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Sensitivity analysis of rent replacement models
production economics
ELSEVIER
Int. J. Production Economics 36 (1994) 267.-279
Sensitivity analysis of rent replacement models K.A.H. Kobbacy”y*,
S.D. Nicolb
Abstract Rent models are a useful class of replacement models which can be used to study the replacement of capital equipment over a long period of time. This paper attempts to quantify the sensitivity of rent models as applied to commercial
vehicles using statistical simulation. A method based on Markov chain to simulate the interest rate and discount factor is presented. Changes in parameters such as capital cost, maintenance cost, resale value, current discount factor as well as tax parameters Key
words:
are made and their effects on the optimal
Maintenance;
Optimisation;
Replacement
In the modern industrial and commercial climate, companies have to be efficient in all aspects of their business, from production to marketing and selling in order to survive. All companies, no matter how large, have limited resources and achieving efficiency means that these resources must be managed with great care. Resource management has become particularly important in the context of replacing plant or capital equipment. As machinery has become more sophisticated, prices have risen likewise. To this end it is obviously in a company’s interests to get as much use out of expensive equipment as possible. This means keeping the equipment on-line and running efficiently. Thus many companies have adopted maintenance
0925-5273/94/$07.00
author ‘Q
1994
SSDI 0925-5273(94)00041-8
age of the trucks are analysed.
of capital equipment;
1. Introduction
*Corresponding
replacement
Simulation;
Sensitivity
analysis
strategies such as planned maintenance and condition monitoring where they aim to maintain their equipment such that a criterion is optimised. For example to maximise availability or to minimise the operating costs. All equipment, though properly maintained, will eventually need to be replaced. There are different reasons for replacing equipment including: declining performance; higher performance required; to offset chance of decreasing performance; obsolescence and considerations of style and image Cl]. With declining performance particularly in mind, and striving to target their resources as carefully as possible, managers now ask ‘when it would be optimal in terms of cost to replace this equipment’?‘. This and similar questions justify the development and use of replacement models. Rent models are used to assess the economic life of industrial plant. A constant rent (or payment) is estimated by equating the NPV (Net Present
Elsevier Science B.V. All rights reserved.
Value) of all costs (or benefits) with the NPV of hypothetical rents (or payments) made at constant intervals (annually). The optimal replacement age is then chosen such that the rent is minimised or the payment is maximised. The term ‘rent model’ was first used in 1987 by Christer and Waller [2] though this type of models was referred to by Churchman et al. [3] and used to solve replacement problems by Russell [4]. Rent models can vary significantly according to the need of analysing a specific problem. The simplest one-cycle rent replacement model assumes that the equipment under consideration is new and that the annual operating cost and rent are incurred at the half year mark. Given that rk is the discount factor for year k, then
(100+jk)
rk= (100 + ik)’ where jk and ik are the inflation rate and interest rate for year k, respectively. The discount factor over n years to end of the year, R,, is given by
Rn= fl rk k=l
and the discount H,, is given by
factor to the midpoint
of year n,
k=l
By equating the total discounted cost with the total discounted rent one can calculate the rent from Rent=k~~P*.HI+(c-S,).R.. :H,
’
k=l
where m = replacement period of equipment in years, Qk = operating cost of equipment in year k, C = capital cost of replacement equipment, S, = resale value of m years old equipment. This basic model can be modified to enable the examination of the effects of important parameters on replacement age such as taxation. The two-cycle rent model is a variation which can help in deciding the optimal replacement age of a currently owned
equipment, i.e. non-new equipment and also the optimal replacement age of its replacement. This is achieved by equating the total discounted cost over the lives of the two equipments with the total discounted rent over the same period. These models are discussed in details elsewhere [2]. In investigating the relative influence of the contributing parameters on the replacement age of commercial trucks, Christer and Waller [Z] used an analysis of variance approach. Their method was based on examining the influence of variation of each of the input parameters on the replacement age and on a penalty measure for sub-optimal choice. So, for each parameter they assumed three levels; low, medium and high. The exact value of each of these levels was chosen arbitrarily. For example for inflation level they were 2, 14 and 26%. Obviously this subjective judgement which might have been relevant at the time of that study may not always remain valid. Furthermore, they assigned one of the selected values to the start year and one to the end year and allowed uniform stepwise change over the study period (10 years). This assumption in particular is far from being realistic when applied to inflation and interest rates. The objective of the current study is to use a more robust approach to examine the sensitivity of the rent model. The selected method was statistical simulation. Our view is that inflation and interest rates and hence discount factor are uncontrolled variables to the study of replacement of capital equipment. In absence of reliable long-term economic forecasting, the simulation of these parameters based on past data is the most realistic choice. Furthermore, simulation by its nature lends itself to sensitivity analysis type of study. The operating costs of commercial trucks used in this study were those published in Christer and Waller [Z] after allowing for inflation. This would enable the comparison between the findings of these two studies. We shall start by introducing the method used to simulate the discount factor, interest rate and inflation rate.
2. Simulation
of economic parameters
Data about interest and inflation rates in the UK were available in published sources [S-7]. The
K.A.H.
Kobbac_v. S.D. Nicol/Inf.
J. Production
shortest time period at which these rates are quoted in any of the sources was one month. This seems quite adequate to the use in the rent model as it will dictate the accuracy of results, i.e. the age of replacing a vehicle will be estimated to the nearest month. It was decided that data over a thirty years period would be sufficient to reflect the performance of the recent British economy. Therefore, the interest and inflation rates were recorded for every month from January 1960 to December 1990. Figs. 1 and 2 show the monthly variation of inflation and interest rates.
T960
1970
1965
Economics 36 (19941
2. I. ModeNing
269
267-279
the discount fuctor
Fig. 3 shows the variation of the calculated discount factor rk. It is clear that the discount factor does not remain constant for any length of time, nor does it behave as a step function, but is rather like a series of peaks and troughs. The aim here was to simulate, not to predict, the discount factor in such a way that the simulation would behave realistically. Based on the observation that discount factor will change gradually from
1960
1975
1985
1990
Year Fig. 1. Variation
of the U.K. inflation
rate (1960-1990).
0
1960
1966
1970
1960
is76
1966
Year
Fig. 2. Actual
and Simulated
U.K. interest
rate (1960-1990).
1990
1996
270
K.A.H.
0.90
1 1960
Kohhuc~~, S.D. Nicol/Int.
I
1965 Fig. 3. Actual
J. Productiotl
I
1970
I
I
1975
and Simulated
one month to the next and drawing on previous experience in simulating weather parameters in the North Sea [S], it was decided that Markov chain approach can be useful here. The underlying assumptions that the discount factor, or indeed any stochastic process, follows discrete-time stationary Markov chain are [9]: The process is a discrete-time process. This assumption suits the simulation of discount factor since the available data only enable calculations of change of discount factor from one month to the next. It has a countable or finite state space. The process satisfies the Markov property, i.e. the probability that the process is in specific state at time period n depends only on its state at time period n - 1 and not on its state at previous time periods. Assumptions 1 and 2 are obviously valid in this situation. However, satisfying the Markov property is a simplifying assumption. It is more likely that the rate of change during the previous month will also influence the discount factor. The crucial test for the validity of this assumptions, though, will be based on the adequacy of the output of the analysis [lo]. By following the change in the discount factor from one month to the next from 1960 to 1990, probability distributions of change from month to month were estimated. The monthly transition probability matrix for the discount factor which is
Econornic.s 36 i 1994) 267-279
1900
I J.K. discount
t
1905
I
1990
I 1995
factor (1960- 1990).
shown in Fig. 4 was then established. This matrix forms an irreducible Markov chain, i.e. all the states can be reached, in more than one step if necessary, from every other state. An element Pij (ith row andjth column) of this transition probability matrix represents the probability of a change from discount factor i to discount factor j in a single transition, i.e. one month. The diagonal clustering of the non-zero probabilities in Fig. 4 reflects the fact that the discount factor varies in small steps each month (maximum monthly variation is 0.04 but mostly within 0.02). To simulate the discount factor we start by selecting an initial value which can be arbitrary or deliberate, e.g. current value of discount factor. We then sample from the distribution represented by the row in the transition probability matrix which corresponds to the selected discount factor. The outcome of sampling will represent the discount factor for month two. This process is repeated by entering the transition probability matrix at the row corresponding to the obtained second month discount factor and then sampling to obtain discount factor of month three. Fig. 3 shows a sample run of the process that simulate the discount factor against the actual values. By making several simulation runs starting with different initial values and comparing the simulated with the actual discount factor over the period 1960- 1990 we observed that they all have comparable means, variances, medians and obviously ranges.
K.A.H.
Kobbacy, S.D. Nirol/lnt. Discount
J. Production
Factor
Economics
36 (1994)
267-279
271
at Month (t+l)
.92 ,93 .94 .95 .96 .97 .98 991.001.01 1.02 1.03 1.041.05 1.061.07108 1.091.10 1.11 1.12 1.13 1.14 115 _
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so 25
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_--.__~-~___~-.___-___~______I____--___-__--___~--~.__-._-_~________-____-____-___________-___---_-
Fig. 4. Transition
2.2. Modelling
probability
the interest rate
Although only the discount factor appears in the simple rent model, the inclusion of tax parameters will also require the interest and inflation rates. It was decided that the interest rate should be simulated. To ensure consistency, the rate of inflation could then be calculated as a function of the simulated discount factor and interest rate. The pattern of interest rate over the period 196OG1990 (Fig. 2) suggests that the interest rate is increasing, all be it erratically, through out this period. To evaluate this trend a linear regression model was fitted to the data, and is shown in Fig. 2. The equation of this straight line is: Interest
Rate = 4.80 + 0.024 x Month
where month 1960.
number
is measured
Number from January
matrix
for the discount
factor.
To be able to make realistic simulation for interest rate the apparent fluctuations in the data should be superimposed on the regression line which represents the increasing linear trend. Having examined several alternatives for modelling these fluctuations it was decided to use a Markov chain to represent the change of the percentage deviations from month to month. Using the history data from 1960 to 1990, probability distributions were calculated. The resulting transition probability matrix is shown in Fig. 5. The interest rate was then simulated using the regression line with a superimposed percentage deviation simulated using the Markov chain method which is explained in the discount factor section. Fig. 2 demonstrates a sample run of the process used to simulate the interest rate. The results seem to be adequate, the mean, median and range of the simulated interest rate are all close to that of the real
272
K.A.H.
S.D. Nicoli ht. J. Productiorl
Kobbucv.
interest rate. It is also apparent that with the identification of trend in interest rate history, its simulation becomes essentially dependent on time and hence becomes a kind of forecasting which has to be viewed with care. However, one should bear in mind that effects of interest and inflation rates on the rent model with tax parameters are relatively minor.
Economics
36 (1994)
267-279
age, mainly unplanned maintenance and the annual vehicle MOT tests. The operating cost shown in Table 1 is essentially an expected or mean value. Due to the nature of unplanned maintenance considerations, actual costs would vary around this mean. Based on the analysis of several data sets, it was suggested [2] that unplanned maintenance costs are approximately normally distributed with a standard deviation of 0.38. Hence, the 95% confidence interval for a vehicle of given age is 0.3 to 1.7 the mean operating cost for a vehicle of that age. To simulate the operating cost for a vehicle a sampled random fraction was used to generate proportionate operating cost in the given range. Table 1
3. Simulation of cost parameters 3.1. The operating cost Table 1 shows the operating cost used in this study which include the costs that vary with vehicle
Percentage Deviation at Month (t+l) -32-30-18-26 -24 -22 -20 -16 -16 -14 -12 -10 -E -6 -4 -2 -32
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Fig.
5. Transition
-
-
-
-
probability
-
-
_
matrix
_
_
_
_
_
_
for the percentage
_
_
_
deviation
_
_
_
_
of interest
_ ,21,65,0,,07
rate
from
the fitted
line.
K.A.H.
Kobbacy,
S.D. Nicol/Inr.
also shows an example for simulating cost.
J. Production
the operating
3.2. Capital cost of new tractor unit The operating cost data used in this study relate to 32.5 tonne tractor units. To remain consistent, the capital cost of a replacement unit used in the simulation is the average price of five models in the 32.5 tonne category which is 534950. The data were obtained from Glass’s Guide for Commercial Vehicles [l l] and CAP Red Book
cm Table Annual Year
3 4 5 6 7 8 9 10
operating
cost with simulation
Operating
3.3. Resale values of second-hand
The purpose of the following analysis is to model the way in which a tractor unit of this type is expected to depreciate. The second hand value of a vehicle is likely to depend on the vehicles’s age, milage and condition in addition to the market forces, i.e. the levels of supply and demand. The Glass’s Guide [l l] gives expected resale values of second-hand commercial vehicles for two situations: (i) Basic Trade, or Basic T, which is the value a dealer would expect to pay for an average second-hand vehicle and (ii) the Guided Condition Retail, GCR, which is the price a commercial
Operating
cost at 1989
Simulation
example
(pounds) Q
1006 1760 2515 3270 4024 2264 3018 3898 5156 8300
800 1400 2000 2600 3200 1800 2400 3100 4100
Random fraction
Proportion of Q to be charged
Simulated (pounds)
0.89 0.15 0.25 0.08 0.60 0.28 0.54 0.99 0.58 0.50
1.546 1.510 0.650 0.412 1.140 0.692 1.056 1.686 1.112 1.000
1555.28 897.60 1634.75 1347.24 4587.36 1566.69 3187.01 6572.03 5733.47 8300.00
1
GCR value
19 784.25 15 154.73 11608.53 8892.13 6811.37 5217.51 3996.6 1 3061.41 2345.04 1796.30
cost
resale value of vehicle Basic T value
Diff.
Simulation Random
2 3 4 5 6 7 8 9 10
tractor units
example
cost at 1984 prices
Table 2 Example for simulating Year
273
1
(pounds)C21
L
Economics 36 (1994) 267-279
16 535.91 12 054.68 8787.86 6406.35 4670.23 3404.60 248 1.95 1809.34 1319.01 961.56
3248.34 3100.05 2820.67 2485.78 2141.14 1812.91 1514.66 1252.07 1026.03 834.74
0.1863 0.1288 0.3948 0.0672 0.6789 0.2418 0.1862 0.8876 0.7254 0.9380
example No.
Resale value 19 179.08 14 755.44 10494.93 8725.09 5357.57 4779.15 3714.58 1950.07 1600.76 1013.31
K.A.H.
274
Kohhacy,
S.D. Nicol,ilnt.
.I. Production
1990, this study was repeated with the program written in FORTRAN [14]. All the results shown below are based on the average results of repeating each simulation experiment 200 times. Preliminary investigations established that after 200 runs the average optimal vehicle replacement age shows negligible variations.
vehicle dealer would expect to be able to sell a vehicle for. Using regression analysis of the data obtained from Glass’s guide, the following relations were obtained: (i) Basic T = 22.683 x 0.729” (ii) GCR = 25.828 x 0.766” Both these models have good fit. These models represent the lower and upper limits for a resale value of a vehicle of age n. A simulated value of the resale price was then sampled from a uniform distribution whose limits are Basic T and GCR. Table 2 shows an example for simulating the resale value of trucks.
4.1. The one-cycle rent model The results obtained while have trends that are in broad agreement with those of Christer and Waller [2], have revealed changes of significantly different magnitudes. The optimal replacement age was about nine years. The main parameters which influence the replacement age are the cost parameters, namely the operating cost and the resale value and to a lesser extent the purchase cost of a new truck. Fig. 6 shows that an increase in the operating cost by 50% would cause a reduction of the optimal replacement age of the truck by around six
4. Simulation and sensitivity analysis A simulation program was written in Pascal 1989 which has the facility to examine the effect many parameters on the optimal replacement age a truck [ 131. To reconfirm some of the results and take into consideration the tax reforms in the UK
Capital
in of of to of
Cost
30,000
0"
-
Operating
0.0
cost
factor
of
a
New
Vehicle
35,000
=
Assumed Estimated
Operating Operating
0.5
cost and capital
Cost
40,000
45,000
1.5
2.0
Cost Cost
1.0 Operating
Fig. 6. Effect of operating
Economics 36 (1994) 267-279
Factor
cost of a new vehicle on its optimal
replacement
age (one cycle model).
K.A.H.
Knhbucy,
S.D. Nicollht.
J. Production
months, almost half that estimated by Christer and Waller [2]. Fig. 6 also shows that an increase in the purchase cost of the vehicle causes minor increase in the optimal replacement age. Fig. 7 demonstrates that the gradual increase of the current resale value of the truck causes slight reduction of the optimal replacement age until a certain critical point is achieved.
275
Economics 36 (1994) 267-279
At around 50% above the current resale value, we observed sudden severe reduction in the optimal age with any further increase of the resale value. This trend was not observed previously and prompted the need to repeat the simulation study to check for any error in the program. The repeated study [14] actually produced exactly the same trend.
10
Value Factor
Resale
0 0.0
1
.
I
.
=
1
Assumed Resale Price Estimated Resale Price I
.I
I
0.5
I1
I
*
1.0
1
.
.
.
.
1.5
Resale Value Factor
Fig. 7. Effect of resale value of vehicle on its optimal
I 0.95
I
I
1.00 Discount
Fig. 8. Effect of selecting
replacement
initial discount
1
1.05 Factor
age (one cycle model).
1.10
seed
factor on optimal
replacement
age (one cycle model)
1.15
276
K.A.H.
Kohhac:,. S.D. Nicol,‘Int.
J. Production
replacement age of the current vehicle tends to decrease with increasing the replacement age of the second vehicle. Figs. 11 and 12 demonstrate the effects of the cost parameters on the optimal replacement age of the current vehicle. The two-cycle model seems to be less sensitive to changes in the operating cost than the one-cycle model. Also the resale value of the first vehicle does not have the dramatic effect on its replacement age displayed in the results of onecycle model. The effect of changes in the trucks purchase cost is shown in Fig. 12 and has similar pattern to that in the one-cycle model. The effects of tax parameters on the optimal replacement age of the initial vehicle are minimal and are of much the same nature as in the case of one-cycle model.
The only economic parameter which could be tested in this study is the initial discount factor which reflects the current economic situation. Fig. 8 shows that the discount factor has no significant effect on the optimal age. Similarly the tax parameters including corporation tax rate, writing down allowance and tax payment delay were found to have minor or slight effect on the replacement age. Some of these influences are depicted in Fig. 9.
4.2. The two-cycle
rent model
In a two-cycle rent model we have two decision variables: K the time to replacement of current vehicle and L the replacement age of the second (replacement) vehicle. The important variable is K, since it represents, the current decision. Fig. 10 shows the effects of age of the first vehicle and the replacement age of the second vehicle. An increase in the age of the current vehicle leads to increase in its replacement age. In contrast, the
Writing
Down
Economic.s 36 (1994) 267-279
5. Conclusions There are two main conclusions for this study. The first is that the approach adopted, i.e. using
Allowance
(a)
6 0
10
20 Corporate
Fig. 9. Effects of corporate
tax rate and writing
down allowance
30 Tax Rate
40
(%)
on the optimal
replacement
age of vehicle (one cycle model).
K.A.H.
Kohhac~, S.D. Nicoli ht.
Replacement 2 i
I
l
0
4 ,
*
1. Pro&rim
Economics
Age of Second vehicle
'
*
s
2
vehicle
and
I”’
*
.
+J
_
8
I’
replacement
s
I
age of second
’
8
6
4
Resale Value
6 *i:
IYears)
6
Age of Initial Vehicle
Fig. 10. Effects of age of initial (two-cycle-model).
277
36 (1994) 267-279
(Years)
vehicle
on optimal
replacement
age of initial
Factor
5 626’ 8
-
2
-
2
.
a ;i
.; -I +J 8*
Resale Value Factor
6
t*ttiII-I‘IttttI*~.~ 0.5
0.0
operating
Assumed Resale Price = Estimated Resale Price
Cost Factor
Assumed Operatins Cost = Estimated Operating Cost
Fig. Il.
Effects of operating
1.5
1.0 Operating
Cost Factor
cost and resale value on the optimal
replacement
age of initial vehicle (two-cycle-model).
vehicle
K.A.H.
278
Kohhoc~~. S.D.
Nic~ol.‘Int. J. Production
Ecorzomics
f 1994)
36
267-279
10
6
I
.,,I,,,,
25,000
Capital
Fig. 12. Effect of capital
I 35,000
30,000
I
statistical simulation in analysing the sensitivity of rent replacement models has shown to be effective. The results of this study, though significantly different in magnitude, are in general agreement with the results of previous study using analysis of variance [2]. This confirms that the simulation of the relevant economic parameters (of the UK) using Markov chains is reasonable and realistic. The simulation approach, unlike AOV, has the advantage of versatility, i.e. the study can be repeated with the cost data of other capital equipment or economical parameters of another country at a minimal effort on the part of the analyst once the approach has been established. The second conclusion of this study concerns the replacement of the type of commercial vehicles studied. Having examined all cost and tax parameter over a relatively wide range, it was found that the optimal replacement age is not significantly sensitive to these parameters with the exception of the resale value of the vehicle. The increase of operating cost has produced moderate reduction in the optimal replacement age. While tax parameters, which are typically of concern to management, are the least influential with almost negligible contribution towards any results. Increasing the estimated resale value of vehicle by 50% or more caused the
I
I,,, 40,000
Cost of A New Vehicle
cost of a new vehicle on optimal
I
45,000
(Pounds)
replacement
age (two-cycle
model).
mean optimal replacement age to drop sharply. However, this effect disappears when the replacement vehicle is considered in two-cycle model. There is also doubt that such large increase in resale value can be realised in reality. In general, the mean optimal replacement age for tractor units of 32.5 tonne articulated lorries is approximately nine years. After nine years, a tractor unit’s resale price is close to its scrape value. This is not a surprising result in that the sales manager of a large commercial vehicle dealer in Scotland has suggested that in most cases firms tend to ‘run their tractor units into the ground before replacing them’.
Acknowledgments The authors wish to express sincere thanks to H.M. Darvesh and A. Davies for their effort in reconfirming some of the results of this study while working in their B.Sc. projects.
References [1 J Christer, A.H., 1984. Maintenance Notes, Umversity of Strathclyde.
and
Replacement
K.A.H. Kobbacy, S.D. Nicolllnt. J. Production Economics 36 (1994) 267-279 [2] Christer, A.H. and Waller, M.W., 1987. Tax-adjusted replacement models. J. Opl Res., 38: 99331006. [3] Churchman, C.W., Ackoff, R.L. and Arnoff, E.L., 1966. Introduction To Operations Research. Wiley, New York. [4] Russell, J.C., 1982. Vehicle replacement: A case study in adapting a standard approach for a large organisation. J. Opl Res., 33: 899991 I. [S] Organisation for Economic Co-Operation and Development, 1976. Interest Rates 1960&1974. [6] Department of Employment. HMSO (1986) Consumer Price Indexes. [7] Annual Abstract of Statistics. HMSO. [S] Kobbacy, K.A.H.. Christer, A.H. and MacCallum, K.J., 1982. Stochastic modelling of the saturation-diving-
279
inspection activities of a North Sea oil platform. Project Mass, Task 3.3, Report No. 2, University of Strathclyde. [9] Isaacson, D.L. and Madsen, R.W., 1976. Markov Chains Theory and Applications. Wiley, New York, p. 12. [lo] White, D.J., 1975. Decision Methodology. Wiley, New York, p. 131. [1 I] Glass’s Guide Commercial Vehicles Values, June 1989 ed. [12] CAP, Current Auto Prices, Red Book Commercial Vehicle Values, June 1989 ed. [13] Nicol, SD., 1989. Sensitivity Analysis of Rent Models. BSc. Project Report, University of Strathclyde. [I43 Davies, A. and Darvesh, H.M., 1991. Sensitivity Analysis of the Rent Replacement Models. B.Sc Project Report, University of Salford.
ELSEVIER
Int. J. Production Economics 36 (1994) 267.-279
Sensitivity analysis of rent replacement models K.A.H. Kobbacy”y*,
S.D. Nicolb
Abstract Rent models are a useful class of replacement models which can be used to study the replacement of capital equipment over a long period of time. This paper attempts to quantify the sensitivity of rent models as applied to commercial
vehicles using statistical simulation. A method based on Markov chain to simulate the interest rate and discount factor is presented. Changes in parameters such as capital cost, maintenance cost, resale value, current discount factor as well as tax parameters Key
words:
are made and their effects on the optimal
Maintenance;
Optimisation;
Replacement
In the modern industrial and commercial climate, companies have to be efficient in all aspects of their business, from production to marketing and selling in order to survive. All companies, no matter how large, have limited resources and achieving efficiency means that these resources must be managed with great care. Resource management has become particularly important in the context of replacing plant or capital equipment. As machinery has become more sophisticated, prices have risen likewise. To this end it is obviously in a company’s interests to get as much use out of expensive equipment as possible. This means keeping the equipment on-line and running efficiently. Thus many companies have adopted maintenance
0925-5273/94/$07.00
author ‘Q
1994
SSDI 0925-5273(94)00041-8
age of the trucks are analysed.
of capital equipment;
1. Introduction
*Corresponding
replacement
Simulation;
Sensitivity
analysis
strategies such as planned maintenance and condition monitoring where they aim to maintain their equipment such that a criterion is optimised. For example to maximise availability or to minimise the operating costs. All equipment, though properly maintained, will eventually need to be replaced. There are different reasons for replacing equipment including: declining performance; higher performance required; to offset chance of decreasing performance; obsolescence and considerations of style and image Cl]. With declining performance particularly in mind, and striving to target their resources as carefully as possible, managers now ask ‘when it would be optimal in terms of cost to replace this equipment’?‘. This and similar questions justify the development and use of replacement models. Rent models are used to assess the economic life of industrial plant. A constant rent (or payment) is estimated by equating the NPV (Net Present
Elsevier Science B.V. All rights reserved.
Value) of all costs (or benefits) with the NPV of hypothetical rents (or payments) made at constant intervals (annually). The optimal replacement age is then chosen such that the rent is minimised or the payment is maximised. The term ‘rent model’ was first used in 1987 by Christer and Waller [2] though this type of models was referred to by Churchman et al. [3] and used to solve replacement problems by Russell [4]. Rent models can vary significantly according to the need of analysing a specific problem. The simplest one-cycle rent replacement model assumes that the equipment under consideration is new and that the annual operating cost and rent are incurred at the half year mark. Given that rk is the discount factor for year k, then
(100+jk)
rk= (100 + ik)’ where jk and ik are the inflation rate and interest rate for year k, respectively. The discount factor over n years to end of the year, R,, is given by
Rn= fl rk k=l
and the discount H,, is given by
factor to the midpoint
of year n,
k=l
By equating the total discounted cost with the total discounted rent one can calculate the rent from Rent=k~~P*.HI+(c-S,).R.. :H,
’
k=l
where m = replacement period of equipment in years, Qk = operating cost of equipment in year k, C = capital cost of replacement equipment, S, = resale value of m years old equipment. This basic model can be modified to enable the examination of the effects of important parameters on replacement age such as taxation. The two-cycle rent model is a variation which can help in deciding the optimal replacement age of a currently owned
equipment, i.e. non-new equipment and also the optimal replacement age of its replacement. This is achieved by equating the total discounted cost over the lives of the two equipments with the total discounted rent over the same period. These models are discussed in details elsewhere [2]. In investigating the relative influence of the contributing parameters on the replacement age of commercial trucks, Christer and Waller [Z] used an analysis of variance approach. Their method was based on examining the influence of variation of each of the input parameters on the replacement age and on a penalty measure for sub-optimal choice. So, for each parameter they assumed three levels; low, medium and high. The exact value of each of these levels was chosen arbitrarily. For example for inflation level they were 2, 14 and 26%. Obviously this subjective judgement which might have been relevant at the time of that study may not always remain valid. Furthermore, they assigned one of the selected values to the start year and one to the end year and allowed uniform stepwise change over the study period (10 years). This assumption in particular is far from being realistic when applied to inflation and interest rates. The objective of the current study is to use a more robust approach to examine the sensitivity of the rent model. The selected method was statistical simulation. Our view is that inflation and interest rates and hence discount factor are uncontrolled variables to the study of replacement of capital equipment. In absence of reliable long-term economic forecasting, the simulation of these parameters based on past data is the most realistic choice. Furthermore, simulation by its nature lends itself to sensitivity analysis type of study. The operating costs of commercial trucks used in this study were those published in Christer and Waller [Z] after allowing for inflation. This would enable the comparison between the findings of these two studies. We shall start by introducing the method used to simulate the discount factor, interest rate and inflation rate.
2. Simulation
of economic parameters
Data about interest and inflation rates in the UK were available in published sources [S-7]. The
K.A.H.
Kobbac_v. S.D. Nicol/Inf.
J. Production
shortest time period at which these rates are quoted in any of the sources was one month. This seems quite adequate to the use in the rent model as it will dictate the accuracy of results, i.e. the age of replacing a vehicle will be estimated to the nearest month. It was decided that data over a thirty years period would be sufficient to reflect the performance of the recent British economy. Therefore, the interest and inflation rates were recorded for every month from January 1960 to December 1990. Figs. 1 and 2 show the monthly variation of inflation and interest rates.
T960
1970
1965
Economics 36 (19941
2. I. ModeNing
269
267-279
the discount fuctor
Fig. 3 shows the variation of the calculated discount factor rk. It is clear that the discount factor does not remain constant for any length of time, nor does it behave as a step function, but is rather like a series of peaks and troughs. The aim here was to simulate, not to predict, the discount factor in such a way that the simulation would behave realistically. Based on the observation that discount factor will change gradually from
1960
1975
1985
1990
Year Fig. 1. Variation
of the U.K. inflation
rate (1960-1990).
0
1960
1966
1970
1960
is76
1966
Year
Fig. 2. Actual
and Simulated
U.K. interest
rate (1960-1990).
1990
1996
270
K.A.H.
0.90
1 1960
Kohhuc~~, S.D. Nicol/Int.
I
1965 Fig. 3. Actual
J. Productiotl
I
1970
I
I
1975
and Simulated
one month to the next and drawing on previous experience in simulating weather parameters in the North Sea [S], it was decided that Markov chain approach can be useful here. The underlying assumptions that the discount factor, or indeed any stochastic process, follows discrete-time stationary Markov chain are [9]: The process is a discrete-time process. This assumption suits the simulation of discount factor since the available data only enable calculations of change of discount factor from one month to the next. It has a countable or finite state space. The process satisfies the Markov property, i.e. the probability that the process is in specific state at time period n depends only on its state at time period n - 1 and not on its state at previous time periods. Assumptions 1 and 2 are obviously valid in this situation. However, satisfying the Markov property is a simplifying assumption. It is more likely that the rate of change during the previous month will also influence the discount factor. The crucial test for the validity of this assumptions, though, will be based on the adequacy of the output of the analysis [lo]. By following the change in the discount factor from one month to the next from 1960 to 1990, probability distributions of change from month to month were estimated. The monthly transition probability matrix for the discount factor which is
Econornic.s 36 i 1994) 267-279
1900
I J.K. discount
t
1905
I
1990
I 1995
factor (1960- 1990).
shown in Fig. 4 was then established. This matrix forms an irreducible Markov chain, i.e. all the states can be reached, in more than one step if necessary, from every other state. An element Pij (ith row andjth column) of this transition probability matrix represents the probability of a change from discount factor i to discount factor j in a single transition, i.e. one month. The diagonal clustering of the non-zero probabilities in Fig. 4 reflects the fact that the discount factor varies in small steps each month (maximum monthly variation is 0.04 but mostly within 0.02). To simulate the discount factor we start by selecting an initial value which can be arbitrary or deliberate, e.g. current value of discount factor. We then sample from the distribution represented by the row in the transition probability matrix which corresponds to the selected discount factor. The outcome of sampling will represent the discount factor for month two. This process is repeated by entering the transition probability matrix at the row corresponding to the obtained second month discount factor and then sampling to obtain discount factor of month three. Fig. 3 shows a sample run of the process that simulate the discount factor against the actual values. By making several simulation runs starting with different initial values and comparing the simulated with the actual discount factor over the period 1960- 1990 we observed that they all have comparable means, variances, medians and obviously ranges.
K.A.H.
Kobbacy, S.D. Nirol/lnt. Discount
J. Production
Factor
Economics
36 (1994)
267-279
271
at Month (t+l)
.92 ,93 .94 .95 .96 .97 .98 991.001.01 1.02 1.03 1.041.05 1.061.07108 1.091.10 1.11 1.12 1.13 1.14 115 _
_
_
_.
0.93 _ ,73 ,16 (09 -
092_1,0_
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.25 .I3 .36 .I3 .13 - JO - .50 _
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03 ,24 ,4, ,29 ,03 _
,15,67,16
_
,03 ,I2 ,62 ,23 _
-
F z =”
101 ,02_
-
:
1 03 104 _
_
1 05 1.06 -
_
I.07 -
_
3.06 _
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109 _
.
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110.
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“e ,’ : .d CI
_ _
_ ,12,70,16
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Ls
_ _
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0.99 1.00
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_ _
0.94 ,09 ,16 ,27 ,46 _
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;;
-
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- .03.31.39.22 .05_ _ ,29 ,43 ,26 _
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,119 .I0 .27 .28 .09 .ot
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_. -
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,on ,34 .25 .25 _ _ ,17 ,56 ,25 _
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113.. _
-
-
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,50
_
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.
_
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_lO__
-
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-
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- .50 _
_.
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- ,67
-
,33
-
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,14,72
-
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-
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-
-
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_
_
,50 _
_
_
.
_
_
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_
_
-
_
_,50__,5O_
IO _
. _
_
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.
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- .50 _ _
_
_
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_
_
50
so 25
75
_--.__~-~___~-.___-___~______I____--___-__--___~--~.__-._-_~________-____-____-___________-___---_-
Fig. 4. Transition
2.2. Modelling
probability
the interest rate
Although only the discount factor appears in the simple rent model, the inclusion of tax parameters will also require the interest and inflation rates. It was decided that the interest rate should be simulated. To ensure consistency, the rate of inflation could then be calculated as a function of the simulated discount factor and interest rate. The pattern of interest rate over the period 196OG1990 (Fig. 2) suggests that the interest rate is increasing, all be it erratically, through out this period. To evaluate this trend a linear regression model was fitted to the data, and is shown in Fig. 2. The equation of this straight line is: Interest
Rate = 4.80 + 0.024 x Month
where month 1960.
number
is measured
Number from January
matrix
for the discount
factor.
To be able to make realistic simulation for interest rate the apparent fluctuations in the data should be superimposed on the regression line which represents the increasing linear trend. Having examined several alternatives for modelling these fluctuations it was decided to use a Markov chain to represent the change of the percentage deviations from month to month. Using the history data from 1960 to 1990, probability distributions were calculated. The resulting transition probability matrix is shown in Fig. 5. The interest rate was then simulated using the regression line with a superimposed percentage deviation simulated using the Markov chain method which is explained in the discount factor section. Fig. 2 demonstrates a sample run of the process used to simulate the interest rate. The results seem to be adequate, the mean, median and range of the simulated interest rate are all close to that of the real
272
K.A.H.
S.D. Nicoli ht. J. Productiorl
Kobbucv.
interest rate. It is also apparent that with the identification of trend in interest rate history, its simulation becomes essentially dependent on time and hence becomes a kind of forecasting which has to be viewed with care. However, one should bear in mind that effects of interest and inflation rates on the rent model with tax parameters are relatively minor.
Economics
36 (1994)
267-279
age, mainly unplanned maintenance and the annual vehicle MOT tests. The operating cost shown in Table 1 is essentially an expected or mean value. Due to the nature of unplanned maintenance considerations, actual costs would vary around this mean. Based on the analysis of several data sets, it was suggested [2] that unplanned maintenance costs are approximately normally distributed with a standard deviation of 0.38. Hence, the 95% confidence interval for a vehicle of given age is 0.3 to 1.7 the mean operating cost for a vehicle of that age. To simulate the operating cost for a vehicle a sampled random fraction was used to generate proportionate operating cost in the given range. Table 1
3. Simulation of cost parameters 3.1. The operating cost Table 1 shows the operating cost used in this study which include the costs that vary with vehicle
Percentage Deviation at Month (t+l) -32-30-18-26 -24 -22 -20 -16 -16 -14 -12 -10 -E -6 -4 -2 -32
-
-
-
_
- 1.0 -
-20
- ,66 -
-
-
-
-
0
2
4
6
8 10 12 14 16 16
_
_
_
_
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_
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_
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_
_ _ _ - - - _ _ - _ - - - - _ - - - - _ _ _ _
-28 .jj .33.,j - - - _ - _ _ _ _ _ _ _ _ - _ _ - _ -24
_
_
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-
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,ofl - - 3 _
.l,,J - ,I,, - ,jJ _ _
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-10
-
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-
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- ,1~.19.6t,O5.05
14 - - _ - - - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ,22,11 ,67 - 16 - - - - _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ,20,&o20 18 - - - - - - - - - - _ _ _ _ _ _ - _ _ _ - - _ - .14.66 12
Fig.
5. Transition
-
-
-
-
probability
-
-
_
matrix
_
_
_
_
_
_
for the percentage
_
_
_
deviation
_
_
_
_
of interest
_ ,21,65,0,,07
rate
from
the fitted
line.
K.A.H.
Kobbacy,
S.D. Nicol/Inr.
also shows an example for simulating cost.
J. Production
the operating
3.2. Capital cost of new tractor unit The operating cost data used in this study relate to 32.5 tonne tractor units. To remain consistent, the capital cost of a replacement unit used in the simulation is the average price of five models in the 32.5 tonne category which is 534950. The data were obtained from Glass’s Guide for Commercial Vehicles [l l] and CAP Red Book
cm Table Annual Year
3 4 5 6 7 8 9 10
operating
cost with simulation
Operating
3.3. Resale values of second-hand
The purpose of the following analysis is to model the way in which a tractor unit of this type is expected to depreciate. The second hand value of a vehicle is likely to depend on the vehicles’s age, milage and condition in addition to the market forces, i.e. the levels of supply and demand. The Glass’s Guide [l l] gives expected resale values of second-hand commercial vehicles for two situations: (i) Basic Trade, or Basic T, which is the value a dealer would expect to pay for an average second-hand vehicle and (ii) the Guided Condition Retail, GCR, which is the price a commercial
Operating
cost at 1989
Simulation
example
(pounds) Q
1006 1760 2515 3270 4024 2264 3018 3898 5156 8300
800 1400 2000 2600 3200 1800 2400 3100 4100
Random fraction
Proportion of Q to be charged
Simulated (pounds)
0.89 0.15 0.25 0.08 0.60 0.28 0.54 0.99 0.58 0.50
1.546 1.510 0.650 0.412 1.140 0.692 1.056 1.686 1.112 1.000
1555.28 897.60 1634.75 1347.24 4587.36 1566.69 3187.01 6572.03 5733.47 8300.00
1
GCR value
19 784.25 15 154.73 11608.53 8892.13 6811.37 5217.51 3996.6 1 3061.41 2345.04 1796.30
cost
resale value of vehicle Basic T value
Diff.
Simulation Random
2 3 4 5 6 7 8 9 10
tractor units
example
cost at 1984 prices
Table 2 Example for simulating Year
273
1
(pounds)C21
L
Economics 36 (1994) 267-279
16 535.91 12 054.68 8787.86 6406.35 4670.23 3404.60 248 1.95 1809.34 1319.01 961.56
3248.34 3100.05 2820.67 2485.78 2141.14 1812.91 1514.66 1252.07 1026.03 834.74
0.1863 0.1288 0.3948 0.0672 0.6789 0.2418 0.1862 0.8876 0.7254 0.9380
example No.
Resale value 19 179.08 14 755.44 10494.93 8725.09 5357.57 4779.15 3714.58 1950.07 1600.76 1013.31
K.A.H.
274
Kohhacy,
S.D. Nicol,ilnt.
.I. Production
1990, this study was repeated with the program written in FORTRAN [14]. All the results shown below are based on the average results of repeating each simulation experiment 200 times. Preliminary investigations established that after 200 runs the average optimal vehicle replacement age shows negligible variations.
vehicle dealer would expect to be able to sell a vehicle for. Using regression analysis of the data obtained from Glass’s guide, the following relations were obtained: (i) Basic T = 22.683 x 0.729” (ii) GCR = 25.828 x 0.766” Both these models have good fit. These models represent the lower and upper limits for a resale value of a vehicle of age n. A simulated value of the resale price was then sampled from a uniform distribution whose limits are Basic T and GCR. Table 2 shows an example for simulating the resale value of trucks.
4.1. The one-cycle rent model The results obtained while have trends that are in broad agreement with those of Christer and Waller [2], have revealed changes of significantly different magnitudes. The optimal replacement age was about nine years. The main parameters which influence the replacement age are the cost parameters, namely the operating cost and the resale value and to a lesser extent the purchase cost of a new truck. Fig. 6 shows that an increase in the operating cost by 50% would cause a reduction of the optimal replacement age of the truck by around six
4. Simulation and sensitivity analysis A simulation program was written in Pascal 1989 which has the facility to examine the effect many parameters on the optimal replacement age a truck [ 131. To reconfirm some of the results and take into consideration the tax reforms in the UK
Capital
in of of to of
Cost
30,000
0"
-
Operating
0.0
cost
factor
of
a
New
Vehicle
35,000
=
Assumed Estimated
Operating Operating
0.5
cost and capital
Cost
40,000
45,000
1.5
2.0
Cost Cost
1.0 Operating
Fig. 6. Effect of operating
Economics 36 (1994) 267-279
Factor
cost of a new vehicle on its optimal
replacement
age (one cycle model).
K.A.H.
Knhbucy,
S.D. Nicollht.
J. Production
months, almost half that estimated by Christer and Waller [2]. Fig. 6 also shows that an increase in the purchase cost of the vehicle causes minor increase in the optimal replacement age. Fig. 7 demonstrates that the gradual increase of the current resale value of the truck causes slight reduction of the optimal replacement age until a certain critical point is achieved.
275
Economics 36 (1994) 267-279
At around 50% above the current resale value, we observed sudden severe reduction in the optimal age with any further increase of the resale value. This trend was not observed previously and prompted the need to repeat the simulation study to check for any error in the program. The repeated study [14] actually produced exactly the same trend.
10
Value Factor
Resale
0 0.0
1
.
I
.
=
1
Assumed Resale Price Estimated Resale Price I
.I
I
0.5
I1
I
*
1.0
1
.
.
.
.
1.5
Resale Value Factor
Fig. 7. Effect of resale value of vehicle on its optimal
I 0.95
I
I
1.00 Discount
Fig. 8. Effect of selecting
replacement
initial discount
1
1.05 Factor
age (one cycle model).
1.10
seed
factor on optimal
replacement
age (one cycle model)
1.15
276
K.A.H.
Kohhac:,. S.D. Nicol,‘Int.
J. Production
replacement age of the current vehicle tends to decrease with increasing the replacement age of the second vehicle. Figs. 11 and 12 demonstrate the effects of the cost parameters on the optimal replacement age of the current vehicle. The two-cycle model seems to be less sensitive to changes in the operating cost than the one-cycle model. Also the resale value of the first vehicle does not have the dramatic effect on its replacement age displayed in the results of onecycle model. The effect of changes in the trucks purchase cost is shown in Fig. 12 and has similar pattern to that in the one-cycle model. The effects of tax parameters on the optimal replacement age of the initial vehicle are minimal and are of much the same nature as in the case of one-cycle model.
The only economic parameter which could be tested in this study is the initial discount factor which reflects the current economic situation. Fig. 8 shows that the discount factor has no significant effect on the optimal age. Similarly the tax parameters including corporation tax rate, writing down allowance and tax payment delay were found to have minor or slight effect on the replacement age. Some of these influences are depicted in Fig. 9.
4.2. The two-cycle
rent model
In a two-cycle rent model we have two decision variables: K the time to replacement of current vehicle and L the replacement age of the second (replacement) vehicle. The important variable is K, since it represents, the current decision. Fig. 10 shows the effects of age of the first vehicle and the replacement age of the second vehicle. An increase in the age of the current vehicle leads to increase in its replacement age. In contrast, the
Writing
Down
Economic.s 36 (1994) 267-279
5. Conclusions There are two main conclusions for this study. The first is that the approach adopted, i.e. using
Allowance
(a)
6 0
10
20 Corporate
Fig. 9. Effects of corporate
tax rate and writing
down allowance
30 Tax Rate
40
(%)
on the optimal
replacement
age of vehicle (one cycle model).
K.A.H.
Kohhac~, S.D. Nicoli ht.
Replacement 2 i
I
l
0
4 ,
*
1. Pro&rim
Economics
Age of Second vehicle
'
*
s
2
vehicle
and
I”’
*
.
+J
_
8
I’
replacement
s
I
age of second
’
8
6
4
Resale Value
6 *i:
IYears)
6
Age of Initial Vehicle
Fig. 10. Effects of age of initial (two-cycle-model).
277
36 (1994) 267-279
(Years)
vehicle
on optimal
replacement
age of initial
Factor
5 626’ 8
-
2
-
2
.
a ;i
.; -I +J 8*
Resale Value Factor
6
t*ttiII-I‘IttttI*~.~ 0.5
0.0
operating
Assumed Resale Price = Estimated Resale Price
Cost Factor
Assumed Operatins Cost = Estimated Operating Cost
Fig. Il.
Effects of operating
1.5
1.0 Operating
Cost Factor
cost and resale value on the optimal
replacement
age of initial vehicle (two-cycle-model).
vehicle
K.A.H.
278
Kohhoc~~. S.D.
Nic~ol.‘Int. J. Production
Ecorzomics
f 1994)
36
267-279
10
6
I
.,,I,,,,
25,000
Capital
Fig. 12. Effect of capital
I 35,000
30,000
I
statistical simulation in analysing the sensitivity of rent replacement models has shown to be effective. The results of this study, though significantly different in magnitude, are in general agreement with the results of previous study using analysis of variance [2]. This confirms that the simulation of the relevant economic parameters (of the UK) using Markov chains is reasonable and realistic. The simulation approach, unlike AOV, has the advantage of versatility, i.e. the study can be repeated with the cost data of other capital equipment or economical parameters of another country at a minimal effort on the part of the analyst once the approach has been established. The second conclusion of this study concerns the replacement of the type of commercial vehicles studied. Having examined all cost and tax parameter over a relatively wide range, it was found that the optimal replacement age is not significantly sensitive to these parameters with the exception of the resale value of the vehicle. The increase of operating cost has produced moderate reduction in the optimal replacement age. While tax parameters, which are typically of concern to management, are the least influential with almost negligible contribution towards any results. Increasing the estimated resale value of vehicle by 50% or more caused the
I
I,,, 40,000
Cost of A New Vehicle
cost of a new vehicle on optimal
I
45,000
(Pounds)
replacement
age (two-cycle
model).
mean optimal replacement age to drop sharply. However, this effect disappears when the replacement vehicle is considered in two-cycle model. There is also doubt that such large increase in resale value can be realised in reality. In general, the mean optimal replacement age for tractor units of 32.5 tonne articulated lorries is approximately nine years. After nine years, a tractor unit’s resale price is close to its scrape value. This is not a surprising result in that the sales manager of a large commercial vehicle dealer in Scotland has suggested that in most cases firms tend to ‘run their tractor units into the ground before replacing them’.
Acknowledgments The authors wish to express sincere thanks to H.M. Darvesh and A. Davies for their effort in reconfirming some of the results of this study while working in their B.Sc. projects.
References [1 J Christer, A.H., 1984. Maintenance Notes, Umversity of Strathclyde.
and
Replacement
K.A.H. Kobbacy, S.D. Nicolllnt. J. Production Economics 36 (1994) 267-279 [2] Christer, A.H. and Waller, M.W., 1987. Tax-adjusted replacement models. J. Opl Res., 38: 99331006. [3] Churchman, C.W., Ackoff, R.L. and Arnoff, E.L., 1966. Introduction To Operations Research. Wiley, New York. [4] Russell, J.C., 1982. Vehicle replacement: A case study in adapting a standard approach for a large organisation. J. Opl Res., 33: 899991 I. [S] Organisation for Economic Co-Operation and Development, 1976. Interest Rates 1960&1974. [6] Department of Employment. HMSO (1986) Consumer Price Indexes. [7] Annual Abstract of Statistics. HMSO. [S] Kobbacy, K.A.H.. Christer, A.H. and MacCallum, K.J., 1982. Stochastic modelling of the saturation-diving-
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inspection activities of a North Sea oil platform. Project Mass, Task 3.3, Report No. 2, University of Strathclyde. [9] Isaacson, D.L. and Madsen, R.W., 1976. Markov Chains Theory and Applications. Wiley, New York, p. 12. [lo] White, D.J., 1975. Decision Methodology. Wiley, New York, p. 131. [1 I] Glass’s Guide Commercial Vehicles Values, June 1989 ed. [12] CAP, Current Auto Prices, Red Book Commercial Vehicle Values, June 1989 ed. [13] Nicol, SD., 1989. Sensitivity Analysis of Rent Models. BSc. Project Report, University of Strathclyde. [I43 Davies, A. and Darvesh, H.M., 1991. Sensitivity Analysis of the Rent Replacement Models. B.Sc Project Report, University of Salford.