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# Open Issues Regarding Use Case Discussions

### Combining Probabilistic and Fuzzy Uncertainty (Mitch/Peter)

Scenario: Make a decision on (1) whether to go to a store today or wait until tomorrow to buy speakers, (2) which speakers to buy and (3) at which store.

1. There is known probability distribution on the availability of particular speaker type in particular stores on a particular day in the future. Say there are two stores (not too close to each other) and the probability that speakers of type X will be available in stores A and B tomorrow are Pr(X, A)=0.4 and Pr(X, B)=0.6. The probabilities for all types of speakers are represented in the same way.

2. The customer has fuzzy preference functions on particular features of speakers. Say f_bass(X) represents customer's assessment of the goodness of particular speakers (the X variable) with respect to the feature "bass". Similarly, the customer may have such functions for other features, like "color", "weight", "high tones", and such.

3. The customer knows both of these types of uncertainty information, probabilistic and fuzzy, since the probabilities and the features (bass, color, weight and so on) are published on the web sites for the stores. The stores use the Uncertainty Ontology for this purpose.

4. The customer has a combination function that computes the decision, d, based upon those types of input. This function is specific to each customer and thus we don't need to worry how the customers do this, but we need to give them the inputs - the probabilities and the features. The features are fuzzified by the customer's client software. The client uses the Uncertainty Ontology to annotate the fuzziness of particular preferences. (Kathy: if the client is making the decision entirely internally, then what is the need for annotating the fuzzy preferences?)

/begin_note_by_Peter Fuzzy part resembless traditional decission support model of modeling objectives and utility functions, here adaptable to different users and utility function need not to be additive, see users_preferences.pdf /end_note_by_Peter

### Alternate Version (Kathy)

1. There is a probability distribution on the availability of particular speaker type in particular stores on a particular day in the future. Say there are two stores (not too close to each other) and the probability that speakers of type X will be available in stores A and B tomorrow are Pr(X, A)=0.4 and Pr(X, B)=0.6. The probabilities for all types of speakers are represented in the same way. This model would be obtained by dynamically updating a statistical model for predicting availability based on historical data on availability.

2. Some attributes of speakers are given as crisp values. For example, numeric values are given for the weight in kilograms, the resistance in ohms, the maximum power in watts. There are also crisp qualitative attributes, such as magnetic shielding (yes/no), or woofer and tweeter composition. There are also features that are expressed with fuzzy membership functions. An example is "color." Some speakers have a membership of 1.0 for a particular color, such as a speaker that is completely black. However, some speakers may have a combination of colors, or may not easily be described with one of the standard color names. Such a speaker might be defined as, say, having a "fuzzy blue value" of 0.8. There are also fuzzy descriptions of the sound quality, such as "base is 0.7 rich, and 0.85 warm," or "treble is 0.2 wobbly."

3. The customer has measurable value functions for particular features of speakers. Say v_bass(X) represents customer's assessment of the goodness of particular speakers (the X variable) with respect to the feature "bass". Similarly, the customer may have such functions for other features, like "color", "weight", "high tones", and such.

4. The crisply defined features (weight, resistance, and so on) are published on the web sites for the stores. The stores use a probabilistic ontology to represent the uncertainty about availability. It uses an ontology for multiattribute utility to represent the value functions for the different attributes. It uses a fuzzy ontology for representing the fuzzy membership functions. The fuzzy memberships are input by third-party reviewers, and are annotated with the reviewers' identification information.

5. The customer has a combination function that computes the decision, d, based upon those types of input. This function is specific to each customer. We don't need to worry about the exact method by which users do this, but we need to know enough about how they do this to make sure they have the inputs they need. Specifically, we are assuming they will need the probabilities, the crisp features, and the fuzzy membership functions. Under item 3, we are assuming that there is some process that elicits measurable value functions from a user, combines them into a multiattribute value function, and assesses risk attitude for a utility function. A good reference for multiattribute utility is Robert Clemen, ''Making Hard Decisions''.

6. The forecasting service that estimates availability uses historical availability data annotated by date and tagged by the product ontology. The statistical model used for forecasting is represented using a probabilistic ontology. A Bayesian learning system combines new data with previous data to obtain current availability forecasts.