A Single Valued neutrosophic Inventory Model with Neutrosophic Random Variable

This paper presents the problematic period of neutrosophic inventory in an inaccurate and unsafe mixed environment. The purpose of this paper is to present demand as a neutrosophic random variable. For this model, a new method is developed for determining the optimal sequence size in the presence of neutrosophic random variables. Where to get optimality by gradually expressing the average value of integration. The newsvendor problem is used to describe the proposed model.

The concept of fuzzy random variable and its fuzzy expectation has been presented by (Kwakernaak H. 1978) and later by Puri and Ralescue (Puri M.L. andRalescu D. A. 1986). Further, recently the notation of a fuzzy random variable has also been considered in (Feng Y., Hu L. and Shu H. 2001). In (Smarandache F. 1998) proposed concept of neutrosophic set which is generalization of classical set, fuzzy set, intuitionistic fuzzy set and so on. In the neutrosophic set, for an element x of the universe, the membership functions independently indicates the truthmembership degree, indeterminacy-membership degree, and false-membership degree of the element x belonging to the neutrosophic set. Also, fuzzy, intuitionistic and neutrosophic models have been studied by (Wang H, Smarandache F, Y, Zhang Q 2010). In a multiple-attribute decision-making problem the decision makers need to rank the given alternatives and the ranking of alternatives with neutrosophic numbers is many is many difficult because neutrosophic numbers are not ranked by ordinary methods as real numbers. However it is possible with score functions , aggregation operation , distance measure and so on. In section 2 of this paper, the neutrosophic random variable and its neutrosophic expectation are defined, a brief overview of the integration of graded mean representation of triangular neutrosophic number discussed later. Next, in section 3 a single-valued neutrosophic inventory problem of neutrosophic random variable demand is formulated and methodology is developed. Section 4 handles the numerical example of the proposed model.

Preliminaries
In this section, the basic definitions involving neutrosophic set, single valued neutrosophic sets and triangularneutrosophic number are outlined. Definition 1. (Smarandache F. 1998 is a special neutrosophic set on the set of real number , whose truth-membership function , indeterminacy-membership function and falsity-membership function are respectively defined by . Clearly, any cut set of a is a crisp subset of a real number set Also any cut set of a SVN-Number for truth-membership function is a closed interval, denoted by .

Definition 7.(Deli I, Subas y) Let be a
Then -cut set of the , denoted by is defined as , , where .
Clearly, any cut set of a is a crisp subset of a real number set Also any cut set of a SVN-Number for indeterminacy-membership function is a closed interval, denoted by .

Definition 8.(Deli I, Subas y) Let be a
Then -cut set of the , denoted by is defined as , , where .
Clearly, any cut set of a is a crisp subset of a real number set Also any cut set of a SVN-Number for indeterminacy-membership function is a closed interval, denoted by .

Cuts and neutrosophicgraded mean integration:
α, β and γ-cuts, expectation of neutrosophic random variable and neutrosophic graded mean are introduced in this section. This type of generalized neutrosophic number is a triangular neutrosophic number, and its denoted by As the neutrosophicdemand is a neutrosophic random variable, so its neutrosophicprofit function is also a neutrosophic random variable and obviously its total neutrosophic expected cost becomes a unique neutrosophic number.Therefore the neutrosophic total expected profit ( ) is determined by,

Mathematical Model with Neutrosophic Random Variable
We consider the problem of stocks described above in a period when demand is seen as a neutrosophic random variable. The data are not accurate for neutrosophic probabilities so for simplicity, all data sets and probabilities are considered as triangular neutrosophic numbers and for , respectively. Moreover the neutrosophic optimal order quantity , we have The result (7) and (8), are gives us the neutrosophic optimal order quantity , which satisfies as following, where, and Therefore, we conclude that the neutrosophic optimal order quantity can be determined over a single-valued neutrosophic inventory model when requirements are mentioned in a mixed environment, it is more accurate and less certain that it is more realistic for an inventory. Then if an inaccurate probability is specified for the search data set, we give a numerical example of how to get the neutrosophic optimal order quantity.

Numerical Example
To illustrate this model, suppose a newsvendor cannot pay in cash for a day if he needs more papers. Let the neutrosophic purchase price , the neutrosophic price , the neutrosophic holding cost , and the neutrosophic shortage price .The daily neutrosophic demand for this section is unknown, but based on experience and previous sales dates, you can set neutrosophic probabilities for different search levels of neutrosophic demand. The neutrosophic demand and neutrosophic probability are given in the first table. Now, using our methodology you can find the neutrosophic optimal order quantity from second table.  Let A and B be the numerator and denominator of the upper limit of (9) respectively. Then the neutrosophic optimal order quantity is gives , this means that for newsvendor it is better to buy about 42 newspapers in order to maximize the expected daily profit.

Conclusion
This paper, introduces the development of stochastic neutrosophic inventory models using neutrosophic random variables to get more realistic information, where fixed value is invalid. Here, a single-valued neutrosophic inventory models are discussed when there are inaccuracies and uncertainties in inventory system. This aggregation can be extended to other neutrosophic inventory models.