More information on this topic HERE.

Calendar plan (schedule) of work production is, of course, a key document of the PPR. The success of the project implementation largely depends on the quality of its development. The calendar plan is a model of construction production, which establishes a rational sequence, priority and timing of work at the facility.

scheduling

The essence of scheduling, its role in construction

Calendar planning is an integral element of the organization of construction production at all its stages and levels. The normal course of construction is possible only when it is thought out in advance in what sequence the work will be carried out, how many workers, machines, mechanisms and other resources will be required for each work. Underestimation of this entails inconsistency in the actions of performers, interruptions in their work, delays in terms and, of course, an increase in the cost of construction. To prevent such situations, a calendar plan is drawn up, which performs the function of a work schedule within the accepted duration of construction. Obviously, the changing situation at the construction site may require a significant adjustment of such a plan, however, in any situation, the construction manager must clearly understand what needs to be done in the coming days, weeks, months.

The duration of construction is assigned, as a rule, according to the norms (SNiP 1.04.03-85 * Standards for the duration of construction ...) depending on the size and complexity of the facilities under construction, for example, the area of ​​hydro-reclamation systems, types and capacities of industrial enterprises, etc. In some cases, the duration of construction may be planned different from the normative (most often in the direction of tightening the deadlines), if required by the needs of production, special conditions, environmental programs, etc. For objects built in difficult natural conditions, an increase in the duration of construction is acceptable, but this must always be properly justified.

In construction practice, simplified planning methods are often used, when, for example, only a list of works is compiled with the deadlines for their completion without proper optimization. However, such planning is permissible only when solving small current tasks during construction. When planning large objects of work for the entire period of construction, careful work is needed to select the most appropriate sequence of construction and installation works, their duration, the number of participants, it is necessary to take into account many factors mentioned above. For these reasons, various forms of scheduling are used in construction, allowing in their own way to optimize the planned course of work, the possibility of maneuvers, etc.

• line calendar charts
• network charts

In addition, depending on the breadth of the tasks to be solved, the required degree of detail of solutions, there are various types of calendar plans that are used at different levels of planning.

When developing schedules in PIC and PPR, the best results are achieved when several options for the schedule are compiled, and the most effective one is selected.

Types of calendar plans (schedules)

There are four types of calendar schedules, depending on the breadth of the tasks to be solved and the type of documentation they are included in. All types of calendar schedules should be closely linked to each other.

Consolidated calendar plan (schedule) in the POS determines the order of construction of objects, i.e. the start and end dates of each object, the duration of the preparatory period and the entire construction as a whole. For the preparatory period, as a rule, a separate calendar schedule is drawn up. The existing norms (SNiP 12-01-2004 instead of SNiP 3.01.01-85) provide for the preparation of calendar plans in the POS in monetary form, i.e. in thousand rubles with distribution by quarters or years (for the preparatory period - by months).

For complex facilities, especially water management and hydraulic engineering, additional summary graphs are compiled, oriented to physical volumes.

When drawing up calendar plans for the construction of hydraulic and water facilities, it is necessary, as already noted, to carefully coordinate the progress of construction work with the timing of water flow in the river, the timing of blocking the channel and filling the reservoir. All these terms should be clearly reflected in the calendar plan. During the reconstruction of such facilities, minimal interruptions in the operation of the hydroelectric complex or hydraulic structure should be ensured.

At the stage of developing a consolidated schedule, the issues of dividing construction into stages, launch complexes, and technological units are being resolved. The calendar plan is signed by the chief engineer of the project and the customer (as a coordinating authority).

Object Schedule in the PPR determines the sequence and timing of each type of work at a particular facility from the beginning of its construction to commissioning. Typically, such a plan is broken down by months or days, depending on the size and complexity of the object. The object calendar plan (schedule) is developed by the compiler of the PPR, i.e. by the general contractor or a specialized design organization engaged for this purpose.

When developing calendar plans for the reconstruction or technical re-equipment of an industrial enterprise, it is necessary to coordinate all the terms with this enterprise.

Working calendar schedules are usually compiled by the production and technical department of a construction organization, less often by line personnel during the construction and installation period. Such schedules are developed not for a week, a month, several months. Weekly charts are most widely used. Work schedules are an element of operational planning, which must be carried out continuously throughout the entire construction period.

The purpose of work schedules, on the one hand, is to detail the object calendar plan and, on the other hand, to respond in a timely manner to all kinds of changes in the situation at the construction site. Work schedules are the most common type of scheduling. As a rule, they are compiled very quickly and often have a simplified form, i.e., as practice shows, they are not always properly optimized. Nevertheless, they usually take into account the actual situation at the construction site better than others, since they are compiled by persons directly involved in this construction. This is especially true for taking into account weather conditions, the specifics of the interaction of subcontractors, the implementation of various rationalization proposals, i.e. factors that are difficult to predict in advance.

Hourly (minute) charts in technological maps and maps of labor processes are compiled by the developers of these maps. Such schedules are usually carefully thought out, optimized, but they are focused only on typical (most likely) operating conditions. In specific situations, they may require significant adjustments.

Simplified scheduling forms

With short-term planning, as already noted, in construction practice a simplified form of scheduling is often used in the form of a list of works with deadlines for their implementation. This form is not visual and is not suitable for optimization, but when solving current problems for the coming days or weeks, it is acceptable due to the simplicity and speed of its compilation. Usually this is the result of an agreement on the timing of work between the performers, which is recorded in the form of a protocol of a technical meeting, an order from the general contractor, or another current document.

The simplified form should also include construction planning in cash. In this case, some optimization is possible, but it solves such issues only in an extremely generalized form, since it relates primarily to construction financing. A calendar plan in terms of money is usually drawn up for especially large amounts of work, when a whole object or complex of objects acts as an element of planning. Such plans are typical, for example, for PIC.

Line calendar charts

A linear calendar chart (Ganga chart) is a table of "work (objects) - time", in which the duration of work is depicted as horizontal line segments.

Such a schedule provides opportunities for optimizing construction and installation works according to a wide variety of criteria, including the uniform use of labor, mechanisms, building materials, etc. The advantage of line charts is also their clarity and simplicity. The development of such a schedule includes the following steps:

• drawing up a list of works for which a schedule is being made
• determining their production methods and volumes
• determination of the labor intensity of each type of work by calculations based on existing time standards, aggregated standards or local experience data
• drawing up the initial version of the schedule, i.e. preliminary determination of the duration and calendar deadlines for the implementation of each work with the display of these terms on the chart
• schedule optimization, i.e. ensuring a uniform need for resources, primarily in the labor force, ensuring the timely completion of construction, etc., establishing the final calendar dates for work and the number of performers.

The results of each stage of development, the calendar plan, must be carefully verified, because errors, as a rule, are not compensated for at subsequent stages. For example, if at the first stage the volume of any work is estimated incorrectly, its duration and deadlines will be incorrect, and the optimization will be imaginary.

When determining the labor intensity of work, it is necessary to pay special attention to the reality of the calculations carried out, taking into account specific working conditions. The latter may differ significantly from those accepted in the norms, so the compiler of the calendar plan must be well acquainted with the actual conditions of construction.

The main disadvantage of linear graphs is the difficulty of their adjustment in case of violations of the initial terms of work or changes in the conditions for their implementation. These shortcomings are eliminated with another form of scheduling - network charts.

network charts

The network graph is based on the use of another mathematical model - a graph. Graphs (obsolete synonyms: network, maze, map, etc.) are called by mathematicians "a set of vertices and a set of ordered or unordered pairs of vertices." Speaking in a more familiar (but less precise) language for an engineer, a graph is a set of circles (rectangles, triangles, etc.) connected by directed or non-directed segments. In this case, the circles themselves (or other figures used) according to the terminology of graph theory will be called "vertices", and the non-directed segments connecting them - "edges", directed (arrows) - "arcs". If all the segments are directed, the graph is called directed; if all segments are undirected, the graph is called undirected.

The most common type of work network diagram is a system of circles and directed segments (arrows) connecting them, where the arrows represent the work itself, and the circles at their ends ("events") - the beginning or end of these works.

The figure shows in a simplified way only one of the possible configurations of the network diagram, without data characterizing the planned works themselves. In fact, the network diagram provides a lot of information about the work being done. Above each arrow is written the name of the work, under the arrow - the duration of this work (usually in days).

The circles themselves (divided into sectors) also contain information, the meaning of which will be explained later. A fragment of a possible network diagram with such data is shown in the figure below.

Dotted arrows can be used in the graphics - these are the so-called "dependencies" (dummy jobs) that require neither time nor resources.

They indicate that the "event" to which the dotted arrow points can only occur after the event from which the arrow originates.

There should be no dead ends in the network diagram, each event should be connected by a solid or dashed arrow (or arrows) to any previous (one or more) and subsequent (one or more) events.

Events are numbered approximately in the order in which they will occur. The initial event is usually located on the left side of the graph, the final event - on the right.

A sequence of arrows in which the beginning of each subsequent arrow coincides with the end of the previous one is called a path. The path is indicated as a sequence of event numbers.

In a network diagram, there can be multiple paths between start and end events. The path with the longest duration is called the critical path. The critical path determines the total duration of activities. All other paths have a shorter duration, and therefore the work performed in them has time reserves.

The critical path is indicated on the network diagram by thickened or double lines (arrows).

Two concepts are of particular importance when drawing up a network diagram:

Early start of work - the period before which it is impossible to start this work without violating the accepted technological sequence. It is determined by the longest path from the initiating event to the start of this work.

Late finish is the latest end date for a job that does not increase the total duration of the job. It is determined by the shortest path from a given event to the completion of all work.

When evaluating time reserves, it is convenient to use two more auxiliary concepts:

Early finish is the deadline before which the work cannot be completed. It is equal to the early start plus the duration of this work.

Late start - the period after which it is impossible to start this work without increasing the total duration of construction. It is equal to the late finish minus the duration of the given work.

If the event is the end of only one job (that is, only one arrow is directed to it), then the early end of this job coincides with the early start of the next one.

The total (full) reserve is the maximum time for which the execution of this work can be delayed without increasing the total duration of the work. It is determined by the difference between late and early start (or late and early finish - which is the same).

Private (free) reserve - this is the maximum time for which you can delay the execution of this work, without changing the early start of the next one. This fallback is only possible when the event includes two or more activities (dependencies), i.e. two or more arrows (solid or dotted) point to it. Then only one of these jobs will have an early finish that coincides with an early start of the subsequent job, while for the others these will be different values. This difference for each work will be its private reserve.

In addition to the described type of network graphs, in which graph vertices ("circles") represent events, and arrows represent jobs, there is another type in which the vertices are jobs. The difference between these types is not fundamental - all the basic concepts (early start, late finish, general and private reserves, critical path, etc.) remain unchanged, only the ways of writing them differ.

The construction of a network diagram of this type is based on the fact that the early start of the subsequent work is equal to the early end of the previous one. If this job is preceded by several jobs, its early download should be equal to the maximum early completion of previous jobs. The calculation of late dates is carried out in the reverse order - from the final to the initial one, as in the "nodes - events" network diagram. For a finishing activity, late and early finish are the same and reflect the length of the critical path. The late start of the next activity is equal to the late finish of the previous one. If a given work is followed by several works, then the minimum value from the late beginnings is decisive.

Network "nodes - work" graphs appeared later than "nodes - events" graphs, therefore they are somewhat less known and relatively less often described in educational and reference literature. However, they have their advantages, in particular they are easier to build and easier to adjust. When adjusting "completed - work" graphs, their configuration does not change, but for "nodes - events" graphs, such changes cannot be excluded. However, at present, the compilation and correction of network graphs are automated, and for a user who only their time reserves, it does not really matter how the schedule is made, i.e. what type it is in. In modern specialized packages of computer programs for planning and operational management, the type of "vertex - work" is mainly used.

Network diagrams are corrected both at the stage of their compilation and use. It consists in optimizing construction work in terms of time and resources (in particular, the movement of labor). If, for example, the network schedule does not ensure the completion of work within the required time frame (normative or established by the contract), it is adjusted in time, i.e. shortening the critical path. This is usually done

due to the time reserves of non-critical work and the corresponding redistribution of resources

by attracting additional resources

by changing the organizational and technological sequence and the relationship of work.

In the latter case, the graphs "vertices - events" have to change their configuration (topology).

Adjustment for resources is made by constructing linear calendar graphs for early beginnings corresponding to one or another variant of the network diagram, and adjusting this variant.

Automated construction management systems usually include computer programs that, to one degree or another, automate almost all stages of compiling and adjusting network schedules.

Reference literature

• SNiP 1.04.03-85 "Standards for the duration of construction and groundwork in the construction of enterprises of buildings and structures";
• MDS 12-81.2007 "Methodological recommendations for the development and execution of a project for the organization of construction and a project for the production of works."

The construction of a network schedule begins with the compilation of a list of operations (works) to be performed (see Table 1). The sequence of operations in the list is arbitrary. The numbering order of operations is carried out in accordance with the sequence of their entry in the list. The list of operations is carefully thought out and, depending on the specific conditions, is detailed to a certain extent. The operations included in the list are characterized by a certain duration, which is established on the basis of current standards or by analogy with previously performed operations. After compiling the list of operations, proceed to the procedure for building a network.

Example. It is necessary to build a network schedule for the implementation of a complex of operations for the reconstruction of the workshop. The list of operations is presented in Table. 1. The final network diagram of the complex of operations is shown in Fig.1.

Solution. Graph operations except for operations 2→3 and 5→6 , are valid. The numbers in brackets assigned to the arcs indicate the duration of the corresponding operations. Operations a1 and a2 do not rely on any operations, so on the graph we will depict them as arcs emerging from the event (1), which means the beginning of the execution of a complex of operations. Operations a3, a5 and a6 rely on surgery a1, therefore, on the graph, these arcs directly follow the arc a1. Event (2) means the end of the operation a1 and the start of the operations represented by the arcs emerging from this event. Operation a4, relies on operations a1 and a2. Graphically, this condition is reflected by sequentially displaying operations 1→3 and 3→4 and introducing a fictitious operation 2→3 . Event (3) is incident to operations 1→3 and 2→3 , therefore, the moment of completion of the event (3) will be the moment by which all the operations included in this event will be completed and the operation reflected by the arc emerging from it can be started. Similarly, taking into account the execution technology, the rest of the operations are shown on the graph. The final event (9) means the moment of completion of the entire complex of operations for the reconstruction of the workshop. Operation codes (see Table 1) consist of the numbers of the initial and final events and are practically entered into the list after the schedule is drawn up.

Table 1 - List of operations for building a network diagram

Operation	Operation code	the name of the operation	Relies on operations	Duration, days
		Preparatory work
		Dismantling of old equipment
		Foundation preparation for new equipment
		Preparing for the installation of new equipment
		Electrical work
		Installation of new equipment
		Connecting equipment to the mains
		Adjustment and technological testing of equipment
		Finishing work
		Acceptance of the shop into operation

The events and arcs of the constructed network graph (see Fig. 7.5) are numbered according to ranks. In practice, in the original network diagram, the elements, as a rule, have an unordered numbering. Therefore, after plotting a graph, it is recommended to renumber its elements using the methods discussed in the previous paragraph.

The construction of network diagrams of transient complexes of operations, when due to lack of time it is not possible to perform optimization calculations, is carried out taking into account technological and resource limitations. The construction of graphs for non-transitory complexes of operations, when there is enough time for their study, is performed only taking into account technological limitations. This approach ensures the minimum duration of the complex of operations. After the graph is built, its time parameters are calculated and optimization is performed for resources or other indicators, for which formal optimization methods are used.

Picture 1

For different levels of management, schedules of varying degrees of detail are drawn up. So in fig. 7.6 shows an enlarged network schedule for the reconstruction of the workshop. For specific performers, private network schedules are compiled with a greater degree of detail.

Assignment for seminar No. 4

Exercise 1. Bring the technological scheme of production developed in the course project in the specialty, list the main technological operations with an indication of the time of their implementation, as a result of which to build a network diagram of the production process and calculate all its time parameters.

Project Management for Dummies Portney Stanley I.

A simple network diagram example

Let's consider the use of a network diagram using the example of organizing a picnic. (I'm not generally suggesting that you plan every picnic using a network diagram, but this example will show you the basic techniques and possibilities.)

Friday night, after a busy week, you and a friend discuss how to make the most of your weekend. The forecast promises good weather, and you decide to go on a picnic in the morning to one of the two nearby lakes. In order to organize a picnic and have fun as best as possible, you decided to make a network schedule.

In table. 4 5 are seven jobs that you think you need to do to prepare a picnic and get to the lake.

Table 4.5. List of picnic activities on the lake

Job number	Job Title	Executor	Duration (in min.)
1	Load things into the car	you and girlfriend	5
2	Get money from the bank	You	5
3	Make egg sandwiches	Friend	10
4	Go to the lake	you and girlfriend	30
5	Choose lake	you and girlfriend	2
6	Fill up the car with petrol	You	10
7	boil eggs (for sandwiches)	Friend	10

In addition, you comply with the following conditions

All work begins on Saturday at 8:00 am at your home. Until then, nothing can be done.

All work on this project must be completed.

You agreed not to change the performers of the planned work.

Both lakes are in opposite directions from your home, so before you set off, you should decide which one to go to.

First, you decide in what order you will do all these jobs. In other words, you need to define for each activity the immediately preceding one. Such dependencies must be taken into account.

A friend has to boil eggs before making sandwiches.

Together you must decide which lake to go to before you set off.

In what order to perform the rest of the work depends on your desire. For example, you accepted such an order.

First of all, you decide together which lake to go to.

Having made a decision about the lake, you go to the bank for money.

After receiving the money in the bank, you fill up the car.

After making a joint decision about the lake, the friend begins to boil the eggs.

After the eggs are cooked, a friend makes sandwiches.

After you've returned from the gas station and your friend has prepared sandwiches, load your things into the car.

After both of you have loaded the car, go to the lake.

Tab. Figure 4.6 illustrates the workflow you have defined.

Table 4.6. The sequence of work for organizing a picnic

To build a network diagram according to this table, follow these steps.

1. Start the project with the Start event.

2. Next, identify all jobs that don't have predecessors. You can start implementing them right away from the moment the project starts.

In our case, this is the only job 5.

3. We begin to draw a network diagram (Fig. 4.5).

Identify all jobs for which job 5 is the immediate predecessor.

Figure 4.5. Start building a picnic network

4. From the table. 4.6 it can be seen that there are two of them: work 2 and work 7. Draw them in the form of rectangles and draw arrows from work 5 to them.

Continue to build a graph in the same way.

For work 6, work 2 will be the previous one, and for work 3 - work 7. At this stage, the graph will look like in Figure 4.6

The table shows that activity 1 is preceded by two activities: activity 3 and activity 6, and activity 4 is preceded by only activity 1. Finally, from activity 4 there is an arrow to the "End" event

Rice. 4.6. Continuation of building a network diagram organizing a picnic

On fig. Figure 4.7 shows the completed network diagram.

Rice. 4.7. The final look of the network diagram for organizing a picnic

Now let's look at a few important questions. First, how long will it take you to pack up and get to the lake?

The upper path, including works 2 and 6, is 15 minutes.

The lower path, including works 7 and 3, is 20 minutes.

The longest in the schedule is the critical path, it includes activities 5, 7, 3, 1 and 4. Its duration is 57 minutes. That's how much you'll need to get to the lake if you follow this network schedule.

Is it possible to delay some tasks and still meet the 57 minute mark? If so, which ones?

The upper path, which includes jobs 2 and 6, is not critical.

It follows from the network that since activities 5, 7, 3, 1, and 4 are on the critical path, they cannot be delayed in any way.

However, jobs 2 and 6 can be done at the same time as jobs 7 and 3. Jobs 7 and 3 take 20 minutes, while jobs 2 and 6 take 15 minutes. Therefore, jobs 2 and 6 have a slack of 5 minutes.

On fig. 4.8 shows the same network diagram, but in the form of "event-work". Event A is equivalent to the "Start" event, and event I is equivalent to the "End" event.

Rice. 4.8. The final view of the network diagram for organizing a picnic in the form of "event-work"

Presented in fig. 4.8 events do not yet have names. You can give them for example:

Event AT, the end of activity 5 ("Select a lake"), can be called "Decision made";

Event FROM, the end of work 2 ("Get money"), can be called "Money received". And so on.

Elementary is called an event that completes one job. Defining elementary events at the end of all activities in the activity-event network diagram makes it easier to track the progress of activities. If activity 1 has multiple predecessors, then instead of navigating multiple arrows to the event after which activity 1 starts, do the following:

End each preceding job with an elementary event;

Connect them with arrows to the next elementary event, from which work 1 will start. The arrows in this case will mean fictitious works.

This is shown in fig. 4.8. You must finish Job 6 "Fill the Car" and your friend should finish Job 3 "Make Sandwiches" before you both start loading things into the car. Instead of leading arrows directly to the event G, end work with event 6 D"The car is fueled", and work 3 event F"The sandwiches are ready." Then mark the fictitious jobs with arrows from the events D and F to the event G, which you can call "Ready to load the machine."

This text is an introductory piece. From the book Fundamentals of Project Management author Presnyakov Vasily Fedorovich

Designing a project network diagram

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With the help of this program you can determine the parameters of the network graph online(calculate the timing of events, time reserves and the critical path), find the tension coefficients. Optimization of the network schedule is carried out according to the following criteria: the number of performers, reserves-costs, reduction of terms.
A network graph can be drawn, as well as set in the form of a matrix or table (Operations menu).

Select the desired vertex type and left-click on the graphics canvas

Graphics canvas dimensions

Width Height

● ■ ▲ ⊗ ↔ ✍ ⊗

network model parameters (critical path, time reserves, build a Gantt chart and much more).

For the generated graph, you can perform the following actions:

Service manual

To add a vertex to the graphics canvas, use the Add button corresponding to the shape. A new object can also be inserted by first selecting it with the left mouse button and then clicking on the workspace. Numbering of vertices can start from 0 , for this you need to uncheck the item Numbering of vertices from No. 1.

1 2 3 4 1 10 30 15

Vertex numbering starting from 0

0 1 2 3 1 10 30 15

To connect the vertices, you must first select them (one click on the object), and then click the Connect button.
The network model can be presented in tabular form and as a weight matrix (distance matrix). To use view data, select the Operations menu.

The constructed graph can be saved in docx or png format.
If a rectangle is used as the vertex shape, then when constructing a pie chart, the Microsoft Visio methodology is used with the display of parameters duration, ES, EF, LS, LF, and slack.

Basic definitions

Directed graph, in which there is only one vertex that has no incoming arcs, and only one vertex that has no outgoing arcs, called a network. The network that models the complex of works is called its network model or network diagram. The arcs connecting the vertices of the graph are oriented in the direction of achieving the result in the implementation of a set of works.
The most common way to represent the simulated complex of works in terms of works and events.
The concept of "work" has the following meanings:

• "actual work" - a process that requires time and resources;
• “dummy job” is a logical relationship between two or more jobs, indicating that the start of one job depends on the results of another. Dummy work does not require time and resources, its duration is zero.

Work on the graph is depicted by an arrow, above which the time spent on it is indicated. The length of the arrow and its orientation on the chart do not matter. It is only desirable to maintain the direction of the arrows so that initial the event to work (denoted by i) was located on the left in the network diagram, and final(indicated by j) - on the right. To display fictitious works, dotted arrows are used, over which the time is not indicated or zero is put down.

On the network model, events correspond to graph vertices.

Rules for building a network model

Rule 1. Each operation in the network is represented by one and only one arc (arrow). None of the operations should appear twice in the model. In this case, one should distinguish between the case when any operation is divided into parts; then each part is represented by a separate arc.

Rule 2. No pair of operations should be defined by the same start and end events. The possibility of ambiguous definition of operations through events appears when two or more operations can be performed simultaneously.

Rule 3. When including each operation in a network model, the following questions need to be answered to ensure proper ordering:
a) What operations need to be completed immediately before the start of the operation in question?
b) What operations should immediately follow after the completion of this operation?
c) What operations can be performed simultaneously with the one under consideration?

When constructing a network diagram, the following rules should be observed:

• there should be no "dead ends" in the network, i.e., events from which no work starts, except for the final event of the chart;
• There should be no "tail" events in the network diagram, that is, events that are not preceded by at least one work, with the exception of the original one.
• the network should not have closed loops (Fig. 1);
• Any two events must be directly related by no more than one work.
• In a network, it is recommended to have one start and one end event.
• The network diagram must be streamlined. That is, events and jobs should be arranged so that for any job, the preceding event is located to the left and has a lower number compared to the event that ends this job.

The construction of the network graph begins with the image of the initial event, which is indicated by the number 1 and circled. Arrows are fired from the start event corresponding to activities that are not preceded by any other activities. By definition, the moment of completion of work is an event. Therefore, each arrow
ends with a circle - an event in which the number of this event is affixed. The numbering of events is arbitrary. At the next stage of construction, we depict works that are preceded by already drawn works (that is, which rely on already built works), etc. At the next stage, we reflect the logical relationships between works and determine the end event of the network diagram, on which no works rely. The construction is completed, then it is necessary to streamline the network diagram.

Network Graph Optimization Methods

The logical-mathematical description, the formation of plans and control actions is carried out on the basis of the use of a special class of models called network models.
After constructing and calculating the network schedule (determining its parameters), performing an analysis of the schedule, which consists in assessing its feasibility and structure, assessing the workload of performers, assessing the probability of the final event occurring within a given period, you should start optimizing the network schedule. The optimization procedure consists in bringing the schedule in line with the given deadlines for the completion of work, the capabilities of contractors, etc. In general, optimization should be understood as the process of improving the organization of work.

To be able to optimize the network model, all initial data are entered in the form of a table (Operations / Add in the form of a table).

• Optimization of the network model according to the "number of performers" criterion. The column Number of performers is being filled H
• Optimization of the network model according to the criterion "time - cost" (time - costs). In the case of known cost factors for work acceleration, only this column h(i,j) is filled. Otherwise, the columns t opt ​​(Normal mode), Minimum work time, t min (Fast mode), Normal cost, Cn and Urgent cost, Cc are filled.

Binding (a) and loading (b) plots before optimization

1,2 6 1,3 1 1,4 5 2,5 3 2,6 1 3,6 8 4,6 4 4,7 2 5,8 6 6,8 1 7,8 3 1 12 2 3 19 4 17 5 6 7 8 18 9 10 11 12 13 14 13 15 16 17 18 10 19 20 21 22 4 23 24 25 26 1 27

Binding (a) and loading (b) plots after optimization

1,2 6 1,3 1 1,4 5 2,5 3 2,6 1 3,6 8 4,6 4 4,7 2 5,8 6 6,8 1 7,8 3 1 12 2 3 11 4 14 5 6 7 15 8 9 10 11 18 12 13 14 15 10 16 17 18 4 19 20 21 22 10 23 24 25 26 7 27

Gantt Chart

1,2 4 1,3 3 1,4 5 2,5 11 2,6 14 3,6 4,6 17 4,7 5,8 19 6,8 27 7,8 25

Examples of network models

Consider options for network graphs from the culinary field using the example of cooking chicken borscht.
a) Cooking in a regular pot

1 2 3 4 5 1 10 30 15 7

Works:

1.3: boil chicken, 30 min.
2.3: put the cabbage and cook for 10 minutes.
3.4: put 1/2 beets, carrots and potatoes. Boil 15 min.
4.5: add the rest of the beets, onions, greens. Boil 7 min.
b) Cooking in a dish with the effect of a Russian stove (three-layer bottom, lid without holes)

1 2 3 4 5 10 10 20 30 60

Works:
1.2: cleaning vegetables (cabbage, carrots, potatoes, beets, onions), 10 min.
1.4: Boil the chicken in a normal pot, 30 minutes.
2.3: put vegetables in a special dish, add 3 tablespoons of water, heat to T=70 C and turn off, 10 min.
3.4: cooking vegetables in their own juice, 20 min.
4.5: Add the cooked vegetables to the chicken. Infused 60 min.

Bibliography

1. Mushik E., Muller P. Methods for making technical decisions. Per. with him. –M.: Mir, 1990.
2. Taha H. Introduction to Operations Research. In 2 books. Book. 2. Per. from English. –M.: Mir, 1985.
3. Management in RAV systems: Textbook. -L .: Military Publishing, 1980.

Vertex Properties

Text

The size Color

Thickness Color

dotted - - - -
Dimensions in px and background

w h

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Connection (arc)

Text (weight)

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dotted - - -
end marker →

To build a network schedule, it is necessary to identify the sequence and interconnection of work: what work needs to be done and what conditions to ensure that this work can be started, what work can and should be performed in parallel with this work, what work can be started after the completion of this work. These questions make it possible to identify the technological relationship between individual works, provide a logical construction of the network diagram and its compliance with the simulated set of works.

The level of detail of the network diagram depends on the complexity of the object under construction, the amount of resources used, the amount of work and the duration of construction.

There are two types of network diagrams:

peaks - works

peaks - events

Network graphs of the "vertex - work" type.

The elements of such a schedule are activities and dependencies. Work is a specific production process that requires time and resources to complete it, and is depicted by a rectangle. Dependence (fictitious work) shows the organizational and technological connection between the works, which does not require time and resources, is depicted by an arrow. If there is an organizational or technological break between jobs, then the duration of this break is indicated on the dependency.

If the work of the "vertex - work" network diagram has no previous work, then it is the original work of this graph. If the job has no subsequent jobs, then it is the final job of the network. There should be no closed contours (cycles) in the "nodes - work" network diagram, i.e. dependencies should not go back into the work they came from.

Network graphs of the "nodes - events" type.

The elements of this type of graphs are activities, dependencies and events. The work is represented by a solid arrow, the dependence is dotted. An event is the result of one or more activities, necessary and sufficient for the start of one or more subsequent activities, and is represented by a circle.

In this type of network diagram, each job is between two events: an initial one, from which it exits, and an end event, into which it enters. Network events are numbered, so each job has a code consisting of its start and end event numbers.

For example, in fig. 6.2 works are coded as (1,2); (2.3); (2.4); (4.5)

If the event of the network graph "vertices - events" has no previous activities, then it is the initial event of this network. The works immediately following it are called the original ones. If an event has no follow-ups, then it is the final event. The works included in it are called final.

To correctly display the relationships between jobs, you must follow the following basic rules for constructing a network diagram "Vertices - Events":

1. When displaying simultaneously or in parallel works (for example, works "B" and "C" in Fig. 6.2), dependence (3.4) and an additional event (3) are introduced.

2. If to start work "D" it is necessary to perform work "A" and "B", and to start work<В» - только работу «А», то вводится зависимость и дополнительное событие (рис.6.З.).

H. There should be no closed loops (cycles) in the network diagram, i.e. a chain of jobs that goes back to the event they came from

4. In the network diagram, with the flow organization of construction, additional events and dependencies are introduced (Fig. 6.5.).

To determine the duration of the critical path and the timing of each activity, the following are determined: time parameters :

Early start -

Early end of work - ;

Late start - ;

Late completion of work

Full slack - R;

Free time reserve

Early start- Earliest start date. The early start of the original network activities is zero. The earliest start of any activity is equal to the maximum early finish of previous activities:

Early end of work- the earliest time the work was completed. It is equal to the sum of the early start and the duration of the work.

Late end of work- the latest end point at which the duration of the critical path does not change. The late completion of finishing activities is equal to the duration of the critical path. The late finish of any job is equal to the minimum late start of subsequent jobs.

Late start- the latest start time at which the duration of the critical path will not change. It is equal to the difference between the late completion of this work and its duration.

Activities on the critical path have equal early and late start and finish dates, so they have no slack. Activities not on the critical path have time reserves .

Full slack- the maximum time by which the duration of the activity can be increased or its start can be postponed without increasing the duration of the critical path. It is equal to the difference between the late and early start or finish dates.

Free time reserve- the time by which you can increase the duration of the work or postpone its start, while not changing the early start of subsequent work. It is equal to the difference between the early start of the next activity and the early end of this activity.

Calculation of the network diagram "tops - work"

To calculate the "tops - work" network graph, the rectangle depicting the work is divided into 7 parts (Fig. 6.6).

The upper three parts of the rectangle record the early start, duration, and early finish of the work; the lower three parts show the late start, time reserves, and late finish. The central part contains the code (number) and the name of the work.

The calculation of the network schedule begins with the definition of early dates. Early starts and finishes are computed sequentially from the original job to the end job. The early start of the original job is 0, the early finish is the sum of the early start and the duration of the job:

The early start of the subsequent activity is equal to the early end of the previous activity. If a given activity is immediately preceded by several activities, then its earliest start will be equal to the maximum of the earliest finishes of the preceding activities:

Thus, the early dates of all network activities are determined and entered in the upper right and left parts.

The early completion of the completion activity determines the length of the critical path.

Late deadlines are calculated in reverse order from final to original work. The late completion of the final work is equal to its early completion, i.e. duration of the critical path.

Late start is defined as the difference between late finish and duration:

The late start of subsequent activities becomes the late finish of previous activities. If a given activity is immediately followed by several activities, then its late completion will be equal to the minimum of the late starts for the following activities:

In a similar way, the late dates of all network activities are determined and recorded in the lower left and right parts.

The full reserve of time, equal to the difference between the late and early dates, is entered in the numerator of the middle of the lower part:

Free slack, equal to the difference between the minimum early start of subsequent activities and the early completion of this activity, is recorded in the denominator of the middle of the lower part:

The free reserve is always less than or equal to the full reserve of work.