fundamentals of engineering supplied-reference handbook - Ventech!

ANALYSIS OF VARIANCE FOR 2 n FACTORIAL
DESIGNS
Main Effects
Let E be the estimate of the effect of a given factor, let L be
the orthogonal contrast belonging to this effect. It can be
proved that
L
E =
n−1
2
m
L = ∑ a(
c)
Y(
c)
c=
1
2
rL
SS L = , where
n
2
m = number of experimental conditions (m = 2 n for n
factors),
a(c) = –1 if the factor is set at its low level (level 1) in
experimental condition c,
a(c) = +1 if the factor is set at its high level (level 2) in
experimental condition c,
r = number of replications for each experimental
condition
Y ( c)
= average response value for experimental condition c,
and
SSL = sum of squares associated with the factor.
Interaction Effects
Consider any group of two or more factors.
a(c) = +1 if there is an even number (or zero) of factors in the
group set at the low level (level 1) in experimental condition
c = 1, 2,…, m
a(c) = –1 if there is an odd number of factors in the group set
at the low level (level 1) in experimental condition
c = 1, 2…, m
It can be proved that the interaction effect E for the factors
in the group and the corresponding sum of squares SSL can
be determined as follows:
L
E =
2
L =
SS
L
n−1
m
∑ a
( c )
c=
1
2
r L
=
n
2
Υ
( )
c
Sum of Squares of Random Error
The sum of the squares due to the random error can be
computed as
SSerror = SStotal – ΣiSSi – ΣiΣjSSij – … – SS12…n
193
INDUSTRIAL ENGINEERING (continued)
where SSi is the sum of squares due to factor Xi, SSij is the
sum of squares due to the interaction of factors Xi and Xj,
and so on. The total sum of squares is equal to
2
m r
2 T
SStotal
= ∑∑Yck −
c=
1k= 1 N
where Yck is the k th observation taken for the c th experimental
condition, m = 2 n , T is the grand total of all observations,
and N = r2 n .
RELIABILITY
If Pi is the probability that component i is functioning, a
reliability function R(P1,P2,..,Pn) represents the probability
that a system consisting of n components will work.
For n independent components connected in series,
R(
P1
, P2
, …Pn
) =
n
∏ Pi
i = 1
For n independent components connected in parallel,
R P , P , …Pn)
= 1−
( 1 2
n
∏
i = 1
( 1−
P )
LEARNING CURVES
The time to do the repetition N of a task is given by
TN = KN s , where
K = constant, and
s = ln (learning rate, as a decimal)/ln 2.
If N units are to be produced, the average time per unit is
given by
K
1+
s)
( 1+
s)
Tavg
= [ ( N + 0.
5)(
− 0.
5 ]
N 1+
s
( )
INVENTORY MODELS
For instantaneous replenishment (with constant demand
rate, known holding and ordering costs, and an infinite
stockout cost), the economic order quantity is given by
EOQ =
2AD
, where
h
A = cost to place one order,
D = number of units used per year, and
h = holding cost per unit per year.
Under the same conditions as above with a finite
replenishment rate, the economic manufacturing quantity is
given by
EMQ =
h
2AD
( 1−
D R)
R = the replenishment rate.
, where
i