On the analysis of network models with random work durations using the Wolfram Mathematica system

As you know, a network model is a plan for performing some complex of
interrelated operations, given in the form of a network, the graphical representation of which is
called a network graph. At the same time, all the interrelationships of the work to be performed
require a clear definition. Network planning is one of the most well-known applications of graph
theory and is widely used in practice.

In English-language literature, this technique is called the Project Evaluation and Review
Technique (PERT). The tasks solved using the network planning method are to reduce the
duration of the entire project to a minimum and rationally allocate labor and other resources
throughout its execution.

Network planning can be used, for example, when solving the problem of creating long-range
radar stations, when planning roadway reconstruction, in construction, in process management
at enterprises, when planning maintenance of communication systems, when planning assembly
operations and when solving a large number of other applied tasks. Thus, the development of
methods for the computer implementation of algorithms for the study of network graphs is a
very urgent task.

In the case when the duration of work is deterministic, the main task is to determine the
critical path - the longest path along the graph. It is the length of this path that determines the
duration of the entire project, and therefore its increase is unacceptable. In this regard, the most
responsible and qualified specialists should be appointed to work on the critical path. However,
in practice, this case is rare. More often there are cases when the actual time of completion of
the work is not known to us exactly (by chance). In this case, the question arises: what is the
probability that the actual execution time of the complex of works will not exceed a given value
or will be in a given interval? The solution of these issues is obviously very important for practice,
when planning real work packages.

Solving problems of this kind presents significant computational difficulties. Solving problems
related to network graphs using the Wolfram Mathematica system turns out to be very effective.
This system, especially its latest versions, contains a number of tools that allow you to study
network graphs with random operation times. The use of these capabilities turns out to be very
useful both in solving real-world applied problems, and in the educational process of economic
and technical universities, in studying disciplines related to operations research and computer
modeling.

This article is devoted to solving problems for the study of network graphs with random
work time using the Wolfram Mathematica system.

Keywords: network model, Wolfram Mathematica system