Design of Mini ROV
ME 6105: Modeling and Simulation in Design
HW5: Preference Modeling and Optimization
Dazhong Wu
Qing Chen
Dec. 3, 2009
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Contents
Task 1: Revisit the Decision Situation identified in HW1 .......................................................................... 3
Task 2: Elicit Preferences -- Utility Function .............................................................................................. 5
Task 3: Explore the Design Space ............................................................................................................... 9
Task 4: Solve the Design Problem Deterministically ................................................................................ 12
Task 5: Solve the Design Problem under Uncertainty .............................................................................. 13
Task 6: Lessons learned ............................................................................................................................. 18
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Task 1: Revisit the Decision Situation identified in HW1
As we go from HW1 throughout to HW4, the modeling and simulation of ROV has changed
considerably. We need to clarify all modifications we have made in the goal and scope of this project.
We revisit our original influence diagram as shown in Fig. 1.
Fig. 1 Original influence diagram
Based on what we have learned in this course so far, we have a better understanding of our system.
The updated influence diagram is shown in Fig.2.
Fig. 2 Updated influence diagram
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The specific changes and reasons behind these changes are described below, which is discussed
regarding decision blocks, chance event, outcomes of calculation and overall utility:
Changes in decisions


Remove material and dimension-we do not consider material and dimension because we do not
consider them as design variables. The material is steel. The weight is constant, which is 30 kg.
Therefore, we remove them from design variables.
Add propeller diameter and front area of ROV-In HW2, according to main effect analysis based
on central composite experiments in Model Center, both propeller diameter and front area have
influence on top speed and power consumed. Since our objectives are to maximize top speed and
minimize power consumed, we add propeller diameter and front area of ROV as design variables.
Changes in chance events

Remove load and water flow chance events-as we have a better understanding of our system, we

find out that strictly speaking, load is not a direct chance event. It can be determined by water
resistance. Water resistance can be determined by front area of ROV and resistance coefficient.
In fact, the resistance coefficient is a direct chance event. In this case, water flow is a redundant
chance event since it is determined by resistance coefficient.
Add thrust coefficient instead of propeller characteristic-after reviewing the literature regarding

thruster force, we find out thruster force can be determined by a formula, in which thrust
coefficient is the direct chance event.
Remove utility rate-since we do not consider cost in this project, we remove this chance event.
Changes in outcomes of calculations




Remove outcomes of calculations associated with cost-since we do not consider cost, we remove
all outcomes of calculations associated with cost, i.e. operating cost, initial cost, operating time,
weight and moment of inertia.
Remove motor speed and motor torque-motor speed and torque are determined by load. We have
already consider load so motor speed and torque are redundant. Therefore, we remove them.
Remove performance and acceleration-we simply ROV performance as its top speed and do not
consider acceleration.
Add power consumption -since one of our objectives is minimize power consumption, we need
to add power consumption as outcomes of calculation. Initially, our objective is to minimize
energy consumption. However, we assume that ROV will accelerate until its top speed and keep
this top speed. Therefore, it is reasonable for us to minimize power consumption instead of
minimizing energy consumption.
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Changes in utility

Reduce original three attributes into two-originally, we have three attributes which are cost, top
speed and acceleration. Since we do not consider cost, we have only two attributes which are top
speed and batter power.
In summary, we formulate our updated design problem as follows:
 Scenario: we are designing a ROV
 Objectives
1) Maximize top speed-attribute=speed: m/s
2) Minimize power consumption -attribute=power: watt

Design variables:
1) Battery voltage: Voltage in [V]
2) Propeller diameter: Diameter in [m]
3) Front area of ROV: Area in [m2]
Task 2: Elicit Preferences -- Utility Function
In order to express our preference with respect to a given design alternative, we need to develop
a utility function that captures the tradeoffs that exist for the attributes of the design alternative. As
discussed in class, go through the elicitation process to determine our utility function. The overall
preference elicitation process is the following:
1.
2.
3.
4.
Verify mutual utility independence
Assess the conditional utility for Y given a value of Z
Assess the conditional utility for Z given a value of Y
Construct the multi-linear utility function
Single-attribute conditional utility elicitations (Top Speed):
Firstly, we need to fit the individual utility function for each attributes. To document the elicitation
process, the elicitation questions for top speed that we asked ourselves are the following:
1.
2.
Determine the best (utility=1) and worst (utility=0) case for top speed. The answer is 1.5 m/s
(utility=1) and 0.3 m/s (utility=0)
50/50 gamble: when are you indifferent between a guaranteed given case or taking a 50/50
between utility=1 case and utility=0 case? Assign this value as utility of 0.5. The answer is 1 m/s
(utility=0.5)
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3.
4.
5.
6.
50/50 gamble: when are you indifferent between a guaranteed given case or taking a 50/50
between utility=0.5 case and utility=0 case? Assign this value as utility of 0.25. The answer is
0.8 m/s (utility=0.25)
50/50 gamble: when are you indifferent between a guaranteed given case or taking a 50/50
between utility=0.5 case and utility=1 case? Assign this value as utility of 0.75. The answer is
1.125 m/s (utility=0.75)
50/50 gamble: when are you indifferent between a guaranteed given case or taking a 50/50
between utility=0.75 case and utility=1 case? Assign this value as utility of 0.875. The answer is
1.225 m/s (utility=0.875)
50/50 gamble: when are you indifferent between a guaranteed given case or taking a 50/50
between utility=0.25 case and utility=0 case? Assign this value as utility of 0.125. The answer is
0.65 m/s (utility=0.125)
The results are summarized in Table 1. Use cubic splines to approximate utility as shown in Fig. 3.
Table 1 Elicited utility for speed
Top Speed
(m/s)
0.3
0.65
0.8
1
1.125
1.225
1.5
Utility
0
0.125
0.25
0.5
0.75
0.875
1
Fig.3 Monotonic utility function for speed
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Interpretation of utility function for top speed:
As shown in Fig. 3, we are risk seeking when speed falls between 0.3 m/s and 0.8 m/s. And then we
tend to nearly risk neutral between 0.8 m/s and 1.125 m/s. At the end, we become risk averse when
speed falls between 1.125 m/s and 1.5 m/s. It makes sense that we are risk seeking when the top
speed is too low between 0.3 m/s and 0.8 m/s. When top speed transitions into the range between 0.8
m/s and 1.125 m/s, we become slightly satisfied. So we are nearly risk neutral. After that, we become
risk averse between 1.125 m/s and 1.5 m/s because we are already slightly satisfied with top speed
between 0.8 m/s and 1.125 m/s, and additional increase in top speed does not increase the utility too
much.
Single-attribute conditional utility elicitations (Power consumption):
We follow the same elicitation process for ROV power consumption as described before. The results
are summarized in Table 2. Use cubic splines to approximate utility as shown in Fig. 4.
Table 2 Elicited utility for ROV power consumption
Power (W)
0
250
500
625
750
1000
1250
1750
2000
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Utility
1
1
0.95
0.75
0.5
0.1
0.02
0
0
Fig.4 Monotonic utility function for power consumption
Interpretation of utility function for power consumption:
As shown in Fig. 4, we are risk seeking when power consumption falls between 1000 w and 2000 w.
We tend to slightly risk neutral between 500 w and 1000 w. After that, we become risk averse when
power consumption falls between 250w and 500w. It makes sense that we are risk seeking when the
power consumption is too high which is between 1000 w and 2000 w. When power consumption
transitions into the range between 500 w and 1000 w, we become slightly satisfied. So we are
slightly risk neural. After that, we become risk averse between 250w and 500w because we are
slightly satisfied with range between 500w and 1000w, and additional reduce in power consumption
does not increase the utility too much.
MAUT elicitations:
1.
2.
3.
Where are you indifferent with the reference point with top speed utility=0.5 and power
consumption utility=0.5? The answer is top speed utility=0.3 and power consumption
utility=0.75
Where are you indifferent with the reference point with top speed utility=0.25 and power
consumption utility=0.75? The answer is top speed utility=0.75 and power consumption
utility=0.25
Where are you indifferent with the reference point with top speed utility=0.75 and power
consumption utility=0.25? The answer is top speed utility=0.25 and power consumption
utility=0.75
The results are summarized in Table 3.
Table 3 Elicitation points for total utility
Question
1
2
3
Elicitation Point
speed
power
Value
Utility
Value
Utility
0.3
0.75
0.75
0.25
0.25
0.75
Reference Point
speed
power
Value
Utility
Value
Utility
0.5
0.5
0.25
0.75
0.75
0.25
Table 4 Least squares solution for multi-linear utility function
Coefficient
K1
K2
K12
value
0.25
0.25
0.5
Interpretation of the coefficients in multi-attribute utility function:
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Utility
Residue
-1.8E-16
-5.6E-17
5.55E-17
Based on the coefficient result K12=0.5>0, it indicates that top speed and power consumption are
complements: our preference for top speed increases as power consumption increases and vice versa.
It makes sense that as top speed increases, my preference for power consumption becomes stronger.
This result pose a tradeoff existed in our objectives as follows:
1. Maximize top speed-attribute=speed: m/s
2. Minimize power consumption -attribute=power: w
Task 3: Explore the Design Space
Design Variables
Uncertain Variables
Propeller Diameter (m^2)
Vehicle front area (m^2)
Battery Voltage (V)
Water resistant coefficient
Propeller coefficient
Objective Attributes
Vehicle Velocity (m/s)
Power consumption (W)
The figure below is the screen-shot of our ModelCenter model, it include one dymola model, 2 utility
functions as excel spreadsheet and one total utility calculating script and one optimizer.
The total utility script is as follows:
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To explore the design space, we did a full factorial exploration by DOE.
Variable
Low
high
Battery voltage (V)
12
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Propeller diameter (m)
0.1
0.2
ROV front area (m^2)
0.4
0.6
It should be noted that front area always converges to the low boundary. The reason is that for both
objectives (maximize speed and minimize power), small front area is always favorable, there’s no
tradeoffs for this variable. So we always set this variable to a fixed minimal value 0.4m^2 in the
DOE simulation plotting.
We set the ROV front area to a constant value of 0.4m^2. The response examined is the total utility.
Figures below shows that there is an optimal point at around (voltage=24V, diameter=0.155). There’s
no other local optimal point except for this one. This point shows the tradeoff between maximizing
speed and minimizing power.
This point (voltage=24V, diameter=0.155) will be our starting point in the coming optimization
iteration.
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The DOE also shows that the largest influence on the total utility is propeller diameter, which is
reasonable since the thruster torque is proportional to D^5 and the thruster force is proportional to
D^4. Increasing D is good for increasing vehicle velocity. But as D increases, the motor torque load
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will also increase exponentially, which will cause current, and power to increase significantly, which
will cause bad power utility. So the selecting of propeller diameter is a tradeoff between speed utility
and power utility. Front area determines the vehicle resistance, thus the speed, we hope that it be as
small as possible. Battery voltage also affects the total utility. Rotating speed of thruster is
proportional to the battery voltage, it determines the output force thus the vehicle speed.
Task 4: Solve the Design Problem Deterministically
First Iteration:
Figure below is the searching range and starting points of the 3 design variables for optimization tool.
Propeller diameter
Battery voltage
Front area
Total utility
Initial value
0.155
24
0.4
0.8374
Optimized value
0.1604
24.44
0.4
0.87332
Second Iteration:
The starting point is the optimal point determined in the first iteration. For the second iteration we
narrowed the searching range for the variables so we could get more precise result.
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Propeller diameter
Battery voltage
Front area
Total utility
Initial value
0.1604
24.44
0.4
0.87332
Optimized value
0.1606
24.43
0.4
0.87349
The table above shows that the utility value for the result of the 2 iterations are almost the same, we
regard this point (propeller diameter= 0.1606, battery voltage = 24.43) as the optimal point for
deterministic optimization, the maximized utility at this point is 0.87349.
Task 5: Solve the Design Problem under Uncertainty
For Monte Carlo simulation, we first elicited 2 CDF curve for uncertain variables, for water
resistance coefficient and thruster coefficient.
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Water resistance coefficient CDF
Propeller thrust coefficient
Mean value of water resistance coefficient is 0.2, and varying between 0.1 to 0.3 by roughly normal
distribution, the mean value of propeller thrust coefficient is 12, varying between 9 to 15.
Below is the screen snapshot of the uncertainty model. This model includes a LHS model which
could generate random samples for uncertain CDF spreadsheet to generate uncertain variables.
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First simulation:
The first simulation includes 5 LHS trials. The range of design variables are the same as before as
listed below, it can also be seen that the convergent point of this first time LHS simulation is
(voltage=30.44, front area=0.4, propeller diameter = 0.135). It should be noted that the front area
always converges to the lowest value of the range because of the reason explained before.
The figure below is the number of runs.
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The optimal point and the max utility and standard deviation is listed below
value
Voltage (V)
30.44
Front area (m^2)
0.4
Propeller diameter (m)
0.135
Average Total Utility
0.8364
Standard Deviation
0.09427
Second simulation:
The 2nd simulation includes 10 LHS trials. The start point is the optimal point determined in the first
simulation. The table below show that the optimal point found in the 2nd simulation is the same with
the 1st simulation. But the total utility is a little small than the former one.
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1st simulation
2nd simulation
Deterministic optimal
Voltage
30.44
30.4
24.43
Front area
0.4
0.4
0.4
Propeller diameter
0.135
0.135
0.1606
Total utility average
0.8364
0.8235
0.87349
Standard deviation
0.09427
0.0856
Comparison between uncertainty two simulations with deterministic simulation results
It can be seen from the above table that there’s no significant difference between the 2 simulations
with uncertain factors; but there’s big difference between the result of deterministic simulation and
that of uncertain simulation. This is mainly caused by the water resistance uncertainty. This variable
has big deviation (value average 0.2, range from 0.1 to 0.3 roughly normal distribution) which
resulted in the big difference in optimal point.
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Task 6: Lessons learned
Qing Chen:
1. When using the spline utility spreadsheet, I was struggling with the problem that when the
input is out of the range, the probability that looked up from the table became wrong. Later
when I defined extra points outside of CDF curve, this problem solved
2. I was struggling with the connection of LHS module with utility script and optimizer. After many
trials I found that the utility script should output the calculated utility value to LHS to calculate
the average utility value, then the LHS feedback this value to the optimizer to evaluate the
optimization.
3. I was struggling with modifying the dymola model, find the appropriate parameters and
importing the modified Dymola model to the Modelcenter.
Dazhong Wu:
1. When I go through this assignment, elicitation of preferences is the difficulty I struggled. When
we originally have three attribute, we always cannot find the right coefficient. We reduce three
attributes to two resulting in a reasonable preference elicitation.
2. I also learned how to formulate entire design problem based on what we have learned in class.
In HW1, the fundamental objective is too vague to model because at that time I did not know
what the dymola can do and how Model Center can be used as a powerful optimization tool for
solving design problem.
3. I also learned how to solve design problem in two ways: one is in deterministic way, the other
is considering uncertainty by maximizing expected utility. The most important thing I learned is
how to do these ways in Model Center.
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