Dear Editor and Reviewer,
We kindly appreciate the thorough and constructive review that has been given by the reviewer on
our manuscript:
Ref. No. : 1696
Title
: Optimal Overhaul-Replacement Policies for Repairable Machine Sold
with Warranty
Suggestions and comments given are very helpful to improve the paper. We have revised
signifantly the old version of the paper by taking into account all of the concerns raised by the
reviewer. Our responses to all concerns are given in the separate pages attached to this letter.
We hope that the revised version of the paper is now acceptable for publication in the journal of ITB
Proceedings.
I am looking forward to receiving a positive response from you.
Yours Sincerely,
Kusmaningrum
Evaluation Result :
Could be published with major revision
Comment :
Overall, the paper is good, but it could still be improved in several areas.
Response:
We thank to the reviewer for reading the manuscript thoroughly and giving helpful
remarks and constructive suggestions that helped us to improve the manuscript.
1.
Firstly, in the paper, terms in a formula are grouped using curly brackets, i.e.
{}. This practice actually confuses the reviewer, as curly bracket is usually used to
denote a set, not to group terms in a formula. It might be better to use the usual
bracket “()” to group terms in a formula.
Response:
We have changed all curly brackets in the formulas to the usual brackets as suggested.
2. Secondly, although equation (4) is used everywhere in the paper, there is no
motivation for the equation. Why are the costs set up that way? Although the
readers might see the reasons from the formula, it is probably better to just
explicitly motivate the equation.
Response:
In accordance with the Reviewer concerns, we have now reconstructed the text by
adding the following paragraph prior to the equation (4) and highlighted in red type on
page 7:
We now obtain the dynamic programming formulation for the overhaul-replacement problem.
Let 𝐹𝑗 (𝑡) be the total cost of running machine age t from j to N. Then 𝐹𝑗 (𝑡) is a summation of
keep, overhaul or replacement cost in j, the associated minimal repair costs, and the best total
ownership costs for the remaining stages (stage j+1 onward). Moreover we define 𝐹𝑗∗ (𝑡) as the
best value of 𝐹𝑗 (𝑡) for the optimal decision 𝑥𝑗∗ . For e(t) is the salvage value of the machine with
age t, by using (1), (2) and (3) the associated costs in j for each 𝑥𝑗 is presented in Table 1.
Table 1. Decision in j, 𝑥𝑗 and the associated costs
for the current stage onward
K
Decision cost and the
associated minimal repair
costs in j
h ( t , t  s )
O
c3 
R
c4  e(t )  h(0, s )
𝑥𝑗
h(t   , t    s )
State in
j+1
t+s
t-+s
s
2
Total costs in j and the remaining stages
h(t , t  s )  F j*1 (t  s )
c3  h ( t   , t    s )  F *
j 1 ( t    s )
*
c4  e(t )  h(0, s )  Fj
1 ( s)
Our purpose is to seek the optimal sequential decisions (Keep, Overhaul, or Replace) which
minimize F0* (t ) . From Table 1 we define F j* (t ) as recursive equation at stage j for of the
machine age t, which minimizes the total ownership costs in j onward by choosing xj. Then the
dynamic programming formulation of optimal overhaul-replacement policies is given by:
3.
Thirdly, because of the keyword dynamic programming, the reviewer
somewhat expected results in the form of:
a. proof of existence of optimal solution (because closed form of solution is hard
to find) either analytically or graphically
b. methods to compute the solution faster, or new algorithms for certain class of
machine models.
Response to point 3.a:
We have rewritten our analysis and highlighted in pharagraph 3.1, 3.2, and 3.3 on page
8-10. We give the analytical proof of existence of the optimal solution and present the
solution graphically in Section 4. Numerical examples. The graphical results of optimal
solution for each numerical examples presented in Figure 3, 4, 5, 6 and 7 (page 13-16).
3.1
Existing of Optimal Solution
For 𝐹𝑗∗ given by (4) there is exists an optimal solution that minimizes F0* (t ) given by the decision
policy  *  {x0* , x1* ,..., x*N 1} . We use an induction proof to show the existence of optimal
solution of our model. First we specify the space state at any stage i.e. machine age at any j
(j=1,2,..,N-1) as t j , t j Tj ,Tj  s,2s,..., js.
We denote G j (t j , x j ) as the current stage costs i.e. decision cost in j and the associated
expected minimal repair costs:

h(t j , t j  s )
,xj  K

G j (t j , x j )  c3  h(t j   , t j    s ) , x j  O
 c  e(t )  h(0, s ) , x  R
j
j
 4
(6)
The state on the successive stage t j 1 depends on the decision taken on the current stage and
we denote:
, t j
, j  0,1,..., N  1
 tj  s ,xj  K

t j 1  t j    s , x j  O , t j  w   , j  w   ,..., N  w  

s
, x j  R ,t j  w
, j  w,.., N

(7)
Since w and  defined as integer multiplication of s then all feasible xj might facilitate every
3
movement from any t j to one t j1 .
By using (6) and (7) we rewrite (4) the best total ownership costs in a particular j as:
Fj* (t )  Fj (t j , x*j ) 
(G j (t j , x j )  Fj*1 (t j 1 ))
min
x j { K , O , R}
(8)
Using a principle of optimality of dynamic programming we can show that, for any j, (j=0,1,
2,…N-1) if there exists x*j for a certain t j that satisfies (8) then we can find x*j 1 .
For j=N;
As N is the last stage, there is no cost to be considered and the machine is sold. To be specific at
the end of planning horizon the warranty should be expired so possible states t N are finite i.e.
t N TN , TN  w, w  s,..., Ns .
As the machine salvage value depends on the machine age then we can rewrite (5) as:
FN*  FN* (t N )  e(t N ), t N TN
(9)
We proceed to show the existence of the optimal solution at j=N-1 by using FN* (t N ) .
For j=N-1;
At j=N-1, the state space t N 1 are finite i.e. t N 1 TN 1 , TN 1  w  s, w,...,( N  1) sBy using (8) the
optimal expected cost at stage N-1 can be expressed as:
FN 1 (t N 1, x *N 1 ) 
min
(G N 1 (t N 1 , x N 1 )  FN* (t N ))
x N 1{ K ,O , R}
(10)
From any t N 1  TN 1 there exists at least one feasible 𝑥𝑁−1 ∈ {𝐾, 𝑂, 𝑅} that facilitates movement
to one t N TN in the subsequent stage. Hence it follows that from all feasible solution 𝑥𝑁−1 there
is at least one x *N 1 that gives the minimum ownership costs for the last period to go. As a result,
at j=N-1 the optimal solution x *N 1 can be obtained for all t N 1  TN 1 .
Continuing the backward process to reach stage 0 we certainly obtain the optimal decision
sequences for the remaining stages i.e. x*N  2 , x*N  3 ,..., x0* . Finally, we have  *  {x0* , x1* ,..., xN* 1}
that minimizes F0* (t ) .
3.2
Necessary Condition for Overhaul
For any j and t (t  w +) we perform overhaul with cost c3 that will reduce the age of the
machine from t to t-. We consider a non-warranted situation (w=0 and c1= c2). Using equation
(4), we obtain the necessary condition in j for doing Overhaul is better than Keep operating the
4
machine (11), and for Overhaul is better than to do Replacement (12). .
(h(t , t  s)  h(t   , t    s))  ( F j*1 (t  s)  Fj*1 (t    s))  c3
(11)
As h(t) and Fj(t) are increasing function in t [20] the left hand side of (11) also increasing in t. The
first term of the left hand side of (11) shows minimal repair cost reduction just in j due to
overhaul, and the second term represents the additional benefits in the remaining stages. Note
when  →0 doing Overhaul or Keep operating the machine in j is equally attractive. To
accomplish (11) the rejuvenation effect  has to be sufficient large to produce a minimal repair
cost reduction in both j and the remaining stages. Using equation (4), we also obtain the
necessary condition in j for doing Overhaul is better than to do Replacement (12).
(c4  e(t ))  (h(t   , t    s )  h(0, s))  ( Fj*1 (t    s)  Fj*1 ( s))  c3
(12)
For  approaches t, the machine condition after overhaul is about as good as new and second
and third terms of the left hand side of (12) nearly turn into zero. In this case, the only factor that
prevents accomplishment of (12) is a high machine salvage value of the machine on hand e(t).
For a non-warranted situation, decision to Overhaul in j is only attractive if the rejuvenation
effect is significant relative to c3. One that might have opposite effect is the resale value of the
second hand machine on hand.
3.3
Interaction Effect of Overhaul and Warranty
The benefit of warranty stimulates early replacement [20]. We develop the necessary condition
in j for doing Replacement is better than Overhaul by using (4) and (2). For w = 1 we obtain (13).
(c2 h(t   , t    s)  c1h(0, s))  ( Fj*1 (t    s)  Fj*1 ( s))  (c4  e(t ))  c3
(13)
The first term of the left hand side of (13) is always non-negative since h(t) and Fj(t) are
increasing function in t, and c2 > c1. The second part also has a positive value. Consequently, the
necessary condition for doing Replacement in j can be accomplished if one of the following
conditions is satisfied.
c4  e(t )  c3  c2 h(t   , t    s )  c1h(0, s )
(14)
c4  e(t )  c3  ( F j*1 (t    s)  F j*1 ( s))
(15)
The distinctive parameter of the model developed are the overhaul rejuvenation effect  and
the warranty benefit that are represented by w and c1 c2 . Therefore, to discuss the underlying
buyer’s decision to replace we consider only equation (14). The value of the right hand side of
(14) is associated with the disparities of c1 and c2, that indicate how significant minimal repair
cost compared to the losses due to the process disruption to the buyer. Reduction of the left
hand side of (14) might happen due to the decrease of c4 and or the increase of e(t), as well as
the increase of c3. This situation shows the underlying buyer’s decision in doing replacement, i.e.
5
a reasonable price of a new machine along with an expensive minimal repair cost and-or a fine
salvage value of the machine. In addition, if the overhaul results are insignificant and costly then
the buyer’s best decision in j is to do replacement.
The graphical results of each numerical example optimal solutions are as follows:
Table 2 The optimal policies 𝑥𝑗 ∗ for  = (1.20, 1.25, 1.35)
Optimal policies in j, 𝑥𝑗 ∗

𝑥1 ∗
𝑥2 ∗
𝑥3 ∗ 𝑥4 ∗
1.20
K
K
K
1.25
K
K
1.35
K
K
𝑥11 ∗
cost
K
K
K
7101.86
K
O
K
K
7725.58
K
R
K
K
8615.43
𝑥6 ∗
𝑥7 ∗
O
K
O
K
O
K
K
R
K
K
R
K
K
R
K
Machine age (years)
Machine age (years)
𝑥10 ∗
𝑥5 ∗
6
 1.2
5
4
3
2
1
Optimal
𝑥8 ∗ 𝑥9 ∗
6
 1.25
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10 11 12
Stage
1
Machine age (years)
6
2
3
4
5
6
7
8
9 Stage
10 11 12
 1.35
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10 11 12
Stage
Figure 3. Machine age at stage j due to decision xj for several value of 
Table 3 The optimal policies 𝑥𝑗 ∗ for =(1, 2, 3)

Optimal policies in j, 𝑥𝑗 ∗
𝑥1 ∗ 𝑥2 ∗
𝑥3 ∗ 𝑥4 ∗ 𝑥5 ∗ 𝑥6 ∗
𝑥7 ∗
𝑥8 ∗
𝑥9 ∗ 𝑥10 ∗ 𝑥11 ∗
Optimal cost
1
K
K
R
K
K
K
K
R
K
K
K
7756.64
2
K
K
K
K
R
K
K
K
O
K
K
7725.58
3
K
K
K
K
O
K
K
O
K
K
O
7611.09
6
6
 1
5
Machine age (years)
Machine age (years)
6
4
3
2
1
0
2
5
4
3
2
1
0
1
2
3
4
5
6 7
8
9 10 11 12
Stage
1
2
Machine age (years)
6
3
4
5
6
7
8
9 10 11
12
Stage
3
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9 10 11 12
Stage
Figure 4. Machine age at stage j due to decision xj for several value of 
Table 4 The Optimal Policies 𝑥𝑗 ∗ for w=(1, 2, 3)
Optimal decisions in j, 𝑥𝑗 ∗
w
𝑥1 ∗
𝑥2 ∗
1
K
2
3
Optimal
𝑥3 ∗ 𝑥4 ∗ 𝑥5 ∗ 𝑥6 ∗
𝑥7 ∗ 𝑥8 ∗ 𝑥9 ∗ 𝑥10 ∗
K
O
K
O
K
O
K
O
K
K
K
K
R
K
K
K
K
K
K
R
K
K
K
R
cost
K
K
7286.32
O
K
K
7725.58
K
K
K
7577.66
6
Machine age (years)
6
Machine age (years)
𝑥11 ∗
w1
5
4
3
2
1
w2
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10 11 12
Stage
1
2
3
4
5
6
7
8
9
10 11 12
Stage
Machine age (years)
6
w3
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
11
12
Stage
Figure 5. Machine age at stage j due to decision xj for several value of w
Table 5 The Optimal policies 𝑥𝑗 ∗ for r=(65%, 70%, 90%)
r
Optimal decisions in j, 𝑥𝑗 ∗
𝑥3 ∗ 𝑥4 ∗ 𝑥5 ∗
𝑥6 ∗
𝑥1 ∗
𝑥2 ∗
65%
K
K
R
K
K
K
K
R
K
K
K
6679.60
70%
K
K
K
K
K
R
K
K
O
K
K
6807.82
90%
K
K
K
O
K
O
K
O
K
K
K
7101.86
7
𝑥7 ∗ 𝑥8 ∗
Optimal
𝑥9 ∗
𝑥10 ∗ 𝑥11 ∗
cost
Machine age (years)
Machine age (years)
6
r  70%
5
4
3
2
1
6
r  65%
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
Machine age (years)
Stage
11
12
Stage
6
r  90%
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
11
12
Stage
Figure 6. Machine age at stage j due to decision xj for several value of r
Table 6 The Optimal policies 𝑥𝑗 ∗ for T= (8, 10, 11, 12)
T
O
K
O
K
--
10
…
K
R
K
K
K
K
--
11
…
O
K
O
K
O
K
K
--
12
…
K
R
K
K
K
O
K
K
6
Machine age (years)
…
Machine age (years)
8
T8
5
4
3
2
1
0
cost
4743.87
6229.75
6977.36
--
7725.58
6
T  10
5
4
3
2
1
0
2
3
4
5
6
7
8
9
6
10 11 12
T  11
5
1
Stage
Machine age (years)
1
Machine age (years)
Optimal
*
*
*
x 4* x 5* x 6* x 7* x8* x 9* x10
x11
x12
x1* ,..., x3*
4
3
2
1
0
2
3
4
5
6
7
8
9
6
10 11 12
Stage
T  12
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10 11 12
Stage
1
2
3
4
5
6
7
8
9
10 11 12
Stage
Figure 7. Machine age at stage j due to different decision xj
at different length of T
8
Response to point 3.b:
As the reviewer wish, we explain methods to compute the solution faster by adding the
following text in the first paragraph of page 8.
“The simple recursive pattern of dynamic programming makes a relative short planning horizon
problem along with a slight range of model parameters can be solved in Microsoft Excel. In
solving a problem with a longer planning horizon and considerable variation of model parameters,
we need to build a computer program to obtain the optimal policy.”
4.
The reviewer did not expect to find an analysis based on dynamic
programming formulation of the problem even though this analysis seems to be the
main results of this paper.
Response:
With all due respect to the reviewer though the analysis based on DP formulation was not
expected, we still show them in our revised manuscript. As we mention in the introduction
section, based on our observation we are not aware of any previous replacement decision
model that considers overhaul option and warranty benefit from the buyer’s point of view.
In addition, the DP fomulation best suits to describe the sequential decisions needed for
the decision problem considered. To be consistent with the model objectives, i.e.
“addressed to find out the interaction effect of overhaul alternative and product warranty
toward buyer’s optimal policy”, then the analysis based on dynamic programming
formulation was provided again in the revised manuscript. However, we have made a
slight modification to make the description clearer than the original one as presented in
section 3.2 and 3.3 that highlighted in red type on page 10-11.
5. It might be beneficial to rearrange the structure of this paper to focus more
on the analysis.
9
Response to point 5:
We examined literatures and concentrated to rearrange the analysis section which has
become the major suggestions. Major changes have been made as follows.
The 1st version
1. Introducion
2. System Characterization and Model
Formulation
4. Analysis
5. Numerical examples
5. Conclusion
6.
The revised version
1. Introducion
2. System Characterization
4. Dynamic Programming Formulation
and Analysis
5. Numerical examples
5. Conclusion
Lastly, the conclusion reads as if it follows from numerical examples instead
from the analysis.
Response to point 6:
We agree with the viewpoint of the reviewer and revised the conclusion as follows (page
17):
5
Conclusions
In this paper, we study a simple optimal overhaul-replacement model for a repairable machine
sold under warranty from the buyer’s point of view. The machine is a means of production that
should be operated for a finite horizon planning. We assumed that during the warranty period the
seller bears the minimal repair cost, so the buyer only experiences a cost due to disruption of
production process.
The buyer wants to minimize the total ownership cost over the planning horizon by considering
overhaul decision to extend the machine life. In the same time, the buyer also wants benefit from
warranty that attached with machine replacement decision. Overhaul is modeled as an action that
decreases the machine’s age with a permanent reduction.
We developed dynamic programming formulation to determine overhaul-replacement policies
that minimize the total ownership cost over the planning horizon considered. The length of
warranty and the rejuvenation effect of overhaul are modeled as integer multiplication of
periodic evaluation span. To have the model solution we execute dynamic programming
algorithm numerically. The minimization equation caried out sequentially for all state in eah
stage for all feasible decision. We solved the model by computer program written in visual basic.
The model is based on some simplified assumptions, but may provide valuable insights about the
10
structure of the optimal solution for machine replacement under warranty in a more complex
situation. We analyzed the model to show the existence of the optimal solution and discuss the
interaction effect of overhaul and warranty to the buyer’s decision. We showed the optimal
solution behaviors by using computational results of some scenarios of model parameter.
Using this model the buyer can find the periodic optimal policy and the intended minimum cost
as well as the replacement schedule and periodic amount of capital required in the planning
horizon. The buyer can apply the model to wide variety of profit generating assets. Further, by
understanding interaction effects of the advantage of warranty and the benefit of overhaul, the
buyer also can attempt to make an overhaul improvement and avoid costly frequent machine
replacement. Manufacturers also can use this model in particular to understand the underlying
customer’s decision in doing replacement under life cycle context and the customer’s purchase
behavior. Then manufacturers can consider improving the offered benefit such as the warranty
length to promote sales.
The replacement overhaul-model for a warranted machine with a Markovian deterioration model
and overhaul rejuvenation being a function of the machine age, are another topics for future
research currently investigated by the authors.
7.
Again, I think this could be corrected by rearranging the structure of the
paper. [Reviewer #1]
Response to point 7:
The manuscript has been reconstructed -the dynamic programming formulation has been
inserted in section 3 and the analysis has been modified. The modification makes the
mathematical model clear and easy to follow.
11