The solution of the transport problem by the method of the smallest element based on the use of complex numbers in the algorithm

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Introduction
Two main types of transport problems are used in economic and mathematical calculations. The first type of tasks solves the problem of minimizing the time for transportation in the second type of tasks solves the problem of imaginary material costs for transportation. According to [1][2][3][4], the transport problem is a kind of linear programming problem. Among the methods used to solve the transport problem, one can distinguish [5][6][7] (Fig. 1). Usually, the following formulation is used for the transport task [8][9][10][11][12][13][14][15]. Have suppliers of a homogeneous product , and each of these suppliers has , respectively , , … . , units of this product . This item needs to be shipped to buyers , in quantity , respectively , , … . , units. The cost of delivery of the goods from the -th seller to the -th buyer is equal to . According to the conditions of the transport task, it is required to determine the schedule of transportation of goods between sellers and buyers in such a way that to deliver all the goods while spending the minimum amount of material resources to move it.
-the quantity of goods transported between by the -th seller and the -th buyer.

Methods
When writing this scientific work, we used an analytical method by which the problems studied were studied in their development and unity.
Taking into account the goals and objectives of the research, a functional and structural method of performing scientific research was used.
This allowed us to consider a number of issues related to the use of complex numbers in algorithms for solving the transport problem.

Simplex method
Method of finding reference plan Using graph theory The smallest element method The Northwest Corner method Let's introduce a new variable exp ( ) = Taking into account the changes, (1) can be written as for the Lagrange function [14 ,15] for the transport problem Substituting (2) and (4) into (3) we get: Since the Hessian of function (5) is positively defined for any positive values , then the condition for having a solution to the transport problem (1) is the presence of a solution to the system of equations (5).
Consider a specific example. Let's say there are four suppliers 1, 2, 3, 4 some product that needs to be delivered to four stores 1, 2, 3, 4 it is required to find a schedule for the transportation of goods at which the cost of transportation is minimal (Table 1). Let 's rank the coefficients in descending order and write them to (Table 2).   We find in Table 3 a complex number with a minimum real part, this number is 1 + i 15 the cost of shipping 15 units of goods from Supplier A3 to buyer B1 will be imaginary The next complex number with the minimum real part is the number 3+i5 , the cost of shipping 15 units of goods from Supplier A1 to buyer B1 will be imaginary (Table 4).  The next complex number with the minimum real part is the number 7+i20 , the cost of delivering 15 units of goods from Supplier A3 to buyer B1 will be imaginary, the remaining 5 units of goods supplier A3 sends to consumer B1 because the imaginary number 8+i20 follows the number 7+i20 (Table5). The remaining 100 units of goods that are available to supplier 2 are distributed among buyers B2 , B3 , B4 (Table 6). It is easy to see that the example considered in this paper corresponds to the system of equations (2) ( Table 7).

Discussion
To find out if there is a solution to the transport problem (1), you can use the Lagrange multiplier method. To do this, you can replace variables so that the objective function is a sum of exponentials. The peculiarity of the sum of exponents is that the principal diagonal minors of the Hessian of the sum of exponents are positive quantities , and therefore the sum of exponents has an extremum and this extremum is the minimum.
Large -scale transport tasks are of great practical importance for optimizing transportation schedules by transport enterprises . There are several algorithms for solving this problem, but the development of other methods for solving the transport problem that would use computing power more efficiently deserves attention.
The method proposed in this paper for solving the transport problem based on the use of complex numbers in the algorithm makes it easier for practical application.

Conclusions
The use of complex numbers in the compilation of algorithms allows them to be made more visual and simple for practical use .
The transport task is used to solve various applied problems related to the transportation of goods and passengers.
In order for a mathematical model based on a transport problem to have practical significance, it is necessary to prove that this problem has a solution.