Chapter 23 Classical Modeling Problems for Merchants and Followers
Looking at the two people in the classroom, Professor Liu Xiangping was slightly stunned, and asked with a smile, "Is there only two people? I heard there is one more."
"...I'll call and ask her." Wang Xiaodong looked embarrassed and went outside with the phone.
The teacher was not in a hurry, he unscrewed the thermos cup, took a sip of hot water, walked slowly to Lu Zhou, and asked with a smile, "What is the configuration of your team?"
Lu Zhou put down his phone and replied with a smile, "I am in charge of modeling, Wang Xiaodong is in charge of programming, and Lin Yuxiang is in charge of writing the thesis and defense."
"Modeling, programming, writing, standard configuration," Mr. Liu smiled and continued, "Speaking of which, you are the Lu Zhou from the first or third year. I read your paper, and it is really good. article."
Lu Zhou's eyes lit up: "Teacher, are you also doing math?"
"Forget it, but my research direction is towards physics, and I rarely publish mathematics papers." Teacher Liu squinted, glanced at the door of the classroom, and continued with a smile, "You lesbians, it seems that you are not very reliable. what."
Lu Zhou smiled awkwardly and politely.
Mr. Liu also smiled and didn't say much about this issue. After a pause, he continued: "As for the configuration of your team, I suggest that you can make some adjustments. Modeling and thesis are handled by the same person, and the rest Two people are in charge of programming. The advantage of this is that the person in charge of modeling has a strong grasp of the overall problem, and the writing ideas are more fluent. However, of course, there is also a disadvantage, that is, your burden will be very heavy. The total game time is only Three days, which means you have to start writing your thesis right after you finish modeling.”
Lu Zhou thought thoughtfully: "...Are two people in charge of programming?"
Teacher Liu took a sip of tea and said slowly, "That's right."
Lu Zhou smiled and didn't give an answer immediately: "I'll think about it again. After all, this matter has to be discussed with my teammates."
At this time, the sound of trotting footsteps came from outside the corridor, and the last teammate finally arrived late.
"I'm sorry, I got up a little late, I'm really, really sorry!" Lin Yuxiang bowed his head to the teacher apologetically as he walked into the classroom with Wang Xiaodong.
"It's okay, since everyone is here, let's start quickly." Teacher Liu Xiangping smiled casually, without blaming her for being late, turned around and walked to the podium, inserted the USB flash drive into the computer, and turned it on. He turned on the projector and released his PPT.
"Before starting the lecture, I hope you will look at an example problem. This problem itself is not difficult, but it can be used as a case of mathematical modeling. I hope that through this case, you can first understand what mathematical modeling is. what?"
Saying that, he pressed the mouse and turned the ppt to the next page.
"The three merchants each take a follower to cross the river by boat. A small boat can only accommodate two people. The followers make a secret agreement that on any bank of the river, once the number of followers exceeds the number of merchants, they will kill and cross the river. But how to cross the river by boat? The power is in the hands of the merchants, and ask: How can the merchants cross the river safely?"
Indeed, this question is not difficult.
Even without the power of the system, Lu Zhou quickly came up with the answer and replied.
"In the first round, two followers went and one came back."
"In the second round, two more followers will pass, and one follower will return."
"In the third round, two merchants went over, and a follower and a merchant came back."
"In the fourth round, two merchants went over and one entourage came back."
"In the fifth round, two followers went and one came back."
"In the sixth round, the last two followers passed by and successfully crossed the river!"
"Clap clap clap!" Lin Yuxiang clapped his little hands and applauded softly, his face full of admiration.
The expression on Wang Xiaodong's face was unmoved, and he looked like an expert outside the world.
In his opinion, this problem is really not difficult, although he didn't use his brain to calculate it, but he believes that his IQ can at most take a little time to solve it.
"Exactly." Mr. Liu smiled and continued, "Even if you don't need any mathematical knowledge, you can solve this problem simply through logical analysis. But what if the problem is extended to N businessmen?"
This problem is indeed difficult, but the difficulty is not in mathematics, but in how to abstract this problem into a mathematical problem to solve.
Lu Zhou thought about it for a while, and he already had a general idea in his mind.
"Can I use the blackboard?"
"Of course," Professor Liu Xiangping made a gesture of invitation with a smile.
Lu Zhou stepped forward, picked up the chalk and started writing on the blackboard.
[①Record the number of merchants on this bank before the kth crossing of the river as Xk. The number of followers is Yk, k=1, 2, ..., Xk, Yk=0, 1, 2, 3. The two-dimensional vector Sk=(Xk, Yk) is defined as the state, and the state set under the condition of safe river crossing is the allowable state set, denoted as S.
Available S={(X, Y)|X=0, Y=0, 1, 2, 3; X=3, Y=0, 1, 2, 3; X=Y=1, 2}
② Record the number of merchants on the kth ferry as Uk and the number of followers as Vk. A two-dimensional vector Dk = (Uk, Vk) is defined as a decision. The allowable decision set is recorded as D, which can be known from the boat capacity: D={(U, V)|1≤U+V≤V, U, V=0, 1, 2}
③ Based on the above conclusions, the change rule of state Sk with Dk is: S(k+1)=Sk+(-1)^k*Dk
】
"It's amazing..." Looking blankly at the blackboard writing on the blackboard, Lin Yuxiang opened his mouth slightly, looked at Lu Zhou who came down from the podium, and asked in a low voice in surprise, "Don't you even need to write a draft?"
"I made it in my heart." Lu Zhou said with a smile.
Wang Xiaodong was also very silent as always and did not speak.
However, judging from his expression, this arrogant scholar was completely convinced of Lu Zhou's mathematical ability.
Looking at the process on the blackboard, Professor Liu Xiangping smiled and nodded: "Exactly! But from the perspective of mathematical modeling, this work is only half completed. After the modeling is completed, we still We need a program to implement this model. Of course, this model is very simple, and I believe that with the ability of our classmates, this level of program is definitely easy to come by, so we won’t waste precious time here.”
Wang Xiaodong pushed his glasses, but the expression on his face was still indifferent, as if it was natural.
Although he didn't have the opportunity to show his programming skills on the spot, it made him a little regretful.
After a pause, Mr. Liu continued, "Data modeling is actually a process of reducing an actual problem to a problem that can be expressed with data and very short language, and then solving this problem with mathematical tools."
"So, the key is what are the actual problems your team is facing?"
"Modeling is not an examination of the programming ability of the team, nor the mathematical ability, but the ability to convert practical problems into data language and solutions. You must remember this."
"As for modeling itself, I don't have much advice. There are no shortcuts other than practice and broadening knowledge. As for programming, I can say a few words."
"There are mainly four kinds of commonly used software, Matlab, Mathematica, lingo and SAS. You don't need to master all of them, at least one of them is enough. The key is to be able to use the software that suits you to solve practical problems." When he said this , Professor Liu Xiangping is looking at Wang Xiaodong.
Because this sentence is mainly for him to listen to.
Wang Xiaodong pushed his glasses, nodded and said, "I have used Matlab and SAS, and it's not a big problem."
Professor Liu nodded: "It's basically enough to know these two softwares. You need to learn these things by yourself. What I can do is to provide guidance when you don't understand."
"Finally, about reference books, I recommend you to read some mathematical modeling cases and other books, such as "Mathematical Modeling", "Mathematical Modeling Algorithms and Applications" and "Mathematical Modeling Basic Course"."
"I still have some internal school handouts and materials here. You can copy them back to read, but remember not to spread them out, let alone upload them to the Internet. Understand?"
"Understood!" The three said in unison.
Professor Liu smiled and said: "Okay, that's all for today, I have nothing else to say, it's mainly up to you. If you need information, come and download it from me, and go back to review English when you're done. ."