful human activity. Computers are useful tools in problem solving,
but it is the human who actually solves the problem.
It is impossible to teach specific facts that will always lead to a solution. The ability to solve problem comes from doing it.
Many things must pull together to solve a problem.
Problem Solving Problem solving is a combination
of experience, knowledge, process, and art
Design process is a series of logical steps that when followed produce an optimal solution given time and resources as two constraints
Problem Solving; cont’
A problem is a situation, quantitative or otherwise, that confronts an individual or group of individuals, that requires resolution, and for which the individual sees no apparent path to the solution.
Problem Solving; cont’
Problem solving is a process, an activity whereby a best value is determined for an unknown, subject to a specific set of conditions. It is a means by which an individual uses previously acquired knowledge, skills and understanding to satisfy the demands of an unfamiliar situation.
What skills must be used when solving a problem?
KnowledgeMotivation Experience Communication Skills Learning Skills Group Skills
Problem Analysis A distinguishing characteristic of a
qualified engineer is the ability to solve technical problems; both art and science Science; knowledge of mathematics,
chemistry, physics, etc Art; proper judgment, experience,
common sense, and know-how; to know when and how rigorously science should be applied and whether the resulting answer reasonably satisfies the original problem is an art
Techniques for Error Free Problem Solving
Always draw a picture of the physical situation,if possible. State any assumptions made. Indicate all given properties on the diagram with their units. Convert units to a given unit system. Label unknown quantities with a question mark.
Techniques for Error Free Problem Solving
From the text, write the main equation which contains the unknown quantity.
Or derive the desire algebraic equation by solving integral or differential equations. Algebraically manipulate the equation to isolate the desired quantity.
Techniques for Error Free Problem Solving
Write subordinate equations for the unknown quantities in the main equation. Indent to indicate that the equation is subordinate. It may be necessary to go through several levels of subordinate equations before all the quantities in the main equation are known. Once all algebraic manipulations and substitutions are made, insert numerical values with their units.
Techniques for Error Free Problem Solving
Insure that all units cancel. Check one last time for sign error. Compute the answer. Clearly mark the final answer. Indicate units! Insure that the final answer makes physical sense! Insure that all questions have been answered.
Skills used in Implementing Problem Solving Strategies
Analysis Use logic to: Identify the system to be analyzed Identify the objective Identify relationships Divide the system into parts
Skills used in Implementing Problem Solving Strategies
SynthesisUse creativity to: Develop ideas via brainstorming Evaluate the ideas by analysis when
enough ideas have been generated
Skills used in Implementing Problem Solving Strategies
Decision MakingUse logic to compare the various ideas and select the “best” one(s)
Generalization - Going from the specific to the broad use abstraction to:Aid in analysis, synthesis, and decision making
3.1 Types of Problems
Research Problems Knowledge Problems Troubleshooting Problems Mathematics Problems Resource Problems Social Problems Design Problems
Types of Problems; cont’
Research Problems A hypothesis be proven or disproved Example; CFC may destroy the earth’s
ozone layer is a hypothesis. Design an experiment that either proves or disproves the hypothesis
Types of Problems; cont’ Knowledge Problems
When a person encounters a situation that he doesn’t understand
Example; A chemical engineer noticed that the
chemical plant produces more product when it rains
Further study showed that heat exchanger cooled by rain increasing product
Types of Problems; cont’
Troubleshooting Problems When equipment or software behaves
in unexpected or improper ways Example During vibration test of an aluminum
beam, the amplitude of the response is higher at all exciting frequencies
Troubleshooting shows that 60 cps of AC current was close to the natural frequency of the beam
Types of Problems; cont’ Troubleshooting Problems; cont’ e.g. an electronic amplifier has a
loud “hum” when it is in a room with fluorescent lights.
Types of Problems; cont’ Mathematics Problems
Describe physical phenomena with mathematical models
Engineers can unleash the extraordinary power of mathematics, with the rigorously proven theorems and algorithms
Example; Isaac Newton’s sine square law can be applied to hypersonic flow
e.g. find x such that 4x + 5 = 0.
Types of Problems; cont’
Resource Problems There is never enough time, money, or
equipment to accomplish the task Engineers who can get the job done in
spite of resource limitations are highly prized and awarded e.g. how will we get the money to build our new factory?
Types of Problems; cont’
Social Problems For example, if a factory is relocated to
where there is shortage of skilled worker, engineers should set up training program for employees
e.g. how can we improve education?
Types of Problems; cont’
Design Problems Require creativity, teamwork, and broad
knowledge Example; design a new car Economy car? SUV? Design goal and parameters
Team Exercise
If you have enough money to buy a car, what kind of car do you like to buy?
If you are a car design engineer, identify design goal and design parameters from your team’s preference
Team Exercise Well Posed Design Problem: Design a
new car that can: 1. Go from 0 - 60 mph in 6 seconds 2. Gets 50 miles/gal 3. Costs less than $10,000 to the
consumer 4. Does not exceed government pollution
standards 5. Appeals to aesthetic tastes
Team Exercise 1. Identify Problem e.g. we need
to build a new car since we are losing market share
2. Synthesis (integrating parts to for a whole) e.g. we can combine an aerodynamic body with a fuel efficient engine to make a new car with very high fuel efficiency
Team Exercise3. Analysis
identify relationships, distinguish fact from opinion, detect logic information, make conclusions from evidence, select relevant information, TRANSLATE REAL-WORLD PROBLEM INTO
MATHEMATICAL MODEL e.g. compare the drag of different body
types and determine if engine can fit under the hood
Team Exercise
4. Application (identify the pertinent information) e.g. What force is required to allow the car to go 60 mph knowing the car has a 30ft2 projected area and a 0.35 drag coefficient based on wind tunnel data?
Team Exercise 5. Comprehension (use the data
and explicit theory to solve the problem) F = 1/2 Cd A V2
F=force Cd=drag coef. =air density
A=protected frontal area V=speed
Difficulties in Problem Solving
Most common difficulty: failure to use known information.
To avoid this problem: Write the problem in primitive form and
sketch an accurate picture of the setup (where applicable).
Transform the primitive statements to simpler language.
Translate verbal problems to more abstract mathematical statement(s) and figures, diagrams, charts, etc.
General Problem Solving Method
Define and understand problem1. Sketch the problem2. Gather information3. Generate and evaluate potential
solutions Use applicable theories and
assumptions4. Refine and implement solution5. Verify and test solution
Define and Understand
Understand what is being asked Describe input/output (I/O)
what are you given knowns
what are you trying to find unknowns
Sketch the problem
Gather Information
Collect necessary data List relevant equations/theories State all assumptions
Generate Solution Methods Apply theories and assumptions. Typically, there is more than one approach
to solving a problem Work problem by hand using the potential
solution methods Break problem into parts; scale it down; etc.
e.g., if the problem was to calculate the average of 1000 numbers, work the problem by hand using, say, 10 numbers, in order to establish a method
Refine and Implement
Evaluate solution methods. accuracy ease of implementation etc.
Implement “best” solution.
Verify and Test
Compare solution to the problem statement Is this what you were looking for? Does your answer make sense?
Clearly identify the solution Sketch if appropriate
CHECK YOUR WORK!!
Don’t stop at getting an answer!! Think about whether the answer makes
physical sense. you are the instructor and you have to turn in
final grades. In your haste, you calculate the average of Susie’s grades (100, 70, 90) to be 78 and give Susie a C...
Getting It Right The problem solving process may
be an iterative process. If at first you don’t succeed (i.e., the
algorithm test fails), try again… The more thorough you are at
each step of the problem solving process, the more likely you are to get it right the first time!!
Team Exercise
Given: A student is in a stationary hot-air balloon that is momentarily fixed at 1325 ft above a piece of land. This pilot looks down 60o (from horizontal) and turns laterally 360o.
Note: 1 acre = 43,560 ft2
Team Exercise; cont’
Required: a) Sketch the problem b) How many acres of land are
contained by the cone created by her line of site?
c) How high would the balloon be if, using the same procedure, an area four times greater is encompassed?
Creative Problem Solving
The nine dots shown are arranged in equally spaced rows and columns. Connect all nine points with four straight lines without lifting the pencil from the paper and without retracing any line.
Individual Exercise (3 minutes)
Creative Problem Solving
Creative Problem Solving If you enjoy solving puzzles, you will enjoy
engineering Crick and Watson figured DNA when they
were young Engineers create from nature what did not
exist before In this creative process, the engineer
marshals skills in mathematics, materials, and other engineering discipline and from these resources create a new solution for a human need
Creative Problem Solving Engineering is not dull or stifling;
send people to moon, communication from battlefield, etc
Creative artists spent many years perfecting their skills
Engineers need patience, practice, and gaining problem-solving techniques by training
Self-Questions for Problem Solving How important is the answer to a given
problem? Would a rough, preliminary estimate be
satisfactory or high degree accuracy demanded?
How much time do you have and what resources are at your disposal? Data available or should be collected,
equipments and personnel, etc
Self-Questions for Problem Solving What about the theory you intend to use?
Can you use it now or must learn to use it? Is it state of the art?
Can you make assumptions that simplify without sacrificing needed accuracy?
Are other assumptions valid and applicable?
Optimize time and resources vs reliability
Engineering Method1. Recognize and understand the
problem (most difficult part)2. Accumulate data and verify accuracy3. Select the appropriate theory or
principles4. Make necessary assumptions5. Solve the problem6. Verify and check results
Engineering Method Perfect solutions to real problems
do not exist. Simplify the problem to solve it; steady state, rigid body, adiabatic, isentropic, static etc
To solve a problem, use mathematical model; direct methods, trial-and-error, graphic methods, etc.
Problem Presentation Problem statement Diagram Theory Assumptions Solution steps Identify results and verify accuracy
Standards of Problem Presentation Engineers should have ability to
present information with great clarity in a neat, careful manner
Poor engineering documents can be legal problems in courts
Follow standard forms such as shown in the textbooks
Team Assignment Page 141 Problem 3.20
Algorithms Algorithm: “a step-by-step
procedure for solving a problem or accomplishing an end” (Webster)
Algorithms can be described by Pseudocode Flowcharts
Pseudocode English-like description of each step
of algorithm Not computer code Example - take out trash barrels
while there are more barrelstake barrel to streetreturn to garage
end
Flowcharts Graphical description of algorithm Standard symbols used for specific
operations
Input/Output
Start/Stop
Branch Test
Process Step
Process Flow
Flowchart ExampleDefine theproblem
Readinput
Solve theproblem
Can Isolve this?
Outputresults
What do I needto know?
Ask formore input
Begin
Can Isolve this?
End
yes
no
yes
no
Top Down Design State problem clearly Sketch problem Describe input/output (I/O) Work problem by hand Algorithm: pseudocode or flowchart
Decomposition - break problem into steps Stepwise refinement - solve each step
Test the algorithm/check your work!!
Example (Team exercise, 15 min) State problem clearly:
Given ax2 + bx + c = 0, find x. Describe I/O:
Input: a, b, c Output: x
Example (cont.) Hand example:
a=1, b=4, c=4 equation? (See Chapter 6,
Mathematics Supplement) x=?
Example (cont.) Algorithm development
write an algorithm in pseudocode to take any set of coefficients (i.e., a, b, c) and give the value of x for each set
Test your algorithm a,b,c = 1,4,4 a,b,c = 1,1,-6 a,b,c = 1,0,1 other good test cases?