Mathematical modeling test paper and reference answer 1. Concept question (***3 small questions, each with 5 points, this big question 15 points) 1. Generally speaking, what steps should be taken to establish a mathematical model? (5 points) A: The general steps of mathematical modeling include: model preparation, model hypothesis, model composition, model solution, model analysis, model verification and model application. 2. What abilities should we pay attention to when learning mathematical modeling? (5 points) A: Observation, association, insight and computer application ability. 3. What are the characteristics of artificial neural network method? (5 points) A: (1) can handle nonlinearity; (2) Parallel structure. (3) Learning and memory ability; (4) High data tolerance; (5) Neural networks can be realized by large-scale integrated circuits. Second, the model verification question (***2 small questions, each small question 10, this big question 20 points) 1. Someone started from Yamashita Hotel at 8:00 in the morning, followed a path up the mountain, arrived at the top of the mountain at 5:00 in the afternoon and checked in. The next day, he went down the mountain along the same road at 8:00 in the morning. Return to the hotel at 5:00 p.m. Prove that this person must pass somewhere on the road at the same time in two days (15 minutes). It is proved that the departure time is t=a, the arrival time is t=b, the distance from the hotel to the top of the mountain is s, and the motion equation of someone's uphill path is f(t), the downhill motion equation is g (t), and t is a time variable in one day. If the auxiliary function F(t)=f(t)-g(t) is continuous, then f(a)=0, f (b) >; 0 and g(a)>0, g(b)=0, we can see f (a).

The reference answer of mathematical model B (electric 05,65438+February), page 2 (***5), shows that the change rule of state ks with kd is:1+ks = ks+() kkd *-1(3 points). It is more convenient to solve this model graphically, as follows: (6 points)

Third, the calculation problem (***5 small questions, 9 points for each small question, 45 points for this big question * * *) 1, one? è? =14/13/1411a try and find the maximum eigenvalue of a, and do consistency test (when n=3, RI=0.58). A: Wandering? è? Standardization of columns in in =14/13/14131a

÷÷÷ ? è? The sum of 8/19/17/18/49/47/38/39/47/3 lines is somewhat awkward. è? 569.0373.1248.1= w 2 minutes and wandering? è? So =328. 1897.4328.4Aw, (1 min), so the maximum characteristic root is123.3) 569.0286 = iiiwawl2, and the consistency index is: ci = 06. 2 points

Mathematical Model B Reference Answer (05,65438+February) Page 3 (***5) 2. A piece of land, if engaged in agricultural production, can receive 100 yuan, if leased to a certain B for industrial production, can receive 200 yuan. If it is rented to a third company to develop tourism, it can be charged according to 300 yuan. When Party C invites Party B to participate in the operation, the income will reach 400 yuan. In order to achieve the highest income, shapley value method is adopted to distribute everyone's income. (9 points) Answer: The income of Party A, Party B and Party C from 250 yuan to 50 yuan should be 100 yuan (the steps are omitted). 3. The daily demand of products is constant r, the preparation cost of each product is C65,438+0, the storage cost of each product is C2, and the loss cost of shortage is C3. Try to make a reasonable assumption, establish a storage model that allows short loans, and work out the total production cycle and output. (9 points) Solution: Model assumption: 1. The daily demand for products is constant r 2. The preparation cost of each production is c 1, and the storage cost of each product is c2 3. The production capacity is unlimited. The loss cost of out-of-stock is C3, and the product has been used up when t=T 1. 4. The production cycle is t and the output is Q (2 points). The total cost of establishing the cycle model is as follows:

2) (2213121ttrcqTCC-++= (2 points) The average cost in a period is

Rt qrtctctctctctqtf 2)(2), (2 322 1-+= (2 points) Model solution: solve the period by differential method.

3 2321) (2crccct+= (1) yield.

)(23223 1 CCCRCQ+= ( 1) 4。 There are three states of people: 1 (health), 2 (illness) and 3 (death). Assuming that people of a certain age are healthy this year, the probability of staying healthy next year is 0.8, and the probability of getting sick is 0. 18, while the probability of getting sick this year is 0.65 and the probability of being healthy next year is 0.25. A Markov chain model is constructed to show that it is an absorption chain, and the average number of transitions from illness to death is sought for health. Solution: State () () (Death, disease, health, 32, 1 = =, what is the transition probability matrix of iii? è? = 065.08.0p025.018.0 ÷ 21.002.02 point () (,32 1 nanan = a, then