Computers & Industrial Engineering 45 (2003) 285–294
www.elsevier.com/locate/dsw
Using simulation software to solve engineering economy problems
Eyler R. Coatesa,*, Michael E. Kuhlb,1
a
School of Engineering Technology, The University of Southern Mississippi, Box 5137, Hattiesburg, MS 39406-5137, USA
b
Industrial and Systems Engineering Department, Rochester Institute of Technology, Rochester, NY 14623-5603, USA
Abstract
Some information required for solving problems in engineering economy problems can be fairly well defined.
Much required information is uncertain, such as the actual cash flows from revenues and costs, the salvage value of
equipment, the interest rate or even the project life. Engineering economy problems with all deterministic inputs
are actually special cases. Probability descriptions of input variables and Monte Carlo sampling together provide a
practical method of finding the distribution of the desired output given the various random and deterministic input
variables. This paper provides three examples that demonstrate how commonly available simulation software
could be used in engineering economy problems. One example is demonstrated that generates the distribution
future worth of an annual series of payments when there is uncertainty about the future earning power (interest
rate) from year to year. Also, an example is demonstrated that models the uncertainty of interest rates and the
uncertainty of project life in order to generate the NPV distribution of a project. Finally, an example is presented to
show the use of simulation in comparing alternative investment opportunities under uncertainty. These examples
can be used to demonstrate how risk is handled in an engineering economy course. The examples can also be used
as additional applications in an industrial simulation course.
q 2003 Elsevier Science Ltd. All rights reserved.
Keywords: Simulation applications—manufacturing, engineering economics, project risk, net present value distributions
1. Introduction
Some information required for an engineering economic problem can be well defined, like the
cost of new machinery or the current tax rate structure. Much required information is uncertain,
such as the actual cash flows from revenues and costs, the salvage value of equipment, the interest
rate or even the project life. In most undergraduate engineering economy courses, the concept of
risk is either not introduced at all or is mentioned briefly at the end of a course and in final
* Corresponding author. Tel.: þ1-601-266-6421; fax: þ1-601-266-5717.
E-mail address: mekeie@rit.edu (M.E. Kuhl).
E-mail address: eyler.coates@usm.edu (E.R. Coates).
1
Tel.: þ1-716-475-2134, fax: þ 1-716-578-2520.
0360-8352/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0360-8352(03)00036-6
286
E.R. Coates, M.E. Kuhl / Computers & Industrial Engineering 45 (2003) 285–294
chapters of a textbook (Goyal, Tien, & Voss, 1997). Yet, engineering economy problems with all
deterministic inputs are actually special cases in practice.
There are a number of approaches for handling economic risk. An easy approach is scenario
analysis. However, as Park (1997) has pointed out, worst-case and best-case scenarios are not easy
to interpret and do not provide probabilities of occurrence of those possibilities nor do they
normally provide additional information such as the probability of losing money on a project or the
probability of other possibilities. Sensitivity analysis and spider plots can provide insight into
engineering economy problems but are not appropriate when there is statistical dependence between
variables (Eschenbach & Gimpel, 1990). Probability descriptions of input variables allow further
refinement of the analysis of economic risk and allow the output of a distribution for the desired
answer.
In his groundbreaking paper, Hillier (1963) proposed the use of probability distributions of the present
worth to properly convey project risk information to augment other methods such as expected value of
the present worth and sensitivity analysis of individual inputs. He also demonstrated that the probability
distribution of the present worth, or annual worth or internal rate of return can under certain assumptions
be derived from yearly cash flows that are themselves random variables. He presented equations for the
NPV distribution parameters when the cash flows were mutually independent random variables and
when the cash flows were random but perfectly correlated to each other. Later, Giaccotto (1984) derived
the distribution parameters for the NPV when the cash flows were correlated by a first order
autoregressive stochastic process (or a Markov process). As projects become complicated, derivations of
the probability distribution of the NPV as a function of the unknown random input variables can be
tedious or impossible (Seila & Banks, 1990). Thus, Monte Carlo sampling provides a practical method
of finding the distribution of the NPV from the various random input variables.
Coats and Chesser (1982) showed that Monte Carlo techniques could be used in the corporate
financial model to produce associated probabilities of occurrence, confidence intervals and standard
deviations in addition to standard financial reports. Seila and Banks (1990) showed that an electronic
spreadsheet could be used to simulate project risk with Monte Carlo techniques. The methods for risk
analysis of projects have been published for a long time and the availability of computers and software is
pervasive. Ho and Pike (1998) report that ‘proponents of risk analysis argue that increased risk
information improves management’s understanding of the nature of risks, helps identify the major
threats to project profitability and reduces forecasting errors’. They also report that ‘the risk analysis
approach provides useful insights into the project, improves decision quality and increases decision
confidence.’ Still, recently, Goyal et al. (1997) have questioned why there is so little exposure to the
stochastic nature of project cash flows and other project variables in undergraduate curricula.
Computer simulation using Monte Carlo techniques has been a part of the industrial engineering and
management science curriculums for many years. Fortunately, simulation packages that are already
available to students are particularly well suited for sampling from various theoretical and data-defined
statistical distributions. In addition, these simulation packages can handle large amounts of sampling
data and they have good output reporting capabilities. Thus, students have access to off-the-shelf
simulation packages either through their school or through student versions of professional simulation
packages that often come with their textbooks. As these students become professionals, they would be
able to access these same simulation packages. The next sections will demonstrate the ease that
engineering economy problems with stochastic input variables can be simulated with industrial
simulation software.
E.R. Coates, M.E. Kuhl / Computers & Industrial Engineering 45 (2003) 285–294
287
2. Problem 1: a future worth problem with uncertain interest rates
Suppose that we want to save $10,000 per year for the next 10 years. We would naturally be interested
in how much worth would be accumulated at the end. If the interest rate was fixed and known, then the
calculation of the future worth would be straightforward. However, there are many investment
possibilities where the interest rate is variable from year to year.
For our example, let us assume a stock market investment where the long-term average return is
12%, but the individual yearly returns are normally distributed with a standard deviation of 6% (i.e.
there could be negative returns in certain years). The standard future worth formula ignores this
variability and can only give a point estimate. Simulation would be the more appropriate approach.
Simulation software such as SLAM II could be used to estimate the NPV distribution by the following
steps:
†
†
†
†
Each entity in the simulation run will represent a replication (scenario) of the future worth calculation.
After an entity is created, the interest rates are selected via sampling from distributions.
The future worth associated with the entity is calculated and stored as an attribute.
Before the entity leaves the system, the future worth is collected as a statistic and tabulated by the
simulation software.
† The network diagram to simulate the problem is given in Fig. 1.
The output of the simulation program is given in Fig. 2. Because each entity was created one time unit
apart, the current time on the SLAM II report also reflects the number of replications in the study (5000).
The summary statistics are automatically generated. The future worth ranges from $109,000 to
$243,000. Based on the histogram, it appears that 90% of the observations fall between $135,000 and
$195,000. Incidentally, if we had used the standard future worth calculation with 10 years at 12% return,
the point estimate would have been $175,487 with no indication of the probable range.
Fig. 1. Network diagram for future worth estimation with variable interest rates.
288
E.R. Coates, M.E. Kuhl / Computers & Industrial Engineering 45 (2003) 285–294
Fig. 2. Simulation output from network in Fig. 1.
3. Problem 2: simulating randomness in project life and interest rates
Using Hillier (1963) original example, Table 1 gives the mutually independent normally distributed
cash flows. Assume that the interest rate, i; is 10%. Hillier has shown that the resultant net present value
(NPV) is also normally distributed with the parameters given by Eqs. (1) and (2).
mNPV ¼
5
X
Eðxj Þ
2400
200
þ ··· þ
¼ 95
j ¼
0
ð1 þ iÞ
ð1:1Þ
ð1:1Þ5
j¼0
ð1Þ
E.R. Coates, M.E. Kuhl / Computers & Industrial Engineering 45 (2003) 285–294
289
Table 1
Estimated net cash flow data
Year, j
Expected value, Eðxj Þ
Standard deviation
0
1
2
3
4
5
2 400
þ 120
þ 120
þ 120
þ 110
þ 200
20
10
15
20
30
50
s2NPV
5
X
Varðxj Þ
ð20Þ2
ð50Þ2
¼
¼
þ
·
·
·
þ
¼ 2247 ) sNPV ¼ 47:4
2j
ð1:1Þ0
ð1:1Þ10
j¼0 ð1 þ iÞ
ð2Þ
However, one can add complications to the problem that will make closed form equations for the NPV
difficult or impossible to derive. Suppose that the interest rate can vary from year to year and the project
life is uncertain. Then, it would be more practical to use simulation to estimate the NPV distribution.
Simulation software such as SLAM II could be used to estimate the NPV distribution by the following
steps:
†
†
†
†
†
Each entity in the simulation run will represent a replication (scenario) of the NPV calculation.
After an entity is created, input data is selected via sampling from distributions.
The sampled data is attached to the entity as attributes.
The NPV associated with the entity is calculated and stored as another attribute.
Before the entity leaves the system, the NPV is collected as a statistic and tabulated by the simulation
software.
For the example, the expected values of each year’s cash flows are stored in an array as follows:
ARRAY(1,1) ¼ 2400, ARRAY(1,2) ¼ 120,…, ARRAY(1,6) ¼ 200. The standard deviations are also
stored in an array as follows: ARRAY(2,1) ¼ 20, ARRAY(2,2) ¼ 10,…, ARRAY(2,6) ¼ 50.
In this model, the interest rate is allowed to take on an initial random starting value (with mean of 10%
and a standard deviation of 0.5%) and each subsequent year’s rate is generated by a first order
autoregressive stochastic process. In this example, the correlation of interest rates from one year to the
next is 80% and the noise component is normally distributed with 0 mean and 1% standard deviation.
From Giaccotto (1984), this type of autoregressive series can be modeled by
X~ t ¼ fX~ t21 þ d þ a~t
where
f is the correlation between adjacent time series data points,
d is a constant such that d=ð1 2 fÞ is the steady state mean of X~ t ;
a~t is a white noise series.
The project life span itself is also allowed to vary from 4 to 6 years with a 25% chance that the
project will last 4 years, a 50% chance that the project will last 5 years and a 25% chance that
290
E.R. Coates, M.E. Kuhl / Computers & Industrial Engineering 45 (2003) 285–294
the project will last 6 years. It takes little effort to build the simulation network to accommodate
variability in yearly cash flows, interest rates over the life of the project and variability in the
project life span itself. This is evident by the elegant structure of the network in Fig. 3. The array
of cash flow parameters is expanded by the addition of ARRAY(1,7) ¼ 200 and ARRAY(2,7) ¼ 50
to accommodate the new possibility of cash flow in the sixth year of the project. The output of
the simulation program with variable cash flows, interest rates and project life span as simulated in
Fig. 3 is given in Fig. 4.
Fig. 3. Network diagram for NPV problem with variable cash flows, interest rate and project life.
E.R. Coates, M.E. Kuhl / Computers & Industrial Engineering 45 (2003) 285–294
291
Fig. 4. Simulation output from network in Fig. 3.
Coates (1999) has already indicated the importance of including the variability of interest rates and
project life when appropriate. He noted that the probability of a negative NPV went from 2.3% in
Hillier’s original problem to 20.9% when the additional variability of the interest rate and the project life
was included. The mean NPV for both problems was essentially unchanged, but the standard deviation
of the NPV distribution had doubled. Thus, using expected values only and ignoring variability of input
data has its perils.
4. Problem 3: comparing alternative investments under uncertainty
The comparison of alternative systems is one of the most important uses of simulation (Banks,
Carson, & Nelson, 1996). In this case of engineering economy problems, the alternative systems often
take the form of alternative projects. Furthermore, these projects are often mutually exclusive. That is,
the decision maker can choose only one of the alternative projects to invest in. Therefore, an analysis of
the alternatives can be conducted to select the best investment. As discussed earlier, as the projects
become more complex, simulation provides a convenient and powerful means of conducting such an
analysis.
The following example illustrates the use of simulation to compare the investments of two alternative
projects based on their net present value. For simplicity, we will let the first alternative be a fixed 5 year
project having the same interest rate and cash flow distributions described in Section 3 where the net
expected cash flows for each period are described in Table 1. The second alternative also has the same
characteristics with the exception of the net expected cash flows which are shown in Table 2.
A common method used for comparing alternatives is to compare the difference in the expected net
present values for the two investments. To do this, a confidence interval can be constructed on
292
E.R. Coates, M.E. Kuhl / Computers & Industrial Engineering 45 (2003) 285–294
Table 2
Estimated net cash flow data for Alternative 2
Year, j
Expected value, Eðxj Þ
Standard deviation
0
1
2
3
4
5
2 600
2 100
þ 220
þ 250
þ 300
þ 450
10
40
50
50
70
100
the difference between population means. Therefore, we can construct a simulation model for each
alternative as before and obtain independent observations of the net present value for each project.
However, since we are using simulation to obtain these observations, variance reduction techniques such
as common random numbers can be used to improve the quality of our estimate (Law & Kelton 1991).
With the introduction of common random numbers, the corresponding independent observations of the
net present value from each project will be treated as matched pairs when constructing the confidence
interval.
The network diagram of the simulation model that incorporates common random numbers is shown in
Fig. 5. Here, as before, the mean and standard deviation of the yearly cash flows are stored in a two
dimensional array. Statistics are collected on the net present value of each alternative. In addition, the
5000 independent observations of NPV are written to the output file ‘NPV.DAT’. These data are then
read into a spreadsheet program to construct the confidence interval. (Note that some simulation
packages provide more elaborate output analysis tools which enable you to conduct this type of
analysis).
Fig. 5. Network diagram for Problem 3.
E.R. Coates, M.E. Kuhl / Computers & Industrial Engineering 45 (2003) 285–294
293
A point estimate of the mean difference in net present value between Alternative 2 and Alternative 1,
respectively, is 65.59 and a 95% confidence interval of the mean difference is
½63:79 # ðm2 2 m1 Þ # 67:39:
From this confidence interval we can conclude that on average investing in Alternative 2 will yield a
significantly higher return than Alternative 1. However, in most investment problems, the decision
maker will only have one opportunity to invest in any particular project. Therefore, perhaps a more
appropriate analysis tool would be to construct a tolerance interval (or prediction interval) on the
difference in NPV for a single investment. That is, one can construct a tolerance interval such that the
probability that a single observation will fall within the interval is at least some specified quantity
(Pritsker, 1995). For this example, a 95% tolerance interval on the difference in net present value on a
single investment is
½261:46 # ðX2 2 X1 Þ # 192:65:
This tolerance interval indicates that a single investment in either Alternative 1 or Alternative 2 may
yield a higher return over the other investment opportunity. Consequently, this method of analysis
provides information to the decision maker on one-time only investments that is often overlooked.
As the alternatives under consideration become more complex in terms of the uncertainty involved in
cash flow characteristics, interest rates, etc. simulation provides a means of analyzing problems when
closed form analytical solutions do not exist.
5. Conclusion
This paper provided three examples that demonstrate how commonly available simulation software
could be used in engineering economy problems. One example was demonstrated that generates the
distribution future worth of an annual series of payments when there is uncertainty about the future
earning power (interest rate) from year to year. Also, an example was demonstrated that models the
uncertainty of interest rates and the uncertainty of project life in order to generate the NPV distribution
of a project. Finally, an example of using simulation for the comparison of alternative investments under
uncertainty was presented. These examples can be used to demonstrate how risk is handled in an
engineering economy course. The examples can also be used as additional applications in an industrial
simulation course.
References
Banks, J., Carson, J. S., II, & Nelson, B. L. (1996). Discrete-event system simulation (2nd ed). New Jersey: Prentice Hall.
Coates, E. R. (1999). Project risk simulation using off-the-shelf simulation software. Proceedings of the 25th International
Conference on Computers and Industrial Engineering, New Orleans, LA, March, 457– 460.
Coats, P. K., & Chesser, D. L. (1982). Coping with business risk through probabilistic financial statements. Simulation, June,
111– 121.
Eschenbach, T. G., & Gimpel, R. J. (1990). Stochastic sensitivity analysis. The Engineering Economist, 35(4), 305– 321.
Giaccotto, C. (1984). A simplified approach to risk analysis in capital budgeting with serially correlated cash flows. The
Engineering Economist, 29(4), 273– 286.
294
E.R. Coates, M.E. Kuhl / Computers & Industrial Engineering 45 (2003) 285–294
Goyal, A. K., Tien, J. M., & Voss, P. A. (1997). Integrating uncertainty considerations in learning engineering economy. The
Engineering Economist, 42(3), 249– 257.
Hillier, F. S. (1963). The derivation of probabilistic information for the evaluation of risky investments. Management Science,
443– 457.
Ho, S. S. M., & Pike, R. H. (1998). Organizational characteristics influencing the use of risk analysis in strategic capital
investments. The Engineering Economist, 43(3), 247– 268.
Law, A. M., & Kelton, W. D. (1991). Simulation modeling and analysis (2nd ed). New York: McGraw-Hill.
Park, C. S. (1997). Contemporary engineering economics (2nd ed). Menlo Park, CA: Addison-Wesley.
Pritsker, A. A. B. (1995). Introduction to simulation and SLAM II (4th ed). New York: Wiley.
Seila, A. F., & Banks, J. (1990). Spreadsheet risk analysis using simulation. Simulation, 57, 163– 170.