Question A: Benefits and risks of investment.

There are n kinds of assets (such as stocks, bonds, …)Si (i= 1, …n) in the market for investors to choose from, and a company has a considerable amount of M funds that can be used for investment in a period. The company's financial analysts evaluate these N kinds of assets, estimate the average rate of return of purchasing Si in this period as ri, and predict the risk loss rate of purchasing Si as qi. Considering that the more diversified the investment, the smaller the total risk. The Company has determined that when using this fund to purchase several assets, the total risk can be measured by the largest risk in the invested Si.

There is a transaction fee for purchasing Si, and the rate is pi. When the purchase amount does not exceed the given value ui, the transaction fee is calculated according to the purchase ui (of course, you don't have to pay if you don't buy Si). In addition, assuming that the bank deposit interest rate is r0 in the same period, there is neither transaction cost nor risk. (r0=5%) The relevant data when n = 4 is known are as follows:

Four days (%) gas (%) skin (%) ui (yuan)

S 1 28 2.5 1 103

S2 2 1 1.5 2 198

S3

S4

Try to design a portfolio plan for this company, that is, with a given fund M, selectively buy several assets or deposit them in the bank for interest, so as to make the net income as large as possible and the overall risk as small as possible. 2. Try to discuss the above problems as a whole and calculate with the following data.

Question B Disaster patrol route

The following figure is a schematic diagram of the township (town) village road network in a county, and the number next to the highway is the kilometers of the road section.

This county was flooded this summer. In order to investigate the disaster situation and organize self-help, the county leaders decided to lead the responsible persons of relevant departments to inspect the townships (towns) and villages in the county. Patrol route refers to the route starting from the seat of the county government, bypassing townships (towns) and villages, and returning to the seat of the county government.

1) If the patrol is divided into three groups (roads), try to design a patrol route with the shortest total distance and as balanced as possible for each group.

2) Assume that the patrol personnel stay in each township (town) for T=2 hours, stay in each village for t= 1 hour, and the vehicle speed for v = 35 km/h. To complete the patrol within 24 hours, at least several groups should be divided; Give the best patrol route you think under this group.

3) Under the above assumptions about T, T and V, if there are enough patrol personnel, what is the shortest time to complete the patrol; Under the requirement of completing the tour in the shortest time, give your best tour route.

4) If the number of patrol groups is fixed (such as three groups), it is required to complete the patrol as soon as possible, and discuss the influence of the change of T, T and V on the best patrol route.