hyper-radial visualization for multi-objective decision

American Institute of Aeronautics and Astronautics

1

Hyper-Radial Visualization for Multi-objective Decision-making Support Under Uncertainty Using

Preference Ranges: The PRUF Method

A. M. Naim* and P.-W. Chiu†. University at Buffalo – State University of New York, Buffalo, New York 14260

C. L. Bloebaum‡ University at Buffalo – State University of New York, Buffalo, New York 14260

and

K. E. Lewis§ University at Buffalo – State University of New York, Buffalo, New York 14260

In this paper, the Preference Range and Uncertainty Filtering (PRUF) Method is proposed for visualizing multi-attribute design problem solutions under uncertainty. This method enables decision makers to perform real-time trade-off investigations by incorporating preference ranges and uncertainty at the input level. In a real world case, designers must consider potentially complex trade-off decisions in order to choose a point or set of points with which to proceed. By using the PRUF method, designers can investigate the performance space under different preference/uncertainty conditions. This allows a more informed and efficient way to find promising solutions for each decision maker, thereby significantly reducing the time required for trade-off analysis in a complex design process. In this paper, an eight objective problem is presented to demonstrate the incorporation of user preferences and uncertainty definition.

hen an engineering model is characterized by more than one objective function, they can be called multi-objective optimization problems (MOPs). The general mathematical form of MOPs is stated in Equation (1)1.

While designers can find multiple solutions to a MOP, only one or few of these solutions hold any practical value for the designer. The true challenge with MOPs is to be able to identify the most desirable solution (or solutions) from amongst the Pareto optimum set.

A feasible solution, X*, is Pareto optimal if and only if there is no other feasible solution such that Fi(X) ≤ Fi(X*) for all objective functions and Fi(X) < Fi(X*) for at least one objective function2. The Pareto definition always gives a set of solutions instead of one solution when there exists some trade-off among the objectives. The region defined by the Pareto optimum solutions is called the Pareto frontier. When MOPs have more than three objectives, the set of Pareto optimum solutions can be called a Hyperspace Pareto Frontier (HPF)3.

Minimize )](),...,(),([)( 21 xfxfxfxF n= where ],...,,[ 21 pxxxx = (1)

* Graduate Research Assistant, Department of Mechanical and Aerospace Engineering; [email protected], Student Member AIAA. † Graduate Research Assistant, Department of Mechanical and Aerospace Engineering; [email protected], Student Member AIAA. ‡ Professor, Department of Mechanical and Aerospace Engineering; [email protected], Associate Fellow AIAA. § Professor for Competitive Product and Process Design, Department of Mechanical and Aerospace Engineering; [email protected], Associate Fellow AIAA.

W

12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference 10 - 12 September 2008, Victoria, British Columbia Canada

AIAA 2008-6087

Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


2

Subject to ;0)( ≤xg j mj ,...,2,1= inequality constraints

;0)( =xhk lk ,...,2,1= equality constraints ;1 u

iii xxx ≤≤ pi ,...,2,1= side constraints In MOPs, the Utopia point is one that corresponds to the minimum value of each individual objective function4.

The Utopia point is a hypothetical concept pertaining to an ideal target of ‘good’, which provides designers a reference point in making decisions during the trade-off process. Since the Pareto optimum solution set typically consists of many feasible solutions, this makes it difficult to choose a single best solution. Moreover, not every objective has the same preference to decision makers in real world engineering problems. Hence, designers need a tool to enable them to choose their final solution(s) from these potential designs – a method that enables intuitive trade-off studies.

The success of a design process can be characterized by being able to identify all the available solutions, and subsequently selecting the best one5. It is, therefore, quite important to assist decision makers in this selection process. However, most design processes are complex in nature, and therefore creates a more complex trade-off process. From Laird’s research6, seventy-five percent of human attention is focused on sight and the remaining twenty-five percent is shared by hearing, smell, touch and taste. Hence, visualization can provide decision makers with qualitative data and then improve the solution selection process. In a two-dimensional visualization environment, Pareto points can be visualized intuitively without any projection process, and designers can make their trade-off decisions corresponding to the level of information available to them. However, most real world problems are multidimensional (greater than 2-D) and the visualization of such a space is a very challenging task.

I. Motivation

Many methods have been developed in order to provide decision makers with a meaningful visual representation. Methods like Physical Programming Visualization8 or Multidimensional Visualization9 allow the designer to focus on a certain region of the design space by either filtering the unfeasible points or by differentiating them by using different colors. However, those methods are limited when dealing with high dimensional problems. Another approach dubbed “design by shopping”10 process first explores the design space to then find the optimal solution. A visual steering command11 can be used in order to help the designers to guide them to a solution in an efficient manner. However, the designers may have no knowledge of their preferences, complicating such an interactive approach.

In order to allow intuitive multidimensional visualization, the Hyperspace Diagonal Counting (HSDC) method was introduced. It is based on a diagonal counting concept that maps the n-dimensional Pareto frontier to 2 or 3-dimensional data12. It can visualize an HPF in an intuitive and succinct way. However, the HSDC method has a drawback in that it does not maintain a neighborhood when creating solutions. That is, when collapsing the n-dimensional data onto two or three dimensions, not all of the n-dimensional neighborhoods are maintained in the lower dimensions. Moreover, different grouping schemes cause different HPF visualizations for the same problem.

All these techniques, which are essentially used as methods of solution validation and subsequent concept selection, sacrifice intuition and efficiency in high dimensional MOPs. Furthermore, those methods fail to incorporate the designer’s preferences as well as uncertainty present for a given problem. A novel method is thus necessary which would allow the visualization of the available alternatives and assisting the designer in his decision making by increasing the level of confidence.

In short, designers need a tool that can visualize all the available options, as well as to study “on the fly” the behavior of these solutions with varying level of information about preference and uncertainty. Hence, the PRUF Method that incorporates preference ranges and uncertainty representation is proposed and studied in the next section.

II. Approach

The PRUF Method is a combination of three different area of research that are combined together in order to provide the designer with a powerful tool to assist in decision making in early design processes. The presented method not only allows the designer to visualize the available alternatives for a given MOP in a two dimensional space, it also provides additional information by incorporating a color-coding scheme that differentiates the alternatives based on preferences and also accounts for the presence of uncertainty for the given MOP. The following sections present the three aspects of the PRUF Method and how they interact with each other in order to assist the designer in making a decision.


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In this paper, the Hyper-Radial Visualization (HRV) Method13 is used to visualize an HPF in a lossless, meaningful and intuitive way. This method enables designers to couple the strengths of the previously defined visualization techniques in order to provide a mechanism to perform an intuitive trade-off analysis.

Furthermore, an uncertainty representation can be incorporated after the generation of attributes in order to account for aleatory and epistemic uncertainty associated with preference choices, thus allowing a more informed and responsible decision-making process. Finally, the possibility of performing changes on-the-fly allows the method to be used as a “steering” paradigm as well as a visual tool to present information relevant to the decision maker. This choice is left to the decision makers, depending on how they choose to interact with the information and capabilities of the method. Figure 1 shows a flowchart of the PRUF Method.

Figure 1. The PRUF Method flowchart The preliminary step is to generate a set of Pareto points using any desirable method. The first step then becomes

a decision on the part of the designer on how to separate the objective ranges into five different degrees of desirability. Here the concept of Physical programming14 is used to incorporate the designer’s preferences.

Having defined the preference ranges, the designer proceeds by defining the uncertainty present. In the proposed method, both the aleatory and epistemic uncertainties are represented. Using the uncertainty information, a filtering process is performed in order to eliminate all the unfeasible alternatives, thus steering the designer into a region of high desirability and robustness toward uncertainty.

A. The Hyper Radial Visualization (HRV) Method

In the HRV Method, a radial calculation concept (Figure 2) is used in which an actual hyper-radius is calculated13. Figure 2 shows that for every point in two-dimensional space there is a corresponding point on a radius. Equation (1) is used to calculate the hyper-radius of each data point, where n represents the dimensionality of the performance space (i.e. the number of objective functions in the MOP). In order to create a meaningful representation for multiple objective functions that have different values, it is necessary to first normalize each objective function by Equation (2). This removes any bias associated with objective functions that have different units.

Figure 2. The concept of the HRV method

The Radial Calculation = ∑

=

n

iiF

1

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min~

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In addition to normalizing each objective function, the result of the hyper-radial calculation also needs to be

normalized. This is done so that the visualization will be associated with axes from [0, 1]. Hence, the Hyper-Radial Calculation (HRC) value is shown in Equation (3).


4

HRC = n

Fn

ii∑

=1

2~

(3)

with ]1,0[~ ∈iF i = 1…n; And ]1,0[∈HRC .

One advantage of the HRV method is that the Utopia point in a HPF visualization is the origin, and it will always correspond to this point for any minimization problem. This is because the minimum value of every objective function must be zero after being normalized by Equation (2). This allows designers to have a reference point in the visualization process. Now, designers can visualize the HPF in a lossless way (i.e. all data is represented) by using the HRV method.

Furthermore, points that are close to the origin have small radius values, where the radius is a function of all the normalized objective function values. Hence, points that have small HRV values correspond to those that have small objective function values. For minimization problems, these are likely to be the Pareto points the designers would be interested in for final selection. The Direct Sorting Method (DSM) uses the distance between the origin (the Utopia point) and each Pareto point in a HRV-based HPF visualization as a judgment to differentiate a relative preference. It is important to note that the unbiased version of the DSM inherently assumes that all objectives are of equal importance. The biased version of the DSM enables the designer to easily incorporate preferences for the objectives. Hence, the DSM is a straight-forward method that designers can use to help identify their final solutions after using Pareto optimization.

Figure 3 shows an n-dimensional Pareto Frontier visualized in a two dimensional figure (i.e. Axis 1 represents some number of original objective functions and Axis 2 represents the rest). The known Pareto points are distributed in the gray area, and the black bold line is the frontier of the HPF visualization. Mathematically, every arc that uses the origin as its circle center can be considered as an indifference curve4. Assume there are four indifference curves (i.e. IC1, IC2, IC3 and IC4). Using the unbiased DSM, the points on IC2 are more desirable than those on IC3, and the points on IC3 are more desired than those on IC4. Therefore, the best solution is the point on the frontier of the HPF representation that is tangent to the indifference curves (Figure 3).

Figure 3. The Pareto Frontier and indifference curves

Again, recall that the unbiased DSM assumes that all objectives are of equal importance. In this way, it is possible to quickly identify ‘good’ solutions in the n-dimensional preference space, regardless of the numbers of objectives or numbers of Pareto points.

B. Preference Incorporation Using Preference Ranges The concept of preference ranges has been introduced in 14 and allows designers to incorporate their preferences

without the explicit definition of weights. The designer creates a mathematical function that directly expresses a priori preferences. A generic objective is divided into five ranges/domains/regions based on the degree of desirability ranging from Highly Undesirable, Undesirable, Tolerable, Desirable, Highly Desirable.

Figure 4 illustrates how a normalized objective range is divided equally into the five different preference ranges where the degree of desirability increases as the objective value decreases. This preference incorporation scheme corresponds to a minimization problem and can easily be modified to correspond to either maximization or constrained optimization problems.

The concept of Physical Programming is used in this paper in order to accurately incorporate the designer’s preferences into each objective together with uncertainty in order to improve the quality of information for the available alternatives. This information is then translated visually to the designer by means of a color-coding that is designer specific. In this paper, an elitist approach is used in order to assign colors based on the degree of desirability for each Pareto point. Table 1 explains the color-coding used.

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8

(Fi_mean, Fi_min, Fi_max). The boundary T is used as the reference point in the cumulative distribution function and its probability is calculated and finally compared to the previously defined probability value PR.

Depending on the objective’s value, all the points will have a different position on the defined preference range. The cumulative distribution function determines the probability of all the points for a given objective and compares it to the one defined by the designer. If PR > PFi for a given point, it will be discarded. If PR <= PFi, this point will be selected.

However, this process does not account for uncertainty in the previously defined objective function preferences, and the designer might be uncertain about the actual extent of those ranges, thus making it necessary to allow the designer in modifying “on the fly” those values in order to provide better information.

Range Uncertainty

The designer could also be uncertain about the range thresholds, and should be able to perform modifications to the threshold values “on the fly”. Figure 10 represents an initial range definition of ratings that was defined by the designer. However, the designer has the possibility to modify the ratings if he/she needs to further investigate its influence on the general behavior of the selection process or is uncertain about the exact value.

The original boundary between “Desirable” and “Tolerable” is represented by T, and the modified one is T’. The probability comparison process is modified since the boundary T is changed by the designer, thus resulting in new probabilities for the objectives. After modifying this value, the process described in the previous section is performed and a new set of probabilities are computed and stored, allowing comparison with the previously generated ones. The designer is also able to define

Figure 10. Threshold changes for preference limits.

upper and lower limits to the thresholds and obtain upper and lower limits on the probabilities to be calculated and provided to the designer in order to have more valuable information.

III. Test Case and Results

An eight objective function problem15 is used to demonstrate the HRV method with uncertainty incorporation.

This sample problem has 635 non-dominated Pareto solutions generated by a Multi-Objective Genetic Algorithm (MOGA)7. The problem statement is shown in Equation (6).

The objectives were separated evenly and, with f1, f2, f3, and f4 on the x axis and f5, f6, f7, and f8 on the y axis. It is to be noted that the grouping order has no influence on the final result as long as the number of objectives in each axis is equal16.

Figure 11 shows how the sorting progress impacts on the visualization of the available options. Figure 11(a) is the HPF visualization without uncertainty. However, as the designer incorporates uncertainty, the filtering process becomes more effective and the number of sorted Pareto points increases.


9

Minimize: 1 2 22 2 1 213 3

1 2 3 2175 2 2 1 217 13

3 1 2 2 4 28 1 2 1 227 15

3 1 2 9 234 1 1 215 29

4 1 2 4 222 2 1 25 17

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17 1 2 3995 8 2 5 1 1 265

7 2 1 5 2 25 2 3 1 2235

(6)

Subject to: 4 4 0 1 0 2 0

Where, 4 4 4 4

Table 2 represents the three different uncertainty definitions used in the test case. Each case has a different

level of uncertainty information. Table 2. Uncertainty information

Case 1 F1 F2 F3 F4 F5 F6 F7 F8 Distribution type N N - - - - - - Selected range D T - - - - - -

Probability definition (PR) 30% 75% - - - - - - Percent deviation (di) 2% 5% - - - - - -

Case 2 F1 F2 F3 F4 F5 F6 F7 F8 Distribution type N N U - U - - - Selected range D T D - T - - -

Probability definition (PR) 30% 75% 25% - 75% - - - Percent deviation (di) 2% 5% 3% - 2% - - -

Case 3 F1 F2 F3 F4 F5 F6 F7 F8 Distribution type N N U U U N - N Selected range D T D T T HD - D

Probability definition (PR) 30% 75% 25% 70% 75% 10% - 25% Percent deviation (di) 2% 5% 3% 2% 2% 2% - 2%


10

(a) No Uncertainty (b) Case 1

(c) Case 2 (d) Case 3 Figure 11. HRV visualization

Figures 11(b) to 11(d) show that the number of candidate Pareto points is reduced from 635 to 38 as the

designer includes more uncertainty into the objective functions, thus allowing him/her to focus on the remaining alternatives.

As the level of uncertainty information increases from case 1 to case 3, more objectives have uncertainty applied to them which allows a more thorough filtering process. The DSM can be applied in order to pick the set of points closest to the origin from the remaining Pareto points. The last step is to include the epistemic uncertainty due to the lack of precise information. In this paper, it is assumed that the designer is unsure of the precise value of the preference range threshold value between “Desirable” and “Tolerable” for the third objective. Table 3 shows the upper and lower values assigned to the threshold value as well as the resulting Pareto point count.

Table 3. Filtered Pareto point count using imprecise probability on F3

Precise boundary value (T)

Lower boundary value (TL)

Upper boundary value (TU)

0.4 0.32 0.45 Pareto Point count 38 22 45

Figure 12(a) and 12(b) illustrate how the epistemic uncertainty impacts on the visualization if the performance

space


11

(a) Lower boundary value (b) Upper boundary value Figure 12. The impact of imprecise probability

This case study illustrates how the PRUF method can effectively illustrate all the available Pareto points for a high dimensional problem. As uncertainty is incorporated into the objectives, the designer is steered to a region of the performance space having the highest available degree of desirability based on their preference definition. In order to make a decision using the PRUF Method, the designer ought to pick the set of points that:

• Is close to the Utopia point, • Has the highest degree of desirability, based on the designer’s preference definition.

IV. Conclusion and Future Work

In this paper, the PRUF Method is presented in order to overcome the common drawbacks identified in current

visualization tools. Using the HRV concept, the designer is able to perform decisions with a higher level of confidence. The multidimensionality aspect of a MOP makes the visualization of the available alternatives cumbersome. Furthermore, the lack of preference and uncertainty incorporation reduces the quality of the decision made. Providing a tool that not only presents the available alternatives in a simple way, but also including the designer’s preferences as well as the uncertainty present in a given MOP allows the design team to reduce the time necessary early on the design process. The method’s application steers the design team into a region of the performance space that has the highest degree of desirability based on the preference definition using a color coding scheme. It also allows the design team to pick a set of alternatives that are robust to both the aleatory and epistemic uncertainty by filtering out the Pareto points that do not satisfy the probability definitions for each objective.

One of the major limitations of the proposed approach is the inability to validate the uncertainty data provided by experts. Future work would involve the linkage of the proposed method with an uncertainty repository. Based on the type of objectives, the method would go through the repository to determine common distribution types as well as standard deviations that best fit the nature of the objective. Further work would also consider using the proposed method in designer’s negotiation. Instead of defining an upper and lower bound for a preference range threshold, those values would be replaced by different bounds for each designer. The PRUF Method would then provide the a set of alternatives that would satisfy all the concerned designers.

Acknowledgments We gratefully acknowledge the support of this research from the National Science Foundation, grant CMMI

0600263.

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