Florentin Smarandache - Neutrosophic Operational Research: Methods and Applications

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The world life of every human being has some problems of uncertainty, imprecise, incomplete, and inconsistent information. Fuzzy set theory is one of the wide frameworks for uncertainty. In the year 1965, A. Zadeh [] defined some similarity measures of single-valued neutrosophic sets in decision-making problems.

Neutrosophic set is an effective and useful tool to describe problems with uncertainty, imprecise, incomplete, and inconsistent information. In this regard, Smarandache and Pramanik [] introduced an ELECTRE approach to finding the best alternative for multi-attributed decision-making problems in a refined neutrosophic environment. Under the neutrosophic environment, every abovementioned researcher founded the best alternatives or the alternatives with the alternatives or the alternatives with the attributes for the entire region (problem). We may here ask some questions Is it possible to find the best alternatives not only in the entire region but in each region? and Can we formulate a new method to identify the alternatives in each and entire region?

The hybridization would cross different plants such as grasses rice, maize, cotton, and wheat for the new hybrid plant which give different increased yield and improved grain quality in both cross- and self-pollinated crops. In the eighteenth century, Mendel [], an Augustinian who is the author of Experiments on Plant Hybridization, produced the F1 hybrid by cross-breeding pea plants. The present chapter formulates a new method to answer the above questions for multi-criteria decision-making problems under neutrosophic environments. The beauty of this chapter is the proposed method uses the neutrosophic set-theoretical operation such as complements, intersections, unions, and single-valued neutrosophic score functions in both direct and reverse direction to identify the best alternatives in each and the entire region. The plant hybridization problem is solved under neutrosophic environments as a real-life application to demonstrate the effectiveness of the proposed method because the hybrid grain yields are varying place to place.

This section presents some of the basic properties of neutrosophic sets and operations on neutrosophic sets which are used for further study.