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Russian Academy of Sciences
Central Astronomical Observatory at Pulkovo

SIXTH US/RUSSIAN
SPACE SURVEILLANCE WORKSHOP

August 22-26, 2005

Proceedings
Edited by P. Kenneth Seidelmann and Victor K. Abalakin

St. Petersburg, 2005


EVOLVING SPACE SURVEILLANCE CAPABILITIES MATHEMATICAL RAMEWORK FOR EVALUATING FUTURE SURVEILLANCE NETWORKS

Susan Andrews
MIT Lincoln Laboratory, 244 Wood Street, S4-600D, Lexington, MA 02420

Abstract. Development of a new space surveillance sensor from basic concept to a fielded system is a process that takes many years. Making changes to overall network capability with the incorporation of several new types of sensors can take over a decade and perhaps two decades. In contrast, the worldwide pace of satellite development is yielding significant changes on a much shorter timeline. The new capabilities brought about by these changes combined with the highly dynamic worldwide socio-political environment yield a variety of potential futures for the use of space in the ten to fifteen year time frame. As a result, architects for future space surveillance networks are being asked to specify network architectures and the sensors within them to meet the needs of an uncertain future.
This paper describes a framework for evaluating complex systems of systems, such as a space surveillance network, in the context of uncertain user needs. The framework is based on utility theory, but classic utility theory (e.g., Ferguson, 1967) would require either a separate specification of utility functions for each distinct vision of a potential future or a set of compromise functions that represent a single blended future. To consider a wide variety of possible futures, the framework described in this paper only requires the user to specify utility functions for a small number of well-defined visions of the future (e.g., 1 - many rogue actors committing hostile acts on spacecraft, or 2 - extensive peaceful use of space requiring traffic management). Consider the set of future visions that can be notionally described as a combination of visions from a smaller select set (e.g., moderate use of space requiring some traffic management plus a small number of hostile acts). In the proposed framework, the utility of a candidate system for a future selected from the larger set is automatically calculated using the utility functions on the smaller select set of visions, and there is no need to employ the manual process of specifying the utility functions for the specific vision. Thus, a modest change to the vision for the future user does not require evaluators to engage in extensive development of new utility functions, and the utility of a system candidate can be considered parametrically over a range of visions of the future. The application of this framework to space surveillance networks is demonstrated through examples.
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+ Note that all examples in this paper are representative of the research area and do not reflect priorities, goals, or capabilities of the current US Space Surveillance Network or planned architectures.
1. INTRODUCTION

The use of Earth-orbital space has changed significantly in the past fifteen years. From a domain where most of the space assets belonged to major international powers during the early 1990s and the assets were predominantly used for national security or national-level scientific investigations, use of space has evolved to include a much broader variety of participants and of functions. Through the development of commercial launch services, many more countries and organizations have access to launch. Technology evolution has enabled development of small and inexpensive satellites, and as access to space has become more available, more reliable, and less expensive, the world of developers has been expanding the services and products offered by space in support of civil, commercial and small-scale scientific users in addition to the national security communities. Given the recent pace of both technology advances in electronics and the addition of developers and functions for space systems, many observers project this fast-paced evolution of space use to continue.

As the diversity of space asset owners and of spacecraft functions increases, the requirements for space surveillance change. Evolution and revolutions in the world socio- political situation also change both the use of space and the requirements on space surveillance. The differences between todays world and that of 1990 have necessitated major changes in space surveillance operations, and architects for the space surveillance networks of the future are projecting that the needs of 2020 will be notably different still. Unfortunately, the time required for a single space surveillance network to develop a significant new capability is much longer than the time between typical significant developments on a worldwide basis. The space surveillance system needed to meet the needs of fifteen to twenty years from now will likely need several new capabilities that are more than simple extensions of existing ones. As such, architects of the future systems are faced with a choice between designing an architecture to meet todays shortfalls, selecting one possible future and designing to that, and projecting a myriad of possible futures and trying to identify a compromise solution.

Given a single projected future, multivariate utility analysis provides an established technique for evaluating the suitability of a system to address a specified set of related tasks and to rank the merits of competing systems. This technique has users define a set of performance parameters to use as the basis for evaluation. Users can prioritize these performance parameters in terms of relative importance (e.g., the importance of timeliness to completion versus accuracy) and specify that set of preferences numerically with a set of numbers referred to as weights. For each parameter, users may also assign increasing merit to improving performance, so that systems that almost meet a requirement threshold show higher merit than those falling far short of the requirement (e.g., if the requirement for timeliness of completion is 30 minutes and completion in 35 minutes is almost as useful, the merit for 35 minutes is close to that of 30 minutes and greater than that of 180 minutes). The merit is specified numerically using a function, called a utility function, that maps performance capabilities to a predefined range of numbers. The specification of prioritization and merit are jointly referred to as user preferences.

The mechanics of multivariate utility analysis are straightforward and easy to automate in software. The technique works well when the desired performance parameters and the trades among them are well understood. The difficult portion of the process is capturing the user preferences. Researchers have developed a variety of tools to help codify and capture these specifications from groups of users, but the process requires time and effort even with the tools. In addition, the process is easier when the participants have a fixed idea of their needs. A requirement to project well into the future for technology research and development goals adds to the difficulty of specifying the user preferences. People looking at systems to meet goals of a fifteen-year future often consider a range of possible futures, and different participants may have different visions. Combining the prospect of multiple and disparate potential futures with the activity of specifying user preferences for a specific future vision just multiplies the difficulty of the process.

Space surveillance is a multi-faceted activity with a variety of tasks, and different world futures will stress different tasks. Even with a disciplined method for determining user preferences, the user community is free to envision multiple future operating environment scenarios for the space surveillance network and will want to know the merit of a candidate architecture for each scenario to see how robust the candidate network is to the operating environment. For example, if space becomes an entirely peaceful realm heavily used for civil, commercial, and humanitarian needs, the requirements for space surveillance are different than a future in which third-party actors are actively trying to upset world stability through offensive acts against space assets.

As new architectures are brought into consideration, it is common for evaluators to want to know how the architecture candidates will fare against new scenarios for the future in addition to those previously considered. Some of the new scenarios may be combinations of previously considered scenarios, such as a high civil and commercial use of space along with a modest threat of offensive acts in space. Multivariate utility analysis will allow a merit to be assessed for each candidate architecture for each envisioned scenario for the future; however it requires that, for each future scenario and for each performance parameter of interest in that scenario, the user community specify a new utility function and weight. Since specification of user preferences is an unwieldy process, evaluators often resort to ad hoc processes to make the overall merit assessments manageable.

The utility framework described in this paper was developed to address the difficulty of eliciting user preferences for a space surveillance network in consideration of multiple visions for the future. It enhances current multivariate utility techniques in the following ways:
  1. It provides the means to derive user preferences for any projected operating environment from the user preferences of a few wisely chosen projected future users.
  2. It reduces the effort required for the remaining manual specification of user preferences by reducing the ambiguity in the nature of the few projected operating environments requiring manual specification.

This utility framework uses the new techniques combined with specific rules for mathematically representing user preferences to accomplish the following:
  1. Provide means to compare candidate systems and to evaluate candidates with respect to an ideal system for any environment within a range of operating environments, each with different user preferences.
  2. Provide basis for sensitivity analysis as a function of changes in operating environment.
  3. Provide means to evaluate and compare systems with respect to individual parameters and groups of related parameters.
  4. Support modular computation, such that selection of a new operating environment does not require reevaluation of system performance.

This paper will define the concepts of utility space and extreme users and describe how they support consideration over a range of operating environments. Section 2 introduces the utility space concept and Section 3 describes the multivariate utility framework for evaluating system merit for a single future user. Section 4 will describe the use of extreme users and the definition of candidate users from the extreme users. Section 5 will define two sets of combination functions for yielding merit for candidate systems for arbitrary users based on the user preferences of the extreme users. Section 6 provides a representative space surveillance architecture example and Section 7 provides a brief summary.
2. UTILITY SPACE CONCEPT

The classic toolset for conducting system trades works from the understanding that each performance capability (e.g., completing a job in x minutes) or outcome (e.g., false alarm) has a benefit and/or cost that can be represented numerically. These numeric representations are relative, and this paper uses the term utility to refer to that representation (also known as cost or value in the literature). A utility function maps performance capabilities into utilities, with better performance mapping to higher utility. When evaluating a complex system, the merit of the system is a function of the utilities of the individual performance capabilities or outcomes. If considering a system with non-deterministic outcomes, then the probability of each possible outcome is folded into the calculus. Depending upon the nature of the evaluation, functions using sums or products of the utilities for individual performance parameters are common ways to create functions yielding overall system merit. The proposed framework uses a weighted sum, and utilities are bounded for both the performance capabilities and on the system as a whole to allow for an absolute assessment of merit with respect to an ideal system.

To reduce the difficulty of specifying user preferences for multiple possible futures, our three goals were to a) reduce the number of weights and utility functions that need to be specified manually while still allowing for a large number of variants on the future, b) find a practical way to specify multiple sets of utility functions for performance parameters for a future space surveillance architecture with respect to different poorly defined future scenarios, and c) provide means to evaluate overall system performance over a range of possible user preferences. The key concept is the creation of a mathematical space of utility functions, where space is meant in the traditional mathematical sense. Every utility function in the space can be formed by an operation performed on a specific small subset of utility functions, and that operation is called combination. For any given evaluation problem and a small set of users that represent the extremes, the space of utility functions for each performance parameter is created by defining extreme user preferences for that parameter. The utility functions specified for the extreme users are the ones used in the combination operation.

The extreme users and their utility functions are created according to the following three guidelines:
  1. Each extreme user represents a future dominated by a single guiding principle (e.g., Earth orbit space will become heavily commercialized and many of the issues driving space surveillance will be traffic management, determination of infractions of commercial rights, anomaly assessment and other support to commercial/civil operators).
  2. Extreme users should be defined, as much as possible, to be orthogonal in needs to one another. This does not imply that two extreme users should not both be interested in success with respect to the same performance parameter (e.g., timely detection of objects), but that their rationale and needs be different.
  3. Together, the collection of extreme users should cover the characteristics of any potential real user under consideration, probably at exaggerated levels. In addition to the functional extreme users, there is one predefined extreme user representing the I dont care user, for whom any system (or no system at all) meets their needs. The intent is that any potential real future user can be represented mathematically as a combination of these extreme users.

For each extreme user and for every performance parameter, utility functions are then defined. Due to the focus of the extreme user, it should be easier to determine the utility functions for these users than for a realistic multi-functional user community. These extreme user utility functions define, for each performance parameter, the extreme points of a convex utility function space. The utility functions for a candidate user are determined by both the extreme user utilities and how the candidate is mathematically represented as a combination of the extreme users. Similarly, the parameter weights are determined by the extreme user weights and the candidates likeness to the extreme users.
3. SINGLE-USER SYSTEM EVALUATION PROCESS

The process for evaluating a candidate system with respect to a single user forms the basis for the multi-user work. We have chosen a utility representation in which the utility for a system as a whole can be represented as a weighted sum of the utility of the individual performance parameters, where the weights are those specifying the relative importance of the individual parameters. Use of weights and parameter-specific utility functions allows for modular computation. Furthermore, the contribution to system merit of a single parameter or group of parameters is easy to determine. This approach does, however, require certain assumptions about the independence of parameters, and it limits the ability to represent the case that a system has no value to a user if minimal performance cannot be met on a single, specific performance parameter. In contrast, a joint non-separable utility function on all of the parameters together can lead to complex calculations and difficulty determining component contributions.

The evaluation process for the single user within our framework has the following steps:
  1. Definition of performance parameters
  2. Specification of weights on performance parameters
  3. Definition of utility functions for each of the performance parameters
  4. Evaluation of candidate system performance
  5. Application of utility functions and weights

Parameters for evaluation are drawn from notional requirements, identified needs, and desired properties. This set defines the measures of performance, and a broad set can be selected to ensure coverage of all key user needs. A performance metric (e.g., time) and units are selected for each capability. To keep the problem bounded and in concert with reality, a finite range of relevant performance values should be identified. Often this range is bounded by a best and worse performance.

Weights are assigned to each parameter to represent the relative importance of one parameter versus another. The proposed framework restricts weight to non-negative numbers and allocates a total weight that must be divided among the performance parameters. In our case, the sum of the weights over all parameters was set to 100, since percents are a natural representation. A user assigns high weight to a performance parameter that has high importance with respect to other performance parameters. Note that setting of weights should be distinct from consideration of the difficulty to achieve a useful level of performance. Consider two users: 1) a man with a car with elite tastes and significant resources; and 2) a young family man with no car. The first man considers only the highest quality luxury train service satisfactory, but does not particularly need train service to be provided. The second finds any train service meeting basic safety, reliability and comfort standards to be acceptable, but requires that such train service be available to enable him to support his family. Weights should capture the need to achieve success in performance (e.g., whether train service is important at the desired quality); utilities should capture the quality of performance (e.g., whether it must be luxury-quality train service).

When performance parameters can be grouped into logical categories (e.g., parameters related to positional accuracy), evaluation teams may find it easier to look at the relative importance of one category with respect to another. Once evaluation teams assign weights to categories, evaluators can assess the relative importance of one parameter in the category to another and allocate individual weights from the group weights. This grouping technique was useful in several studies evaluating space surveillance architectures.

The utility function is defined for each performance parameter based on how the user wants the system to perform on that parameter. For this work, all utility functions have a common minimum of zero and a common maximum value, and the maximum value should be achieved over the range of performance values. In the studies conducted using this framework, evaluators specifying utility functions appeared to find that a value of 5 was a comfortable maximum. For every performance parameter, the utility function is defined over the range of performance values bounded by the previously selected performance parameter range.

Note that utility does not need to be an increasing function in the metric selected (e.g., if the performance parameter is the time to find an object in low earth orbit and the unit is minutes, higher utility is assigned to lower numbers of minutes). Utility functions do not need to be linear in the units of performance. Concave-up monotonic functions represent users who require near the best possible level of performance to yield a high utility (a stressing utility function). Monotonic functions that are concave-down will grant a relatively high utility to modest performance. Utility functions do not need to be monotonic, however care must be taken to specify the range of performance values to be considered in such cases.

The evaluation community found it useful to have a set of parameterized functions to draw from that was appropriate to representing performance parameter utilities. The set of functions offered a variety of shape options (segmented linear, segmented concave, or step) and input parameters would let them set transition points (e.g., points for steps or inflections) as well as slope or curvature. This reduced the overhead of having evaluators define functions from scratch.

After the performance parameters have been specified, the performance of the candidate system is assessed versus each performance parameter and recorded. For each performance parameter, the candidate system performance is scored according to the utility function for that user and performance parameter. The total score for a system is then determined by multiplying the utility assessed for the candidates performance in each parameter by the corresponding user weight and summing this product over all performance parameters. We found renormalizing the total system score to a scale of 0-100 produced results on a commonly used reference scale. Figure 1 shows an example of the evaluation mechanics.

Fig. 1. Example illustrating single-user evaluation mechanics

To summarize the basic mechanics:
For a given user, the utility of a candidate system is the weighted sum of the utilities for the performance parameters. The maximum possible system utility is the maximum value of the utility function times the constant that is the sum of the weights over the set of parameters. The minimum utility of a system is zero. This construct supports assessment of system merit with respect to a given user in absolute terms, as well as its merit in comparison with another system. The process of assessing utilities of system performance on a parameter-by-parameter basis, multiplying by the corresponding weights, and summing is used for merit evaluation in the utility space concept. The primary difference is the source of the utility functions and weights.
4. SYSTEM EVALUATION PROCESS USING EXTREME USERS

The process for evaluating a candidate system under the proposed framework is similar to that for the single user case.
  1. Identify the extreme users and specify their characteristics
  2. Determine the set of performance parameters that capture the needs of all extreme users
  3. .
  4. For every extreme user, allocate weights to performance parameters.
  5. For every extreme user, for each performance parameter, specify a utility function.
  6. Evaluate performance of candidate system.
  7. For every extreme user, for each parameter, determine the utility of the candidate system.
  8. Select candidate user and determine characteristics in terms of extreme users.
  9. Apply the weight combination function to extreme user weights.
  10. Apply the utility combination function to system utility scores for extreme users.
  11. Multiply resulting utilities by corresponding result weights and sum to yield score.

As described previously, each extreme user represents a future dominated by a single guiding principle. The Traffic Management and Space Threat Awareness users are two examples. The set of extreme users should be selected to represent the most stressing user with that focus envisioned within the given timelines, since any less stressing user of the same focus can be created from a combination of the extreme user and the I dont care user. The extreme user set selected should be the smallest set possible that still allows for representation of any potential user group within the projected timeframe as a combination of the extreme users. Characteristics and goals of these extreme users should be laid out clearly.

Definition of performance parameters follows as with the single user case, however all parameters appropriate to any projected user should be included. A parameter that is irrelevant to an extreme user should be weighted zero by that user and assigned a uniform maximum utility. The range of performance values under consideration should include all performance values of interest to any user. For each parameter, the utility function for each extreme user will be defined over this entire range of performance values, although some extreme users may only assign non-maximum, non-minimum utilities to a portion of the range.

For each extreme user, assignment of weights and utility functions follows as with the single user. For each extreme user, the total weight allocation is the same (e.g., weights sum to 100). For each extreme user, for every parameter, the maximum value of the utility function is the same and the minimum value is no less than zero.

The candidate user is characterized by assessing its similarity to each functional extreme user. The similarity to the I dont care extreme user is captured through the similarities to the set of functional extreme users. A number in the range [0,1] represents the similarity to each extreme user, with 1 implying that the candidate user has the full characteristics of that extreme user, or full commitment to that extreme user. An assignment of 0 implies no similarity to the focus of that user. The number assigned is referred to as the influence of the extreme user on the candidate user. It is possible to have a candidate user that has the full characteristics of each of the extreme users (representing a very assertive multi-mission user community). Similarly, it is possible to have a candidate user that has no characteristics of any extreme user. If the extreme user set was properly selected, this no characteristics case should represent a user with no real interest in how well the system performs against the performance parameters.

The candidate user specification is not an allocation of a fixed total weight among extreme users such that the sum of the influence numbers is one. The level of similarity to each user is independent of the similarity to other users. Suppose a candidate user is equally similar to each of n extreme users. If the common similarity to all users was represented by 1/n, then there is no way to differentiate between a strong, proactive user community willing to dedicate a large amount of funding and a get a reasonable compromise community with modest funding. The candidate user can be represented as a coordinate in user space, with the focus for each extreme user defining an axis and the extreme user at the unit point on the corresponding axis.

Once the utilities for a candidate system are determined for every extreme user and the influence vector defining the candidate user is specified, the extreme user utilities and candidate influence vector are fed to the utility combination function to yield the utility vector for the candidate user. Similarly, the extreme user weights and candidate influence vector are fed to the weight combination function to yield a weight set for the candidate user. As with the single-user case, the system score for the candidate system is created by multiplying the candidate user utility for each system performance parameter by the corresponding candidate weight and summing over the performance parameters.
5. WEIGHT AND UTILITY COMBINATION FUNCTIONS

There are a variety of weight and utility combination functions that will achieve the properties of a bounded space of utilities, where the bounds are determined by the extreme user utility functions. This section describes one weight combination function and two utility combination functions along with the rationale for selecting them. The notation below is used in their specification:
Mi Performance parameter i
Sk Candidate system k
pki Performance of candidate system Sk on parameter Mi
EUj Extreme user j
wij Weight extreme user j (EUj) assigns to performance parameter Mi
wsum Sum of weights over all of the performance parameters (constant, we use wsum=100)
uij Utility EUj assigns to performance parameter Mi
umax Maximum allowed utility for any utility function
C, Ca Candidate user, Candidate user a
eaj Influence (level) of extreme user j (EUj) in candidate user Ca. (no superscript for single candidate)
wiCa Weight candidate user Ca assigns to Mi
uiCa Utility candidate user Ca assigns to performance parameter Mi

A natural combination function to generate weights for a candidate user is defined by Equation 1. This function yields a valid weight set such that the sum over all parameters equals wsum. Since one parameter can only be prioritized at the cost of others, this function represents the compromise of two users with different priority distributions. The compromise takes the influence of EUj into account by scaling the impact of EUjs weight by ej,
          (1)

It is natural to try a function of the same nature to combine utility functions; creating a new utility by a weighted averaging of the extreme user utilities. We argue that approach has the following shortcomings:
  1. The utility for a candidate user represented by ej ≡ 1, for all j, would be identical to the utility represented by ej ≡ 1/n, for all j, where n is the number of extreme users. The latter candidate should be more willing to accept a lesser system (i.e., give higher utility score to) than the former.
  2. The similarity to the I dont care extreme user is defined implicitly by the candidate vector. A single-focus candidate user (ej = x < 1 for j = m, ej = 0 otherwise) that does not have the strength of interest of the full-focus user would have the same utility as the full focus user despite a lower commitment to the system.
  3. If a candidate user includes a fully committed user (em = 1) along with partially committed users (ej > 0 for all jm), the averaging could yield a high score on a system that does not meet a critical need of a fully dedicated advocate.
  4. The weighted averaging does not reflect the fact that two modestly invested users who want a capability should jointly be less happy with a compromise on that capability than one modestly invested user with the same compromise.

It is possible to add the I dont care user into such a weighted averaging by assigning an influence to that user equal to the result of summing over all functional extreme users the difference between 1 and the influence number of each extreme user. This is equivalent to assigning the difference between the number of functional users and the sum of their influence numbers. The total influence, including the I dont care user (the denominator in the weighted sum) would be equal to the number of functional extreme users, and the resulting utility would be uic = 1/n [umax(n ej) + uijej]. What this option fails to represent is the case where a candidate user has full influence of each functional extreme user (influence number ej = 1 for each extreme user). The weighted sum would average the utilities over all extreme users, and we assert that given equivalent parameter weightings among the extreme users, the utility score should be no higher than that of any extreme user to whom the candidate is fully committed. In other words, if a stressing user is willing to bring his full force (and funding) to the architecture specification, then his full needs should be met for the system to receive a good score.

In consideration of a need to represent varying levels of influence of multiple types of users, the following are the desired properties for a utility combination function:
  1. The combination function should create a valid utility function within the utility space.
  2. The utility function for each performance parameter, Mi, should be based on the utilities of the extreme users and the level to which each of those users is represented in the candidate user.
  3. The combined utility function should represent the actual strengths of commitment to each extreme user and not just relative commitment among extreme users.
  4. For a given performance capability, gradual changes in the influence numbers for a candidate user should produce smooth changes in the utility.
  5. The combination function should represent the greater need of multiple partial masters over a single partial master.
  6. If the candidate user has a full commitment to the extreme user EUj with the lowest utility for the system performance, pki, on Mi and EUj weights Mi no lower than other extreme users, than the combined utility should be no higher than awarded by EUj.
  7. The function should be symmetric in the ordering of extreme users.

Both of the combination functions proposed in this paper are determined on a parameter-by-parameter basis and for each parameter are based on the evaluated extreme user utilities for a specific performance value, pki. Each combination function requires that the resulting extreme user utilities uij(pki) be ordered from minimum to maximum utility, which is referred as most stressing to least stressing. This ordering will shift with respect to the order of the definition of the extreme users as a function of performance parameter and even values of performance within a parameter. Therefore, the notation uij(l)(pki) represents the extreme user utilities on pki in ascending order (i.e., uij(l)(pki) ≤ uij(l+1)(pki)). An additional term and notation are required for the definition of both functions. The discount, dij(l), represents the amount of discount applied to the influence of extreme user j(l) on the combined utility for parameter i based on the influence of extreme users with more stressing utility functions. The discount is computed recursively, as shown in Equation 2, with dij(1)=0.
          (2)

Equation 3 shows utility combination function A, and it is shown in expanded form for the four-user case in Equation 4. It meets all of the desired properties and is a good example of how utility functions for multiple users might be combined when there is only one performance parameter of interest. If the hypercube bounded by the points (0,0,,0), (0,0,,1), , (0,1,,0), (1,0,,0), and (1,1,,1) is considered (representing the n functional extreme users, the I dont care user, and all of the extreme users at full influence level), then the effect on utility for the most stressing user, j(1), is represented by the volume within the hypercube bounded by xj(1) ≤ ej(1). The effect of the next most stressing extreme user, j(2), is the volume xj(2) ≤ ej(2), xj(1) > ej(1), which is proportional to ej(2)*(1-ej(1)) and the effect of the second most stressing user is reduced by the volume already considered by the first. Figure 2 shows this in two dimensions. Equation 4 shows that if ej(l)=1, the terms for less stressing extreme users zero out. Table 1 demonstrates a sample calculation for one parameter, where columns 2-4 show the utilities for the extreme users.
          (3)

  (4)

Fig. 2. Effect of ordered extreme users on candidate utility

Table 1. Extreme user (columns 2-4) and candidate user (columns 5-8) utilities, function A

  All EU1 All EU2 All EU3 All EUs No EU1 Half EU1 Half each EU
EU1 1 0 0 1 0 .5 .5
EU2 0 1 0 1 1 0 .5
EU3 0 0 1 1 1 0 .5
Utility of combination 1 2.5 4 1 2.5 3 2.25

Figure 3 shows the progression of utility curves for candidate users based on a single functional extreme user and the influence number for that extreme user set at 1.0, 0.8, 0.6, 0.4, 0.2, and 0.0. Note that the characteristic curvature of the original utility function is retained and the primary change is that of raising the minimal value. This is consistent with the notion that the I dont care user will grant a maximum score to any level of performance. Figure 4 shows an example of a case with two extreme users and combinations with various influence vectors. Note how the most stressing user affects the curves, even when which user is most stressing changes at 800 m.

Fig. 3. Graduated utility with changing influence from one user

Fig. 4. Combination with two users and crossing utility functions

Combination function A provides reasonable results when the most stressing utility for a given performance corresponds to the extreme user with the highest weight among the users for that parameter. It produced counterintuitive results when the ordering of weights did not map to the same set of extreme users as the ordering of utilities and the differences in the weights was significant. The utility of the stressing user dominates even if the user has a low priority for the performance parameter. As a result, the network scores may underrate total satisfaction of the multiple users of the candidate system.

Combination function B adjusts for the low weight/high stress discrepancy of function A by modifying the discount to include a term related to the extreme user weights. The term fij(l) is between zero and one, and it scales the discount dij(l). Equation 5 defines fij(l), where wi max is the maximum weight for performance parameter Mi over the extreme users. Equation 6 shows the adjustment to the discount, dij(l), and Equation 7 shows combination function B. Note that if the weights for the extreme users are the same, then combination functions A and B are the same. If the weights are different, then the effect of an extreme user is scaled down based on the relative weight difference between that user and the highest weighting user. Since a users raw impact is based on how stressing it is compared with other users, the effect of the fij(l) scaling is greater when a more stressing user has a lower weight than a less stressing user. Normalizing by wsum means that small differences in weights relative to the total weight allocation yield little effect.

          (5)

          (6)

          (7)
6. EXAMPLE WITH MULTIPLE PARAMETERS AND BOTH COMBINATION FUNCTIONS

The following example is again drawn from the problem of developing an architecture to support space surveillance in the ten to fifteen year future. The performance parameters and extreme users are realistic but simplified. The utility functions and weights are selected strictly for the purposes of illustrating the two combination functions (and their differences) and are not representative of US government needs or priorities. The utility scores for parameters are out of a maximum of 5, weights for a user sum to 100, and system scores are normalized to 0-100 scale.

The two extreme users selected for this example are; a) space threat awareness user, who wants to know what is in earth orbit on a very timely basis to determine whether there is something potentially hostile, and b) the space traffic management user, whose job it is to maintain easy use of space with limited risk of collision or interference for all operators. Jointly, the two users are interested in five performance parameters: 1) the time for access to track a satellite with maximum altitude less than 6000 km; 2) the minimum size of object that can be tracked by the system; 3) the accuracy to which an objects location can be estimated in low Earth orbit; 4) the minimum size of an object detectable at GEO; and 5) the capacity of the system in low Earth object tracks/day. Figure 5 (next page) shows the five sets of utility curves, with filled squares for the awareness user and X-markers for the traffic management user. The figure includes a table with the weights for each extreme user. Two candidate users are selected; C1 has half similarity to each of the extreme users, C2 has full similarity to the awareness user and half similarity to the traffic management user.

Table 2 shows a notional performance level of a candidate system in each parameter, the utility assigned by each extreme user to each parameter, the utility for each performance value for each of the two candidate users under combination function A and the corresponding set of utilities under combination function B. The rows are shaded gray for the parameters for which the most stressing extreme user is not the one who weights the parameter highest. Note that the parameter scores in the non-shaded rows are similar for the two combination rules, while they are notably higher for combination function B in the shaded rows. In particular, even though C2 is fully committed to the awareness user, C2 utilities do not default to the more stressing awareness utilities when the awareness weights are much lower than those for the traffic user. The overall score is higher for Combination B.

Table 2. Candidate System Utilities for Multi-parameter Example

 Combination A Combination B
Performance
parameter
Performance Utility
Awareness
Utility
Traffic
C1 C2 C1 C2
Time 180 3.13 4.38 3.90 3.13 3.94 3.13
Size (LEO) 90 2.44 3.33 3.30 2.44 3.43 2.70
Position 6000 1.13 4.56 2.95 1.13 3.22 1.67
Size (GEO) 200 1.08 3.00 2.54 1.08 2.54 1.08
# tracks (LEO) 100000 2.44 1.19 2.45 1.81 2.58 1.87
System  44 70 61 39 63 43

Weights (100 total) Aware Traffic
Time to track 30 10
Minimum LEO Size 15 30
Position Accuracy 15 30
GEO size 20 20
# tracks/day 20 10

Fig. 5. User preferences for multi-parameter example
7. SUMMARY AND FUTURE WORK

This paper describes a framework for multivariate utility analysis that supports evaluation of multi-function systems in the context of a future user when the nature of that future user is uncertain. The problem of specifying utilities for arbitrary users is reduced to a parametric one, where only a small set of single-minded users need to be characterized in terms of weights and utilities, and the preferences for all other users can be generated mathematically from the preferences of the few. In addition, the use of extreme users provides a measure of worst-case utility over all envisioned users.

Two combination functions were proposed for generating user preferences from the preferences of extreme users. Each function represents the additive effects of multiple user needs on the development of a multi-functional system-of-systems, as well as the differences between a seriously interested user and a moderately interested user with the same focus. The first function provides more stressing utility functions, dominated by the more stressing user for each parameter. The second function reduced the impact of users who place a lower priority on a parameter than other users.

Future work will include research into other combination functions. Two specific areas have been identified for exploration. The first is based on an observation that if an extreme user assigns a utility of zero to a range of performance capabilities, when the influence number for that extreme user is less than one and no other extreme users are considered, the utility for that entire range is raised to a uniform value. It may be more representative to have some ramping up in utility over that originally zeroed range. The second returns to the earlier observation that this framework does not provide the means to represent the case in which a user has one or more key parameters such that the candidate system should be assigned zero utility if its performance in the any of the key parameters is not better than the corresponding threshold. A multiplicative component to the calculation of system utility might be appropriate (as opposed to the current weighted sum), but work is needed to identify ways to incorporate such a component while maintaining the desired utility and combination properties.

The algorithms are relatively simple to program and were initially implemented as Visual Basic macros in combination with a Microsoft Excel workbook. This implementation supported evaluation of more than fifty candidate space surveillance architectures against fifty parameters for four extreme users. To support analysis of several thousand candidates and a larger number of extreme users, a MATLAB version of the code was developed. Since both implementations were developed in support of specific evaluation efforts, user interfaces were cumbersome and required detailed knowledge of the program and data entry interface. While it was easy for an experienced user to add new candidates for evaluation, it was a highly manual process to add the results to charts and graphs that compared candidates. Work is currently in progress to implement the algorithms in a user-friendly package that does not require extensive knowledge of MSExcel and that provides easy commands to generate commonly desired charts and graphs.

Acknowledgements. This work was sponsored by the Dept. of the Air Force under Air Force Contract NoFA8721-05-C-0002. Opinions, interpretations, conclusions and recommendations are those of the author and are not necessarily endorsed by the United States Government.
REFERENCES

28 2006

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