A modification of Dempster's and Pawlak's constructs forms a new foundation for the identification of upper and lower sets formulas. Also, in this modified Dempster-Pawlak construct we require that subsets of the power set be restricted to the well-known information granules of the power set. An aggregation of upper information granules amongst each other and lower information granules amongst each other determine upper and lower set formulas for both crisp and fuzzy sets. The results are equivalent to the Truth Table derivation of FDCF and FCCF, Fuzzy Disjunctive Canonical Forms and Fuzzy Conjunctive Canonical Forms, respectively. Furthermore, they collapse to DNF \equiv CNF, i.e., the equivalence of Disjunctive Normal Forms and Conjunctive Normal Forms, in the combination of concepts once the LEM, LC and absorption, idempotency and distributivity axioms are admitted into the framework. Finally, a proof of the containment is obtained between FDCF and FCCF for the particular class of strict and nilpotent Archimedian t-norms and t-conorms.
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This survey presents crisp and fuzzy models developed for the Supply Chain (SC). To this end, fuzzy concepts are first briefly reviewed. Then the structure of a supply chain system is explained. Recent challenges and ideas on these systems are also surveyed. Then, with reference to different aspects of an SC system, we classify the recent fuzzy models developed for different SC systems and compare them with related crisp models. Applications of the SC in manufacturing and service industries are investigated in each case.
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