On Some Bipolarity Related Conjunctive and Disjunctive Type Aggregation Operators: Can Z-Numbers Provide a New Functionality?

We take as a point of departure our approach (Kacprzyk and Zadrożny, 2005–2023) to the so-called bipolar queries to (relational) databases in which the user’s intentions and preferences concerning what should be retrieved are first to be found and then formally represented. The fuzzy query is extended to the so-called bipolar query in which the user can specify the query via two conditions, one which should be necessarily satisfied, called a mandatory, required, obligatory, etc. condition, and one which is optional, or desired, the satisfaction of which is not necessary, or mandatory, but which should be satisfied, if possible. Such a bipolar query may be exemplified by, in the context of real estate, as that the user looks for “an inexpensive house and, if possible, close to public transportation”. In this case, the condition “inexpensive house” is meant to be mandatory because the person can just only afford such a house or is not willing to spend much money. The second condition, “close to public transportation”, is meant to be optional because its satisfaction would be good but a failure to satisfy it would not make the use of the house impossible as opposed to the mandatory condition for which, if the house is not inexpensive, then the user cannot or does not want to purchase it. Therefore, such bipolar database queries involve a non-conventional conjunctive type of aggregation operator “and, if possible” which is clearly not the same as the traditional and widely employed “and”. The representation of such bipolar queries and their processing has been proposed using various tools and techniques, notably some fuzzy logic based propositional calculi, some specific database operators like the winnow, etc. In a similar way this idea has been extended by Kacprzyk (2017–2023) to a related case in which the conjunctive type “and, if possible” aggregation operator is replaced by a disjunctive type operator, “or, if impossible” the essence of which is that the two conditions are accounted for but if the first condition is not satisfied, then it is dropped and the second condition is just taken. For example, if the query is “an inexpensive house or, if impossible, well located”, that is, we want to have an inexpensive house but if, e.g., in this city or its part there are no inexpensive houses, then “well located” is the key condition to be taken. We show that these two types of bipolar queries open new vistas in getting a more adequate response to the users’ interests and intentions and can also help better reflect contexts in which querying can proceed. Moreover, virtually all real-world complex decision making problems boil down to the choice of an option from a finite, usually not too large, set of feasible, satisfactory, possible, etc. options, exemplified by scenarios that dominate all kinds of non-trivial real world economic, business, etc. problems. This is, in our perspective, essentially the same as querying a (relational) database in which the tuples correspond to options described via some attributes. We present the representation and handling of such querying (choice) problems using mainly our fuzzy logic-based approach which involves various aggregation operators, notably the triangular norms (t-norms) and triangular conorms (t-conorms). Now we consider an extended case in which the usual satisfaction degrees of the mandatory and optional condition are not degrees in the unit interval but are specified as Zadeh’s Z-numbers defined in [0,1]. Then, we use some theoretical and algorithmic elements employed in the field of Z-numbers, notably those related to the handling of various aggregation operators, in particular the t-norms and t-conorms (cf. Aliev et al., 2016), and show how the new forms of the “and, if possible” and “or, if impossible” aggregation operators can be represented by using the Z-numbers. We analyze possible computational difficulties while using some forms of Z-number based aggregation operators and propose the use of some simplified algorithmic versions which would be useful for larger sets of possible options. We finish with some general remarks on the use of Z-numbers in various aggregation related contexts.