Techniques for Engineering Decisions Using Data

ECE 307 – Techniques for Engineering
Decisions
Using Data

George Gross
Department of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign

© 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

1


FOCUS
‰ Use of historical data to obtain probability
distributions
‰ The interpretation of probability information
‰ Use of estimators
‰ Application example
© 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

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EXAMPLE
‰ Consider the interpretation of the statement

P { sunny day in June in Champaign} = 0.53
‰ June weather patterns in Champaign for the past
20 years are collected and every day is classified
as either sunny or not sunny

‰ 600 days of June data are available with 318 or
53% of these days classified as sunny
‰ Given the long – term historical behavior, the
probability of 0.53 makes sense
© 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

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relative frequency (days/3650)

USE OF HISTOGRAMS

rated capacity
0 outage

high
derated
capacity

low
derated
capacity

full outage
capacity

outage capacity of a generating plant (MW )
© 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

4


CONSTRUCTION OF THE c.d.f.

1.0

P{ X ≤ a } = p

p

a
© 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

x
5


STATISTICAL PARAMETER
ESTIMATORS
‰ Estimator of the mean
n

∑x

mean of the

i

i=1

distribution

x =

n
‰ Estimator of the variance
n

∑( x
s

2

i

− x)

2

variance of the

i=1

=

n −1

distribution

© 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

6


STATISTICAL PARAMETER
ESTIMATORS

{

‰ We use a set of random samples x 1 , x 2 , . . ., x n

}

of a r.v. X : these are n randomly picked values
from the sample space of X
‰ The estimator x computed with the set of random
samples provides an estimate of

μ = E {X}
‰ The estimator s 2 computed with the set of random
samples provides an estimate of

σ 2 = var { X }
© 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

7


EXAMPLE: TACO SHELLS

‰ This application example focuses on taco shells
and is concerned with the high breakage rate in
the shipment of most taco shells: typical rate is
10 – 15 %
‰ A company with a new shipping container claims
to have a lower, approximately 5 % breakage rate
‰ This company’s price is $ 25 for a 500 – taco shell
box vs. $ 23.75 for a 500 – taco shell box of the
current supplier
© 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

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EXAMPLE: TACO SHELLS
‰ A test run using 12 boxes from the new company
and 18 boxes from the current company is
performed and used for comparison purposes: in
other words, we pick randomly

{x

1

, x 2 , . . . , x 12

}

from the sample space of the r.v. X describing the

{

}

new company shells and y 1 , y 2 , . . . , y 18 from the
sample space of the r.v. Y describing the current
company shells
‰ The data of the useable shells from the two
suppliers are tabulated
© 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

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EXAMPLE: TACO SHELLS
useable shells
new supplier

current supplier

468

467

444

441

450

474

469

449

434

444

474

484

443

427

433

479

470

440

446

441

482

463

439

452

436

478

468

448

442

429

© 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

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EXAMPLE: TACO SHELLS

r
e
i

pl
p
u
e
s
s
a
new 5.00/c
$2
cur

ren
$ 23 t sup
plie
.75
/cas
r
e

costs per
unbroken shell

i
ii

number of unbroken
shells (x)

i
ii

number of unbroken 23.75
shells (y)
y

© 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

25
x

11


c.d.f.s CONSTRUCTED FOR THE TWO
SUPPLIERS
1
0.9
0.8
current supplier
0.7
0.6
0.5
0.4
0.3
0.2
441
0.1
0
450
460

430
440
420

new supplier

473
470

unbroken shells per box

480

© 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

490
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c.d.f.s OF THE TWO SUPPLIERS
‰ Clearly, the new supplier has the higher expected
number of useable shells per box; the two
distributions, however, are highly similar
‰ The mean number of useable shells for the new
supplier is 473 and so the expected costs per
© 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

13

c.d.f.s OF THE TWO SUPPLIERS
useable shell is $0.0529; the minimum (maximum)
number of useable shells is 463(482)
‰ The mean number of useable shells for the
current supplier is 441 and so the expected costs
per useable shell is $0.0539; the minimum
(maximum) number of useable shells is 429(452)
© 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

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EXAMPLE: TACO SHELLS
number of usable shells cost per usable
shell ($)
462
0.185
0.0541
er
i
l
p
p
su /box
w 00
e
n 25.
$

cu

rr e
nt
$23 sup
.75
/bo plier
x

472

0.630

0.0530

485

0.185

0.0515

427

0.185

0.0556

442

0.630

0.0537

452

0.185

0.0525

© 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

15


COMMENTS
‰ We use the c.d.f.s to estimate the means of the

two populations of suppliers
‰ Typically, the function
−1
⎧1⎫
E ⎨ ⎬ ≠ ⎡⎣ E { X }⎤⎦
⎩X ⎭
© 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

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COMMENTS
and so we cannot use the approximation

⎧ 25 ⎫

25
E⎨ ⎬≈
⎩ X ⎭ E {X}
‰ This example demonstrates the usefulness of the
c.d.f.s in applications even when they can only be
approximated for the available data
© 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved.

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