Advanced computing with words using syllogistic reasoning and arithmetic operations on linguistic belief structures

In this paper, we present solutions to an Advanced Computing with Words problem that is equivalent to one of Zadeh´s challenge problems on linguistic probabilities. We use a syllogism based on the entailment principle to interpret the problem so that it yields two linguistic belief structures. Then we perform an addition of those linguistic belief structures to obtain a belief structure on the variable about which linguistic probabilities have to be inferred. We show that pessimistic (lower) and optimistic (upper) probabilities can be inferred from such a belief structure using Linguistic Weighted Averages and pessimistic and optimistic compatibility measures. Then, we choose vocabularies for linguistic attributes (lifetimes of products) and linguistic probabilities that are involved in the problem statement. The vocabularies are modeled using interval type-2 fuzzy sets. We calculate optimistic (upper) and pessimistic (lower) probabilities, and map them into words present in the vocabulary of linguistic probabilities, so that the results can be comprehended by a human.