Open Journal of Applied Sciences, 2012, 2, 236-240
doi:10.4236/ojapps.2012.24035 Published Online December 2012 (http://www.SciRP.org/journal/ojapps)
The Principle of Mathematical Induction Applied to the
Generalized Model for the Economic Design of
x
-Control Charts
Mohamed E. Seliaman1, Salih O. Duffuaa2
1Department of Information Systems, King Faisal University, Al-Ahsa, Saudi Arabia
2Department of Systems Engineering, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia
Email: mseliaman@gmail.com
Received August 4, 2012; revised September 6, 2012; accepted September 25, 2012
ABSTRACT
Rahim and Banerjee [1] developed a general model for the optimal design of
x
-control charts. The model minimizes
the expected cost per unit time. The heart of the model is a theorem that derives the expected total cost and the expected
cycle length. In this paper an alternative simple proof for the theorem is provided based on mathematical induction.
Keywords: Discrete Mathematics; Mathematical Induction; Statistical Quality Control; Design of
x
-Control Charts;
Hazard Rate Function
1. Introduction
Quality control charts are graphical statistical tools used
for process control. The first control chart was developed
by Walter A. Shewart [2]. Since then, these charts have
been widely applied in industry and also received inten-
sive attention from researchers. The
x
-control charts are
the most used statistical control charts when continuous
variables are used to measure quality characteristics [3].
Generally, control charts are designed using four ap-
proaches: heuristics, statistical design, economic design
and economic statistical design [4]. For more information
on the design of control charts, interested readers are re-
ferred to [1-7].
The economic design of
x
-control charts was origi-
nated by Duncan [8]. In his model, Duncan considers a
production process that is subject to an occurrence of an
assignable cause that will drive the process out of control.
The output of the process is measurable on a certain scale
and is normally distributed with mean 0
and standard
deviation 0
. The assignable cause is assumed to occur
according to a Possion process with intensity
and
causes a shift in the process mean to 0
, where
is a positive parameter. The control limits of the
x
-control chart are set at 00
k
, where 0
is the
standard deviation of the process. A sample of size n is
taken from the output of the process every h hour, and
used to decide whether the process is in control or not.
Banerjee and Rahim [9] generalized Duncan’s model [8]
for the economic design of
x
-control charts by relaxing
Duncan’s assumption that the in-control period follows an
exponential distribution. Instead, they assumed it follows
a general probability distribution having an increasing
hazard rate function. The increasing hazard rate assump-
tion resulted in the modification of the fixed sampling
interval to a variable sampling interval, dependent on the
process age. In both models, if the sample falls outside the
control limits, a search is initiated to locate the assignable
cause. If the search indicates that there is a false alarm,
the process continues. On the other hand, if the alarm is
true, repair or replacement is undertaken to bring the
process in control. Later, Rahim and Banerjee [1] ex-
tended the model in Banerjee and Rahim [9] by assuming
a general distribution of in-control periods with increasing
failure rate and considering an age-dependent repair be-
fore failure. The objective of Rahim and Banerjee’s
model [1] is to find the parameters n, hj, and k. Under the
assumptions described for this generalized model, they
derived expressions for the expected cycle length and the
total expected cost per cycle. They also presented proofs
for these expressions using a recursive argument. In this
paper a simpler and shorter mathematical induction proof
of these results is presented. Thus, our alternative proof
may make it easier for students and professors interested
in the topic within graduate coursework or further re-
search on extending the model. In this paper, we adopt the
same notations and model descriptions as defined in Ra-
him and Banerjee [1].
In order to make the paper self-contained, we describe
the main elements involved in the design of control charts,
Copyright © 2012 SciRes. OJAppS
M. E. SELIAMAN, S. O. DUFFUAA 237
as modeled in [1].
2. Model Development
In [1], the following assumptions are made:
1) The duration of the in-control period is assumed to
follow an arbitrary probability density function, f(t),
having an increasing hazard rate, r(t), and F(t) as its cu-
mulative density function.
2) The process is monitored by drawing random sam-
ples of size n at times h1, h1 + h2, h1 + h2 + h3··· and so
on. Further, hj satisfies: (i) h1 h2 h3···, and (ii)

lim 1
m
m
F
 .
3) A production cycle begins with a new machine and
ends either with a true alarm or at time m
, whichever
occur first. In other words, if no true alarm is observed
by the time 1m
, then the cycle is allowed to continue
for an additional time m; at time m
h
, the old compo-
nent is replacement by a new one. Thus, there is no cost
of sampling and charting during the mth-sampling inter-
val.
4) For mathematical simplicity, it is assumed that the
production ceases during search and repair.
In their paper, Rahim and Banerjee [1] state the fol-
lowing theorem and provide a proof for it based on re-
cursive relationships. Below, we first present their result
and then provide a simpler mathematical induction proof.
Theorem 1
Under the assumptions (1)-(3) described above the fol-
lowing is true:

 

1
10
11
11
1
11
mm
j
jj
jj
mm
ij
ji
jij
EThFZ F
Fh
 
 




 


Z
and
(1)
 
 
 




 


22
11
111
1
01 10
1
0
11
011
111
1
1
1
11
d
m
mm-
jj
jj
mj
im-j
i
ωm
jj
j
mm-m
i- j
jj-j i
jjij
m-
jmm
j
ECa bnFwβFω
βiβmjβ
DD xfxxDDωFω
DhFωWβDFωβh
αYFωFωSω.

 

 






 









(2)
3. The Mathematical Induction Proof
or Theo-
n induc-
del with one interval. Thus, E(T) = h1 +
Z1 in Equation (1) (the basis
st
vals less than m. Then we show that it is true for a
m
ach possible
st
ing data.
In this section, an alternative proof is provided f
rem 1. The proof consists of the two well-know
tion steps: the basis step and the inductive step. For this
purpose, we consider the case after the first sampling
interval as a new model with a smaller number of inter-
vals and as having a new density function for the in-
control period.
1) Expected cycle length:
a) Let M be a mo
, obtained by letting m = 1
ep).
b) Assume that (1) is true for a model with a number
of inter
odel with m intervals (the inductive step).
As in [7], let us view the possible states of the system
at the end of the first sampling interval. For e
ate of the system, the expected residual times in the
cycle and the associated probabilities are presented in
Table 1.
E(T*) is the expected cycle length for a model M* with
the follow
1) M* has m 1 intervals with *
1
j
j
hh
for 1,j
2, ,1m
.
2) In M*, the duration of the in-coeriod is as-
oll
ntrol p
sumed to fow an arbitrary probability density function,
*
f
t, and
*
F
t as its cumulative density function
where
 
1
*
1
1
,
1
ot
ft ft t
p
(3)
and
**
ET
with th
is the expected cycle length for a model
M** e following data.
M** h1. as m 2 intervals with **
2
j
j
hh
for
1, 2,,2jm
.
2. In M**, the duration of the in-contro as-l period is
sumed to follow an arbitrary probability density function,
**
f
t, and
**
F
t as its cumulative density function
where

1
**
1
1
0.
ft t
ft p
t
(4)
Table 1. The expected residual times.
State Probability
dual time
Expected
resi
In control and no alarm (1) – p1) (1 –
E(T*)
In control and false alarm (1p)
arm h)
m
1
p
Z0 + E(T*)
Out of control but no al1
p1 (1 –
)
2
Z1
+ E(T**
Out of control and true alar
Copyright © 2012 SciRes. OJAppS
M. E. SELIAMAN, S. O. DUFFUAA
Copyright © 2012 SciRes. OJAppS
238
Thus,
 








****
1110 121
***
11101121 1
11 11
11 1
EThpETpZETph ETpZ
hpETpZpETphp Z
 
 

 

 
1
(5)
Thus, according to the induction hypothesis along with the relations (3) and (4)
 


  
1221mmmm
**** ****1
10 1
1111
11
11
01
2221
111
,
111
ij
jj j ji
jjjij
mmmm
jjj
ij
ji
jjjij
EThFZ FFhZ
FFF
hZ hZ
ppp
 

 






 
 



(6)
and
 

2332
**** **********1
10 1
1111
()
mmmm
ij
jjj ji
jjjij
EThFZFFhZZ
 



 

1
. (7)
Substituting (6) and (7) into (5), we obtain



 
  
11
1
1110012
222
11 1
10 1
1111
1
,
1
1
mmm
ij
jjjj i
jjjij
mmmm
ij
jjjj i
jjjij
EThhFpZZFp hFhZ
hFZFFhZ

 



 

 
 


hich gives the proof for Equation (1) in the theorem.
he
ex
costs for M* and M** respectively, from the above table
m
w2) The Expected Total Cost
ost E(C), we consider t
we obtain
 

 



10
1
101111
**
11211
1
1
pa bnDhYC
abn Dh
To obtain the expected total c*
EC
.
p
DhpE CpW

 
 
 
 
pected residual cost beyond time h1 as the expected
total cost for a model with less than m intervals. For each
possible state of the system at the end of the first sam-
pling interval, the expected residual costs in the cycle
(8)
Thus, employing the induction hypothesis along with
th
and the associated probabilities are presented in Table 2.
Where E(C*) and E(C**) are the two expected total e relations (3) and (4),



 



 

1
11
22 1
11
1
01 100
22
111
11
1
21
1
111
11
d
111
1
m
jj
*im
j
jj i
wmm
jj
jj
jj
w
mm
ji- j
i
jij
EC a bnββiβmjβ
pp
FωFω
xf xx
DDDD ωDh W
-pp p
FωF
βDβhαY
p

 


1mj
22mm
FωFω


 




 







 





1
211
11
mjm
m
j
ωFωSω
pp

(9)
and


**
ECabnW
. (10)
Substituting (9) and (10) into (8), we obtain
M. E. SELIAMAN, S. O. DUFFUAA 239
Table 2. The expected residual costs.
State Probability Expected Residual Cost Current Cost
In control and no alarm 1
) 1
(1 – p) (1 – (a + bn) + D0h E(C*)
In control and false alarm 0h1
h1
1) 2E(C**)
)
(1p1)
(a + bn) + Y + D E(C*)
Out of control but no alarm p1
(a + bn) + D01 + D1(D1h +
Out of control and true alarm p1(1 –(a + bn) + D01 + D1(h1
1) W
 
 
 





 
1
1
22 11
11 1
01101111 10
2
11
10101121
22
111
d
1
m
mj
mm im
jj
jj i
ωm
jj
j
ω
mm
ij
1
j
m
j
jj
jj
ECa bnFωβ Fωβiβmjβ
DDτDDxfxxpDh DDωFω
pDh DhFωWβDph βDFωβh

 
 


i
ij



 
 




1
1
m
jmm
j
αYFωFωSω .

(11)
By adding and subtracting the term p1D0h1 to the right hand side of (11) and substituting

1
1d
m
x
fx x
,
we obtain
 
 
 



 
 

1
22 11
11 1
011001
11
0
11
1
1
11 1
111
d
m
mj
mm im
jj
jj i
ωmm
jj jj
jj
mm m
ij
ji jmm
jijj
ECa bnFωβ Fωβiβmjβ
DD xfxxDDωFωDhFωW
βDFωβhαYFωFωSω .

 
 



 

j





 


 





 
This completes the proof.
4. Acknowledgements
cknowledge the support for
REFERENCES
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The authors would like to a
this research provided by the King Fahd University of
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240
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Notation
Z0 = expected search time associated with a false alarm.
Z1 = expected search time to discover the assignable
cause and repair.
a = fixed sampling cost.
b = sampling cost per unit sampled.
Y = cost per false alarm.
W = cost to locate and repair the assignable cause.
D0 = quality cost per hour while producing and the pro-
cess is in control.
D1 = quality cost per hour while producing and the pro-
cess is out of control.
= probability of Type I error.
= probability of Type II error.
h = the sampling interval for a uniform sampling
scheme.
j
h = the jth sampling interval for a non-uniform sam-
pling scheme.
j
= h1 + h2 +···+ hj.
M = a specified number of sampling intervals.
n = sample size.
E(T) = expected cycle length.
E(C) = expected cost.
Pj = the conditional probability that process goes out of
control between 1
j
and
j
, given that it was at the
in-control state at time 1
j
.
Copyright © 2012 SciRes. OJAppS