fundamentals of engineering supplied-reference handbook - Ventech!

Standard Deviation Charts
n A3 B3 B4
2
3
4
5
6
7
8
9
10
2.659
1.954
1.628
1.427
1.287
1.182
1.099
1.032
0.975
0
0
0
0
0.030
0.119
0.185
0.239
0.284
3.267
2.568
2.266
2.089
1.970
1.882
1.815
1.761
1.716
UCLX
= X + A3S
CLX
= X
LCL X = X − A3S
UCLS
= B4
S
CLS
= S
LCLS
= B3S
Approximations
The following table and equations may be used to generate
initial approximations of the items indicated.
n c4 d2 d3
2
3
4
5
6
7
8
9
10
0.7979
0.8862
0.9213
0.9400
0.9515
0.9594
0.9650
0.9693
0.9727
σ= ˆ R/d
σ= ˆ S/c
3
2
4
σ = d σˆ
R
2
4
1.128
1.693
2.059
2.326
2.534
2.704
2.847
2.970
3.078
σ =σˆ1− c , where
s
ˆσ = an estimate of σ,
0.853
0.888
0.880
0.864
0.848
0.833
0.820
0.808
0.797
σR = an estimate of the standard deviation of the ranges
of the samples, and
σS = an estimate of the standard deviation of the standard
deviations of the samples.
Tests for Out of Control
1. A single point falls outside the (three sigma) control
limits.
2. Two out of three successive points fall on the same side
of and more than two sigma units from the center line.
3. Four out of five successive points fall on the same side
of and more than one sigma unit from the center line.
4. Eight successive points fall on the same side of the
center line.
190
PROCESS CAPABILITY
C
pk
⎛µ −LSL USL −µ ⎞
= min ⎜ , ⎟ , where
⎝ 3σ 3σ
⎠
INDUSTRIAL ENGINEERING (continued)
µ and σ are the process mean and standard deviation,
respectively, and LSL and USL are the lower and upper
specification limits, respectively.
QUEUEING MODELS
Definitions
Pn = probability of n units in system,
L = expected number of units in the system,
Lq = expected number of units in the queue,
W = expected waiting time in system,
Wq = expected waiting time in queue,
λ = mean arrival rate (constant),
λ ~ = effective arrival rate,
µ = mean service rate (constant),
ρ = server utilization factor, and
s = number of servers.
Kendall notation for describing a queueing system:
A / B / s / M
A = the arrival process,
B = the service time distribution,
s = the number of servers, and
M = the total number of customers including those in
service.
Fundamental Relationships
L = λW
Lq = λWq
W = Wq + 1/µ
ρ = λ /(sµ)
Single Server Models (s = 1)
Poisson Input—Exponential Service Time: M = ∞
P0 = 1 – λ/µ = 1 – ρ
Pn = (1 – ρ)ρ n = P0ρ n
L = ρ/(1 – ρ) = λ/(µ – λ)
Lq = λ 2 /[µ (µ– λ)]
W = 1/[µ (1 – ρ)] = 1/(µ – λ)
Wq = W – 1/µ = λ/[µ (µ – λ)]
Finite queue: M < ∞
P0 = (1 – ρ)/(1 – ρ M+1 )
Pn = [(1 – ρ)/(1 – ρ M+1 )]ρ n
L = ρ/(1 – ρ) – (M + 1)ρ M+1 /(1 – ρ M+1 ~
λ = λ(
1−
Pn
)
)
Lq = L – (1 – P0)