I need help please.
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Discuss what you learned about Practice Management Integration and the symposium on the promises and challenges of realizing the value of EHR (Electronic Health Record). What role can health informatics play in realizing these values?
2. As with all reaction essays, please read and follow the generic instructions on the
here.
For this essay, please react to Viewpoint article on “Back to the Future: Achieving Health Equity Through Health
Informatics and Digital Health ”
https://mhealth.jmir.org/2020/1/e14512/pdf (Links to an external site.)
· Do you agree with their assessment of the unintended consequences of advancement in health technologies?
· Discuss in particular an example in the article that illustrates these consequences?
· Which of the approaches discussed in the article to foster equity and reduced divide in digital health would be suitable for an application development project for a population diverse in race and socioeconomic status?
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Optimal Allocation of Students to Naval Nuclear-Power
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Michael R. Miller, Robert J. Alexander, Vincent A. Arbige, Robert F. Dell, Steven R. Kremer,
Brian P. McClune, Jane E. Oppenlander, Joshua P. Tomli
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Oppenlander, Joshua P. Tomlin (2017) Optimal Allocation of Students to Naval Nuclear-Power Training Units. Interfaces
47(4):320-335.
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INTERFACES
Vol. 47, No. 4, July–August 2017, pp. 320–335
http://pubsonline.informs.org/journal/inte/ ISSN 0092-2102 (print), ISSN 1526-551X (online)
Optimal Allocation of Students to Naval Nuclear-Power
Training Units
Michael R. Miller,a Robert J. Alexander,a Vincent A. Arbige,a Robert F. Dell,b Steven R. Kremer,a Brian P. McClune,a
Jane E. Oppenlander,c Joshua P. Tomlina
a Naval Nuclear Laboratory, Kesselring Site, Schenectady, New York 12301; b Operations Research Department, Naval Postgraduate School,
Monterey, California 93943; c School of Business, Clarkson University, Schenectady, New York 1230
8
Contact: michaelr.miller@unnpp.gov (MRM); bobby.j.alexander@gmail.com (RJA); varbige@gmail.com (VAA); dell@nps.edu (RFD);
skremer@nycap.rr.com (SRK); bpmcclune@gmail.com (BPM); joppenla@clarkson.edu, http://orcid.org/0000-0001-8778-6461 (JEO);
JoTomlin509@gmail.com (JPT)
Received: November 12, 2015
Revised: July 11, 2016; December 1, 201
6
Accepted: March 24, 201
7
Published Online in Articles in Advance:
June 27, 2017
https://doi.org/10.1287/inte.2017.0905
Copyright: This article was written and prepared
by U.S. government employee(s) on official time
and is therefore in the public domain.
Abstract. The U.S. Navy operates an impressive fleet of nuclear-powered submarines and
aircraft carriers and has safely operated its nuclear fleet for more than 60 years, while steam-
ing over 154 million miles. Rigorous training has been key to maintaining such an impres-
sive record. The U.S. Naval Nuclear Propulsion Training Program develops, certifies, and
delivers the nuclear-operator qualification training for enlisted and officer personnel oper-
ating its nuclear fleet. This training finishes at one of four nuclear-power training units
(NPTUs), operates under a complex set of hard and soft constraints, varies depending on
the type of student, and requires significant personnel and equipment resources. We devel-
oped and implemented a mixed-integer linear program (MILP) that prescribes how many
students of each type to allocate to each NPTU at the start of each class (a group of stu-
dents who train together) and how allocated students complete NPTU training. The use of
MILP has improved student allocation by an estimated eight percent and led to significantly
improved use of both NPTU personnel and equipment resources. In this paper, we describe
this unique optimization application, the MILP formulation, its path to adoption, its user
interface, and impacts from its development and use over the past three years.
History: This paper was refereed.
Funding: The submitted manuscript has been authored by contractor of the U.S. Government [Contract
DE-NR-0000031]. Accordingly, the U.S. Government retains a non-exclusive, royalty-free license to
publish or reproduce the published form of this contribution, or allow others to do so, for U.S.
Government purposes.
Keywords: military • personnel: programming • integer • applications: education systems • planning: decision analysis • applications
Nuclear-powered submarines and aircraft carriers (Fig-
ure 1) are key elements for the defense of the United
States and for the maintenance of free and open com-
merce across the world’s oceans (Department of the
Navy and Department of Energy 2014). These vessels
are staffed by highly trained enlisted and officer per-
sonnel who operate and maintain the power-generation
and propulsion systems capable of extended unsup-
ported operations. The U.S. Naval Nuclear Propulsion
Training Program develops, certifies, and delivers the
nuclear-operator qualification training for enlisted and
officer personnel who operate its nuclear fleet. This
training finishes at one of four Nuclear Power Train-
ing Units (NPTUs). This paper describes the benefits
achieved by using a MILP to prescribe the number of
students of different types to allocate to each NPTU at
the start of each class and the activity sequence for allo-
cated students to complete NPTU training.
Certifying Nuclear Operators
Certification as a naval nuclear operator requires rig-
orous training that lasts at least one year for each of
five student types referred to by the name of the certi-
fication: electrician’s mate; machinist’s mate; electron-
ics technician; engineering laboratory technician; and
engineering officer of the watch. Each student type
completes a unique training track consisting of knowl-
edge and hands-on requirements. A student is cer-
tified (i.e., qualified) to operate a specific area of a
naval nuclear-propulsion plant only after demonstrat-
ing mastery of propulsion plant equipment.
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mailto:michaelr.miller@unnpp.gov
mailto:bobby.j.alexander@gmail.com
mailto:varbige@gmail.com
mailto:dell@nps.edu
mailto:skremer@nycap.rr.com
mailto:bpmcclune@gmail.com
mailto:joppenla@clarkson.edu
http://orcid.org/0000-0001-8778-6461
mailto:JoTomlin509@gmail.com
Miller et al.: Optimal Allocation of Students to Naval NPTUs
Interfaces, 2017, vol. 47, no. 4, pp. 320–335 321
Figure 1. The Nuclear-Powered Aircraft Carrier USS JOHN C STENNIS (CVN 74), with Destroyer Escort (to Left), and the
Nuclear-Powered Submarine USS SEAWOLF (SSN 21) Operates on Deployment in the Pacific Ocean
Notes. Operating the nuclear power plants on these ships requires highly trained enlisted and officer personnel. (Photo from http://navy.mil.)
Depending on the student type, students attend one
or more schools prior to beginning NPTU training.
While required schools vary by student type, all stu-
dent types must satisfactorily complete a six-month
program (i.e., nuclear-power school), consisting pri-
marily of classroom instruction, prior to six additional
months of NPTU training. Ideally, upon completion of
this classroom instruction, students immediately begin
training at one of four NPTUs at one of two training
sites; each site has two units. During NPTU training,
students engage in a mix of classroom, simulator, and
hands-on training. Delays in starting NPTU training
often occur due to limited resources; this produces a
backlog of students waiting to begin NPTU training.
Navy leadership carefully monitors this backlog.
Each NPTU is a self-contained training facility com-
posed of a nuclear reactor, simulators, classrooms, staff
instructors, and other training assets. Each NPTU class
consists of a group of students who train together,
which is designated by a sequential number based on
the fiscal year. For example, class 1501 corresponds to
the first class started in fiscal year 2015. The starting
week of each NPTU class is known, and all plants start
classes on the same day; therefore, each plant runs
classes with the same class number. With rare excep-
tions, a new class starts every eight weeks and three
classes normally train simultaneously at each NPTU.
Students are assigned (i.e., allocated) to a NPTU class
and train together as a class.
NPTU training consists of a classroom phase (seven
weeks) followed by a hands-on phase (17 weeks).
Hands-on activities at the NPTU are critical because
they teach students to perform the tasks that are
required to safely operate nuclear reactors. Much of
this hands-on training (referred to as “watchstand-
ing”) takes place at “watchstations” located in either
a plant that contains a nuclear reactor or at a simula-
tor. Training conducted outside of the plant is referred
to as “off-watch” training. Although extremely real-
istic, simulator training can only satisfy a fraction of
the required watchstanding. A qualified staff instruc-
tor must be present at each watchstation in both the
plant and a simulator to ensure proper plant operation.
During operations, students are able to perform watch-
standing. Simulators require staff time only when oper-
ating for training.
There are five types of staff instructors, each per-
forming different training functions. Whereas students
arrive in batches at defined intervals as part of a class,
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Miller et al.: Optimal Allocation of Students to Naval NPTUs
322 Interfaces, 2017, vol. 47, no. 4, pp. 320–335
staff instructors continuously flow in and out of a
NPTU based on their individual military assignments.
Typically, a staff instructor is assigned to a NPTU for
three years consistent with the typical length of other
Navy assignments. In addition to supporting hands-
on training in the simulator and plant, staff instruc-
tors engage in a variety of other duties, including
plant operation, providing classroom instruction, and
administrative tasks. The amount of time devoted to
each duty depends on staffing levels and the number
of students being trained.
It is generally desirable to allocate as many students
as possible to a class, while satisfying a variety of con-
straints and assuring the efficient use of limited staff
instructors, equipment, and facilities. The number and
type of students impacts the use of resources; each stu-
dent type has a different set of qualification require-
ments, some that are unique and others that are com-
mon to other student types. Assigning the number and
type of students to a training class is referred to as
“student allocation” (or simply “allocation”). The train-
ing capability model (TCM), a MILP, prescribes how
many students of each class and type to allocate to each
NPTU; prescribes weekly staff-instructor assignments;
and prescribes weekly student watchstanding and off-
watch training.
Historical Approach
For more than 20 years prior to the adoption of the
TCM, a single training analyst made student alloca-
tions using an iterative process, with expert judgment
Figure 2. In the 1980s, the Number of NPTUs (Solid Line Graph) Was Eight; It Is Four Today and We Expect It to Decrease to
Three After 2017, While the Number of Students Requiring Training, Which We Express as a Percentage of the Peak Number
Trained in 1983 (Dotted Line Graph), Has Increased in Recent Years
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applied at each iteration. Because of the importance
of the allocation, a second person verified calculations
to ensure potential errors were minimized. The ana-
lyst used a spreadsheet application to store data and
assist with calculations. Over time, this spreadsheet
application grew to more than 100 worksheets, which
included numerous formulas and calculations, aided
by Visual Basic for Applications (VBA) code (Microsoft
2015). The analyst required days to plan a single stu-
dent allocation and the iterative effort was difficult to
duplicate for any what-if analyses. In addition, several
simplifying assumptions were employed for staff and
simulator availability.
The Need for Optimization and an Expanded Model
Figure 2 shows the relationship between the num-
ber of nuclear operators qualifying each year and
the available number of NPTUs. In recent years, the
annual number of students requiring training has
stayed near historic highs, while the number of NPTUs
has decreased. This has necessitated the increased use
of simulators and increased staffing levels. This in turn
has complicated the task of determining student alloca-
tions. An improved student-allocation method capable
of being used by more than a single analyst and capa-
ble of rapid what-if analysis was considered essential
to best utilize the few remaining NPTUs.
Literature Review
Naval nuclear operator training is unique, but it
shares much in common with other military training.
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Miller et al.: Optimal Allocation of Students to Naval NPTUs
Interfaces, 2017, vol. 47, no. 4, pp. 320–335 323
Military training often involves the need to complete a
sequence of qualification activities, which take place at
“schools.” Each school lasts a number of weeks, some-
times occurs at different locations, and is only avail-
able periodically. Each military specialty typically has
unique schools and schools common to other military
specialties. Waiting often occurs between the end of
one school and the start of another. Minimizing this
waiting, or minimizing the backlog in the case of the
TCM, is desirable. Grant (2000) minimizes waiting time
for Marines by selecting the military occupational spe-
cialty (MOS) for each graduate of a common begin-
ning school. Here, each MOS has its own series of
schools, the timetable of when each school class begins
is given, and capacity is simply the maximum num-
ber of students allowed in any class. Detar (2004) and
Whaley (2001) also seek to minimize the waiting time
of Marines between schools by prescribing a timetable
of when each school course should start and how many
students should be allocated to each class, with each
capacity again simply the maximum number of stu-
dents allowed in any class. Capacity constraints for
the TCM are more complex because each student type
impacts various training resources in different ways.
There is substantial operations research literature
on military manpower planning as it relates to man-
aging and growing military services. Early published
work on hierarchical organizations can be found in
Seal (1945) and Vajda (1947). The military services
employ various models to determine recruiting, pro-
motion rates, and retirement (Ginther 2006, Gibson
2007, Workman 2009). Wang (2005) provides a review
of operations research applications in manpower plan-
ning, mostly with a focus on military training. His
review includes applications that address optimization
in the areas of: cost minimization for hiring and rede-
ployment, personnel promotion, recruitment, and the
mix and frequency of training modes (e.g., simulators,
training aids) to maintain force proficiency. In general,
these military manpower planning models have little
in common with the TCM.
There is also substantial operations research lit-
erature on the related problem of course schedul-
ing, where prescriptions assign students and instruc-
tors to classes, and classes to rooms and times;
examples include de Werra (1985), Bonutti et al.
(2012), and their extensive reference lists. The TCM
primarily differs from these course-scheduling appli-
cations because different student types train simulta-
neously and impact resources in different ways.
Similar resource-allocation problems can be found
in healthcare literature. Caunhye et al. (2012) and
Cardoen et al. (2010) provide reviews of the literature
for emergency logistics and operating room planning,
respectively. Examples of the allocation of operating
room capacity using mixed-integer programming can
be found in Zhang et al. (2009) and Blake and Donald
(2002). These applications do not include prerequisite
events, as required in the naval nuclear operator train-
ing environment. Integer programming and simula-
tion are used in decision support systems that allo-
cate medical assets during public health emergencies
(Lee et al. 2009) and improving emergency department
operations (Lee et al. 2015). Ernst et al. (2004) give an
annotated bibliography of over 700 papers on person-
nel scheduling and rostering in a variety of applica-
tion areas.
The TCM employs an elastic MILP to determine the
number of students of each type to assign to each
NPTU to comprise each training class over multiple
years. Additionally, the TCM determines the number
of simulator sessions each week at each NPTU. It does
not explicitly plan the watchstanding sequence for each
individual student. Instead, for each week, it plans the
off-watch hours, plant watchstanding hours for each
watch, and simulator watchstanding hours for each
watch for all students of each student type and class
at each NPTU. It establishes a preferred watchstanding
window for each class at each NPTU with soft con-
straints that limit the number of watchstanding hours
occurring very early, early, and late with respect to this
preferred watchstanding window (Figure 3).
The TCM MILP models a number of practices that
balance the competing needs for supplying qualified
operators to the fleet and providing for plant main-
tenance periods and limits on staff working hours.
An important objective is to minimize the number
of students who must wait to receive NPTU train-
ing. Consequently, the first (and primary) term of the
TCM objective function, which we show in Section A.5,
Equation (A.1) in the appendix, imposes a penalty for
each student in the training backlog (i.e., each student
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Miller et al.: Optimal Allocation of Students to Naval NPTUs
324 Interfaces, 2017, vol. 47, no. 4, pp. 320–335
Figure 3. The Preferred Watchstanding Window for a Class at a NPTU Occurs Between Its 14th and 21st Training Week
Preferred weeks
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Very early weeks Early weeks Late weeks
Watchstanding weeks
All training weeks
Notes. Up to five percent of the total watchstanding requirements can occur in weeks 8 and 9 and up to 50 percent of the requirements in
weeks 10–13; however, any early watchstanding incurs a penalty. Late watchstanding is allowed, with increasing penalty severity, beyond
week 21.
waiting to begin training). In some situations related to
plant availability, a class of students is not assigned to
a NPTU. This is referred to as “skipping a class” and
should be avoided if at all possible. The second term of
the objective function penalizes skipping classes. The
third term of the objective function limits staff time
to only what is needed to provide student training.
Finally, a set of elastic penalties guide a solution to
numerous goals.
Next, we give a summary of the primary prescrip-
tions and constraints to provide a general understand-
ing of the richness of the TCM MILP. Mathematical
details can be found in the appendix.
The primary TCM variables are as follows.
• The integer number of each student type to start
each class at each NPTU;
• The integer number of each student type waiting
for training after the start of each class;
• The integer number of simulator sessions for each
simulator at each NPTU each week;
• The number of hours each student type in each
class performs each watchstanding requirement in its
NPTU plant each week;
• The number of hours each student type in each
class performs each watchstanding requirement in its
NPTU simulator each week;
• The number of hours each student type in each
class at each NPTU performs off-watch training each
week; and
• The number of hours each staff-instructor type at
each NPTU is assigned for off-watch and simulator
watch instruction each week.
Weorganizedthehardandsoftconstraintsforassign-
ing students to a training class and NPTU into five
groups. In the following description, we use “goal” for
a soft constraint (a constraint that can be violated at a
cost), “limit” for a hard constraint, “each” for a con-
straint that exists for each permitted value of an index,
and “all” when summing over all permitted index val-
ues. In Sections A.3 and A.5 in the appendix, we give the
individual constraint sets and associated mathematical
details for each group in the order presented here.
The class-composition constraint group establishes
goals and limits on class size and distribution of stu-
dent types; see constraints (A.2)–(A.7) in Sections A.3
and A.5 in the appendix. These constraints include:
• Bookkeeping to keep track of the backlog of each
student type waiting for training after the start of each
class;
• A lower goal and an upper goal for each student
type in each class at each NPTU;
• A lower limit and an upper goal on all students in
each class at each NPTU;
• An upper limit on all students at each NPTU
across all simultaneous classes;
• An upper limit on all students in all simultaneous
classes at each site; and
• An upper limit on all students of each student
type in each class at each site.
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Miller et al.: Optimal Allocation of Students to Naval NPTUs
Interfaces, 2017, vol. 47, no. 4, pp. 320–335 325
The student-training constraint group establishes
goals and limits on watchstanding in each NPTU
simulator and plant; see Constraints (A.8)–(A.14) in
Sections A.3 and A.5 in the appendix. These constraints
include:
• A lower limit on required training hours for each
student type in each class at each NPTU and each
watchstation;
• An upper limit on the total training hours for each
student type in each class at each NPTU for each watch
in each week;
• An upper goal on total training hours for each
student type in each class at each NPTU each week;
• An upper limit on total simulator and off-watch
hours each week for all simultaneous classes for each
student type at each NPTU; the upper limit is set by
TCM decisions on staff-instructor assignments;
• An upper limit on total simulator watchstanding
for each watch across all weeks for each student type
in each class at each NPTU;
• An upper limit on simulation watchstanding for
each watch across all classes and all student types for
each week at each NPTU; and
• An upper limit on each NPTU plant’s watch hours
each week for all simultaneous classes and all student
types.
The staff-instructor work constraint group estab-
lishes goals and limits on the assignment of staff hours;
see constraints (A.15)–(A.17) in Sections A.3 and A.5 in
the appendix. These constraints include:
• An upper goal on staff-instructor hours available
for simulator and off-watch instruction for each staff
type at each NPTU each week; both weekly control lim-
its and cumulative sustained limits on staff-instructor
availability are set;
• A lower limit on the number of staff hours re-
quired for each simulator session watch at each NPTU
each week; the lower limit is set by a TCM decision on
the number of simulator sessions; and
• A lower goal and an upper goal on simulator ses-
sions for each simulator at each NPTU each week.
The watch placement constraint group establishes
goals and limits on the pace of student training rela-
tive to the preferred watchstanding window; see con-
straints (A.18)–(A.21) in Sections A.3 and A.5 in the
appendix. These constraints include:
• An upper goal on the percentage of watchstand-
ing to be completed prior to a very early week (and an
early week) for each student type in each class at each
NPTU and each watch; and
• An upper limit on the percentage of student off-
watch hours relative to watchstanding hours for each
student type in each class at each NPTU during early
and late weeks.
Finally, the persistent (Brown et al. 1997) con-
straint group establishes goals and limits on adher-
ence to a desired partial solution; see constraints (A.22)
and (A.23) in Sections A.3 and A.5 in the appendix.
TCM Implementation Features
Personnel impacted by TCM prescriptions drive many
of the features of TCM and its objective function. For
any student allocation, the throughput of students
(and the expedient reduction of any student backlog)
are of primary concern, but secondary considerations,
such as the efficient use of staff and plant training
resources, are also considered. The TCM reflects this
tiered priority structure with different penalty values
for its objective function terms. The implementation
also includes a time-based “reverse” discount factor
applied to encourage students to complete training ear-
lier rather than later in the planning horizon.
Implementers and approvers of the TCM prescrip-
tions expect that small perturbations in inputs to the
TCM (e.g., a small adjustment in staff-instructor hours
available for weeks during a class) will yield no or at
most only small changes in the TCM student-allocation
prescriptions. We address this expectation by imple-
menting persistence (Brown et al. 1997) as a feature,
allowing a previous student allocation to be referenced
as a preferred target.
Additional features are driven by practical consid-
erations. Although a typical planning horizon spans
two to four years, 10-year student allocations are
often required to evaluate the expected long-term per-
formance of the training program. For such long-
term instances, the TCM employs another time-based
discount that ensures near-term constraint violations
(excluding those for classes already in training at the
time of the allocation, which we designate as “fixed”
in the appendix) are penalized more than violations
that occur further in the future. Solutions over longer
horizons can quickly stretch both the bounds of run-
time practicality and the limits of desktop computer
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resources. To ensure the TCM provides long-term pre-
scriptions on standard desktop machines, we imple-
ment a solution cascade or receding-horizon solution
(Brown et al. 1987, Baker and Rosenthal 1998). This
method uses a rolling horizon to solve overlapping
subsets of the planning horizon, thus reducing solution
time. While this method has no guarantee of optimal-
ity, in practice, the TCM prescriptions are face valid
using a solution cascade, and TCM users now prefer
this solution method.
We implement the TCM using the General Algebraic
Modeling System (GAMS) (GAMS 2015a) and solve
it using CPLEX (GAMS 2015b). For a typical cascade
subset of the planning horizon, about 1.5 years, an
instance of the TCM has approximately 500,000 con-
straints and 950,000 variables (950 of them integer). The
cascade subsets typically overlap by 0.5 years. Solution
time per cascade is approximately five minutes using a
Windows 7 workstation with two 3.33 GHz Intel Xeon
X5680 processors and 8 GB RAM. Typical TCM plan-
ning horizons of 2.5 years, for example, require two
cascade iterations and take approximately 10 minutes
to solve, while long-term student allocations of 10 years
typically require about 50 minutes.
TCM users find solution times without the use of a
rolling horizon undesirable. For example, we recently
conducted some experiments with real instances that
have a planning horizon of 3.5 years. Using the pre-
ferred method of three cascade iterations, these took
between 10 and 15 minutes to solve. Solving these
instances without a rolling horizon took between 30
and 45 minutes. An examination of their respec-
tive solutions showed the results obtained using both
methods were almost identical; the distributions of stu-
dents to classes and weekly instructor staff workload
varied only slightly. Empirical results such as these
have led TCM users to almost exclusively use solution
cascades. Despite this strong preference, we maintain
an interface option for TCM users to easily disable solu-
tion cascades or adjust the subset of the planning hori-
zon considered for each cascade subset.
For improved solution time, TCM users are also
often willing to accept a solution that is only guaran-
teed to be within approximately one percent of opti-
mality. With such a permitted gap, staff assigned hours
(the third term of the objective function) may exceed
the number required for a given allocation. This cos-
metic annoyance is corrected by solving a revised MILP
that fixes the student allocation, thereby fixing the first
two terms of the objective function, and minimizes the
third term of the objective function.
TCM Testing Before Adoption
Several management layers were needed to approve
the adoption of the TCM; therefore, we designed a
rigorous test program. At the time of the TCM devel-
opment, decision makers had no approved and uni-
versally applied criteria when evaluating allocations.
Management judged new student allocations primar-
ily based on the backlog and how they compared with
historic allocations. Given this history, the first phase
in the test program was to engage a broad range of
stakeholders to develop objectives, criteria, and met-
rics for allocations using value-focused thinking tech-
niques (Keeney 1994). Ewing et al. (2006) and Parnell
and West (2011) provide examples of applying value-
focused thinking. This resulted in the following overar-
ching objective statement: The allocation process seeks
to maximize student throughput with on-time training
completion while efficiently utilizing staff and facili-
ties. The characteristics of a good allocation were iden-
tified as equity in class assignments across NPTUs,
on time completion, and full utilization of training
resources. We then defined a set of 11 questions reflect-
ing these characteristics for use by training experts in
evaluating an allocation.
Simultaneously, training experts established 13
benchmark test cases representing a range of routine
and abnormal scenarios. For each test case, several
training experts were asked to independently evalu-
ate TCM results using the 11 established questions,
each rated on a four-point Likert scale (i.e., unsatisfac-
tory, marginal, good, excellent). Evaluations, including
open-ended comments, were submitted to a database
created to enable long-term collection and analysis of
TCM results. Once all test cases had been evaluated,
statistics were computed quantifying the acceptability
of the allocation and the associated inter-rater reliabil-
ity using the method from Fleiss (1971).
Each testing round concluded with a meeting of
the evaluators and model developers to discuss the
results and agree on model refinements, if needed. For
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example, one round resulted in changes to the staff-
instructor workload model. The evaluators found it
unacceptable to expect sustained periods of high staff-
instructor workload, even in theoretical cases of low
instructor staffing with high demand for qualified stu-
dents. A new TCM constraint was implemented to con-
trol sustained staff-instructor workload. This feature
allows a periodic surge of high workload if it is bal-
anced by a period of lower workload.
In addition to benchmark testing, a series of retro-
spective tests were performed using the most recent
two years of data. In the first test, the TCM prescribed
student allocations, given actual plant availability and
staffing levels. In the second test, actual student allo-
cations were also fixed, with the objective of solving
for staff workload. The TCM has elastic constraints that
allow minor violations, and we experimented exten-
sively with penalties for these violations to ensure that
all penalties were scaled, such that each was mean-
ingful, and to capture the trade-off between exces-
sive staff workload and having a backlog of students
waiting for training. Following this experimentation,
default penalty values were established where the stu-
dent throughput was less than a five percent difference
from historical values and staff workload was within
acceptable ranges.
Near the end of development, the TCM was run in
parallel with the legacy model for several allocations.
In the parallel operations, the primary training plan-
ner found the results of the TCM acceptable. In addi-
tion, the TCM provided insights that were previously
unavailable. These insights, coupled with the results
from the benchmark and retrospective testing, were
reviewed with training management who requested
immediate adoption of the TCM.
Interface Design
One principle analyst and several assistants are respon-
sible for using the TCM to produce student allocations.
Preparing TCM input requires considerable knowledge
of student-training database systems and the TCM is
run many times each week. A broad range of deci-
sion makers rely on its prescriptions to both deter-
mine the student allocation and to plan (e.g., staff-
instructor schedules and plant maintenance periods).
In addition, decision makers frequently request what-if
analyses representing different training scenarios. All
these decision makers require graphical displays of the
prescriptions.
A Microsoft Excel VBA application serves as the
TCM interface. It contains all TCM documentation and
input spreadsheets, serves to obtain input data from
several independent databases, calls GAMS, produces
all the TCM output reports, and displays all the graph-
ics. The TCM produces 15 standard graphs; Figures 4–9
show examples. The TCM replicated the basic look and
feel of all legacy graphics, while also providing new
visualizations to display information not previously
available in the legacy application.
Results and Impact
Over three years, the insights gained from the use
of the TCM have increased the number of students
trained by an estimated eight percent (when compared
with the legacy model). This improvement stems pri-
marily from a holistic understanding of how student
training is impacted by the interrelationships between
plant availability, staffing, facility availability, and sim-
ulator utilization. The ability to rapidly conduct what-
if analysis that explicitly considers these interrelation-
ships has led to these new insights; as a result, decision
makers have altered their allocation decisions to better
balance available resources.
The key to communicating the TCM prescriptions
is effective visual displays. The primary display for
Figure 4. The Student Backlog (Number and Type
Represented by the Different Shades in the Bar Graph) as a
Function of Time (the Horizontal Axis) Provides
Executive-Level Decision Makers with Key Information
Concerning the Flow of Students through the Training
Program, and Forms the Basis for Comparison of What-If
Scenarios Involving the Allocation of Resources
4
50
400
350
300
250
200
S
tu
d
e
n
ts
Time (weeks)
Student backlog
150
100
50
0
Student type 1 Student type 2
Student type 3 Student type 4
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Figure 5. Student Backlog Is Presented for Three Cases Based on the Placement of an Extended Maintenance Period, Which
Would Result in Skipping Several Classes of Students at One NPTU
0
100
200
300
400
500
600
700
800
900
2018 2019 2020 2021 2022 2023
S
tu
d
e
n
t
b
a
ck
lo
g
Year
Impact of maintenance alternatives
Baseline
Alternate 1
Alternate 2
executive-level decision makers is the expected stu-
dent backlog. Figure 4 shows an example of the pro-
jected backlog of four student types waiting for NPTU
training over time. This figure shows that initially
the student backlog is large because of resource con-
straints (i.e., qualified staff instructors, plant availabil-
ity, or facilities). As student-training resources become
available with each succeeding class, the backlog of
students reduces. Subsequently, resources are again
constrained resulting in a rise in the backlog. These
fluctuations continue in response to changing resource
profiles.
Figure 5 shows an example of a what-if scenario.
The scenario is to determine the impact on student
training based on the placement of a required mainte-
nance period, which will make the plant unavailable
Figure 6. For Each Staff-Instructor Type, Workload Is
Controlled Within the Desired Level, Sustained Limit, and
Control Limit by Varying the Number and Type of Students
Allocated to Each Class
Time (week)
Workload per staff instructor
Fixed staff work
W
o
rk
lo
a
d Variable staff work
Desired limit
Control limit
Sustained limit
Allocation
Window
for student training for an extended period and result
in several skipped classes. The baseline indicates the
current schedule for the maintenance period; alter-
nate 1 places the maintenance period in the schedule
one year later; alternate 2 places it two years later.
Schedules for the other plants are unaffected and all
other resources are fixed; the only resource affected is
the availability of the plant at which the maintenance
is taking place. The comparison clearly shows that the
baseline generates the fewest number of backlogged
students; however, if the maintenance is moved from
the current placement, alternate 2 would be the pre-
ferred option.
The TCM prescribes weekly staff-instructor variable
workload across the planning horizon for each of the
five types of staff. Figure 6 shows a staff-instructor
workload display for one instructor type. It displays
the fixed workload (duties required to keep the plants
operational) and the variable workload for simulator
Figure 7. Each Bar Represents the Total Number (and with
Shading the Actual Number) of Students Allowed to
Simultaneously Train at the Start of Class at a NPTU
Facilities utilization
U
til
iz
a
tio
n
Time (weeks)
Note. The student limits are a proxy for facility capacities, such as
classrooms, study cubicles, and computer stations.
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Figure 8. TCM Output Provides a Graphical Representation of Plant Availability in Conjunction with the Watchstanding
Timeline Associated with Each Class
Notes. The figure presented is simplified and truncated for readability. Plant availability, student allocations, and watchstanding preferences
by class are shown. A box in each plant’s row indicates the week when a class completes watchstanding. For example, at Plant 1, Class 2
completes watchstanding in Week 65.
watchstanding and off-watch training as prescribed by
the TCM. Figure 6 also displays the desired staff work-
load level, the maximum sustained workload limit, and
the maximum control limit—a surge limit that must be
offset by reduced hours a few weeks prior to or after
the surge.
Insights provided by the TCM into the details of
staff-instructor workload for each staff type yielded
one of the greatest early benefits from adopting TCM,
in part because the legacy model did not explic-
itly consider instructor workload. By using TCM, it
became clear that some staff types were working at
capacity (and thereby preventing additional student
allocations), while other instructor types had hours
available. Equipped with this new insight (and verifi-
cation by staff instructors), changes were implemented
to better balance workload among staff-instructor
types and thereby increase student allocations. These
changes included modifications of work assignments
and adjustments in the number and type of instructors
assigned to the NPTUs.
Optimally planning staff-instructor workload also
allowed increased student allocations when part of
training for a class coincided with a plant shutdown.
Before the TCM, a simple rule of thumb dictated that
only a few or no students would be allocated to a plant
when its availability prevented watchstanding comple-
tion before a standard number of weeks. Using the
TCM, it became apparent that this rule of thumb was
too restrictive. The TCM showed how to maintain near-
normal student allocations by using increased staff-
instructor hours during the shutdown for off-watch
and simulator training. Sometimes this needs to be
coupled with a modest extension beyond a normal
training deadline, with the TCM providing the details
used to obtain permission for such an exception to nor-
mal operations.
The TCM also ensures that student loading for each
class fits within the capacity limits of the training facil-
ities. This includes limits for individual classes (e.g.,
classroom number and size, study cubicles, computer
stations), and cumulative NPTU and site-loading lim-
its for all classes projected to be at a facility at one time.
TCM does this by limiting the number of students in
each NPTU and each site for all simultaneous classes.
Figure 7 shows an example of student capacity of the
training facilities over time.
Figure 8 shows a simplified summary of the TCM
results for a student allocation at the four NPTUs; these
results replicate much of a legacy report used by man-
agement. The timeline shows how the allocation fits
within each plant’s operating schedule, with no plant
watchstanding occurring during maintenance periods.
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Figure 9. TCM Provides Detailed Information for Each Plant and Class (with Only Plant 1 Shown for Readability)
Notes. For example, in Week 66, Class 2 has one simulation session and Class 3 has 20. The last row reports the percentage of plant watch-
standing completion for each class. For example, in Week 64, Class 2 has 92 percent of its watchstanding complete and Class 3 has four
percent.
The relationship between the preferred watchstanding
periods for the three classes appears in the top por-
tion of the timeline. The timeline view includes the size
and composition of each class (represented in Figures 8
and 9 by the “# Students” in each of the class boxes),
simulator utilization, and class completion. Figure 9
magnifies the timeline to show detailed information for
each class. The timeline view is particularly useful to
those making short-term training decisions.
In addition to the what-if scenario presented in
Figure 5 (i.e., variation in the placement of planned
maintenance periods), other typical scenarios involve
changes to the number of staff instructors, additional
facilities to expand student capacity, and the type and
quantity of simulation equipment that could be inte-
grated into the training program. In each of these
what-if scenarios, the key element is the number of
students that can be trained and the cost of training
them, including personnel, facilities, and (or) equip-
ment costs, as compared to the required number of
trained students that must be transferred to the fleet
to maintain the desired staffing levels onboard sub-
marines and aircraft carriers. These what-if scenarios
have driven changes to the number of assigned instruc-
tor staff, placement of maintenance periods, and strate-
gic decisions concerning future investments in equip-
ment and recapitalization.
The use of TCM, with its ability to conduct mul-
tiple what-if scenarios that produce accurate and
repeatable results in the current resource-constrained
environment, has enabled both tactical and strategic
decisions to ensure fleet staffing needs are continually
met to satisfy U.S. Navy operational and strategic
requirements.
Conclusions
The TCM provides an optimal use of the key resources
needed to qualify naval nuclear operators in a chal-
lenging operational and budgetary environment. The
TCM has helped increase the number of students
trained by showing how to better employ these key
resources, especially staff instructors. It eliminated the
long-standing dependence on the few (i.e., only one or
two) experts who can perform capacity analysis. Four
analysts now routinely use the TCM.
The TCM’s prescriptions were quickly found to be
superior to the legacy model. Today, decision makers
frequently request a broad range of what-if analyses
that rely on using the TCM.
Acknowledgments
The authors thank Martin Andrew, Fred Lanou, and Paul
Zanella for their executive support throughout the devel-
opment and deployment of the TCM, Scott Ciampa (Naval
Nuclear Laboratory), Dan Arguello (Naval Nuclear Labo-
ratory), Karina Rodriguez (Naval Nuclear Laboratory), and
Anton Rowe (Naval Postgraduate School) for their techni-
cal contributions, Torre Bissell (retired) for his mentoring
and software development expertise, and LT Kai Seglem
(USN) for his many contributions to initial TCM develop-
ment. Additionally, the authors thank the numerous Navy
junior officers whose contributions at different stages of the
project ensured a successful product delivery. Finally, the
authors thank the Interfaces editors and referees for their
thoughtful reviews that helped improve this article.
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Appendix
We present a representative TCM formulation using
o
≥,
o
≤,
and
i
≤ as shorthand for elastic constraints (i.e., constraints
that can be violated at a cost per given unit of violation), as
discussed in Brown et al. (1997). The elastic constraints rep-
resented by
o
≥ and
o
≤ can be violated in any period, while
the elastic constraints represented by
i
≤ can only be violated
for periods corresponding to the first τ classes, whose mem-
bers have already begun training and are considered “fixed.”
For example, if τ equals 2, with Class 1 allowed to stand
watch for weeks t ∈ {1, 2} and Class 2 weeks t ∈ {1, 2, . . . , 9},
elastic constraints represented by
i
≤ could only be violated
where c ∈ {1, 2} and t ∈ {1, 2, . . . , 9}. Let c (or alias c′) be the
index for class number; then the allocation for classes c ∈
{1, 2, . . . ,τ} are fixed as starting conditions for the TCM and
c ∈{τ + 1,τ + 2, . . .} are not fixed. Because these fixed classes
may violate desired planning, we must have the ability to
violate constraints for these fixed classes. Some of the terms
in the constraints are applicable for only a single student type
(referred to as rate r � r3).
In the following Sections A.1–A.6, we present the TCM for-
mulation for indices, index sets, parameters, variables, objec-
tive function, and constraints, and then conclude with a brief
description. Beale et al. (1974) originally documented this
ordering, which was adhered to for decades under the label
NPS format in hundreds of published theses and papers by
Naval Postgraduate School (NPS) students and faculty, with
acknowledgment of the original Beale et al. reference (Brown
and Dell 2007), and recently reintroduced with additional
guidance by Teter et al. (2016).
A.1. TCM Indices [∼Cardinality]
c, c′ class [6 per year];
d training site [2];
j simulator type [2];
p NPTU [4];
r student type or rate [7];
s staff-instructor type [5];
t week of the planning horizon [208];
w, w′ watch to stand [25].
A.2.
p ∈ TSd set of NPTU p at training site d;
r ∈ RWw set of all student types r that stands watch w;
s ∈ SWw set of staff s that can stand watch w;
t ∈ AWcpr set of all possible watch weeks t for class c, student
type r, at NPTU p;
t ∈ EWc r set of all early watch weeks t for class c for student
type r;
t ∈ FWcpr set of weeks t when class c can finish at NPTU p
for student type r;
t ∈ LWcpr set of all late weeks t at the end of class c at NPTU
p for student type r;
t ∈ V Wc r set of all very early watch weeks t for class c for
student type r;
w ∈ AS set of all simulator watches w;
w ∈ OW set of watches w satisfying off-watch require-
ments;
w ∈ PT set of watches w that require a plant;
w ∈ SS set of all watches w where a simulator can substi-
tute;
w′∈ SBw set of watches w′ that can substitute for watch w;
w′∈ SMw set of simulator watch w′ that can be substituted
for watch w.
A.3.
Parameters for the Objective Function
pcrc r penalty for student type r student waiting for training
after the start of class c [penalty per student];
pcpc p penalty for not starting class c at NPTU p [penalty per
class];
psws w penalty for staff s standing watch w [penalty per
hour].
Parameters for Constraints (A.2)–(A.7) (The
Class-Composition Group)
rollr number of student type r students waiting for
training at the start of planning [students];
newc r number of newly arriving student type r for
class c [students];
gcprcpr, gcprcpr lower and upper goal on the number of class c
students of student type r desired at NPTU p
[students];
lcpc p, gcpc p lower limit and upper goal on the number of
class c students (excluding student type 3) at
NPTU p [students];
lcpc p upper limit on the number of students
NPTU p facilities can support during the first
week class c is held [students];
lcdc d upper limit on number of students Site d facil-
ities can support during the first week class c
is held [students];
lcdrcdr upper limit on number student type r site d
facilities can support during the first week for
class c [students].
Parameters for Constraints (A.8)–(A.14) (The
Student-Training Group)
reqcprw hours of class c watch w training required by student
type r at plant p [hours per student];
donecprw hours of watch w already completed at the start of
planning by class c at NPTU p by student type r
[hours];
nwwcrw minimum number of weeks per class c, student
type r, and watch w that must contain some watch-
standing (divisor used to provide an upper bound
on weekly training for each watch);
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grtr t upper goal on the number of hours student type r
can work in week t [hours per student];
lcrwcrw upper limit on the number of class c watch hours
allowed in simulators by student type r for watch w
[hours per student];
ljpwjpw hours available for simulator j, watch w for each
simulator watch session at NPTU p [hours per
session];
lptwptw upper limit on hours of watch w available at NPTU p
in week t [hours].
Parameters for Constraints (A.15)–(A.17) (The
Staff-Instructor Group)
gpstpst upper goal on hours of staff s for assignment at
NPTU p in week t [hours];
hfixpst fixed hours for staff s at NPTU p in week t
[hours];
hsimjpw staff hours required for each j simulator
watch w session at NPTU p [hours per session];
gjptjpt, gjptjpt lower and upper goal on the number of sim-
ulator j watch sessions in NPTU p in week t
[sessions].
Parameters for Constraints (A.18)–(A.21) (The Watch
Placement Group)
grvr upper limit on the fraction of student type r very early
training allowed [hours per hours];
grer upper limit on the fraction of student type r early train-
ing allowed [hours/hours].
Parameters for Constraints (A.22) and (A.23) (The
Persistent Group)
τ maximum class for index c for which allocations Xcpr
are fixed for all p, r;
fixXcpr allocation of student type r from class c fixed to train
at NPTU p [students];
oldXcpr prior value of Xcpr used to guide a persistent solution
[student].
A.4.
Xcpr integer number of student type r to start class c at
NPTU p;
Kc r integer number of student type r waiting for training
after start of class c;
Ijpt integer number of simulator j sessions at NPTU p in
week t;
Gc p binary variable with value one if class c is in session at
NPTU p and zero otherwise;
Fcprtw plant watch w training hours from fixed plant opera-
tions assigned; to student type r in class c at NPTU p
in week t;
Ucprtw simulator watch w hours assigned to student type r in
class c at NPTU p in week t;
Vcprtw Off-watch w hours assigned to student type r in class c
at NPTU p in week t;
Hpstw staff s hours assigned at watch w (includes simulator
and off-watch instruction only) at NPTU p in week t.
A.5. TCM Formulation
Minimize
{∑
c r
pcrc r Kc r
+
∑
c p
pcpc p(1−Gc p)
+
∑
ptw, s∈SWw
psws w Hpstw + elastic penalties
}
(A.1)
Subject to:
Kc r �rollr |c�1 +Kc−1,r |c>1 +newc r−
∑
p
Xcpr, ∀c, r, (A.2)
gcprcpr
o
≤Xcpr
o
≤gcprcprGc p, ∀c, p, r, (A.3)
lcpc p Gc p
i
≤
∑
r,r3
Xcpr
o
≤gcpc p, ∀c, p, (A.4)
c∑
c′�c−2
∑
r,r3
Xc′p r +
c∑
c′�c−1
∑
r�r3
Xc′p r
i
≤lcpc p, ∀c, p, (A.5)
c∑
c′�c−2
∑
r,r3
∑
p∈TSd
Xc′p r +
c∑
c′�c−1
∑
r�r3
∑
p∈TSd
Xc′p r
i
≤lcdc d,∀c, d, (A.6)∑
p∈TSd
Xcpr≤lcdrcdr, ∀c, d, r, (A.7)∑
t,w′∈SBw
(Fcprtw′ +Vcprtw′ +Ucprtw′)
≥reqcprwXcpr−donecprw, ∀c, p, r, w, (A.8)∑
w′∈SBw
(Fcprtw′ +Vcprtw′ +Ucprtw′)
≤
(reqcprwXcpr−donecprw)
nwwcrw
, ∀c, p, r, t, w, (A.9)∑
w
(Fcprtw +Vcprtw +Ucprtw)
o
≤grtr t Xcpr, ∀c, p, r, t, (A.10)∑
c,r∈RWw
(Vcprtw +Ucprtw)≤
∑
s∈SWw
Hpstw, ∀p, t, w, (A.11)∑
t∈AWcpr,w′∈SMw
Ucprtw′≤lcrwcrwXcpr, ∀c, p, r, w∈SS, (A.12)∑
c |t∈AWcpr,r∈RWw
Ucprtw≤
∑
j
ljpwjpwI j p t , ∀p, t, w∈AS, (A.13)∑
c,r∈RWw
Fcprtw
i
≤lptwptw, ∀p, t, w∈PT, (A.14)∑
w o s∈SWw ∑ gjptjpt o t∈V Wc r
( ∑ Ucprtw′ o t∈EWc r
( o D Miller et al.: Optimal Allocation of Students to Naval NPTUs ∑ w∈OW reqcprw ∑ w∈PT reqc p r w
+ ∑ w∈PT,w′∈SBw∩AS reqc p r w′ w∈OW Vcprtw∑ ≥ w∈PT Fcpr,t,w∑ + ∑ , ∀c, p, r, t∈FWc p r , (A.21) o Xc p r ≥0 and integer, ∀c, p, r, (A.24)
Gc p ∈{0,1}, ∀c, p, (A.25) (A.26)
The objective function (A.1) expresses the total penalty each watch. Constraint set (A.10) restricts total weekly stu- A.6. The TCM formulation has many elastic constraints that we w In most cases, the addition of such an elastic variable (and Opst ≤ overpst, ∀ p, s, t, (F.1a) t∈AWcpr References programs. Technical Report NPS-OR-98-004, Naval Postgradu- Beale EML, Beare GC, Tatham PB (1974) The DOAE reinforcement D Miller et al.: Optimal Allocation of Students to Naval NPTUs Blake JT, Donald J (2002) Mount Sinai Hospital uses integer program- Bonutti A, De Cesco F, Di Gaspero L, Schaerf A (2012) Benchmark- Brown GG, Dell RF (2007) Formulating integer linear programs: Brown GG, Dell RF, Wood RK (1997) Optimization and persistence. Brown GG, Graves GW, Ronen D (1987) Scheduling ocean transporta- Cardoen B, Demeulemeester E, Beliën J (2010) Operating room plan- Caunhye AM, Xiaofeng N, Pokharel S (2012) Optimization models in de Werra D (1985) An introduction to timetabling. Eur. J. Oper. Res. Department of the Navy and Department of Energy (2014) The United Detar PJ (2004) Scheduling Marine Corps entry-level MOS schools. Ernst AT, Jiang H, Krishnamoorthy M, Owens B, Sier D (2004) An Ewing PL Jr, Tarantino W, Parnell GS (2006) Use of decision anal- Fleiss JL (1971) Measuring nominal scale agreement among many GAMS (2015a) GAMS: Cutting edge modeling. Accessed July 15, GAMS (2015b) CPLEX 12. Accessed July 15, 2015, http://www.gams Gibson HO (2007) The total Army competitive category optimization Ginther TA (2006) Army Reserve enlisted aggregate flow model. Grant JM (2000) Minimizing time awaiting training for graduates of Keeney RL (1994) Creativity in decision making with value-focused Lee EK, Chen C-H, Pietz F, Benecke B (2009) Modeling and opti- Lee EK, Atallah HY, Wright MD, Post ET, Thomas IV C, Wu DT, Microsoft (2015) Introduction to the Visual Basic programming lan- Parnell GS, West PD (2011) Systems decision process overview. Par- Seal HL (1945) The mathematics of a population composed of k Teter MD, Newman AM, Weiss M (2016) Consistent notation for Vajda S (1947) The stratified semi-stationary population. Biometrika Wang J (2005) A review of operations research applications in Whaley DL (2001) Scheduling the recruiting and MOS training of Workman PE (2009) Optimizing security force. Unpublished master’s Zhang B, Murali P, Dessouky M, Belson D (2009) A mixed-integer Verification Letter nology Naval, Training and Simulation, Naval Nuclear Lab- “This is to verify that the claims made in the manuscript Michael R. Miller is an engineer with the Naval Nuclear Robert J. Alexander works at the Naval Nuclear Lab- Vincent A. Arbige works at the Naval Nuclear Laboratory Robert F. Dell is a professor of operations research (OR) D http://www.gams.com http://www.gams.com http://www.gams.com https://msdn.microsoft.com/library/xk24xdbe(v=vs.90).aspx https://msdn.microsoft.com/library/xk24xdbe(v=vs.90).aspx Miller et al.: Optimal Allocation of Students to Naval NPTUs an OR assistant professor in 1990 and served as chairman Steven R. Kremer works at the Naval Nuclear Labora- Brian P. McClune is a scientist at the Naval Nuclear Jane E. Oppenlander teaches statistics in the School of Joshua P. Tomlin works at the Naval Nuclear Laboratory D TCM Index Sets
≤gpstpst, ∀p, s, t, (A.15)∑
Hpstw≥
j
hsimjpwIjpt, ∀p, t, w∈AS, (A.16)
o
≤Ijpt
≤gjptjpt, ∀ j, p, t, (A.17)∑
Fcprtw +
w′∈SBw∩AS
)
≤grvrreqcprwXcpr, ∀c, p, r, w
Fcprtw +
∑
w′∈SBw∩AS
Ucprtw′
)
≤grerreqcprwXcpr, ∀c, p, r, w
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Interfaces, 2017, vol. 47, no. 4, pp. 320–335 333
t′≥t,w∈OW Vcprt’w∑
≥
w∈PT Fcpr,t−1,w∑
w∈PT,w′∈SBw∩AS Ucpr,t−1,w′∑
, ∀c, p, r, t∈LWcpr, (A.20)∑
w∈OW reqcprw
∑
w∈PT reqcprw
w∈PT,w′∈SBw∩AS Ucpr,t,w′∑
w∈PT,w′∈SBw∩AS reqcprw′
Xcpr �fixXcpr, ∀c≤ f c, p, r, (A.22)
Xcpr
≥oldXcpr, ∀c, p, r, (A.23)
I j p t ≥0 and integer, ∀ j, p, t,
Kc r ≥0 and integer, ∀c, r,
Fcprtw≥0, ∀c, p, r, t, w∈PT,
Ucprtw≥0, ∀c, p, r, t, w∈AS,
Vcprtw≥0, ∀c, p, r , t, w∈OW,
Hpstw≥0 ∀p, s, t, w
value. The first term of the objective function is the weighted
penalty of the student backlog; the second term is the
weighted penalty for skipped classes; the third term is the
weighted penalty for staff workload, and the last penalty
term includes all elastic constraint violations (Section A.6
in this appendix). Constraint set (A.2) tracks student back-
log (i.e., inventory of students by student type waiting to
start training after the start of a class). For the first period
(c � 1), the backlog is initial backlog (rollr). The student back-
log increases for any newly arriving students who cannot be
trained in the current class, and decreases whenever more
students may be trained than those newly arriving for the
current class. Constraint set (A.3) measures deviation from
desired lower and upper limits for each student type in each
class at each plant, and ensures that no student of any student
type is assigned to a skipped class. Constraint set (A.4) mea-
sures deviation from the desired upper and lower bounds
for the total number of students in a class at each NPTU
(excluding student type r3). Constraint set (A.4) removes
the lower bound for skipped classes to ensure feasibility.
Constraint set (A.5) sets an upper bound on the number
of students in all simultaneous classes (classes c − 2 to c
for all student types, excluding student type r3 and classes
c − 1 to c for student type r3). Similarly, constraint set (A.6)
sets an upper bound for total number of students a train-
ing site can support across all simultaneous classes. Con-
straint set (A.7) limits the number of each student type for
a class at a site. Constraint set (A.8) ensures sufficient train-
ing hours are assigned for each student type at each watch
station (watch station includes plant and off-watch require-
ments). Constraint set (A.9) limits weekly training hours for
dent work hours. Constraint set (A.11) limits student and off-
watch hours to those with assigned staff. Constraint set (A.12)
restricts simulation training to be no more than a user input
fraction of the total for training that can be conducted using
simulation. Constraint set (A.13) restricts simulator hours to
those with assigned personnel. Constraint set (A.14) restricts
plant hours. Constraint set (A.15) limits staff hours available
for watch and off-watch duties. Constraint set (A.16) ensures
adequate personnel for each simulation session. Constraint
set (A.17) restricts weekly simulator sessions. Constraint
set (A.18) allows no more than a user input fraction of the
total plant-based training to be completed very early. Con-
straint set (A.19) allows no more than a user input fraction
of the training of the total plant-based training to be com-
pleted early. Constraint sets (A.20) and (A.21) ensure suffi-
cient off-watch training in the last weeks. Constraint (A.22)
fixes the initial student allocation for the first classes. Con-
straint (A.23) measures negative deviation from a prior solu-
tion. Constraints (A.24)–(A.26) declare variable types.
have already introduced using notation shorthand. Here are
the details for elastic constraint set (A.27), which controls the
weekly and sustained staff-instructor workload by adding a
new “elastic” variable Opst for the overtime worked at NPTU
p by staff type s in week t:∑
its corresponding penalty term in the objective function) is
all that is required to convert an elastic constraint from short-
hand to more traditional notation. In this case, there are addi-
tional constraints on the elastic variable. Constraint set (F.1a)
limits the extent of overtime allowed at NPTU p by staff type s
in week t:
and constraint set (F.1b) limits the sustained overtime
allowed at NPTU p by staff type s over all watch weeks for
each student type r and each class c:∑
Opst ≤ overap s , ∀ c, p, r, s. (F.1b)
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Geoffrey Guido, Manager, Training Programs and Tech-
oratory, Kesselring Site, Schenectady, NY 12301, writes:
‘Optimal Allocation of Students to Naval Nuclear Power
Training Units’ are accurate. This manuscript describes the
development and use of an application that optimizes the
number of students assigned to Naval Nuclear Power Train-
ing Units. Since its adoption two years ago, we have realized
a gain in student throughput of approximately eight percent
and have significantly improved the deployment of our per-
sonnel and equipment.”
Laboratory. He spent 12 years in the Naval Nuclear Labora-
tory’s Nuclear Operations Program training naval personnel
on an operating reactor plant. He leads a software develop-
ment group dedicated to providing analytical solutions for
the Naval nuclear training community. He holds a Bachelor
of Science in Mechanical Engineering from the University at
Buffalo.
oratory developing analytic methods to support engineer-
ing and training efforts. Robert earned his Bachelor of Sci-
ence degree in chemical engineering from Brigham Young
University.
developing analytical data models to support allocation of
training resources. Vincent earned his Bachelor of Science
degree in chemistry from the University of Rhode Island and
has worked in the Naval Nuclear Training Program for 36
years. He spent 10 years in various training positions and
then built the first resource allocation models for the training
program, which he has supported for the past 26 years.
at the Naval Postgraduate School (NPS). He joined NPS as
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Interfaces, 2017, vol. 47, no. 4, pp. 320–335 335
of the OR Department from 2009 to 2015. During his tenure
as chairman the department received the 2013 INFORMS
Smith Prize. Professor Dell has been awarded the Barchi,
Koopman, and Rist prizes for military operations research.
He has also received two Department of the Army Payne
Memorial Awards for Excellence in Analysis and two Depart-
ment of the Navy Superior Civilian Service Awards. Profes-
sor Dell is editor-in-chief of the Military Operations Research
Journal.
tory developing analytic methods to support training. He is
a retired Navy Captain who commanded USS ARCHERFISH
(a nuclear powered submarine) and Naval Station Bremer-
ton. He graduated from the United States Naval Academy
with a Bachelor of Science degree in mechanical engineering
and holds a master’s degree in political science from Auburn
University at Montgomery.
Laboratory. Brian divides his time between development of
optimization software for the nuclear training community
and development of high performance computing applica-
tions in support of reactor design. He earned Bachelors of Sci-
ence degrees in mathematics and physics and a Master’s of
Science degree in computer science from Clarkson University.
Business and the Bioethics Program at Clarkson University.
She earned her PhD in engineering and administrative sys-
tems from Union College. Jane recently retired after a 35-year
career as a statistician at the Naval Nuclear Laboratory.
programming systems to support engineering and training
efforts. Along with his programming support, Joshua main-
tains and develops processes for the Learning Management
System. He earned his master’s degree in computer science
from The College of Saint Rose.
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TCM Parameters [Units]
TCM Variables
TCM Formulation
Elastic Constraints
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