GROUP VERSUS STAGERED REPLACEMENT POLICY
STRATEGIC REPLACEMENT DECISIONS
Except where reference is made to the work of others, the work described in this thesis is
my own or was done in collaboration with my advisory commite. This thesis does not
include proprietary or clasified information.
_____________________________________
Kyongsun Kim
Certificate of Approval:
Jorge Valenzuela Chan S. Park, Chair
Asociate Profesor Ginn Distinguished Profesor
Industrial and Systems Engineering Industrial and Systems Engineering
Ming Liao Joe F. Pitman
Profesor Interim Dean
Mathematics and Statistics Graduate School
GROUP VERSUS STAGERED REPLACEMENT POLICY
STRATEGIC REPLACEMENT DECISIONS
Kyongsun Kim
A Thesis
Submited to
the Graduate Faculty of
Auburn University
in Partial Fulfilment of the
Requirements for the
Degre of
Master of Science
Auburn, Alabama
August 9, 2008
ii
GROUP VERSUS STAGERED REPLACEMENT POLICY
STRATEGIC REPLACEMENT DECISIONS
Kyongsun Kim
Permision is granted to Auburn University to make copies of this thesis at its discretion,
upon the request of individuals or institutions and at their expense.
The author reserves al publication rights.
________________________
Signature of Author
________________________
Date of Graduation
iv
VITA
Kyongsun Kim, the second daughter of JaeChon Kim and YoungJa Kim, was
born on March 4th, 1977 in Seoul, Republic of Korea. She entered the Korea Military
Academy in January, 1998, and earned a Bachelor of Science degre in Information
System Analysis in March, 2002. In August 2006, she entered Auburn University to
pursue a Master of Science degre in Industrial and Systems Engineering.
v
THESIS ABSTRACT
GROUP VERSUS STAGERED REPLACEMENT POLICY
STRATEGIC REPLACEMENT DECISIONS
Kyongsun Kim
Master of Science, August 9, 2008
(B.S.Statistical Information Analysis, Korea Military Academy, South Korea 2002)
102 typed pages
Directed by Chan S. Park
Replacement decisions are critical in most busineses, because asets are subject
to deterioration or obsolescence with usage and time. In addition, technological
improvement afects the replacement cycle of asets. In our paper, we focus on a flet
replacement problem with a singleunit. The main problems of flet replacement
decisions are first, when we should replace existing asets with new asets, and second,
how many asets to replace at once. To solve these problems, we introduce two policies
for flet replacement: group replacement and staggered replacement. To addres these
isues, we develop mathematical models and analyze results to find the preferable policy
under certain conditions.
vi
ACKNOWLEDGEMENTS
I would like to thank my advisor, Dr. Chan S. Park, for his constructive advice,
encouragement and complete support through this research, and Dr. Jorge Valenzuela and
Dr. Ming Liao for their kind consent to serve on my thesis commite. Thanks are given
to my parents and family for their support. Particularly, I would like to thank Mr. Hwan
Sic Le for his advice and discussions, and other faculty and graduate students who have
helped for this research.
vii
Computer software used: Microsoft Word and Excel 2003, 2007
Palisade @ Risk 5.0
vii
TABLE OF CONTENTS
LIST OF TABLES???????????????????????... xi
LIST OF FIGURES???????????????????????.. xii
CHAPTER 1. INTRODUCTION??????????????..???.. 1
1.1 BACKGROUND???????????????????? 1
1.2 PROBLEM STATEMENT???????????????? 2
1.3 RESEARCH METHODOLOGY?????????????.. 3
1.4 RESEARCH PLAN??????????????????? 3
CHAPTER 2. LITERATURE REVIEW???????????????.. 5
2.1 GENERAL REPLACEMENT?????????????.?.. 5
2.2 GROUP REPLACEMENT???????????????... 6
2.2.1 TAGE REPLACEMENT POLICY????????. 7
2.2.2 MFAILURE REPLACEMENT POLICY?????? 7
2.2.3 (M, T) REPLACEMENT POLICY???.?????. 8
2.3 STAGERED REPLACEMENT????????.?????. 9
2.4 SUMARY?????????????????????.. 9
CHAPTER 3. BASIC MODEL???????????????????. 11
3.1 ASUMPTION????????????????????.. 11
3.2 REPLACEMENT MODELS???????????????. 12
ix
3.2.1 GROUP REPLACEMENT MODEL???????? 13
3.2.2 STAGERED REPLACEMENT MODEL?????.. 17
3.3 ECONOMIC ANALYSIS????????????????.. 22
CHAPTER 4. MODEL UNDER TECHNOLOGICAL PROGRES????.. 31
4.1 CONSIDERING TECHNOLOGY IMPROVEMENT IN
REPLACEMENT DECISIONS???????????????..
31
4.2ASUMPTION????????????????????.. 32
4.3 REPLACEMENT MDELS???????????????? 32
4.3.1 GROUP REPLACEMENT MODEL???????? 34
4.3.2 STAGERED REPLACEMENT MODEL?????.. 38
4.4. ECONOMIC ANALYSIS????????????????. 43
4.4.1 AN ILUSTRATIVE CASE EXAMPLE??????. 43
4.4.2 ECONOMIC INTERPRETATION OF THE
NUMERICAL RESULTS??????????????..
44
4.4.3 SENSITIVITY ANALYSIS???????????.. 45
CHAPTER 5. MODEL UNDER ISK????????????????. 53
5.1 RISK SIMULATION PROCEDURES USING @RISK????.. 53
5.2 DEVELOPING A SIMULATION MODEL?????????.. 56
5.2.1 PURCHASE COST (P) AS A SINGLE RANDOM
VARIABLE????????????????????
56
5.3 MULTIPLE RANDOM VARIABLES???????????.. 64
5.3.1 CASE 1 AL RANDOM VARIABLES ARE
x
MUTUALY INDEPENDENT????????????. 64
5.3.2 CASE 2CONSIDERING CORELATION AMONG
RANDOM VARIABLES??????????????..
72
CHAPTER 6. SUMARY AND CONCLUSION???????????.. 77
REFERENCES?????????????????????????.. 80
APENDIX 1. OUTPUT DATA OF UNCERTAIN PARAMETER P??. 83
APENDIX 2. OUTPUT DATA OF AL UNCERTAIN PARAMETERS??. 86
xi
LIST OF TABLES
Table 3.1 Summary of example data.......................................23
Table 3.2The diference betwen the NPW of group and staggered model under vary N.
...................................................................25
Table 3.3 The diference betwen the NPW of group and staggered model under N=5..27
Table 4.1Sumary of example data........................................44
Table 4.2The summary of service life......................................44
Table 4.3 The diference betwen the PW of group and staggered model under variable N
...................................................................46
Table 4.4 The diference betwen the PW of group and staggered models under various a
...................................................................48
Table 5.1Sumary of PW Cost for the Group and Staggered Replacement Policies...59
Table 5.2Sumary of PW Cost Distribution with Random Variable of P (GS).....62
Table 5.3 Simulation Output Data and Summary Measures for Comparing Two Models.
...................................................................63
Table 5.4 ThrePoint Estimates for Key input Variables.......................65
Table 5.5 Beta Distribution Functions for Key input Variables...................66
Table 5.6 Summary of the PW Cost Distributions for Group and Staggered Replacement
Policies.............................................................68
Table 5.7 Summary of PW Diferential Cost Distributions (GS).................70
xii
Table 5.8 Summary of Simulation Output Data with Multiple Random Variables.....71
Table 5.9 Matrix of Correlation Coeficients with @RISK......................72
Table 5.10 Summary of the PW Cost Distribution Statistics.....................76
xii
LIST OF FIGURES
Figure 2.1 the summary of replacement research...............................6
Figure 3.1 A graphical representation of one N
th
staggered replacement policy......19
Figure 3.2 under N=5 (base case).........................................24
Figure 3.3under N=7...................................................26
Figure 3.4 under d=5%.................................................27
Figure 3.5 under d=10%................................................28
Figure 3.6 under 15%..................................................28
Figure 3.7under d=20%.................................................29
Figure 3.8 Sensitivity graph for the PW cost diferential betwen Group and Staggered
replacement policies...................................................30
Figure 4.1under N=3(under ongoing technological model)......................45
Figure 4.2 under N=5(under the ongoing technological model)...................47
Figure 4.3under N=7...................................................47
Figure 4.4 under a=85%................................................49
Figure 4.5 under a=90%................................................49
Figure 4.6 under a=95%................................................50
Figure 4.7 Sensitivity graph for the PW cost diferential betwen Group and Staggered
replacement policies...................................................51
xiv
Figure 4.8 Sensitivity graph (Cont?) for the PW cost diferential betwen Group and
Staggered replacement policies...........................................52
Figure 5.1Selecting a distribution function in @Risk..........................55
Figure 5.2Displaying the simulation result..................................56
Figure 5.3 The PW cost for the group replacement model as a function planning horizon
...................................................................58
Figure 5.4 The PW cost for staggered replacement model as a function of panning
horizon.............................................................58
Figure 5.5 The cumulative ascending graph for group replacement model...........60
Figure 5.6 The cumulative ascending graph for staggered replacement model........60
Figure 5.7 The trend of PW for the diference betwen two models...............61
Figure 5.8The trend of PW for group replacement model under al the beta distribution67
Figure 5.9The trend of PW for group replacement model under al the beta distribution68
Figure 5.10 The cumulative ascending graph for group replacement model..........69
Figure 5.11The cumulative ascending graph for staggered replacement model.......69
Figure 5.12 The trend of the diference for two models under al uncertain condition..70
Figure 5.13 Cross Plots of Simulated Dependent Random Deviates (a vs q).........73
Figure 5.14 Cross Plots of Simulated Dependent Random Deviates (q vs. s).........73
Figure 5.15 Cross Plots of Simulated Dependent Random Deviates (a vs. s).........74
Figure 5.16 PW Cost Distributions as a Function of Planning Horizon (Group Policy).74
Figure 5.17 PW Cost Distributions as a Function of Planning Horizon (Staggered Policy)
...................................................................75
Figure 5.18 PW Diferential Cost Distribution (G ? S).........................75
1
CHAPTER 1. INTRODUCTION
Replacement decisions are critical in most busineses, because asets are subject
to deterioration or obsolescence with usage and time. In addition, technological
improvement afects the replacement cycle of asets. In this research, we focus on a flet
replacement problem concerning a singleunit. The main problems of flet replacement
decisions are: 1) when we should replace existing asets with new asets, and 2) how
many asets should be replaced at one time. To solve these problems, we introduce two
types of flet replacement policies: Group replacement and Staggered replacement. To
addres these isues, we develop mathematical models and analyze the results to find the
preferable policy under certain conditions.
1.1. BACKGROUND
Replacement is inevitable in busines. Replacement costs consist of thre main
components: 1) the initial costs, 2) operating and maintenance costs, and 3) resale values.
As equipment ages, operating and maintenance costs gradualy increase, and resale
values gradualy decrease. The initial costs are also afected by technological
improvement. Therefore, at some point in time, the retention costs for old asets may
exced the costs of purchasing and operating new asets.
2
We focus on a flet replacement problem in this research. In practice, there are
two replacement strategies for this problem. One is Group replacement, which replaces
al asets at once during each service life cycle of asets. The second design is Staggered
replacement, which replaces an equal portion of the flet every year. We chose
replacement of the same numbers of asets every year instead of replacing a diferent
number of asets every two or thre years. Acording to Jones and Zydizk (1993), their
main result suggests that flet operators would want an equal number of asets in each
replacement group.
Industry under ongoing technological improvement makes new products which
may be cheaper and more eficient. When we are faced with replacing equipment which
has technological improvement over time, the important questions are ?should we change
al equipment at once or follow a Staggered policy?? The purpose of our research is to
suggest a decision tool for replacement decisions in specific cases.
1.2. PROBLEM STATEMENT
This research presents an analysis of the replacement decisions of a company that
has single asets to replace periodicaly in an infinite horizon period. Also, it considers
the technological progres of asets, which changes the costs of new investment and
operating and maintenance. Productivity is also afected by technological progres.
Acording to a recent survey that the research firm Gartner conducted with 177 large
busineses, the average life span of a PC is 3643 months. Traditionaly, many busineses
replaced their PCs in staggered, onethirdperyear increments over a threyear cycle.
More recently, large companies are replacing al their PCs at once rather than in
3
staggered cycles (Dunn, 2005). The main inspiration for CIOs to make this change is the
benefit, which includes reduced maintenance costs. Considering the limited budgets of
companies, we need to know exactly how much benefit is possible.
1.3. RESEARCH METHODOLOGY
Our research presents the procedure for finding the optimal replacement policy in
flet replacement. We use the net present value decision as a cost comparison approach,
because modern replacement theory is based on discounted cashflow. Besides, the
problem is solved in a costminimizing framework: we find the minimization of total
present worth costs.
To find the optimal replacement policy in a flet replacement problem, we
construct closed form mathematical models: the Group replacement policy and the
Staggered replacement policy in both the basic model and the model under technological
improvement. These models apply an exponential form of technological progres to show
ongoing technological progres. Furthermore, we simulate our models under ongoing
technological progres to ilustrate and analyze the uncertain situation using @Risk, a
risk analysis plugin for Microsoft Excel.
1.4. RESEARCH PLAN
This research proceds as follows. Section 2 describes the literature review of
general, group and staggered replacement. Section 3 constructs the basic mathematical
models of two policies: group and staggered replacement without considering
technological changes. We also examine numerical examples to demonstrate our models.
4
Section 4 develops mathematical models for each replacement policy under ongoing
technological progres. Here we analyze the models with the same numerical example as
the example in Section 3. Section 5 ilustrates the uncertain situation of these models,
which we intend to acount for using @Risk software. Section 6 summarizes and
presents conclusions, including contributions of the proposed research.
5
CHAPTER 2. LITERATURE REVIEW
When we place an aset in service, we need to replace the aset at some point in
the future. Obsolescence and deterioration are the two major reasons for considering the
replacement of an existing aset. The isue of when to replace an existing aset is one of
the critical operating decisions in busines. Consequently, many researchers have
investigated a variety isues related to aset replacement. However, our literature review
wil focus only on two types of replacement policies?Group replacement and Staggered
replacement, as our ultimate goal is to examine which replacement policy is more cost
efective.
2.1. GENERAL REPLACEMENT
James S. Taylor (1923) and Harold Hoteling (1925) developed a mathematical
theory of depreciation for an aset which loses value over time. Roos (1928) is one of the
early researchers who studied replacement decision problems in a systematic way for a
single machine by considering the cost of production and the changing market value of
the machine. Preinreich (1940) recognized the importance of depreciation in finding the
optimum economic life of a machine. ?Al rules of economic life are also rules of
depreciation.? Terborgh (1949), considered the father of modern replacement theory,
developed a simple and complete rule prescribing the time at which existing production
6
equipment should be replaced. His esential contribution is the integration of
obsolescence into the applied theory of replacement as detailed in Dynamic Equipment
Policy.
2.2. GROUP REPLACEMENT
When replacing identical asets placed in service, one policy to follow is to
replace al asets together at the end of their economic service life. This is known as a
group replacement policy. As outlined in Figure 2.1, group replacement policies are
clasified further into thre major clases acording to when units are replaced. The first
clas, the Tage policy, says that when the age of a unit reaches a prescribed point, units
are replaced periodicaly. In the second clas, MFailure, units are replaced when the
number of failure reaches a prescribed number, m. The third clas, (m, T), considers both
Tage and mfailure.
Figure 2.1 Summary of replacement research
Replacement
Group
Tage Policy
MFailure Policy
(M, T) Policy
Staggered
Equal number policy
Random number policy
Mathematical Theory
7
2.2.1. TAGE REPLACEMENT POLICY
Barlow and Hunter (1960) first introduced the periodic replacement policy with
minimal repair at failure, which takes a negligible amount of time. They further
considered two preventive maintenance policies: one for simple equipment which
operates continuously without failure, and one for complex systems which operate with
minimal repairs. In this model it is asumed that the failure rate of a unit or system is not
changed after repair.
Tahara and Nishida (1973) introduced a preventive maintenance policy that
considers repairable systems. The failure rate of systems in their models increases
because the system is not able to recover completely after repair, and the service life of
the system decreases after repair. Okumoto and Elsayed (1983) extended Barlow and
Hunter?s model, which is basicaly the optimal scheduled time for preventive
maintenance. They further provided an optimal group replacement policy of single units
with an exponential failure distribution during a given interval. Recently, Park and Yoo
(2004) considered the same replacement problem under minimal repair. Then they
compared thre types of replacement policies. First, al units are replaced periodicaly.
Second, the group replacement interval considers both repair and waiting times. Third,
the minimal repair for each unit is conducted during the repair interval. They
recommended the third policy to be most economical among the thre policies.
2.2.2. MFAILURE REPLACEMENT POLICY
Gertsbakh (1984) provided an optimal repair policy: repair is conducted when the
number of failed machines reaches some prescribed number. Gertsbakh asumed that a
8
group of machines has n independent but identical machines, and each machine has an
exponential lifetime.
Asaf and Shanthikumar (1987) proposed the group maintenance policy under
continuous and periodic inspections with stochastic failures. The idea is that they decided
to repair the failed machine after inspection. Asaf and Shanthikumar examined the
optimal repair policy: if the number of failed machines reaches a prescribed number n, m
machines are repaired (m < n).
Wilson and Benmerzouga (1990) extended Asaf and Shanthikumar?s optimal m
failure replacement policy. They asumed the failure times of n machines are
independent but identicaly distributed exponential random variables. They developed a
cost function to use in acordance with the behavior of the optimal policy.
More recently, Liu (2004) developed an mfailure group replacement policy for
M//N queuing systems which are unreliable with identicaly exponential failure times.
They formulated a matrixgeometric model to consider the steadystate situation.
2.2.3. (M, T) REPLACEMENT POLICY
Morimura (1970) introduced an (m, T) policy which combined two policies: m
failure replacement and Tage replacement. They considered the number of failed
machines and the operating time of the machines to find a minimum replacement cost. If
the number of failed machines reaches a prescribed number m before the Tage of a
machine, or the Tage comes before the mfailure for the machines, they are repaired.
Nakagawa (1983) considered counting the number of failed machines and
recording the age of machines over a fixed replacement time and then repairing the failed
9
machines if either the replacment age T or the number N of a predetermined number of
machines fail first.
More recently, Ritchken and Wilson (1990) considered the same problem with
two decision variables: a fixed time interval and a fixed number of failed units. If one of
the two variables occurs, al failed machines are replaced with new ones that perform
perfectly. They provided an algorithm to obtain the optimal (m, T) policy and
demonstrated it with a numerical example.
2.3. STAGERED REPLACEMENT
The term ?Staggered replacement? was first mentioned by Cook and Cohen
(1958). Although they did not outline any specific replacement policy, the purpose of the
Staggered replacement policy is to smooth out the required lump sum capital outlay over
time. The Staggered replacement policy is comonly practiced in many industrial
setings.
Jones and Zydiak (1993) formaly considered Staggered replacement by comparing two
prevalent replacement designs; one replaces an equal portion of a flet every year, and
the other replaces larger bunches les frequently in order to acount for the flet
management problem. In their paper, they defined that the second case is a Staggered
replacement policy. They concluded that the first policy is beter than the other.
2.4. SUMARY
Many researchers have been studied to find optimal replacement policies of the
group replacement models for singleunit systems. There are thre main types of group
10
replacement. The first is the age replacement policy, the second is the mfailure (failure
number) policy, and the third combines (m, T) policies. While extensive research has
been done for Group replacement policy, not much work has been done to determine the
efectivenes of a Staggered replacement strategy. Therefore, our research focus is to
compare the Group replacement policy with a Staggered replacement policy to find
which policy is more strategicaly costefective. We wil focus mainly on the Tage
Group replacement policy and the equal number Staggered replacement policy.
11
CHAPTER 3. BASIC MODEL
We wil first examine replacement problems without technological changes in
asets being considered for replacement. This basic model focuses on flet replacement
decisions about identical asets such as PCs, delivery trucks, buses and airplanes. Two
types of replacement policy wil be examined?Group replacement and Staggered
replacement. The Group replacement policy cals for replacing al asets once at the end
of the economic service life of the each of asets. On the other hand, the Staggered
replacement policy recommends that busineses replace a predetermined number of
asets during a specified time interval (possibly every year). In this chapter we wil
present mathematical models for each replacement policy and give numerical examples
to demonstrate how the models work in a specific replacement environment. We wil also
interpret the results to determine which policy is more economicaly preferable.
3.1. ASUMPTION
In order to decide whether to replace existing asets, we asumed the following
factors: First, we chose the infinite planning horizon for a corporation whose busines
requires the same type of asets for an indefinite period. Second, we used the net present
value of the total cost for the entire planning horizon as a decision criterion to compare
the results betwen the Group and Staggered replacement models. Third, we
12
considered a replacement policy under a stable economy, meaning that the aset prices
and operating and maintenance cost would remain constant in the absence of inflation.
The concept of an infinite sequence of replacements can be generalized to the situation in
which the life of an aset is a decision variable. A common example of this type of
problem is deciding on the replacement interval for an automobile.
3.2. REPLACEMENT MODELS
Thre types of cash flows are considered in developing a basic replacement
model: First is the sequence of aset purchases over the planning horizon (
1
PW')i).
Second is the sequence of salvage values for the asets purchased at the time of each
replacement cycle (
2
PW')i). Third is the sequence of cash flows related to the operating
and maintenance costs of the asets over the entire planning horizon (
3
P')i). Since we
are dealing with cash flows over an indefinite period, these cash flows must be
discounted at an inflationfre interest rate. The total present cost of a typical replacement
policy is then simply the sum of these thre present values (
123
PW')(')P(')iii+).
We wil use the folowing set of notations in describing the replacement models:
P = purchase price of a new aset without volume discount at time 0, the cost per unit
multiplied by number of asets
P
n
= purchase price of the aset at time n
d = volume discount multiplier for purchase cost, where d < 1
i? = inflationfre (real) interest rate
13
b = multiplier for endofyear1 salvage value, where b < 1
c = annual multiplier for subsequentyear salvage values, c < 1
A = firstyear O&M costs for asets purchased at time 0
p = annual multiplier for O&M costs for given asets, where p > 1
3.2.1. GROUP REPLACEMENT MODEL
In this section, we wil develop the group replacement model where al asets are
replaced in a group when they reach the end of their economic service life, N. One of the
advantages of the group replacement policy is to obtain some form of volume discount
when purchasing the new asets. The degre of volume discount depends on the nature of
asets, but these savings must be considered in the model.
The Purchase Cost
Suppose we begin in year 0. Asets cost P
0
at time 0; the cost includes discounts
for volume of purchased asets as follows:
0
(1)Pd=! (1)
Under the inflationfre environment with no technological improvement in future asets,
we can asume the purchase cost at times N, 2N, ?, kN wil be the same as the initial
purchase cost at time 0.
20kNN
PP= (2)
Here we also further asume that the volume discount for future replacements would
remain the same. Then, we determine
1
W(')i by discounting the initial cost cash flow
streams when asets are purchased every Nyear as follows:
14
2
10
0
2
0
0
PW(')
(')(1')
''
1
(')
1
1' '
NN
N
P
i
ii
P
i
ii
=++
!"
#$
=
%
+
&?
!"
()
<
*+#$
,
(3)
Salvage Values
While we retain the aset for N years, the value of the aset wil continue to
decrease over the holding period. Let?s consider a sequence of salvage values. If we sel
the aset purchased at time 0 after one year, we would receive
10
SbP= (4)
If we sel the aset after two years of use, we would receive
20
c (5)
Here, the parameter c, which les than 1, represents the scaling factor related to
calculating subsequentyear salvage values. The main logic for introducing a new
multiplier (c) is that most asets lose a greater portion of their values during the first year
of ownership, implyingbc!. This asumption is also considered by Park and Gunter
(1990). If we consider a sequence of aset retirements for subsequent replacement cycles,
the salvage value of the kth replacement cycle can be expresed by
11
0
NN
kk
SbP
!!
= (6)
15
The contribution to the total PW of costs from a sequence of aset replacements every N
year is
11
2
2
111
000
2
1
0
1
0
P()
')(')
(')(')(')
(')
'
(')()
'
NN
NNN
N
NN
N
bc
i
ii
Pbc
iii
i
bci
i
P
i
!!
!!!
!
=+
"#
$%
=
+
&?
"#+
=
$%
&?
(7)
O&M Cost
Firstyear operating and maintenance costs (O&M costs) often are considered to
follow a negative exponential curve based on the O&M costs of the current year?s model.
Then, the O&M costs of future models are usualy asumed to follow the same patern as
that of the O&M costs of the curent year?s model. With this asumption, the expresion
for the O&M costs is more tedious, but it follows along similar lines as the previous two
cash flow sequences. Recal that each replacement cycle contributes Nyear O&M cost
terms, with each succesive year showing a higher cost than that of the previous year.
Once again, with no technological improvement in future asets, the O&M cost series in
the first replacement cycle wil repeat in future replacement cycles. The first O&M cost
term in each cycle is
1
A= (8)
The second O&M cost term wil be higher than the first cost by
16
2
Ap= (9)
The O&M cost term in N
th
period wil be
1!
(10)
As mentioned previously, the O&M cost terms in the second replacement cycle wil be
the same as those during the first cycle.
1
2
1
2
N
N
N
A
p
+
!
=
!
(11)
As this O&M cost series repeats for subsequent replacement cycles, the closed form
expresion for the PW of the O&M costs is
1 1
3
2 1
1
1
()
')(')(')(')(')
'
'
(')()
'
NN
N
k
N
kN
k
N
k
ApApA
i
iiiii
i
pi
i
i
!!
+
!
=
=
+ +
+
"#
$%
&?
*+$%%
,
*++
!
()
.
(12)
The Total PW Cost Function for the Group Replacement Policy
The total PW cost function for the group replacement policy can be summarized
as follows:
Group123
11
0
0
(')= (') (')P(')
(')
''(')1
NNkNN
k
iiii
bcpi
PA
iii
!!
=
+
"#$%$%+
&?&?
* ,,
.
(13)
17
Note that the second PW term needs to be subtracted from the total cost function, as these
are salvage values, which reduce the total cost of the replacement cycles.
3.2.2. STAGERED REPLACEMENT MODEL
While a Group replacement policy replaces al asets once, a Staggered
replacement policy replaces an equal number of asets during the economic service life of
asets. As with the Group replacement policy model, we have thre types of cash flow
streams, and the modeling scheme is quite similar to the Group replacement model.
The Initial Cost Stream
With the Staggered replacement policy, we have many diferent ways of
staggering the replacement asets. Staggering options themselves lead to another
optimization problem, that is, what the best way to stagger the replacement asets is.
However, as shown in Jones and Zydiak (1993), we wil asume that one N
th
of asets is
purchased every year over the economic service life. We also can consider some form of
volume discount for the smaler scale of purchases, even though the discount may not be
as high as with group replacement; if we asume a uniform rate of volume discount, the
periodic purchase cost can be expresed as follows:
()
0
1Pd=! (14)
With one N
th
replacement each period, the initial cost stream wil be
21k
P
N
!"
#
$%
&?
(15)
The closed form expresion for the PW cost of purchase streams is
18
1211
10 0
2
1 1
0 001
PW(')
(')'(')'
'
(')(')'
(')
PP
i
iiii
i
i i
i
=++=+
!"
#$
!"%&
=
?(
#$
*+,

+
*
(16)
Certainly, if the volume discount itself is a function of volume, then we need to adjust the
scaling factor 1
N
!"
#
$%
&?
as
N
!"
$%
&?
. However, for the sake of simplicity, we wil consider
a uniform discount.
Salvage Values
Unlike the group replacement model, we need to estimate the salvage value of the
aset as a function of the aset age until it reaches the end of its economics service life.
The diference betwen the resale price of the asets under Group and Staggered models
wil be the same after the N
th
year.
If we sel the one N
th
aset purchased at time 0 after oneyear, we could receive
0
1
P
Sb
!"
=
#$
%&
(16)
Then in the second year, we also sel the one N
th
aset purchased at time 0.
0
2
Sb
N
!"
=
#$
%&
(17)
In the third year,
20
3
P
Sbc
!"
=
#$
%&
19
When we get to the N
th
year, the aset group purchased at time 0 has been completely
replaced.
10N
P
Sbc
!
"#
=
$%
&?
(18)
After the N
th
year, the salvage value of the Staggered model wil be the same as the
salvage value of the N+1th year.
1
1
N
SbcP
!
"
= (19)
Time 0 1 2 3 4  ?
1/3
(2 year)
(2 year) (2 year)
2/3
(1 year) 1/3
(1 year)
(1 year) (1 year)
Rate of
Quantity
1
(New)
1/3
(New)
1/3
(New)
(New) (New)
First aset Second aset Third aset
Figure 3.1 A graphical representation of one N
th
Staggered replacement policy
For example, suppose that the service life is 3 years. Figure 3.1 ilustrates proces of
staggering the replacement asets. New asets wil be placed in service in year 0. Then, at
the end of year 1, one third of the asets placed in service in year 0 wil be replaced. At
year 2, another one third of the asets placed in service at period 0 wil be replaced. Then,
in period 2, we wil have thre types of asets: one third of the old asets placed in service
at period 0, onethird of new asets placed in service at period 1, and onethird of new
asets purchased in period 2. Finaly at period 3, the old asets placed in service in period
0 wil be completely gone. The composition of the asets includes (1) onethird of the
aset group purchased at period 1, (2) onethird of the aset group purchased at period 2,
20
and (3) onethird of new aset group purchased at period 3. After year 3, the aset
composition wil be the same as that of year 3?that is 1/3 of asets are two years old, 1/3
of the asets are one year old, and 1/3 of the asets are brand new.
In terms of the sequence of salvage value, Figure 3.1 can be translated as follows:
0
3
P
b
!"
#$
%&
0
c
!"
#$
%&
20
3
P
b
!"
#$
%&
2
1
b
2
1
P
=
?
2
3
1
n
bcP
!
=
0 1 2 3 4 5 ? n
The PW of the salvage value stream is
211
0000
2
123
1
0
1
(')
(')(')(')(')(')
.
'''
NN
NkN
bcPbbcP
i
iiiii
iii
!!
+
!!
+
=
+
+
"#
$%
&?
(20)
O&M Cost Streams
The O&M cost stream is a bit more involved. Refering to Figure 3.1, the O&M
costs in each year must acount for the composition of asets placed in service at
diferent points in time. For example, with N = 3 years, the O&M costs at period 1
consist of only asets purchased in year 0:
1
A=
The O&M costs at period 2 consist of two diferent asets: twothirds is from the first
aset group, and onethird is from the second aset group.
21
2
1
3
ApA
!"
=+
#$
%&
Then, the O&M costs at period 3 wil be
2
3
11
3
p
!"!"
=+
#$#$
%&%&
After the N
th
period, say year 4 in our example, the O&M costs would take the following
expresion:
2
4
11
33
ApA
!"!"
=+
#$#$
%&%&
Then, the O&M costs beyond the N period would be exactly the same as those of Nth
period, because the composition of the aset groups is exactly same?the one year old
aset group, the two year old aset group, and brand new aset group. Therefore, we can
generalize the O&M cost at time N as
()
123
1
1
.
N
N
N k
ApAppA
N
!!!
=
"#"#"#"#
=++
$%$%$%$%
?&?&?&?
&
(
(21)
And
1
.
NN
A
+!+
= (22)
The equivalent total PW cost of the O&M cash flows is as follows:
22
2
3
12
11
1
PW(')
')(')(')
(')(')
(')
NN
kk
AA
pp
A N
i
ii i
ii
A
p
A
i
!!
==
+
"#"##"#
++
$%$%%$%
&?&??&?
=
+
"#"#
$%$%
&?&?
"#"#
$%$
&?&
=+
1
2
'
(')(')
N
k
p
i
ii
!
=
"#
%$%
+
*
?&?
,
.
(23)
The Total PW Cost for the Staggered Replacement Policy
We obtain the total PW cost expresion for the Staggered replacement policy
without technology improvement as follows:
11
0
Stagerd01
1
2
'
P(')
'(')(')
'
(')(')(')
NkN
Nk
bbcPi
i
iii
A
pp
Ai
iii
!!
+
=
!
=
"#$%
=+
&?
)
+
* ,
"#"#
&?&?
$%
**
()
,
.
(24)
We wil use Equations (13) and (24) to compare the efectivenes of a given replacement
policy by minimizing the equivalent total PW cost of entire replacement cycles.
3.3. ECONOMIC ANALYSIS
To compare the efectivenes of the two diferent replacement policies, we wil
consider a case example and give an economic interpretation of the results. We wil
further conduct a series of sensitivity analyses for the key input parameters.
23
3.3.1. An Ilustrative Case Example
The KCompany is considering replacing their old copy machines with new ones.
The unit price of a new copy machine is $500, and they have 100 machines. If they buy
100 machines once, they can get 10% discount, and for each 10 machines the volume
discount is 1%. The value of each machine decreases to 60% of the original purchase cost
after using it for 1 year. Then, the value wil decease 20% each subsequent year. O&M
costs in the first year are $50 per unit, and O&M costs wil increase 25% each year. Each
copy machine has an economic service life of five years. The interest rate is 10%. A
summary of key input parameters follows:
Parameter Value
P $50,000
d 10%
i? 10%
b 60%
c 80%
A $5,000
p 125%
Table 3.1 Summary of example data
It is asumed that the KCompany has enough money to replace al the asets at once if
the group replacement policy is considered to be more economical. Here, al analysis is
done on a beforetax basis.
24
3.3.2. Economic Interpretation of the Numerical Results
With the set of parameters asumed in the previous section, the Group
replacement policy appeared more cost efective when compared with the Staggered
replacement policy. Figure 3.2 ilustrates the incremental cost of the Staggered
replacement policy over the Group replacement policy. Note that the Group replacement
policy takes a stairshaped curve because of the chunk of costs occurs at the replacement
period. In terms of the PW cost of the entire cash flow stream, Group replacement
results in $176,318, while Staggered replacement costs $189,030 with a planning horizon
of 41 years and an economic service life of the asets at N = 5 years. The incremental cost
of choosing the Staggered replacement policy over Group replacement is $12,712 in
present value. The cost diferentials wil vary as a function of planning horizon, but the
Group replacement policy wil be more cost efective for a widerange of planning
horizons.
Basic N=5
$0
$50,000
$100,000
$150,000
$200,000
$250,000
1 11 21 31 41
year
P
W
G
S
Figure 3.2 under N=5 (base case)
25
Certainly, we should not conclude that the Staggered replacement policy is always
inferior to the Group replacement policy. To answer this question, we need to conduct a
series of ?what if? analyses.
3.3.3. Sensitivity analysis
To determine what conditions make Group replacement cost efective, we wil perform a
series of sensitivity analyses on the key input variables. The thre key input parameters
considered are (1) economic service life, which dictates the replacement intervals for the
Group replacement policy, (2) the amount of volume discount with Group replacement as
wel as staggered replacement, and (3) the discount rate used in comparing the two
replacement policies.
Replacement Interval N
Idealy the best replacement interval is the economic service life of the aset.
However, as we vary the replacement interval from N = 3 years to N = 10 years, we
obtain the present worth cost of each replacement policy as follows:
Type N=3 N=5 N=7 N=10
Diference (GS) $11,504 $12,712 $4,721 $8,889
Table 3.2 The Diference betwen the NPW of Group and Staggered models under
varying N.
As expected, with the replacement interval set at the economic service life of the aset (N
= 5 years, which is our base case), the Group replacement policy is more cost efective.
26
As we deviate from this base further out, the cost diferential gap betwen the two
policies narows.
If we further examine the PW cost diferential with N = 7 (that is, we keep the asets
two more years beyond their economic service life), we observe that the Staggered policy
turned out to be more cost efective, as depicted in Figure 3.3.
Basic N=7
$0
$50,000
$100,000
$150,000
$200,000
$250,000
$300,000
1 11 21 31 41
year
P
W
G
S
Figure 3.3 under N=7
This is simply because the replacement cost for the Group model increases if the asets
are replaced at an interval other than the economic service life, which minimizes the total
equivalent cost. This also clearly ilustrates that if we go with the Group replacement
policy, the asets must be replaced at their economic service life.
27
Volume Discount (d)
The amount of volume discount available wil be an important parameter, as the
volume discount reduces the capital cost for the replacement chains for both Group and
Staggered policies. The Group policy wil enjoy a higher volume discount as compared
with the Staggered policy where the purchased amount is spread over the Nperiod. As
we vary the volume discount from 5% to 20%, the preference for the group policy is
furthered evidenced.
N=5(d) 5% 10% 15% 20%
Diference(GS) $8,363 $12,712 $17,061 $21,410
Table 3.3 The diference betwen the NPW of Group and Staggered models under N=5.
Figures 3.4  3.7 ilustrate how the total present worth cost functions acording to the
Group and Staggered policies over a wide planning horizon. As expected, the gap
betwen the two policies widens as we increase the volume discount, which favors the
Group policy.
Basic N=5(d=5%)
$0
$50,000
$100,000
$150,000
$200,000
$250,000
1 11 21 31 41
year
P
W
G
S
Figure 3.4 under d=5%
28
Basic N=5(d=10%)
$0
$50,000
$100,000
$150,000
$200,000
$250,000
1 11 21 31 41
year
P
W
G
S
Figure 3.5 under d=10%
Basic N=5(d=15%)
$0
$50,000
$100,000
$150,000
$200,000
$250,000
1 11 21 31 41
year
P
W G
S
Figure 3.6 under 15%
29
Basic N=5(d=20%)
$0
$50,000
$100,000
$150,000
$200,000
$250,000
1 11 21 31 41
year
P
W
G
S
Figure 3.7 under d=20%
Obtaining an Overall Sensitivity Graphs
Figure 3.8 shows the sensitivity graphs for seven of the key input variables. The
basecase PW cost diferential (Group ? Staggered) is plotted on the ordinate of the graph
at the value of 0 (0% deviation) on the abscisa. Next, the value of volume discount is
reduced to 80% of its basecase value, and the PW cost diferential is recomputed, with
al other variables held at their basecase value. We repeat the proces by either
decreasing or increasing the relative deviation from the basecase. The lines for the
variable interest rate (i?), purchase price (P), and other parameters such as b, c, A, and q
are obtained in a similar manner.
30
Figure 3.8 Sensitivity graph for the PW cost diferential betwen Group and Staggered
replacement policies
In Figure 3.8, we se that the group replacement policy is quite cost efective for the
range of values examined. In particular, the cost diferential is (1) most sensitive to
change in purchase price (P) and the firstyear?s loss of market value of the aset (b), (2)
fairly sensitive to changes in the volume discount (d) and the scaling factor of the market
value of the aset (c) , and (3) relatively insensitive to changes in the interest rate (i?),
initial O&M cost (A) and the scaling factor of the future O&M cost (p).
31
CHAPTER 4. MODEL UNDER TECHNOLOGICAL PROGRES
In Chapter 3, we presented two types of replacement models (Group and
Staggered) without considering any technological changes in future replacement asets.
However, technology improvement is one of the critical factors that can change the
purchase prices and operating and maintenance costs of future asets in years to come. In
this chapter, we wil develop mathematical models for each replacement policy and
examine which replacement policy is more cost efective when we experience
technological progres in future replacement asets. To compare the results with the
basic models, we wil use the same numerical example as used in Chapter 3.
4.1. CONSIDERING TECHNOLOGY IMPROVEMENT IN REPLACEMENT
DECISIONS
Technological improvement in future asets is one of the main reasons for
replacing existing asets, since the future asets should be more eficient in many aspects:
improved eficiency (productivity), reduced operating and maintenance costs, and lower
purchase costs. However, it is rather dificult to predict the trend of eficiency and the
price of asets over several years in any precise fashion. The problem of replacement
under technological progres has been studied by many researchers. Grinyer (1973) and
32
Bethuye (1998) examined the influence of technological progres and concluded that
technology may lead to an increase of the economic service lives of asets in some cases.
In contrast, Howe and McCabe (1983) and Rogers and Hartman (2005) explained that
technological change makes the replacement cycle of asets shorter than in a stationary
situation. In practical models of replacement under technological progres, Terborgh
(1949), a previous researcher, applied a linear form for technological change, but Grinyer
(1973) recommended a geometric form after comparing the linear form with the
geometric form.
4.2. ASUMPTION
In our basic case, we asume thre factors: (1) an infinite planning horizon, (2) the
PW of the total cost as a decision criterion to compare both models, and (3) the aset
prices and operating and maintenance costs remain constant in the absence of inflation. In
order to compare with the basic models, we wil further asume thre additional factors.
First, the aset price keeps decreasing (or remains relatively stable) due to technological
progres. Second, the operating and maintenance costs for the future replacement asets
wil continue to decrease compared with those asets purchased in the previous
replacement cycles, but they increase each year during the holding period. Third, the
productivity of asets decreases every year during the holding period.
4.3. REPLACEMENT MODELS
As with the basic model, we wil consider thre types of cash flows: first is the
sequence of aset purchases (
1
PW')). Second is the sequence of salvage values
33
(
2
PW')i). Third is the sequence of the operating and maintenance costs (
3
PW')i). The
total present worth cost, which is obtained by discounting the combined cash flows at an
inflationfre interest rate, is then simply the sum of these thre present values
(
123
P')(')P(')+). We wil use the same notations as in the basic models,
while introducing thre additional variables (a, q, s), in developing the replacement
models:
P = purchase price of a new aset without volume discount at time 0, the cost per unit
multiplied by number of asets
P
n
= purchase price of the aset at time n
d = volume discount multiplier for purchase cost, where d < 1
i? = inflationfre (real) interest rate
b = multiplier for endofyear1 salvage value, where b < 1
c = annual multiplier for subsequentyear salvage values, where c < 1
A = firstyear O&M costs for asets purchased at time 0
p = annual multiplier for O&M costs for given asets, where p > 1
a = annual multiplier to calculate purchase price, where a < 1
q = annual multiplier to calculate firstyear O&M costs for an aset purchased after
time 0
s = productivity loss multiplier for O&M costs, where s < 1
34
4.3.1. GROUP REPLACEMENT MODEL
In this section, we wil first develop a group replacement model when asets are
replaced in a group when they reach the end of their economic service life, N. In general,
technological progres leads to a reduction in the purchase costs and operating and
maintenance costs of future asets, even though the operating and maintenance costs
increase as asets age during the replacement cycle. Further, as new asets tend to have a
higher productivity rate, keeping existing asets longer implies a productivity loss. We
wil consider al these factors in developing the group replacement model.
The Purchase Cost
Suppose that the firm purchases brand new asets at period 0, meaning that there
are no existing asets to consider at time 0. Let?s asume that asets cost P
0
at time 0:
0
(1)Pd=! (1)
Since the purchase cost of subsequent asets decreases over time, the purchase cost in
year one is
10
Pa= (2)
Then, if we make a new purchase when the aset placed in service in year 0 reaches its
economic service life of N years, the aset costs would be
0
P= (3)
In group replacement, we purchase asets every Nyear, therefore
0
kN
(4)
35
The contribution to the total PW of costs from a sequence of aset replacements every N 
year is
2
10
00
2
00
(')
(')(1')
''
1(1')
'
NN
N
N
i
ii
aP
ii
i
a
a
i
=++
!"
#$
!"+
==
#$
%
&?
%
*+
,
()
(5)
Salvage Values
A sequence of salvage values over an aset?s economic service life is the same as
in the basic Group replacement model.
10
2
1
0
N
SbP
c
!
=
(6)
However, as the purchase cost decreases in the second replacement cycle, the sequence of
salvage values during the second replacement cycle also decreases in the following
fashion:
10
1
2 0
N
N
SbaP
ca
+
!!
=
(7)
We can obtain the closed form expresion for the PW of the salvage values over the
infinite planning horizon as follows:
36
11
200
2
2
1
0
1
0
1
0
PW(')
')(')(')(')
(')
'
()
(')'
'
NN
NN
NN
NN
SbcPa
i
iiii
bcP
i
a
i
ii
bcP
i
!!
!
!
=+=+
"#
$%
&?
+
*
,
"#
=
$%
+a
(8)
O&M Cost
The trend of operating and maintenance costs for the asets purchased in year 0 is
quite similar to the basic Group replacement model. In the basic Group replacement
model, the operating and maintenance cost follows a negative exponential curve over its
service life of the asets. With technology improvement, we need to consider another
factor productivity loss. Since brand new asets tend to have a higher productivity rate
(they produce more with the same amount of operating hours), we wil experience some
sort of productivity loss as we delay replacing the old asets. This productivity loss needs
to be captured in terms of operating cost as wel. In other words, if we retain the asets
longer, the O&M costs increase on two fronts: requiring more frequent maintenance, and
increasing productivity loss due to aging asets. Recal that each replacement cycle
contributes Nyear O&M cost terms, with each year showing a higher cost than that of the
previous year because of aging asets. To reflect the two diferent sources for acounting
for O&M costs, we wil introduce an additional factor, productivity loss (s), in our Group
replacement model.
37
The first O&M cost term during the first replacement cycle is
1
A= (9)
The second O&M cost term wil be higher than the first cost by
2
()ps+ (10)
The O&M cost term in N
th
period wil be
1
()As
!
= (11)
Now we enter the second replacement cycle with brand new asets which wil have les
O&M costs compared with the asets placed in service during the first cycle. By
introducing a new annual multiplier q, which is les than 1, the O&M cost term at time
N+1 wil be
1
N
Aq
+
= (12)
The sequence of O&M costs during the second replacement cycle is
1
2
1
2
()
N
N
pA
Asq
+
!
=
(13)
As this O&M cost series repeats for subsequent replacement cycles, then we determine
3
PW(')ias follows:
1212
3
1 1
12 12
1
'
(')(')(')(')(')
'' '' '
()'
'
NN
N
k
N
k
AAA
i
iiiii
pspsqpsq
iiiii
si
A
q
i
+
! !
+
=
=+ +
"#
$%
&?
)
+
*+
,
!
.
()
1
1
(')
'
k
NN
k
s
i
q
i
!
=
$%
$%
*+
*+
(14)
38
The Total PW Cost Function for the Group Replacement Policy
The total PW cost function for the Group replacement policy with technology
improvement can be summarized as follows:
Group123
11
0
0
1
P(')=(')'(')
()(')
' ' '
NN kNN
k
i
bcPpsi
A
iaiaiq
!!
=
+
"#$%$%+
&?&?
* ,,
.
(15)
Recal that the second PW term needs to be subtracted from the total cost function, as
these are salvage values which reduce the total cost of the replacement cycles.
4.3.2. STAGERED REPLACEMENT MODEL
As we mentioned in the basic model, the Staggered replacement policy cals for
the replacement of an equal number of asets during the economic service life of the
asets. This model also has thre types of cash flow streams, and we wil follow the same
modeling scheme as in the Group replacement model.
The Initial Cost Streams
Although the Staggered replacement policy replaces an equal number of asets in
each period, the amount of asets purchased in year 0 wil be the same as with the Group
replacement policy. That is, we start with the same number of asets. The first purchase
cost can be expresed as follows:
()
0
1Pd=! (16)
With one N
th
replacement in each period, the initial purchase cost at time 1 wil be
39
1
daP
N
!"
=#
$%
&?
(17)
The sequence of the initial purchase cost stream wil be
2
21
3
31
1
(2)
k
d
P
aP
a
N
Pk
!
"#
=!
$%
&?
=
(18)
The closed form expresion for the PW cost of the initial purchase cost stream is
1211
10 0
2
1 1
0 0
(')
(')(')(')(')
(')(')
'
P
i
iiii
P
ai ia
i
++=+
!"
#$
=
%&?
%
()
+
*
,
!
(19)
Salvage Values
As we expres the salvage value as a function of the initial purchase cost, the
salvage value would be the same after the N
th
year in the basic model. Although the
scheme of salvage values under Staggered replacement is quite similar to the Group
replacement model, unlike in the basic model, the salvage values considering
technological change are smaler than they are under the basic model because the initial
purchase cost continues to decrease under ongoing technological progres. As we wil
se, the sequence of salvage values during the first replacement cycle is the same as the
basic Staggered replacement model in Chapter 3.
40
0
1
0
2
0
3
10N
P
Sb
N
c
Sb
P
c
!
"#
=
$%
&?
"#
=
$%
&?
"#
$%
&?
!
(20)
After the N
th
year, we start replacing the asets purchased during the first cycle. The
salvage value stream, after the N
th
year, wil be
1
1
1
22
12
33
1
()
N
N
Nkk
SbcP
a
Sa
!
+
!
+!
=
="
(21)
The PW of the salvage value stream is
111
0002
2
12
11
2
0
(')
')(')(')(')(')
('' '''
NNN
NN
k
bPcbcPbcP
i
Niiiii
a
iiiii
bPc
N
!!!
++
=++
=
11
11
0
(')(')
'
(')(')')
N
Nk
bP
aii
i
c
ii
!!
+
!!
=
"#
$%
&?
(
*+
,
.
(22)
41
O&M Cost Streams
Refering to Figure 3.1, the composition of asets in each year is composed of
several groups of asets which were purchased at diferent points in time. This implies
that each group of asets within the same period has diferent O&M costs. Therefore, we
need to consider al these variations of the O&M costs in each year. Although the O&M
cost streams are similar to those of the basic model, the scale of the O&M costs can
change due to the productivity loss factor that we have mentioned earlier. For example,
with N = 3 years, the O&M costs at period 1 consist of only asets purchased in year 0:
1
A= (23)
The O&M costs at period 2 consist of two diferent asets: twothirds is from the first
aset group, and onethird is from the second aset group.
2
1
()
33
ApqA
!"!"
=+
#$#$
%&%&
(24)
The O&M costs at period 3 wil be
22
3
111
()()
33
pspsq
!"!"!"
=++
#$#$#$
%&%&%&
(25)
After the N
th
period, year 4 in our example, the O&M costs would take the following
expresion:
212
4
3
1
()()
33
ApsqpsqA
!"!"!"
=++
#$#$#$
%&%&%&
(26)
The general form of the O&M cost stream is then as follows:
42
2
1
1
1
()
()
N
k
k
k
ApsAq
s
Aq
!
=
+!
"#"#
!+
$%$%
&?&?
"#
$%
*+
&?
,

(27)
The PW cost expresion for the entire O&M cost stream over the planning horizon is
obtained as follows:
12
3
1
12
12
1
P(')
')(')(1')(')
(')(')(')
(')
(')(')(')'
NN
N
N
AqA
i
iiii
q
iii
i
AA
iiiia
+
!
=+
+
"#
$%
&?
=+
+
(28)
The Total PW Cost of the Staggered Replacement Policy
We obtain the total PW cost expresion for the Staggered replacement policy
under technology improvement as follows:
11
01
Stagerd0
12
1
P(')=
(')(')(')')
(')')''
NkN
N
bbcPc
i
iiiia
AA
ii a
!!
=
!
++
+
"
(29)
Since we have developed the total PW cost expresions for both Group and Staggered
models considering technology improvement in future replacement asets, we wil
examine the efectivenes of each policy with the ilustrating case example.
43
4.4. ECONOMIC ANALYSIS
In the basic model, we compared the results betwen the Group and Staggered
replacement policies through a case example. In this section, using the same case
example, we wil follow a similar scheme. Further, we wil compare the results for both
replacement models under the basic replacement model and after considering
technological progres.
4.4.1. AN ILUSTRATIVE CASE EXAMPLE
Recal the case example in chapter 3. The owner of KCompany gets some
information about technological improvement in the copy machine market. The price and
operating and maintenance costs of the copy machine wil decrease 10% each year.
Further, the speed of the new machine wil increase 5% each year. The parameter a
represents the annual multiplier for the purchase cost, and parameters q and s represent
the multipliers for operating and maintenance cost and productivity loss respectively. The
summary of the case example is as follows:
Parameter Value
P $50,000
d 10%
i? 10%
b 60%
c 80%
Previous
information
44
A $5,000
p 125%
a 90%
q 90%
s 5%
Additional
information
Table 4.1 Summary of example parameters and values.
4.4.2. ECONOMIC INTERPRETATION OF THE NUMERICAL RESULTS
Recal that we considered additional parameters to reflect technological progres.
As we mentioned, technological improvement can afect the economic service life of the
aset. Therefore, we should check whether the economic service life of asets is stil N=5
under ongoing technological progres. Table 4.1, the PW cost of Group replacement
under ongoing technological progres, indicates N=3 is the economic service life in this
case.
Service life (N) The PW cost of group replacement
2 $112,125
3 $106,752
4 $107,500
5 $111,736
7 $111,963
Table 4.2 The summary of service life
Figure 4.1 ilustrates the PW cost of the Group and Staggered replacement policies when
the economic service life is N=3. Figure 4.1 indicates that the incremental cost of the
45
Staggered replacement policy is greater than the Group replacement policy along the
entire cash flow stream. The PW cost of Group replacement is $108,346, while Staggered
replacement costs $114,102 with a planning horizon of 41 years. The incremental cost of
choosing Staggered replacement is $5,756 in present value.
Figure 4.1 under N=3(under ongoing technological progres model)
When we adjust the service life to acount for replacement under technological progres,
Group replacement is more cost efective when compared with Staggered replacement.
To verify, we wil conduct sensitivity analyses.
4.4.3. SENSITIVITY ANALYSIS
We wil perform a series of sensitivity analyses to se the results under diferent
conditions, as we did in chapter 3. The procedure for this sensitivity analysis is the same
as it is for the basic model. The key input parameters are as follows: (1) service life, (2)
46
the annual multiplier for the purchase cost (a), and (3) overal sensitivity graphs of the
diference betwen Group and Staggered replacement models.
Replacement Interval, N
For this replacement problem under ongoing technological change, we found that
the economic service life is N=3 years. However, we observe the diference betwen two
PW costs when the replacement intervals change from N=2 years to N=7 years in table
4.2.
Type N=2 N=3 N=5 N=7
Diference (GS) $14,116 $5,756 $2,131 $6,438
Table 4.3 The diference betwen the PW of Group and Staggered model under N change
In this case, the Group replacement policy stil has slightly les total cost than the
Staggered replacement policy. Here, let?s consider N=5?that is, we keep the asets two
more years beyond its economic service life. Figure 4.2 ilustrates that the total cost of
the Staggered replacement policy is higher than the Group replacement policy during
most periods when N=5 years. With a planning horizon of 41 years and an economic
service life for asets of 5 years, the Group replacement policy costs $116,528, while the
Staggered replacement policy costs $118,658. The cost diference of both of the policies
is $2,131 in present value.
47
Figure 4.2 under N=5(under the ongoing technological model)
However, the Staggered replacement policy appears to be more cost efective when
compared with the Group replacement policy betwen the fifth and seventh year. That
means that if Kcompany keeps the asets from five to seven years, Staggered
replacement is more cost efective than Group replacement. Figure 4.3 ilustrates that
Staggered replacement is more eficient when N=7.
Figure 4.3 under N=7
48
This result explains that if we keep the asets beyond their economic service life, the total
cost of Group replacement increases.
Anual Multiplier for the Purchase Cost (a)
We added thre parameters (a, q, s) to reflect technological progres in our
replacement problem. The annual multiplier a is the most significant factor among them.
It afects the total PW cost of both replacement models; a relatively smal a means that
the new purchase cost wil decrease because technological improvement of asets
relatively much increases in market. The parameter a reduces the capital cost for both
Group and Staggered replacement models. As we vary the volume discount from 85% to
95%, the trend is demonstrated in table 4.3.
Under N=3 a=85% a=90% a=95%
Diference(GS) $4,724 $5,956 $7,569
Table 4.4 The diference betwen the PW of Group and Staggered models under various a
Figures 4.4  4.7 ilustrate more detail for the results presented in table 4.3; the gap
betwen the Group and Staggered replacement policies increases acording to an increase
in the annual multiplier a.
49
Tech N=3(a=85%)
$0
$30,000
$60,000
$90,000
$120,000
$150,000
1 11 21 31 41
year
P
W
G
S
Figure 4.4 under a=85%
Tech N=3(a=90%)
$0
$30,000
$60,000
$90,000
$120,000
$150,000
1 11 21 31 41
year
P
W
G
S
Figure 4.5 under a=90%
50
Tech N=3(a=95%)
$0
$30,000
$60,000
$90,000
$120,000
$150,000
1 11 21 31 41
year
P
W
G
S
Figure 4.6 under a=95%
Obtaining an Overall Sensitivity Graphs
Figure 4.7 ilustrates the sensitivity analysis for seven of the key input variables,
which are the same variables as in the replacement problem under no technological
progres. Figure 4.8 shows the sensitivity analysis for the thre key input variables we
added because of technological progres. In the first sensitivity analysis, the values of
variables change plus or minus 20% from the basecase value. In the second sensitivity
analysis, the range of variables is plus or minus 10% from the basecase because when
we increase the value of a to 20%, it is possible for the value of a to go over 1, and this
would violate the restriction that a is les than 1.
51
Figure 4.7 Sensitivity graph for the PW cost diferential betwen Group and Staggered
replacement policies.
In figure 4.7, we se that the total cost of the Staggered replacement policy is higher than
the Group replacement policy for the range of values examined. In particular, the first
year?s loss of market value of the aset (b) is the most sensitive variable, and the initial
52
O&M cost (A) and the scaling factor of the future O&M cost (p) are relatively insensitive
variables.
Figure 4.8 Sensitivity graph (Cont?) for the PW cost diferential betwen Group and
Staggered replacement policies.
Figure 4.8 also shows that the Group replacement policy is quite cost efective for the
range of values examined. The annual multiplier for the purchase cost (a) is the most
sensitive variable among them.
53
CHAPTER 5. MODEL UNDER ISK
In Chapter 4, we developed two replacement models (Group and Staggered) under
ongoing technological progres. In those models, we asumed al parameters to be known
with reasonable certainty. However, this certainty asumption is rather na?ve, as it is very
dificult to predict the price or operating and maintenance costs of asets in any precise
fashion. One practical way to estimate these parameters is to observe the trend of costs of
similar asets during past periods. To introduce possible variations in our decision
parameters, we wil treat some key input parameters as random variables. Since we are
not likely to atain an analytical solution, we wil rely on computer simulation using
@RISK
5.1. RISK SIMULATION PROCEDURES USING @ RISK
To conduct a risk simulation, we wil use a Microsoft Excel plugin known as @
Risk. Al we have to do is to develop an Excel worksheet to calculate the net present cost
of either the Group or Staggered replacement policy over the planning horizon. These
worksheets were already developed as functions of key input parameters presented in
Chapters 3 and 4. Then, we need to identify the random variables in the replacement
models. With @RISK, we have a variety of probability distributions to choose from to
54
describe our beliefs about the random variables of interest. Running an analysis with
@RISK involves five steps:
Step 1: Create a cash flow statement within Excel in which the cash flow entries are a
function of the input variables.
Step 2: Define Uncertainty. Here we start by replacing uncertain values in our
spreadsheet model with @RISK probability distribution functions. As shown in Figure
5.1, @RISK provides a wide range of probability functions to choose from. In our
demonstration, however, we wil asume a Beta distribution for each random variable.
The Beta distribution has been chosen primarily for convenience, as we can easily make
threpoint estimates: an optimistic estimate, a pesimistic estimate, and a most likely
estimate. These thre estimates are used as the upper bound, the lower bound, and the
mode of the corresponding input parameter distribution. Then, the probability distribution
itself is asumed to be a Beta distribution with a standard deviation of onesixth for the
spread betwen the upper and lower bounds (Park and Gunter, 1990).
Step 3: Pick Your Bottom Line. With @RISK, we need to designate our output cels,
which are the bottom line cels whose values we are interested in. In our case, this is the
total present worth cost of each replacement policy.
55
Figure 5.1 Selecting a distribution function in @Risk
Step 4: Simulate. Once we have completed Step 3, we are ready to simulate. There is no
limit to the number of diferent scenarios we can look at in our simulations. Each time,
@RISK samples random values from the @RISK functions we entered in Step 2 and
records the resulting outcome (present worth cost of adopting a Group (or Staggered)
replacement policy). With 100 iterations for each scenario (or any number of iterations),
we obtain the probability distribution of the present worth cost function for each
replacement strategy.
Step 5: Analyze the Simulation Results. Once we obtain the probability distribution for
each replacement policy, we have a way to compare the efectivenes of one policy over
the other. As shown in Figure 5.2, @RISK provides a full statistical report with a wide
range of graphing options for interpreting and presenting the simulation results.
56
.
Figure 5.2 Displaying the simulation results
5.2. DEVELOPING A SIMULATION MODEL
To ilustrate the proces of developing a simulation model and the impact of
uncertainty in choosing a replacement policy, we could consider one critical random
variable (such as purchase cost, P) at a time, just like conducting a sensitivity analysis.
Since we consider one single random variable at a time, we don?t need to define any
statistical relationship with other input parameters. In Section 5.3, we wil extend our
basic simulation model by considering al input parameters to be random variables.
5.2.1. PURCHASE COST (P) AS A SINGLE RANDOM VARIABLE
Let?s asume that the purchase cost (P) is the only random variable among the
input parameters. In that case, we select a Beta distribution to describe the nature of
57
uncertainty asociated with the purchase cost. Using the @RISK distribution function, the
Beta probability distribution for P looks like the folowing:
Name Function Min Mean MAx
P
n
RiskBetaGeneral(2,2,4500,500,RiskStatic(500) 4500 500 500
Table 5.1 Beta Distribution Function for Purchase Cost, P
With 100 iterations, @RISK produces the simulation outputs as summarized in Figures
5.3 and 5.4. Recal that the economic service life for group replacement was 3 years (N =
3) in Chapter 4. In Figures 5.3 and 5.4, we se how the mean, standard deviation and
percentiles of the PW cost for each year change over the planning horizon. Diferent
colors were used to display the mean value in yelow, ?1 standard deviation in red and
the range betwen the lower 5
th
percentile and upper 5
th
percentile in gren. Clearly, as
we further extend the planning horizon, the variability of the PW cost continues to
increase, but eventualy it reaches some form of steady state after 40 years. Recal that
the PW cost function we have developed in Chapters 3 and 4 was based on the infinite
planning horizon. Therefore, the steadystate results are more important for replacement
decisions.
58
Figure 5.3 The PW cost for Group replacement as a function of the planning horizon
Figure 5.4 The PW cost for Staggered replacement as a function of the planning horizon
Table 5.2 summarizes the trend of the PW cost for both models. Although the mean PW
of the group replacement policy is smaler than that of the Staggered replacement model,
59
it has yet to be verified for any instance of clear stochastic dominance betwen the two
policies, which wil be shown in Section 5.2.1.2.
95 Per. +1 Std. Dev. Mean 1 Std. Dev. 5 Per.
Group $ 111,854 $ 109,982 $ 106,789 $ 103,596 $ 101,528
Staggered $ 118,278 $ 116,260 $ 112,749 $ 109,238 $ 106,844
Table 5.2 Summary of PW Cost for the Group and Staggered Replacement Policies
Figures 5.5 and 5.6 show the cumulative probability distribution charts for the two
policies. With these cumulative probability distributions, we can ases the likelihood of
incurring a certain level of replacement cost over the infinite planning horizon. For
example, if a firm targets the total replacement cost for a certain aset group at $110,000,
we se that the Group replacement policy wil met this target level with an 82%
probability, whereas the Staggered replacement policy mets this target level with only a
25% probability. Even though we cannot say in an absolute sense that the Group policy
dominates the Staggered policy, we can say clearly that the Group policy appears to be
more cost efective in a general sense.
60
Figure 5.5 The cumulative ascending graph for Group replacement
Figure 5.6 The cumulative ascending graph for Staggered replacement
61
Incremental Cost of Selecting the Group Policy
One practical way to compare these two policies is to develop the incremental
cost betwen the two policies as shown in Figure 5.7. We can se that the diferential
cost (GS) is in wide swing in either direction (positive or negative) until it reaches the
steadystate. As mentioned earlier, however, we only consider the results in the steady
state condition, because we asumed the study period of an infinite planning horizon. A
negative diference betwen the PW for the Group and Staggered replacement policies
implies that the company can benefit from choosing the Group replacement policy. Table
5.3 shows with certainty that the Group replacement policy is more cost efective than the
Staggered replacement policy under an infinite planning horizon. With the data set
asumed in our model, the company would save $6,013 on average by choosing the
Group replacement policy.
Figure 5.7 The PW trend for the diference betwen the two replacement models
62
95 Per. +1 Std. Dev. Mean 1 Std. Dev. 5 Per.
GS $5,494 $5,684 $6,013 $6,342 $6,550
Table 5.3 Summary of PW Cost Distribution with Random Variable of P (GS)
Validation of the Simulation Results
Since the risk simulation model contains random elements (such as the purchase
cost, P), outputs from the simulation are limited to the number of observed samples of
this random variable. As a consequence, any decisions made on the basis of simulation
results should consider the variability of the simulation outputs. Our ultimate question is
how close an estimator (e.g., mean value of the diferential PW cost) is to the true
measure. The common approach to asesing the acuracy of an estimator is to construct
a confidence interval for the true measure?we determine an interval about the mean
within which the true value may be expected to fal with a certain probability.
As we have sen in Figure 5.7, the results of the simulation show that the PW cost
of Staggered replacement is higher than Group replacement in most periods. To analyze
the output data, we wil use the method of replication. Our goal is to obtain point and
interval estimates of the diference in mean performance. Table 5.4 gives the summary of
simulation output data for a random sample from a Beta distribution and the sample mean
and variance from 100 iterations.
63
Iteration
Model
1 2 3 ? 100
Sample
Mean(Y)
Sample
Variance
Group
(?
1
)
$106,130 $109,836 $107,126 ? $108,930 $106,734 3,185
2
Staggered
(?
2
)
$112,080 $116,170 $113,181 ? $115,170 $112,747 3,515
2
Table 54 Simulation Output Data and Summary Measures for Comparing Two Models.
To define a confidence interval, we denote
12
!=" which is an interval estimate of
the diference in mean performance. A twosided 100(1 ? ?) confidence interval for !
wil take the following form:
/2,
?
v
Yt
!
"#
/2,
?
v
Yt
!
#$+ (5.1)
where
12
= and
2
1
?
S
n
=.
Then, the sample mean and variance based on 100 iterations are
12
$06,73412,7$6,013Y=!!=
12
,85,
? 5.9
S
n
++
The degre of fredom is
2 2
2 2
1
0 2 2
1
3,185,
0
196.
(/)(/)(,/)(,/0)
S
n
v
!"!"
++
#$#$
%&%&
===
???
64
0
2196.4.1v=!=
Finaly, a 95% confidence interval is
$,3.52 $6,03+1.952
6,9 5,4
!"#"
which indicates that the confidence interval for is totaly to the left of zero, so we
may conclude
12