شبكات الأعمال| أساليب كمية وبحوث العمليات

Hinzugefügt: 2026-06-03

[00:00:00]Peace, mercy, and blessings of God be upon you. Welcome to the North Center for Educational Services. I'm Mahmoud Gharib. In this video, we'll explain a very important topic in the subject of Quantitative Methods, or its former name, Operations Research. So, what topic will we be explaining? We'll be explaining the topic of business networks. In the topic of business networks, the professor gives me a set of activities related to implementing a project. Each activity has a start point and an end point, and each activity has an expected or necessary time for performing the activity. Each activity also has a pessimistic time and an optimistic time. Now, I have a set of activities necessary to implement a project. The projects or activities are A, B, C, D, E, F, and Y. The facts related to the activity, meaning the start and end points of each activity, are as follows: A starts from one and ends at where? B from one to three? C from one to four? D from where to five? E from three to five? F from four to five? G from five to six? In short, the project as a whole starts from one and ends at six. And then... Give me something called the time required to perform the activity. We call it the expected time to perform the activity.

[00:01:19]Also, give me the pessimistic time and the optimistic time for each activity.

[00:01:26]Then, below, it tells me a very important piece of information. It says, " If you know that the corresponding values ​​in the project execution probability table were as follows: the value 0 corresponds to 50%, the value 0 corresponds to 72%, and the value 08 corresponds to 88%."

[00:01:45]Always, always, before looking at the multiple-choice questions, you must first do two very important things: first, draw the network, and second, calculate or define the project paths. Let's start, with God's blessing, by drawing the network. In drawing the network, you look at the exercise and see that the project starts from circle number one, from event number one, and ends at event number six. So, I'll take a small piece of paper and start by saying, "Here's a circle at the beginning of the paper, and I'll call it number one." I draw a circle at the end of the paper and call it number six. Then I start going from one to what? From one to six. Okay, let's see how this works. He tells me that activity A is from 1 to 2, activity B is from 1 to 3, and activity C is from 1 to 4. Is there anything else from one to something else? No, there isn't one. It's from one to two, from one to three, and from one to four. So, from circle number one, I'll draw three arrows: one towards circle number two, one towards circle number three, and one towards circle number four. Now, let's continue.

[00:02:56]After 1A, 1A, 3, and 1A, we found "where" to five, "from there" to five, and "from four" to five. So, I'll draw a circle in the middle and call it number five, and start connecting "where" to five, "from there" to five, and " from four" to five. The last activity, which is activity H, connects "from five" to six. Let's say we've reached the last activity, "from five" to six. Let's go back to the beginning. We said activity A's start time is from one to two, and its expected duration was four weeks. So, "from one to two" is labeled "A," and its expected duration is four weeks. Now, "from one to three" is called "B," and its expected duration is " where." So, I'll label "from one to three" B and its expected time. Okay, from one to four, its name is C and its expected time is 3, so I'll write on it: From one to four, C and its expected time is 3. Now let's go to the one from two to five, its name is D and its expected time is 5, so we'll take the one from two to five, call it D, and say that its expected time equals 5. Now the one from one to five is called E and its expected time is 6, so I'll write on it: From three to five, E and its expected time is 6.

[00:04:18]Okay, the one from four to five is called Y and its expected time is 7, so I'll write: From four to five and its expected time is 7. Okay, the one from five to six is ​​called Y and its expected time is 4, so I'll write: From five to six, Y and its expected time is 4. So, I've drawn the network of work. Now, the next step after drawing the network of work is what we determine. We said we determine the project paths.

[00:04:46]What does the word path mean? The word path is the way that takes me from the beginning of the project to its end. Now, I want to get from circle number one to circle number six, can I take How long did it take me on this road? I took 4 and 4, so I'll say the first path is path A, D, and its time is 4 + 5 + 4, which equals 13. Now, for the second path, it's called B, W, and Y, and its time is 2, 6, and 4. So, the second path is B, W, and Y, and its time is 2, 6, and 4, which equals 12. Now, the third path is C, W, and Y, and its time is 3, 7, and 4, which equals 14. So, we'll say the third path is C, W, and Y, and its expected time was 14. Now, is there a fourth path?

[00:05:56]Look closely, no, there isn't a fourth path. So, how many paths does the business network in this project contain?

[00:06:02]Four paths.

[00:06:06]Now, let's also determine point number three, which is identifying the critical path. Which of these three paths is the critical path? Which path is the longest?

[00:06:19]I have path B with 13, path B with 12, and path B with 14. The path with the longest time is path number 3, C, and Y. Therefore, if someone asks me how long the project will take to complete, I'll tell him the project implementation time is 14 weeks. Now, if you don't mind, let's look at a set of multiple-choice questions. We've already defined and drawn the network of work, identified the project paths, and also identified the critical path.

[00:06:50]Let's start solving some of the multiple-choice questions. He asks me the number of weeks in the first path. I'll tell him the number of weeks in the first path was 13 weeks. He asks me the number of weeks in the second path. I'll tell him the number of weeks in the second path was 12 weeks. He also asks me the number of weeks in the third path. I'll tell him the number of weeks in the third path was 14 weeks. He also asks me the critical path. I'll tell him the critical path is the path with the longest duration, and it was C and Y. The answer is C and Y. After that, he asks me for the early start time of activity A, the early start time of activity D, the early end time of activity E, and the early end time of activity Y. What does this mean? It means that I need to go back to the drawing again. Here, everyone, is my drawing that I drew before. You are asking me early.

[00:07:50]What does the word early mean? It means that you will come from above each of these arrows and put two brackets for me, and inside each bracket you will put a comma or an 'and' mark to divide the bracket into two ends. So at A I will put an bracket, at B an bracket, at C an bracket, at D an bracket, at E an bracket, at F an bracket, and at Z an bracket. And each bracket is divided into two ends, a beginning and an end. The beginning and end of each bracket is divided. Where is the beginning and end? Now let's see the beginning of activity A. Activity A is outside of what number circle? Outside of circle number one, any activity outside of circle number one, its beginning is zero. I am still in the name of God, I am still saying the name of God in the project, so I always start from where? Starting from zero, the activity will be called A because it is outside of circle number one, so its beginning will be zero. What is its end? The end of any activity always equals its beginning plus its time. What will the end of the activity equal? 0 + 4 4. Now let's go to activity B. It also comes out of circle number one, meaning its start is zero. And its end is 0 + 2, so now it's 2. Now let's go to activity C. It also comes out of circle number one, meaning its start is zero. And its end is 0 + 3, which equals 3. Now let's go to activity D. Activity D starts from circle number two, so its start isn't zero.

[00:09:32]So what will its start be? See what the previous one ended at. The previous one, which is activity A, ended at four, so D starts at four. And its end is 4 and five, which equals t. Now let's go to activity E. Its start is from circle number 1, and circle number 1 before it ended at where? So circle number 3 starts at two. And its end is 2 and 6, which equals 8. Now let's go to activity E. Its start is from circle number 4. The one before circle number 4 was 3, so it starts and will start from 3. So 3 and s add up to 10. Now let's go to activity Y. It starts from circle number five. What is the Before circle number five, the one entering circle number five has an arrow entering at nine, an arrow entering at eight, and an arrow entering at 10. Which end will I take?

[00:10:37]I'll take the larger limit. So if I have a circle with an arrow pointing into it, I'll take the larger limit of the arrow. Is the largest for me nine, eight, or ten? Let's say ten. So what is the limit of Y?

[00:10:51]Ten. And what will the limit of R be? Fourteen. And if you pay attention, there's a very important piece of information that confirms that everything I did is correct. How do you know that the activity ended with 14? The activity is about time, the critical path. Go back to the critical path, and you'll find that the critical path, which is the path R, the longest in time, had a time equal to what? 14.

[00:11:22]When I calculate the early times, I need to know that the last activity in the diagram, which is activity Y, will always have an end time equal to the critical path time that I got. B14, in the information I want to tell you, did you notice that when we set the early times we went by the order of the activities? What does that mean? I mean, I went A B C D E F Y, I didn't go by sequencing the drawing. I didn't go, I mean, I didn't say A and then D and then Y or B and then E and then Y. I didn't say that. I said A B C, then D E and then Y. Okay, let's answer some of the questions that are in the question paper. In the question paper, he was telling me, "I want the early time for the start of the activity." The early time is the one above.

[00:12:16]What is the beginning of the arc above? If the start time is zero, then I'll tell him that activity A's start time is zero. Now, for the early start time of activity D, the early start time of activity D is four. I'll tell him that the start time of D is four. Now, for the early end time of activity E, I'll tell him that the end time of activity E was 8 seconds.

[00:12:42]The answer is 8 seconds. Now, for the early end time of activity Y, the end time of activity Y was 14. Now, continue with me.

[00:12:54]Next, in questions 9, 10, 11, and 12, I want the late end time.

[00:13:02]We've already calculated the early end time, and now we want to calculate the late end time.

[00:13:09]You said that the early end time is represented by arcs above the arrow. So, for the late end time, we'll draw arcs below the arrow, and inside each arc, we'll put an 'and' mark and a comma. So, at A, I'll put an arc below B, an arc below C, an arc below D, and an arc below E. Then I'll start determining the late end times for each activity, just like we did before. You see the late times are also at the beginning and the end, but we should pay attention to something. When I was calculating the early times, which are the brackets above the arrow, I was saying A B C D E W Y. So when I was calculating the late times, I would go in reverse, meaning I would say Y W E D C B A. I would go the drawing from back to front, or the activities from back to front: Y W W A W E D C B. And when I was calculating the early times, I was adding. When I was calculating the late times, I would subtract. So let's look at this quickly. The late time to the end of Y, I would say the beginning of Y. Its early time is above its end.

[00:14:22]What is it? 14 remains. It ends at 14. Subtract 14, what does that equal? 10. Come back with me from Y to Ali. And when you come back from Y to Ali, how many are with you here?

[00:14:36]I have 10, so it will end with 10 - 7. So let's go to it. It had a limit at circle number five, and circle number five is related to activity Y. How much was it coming back with? He was returning with 10. I will write here 10 - 6, which equals 4. Activity D was hitting circle number five, and circle number five ends with what?

[00:15:06]It ends with 10, so here I will say that 10 is the end of activity D and 10. 10 - 5 will equal five for me.

[00:15:17]Okay, let's move on to activity C. Activity C was confused with circle number four, and circle number four had a beginning, so C ends in 3, and 3 - 3 equals zero.

[00:15:32]Activity B was hitting circle number 3, and circle number 3 had a beginning below 4, so B's end will be 4, so 2 will equal 2.

[00:15:46]Activity A was hitting circle number Ain, and circle number Ain had a beginning below 5, so activity A's end will be 5, and 5 - 4 will equal what? It will equal one. Please note something very, very important that confirms that I solved it correctly: what was the name of the critical path? Its name was C and Y, which is the longest and most frequent path. If we look at the activities of the critical path, which is C, W, and Y, you'll find that the numbers above the arrow are the same as those below the arrow. Early is late. Any activity not on the critical path will have a different early and late time. The early one above the arrow is different from the late one below the arrow.

[00:16:33]All activities outside the critical path have different early and late times. Now let's answer questions 9 to 13.

[00:16:47]Question 9 asked for the late start time of activity C. I would say the late start time of C.

[00:16:53]The word "late" means you look under the arrow; the start time of C was zero. It also asked for the late start time of activity D. I would say the late start time; I would look under the arrow; the start time of D was five. The answer is five. It asked for the late end time; I would say the late end time; I would look under the arrow; the end time was 10. So my answer is 10. I would say the late end time of activity Y. The late end of activity Y from below Y, the late end time equals 14. The answer is 14. He also asks me for the surplus time for activity B. I will tell him that the surplus time for activity B equals the beginning of the late one minus the beginning of the early one, the beginning below minus the beginning above. If he asks me about B, I will tell him 2 below minus 0 above.

[00:17:56]What will it equal? It will equal two. If he asks me about the activity and I tell him the bottom start minus the top start, 3 below minus 3 above, the result will equal zero. If he tells me the excess time for the critical path, this is constant information, the excess time for the critical path always equals zero. If he tells me he wants the variance for the critical path, I will tell him this variance has its own law.

[00:18:24]What does the law of variance say? The law of variance says you say the pessimistic time minus the optimistic time divided by six, and six is ​​a constant number, and in parentheses, all squared. So when I apply the variance, what do I apply it to? Applying it to the critical path, what is the critical path? It's C and W. Now, if you don't mind, let's apply the variance law. I'll say here where the pessimistic time and the optimistic time are in the exercise. I want C, W, and Y. So, I'll say the pessimistic time at C is 14 - the optimistic time is 10, divided by the constant we said to memorize, which is 6 in parentheses, all squared. The result is 444. Now, for activity W, I'll say the fractional part for activity W is 16 - 8 divided by the constant I memorized, which was 6, all squared. The result is 1.77777.

[00:19:33]Now, for activity Y, I'll say the fractional part for activity Y is 7 pessimistic - 3 optimistic, divided by the constant I memorized, which was 6, all squared. The variance for activity Y is Y = W, 444.

[00:19:51]After finding the variance for each activity, I'll find the sum of the variances, meaning I'll add the variance of activity C plus the variance of activity W plus the variance of activity C. The sum of the variances I get is 2/665.

[00:20:08]What did the question after the variances say? He was asking me to calculate the standard deviation of the critical path. I would tell him, sir, the standard deviation of the critical path was equal to the square root of the variance. So, I would take the calculator and say the square root of 2 divided by 665,000, and the standard deviation would equal 1 divided by 632.

[00:20:28]The answer is number C. He also asked me to calculate the probability of completing the project on time. I would tell him that the probability of completing the project on time is always equal to a constant value: 50%, 5%, or 0%. These three values ​​are constant. See which one you get in the choices: 50%, 60%, 70%, 72%, or 78%.

[00:20:54]So, the correct answer is 50%, since the probability of completing the project on time has a constant answer of 50%. Now, the last question we have asks me to calculate the probability of completing the project within 15 weeks.

[00:21:09]I would tell him that the probability of completing the project within 15 weeks is equal to the fraction of the target time minus the time of the critical path divided by the standard deviation. What is the target time that he is asking for? 15 weeks minus the critical path time. The Hajj path time, which we calculated before, came out to 14 weeks on the standard deviation. We just calculated the standard deviation in the previous step, and the result we got was 1/632.

[00:21:41]When I calculate this on the calculator, 15 - 14/1/632, the result I get is 0. So where is this 0? Let me tell you something very important that the doctor told me. He said that the corresponding values ​​in the project execution probability table were zero equal to 50%, 6 equals 72%, and 8 equals 88%. What value did I get? I got 0.66, so what was the corresponding value in the data that Dr. Madihani gave me? 72% means the probability of completing the project within 15 weeks is equal to 72%, which is answer C. That brings us to the end of the video. May God Almighty benefit you with what we have said. Don't forget to share this video, as it is very important. May God Almighty grant you success and improve our situation and yours. We hope to see you all in good health. Peace, mercy, and blessings of God be upon you.