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Deployment-to-dwell metrics and supply-based force sustainment


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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.
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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
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 members 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 membersquality 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 completionis often fraught with unknowns. Furthermore, the business of war
itself is constantly changing (Votel et al.,2016). Therefore, warghters 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
conicts, 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
Received 22May 2017
Revised 3 October2017
28 February 2018
29 March 2018
Accepted 4 April2018
Journal of Defense Analytics and
Vol. 2 No. 1, 2018
pp. 2-21
Emerald Publishing Limited
DOI 10.1108/JDAL-05-2017-0009
The current issue and full text archive of this journal is available on Emerald Insight at:
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 memberscombat
readiness and quality of life (Losey, 2017b;Woody, 2017).
For the purposes of this paper, an individualsquality of lifeis 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
SOFs 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
As early as May 2005, in guidance on Global Force Management, the SecDef expressed
concerns about US military forcesoperations 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 individuals
operations tempo: personnel tempo and D2D. Personnel tempo measures an individuals
operations tempo based on total days away from home station for any duty-related purpose,
whereas D2D measures an individuals 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 SOFs roles and responsibilities (Robinson, 2013).
SOFs are faster and more exible 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 demandor,
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.
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 rst
ag ofcer 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
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 xed. 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 unfullled demand for SOF, risk increases in the eld. 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-ghting capability. We model the problem using a
continuous time Markov chain (CTMC) and use the chains 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 dening D2D. Second, it develops a method that
enables a decision-maker to efciently 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 lls a gap in the literature because, to
our knowledge, we are the rst 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 Commands (AFSOCs) active duty forces. Following this, Section 5
states the ndings 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 dened 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 gure
need to be determined.
The rst 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 efciencies.
However, the tradeoff for these efciencies manifests itself in decreased exibility. Smaller
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 capabilitys 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 rst 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 quantiable 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
Based on the denitions 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
individuals 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. Xs 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
Figure 3.
Historic D2D example
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 rst dwell period.
Though D2D ratios are individual metrics, a groups 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 groups 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 groups 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 groups 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 groups
average historic D2D ratio is 1: 1þ3
¼1: 2:In the holistic average, rather than weighting
each individuals D2D ratio equally, the individual ratios are weighted based on their
lengths. The groups 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 USAs 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
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 fulllment 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 peopleslives
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
command Nomenclature Directive Notable characteristics
Air Force Air
Force Cycle
Air Force Instruction
3-phase, demand-driven cycle. Surge above 12
months may require signicant actions to
reconstitute the force (AFI10-244, 2002).
Disproportionate deployment burdens have
been a problem (Losey, 2016)
Army Army Force
Army Regulation
3-phase, demand-driven cycle. Will be replaced
by Sustainable Readiness Model in scal year
2017 because of increased demand in
conjunction with decreasing resources (Army
Readiness Guidance, 2017)
Navy Optimized
Fleet Response
Ofce 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 eets converted to
OFRP, many have had difculty maintaining
the cycle schedule because of maintenance
overrun (GAO, 2016)
USSOCOM Directive
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)
Table III.
Academic literature
on personnel
Reference Research question Method Application
Mitropoulos (1983) What metrics best describe members
professional evolution in hierarchical
Markov model General
Weigel and Wilcox
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
Markov model European Union/
Kinstler et al. (2008) How do different policies impact
balance in the rank structure and
Markov model US Navy Nurse Corps
Filinkov et al. (2011) What force strength is required to meet
operational demands, while considering
individualscareer progression?
Markov model with
hierarchical classes
Australian Army
Richmond et al.
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
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
Discrete event simulation Canadian Forces
Pall et al. (2007) What is the most efcient way to
provide and sustain qualied 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
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
Human Capital Supply
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
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
What are the impacts of personnel
management policies on the US Armys
enlisted force?
Decision analysis United States Army
Edwards (1983) What manpower models exist and to
what extent have these been effective in
Qualitative assessment Industry
Whitney et al.
What is the Australian Armys ability
to undertake new or existing
Qualitative assessment Australian Army
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 lls 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)
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 lling
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 packages 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 rst 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
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
length, RL,plusthexed 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 xed 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
The steady-state probabilities are derived using the balance equations which, based on the
transition rates, are:
RL *fm 1
 (1)
The D2D ratio is now simply stated as:
which can be expressed by rewriting equation (1) as:
¼RL þT
RL *fm 1
Normalizing the length of the deployment and writing D2D in the standard format yields:
D2D¼1: RL þT
RL *fm 1
To maintain a D2D ratio of 1:D:
RL *fm 1
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
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
RL *fm 1
RL þT;
from equation (3) is useful because it yields a force packages 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 nal 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
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 signicantly. 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 denition 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
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:Authorized ¼85%;
No:Assigned ¼91%;and No:Available
No:Authorized ¼77%:
As, historically at AFSOC A= 0.91 and as, for AFSOCs active duty force, the goal is 1:2
D2D ratios, the force multiplier is given as:
fm 3
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 xed amount of
changeover and transit time. As a side note, because each rotation requires a xed 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
(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)
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 sufcient, 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 ndings of this research is dening what is meant by
maintaining a D2D ratio of 1:D.Here, maintainingcould 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)
As this research has shown, the third denition is the most restrictive and, consequently,
will require the most manpower and the largest force multiplier. However, the third
denition provides service members with the most predictability and, arguably, the best
quality of life. If the intent is to satisfy either of the rst two denitions, then equation (5) is
sufcient to determine the required force multiplier. If, however, the intent is to satisfy the
third denition, then the results from equation (5) should be rounded up with consideration
for associated risks.
The following example helps to explain some of the ner 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
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
rst 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 specically 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
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
becomesmoredifcult 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 ll 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 justied. As the preponderance of AFSOCs 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,
specic 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 rst determine which of the above denitions 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 specied 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 efciency 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 denitive 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 unitsD2D ratios.
Another signicant 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
connects force structure decisions with the implications for service memberscombat
readiness and quality of life, thus associating the service memberscost 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
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Recent studies have found that longer dwell times, or the period of time between deployments, may be protective against combat-related psychological outcomes. The purpose of this study was to examine the association between dwell time and psychological morbidity, while accounting for combat exposure. U.S. Marines with two combat deployments between 2005 and 2008 were identified from electronic deployment records. Those who screened positive for post-traumatic stress disorder and depression, and who were referred for mental health services were identified from the Post-Deployment Health Assessment. For the final study sample of 3,512 Marines, dwell time was calculated as time between deployments, and was analyzed as a ratio over length of first deployment. After adjustment for all covariates, there was an interaction (p = 0.01) between dwell time and combat exposure on mental health referral outcome. For personnel with maximum reported combat exposure, longer dwell times were associated with a 49% to 92% reduced odds of mental health referral. Longer dwell times may be protective against combat-related psychological outcomes. Because multiple deployments are likely to be the norm in future military operations, regulating dwell time, particularly for those with greater risk of combat exposure, should continue to be explored.
Personnel retention is one of the most significant challenges faced by the US Army. Central to the problem is understanding the incentives of the stay-or-leave decision for military personnel. Using three years of data from the US Department of Defense, we construct and estimate a Markov chain model of military personnel. Unlike traditional classification approaches, such as logistic regression models, the Markov chain model allows us to describe military personnel dynamics over time and answer a number of managerially relevant questions. Building on the Markov chain model, we construct a finite-horizon stochastic dynamic programming model to study the monetary incentives of stay-or-leave decisions. The dynamic programming model computes the expected pay-off of staying versus leaving at different stages of the career of military personnel, depending on employment opportunities in the civilian sector. We show that the stay-or-leave decisions from the dynamic programming model possess surprisingly strong predictive power, without requiring personal characteristics that are typically employed in classification approaches. Furthermore, the results of the dynamic programming model can be used as an input in classification methods and lead to more accurate predictions. Overall, our work presents an interesting alternative to classification methods and paves the way for further investigations on personnel retention incentives. © 2015 This work was authored as part of the Contributor's official duties as an Employee of the United States Government and is therefore a work of the United States Government. In accordance with 17 U.S.C. 105, no
Szivak, TK and Kraemer, WJ. Physiological readiness and resilience: Pillars of military preparedness. J Strength Cond Res 29(11S): S34-S39, 2015-Warfighters require a range of physical capabilities to meet the demands of the military profession, and physical training must address performance along an entire continuum, depending on individual needs and mission requirements. Strength and power capabilities are needed for optimal performance of anaerobic tasks such as heavy load carriage, sprinting under load, and maneuvering over uneven terrain. For optimal performance, soldiers must also be able to recover from mission demands and strenuous training. The demands placed on a soldier can result in a chronic stress, leading to decreased mission performance, increased injury risk, and increased susceptibility to illness. These factors are exacerbated by inappropriate training strategies such as overemphasis on endurance exercise combined with other stressors such as lack of sleep or inadequate nutrition. Chronic stress has been linked to overreaching/overtraining and to the development of comorbidities such as metabolic syndrome, insulin resistance, and hypertension and has adverse effects on memory and cognitive function. Resistance exercise is an effective method to improve warfighter physical performance and resilience to stress, thereby impacting mission readiness. Resistance exercise in particular confers many benefits to include increased strength and power, improved body composition, and protective effects on tendons, ligaments, and bone. Physically fit individuals not only benefit from improved mission performance but also are more resilient to operational stressors faced during combat. Ultimately, resilient soldiers are better able to cope with the physical and mental demands of the military profession and over the long term will perform better while maintaining health and well-being.
Managing knowledge workers is highly complex because of the need to balance the costs associated with their training and holding costs against the need to meet market demand as quickly as possible. Unlike previous approaches to this problem in the workforce management literature, this paper develops a stochastic optimization model to examine the impact of not only uncertainty of the demand of knowledge services but also that of the supply of knowledge workers on a recruiting strategy. Hypotheses on optimal recruiting decisions that this paper suggests include: (1) high holding and training costs decrease recruitment of apprentices; (2) high mobility of skilled workers decreases recruitment of apprentices; (3) high elasticity of the supply of skilled workers decreases recruitment of apprentices; (4) high volatility of the demand of knowledge services decreases recruitment of apprentices; and (5) in high seasonal effect on the demand, the decision policy based on decision thresholds proportional to seasonal demands outperforms the decision policy based on a constant decision threshold. Suggested hypotheses are supported by a simulation of the model. Model parameters in the simulation are estimated based on the survey of information security consulting service companies in South Korea.