Decision Rules and Decision Analysis

AuthorSalvatore Greco, Benedetto Matarazzo, Roman Slowinski, Wendy Mason

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A decision rule is a logical statement of the type "if [condition], then [decision]." The following is an example of a decision rule experts might use to determine an investment quality rating:

If the year's margin is at least 4.27 percent and the year's ratio of shareholder funds to fixed assets is at least 35.2 percent, then the class of rating is at least lower investment grade (LIG).

The condition in this decision rule is "the year's margin is at least 4.27 percent and the year's ratio of shareholder funds to fixed assets is at least 35.2 percent," while "the class of rating is at least lower investment grade" is the decision part of the rule.

Decision rules give a synthetic, easily understandable, and generalized representation of the knowledge contained in a data set organized in an information table. The table's rows are labeled by objects, whereas columns are labeled by attributes; entries in the body of the table are thus attribute values. If the objects are exemplary decisions given by a decision maker, then the decision rules represent the preferential attitude of the decision maker and enable understanding of the reasons for his or her preference.

People make decisions by searching for rules that provide good justification of their own choices. However, a direct statement of decision rules requires a great cognitive effort from the decision maker, who typically is more confident making exemplary decisions than explaining them. For this reason, the idea of inferring preference models in terms of decision rules from exemplary decisions provided by the decision maker is very attractive. The induction of rules from examples is a typical approach of artificial intelligence. It is concordant with the principle of posterior

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rationality, and with aggregation-disaggregation logic. The recognition of the rules by the decision maker justifies their use as a powerful decision support tool for decision making concerning new objects.

There are many applications of decision rules in business and finance, including:

Credit card companies use decision rules to approve credit card applications.

Retailers use associative rules to understand customers' habits and preferences (market basket analysis) and apply the finding to launch effective promotions and advertising.

Banks use decision rules induced from data about bankrupt and non-bankrupt firms to support credit granting decisions.

Telemarketing and direct marketing companies use decision rules to reduce the number of calls made and increase the ratio of successful calls.

Other applications of decision rules exist in the airline, manufacturing, telecommunications, and insurance industries.


The examples (information) from which decision rules are induced are expressed in terms of some characteristic attributes. For instance, companies could be described by the following attributes: sector of activity, localization, number of employees, total assets, profit, and risk rating. From the viewpoint of conceptual content, attributes can be of one of the following types:

Qualitative attributes (symbolic, categorical, or nominal), including sector of activity or localization

Quantitative attributes, including number of employees or total assets

Criteria or attributes whose domains are preferentially ordered, including profit, because a company having large profit is preferred to a company having small profit or even loss

The objects are compared differently depending on the nature of the attributes considered. More precisely, with respect to qualitative attributes, the objects are compared on the basis of an indiscernibility relation: two objects are indiscernible if they have the same evaluation with respect to the considered attributes. The indiscernibility relation is reflexive (i.e., each object is indiscernible with itself), symmetric (if object A is indiscernible with object B, then object B also is indiscernible with object A), and transitive (if object A is indiscernible with object B and object B is indiscernible with object C, then object A also is indiscernible with object C). Therefore, the indiscernibility relation is an equivalence relation.

With respect to quantitative attributes, the objects are compared on the basis of a similarity relation. The similarity between objects can be defined in many different ways. For example, if the evaluations with respect to the considered attribute are positive, then the following statement may define similarity:

For instance, with respect to the attribute "number of employees," fixing a threshold at 10 percent, Company A having 2,710 employees is similar to Company B having 3,000 employees. Similarity relation is reflexive, but neither symmetric nor transitive; the abandon of the transitivity requirement is easily justifiable, remembering, for example, Luce's paradox of the cups of tea (Luce, 1956). As for the symmetry, one should notice that the proposition yRx, which means "y is similar to x," is directional; there is a subject y and a referent x, and in general this is not equivalent to the proposition "x is similar to y."

With respect to criteria, the objects are compared on the basis of a dominance relation built using out-ranking relations on each considered criterion: object A outranks object B with respect to a given criterion if object A is at least as good as object B with respect to this criterion; if object A outranks object B with respect to all considered criteria then object A dominates object B. An outranking relation can be defined in many different ways. Oftentimes, it is supposed that outranking is a complete preorder (i.e., transitive and strongly complete). For each couple of objects, say object A and object B, at least one of the following two conditions is always verified: object A outranks object B and/or object B outranks object A. A dominance relation, built on the basis of the outranking relation being a complete preorder, is a partial pre-order (i.e., it is reflexive and transitive, but in general not complete).


The syntax of decision rules is different according to the specific decision problem. The following decision problems are most frequently considered:





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Following is a presentation of the syntax of decision rules considered within each one of the above decision problems.


Classification concerns an assignment of a set of objects to a set of predefined but non-ordered classes. A typical example of classification is the problem of market segmentation; in general there is no preference order between the different segments. The objects are described by a set of (regular) attributes that can be qualitative or quantitative. The syntax of decision rules specifies the condition part and the decision part.

With respect to the condition part, the following types of decision rules can be distinguished:

Decision rules based on qualitative attributes: "if the value of attribute q1 is equal torq1 and the value of attribute q2 is equal to rq2 and … and the value of attribute qp is equal to rqp, then [decision]," where rq1, rq2, …, rqp are possible values of considered attributes.

Decision rules based...

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