2020 Mathematical modeling competition B Title :

Consider the following games : Players rely on a map , Use the initial funds to buy a certain amount of water and food ( Including food and other daily necessities ), Starting from the starting point , Walking in the desert . There will be different weather on the way , It can also be found in mines , Village supplementary funds or resources , The goal is to reach the destination within the specified time , And keep as much money as possible .
The basic rules of the game are as follows :
(1) The basic time unit is day , The start time of the game is No 0 day , Player at starting point . Players must reach the finish line on or before the deadline , At the end of the game, the player's game ends .
(2) Two kinds of food and water are needed to cross the desert , Their minimum unit of measurement is the box . The sum of the quality of water and food that players have every day can't exceed the weight-bearing limit . If the end point is not reached and water or food is exhausted , As a game failure .
(3) The weather is “ sunny ”,“ high temperature ”,“ Sandstorm ” One of the three conditions , The weather is the same in all regions of the desert .
(4) Every day players can move from one area of the map to another adjacent area , You can also stay where you are . Sandstorm days must stay where they are .
(5) The amount of resources consumed by players staying in place for one day is called basic consumption , The amount of resources consumed in one day is the basic consumption times .

(6) Player No 0 Days can use initial funds to buy water and food at a benchmark price at the starting point . Players can stay at or return to the starting point , But you can't buy resources at the starting point many times . Players can return the remaining water and food when they reach the destination , The return price of each case is half of the base price .
(7) When players stay in the mine , Funds can be obtained through mining , The amount of money obtained in one day of mining is called basic income . If mining , The number of resources consumed is the base consumption
times ; If you don't dig , The amount of resources consumed is the base consumption . No mining is allowed on the day of arriving at the mine . Mining in sandstorm days .
(8) When players pass through or stay in the village, they can purchase water and food at any time with the remaining initial funds or mining funds , The price per case is the base price 2 times .
Please set it according to the game , Establish mathematical model , Solve the following problems .

*
Suppose there is only one player , All weather conditions are known in advance throughout the game , Try to give the player's optimal strategy in general . Solve the “ The first level ” and “ The second level ”, And fill in the corresponding results respectively Result.xlsx.
* Suppose there is only one player , Players only know the weather of the day , On this basis, the action plan of the day can be decided , Try to give the best strategy for players in general , And the “ The third level ” and “ The fourth level ” Discuss in detail .
* existing Players , They have the same initial funding , And start from the starting point at the same time . If one day any of them Players from the zone A Walk to area B(
), Then the amount of resources consumed by any one of them is equal to that of the base consumption times ; If one day any of them Players mining in the same mine , Then the amount of resources consumed by any one of them is equal to that of the base consumption
times , And the funds that each player can obtain through mining in one day is the basic income ; If one day any of them Players buy resources in the same village , The price of each case is the base price
times . In other cases, the amount and price of resources consumed are the same as those of single player games .
(1) Suppose that all weather conditions are known in advance throughout the game , Each player's action plan should be in the
Days are determined and cannot be changed thereafter . Try to give the strategy that the player should adopt in general , And the “ The fifth level ” Discuss in detail .
(2) Suppose all players know only the weather conditions of the day , From
Tianqi , After the end of the day's action, each player knows the plan of action and the amount of remaining resources of other players , Then determine their action plans for the next day . Try to give the strategy that the player should adopt in general , And the “ The sixth level ” Discuss in detail .
notes 1: In the attached map , Two areas with a common boundary are called adjacent , Two regions with only common vertices and no common boundary are not considered adjacent .

notes 2:Result.xlsx Amount of remaining funds in ( Surplus water , Food surplus ) Refers to the amount of funds after all the resources required for the day have been consumed ( Water quantity , Amount of food ). If there is any purchase on the same day , It refers to the amount of funds after the purchase ( Water quantity , Amount of food ).

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