An Investigation in the Initial Solution for Neutrosophic Transportation Problems (NTP)

Transportation issues arise often in everyday life. To ensure that the regions' needs for transported material are met at the lowest feasible cost, materials must be carried from production centers to consuming centers as quickly as possible. Operations research approaches, notably mathematical programming, are utilized to solve these recurring and daily challenges. The problem's data is transformed into a mathematical model, and then the best solution is discovered using the proper procedures. When dealing with transportation issues, we arrive at a linear mathematical model, which can be solved using the direct simplex method and its modifications. However, because of the clarity and specificity of the transportation model, scholars and researchers were able to find other methods that were easier than the simplex method. Whatever method is used, the goal is to determine the number of units transferred for any material from the production centers to the consumption centers in order to minimize transportation costs, keeping in mind that each export center has its own capacity and cannot supply quantities of the material greater than that capacity. Furthermore, each import center has a certain requirement for which it makes a request and for which it is unable to consume further quantities. In this manuscript, the researchers will use the North-West Corner approach, the Least-Cost method, and Vogel's approximation method to discover an initial solution to the balanced neutrosophic transport problems. The term "neutrosophic transportation problems (i.e. NTP)" refers to the transportation problems in which the required and available quantities have neutrosophic values of the form = + , where is the indeterminacy in the produced quantities, and it is either = [1, 2] or = {1, 2 }. While the required quantities are also neutrosophic values of the form = + , here is the indeterminacy on the required quantity, and it is either = [1, 2] or = {1, 2}. It is worthy to mention that when the problem is unbalanced (i.e. the summation of the required quantities is not equal to the produced quantities), then firstly will convert the problem to a balanced one. In this article, the authors assumed the representation of the neutrosophic numbers as intervals such as = [1 , 2 ], = [1 , 2 ]. It is important to notice that the authors did not adopt (trapezoidal numbers, pentagonal numbers, or any other neutrosophic numbers which need to specify using the membership functions, this kind of neutrosophic numbers or parameters represented by intervals have been firstly introduced by Smarandache F. in his main published books [32,33]. The sections of this manuscript has been organized as follow: the introduction is the inception of this article, section one has been dedicated to the north-west corner method containing three subsections for three case studies depending upon the existence of the indeterminacy in the problem, the least- cost method represents section two, while section three has been devoted to Vogel's approximation method, section four is conclusion and results