Network and cpm / pert project management. Operative administration

INTRODUCTION TO ACTIVITY NETWORKS (INDUSTRIAL ENGINEERING)

One-time large-scale projects have been around since ancient times; This fact is attested by the construction of the pyramids of Egypt and the aqueducts of Rome. But only recently have the operational problems associated with such projects been analyzed by operational researchers.

The problem of project management arose with the Polaris armaments project, beginning in 1958. With so many components and subcomponents together produced by various manufacturers, a new tool was needed to program and control the project. The PERT (program evaluation and review technique) was developed by scientists from the Naval Office of Special Projects. Boaz, Allen and Hamilton and the Armaments Systems Division of the Lockheed Aircraft Corporation. The technique proved so useful that it has gained wide acceptance in both government and the private sector.

network-operations-research-and-management-of-projects-cpm-and-pert

Around the same time, the DuPont Company, in conjunction with the Remington Rand's UNIVAC Division, developed the Critical Path Method (CPM) to control the maintenance of DuPont chemical plant projects. The CPM is identical to the PERT in concept and methodology. The main difference between them is simply the method by which time estimates are made for project activities. With CPM, activity times are deterministic. With PERT, activity times are probabilistic or stochastic.

PERT / CPM was designed to provide various useful pieces of information for project managers. First, PERT / CPM exposes the "critical path" of a project. These are the activities that limit the duration of the project. In other words, to get the project done soon, the critical path activities must be done soon. On the other hand, if an activity on the critical path is delayed, the project as a whole is delayed by the same amount. Activities that are not on the critical path have a certain amount of slack; that is, they can be started later, and the project as a whole can be allowed to stay on schedule. PERT / CPM identifies these activities and the amount of time available for delays.

The PERT / CPM also considers the resources necessary to complete the activities. In many projects, labor and equipment limitations make scheduling difficult. The PERT / CPM identifies the moments in the project when these restrictions will cause problems and, according to the flexibility allowed by the slack times of non-critical activities, it allows the manager to manipulate certain activities to alleviate these problems.

Finally, PERT / CPM provides a tool to control and monitor the progress of the project. Each activity has its own role in it and its importance in the completion of the project is immediately manifested for the director of the project. The activities of the critical path, therefore, allow receiving most of the attention, since the completion of the project depends heavily on them. Non-critical activities will be manipulated and replaced in response to the availability of resources.

BACKGROUND

There are two origins of the critical path method: the PERT (Program Evaluation and Review Technique) method developed by the United States Navy in 1957 to control the execution times of the various activities involved in space projects, for the need to finish each one of them within the available time intervals. It was originally used by the Polaris project time control and is currently used throughout the space program.

The CPM (Critical Path Method) method, the second origin of the current method, was also developed in 1957 in the United States of America, by an operations research center for the firm Dupont and Remington Rand, seeking control and optimization of operating costs through proper planning of project component activities.

Both methods provided the administrative elements necessary to form the current critical path method, using control of execution times and operating costs, to seek that the total project be executed in the shortest time and at the lowest possible cost.

Definition: The critical path method is an administrative process of planning, programming, execution and control of each and every one of the component activities of a project that must be carried out within a critical time and at the optimal cost.

Uses: The scope of this method is very wide, given its great flexibility and adaptability to any large or small project. To obtain the best results, it should be applied to projects that have the following characteristics:

1. That the project is unique, not repetitive, in some parts or in its entirety.

   b. That the entire project or part of it must be executed, in a minimum time, without variations, that is, in critical time. That the lowest possible cost of operation is desired within an available time. Within the scope of application, the method It has been used for planning and control of various activities, such as dam construction, road opening, paving, house and building construction, ship repair, market research, settlement movements, regional economic studies, audits, planning of university careers, distribution of operating room times, factory extensions, planning itineraries for collections, sales plans, population censuses, etc., etc.

DIFFERENCES BETWEEN PERT AND CPM

As indicated before, the main difference between PERT and CPM is the way time estimates are made. E1 PERT assumes that the time to carry out each of the activities is a random variable described by a probability distribution. The CPM, on the other hand, infers that the times of the activities are known in a deterministic way and can be varied by changing the level of resources used.

The time distribution assumed by PERT for an activity is a beta distribution. The distribution for any activity is defined by three estimates:

(1) the most probable time estimate, m;

(2) the most optimistic time estimate, a; and

(3) the most pessimistic time estimate, b.

The shape of the distribution is shown in the following Figure. The most likely time is the time required to complete the activity under normal conditions. Optimistic and pessimistic times provide a measure of the uncertainty inherent in the activity, including equipment failure, labor availability, material delay, and other factors.

With the defined distribution, the mean (expected) and standard deviation, respectively, of activity time for activity Z can be calculated using the approximation formulas.

The expected time of completion of a project is the sum of all expected times of activities on the critical path. Similarly, assuming that the activity time distributions are independent (realistically, a strongly questionable assumption), the project variance is the sum of the activity variances on the critical path. These properties will be demonstrated later.

In CPM only a time estimate is required. All calculations are made with the assumption that activity times are known. As the project progresses, these estimates are used to control and monitor progress. If any delay occurs in the project, efforts are made to get the project back on schedule by changing the resource allocation.

CPM METHODOLOGY (CRITICAL PATH METHOD)

The Critical Path Method consists of two cycles:

1. Planning and Programming.

1.1.- Project definition 1.2.-

Activities List

1.3.- Sequence

Matrix 1.4.- Time Matrix 1.5.-

Activity Network

1.6.- Costs and pending

1.7.- Network compression

1.8.- Time limitations, of resources and economic

1.9.- Elasticity matrix

1.10.- Probability of delay

2. Execution and Control.

2.1.- Project approval

2.2.- Work orders

2.3.- Control

charts 2.4.- Progress reports and analysis

2.5.- Decision making and adjustments

Activity Network: The graphic representation of the activities that show their events, sequences, interrelationships and the critical path is called a network. Not only is the critical path called the method but also the series of activities counted from the start of the project to its completion, which do not have flexibility in their execution time, so any delay suffered by any of the activities in the series would cause a delay in the entire project.

From another point of view, critical path is the series of activities that indicates the total duration of the project. Each of the activities is represented by an arrow that starts at one event and ends at another.

An event is called the time of initiation or termination of an activity. It is determined in a variable time between the earliest and the latest possible, of initiation or termination.

Events are also known as node names.

Event event

I j

The initial event is called i and the final event is called j. The final event of an activity will be the initial event of the next activity.

The arrows are not vectors, scalars, nor do they represent any measurement. The shape of the arrows does not interest, since they will be drawn according to the needs and comfort of the network presentation. They can be horizontal, vertical, upward, downward curved, straight, broken, etc.

In cases where there is a need to indicate that an activity has an interrelation or continuation with another, a dotted line, called a league, will be drawn between them that has a duration of zero.

The league can sometimes represent a waiting time to start the next activity.

Several activities can end in an event or start from the same event.

(a) Incorrect, (b) Correct .

When building the network, the following should be avoided:

1. Two activities that start from the same event and arrive at the same event. This produces confusion of time and continuity. The opening event or the ending event must be opened in two events and linked with a league.

2. Starting an activity from an intermediate part of another activity. All activity must invariably start in one event and end in another. When this case arises, the base or initial activity is divided into events based on percentages and the secondary activities are derived from them.

(a) Incorrect, (b) Correct .

1. Leave events loose at the end of the network. All of them must be related to the initial event or the final event.

PROCEDURE FOR MAPPING THE MEASURED NETWORK

To draw the measured network, graph paper is used, indicating at the top the scale with the chosen time units, in a reasonable interval for the execution of the entire project. As its duration is not known at this time, since one of the objectives of the network is to know it, this interval is only approximate.

The network is then started by drawing the activities that start from event zero. Each of them must be drawn in such a way that event j ends, according to the standard duration, in the time indicated on the upper scale. Now we will show the initiation of activities 1, 2, 3, and 4 with duration of three, two, three, and five days respectively.

In the case of the factory expansion, the initial activities are those shown in the figure below, since the three activities that start from scratch are three days long each.

Next, the progressive numbering of the sequence matrix should not be taken to draw the network, but the terminals of the activities, from top to bottom and from left to right, as events j appear.

In the previous case we look for the sequences of activity 1, after 12 and the last one of 18. In order, we look for the sequences of 2, 13 and 19. If an activity has zero duration, draw vertically, either ascending or descending, so that it does not take up time within the network.

Rigorously, an activity cannot have time of zero duration, since it would not exist; however, some activities are so short in duration that it is negligible and should not be considered a unit of time. For example, if the unit you work with for one day and the duration of the activity is five or ten minutes, there is no reason for this activity to be assigned a work day. In the case that it develops, the approval of the budgets is supposed to take from half an hour to an hour for their execution; but since the unit taken in the project is one day, the execution time is considered zero.

According to the annotations in the sequence matrix, activities 3, 14 and 20 must be simultaneous, so we need a common event to finish all three. Due to the need for construction, Activity 14 will only be indicated with the number in parallel to Activity 3, which also has a duration of zero. It can also appear parallel to Activity 20.

In this type of network there is no need to indicate the activities with arrows, but only with lines, except for the links that will indicate the direction of continuity. To continue with the drawing of the network, it should be remembered that activities 3, 14 and 20 converge to the common event and therefore we must look for the sequences of these three activities, which will logically start from the same event. We continue to lengthen terminals 15,4,21 and 9, in this order precisely, according to the method adopted.

Thus we find that after activity 15, the 16th continues with a duration of six days; after activity 4, follow 5 with a duration of six days; after activity 21, follow 23 with a duration of three days and also 5 with a duration of six days; and after activity 9, 10 follows for two days.

When an activity is a sequence of two or more previous activities, it must be placed in the network after the most advanced antecedent activity. That is why it is convenient to make the network with a pencil to be able to erase the activities and easily change them. In this way, the diagram in the previous figure must be modified, since activity 5 is subsequent to activity 4 and activity 21; We remove it from the place that ends on an earlier date and place it after 21 that appears on an earlier date. However, so that the sequence from 4 to 5 is not lost, a link is placed between the two. We looked for the continuation of the terminals of activities 16, 5, 23 and 10, finding that they are respectively 17 with two days; 6 with four days; 22 with four days and 11 with twelve days.

The activities corresponding to 17, 6, 22 and 11 are respectively 6 with four days; 7 with six days and none for 11, so in the network we only put a link between the termination of 17 and the initiation of 6 to indicate continuity and another one between the termination of 22 and the initiation of 7 with the same object of continuity. Now we put the sequence of 6 only, since we have already seen that 11 is the end of the process. The sequence of activity 6 is 7 with six days and the sequence of activity 7 is 8 with a duration of zero. As there is no other activity subsequent to the network terminals, it should be considered that the project has been completed, so its duration is 26 days.

Since no loose events should be left, a link is placed between the terminal on 11 and the final event of the project, leaving the entire network as follows and showing the following features:

1. a) Activities with zero duration are indicated vertically, either ascending or descending, such as those corresponding to activities 3, 20 and 8.

   b) Activity 14 with zero duration is not drawn on the network for construction reasons and it is only indicated together with activity 20 that has the same characteristics.

   c) The activities that are consequent to two or more previous activities appear drawn after the antecedent that has the highest date in its final event. Like activity 5, which is a consequence of activities 4 and 21. Activity 4 ends on day 6 and activity 21 ends on day 10. Activity 7 is a sequence of activities 6 and 22 and is placed in front of the one with the oldest date. discharge at the end, that is activity 6. This same activity 6 is subsequent to activities 17 and 5 and is placed after 5 for the reason already given.

   d) The links that appear in the graph mean the following: Activity 5 is a continuation of 4; 6 is a continuation of 17; 7 continues from 22 and 11 will end at the end of the project.

   e) The critical path is the series of activities that begin in event i of the project and end in event j of the project, without being interrupted by what the size or duration of the project indicate, and is represented by activities 12, 13, 21, 5, 6, 7 and 8 drawn with a double line.The above network can be drawn with colors to indicate different responsibilities: for example, the responsibility of the electrical engineer is drawn in red, that of the civil engineer with green and the of plant engineer with blue.

COSTS AND EARRINGS

In this step the costs of each activity carried out in standard time and in optimal time will be requested. Both costs must be provided by the persons responsible for the execution, in accordance with the budgets already provided by them. These costs must be noted in the information matrix.

Activities	Normal	Limit
A. From the Plant Engineer
1. Project	600.00	800.00
2. Cost	100.00	100.00
3. Approval	-	-
4. Unpacking	200.00	200.00
5. Placement	600.00	800.00
6. Installation	1,400.00	2,800.00
7. Evidence	6,100.00	6,300.00
8. Start	-	-
9. Review	2,100.00	2,800.00
10. Machine Painting	960.00	960.00
11. Building Paint	3,160.00	3,520.00
	15,220.00	18,280.00
B. Of the Electrical Engineer
12. Project	6,000.00	6,500.00
13. Cost	100.00	100.00
14. Approval	-	-
15. Transformer	18,600.00	19,000.00
16. Lighting	8,900.00	9,300.00
17. Switches	4,100.00	4,400.00
	37,700.00	39,300.00
C. Of the Contractor Engineer
18. Project	4,000.00	4,600.00
19. Cost	100.00	100.00
20. Approval	-	-
21. Foundation	3,400.00	3,800.00
22. Floors	2,800.00	3,200.00
23. Windows	1,900.00	2,200.00
	12,200.00	13,900.00
Total of the Three Budgets	65,120.00	71,480.00
Buy New Machinery	80,000.00	80,000.00
Totals ………………………….	145,120.00	151,480.00

In the previous table we see the budgets with the normal cost for activities carried out in standard time and the limit cost for activities carried out in optimal time. The totals in the normal cost column indicate the direct costs of the project executed in standard times, however, the limit cost totals do not indicate a real cost, since it will not be necessary for all activities to be carried out in optimal time, but rather just some of them.

1. a) Activities with zero duration are indicated vertically, either ascending or descending, such as those corresponding to activities 3, 20 and 8.

   b) Activity 14 with zero duration is not drawn on the network for construction reasons and it is only indicated together with activity 20 that has the same characteristics.

   c) Activities that are sequential to two or more previous activities are drawn after the antecedent that has the highest date in your final event. Like activity 5 that is sequential to activities 4 and 21. Activity 4 ends on day 6 and 21 ends on day 10. Activity 7 is a sequence of activities 6 and 22 and is placed in front of the one with the oldest date. discharge at the end, that is activity 6. This same activity 6 is subsequent to activities 17 and 5 and is placed after 5 for the reason already given.

   d) The links that appear in the graph mean the following: Activity 5 is a continuation of 4; 6 is a continuation of 17; 7 continues from 22 and 11 will end at the end of the project.

   e) The critical path is the series of activities that begin in event i of the project and end in event j of the project, without being interrupted by what the size or duration of the project indicate, and is represented by activities 12, 13, 21, 5, 6, 7 and 8 drawn with a double line.The above network can be drawn with colors to indicate different responsibilities: for example, the responsibility of the electrical engineer is drawn in red, that of the civil engineer with green and the of plant engineer with blue.

Costs and Earrings

NETWORK COMPRESSION

Compressing a network will help us determine what activities will be optimized in time.

ELASTICITY MATRIX

In order to make quick and effective decisions during the execution of the project, it is necessary to have at hand the data on the probabilities of delay or advancement of work for each of the activities, that is, the elasticity of the same.

Let us first examine the procedure for calculating the gaps provided by the possibility of delaying an activity without consequences for other jobs.

Slack is the freedom that an activity has to lengthen its execution time without harming other activities or the total project. Three classes of clearances are distinguished:

a) Total clearance; does not affect the completion of the project;

b) Free slack; does not modify the termination of the process; and

c) Independent clearance; It does not affect the termination of previous activities or the initiation of subsequent activities.

Total slack is of importance to the project manager, who has the responsibility to finish it on time; free play is in the interest of the head of a process because of his responsibility for it; and the independent slack is information that is useful to the person who will coordinate the work of the project.

To calculate the clearances, the approved network is measured in the forward direction, as the first reading and then in the opposite direction as the last reading. The first reading will be indicated in each event within a circle and the last reading will also be indicated in each event within a square. It begins with the zero time indicated on the initial event and the standard duration of each activity is added, accumulating in each event.

When two or more activities converge in an event, the longest duration will be taken to indicate the event. For example, in activities 4 and 2 with a duration of two and six days respectively, the duration greater than six will be noted, which added to the previous four will give a time of ten in the referred event. Note these same indications in the events found on days 15, 19 and 21.

When you have a league that indicates completion of the process, the same amount accumulated in the final event will be run towards the initial event. When the league does not indicate completion of the process, but only continuity between two processes, the accumulated amounts should not be modified even if the league has different starting and ending dates.

Then the last reading begins in the final event, scoring the same amount of 21 within a square; then the duration of each activity is subtracted and the difference is indicated in the following event. When two or more activities converge in an event, the lowest reading of them should be noted in it. In the initial events of the end-of-process leagues, the same amount noted in the final event must appear, but in the continuity leagues, the smallest number of converging activities will be placed.

In the figure it can be seen that in each activity of the network there are four readings; the first and last of event i and the first and last of event j. Where: Pi means the earliest the activity can start. Ui means the latest that can be started. Pj means the earliest it can be finished. Uj means the latest that can be finished. The difference between the earliest start date and the latest end date produces the longest available time interval and is based on the project count.

Subtracting the duration t from this interval produces the total clearance:

The difference between the earliest start date and the earliest end date indicates the available interval depending on the process, And by subtracting the duration t from this interval, the free clearance remains:

The difference between the latest start date and the earliest end date indicates the shortest possible time interval and this depends on the previous and subsequent activities, and by subtracting the time t from this interval the independent clearance is obtained:

The readings of the events and the results of the application of the slack formulas are passed to the information matrix.

In column 6, the standard time t was changed to the scheduled execution time e. The expansion percentage (column 15) is calculated by dividing the number of days of total slack by the standard time of each activity.

The activity class (column 16) is graduated taking the previous percentage from lowest to highest, being those of zero percentage of critical class those that require the most attention and control. The days that activities can be compressed (column 19) are obtained by subtracting the optimal time from the standard time. The compression percentage (column 20) is equal to the compressed days divided by the standard time of each activity.

The standard deviation (column 21) that represents the probability of delay or advance on average, is equal to the lousy time minus the optimal time divided by 6.

By definition it represents 68% security. If greater security in the result is desired, 95% will take the equivalent of two standard deviations and if 99% security is desired in the duration of the activity, three standard deviations will be taken. In this way, we can see that Activity 5 has a standard time of six days and a standard deviation of one day. This means that it can run between five and seven days with 68% security; between four and eight days with 95% security; and between three and nine days with 99% security. The greater the interval mentioned for execution, the greater the security of hitting. The standard deviation of the project is equal to the sum of the standard deviations of the critical path:

This deviation will be the probability of delay of the entire project. Of course it is the same probability of advancement of the same. If there are several critical paths within the project, the major deviation of them will be taken as the project standard deviation. In the previous case, the critical path is given by:

This means that the project is going to run between

that is, between 21 and 25 days, with 68% security. There is no probability of advancement in this project since its execution time is already compressed. The standard deviation can be pointed out as tolerance in the development of the project.

DELAY CHANCES

To determine the probability that an activity or the entire project will be delayed, the corresponding amount of standard deviation for the desired days of delay is calculated and the following table is prepared:

PERT GRAPHICS

The PERT Technique (Program Evaluation and Review Technique) is an instrument

specially designed for management, allowing you to plan, program and control the resources available to you, in order to obtain the desired results.

It is a technique that provides management with information on the actual and potential problems that may arise in the completion of a project, the current condition of a project in relation to the achievement of its objectives, the expected date of completion of the project and the possibilities of achieving it, and where are the most critical and least critical activities in the total project.

PERT does not attempt to usurp the functions of management, but to help it carry out its

activities more successfully. Nor, naturally, does it lead by itself, but it can be said that it depends on the skill with which the management uses these techniques, the one that discovers and solves the problems that arise most effectively.

At the time of making a decision, all the complexity and

consequences that it can have are rarely known. However, the PERT Technique outlines an effective method to reduce risks by making those decisions that are most likely to succeed.

We all know that there are different levels of management: managing director,

department head, division head, center head, etc. But at all levels,

three activities are carried out fundamentally and they need tools such as the PERT method, in order to effectively carry out their activities.

1. Set the objectives Seek and organize the necessary means to achieve the objectives previously Check the existing agreement between the set plan and what is being carried out,

   in order to be able to act on the resources and face the conditions

PERT NETWORK: A PERT network is the graphic and symbolic representation of the tasks to be

carried out to carry out a proposed goal.

The PERT graph is an original graph of unmeasured networks that contains the data of the activities represented by arrows that start from an event i and end in an event j.

At the top of the arrow is the identification number, usually the event numbers (ij). The standard duration (t) of the activity appears in a rectangle at the bottom. The progressive number is noted in the upper half of the event, the last reading of the project in the lower left room and the first reading of the project in the lower right room.

This graph has the advantage of reporting the earliest and latest start and end dates of each activity, without having to resort to the gap matrix.

Let's see how the factory expansion is presented using a PERT chart.

EXECUTION AND CONTROL OF THE PROJECT

Project approval

When the people involved in the execution of the project are fully satisfied with the times, sequences, costs and distribution of human and material resources, it must be approved. At this time, the work program should be completed with the following:

a) The list of activities

b) The general budget

c) The activity specifications

d) The identification of positions and responsibilities and organization of command

e) The network of activities

f) The limiting conditions of work

g) The work procedures

h) The necessary equipment

i) The plans and itinerary and schedule scheme

j) The information matrices

Work orders

Work orders are prepared based on activity specifications, limiting conditions, work procedures, necessary equipment, and process, itinerary, and schedule schemes, as well as information matrix help.

In them the precise indications must be given so that the activity is carried out by the person or group of persons responsible, according to the general plans, in time, in the quantity and of the desired quality.

CONTROL GRAPHICS

In the control of the project it is necessary to determine with precision both the progress of each of the activities and that corresponding to the total project. An effective form of control is the use of graphs that allow to visually monitor the development of activities, and for this purpose two types of graphs will be used:

1. a) The progress graph

   b) The performance graph The progress graph contains, in addition to the network, a strip at the bottom that shows the percentage of progress achieved in each time unit. The ordinates found in the divisions of time mark the schedule for each activity, for each process and for the entire project.To calculate the scheduled percentage of progress, we proceed as follows: a) Divide the total percentage of progress (1.00) by the number of activity days that the draft. This number is the sum of column "e" of the information matrix (66).

Naturally, if the unit of time does not represent days but hours, the unit of advance will be Ha (activity-hours).

b) The advance units (Da) that appear in the network on each scheduled day are counted. In each of the first four days we found 3 activities; in the fifth and sixth there are 4 activities; from the seventh to the tenth we find 3 activities, etc.

c) The advance units are accumulated in each elapsed day.

d) The accumulated advance units are multiplied by the advance factor calculated in part a.

In this way and for our base example, we have the following results:

The amounts appearing in column 4 of this table are noted at the bottom of the progress grid. It is sufficient to indicate two decimal places.

If greater precision is desired in the drawing and the size of the graph allows it, divisions can be made in the daily sections to show the progress of one by one percent.

Note that the scales are different in the sections that contain unequal amounts of (Da).

With the above, the progress graph is ready to receive the information.

Now let's prepare the performance graph that will help us to observe the pace or speed of work at the same time as the partial goals that are achieved over time.

In the ordinate we present a scale with percentages and in the abscissa the days of the project duration plus the calculated tolerance.

This graph indicates the final goal that is on the line of 100% efficiency and the coordinate of the final time of the project.

Now we can calculate the progress achieved daily in the project and present it in the previous graphs. The progress of the project is the sum of the progress achieved by each of the component activities. The following table shows the daily reports of actual progress in each activity.

This information is processed in the project progress chart shown below:

The columns of this table are filled as follows:

A. At the time of receiving the actual progress information:

1. The day of the information is recorded.

2. The numbers of the reported activities are expressed. A T will be noted first to indicate the activities completed previously

7. The percentages, as much as one, of the work carried out up to the day of the information, for each one of the activities scheduled on the indicated day are noted.

10. The accumulated total of the activities completed previously is noted.

B. After making the previous entry, the following columns are calculated:

3. Indicate the scheduled days of execution for each activity reported in accordance with column e of the information matrix. In the base example, the matrix is ​​found in the table above.

4. The reciprocals of the previous times are determined to indicate the volume of work or load that corresponds to each day. For example, if an activity must be done in 3 days, each day corresponds to 1/3 of work, that is, in decimal 0.33. The reciprocal is obtained by dividing the unit by the number of days programmed and expressing this result in decimals.

5. The days elapsed in each activity are indicated in accordance with the program, and not with the days elapsed in advance. Verify that these quantities are not greater than those indicated in column 3 of the table, since it is not possible to schedule more than 100% work of an activity.

6. The values ​​in columns 4 and 5 are multiplied to obtain the percentage of work that must be completed according to the program, for each activity, as of the day of the information. This corresponds to the daily work load for the days spent in the reported activity.

8. The total progress factor by activity (fa) is calculated by multiplying the advance unit factor (Da) by the number of days programmed in column 3 of this table. In our example, we must remember that Da = 1.00 / 66 = 0.0151. This column indicates the progress of the project with the work carried out in its entirety of the indicated activity.

9. The previous percentage of progress in the project is adjusted with the actual percentage of the activity. For this, the percentage of activity in column 7 is multiplied by the percentage in column 8.

11. Since the progress of the project is the sum of the partial progress achieved by the activities, the amounts appearing in column 9 corresponding to the activities in operation are added together with the total accumulated in column 10 for the activities already completed. This sum represents the actual progress of the project up to date with the information.

12. Now check the programmed progress scale in the progress graph to know the percentage corresponding to the day of the information. Once found, it will be indicated in this column. This data can also be found in column 4 of the table.

13. The percentage of performance, productivity, speed or efficiency of the project is equal to the amount of progress achieved. Divided by the percentage of programmed progress. In this column, the result of dividing the quantities that appear in column 11 by the quantities in column 12 is recorded.

The following notes are made in the progress graph:

a) The scheduled day, according to column 1. Fill or color the rectangle corresponding to this day.

b) The progress of the three activities in operation, as indicated in column 7. For activity 1 the scheduled work is 0.33 according to column 6, so the coordinate marks this amount. As the work accomplished is the same as scheduled, the advance reaches the same coordinate. If this had not been the case, the annotation would have been made up to the proportional part.

c) The progress of the project according to column 11. The lower stripe must be filled in with color to make this annotation.

d) Unite the programmed percentage and the percentage achieved in the deviation area. If there is no angle, it means that it works according to what is programmed; otherwise it may indicate delay or advance. The angle measurement is not related to the percentage of delay or advance because the advance scale is irregular. It is only a visual call for attention to non-compliance with the program.

Note that the coordinate corresponding to the scheduled days has different values ​​for the activities and for the project. Furthermore, it can present different values ​​for each activity. The values ​​it takes for each activity must be consulted in the project progress chart and the project values ​​must be observed in column 12 of said chart.

Next we will proceed to make the annotation on the performance graph:

a) enter the elapsed day in the lower band, according to column 1 of the project progress chart.

b) Record the efficiency percentage according to column 13.

If there is a deficiency, an area will appear that should be colored below the 100% level.

c) Indicate the percentage of progress, according to the amount that appears in column 11 of the table. The advance area should be colored.

The progress of the project suffered a delay of 0.2426 - 0.2155 = 0.0271 (2.71%), reducing its efficiency or performance to 89% of the program, due to the fact that some of the activities were delayed. Activity 4 did not start because the machinery did not reach the warehouse. Activity 9 corresponds to the critical process. It has the maximum advance control and was carried out according to the program. Activity 15 is late; it had to move forward; it had to advance 30% and only reached 10%. Activity 21 was also delayed, although very little, perhaps only a mistake of appreciation by the supervisor. In any case, the delay is recorded.

The project suffered a greater delay as a consequence of not having started activity 4. Now the delay is 0.3032 - 0.2488 = 0.544 (5.44%) with an efficiency of 83%. Activity 9 is carried out according to the program. Activity 15 with a long delay and activity 21 with a small delay.

The delay of the project was reduced, thanks to the initiation of activity 4. Now we have a delay of 0.3487 - 0.3246 = 0.0241 (2.41%) with 93% efficiency. Critical Activity 9 continues on schedule. Activities 15 and 21 accelerated the pace of work. The 21 managed to reach the scheduled quota.

Again, although small, a reduction in project delay was achieved. Activities 4 and 15 were completed. Activities 9 and 21 were carried out on time. Activity 16 cannot be indicated due to the delay of 15.

The project is almost in time, as its efficiency reaches 99%. Activities 9 and 21 are finished and 16 is late. Activity 21, on the other hand, ended, but anticipating the program.

This allows activities 5 and 23 to start, which are sequential to activities 4 and 21, already completed.

The project has a small delay: 0.4852 - 0.4731 = 0.0121 (1.21%) with 97% efficiency. Activity 5 started one day in advance. On the other hand, 23 could not be started in advance, so the initiation will be normal. Activity 10, which is critical, was performed normally. Activity 16 continues with a strong delay due to lack of materials.

The work rate of the project was maintained at 97% efficiency. Activity 10 was completed on time. Activity 5 normally runs one day in advance of the program. Activity 16 is still late. Activity 23 on time.

The project presents the same small delay. Activities 5 and 11 are carried out on time. Activities 16 and 23 late.

He slightly sped up the project. The same situation in general, as in the previous day.

The project continues with the same small delay. Process A was completed in its entirety.

Activity 22 is the only one delayed.

The same comment as the previous day.

Process B was completely finished. The project on time.

The project on time.

The project and activities on time.

Processes C and D were completed. The project was completed on schedule.

Now let's see how the progress and performance graphs of the project were:

IMPLEMENTATION AND CONTROL OF PROCESSES

Since each of the project's component processes is conducted by different people who have the responsibility to start and finish their activities on time, they need to have their control chart where they can observe both the progress of their process and their performance.

This graph is similar to the performance graph used in the project.

A diagram of the sequences of the activities showing where the total gaps are can be added at the top, so that the person in charge of the process has a precise idea of ​​their time availability.

We also need a progress chart of the process with the following data and it is filled as follows:

A. With the original information from the supervisor:

1. Record the day of the information

2. Indicate the number of the activity reported

3. Express, as one, the progress of the same.

B. The above data is then processed in the following columns:

4. Take the percentage from column 9 of the project progress chart and record it in this column.

5. Make the conversion with the factor (fa) previously calculated.

6. Record the accumulated total of the activities completed.

7. Sum of columns 5 and 6 representing respectively the progress of the activity in operation and the accumulated total of activities completed in the process. This column indicates, therefore, the total progress in the process on the day of the information.

8. Calculate the scheduled daily progress, dividing the unit by the total number of days of the component activities of the process and accumulating said result.

9. Divide the progress achieved by the scheduled progress to measure the performance of the process. Column 7 between column 8.

Let's see, in the base example, how the activities of process A are carried out.

Process A

This process will consist of five activities that last 15 days. If we remember that the value of the project progress unit (Da) is equal to = 0.01515, then this process represents 15 x 0.01515 = 0.2272 (22.72%) of progress in the project. Since this amount 0.2272 represents 100% progress of the process, then the conversion factor of the progress percentage of the project to process (fa) will be:

0.2272: 1.00:: n: fa

In this way, the percentage that appears in column 9 of the project progress chart and transferred to column 4 of the process progress chart, can become, with this factor, the progress achieved in the activity based on this process..

This process A consists of five activities with a duration of 15 days. Your programmed advance unit will therefore be

Since only one unit of progress is worked per day, this will be the daily accumulated progress programmed in column 8 of the process progress chart.

Process B

This process consists of five activities with a total duration of 17 days, so its contribution to the progress of the project is 17 x 0.01515 = 0.2576.

The conversion factor (fa) from the project progress percentage to the process progress percentage is:

Fa = = 3.88

The unit of daily progress of this process will be:

Da = = 0.05882, What accumulated will be used to make the annotations in column 8 of the process progress chart.

Process C

Process C is made up of six activities with a total duration of 17 days and, therefore, the conversion factor (fa) and the programmed daily advance factor (Da) are the same as those of process B above.

Fa = = 3.88

Da = = 0.05882, The account of the programmed advance was interrupted on day 6 with 0.3533 until day 11, when activity 5 continues.

Process D

This process D, with activities 9, 10 and 11 has, like the two previous processes, a duration of 17 days, so the conversion and progress factors are the same.

Fa = = 3.88

Da = = 0.05882, The process progress chart appears in the process progress chart D table.

EVALUATION PROCEDURE

When the activities are advanced in their execution to the scheduled dates, they generally do not modify their direct costs and instead they do decrease the indirect costs. In general terms we can say that they benefit the results of the budgets when finishing the activities before the scheduled date. The decision to advance the activity following that completed in advance is also simple and the possibility of doing so should only be investigated in terms of having the required human and material resources at that time.

In the case of delays, the evaluation and the decision are not so simple because, as a general rule, costs are modified, the sequences are upset and the availability of time is lost, so there is a need to have an evaluation procedure that allows determining all the consequences of a delay in a project activity.

The delays must be absorbed by the slacks and in the event that they do not exist, they must be neutralized by means of compressions in the activities.

CLEARANCE ABSORPTION

Multiply the scheduled execution time e therefore by one of the amount of work that remains to be done. The result is the time it takes to finish the activity normally. The time available is subtracted from the previous time and the difference represents the delay, which must be absorbed by the total clearance. If this is not possible, you should proceed as follows:

COMPRESSION ABSORPTION

The optimal time is multiplied or therefore by one of the volume of work pending execution. The product represents the time required to complete the activity in optimal conditions, that is, with maximum acceleration. If this time is less than the available time, it means that the project will not be delayed, but if it is greater, the difference will be the amount of time that the project will delay, except that an activity can be compressed after the delayed activity within the process.

EVALUATION TABLE

All activities that are delayed or that have changed in any way the scheduled start or finish times should be analyzed using an evaluation chart such as the following:

The columns of this table will be filled with the following data:

1. Record the day of the information.

2. Indicate the numbers of the activities that suffer variations in the program.

3. Percentage of work advanced by the activity on the day reported, expressed as one.

4. Percentage of work pending, equal to the quantity minus the quantity noted in column e.

5. Execution time and programmed by the activity, according to the approved network.

6. Real time elapsed from the date scheduled for its initiation.

7. The normal time needed to finish the activity is equal to the product of multiplying the execution time (5) by both the missing work time (4).

8. The time available to execute the activity is the difference between the programmed time (5) and the elapsed time (6).

9. The missing time is equal to the necessary time (7) minus the available time (8).

10. Record the days of total slack calculated for the activity.

11. Determine the number of days of slack that will be needed to fill the time gap in column 9. Full days will always be used to fill in the missing time fractions. It is convenient to make the modification in the information matrix. The amount of time used to absorb the delay will be increased to the available time 8 in the following days until the end of the activity.

12. Available clearance is the difference between the original quantity (19) and the used clearance (11).

13. Write down the optimal time or the activity in progress.

14. The optimal time needed is equal to the product of multiplying the goal by one of the missing work (4) by the optimal time (13).

15. If, when compressing the activity, the time required (14) to complete the activity is less than the available time (8), a zero will be entered in this column; otherwise the difference representing the missing time to finish the activity will be recorded even after compression.

16. Record the slope of the activity, taken from the information matrix.

17. The compressed time is equal to the programmed time (5) minus the optimal time (13).

18. Enter the same amount that appears in column 4.

19. The cost of compression of the activity is equal to the product of multiplying the slope (16) by the compressed time (17) and by the volume of work that remains to be done (18). This cost will be increased to the normal cost to obtain the total cost of the activity.

20. If there is a lack of time (15) after compressing the delayed activity, a later activity should be used in the same process. In this case, the number of the activity affected should be noted in this column.

21. Record the slope of the affected activity taken from the information matrix.

22. Record the scheduled time and activity affected according to the information matrix.

23. Determine the amount of understanding of the affected activity needed to absorb the time gap in column 15. The maximum compression of the affected activity should be obtained from the information matrix. In the event that this compressed time is not enough, another or other activities of the same process must be compressed and if they are not available, this shortage represents the amount of time that will delay the completion of the entire project.

24. The cost of compression of the affected activity is equal to the product of multiplying the slope (21) by the compressed time (23).

25. The total cost resulting from compressions is equal to the sum of columns 19 and 24.

26. Record in this column the modifications that must be made to the program. The following symbols are suggested:

a) HT-2 (14)

Occupy two days of total slack to finish activity 14.

b) HT-1 (18) (23) (25)

Occupy one day of total slack to finish activity 18 and subtract one day from total clearance, in the information matrix, to activities 23 and 25.

c) Co-1 (5)

Compress activity 5 one day. In any case, the activity will be carried out in optimal conditions to accelerate the missing work. The indicated time only serves for programming, but it is difficult to indicate the degree of acceleration, so it is preferable to apply the maximum.

d) Co-2 (7) (15)

Compress the missing work from Activity 7 into two days and the same amount for Activity 15. In this case Activity 15 will start two days after the scheduled date to end on the scheduled day.

e) Co-1 (10) 3 (12)

Compress activity 10 one day and activity 12 three days.

CONCLUSIONS

PERT and CPM have been applied to numerous projects. Starting with their initial application to the Polaris project and to the maintenance of chemical plants, today they (and their variants) are applied to the construction of roads and buildings, and to the development and production of high-tech items such as airplanes, space vehicles, boats and computers.

The PERT was developed for projects where there was uncertainty in the timing of the activities (usually because the project had never been attempted before and therefore there were no databases for the timing of the activities). This led to the probabilistic approach that was taken. While in PERT time estimates and their distributions have been controversial, PER'I 'has been a useful tool for project management. The main disadvantage is that it is not functional for large projects, due to the three time estimates that are required in each activity and the limited capacity of current computers to store this vast amount of data. Also, the cost of updating and maintaining project information over time in such dynamic environments,it can be excessively prohibitive.

Furthermore, the CPM was developed to handle repetitive or similar projects (eg, maintenance of chemical plants). Obviously, a great deal of experience is gained over time in such circumstances, even though two projects may not be the same. This experience led to the analysis of collision techniques used in CPM networks.

While CPM and PERT are essentially the same, their nuances make each more applicable than the other in different situations. In both methods the desired essential information is the critical path and the gaps. These allow the project manager to make decisions based on information, based on the principle of administration by exception, on the plans and projects of the current work and monitor the progress of the project.

SUMMARY

PERT / CPM was designed to provide various useful pieces of information for project managers. First, PERT / CPM exposes the "critical path" of a project. These are the activities that limit the duration of the project. In other words, to get the project done soon, the critical path activities must be done soon. On the other hand, if an activity on the critical path is delayed, the project as a whole is delayed by the same amount. Activities that are not on the critical path have a certain amount of slack; that is, they can be started later, and the project as a whole can be allowed to stay on schedule. PERT / CPM identifies these activities and the amount of time available for delays.

OPERATIONAL RESEARCH FUNDAMENTALS - PERT AND CPM (PROPOSED PROBLEMS)

1. Before being able to introduce a new product to the market, all the activities shown in the table must be carried out (all the times are in weeks).

Exercise	Description	Predecessors	to	b	m
TO	Product design	-	two	10	6
B	Market research	-	4	6	5
C	Issue material orders	TO	two	4	3
D	Receive materials	C	one	3	two
AND	Build prototype	A, D	one	5	3
F	Development and promotion	B	3	5	4
G	Commissioning plant for mass production	AND	two	6	4
H	Distribute products to warehouses.	G, F	0	4	two

Draw the mesh of the project and determine the critical path. Interpret your results. Make a linear programming model to determine the minimum duration of the project.

What is the probability that the product will be on the market before Easter?

1. There is the following schedule of activities:

Exercise	Predecessor	to	m	b
TO	-	two	6	10
B	-	4	5	6
C	TO	two	3	4
D	C	one	two	3
AND	A, D	one	3	5
F	B	3	4	5
G	AND	two	4	6
H	F, G	0	two	4

Determine the minimum project duration, critical path, and interpret slack time, run a program to determine the minimum project duration. Lastly, suppose today is July 15 and the project begins, determine the probability that the project is ready by December 18.

1. There is the following schedule of activities:

Exercise	Predecessor	Expected time	Accelerated time	Variance	cost	Accelerated cost
TO	-	3	two	0.3	6000	8000
B	-	5	one	0.5	5000	7000
C	TO	4	two	two	16000	25000
D	B	3	two	one	18000	26000
AND	B	one	one	0.2	20000	20000
F	C, D, E	4	two	0.4	16000	18000
G	C, D	two	one	0.1	2000	4000
H	F, G	two	one	one	6000	10000
I	F	3	two	0.6	9000	12000

Determine the minimum project duration, critical path, and interpret slack time, run a program to determine the minimum project duration. Also consider the new accelerated times and the respective costs. Based on this, make a linear programming model that allows determining which activities should be accelerated and how much to complete the project in a maximum time of T weeks, incurring a minimum cost.

1. There is the following schedule of activities:

Exercise	Predecessor	hope	Variance	Budget
TO	-	3	0.3	6000
B	TO	two	0.5	4000
C	-	8	2.0	16000
D	B, C	6	1.0	18000
AND	C	4	0.2	20000
F	OF	5	0.4	15000
G	OF	one	0.1	2000
H	F	5	1.0	5000
I	G	6	0.6	12000

Determine the minimum project duration, critical path, and interpret slack time. Make a linear programming model to determine the minimum duration of the project.

1. There is the following schedule of activities:

Activity code	Name of the activity	Days required	Previous immediate tasks
TO	Disconnect and move	0.2	-
B	Connect to the current and do a test	0.2	TO
C	Remove electrical units	0.2	B
D	Clean the machine	0.3	C
AND	Remove and take apart the mechanical units	0.2	C
F	Clean machine parts	0.4	D
G	Order a list of mechanical parts	0.5	F
H	Sort the machine parts	0.5	G
I	Receive the machine parts	1.0	H
J	Paint the cross cursors	25.0	I
K	Machining the pieces	1.5	G
L	Inspect and order a list of electrical parts	1.0	K
M	Paint the motor	1.0	L
N	Assemble the motor	0.8	P, Q, R
OR	Machining the bank	2.5	H
P	Machining the cursors	2.0	V
Q	Machining the table	2.0	L
R	Paint machine	2.0	M
S	Clean the cursors	1.0	N
T	Clean the table	1.0	G
OR	Clean banks	0.5	AND
V	Machining the jaws	2.0	K
W	Install the shaft	1.0	J, O, T
X	Assemble the pieces	1.0	J, S
AND	Clean the jaws	0.5	OR
Z	Assemble the head	1.0	J, O, T
AA	Install the motor and electrical parts	0.3	AND
AB	Assemble the motors	0.4	J, O, T
AC	Connect to the current and test	0.5	AA, AB, Z, W, X
AD	Retouch, move, reinstall	0.3	AC

Suppose you are on day 29 and the current situation that is recorded is:

Exercise	OR	I	T	N	S	J	OR	AND
% Finished	100	100	100	100	30	80	70	10

• Determine when the project will be completed and what critical activities remain. What is the probability of completion within 10 days? And within 29 days? Formulate a linear programming model that determines how much of the project remains and the critical path.

1. A certain company presents the following schedule of activities in carrying out a project. The detail of the activities and their respective predecessor activities, the pessimistic, optimistic and most probable times in weeks and the normal cost of each activity associated with each normal time are given. In addition, the percentage by which the normal time of each activity can be decreased and the respective cost are given.

Exercise	Predecessor	to	m	b	Normal cost	Percentage that decreases	Accelerated cost
TO	-	8	12	16	800	twenty	960
B	-	6	8	10	600	fifty	900
C	TO	7	10	13	200	30	340
D	B	fifteen	twenty	25	600	10	660
AND	B	one	4	7	500	0	500
F	AND	two	5	8	300	60	480
G	C, D	6	10	14	1000	10	1100
H	C, D	10	12	14	1000	30	1300
I	G	5	6	7	500	fifteen	650
J	H	two	4	6	650	twenty	780
K	I	4	9	14	200	fifty	300
L	I	two	4	6	800	35	1080
M	J, K	two	3	4	600	10	660

From the above data, it is requested:

1. Draw the network associated with the project. Considering the normal duration of each activity, determine the critical path and the minimum duration of the project. If there is more than one critical path, determine which one you would recommend and why. Formulate a linear programming model that minimizes the duration of the project, considering the normal time of each activity. Consider that each activity can be carried out. carried out at any time between its normal duration and its accelerated duration. Formulate a programming model that allows determining the activities that must be accelerated to complete the project in a maximum of 45 weeks, incurring a minimum cost. Suppose you are at week 52 and that activities F,I and J have 50% of their time left to finish (which means that the previous activities have already finished). What is the probability of finishing the project before 4 weeks? The following network represents a project made up of activities whose Features are presented in the following table:

Exercise	Normal	Accelerated
Duration	Variance	cost	Duration	cost
TO	4	0.4	200	4	200
B	7	0.6	500	6	650
C	3	0.2	400	two	450
D	5	0.8	400	3	600
AND	4	0.3	200	4	200
F	6	1.1	300	4	700
G	8	1.5	600	5	900
H	9	2.0	700	8	900
I	3	0.4	300	3	300
J	6	0.6	500	6	500

1. Determine the critical path, expected duration, and total cost of the project, considering the normal duration of activities. Determine the probability of completing the project within 20 days, within 26 days, and after 28 days from the start date. Determine the probability that event 5 will occur within 7 days after the project begins. What would be the minimum duration in which you would commit to complete the project expecting to have a 0.9 probability of actually completing it at that time? Determine the minimum duration of the project and the associated minimum total cost, considering the possibility of accelerating activities.What would be the critical path? Formulate a linear programming model that allows determining which activities should be reduced and by how much in order to finish the project within T days, minimizing the total cost of the project.

1. A company is planning the development of a project considering the following information:

Exercise	Predecessor	Expected time (weeks)	Variance	Cost (US $)
TO	-	3	0.3	6,000
B	-	5	0.5	5,000
C	TO	4	2.0	16,000
D	B	3	1.0	18,000
AND	B	one	0.2	20,000
F	C, D, E	4	0.4	16,000
G	C, D	two	0.1	2,000
H	F, G	two	1.0	6,000
I	F	3	0.6	9,000

1. Build the network associated with the project. Determine the probability of completing the project at:

• More than 18 weeks. Exactly 16 weeks.

1. Suppose the following updated project information is available:

Exercise	Percentage finished	Current cost (US $)
TO	100%	8,000
B	100%	5,000
C	fifty%	8,000
D	33%	9,000
AND	100%	25,000
F	0%	6,000
G	0%	0
H	0%	0
I	0%	0

Determine:

• What are the critical activities remaining? When do you really expect to complete the project? What is the probability that the remainder of the project will take less than 5 weeks?

1. If you are offered $ 1,000 to reduce the time to complete the project in 2 weeks, formulate a linear programming model considering the following costs with crashing:

Exercise	Expected time (weeks)	Crashing cost (US $)
TO	two	8,000
B	one	7,000
C	two	25,000
D	two	26,000
AND	one	20,000
F	two	18,000
G	one	4,000
H	one	10,000
I	two	12,000

1. Consider the project represented by the following network:

1. Determine the critical path and minimum duration of the project. Considering that each activity can be accelerated by up to 20% and that the duration of activities A, B and C can be decreased by one unit at a cost of US $ 100, the duration of activities D, E and F can be decreased by one unit at a cost of US $ 200 and the duration of the remaining activities can be decreased by one unit at a cost of US $ 300, which activities should be accelerated to complete the project three days before what is determined in a) minimizing costs? Considering the acceleration conditions of the silver activities in b), formulate a linear programming model to determine which activities should be accelerated and by how much, in order to complete the project and 75% of the time determined in a) minimizing the resources invested.Clearly define variables, objective function, and constraints.

CASE STUDY OF INDUSTRIAL ENGINEERING: WORLD OIL

World Oil receives crude oil from the Middle East at its Marseille and Venice facilities. The oil is then shipped via a pipeline with pumping stations in Dijon, Bern, Reims and Luxembourg to storage tanks in Paris, Cologne and Brussels. The approximate distances in kilometers between the points connected by a pipe segment are provided in the following tables.

	TOWARD
SINCE	DIJON	BERN
MARSEILLES	475	450
VENICE	-–	425

	TOWARD
SINCE	REIMS	LUXEMBOURG
DIJON	240	275
BERN	375	325

		TOWARD
SINCE	PARIS	BRUSSELS	SUBURB
REIMS	130	175	-–
LUXEMBOURG	-–	150	140

This month 250,000 barrels of oil are available in Marseille and 150,000 barrels are in Venice. The Paris storage facility needs to receive 200,000 barrels, and the Brussels and Cologne facilities each need to receive 100,000 barrels.

1. Draw a distribution network indicating appropriate supplies, demands, and other relevant data (when appropriate, add figured arcs and nodes for a balanced problem). Formulate a mathematical model to determine how oil should be shipped from these facilities to minimize the total kilometers that the oil travels (that is, the sum of the number of barrels of oil times the number of kilometers traveled).

BIBLIOGRAPHY

TAHA, Hamdy A. Operations Research, An Introduction. 1989. Ediciones Alfaomega, SA México. DF Mexico.

MOUNTAIN, Agustín. Initiation to the Critical Path Method. 1972. Editorial Trillas, SA México. DF Mexico.

MOSKOWITZ, Herbert and Gordon P. Wrigth. Operations research. 1982. Prentice Hall Hispanoamericana, SA Naucalpan de Juárez. Mexico.

REFERENCES AND WEB LINKS:

Industrial Engineering Published Work (UPIICSA - IPN)

Engineering of Work Methods

http://www.monografias.com/trabajos12/ingdemet/ingdemet.shtml

Work Measurement Engineering

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Quality Control - Its Origins

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Market research

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Method Engineering - Production Analysis

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Measurement Engineering - Standard Time Applications

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Chemistry - Atom

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Plant Distribution and Materials Management (UPIICSA)

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University Physics - Classical Mechanics

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UPIICSA - Industrial Engineering

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Mechanical Tests (Destructive Tests)

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Classical Mechanics - One-dimensional movement

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Quality Control - Shewhart Control Charts

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Chemistry - UPIICSA Physicochemistry Course

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Method Engineering - Work Sampling

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Biology and Industrial Engineering

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Linear Algebra - UPIICSA Exams

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Laboratory Practices of Electricity (UPIICSA)

http://www.monografias.com/trabajos12/label/label.shtml

Practices of the UP Chemistry Laboratory

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Physics Problems by Resnick, Halliday, Krane (UPIICSA)

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Biochemistry

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Theory of the Company

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Code of ethics

http://www.monografias.com/trabajos12/eticaplic/eticaplic.shtml

Method Engineering: Systematic Analysis of Production 2

http://www.monografias.com/trabajos12/igmanalis/igmanalis.shtml

University Physics - Oscillations and Harmonic Movement

http://www.monografias.com/trabajos13/fiuni/fiuni.shtml

Chemical Production - The world of plastics

http://www.monografias.com/trabajos13/plasti/plasti.shtml

Plastics and Applications - Case Study at UPIICSA

http://www.monografias.com/trabajos13/plapli/plapli.shtml

Production Planning and Control (PCP - UPIICSA)

http://www.monografias.com/trabajos13/placo/placo.shtml

Operations Research - Linear Programming

http://www.monografias.com/trabajos13/upicsa/upicsa.shtml

Legislation and Mechanisms for Industrial Promotion

http://www.monografias.com/trabajos13/legislac/legislac.shtml

Operations Research - Simplex Method

http://www.monografias.com/trabajos13/icerodos/icerodos.shtml

Published Pneumatics in Industrial Engineering

UPIICSA compressed air

http://www.monografias.com/trabajos13/compri/compri.shtml

Pneumatics and Industrial Engineering

http://www.monografias.com/trabajos13/unointn/unointn.shtml

Pneumatics: Air Generation, Treatment and Distribution (Part 1)

http://www.monografias.com/trabajos13/genair/genair.shtml

Pneumatics: Air Generation, Treatment and Distribution (Part 2)

http://www.monografias.com/trabajos13/geairdos/geairdos.shtml

Pneumatics - Introduction to Hydraulic Systems

http://www.monografias.com/trabajos13/intsishi/intsishi.shtml

Structure of Hydraulic Circuits in Industrial Engineering

http://www.monografias.com/trabajos13/estrcir/estrcir.shtml

Pneumatics and Hydraulics - Power Generation in Industrial Engineering

http://www.monografias.com/trabajos13/genenerg/genenerg.shtml

Pneumatics - Pneumatic Valves (Industrial Engineering applications) Part 1

http://www.monografias.com/trabajos13/valvias/valvias.shtml

Pneumatics - Pneumatic Valves (applications in Industrial Engineering) Part 2

http://www.monografias.com/trabajos13/valvidos/valvidos.shtml

Pneumatics and Hydraulics, Hydraulic Valves in Industrial Engineering

http://www.monografias.com/trabajos13/valhid/valhid.shtml

Pneumatics - Pneumatic Auxiliary Valves (Applications in Industrial Engineering)

http://www.monografias.com/trabajos13/valvaux/valvaux.shtml

Industrial Engineering Problems in Pneumatics (UPIICSA)

http://www.monografias.com/trabajos13/maneu/maneu.shtml

Solenoid Valves in Control Systems

http://www.monografias.com/trabajos13/valvu/valvu.shtml

Pneumatics and Industrial Engineering

http://www.monografias.com/trabajos13/unointn/unointn.shtml

Structure of Hydraulic Circuits in Industrial Engineering

http://www.monografias.com/trabajos13/estrcir/estrcir.shtml

Energy saving

http://www.monografias.com/trabajos12/ahorener/ahorener.shtml

Published Law Work of the Atoyac School Center

Notions of Mexican Law

http://www.monografias.com/trabajos12/dnocmex/dnocmex.shtml

Notions of Positive Law

http://www.monografias.com/trabajos12/dernoc/dernoc.shtml

Civil Family Law

http://www.monografias.com/trabajos12/derlafam/derlafam.shtml

Amparo trial

http://www.monografias.com/trabajos12/derjuic/derjuic.shtml

Property crimes and Professional Responsibility

http://www.monografias.com/trabajos12/derdeli/derdeli.shtml

Individual Employment Contract

http://www.monografias.com/trabajos12/contind/contind.shtml

The Family in Mexican Civil Law

http://www.monografias.com/trabajos12/dfamilien/dfamilien.shtml

The Family in Positive Law

http://www.monografias.com/trabajos12/dlafamil/dlafamil.shtml

Articles 14 and 16 of the Mexican Constitution

http://www.monografias.com/trabajos12/comex/comex.shtml

Individual guarantees

http://www.monografias.com/trabajos12/garin/garin.shtml

The Family and the Law

http://www.monografias.com/trabajos12/lafami/lafami.shtml

Published work of History and Philosophy

Understanding Today's World by Ricardo Yépez Stork

http://www.monografias.com/trabajos12/entenmun/entenmun.shtml

The Power of Self Esteem

http://www.monografias.com/trabajos12/elpoderde/elpoderde.shtml

Mexico from 1928 to 1934

http://www.monografias.com/trabajos12/hmentre/hmentre.shtml

Mexican Independence Stage

http://www.monografias.com/trabajos12/hmetapas/hmetapas.shtml

Vicente Fox

http://www.monografias.com/trabajos12/hmelecc/hmelecc.shtml

The Profile of man and Culture in Mexico

http://www.monografias.com/trabajos12/perfhom/perfhom.shtml

Religions and morals

http://www.monografias.com/trabajos12/mortest/mortest.shtml

Moral - Salvifichi Dolorishttp: //www.monografias.com/trabajos12/morsalvi/morsalvi.shtml

The government of General Manuel González

http://www.monografias.com/trabajos12/hmmanuel/hmmanuel.shtml

José López Portillo

http://www.monografias.com/trabajos12/hmlopez/hmlopez.shtml

Museum of Cultures

http://www.monografias.com/trabajos12/hmmuseo/hmmuseo.shtml

Man and the Robot: In search of harmony

http://www.monografias.com/trabajos12/hommaq/hommaq.shtml

History of Mexico - The Laws of Reform

http://www.monografias.com/trabajos12/hmleyes/hmleyes.shtml

History of Mexico - Inquisition in New Spain

http://www.monografias.com/trabajos12/hminqui/hminqui.shtml

History of Mexico - The French Intervention

http://www.monografias.com/trabajos12/hminterv/hminterv.shtml

History of Mexico - First Centralist Government

http://www.monografias.com/trabajos12/hmprimer/hmprimer.shtml

History of Mexico - El Maximato

http://www.monografias.com/trabajos12/hmmaximt/hmmaximt.shtml

History of Mexico - The War with the United States

http://www.monografias.com/trabajos12/hmguerra/hmguerra.shtml

Mexico: Adopting New Culture?

http://www.monografias.com/trabajos12/nucul/nucul.shtml

Ranma Manga (English only)

http://www.monografias.com/trabajos12/ranma/ranma.shtml

Fraud of the Century

http://www.monografias.com/trabajos12/frasi/frasi.shtml

Jean Michelle Basquiat

http://www.monografias.com/trabajos12/bbasquiat/bbasquiat.shtml

The Sense of Humor in Education

http://www.monografias.com/trabajos12/filyepes/filyepes.shtml

Engineering Teaching vs. Privatization

http://www.monografias.com/trabajos12/pedense/pedense.shtml

Learning process

http://www.monografias.com/trabajos12/pedalpro/pedalpro.shtml

Giovanni Sartori, Homo videns

http://www.monografias.com/trabajos12/pdaspec/pdaspec.shtml

Life: Things are known for their operations

http://www.monografias.com/trabajos12/lavida/lavida.shtml

What is Philosophy?

http://www.monografias.com/trabajos12/quefilo/quefilo.shtml

Sensitive knowledge

http://www.monografias.com/trabajos12/pedyantr/pedyantr.shtml

Comparison of authors and schools

http://www.monografias.com/trabajos12/pedidact/pedidact.shtml

Philosophy of education

http://www.monografias.com/trabajos12/pedfilo/pedfilo.shtml

Analysis of the Psychopathology of memory

http://www.monografias.com/trabajos12/pedpsic/pedpsic.shtml

Company and family

http://www.monografias.com/trabajos12/teoempres/teoempres.shtml

Philosophical Anthropology

http://www.monografias.com/trabajos12/wantrop/wantrop.shtml

Definition of Philosophy

http://www.monografias.com/trabajos12/wfiloso/wfiloso.shtml

Review of the Magna Didactic Book

http://www.monografias.com/trabajos12/wpedag/wpedag.shtml

The man before the problems and limits of Science

http://www.monografias.com/trabajos12/quienes/quienes.shtml

Review of the book Froebel. The education of man

http://www.monografias.com/trabajos12/introped/introped.shtml

Philosophical Anthropology

http://www.monografias.com/trabajos12/antrofil/antrofil.shtml

Technical calculation report

http://www.monografias.com/trabajos12/electil/electil.shtml

Calculation memory

http://www.monografias.com/trabajos12/elplane/elplane.shtml

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