Download III. Single Objective Optimization

Optimization of a Heterogeneous Unmanned Mission
Yves Boussemart1, Anna E. Massie2, Brian Mekdeci3
Massachusetts Institute of Technology, Cambridge, MA, 02139
Decision makers are often confronted with the problem of determining the number and
type of unmanned vehicles to allocate to a particular mission and determining the operator
strategies of commanding those vehicles given the constraints of finite resources and human
cognitive limitations. This paper probes this optimization problem using a predictive model
of a human operator controlling multiple, heterogeneous, unmanned vehicles. Using a set of
mission paramenters, a pareto frontier is found between the competing performance
criteria in the model. By analyzing this frontier, general guidelines are outlined as to
maximize performance while operating within a human’s cognitive capabilities.
Nomenclature
HALE
MALE
UUV
NH
NM
NU
RS
SS
TNH
TNM
TNU
=
=
=
=
=
=
=
=
=
=
=
High Altitude Long Endurance vehicle
Medium Altitude Long Endurance vehicle
Unmanned Underwater Vehicle
Number of HALEs
Number of MALEs
Number of UUVs
Re-plan Strategy
Switching Strategy
Maximum number of HALEs
Maximum number of MALEs
Maximum number of UUVs
CNH
CNM
CNU
λi
μi
ρ
T
S
U
C
MT
=
=
=
=
=
=
=
=
=
=
=
Cost of each HALE
Cost of each MALE
Cost of each UUV
arrival rate of event i
service rate of event i
operator utilization
Mission Time
Score
Utilization
Total Mission Cost
Mission Time
I. Introduction
I
NTEREST in unmanned vehicles (UV) has increased significantly in the last several years for both military and
civilian applications. Due to the increase in levels of automation used, there is a shift in UV research away from
manually controlling vehicles to managing vehicles at the supervisory control level. At the supervisory control
level, a single human operator will be able to command multiple, heterogeneous unmanned vehicles simultaneously.
A current area of research is to determine how teams of UVs should be allocated to various missions and what
strategies operators should employ to command those UVs. The trade-offs are that UVs which are more highly
automated tend to perform better than less automated UVs (due to less operator involvement), although they tend to
be the most expensive. Allocating more UVs to a particular mission also increases operational costs, but often also
increases the probability of the mission being successful. Having too many UVs, however, may decrease
performance overall since the operator may lose situational awareness as a result of the human cognition limitations
under increased workload. In addition to UV allocation, the operator strategies also have a large effect on
performance. Determining the order in which vehicles are serviced or deciding whether to perform optional replanning tasks has a mixed effect on overall performance since the benefits or disadvantages in these strategies are
highly dependent on the situation upon which they are executed. In essence, this is an optimization problem, where
the objective is to determine the optimal UV allocation and operator strategies to maximize mission performance
and minimize mission cost.
1
Doctoral Candidate, Engineering Systems Division, 77 Massachusetts Ave. Room 33-407
Master’s Candidate, Aeronautical and Astronautical Engineering, 77 Massachusetts Ave. Room 35-220, AIAA
student member.
3
Doctoral Candidate, Engineering Systems Division, 77 Massachusetts Ave. Room 33-407
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II. Problem Formulation
A. Underlying Model Description
A model is necessary to predict the performance and cost associated with a particular mission for a given set of UV
allocation and operator strategies. Based on cognitive principles, the attention of a human operator can be viewed
as a server and the tasks to be performed can be considered as a sequence of events that need servicing from the
server. This model can then be formulated as queuing theory problem. Thus, a discrete event simulator was used to
implement a model that predicts the cost and performance of an unmanned vehicle operation for a given set of
constraints1.
At the core of our model are the operator and the events that the operator perceives. Everything that requires the
operator’s attention is considered to be an event arriving in the operator’s mental queue. Each event arrives based
on stochastic distribution whose parameters (e.g. the average time between events) are determined by the specific
mission scenario. These events can either be a mission generated task such as a vehicle requesting to be attended by
the operator, or they can be an operator generated task such as a re-plan of a vehicle’s existing route. The operator
can only handle (or service) one event at a time, so when more than one event is in the operator’s queue, the
operator much choose the order in which he or she will handle the events. The order in which the events are
serviced when is known as the switching strategy (SS) and is set at the beginning of the scenario based upon the
preferences of the operator. Acceptable switching strategies are first-in, first-out (FIFO) or a priority system based
upon whether the vehicle is aerial or undersea. The time it takes for an operator to service a particular event is
known as the service time and is drawn from a random distribution whose parameters (e.g. the average service time)
are also determined by the specific mission scenario. A structural overview of the model is presented in Figure 1,
where the mission parameters influence the human operator’s situational awareness and ultimately mission
performance. Our model therefore encompasses multiple disciplines such as cognitive psychology, queuing theory,
discrete event simulation and unmanned vehicle operation. The different objectives, parameters, variables and
constraints of the model are described in more detail below.
Figure 1: Structural Overview of the Cognitive Model
B. Objectives (J)
The objectives of this optimization problem are to maximize performance and minimize cost.
1) Score – the total number of points the operator has earned for completing tasks during the mission. Score
is a measure of the operator’s performance. The higher the score, the better the performance of the
operator.
2) Cost – the total cost of the mission (in $).
3) Utilization – The percentage of time during which an operator is busy. Because overworked operators are
more susceptible to human error, it is important to minimize this metric.
C. Parameters (p)
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The parameters of this optimization problem are properties of the mission scenario itself that remained fixed
over the duration of the discrete event simulation, but can vary between simulations if the scenario changes.
1) TNH – total number of HALE UAVs available for the mission. HALE UAVs are high altitude, longendurance planes suitable for very large range, coarse surveillance. HALEs have the highest level of
automation, the best performance but are the most costly unmanned vehicles looked at in this problem.
2) TNM – total number of MALE UAVs available for the mission. MALE UAVs are medium altitude, longendurance planes suitable for large range, detailed surveillance. MALEs have a medium level of
automation, performance and cost.
3) TNU– total number of unmanned undersea vehicles (UUV) available for the mission. UUVs are the least
automated, lowest performing and least costly of the unmanned vehicles looked at in this problem.
4) Inter-arrival times of mission generated tasks. Inter-arrival times take into account the level of automation
of each vehicle, where the vehicles with the higher level of automation have shorter inter-arrival times.
5) Service times of mission generated tasks and re-plan events, per vehicle type. This is how long it takes the
operator to service an event, taken from a random distribution. Service times take into account the level of
automation of each vehicle, where the vehicles with higher levels of automation have shorter service times.
In order of highest-to-lowest automation (lowest-to-highest service times), the vehicles types are ranked
HALE, MALE and UUV.
6) Switching penalty– A penalty that increases service time of the next event due to the increased cognitive
workload of reorientation required to switch from operating one type of vehicle to another. If the next
vehicle being serviced is the same as the last, then the switching penalty is zero.
7) Cost of vehicle – the operating cost (in $) per vehicle of a specific type used in the mission. In order of
increasing cost, the vehicles types are ranked UUV, MALE and HALE. 4
8) Total operating cost – the total operating cost of the mission which equals the sum of all the costs of all the
vehicles used in the mission.
9) Maximum operating cost – the maximum allowable operating cost of the mission.
10) Points per task completed – the number of points awarded towards the score for servicing an event.
11) Mission time – the total time allotted for the mission (and simulation).
D. Design Variables (X)
Decision makers can vary the number and type of UVs allocated to a mission along with the operator strategies
used to command those vehicles for a given mission.
1) NH – actual number of HALE (High Altitude, Long Endurance) UAVs used in the mission,
2) NM – actual number of MALE (Medium Altitude, Long Endurance) UAVs used in the mission.
3) NU – actual number of UUVs used in the mission.
4) RS – the re-plan strategy which is the amount of re-planning an operator does during a mission. A re-plan
is defined as an operator generated event where the operator is choosing to re-route a vehicle’s path in
order for the vehicle to reach its destination sooner, thus increasing performance. RS is a continuous
variable measured in seconds, representing the time in-between re-planning events
5) SS – the switching strategy which is how the operator decides the order in which events are serviced from
the queue. SS is a discrete variable that can be FIFO (first in, first out), UAV First which gives priorities to
UAVs over UUVs or UUV First which gives priority to UUVs over UAVs.
In the rest of the paper, the design vector will be noted as follows: [NH NM NU RS SS]. Each vector element
maps directly to the design variable space with the exception of SS, the switching strategy, which was encoded
as a single digit: SS=1: UAV are prioritized, SS=2: UUV are prioritized, SS=3: FIFO.
E. Constraints (q, g)
Decision makers are constrained in their decision making by various operational, cognitive and physical
limitations.
1) Operator utilization (ρ) must be < 1. The operator utilization is the fraction of time (between 0 and 1) of
how much the operator is servicing any vehicle over the total time of the mission.
2)
.
4
Note that the prices are based on the current purchase prices of the systems: the HALEs are modeled after the
Global Hawk (about $32mi per unit), the MALE are modeled on the MQ-9 Reaper (about $8mi per unit), and the
UUVs are modeled according to the Office of Naval Research UUVs projects (estimated $3mi per unit).
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3)
4)
5)
6)
0≤ RS ≤ 100
0 ≤ WV ≤ TM. Mission. Average vehicle wait time is the average amount of time a vehicle is idle (waiting
for orders from the operator) during a mission.
F. Model Validation
The model was validated against experimental data obtained by having human subjects perform a human
supervisory control task of a multiple heterogeneous unmanned vehicles. The results generated by the model were
found to be, in most cases, within 10% of the experimental data (more details have been about the model
validation have been published in previous work1). Furthermore, the experimental data obtained also provided an
estimate of the different stochastic distribution parameters of the different event arrivals probabilities.
III. Single Objective Optimization
A. Gradient and Heuristic Based Single Objective Optimization
In order to get an overall sense of the impact of the different design variables on the objective function vector, we
conducted a coarse One Factor At a Time (OFAT) state space exploration. In particular, we wanted to see how the
heterogeneity of the vehicle would impact the results both in terms of score and utilization. We did not concentrate on cost
since it is a simple linear function of the number of unmanned vehicle. The OFAT allowed us to formulate the hypothesis that
the number of HALEs would have a large impact on performance.
We then focused on single objective optimization –in this case, score- was conducted in two different ways. The first
method was gradient-based and relied on the Matlab implementation of the Sequential Quadratic Programming (SQP)
algorithm. The second method was heuristics-based and relied on the Simulated Annealing paradigm. The results of the
optimization for each method are shown below in Figures 1 and 2. The optimal vector found with SQP was: [20 12 12 53.0 1]
for an optimal score of 159.39, and the one found through Simulated Annealing was: [19 5 1 88.06 1] for a score of 276.0.
The score obtained with the Simulated Annealing methods were about 70% better than that of SQP. This is explainable by
the fact that gradient based methods are getting stuck in local optima of the design space. Furthermore, because most of our
values are inherently discrete (one cannot have fractions of unmanned vehicles) or categorical (i.e. the switching strategy), it is
quite probable that the computations of the Hessian and Jacobian matrices by finite difference contain errors in the singular
regions of our design space. In contrast, the Simulated Annealing method is able to deal with such discontinuities and arrive at a
much better optimal score. In terms of computing power, the cost of getting better results was quite significant: where the SQP
runs in about 20 seconds, the Simulated Annealing would take about 220 seconds, about an order of magnitude increase in
computation time. As post-optimality validation, it is interesting to note that both gradient and heuristic based solution favor
HALEs both in numbers and switching strategy priority. This is intuitive since HALEs are, as a whole, better performing
systems than MALEs or UUVs. However, HALEs are significantly more expensive to operate than the other 2 types of vehicle,
but since cost was not an optimization target, this factor was not taken into account when computing the optimal solution. The
importance of HALEs was confirmed by performing a computational sensitivity analysis. We realized that the number of
HALEs employed was a significant driver of performance, with each HALE increasing the score by about 10 points. This result
also explains the large influence of the switching strategy: with a large number of HALEs, it is more beneficial to prioritize air
vehicle as opposed to underwater vehicle or having no priority scheme at all (FIFO queuing policy). In a situations where the
operator has to manage a large number of vehicles, prioritizing the UUVs over the UAVs leads to a 60% drop in score while
having a FIFO policy leads to a 40% decrease in score.
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Figure 2: SQP Optimization: Optimal Score=159.39
Figure 3: Simulated Annealing. Optimal Score = 276.0
IV. Multi-Objective Optimization
A. Pareto Front Determination through Weighted Sum
An estimation of the pareto front was generated with the Weighted Sum method. The weighted sum method
consists of attributing different weights to each objective function and repeatedly optimizing the resulting function.
With the weighted sum method, a function weights,  ’s, are chosen to sum up to 1 in order to maintain a correct
ratio. In our case, because we only analyzing 2 objective functions, we use the pair of scalars  and (1   ) as our
objective weights. We can then vary  from 0 to 1 to vary the importance of either score or cost. The scaling factors
sfscore and sfcost were chosen to so that both score and cost metric become equal at the optimal point as determined by
the single-objective optimization.
min J mo 
1 Score
sf score

2 Cost
sf cos t
2
, s.t.  i  1
i 1
By plotting cost against score we obtain a nearly linear pareto front as indicated by the green dots in Figure 5. It
appears that the pareto front could have a slight concavity, but this may also simply be due to the low number of
points that were assessed.
Figure 4: Yerkes-Dodson Cost/Score Curve
Figure 5: Experimental Cost/Score Curve
Cost is directly proportional to the number of vehicle used. A high cost necessarily implies a large number of
vehicles. Previous psychological studies have hypothesized that human performance followed an inverted U curve
with respect to the level of arousal2. In essence, a high level of arousal leads to bad performance due to cognitive
overload while low levels of arousal lead to complacency. With respect to our model, we posit that very low costs
(i.e. very low number of unmanned vehicle) and very high costs (i.e. large number of vehicle) both could lead to
poor score (see Figure 4). Experimental results confirmed this intuition and we can see that the red interpolated
curve on Figure 5 mirrors the generic expected behavior depicted in Figure 4. This result is explainable because the
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most autonomous UVs (HALEs) perform the best, generating the highest scores but also cost the most. Similarly
the least autonomous UVs (UUVs) perform the worst, generating the lowest scores but cost the least. However,
because a managing a high number of systems impose a high cognitive load on the operator, a number of vehicles
that exceeds the operators’ cognitive capacity will be detrimental to the score. Conversely, a low number of UV (and
therefore a low cost) will not allow an operator to process enough targets to get a good score. Since we are trying to
maximize the performance (score) and minimize cost, we are really trying to identify the location of the inflexion
point on the cost/score curve.
B. Full Factorial Design
Thus far, an understanding of the trade off between cost and score has been obtained, yet the pareto frontier that
was found is sparse and the methodology to obtain these data points was not ideal for our model simply due to the
integer and categorical variables in the design vector. In order to more accurately determine a pareto frontier, either
a heuristic based search needed to be used to supplement or replace the information from the gradient based
calculations, or a full factorial design space search had to be conducted. Through using heuristic based search
previously, it was discovered that Matlab could only call the simulation model a finite number of times before
crashing. Therefore, it was determined that implementing a weighted sum optimization with a heuristic search
would limit the number of design points that could be found. However, any number of points could be found in a
full factorial setting as these design points could be found in blocks to be added together.
Each of the five design variables was iterated over a number of times in loops to ensure at least one design point
was obtained for each of the design variable combinations. The number of iterations for each variable was limited
by the expected time to complete such an exhaustive search. Thus, the numbers of each type of vehicle were
iterated over the values: 1, 5, 10, 15 or 20. The time between replans, or replan strategy were iterated over one of
seven values ranging evenly from 1 second to 100 seconds (1, 17.5, 34, 50.5, 67, 83.5, and 100 seconds). Switching
strategy could be either one of the three categorical numbers representing a specific strategy. In total 3061 points
were collected. Using a pareto filter, 502 of these points were found to be non-dominated solutions and thus part of
the pareto frontier.
C. Design Space Characteristics
Many of the same trends may be seen in the results from the full-factorial exploration as the weighted sum and
gradient based optimization. The cost-score boundary is nearly a straight line as shown in Figure 5, while other
design space points are curved, indicating the human cognition boundary is being found as previously discussed in
the weighted sum optimization. In terms of utilization, most design points have a utilization greater than 0.7 as
shown in Figure 6. In addition, the boundary between score and utilization appears to be a straight line which is
intuitive given the simulation’s parameters. For example, a smaller number of vehicles is more likely to yield a
smaller utilization than a larger number of vehicles as the operator is not very busy, but the smaller number of
vehicles will also yield a smaller score as most of the potential targets cannot be reached. In contrast, a larger
number of vehicles will lead to higher utilization, but will also yield a higher score.
Figure 5. Full Factorial Cost and Score.
Figure 6. Full Factorial Score and Utilization.
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The same trends viewed in two dimensions may also be
seen in three dimensional space as shown in Figure 7.
Here all pareto frontier points have been highlighted with
green “+” symbols. The full design space is more or less a
plane which is twisted in score and utilization. This twist
increases as cost increases. This trend can be seen more
readily in Figure 8, which shows Utilization vs Score with
cost color coded from blue (low cost) to red (high cost).
Here as the cost increases there is a decrease in the
utilization range, but a larger score range. It appears as if
the system is maximized out as well when the highest costs
are achieved, as the score range begins to diminish. This
trend can be verified by inspecting Figure 3, where indeed
none of the greatest values of cost are on the pareto
frontier. This limitation likely shows that giving more
tasks to an operator after his cognitive capabilities are
reached will cause a degradation in performance.
Previously this trend could only be seen in two
dimensions, but here it can be seen with all three of the
objective functions.
By analyzing the objective range and averages in Table
1, it can be seen that the objective score retains the same
range, while both utilization and cost have a lower
maximum value. The non-dominated points yield a higher
average score, and lower average utilization and cost
compared to when all design points are considered. A
further disparity between averages and ranges would be
seen if the non-dominated design points were compared to
those points not in the frontier as opposed to all design
points regardless of if they are or are not dominated.
Figure 7. Full Factorial Design Space with Pareto Frontier.
Figure 8. Utilization vs Score with cost color gradation.
Average
Min
Max
Table 1: Full Factorial Objective Range and Average
Score ScoreUtilizationUtilizationfull
pareto
full
pareto
Cost-full
106.38
117.54
0.78
0.54
45298.73
4.89
4.89
0.10
0.10
4300
298.38
298.38
0.93
0.86
86000
Costpareto
29208.76
4300
68300
D. Design Trade Offs
Now that trends within the three objective functions have been analyzed it is important to determine how
changes within the design vector variables are affecting these trends, or in essence, what the design trade of the
system are. Analytical, visual and statistical investigations were made in order to determine these design trade offs.
Analytically comparing the ranges and averages of the non-dominated points to all points in the full factorial, as
shown in Table 2, yields some of the main trends for what variable values had a higher likelihood of being nondominated points. For example, for NU, or number of underwater vehicles, the average for the full set of design
variables is 10.66 with a range between 1 and 20 vehicles, whereas for the pareto frontier points the average is 1.90
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vehicles with a range between 1 and 10 vehicles. It can be easily seen that not only is the average number of
vehicles much less, but there are no non-dominated solutions that have more than 10 underwater vehicles. In a
similar manner, there is a smaller average for total number of unmanned vehicles as well as a smaller range. On the
other hand, there is little to no apparent difference when analyzing replan strategy time as well as switching strategy.
Table 2: Full Factorial Design Vector Variables
Average
Min
Max
Total
UV full
30.97
3
60
Total
UV pareto
16.28
3
31
NH full
10.66
1
20
NH pareto
7.14
1
20
NM full
10.66
1
20
NM pareto
7.24
1
20
NU full
10.66
1
20
NU pareto
1.90
1
10
RS full
47.92
1
100
RS pareto
54.05
1
100
SS full
1.86
1
3
SSpareto
1.85
1
3
Visually, the greatest trends with objective performance and design vector variables can be seen with the number
of vehicles and switching strategy. Looking at a two dimensional representation of cost versus score with a color
gradation on number of vehicles, an increase in the number of each type of vehicle yields a different result as shown
in Figure 9. For the number of HALEs (right), or those vehicles which are highly autonomous, an increase in the
number of vehicles generally results in obtaining a higher score. For the number of MALES (center), or those
vehicles which are moderately autonomous, an increase in the number of vehicles increases the score based upon the
number of HALEs which are being used. For the number of underwater vehicles (left), or those vehicles which are
not very autonomous, an increase in the number of vehicles tends to decrease the score.
Figure 9: Score vs Cost. Shown with color coding for number of HALES (left), number of MALES(center) and
number of underwater vehicles(right)
Further inspection of the relationship between the number of highly and moderately autonomous vehicles reveals
a microcosm of the Yerkes-Dodgson curve. As shown in Figure 6, at a higher number of HALEs, increasing the
number of MALEs no longer increases score, but instead begins to decrease the score. As the number of HALEs
increasing further, this turning point occurs earlier. For instance, looking at the curve of MALEs that correspond to
having 10 HALE vehicles (green), when more than 15 MALEs (yellow) are introduced than score begins to
decrease. However, when looking at the curve for 20 HALE vehicles (red), when more than 5 MALEs (light blue)
are introduced than score begins to decrease. Because the unmanned vehicles require much more attention than the
other two vehicles, this Yerkes-Dodgson microcosm can only be seen in those instances when the number of
unmanned vehicles equals 1. Any more unmanned vehicles causes the operator to reach his cognitive capacity much
faster, thus most design points containing more unmanned vehicles lie to the right hand side of the curve.
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Figure 10: Yerkes-Dodgson Microcosm. View of how increasing the number of HALEs causes the operator to
reach the peak in the Yerkes-Dodgson curve earlier when more MALE vehicles are added
Statistical analysis can give help generalize the design trade off further. Using the design points from the fullfactorial investigation, a 5x5x5 analysis of variance (ANOVA) was ran for each of the objective parameters using all
design points and only non-dominated design points to determine if changing the number of each type of vehicle
caused each objective to change. In all three tests, the number of all types of vehicles created a statically significant
difference for the three objective functions. All p values were less than 0.001 hence p and F values are omitted here.
A 7x3 ANOVA was then ran for score and utilization for all design points and only non-dominated design points.
Cost was omitted here as the replan strategy and switching strategy have no bearing on the cost of the mission as
cost is defined solely by which vehicles are being used. The summary of the findings are shown in Table 3. All
parameters except two combinations had p<0.05, therefore were statistically significant. When all design points
were analyzed it was found that the replan strategy was not statistically significant for score and when only the
pareto frontier design points were analyzed it was found that the switching strategy was not statically significant for
utilization.
Score
Utilization
F
p
F
p
Table 3: Full Factorial Design Vector Variables
Pareto Frontier Design
Points
All Design Points
RS
SS
RS
SS
3.644
6.793
0.132
8.19
0.002
0.001
0.992
<0.001
10.937
2.261
35.098
7.119
<0.001
0.105
<0.001
0.001
More interesting than the pure statistics are the estimated marginal means that illustrate how changes in the
design vector can affect the objectives. Inspecting all design points, Figure 7 shows that switching strategy 1,
priority given to the HALEs and MALEs over the underwater vehicles, increases the score significantly whereas the
other two strategies are essentially equivalent with no priority obtaining a slightly higher score than a priority given
to the underwater vehicles over the HALEs and MALEs. Inspecting the pareto frontier design points, Figure 8
shows that a replan switching time less than twenty seconds will have a much higher utilization than those above
twenty seconds, but that between twenty and 100 seconds does not have further effect on utilization. In addition, the
switching strategy with priority is given to the HALEs and MALEs over the underwater vehicles gives a slight
increase in utilization whereas the other two strategies are essentially equivalent.
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Figure 7: Marginal Means of Score
Figure 8: Marginal Means of Utilization
E. Optimal Design Choice
Through the design vector trade offs, a list of optimal design choices can be generalized.
1) Switching Strategy: Use a preemptive priority, giving vehicles with faster service times priority. Here
those vehicles were the HALEs and MALEs over the underwater vehicles
2) Replan Strategy: Do not service vehicles more than approximately every 20 seconds. Between 1 and 20
seconds a utilization increase will occur as will a potential decrease in score. Replanning between every 20
and 100 seconds will likely not have a major effect on either score or utilization, so these replans can occur
when desired within that time frame
3) Number of HALEs: Increasing the number of HALEs will increase score, utilization and cost. If there is
enough money, than HALEs should be chosen for missions over any other vehicle as they have faster
average service and flight times.
4) Number of MALEs: Increasing the number of MALEs will increase score, utilization and cost up to a
certain point depending on the number of other vehicles being used. MALEs are the next best vehicle to
use if HALEs cannot be purchased.
5) Number of underwater vehicles: Increasing the number of underwater vehicles will not likely increase
score, but will increase utilization and cost so they should be considered a last option after using MALEs
and HALEs.
V. Lessons Learned
Through developing and running our optimization we learned many things regarding human-machine interaction
in an unmanned heterogeneous mission, but also learned many things about optimization itself and the
methodologies behind multi-disciplinary optimization.
A. Matlab – Java Call Limitations
The main obstacle that we faced while completing our optimization was calling the simulation code which was
written in java. This code was bundled as a .jar file and had to be called each time the function was analyzed.
Especially for the Simulated Annealing and full factorial analysis this resulted in many calls to the java file.
However, depending upon the version of Matlab, and the java platform installed, the number of this runs was limited
before the code would crash. Usually the crash would be an out of memory error, but other more cryptic errors
occurred as well. For this reason, the number of iterations for the Simulated Annealing had to be limited. In order
to get around this problem in the full factorial analysis, blocks of the design space were analyzed and added to a
Matlab matrix. Whenever the program crashed, the matrix was saved and a new program started where the last
ended. This led to some design points being visited more than once, but allowed a larger number of design points to
be analyzed. On average the program would crash after a thousand calls to the simulation were made.
B. Interval and Categorical Data
From the beginning of the project it was understood that there would be some difficulty due to the fact that only
one of the five design vector factors was a continuous variable. The number of each type of vehicles was an interval
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value which could use a continuous approximation, but the switching strategy is a discrete value likely making it the
largest hindrance to successfully completing the gradient based search. This problem was bypassed by relying
primarily on heuristic algorithms and the full factorial search. Heuristic algorithms are able to bypass this problem
by constraining variables. Likewise, the full factorial search only allowed the discrete and categorical variables to
be whole numbers.
C. Limitations to Heuristic and Gradient Based Search
With a single objective search, the gradient based algorithm was able to obtain a maximum score of 170. Due to
the interval and categorical data, it was assumed that this was not the greatest score that could be achieved. Using a
heuristic method, simulated annealing, a maximum score of 276 was found. More confidence was placed in this
answer and it was assumed to be close to the maximum obtainable score. However, in the full factorial design a
maximum score of 298 was found. This corresponds to a design vector of 20 HALEs, 5 MALEs, 1 underwater
vehicle, a replan switching time of 100 seconds and switching strategy of 1. The heuristic search may not have
found this design vector as it is at the upper boundaries of many of the design variables.
D. Stochastic Simulation
Finally, it is important to note the difficulty of finding an optimal solution to a problem which is backed by a
stochastic simulation. Even though the simulation module ran 1,000 times for each design point analyzed, there was
some modulation in the final objective functions that were obtained for a given design vector. The simulation
imitates the variance a normal population would have while completing this task, and thus not always the same
solution will be found every time. This is a key element of the simulation, but both the heuristic and gradient based
optimization required information regarding the other points sampled in the design space to determine which
direction to best search in. Because some points were analyzed multiple times in the full factorial analysis these
differences can be seen. For example the same design vector gives all five objective solutions in Table 3. Cost here
is constant as the cost is correlated to the number of each vehicle in the design vector. Score varies by
approximately 0.6, and utilization by 0.0012, so neither varies greatly but it is important to mention that it is possible
that the optimization algorithms used could have been influenced by these variations.
Table 4: Objective Variation from a Single Design Vector
Score
Utilization Cost
179.0729
0.7454
49100
179.0916
0.7442
49100
179.2120
0.7449
49100
179.4553
0.7449
49100
179.6399
0.7452
49100
VI. Conclusion
In conclusion, various optimization techniques can be used to optimize the output of a human-system unmanned
vehicle simulation with varying degrees of success. Gradient based algorithms lack the capacity to handle discrete
and interval design variables, thus are unsuitable to be used with this optimization problem. Heuristic algorithms
have the capacity to handle the necessary design vector, but due to the stochastic nature of the simulation, or the
design vector constraints, these algorithms may not be robust enough to find the global maxima and minima. Rough
approximations of the design space and pareto frontier can be found with either a gradient based method, or through
heuristic algorithms, however, a near-optimal pareto frontier of the simulation’s three objective functions can be
found through full factorial exploration. Through this design space exploration, design trade offs can be identified.
Further, an average human’s cognitive capacity can be predicted by determining where these design trade offs occur.
With this information, future missions and systems can be built to maximize overall performance for short duration
missions. For longer mission, additional variables such as operator fatigue will also have to be considered.
Therefore, based upon a certain set of parameters, the human-system interaction can be optimized in the context of
heterogeneous unmanned vehicle control.
VII. References
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1. Nehme, C., Crandall, J. W. & Cummings, M. L. in Modeling, Analysis and Simulation Center Capstone
Conference (Norfolk, VA, 2008).
2. Yerkes, R. M. & Dodson, J. D. The Relation of Strength of Stimulus to Rapidity of Habit-Formation. Journal of
Comparative Neurology and Psychology 18, 459-482 (1908).
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