ASA_0.1.ps
Real Options for Project Schedules (ROPS)
Lester Ingber
Lester Ingber Research
Ashland Oregon
ingber@ingber.com, ingber@alumni.caltech.edu
http://www.ingber.com/
Abstract
Real Options for Project Schedules (ROPS) has three recursive sampling/optimization shells. An outer
Adaptive Simulated Annealing (ASA) optimization shell optimizes parameters of strategic Plans
containing multiple Projects containing ordered Tasks. A middle shell samples probability distributions
of durations of Tasks. An inner shell samples probability distributions of costs of Tasks. PATHTREE is
used to develop options on schedules. Algorithms used for Trading in Risk Dimensions (TRD) are
applied to develop a relative risk analysis among projects.
KEYWORDS: options; simulated annealing; risk management; copula; nonlinear; statistical
† L. Ingber, “Real Options for Project Schedules (ROPS)”, Report 2007:ROPS, Lester Ingber Research,
Ashland, OR, 2007. URL http://www.ingber.com/markets07_rops.pdf.
$Id: markets07_rops,v 1.22 2007/04/01 14:32:24 ingber Exp ingber $
Lester Ingber - 2 -  Real Options for Project Schedules (ROPS)
1. Introduction
This paper is a brief description of a methodology of developing options (in the sense of financial options,
e.g., with all Greeks), to be applied in collaboration with Michael Bowman, as a first example to
scheduling a massive US Army project, Future Combat Systems (FCS) [1].
The major focus is to develop Real Options for non-financial projects, as discussed in other earlier
papers [3,4,12]. Data and some guidance on its use has been reported in a previous study of FCS [2,5].
The need for tools for fairly scheduling and pricing such a complex project has been emphasized in
Recommendations for Executive Action in a report by the U.S. General Accounting Office (GAO) on
FCS [14], and they also emphasize the need for management of FCS business plans [13].
2. Goals
A giv en Plan results in S(t), money allocated by the client/government is defined in terms of Projects
Si(t),
S(t) =
Σ Si(t)
where ai(t) may be some scheduled constraints. PATHTREE processes a probability tree developed over
the life of the plan T , divided into N nodes at times {tn}, each with mean epoch length dt [11]. Options,
including all Greeks, familiar to financial markets, are calculated for quite arbitrary nonlinear means and
variances of multiplicative noise [6,9]. This ability to process nonlinear functions in probability
distributions is essential for real-world applications.
Each Task has a range of durations, with nonzero Ai , with a disbursement of funds used, defining Si(tn).
Any Task dependent on a Task completion is slaved to its precursor(s).
We dev elop the Plan conditional probability density (CPD) in terms of differenced costs, dS,
P(S ± dS; tn + dt |S; tn)
P is modeled/cast/fit into the functional form
P(S ± dS; tn + dt |S; tn) = (2π g
2 exp(−Ldt)
(dS − fdt)2
(2g2dt2)
where f and g are nonlinear function of cost S and time t. The g2 variance function absorbs the multiple
Task cost and schedule statistical spreads, to determine P(dS, t), giving rise to the stochastic nature of
dollars spent on the Plan.
A giv en Project i with Task k has a mean duration iik , with a a mean cost Sik . The spread in dS has two
components arising from: (1) a stochastic duration around the mean duration, and (2) a stochastic spread
of mean dollars around a deterministic disbursement at a given time. Different finite-width asymmetric
distributions are used for durations and costs. For example, the distribution created for Adaptive
Simulated Annealing (ASA) [8], originally called Very Fast Simulated Re-annealing [7], is a finite-ranged
distribution with shape determined by a parameter “temperature” q. For each state (whether duration or
cost): (a) A random binary choice can be made to be higher or lower than the mean, using any ratio of
probabilities selected by the client. (b) Then, an ASA distribution is used on the chosen side. Each side
has a different q, each falling off from the mean. This is illustrated and further described in Fig. 1.
At the end of the tree at a time T (T also can be a parameter), there is a total cost at each node S(T ),
called a final “strike” in financial language. (A final strike might also appear at any node before T due to
cancellation of the Project using a particular kind of schedule alternative.) Working backwards, options
are calculated at time t0. Greeks (functional derivatives of the option) assess sensitivity to various
variables, e.g., like those discussed in previous papers [12], but here we deliver precise numbers based on
as much real-world information available.
Lester Ingber - 3 -  Real Options for Project Schedules (ROPS)
-1 -0.5  0  0.5  1
ASA (q = 0.1)
1/(2 * (abs(y) + q) * log(1 + 1/q))
Fig. 1. The ASA distribution can be used to develop finite-range asymmetric distributions
from which a value can be chosen for a given state of duration or cost. (a) A random binary
distribution is selected for a lower-than or higher-than mean, using any ratio of probabilities
selected by the client. Each side of the mean has its own temperature q. Here an ASA distri-
bution is given for q = 0.1. The range can be scaled to any finite interval and the mean
placed within this range. (b) A uniform random distribution selects a value from [-1,1], and
a normalized ASA value is read off for the given state.
3. Data
The following data are used to develop Plan CPD. Each Task i has
(a) a Projected allocated cost, Ci
(b) a Projected time schedule, Ti
(c) a CPD with a statistical width of funds spent, SWSi
(d) a distribution with a statistical width of duration, SWTi
(e) a range of durations, RTi
(f) a range of costs, RSi
Expert guesses need to be provided for (c)-(f) for the prototype study.
A giv en Plan must be constructed among all Tasks, specified the ordering of Tasks, e.g., obeying any
sequential constraints among Tasks.
4. Three Recursive Shell
4.1. Outer Shell
There may be several parameters in the Project, e.g., as coefficients of variables in means and variances of
different CPD. These are optimized in an outer shell using ASA [8]. This end product, including
MULTI_MIN states returned by ASA, gives the client flexibility to apply during a full Project [12]. We
may wish to minimize Cost/T , or (CostOverrun - CostInitial)/T , etc.
Lester Ingber - 4 -  Real Options for Project Schedules (ROPS)
4.2. Middle Shell
To obtain the Plan CPD, an middle shell of Monte Carlo (MC) states are generated from recursive
calculations. A Weibull or some other asymmetric finite distribution might be used for Task durations.
For a giv en state in the outer middle, a MC state has durations and mean cost disbursements defined for
each Task.
4.3. Inner Shell
At each time, for each Task, the differenced cost ((Sik(t + dt) − Sik(t))) is subjected to a inner shell
stochastic variation, e.g., some asymmetric finite distribution. The net costs dSik(t) for each Project i and
Task k are added to define dS(t) for the Plan. The inner shell cost CPD is re-applied many times to get a
set of {dS} at each time.
5. Real Options
5.1. Plan Options
After the Outer MC sampling is completed, there are histograms generated of the Plan’s dS(t) and
dS(t)/S(t − dt) at each time t. The histograms are normalized at each time to give P(dS, t). At each time
t, the data representing P is “curve-fit” to the form of Eq. (0), where f and g are functions needed to get
good fits, e.g., fitting coefficients of parameters {x}
f = x f 0 + x f 1S + x f 2S
2 + . . .
g = xg0 + xg1S + xg2S
2 + . . .
At each time t, the functions f and g are fit to the function ln((P(dS, t)), which includes the prefactor
containing g and the function L which may be viewed as a Padé approximate of these polynomials.
Complex constraints as functions of Sik(t) can be easily incorporated in this approach, e.g., due to regular
reviews by funding agencies or executives. These P’s are input into PATHTREE to calculate options for a
given strategy or Plan.
5.2. Risk Management of Project Options
If some measure of risk among Projects is desired, then during the MC calculations developed for the top-
level Plan, sets of differenced costs for each Project, dSi(t) and dSi(t)/Si(t − dt), stored from each of the
Project’s Tasks. Then, histograms and Project CPDs are developed, similar to the development of the
Plan CPD. A copula analysis, coded in TRD for risk management of financial markets, are applied to
develop a relative risk analysis among these projects [10]. In such an analysis, the Project marginal CPDs
are all transformed to Gaussian spaces, where it makes sense to calculate covariances and correlations.
An audit trail back the original Project spaces permits analysis of risk dependent on the tails of the Project
CPDs.
6. Generic Applications
ROPS can be applied to any complex scheduling of tasks similar to the FCS project. The need for
government agencies to plan and monitor such large projects is becoming increasingly difficult and
necessary [15]. Many large businesses have similar projects and similar requirements to manage their
complex projects.
Lester Ingber - 5 -  Real Options for Project Schedules (ROPS)
References
[1] M. Bowman and L. Ingber, ‘‘Real Options for US Army Future Combat Systems,’’ Report
2007:ROFCS, Lester Ingber Research, Ashland, OR, 2007.
[2] G.G. Brown, R.T. Grose, and R.A. Koyak, ‘‘Estimating total program cost of a long-term, high-
technology, high-risk project with task durations and costs that may increase over time,’’ Military
Operations Research 11, 41-62 (2006). [URL
http://www.nps.navy.mil/orfacpag/resumePages/papers/Brownpa/Estimating_total_
program_cost.pdf]
[3] T.E. Copeland and P.T. Keenan, ‘‘Making real options real,’’ McKinsey Quarterly 128-141 (1998).
[URL http://faculty.fuqua.duke.edu/˜charvey/Teaching/BA456_2006/McK98_3.pdf]
[4] G. Glaros, ‘‘Real options for defense,’’ Tr ansformation Trends June, 1-11 (2003). [URL
http://www.oft.osd.mil/library/library_files/trends_205_transforma-
tion_trends_9_june%202003_issue.pdf]
[5] R. Grose, ‘‘Cost-constrained project scheduling with task durations and costs that may increase
over time: Demonstrated with the U.S. Army future combat systems,’’ Thesis, Naval Postgraduate
School, Monterey, CA, 2004. [URL http://www.stormingmedia.us/75/7594/A759424.html]
[6] J.C. Hull, Options, Futures, and Other Derivatives, 4th Edition (Prentice Hall, Upper Saddle River,
NJ, 2000).
[7] L. Ingber, ‘‘Very fast simulated re-annealing,’’ Mathl. Comput. Modelling 12, 967-973 (1989).
[URL http://www.ingber.com/asa89_vfsr.pdf]
[8] L. Ingber, ‘‘Adaptive Simulated Annealing (ASA),’’ Global optimization C-code, Caltech Alumni
Association, Pasadena, CA, 1993. [URL http://www.ingber.com/#ASA-CODE]
[9] L. Ingber, ‘‘Statistical mechanics of portfolios of options,’’ Report 2002:SMPO, Lester Ingber
Research, Chicago, IL, 2002. [URL http://www.ingber.com/markets02_portfolio.pdf]
[10] L. Ingber, ‘‘Trading in Risk Dimensions (TRD),’’ Report 2005:TRD, Lester Ingber Research,
Ashland, OR, 2005.
[11] L. Ingber, C. Chen, R.P. Mondescu, D. Muzzall, and M. Renedo, ‘‘Probability tree algorithm for
general diffusion processes,’’ Phys. Rev. E 64, 056702-056707 (2001). [URL
http://www.ingber.com/path01_pathtree.pdf]
[12] K.J. Leslie and M.P. Michaels, ‘‘The real power of real options,’’ McKinsey Quarterly 4-22 (1997).
[http://faculty.fuqua.duke.edu/˜charvey/Teaching/BA456_2006/McK97_3.pdf]
[13] General Accounting Office, ‘‘Future Combat System Risks Underscore the Importance of
Oversight,’’ Report GAO-07-672T, GAO, Washington DC, 2007. [URL http://www.gao.gov/cgi-
bin/getrpt?GAO-07-672T]
[14] General Accounting Office, ‘‘Key Decisions to Be Made on Future Combat System,’’ Report
GAO-07-376, GAO, Washington DC, 2007. [URL http://www.gao.gov/cgi-
bin/getrpt?GAO-07-376]
[15] B. Wysocki, Jr, ‘‘Is U.S. Government ’Outsourcing Its Brain’?,’’ Wall Street Journal March 30, 1
(2007).

  Real Options for Project Schedules (ROPS) has three recursive
sampling/optimization shells. An outer Adaptive Simulated Annealing (ASA)
optimization shell optimizes parameters of strategic Plans containing multiple
Projects containing ordered Tasks. A middle shell samples probability
distributions of durations of Tasks. An inner shell samples probability
distributions of costs of Tasks. PATHTREE is used to develop options on
schedules.. Algorithms used for Trading in Risk Dimensions (TRD) are applied to
develop a relative risk analysis among projects.

Introduction
This paper is a brief description of a methodology of developing options (in the sense of financial options,
e.g., with all Greeks), to be applied in collaboration with Michael Bowman, as a first example to
scheduling a massive US Army project, Future Combat Systems (FCS) [1].
The major focus is to develop Real Options for non-financial projects, as discussed in other earlier
papers [3,4,12]. Data and some guidance on its use has been reported in a previous study of FCS [2,5].
The need for tools for fairly scheduling and pricing such a complex project has been emphasized in
Recommendations for Executive Action in a report by the U.S. General Accounting Office (GAO) on
FCS [14], and they also emphasize the need for management of FCS business plans [13].
2. Goals
A giv en Plan results in S(t), money allocated by the client/government is defined in terms of Projects
Si(t),
S(t) =
Σ Si(t)
where ai(t) may be some scheduled constraints. PATHTREE processes a probability tree developed over
the life of the plan T , divided into N nodes at times {tn}, each with mean epoch length dt [11]. Options,
including all Greeks, familiar to financial markets, are calculated for quite arbitrary nonlinear means and
variances of multiplicative noise [6,9]. This ability to process nonlinear functions in probability
distributions is essential for real-world applications.
Each Task has a range of durations, with nonzero Ai , with a disbursement of funds used, defining Si(tn).
Any Task dependent on a Task completion is slaved to its precursor(s).
We dev elop the Plan conditional probability density (CPD) in terms of differenced costs, dS,
P(S ± dS; tn + dt |S; tn)
P is modeled/cast/fit into the functional form
P(S ± dS; tn + dt |S; tn) = (2π g
2 exp(−Ldt)
(dS − fdt)2
(2g2dt2)
where f and g are nonlinear function of cost S and time t. The g2 variance function absorbs the multiple
Task cost and schedule statistical spreads, to determine P(dS, t), giving rise to the stochastic nature of
dollars spent on the Plan.
A giv en Project i with Task k has a mean duration iik , with a a mean cost Sik . The spread in dS has two
components arising from: (1) a stochastic duration around the mean duration, and (2) a stochastic spread
of mean dollars around a deterministic disbursement at a given time. Different finite-width asymmetric
distributions are used for durations and costs. For example, the distribution created for Adaptive
Simulated Annealing (ASA) [8], originally called Very Fast Simulated Re-annealing [7], is a finite-ranged
distribution with shape determined by a parameter “temperature” q. For each state (whether duration or
cost): (a) A random binary choice can be made to be higher or lower than the mean, using any ratio of
probabilities selected by the client. (b) Then, an ASA distribution is used on the chosen side. Each side
has a different q, each falling off from the mean. This is illustrated and further described in Fig. 1.
At the end of the tree at a time T (T also can be a parameter), there is a total cost at each node S(T ),
called a final “strike” in financial language. (A final strike might also appear at any node before T due to
cancellation of the Project using a particular kind of schedule alternative.) Working backwards, options
are calculated at time t0. Greeks (functional derivatives of the option) assess sensitivity to various
variables, e.g., like those discussed in previous papers [12], but here we deliver precise numbers based on
as much real-world information available.
Lester Ingber - 3 -  Real Options for Project Schedules (ROPS)
-1 -0.5  0  0.5  1
ASA (q = 0.1)
1/(2 * (abs(y) + q) * log(1 + 1/q))
Fig. 1. The ASA distribution can be used to develop finite-range asymmetric distributions
from which a value can be chosen for a given state of duration or cost. (a) A random binary
distribution is selected for a lower-than or higher-than mean, using any ratio of probabilities
selected by the client. Each side of the mean has its own temperature q. Here an ASA distri-
bution is given for q = 0.1. The range can be scaled to any finite interval and the mean
placed within this range. (b) A uniform random distribution selects a value from [-1,1], and
a normalized ASA value is read off for the given state.
3. Data
The following data are used to develop Plan CPD. Each Task i has
(a) a Projected allocated cost, Ci
(b) a Projected time schedule, Ti
(c) a CPD with a statistical width of funds spent, SWSi
(d) a distribution with a statistical width of duration, SWTi
(e) a range of durations, RTi
(f) a range of costs, RSi
Expert guesses need to be provided for (c)-(f) for the prototype study.
A giv en Plan must be constructed among all Tasks, specified the ordering of Tasks, e.g., obeying any
sequential constraints among Tasks.
4. Three Recursive Shell
4.1. Outer Shell
There may be several parameters in the Project, e.g., as coefficients of variables in means and variances of
different CPD. These are optimized in an outer shell using ASA [8]. This end product, including
MULTI_MIN states returned by ASA, gives the client flexibility to apply during a full Project [12]. We
may wish to minimize Cost/T , or (CostOverrun - CostInitial)/T , etc.
Lester Ingber - 4 -  Real Options for Project Schedules (ROPS)
4.2. Middle Shell
To obtain the Plan CPD, an middle shell of Monte Carlo (MC) states are generated from recursive
calculations. A Weibull or some other asymmetric finite distribution might be used for Task durations.
For a giv en state in the outer middle, a MC state has durations and mean cost disbursements defined for
each Task.
4.3. Inner Shell
At each time, for each Task, the differenced cost ((Sik(t + dt) − Sik(t))) is subjected to a inner shell
stochastic variation, e.g., some asymmetric finite distribution. The net costs dSik(t) for each Project i and
Task k are added to define dS(t) for the Plan. The inner shell cost CPD is re-applied many times to get a
set of {dS} at each time.
5. Real Options
5.1. Plan Options
After the Outer MC sampling is completed, there are histograms generated of the Plan’s dS(t) and
dS(t)/S(t − dt) at each time t. The histograms are normalized at each time to give P(dS, t). At each time
t, the data representing P is “curve-fit” to the form of Eq. (0), where f and g are functions needed to get
good fits, e.g., fitting coefficients of parameters {x}
f = x f 0 + x f 1S + x f 2S
2 + . . .
g = xg0 + xg1S + xg2S
2 + . . .
At each time t, the functions f and g are fit to the function ln((P(dS, t)), which includes the prefactor
containing g and the function L which may be viewed as a Padé approximate of these polynomials.
Complex constraints as functions of Sik(t) can be easily incorporated in this approach, e.g., due to regular
reviews by funding agencies or executives. These P’s are input into PATHTREE to calculate options for a
given strategy or Plan.
5.2. Risk Management of Project Options
If some measure of risk among Projects is desired, then during the MC calculations developed for the top-
level Plan, sets of differenced costs for each Project, dSi(t) and dSi(t)/Si(t − dt), stored from each of the
Project’s Tasks. Then, histograms and Project CPDs are developed, similar to the development of the
Plan CPD. A copula analysis, coded in TRD for risk management of financial markets, are applied to
develop a relative risk analysis among these projects [10]. In such an analysis, the Project marginal CPDs
are all transformed to Gaussian spaces, where it makes sense to calculate covariances and correlations.
An audit trail back the original Project spaces permits analysis of risk dependent on the tails of the Project
CPDs.
6. Generic Applications
ROPS can be applied to any complex scheduling of tasks similar to the FCS project. The need for
government agencies to plan and monitor such large projects is becoming increasingly difficult and
necessary [15]. Many large businesses have similar projects and similar requirements to manage their
complex projects.
Lester Ingber - 5 -  Real Options for Project Schedules (ROPS)
References
[1] M. Bowman and L. Ingber, ‘‘Real Options for US Army Future Combat Systems,’’ Report
2007:ROFCS, Lester Ingber Research, Ashland, OR, 2007.
[2] G.G. Brown, R.T. Grose, and R.A. Koyak, ‘‘Estimating total program cost of a long-term, high-
technology, high-risk project with task durations and costs that may increase over time,’’ Military
Operations Research 11, 41-62 (2006). [URL
http://www.nps.navy.mil/orfacpag/resumePages/papers/Brownpa/Estimating_total_
program_cost.pdf]
[3] T.E. Copeland and P.T. Keenan, ‘‘Making real options real,’’ McKinsey Quarterly 128-141 (1998).
[URL http://faculty.fuqua.duke.edu/˜charvey/Teaching/BA456_2006/McK98_3.pdf]
[4] G. Glaros, ‘‘Real options for defense,’’ Tr ansformation Trends June, 1-11 (2003). [URL
http://www.oft.osd.mil/library/library_files/trends_205_transforma-
tion_trends_9_june%202003_issue.pdf]
[5] R. Grose, ‘‘Cost-constrained project scheduling with task durations and costs that may increase
over time: Demonstrated with the U.S. Army future combat systems,’’ Thesis, Naval Postgraduate
School, Monterey, CA, 2004. [URL http://www.stormingmedia.us/75/7594/A759424.html]
[6] J.C. Hull, Options, Futures, and Other Derivatives, 4th Edition (Prentice Hall, Upper Saddle River,
NJ, 2000).
[7] L. Ingber, ‘‘Very fast simulated re-annealing,’’ Mathl. Comput. Modelling 12, 967-973 (1989).
[URL http://www.ingber.com/asa89_vfsr.pdf]
[8] L. Ingber, ‘‘Adaptive Simulated Annealing (ASA),’’ Global optimization C-code, Caltech Alumni
Association, Pasadena, CA, 1993. [URL http://www.ingber.com/#ASA-CODE]
[9] L. Ingber, ‘‘Statistical mechanics of portfolios of options,’’ Report 2002:SMPO, Lester Ingber
Research, Chicago, IL, 2002. [URL http://www.ingber.com/markets02_portfolio.pdf]
[10] L. Ingber, ‘‘Trading in Risk Dimensions (TRD),’’ Report 2005:TRD, Lester Ingber Research,
Ashland, OR, 2005.
[11] L. Ingber, C. Chen, R.P. Mondescu, D. Muzzall, and M. Renedo, ‘‘Probability tree algorithm for
general diffusion processes,’’ Phys. Rev. E 64, 056702-056707 (2001). [URL
http://www.ingber.com/path01_pathtree.pdf]
[12] K.J. Leslie and M.P. Michaels, ‘‘The real power of real options,’’ McKinsey Quarterly 4-22 (1997).
[http://faculty.fuqua.duke.edu/˜charvey/Teaching/BA456_2006/McK97_3.pdf]
[13] General Accounting Office, ‘‘Future Combat System Risks Underscore the Importance of
Oversight,’’ Report GAO-07-672T, GAO, Washington DC, 2007. [URL http://www.gao.gov/cgi-
bin/getrpt?GAO-07-672T]
[14] General Accounting Office, ‘‘Key Decisions to Be Made on Future Combat System,’’ Report
GAO-07-376, GAO, Washington DC, 2007. [URL http://www.gao.gov/cgi-
bin/getrpt?GAO-07-376]
[15] B. Wysocki, Jr, ‘‘Is U.S. Government ’Outsourcing Its Brain’?,’’ Wall Street Journal March 30, 1
(2007).
