Universal Journal of Industrial and Business Management 1(2): 50-53, 2013
DOI: 10.13189/ujibm.2013.010205
http://www.hrpub.org
A Comment on “The Correct Solution Procedure on the
Replenishment Run Time Problem with Machine
Breakdown and Failure in Rework”
Jones P. C. Chuang
Department of Traffic Science, Central Police University, Taiwan, R. O. C.
*Corresponding Author: [email protected]
Copyright © 2013 Horizon Research Publishing All rights reserved.
Abstract Recently, Chung published a paper in Expert
Systems with Applications providing a correct solution
procedure to the inventory system for the replenishment run
time problem with machine breakdown and failure in
rework in Lin and Chiu 〔2〕. In this short note, we provide
a detailed discussion for the convex property in Chiu et al.
〔1〕, Lin and Chiu〔2〕and Chung〔3〕. Moreover, we will
put forth an exact expression for the concave up and concave
down sub-intervals to clearly solve the non-convex
phenomenon. Our results will help researchers realize the
importance of the findings of Chiu et al.〔1〕, Lin and Chiu
〔2〕and Chung〔3〕.
Keywords
Expert System
Inventory System, Machine Breakdown,
convex and/or non-convexity property for the objective
function. Our findings will help researchers enjoying the
significant results of Chiu et al. 〔1〕, Lin and Chiu〔2〕 and
Chung〔3〕.
2. Literature Review and Key Concepts
In this section, we will provide a brief review of the
derivations of Chiu et al.〔1〕, Lin and Chiu〔2〕and Chung
〔3〕 related to the convexity problem. We directly cite the
expected total inventory cost per unit time proposed by Chiu
et al. 〔1〕 as follows
E TCU ( t1 )  =
K λβ
M λβ


 P 1 − e − β t1 + P + C λ 
1
(
)

1 − θ1 E [ x ] 

 +Ck E [ x ] λ + Csθ1 E [ x ] λ 
1. Introduction
Every year thousands papers related to inventory systems
are published. Most of these papers concern themselves
with the construction of new models with more controlled
variable or the application of previous model to new fields.
Very few papers actually try to improve existing solution
procedures through the verification of the existence and
uniqueness of the optimal solution. Chiu et al. 〔 1 〕
developed a new inventory model, showing that under
certain conditions the objective function is convex. Lin and
Chiu〔2〕claimed that without any extra restrictions, the
objective function of Chiu et al.〔1〕 is convex for the entire
domain. By analyzing the second derivation of the
objective function, Chung〔3〕 was the first pointed out the
convex property proposed by Lin and Chiu 〔 2 〕 is
questionable. He then went on to provide an improved
solution method to locate the optimal solution. The purpose
of this communication is to present a detailed explanation for
derivations of Chiu et al.〔1〕, Lin and Chiu 〔2〕 and Chung
〔3〕after which we will offer a complete analysis for the

e − β t1
1
− β t1
−
+
 − t1 e
β
β
+γ
 (1 − θ1 E [x ]) 1 − e − β t1


(
)






(1)
with
γ = hP (1 − 2θ 1 E [x ] + θ 12 E [x 2 ]) − hλ + 2hλθ 1 E [x ]
[ ]
PλE x 2 (h1 − h(1 − θ 1 ))
+ hθ 1 E [x ]gλβ .
P1
In Chiu et al.〔1〕, Lin and Chiu〔2〕, they used γ , whereas
+
in Chung〔3〕, he used π . Seeing that Chiu et al.〔1〕is
the original paper, we used the same notation as Chiu et al.
〔1〕.
For a detailed representation of all the notation and
expressions, please refer to Chiu et al.〔1〕. They had derived
the first and the second derivatives for
E [TCU (t1 )] as
Universal Journal of Industrial and Business Management 1(2): 50-53, 2013
follows
β t1 + e − β t =
1
d
E TCU ( t1 )  =
d t1 
e
that is
− β t1
 − K λβ

− β t1
 P + γ ( e − 1 + β t1 ) 


2
(1 − θ E [ x ]) (1 − e )
− β t1 2
1
(2)
t1 =
(
α (t1 ) > t1
they concluded that
domain
)
(1 − θ E [ x ]) (1 − e β )
− t1 3
1
0 < t1 < ∞ .
(
(
Kλβ 2 1 − e − β t1
+
Pγ
β 1 + e − β t1
for their next theorem.
Theorem 1 of Chiu et al. 〔1〕.
0 < t1 < ϕ (t1 ) .
In
Lin
and
β e − β t (1 + e − β t
1
1
)
(1 − θ1 E[x])(1 − e − β t
1
3
)
)
then he found that
they
knew
that
> 0 , and so by verifying that
d2
E [TCU (t1 )] > 0 is equivalent to showing that
d t12
(
(
− β t1
)
)
 Kλβ 2 1 − e

+
− t1  > 0 . They had used a

− β t1
β 1+ e
 Pγ

different expression, α (t1 ) , for their auxiliary function
α (t1 ) =
(
(
Kλβ 2 1 − e − β t1
+
Pγ
β 1 + e − β t1
)
)
(5)
to mention that if
1− e 1
< 1 to imply that if
1 + e − β t1
 Kλβ
2
t1 > 
+ 
P
γ
β


(4)
(10)
(
(
)
)
 Kλβ
2 1 − e − β t1 
+
t1 > 

β 1 + e − β t1 
 Pγ
E [TCU (t1 )] is convex
Chiu〔2〕,
)
E [TCU (t1 )] is convex in the entire
In Chung〔3〕, he used
Chiu et al. 〔1〕had defined an auxiliary function
if
(9)
− β t1
is monotone and
 K λβ 2 2γ (1 − e − β t1 )
 Lin and Chiu mentioned that β t1 + e

+
− γβ t1  will imply that the solution for the critical points (root of the
1 + e − β t1
 P
 first derivative) is at most one.
−β t
(3)
ϕ (t1 ) =
)
and then under the condition of equation (6)
d2
E TCU ( t1 )  =
d t12 
1
(8)
Lin and Chiu〔2〕compared equations (7) and (9), implying
that
1
Kλβ 2
+1
Pγ
Kλβ
1 − e − β t1 .
+
Pγ
β
and
β e − β t (1 + e − β t
51
(11)
d2
E [TCU (t1 )] < 0 invalidating Lin and Chiu〔2〕’s
d t12
claim of the convexity of E [TCU (t1 )] on the entire
domain, 0 < t1 < ∞ .
Chung〔3〕 provided an outstanding solution method to
[
( )]
examine the first derivative of E TCU t1 to prove the
existence and uniqueness of the minimum solution.
However, he did not provide any further analysis for
d2
E [TCU (t1 )].
d t12
3. Research Model and Hypothesis
First, we recall the Theorem 1 of Chiu et al. 〔1〕. It
α (t1 ) > t1
(6)
d2
then
E [TCU (t1 )] > 0 . Owing to the fact that
d t12
2 > 1 + e − β t for t1 > 0 , they derived that
1
(
)
Kλβ 1 − e − β t1
.
+
α (t1 ) >
Pγ
β
(7)
They considered the solution for the first derivative of
equation (2) as
reveals
that
0 < t1 < ϕ (t1 )
if
and
only
if
d2
E [TCU (t1 )] > 0 . However, Chiu et al. 〔1〕 did not
d t12
solve the set that satisfies the condition.
{t1 : 0 < t1 < ϕ (t1 )}
(12)
As such, the above relationship was not explained in Chiu
et al. 〔1〕.
Second, we will provide a revision for the discussion of
Lin and Chiu〔2〕. The result of equation (9) should be
revised to
52
A Comment on “The Correct Solution Procedure on the Replenishment Run Time Problem
with Machine Breakdown and Failure in Rework”
(
*
Kλβ 1 − e − β t1
t =
+
Pγ
β
*
1
)
*
where t1 is the minimum solution guaranteed by Chung
〔3〕. In Lin and Chiu〔2〕, they mistakenly accepted
that equation (9) is valid for all t1 with 0 < t1 < ∞ .
If we follow the approach of Lin and Chiu 〔2〕, where
they derived that
α (t1* ) > t1*
(14)
2
d
E [TCU (t1 )] > 0 at point t1* . Due to
d t12
such that
the fact that
d2
E [TCU (t1 )] is a continuous function,
d t12
d2
E [TCU (t1 )] > 0 for a neighborhood of t1* implies
d t12
*
1
that t is a local minimum point. However, it is impossible
to prove that the local minimum is also the global minimum
from the above approach.
They should have define an auxiliary function
f (t1 ) = β t1 + e − β t1 to consider the values of lim f (t1 )
t1 →0
and
lim f (t1 ) , and then applied the Intermediated Value
t1 →∞
Theorem, like Chung 〔3〕, for a similar function
e − β t1 − 1 + β t1 . Therefore, the form of f (t1 ) more
accurate, from monotone to increasing such that the critical
point not only exist but is also unique.
Third, we consider the discussion of Chung〔3〕. His
E [TCU (t1 )] is concave down for
finding pointed out that
 Kλβ 2 
t1 > 
+  . However, he did not provide a
β
 Pγ
complete
discussion
t1 .
E [TCU (
for
)]
the
second
derivative
of
4. Result and Discussion
From equation (3),
d2
E [TCU (t1 )]
d t12
(
is
to
know that the
equivalent
Owing
− β t1
1− e
1 + e − β t1
to
to
the
sign
of
the
( e −1)
− βγ
g′ (t ) =
( e + 1)
1
such
2
β t1
2
<0
(16)
g (t1 ) is a decreasing function from
that
g (0 ) =
β t1
Kλβ 2
to lim g (t1 ) = −∞ . Hence, there is a
t1 →∞
P
∆
point, say t1 such that
 > 0, for 0 < t1 < t1∆ ,
g (t1 )
∆
< 0, for t1 < t1 < ∞.
(17)
Consequently, we know that
{t : 0 < t
1
1
< φ ( t1 )} = (0, t1∆ )
E [TCU (t1 )] is convex (concave up) for
(0, t ) . On the other hand, E[TCU (t1 )] is concave down
∆
for (t1 , ∞ ) .
which means that
∆
1
Based on the above discussion, we will provide a complete
analysis for the concave up and/or concave down for the
objective function of Chiu et al.〔1〕, Lin and Chiu 〔2〕
and Chung 〔3〕.
Based on the data set of Chiu et al. 〔1〕, Lin and Chiu
〔2〕 and Chung 〔3〕, we considered the next numerical
K = 450 , λ = 4000 , β = 0.5 ,
example:
P = 10000 , θ1 = 0.1 , E [x ] = 0.1 and γ = 3760.13 .
*
∆
We found that t1 = 0.3175 , t1 = 1.0570 and
 Kλβ 2 

+  = 4.0239 such that
β
 Pγ
t1∆ 1.0570 <
t1* = 0.3175 <=
 Kλβ 2 

+  = 4.0239
β
 Pγ
(18)
as we expected. If we compare our finings with that of Chung
〔3〕 then the concave down domain is revised from
fact
that
2 e − β t1
2
, hence, we
= 1−
= 1 − β t1
− β t1
1+ e
e −1
assume an auxiliary function as follows
(15)
We derive that
sign of
)
 Kλβ 2 2γ 1 − e − β t1

− γβ t1 
+

− β t1
1+ e
 P

 Kλβ 2

4γ
+ 2γ − β t1
− γβ t1 
g (t1 ) = 
e +1
 P

(13)
 Kλβ 2 

+ , ∞  = (4.0239, ∞ )
β 
 Pγ
to the best expression
(19)
Universal Journal of Industrial and Business Management 1(2): 50-53, 2013
, ∞ ) (1.0570, ∞ )
(t=
∆
1
In this short communication, we found the exact
expression for the convex domain for the objective function
of Chiu et al.〔1〕. Moreover, we also discovered the
concave down domain for the objective function. Our patch
work provided a clear vision for the concave up and/or
concave down problem for the objective function of Chiu et
al.〔1〕, Lin and Chiu〔2〕and Chung〔3〕.
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Replenishment run time problem with machine breakdown
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39(1), 1291-1297.
(20)
5. Conclusions
Chiu, Y. S. P., Chen, K. K., Ting, C. K., (2012),
53
[2]
Lin, H. D., Chiu, Y. S. P., (2012), Note on “replenishment run
time problem with machine breakdown and failure in rework.
Expert Systems with Applications,
http://dx.doi.org/10.1016/j.eswa.2012.05.001.
[3]
Chung, K-J., (2012), The correct solution procedure on the
replenishment run time problem with machine breakdown
and failure in rework, Expert Systems with Applications, doi:
http://dx.doi.org/10.1016/j.eswa.2012.07.027.
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