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Deployment-to-dwell metrics and
supply-based force sustainment
Sarah E. Evans
US Air Force Special Operations Command, Hurlburt Field, Florida, USA, and
Gregory Steeger
US Air Force Special Operations Command, USAFA, Colorado, USA
Abstract
Purpose –In the present fast-paced and globalized age of war, special operations forces have a comparative
advantage over conventional forces because of their small, highly-skilled units. Largely because of these
characteristics, special operations forces spend a disproportionate amount of time deployed. The amount of
time spent deployed affects service member’s quality of life and their level of preparedness for the full
spectrum of military operations. In this paper, the authors ask the following question: How many force
packages are required to sustain a deployed force package, while maintaining predetermined combat-
readiness and quality-of-life standards?
Design/methodology/approach –The authors begin by developing standardized deployment-to-dwell
metrics to assess the effects of deployments on service members’quality of life and combat readiness. Next,
they model deployment cycles using continuous time Markov chains and derive closed-form equations that
relate the amount of time spent deployed versus at home station, rotation length, transition time and the total
force size.
Findings –The expressions yield the total force size required to sustain a deployed capability.
Originality/value –Finally, the authors apply the method to the US Air Force Special Operations
Command. This research has important implications for the force-structure logistics of any military force.
Keywords Readiness, Continuous time Markov chains (CTMC), Deployment-to-dwell (D2D),
Force sustainment, Personnel tempo (PERSTEMPO), Special operations forces (SOF)
Paper type Research paper
1. Introduction
When embarking upon any long-term endeavor, it is important to count or at least estimate
the cost necessary to ensure completion. The trouble with counting the cost of wars is that
the path to “completion”is often fraught with unknowns. Furthermore, the business of war
itself is constantly changing (Votel et al.,2016). Therefore, warfighters and planners must do
their best, despite all the unknowns, to ensure the provided military capability is sustainable
with the resources on hand.
Special operations forces (SOFs) are uniquely capable to adapt to the changing rigors of
war. SOFs are small highly skilled units designed to respond quickly to emerging crises in
any part of the world, but they are not designed to conduct a major combat operation
without conventional forces (Bucci, 2015;Spulak, 2007). As evidenced by recent global
conflicts, SOF use has increased, perhaps to the detriment of their future use (Hennigan,
© In accordance with section 105 of the US Copyright Act, this work has been produced by a US
government employee and shall be considered a public domain work, as copyright protection is not
available. Published in Journal of Defense Analytics and Logistics. Published by Emerald Publishing
Limited.
JDAL
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Received 22May 2017
Revised 3 October2017
28 February 2018
29 March 2018
Accepted 4 April2018
Journal of Defense Analytics and
Logistics
Vol. 2 No. 1, 2018
pp. 2-21
Emerald Publishing Limited
2399-6439
DOI 10.1108/JDAL-05-2017-0009
The current issue and full text archive of this journal is available on Emerald Insight at:
www.emeraldinsight.com/2399-6439.htm
2017;Robinson, 2013;Watson, 2017). Employing SOF in a manner that puts production
capacity in peril leads to decreasing capability (JP3-05, 2014). Logically, identifying and
studying examples of decreasing capability can help reduce the risk of improper
employment. Additionally, depleting capability negatively impacts members’combat
readiness and quality of life (Losey, 2017b;Woody, 2017).
For the purposes of this paper, an individual’s“quality of life”is based on a multitude of
factors including mental, physical, social and spiritual well-being. Combat readiness and
quality of life are related because physical and mental well-being affect ability to deploy, in
addition to job-related competencies and regular training (Tucker et al.,2005;Rounds, 2010;
Szivak and Kraemer, 2015). Consequently, training, deployment and recovery compose the
readiness cycle.
One way to measure the readiness and quality of life of an individual or an organization
is to look at deployment-to-dwell (D2D) ratios (Dabkowski et al.,2009;Langstroth, 2013;
MacGregor et al., 2014;Trautmann et al.,2015). A D2D ratio is the amount of time an
individual (or group) is operationally deployed to the time the individual (or group) is not
deployed (PM 15-37, 2016). D2D ratios are typically normalized to the length of
a deployment so that they are reported as 1:days in dwell divided by days away on the
deployment. The recent necessity to obtain waivers for breaching D2D thresholds, at the US
Secretary of Defense (SecDef) level, along with low D2D ratios are possible indications of
SOF’s depleting readiness capacity (Losey, 2017a). These occurrences motivate the
overarching research question: How many force packages are needed to sustain a deployed
force package, while maintaining predetermined combat-readiness and quality-of-life
standards?
As early as May 2005, in guidance on Global Force Management, the SecDef expressed
concerns about US military forces’operations tempos and their impacts on the troops
(SecDef, 2007;USD, 2005;Chamberlain et al.,2005). In this context, operations tempo refers
to how often an individual is away from home because of combat-related deployments or
temporary duties. Later, SecDef (2007) established two metrics to measure an individual’s
operations tempo: personnel tempo and D2D. Personnel tempo measures an individual’s
operations tempo based on total days away from home station for any duty-related purpose,
whereas D2D measures an individual’s operations tempo based solely on combat-related
deployments. An operational deployment begins when a member departs his or her home
station, or en route training location, to meet a SecDef-approved operational requirement
(PM 15-37, 2016). An operational deployment ends when the individual arrives back at his or
her home station (PM 15-37, 2016). The SecDef imposed restrictions on force supply, for both
active and reserve forces, by establishing both personnel tempo and D2D goals and
limitations (SecDef, 2007). As originally stated, “the planning objective for the Active Force
remains one year deployed to two years at home station”(i.e. 1:2 D2D). The SecDef directive
also required that members maintain D2D ratios above 1:1. Though this policy was set
nearly 10 years ago because of insatiable demand and little to no enforcement of supply
restrictions, SOFs have operated at close to 1:1 D2D ratios (Copp, 2018;Losey, 2018).
There are many debates on the scope of SOF’s roles and responsibilities (Robinson, 2013).
SOFs are faster and more flexible than conventional forces, and SOFs have the skills to
address almost any mission. However, it is important to realize that the very characteristic
that gives SOF the ability to do what SOF does is the very thing that limits how much they
can do: their small size. Many SOF capabilities worldwide are “low density high demand”or,
in other words, they are limited assets or forces with unique mission capabilities stressed by
continual high requirements for their capabilities. The reality is that as a low-density high-
demand force, SOF cannot do everything demanded because their resources are limited.
Supply-based
force
sustainment
3
Therefore, it is in the best interest of the organization to objectively determine the resources
that are needed to sustain enduring deployment requirements.
In a resource-constrained environment, protecting production capacity is paramount. In
this context, protecting production capacity means producing a sustainable capability by
ensuring the availability of training, resources and equipment. Arguably, the most valuable
resource is personnel. To protect production capacity in this environment, with less than 100
per cent manning, policies must be put in place to sustain supply. Even with manning at 100
per cent, there can still be readiness problems, especially in the context of insatiable demand.
Typically, SOF responds to increasing demands by increasing the quantity supplied. For
instance, for US SOF, it is okay to break the SecDef mandate to maintain a D2D of at least
1:1 so long as those breaking the mandate are volunteers and have approval from the first
flag officer in their chain of command (PM 14-07, 2014). As a result, supply has increased
with demand. An illustration of this paradigm, commonly known as perfectly elastic supply
in economics, is provided in Figure 1(a).
In general, if supply exceeds demand and there are no other limiting factors in the
system, the number of transactions in the system are limited by the demand. Conversely, a
system with insatiable demand, but constrained supply, is limited to the number of
transactions possible given the existing supply. In actuality, SOF is ultimately a system
Figure 1.
(a) Perfectly elastic
supply; (b) perfectly
inelastic supply; and
(c) SOF supply
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with limited supply under insatiable demands.In economics, this situation is called perfectly
inelastic supply and is shown in Figure 1(b). In this case, the price could be viewed as
opportunity cost or risk because both increase as demand increases, though the quantity
supplied remains fixed. To prioritize one operation over another, in the context of limited
supply, is to do that operation at the opportunity cost of the other. Additionally, as there is
more unfulfilled demand for SOF, risk increases in the field. Structuring the forces in a
manner, which protects production capacity, becomes all the more important in a resource-
constrained environment because the parties making demands are not necessarily affected
by the costs, either in the long or short term. Rather the service members assume the costs as
their quality of life and combat readiness are affected. Therefore, to account for supply
limitations, the capabilities should be presented as in Figure 1(c).Figure 1(c) demonstrates
how capability supply ceilings allow increases in demand to a predetermined point, after
which all increases in demand have no effect on the quantity supplied.
This paper presents a supply-based model for determining the required force strength
necessary to sustain an enduring war-fighting capability. We model the problem using a
continuous time Markov chain (CTMC) and use the chain’s limiting behavior to determine
steady-state equations for D2D ratios. The equations yield the relationship between the force
multiplier (i.e. the number of identical force packages used to sustain one that is deployed),
deployment rotation length, transition time and non-availability of forces.
This research is valuable for several reasons. First, it standardizes an objective
measurement of readiness by clearly defining D2D. Second, it develops a method that
enables a decision-maker to efficiently allocate resources, based on existing supply, while
preserving readiness for the long term. Finally, to mathematically justify current and future
force employment decisions, this research derives an equation that relates D2D to key
deployment planning factors. Our work is unique and fills a gap in the literature because, to
our knowledge, we are the first to relate force sustainment to a quality-of-life metric. This
work has undeniable applications to and implications for SOF worldwide and military
forces in general.
The remainder of this paper is organized as follows: Section 2 provides the necessary
background and framework for the problem. Next, Section 3 logically frames and then
presents the methodology in general terms. Section 4 applies the method to US Air Force
Special Operations Command’s (AFSOC’s) active duty forces. Following this, Section 5
states the findings and recommendations.
2. Background
Before attempting to answer the overarching research question, it is essential to properly
frame the problem. Any attempt to structure forces, whether conventional or SOF, should
begin with determining the capability that is needed. Properly sizing and structuring each
force package, or all the equipment and personnel associated with a defined military
capability, is important for creating measurable capabilities and reasonable expectations.
2.1 The big picture: capability-based force structuring
Figure 2 explains how the force should be built and structured based on supply. Three
phases are used to describe the process: building the force, employing or deploying the force
and sustaining the force. In each of the phases, answers to the questions shown in the figure
need to be determined.
The first step in the process is to determine the capability or capabilities to be provided.
It is important to understand that larger force packages gain maintenance efficiencies.
However, the tradeoff for these efficiencies manifests itself in decreased flexibility. Smaller
Supply-based
force
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5
force packages allow for enhanced projection of the capability to more locations. When
creating a force package, it is important to make its capability objectively measurable to
effectively communicate the capability’s readiness.
We assume initiation in the third phase of the process; an appropriately sized force
package has been built and employed, and now it is necessary to determine the number of
identical force packages required to sustain each one that is deployed long term. To sustain
the force package, we must first determine how to quantify force sustainment. Force
sustainment is achieved when force packages are used to provide an enduring capability
and that all the personnel involved are healthy. We measure personnel health via D2D.
2.2 Measuring and standardizing operations tempo
The D2D metric we develop meets all of the standards listed in Harrison (2014). The D2D
metric measures outputs rather than inputs, is linked to strategy, is quantifiable and avoids
subjective assessments. D2D metrics can be used in two separate ways to measure either the
historical health or the combat capability (or availability) of the force. We focus on the
former.
Based on the definitions provided for dwell and an operational deployment (Section 1;
PM 15-37, 2016), individual historic D2D ratios can only be computed for individuals with at
least two deployments. To be counted, a dwell period must be bookended with an
operational deployment return date and an operational deployment departure date. Each
individual’s historic D2D ratio is based solely on the length of their penultimate operational
deployment and the length of the dwell period immediately following it. Of the three
individuals shown in Figure 3, a historic D2D ratio can only be computed for X. X’s historic
D2D ratio of 1:2 is based on their deployment from 1 January to 31 March and their dwell
period from 1 April to 30 September. A historic D2D ratio cannot be computed for Y because
Figure 2.
Force structure
overview
Figure 3.
Historic D2D example
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they have not departed on their second deployment and, consequently, their dwell period
has not ended. Finally, a historic D2D ratio cannot be computed for Z because they are yet to
start their first dwell period.
Though D2D ratios are individual metrics, a group’s average historic D2D ratio can be
used to answer health-of-the-force questions for different groups of individuals. In an effort
to capture the most representative metric, a group’s average historic D2D ratio is computed
holistically using historic D2D ratios, when they can be computed, from everyone assigned
to the group. The holistic average is used to place less emphasis on individual historic D2D
ratios that may be considered outliers; an individual with a short deployment followed by a
long period of dwell may skew the group’s average. To compute the holistic average dwell
ratio, one divides the sum of the dwell lengths (in days) for all the members in the group, for
which a D2D ratio can be computed, by the deployment lengths (in days) for all the members
in the group, or mathematically (for npersonnel with valid D2D ratios in a group):
average historic dwell ¼dwell1þdwell2þ... þdwelln
deploy1þdeploy2þ... þdeployn
:
The average historic D2D is then reported as 1: average historic dwell.
An example showing how to compute a group’s average historic D2D ratio is given in
Table I. Assume that the entire group consists of two members, A and B, with penultimate
deployment and last dwell lengths shown. Weighting each deployment equally, the group’s
average historic D2D ratio is 1: 1þ3
1þ1
¼1: 2:In the holistic average, rather than weighting
each individual’s D2D ratio equally, the individual ratios are weighted based on their
lengths. The group’s holistic historic D2D ratio is 1: 60 þ90
60 þ30
¼1: 1:67:Having developed a
standardized way to measure the health of the force, this paper now reviews force
sustainment models, both in practice and in the academic literature.
2.3 Force sustainment approaches
Each of the USA’s services has a unique way of sustaining their forces via readiness cycles.
The intent of each plan is to rotate personnel and equipment in such a way that training,
quality of life, maintenance and deployment requirements are all met. Table II describes the
methods of the US Air Force, Army, Navy and USSOCOM. As shown in Table II, none of
these plans are functioning as perfectly as intended.
2.4 Literature review
More often than not, work in this area views force sustainment from the perspective of
readiness measurement or manpower requirements. There is a wide variety of approaches
to measuring readiness (Harrison, 2014;Freeman et al., 2014;Barzily et al.,1979;Scales et al.,
2011). Our review of the literature on personnel modeling is summarized in Table III.Note
that the literature is sorted by method and then by publication date. In an effort to be
concise, we only discuss a sample of the literature from each of the methods used and focus
Table I.
Example historic
D2D calculation
Deployment length (in days) Dwell length (in days) Individual D2D ratio
A 60 60 1:1
B 30 90 1:3
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on the literature that is closely related to our work. More comprehensive literature reviews
and surveys can be found in the studies of Gass (1991),Wang (2005),Guerry and Feyter
(2009) and Parlier (2016).
Kinstler et al. (2008) use a Markov model to rectify rank imbalances in the Navy Nurse
Corps. Filinkov et al. (2011) create a software tool to test the personnel sustainability of a
land force structure in terms of career progression and operational considerations for the
Australian Army. Richmond et al. (2012) model the population of ground forces to manage
personnel and major system sustainability with Markov techniques for the Australian
Army. Zais and Zhang (2016) examine stay or leave decisions in the US Army using a
Markov chain model. Mitchell (1993) uses a simulation to estimate the impact on training
requirements of force structure decisions for the US Air Force. Pall et al. (2007) use a
simulation to examine personnel and materiel policies for the Canadian Army. Kim et al.
(2012) applied a stochastic optimization model to manage the uncertainty of demand and
supply for knowledge workers. Durbin and Wright (1967) use linear programing to manage
overseas tour lengths for high-demand positions in the US Air Force. Whitney et al. (2013)
consider the effects of force organization on capability fulfillment using a qualitative
methodology for the Australian Army.
While the literature above has important implications, many are impractical for
informing the day-to-day decision-making that impacts lives, such as how many force
packages to deploy. The effects of military organization decisions on the lives of the people
which compose it are important considerations for force management because people’slives
are fundamentally entwined with their combat readiness (AFI90-506, 2014). While studies
show that deployments have both positive and negative effects on retention, a reoccurring
top concern for service members is the amount of time they spend separated from their
families (Fricker, 2002;Badger, 2004). Additionally, deployment duration has been
Table II.
Deployment cycle
plan comparison
Service/
command Nomenclature Directive Notable characteristics
Air Force Air
Expeditionary
Force Cycle
Air Force Instruction
10-244
3-phase, demand-driven cycle. Surge above 12
months may require significant actions to
reconstitute the force (AFI10-244, 2002).
Disproportionate deployment burdens have
been a problem (Losey, 2016)
Army Army Force
Generation
Army Regulation
525-29
3-phase, demand-driven cycle. Will be replaced
by Sustainable Readiness Model in fiscal year
2017 because of increased demand in
conjunction with decreasing resources (Army
Readiness Guidance, 2017)
Navy Optimized
Fleet Response
Plan
Office of the Chief of
Naval Operations
Instruction 3000.15A
4-phase, supply-driven cycle. Maintains the
capacity to rapidly increase forward presence as
world events dictate and additional funding
becomes available (OPNAV Instruction
3000.15A, 2014). Of the fleets converted to
OFRP, many have had difficulty maintaining
the cycle schedule because of maintenance
overrun (GAO, 2016)
USSOCOM Special
Operations
Force
Generation
USSOCOM Directive
5225-10
3-phase, demand-driven cycle. More of a
guideline than a policy because each SOF
component must also answer to their respective
service (USSOCOM Directive 5225-10, 2013)
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Table III.
Academic literature
on personnel
modeling
Reference Research question Method Application
Mitropoulos (1983) What metrics best describe members’
professional evolution in hierarchical
organizations?
Markov model General
Weigel and Wilcox
(1993)
How do high-level personnel planning
decisions impact troops at the
occupational specialty level?
Markov, network, linear
programming and goal-
programming models
US Army
Georgiou and
Tsantas (2002)
What is the optimal way to minimize
cost, in a k-classed hierarchical system,
while meeting workforce demand and
satisfying government constraints and
regulations?
Markov model European Union/
Workforce
Kinstler et al. (2008) How do different policies impact
balance in the rank structure and
retention?
Markov model US Navy Nurse Corps
Filinkov et al. (2011) What force strength is required to meet
operational demands, while considering
individuals’career progression?
Markov model with
hierarchical classes
Australian Army
Richmond et al.
(2012)
How will personnel and/or major
systems populations change over time?
Markov model Australian Army
Zais (2014) What are the workforce requirements
based on uncertain demand?
Markov model, simulation U.S Army
Zais and Zhang
(2016)
What incentives have the greatest
impact on personnel retention?
Markov model, stochastic
dynamic programming
US Army
Bender and
Isbrandt (1991)
What are the effects of different policy
options on career progression?
Discrete event simulation Canadian Forces
Mitchell (1993) How many individuals need to be
trained annually, for every occupation?
Discrete event simulation United States Air Force
Zegers and
Isbrandt (2006)
What simulation tool most accurately
depicts military training and career
progression?
Discrete event simulation Canadian Forces
Pall et al. (2007) What is the most efficient way to
provide and sustain qualified units for
operational tasks?
Discrete event simulation Canadian Army
Moorhead and
Halbrohr (2010)
What simulation tool best determines
the ability of the Canadian Forces to
meet the personnel demands of
operations?
Discrete event simulation Canadian Forces
Cao et al. (2010) How can human capital supply chain
decisions be improved to drive better
business performance with integrated
OR models?
Stochastic modeling and
optimization
Human Capital Supply
Chains
Kim et al. (2012) How does uncertainty of demand for
knowledge services as well as the
supply of knowledge workers impact
recruiting strategy?
Stochastic optimization model Korean Security
Consulting Companies
Durbin and Wright
(1967)
How many personnel are needed to meet
the requirements associated with
rotating personnel between state-side
and overseas locations?
Linear programming United States Air Force
Durso and Donahue
(1995)
What are the impacts of personnel
management policies on the US Army’s
enlisted force?
Decision analysis United States Army
Edwards (1983) What manpower models exist and to
what extent have these been effective in
application?
Qualitative assessment Industry
Whitney et al.
(2013)
What is the Australian Army’s ability
to undertake new or existing
contingencies?
Qualitative assessment Australian Army
Supply-based
force
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9
associated with negative effects on psychological and physical health (Meadows et al.,2017;
Mulligan et al., 2012;Szivak and Kraemer, 2015). Our work establishes an easy-to-
understand feedback loop for decision-makers which reconnects the costs associated with
these effects on the capability demanded. Additionally, this work fills a gap in the existing
literature by connecting an objective and mathematically rigorous force sustainment model
to a quality-of-life metric.
3. Methodology
For the intents and purposes of this research, force readiness and quality of life are
measured solely via historic D2D metrics. We use a 1:2 D2D ratio to logically frame and
build the method. After making the necessary logical arguments, we present the resulting
formulas in general terms assuming one wishes to maintain a 1:DD2D ratio, as opposed
to 1:2.
To perpetually sustain one deployed force package and maintain a 1:2 D2D ratio for all of
the associated personnel, at least three force packages are required. The three force
packages will rotate through equal periods of deployment, recovery and preparation
(Figure 4). As they rotate, the D2D ratios for the personnel assigned to each force package
will change as shown. To maintain D2D ratios of 1:2, the preparing forcepackage will not be
ready to deploy until all of its associated personnel have stayed in dwell long enough so that
their individual D2D ratios are 1:2 or better.
The three-force package model assumes 100 per cent manning and availability in all
positions, and instantaneous changeover and transit. In reality, manning shortfalls (in one or
multiple crew positions), manning unavailability (in one or multiple positions) and the
reality of transit time and responsibility changeover make it so the three-force package
model does not provide adequate manpower to maintain 1:2 D2D ratios.
Based on manning shortfalls and non-availability, it may seem reasonable to suggest a
force multiplier of four [Figure 5 (left)].
Figure 5.
Four- force package
model (left) and four-
force package model
with overlap (right)
Figure 4.
Three-force package
model (left) and three-
force package model
accounting for
changeover and
transit time (right)
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There are several problems with suggesting a force multiplier of four based on the
arguments posed thus far. First, the four-force package model compensates for low manning
with additional force package-provided billets. Increasing the force multiplier (i.e. adding
billets) to compensate for manning shortfalls is nonsensical. The focus should be on filling
the empty billets as opposedto askingfor more billets.
Second, the idea assumes that all the non-available personnel are entirely separate,
or mutually exclusive, from any of the other categories. In reality, non-availability
occurs in all the categories; one may become unavailable just prior to their
deployment, or they may not be able to accomplish the necessary tasks associated
with either the recovery or preparing phases which will create a chain reaction
resulting in the individual not deploying on time later in the cycle. Figure 5 (right)
illustrates how personnel who are not available can also be in one of the other three
categories. Arrows to and from the non-available bucket have been removed to avoid
cluttering the diagram.
Figure 5 (right) still does not address the concern about compensating for low
manning (i.e. empty billets) with additional force package-provided billets.
Consequently, the non-available bucket should only account for the number of non-
available personnel of those assigned, as opposed to those authorized. Even so, Figure 5
(right) suggests that the necessary force multiplier is a number between three and four.
Third, because changeover and transit time are still not considered, incorporating them
is necessary to ensure that there are no manning gaps. In other words, individuals do not
leave the area of responsibility until changeover. The time required for changeover and
transit is an important consideration for low-density high-demand assets because, with
higher operations tempos, it makes deployment periods longer and dwell periods shorter.
For conventional forces, 1:4 D2D rates are more typical so changeover and transit times are
effectively negligible.
Figure 4 (right) shows the resulting D2D ratios using the three-force package model and
assuming 14 days for transit and changeover with a rotation length of 120 days. In this
scenario, the force package’s D2D never reaches 1:2. Based on the arguments posed thus far,
it seems a force multiplier between three and four is necessary to maintain a 1:2 D2D.
However, this conclusion is an impasse; to move forward and better quantify the required
force multiplier, a more sophisticated model is necessary.
Therefore, we first model the deployment and dwell periods for the force package
using a CTMC and then compute the steady-state probabilities for each state in the
chain. Figure 6 depicts the transition diagram. The steady-state probabilities are used
to compute historic D2D ratios because they yield the proportion of time a force
package is in dwell versus the proportion of time the force package is deployed. With
an expression for steady-state probabilities, rearranging reveals the dependence of
the force multiplier on rotation length, as well as the amount of changeover and
transit time. This CTMC is a birth-and-death process with two states “deployed”
(state 0) and “in dwell”(state 1).
Figure 6.
Transition diagram
for one force package
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11
To solve for the steady-state probabilities, the transition rates q01 and q10 must be
found. q01 is the rate at which a force package transitions from being deployed to being
indwell.Thenumberofdayseachforcepackageisdeployedisequaltotherotation
length, RL,plusthefixed number of changeover and transit time, T.Aforcepackage
transitions from deployed to in dwell once every RL þTdays. In other words,
q01 ¼1
RL þT:
Similarly, q10 is the rate at which a force package transitions from being in dwell to being
deployed. This rate is equivalent to one over the length of time spent in dwell. The length of
time spent in dwell equals the rotation length, RL, times the number of rotations the force
package remains in dwell, minus the fixed number of days spent in changeover and transit,
T. The number of rotations the force package remains in dwell is equal to the number of
force packages that are not deployed (i.e. in dwell) or the force multiplier, fm, minus one.
Changeover and transit days, T, are considered deployment days and, thus, do not count as
dwell. Putting all of this together:
q10 ¼1
RL *fm 1
ðÞ
T:
The steady-state probabilities are derived using the balance equations which, based on the
transition rates, are:
p
0
1
RL þT
¼
p
1
1
RL *fm 1
ðÞ
T
(1)
p
0þ
p
1¼1:(2)
The D2D ratio is now simply stated as:
p
0
p
1
;
which can be expressed by rewriting equation (1) as:
p
0
p
1
¼RL þT
RL *fm 1
ðÞ
T:
Normalizing the length of the deployment and writing D2D in the standard format yields:
D2D¼1: RL þT
RL *fm 1
ðÞ
T:
To maintain a D2D ratio of 1:D:
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RL *fm 1
ðÞ
T
RL þTD:(3)
Finally, to determine how the force multiplier depends on rotation length, as well as
changeover and transit time, solve equation (3) for fm to obtain:
fm Dþ1
ðÞ
RL þT
RL
:(4)
Equation (4) yields the force multiplier necessary to maintain long-term 1: DD2D ratios
based on rotation length, as well as changeover and transit time. Note that the
expression:
RL *fm 1
ðÞ
T
RL þT;
from equation (3) is useful because it yields a force package’s long-term dwell based on
rotation length, a given force multiplier, as well as changeover and transit time. Non-
availability is still not part of the equation.
To account for non-availability of the assigned manpower, divide the number produced
in equation (4) by A, where:
A¼Number Available
Number Assigned :
The final relationship that determines the force multiplier needed to maintain 1: DD2D and
accounts for rotation length, changeover, transit time and manpower non-availability is:
fm Dþ1
A
RL þT
RL
:(5)
4. Application
To show the value of this work, our method is applied to the AFSOC. To begin, we discuss
the importance of the holistic D2D average.
Figure 7 depicts individual D2D ratios for members of an AFSOC operations group.
In this particular group, as the histogram on the left shows, approximately 4 per cent of
Airmen have dwell rates greater than 19.5 (i.e. D2D ratios less than 1:19.5). The high dwell
rates skew the average dwell rate significantly. In this case, all of the high dwell rates are
based on deployments that lasted less than 30 days. Often times, because of database and
administrative limitations, short trips overseas are coded as deployments when they do not
actually meet the definition of an operational deployment. Once the short trips are removed,
the mean dwell rate is a more representative metric. However, distinguishing short trips
from actual deployments by removing anything that lasts less than 30 days is completely
subjective because the 30-day cutoff was chosen arbitrarily. In truth, some deployments last
less than 30 days. Typically, when a statistician does not want the measure of central
tendency to be impacted by outliers, they will use the median instead of the mean. However,
outliers (or short deployments) should not be completely ignored but instead should be
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weighted appropriately. The holistic average takes short deployments into account but
weights them according to their length.
Historically, at AFSOC, based on personnel data from the Military Personnel Data
System collected from 2011 through 2014, 85 per cent of authorized billets actually have
personnel assigned and 91 per cent of those assigned are available. Therefore, AFSOC tends
to have 77 per cent of authorized personnel available. Mathematically:
No:Assigned
No:Authorized ¼85%;
No:Available
No:Assigned ¼91%;and No:Available
No:Authorized ¼77%:
As, historically at AFSOC A= 0.91 and as, for AFSOC’s active duty force, the goal is 1:2
D2D ratios, the force multiplier is given as:
fm 3
0:91
RL þT
RL
:
Figure 8 plots the required force multipliers, as a function of rotation length, based on just
the combined changeover and transit time (solid line) and based on the combined
changeover and transit time plus the non-availability of assigned personnel (dashed line).
Note that the required force multiplier decreases, but not linearly, as the rotation length
increases. This inverse relationship is expected, as the number of rotations required
decreases as the rotation length increases and as every rotation requires a fixed amount of
changeover and transit time. As a side note, because each rotation requires a fixed amount
of changeover and transit time, as rotation length increases, the total number of days away
Figure 7.
Left: Dwell rates
histogram (all data)
and right: Dwell rates
histogram
(no deployments
#30 days)
Figure 8.
Force multiplier
needed to maintain
1:2 D2D based on
rotation length (14
days of changeover
and transit time)
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decreases. This relationship creates a tradeoff between rotation length and total number of
days away.
Basedonthemath,Figure 8 plots non-integer-valued force multipliers. Fractions of force
packages are permissible if compensating for personnel non-availability; however, a fraction of
a force package cannot cover a deployment requirement for a full force package. The question
becomes “since it is not possible to deploy fractions of people or crews, is it possible to use
fractions of force multipliers to determine the correct force strength or force package size?”
The answer is, it depends. If the goal is for personnel to maintain an average D2D of 1:2
over multiple deployments and dwell periods, then the answer is yes. If, however, the goal is
to have the manpower necessary so that each dwell period, for each individual, is twice as
long as the deployment preceding it, then the answer is no. The latter requires rounding up
to the next integer-valued force multiplier. The solid line in Figure 8 shows that rounding up
to a force multiplier of four provides the manning necessary to account for changeover,
transit time and non-availability for rotations longer than 90 days. Because of rounding up,
a force multiplier of four will result in individual and average D2D ratios better than 1:2. The
margin by which the D2D ratios are better than 1:2, as shown in Figure 8, increases as
rotation length increases.
To determine how sensitive the results in Figure 8 are to changeover and transit time,
this paper will now examine how the force multiplier is affected by an increase or decrease
in the number of changeover and transit days. Originally the assumption was a total of
fourteen days of changeover and transit time, seven days on either end of each rotation.
Decreasing the number of changeover and transit days to six (three on either end of each
rotation) yields the force multipliers plotted at the bottom of Figure 9. These can be thought
of as a lower bound on the force multiplier (i.e. a best-case scenario). On the other hand,
increasing the number of changeover and transit days to 20 (ten on either end of each
rotation) produces the force multipliers plotted at the top of Figure 9. Ten days of
changeover and transit time on either end of each rotation is sufficient, in most cases, to
place an upper bound on the force multiplier. Note that, for rotations of 60 days or longer, the
force multiplier is generally bounded between three and four.
The key to effectively using the findings of this research is defining what is meant by
“maintaining a D2D ratio of 1:D.”Here, “maintaining”could mean:
Maintaining an average D2D of 1:Dover all members in a group.
Maintaining an average D2D of 1:D, for each individual in the group, over all of their
individual deployments.
Maintaining a D2D for each individual in the group, after each deployment.
Figure 9.
Sensitivity analysis:
Force multiplier
bounds (6-20 days
changeover and
transit time)
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As this research has shown, the third definition is the most restrictive and, consequently,
will require the most manpower and the largest force multiplier. However, the third
definition provides service members with the most predictability and, arguably, the best
quality of life. If the intent is to satisfy either of the first two definitions, then equation (5) is
sufficient to determine the required force multiplier. If, however, the intent is to satisfy the
third definition, then the results from equation (5) should be rounded up with consideration
for associated risks.
The following example helps to explain some of the finer points of this analysis. Assume
the 123rd Special Operations Group has a requirement to deploy an enduring force package
of ten individuals for 120-day rotations. Also, assume that there are 14 total days of
changeover and transit associated with every rotation, seven days on either end of each
rotation. According to Figure 8, not considering availability, and not rounding up to the
nearest integer, the 123rd Special Operations Group requires a total of 34 personnel.
Figure 10 shows how different groups of personnel will transit and rotate in and out of
theater and dwell over a 494-day period (four full rotations plus changeover and transit
time).
Personnel are gone for 134 days on each deployment; 120 days for the rotation and 14
days for changeover and transit time. However, as there are not four complete force
packages (or groups of ten personnel), dwell periods are neither the same length for each
group nor the same length for each individual after each of their deployments. After their
first rotation, Personnel 1-6 are in dwell 226 days after their deployment of 134 days and
have a D2D ratio of 1:1.69. Personnel 7-10 have a longer dwell period of 346 days and,
consequently, a D2D ratio of 1:2.58. Personnel 1-6 have fewer dwell days than Personnel 7-10
because Personnel 1-6 must depart to cover rotation four with Personnel 31-34. The groups
of personnel, as listed in the rows, have alternating D2D ratios of 1:1.69 (shown in grey) and
1:2.58 (shown in white). The average D2D ratio, over all personnel or for each individual,
over multiple complete cycles, is 1:2.
Two main conclusions can be drawn from this analysis and specifically from the
example above:
(1) Average 1:2 D2D ratios, over multiple individuals or over multiple deployments
for each individual, can be achieved with the force multipliers shown in
Figure 8.
(2) Individual D2D ratios of 1:2 for each individual and following each deployment
cannot be achieved unless the force multipliers shown in Figure 8 are rounded up
to four.
Figure 10.
Example: 123rd
special operations
group changeover,
transit, rotation, and
dwell timeline
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Additionally, note that the example does not account for non-available personnel and,
therefore, is a best-case scenario. When personnel become non-available, variability
within the groups shown in Figure 10 is introduced and the required force multiplier
becomesmoredifficult to compute. Because of non-availability, a force multiplier of
four does not necessarily guarantee that individuals will not drop below 1:2 D2D. This
can happen when someone who is supposed to deploy as part of their normal rotation
becomes non-available just prior to their scheduled departure date, and someone else on
a separate rotation has to fill in and break their 1:2 D2D. If it is unacceptable for
individuals to drop below 1:2 D2D, on such occasions, increasing the force multiplier to
a number larger than four may be justified. As the preponderance of AFSOC’s rotations
are between 90 and 150 days, and as the goal is to generally maintain individual D2D
ratios of 1:2 for each deployment, a force multiplier of four is recommended, for every
persistently deployed force package.
The case study above highlights a limitation to this research. Many of the results
discussed in this section are based on analyses we accomplished for AFSOC and, thus,
specific to our application. That said, similar takeaways to those above can be made, for a
given organization, if adequate analysis is done up front. At the least, to successfully apply
this work, an organization must first determine which of the above definitions is appropriate
and study the organization of interest to determine typical deployment lengths and
availability of personnel.
5. Conclusion
This work derives closed-form equations for determining the force multiplier required to
maintain a specified deploy-to-dwell ratio (i.e. a quality-of-life metric). The equations relate
the amount of time spent deployed versus at home station, rotation length, transition time
and the total force size, making it possible to analyze the relationships among these factors.
Our methodology provides a way to mathematically justify protecting force production
capacity to sustain enduring deployments while maintaining predetermined combat-
readiness and quality-of-life standards.
The relationships among these factors have important implications for force-
structure logistics. Shorter, more frequent force rotations result in greater amounts of
time deployed overall because of the transition time incurred for each rotation.
Increasing the efficiency of force movements, by making transition times shorter,
decreases the overall force size necessary to sustain an enduring deployed capability.
Incomplete force packages and unexpected unavailability lead to uneven
distributions of D2D ratios within units. Therefore, to improve the equitable
distribution of D2D ratios in units, commanders should consider a full force package
committed if any part of it is used to meet demand. Objectively calculating the force
structure necessary to sustain a deployed capability establishes a definitive supply
cutoff, at which the decision maker risks damaging the production capacity of the
capability. Additionally, using the simple equations derived above, decision-makers
can readily assess force structure logistics decisions in terms of the effects they will
have on units’D2D ratios.
Another significant contribution of this work is the development of a standardized
method for calculating the D2D metric. While there are current policies governing the
management of health of the force via D2D, different interpretations can result in different
calculation methods and thus inaccurate comparisons. Standardizing the calculation
facilitates objective equitable comparisons and more effective decision-making.
Furthermore, using the D2D metric for determining deployed capability sustainment
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connects force structure decisions with the implications for service members’combat
readiness and quality of life, thus associating the service members’cost with the quantity of
capability demanded.
Although this research focuses on one aspect of readiness, for a single military
organization, over a limited time period, future work could apply the methodology to
other military organizations and evaluate the effectiveness of the implementation of
this research over time. The method could also be extended to determine the force
multiplier based on different readiness or quality-of-life metrics. Additionally, one
could analyze the effect of randomly occurring schedule changes on the distribution of
D2D throughout the units. Finally, creating a simulation to evaluate the long-term
effects of differing types and levels of non-availability on the system, as a whole,
provides another opportunity for future work.
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Corresponding author
Sarah E. Evans can be contacted at: sarah.evans.7@us.af.mil
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