A Neutrosophic ; new view of modeling , designed to effectively address the uncertainties inherent in the real world, It came to replace the binary logic that admits right and wrong by introducing a neutral third state that can be interpreted as undetermined or uncertain. It was founded by the American mathematician Florentine Smarandache [2,4,5,6,8] . Where he presented it in 1995 as a generalization of fuzzy logic and an extension of the theory of fuzzy sets [3] presented by Lotfi Zadeh in 1965 [1]. In addition, Ahmed Salama presented the theory of classical neutrosophic sets as a generalization of the theory of classical sets [7,15,16], and he developed, introduced and for mulated new concepts in the fields of mathematics, statistics, computer science and others through neutrosophic  [17,25]. 

Neutrosophic has grown significantly in recent years, through its application in measurement, sets, graphs, and many scientific and practical fields [10,11,12,13,14,19,20,21,22].

In this paper, we highlight the application of neutrosophical logic to one of the inventory management models, which is the static model without deficit and for a single substance, which will open the way for us to deal with inventory management models whose formulas are expressed in a way that is not precisely defined or uncertain, as if We take ranges of values ​​instead of a single value, as is known in the classic, and in particular we will present the static model without a deficit for a single substance according to the neutrosophic logic ,when the rate of demand for stock is defined in an undefined way, and then we will present an applied example that shows how to deal with this case.

2. Discussion:

Inventory management plays an important role in production and marketing operations, especially in production facilities and commercial establishments that have warehouses where they keep their equipment and goods[9].

 Inventory management is one of the most important functions of management, in terms of determining the ideal size of inventory and calculating its costs, as this affects the efficiency of the facility and achieves either great profits or causes it to suffer heavy losses.

It is often the responsibility of warehouse managers in production facilities or enterprises to determine the appropriate and ideal volume of Inventory of each material, to secure the demand in a certain time. We know that if the volume of the Inventory is very large, this guarantees the provision of the material on the one hand, but in return it may cause the institution losses because the value of the Inventory is frozen capital, and this large quantity needs a marketing period that depends on the rate of demand.

But if the inventory quantity is small, this may lead to a stalemate in securing materials and to various disturbances, such as price hikes and others.

Therefore, it was necessary to present a study to help us determine the ideal size of the inventory  at the lowest possible cost.

When studying static inventory models in classical logic, the rate of demand for inventory was limited, and we found this rate is subject to a uniform probability distribution. Thus when the rate of demand for inventory is undetermined with the time, we use the neutrosophic uniform probability distribution, which was studied previously [18 ].

2.1. The main hypotheses of the study:

1.     order size   Q .

2.     The rate of demand for inventory during a given time l_N (unspecified) where  l _N = {l₁, l₂}  or                         l _N = [l₁, l₂]   or … .

so that:    l₁ minimum rate of demand for inventory.

                             l₂ Maximum rate of demand for inventory.

3.     Fixed cost of order preparation  C₁ = K.

4.     Cost of purchase, delivery and pickup  C₂ = C. Q  .

5.     Storage cost for the remaining quantity in the warehouse during a certain time C₃.

6.     inventory run out time is    ( or the duration of the storage cycle).

-        We denote by q_t the quantity remaining in the warehouse at moment  t  during the period    

               (the duration of the storage cycle) , q_t is given as:   

                                                               q_t = Q - l_N t 

-       To calculate C₃ the "storage cost", we divide the period into  n  partial periods,  of length Dt.

If  t_i  is a point of partial period i  , then the remaining quantity of inventory corresponding to that partial period is equal to:

                                               q_i = Q - l_N t_i   

            Thus the storage cost in the partial period is:

                                              C_i = h .q_i . Dt = h(Q - l_N t_i) Dt

        Therefore, the total storage cost during the period    is the sum of the partial costs,    

          ie:                

          When Dt ® 0 , the number of partial periods is n ® ¥, then:

                                     = 

   That is, the cost of storage during    is:  

The total cost is:   

                                        TC(Q) = C₁ + C₂ + C₃

                                            = K + CQ + 

      To get the cost of storage at a given time, we divide the total cost TC(Q) by the length of the period , 

          denoted by  the symbol C(Q):

       To determine the maximum values of the function C(Q), with the condition Q > 0, we find:

       To specify the type of this maximum value:   

So the function C(Q) achieves a minimum value when           , The ideal size for the demand is:

                                                                                ……(**)

3. Application example:

   Assuming we have a production facility, which stores spare pieces  for its production lines, and that the rate of demand for these pieces  in the warehouse is not absolutely specified, but ranges between [2.4] pieces per month, According to the conditions of the facility such as working pressure, operating pressure, weather conditions, etc., and that the cost of Preparing a single demand of Q Pieces is $100, and the cost of storing one piece  for a month is $20.

Let us calculate the ideal size of the demand , which makes the cost of storage as small as possible, taking into account that the rate of demand for Pieces is an indeterminate case and ranges between  l_N = [2, 4].

And :

K = $100 (the cost of preparing the demand ).

h = $20 (the cost of storing one piece for a month).

We find, based on (**):

That is, the ideal size of the demand  ranges from 4 to 6 spare Pieces per month, so that we remain in the safety of Inventory management and avoid reaching the stage of deficit.

-        Calculating the duration of the storage cycle in hours, knowing that the number of working days per month is 23 days and the number of working hours per day is only 8 hours. We find:

The time period is in hours:

[1.1, 3.2] ´ 23 ´ 8 = [202.4 , 588.8]

That is, between 8 to 24 days.               

4. Conclusion and results:

We conclude that dealing with inventory management models, under the neutrosophic  logic, provides us with a more general and comprehensive study than the well-known classical study, so that it does not neglect any data just because it is not explicitly specified, especially in our dealing with the static model without a deficit , we found that the unspecified value of demand for inventory has affected the ideal size of the inventory , and it was given approximatly,  so that it gives the minimum ideal size of the inventory  that protects from reaching the stage of deficit and loss, and gives the maximum ideal size of the inventory  so that it makes the inventory available in the best quantity without cause loss for the facility.

Thus, we find that the  indeterminacy actually affects the results, and these unspecified values ​​cannot be ignored and removed from the study framework , in order to obtain the most accurate results as possible, and thus obtain the ideal volume of inventory that meets the needs of the facility , during the period of the storage cycle at the lowest cost.

We find that working within the classical logic is no longer sufficient at the present time, as the development of science has put a large number of new problems that need more general and accurate results than the results we got it using classical logic and fuzzy logic, and here comes the role of neutrosophic logic, which provides us with more comprehensiveness to  Interpret the study data and get the most accurate results possible.

In the near future, we look forward to studying the rest of the inventory management models according to the neutrosophic logic , such as inventory models with a deficit , dynamic models, and others..

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