preference modeling and optimization€¦ · our model is a suspension system utilizing a...

Georgia Institute of Technology

Spring 2008

Preference Modeling and Optimization ME6105 Homework 5 Jonathan Bankston, Yavuz Mentes, Alex Ruderman

Objectives

To develop a deeper understanding of decision making in design

To gain experience modeling preferences in a utility function

To explore a design space and model an optimization problem

To solve design problems computationally and gain a deeper understanding of the computational trade offs.

To develop a deeper understanding of how uncertainty plays a role in design

Background

In HW2, you have identified a particular system for which you would like to solve a design problem. You modeled a

physics-based relationship between design alternatives and design objectives in HW3, and explored the influence of

uncertainty on the system performance in HW4.

You are now ready to solve the entire design problem formulated in HW1. In this final assignment, you will use your

Modelica model and combine it with models for uncertainty and preferences to formulate a complete decision problem.

You will do so in two ways: first, deterministically, to establish a base-line solution, and secondly, considering

uncertainty, i.e., solving the same problem by maximizing the expected utility.

Both the deterministic decision problem and the decision problem under uncertainty will be modeled and solved in the

software ModelCenter by Phoenix Integration.

Tasks

You are asked to write a report on your simulation-based design study. I suggest that you address each of the topics below

in a separate section of your report. Each section should describe your solution to that particular part of the design study,

list the important results (if appropriate), and provide an interpretation of the results.

Important Note: I’m not looking for a 200 page novel... Also, do not hesitate to include screenshots -- a picture is worth

a thousand words. I should be able to grade your assignment based solely on the information in your report without

having to open/run any of your models.

http://www.phoenix-int.com/

Task 1: Revisit the Decision Situation identified in HW1

Based on what you have learned in this course so far and now that you have a better understanding of the complexity

of your system, briefly revisit the goals and scope of your simulation-based design study.

Include an updated influence diagram that specifically lists the design variables (parameterization of the design

alternatives) and the measures of effectiveness that you will be using in this assignment.

Interpretation: Provide an assessment of what you proposed in HW2 and justify any changes.

The problem we have devised for the semester is slightly different from that described in Homework 2. One

change is the removal of the fluid damper. This is because a traditional damper connected in parallel will

reduce the power output of the system, making it less efficient. We also made some changes to our means and

fundamental objectives hierarchy. The new hierarchies are shown below.

Figure 1.1: Fundamental Objective Hierarchy for Automobile Suspension

Maximize Customer

Satisfaction

Maximize energy recovered

Maximize Safety

Maximize Roll Stability

Maximize Reliability

Maximize Ease of Installation

Minimize Overall Size of Geometry

Maximize Ride Comfort

Minimize Cost

Figure 1.2: Means-Objectives Network for Automobile Suspension

Using the relations from the modified objectives networks above, we made modifications to our original

influence diagram, which is shown below.

Minimize Damping Response Time

Minimize Settling Time

Maximize Power Capability of

LMES

Maximize Energy Recovered

Maximize Ride Comfort

Maximize Customer Satisfaction

Maximize Safety

Maximize Roll Stability

Larger Stator Coil

Thicker Wire More Turns of

the Wire

Maximize number of magnets

Minimize Vertical Acceleration

Minimize Cost

Minimize Cost of PGSA

Figure 1.3: Influence Diagram for Suspension Design

Strength of Magnets

Cross-Sectional Area of wire

# of turns of stator coil wire

Speed of Car

Spring Stiffness

PGSA Capability

Force from Road

Spring Constant

Force to PGSA

Road Condition

Force to Spring

Force to Chassis

Energy Generated

Cost

Comfortable Ride

Vertical Acceleration of Chassis

Mass of Car

Our model is a suspension system utilizing a power-generating shock absorber (PGSA) that we introduced in

Homework 2. The system consists of a spring to counter any vertical motion of the car and the PGSA to soften

the recoil of the spring. The PGSA uses electromagnetic induction to dampen the amount of recoil instead of a

traditional fluid-based damper which uses the compression of a fluid. The PGSA converts kinetic energy from

suspension travel into electrical energy that supplements the electrical system of the car.

The original scenario for this energy-based simulation was the simple vertical motion of the suspension system

installed in a car. The simulations included a displacement representing a single and double bump. Our

original model is shown below in Figure 1.4.

Figure 1.4: Original dymola model of car Suspension with PGSA.

After revisiting this model, we realized a couple flaws. The model we created represents the entire mass of the

car acting on one suspension system. Furthermore, there is no gravity acting on car’s mass. In addition after

reading comments from Chris, we decided that a tire should be included to better represent the system and only

the single bump simulation is necessary for our purposes. With these changes, our models become the figures

below. Figure 1.5 is our new model of the suspension system, and Figure 1.6 is the model of the car attached

four suspension systems and four tires excited at the same time by the same size bump.

Figure 1.5: Existing Dymola Model of the Suspension System

Figure 1.6: Dymola model of the entire car system.

Our objectives are to maximize the ride comfort, maximize electrical energy recovered, and minimize the

overall cost of the system. The ride comfort is characterized by the vertical acceleration of the car where lower

acceleration means a more comfortable ride. The recovered energy from the PGSA is the amount of energy

contributed to the electrical system of the car. We measured AC power from the terminals of the PGSA. The

cost is a relative measurement based on researched costs of the components of the PGSA (i.e. the length of the

copper wire, the cross-sectional area of the copper wire, and the magnetic field for the magnets). These three

objectives are obviously dependent upon one another. Therefore, the attributes were adjusted to find the

optimal system.

Task 2: Elicit Preferences -- Your Utility Function

In order to express your preference with respect to a given design alternative, you 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 your utility function. This process becomes quite tedious for more than 4 attributes --

limit yourself to 4.

To document the elicitation process, briefly describe in your report the elicitation questions that you asked yourself,

the fitting of the individual utility functions, the computation of the coefficients in the multi-linear utility function.

Briefly describe how the coefficients in the multi-linear utility function are computed from elicitation questions. In

other words, based on your example, explain the computations that take place in the MAUT spreadsheet.

Interpretation: Provide an interpretation of each of the single-attribute utility functions: Are you "risk averse," "risk

seeking" or both? Explain why? Also provide an interpretation of the coefficients in your multi-attribute utility

function.

As described above, the attributes of our system are the vertical acceleration of the car (ride comfort), the

energy in Joules generated by the PGSA, and the cost in dollars of the PGSA. In order to come up with the

individual utility functions for each of these attributes we used the spline spreadsheet provided by Chris. The

elicitation question that we chose to ask ourselves is shown below.

“At what X would you be indifferent to accepting this X value

or taking a 50/50 chance between X(low) and X(high)?”

In the above elicitation question, X(low) represents the changing value of our attribute corresponding to a

utility value of zero and X(high) represents the changing value corresponding to a utility of 1. The values of X

correspond to the utility value of 0.5 for the different X(low) and X(high) values. In order to change the values

of X(low) and X(high), we made the indifferent X value equal to either X(low) for utilities 0.5-1 and X(high)

for utilities 0-0.5.

Figure 2.1: Monotonic Utility Function for the Total Cost of the PGSA

The curve resulting from the elicitation of our preferences for the total cost of PGSA is Figure 2.1. It was

plotted using the provided Excel spline template. This curve is the monotonic utility function for the total cost.

From the figure, it is evident that we are risk seeking throughout the range of PGSA costs. This means that we

are willing to spend more on the PGSA for increases in performance. It is also notable that the PGSA cost

curve is much smoother than the other two elicitations. We assume that this results from more experience

imagining monetary costs than acceleration or energy generation.

The monotonic utility function for vertical acceleration was elicited using the same question above. This

acceleration acts on the car body and is due to excitation of the lower suspension arms and the weight of the

car. This function was plotted using the same Excel template and presented in Figure 2.2. The general trend of

our preferences towards vertical acceleration appears to be risk-neutral. More specifically, we are slightly risk-

averse in the middle of our range but more risk-seeking toward the end points. This makes sense considering

changes in acceleration near the outer edges do not appear as drastic as changes in acceleration in the mid-

range.

Figure 2.3 is a plot of the monotonic utility function for the energy generated by the movement of the PGSA.

We elicited our preferences for energy generation using the question posed earlier. This elicitation follows the

same pattern as our elicited preferences for the vertical acceleration. Again, we felt more susceptible to change

in the midrange, making us more risk-averse, while we were risk-seeking toward the outer edges of the range.

Figure 2.2: Monotonic Utility Function for the Vertical Acceleration (Comfort) of the Car

Figure 2.3: Monotonic Utility Function for the Energy Generated by the PGSA

We then used the MAUT spreadsheet to generate our multi-linear utility function. To determine the minimum

and maximum utility values for each of the attributes we used the Latin-Hypercube Sampling results from

Homework 4. Using these results helped us estimate acceptable ranges of values for these attributes. Since the

values in the LHS were derived from our input variables that we are confident with, these minimum and

maximum values represent a satisfying range that should not restrict any useful data. We want to use a wide

range of values in order to better gauge the possibilities of our system. From the notes on our graded

Homework 4, we realized that the upper bound of our magnetic field strength distribution values was too large.

We performed the LHS again with a more reasonable range of 0.5 to 2 Tesla for the magnetic field. The

results from the LHS and the elicitations were combined to come up with the multi-attribute utility function.

These process and the results are shown below.

Table 2.1: MAUT Spreadsheet Elicitation

Table 2.1 shows the elicitation section of the MAUT spreadsheet (for three variables) where we elicited our

preferences to calculate the coefficients of our multi-linear utility function. The spreadsheet uses reference

points and elicitation points to determine relationships between the utilities of all three attributes. Our first

step was to develop the list of reference points. A reference point is described by the set of values for each

attribute at that point as well as the corresponding utilities for those values. We retrieved the utility and value

combinations from the monotonic utility functions.

Then, we developed the corresponding list of elicitation points. For each question, we defined two of the

attributes by selecting utilities that were sufficiently far from the reference point. We then found values for

those utilities using the monotonic utility functions. The third attribute was defined by answering the elicitation

question:

“Keeping in mind the range of each attribute and the available

PGSA given by the reference point, what value must the

attribute in question be for you to accept this design given

the predefined values of the other attributes?”

As shown in Table 2.1, we began by repeating the same reference point for sets of three elicitation questions.

This way a different attribute was elicited for each reference point using varied predefined values. If the utility residue or inv(ATA) intermediate results were unacceptable, we adjusted these utilities and elicited the attribute

again.

Elicitation

Elicitation Point Reference Point

Utility

Residue

Attribute #1 Attribute #2 Attribute #3 Attribute #1 Attribute #2 Attribute #3

Question Value Utility Value Utility Value Utility Value Utility Value Utility Value Utility

1 1.75 0.57 1300 0.75 29.7 0.2 2 0.5 850 0.5 26.5 0.5 0.01738

2 0.25 0.875 400 0.017 24.5 0.875 2 0.5 850 0.5 26.5 0.5 0.01589

3 3.25 0.125 1475 0.875 29 0.25 2 0.5 850 0.5 26.5 0.5 -0.01107

4 2.5 0.33 1173 0.675 24.5 0.875 3.25 0.125 1475 0.875 26.5 0.5 0.0461

5 2.75 0.25 1200 0.69 25 0.75 3.25 0.125 1475 0.875 26.5 0.5 0.00903

6 2 0.5 355 0 24.5 0.875 3.25 0.125 1475 0.875 26.5 0.5 -0.02367

7 3.125 0.16 1550 1 34 0 1 0.75 500 0.125 31 0.125 0.01844

8 0.25 0.875 1475 0.875 33.5 0.02 1 0.75 500 0.125 31 0.125 -0.02174

9 1.5 0.64 550 0.25 25 0.75 2.75 0.25 1550 1 24 1 -0.01903

10 2 0.5 900 0.53 29 0.25 2.75 0.25 355 0 24 1 -0.00918

11 0.25 0.875 1300 0.75 30 0.18 2.75 0.25 1550 1 24 1 0.02497

On the spreadsheet, elicitation and reference points represent two design alternatives with equal utility.

Formulas in the spreadsheet are used to exploit this quality to develop a system of equations to describe the

constant utility contours of the multi-linear utility function. Excel functions are then used to solve this system,

producing the multi-attribute utility function coefficients shown in Table 2.2.

Table 2.2: MAUT Spreadsheet Utility Function Coefficients

kVA 0.862892

kGE 0.67316

kC 0.575632

kVA,GE -0.95555

kVA,C -0.84297

kGE,C -0.71852

kVA,GE,C 1.40536

Analysis of the coefficients provides information about the interaction among the design variables.

Specifically, the coefficients representing multiple attributes are important. For example, imagine a coefficient

kXY, developed from random attributes X and Y. If kXY is positive, then the attributes X and Y are

complimentary so our preference for one will increase as the other attribute increases. If kXY is negative, then

the attributes are substitutive, which means our preferences for each change in an inverse fashion. Finally, in

the case of kXY = 0, the attributes do not depend on one another and are therefore considered additive.

From Table 2.2, the values for kVA,GE, kVA,C, and kGE,C are all negative, so it is evident that all of our attributes

are substitutes. This means our attributes are alternates to one another. Our preferences allow for the vertical

acceleration to increase when electrical energy output increases or when the PGSA cost decrease.

Additionally, our preferences accept a lower energy generation for reduced cost and vice versa.

For our PGSA, the multi-attribute utility function was given the coefficient labels in Equation 1. The

coefficients calculated from the preference elicitations in the MAUT spreadsheet were substituted, resulting in

Equation 2.

(1)

(2)

Task 3: Explore the Design Space

Within ModelCenter, implement the entire utility function by combining your Modelica model with "script" models or

Excel spreadsheets. The script models can be used to model measures of effectiveness that cannot be derived from

your Modelica models (e.g., cost models or simple algebraic performance models). The scripts also include the

mappings from individual utility functions into a multi-attribute utility function. The mappings from measures of

effectiveness into single-attribute utility functions is best implemented using the spreadsheets provided.

Explore this design space by using the appropriate exploration tools (e.g., a space-filling DOE) -- the goal is to

narrow down the design space that you will consider for optimization, e.g., to avoid getting stuck in local minima.

These parameter studies will also serve as a validation of your behavior models and utility functions (e.g., it is not

uncommon that a bug in your model surfaces when using parameter values that are different from the ones you

started with when building the model).

For instance, you may do a full factorial exploration of your design space; among the alternatives evaluated in this

full factorial design, find the design alternative that gives you the largest (deterministic) utility. This point could

serve as a starting point for further optimization.

It is also important to develop an idea of how many local minima your utility surface has, whether it is smooth, etc. so

that you can decide on the appropriate optimization algorithm in the next step.

Interpretation: Describe briefly the design exploration steps you performed. What did you learn about your design

problem?

We created a ModelCenter model to explore the design space of our system based on our design variables. As

we show in Figure 3.1, the exploration model consists of two different assemblies: a behavioral model and a

utility model. The behavioral model represents the behavior of our suspension system using the Dymola

model. The design objectives were linked to the corresponding excel spreadsheet for the monotonic utility

functions. The spreadsheet accepts values for vertical acceleration, generated energy, or PGSA cost and

outputs the utility based on the spline curve equation. These individual utility outputs were substituted into the

multi-attribute utility function from Task 2 to calculate the total utility of the given set of input design

variables.

Figure 3.1: (a) Exploration Model, (b) Behavioral model, (c) Utility Model

To explore the entire design space and lower the probability of getting stuck in local maxima during

optimization, we ran a space filling full-factorial DOE with 10 levels. This DOE allowed us to discover the

relationships between the main design variables and the total utility of the system. Evaluating a 10 level full-

factorial created enough runs to generate a smooth surface plot which was used to visualize any local maxima

and determine a good starting point for the optimization process. Since we have three design variables, the

surface plots showed the effect of the magnetic field and wire length on the total utility for each value of the

cross-sectional area, resulting in 10 surface plots. To account for the uncertain variables, we kept the car mass

constant during the full-factorial analysis. We ran the simulation five times to explore design space based on

changes in the car mass (one run for each number of passengers in the car, i.e. 1, 2, 3, 4, or 5 passengers). The

spring constant was kept at the expected value for all five simulations because it was determined previously

that this value did not have a large effect on the design objectives.

Through the design exploration described above, there were five different full-factorial simulations and 10

surface plots per simulation resulting in 50 figures. These plots, along with the individual spreadsheets, were

analyzed to find values of the design variables in a global maximum. In all five simulations, a commonality

with the cross-sectional area was the first few values generated the largest utility. The surface plots for these

cross-sectional area values are shown below.

(b)

(a)

(c)

Figure 3.2: Utility Surface Plot with CarMass=1504 kg and Cross-Sectonal Area=1.0 x 10-6 m2


Figure 3.4: Utility Surface Plot with CarMass=1567.45 kg and Cross-Sectonal Area=5.0 x 10-7 m2




We learned a lot about our design problem from the design exploration. By separating the analyses by cross-

sectional area, we realized that the utility does not change much for higher values. The utility was greatest for

a smaller range of cross-sections allowing the exploration to become dependent on two design variables rather

than three. We also discovered that there is a significant change in the utility when the mass of the car is

altered due to the number of passengers. However, the largest change was between 1 and 2 passengers and the

variation in utility between 2, 3, 4, and 5 passengers is marginal.

In addition to the trend in utility based on cross-sectional area, we noticed that the global maximum on these

plots was occurring at the maximum bounds of our Magnetic field and Length of Wire design variables. This

meant that our maximum may be too conservative. We were confident with our upper and lower bounds for

the length of wire; however, we were not as confident of our upper bound for the magnetic field. Due to this

we chose to run another full-factorial with a much larger number of levels overnight due to the relatively short

length of time it took to run the 10 level simulation. We chose to increase the upper bound of the magnetic

field to 5 Tesla and run the full-factorial with 40 levels to see if we found a drop in expected utility at the upper

bounds. The increase in levels resulted in 64000 runs taking approximately 12 hours to run. From the

previous exploration, we chose to keep the car mass and spring constant at a their expected values of 1630.9 kg

and 87564 N/m to limit the computation time. The full-factorial generated 40 surface plots (one for each value

of cross-sectional area) and the most promising results are shown below in Figure 3.15-3.17.

(a)

(b)

Figure 3.15: Two view of the surface plot for cross-sectional area=5.0 x 10-7

m2

(a)

(b)

Figure 3.16: Two views of the surface plot for cross-sectional area=6.15 x 10-7

m2

(a)

(b)

Figure 3.17: Two views of the surface plot for cross-sectional area=7.31 x 10-7

m2

We noticed from these plots that there is a drop in total utility resulting in a global maximum (highlighted with

a blue ring) due to our increase in the upper bound of the magnetic field. From this more detailed plot, we

found that the utility increases along the “shelf” up to our upper bound of the wire length. Due to the lack of

time left in the semester, we could not perform a more in-depth analysis to find better bounds for the length of

the wire. Therefore, we used the existing results to determine an optimal design for our PGSA.

From the excel spreadsheet generated with this analysis, we discovered there is a global maximum for our data.

This occurs when the cross-sectional area is 7.31 x 10-7

m2, a magnetic field of 1.88 Tesla, and a wire length of

150 m. We used this as a starting point for our optimization to reduce the possibility of getting stuck at a local

maximum. To attempt to find the global maximum without re-exploring the entire design space, we increased

the bounds on the optimization for the wire length to 500 m.

We were surprised to learn from the exploration that our upper bounds for our design variables were too

conservative. We were confident throughout the semester from our knowledge of the system and the

information gathered from the internet that our bounds were reasonable. After performing an in-depth

exploration of our design space, we felt that an optimum design was possible. The surface plots have clear

local maxima and converge to a global maximum. We were pleased to find from these results that of our

model and utility analysis are accurate. We think that for future work it may be interesting to gain preference

input from more than three people and involve people who have more of experience with this type of

equipment to view the accuracy of our design for an industrial application.

After the design space was explored, we found the optimization algorithm that best fits our system. To do so

we answered the list of questions shown below as presented by Chris in class.

Is the design space continuous or discrete?

To determine if the design space was continuous or discrete, a close look at the design variables and how they

relate to the purchase of the PGSA was needed. The values for the cross sectional area are discrete because the

cross-sectional area of a piece of wire is based on gauge which has set values. The length of the wire, on the

other hand, is continuous since any length of wire can be purchased. The magnetic field based on the method

of purchasing a magnet would be considered discrete because magnets tend to only be sold in discrete tesla

values. Even though two of the three design variables can be considered discrete, we considered the design

space to be continuous to simplify the optimization. We believe this is reasonably accurate because the

magnetic field and cross-sectional area can be rounded to discrete values with marginal effect on the utility.

How many design variables are there?

For the PGSA, we have three design variables: magnetic field, length of wire, and cross-sectional area. In

Homework 4, we determined that there were 2 uncertain design variables that may affect the results: the mass

of the car and the spring constant for the suspension spring.

Is the object function linear or nonlinear?

The objective function we are optimized is nonlinear.

Does the objective function (i.e. expected utility) have local maxima? If so, how many local maxima?

We found from the surface plots that there are local maxima within the design space. From the many surface

plots we generated, it seems there are 3 local maxima.

Is the objective function smooth? Do we have analytical expressions for the derivatives?

The objective function that will be optimized is smooth and the derivative should be relatively easy to

determine.

Are there constraints? If so, are the constraints smooth and convex?

There are no constraints on our model.

From the evaluation of our system design space described above, a good optimization algorithm for our system

was either the Pattern Search or Direct Search algorithm or the Quasi-Newton Methods. These two algorithms

work well for unconstrained objective functions and are meant for local optimization. We believed that the

results would be accurate for these local optimizers as long as we specified starting points close to the global

maxima. Difficulty could have arisen using a Pattern Search or Direct Search algorithm because the local

maxima in our system were not “small” relative to the global maxima.

Task 4: Solve the Design Problem Deterministically

Apply one of the ModelCenter optimization tools to determine the design alternative for which the value is maximized.

Interpretation: Describe, interpret and critically evaluate your result: does the "optimum" seem reasonable? If not,

revisit your models and utility functions.

We used a ModelCenter optimizer to determine which design alternative maximizes the value of our system.

Our ModelCenter license offers two optimization tools: DOT Optimization and Darwin Genetic Optimization.

We chose to use the DOT Optimization because it uses a quasi-Newton method and works well for smooth

constrained and unconstrained optimization problem. Starting with the initial design variable values described

in the previous task, we ran the optimizer for 8 iterations, requiring 61 function calls. Due to our choice to

drastically increase the upper bound of the wire length, we expected a global maximum to be found with at a

higher length than 150 meters. We did not expect much change in the magnetic field or the cross-sectional

area due to the convergence we noticed in the surface plots to optimal values. Our expectations were

confirmed when we used the optimizer to evaluate the design space and showed a maximum utility of 0.8475

to occur for design variable values of 1.853 Tesla for magnetic field, 160.76 meters for wire length, and 6.898

× 10-7

m2

for cross-sectional area. The optimum design is reasonable because it closely reflects the expected

global maximum that was determined from the surface plots in the design exploration. The increase of wire

length above 150 meters verified our belief that our previous upper bound was too conservative. As mentioned

previously future work including a design exploration with the new upper bound for the wire length would

provide more accurate verification of the global maximum

Task 5: Solve the Design Problem under Uncertainty

Create a ModelCenter model that solves the design problem under uncertainty. This involves a nested loop in which

you optimize the results from a probabilistic analysis -- the computation of the expected utility.

Note: this can be very computationally expensive. A few hints to help you reduce the computational complexity:

Use the LatinHypercubeSampling driver rather than a Monte Carlo driver. The LHS driver converges

faster and has the option not to reinitialize the sample between runs. Although a fixed sample will introduce

a bias, it will make it much easier for the optimizer to converge (the objective function in now deterministic)

Solve the optimization multiple times with an increasing number of samples. For instance, start with only 5

samples in your LHS. Once you have reached an optimum, increase the number of samples and restart the

optimization with the previous solution as a starting point. Repeat until the optimum no longer changes

significantly.

Interpretation: Describe, interpret and critically evaluate your results: How is this solution different from the

deterministic solution determined in Task 4? Explain why there is a difference.

We included uncertainty in our ModelCenter model with a Latin-Hypercube Sampling driver. The optimizer

was used to generate values for our design variables while the LHS driver was used to generate a random

sample of our uncertain variables, car mass and the four spring constants. The output of the LHS is the total

utility determined from the random sample of uncertain variables. The utility value from this nested loop is

the expected utility. The optimizer was used to maximize the average value of the generated distribution of

expected utility. The modified suspension model is shown in the figure below.

Figure 5.1: Optimization ModelCenter Model of Suspension System with uncertainty

The design variables for the DOT Optimization tool were the same as in Task 4 (magnetic field, wire length,

and cross-sectional area). The bounds for these design variables were the new bounds described in Task 3.

The bounds for the uncertain variables were the same as described in the previous homework assignments.

The car mass was varied from 1504 kg to 1758 kg based on the possibility of multiple passengers and the

spring constants were varied between 85813 N/m and 89315 N/m based on expected manufacturing defects.

The hypothesis was that the use of these variables in the model optimization loop would determine the set of

our design variables that gives the highest utility for the system and is the optimal design alternative of our

suspension system when considering the effects of uncertainty.

The process of determining this optimal design alternative required many iterations of our design problem

uncertainty model. The optimizer was first run using the optimal result from Task 4 as the starting values for

our design variables with 10 runs for the LHS driver in the nested loop. The optimal design results of the first

run were used as the starting values for the next iteration and the number of LHS runs was increased. This

process was repeated until the variation of the optimal design variables was minimal. The spreadsheet below

is a list of the number runs in the LHS and the corresponding values for each design variables and utility of the design alternative.

Table 5.1: Initial optimization run under uncertainty

#LHS Runs

Magnetic Field (Tesla)

(Start=1.853)

Wire Length (m)

(start=160.756)

Cross-Sectional Area (m

2)

(Start=6.898e-7) # Iterations # Function Calls Utility

10 1.047664 256.2986 1.27700E-06 12 78 0.868

20 0.540463 271.018 1.17105E-06 7 45 0.849

30 0.563595 270.7991 1.18125E-06 4 22 0.8498

40 0.55795 269.9709 1.17992E-06 3 15 0.848715

50 0.556836 268.6767 1.17625E-06 3 15 0.849019

80 0.555835 268.4865 1.17602E-06 3 10 0.849287

As seen in Table 5.1, our use of the optimization process helped our design variables to converge to an optimal

design alternative. The most valuable combination of design variables is a magnetic field of 0.556 Tesla, a

wire length of 268.49 meters, and a cross-sectional area of 1.176 x 10-6

m2. The corresponding utility value for

the nondeterministic case is 0.8493 which is slightly higher than that of our optimal design alternative in the

deterministic case from Task 4.

The values determined for the optimal design of our Power-Generating Shock Absorber when accounting for

uncertainty are well within our bounds. The value for the magnetic field came out to be lower than our

original upper bound of 2 Tesla confirming that our original thoughts were actually correct. The cross-

sectional area value is approximately in the middle of our bounds confirming our value predictions. The wire

length value is within our predicted bounds proving that the values were reasonable. Therefore it is acceptable

to believe that this is a global maximum. However, the design exploration could not be completed to its full

extent so the values actually lie outside of the explored design space due to our change in the upper bound of

the magnetic field. We are confident that the values are preferable due to the expected location of the global

max in the design space exploration and to the high utility value. To better determine the accuracy of this

value would require a larger design exploration to determine that there is not another maximum with the new

bounds.

The optimal design alternative for this nondeterministic case was significantly different than that of the

deterministic case. The large change was from the introduction of the uncertain variables and more

particularly the mass of the car. We knew from both our previous knowledge of electromagnetism and our

sensitivity analysis performed in Homework 4 the car mass had a profound effect on the results of our

objectives and hence a huge influence on our utility. For this reason the large change was not unexpected.

More specifically, the importance of the car mass makes this nondeterministic optimum much more accurate

than that of the deterministic because the car mass based on the number of passengers was not a variable we

could control. Therefore our design needs to be robust for any variation in the number of passengers in the car

to allow for the most energy to be generated and the most comfortable ride created for any situation possible.

Based on the overall design process undertaken, the optimal values were acceptable for the assumptions that

were made. We believe that a more accurate evaluation of the system could be made by expanding the number

of controllable design variables and taking into account other uncertain variables that were not considered.

Due to the lack of high level knowledge and attention required in this course and the short period of time, the

beginning of the problem selection and analysis began a rode of a number of large assumptions that limited the

possibilities of a truly accurate design. Even though the design space was limited from the beginning, the

overall process was followed and internalized by all of us relating in a comparable increase in knowledge of

the work necessary to solve a design problem under uncertainty using the modeling and simulation tools from

this course.

Task 6: Perform Sensitivity Analysis

Finally, perform a sensitivity analysis to see whether your solution changes significantly when you consider epistemic

uncertainty. Select one or two parameters for which you lack confidence in your uncertainty assessment and which at

the same time you expect to influence the overall utility significantly. Solve the design problem again with the

extreme upper and lower bounds for your epistemic uncertainty -- one factor at a time.

Interpretation: Interpret and critically evaluate your results. Is your design decision robust with respect to your

epistemic uncertainty?

To begin the sensitivity analysis, we chose to perform a central composite analysis to determine which

variables have the largest effect on the total utility of the design.

Figure 6.1: Central Composite results for sensitivity analysis

We deduced from the above results that all three design variables and the car mass had a similar level of

significant to the utility. We were confident that our uncertainty analysis of the cross-sectional area was

accurate. We were also confident of the accuracy of our chosen range of values for the car mass, even though

these values were strictly application based because of the large range of vehicle types in the world. We were

not fully confident of our bounds for wire length and magnetic field. For this reason we chose to evaluate the

two variables separately for epistemic uncertainty constraints.

10095908580757065605550454035302520151050

WireLength

MagneticField

CarMass

XSectionalArea

SpringConstant2

SpringConstant

SpringConstant3

SpringConstant1

29%

28%

28%

14%

0%

0%

0%

0%

To do this sensitivity analysis, the optimization procedure described in Task 5 was used while keeping one of

our two epistemic uncertainty variables constant. The design problem was solved once for both the extreme

upper and lower bound of each uncertain variable. This resulted in four spreadsheets that we used to represent

the convergence of the design variables towards an optimum for the magnetic field bounds of 0.5 and 5 Tesla

and the wire length bounds of 20 and 500 meters.

Table 6.1: Sensitivity Analysis for epistemic uncertainty of Magnetic Field

Magnetic Field=0.5 #LHS Runs


(Start=0.5)

Wire Length (m) (start=160.756)


2)

(Start=6.898e-7) # Iterations

# Function Calls

Utility

10 0.5 274.8975 1.25300E-06 5 27 0.8466

20 0.5 281 1.28999E-06 4 18 0.8478

30 0.5 283.391 1.30160E-06 3 13 0.8464

40 0.5 281.295 1.28950E-06 3 12 0.8458

50 0.5 282.456 1.29210E-06 3 12 0.8461

80 0.5 282.456 1.29210E-06 3 9 0.8461

Magnetic Field=5.0 #LHS Runs


(Start=5)

Wire Length (m) (start=160.756)


2)


# Function Calls

Utility

10 5 500 5.000000E-06 4 16 0.2155

20 5 500 5.000000E-06 2 3 0.253

30 5 500 5.000000E-06 2 3 0.2329

40 5 500 5.000000E-06 2 3 0.2296

50 5 500 5.000000E-06 2 3 0.2344

80 5 500 5.000000E-06 2 3 0.2328

Table 6.2: Sensitivity Analysis for epistemic uncertainty of Wire Length

Wire Length=20 #LHS Runs


(Start=1.853)

Wire Length (m) (start=20)


2)


# Function Calls

Utility

10 2.5856 20 9.24519E-07 4 20 0.784

20 2.5436 20 9.55359E-07 3 13 0.78436

30 2.4815 20 9.66789E-07 3 13 0.78419

40 2.4815 20 9.66789E-07 3 9 0.78449

50 2.4815 20 9.66789E-07 3 9 0.78449

80 2.4815 20 9.66789E-07 3 9 0.78449

Wire Length=500 #LHS Runs


(Start=1.853)

Wire Length (m) (start=500)


2)


# Function Calls

Utility

10 1.24735 500 5.000000E-07 5 24 1.18099

20 1.23832 500 5.000000E-07 3 12 1.18486

30 1.24423 500 5.000000E-07 3 13 1.18914

40 1.237488 500 5.000000E-07 3 13 1.1894

50 1.24073 500 5.000000E-07 3 9 1.188

80 1.23045 500 5.000000E-07 3 12 1.1889

By fixing these two variables at their extreme bounds we were able to see the robustness in our design based

on our lack of knowledge of the uncertainty of these variables. As seen in Table 6.1, there was no real change

in the design variables at the lower bound of the magnetic field due to the closeness of the optimal value to this

bound. At the upper bound, the other two variables also converged to their upper bound. The optimal utility is

also much lower. This drastic change in optimal values was due to the high increase in cost for a higher power

magnet causing a requirement of a high amount of generated energy and a lower vertical acceleration. We

concluded from these results that the overall design of the PGSA was not robust to major changes in the

magnetic field. This conclusion was expected and is always a concern when designing systems that utilize

magnets. The magnet was usually the driving force for any fluctuation in cost.

The epistemic uncertainty evaluation of the wire length is shown in Figure 6.2. When the wire length was set

to the lower bound, the optimal cross-sectional area did not change much, but there was a large jump in

magnetic field due to the large drop in wire length. From our knowledge of electromagnetism, we suspect that

there is a direct relationship between both wire length and magnetic field and the amount of energy generated.

Therefore we expect that a drastic drop in the wire length requires a much more powerful magnet to have a

comparable utility for the design. At the upper bound, the magnetic field increased to improve the utility of the

system. The cross-sectional area dropped to its lower bound which indicated an attempt to lower cost due to

the increased cost of the longer wire. The utility of this optimum was also much higher than the preferred

design alternative derived in Task 5.

From this sensitivity analysis, it was clear that the preferred alternative was highly sensitive to the epistemic

uncertainty due to both the dramatic changes in the design variables and the total utility when the uncertain

variables were changed. We determined through this evaluation that the system is not robust relative to the

epistemic uncertainty. This is the same case in fluid damper suspension system. The design specifications are

specific to the car used. In the future, more and more cars will benefit from such a power-generating source as

our shock-absorber, so a wide range of cars and trucks can use this design. For these reasons we expect that

the design of the PGSA is application based.

Task 7: Lessons learned

In a few paragraphs, identify what you learned in this assignment. Where did you struggle? Where did you go down the

wrong path? How did you revise your approach? If you were to solve another design problem, how would you do things

differently? Did you learn anything new about the trade-offs in the system you were designing?

Jonathan Bankston

In this homework assignment, I was introduced to the amount of work involved in the optimization of

variables in an engineering design. Many people, including myself, like the throw out the term optimization

loosely, as if it was a simple process with clear-cut results. I’ve become very interested in sensitivity analysis

and their possible application in design coordination. Completing this assignment helped me learn more about

performing a sensitivity analysis. I also learned what can and cannot be learned through behavioral modeling. I

want to explore the use of ModelCenter in my future research, particularly in linking ModelCenter with the

informational databases available in electrical CAD software.

Yavuz Mentes

After finishing all the assignments in ME 6105, I am delighted to easily say that I gained at least the basic

knowledge about modeling and simulation which will help me in my Structural Engineering career. My goal

was getting familiar with Mechanical Engineering and systems realization by taking modeling and simulation

course so I ended up taking ME 6105 after I checked its course content. As we started working on assignments

1 and 2, I learned how to use Dymola and started thinking beyond my original civil engineering view.

Actually, the topics we covered after we started working on our group project are going to be useful for my

future career. Especially, uncertainty analysis and sensitivity analysis were the most amazing topics. I learned

how uncertainty of different variables could affect the whole system. In addition to this, in this last homework,

I learned how we can reach an optimal value for different variables that would affect the whole system. The

effect of uncertainty on optimization was an advanced learning experience for me. Overall, I gained a great

amount of value from ME 6105 and I am sure it will pay off in the future...

Alex Ruderman

By performing all of the steps required in the complete design of our system I learned a great deal about utility

properties and the process of optimization. Due to the relative nature of this process to my thesis research, I

gain much value from the lessons learned about the iterative process of design space exploration and

simulation iterations. I realized in Task 3 that the design exploration portion of the process can be very

tedious. I know in our exploration we ran into problems with our bounds and the fact that we had three design

variables. Since a surface plot can only show the relationship between 2 variables and utility, it became very

cumbersome to have to extract plots for all of the variations in the third parameter and interpret them. We

ended up going down the wrong path at first because we did not take into account that the bounds could be too

conservative. When I realized this after talking to Chris, we were forced to redo the design exploration at the

last minute. For me this part became very frustrating because of the lack of time to generate the most accurate

results. I think that by simply increasing the bounds on the wire length without actually performing the

exploration cut a lot of the necessary time down and gave a reasonably accurate result. If I had to do it over

again I think we should have ran smaller full-factorials (maybe level 5 or 10) that would give a small picture

and then increase the bounds and run a higher level analysis later. Overall I feel that I gain much value from

this course project for my research and enjoyed using Dymola and ModelCenter for the design problem

evaluation.

Task 8: Project Web-Page

Add a short paragraph to your project web-site, maybe with a screenshot of your ModelCenter model, illustrating the

outcomes of your simulation-based design study. You should also upload, in a zip folder, all the files you need to run the

ModelCenter model: all the Excel, *.pxt and *.pxc files, and the folder with the final file wrapper and corresponding .mo

file.

Final Project Presentation: Thursday May 1st at 11:30AM.

During the final exam period, we will provide each (group of) student(s) with an opportunity to make a 10 minute

presentation about their project.

The goal here is to learn from each other. Shortly introduce your application domain and then highlight the most

interesting aspects of your assignments/project: Where did you encounter unanticipated obstacles, and how did you

overcome them? Did you learn any neat modeling tricks you want to share with your colleagues? Did you obtain some

interesting results? Can you provide us with some interesting insights based on your models?

So, the focus should not necessarily be on the results of your design study, but on the interesting issues that you

encountered in the process of solving your design problem. This Project Presentation is meant to be a fun learning

experience -- don't hesitate to entertain some also.

Evaluation Criteria

Is the design problem well-defined?

Are the individual steps in defining the design problem performed correctly?

How well are the results interpreted and critically evaluated?

Project presentation: contents and presentation (only small portion of grade)

Organization and clarity of thoughts and report

Submission

1. Hard copy in class (for on-campus students only)

2. T-square: upload your report and all your Modelica (*.mo), Excel (*.xls) and ModelCenter files (*.pcx,

.fileWrapper) to t-square. All your files should be included in one zip file. Name the zip file:

ME6105.HW4.yourfamilyname.zip. (for all students)

Distance Learning Students: There is no need to fax your hard copy in to DLPE. It is easier for me to just download it

from t-square.

preference modeling and optimization€¦ · our model is a suspension system utilizing a...

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