Calculation of the critical path of the network diagram conclusion. Calculation of a network diagram using the sector method

Let's imagine the situation of developing a capital construction project at a manufacturing enterprise. The project has been successfully initiated and planning work is in full swing. Formed and approved, the milestone plan has been adopted. A primary version of the calendar plan has been developed. Since the task turned out to be quite large-scale, the curator decided to develop a network model as well. The calculation of a network diagram in the applied aspect of its execution is the subject of this article.

Before starting the simulation

The methodological basis of network project planning is presented on our website in several articles. I will just refer to two of them. These are materials devoted in general and directly. If during the course of the story you have questions, review the previously presented understandings; the main essence of the methodology is outlined in them. In this article we will look at a small example of a local part of a complex of construction and installation works as part of a significant project implementation. We will perform calculations and modeling using the “vertex-work” method and the classic tabular method (“vertex-event”) using the MKR (critical path method).

We will begin constructing the network diagram based on the first iteration of the calendar plan, made in the form of a Gantt chart. For clarity purposes, I propose not to take precedence relations into account and simplify the sequence of actions as much as possible. Although this rarely happens in practice, let’s imagine in our example that the operations are arranged in a “finish-start” sequence. Below you will find two tables: an extract from the list of project works (a fragment of 15 operations) and a list of network model parameters necessary for presenting formulas.

An example of a fragment of a list of operations of an investment project

List of network model parameters to be calculated

Don't be intimidated by the abundance of elements. Building a network model and calculating parameters is quite simple. It is important to prepare thoroughly, to have at hand a hierarchical structure of work, a linear Gantt chart - in general, everything that makes it possible to determine the sequence and interrelations of actions. Even the first time you run a graph, I recommend having formulas for calculating the required values ​​in front of you. They are presented below.

Formulas for calculating network diagram parameters

What do we need to determine when constructing the graph?

1. An early start to ongoing work that includes multiple connections from previous operations. We select the maximum value from all early endings of previous operations.
2. Late ending of the current activity from which multiple links exit. We select the minimum value from all late starts of subsequent actions.
3. The sequence of activities that form the critical path. For these actions, early and late starts are equal, as are early and late finishes, respectively. The reserve for such an operation is 0.
4. Full and private reserves.
5. Work intensity coefficients. We will consider the logic of the formulas for reserves and the work intensity coefficient in a special section.

Sequence of modeling actions

Step one

We begin constructing a network diagram by placing task rectangles sequentially from left to right, applying the rules described in previous articles. When performing modeling using the “vertex-work” method, the main element of the diagram is a seven-segment rectangle, which reflects the parameters of the beginning, end, duration, time reserve and name or number of operations. A diagram of its parameters is shown below.

Diagram of the work image on the network diagram

The result of the first stage of constructing a network diagram

In accordance with the logic of the sequence of operations, using a specialized program, MS Visio or any editor, we place images of work in the format specified above. First of all, fill in the names of the actions to be performed, their numbers and duration. We calculate the early start and early end taking into account the formula for the early start of the current action in the conditions of several incoming connections. And so we go until the final fragment of the operation. At the same time, in our example project, the same Gantt chart does not provide for outgoing connections from operations 11, 12, 13 and 14. It is unacceptable to “hang” them on the network model, so we add fictitious connections to the final work of the fragment, highlighted in blue in the figure.

Step two

Finding the critical path. As you know, this is the path that has the longest duration of the actions that are included in it. By looking through the model, we select connections between jobs that have the highest early finish values ​​for activities. The designated critical path is highlighted with red arrows. The result obtained is presented in the intermediate diagram below.

Network diagram with a highlighted critical path

Step three

Fill in the values ​​for late finish, late start and full work reserve. To perform the calculation, we go to the final work and take it as the last operation of the critical path. This means that the later end and start values ​​are identical to the earlier ones, and from the last operation of the fragment we begin to move backwards, filling in the bottom line of the action diagram. The calculation model is shown in the diagram below.

Scheme for calculating late starts and finishes outside the critical path

The final view of the network diagram

Step four

The fourth step of the network modeling and calculation algorithm is the calculation of reserves and tension coefficient. First of all, it makes sense to pay attention to the total reserves of paths of non-critical directions (R). They are determined by subtracting from the duration of the critical path the time duration of each of these paths, numbered on the final network diagram.

• R path number 1 = 120 – 101 = 19;
• R path number 2 = 120 – 84 = 36;
• R of path number 3 = 120 – 104 = 16;
• R path number 4 = 120 – 115 = 5;
• R path number 5 = 120 – 118 = 2;
• R path number 6 = 120 – 115 = 5.

Additional model calculations

The calculation of the total float of the current operation is carried out by subtracting the early start from the late start value or the early finish from the late finish value (see the calculation diagram above). The general (full) reserve shows us the possibility of starting the current work later or increasing the duration by the duration of the reserve. But you need to understand that you should use the full reserve with great caution, because the work that is farthest from the current event may end up without a reserve of time.

In addition to full reserves, network modeling also operates with private or free reserves, which represent the difference between the early start of subsequent work and the early completion of the current one. The private reserve shows whether it is possible to move the earlier start of the operation forward without affecting the start of the next procedure and the entire schedule. It should be remembered that the sum of all partial reserve values ​​is identical to the total reserve value for the path in question.

The main task of performing calculations of various parameters is to optimize the network schedule and assess the probability of completing the project on time. One of these parameters is the tension coefficient, which shows us the level of difficulty in completing the work on time. The coefficient formula is presented above as part of all calculation expressions used to analyze the network diagram.

The tension coefficient is defined as the difference between one and the quotient of the total reserve operating time divided by the difference in the duration of the critical path and the special design value. This value includes a number of segments of the critical path that coincide with the maximum possible path to which the current operation (i-j) can be assigned. Below is the calculation of private reserves and work intensity factors for our example.

Table for calculating private reserves and tension coefficient

The tension coefficient varies from 0 to 1.0. A value of 1.0 is set for activities on the critical path. The closer the value of a non-critical operation is to 1.0, the more difficult it is to stay on schedule for its implementation. After the coefficient values ​​for all chart actions are calculated, operations, depending on the level of this parameter, can be categorized as:

• critical zone (Kn more than 0.8);
• subcretic zone (Kn more than or equal to 0.6, but less than or equal to 0.8);
• reserve zone (Kn less than 0.6).

Optimization of the network model, aimed at reducing the overall duration of the project, is usually achieved by the following activities.

1. Redistribution of resources in favor of the most stressful procedures.
2. Reducing the labor intensity of operations located on the critical path.
3. Parallelization of critical path activities.
4. Redesign of the network structure and composition of operations.

Using the table method

Well-known scheduling software (MS Project, Primavera Suretrack, OpenPlan, etc.) are capable of calculating key parameters of the project network model. In this section, we will use the tabular method to configure such a calculation using conventional MS Excel tools. To do this, let’s take our example of a fragment of project operations in the field of construction and installation work. Let's arrange the main parameters of the network diagram in the columns of the spreadsheet.

Model for calculating network diagram parameters in a tabular way

The advantage of performing calculations in a tabular manner is the ability to easily automate calculations and avoid a lot of errors associated with the human factor. We will highlight in red the numbers of operations located on the critical path, and in blue we will mark the calculated positions of private reserves that exceed the zero value. Let us analyze step by step the calculation of network diagram parameters for the main positions.

1. Early starts of operations following current work. We configure the calculation algorithm to select the maximum value from the early end time of several alternative previous actions. Take, for example, operation number 13. It is preceded by operations 6, 7, 8. Of the three early finishes (71, 76, 74, respectively), we need to select the maximum value - 76 and set it as the early start of operation 13.
2. Critical path. Carrying out the calculation procedure according to point 1 of the algorithm, we reach the end of the fragment, finding the value of the duration of the critical path, which in our example was 120 days. The highest early completion values ​​among the alternative actions indicate operations on the critical path. We mark these operations in red.
3. Late completions of activities preceding the current job. Starting from the end work, we begin to move in the opposite direction from actions with higher numbers to operations with lower ones. In this case, from several alternatives for outgoing work, we choose the least knowledge of the late start. Late starts are calculated as the difference between the selected values ​​of late finishes and operation durations.
4. Operation reserves. We calculate total (total) reserves as the difference between late starts and early starts or between late finishes and early finishes. The values ​​of private (free) reserves are obtained by subtracting the early start of the next operation from the early end of the current one.

We examined practical mechanisms for drawing up a network schedule and calculating the main parameters of the project's time duration. Thus, we have come close to exploring the possibilities of analysis carried out with the aim of optimizing the network model and directly forming an action plan to improve its quality. This topic takes up little space in the project manager’s body of knowledge and is not that difficult to understand. In any case, each PM must be able to reproduce the visualization of the graph and perform the accompanying calculations at a good professional level.

When drawing up a network schedule, time is estimated based on the assumption that all available resources can be used to complete each job based on work plans and technological maps. This time estimate is then refined by combining individual jobs based on the principles of optimal use of available labor and other resources. Due to the fact that the labor intensity of work is usually expressed in man-days, it is enough to divide data taken from technological maps or regulatory reference books by the number of workers available to construction management in order to determine the total duration of work expressed in days. The unit of time used in network schedules must be the same for all types of work included in the network.

Based on the duration of each type of work, the total construction period is determined, which, after linking to the calendar, represents the construction calendar plan. Taking the duration of individual work according to the data shown in Fig. 121, you can find the critical path along the grid in order to determine the earliest and latest completion dates for each job.

Rice. 121. Network diagram with a critical path.

The critical path begins with the initial event and proceeds through the network from left to right until the final event. In this case, the earliest dates for the start and completion of work are determined by summing the duration of all works on which this work depends, starting from the initial event. This data is entered in the boxes located next to the event circles.

Thus, the calculation of the network schedule comes down to determining the time reserves of individual activities and, based on them, the total duration of the critical path.

For a small number of events, this calculation is not difficult. However, if we consider that network graphs of launch complexes of industrial enterprises usually cover hundreds and even thousands of events, then it takes considerable time to count them. In such cases, the calculation of the network diagram is carried out sequentially using appropriate formulas and tables manually for a number of events up to 500 or using a computer for a larger number. To understand the methodology for these calculations, you can use the data shown in Fig. 121.

If we accept the letter designations of the initial event of any of the works - m, the final n and the final event of the work following it - k, then these works can be denoted by the indices m - n and n - k.

It was said earlier that all activities that are not on the critical path have time reserves, and for them two start and end dates can be determined, the earliest and the latest, respectively.

Taking the notation:

The calculation begins with determining the early dates of work, i.e.

The early start of the first jobs 1-2 and 1-3 coming from start event 1 is zero, or

i.e. if event m is initial, then the early start of work m - n will be

Earliest start of work

determined by the duration of the longest path from the initial event to the previous event of this work.

For example, for job 7 - 8, the early start on chain 1 - 2 - 7 is:

However, according to the technological dependence of the work, it follows that it is impossible to start work 7 - 8 before finishing work 2 - 7, therefore the early start of work 7 - 8 should be accepted after 9 days, i.e. work can start on the 10th day.

By analogy, we determine the early start for the rest of the work:

Early start 5 - 9:

Since work 5 - 9 cannot begin until completion of 7 - 8, it should be assumed to begin according to the calculation of chain 1 - 2 - 7 - 8, i.e. 14 days after the start of construction. For the same reasons, the early start of work 8 - 9 should be taken along the chain 1 - 2 - 7 - 8, i.e.

Early start 9 - 10:

Should be accepted

18 days, since this work cannot be completed until the end of work 7 - 8.

Early completion dates for work are determined by adding its specified duration to the early start date of work using the formula:

Obviously, the early start of subsequent work is determined by the early completion of previous work, i.e.

If a given job is preceded by several jobs, then its Trn will be the maximum of the values ​​of the early completions of previous jobs:

Equality is a direct consequence of the fact that it is impossible to start any work unless the previous work has been completed or a series of works have not been completed, converging on one event and having different completion dates.

Finish work early

determined by the formula:

In the example under consideration these terms will be:

As can be seen from the above calculation, the early start and finish are determined for all work on the schedule sequentially from the initial event. The calculation of determining the early completion dates of work is always based on the longest values ​​of the duration of work.

The maximum value of the sum of early completions of a technologically connected chain of work, ending with the final event of the entire schedule (in our case, chains 1 - 2 - 7 - 8 - 9 - 10), determines the duration of the critical path and the construction period. In the example under consideration, Pcr = 23 days.

The latest start of work that will not delay the completion of construction of the entire facility is determined by the difference between the duration of the critical path and the longest path from the previous event of this work to the final event. ^

For example, for work 7 - 8 (Fig. 121), the late start will be equal to:

It is somewhat more difficult to determine the late start of work 2 - 7 or the latest occurrence of event 2, on which the beginning of subsequent works 2 - 7, 2 - 8, 2 - 9, etc. depends. Work 2 - 7 can be approached from the final event 10 to the considered 2 in several ways:

path 1 (10 - 9 - 2) duration L1 = 5+ 10 = 15 days;

path 2 (10 - 9 - 8 - 2) duration L2 = 5 + 4 + 8 = 17 days;

path 3 (10 - 9 - 8 - 7 - 2) duration L3 = 5 + 4 + 5 + 6 = 20 days.

According to these paths, the late start dates for work

will be equal:

Obviously, in order not to cause delay in subsequent work and other work, the minimum value should be taken

i.e. start work 2 - 7 no later than 3 days after the start of construction. If we take a longer period of late start of work 2 - 7, then all subsequent work will also be carried out later, which will cause an overall delay in the completion of construction.

The latest completion of the last job 9-10 in the network diagram under consideration will be the completion of event 10, the duration of which is determined by the duration of the critical path, i.e., the earliest completion date of the jobs lying on the path 1 - 2 - 7 - 8 - 9 - 10. B in our case, Pcr = 23 days and

23 days, so

or in general

The late completion of other jobs in the chain under consideration is determined by the sum of the late start and duration of this job.

For work 7 - 8:

For work 2 - 7:

In general, the late deadline for completing a job can be determined as follows. Late start of work

equal to the difference of the late end

and duration of work m - n, i.e.

Further analysis of the network diagram is carried out by comparing early and late characteristic activities to identify the critical path and determine time reserves. Those jobs whose early starts and finishes are equal to late starts and finishes have no slack, and therefore they are on the critical path. If this match is not established, then the work in question has a certain reserve of time.

As mentioned earlier, a distinction is made between the total time reserve of the path (circuit) under consideration, private and general operating time reserves.

The total time reserve for a given chain of work is the difference in time between the total duration of the work lying on the critical path and the duration of the work of the chain (path) in question, i.e.

where Pkr is the total duration of work lying on the critical path;

Pc - the same, lying on the circuit in question.

In our example, the value of the total slack between the critical path 1-2-7-8-9-10, equal to 23 days, and the chain 1-3-4-5-9-10, equal to 2+4+3 + 3+5= 17 days, there will be Rpol = 23-17 = 6 days.

Thus, the total reserve Ppol of a given chain (path) is equal to the sum of the private (free) reserves of work lying on it

In our example:

The total or total reserve time Р° of work m - n is defined as the reserve time of the maximum of the paths passing through this work.

Magnitude

shows by what time the duration of an individual job m n can be increased so that the length of the maximum path passing through this job does not exceed the length of the critical path.

The total time reserve is determined by the difference between the late and early start of time or the late and early finish of work.

For example, the total time reserve for work 7 - 8 is

The consolidated or private time reserve determines the amount of time by which the start of work can be delayed or its duration can be increased without changing the early start of the following work.

Such a reserve may be revealed when an event is the result of two or more activities. It is determined by the difference between the early start of subsequent work and the early end of this work.

For example, the private time reserve for work 2 - 8 is:

In general, the private RF time reserve is determined by the formula:

After completing the calculation of the network diagram, it is easy to determine the critical path by types of work for which P° = 0; the critical path includes all activities (arrows) located sequentially one after another, i.e., it indicates the activities that require the greatest amount of time to complete.

The concept of critical work covers both basic construction and installation work and auxiliary work. For example, delivering construction parts or process equipment to a construction site may be a critical job.

In addition to the critical path, the so-called critical zone is of interest, which determines the set of activities that have small reserves of time. Activities in the critical zone that do not lie on the critical path may end up on it even if the duration of some activities changes slightly. Such work is called subcritical. There is also a reserve zone, the totality of work of which has significant time reserves.

By summing up the time required to complete all work located on the critical path, the duration of construction of the facility is determined

Purpose of the service. The online calculator is designed to find network model parameters:

• early date of the event, late date of the event, early start date of the work, early end date of the work, late start date of the work, late end date of the work;
• time reserve for the event, full time reserve, free time reserve;
• duration of the critical path;

and also allows you to estimate the probability of completing the entire complex of work in d days.
Instructions. The online solution is carried out analytically and graphically. Formatted in Word format (see example). Below is a video instruction.

Number of vertices Numbering of vertices from No. 1.

The initial data is usually specified either through a distance matrix or in a tabular manner.
Data entry Distance matrix Tabular method Graphical method Number of lines
Analyze the network model: t min and t max are given t min , t max , m opt are specified
Optimization according to the criterion number of performers reserves-costs reduction of deadlines
",0);">

Example. A description of the project in the form of a list of operations performed, indicating their relationship, is given in the table. Construct a network diagram, determine the critical path, construct a calendar schedule.

Work (i,j)	Number of previous works	Duration t ij	Early dates: beginning t ij R.N.	Early dates: end t ij R.O.	Late dates: beginning t ij P.N.	Late dates: end t ij P.O.	Time reserves: full t ij P	Time reserves: free t ij S.V.	Time reserves: events R j
(0,1)	0	8	0	8	0	8	0	0	0
(0,2)	0	3	0	3	1	4	1	0	1
(1,3)	1	1	8	9	8	9	0	0	0
(2,3)	1	5	3	8	4	9	1	1	0
(2,4)	1	2	3	5	13	15	10	10	0
(3,4)	2	6	9	15	9	15	0	0	0

Critical path: (0.1)(1.3)(3.4) . Critical path duration: 15.

Independent operating time reserve R ij N - part of the total time reserve if all previous work ends at a late date, and all subsequent work begins at an early date.
The use of an independent time reserve does not affect the amount of time reserves for other activities. They tend to use independent reserves if the completion of the previous work occurred at a late acceptable date, and they want to complete subsequent work at an early date. If R ij Н ≥0, then such a possibility exists. If R ij Н<0 (величина отрицательна), то такая возможность отсутствует, так как предыдущая работа ещё не оканчивается, а последующая уже должна начаться (показывает время, которого не хватит у данной работы для выполнения ее к самому раннему сроку совершения ее (работы) конечного события при условии, что эта работа будет начата в самый поздний срок ее начального события). Фактически независимый резерв имеют лишь те работы, которые не лежат на максимальных путях, проходящих через их начальные и конечные события.

The network diagram is calculated in a tabular manner using the formulas previously set out in Section 4 (1-10). When determining the parameters of network models analytically, the calculation is performed in the form of a table. Let us consider the features of calculating network models using this method (application 1) using the example of calculating the parameters of the network diagram depicted in the assignment for this course work (option 15).

At the initial stage, it is necessary to describe the initial network model. In this case, the codes of all jobs and dependencies are entered in the first column of the table, starting with the job coming out of the first event. Job codes must be included in the table sequentially; arbitrary order of inclusion of jobs and dependencies in the table is unacceptable. The second column of the table contains the durations of all activities and dependencies.

The calculation of the network diagram begins with determining the values ​​of the early work parameters. The early start of work 1-2 is equal to zero (formula 1), and its early end according to formula 2.

The early start of jobs 2-6 and 2-7 (in accordance with formula 3) is equal to the early finish of jobs 1-2.

The maximum early finish value of job 19-21, equal to 36, determines the duration of the critical path and, therefore, the total duration of execution of all jobs in the original network model. The resulting value of the early completion of this work 19-21 = 36 is transferred to the late completion column of the final work 20-21.

Late start of work 20-21 is determined in accordance with formula 5 (= 34)

The late start of work 20-21 is the late finish of the preceding work 15-20 (=).

Further, the calculation of later parameters is performed in the same way, except for cases when the job has several subsequent jobs (for example, job 6-9 has two subsequent ones - 9-10 and 9-14). In this case, in accordance with formula 4, the late finish of work 6-9 is equal to the minimum value of the late start of subsequent works 9-10 and 9-14.

To find the position of the critical path, it is necessary to determine the values ​​of the total and private slack time for each job and dependency of the network diagram and enter their values, respectively, into columns 7 and 8 of the calculation table.

The total work time reserve, according to formulas 8-9, is defined as the difference between the late and early finishes or as the difference between the late and early starts of the corresponding work. It is useful to determine the value of the total slack using both methods; the coincidence of the obtained values ​​can be considered as an additional check. For example, for work 6-7:

The partial work time reserve, according to formula 10, is defined as the difference between the early start value of the subsequent work and the early finish value for this work. For example, for work 6-7:

The critical path is characterized by zero slack time. A comparison of the network model parameters obtained by sector and tabular methods should reveal their complete identity; the presence of discrepancies indicates that the calculations are erroneous.

Graphical method for calculating network diagrams

Calculation of a network diagram graphically is carried out similarly to the tabular method (formulas 1-10), however, the graphical or sector method of calculating network diagram parameters involves recording them directly on the model (Appendix 2). In this case, each event (circle) is divided into four sectors. The designation of the sectors is shown in the following figure:

For activities on the critical path, the values ​​of the total and private float are equal to zero; it is highlighted on the network diagram by a double line.

To check the correctness of the calculations performed, you should make sure that:

• * a continuous critical path has been identified;
• * calculated time reserves have a non-negative value;
• * the value of the private time reserve for all jobs is less than or equal to the value of the general time reserve for these jobs;
• * at least one late start value of jobs (jobs) coming from the first event is zero.

Practical lesson No. 2

Network model parameters

1. Network planning procedure

1. Establishing a complete list of work that must be performed when planning a set of works.

2. Drawing up a network topology - a clear sequence and interconnections of all works and constructing a network diagram.

3. Estimation of the duration of individual work.

4. Calculation of network diagram parameters.

5. Analysis and optimization of the network diagram.

6. Managing the progress of work according to the network schedule.

Network model parameters

In PCS systems, various types of network models are used, differing in the composition of information about the complex of works.

There are models with deterministic and probabilistic network structure, with deterministic and probabilistic estimates of the duration of network operation. When choosing a model, the project manager has to make a compromise decision: on the one hand, the network model must be simple, and on the other, adequate to the object.

The network model of PDV (the simplest deterministic time model), which is characterized by the following three points, has been widely used:

a) there is a network with a single initial and a single terminating event;

b) duration of all work t ij known, unambiguously defined (remember from mathematics: determinant - determinant) and indicated on the graph (usually in days, in foreign practice - more often in weeks);

c) the moment of start of execution of the complex is specified T 0, and also sets (but not necessarily) a deadline T dir the arrival of the final event.

Let's consider the time parameters of this model.

Based on the known durations of work, it is easy to determine the duration of each path - t(L). The duration of any path is equal to the sum of the durations of the work that compose it:

For clarification, let's look at Fig. 1. The graph above the arrows shows the duration of the work in days (recall that the duration of the fictitious work is zero).

Let's find the complete paths on the graph and determine their duration (by event numbers):

L 1 1 – 2 – 5 – 7 – 8 t(L 1)= 14 days

L 2 1 – 2 – 4 – 5 – 7 – 8 t(L 2)= 12 days

L 3 1 – 3 – 4 – 5 – 7 – 8 t(L 3)= 13 days

L 4 1 – 3 – 6 – 7 – 8 t(L 4)= 16 days

There is always a path that has the longest duration, it is called critical – L cr. Its duration received a special designation:

t(L cr) = T cr.

The concept of the critical path is a central concept in the SPU system. Meaning L cr The first is that it is the longest path in the network and is thus the only path that determines the total duration of the process. Therefore, if we want define the total duration of the process must be determined T cr, and determine all the rest for this purpose t(L) doesn't make sense. Secondly, if we want reduce duration of the process, it is necessary first of all to reduce the duration of work belonging to L cr. Thus, the logic of network planning leads us to the need to find critical paths in networks and determine their duration.

On the graph in Fig. 1 way L 4 has the longest duration, equal to 16 days, and is therefore critical. Usually the critical path is highlighted on the graphs (colored, double, bold, etc. arrows).

Please note that there may be several critical paths in the network (from the point of view of resource use - the more critical paths in the graph, the better).

Usually to L cr owns 10-15% of the works. The more complex the network, the fewer such jobs (it is believed that in a network of average complexity the number of jobs is 1.7 times greater than the number of events).

Other complete paths of the network in question may either lie entirely outside the critical path ( L 1 And L 2), or partially coincide with it ( L 3). These paths are called relaxed : in areas that do not coincide with the critical sequence of work, they have time reserves. The delay in the occurrence of events lying in these areas, up to a certain point, does not affect the completion date of the entire complex.

Of the non-stressed paths, the least stressed and subcritical ones attract the most attention. Subcritical paths have a duration close to T cr(different from T cr by a certain amount set by the project manager). These paths may become critical as a result of delays in their activities or as a result of shortening the duration of activities on the critical path, and are therefore potentially dangerous in terms of meeting project completion deadlines.

For example, if the execution time of work 2-5 (Fig. 1) is increased by 2 days, this will lead to t(L 1)= 16 days = T cr. Then L 1 will also become critical and will determine the deadline for completing the entire complex.

The least stressful paths can be considered from the point of view of the possibility of using resources (labor, equipment, funds). The possible lengthening of these paths caused by the transfer of resources, up to certain limits, is not dangerous for the timing of the project.

The activities belonging to the critical and subcritical paths constitute critical zone complex (15-20% of all work).

Knowing the duration of all work, you can also determine the timing of all network events. For each event, the early and late dates of its occurrence are determined.

Early date of the event – this is the minimum possible moment of its occurrence, when all the work preceding this event will be completed. It is determined by the maximum of the durations of all paths preceding a given event:

where is the path preceding this event i;

Let us explain this using the example of Fig. 1. Event 5 is preceded by three paths: 1-2-5 with a duration of 7 days, 1-2-4-5 with a duration of 5 days. and 1-3-4-5 with a duration of 6 days. Event 5 cannot occur earlier than 7 days, because only during this period will all preceding works 2-4, 3-4 and 2-5 be completed.

It is easy to see that for event 3 the early period of its occurrence = 4 days, because it is preceded by only one path 1-2, consisting of one job.

Late occurrence of the event - this is the maximum permissible moment of its occurrence, at which the total deadline for completing the entire complex does not yet change. The late date is determined by the difference between T cr and the longest duration of the paths following the event i:

(3)

where is the path following the event i;

The maximum of these paths.

Let's continue looking at Fig. 1. Event 5 is followed by only one path 5-7-8 lasting 7 days. Hence,

16 – 7 = 9 days.

Event 3 is followed by two paths: 3-4-5-7-8 with a duration of 9 days. and 3-6-7-8 with a duration of 12 days. Therefore, = 16 – 12 = 4 days, i.e. event 3 cannot occur later than 4 days from the start of work, otherwise it will affect the change in the period of the entire complex.

Since by definition of the critical path

, (4)

then for all events belonging to the critical path the equality is true:

We have already been convinced of the validity of this from the considered example for event 3. It lies on the critical path, therefore

Knowing the timing of events, it is possible to determine the start and end dates for each network operation, thereby identifying the possibility of shifting deadlines. Four deadlines are considered for each job:

Early start date; (6)

Early completion date; (7)

Late start date; (8)

Late work completion date. (9)

Taking into account equality (5) for events lying on the critical path, we can conclude that for activities on the critical path, the early and late start or finish dates coincide:

The next important parameter is the time reserve - in relation to the path, event and work.

The critical path is the longest on the network. The difference between the duration of the critical path T cr and the duration of any other journey t(L) called travel time reserve L and is denoted by:

(11)

The shorter the path L, the more time it does not coincide with the critical one, the greater the time reserve it has. The physical meaning of this parameter is: travel time reserve shows how much in total the duration of work belonging to the path can be increased L so that the overall deadline for completing the entire set of works does not change.

So, L 1(see Fig. 1) does not coincide with the critical one in the network section between 1 and 7 events. Its duration, as shown above, is 14 days, and, therefore, the reserve is equal to two days. The managers of all three works have only two days in the event of an unforeseen delay in their completion.

All events that are not on the critical path have a reserve of time, which is defined as the difference between the late and early dates of its occurrence:

Event time reserve shows by what maximum permissible period the onset of this event can be delayed without causing an increase in the time required for completing the entire complex of work. With a larger delay, the critical path will move to the maximum of the paths passing through this event i.

So, for event 5 (Fig. 1) = 9 – 7 = 2 days. If this event is delayed by 2 or more days, the critical path will move to the maximum path L 1, passing through event 5.

Events lying on the critical path have zero slack time, including the initial and final events.

For network model operations, two time reserves are defined: full and free.

Full operating time reserve- this is the reserve of the maximum of the paths passing through the work i,j

, (13)

where is the late date of the final event of this work;

The early date of the initial event of this work;

Duration of the work.

The physical meaning of this parameter is as follows: this reserve shows how much you can delay the start or increase the duration of a separate job without changing the directive (or early, if the directive is not specified) deadline for the completion event. In the latter case (if the deadline is not specified) - without changing T cr.

Let us pay attention to the following important point: the full reserve belongs not to one job, but to all the paths that pass through this job. Therefore, using it entirely on one of the work paths L cancels full time reserves all works belonging to this path.

For example, = 2 days (see Fig. 1), because it is determined by the path reserve L 1. If you use it completely at work 5-7, then other works on this path (1-2, 2-5) will be left without time reserves.

Total time reserves take a minimum value for activities lying on the critical path. This property is a necessary and sufficient condition for the work to belong to the critical path and is used to find it when calculating the network. The minimum value of total reserve is zero if T dir not specified or exceeds the start time of work T 0 by the amount T cr. In general, it is equal to the difference ( T cr - T dir).

Free reserve work time represents the maximum time by which the start or duration of an operation can be delayed i,j provided that all network events occur at their earliest times:

. (14)

A free reserve is not formed for all jobs, but only for jobs that directly belong to events through which paths of varying durations pass. This should be understood this way: if an event is preceded by one job (for example, job 1-2 in Fig. 1), then for it the free reserve is equal to zero by definition ( = 0), in other cases - 0.

The free reserve is part of the full reserve, and therefore another formula is more often used in practice:

where is the reserve of the final work event i,j.

Free float shows what part of the total float of operating time can be used to increase its duration, provided that this does not cause a change early date the arrival of its final event. Free reserve is an independent reserve, i.e. using it on one of the jobs does not change the amount of free time reserves for the remaining jobs in the network. Using the free time reserve, the responsible executive can maneuver within its limits the start time of this work, its end or its duration, without affecting the interests of other work managers.

Practical lesson No. 3

Calculation of network diagram parameters.

"Graphic" method

A number of methods have been proposed to calculate the parameters of network diagrams:

a) directly on the chart itself (the so-called “graphical” method);

b) tabular method;

c) matrix method;

d) based on machine algorithms.

In medium and large complexes, such work is performed by specially designated employees who are part of the SPU service. Currently, many enterprises and organizations have standard and proprietary programs for calculating network parameters on a computer.

"Graphic" method

Calculation of parameters and recording of results are carried out on the graph itself. To do this, a network graph, preferably without intersections, is drawn on a larger scale: the diameter of the circles depicting events on the graph is 15-25 mm. The circles are divided into 4 sectors.

The “key” to reading such a graph is shown in Fig. 2: in the lower sectors we will display the event number; in the left sectors - early dates of events; on the right - late dates of events; in the upper ones - event time reserves; in square brackets under the arrow - the full and free reserve of each job; above the arrow is the duration of work.

First, the graph is redrawn on a larger scale (Fig. 4). Recall that the duration of the fictitious work is zero. And one more thing: it doesn’t matter which sector of the circle the arrow is pointing to.

The chart parameters are calculated in the following order.

1. Determination of the early date of occurrence of each event.

For the initial event 1 we have = 0 and this is indicated in the left sector. For other events in accordance with formula (2).

This means that if an event includes one arrow (for example, event 2), then the duration of work 1-2 is added to the early date of occurrence of the previous event 1, and the result is recorded in the left sector of event 2.

Event 3 includes two jobs: 1-3 and 2-3. Therefore, first we get two values: 0 + 4 = 4 and 2 + 7 = 9. The larger value (9 days) is the early date of occurrence of event 3, which is noted in its left sector.

Since the final event is always on the critical path, we can say that = = 19 days. We don’t know what jobs and events the critical path will go through, but its duration has already been determined when calculating the first network parameter.

2. Determination of late dates for the occurrence of events.

Calculation is carried out from final event(from the end of the graph) in strictly reverse order. Since the events lying on the critical path have the same early and late dates, then for the final event = = 19 days, which is noted in the right sector (Fig. 5).

For other events, in accordance with formula (3), we can write . For event 5 we have = 19 – 4 = 15 days, for event 6 = 19 – 2 = 17 days, for event 4 = 15 – 0 = 15 days.

These events, coming from the end of the graph, can only be reached in one way, so there is no need to determine the minimum value, as, for example, for event 3. Works 3-4 and 3-6 come out of it, so first we get two values: 15 – 6 = 9 and 17 – 3 = 14. The smaller value (9 days) is a late date for the onset of event 3, which is noted in its right sector.

3. Determination of event time reserves.

The calculation can be carried out either from the beginning of the chart or from its end. For any event. This means that for each event, the value of the left sector must be subtracted from the value of its right sector, and the result must be placed in its upper sector (Fig. 6).

4. Finding the critical path on the graph, i.e. there are events and activities lying on the critical path.

The procedure can be carried out from the beginning or end of the chart.

A) A necessary condition for an event to belong to the critical path:, i.e. sequentially search for events with zero time reserves.

b) If from an event with zero float several jobs come out that have a zero float of the final event, then it is checked a sufficient condition for the work to belong to the critical path:

5. Determination of the full operating time reserve.

We find full reserves only for jobs that do not lie on critical paths and are not fictitious, using formula (13) . The result is written in square brackets below or next to the arrow. So, for work 1-3 the total reserve time is 9 – 4 – 0 = 5 days, for work 2-5 we have 15 – 6 – 2 = 7 days, etc.

Let us remind you that if a job has , then it is necessarily on the critical path (this is for self-testing).

6. Determination of free reserve operating time.

Free time reserve is part of the total time, therefore it is determined for the same jobs that are not on the critical path and are not fictitious, according to formula (15). Calculation using this formula is simpler than using formula (14), because By this time, the full work reserves and event reserves have already been calculated. So, to calculate, you need to take the value of the total reserve of work 2-5 (7 days) and subtract from it the reserve of the final event of this work (0 days), indicate the result under the arrow and close the square bracket. Free reserves for other work are quickly calculated in the same way.

Practical tips:

b) to speed up the process of calculating parameters, it is advisable to combine stage 6 with stage 5, because in complex networks, it is difficult to find the same work on the graph every time.

Practical lesson No. 4