Description Usage Arguments Details Value References See Also Examples
Calculate the optimum parameters of n(sample size), h(sampling interval), k(reference value) and H(decision interval) for Economic Design of the CUSUM control chart. For more information about the reference value see 'Details'.
1 2 3 4 5 6 7 8 9 10 11  ecoCusum(h, H, n, delta = 2, lambda = 0.01, P0 = NULL,
P1 = NULL, C0 = NULL, C1 = NULL, Cr = 20, Cf = 10,
T0 = 0, Tc = 0.1, Tf = 0.1, Tr = 0.2, a = 0.5, b = 0.1,
d1 = 1, d2 = 1, nlevels = 30, sided = "one",
par = NULL, contour.plot = FALSE, call.print = TRUE,
...)
echCusum(h, H, n, delta = 2, lambda = 0.01, P0 = NULL,
P1 = NULL, C0 = NULL, C1 = NULL, Cr = 20, Cf = 10,
T0 = 0, Tc = 0.1, Tf = 0.1, Tr = 0.2, a = 0.5, b = 0.1,
d1 = 1, d2 = 1, sided = "one")

h 
sampling interval. It can be a numeric vector or left undefined. See 'Details' 
H 
decision interval. It can be a numeric vector or left undefined. See 'Details' 
n 
sample size. It can be an integer vector or left undefined. See 'Details' 
delta 
shift in process mean in standard deviation units when assignable cause occurs (delta = mu1  mu0/sigma), where sigma is the standard deviation of observations; mu0 is the incontrol process mean; mu1 is the outofcontrol process mean. Default value is 2. 
lambda 
we assume the incontrol time follows a exponential distribution with mean 1/lambda. Default value is 0.05. 
P0 
profit per hour earned by the process operating in control. See 'Details'. 
P1 
profit per hour earned by the process operating out of control 
C0 
cost per hour due to nonconformities produced while the process is in control. 
C1 
cost per hour due to nonconformities produced while the process is out of control.(C1 > C0) 
Cr 
cost for searching and repairing the assignable cause, including any downtime. 
Cf 
cost per false alarm, including the cost of searching for the cause and the cost of downtime if production ceases during search. 
T0 
time to sample and chart one item. 
Tc 
expected time to discover the assignable cause. 
Tf 
expected search time when false alarm occurs. 
Tr 
expected time to repair the process. 
a 
fixed cost per sample. 
b 
cost per unit sampled. 
d1 
flag for whether production continues during searches (1yes, 0no). Default value is 1. 
d2 
flag for whether production continues during repairs (1yes, 0no). Default value is 1. 
nlevels 
30. It works only when 
sided 
distinguish between one, twosided and
Crosier's modified twosided CUSUM scheme by choosing
"one", "two", and "Crosier", respectively. See details in

par 
initial values for the parameters to be optimized over. It can be a vector of length 2 or 3. See 'Details' 
contour.plot 
a logical value indicating wether a contour plot should be drawn. Default is FALSE. 
call.print 
a logical value indicating whether the "call" should be printed on the contour plot. Default is TRUE 
... 
other arguments to be passed to 
When parameter par
is specified, optimization
algorithms would be used as default. par
can be
specified as: par = c(h, H)
where h
and
H
are the initial values of smapling interval and
decision interval when n
is specified; or
par = c(h, H, n)
. Good inital values may lead to
good optimum results.
When parameters h
, H
, n
are all
undefined, ecoCusum
function will try to find the
global optimum point to minimize the ECH (Expected Cost
per Hour) using optimization algorithms (optim
function), but in this case n
would not be
integer. It is usually helpful for the experimenter to
find the region where the optimum point may exist
quickly. When h
and H
are undefined but
n
is given as an integer vector, ecoCusum
function will try to find the optimum point for each
n
value using optimization algorithms. When
h
, H
and n
are all given,
ecoCusum
function will use a "grid method" way to
calculate the optimum point, that is ECH for all the
combinations of the parameters will be calculated. The
"grid method" way is much slower than using optimization
algorithms, but it would be a good choice when
optimization algorithms fail to work well.
There is strong numerical and theoretical evidence that
for given L1, the value of L0 approaches its maximum when
k(reference value) is chosen midway the between AQL and
the RQL: $k = mu0 + 0.5*delta*sigma (Appl. Statist.(1974)
23, No. 3, p. 420). For this reason we treat k as a
constant value and optimize n, h and H. For cost
parameters either P0, P1 or C0, C1 is needed. If P0
and P1 are given, they will be used first, else C0 and C1
will be used. For economic design of the CUSUM chart,
when d1
and d2
are both 1, only if the
difference between P0 and P1 keeps the same, the results
are identical. If the difference between C0 and C1 keeps
the same, the optimum parameters are almost the same but
the ECH(Expected Cost per Hour) values will change.
echCusum
is used to calculate the ECH (Expected
Cost per Hour) for one given design point.
The ecoCusum
function returns an object of class
"edcc", which is a list of elements optimum
,
cost.frame
, FAR
and ATS
.
optimum
is a vector with the optimum parameters
and the corresponding ECH value; cost.frame
is a
dataframe with the optimum parameters and the
corresponding ECH values for all given n
(if
n
is not specified, cost.frame
won't be
returned); FAR
indicates the false alarm rate
during the incontrol time, which is calculated as
lambda*(average number of false alarm); ATS
indicates the average time to signal after the occurrence
of an assignable cause, calculated as h*ARL2  tau, where
tau is the expected time of occurrence of the assignable
cause given it occurs between the ith and (i+1)st
samples. The echCusum
function returns the
calculated ECH value only.
Weicheng Zhu, Changsoon Park (2013), edcc: An R Package for the Economic Design of the Control Chart. Journal of Statistical Software, 52(9), 124. http://www.jstatsoft.org/v52/i09/
Lorenzen and Vance (1986). The economic design of control charts: a unified approach, Technometrics, 28. 310.
Chiu, W.K. (1974). The economic design of CUSUM charts for controlling normal means, Journal of the Royal Statistical Society. Series C (Applied Statistics), 23(3), 420433.
ecoXbar
, ecoEwma
,
xcusum.arl
,
optim
, update.edcc
,
contour
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23  #Chiu, W.K. (1974). Applied Statistics, 23, p427 Table3, row 12,14
## LINE 1
## global optimization to h, H and n, when lambda = 0.01, "NelderMead" optimization algorithm doesn't work
#(y < ecoCusum( P0=150,P1=50,Cr=30,d1=0,d2=0))
## we can try other algorithms:
(y1 < ecoCusum( P0=150,P1=50,Cr=30,d1=0,d2=0,method="BFGS"))
# Based on the global optimum above, we specify the range of the
# parameters like this
(yy1 < ecoCusum( h=seq(1.3,1.45,by=.01), H=seq(.5,0.6,by=.01),n=4:6,
P0=150,P1=50,Cr=30,d1=0,d2=0))
## LINE 2
(y2 < ecoCusum( P0=150,P1=50,Cr=30,d1=0,d2=0,lambda=0.05))
(yy2 < ecoCusum( h=seq(.6,0.7,by=.01), H=seq(.5,0.6,by=.01),n=3:6,
P0=150,P1=50,Cr=30,d1=0,d2=0,lambda=0.05))
contour(yy2)
## LINE 14
(y14 < ecoCusum(n=30,P0=150,P1=50,Cr=30,delta=0.5,d1=0,d2=0,method="LBFGSB"))
(yy14 < ecoCusum(h=seq(2.55,2.65,by=0.01),H=seq(0.3,0.4,by=0.01),
n=28:30,P0=150,P1=50,Cr=30,delta=0.5,d1=0,d2=0))
#Douglas (2009). Statistical quality control: a modern introduction, sixth edition, p470.
ecoCusum(lambda=.05,P0=110,P1=10,Cr=25,Cf=50,Tr=0,Tf=0,Tc=1,T0=.0167,a=1)
ecoCusum(h=seq(0.75,0.85,by=.01),H=seq(.55,0.65,by=.01),n=4:6,lambda=.05,
P0=110,P1=10,Cr=25,Cf=50,Tr=0,Tf=0,Tc=1,T0=.0167,a=1)

Loading required package: spc
$optimum
Optimum h Optimum H Optimum n ECH
1.4074981 0.5563472 5.0000000 2.2608118
$FAR
[1] 0.001865727
$ATS
[1] 0.7728745
$optimum
Optimum h Optimum H Optimum n ECH
1.41000 0.56000 5.00000 2.26082
$cost.frame
Optimum h Optimum H Optimum n ECH
1.33 0.60 4 2.275288
1.41 0.56 5 2.260820
1.45 0.50 6 2.281802
$FAR
[1] 0.001841721
$ATS
[1] 0.7747774
$optimum
Optimum h Optimum H Optimum n ECH
0.6439739 0.5488288 5.0000000 6.9732583
$FAR
[1] 0.004134648
$ATS
[1] 0.3540967
$optimum
Optimum h Optimum H Optimum n ECH
0.640000 0.550000 5.000000 6.973315
$cost.frame
Optimum h Optimum H Optimum n ECH
0.60 0.60 3 7.199799
0.61 0.60 4 7.000770
0.64 0.55 5 6.973315
0.68 0.50 6 7.019643
$FAR
[1] 0.00414591
$ATS
[1] 0.3519764
$optimum
Optimum h Optimum H Optimum n ECH
2.6204264 0.3689877 30.0000000 4.2078412
$cost.frame
Optimum h Optimum H Optimum n ECH
2.620426 0.3689877 30 4.207841
$FAR
[1] 0.01577703
$ATS
[1] 1.798679
$optimum
Optimum h Optimum H Optimum n ECH
2.620000 0.370000 30.000000 4.207842
$cost.frame
Optimum h Optimum H Optimum n ECH
2.55 0.39 28 4.209709
2.58 0.38 29 4.208015
2.62 0.37 30 4.207842
$FAR
[1] 0.01574711
$ATS
[1] 1.799209
$optimum
Optimum h Optimum H Optimum n ECH
0.8207753 0.6078121 5.0000000 10.2650248
$FAR
[1] 0.002694194
$ATS
[1] 0.4570376
$optimum
Optimum h Optimum H Optimum n ECH
0.82000 0.61000 5.00000 10.26503
$cost.frame
Optimum h Optimum H Optimum n ECH
0.79 0.65 4 10.31297
0.82 0.61 5 10.26503
0.85 0.55 6 10.32087
$FAR
[1] 0.002678567
$ATS
[1] 0.4568029
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.