Ridvan BOZKURT

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INTRODUCTION TO ENGINEERING IE 101 ATILIM
UNIVERSITY FACULTY OF ENGINEERING DEPARTMENT OF
INDUSTRIAL ENGINEERING 2009 2010 FALL SEMESTER

• Ridvan BOZKURT

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TABLES AND GRAPHS

• Engineers must write and speak well and use
  graphical and usual communications to convey
  complex engineering information.
• A well prepared graph can accurately communicate
  information that would require many pages of
  written text
• Graphs are prepared from tabulated data
• Tables goes hand in hand with understanding graphs

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Dependent and independent variables

• An engineer is studying an automobile (the
  system) and interested in the factors that affect
  its speed. The dependent variable is speed (s).
  Independent variables that affect speed are the
  rate of fuel entering the engine f, time pressure
  p, air temperature T, air pressure P, road grade
  r, car mass m, frontal area A, and drag
  coefficient Cd).
• s s(f, p, T, P, r, m, A, Cd)

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TABLES

• A way to list dependent and independent variables
• Independent variable(s) are usually listed in the
  left column(s) and the dependent variable(s) are
  usually listed in right columns.
• The values in a given row correspond to each
  other.

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Effect of Fuel Rate and Road Grade on Car Speed
(TITLE)
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GRAPHS

• It is very difficult to interpret tabulated data
• Graphs are much more suited
• A wide variety of graphs are available to help
  visualize data
• Graphs must communicate information accurately
  and rapidly
• Descriptive titles
• Axis labels (including units)
• Readeable fonts
• Legible symbols

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GRAPHS
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GRAPHS

• Dependent variable is plotted on the ordinate
  (x-axis)
• Independent variable is plotted on the abcissa
  (y-axis)
• dependent variable is plotted versus the
  independent variable
• Ordinate and abcissa must have labels with the
  units

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GRAPHS

• Each axis is graduated with tick marks. They are
  preferred to appear outside the graph field so
  that they do not interfere with the data.
• 1 2 3
• 1 2 3

avoid
prefer
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GRAPHS

• The numbers on the axes should be spaced so they
  can be easily read.
• 1 2 3 4 5 6 7 8 910111213141516171819
  20 30
• 10 20 30

avoid
prefer
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GRAPHS

• The smallest graduations on the scale are
  selected to follow the 1,2,5 rule (if the number
  were written in scientific notation, the mantissa
  would be a 1,2 or 5).

acceptable
1
2
5
10
10
10
2,5
3,33
10
10
Not acceptable
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GRAPHS

• The allowable exceptions to the 1,2 and 5 rule
  include units of time (days, weeks, years, etc),
  because these are not decimal numbers.

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GRAPHS

• Problems can result if the numbers on the axis
  are extremely large or small, because they will
  crowd.

10,000
20,000
30,000
40,000
50,000
0
Voltage (V)
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GRAPHS

• Problems can result if the numbers on the axis
  are extremely large or small, because they will
  crowd.

0,0001
0,0002
0,0003
0,0004
0,0005
0
Voltage (V)
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GRAPHS

• In order to solve this problem, use SI system
  multipliers (K means 1,000x, and m means 0.001x).

10
20
30
40
50
0
Voltage (kV)
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GRAPHS

• In order to solve this problem, use SI system
  multipliers (K means 1,000x, and m means 0.001x).

0.1
0.2
0.3
0.4
0.5
0
Voltage (mV)
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GRAPHS

• IT IS IMPORTANT TO OBSERVE THE CASE OF UNITS AND
  MULTIPLIERS
• M means 1,000,000 x
• SI multipliers is convenient for solving of
  numbers that crowd together
• Some SI units do not have multipliers (e.g.0C)

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GRAPHS
0
1000
2000
3000
4000
5000
Volume (gal)
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GRAPHS
0
1
2
3
4
5
Volume (1000 gal) Volume (103 gal) Volume
(thousand gal)
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GRAPHS
0
1
2
3
4
5
Volume (10-3 gal)
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GRAPHS

• How to plot numbers that span many orders of
  magnitude?
• What if you wanted to have the following numbers
  on the abcissa (x-axis)
• 2, 23, 467, 3876, 48,967

USE A LOGARITHMIC SCALE
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GRAPHS
Linear scale
0
5
15
20
10
10
100
1
50
Logarithmic scale
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GRAPHS

• This logarithmic scale has 2 orders of magnitude
  and is called two cycle log scale
• If it had 3 orders of magnitude then it would be
  called three cycle log scale
• Log scale has no zero, it occurs at minus
  infinity on linear scale.

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GRAPHS

• Data points are plotted with symbols
• Each symbol should be carefully distinguished
• Size of the symbols should be large enough to
  easily distinguish, but not so large that they
  run into each other
• Different symbols is used for each data set

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GRAPHS

• Data points are often connected together with
  lines
• Line style may also be used to differentiate data
  sets with different data points with solid lines
  of uniform width

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GRAPHS

• The lines must not penetrate into open symbols,
  because they could easily be mistaken for closed
  symbols

YES
NO
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GRAPHS

• The meanings of the symbols or lines must be
  identified on the graph
• In the figure title
• In a legend
• Adjacent to the lines (preferred)

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GRAPHS

• Data may be categorized as observed, emprical or
  theoretical.
• Observed data presented without an attempt to
  smooth them or correlate them with a mathematical
  model.
• Emprical data presented with a smooth line which
  may be determined by a mathematical model (or
  where the data points would have fallen had there
  been no error in the experiment)

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GRAPHS

• Theoretical data generated by mathematical
  models. No data points are indicated with
  theoretical data, because the calculated points
  are completely arbitrary and of no interest to
  the reader.

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GRAPHS
31
GRAPHS
32
GRAPHS
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LINEAR EQUATIONS
y2
(x2,y2)
x,y
(y2 y1)
y
(x1,y1)
y1
(x2 x1)
b
a
x1
x
x2
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LINEAR EQUATIONS

• Two distinct points (x1, y1) and (x2, y2)
  establish a straight line.
• (x,y) is an arbitrary point on the line.
• The slope, m, of this line is defined as rise
  over run or
• m (y y1) / (x x1)

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LINEAR EQUATIONS

• Multiply both sides of the equation by (x2 x1)
• y y1 m (x x1) mx - m x1
• y mx y1 - m x1
• if b is defined as (y1 - m x1) then
• y mx b

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LINEAR EQUATIONS

• b is interpreted as the y-intercept, because x 0
  when yb
• The x-intercept, a, is where y0.
• a- b / m

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POWER EQUATIONS

• y k xm
• log y log (kxm) log (xmk)
• log y log xm log k
• log y mlog x log k
• A plot of log y versus log x gives a straight
  line with a slope m and y-intercept log k, which
  is analogous to b. You can derive this equation
  using any desired base (2, e or 10)
• If the exponent m is positive, then the power
  equation plots as a parabola.

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EXPONENTIAL EQUATIONS

• y k Bmx
• B desired base (2, e or 10)
• assuming base 10 is used,
• y k 10mx
• logatihms are taken of both sides to give a
  linear equation
• log y log (k 10mx) log (10mx k)
• log 10mx log k
• log y mx log k

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Interpolation Extrapolation

• Interpolation extending between data points
• Extrapolation extending beyond data points
• The smooth curve drawn between data points is
  actually an interpolation, because there are no
  data between the data points
• Extrapolation can be quite risky, particularly
  if one extrapolation extends far beyond data.

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Interpolation Extrapolation
interpolation
extrapolation
y
x
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Linear Extrapolation

• Approximates a curve with a straight line
• A straight line passes through the points (x1,
  y1) and (x2, y2) which are on the curve.
• Providing those points are close to each other
  and the curve is continuous, the straight line
  approximates the curve well.

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Linear Extrapolation

• For an arbitrary value of x that lies between x1
  and x2, the corresponding value of y that lies on
  the curve (yc) is very close to the corresponding
  value of y that lies on the straight line (ys)
• ys mx b
• m (y2 y1) / (x2 x1)
• y1 (y2 y1) / (x2 x1)x1 b
• b y1 - (y2 y1) / (x2 x1)x1

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Linear Extrapolation

• b y1 - (y2 y1) / (x2 x1)x1
• ys (y2 y1) / (x2 x1)x y1 - (y2 y1)
  / (x2 x1)x1
• ys y1 (y2 y1) / (x2 x1) (x - x1)
• ys is app equal to the desired value yc

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Linear Extrapolation (Problem)

• The steam tables list properties of water
  (steam) at different temperatures and pressures.
  Because water is one of the most common materials
  on earth, these tables are widely used by a
  variety of engineers. Among the properties listed
  in the steam tables is the vapor pressure of
  water at 114.7 0F. Altough this temperature is
  not explicetly listed on the table, estimate it
  by linear extrapolation.

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Linear Extrapolation (Problem)
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Linear Extrapolation (Problem)

• P P1 (P2 P1) / (T2 T1) (T - T1)
• 114(1.5121.429)/(116-114)(114.7114)
• P 1.458 psia

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Linear Regression

• In mathematics, we are normally given a formula
  from which we calculate numbers
• If we determine the formula from the number, this
  process is called regression (going backward)
• If the formula we seek is the equation of a
  straight line, then the process is called linear
  regression.

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Linear Regression

• Given a data set, what are the slope and
  y-intercept that best describe these data.
• The method of selected points
• Least squares linear regression

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method of selected points
(x2,y2)
y
m
(x1,y1)
b
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method of selected points

• Plot the data and then manually draw a line that
  best describes the data as judged by the person
  analyzing the data. Two arbitrary points at
  opposite ends of the line are selected (not
  necessarily data points, they are merely two
  convenient points that fall on the line. These
  two selected points (x1,y1) and (x2,y2) are
  substituted into following equations so that the
  slope and y-intercept may be calculated).

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method of selected points

• m (y2-y1) / (x2-x1)
• y1 (y2-y1) / (x2-x1)x1 b
• b y1 - (y2-y1) / (x2-x1)x1
• The problem with the selected points is that it
  relies on the judgement of the person analyzing
  the data.

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Least squares linear regression

• Not affected by personal bias
• Produce the best line, slope and y-intercept
• Uses a rigorous mathematical procedure to find a
  line that is close to all the data points

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Least squares linear regression
y4
d4
y2
d2
d3
m
y3
d1
y1
b
x1
x2
x3
x4
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Least squares linear regression

• di yi ys
• di residual
• yi actual data point
• ys point predicted by the straight line
• di yi (mxi b)
• Find m and b such that the sum d2i is minimum.

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Least squares linear regression

• sum ?ni1 d2i ?ni1 yi (mxi b)2
• n number of data points
• Best line y mx b
• mn(?xiyi)-(?xi)(?yi) / n(?xi2) (?xi)2
• b (?yi) - m(?xi) / n
• Best line through the origin
• y mx
• m (?xiyi )/ ?xi2

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Least squares linear regression

• correlation coefficient, r, is used to determine
  how well data fit on a straight line.
• r 1 - (?yi ys)2 / (?yi y)2 1/2
• y (? yi )/ n

(Mean value of y)
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Least squares linear regression

• If all data lie precisely on the line, then (?yi
  ys)2 0 and r1 (positively sloped line) or r
  -1 (negatively sloped line).
• If the data are scattered randomly and do not fit
  a straight line, then
• ?(yi ys)2 (?yi y)2
• r 0

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Least squares linear regression
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Least squares linear regression (example problem)

• A group of engineering students who study
  together decide they must use their time more
  efficently. Because they have many classes, they
  must allocate their study time to each class in
  an optimal manner. They decide that an equation
  that predicts theri exam grade on the basis of
  number of hours studied will help them allocate
  their time more efficiently. They prepare the
  following table on the basis of their performance
  on the last exam. What linear euation correlates
  their performance? What is the correlation
  coefficient?

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Least squares linear regression (example problem)
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Least squares linear regression (example problem)

• m9(4301)-(55)(596) / 9(453)-552
  5.64
• b (596) 5.64 (55) / 9 31.8
• r 9(4301)-55(596) / 9(453)-5521/29(43,266)-
  59621/2
• 0.98878