Determining the Required Size of a Military Training Pipeline

Etienne Vincent

and Michelle Straver

Director General Military Personnel Research and Analysis, Department of National Defence,

101Colonel By Drive, Ottawa, Canada

Keywords: Markov Manpower Model, Monte Carlo Simulation, Training Pipeline, Military Establishment, Personnel

Operations Research, Workforce Analytics.

Abstract: This paper addresses the problem of deciding how many positions to set aside, in a military establishment,

for recruits undergoing training. We assume a cap on total strength, and thus must select a ratio between

positions in the force’s training pipeline versus its trained establishment. We develop a Markovian model of

the training pipeline, with parameters derived from historical Human Resources data. Through Monte Carlo

simulation we may then predict how often a given ratio will be sufficient to generate the required trained force,

as well as how much surplus trained personnel it is expected to generate. Our modelling results have informed

ongoing initiatives to optimize the force mix and structure of the Canadian Armed Forces.

1 INTRODUCTION

This paper describes the approach taken to solve a

challenging Human Resource problem faced by the

Canadian Armed Forces. This problem concerns how

many positions must be set aside for Regular Force

recruits undergoing their training. We present a

solution based on a stochastic simulation of the

training pipeline. Through simulation, we estimated

how often a given number of positions reserved for

trainees will be sufficient to generate the desired

trained force. Simulation also gives us an estimate of

the number of surplus trained personnel that is to be

expected. Armed with these results, departmental

decision makers will be able to select a training

structure that balances the risk of not meeting

operational requirements against the costs from

pipeline overcapacity.

The scheme presented in this paper is simple and

effective. It is presented in the hope that it will be

appreciated as a practical application of military

Operations Research. Our solution was developed in

the context of designing a future fighting force, but is

also relevant to examining the current force structure.

Regularly revisiting the ratio of trainee-to-trained

positions will be necessary to preserve a force where

https://orcid.org/0000-0002-6877-2379

https://orcid.org/0000-0002-4130-9376

all units can be sufficiently manned. Our model offers

a way to inform this rebalancing.

2 BACKGROUND

In 2017, the Government of Canada issued Strong

Secure and Engaged, the latest Canadian Defence

Policy. Notably, this policy mandates the capability

to conduct a defined set of concurrent operations,

including both emergency responses and planned

deployments, and ranging from limited to sustained

commitments. To ensure that the Canadian Armed

Forces have the right mix of military personnel to

satisfy these requirements, the Force Mix and

Structure Design initiative was launched. This

initiative is in the process of designing, from the

ground up, the required military establishment.

Operations Research analyses in support of the

Force Mix and Structure Design initiative are not the

first to be conducted in support of establishment

reviews. For example, both Bender (2005) and

Couillard et al (2015) present stochastic simulations

that quantify to what extent given force mixes (mixes

of military personnel from various occupation

categories) meet contingent operational

358

Vincent, E. and Straver, M.

Determining the Required Size of a Military Training Pipeline.

DOI: 10.5220/0010249503580365

In Proceedings of the 10th International Conference on Operations Research and Enterprise Systems (ICORES 2021), pages 358-365

ISBN: 978-989-758-485-5

requirements. Filinkov et al (2011) similarly present

a model designed to study the Australian Army.

A first phase of Force Mix and Structure Design

strived to determine the Force Employment

requirement – the required number of “boots on the

ground” conducting operations. The second phase

then looked at supporting elements involved in force

generation and the institution of the Canadian Armed

Forces. It is in this context that the required size of the

combined Basic Training List (recruits in their initial

phases of training) and Supplementary University

Training List (recruits requiring training in higher

education institutions) had to be determined. For

simplicity, we will thereafter refer to the positions

reserved for members on these lists as the training

pipeline. Regular Force recruits remain in this

pipeline until they reach an Operationally Functional

Point – at which point they are considered trained and

can occupy trained effective positions.

Straver and Christopher (2015) have conducted a

study based on stochastic simulation to determine the

sustainable composition of the Regular Force,

including the size of the training pipeline.

Nevertheless, it is now understood that the data

underpinning that study were problematic. As such,

our model is a successor to Straver and Christopher’s

that now focuses exclusively on the training pipeline,

and that was built to work with new and improved data.

3 MARKOV MANPOWER

MODELS

Many approaches have been used to model personnel

systems. For example, Wang (2005) categorizes these

approaches into Markov Chain models, Computer

Simulation models, Optimisation models and System

Dynamics. The approach that we describe is a hybrid

of the first two categories, being a discrete-time

Markovian model serving as a basis for stochastic

simulation.

The earliest described application of Markov

Chains to a personnel systems is found in (Seal,

1945), whereas a general overview of their use in this

context is found in (Guerry and De Feyter, 2009).

Guerry and De Feyter define Markov Manpower

models as satisfying four assumptions:

 They are memory-less (the usual Markovian

assumption);

 Their flow rates are time-independent;

We used 14 years of historical data. This goes beyond the

maximum length of training, but excludes earlier years,

when the system may have behaved differently.

 They are discrete-time, with fixed-increment

intervals;

 Their stocks describe homogeneous populations.

Models developed to study the Canadian Armed

Forces have generally obeyed the first two of these

assumptions, but not always the last two. In fact,

Discrete Event Simulation has been the most-

employed paradigm (Okazawa, 2013), but is based on

next-event time progression rather than fixed

increments.

The homogeneous stocks assumption requires that

populations be broken down into homogeneously

behaving subsets. To do this, regression is often used

to identify the variables that most-affect behaviour. In

our case, this would pose a problem. The Regular

Force employs members of various ranks, and who

have completed different periods of service – factors

that are closely tied to the propensity to release (to

leave the forces). Also, they belong to many different

occupations, and are recruited through a number of

different entry plans, implying widely different

durations of training. Given this wide spectrum of

factors, and given the need to use data that remain

representative of the current system,

we would not

have sufficient data available to accurately estimate

all the parameters that would be associated with the

multiple resulting homogeneously behaving subsets

(groups of members of similar ranks, years of service,

occupations and entry plans).

We thus build a model that diverges from Markov

Manpower models with respect to the homogeneous

stock assumption. We group in our stocks cohorts of

members who behave diversely, but who when taken

collectively, have aggregate flow probabilities that are

nevertheless fairly consistent from year to year.

Therefore, the historically observed proportion of

members from a stock who flow in a given direction

cannot be interpreted as a probability applying to

individuals, but it can still be interpreted as the

expected proportion of individuals who will transition.

4 TRANSITION PROBABILITIES

For Markov Manpower models, Anderson and

Goodman (1957) demonstrate that the maximum

likelihood estimator for transition probabilities is given

by the numbers of employees that underwent that

transition divided by the total number of eligible

employees in the relevant sub-periods (e.g. each year,

Determining the Required Size of a Military Training Pipeline

359

if looking for an annual probability). Under the

homogeneous stock assumption, the probability for the

total number of employees undergoing that transition

is then binomial. Binomial distributions are thus

commonly used in stochastic interpretations of Markov

Manpower models, but this would be inappropriate in

our case, as our stocks are not homogeneous.

In the absence of a priori knowledge of the shape

of the flow probabilities in our model, we fit Gaussian

distributions. For example, Figure 1 shows the

distribution used for members graduating from the

training pipeline within the year after they had first

appeared in it. The 14 years of available annual

observations are shown as a bar histogram. The

Gaussian that was fit, with mean 45.2% and standard

deviation 2.1% is shown as the dashed line.

Figure 1: Distribution for the proportion of members who

graduate within the next year, among members who first

appeared in the training pipeline, 2006-2020. A Gaussian

fit is also shown as the dashed line.

When we get to our simulation, the proportion of

members who graduate from the training pipeline in

a given year will thus be drawn from this and similar

distributions in each annual iteration. In order to

avoid unrealistically extreme outcomes, we only draw

within three standard deviations of the mean.

Previous efforts to model the Regular Force

training pipeline, such as (Bender, 2005) and (Straver

and Christopher, 2015) have sought to explicitly

model the main specific processes of the human

resources system. For example, graduation from the

training pipeline would be modelled as the result of

separately considering graduations of Officers and

Non-Commissioned Members, further split according

to their entry plan, as various categories of trained,

semi-trained and untrained recruits. Instead, our

approach considers no more than a single flow

between each model stock, consolidating a number of

sub-flows. This greatly simplified our task of

historical data analysis for estimating model

parameters, as we did not need to categorize the data

associated with past recruits. We also expect the

resulting model to be more reliable, as covariance

between sub-flows would have been impossible to

determine from our limited data, but is avoided by

directly estimating the aggregate flows.

5 THE MODEL

Figure 2 depicts our model. The percentages shown

on arrows correspond to the mean proportions for

certain modelled flows. The model has two parts: A

stock for the Trained Effective Strength (TES), and a

set of stocks for the training pipeline. The TES

encompasses all members who have completed their

training up to the Occupationally Functional Point,

and are not in certain operationally unavailable

categories. Members who change occupation

generally require re-training – these correspond to the

0.8%

arrow

flowing

toward

the

training

pipeline.

Figure 2: Illustration of our model of the Canadian Armed Forces training pipeline.

ICORES 2021 - 10th International Conference on Operations Research and Enterprise Systems

360

Others leave the system, corresponding to the 7.2%

arrow. These leaving members may be leaving the

forces completely through release or death, but could

also be moving to the Reserve Force, or to a non-

effective status (ill, injured or pre-release).

The training pipeline is divided into cohorts. After

having first appeared in the pipeline, members may

graduate (45.2%), remain for at least another year

(46.3%), or leave the system (8.6%). A total of 12

such similar stocks are modelled, with any remaining

trainees graduating after the twelfth year. In our

historical record, only one member was in the training

pipeline for 12 consecutive years. No occupation

requires this much training, but delays can result from

changes in occupation, or pauses in training. The

most frequent type of pause is parental leave – an

entitlement for new parents.

The remaining important flow is intake, which

here includes recruitment, but also return from ill or

injured status. On average, 19.9% of intake go

straight to the TES. This includes trained recruits (re-

hires or transfers from the Reserve Force), but also

recruits requiring less than a year of training, who

joined the Regular Force and move on to the TES

within the year (our model being based on annual

iterations).

Markov Chains can be treated as deterministic or

stochastic. Davies (1982) introduced a partially

stochastic Markov model. In that model, attrition is

considered an uncontrollable flow, and treated as

stochastic, whereas promotions are decided by

management, and thus treated as deterministic. Our

model does not consider promotions, but does treat

the magnitude of total intake deterministically, and is

thus also partially stochastic. Intake is set to re-fill the

training pipeline each year (with a hard cap on total

strength, i.e. the total Regular Force population)

rather than varying stochastically. This intake is also

the only pull flow in our model. It is generated by

vacancies in the destination (pull), rather than arising

spontaneously from the source (push), as defined by

Bartholomew et al (1991).

Although we set the magnitude of intake

deterministically, we vary the proportion going to the

TES versus the training pipeline stochastically. This

treatment of TES intake as a direct proportion of total

intake resembles the proportionality constraint

introduced by Nilakantan and Raghavendra (2005).

Their constraint requires that a fixed proportion of

vacancies in a given grade be filled externally. Our

model is however different in that our proportion

varies according to the observed historical

distribution.

6 MODEL LIMITATIONS

We will now highlight three limitations of our model.

We do not believe that these limitations invalidate our

results, but they should be kept in mind when

interpreting them. A first limitation has to do with

using historical data to estimate the rates of flow out

of the training pipeline. Currently, delays result from

limitations on training institution capacity or from

their sub-optimal organisation. However, our results

are meant to be applicable to future force structures,

where sources of delay will hopefully have been

reduced. Historical observation could therefore

overestimate future training durations, and

consequently, over-estimate the number of required

training pipeline positions.

A second limitation of our model is that it is based

on annual-duration iterations (taken at fiscal year-

end: 31 March). However, that day does not

correspond to the annual peak for the training

pipeline. Typically, the peak will be in summer, when

more recruits begin their training. As such, slightly

more training pipeline positions are likely to be

required than is determined by our model. It would

however be possible to introduce a correction factor

for our results based on the historical differences

between end fiscal year and annual peaks. Finally, our

model’s last important limitation is that it only

considers an overall TES target, ignoring its

composition in terms of ranks and occupations. This

will mask specific gaps in trained personnel. In the

normal course of business, retention encounters ups

and downs at various ranks and occupations, leading

to local gaps. Certain positions can be filled from a

range of different ranks and occupations, but others

cannot, and a larger training pipeline cannot address

gaps in senior or specialized positions in the short

term. It should therefore be understood that some

vacancies in the establishment are to be expected,

even when the pipeline trains enough members to

counter the raw number of departures.

Fully addressing these three limitations with an

enhanced model is likely impossible, given data

constraints. For example, given that there are only so

many members in each occupation, and that it is only

relevant to look back so many years in the data record,

accurately estimating occupation-specific training

and attrition model parameters would not be feasible.

However, decision makers can appreciate the

constraints’ impact on our modelling results, and

consider them in developing policy. Overall, our

model outputs remain informative, especially if

interpreted as slightly under-estimating true training

pipeline requirements.

Determining the Required Size of a Military Training Pipeline

361

Figure 3: Five hundred years of simulation result showing excess TES – trained effective personnel generated above the set

requirement.

7 SIMULATION

We derived Monte Carlo simulation results using a

spreadsheet. Each line of the spreadsheet recorded the

state of the stocks in a given year. Subsequent years

were then computed from the line above, with the

annual flows drawn from specified Gaussian

distributions. The number of lines in the spreadsheet

then corresponds to the number of simulated years.

Our goal was to assess the adequacy of various

force structures by considering ratios of the number

of positions allocated to the TES versus the training

pipeline. From year to year, the simulated TES varied

with the simulated stochastic flows, and could either

be driven above or below the set objective. Each year,

the intake was set to re-fill the training pipeline, but

subject to a cap of total strength, which came into play

when the number of trained personnel exceeded the

target TES.

Each simulation began with a fully manned TES,

and a training pipeline full of fresh recruits. We

would then simulate 200 years, in order to allow the

composition of the pipeline to stabilize. Each

subsequent year was then captured toward the output.

As an example, Figure 3 shows 500 years of

simulation, with a ratio of 1,547 training pipeline

positions to 10,000 required TES. The graph shows

the resulting annual variation in excess TES. We see

that the excess TES tends to vary within a set range.

Whenever the excess is too great, the cap on total

strength means that the training pipeline cannot be

filled to capacity, eventually leading to fewer

graduations into the TES, and thereby eventually

reducing the excess. Conversely, when the excess is

negative, the training pipeline is filled to capacity,

allowing for eventual growth of the TES. For the ratio

of training pipeline positions to required TES used in

this example, we see that the TES meets (or exceeds)

the requirement approximately 95% of the time.

All other results presented in this paper are based

on 100,000 simulated years, which empirically

proved to be enough for very stable results.

8 RESULTS

The main consideration, in fixing the capacity of the

Regular Force’s training pipeline is ensuring that it is

sufficient to generate the required TES. Figure 4 was

obtained by simulating various ratios of training

pipeline positions to required TES. It shows how

often each ratio is expected to fail to yield the required

TES. The failure rate decreases as the capacity of the

training pipeline increases.

Because a small number of unfilled TES positions

might be a tolerable outcome, Figure 4 also shows an

alternative measure of effectiveness. This alternative

measure records how often the shortage is above 1%

of the required TES.

Figure 4: Fraction of simulated years where a TES shortage

is observed, as a function of the number of training pipeline

positions, given a 10,000 TES requirement.

However, there is a trade-off in increasing the

number of training pipeline positions. The larger

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362

Figure 5: Mean TES above the requirement in those

simulated years when the TES requirement is met, as a

function of the number of training pipeline positions, given

a 10,000 TES requirement.

pipeline risks generating more TES than required.

This over-generation would be costly in and of itself,

but would also burden the forces with more trained

personnel than required, leading to persistent excess

costs. Figure 5 shows how this expected over-

generation increases with the size of the pipeline.

To highlight the interplay between the two metrics

shown in Figures 4 and 5, the results were combine

into Figure 6. This depiction of the results was

highlighted to departmental decision makers, as it

quantifies the trade-off involved in right-sizing the

training pipeline.

Finally, Table 1 was disseminated as our principal

deliverable. It essentially displays four scenarios from

Figure 6 corresponding to the TES requirement being

met 80%, 90%, 95% and 99% of the time. It was

obtained by manually adjusting the training pipeline

to TES ratio up or down until the simulation returned

round number frequencies of TES shortages. Table 1

can be used as a starting point on the way to settling

on a preferred training pipeline to required TES ratio,

while considering the limitations set out earlier in this

paper.

9 DISCUSSION OF MODELLING

ERROR

Guerry and De Feyter (2009) distinguish three types

of error that apply to Markov Manpower models:

statistical, estimation and specification. Statistical

error results from the stochastic nature of models and

may be reduced by increasing the number of

simulation iterations. In our case, we expect this error

to be small, given the 100,000 simulated years per

scenario, and because that increasing that number

minimally changes our results.

Estimation error is related to the accuracy to

which parameters are estimated. In our case, we relied

on 14 years of available Human Resources data to

estimate

parameters.

Older

data,

data

from

other

Figure 6: Combination of our two metrics into a single chart.

Table 1: Four potential choices of training pipeline to TES ratios, with corresponding metrics obtained from simulation.

Training pipeline to TES requirement 1,489:10,000 1,519:10,000 1,547:10,000 1,598:10,000

Frequency of TES shortage 80% 90% 95% 99%

Excess TES personnel 0.69% 0.95% 1.19% 1.64%

Determining the Required Size of a Military Training Pipeline

363

organisations would likely have been less

representative of the current system, and therefore of

little use in further reducing estimation error.

Therefore, although we expect our estimation error to

be substantial, there is no viable path to reducing it.

Lastly, specification error results from the model

being an inaccurate representation of reality.

Reducing specification error eventually involves

increasing the complexity of the model, thereby

increasing the number of parameters. This introduces

a trade-off with estimation error, since estimation

error increases with the number of parameters that

must be estimated. In our case, it would be

unadvisable to attempt reducing specification error by

introducing more parameters. We believe that our

model design strikes an appropriate balance between

estimation and specification error.

10 CONCLUSIONS

This paper described an application of Monte Carlo

simulation on a straightforward Markovian model of

the Canadian Armed Forces’ Regular Force training

pipeline. The results of this simulation were provided

to military staff tasked with redefining the

organization’s force mix. An advantage of the method

employed in deriving our results is that it was

straightforward and easy to follow. Despite the

complexity of the Canadian Armed Forces’ personnel

system, our modelling results were readily trusted and

appreciated.

Although the Regular Force training pipeline has

been undersized in recent years, efforts have been

directed to bringing awareness to the issue and to

improving the sustainability of the establishment. It is

hoped that our model will support this continuing

effort.

Under existing total strength caps, sustainability

can be improved by converting some TES positions

into positions for trainees, but this comes at the

expense of military capability. Alternatively, the

current TES requirement can be maintained if the

training pipeline (and by extension the total strength)

are increased.

Our model was nevertheless developed to support

the currently ongoing Force Mix and Structure

Design initiative. In that context, we have informed

the future force structure. Subject to the limitations

presented in this paper, our simulation results have

allowed initiative staff to foresee the range of possible

outcomes from different proposed force structures.

ACKNOWLEDGEMENTS

The authors would like to thank our colleague,

Dragos Calitoiu, for helpful modelling advice.

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