Regular Paper
Modeling Complex Systems of Systems with
Phantom System Models
Yacov Y. Haimes*
Center for Risk Management of Engineering Systems, University of Virginia, 112A Olsen Hall, Charlottesville, VA 22903
MODELING COMPLEX SoSs WITH PHANTOM SYSTEM MODELS
Received 16 February 2011; Revised 23 July 2011; Accepted 20 September 2011, after one or more revisions
Published online in Wiley Online Library (wileyonlinelibrary.com).
DOI 10.1002/sys.21205
ABSTRACT
Complex systems are commonly composed of interconnected and inter- and intradependent subsystems,
which in their essence constitute systems of systems with multiple functions, operations, and stakeholders.
Phantom System Models (PSM) is a modeling methodology inspired by philosophical and conceptual
thinking from the arts, and is driven and supported by systems engineering theory, methodology, and
practice. The PSM is designed to model inter- and intradependencies between and among the subsystems
of a complex system of systems by exploiting vital knowledge and information embedded in the intrinsic
and extrinsic common and uncommon state variables among the subsystems. Among the several systems
engineering theories and methodologies, the PSM in particular builds on the centrality of the states of the
system in modeling and in risk analysis; fundamentals in system identification (the inverse problem);
hierarchical holographic modeling; coordinated hierarchical Bayesian model; and hierarchical decompo-
sition and higher-level coordination. An example problem of a PSM-based modeling of a prototype system
of systems is presented. © 2012 Wiley Periodicals, Inc. Syst Eng:15
Key words: systems of systems; Phantom System Models; state variables; system identification; meta-
modeling
(e.g., random, decision, and exogenous variables), quantify-
PREFACE: ON SYSTEM S MODELING, SYSTEM
ing intra- and interdependencies within and among its various
IDENTIFICATION, AND THE INVERSE
components and subsystems, and determining the appropriate
PROBLEM
model topology (structure) and parameters that best represent
its essence and functionality. To achieve this, modelers rely
Modeling a simple system, or a complex system of systems
extensively on data collection through testing, observation,
(SoS), necessarily implies determining its properties, con-
experimentation, and measurement, and through a tedious
structing the relationships among its inputs and outputs
learning process about the  system, including the use of
through its state variables and other variables and parameters
simulation. In this context, a  system may connote an exist-
ing or newly reconfigured multiple system that may span
physical, organizational, societal, and political entities. (In
*E-mail: [e-mail]
principle, not a dissimilar process may be followed for a
Contract grant sponsor: National Science Foundation (Award No. 0928550:
newly planned or to be constructed system.)
Adaptive Systems-Based Prioritization of Bridge Infrastructure Mainte-
There is a fundamental difference between the complexity
nance: Integrated Modeling of Technical, Socio-Economic, and Normative
and challenges associated with modeling physical systems
Dimensions).
which in their essence are controlled and driven by physical
laws and the challenges that characterize organizational and
Systems Engineering
© 2012 Wiley Periodicals, Inc. social systems, as well as combinations of multiple types of
1
2 HAIMES
systems. For example, to model groundwater systems, model- could lead to optimizing a system with a poorly constructed
ers build on the basic flow equation and start with Darcy s or misrepresentative model. This reality was recognized and
Law, which characterizes the slow flow of a compressible gained the interest and contributions of many researchers in
fluid through porous media. Indeed, a plethora of models of the 1960s and 1970s in books, technical reports, and archival
two- and three-dimensional partial differential equations have papers on system identification often termed as the  inverse
been successfully developed and deployed worldwide [NRC, problem. For example, see Eykhoff [1974], Graupe [1972],
1984]. However, entirely different challenges face modelers and Haimes [1970]. In system optimization we assume
who attempt to model a complex system of systems that knowledge of the system model, under specific assumptions,
represents a combination of interconnected physical, organ- where, for each set of inputs, we can generate, or prob-
izational, social, and political systems. For example, the Fed- abilistically estimate, the outputs. For example, in the context
eral Aviation Administration (FAA) is developing the of risk management, no effective risk management policy
 NextGen [next generation], whose mission involves myriad options can be developed; nor can the associated tradeoffs
technology-based systems and dozens of U.S. and interna- among all critical costs, benefits, and risks be evaluated; and
tional agencies and organizations, in order to develop tech- neither can the impacts of current decisions on future options
nologies and procedures to improve airspace redesign to be assessed, without having constructed a model, or a set of
enable more direct routes and more efficient operations, ex- interdependent models, that represent the essence of the sys-
pand satellite-based surveillance, improve airport runway tem.
access, increase safety and efficiency on the ground, enhance The fact that modeling is as much an art as a science a
airspace safety and operations, use less fuel and reduce emis- tedious investigative trial-and-error, learn-as-you-go proc-
sions and air pollution, and enable more direct routes, among ess means that an equally imaginative approach is necessary
other goals and objectives. to discover the inner functionality of complex systems
Among the many challenges facing modelers is the need through modeling. In this context, this paper (i) addresses the
to determine the ways and means with which to enhance their inverse problem, or the system identification problem,
knowledge about the system, discover its dynamic behavior, through the Phantom System Models (PSM); (ii) analyzes the
and identify the intra- and interdependencies among its sub- contributions of PSM as a modeling mechanism through
systems and its environment all by adhering to physical and which to experiment with creative approaches to modeling
other natural laws, basic principles in economics, and social complex SoS; and (iii) relates (at the meta-modeling level) the
and organizational behavior, among others. In many ways, intrinsic common state variables among the subsystems of the
modeling is the ultimate trial-and-error interplay between (i) SoS, thereby offering more insight into the intra- and interde-
theory and prior knowledge about the system and (ii) experi- pendencies among the subsystems.
mentation, measurement, and estimation, guided by a learn-
as-you-go inquisitive and exploratory process. Models are
2. WHAT HAVE WE LEARNED FROM OTHER
built to answer specific questions; they must be as simple as
possible but as complex as required. This tradeoff is at the CONTRIBUTORS?
heart of model building, given that overcomplexity within a
model is likely to impair its usefulness. Furthermore, the fact Reflecting on the history of modern systems theory, and its
that all systems are ultimately affected by human actions close ties to the Gestalt psychology first introduced in 1912,
(among others) implies the necessity of recognizing and ac- we cannot underestimate the intellectual power of this mul-
counting for human cognition, perception, and behavior. tidisciplinary field and the holistic philosophy that has sus-
tained it, allowing it to transcend the arts, the humanities, the
The nontrivial challenge associated with the modeling
natural, social, and physical sciences, as well as engineering,
process of one system is magnified when modeling complex
medicine, and law. The fact that systems engineering and
systems of systems. This challenge, which constitutes the
systems analysis have continued to grow and infiltrate other
theme of this paper, is addressed by exploring the centrality
fields of study over the years can be attributed to the funda-
of the states of a system as the major building blocks of
mental premise that a system can be understood only if all the
models. This modeling process also represents the mechanism
intra- and interdependencies among its parts and its environ-
with which to model each subsystem within the SoS, and to
ment are also understood and accounted for. For more than a
relate, through meta-modeling, the intrisic interdependencies
of common and uncommon state variables among the subsys- century, particular mathematical models, upon which sys-
tems-based theory and methodologies were developed, have
tems, which in turn enables a deeper understanding of the
been deployed in myriad large-scale projects in the natural
entire SoS.
and constructed environments. Moreover, if we were to iden-
tify a single concept that has dominated systems thinking and
1. INTRODUCTION modeling, it would be the state space. Indeed, the centrality
of state variables in this context is so dominant that no
There is an unfortunate imbalance in the curricula of most meaningful mathematical model of a real system can be built
undergraduate and graduate programs in systems and indus- without identifying the states of that system and relating all
trial engineering and in operations research devoted to system other building blocks of the model to them (including deci-
modeling versus system optimization (whether modeling sys- sion, random, and exogenous variables, and inputs and out-
tems with single or multiple objectives). Such imbalance in puts). (More will be discussed on the centrality of state
education and experience, and possibly in knowledge as well, variables in modeling as it relates to the entire theme of this
Systems Engineering DOI 10.1002/sys
 MODELING COMPLEX SoSs WITH PHANTOM SYSTEM MODELS 3
paper.) In this respect, the art and science of systems modeling (v) Evolutionary Development. A system of systems is
has served, in many ways, as the medium through which the
never fully formed or complete. Development of
holistic systems philosophy has informed the practice not
these systems is evolutionary over time and with
only of engineering, but of a broad range of other fields. As
structure, function and purpose added, removed, and
the discipline of systems engineering continues to develop
modified as experience with the system grows and
and expand its domains of application, the need for new
evolves over time.
organizational and modeling paradigms to represent complex
systems has emerged, and has ultimately led to the study of
Building on the above five principles, this paper attempts to
systems of systems.
improve our understanding of systems of systems by extend-
Complex systems are commonly composed of myriad
ing the multiperspective modeling schema (through hierarchi-
subsystems, which in their essence constitute systems of sys-
cal holographic modeling (HHM) [Haimes 1981, 2009]) into
tems. Each complex system is characterized by a hierarchy of
the phantom system models.
interacting components, with multiple functions, operations,
Several modeling philosophies and methods have been
efficiencies, costs, and stakeholders. Clearly, no single model
developed over the last five decades to address the complexity
can ever attempt to capture the essence of such systems their
of modeling complex large-scale systems and to offer various
multiple dimensions and perspectives. Indeed, almost every
modeling schema. They are included in the following vol-
living entity, all infrastructures, and both the natural and
umes: New Directions in General Theory of Systems [Me-
constructed environment, are systems of systems [Haimes
sarovic, 1965]; General Systems Theory [Macko, 1967];
2008, 2009a]. For example, different organs and parts of the
Systems Theory and Biology [Mesarovic, 1968]; Advances in
human body, as a system of systems, are continuously bom-
Control Systems, [Leondes, 1969]; Theory of Hierarchical
barded by a variety of bacteria, viruses, and other pathogens;
Multilevel Systems [Mesarovic, Mako, and Takahara, 1970];
however, only a subset of the (states of the) human body is
Methodology for Large Scale Systems [Sage, 1977]; Systems
vulnerable to the threats from yet another subset of the would-
Theory: Philosophical and Methodological Problems
be attackers, and due to our immune system, only a smaller
[Blauberg, Sadovsky, and Yudin, 1977]; Hierarchical Analy-
subset of the human body would experience adverse effects.
ses of Water Resources Systems: Modeling and Optimization
Thus composites of low-level, measurable states integrate to
of Large-Scale Systems [Haimes, 1977]; and Multifaceted
define higher-level fundamental state variables that charac-
Modeling and Discrete Event Simulation [Zigler, 1984].
terize the system. Indeed, the vulnerability of a system is a
Synectics, the Development of Creative Capacity [Gordon,
manifestation of the inherent states of that system, and each
1968] introduced an approach that uses metaphoric thinking
state of a system can be dynamic and change in response to
as a means to solve complex problems. Gheorghe [1982]
inputs, other random variables, and the building blocks of
presented the philosophy of systems engineering as it is
mathematical models (as discussed in the next section).
applied to real-world systems. Hall [1989] developed a theo-
The precise definition of SoS, however, is more elusive. In
retical framework to capture the multiple dimensions and
a seminal paper, Sage and Cuppan [2001] directly ask,  What
perspectives of a system. Other works include Sage [1977,
is a system of systems? They conclude,  Unfortunately, there
1992, 1995], Shenhar [1994], and Sage and Rouse [1999].
is no universally accepted definition of these  super systems.
Eisner [1993], Maier [1998], and Sage and Cuppan [2001]
What distinguishes a system of systems from other systems
provide valuable insight into systems of systems and defini-
does not at this point have a definitive answer. In a more
tions of emergent behavior of complex systems in the context
recent paper, Sage and Biemer [2007] provide the following
of systems of systems.
answer to the same question:  No universally accepted defi-
Most of the works on systems of systems have been
nition of an SoS is available at this time. To address this
devoted to their organizational, functional, and structural
problem, Sage and Cuppan [2001] build on the following five
nature; on the other hand, there has been comparatively less
properties of systems of systems suggested by Maier [1998]:
inquiry into the problem of modeling systems of systems, and
much of it has emerged within the last decade. For example,
(i) Operational Independence of the Individual Sys-
Ottino [2003] reviews three major tools for quantitative mod-
tems. A system of systems is composed of systems
eling and studying complex systems: nonlinear dynamics,
that are independent and useful in their own right.
agent-based models, and network theory. Shalizi [2006] also
(ii) Managerial Independence of the Systems. The com-
reviews the main methods and techniques of complex sys-
ponent systems not only can operate independently;
tems, which include tools for analyzing data, constructing and
they generally are operated independently to achieve
evaluating models, and measuring complexity. Chang and
an intended purpose.
Harrington [2005] provide a comprehensive description of
(iii) Geographic Distribution. Geographic dispersion of
agent-based models of organizations. Amaral and Ottino
component systems is often large. Often, these sys-
[2004] describe network theory and its importance in aug-
tems can readily exchange only information and
menting the framework for the quantitative study of complex
knowledge with one another, and not substantial
systems. Lloyd and Lloyd [2003] present a general method
quantities of physical mass or energy.
for modeling complex systems in terms of flows of informa-
(iv) Emergent Behavior. The system of systems performs
tion. Page [1999] discusses robust computational models. In
functions and carries out purposes that do not reside
an analysis of the challenges associated with complex systems
in any component system. engineering, Johnson [2006] provides a comprehensive re-
Systems Engineering DOI 10.1002/sys
4 HAIMES
view of emergent properties and how they affect the engineer- body may be represented by state and substate variables.
ing of complex systems. Bar-Yam [2003a] reviews past les- Consider the state of the heart of the human body and its
sons learned from problems with systems engineering over components, muscles, compartments, and so forth. With any
the past and suggests adopting an evolutionary paradigm for complex system, the most critical fact to note is the intra- and
complex systems engineering. Within the application of com- interdependencies that exist among the states of the system,
plex system theory, in a multiscale analysis of military littoral which necessarily overlap the multiple perspectives of the
warfare, Bar-Yam [2003b] suggests the necessity of consid- system represented by the multiple models. In other words, a
ering the specific organizational and technological require- central role of modeling systems of systems is to coordinate,
ments needed to perform effectively in a high-complexity to integrate, or to  make a whole of the various systems
environment. In health care, Funderburk [2004] presents a perspectives represented by the multiple models through the
brief survey of several formal dynamic and/or network-based states of the systems. This important task cannot be achieved
models that are relevant for health-care policy development without carefully identifying and discovering those states that
and evaluation. Tivnan [2007] describes the formulation, suc- characterize the most important perspectives of the system.
cessful replication, and critical analysis of Levinthal s model The fact that all state variables are uncertain functions of
of emergent order for economic firms. Most recently, Jam- uncertain initiating events requires that modeling efforts take
shidi [2009a, 2009b] edited two volumes on systems of sys- into account both epistemic and aleatory uncertainties [Paté-
tems engineering. In the preface of the first volume [2009a], Cornell, 1996].
he writes:  The SoS [Systems of Systems] concept presents a Consider the following definitions of the vulnerability and
high-level viewpoint and explains the interactions between resilience of a system [Haimes [2007, 2009]:
each of the independent systems. However, when it comes to
engineering and engineering tools of SoS, we have a long way Vulnerability is the manifestation of the inherent states of
to go. This is the main goal of this volume. Indeed, Jamshidi the system (e.g., physical, technical, organizational,
confirms the need for concerted efforts in modeling complex and cultural) that if exploited by an adversary, or
systems of systems. affected by a harmful initiating event, can result in
adverse consequences to that system.The vulnerability
of a system is multidimensional, a vector that is time-
3. THE CENTRALITY OF THE STATES OF THE
and threat-dependent (initiating event).
SYSTEM IN MODELING AND IN RISK ANALYSIS The resilience of a system is also a manifestation of the
states of the system and it is a vector that is time- and
Chen [1999] offers the following succinct definition of state threat (initiating event)-dependent. More specifically,
variable:  The state x(t0) of a system at time t0 is the informa- resilience represents the ability of the system to with-
tion at time t0 that, together with the input u(t), for t e" t0, stand a major disruption within acceptable degrada-
determines uniquely the output y(t) for all t e" t0. The states tion parameters and to recover within an acceptable
of a system, commonly a multidimensional vector, charac- cost and time. In other words, resilience is a vector state
terize the system as a whole and play a major role in estimat- of the system that is neither abstract or static, nor
ing its future behavior for any given inputs. Thus, the behavior deterministic. Moreover, resilience is similar to vulner-
of the states of the system, as a function of time, enables ability in that it cannot simply be measured in a single
modelers to determine, under certain conditions, its future unit metric; its importance lies in the ultimate multidi-
behavior for any given inputs, or initiating events. For exam- mensional outputs of the system (the consequences) for
ple, to determine the reliability and functionality of a car, one any specific inputs (threats).
must know the states of the fuel, oil, tire pressure, and other
mechanical and electrical components. In other words, all The question  What is the resilience of a specific system
systems are characterized at any moment by their respective X? is unanswerable. This question cannot be answered with-
state variables and the conditions thereof, and these condi- out reverting to the states of the system and to the specific
tions are subject to continuous change. In addition, a modeler threat and its timing. Furthermore, the answer implicitly
who has determined to select only those state variables that depends upon knowing whether system X would recover
represent the  essence of a system must decide whether its following any attack Y within an acceptable time, taking into
state variables should be modeled as static (constant) or account the associated costs and other risks. Thus, such a
dynamic (time dependent), deterministic or stochastic, etc. question can be answerable only when the threat (initiating
Given that all systems large and small can be characterized event) scenario (or a set of scenarios) is specifically identified,
by their state variables, recognizing the hierarchy of states, and the essential states of the system at the initiating event
substates, and subsubstates is crucial to system modeling. For (threat) are known. Resilience is not merely an abstract con-
example, a simplified water resources system that supplies cept; it is a state of the system (composed of a vector of
water to a large community can be characterized by the states substates) that may have different responses to different inputs
of the water distribution (groundwater and surface water) (threat scenarios) from any specific substate within the hard-
storage, purification, and sewer systems. The data for each of ware, software, policies and procedures, or connections to the
the states can be further presented by substates. For example, Internet.
the states of the water distribution system may be represented This discussion of the centrality of states of the system in
by the status of the main carriers, local pipes, pumps, and modeling will be further explored and will be related to the
storage tanks. Similarly, the status of each organ of the human intrinsic meta-modeling coordination and integration of the
Systems Engineering DOI 10.1002/sys
 MODELING COMPLEX SoSs WITH PHANTOM SYSTEM MODELS 5
multiperspective models and the necessity of relying on the (structure) (e.g., linear for a highly nonlinear system, its
states of the system. This is in contrast to relying solely on the parameters, data collection, and the employed processing
extrinsic outputs-to-inputs model coordination and integra- techniques). Model uncertainties will often be introduced
tion, which does not build explicitly on the common and through human errors of both commission and omission.
overlapping states among the submodels. The multidimensional probabilistic consequences result-
ing from an initiating event yield a multidimensional risk
function whose modeling and quantification complexity pre-
4. THE CENTRALITY OF TIME IN MODELING
sent considerable challenges. The selection of appropriate
MULTIDIMENSIONAL RISK, UNCERTAINTY,
models to represent the essence of the system s multiperspec-
AND BENEFITS tives determines the effectiveness of the entire risk assess-
ment, management, and ultimately communication process.
The time frame is central to all decisions, whether implicitly In particular, the scope and effectiveness of strategic risk
or explicitly. For a pilot, the time frame may be measured in management options are implicitly and explicitly dependent
mere seconds; for a planner, it may be years or decades. For on the perspectives of the system that are included or excluded
example, all real-world systems are characterized by multiple in the ultimate modeling efforts. In particular, a probable
objectives (often noncommensurate, competing, and in con- initiating event would necessarily affect only substates of a
flict with each other); thus, Pareto-optimal policies associated subsystem, but not necessarily the entire system of systems.
with such system models are achieved through the manipula- Thus, one must model the different probability distribution
tion of the appropriate states of the system; and since the latter functionsof consequences affecting each subsystem resulting
are a function of time, the time frame is thus critical for from the same initiating event. Each perspective of a system
modeling all systems. Models, which are built to answer manifested through its structure, functionality, the services it
specific questions, must also be constructed to address the provides, the customers it supports, the other systems on
following basic question: What are the impacts of current which it depends will experience specific, and likely,
decisions on future options, given the inevitable occurrence unique consequences resulting from the same initiating event.
of emergent forced changes? (The term emergent forced
changes connotes external or internal trends in sources of risk
5. EXTENSION OF HIERARCHICAL
and uncertainty to a system that may adversely affect or
enhance specific states of that subsystem and consequently HOLOGRAPHIC MODELING (HHM) TO
affect the entire system of systems.) Unanticipated, unde- PHANTOM SYSTEM MODELS (PSM)
tected, misunderstood, or ignored emergent forced changes,
whether they originate from within or from outside a subsys- Hierarchical holographic modeling is a holistic philoso-
tem, are likely to affect a multitude of states of that system phy/methodology aimed at capturing and representing the
with potentially adverse consequences to the entire system of essence of the inherent diverse characteristics and attributes
systems. Therefore, it is imperative to be able through sce- of a system its multiple aspects, perspectives, facets, views,
nario structuring, modeling, and risk analysis to envision, dimensions, and hierarchies [Haimes 1981, 2009a]. In the
discover, and track emergent forced changes. Consider, again, abstract, a mathematical model may be viewed as a one-sided
the FAA NextGen, with its multiple goals and objectives, image of the real system that it portrays. With single-model
agencies, functionality, geographic dispersion, and stakehold- analysis and interpretation, it is virtually impossible to repre-
ers. This multibillion-dollar, decade-effort system of systems sent the multiple perspectives of the system.
enterprise will be subjected to emergent changes in technol- The term holographic refers to the desire to have a mul-
ogy spanning satellite communication, airspace congestion, tiview image of a system. A hologram captures the multiple
trends in air traffic, and pollution emission, among myriad features of an object through multiple scattered light fields.
other changes. In our attempt to model a system, each model represents either
These emergent forced changes may be characterized, as one or limited aspects, dimensions, or perspectives of the
appropriate, through uncertainty and through risk analysis. system. The term hierarchical refers to the desire to under-
Uncertainty, commonly viewed as the inability to determine stand the intricacy that characterizes the many different levels
the true state of a system, can be caused by incomplete of the system s organizational, temporal, functional, and de-
knowledge, and/or by stochastic variability. Two major cision-making hierarchy.
sources of uncertainty in modeling affect risk analysis [Paté- HHM has turned out to be particularly useful in modeling
Cornell, 1990, 1996; Apostolakis, 1999]. Knowledge large-scale, complex, and hierarchical systems, such as de-
(Epistemic) Uncertainty manifests itself in the selection of fense and civilian infrastructure systems. The multiple visions
model topology (structure) and model parameters, among and perspectives of HHM add strength to risk analysis. It has
other sources of ignorance (e.g., lack of knowledge of impor- been extensively and successfully deployed to study risks for
tant interdependencies within the states of the system and government agencies such as the President s Commission on
among other systems). Variability (Aleatory) Uncertainty in- Critical Infrastructure Protection (PCCIP), TRW, the FBI,
cludes all relevant and important random processes, and other NASA, the U.S. Army, the U.S. Army Corps of Engineers, the
random events. Uncertainty dominates most decision-making U.S. Department of Homeland Security, the FAA, the Virginia
processes and is the Achilles heel for all deterministic and Governor s Office for Preparedness, the Virginia Department
most probabilistic models. This uncertainty is commonly of Transportation (VDOT), and the National Ground Intelli-
introduced through the selection of incorrect model topology gence Center, among others [Haimes, 2009a]. The HHM
Systems Engineering DOI 10.1002/sys
6 HAIMES
methodology/philosophy is grounded on the premise that in fested in the quest to achieve a level of risk that is deemed
the process of modeling large-scale and complex systems, acceptable when the tradeoffs among all the costs, benefits,
more than one mathematical or conceptual model is likely to and risks are considered. By virtue of the existence of multiple
emerge. Each of these models may adopt a specific point of
subsystems, hierarchical systems commonly have multiple
view, yet all may be regarded as acceptable representations of
noncommensurate and often competing and conflicting ob-
the complex system. Through HHM, multiple models can be
jectives, and multiple decision-makers and stakeholders (e.g.,
developed and coordinated to capture the essence of many
departments in a factory or subregions in a regional planning
dimensions, visions, and perspectives of infrastructure sys-
problem).
tems.
Haimes and Macko [1973] have identified four major
To present a holistic view of the elements that must be
decomposition structures in water resources systems on the
included in the model, the HHM approach involves organiz-
basis of political-geographical, hydrological, temporal, and
ing a team of experts with widely varied experiences and
functional considerations. The decomposition of a regional
knowledge bases (technologists, psychologists, political sci-
area into subregions depends on the viewpoint and aims of the
entists, criminologists, and others). The broader the base of
analyst. One decomposition may be performed with respect
expertise that goes into identifying potential risk scenarios,
to the region s hydrology. The region would be decomposed
the more comprehensive is the ensuing HHM.
into subregions, such as river basins and subbasins, having
This phenomenon is particularly common in modeling
topographical divisions as their boundaries. A second decom-
hierarchical complex systems of systems. For example, an
position might be with respect to political boundaries. The
economic system may be decomposed into, or represented
regional area would be decomposed into political subregions
through, geographic regions or activity sectors. An electric
such as townships, municipalities, counties, and so forth. A
power management system may be decomposed according to,
third decomposition might be with respect to regional goals
or represented through, the various functions of the system
and functions. A fourth decomposition might address the time
(e.g., power generation units, power transformer units, and
frame and resource allocation that would affect the planning
transmission units) or along geographic or political bounda-
for irrigation, navigation, hydroelectric power generation,
ries. Another decomposition might be a timewise decomposi-
recreation, and so forth. In regional water resource manage-
tion into various planning periods. If several aspects of the
ment, the major aspects of the regional area cannot be di-
system are to be dealt with, such as the geographic regions
vorced from each other. The decompositions just cited overlap
and activity sectors of an economic system, it could be advan-
one another. Hydrological subregions can easily overlap or
tageous to consider several decompositions or to model rep-
span political boundaries; and hydroelectric generating sta-
resentations of the multiple perspectives and functionalities
tions may be dispersed through a region and not be confined
of the system. For example, four major decomposition struc-
to anyone political or hydrological sub-region. Indeed, the
tures may be identified for water resources systems on the
subregional boundaries in hydrological decomposition gener-
basis of political-geographical, hydrological, temporal, and
ally do not coincide with the subregional boundaries in geo-
functional considerations.
graphical decomposition. Since multiple models are required
The multiple perspectives of complex systems have been
when modeling complex systems of systems, which are com-
often characterized and represented through the hierarchical
mon in hierarchical multilevel modeling, hierarchical over-
nature of the system. Indeed, many organizational as well as
lapping coordination between two or more hierarchical
technology-based systems are hierarchical in nature, and most
structures has been proven to serve as an effective schema to
states of a system (state variables) are hierarchical with sub-
supplement and complement the knowledge and information
states and subsubstates (e.g., any organ of the human body,
provided by each structure separately [Haimes and Macko,
and any physical or cyber infrastructure); thus, the modeling
1973; Macko and Haimes, 1978; Haimes et al., 1990; Yan and
of such systems has been driven by and responsive to this
Haimes, 2010]. PSM builds on and takes advantage of hierar-
hierarchical structure. This hierarchical structure of the sub-
chical overlapping coordination.
systems and subsubsystems, when it is understood and taken
The principal advantage of hierarchical multilevel model-
advantage of, can simplify the modeling process and the
ing is that it breaks down a large complex system into its
ultimate management of the system as a whole [Haimes et al.,
component subsystems. It allows each subsystem to be stud-
1990]. Hierarchical modeling makes it possible to decompose
ied, analyzed, understood, and possibly managed at a lower
an overall system into smaller subsystems, which are easier
level of the hierarchy independently of the other levels, and
to model, analyze, and subsequently integrate with other
coordinated at a higher level of the hierarchy. It might be
subsystem models. The decomposition can be based on func-
argued that decomposition is fairly easy; the real challenge is
tional, technical, geographical, organizational, political, so-
resolving the conflicts and interactions between and among
cial, and myriad other perspectives of a system, and especially
the subsystems and ensuring that the submodels account for
of systems of systems. Hierarchical modeling also has signifi-
all critical states of the system, as well as for the specified
cant implications for risk modeling, assessment, and manage-
system s overall objectives and constraints. The hierarchical
ment [Tarvainen and Haimes 1981]. For example, the risks
associated with each subsystem within the hierarchical struc- approach meets these requirements via higher-level coordina-
tion. For example, general coordination methodologies [Las-
ture may contribute to and ultimately determine the risks of
don, 1964, 1970, 2002; Haimes, 1977; Singh, 1987; Haimes
the overall system. Furthermore, the distribution of risks
within critical subsystems often plays a dominant role in the et al., 1990] distribute the total planning and management task
allocation of resources for the entire system. This is mani- among the component subsystems.
Systems Engineering DOI 10.1002/sys
 MODELING COMPLEX SoSs WITH PHANTOM SYSTEM MODELS 7
PSM builds on and extends the basic theory and philoso- from multiple disciplines with varied perspectives, experi-
phy of HHM by offering operational guidelines and principles ence, skills, and backgrounds.
on the basis of which to model systems of systems; one of its It is not unrealistic to compare the evolving process of the
most salient features is that it offers modelers a 4-decade-old PSM to the  modeling experience of children at play. They
tested approach to learning the inherent characteristics and
experiment and explore their uncorrupted imaginative emer-
interdependencies of systems of systems. In his book Me- gent world with Play-Doh® and LEGO®, while patiently
tasystems Methodology, Hall [1989] states:  In this way,
embracing construction and reconstruction in an endless trial-
history becomes one model needed to give a rounded view of
and-error process with great enjoyment and some success.
our subject within the philosophy of hierarchical holographic
The innovation, imagination, and initiatives of modelers ex-
modeling [Haimes, 1981] being used throughout this book,
perimenting with the PSM on systems of systems can be
defined as using a family of models at several levels to seek
instrumental in creating a learning process that can benefit
understanding of diverse aspects of a subject, and thus to
decision-makers.
comprehend the whole.
Modeling emerging unprecedented and complex systems
(e.g., a new national electric-power grid system, a new and
safe generation of cars fueled by hydrogen, or a human space
6. PHANTOM SYSTEM MODELS (PSM) AND
mission to Mars and back), which are inherently elusive and
META-MODELING
visionary, as well as modeling existing large-scale systems of
systems, by and large involve phantom entities of multiple
6.1. Philosophical-Conceptual Foundations perspectives. This modeling effort is driven and constrained
by a mix of evolving future needs and available resources,
Architects, painters, and music composers share similar chal-
technology, emergent forced changes and developments, and
lenges with analysts who are involved in the art and science
myriad other unforeseen events.
of systems modeling. The similarities are manifested in a
Consider the tradeoffs between (i) the relatively low cost
seemingly endless process of discovery and creativity and in
of modeling a complex system of systems and the inher-
continuous learning through experimentation, measurement,
ently invaluable, often unrecognized and unappreciated effi-
assessment, and trial and error. Creative artists invariably start
cacy that such modeling generates or offers; and (ii) the cost
with a visionary theme through which they deliver one or
(higher by many orders of magnitude) associated with the
multiple messages. Through their creative artistic talent and
conception, development, construction, and planning for op-
capability, and by intuitive inquiries and exploration of a
eration of a new generation of physical infrastructures (e.g.,
variety of motifs, artists and composers strive to express their
water and sewers, electric power grids, transportation sys-
visionary themes by answering imaginary or invisible ques-
tems, communications, public support buildings, etc.) Indeed,
tions (at least to the layperson).
the cost associated with bringing to life complex infrastruc-
Artists, as the quintessential modelers, represent through
ture systems could be in the billions of dollars, while the
their artwork the influence of the culture and social environ-
associated modeling cost would be in the millions of dollars.
ment within which they live. In an analogous way, systems
Thus, a ratio of 3 orders of magnitude ought to encourage and
modelers attempt to represent the multiple perspectives and
justify essential investments in modeling.
facets of the system under study so that they may gain a better
Models enable us to experiment and test hypotheses and
understanding of the composition of its inherent intra- and
different designs options, or to generate responses to or im-
interconnectedness and interdependencies, and thus be able
pacts on varied policy options. Inversely, by their nature,
to answer specific questions relevant to the system. Thus, both
complex systems constitute, in many respects, black holes to
artists and system modelers assume a similar creative, sys-
modelers that can be penetrated only by acknowledging our
temic, and challenging task of representation. Finally, not
inability to directly uncover, understand, or predict their
dissimilar to an artistic composition, models ought to be as
behaviors under different scenarios of disturbances (in-
simple as possible but as complex as required resulting in a
puts).We commonly lack sufficient knowledge to assess the
model that offers an acceptable representation of the system
causal relationships among the subsystems, and to compen-
and is capable of providing answers and clarifications to the
sate for this shortfall, we revert to multiperspective experi-
important questions that the model was designed to address.
mentation aided by the ingenuity, creativity, and domain
Indeed, models must represent broad perspectives, and
knowledge of experts, supported by the availability of data-
modelers must possess matching capabilities, wisdom, and
bases. There is no assurance that modelers would be able to
foresight for futuristic and out-of-the-box thinking. Emergent
explain the reasons behind any variability among submodels;
forced changes, the need for agile and flexible multiplicity of
nevertheless, the very process of modeling such variability
models, building on the human systems engineering experi-
may highlight limited databases, inconsistent assumptions,
ence, expertise, and capabilities together contribute to the
unrecognized epistemic and aleatory uncertainties, and a host
need for the PSM. In this sense, the PSM constitutes a
real-to-virtual laboratory for experimentation, a learn-as-you- of other technical or perceptual reasons that ought not to be
dismissed. For example, in a closed-loop process control of a
go facility, and a process  for exploring existing or emergent
systems that are not yet completely designed and devel- system in operation, the automatic controller adjusts the pa-
oped [Horowitz and Lambert, 2006]. The Human Genome rameters of the system in response to internal or external
project may be considered an audacious complex system of disturbances or initiating events. In contrast, the adjustment
systems, fraught with uncertainties and involving participants of the parameters in an open-loop process (in response to the
Systems Engineering DOI 10.1002/sys
8 HAIMES
initiating events), is made by the system s operator or engi- representing specific aspects of the subsystem for the
neer. purpose of gaining knowledge and understanding of
In the meta-model coordination and integration of the the multiple interdependencies among the submodels,
multiple submodels (to be discussed subsequently), the task
and thus allowing us to comprehend the system of
is exceedingly more complicated, because the modeler as-
systems as a whole.
sumes the roles of both the closed-loop controller and the
iii. The essence of each subsystem can be represented by
open-loop controller. More specifically, the modeler at the
a finite number of essential state variables. (The term
meta-modeling level makes extensive use of the knowledge
essence of a system connotes the quintessence of the
generated through lessons learned from: (i) the subsystems
system, the heart of the system; that is, everything
coordination; (ii) interdependencies within and among the
critical about the system.) Given that a system may
states of the subsystems; (ii) innovation and creativity in
have a large number of state variables, the term essen-
model experimentation; and (iv) intrinsic overlapping and
tial states of a system connotes the minimal number of
mutual characteristics, functionality, objectives, and states
state variables in a model with which to represent the
that combine to make all the subsystems a system of systems.
system in a manner that permits the questions at hand
to be effectively answered. Thus, these state variables
become fundamental for an acceptable model repre-
6.2. Meta-Model Coordination and Integration
sentation.
iv. For a properly defined system of systems, any intercon-
6.2.1. Methodological Approach
nected subsystem will have at least one (typically
The essence of meta-model coordination and integration is to
more) essential state variable(s) and objective(s) shared
build on all relevant direct and indirect sources of information
with at least one other subsystem. This requirement
to gain insight into the interconnectedness and intra- and
constitutes a necessary and sufficient condition for
interdependencies among the submodels and, on the basis of
modeling interdependencies among the subsystems
this insight, to develop representative models of the system of
(and thus interdependencies across a system of sys-
systems under consideration. The coordination and integra-
tems). This ensures an overlapping of state variables
tion of the results of the multiple models are achieved at the
within the subsystems. Of course, the more we can
meta-modeling phase within the PSM, thereby yielding a
identify and model joint (overlapping) state variables
better understanding of the system as a whole. More specifi-
among the subsystems, the greater is the repre-
cally, modeling the intra- and interdependencies within and
sentativeness of the submodels and the meta-model of
among the subsystems of complex SoSs requires an under-
the system of systems.
standing of the intricate relationships that characterize the
v. The importance of the availability of multiple, albeit
dynamics within and among the states of the subsystems. This
overlapping, databases can be effectively utilized by
very important task is achieved at the meta-modeling level of
multiple submodels, each of which is built to answer
the PSM by observing, estimating, and assessing the outputs
the specific questions for which it is built. Furthermore,
for given inputs, and by building on the intrinsic common
each submodel s characterization, whether modeled
states within and among the subsystems. Note that although
separately or in groups, is likely to share common state
the intrinsic common states constitute a key element of the
variables a fact that facilitates the ultimate coordina-
PSM, the extrinsic (input-output) relationships are also very
tion and integration of the modeled multiple submodels
important and support the intrinsic one. Indeed, the selection
at the meta-modeling level. Thus, a common database
of the trial inputs to the model and the inquisitive process of
that supports the family of systems of systems must be
making sense of the corresponding outputs are at the heart of
available.
system identification and parameter estimation. This is not a
vi. The fusion of multiple submodels via the intrinsic
one-shot process; rather, it can be best characterized by tire-
meta-modeling coordination and integration enhances
less experimentation, trial and error, and parameter estimation
our understanding of the inherent behavior and inter-
and adjustments, as well as by questioning whether the as-
dependencies of existing and emergent complex sys-
sumed model s topology is representative of the system being
tems.
modeled.
The PSM-based intrinsic meta-modeling of systems of
6.2.2. PSM-Based Modeling of a Prototype System of
systems stems from the basic assumption that some specific
commonalities, interdependencies, interconnectedness, or Systems
other relationships must exist between and among any two This subsection, which focuses on saltwater intrusion into
systems within any system of systems. More specifically: groundwater systems and seawater rise due to climate change,
explores and highlights some concepts associated with mod-
i. A system of systems connotes a specific group of eling a real system of systems with PSM, albeit not suffi-
subsystems. A subsystem will denote any system mem- ciently developed to generate results. Figures 1 graphically
ber of the system of systems. A model of a subsystem depict the commonly used extrinsic nonreliance on state
will be denoted as a submodel. variables in systems integration (by using inputs from sub-
ii. A meta-model represents the overall coordinated and models as inputs to others). In contrast to Figure 1, Figure 2
integrated submodels of the system of systems. We depicts the intrinsic reliance on shared and unshared state
define a meta-model as a family of submodels, each variables for meta-modeling coordination and integration.
Systems Engineering DOI 10.1002/sys
 MODELING COMPLEX SoSs WITH PHANTOM SYSTEM MODELS 9
iv. Meta-modeling of the groundwater system serves as
the coordinator and integrator of the multiple models,
building on the shared and unshared state variables.
Let c(t) represent an initiating event of climatological input
that impacts seawater level and temperature; let s1(t) represent
seawater level at time t; and let s2(t) represent the temperature
Figure 1. Extrinsic input-output submodel coordination and integra-
at time t. Note the common and uncommon state variables in
tion.
the following functional relationships:
a. Groundwater salinity level s3(t) = s3(t, c(t), s1(t), s2(t))
Fresh water has been and continues to be a scarce resource,
b. Groundwater yield s4(t) = s4(t, c(t), s1(t), s2(t))
and groundwater plays a major role in the overall water supply c. Crop quality and variety s5(t) = s5(t, c(t), s3(t), s4(t))
of the United States and around the world. Many models d. Income to farmers s6(t) = s6(t, c(t), s5(t))
predict a significant seawater rise due to climate change e. Regional viability of farms s7(t) = s7(t, c(t), s5(t) s6(t)).
[USDOT, 2008], which would cause saltwater intrusion into
coastal groundwater aquifer systems. We consider three sub- Figure 3 depicts the PSM-based meta-system intrinsic
coordination via the shared and nonshared state variables of
system models: hydrologic, agricultural-social, and regional
the system. The knowledge and information provided by the
economic models, where the only inputs are provided from
state variables (s1  s7) enable modelers to learn and better
external climatological models.
understand the interdependencies among the different sub-
The role of the meta-model, which is composed of the
models. For instance, the following set of intersections of state
above submodels, is to explore and learn about the intra- and
variables s1 )" s2, s2 )" s3, s3 )" s4, s4 )" s5, and s5 )" s1 can help
interdependencies among the submodels and to evaluate the
modelers to identify causal relationships among the multiple
information necessary to assess the ultimate multiple impacts
perspectives of the groundwater system.
of the rise of groundwater salinity (due to the expected seawa-
The effectiveness of the PSM-based meta-model intrinsic
ter rise resulting from climate change) on crop yield and
coordination and integration is grounded on: (i) the number
variety, drinking water quality, farmers economic well-being,
of common state variables shared between two or more sub-
and the regional economy. We envision the following multiple
systems models (a minimum of one shared state is required;
models:
otherwise, modelers can reasonably assume that a subsystem
i. Hydrologic modeling effort can focus on a repre- without any shared state variable is completely independent
sentative set of scenarios of climate change and seawa- of the other subsystems); (ii) the domain knowledge of each
ter rise, and can address the questions regarding the of the subsystems perspectives to ensure proper and effective
modeling of the corresponding subsystems; (iii) the appropri-
resulting consequences of seawater intrusion into the
ate modeling efforts, skills, and expertise invested in model-
groundwater system.
ing each subsystem, including, most importantly, the skill and
ii. The agricultural-social model can focus on the impacts
ability of modelers to learn through the  mixing bowl of
of increased groundwater salinity on (a) agriculture,
infused knowledge, information, and learning generated
affecting the quality and yield of crops that are heavily
through the meta-model at higher-level model coordination
dependent on groundwater, and (b) domestic water
and knowledge integration; (iv) the appropriate modeling
supply.
methodologies and tools (e.g., analytical, simulation) devoted
iii. The regional economic model can focus on the regional
in modeling each subsystem [which entails the proper selec-
economic impacts of the above on the agricultural and
tion of model topology/structure and parameters and the
domestic use of groundwater.
incorporation (through the states of the system of systems) of
Figure 2. Intrinsic submodel, coordination, and integration via Figure 3. PSM-based meta-system intrinsic coordination via the
system state variables. shared and nonshared state variables of the system.
Systems Engineering DOI 10.1002/sys
10 HAIMES
the intra- and interdependencies within and among the sub- Lasdon [1970] and Haimes [1975]. When observing (meas-
systems]; (v) the availability of proper databases with which uring) different values of shared states or outputs between two
to calibrate, test, validate, and verify the model of each subsystems, and when there are sufficient reasons to believe
subsystem (submodel) under varied conditions; and (vi) the that the outputs associated with the two subsystems (corre-
availability of an appropriate computing laboratory that sup- sponding to the common states) ought to be the same or with
ports all of the above modeling efforts. an acceptable difference, then the use of pseudovariables can
The intrinsic shared states provide a powerful mechanism become a useful instrument in the system identification and
for understanding and exploiting the strong interdependen- parameter estimation process within the PSM. On the other
cies among the subsystems of systems of systems. The efficacy hand, differences between state variables representing a com-
of shared states among subsystems may be manifested mon perspective of two subsystems could also be due to our
through (i) sharing databases; (ii) realizing that decisions lack of understanding of the interdependencies between the
made by the stakeholders of subsystem I can have direct two subsystems. More specifically, in intrinsic meta-model-
impact on subsystem II; and (iii) encouraging and enticing ing, we aim to reconcile the differences between common
stakeholders of different subsystems to collaborate on inputs, state variables to compensate for our ignorance. The availabil-
constraints, and decisions that affect the shared states for ity of sufficient time-variant database is a requisite for an
win-win outcomes. On the other hand, understanding the effective PSM modeling effort, given that most states of the
potential adverse organizationally induced consequences re- system are time-variant, and comparing the differences of [s1(t
sulting from unshared states, due to competitiveness among + 1)  Ã1(t + 1)] over time can shed more light on the system s
subsystems; and thus, by exploiting unshared states could, for behavior.
example, (i) defuse potential conflicts among the subsystems
6.3.2. Coordinated Hierarchical Bayesian Model (CHBM)
and (ii) generate collaboration in the face of joint budgetary
constraints or unfavorable policies affecting the subsystems. The reliance of direct and indirect information and database
is common in system s modeling with sparse database and
when empirical data are usually either sparse or lacking, in
6.3. Systems-Based Theoretical and
particular in risk of extreme events [Yan, 2007; Yan and
Methodological Foundations
Haimes, 2010]. Furthermore, with sparse data, important
The following is a sample of tested systems-based method- model parameters may not be estimated and tested within an
ologies that support the PSM. acceptable level of significance. When a large database is
available, standard statistical techniques can be applied to
6.3.1. Decomposition and Hierarchical Coordination estimate the parameters and create a fairly accurate and well-
Hierarchical decomposition of complex large-scale systems parameterized model. Researchers and practitioners in sys-
enables modelers and systems analysts to use the decentral- tems engineering and risk analysis are commonly plagued by
ized approach to analyze and comprehend the behavior of the data scarceness problem, which can be prevalent in mod-
subsystems at the lower level of the hierarchy and to transmit eling complex systems of systems. On the other hand, it is
the information gained to fewer subsystems at the higher well known that when estimating the parameters of a model
level. More specifically, the system s model is decomposed at by traditional statistical methods using relatively small
the lower level of the hierarchy into  independent subsys- datasets, those methods generate  unstable results with large
tems (using pseudovariables) and the interdependencies are estimation variance [Farrell, MacGibbon, and Tomberlin,
coordinated at a higher level. This system s decomposition 1997; Assuncao and Castro 2004]. Consequently, important
and hierarchical coordination methodology, which is well model parameters cannot be estimated and tested within an
documented in copious books and archival papers, has been acceptable level of significance. For example, Ferson [1997]
successfully deployed for modeling and optimizing hierarchi- argues that  problems in risk analysis often involve extreme
cal complex systems, and it constitutes one of the methodolo- events, which rarely happen, or are even hypothetical at the
gies that supports PSM [Dantzig and Wolf, 1961; Bauman, time of the assessment.
1966; Lasdon and Scheffler, 1966; Lasdon, 1970; Wismer, In this paper we adopt an alternative approach to address
1971; Haimes, 1977; Haimes et al., 1990]. For example, this problem at the meta-modeling level, through borrowing
consider a system composed with two subsystems that are strength from indirect but relevant data from one subsystem
coupled by one state variable (s1). The system can be decom- and applying it to another. Strength-borrowing methods aim
posed into two  independent subsystems by assuming at the to borrow strength from indirect data to compensate for the
lower level of decomposition a pseudovariable (Ã1) as a sparseness of direct data. Subjective methods include expert
surrogate for the state variable (s1) of one subsystem, and evidence solicitation and Bayesian analysis; the latter pro-
keeping (s1) for the state variable for the second subsystem. vides a natural way to combine expert evidence with limited
Then, the sources of the difference [(s1)  (Ã1)] must be direct data.
investigated, understood, and, if possible, minimized at the We decompose the term  data into three parts: direct data,
second level of the hierarchy. Several higher-level coordina- indirect data, and expert evidence: (i) Direct data may repre-
tion methods for different types of decompositions, such as sent testing, experimentation, measurements, observations
the feasible and nonfeasible decompositions, have been de- from a system (or a subsystem) with unknown parameters; (ii)
veloped and successfully deployed. This approach is applica- indirect data represent observations from different but related
ble to any number of coupling state variables with complex (or similar) subsystems; and (iii) expert evidence is informa-
interdependencies among the subsystems. See, for example, tion received by soliciting evidence from one or multiple
Systems Engineering DOI 10.1002/sys
 MODELING COMPLEX SoSs WITH PHANTOM SYSTEM MODELS 11
experts. The Hierarchical Bayesian Models (HBM),which has accommodate multiple dimensional cross-classified random
been applied in the reliability, risk, and system safety fields, effects as multiple dimensions presenting in a system.
is an objective method suitable for addressing the data sparse-
6.3.3. Influence Diagrams
ness problem [Ghosh and Rao, 1994; Ghosh and Meeden,
The combined art and science of systems modeling builds on
1997; Carlin and Louis, 2000; Gelman et al., 2004]. Coordi-
diverse philosophies, theories, tools, and methodologies.
nated Hierarchical Bayesian Models (CHBMs), which bor-
Probably the most basic, logical, and intuitive of all are
row strength from indirect data or expert evidence to
influence diagrams [Oliver and Smith, 1990]. They are effec-
compensate for the sparseness of direct data [Yan, 2007; Yan
tive because they enable systems engineers and decision-mak-
and Haimes, 2010], can provide valuable support to the meta-
ers alike to represent the causal relationships among the large
modeling process.
number of variables affecting and characterizing the system.
The structures of HBM and CHBM are described in Fig-
Furthermore, through the use of conventional symbols, such
ures 4(a) and 4(b), respectively. In HBM, y represents the data
as decision nodes and chance nodes, influence diagrams
set observed from subsystem i, ¸ represents the parameter for
capture the probabilistic nature of the randomness associated
the subsystem i, and · represents the hyper-parameter. In
with the system. Consequently, the quantification of risks and
CHBM, y represents the data set observed from scenario (i,
benefits can be performed on sound foundations.
j), ui represents the fixed effect of perspective i, Ä…ij represents
The most effective deployment of influence diagrams is
the cross-classified random effects from scenario (i, j), and
through brainstorming sessions with all principal parties in-
ÄÄ…i represents the variance of the hyperdistribution of the
volved with the system. In this setting, the varied expertise of
random effects in perspective i.
the study team members produces a deeper understanding of
As opposed to HBM, where there is only one dimension
the interactions between and among the subsystems. Similar
and a single hierarchy, the CHBM has two cross-hierarchies,
to an engineering design project, the initial phase of construct-
each corresponding to one dimension of strength borrowing.
ing an influence diagram may result in an unwieldy  mess
Note that the bidimensional model can be easily extended to
chart that includes trivial, as well as critical, components.
Through an open and constructive dialogue among the ana-
lyst(s) and decision-maker(s), the  mess chart becomes more
coherent and includes what are deemed to be only essential
variables and building blocks of the system s model.
6.3.4. Summary
The systems-based approaches presented in Section 6.3 con-
stitute only a sample of methodologies that support the mod-
eling of complex systems of systems through the PSM. The
challenges associated with modeling systems of systems nec-
essarily require the reliance on every applicable theory and
methodology that can support this effort.
7. PHANTOM SYSTEM MODELS LABORATORY
There is a need for a PSM laboratory (PSML) to support,
coordinate, and integrate results from a plurality of computer-
based analytical (and simulation models), each providing a
unique system perspective, with the outlook that the combi-
nation of such results can improve our learning and ability to
gain knowledge. A PSML configuration can make available
to the group of modelers: (i) desired software-based models
for a particular analysis; (ii) an array of data sources to support
the desired modeling activities; (iii) tools for organizing the
components of the modeling system so as to achieve the
desired model relationships; and (iv) support for the intrinsic
meta-modeling coordination and integration.
To perform these functions, the PSML ought to provide a
structure based upon Service-Oriented Architecture that will
enable the user to perform desired modeling efforts. Services
may include:
" Modeling Services that determine which models will be
Figure 4. (a) Structure of HBM; (b) structure of CHBM. executed
Systems Engineering DOI 10.1002/sys
12 HAIMES
" Data Services that include organizing the needed data tion, imagination, and initiatives of modelers experimenting
inputs, collecting the desired data outputs, and perform- with the PSM on systems of systems can be instrumental in
ing needed data conversions the creation of a learning process that can ultimately benefit
" Analysis Services that provide postmodeling analysis decision-makers.
that compares results from different models and that In sum, this paper advances the following premises: (i) The
assesses the sensitivity of results derived from the vary- emergent and dynamic nature of systems of systems neces-
ing analyses performed by a given model sarily render their models to be ephemeral and visionary,
" Data Presentation and Visualization Services that in- building on the intrinsic relationships among the states of the
subsystems; (ii) this modeling process benefits from a well-
clude composing the analytical results to aid the
designed and executed learn-as-you-go process; (iii) system
modeler and decision-makers in interpreting results,
models are likely to build on hierarchical and overlapping
presenting the results for different presentation media
structures; (iv) since the vulnerability and resilience of a
(print, small/large screen display, etc.)
system are manifestations of the states of the system, then the
" Workflow Services that determine the sequencing of
above points have important implications for system engi-
services that might be executed (e.g., determining
neering, particularly for identifying sources of risk and under-
which models can be run in parallel and which must be
standing system vulnerability and resilience; (v) Hierarchical
run sequentially) and providing the necessary data to a
Bayesian Models (HBMs) and Coordinated Hierarchical
hardware organizing service to allow proper physical
Bayesian Models (CHBMs), which borrow strength from
configuration to support the desired modeling effort
indirect data or expert evidence to compensate for the sparse-
" Library Services to provide model descriptions, histori-
ness of direct data, can provide valuable support to the meta-
cal model results, model software designs, and varying
modeling process; (vi) other systems engineering methods,
model configuration descriptions
such as hierarchical decomposition and higher-level coordi-
" Configuration Management and Control Services to
nation, influence diagrams, and others can be instrumental in
help manage new model development, model modifi-
the meta-modeling process; and (vii) building on the intrinsic
cations, and existing model integration for the overall
interplay among the shared and unshared state variables
PSML system.
among the subsystems, the philosophy and theory of the
phantom system models provide a modeling paradigm that
The above constitute representative services that a PSML
complements and supplements the commonly used extrinsic
ought to provide modelers in support of their modeling ef-
(input-output-based) modeling approach.
forts.
ACKNOWLEDGMENTS
8. EPILOGUE
In an introduction to the history of European art, William The constructive comments and suggestions received from
Kloss [2005] writes:  We will place these artists and their my colleagues Barry Horowitz and Jim Lambert and from my
masterpieces in the political, religious, and social context of Ph.D. students Zhenyu Guo, Sung Nam Hwang, and Eva
their time, so that we have a profound understanding of both Andrijcic are most appreciated. The research reported in this
paper was in part supported by a grant from the National
why an artwork was created and how it responded to a
particular set of historical circumstances. The creative work Science Foundation (Award No. 0928550: Adaptive Systems-
of a system s modelers is not dissimilar from that of artists. Based Prioritization of Bridge Infrastructure Maintenance:
Just as no single model is capable of representing the multiple Integrated Modeling of Technical, Socio-Economic, and Nor-
mative Dimensions).
perspectives of a complex system, whether in harmony or in
juxtaposition, no symphony by Beethoven could have been
composed using one instrument or one motif or theme. The
REFERENCES
similarities among artists and analysts involved in the art and
science of systems modeling are manifested in a seemingly
G. Apostolakis, The distinction between aleatory and epistemic
endless process of discovery and creativity and in a continu-
uncertainties is important: An example from the inclusion of
ous process of learning through experimentation and trial and
aging effects into probabilistic safety assessment, Amer Nucl Soc
error. Thus, similar to the challenge facing the composer of a
Proc PSA 99, Washington, DC, August 22 25, 1999, pp. 199
symphony and ultimately the conductor of the orchestra per-
211.
forming the piece, the challenge facing the modeler is one of
L. Amaral and J.M. Ottino, Complex networks: Augmenting the
amalgamating and assembling the multiple system models to
framework for the study of complex systems, Eur Phys J B 38(2)
yield the development of a harmonious multimodel repre-
(2004), 147 162.
sentation of the multiperspective system.
R.M. Assuncao, and M.S. Castro, Multiple cancer sites incidence
It is not unrealistic to compare the evolving process of the
rates estimation using a multivariate Bayesian model, Int J
Phantom System Models to the  modeling experience of
Epidemiol 33 (2004), 508 516.
children at play. In this sense the PSM is a real-to-virtual
laboratory for experimentation, a learn-as-you-go facility, and Y. Bar-Yam, When systems engineering fails toward complex
a process for existing and emergent systems that are not yet systems engineering, Int Conf Syst Man Cybernet 2 (2003a),
completely designed, developed, or understood. The innova- 2021 2028.
Systems Engineering DOI 10.1002/sys
 MODELING COMPLEX SoSs WITH PHANTOM SYSTEM MODELS 13
Y. Bar-Yam, Complexity of military conflict: Multiscale complex Y.Y. Haimes, Hierarchical modeling for the planning and manage-
systems analysis of littoral warfare, Chief of Naval Operations, ment of a total regional water resource system, Automatica, 11
Report for Contract F30602-02-C-0158, Multiscale repre- (January) (1975), 25 36.
sentations phase II. Task 2: Multiscale analysis of littoral warfare, Y.Y. Haimes, Phantom System Models for emergent multiscale
U.S. Department of the Navy, Washington, DC, 2003b. systems, J Infrastruct Syst 13(2) (2007), 81 87.
E.J. Bauman,  Multilevel optimization techniques with application Y.Y. Haimes, Models for risk management of systems of systems,
to trajectory decomposition, C.T. Leondes (Editor), Advances Int J Syst Syst Eng 1(1/2) (2008), 222 236.
in Control Systems, 4th edition, Vol. 6, Academic Press, New Y.Y. Haimes, Risk modeling, assessment, and management, 3rd
York, 1968. edition, Wiley, Hoboken, NJ, 2009a.
I.V. Blauberg, V.N. Sadovsky, and E.G. Yudi, Systems theory: Philo- Y.Y. Haimes, On the complex definition of risk: A systems-based
sophical and methodological problems, Progress Publishers, approach, Risk Anal 29(12) (2009b), 1647 1654.
New York, 1977, p. 132. Y.Y. Haimes and D. Macko, Hierarchical structures in water re-
B.P. Carlin and T.A. Louis, Empirical Bayes: Past, present, and sources systems management, IEEE Trans Syst Man Cybernet
future, J Amer Statist Assoc 95 (2000), 1286 1289. 3(4) (1973), 396 402.
M. Chang and J. Harrington,  Agent-based models of organiza- Y.Y. Haimes, K. Tarvainen, T. Shima, and J. Thadathil, Hierarchical
tions, Handbook of computational economics, Vol. 2: agent- multiobjective analysis of large-scale systems, Hemisphere, New
based computational economics, K. Judd and L. Tesfatsion York, 1990.
(Editors), Agent-Based Computational Economics (ACE), Iowa A.D. Hall, III, Metasystems methodology: A new synthesis and
State University, Ames, 2005, pp. 1 66. unification, Pergamon, New York, 1989.
C. Chen, Linear system theory and design, 3rd edition, Oxford B.M. Horowitz and J.H. Lambert. Learn as you go systems engineer-
University Press, New York, 1999. ing, IEEE Trans Syst Man Cybernet Part A 36(2) (2006), 286
G.B. Dantzig and P. Wolfe, The decomposition algorithm for linear 297.
programs, Econometrica 29(4) (1961), 767 778. M. Jamshidi (Editor), Systems of systems engineering: Innovations
H. Eisner, RCASSE: Rapid Computer-Aided Systems of Systems for the 21st century, Wiley, Hoboken, NJ, 2009a.
Engineering, Proc 3rd Int Symp Natl Council Syst Eng, INCOSE, M. Jamshidi (Editor), System of systems engineering: Principle and
1993, Vol. 1, pp. 267 273. applications, CRC Press, Taylor & Francis Group, New York,
P. Eykhoff, System identification: parameter and state estimation, 2009b.
Wiley, New York, 1974. C. Johnson, What are emergent properties and how do they affect
P. Farrell, B. MacGibbon, and T. Tomberlin, Empirical Bayes esti- the engineering of complex systems? Reliab Eng Syst Safety
mators of small area proportions in multistage designs, Stat Sin 91(12) (2006), 1475 1481.
7 (1997), 1065 1083. W. Kloss, History of European art, 48 lectures for The Great Courses,
S. Ferson, Bayesian methods in risk assessment, Technical report, The Teaching Company, Chantilly, VA, 2005.
Applied Biomathematics, Setauket, NY, 1997, http://cite- A. Kossiakoff and W.N. Sweet, Systems engineering principles and
seerx.ist.psu.edu/viewdoc/download?doi=10.1.1.87.4577&rep practice, Wiley, Hoboken, NJ, 2002.
=rep1&type=pdf. L.S. Lasdon, A multi-level technique for optimization, Ph.D. Thesis,
F. Funderburk, Organizational culture from a complex dynamic Case Institute of Technology, Cleveland, OH, 1964.
systems perspective: Moving from metaphor to action in health- L.S. Lasdon, Optimization theory for large systems, Macmillan,
care, Syst Models Org Behav (2004), http://www.necsi.edu/ New York, 1970.
events/iccs/openconf/author/papers/f183.pdf, accessed February L.S. Lasdon, Optimization theory for large systems, 2nd edition,
14, 2011. Dover, New York, 2002.
A. Gelman, J. Carlin, H. Stern, and D. Rubin, Bayesian data analysis, L.S. Lasdon and J.D. Scheffler, Decentralized plant control, ISA
Chapman & Hall, New York, 2004. Trans 5(2) (1966), 175 183.
A.V. Gheorghe, Applied systems engineering, Wiley, New York, C.T. Leondes (Editor), Advances in control systems, Academic, New
1982. York, 1969, Vol. 6.
M. Ghosh and G. Meeden, Bayesian methods for finite population S. Lloyd and T. Lloyd. Bits and bucks: Modeling complex systems
sampling, Chapman & Hall, London, 1997. by information flow, Working Paper Series ESD-WP-2003-
M. Ghosh and J. Rao, Small area estimation: An appraisal, Statist 01.17, MIT Engineering Systems Division, Cambridge, MA,
Sci 9 (1994), 55 76. 2003.
W.J. Gordon, Synectics: The development of creative capacity, D. Macko, General system theory approach to multilevel systems,
Collier Books, New York, 1968. Report SRC 106-A-67-44, Systems Research Center, Case West-
D. Graupe, Identification of systems, Van Nostrand Reinhold, New ern Reserve University, Cleveland, OH, 1967.
York, 1972. D. Macko and Y.Y. Haimes, Overlapping coordination of hierarchi-
Y.Y. Haimes, The integration of system identification and system cal structures, IEEE Trans Syst Man Cybernet 8(10) (1978),
optimization, UCLA-ENG-7029, University of California, Los 745 751.
Angeles, 1970. M.W. Maier, Architecting principle for systems-of-systems, Syst
Y.Y. Haimes, Hierarchical analyses of water resources systems: Eng 1(4) (1998), 267 284.
Modeling and optimization of large-scale systems, McGraw- M.D. Mesarovic, Multilevel concept of systems engineering, Proc
Hill, New York, 1977. Syst Eng Conf, Chicago, IL, 1965, pp. 111 124.
Y.Y. Haimes, Hierarchical holographic modeling, IEEE Trans Syst M.D. Mesarovic (Editor), Systems theory and biology, Springer,
Man Cybernet 11(9) (1981), 606 617. New York, 1968.
Systems Engineering DOI 10.1002/sys
14 HAIMES
M.D. Mesarovic, D. Macko, and Y. Takahara, Theory of hierarchical, C. Shalizi, Methods and techniques of complex systems science: An
overview, March 2006, http://www.scribd.com/complex-over-
multilevel systems, Academic, New York, 1970.
view/d/28107.
National Research Council (NRC), Groundwater contamination,
A. Shenhar, A new systems engineering taxonomy, Proc 4th Int
Studies on Geophysics, National Academy Press, Washington,
Symp Natl Council System Eng, NCOSE, 1994, Vol. 2, pp.
DC, 1984.
261 276.
R.M. Oliver and J.Q. Smith (Editors), Influence diagrams, belief nets
M.G. Singh, Systems and control encyclopedia: Theory, technology,
and decision analysis, Wiley, New York, 1990.
applications, Pergamon, New York, 1987.
J. M. Ottino, Complex systems, AIChE J 49(2) (2003), 292 299.
K. Tarvainen and Y.Y. Haimes,  Hierarchical-multiobjective frame-
S.E. Page, Computational models from A to Z, Complexity 5(1)
work for energy storage systems, Organizations: Multiple
(1999), 35 41.
agents with multiple criteria, 190, J. Morse (Editor), Springer-
E. Paté-Cornell, Uncertainties in risk analysis: Six levels of treat-
Verlag, Berlin, 1981, pp. 424 446.
ment, Reliab Eng Syst Safety 54(2 3) (1996), 95 111.
B. Tivnan, Modeling organizational adaptation: A replication of
E. Paté-Cornell, Organizational aspects of engineering system
Levinthal s model of emergent order, Proc 2007 Winter Simula-
safety: The case of offshore platforms, Science 250 (1990), tion Conf, S.G. Henderson, B. Biller, M.H. Hsieh, J. Shortle, J.D.
1210 1217. Tew, and R.R. Barton, (Editors), 2007, pp. 1241 1246.
U.S. Department of Transportation (USDOT), Impacts of climate
A.P. Sage, Systems management for information technology and
change and variability on transportation systems and infrastruc-
software engineering, Wiley, New York, 1995.
ture: Gulf Coast study, Phase I, Washington, DC, 2008.
A.P. Sage, Systems engineering, Wiley, New York, 1992.
D.A. Wismer (Editor). Optimization methods for large-scale sys-
A.P. Sage, Methodology for large scale systems, McGraw-Hill, New
tems: With applications, McGraw-Hill Companies, New York,
York, 1977.
1971.
A.P. Sage and S.M Biemer, Processes for system family architecting,
Z. Yan, Risk assessment and management of complex systems with
design and integration, IEEE Syst J 1(1) (2007), 5 16.
hierarchical analysis methodologies, Ph.D. Thesis, University of
A.P. Sage and C.D. Cuppan, On the systems engineering and man-
Virginia, Charlottesville, 2007.
agement of systems of systems and federation of systems, Inform
Z. Yan and Y.Y. Haimes, Cross-classified hierarchical Bayesian
Knowledge Syst Management 2(4) (2001), 325 345.
models for risk-based analysis of complex systems under sparse
A.P. Sage and W.B. Rouse (Editors), Handbook on systems engi-
data, Reliab Eng System Safety 95 (2010), 764 776.
neering and management, Wiley, New York, 1st edition, 1995,
B.P. Zigler, Multifaceted modeling and discrete event simulation,
2nd edition, 1999. Academic, New York, 1984.
Yacov Y. Haimes is the L.R. Quarles Professor of Systems and Information Engineering, and Founding Director (1987)
of the Center for Risk Management of Engineering Systems at the University of Virginia. He received his M.S. and
Ph.D. (with Distinction) degrees in Systems Engineering from UCLA, and his B.S. degree in Mathematics, Physics,
and Chemistry from the Hebrew University, Jerusalem. On the faculty of Case Western Reserve University (1970 1987),
he chaired the Systems Engineering Department. As AAAS-AGU Congressional Science Fellow (1977 1978), Dr.
Haimes served in the Office of Science and Technology Policy, Executive Office of the President, and on the Science
and Technology Committee, U.S. House of Representatives. Since 1990 he has served as a consultant to the Software
Engineering Institute, Carnegie Mellon University, and for the last decade as a visiting scientist. He is a Fellow of seven
societies: ASCE, IEEE, INCOSE, AWRA, IWRA, AAAS, and Society for Risk Analysis (SRA), (where he is a past
President). The third edition of his most recent book, Risk Modeling, Assessment, and Management, was published by
John Wiley & Sons in 2009 (the first two editions were published in 1998 and 2004). Professor Haimes is the recipient
of the 2010 Distinguished Educator Award, presented by SRA; the 2007 Icko Iben Award, presented by AWRA; the
2001 Norbert Weiner Award, presented by IEEE-SMC; the 2000 Distinguished Achievement Award, presented by SRA;
the 1997 Warren A. Hall Medal, the highest award presented by Universities Council on Water Resources; the 1995
Georg Cantor Award, presented by the International Society on Multiple Criteria Decision Making; and the 1994
Outstanding Contribution Award presented by the IEEE-SMC, among others. He is a registered Professional Engineer
in Ohio and Virginia; Diplomate of the American Academy of Water Resources Engineers (and a Founding Trustee of
the AAWRE); the Past Engineering Area Editor of Risk Analysis: An International Journal. He has authored (and
co-authored) six books and 300 technical publications, over 200 of which were published in archival refereed journals.
He has served as dissertation/thesis advisor to 36 Ph.D. and 80 M.S. students.
Systems Engineering DOI 10.1002/sys