Entropy of the system of human knowledge

14 Dec
  1. System of knowledge and skills of a person can be represented as the sum of knowledge and skills relating to different areas of activity.
  2. During daily activities a person can use the knowledge and skills from different areas. This means that human knowledge is a system that can be in different states and at the same time randomly move from state to state.
  3. For example, an electronics engineer can use the knowledge from the fields of physics, mathematics, electronics, and other areas that may be considered different states in which the system human knowledge can be.
  4. In turn, each area of ​​knowledge can have its sub states or micro states, and so on.
    Thus the system of human knowledge can be represented as a hierarchy of different states and micro states. Depending on the type of human activity the system of knowledge can be in different states and micro states with certain probabilities.
  5. In accordance with this the thermodynamic entropy of a system of human knowledge can be defined by the following known formula [1].


where pi – is the probability of the microstate i and k – is the Boltzmann’s constant.

6. If all microstates have equal probabilities the statistical thermodynamic entropy reduces to the famous Boltzmann’s formula

S = k log W

where W is the number of micro states.

7. During its activities a person can build up knowledge and skills, which will lead to an increase in the entropy of his knowledge due to the growing number of microstates. But also people can forget their knowledge, which will reduce the number of possible microstates of the system of knowledge and, consequently, will lead to a decrease in entropy.

8. To eliminate the additional uncertainty in the reasoning, we assume that we are dealing with the knowledge of a person in a state of isolation. So we have a closed system, to which the second law of thermodynamics can be applied. Naturally, in a state of isolation people will gradually forget the accumulated knowledge, which would reduce the number of possible microstates of its knowledge system and correspondingly reduce the entropy.

9. That is, it turns out that the second law of thermodynamics is not applicable for the representation of human knowledge, which in turn contradicts the universality of the second law of thermodynamics.

What can be the most acceptable explanation of this phenomenon?


  1. Entropy in thermodynamics and information theory. Wikipedia. http://en.wikipedia.org/wiki/Entropy_in_thermodynamics_and_information_theory

Quantitative PM: State of the art and the new problems of development

8 Dec
  1. The field of project management (PM) needs new methods to cope with the difficulties connected to the problems of protracted project failure,
  2. One of the main reasons of project failure is the lack of reliable project estimation methods for the whole project and for its separate portions during the execution phase,
  3. This state of affairs is directly related to the underdevelopment of the methods of quantitative project management and can be described as very close to a crisis,
  4. The main missing link in the quantitative project management are the reliable functional relationships between the parameters of the projects and their separate parts,
  5. Existing techniques designed to solve this problem are based on the statistical approaches and mental models that have low reliability, and therefore do not meet the requirements of the industry,
  6. Attempts to solve this problem within the framework of the existing approaches in the field of quantitative management of projects are doomed to failure because of the narrowness of the conceptual foundations of these approaches,
  7. Solution of  this problem requires a paradigm shift in the quantitative project management, which aims to move this field of knowledge from empirical and intuitive development phase to the methods and theories built on the basis of general principles and balance conditions,
  8. This approach is common in highly developed quantitative sciences such as physics [1], mathematical biology [2], mathematical economics [3], and so on,
  9. This is the only way that may allow finding reliable alternatives to the statistical and mental models and building an adequate mathematical theory of project management.

An example of the functional relationship [4] between project total effort and project planned duration (Fig.1).


Fig.1 Project effort vs. duration derived on the basis of the mathematical theory of PM (each curve in this family corresponds to a constant value of project complexity)

Analysis of this family of curves indicates:

a)      The increase in the duration of the project leads to a reduction in project effort due to reducing the number of people working on the project and corresponding increase of team productivity.

b)      The greater decline in effort (top curve) corresponds to the bigger project because of the relatively large change of the number of working people with the change of the planned duration.

c)       Relative constancy of the effort value corresponds to the larger project duration because of the relative constancy of the productivity for the larger number of working people.


  1. Physics, Wikipedia: http://simple.wikipedia.org/wiki/Physics
  2. Mathematical and theoretical biology, Wikipedia: http://en.wikipedia.org/wiki/Mathematical_biology
  3. Mathematical economics, Wikipedia: http://en.wikipedia.org/wiki/Mathematical_economist
  4. Pavel Barseghyan (2009) Principles of Top-Down Quantitative Analysis of Projects: Part 2  Analytical Derivation of Functional Relationships between Project Parameters without Project Data. PM World Today – June 2009 (Vol XI, Issue VI). 16 pages. http://www.scribd.com/doc/114484663/Principles-of-Top-down-Quantitative-Analysis-of-Projects-Part-2-Analytical-Derivation-of-Functional-Relationships-Between-Project-Parameters-Without-P

Reflections on the functional relationship between project effort and its complexity

30 Nov
  1. If the complexity of the project increases and approaches to the upper limit of its feasible value Wf, then the effort consumed on the project increases sharply because of the increasing number of errors and endless design iterations. For that reason the work of the project team gradually becomes unfeasible (Fig.1).


Fig.1. Functional relationship between project effort E and the complexity W of the project

  1. Each specific project is a trade-off between competitiveness of the project (which requires complexity increase) and the feasibility of the project (which requires complexity reduction).  Therefore each realm of human activity has its own trade-off range (Fig.2) (http://www.scribd.com/doc/113818423/Human-Effort-Dynamics-and-Schedule-Risk-Analysis).
  2. It is important to note that the majority of human actions and activities take place in the linear range of this relationship but the most critical actions, activities and decisions take place in the nonlinear range. In other words the whole progress of human society is concentrated in the nonlinear realm including advanced project work and research projects of fundamental nature.
  3. Traditional methods of project effort estimation do not take into account this fact which might be one of the reasons of big effort estimation errors especially for high end projects complexities of which are close to the upper limits (Fig.3).


Fig.2 Advanced project works are concentrated in the nonlinear range of the curve


Fig.3 Effort estimation error of traditional methods increases sharply close to the upper limits of feasible complexity of the project

Because of its practical importance and many failed attempts to solve the problem by constructing simple mathematical models, the solution of this problem will require a more thorough theoretical coverage that accounts for the nonlinear effects discussed above and another non-linear dependence of team productivity on the size and objectives of the project.



Missing nonlinearities in quantitative project management

11 Nov

The most important problem of modern methodologies in project management is first of all, the need for improvements of the situation with the massive failures of projects and the related huge financial losses.
It should be noted that, despite great efforts to develop sophisticated new quantitative methods in this area, such as System Dynamics (SD), Earned Value Management (EVM) and others, the situation with massive failures of projects did not change significantly for the better over the past twenty years. But these are the years during which there was a serious and important progress in the modern methodologies of quantitative project management, including SD and EVM.

Thus, on the one hand we see a real progress and prospects in the quantitative project management, on the other hand – almost unchanged statistics of project failures, suggesting possible serious drawbacks of these methods.
This is a serious challenge for developers and users of modern PM methodologies, because the main purpose of the development of quantitative methods in this realm is to increase the level of controllability of projects, and as a consequence, reduce the number of failures of projects.

Since, in fact, this goal was not achieved, or the achieved results were so modest that they do not justify the huge amounts of money spent, then the question naturally arises about the analysis of the causes of this state of affairs.

Analysis of the extensive literature on this subject and many years of experience with project data analysis to create new methodologies in project management indicate that one of the biggest reasons for the failure of projects, along with other no less serious causes, is the disadvantages and undeveloped quantitative methods in project estimation.

A more detailed analysis of this problem shows that these shortcomings of modern quantitative methodologies of project management related to the fact that they do not take into account a number of nonlinear relationships between project parameters, which are necessary to reflect adequately the essence of the project and the behavior of the team of performers.

The main sources for obtaining these functional relationships between project parameters in contemporary PM are the statistical project data mining, mental models and expert information.

Statistical project data mining results have low accuracy for project estimation and other purposes. Therefore, without significant improvements in statistical methods, their use to assess projects just does not make sense because of the large estimation errors.

Mental models contain a considerable portion of subjectivity and consequently estimation risks are high even for the short term project report generation purposes because of accuracy problems and qualitative nature of mental models. Therefore, it is advisable in the current quantitative methodologies to find a replacement of mental models through the development of more adequate models in the form of reliable and data independent functional relationships between the parameters characterizing the process of human labor.

As for the expert methods, traditionally they were not able to predict correctly major delays during the project execution. The situation with these predictions is interesting because, if the duration of a typical task according to the expert’s opinion has a natural upper limit, then none of the experts as an estimate of the duration will specify a value that exceeds the natural limit a number of times, since such a decision has no justification. But in reality, very often the actual duration of work exceeds the expected time a few times. This is the phenomenon of the delay of human work, which is very difficult to explain and manage.

Also the situation with expert estimates can help to explain another phenomenon, which is the relative constancy of the percentage of failed projects during the last twenty years, as the expert estimates have a dominant role in both old and new PM technologies.

Analysis indicates that one of the main reasons for this disadvantageous situation in quantitative project management is the missing nonlinearities in human labor description and mathematical modeling. This suggests that the leading methodologies in the area of quantitative project management, such as SD and EVM, despite of their great positive role in this area,are in need of further improvements of a fundamental nature. In particular, these improvements may involve consideration of various nonlinearities inherent to the behavior of both the projects, regardless of their size and complexity, as well as development teams again, regardless of the number of people in the team.

These nonlinear relationships that accompany the work of people and need to be described quantitatively can be divided into three following groups.

  1. Nonlinear relationships between project parameters that arise as a consequence of the balance between complexity of work, objectives of work and productivity of work performers.
  2. Nonlinearities that arise as a consequence of the limited capabilities of work performers and limitations that are connected with technological feasibility of work
  3. Nonlinear relationships that characterize communication and contacts between people, and , as a consequence, team productivity



First group of nonlinearities:

Nonlinearities that arise as a consequence of the balance between complexity of work, objectives of work and productivity of work performers
Main source of nonlinear functional relationships between the parameters characterizing the process of human labor, it is a natural balance between the three following group of factors:

1. Complexity of the work that includes the size and the difficulty of work,

2. Goals and objectives of work,

3. Professional capabilities of the work performers.

Each specific combination of these three components determines a particular state of human work as a system. The quantitative reflection of the balance between the complexity of work, the objectives of work and team productivity is the equation of state that reflects the equilibrium of the process of human labor. Any project as a specific kind of human work can have its own equation of state too.

Any change in work parameters leads to the transition of the work from one state to another which occurs at predictable trajectories. These transition trajectories in the project space are the nonlinear functional relationships between the parameters of work.

State equations contain all possible functional relationships between the parameters of work (project) therefore it cannot be used directly for project estimation. But in combination with the objectives of work or project’s goals equation of state can serve as a basis for deriving the above mentioned nonlinear functional relationships suitable for project estimation purposes.

Second group of nonlinearities:

Limited capabilities of work performers and limitations that are connected with technological feasibility of work

The main sources of these nonlinearities are the limited capabilities of the work performers (development team and individuals in it), as well as limited technical and technological feasibility of the projects in the specific area of ​​industry.

If the complexity of the project of specific technical product is close to the limiting possibilities of engineering and technology at that time, then this may give rise to a number of nonlinearities in the sense of the technical feasibility of project.

If the complexity and, therefore, the difficulty of the project are close to the professional limits of the development team, then regardless of the absolute complexity of the project more nonlinearities can arise between the parameters of human work, causing delays and failures of projects.

If in addition we take into account that for the economic reasons both projects and project teams should be close to the upper limits of their capacities, it is clear that people involved in such projects are almost always working in the field of double nonlinearity.

Third group of nonlinearities:

Nonlinear relationships in communication and contacts between people

The work of any human group is impossible to imagine without communication within the team and communication of the team with the outside world. Communication is literally the core of any organization of human labor and is taking place through the contacts between people, the intensity and effectiveness of which have direct influence on the productivity of human labor.

Communications have a dual effect on the productivity of human groups and the size of the groups in this sense is important. On the one hand communication through discussions and exchange of ideas enhances productivity of working groups. On the other hand communication reduces the labor productivity because of the wasted extra time needed for the contacts between people. Therefore, the nonlinear dependence of productivity on the number of people must serve as a basis for human work organization both for small and large development teams.

Typical nonlinear communication characteristics of the group of people are the dependency of the number of internal contacts between group members on the group size, the dependency of the number of external contacts of group members on the group size, and so on.

The same nonlinear relationship between the productivity of the team and its size should be the basis for constructing a hierarchical cell structure of organizations.

In such organizational structures the roles of cells are played by human groups, which are characterized by the dynamics of their internal and external contacts. In turn, the quantitative description of such hierarchical structures is based on nonlinear communication characteristics of individuals and groups.

All of the mentioned nonlinearities are very important for an adequate description of human labor, and in particular for the quantitative description of the project works. Therefore, these nonlinearities must be an organic part of any model to represent the total effort of human labor, its duration, cost, and the various risks associated with successful job performance.

Currently, the existing quantitative techniques in project management do not take into account these nonlinearities in the mathematical models of effort, duration and other parameters of human labor.

To fill this gap in the quantitative project management it is necessary to develop fundamentally new methods of mathematical description of human labor. In other words it is necessary to change the paradigm in this field and make a transition from primitive empiricism and fragmentary mathematical models to a more fundamental quantitative description of human labor. To do that it is advisable to use the existing more advanced methods and techniques that are developed in quantitative fields of knowledge such as physics, mathematical biology, mathematical economics, etc., as well as to develop new methods for problems related to specific tasks of the quantitative description of human labor.