2020 Mathematical Modeling National Contest B questions:
Consider the following mini game: Players rely on a map, use initial funds to buy a certain amount of water and food (including food and other daily necessities), start from the starting point, and walk in the desert. You will encounter different weather on the way, and you can also replenish funds or resources in mines and villages. The goal is to reach the end within the specified time and retain as much funds as possible.
The basic rules of the game are as follows:
(1) Day is the basic time unit, the start time of the game is the 0th day, and the player is at the starting point. The player must reach the end on or before the deadline, and the player’s game ends after reaching the end.
(2) Two resources, water and food, are needed to cross the desert, and their smallest unit of measurement is a box. The sum of the quality of water and food that the player has every day cannot exceed the upper limit of weight. If the end is not reached and the water or food is exhausted, the game is deemed to have failed.
(3) The daily weather is one of three conditions: "clear", "high temperature", and "sandstorm". The weather in all areas of the desert is the same.
(4) Every day, the player can reach another area adjacent to it from an area on the map, or stay in place. You must stay in place on a sandstorm day.
(5) The amount of resources consumed by the player staying in place for a day is called the basic consumption, and the amount of resources consumed for a day of walking is times the basic consumption.
(6) Players can purchase water and food at the base price with initial funds at the starting point on day 0. Players can stay at or return to the starting point, but cannot purchase resources at the starting point multiple times. After the player reaches the end, the remaining water and food can be returned. The return price of each box is half of the base price.
(7) When the player stays in the mine, he can obtain funds through mining. The amount of funds obtained in a day of mining is called the basic income. If mining, the amount of resources consumed is a multiple of the basic consumption; if not mining, the amount of resources consumed is the basic consumption. You cannot mine on the day you arrive at the mine. Mining is also possible on sandstorm days.
(8) Players can use the remaining initial funds or the funds obtained from mining to buy water and food at any time when passing through or staying in the village. The price of each box is twice the base price.
Please establish a mathematical model according to the different settings of the game to solve the following problems.
- Assuming that there is only one player and the weather conditions are known in advance during the entire game period, try to give the player's optimal strategy under normal circumstances. Solve the "first level" and "second level" in the attachment, and fill in the corresponding results in Result.xlsx.
- Assuming that there is only one player, and the player only knows the weather conditions of the day, he can decide the action plan for the day based on this, try to give the player's best strategy under normal circumstances, and check the "third level" and "fourth level" in the attachment "Conduct specific discussions.
- Existing players, they have the same initial capital, and start from the starting point at the same time. If any player among them walks from area A to area B () on a certain day, the amount of resources consumed by any of them will be times the basic consumption; if any player among them digs in the same mine on a certain day Mine, the amount of resources consumed by any of them is twice the basic consumption, and the funds that each player can obtain through mining in a day are the basic income; if any player among them buys in the same village on a certain day Resources, the price of each box is times the benchmark price. In other cases, the amount of resources consumed and the price of resources are the same as the single player game.
(1) Assuming that all weather conditions are known in advance during the entire game period, the action plan of each player must be determined on the first day and cannot be changed thereafter. Try to give a strategy that players should adopt under normal circumstances, and discuss the "fifth level" in the attachment.
(2) Assuming that all players only know the weather conditions of the day, from the first day, every player knows the action plan of the other players and the amount of remaining resources after the end of the day's action, and then determines their action plan for the next day. Try to give a strategy that players should adopt under normal circumstances, and discuss the "sixth level" in the attachment.
Note 1: In the map given in the attachment, two areas with a common boundary are called adjacent, and two areas with only a common vertex but no common boundary are not considered adjacent.
Note 2: The remaining funds (remaining water and food) in Result.xlsx refer to the funds (water and food) after all the resources required for the day are consumed. If there are still purchases on the day, it refers to the amount of funds (water and food) after the purchase.
