In the late 50s in the United States, for the implementation of a program of research and development work on the creation of the Polaris rocket, the method of planning and control was first used, based on the idea of determining, estimating the likely timing and control of the so-called "critical path" of the entire complex of works.

The results exceeded all expectations: first, the number of failures in work due to the inconsistency of the resources used was significantly reduced, the total duration of the entire complex of works was sharply reduced, a huge effect was obtained due to a decrease in the total need for resources and, accordingly, a decrease in the total cost of the program. Soon after the results of the Polaris program became public, the whole world started talking about the Pert (Project Evaluation and Review Technique) method as a new approach to the organization of management.

Since then, the "critical path" method has not only been widely used in everyday management practice, but has also led to the emergence of a special scientific and applied discipline - project management. The focus of this discipline is on the issues of planning, organization, control and regulation of the implementation of projects, the organization of material and technical, financial and personnel support for projects, the assessment of the investment attractiveness of various options for the implementation of projects.

In the modern business environment, the relevance of project management as a method of organizing and managing production has increased significantly. This is due to objective trends in global business restructuring. The principle of concentration of production and economic potential gave way to the principle of focusing on the development of the organization's own potential. Large production and economic complexes of a conglomerative type are quickly replaced by flexible network structures, among the participants of which the principle of preference for the use of external resources by internal (outsourcing) dominates. Therefore, production activities are increasingly turning into a complex of works with a complex structure of resources used, a complex organizational topology, a strong functional dependence on time and a huge cost.

Project Management Object

The term project, as you know, comes from the Latin word projectus, which literally means "thrown forward". Thus, it immediately becomes clear that the object of management, which can be represented in the form of a project, is distinguished by the possibility of its prospective deployment, i.e. the ability to foresee its states in the future. Although different official sources interpret the concept of a project in different ways2, all definitions clearly show the features of the project as an object of management, due to the complexity of tasks and works, the clear orientation of this complex to achieve certain goals and restrictions on time, budget, material and labor resources.

However, any activity, including the one that no one is going to call a project, is carried out for a certain period of time and is associated with the costs of certain financial, material and labor resources. In addition, any reasonable activity, as a rule, is expedient, i.e. aimed at achieving a certain result. And yet, in some cases, the management of activities is approached as project management, and in other cases it is not.

Activity as an object of management is considered in the form of a project when

it is objectively complex in nature and for its effective management it is important to analyze the internal structure of the entire complex of works (operations, procedures, etc.);

transitions from one job to another determine the main content of all activities;

the achievement of the objectives of the activity is associated with the sequential and parallel implementation of all elements of these activities;

restrictions on time, financial, material and labor resources are of particular importance in the process of performing a set of works;

the duration and cost of activity clearly depends on the organization of the entire complex of works.

Therefore, the object of project management is considered to be a specially organized set of works aimed at solving a certain task or achieving a certain goal, the implementation of which is limited in time, and is also associated with the consumption of specific financial, material and labor resources. At the same time, "work" is understood as an elementary, indivisible part of this set of actions.

The elementality of work is a conditional and relative concept. What is impractical to divide in one system of action is useful to disaggregate in another. For example, if a technological operation is taken as an element of the complex of works on the assembly of a car, then one of the "works" can be considered the installation of a headlight assembler. This "work" in this case is indivisible, since its factors remain unchanged - the performer, the object and the object of action. But, as soon as we begin to consider the performance of this work as a separate task, it itself turns into a complex.

However, if a problem arises regularly, and its solution turns into a routine activity brought to automatism, then there is no particular point in considering and modeling its complex structure every time we begin to solve it. The result is known in advance and the time spent on planning will simply be lost. Therefore, the object of project management is, as a rule, a complex of interrelated works aimed at solving some original task. But, the fact is that in the modern business environment, with the rapid development of equipment, technology and organization of production, with the rapid change of types and varieties of goods and services in the markets, the appearance of original tasks before the manager has actually become a common situation. If in the late fifties, at the dawn of the birth of project management, only research and development programs acted as objects of such management, then nowadays few people can be surprised by technical, organizational, economic and even social projects. Already in the very definition of the type of project there is a characteristic of the scope of its application.

Theoretical foundations of project management

For the description, analysis and optimization of projects, the most suitable were network models, which are a kind of directed graphs.

In a network model, the role of the vertices of the graph can be played by events that determine the beginning and end of individual works, and the arcs in this case will correspond to the work. Such a network model is called a network model with work on arcs (Activities on Arrows, AoA). At the same time, it is possible that in the network model, the role of the vertices of the graph is played by the work, and the arcs reflect the correspondence between the end of one work and the beginning of another. Such a network model is called a network model with works in nodes (Activities on Nodes, AoN).

Let be the set A={a1, a2, a3, ... an} is a set of works that are required to solve a certain task, for example, the construction of a house. Then, if the set V={v1, v2, v3, ..., vm} represents a set of events that occur during the execution of the complex of works, then the network model will be given by the directed graph G = (V, A), in which the elements of the set V play the role of vertices, and the elements of the set A play the role of arcs connecting the vertices, and each arc ai can be set in a one-to-one correspondence of a pair of vertices (vsi, vfi), the first of which will determine the start of the ai work, and the second - the moment of completion of this work. Such a network model would be a network model with arc work.

Now let the set A={a1, a2, a3, ... an} – will still be considered as a set of works that are required to solve a certain task, for example, the construction of a house. Then, if the set V={v1, v2, v3, ..., vm} will represent a complex of relations of precedence-following of works in the process of their execution, then the network model will be given by the directed graph G = (A, V), in which the elements of the set A play the role of vertices, and the elements of the set V play the role of arcs connecting the vertices, and each arc vi can be put in a one-to-one correspondence to a pair of vertices (asi, afi), the first of which will be the immediately prior work in this pair, and the second - immediately the next. Such a network model will be a network model with work in nodes.

A network model can be represented by: 1) a network graph, 2) in tabular form, 3) in matrix form, 4) in the form of a diagram on a timeline. As will be shown below, the transition from one form of presentation to another is not difficult.

The advantage of network graphs and time diagrams over tabular and matrix presentation forms is their visibility. However, this advantage disappears in direct proportion to how the size of the network model increases. For real-world network modeling problems, which involve thousands of papers and events, drawing network graphs and diagrams makes no sense.

The advantage of tabular and matrix forms over graphical representations is that they are used to analyze the parameters of network models; in these forms, algorithmic analysis procedures are applicable, the implementation of which does not require a visual display of the model on the plane.

Network graphics refer to the complete graphical display of the structure of a network model on a plane.

If the network diagram displays a network model of type AOA on the plane, then all works and all events of the model should receive a one-to-one view. However, the network diagram structure of the AOA model may be more redundant than the structure of the network model itself. The fact is that according to the rules for constructing a network diagram, for the convenience of its analysis, it is necessary that the two events be connected only by a single work, which in principle does not correspond to the real circumstances in the reality around us. Therefore, it is customary to introduce into the structure of a network diagram an element that does not exist either in reality or in the network model. This element is called fictitious work. Thus, the structure of a network diagram is formed from three types of elements (in contrast to the structure of a network model, where there are only two types of elements):

events - moments in time when the beginning or end of the performance of any work (work) occurs;

works - indivisible parts of a set of actions necessary to solve a certain task;

fictitious works are conditional elements of the structure of a network diagram used solely to indicate the logical connection of individual events.

Graphically, events are depicted in circles divided into three equal segments (radii at an angle of 120°); works are represented by solid lines with arrows at the end, oriented from left to right; fictitious works are represented by dotted lines with arrows at the end, oriented from left to right. An example of a network diagram of the AoA model is shown below in Fig. 1.

Note that the indexing of works is carried out next to the corresponding arrows; fictitious works are not indexed; event indexes are placed in the lower segment of the corresponding circle. Filling in the remaining segments is discussed below.

If the network diagram displays an AoN model, structure redundancy can be avoided. There is no need to introduce fictitious works as an additional structural element, since there are no structural elements that they are designed to serve, namely, events. In the network diagram of an AoN model, there are only nodes (or vertices) that denote work and arcs (solid lines with arrows oriented from left to right) that denote work precedence-sequence relationships. No events and no fictitious works! Note that in the most famous program for project management Microsoft Project, this type of model is implemented.

Here, the nodes of the network corresponding to the works are usually depicted in rectangles divided into 5 sectors. In the central sector, an index is put (or the name of the work is recorded). Filling in the remaining sectors is discussed below. An example of a network diagram for an AoN model is shown below in Figure. 2.

In tabular form, the network model is given by the set {A, A(IP)}, where A is the set of indices of works, and A(IP) is the set of combinations of works immediately preceding the work of A. For the example discussed above, the tabular form of the network model will be as presented in Table. 1.

The matrix form of the description of the network model is given as a relationship between events (ei, ej), which is equal to 1 if there is work between these events (either real or fictitious) and 0 - otherwise. The matrix form for describing the network model from the above example is shown below in Table. 2:

The description of the network model in the form of a time diagram (or Gantt graph) involves the placement of work in a coordinate system, where time (t) is deposited along the abscissa (X) axis, and work is deposited along the ordinate (Y) axis. The starting point of any of the works will be the moment of completion of all its previous works. If nothing precedes the work, it is deferred from the beginning of the timeline, i.e. from the leftmost edge of the chart. In Fig. Fig. 3 shows the Gantt graph for the network model according to table data. 1 with the addition of information about the duration of the work.

Since in the network graphs of models of the AoA type, the vertices correspond to events, insofar as these elements of the structure have the property of "cross-linking" previous works with subsequent ones. In other words, any event occurs only when all the works preceding it are completed. On the other hand, it is a prerequisite for the start of the work that follows it. The event has no duration and occurs instantly. In this regard, special requirements are imposed on its definition.

Thus, each event included in the network diagram should be fully, clearly and comprehensively defined, its wording should include the result of all work immediately preceding it. And until all the work immediately preceding this event has been completed, the event itself cannot occur, and, therefore, none of the work immediately following it can be started. Moreover, if an event occurs, it means that the work that follows it can be immediately and realistically started. If, for any reason, at least one of these works cannot be started, therefore, this event cannot be considered to have occurred.

The following varieties of network diagram events of the AoA model are distinguished:

initial event - a result in respect of which it is conditionally assumed that it has no previous work;

the final event is the result for which it is assumed that no work follows it; this is the ultimate goal of performing the whole complex of works or solving the problem;

an intermediate event or just an event. This is any achievable result in the performance of one or more works, which makes it possible to begin subsequent work;

initial event – the event immediately preceding this particular work;

the final event is the event immediately following this work.

The time parameters (or temporal characteristics) of the network model are the main elements of the analytical system of project management. It is for their definition and subsequent improvement that all preparatory, auxiliary work is carried out to compile a network model of the project and its subsequent optimization.

There are the following time parameters:

duration of work;

early start times

early finishing time;

a late start time.

late end time;

early time of occurrence of the event;

late time of occurrence of the event;

the duration of the critical path;

reserve time of occurrence of the event;

full reserve of work time;

free reserve of work time;

independent reserve of work time.

Duration of work (ti) – the calendar time it takes to complete the work.

The Early Start Time (ESTi) is the earliest possible start date.

Early Work End Time (EFTi) is equal to the early start time plus the duration of the work.

The Late End Time (LFTi) is the latest allowable finish time.

The Late Start Time (LSTi) is equal to the late start time of the work minus the duration of the work.

Early time of occurrence of an event (EETj) - characterizes the earliest possible time for the occurrence of an event. Since each event is the result of the accomplishment of one or more works, and they, in turn, follow any previous events, the timing of its occurrence is determined by the magnitude of the longest segment of the path from the original event to the one in question.

The late time of occurrence of the event (LETj) - characterizes the latest of the permissible dates for the commission of an event. If the deadline for the occurrence of the final event, which is the result of the entire complex of work carried out, is established, then each intermediate event must occur no later than a certain period. This period is the maximum permissible date for the occurrence of the event.

Any sequence of directly following works in a network model is called a path. There can be many paths in a network model, but the paths that link the initial and final events of the network model are called complete, and all others are called incomplete. The sum of the duration of the work that makes up a particular path is called the duration of this path.

The longest of all full paths is called the critical path of the network model. Thus, the duration of the critical path is equal to the sum of the durations of all the works that make up this path.

Works that lie on a critical path are called critical works, and events are called critical events.

The very definition of the critical path of the project network model is enough to organize the management of the entire complex of works. Strictly controlling the calendar deadlines for the performance of critical work, you can eventually avoid losses. Work that is not on a critical path usually has time reserves that allow you to postpone their implementation for some time, if necessary.

The time reserve of the occurrence of an event is the difference between the late and early dates of the occurrence of this event.

A full reserve of work time (TFi) is the maximum possible margin of time to perform this work beyond the duration of the work itself, provided that as a result of such a delay, the final event for this work will occur no later than at its later date.

A free margin of work time (FFi) is a margin of time that can be available when performing a given work, assuming that the preceding and subsequent events of this work occur at their earliest dates.

An independent work time reserve (IFi) is a margin of time that can be delayed from starting work without the risk of affecting any timing of any events in the model at all.

The parameters of the early and late time of the event occurrence are used in marking the vertices of the network graph of the AoA type model. The left segment records the early time of occurrence of the corresponding event (EETj), and the right segment is later (LETj), as shown in

Figure 4.

In marking the vertices of the network graph of the AoN model, in addition to the work index, parameters are used (see Fig. 5):

Early start time (ESTj), which is written to the upper-left sector of the rectangle marking the top of the work.

Late work start time (LSTj), which is written to the upper-right sector of the rectangle marking the top of the work.

the duration of the work (tj), which is written to the lower left sector of the rectangle marking the vertex of the work.

full work time reserve (TFi) – which is written to the lower right sector of the rectangle marking the top of the work.

Methods for calculating the time parameters and critical path of the project network model

If the dimensions of the network graph are small, then its time parameters and critical path can be found by directly examining the graph vertex by vertex, work by work. But, of course, as the scale of the model increases, the probability of an error in the calculations will increase exponentially. Therefore, even with small model sizes, it is advisable to use one of the most suitable algorithmic calculation methods that allow you to approach this task formally.

Figure 5. Example of marking the vertices of a network diagram of an AoN type model

The most common methods for calculating the temporal parameters of a network model are tabular and matrix. Therefore, even if the original information on the network model is presented in the form of a network graph or a time diagram, when it begins to analyze, it should be brought to tabular or matrix form.

As an example, we will consider the model specified initially by the network diagram shown in Fig. 6.

Figure 6. Example of a network diagram to illustrate methods for calculating time parameters

Both the tabular and matrix method of calculating the temporal parameters of the network model is based on the following relationships arising from the definitions of time parameters. For ease of understanding, the work index usually consists of two letters, for example, [ij], the first of which corresponds to the index of the initial work event, and the second to the index of the final work event. Subject to this observation:

The early start time [ij] coincides with the early start time of the event [i], i.e.

ESTij = EET [i].

The late end time of [ij] coincides with the late time of the occurrence of the event [j], i.e.

LFTij = LET [j].

Early end time [ij]:

EFTij = ESTij + tij.

Later start time [ij]:

LSTij = LFTij – tij.

The early occurrence time of event [j] coincides with the latest (maximum) early end time of all those works for which this event is finite, i.e.

EET[j] = max { EFTrj, EFTnj, ..., EFTmj}, where [rj], [nj], ..., [mj] are the indexes of work for which event [j] is finite.

The late time of occurrence of event [j] coincides with the earliest (minimum) late start time of all those works for which this event is the initial, i.e.

LET[j] = min { LSTjr, LSTjn, ..., LSTjm}, where [jr], [jn], ..., [jm] are indexes of works for which event [j] is initial.

For the initial and final event of the network model, the following is true:

EET[s] = LET[s]

But if, as a rule, a time point of 0 is taken for the initial event, then for the final event it appears as a result of calculations and the duration of the critical path can be judged by it. So, for the final event, it is true:

EET[f] = LET[f] = TK, where TK is the duration of the critical path.

Full work time reserve [ij]:

TFij = LET[j] – EET[i] – tij.

Free work time reserve [ij]:

FFij = EET[j] – EET[i] – tij.

Independent work time reserve [i]:

IFi = EET[j] – LET[i] – tij.

Let's first consider the matrix method for determining time parameters.

First of all, you need to make a square matrix (see Fig. 7), the number of columns and rows, in which the number of events of the network model is equal. Rows and columns are indexed in the same order by event indexes. The cells obtained at the intersection of rows and columns are divided into two parts diagonally from the bottom from left to right. The upper left part of the cell is called its numerator, the lower right is called the denominator.

The first step in filling the matrix is as follows. If the events [i] and [j] are joined by some work, then the duration of this work tij is recorded in the numerators of two cells: the cell lying at the intersection of the i-th row and the j-th column, and the cell lying at the intersection of the j-th row and the i-th column. These actions are performed for all the works of the network model, and the numerators of all other cells, except for the cells lying on the main (from top to right to bottom) diagonal of the matrix, are filled with zeros or not filled at all.

The next step in filling the matrix initially involves entering the value 0 into the numerator of the first cell of the principal diagonal. This is equivalent to the fact that we assume that the early time of occurrence of the original event of the network model is 0. Then we fill in the denominators of those cells of the first line lying to the right of (or above) the main diagonal, whose numerators contain values greater than 0. In this case, the values that are put in the denominators are calculated as the sum of the numerator of the cell of this line lying on the main diagonal and the numerator of the filled cell. Thus, we calculate the early end time of the relevant work. The result of these steps is shown in Figure. 8.

Figure 7. Matrix markup when determining the temporal parameters of a network model by the matrix method

Figure 8.

It is not difficult to verify by formulas that the early end time of work 1-2 is 4, and work 1-4 is equal to 7.

The next step of filling the matrix begins with the fact that we must decide what value should be in the numerator of the diagonal cell of the second row. By definition, this should be a value corresponding to the early onset of event 2. The early beginning of an event that is the final one for several works is equal to the early end of the latest of the works that end with this event. So, you just need to view the denominators of the cells of column 2 from top to bottom to the main diagonal and select the maximum value, and then write it to the numerator of the diagonal cell 2. In our example, this would be the cell denominator 1-2, which is 4.

After that, as well as the denominators in the first line above the diagonal were calculated, the denominators of the cells of the second line above the diagonal are counted.

The procedures described above are repeated until the numerator of the last diagonal cell is found.

Having reached the last diagonal cell (see Fig. 9), we obtained the value of the early time of occurrence of the final event of the network model (36), which determines the duration of the critical path. At the same time, for the final event, as is known, the early time is equal to the late time of its occurrence, therefore, the denominator of this cell will be equal to its numerator. Let's write it down.

By obtaining the value of the denominator of the last diagonal cell, it is possible to calculate the values of the denominators of the cells (whose numerators are greater than 0) located in the same line to the left (below) of the main diagonal. They will be equal to the difference in the value of the denominator of the corresponding diagonal cell and the value of the numerator of the cell for which the calculation is made. For example, the value of the denominator of cell 8-7 will be equal to 36-5 = 31, and cells 8-4 will be equal to 36-6 = 30.

After counting all the denominators in the last line, you can find the value of the denominator in the diagonal cell on the penultimate line. It will be equal to the minimum value of the denominators of all cells lying in this column below the main diagonal, i.e. 31.

Then, in the same way, we calculate the penultimate line and find the denominator of the third from the end of the diagonal cell.

From the filled matrix, it is not difficult to see not only the duration of the critical path (the numerator or denominator of the last diagonal cell), but also the critical path itself. It passes through events in which the early and late time of onset are equal, i.e. through events in which numerators and denominators coincide in the corresponding diagonal cells. In our example, these would be events 1, 2, 4, 6, 8 (see Figure 9).

In accordance with the calculated formulas of time reserves that have been given above, the total time reserve of the work between events i and j is determined by the difference in the values of the denominator of the diagonal cell j-j and the denominator of cell j in line i above the main diagonal. To find a free reserve of the time to perform the work between events i and j, it is necessary to subtract the numerator of the diagonal cell i-i and the numerator of the cell i-j from the numerator of the diagonal cell j-j. To find an independent reserve of the execution time of the work between events i and j, it is necessary to subtract the denominator of the diagonal cell i-i and the numerator of the cell i-j from the numerator of the diagonal cell j-j.

So, for work 3-5, the full reserve will be equal to 29-9 = 20, free - 17-2-7 = 8, and independent - 17-22-7 = -12 (taken as equal to 0). For work 2-6, the full reserve will be equal to 26-12 = 14, free - 26-4-8 = 14 and independent - 26-4-8 = 14.

In Fig. Fig. 10 shows the results of calculations of all time reserves based on the data from the table in

Tabular method. A table is compiled, the number of rows in which is equal to the number of works, which includes the following columns (in the order of their sequence from left to right):

- work index;
- indexes of immediately preceding works;
- indexes of the following works;
- the duration of the work;
- early start time of work;
- a late start time
- early finishing time of work;
- late time to finish the work;
- full reserve of operating time;
- free reserve of working time;
- independent reserve of operating time.

The background information associated with describing the topology of the network model is contained in columns (1), (2), and (4). The essence of the tabular method for calculating the temporal parameters of the network model is to sequentially populate the remaining columns of this table.

The tabular method algorithm performs the following sequential steps:

### STEP 1. Define indexes directly in the following works.

We consider working with the index [i]. The works immediately following it are those for which the work [i] is immediately preceding. Therefore, the indexes of the following works are indexes of those works that have an index of work [i] in column (2).

### STEP 2. Determine the early start time and early finish time of work.

Determining the early start time and early completion of work, i.e. filling in columns (5) and (7) of the table should be carried out simultaneously, because the start time of some works depends on the time of completion of others.

These columns are filled in sequentially from the beginning of the network model to its end, i.e. from top to bottom. The following rules apply:

The early end time of the work in question is equal to the early start time (from column (5)) plus the duration of the work (from column (4)).

The early start time of the work is 0 if the work is not directly preceded by any of the works in the network model, or equal to the maximum early finish time among all the work immediately preceding it (from column (7)).

The duration of the critical path is equal to the maximum value in the column (7).

### STEP 3. Determine the late end time and late start time of work.

Determining the late finish time and late start of work, i.e. filling in columns (6) and (8) of the table should also be carried out simultaneously, because the start time of some works depends on the time of completion of others.

These columns are populated sequentially from the end of the network model to its beginning, i.e. from the bottom up. The following rules apply:

The late start time of the work in question is equal to the late start time (from column (8)) minus the duration of the work (from column (4)).

The late end time of a job is equal to the length of the critical path if there is no immediately following work (from column (3)) of the network model, or is equal to the minimum late start time among all the work immediately following the work (from column (6)).

### Step 4. Determine the full time reserve of work.

The full reserve of run time [i] is found as the difference in the values of its late and early end times (columns (8) and (7), respectively), or as the difference in the values of its late and early start (columns (6) and (5), respectively).

### Step 5. Determination of the free reserve of work time.

The free reserve of work time [i] is defined as the difference between the early start time value of any of the work immediately following it and the sum of the early start time [i] and its duration.

### Step 6. Define an independent reserve of work time.

The independent reserve of operating time [i] is defined as the difference between the early start time value of any of the work immediately following it and the sum of the late time of occurrence of the initial work event [i] and its duration. The late time of occurrence of the initial event of work [i] by tabular means is defined as the minimum late start time of those works that have the same composition of immediately preceding work with the work [i].

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