Abstract: According to the meaning of the problem and the actual situation in real life, ignoring some secondary factors, this paper puts forward a mathematical model to solve the problem of fish farming. Four basic fish farming modes are described and designed from several simple aspects:
Mode ⅰ: basic culture mode, which is bought once a year and put a certain amount of fish to make the fish grow into adult fish;
Because of the complexity and variability of fish farming, some rearing factors are ignored. Linear programming and dynamic programming models are applied to solve the problem of fish farming.
Keywords: fish farming model, dynamic programming, linear programming, maximum profit
Paper text:
⑴ Question: Suppose there is a pond in a certain place, and its water surface area is about 100× 100㎡, which is used to cultivate certain fish. Under the following assumptions, design a three-year fish culture plan that can make greater profits.
① The living space of fish is1kg/㎡;
② Each fish 1kg needs 0.05kg of feed every day, and the market fish feed price is 0.2 yuan /kg.
③ The fry price is neglected, about 500 fry per 1kg;
④ Fish grow all the year round, and the daily growth weight is directly proportional to the fish's own weight. After 365 days, it became an adult fish, weighing 2 kg.
⑤ ignoring the reproduction and death of fish in the pond;
6. If m is the weight of the fish, the price of the second fish is:
Q = 0 yuan/kg m.
Q = 6 yuan/kg 0.2 ≦ m.
Q =8 yuan/kg 0.75 ≦ m.
Q = 10 yuan/kg 1.5 ≦ m.
⑦ Only fry can be put into the pool.
(2) problem analysis:
This paper mainly designs an optimal fish culture scheme. We know the area of fish ponds, the storage space of fish, the daily feed required per 1kg fish, the time for fish to grow into mature fish and the prices of fish with different qualities. We link the price of fish with the culture time of fish, construct a price system, draw the growth curve of fish, and analyze the value orientation of fish to consider and design the optimal fish culture scheme. However, due to the complexity of fish farming, some factors affecting fish farming are ignored. Linear programming and dynamic programming models are applied to solve the problem of fish farming.
⑶ Model hypothesis
Only fry are put in the pond. Regardless of the reproduction and death of fish;
(2) Fish can grow all year round, and the daily growth weight is directly proportional to the fish's own weight. The adult fish was 365 days old and weighed 2kg.
③ The stock space of fish is1kg/㎡; Every 1kg fish needs 0.05kg of feed every day, and the market price of fish feed is 0.2 yuan/kg; The price of fry is neglected, about 500 fry per kilo-0;
(4) Assuming that there is no variation in the growth process of fish, the growth of each fish obeys the growth coefficient.
⑤ Suppose that all the fish are fresh during fishing and can be sold at the price given in the title.
⑥ Assuming that the fish caught every day can be sold normally, there is no fish left.
All landowners stocking fry and fishing are carried out all year round, regardless of time and season.
⑧ The released fry can grow according to the conditions given in the topic without the influence of individual differences, and the released fry can also grow to the same size in the same time.
The prices of fish and feed in the market have not changed for three years.
(4) Model design:
The following symbols are used in this article:
1, A0-the number of fish initially released.
2, β-the proportion of fish gaining weight every day.
3. mt —— the weight of each fish under t-day culture conditions.
4. CT—— the feed cost required for each fish under the condition of T days.
5, m-the total income for three years.
6.w—— the average daily profit generated after each fish is cultured for t days.
7.a—— Number of fry released every day.
8. Qt —— the weight of each fish under t-day culture conditions.
(5) Solution of the model:
Model 1 (basic breeding model)
It is assumed that the fry will be put into the fish pond at one time at the end of the year and sold at one time, and the second and third years will follow the plan of the first year respectively.
According to the capacity of the fish pond, when the fish grow into adult fish, the mass is 2kg, and the storage space of each fish is 1 kg/m2, so the number of fish initially put in is a0, a0= 1000/2=500 (fish). Let the proportion of fish gaining weight every day be t, then there are:
1000/500×( 1+β)365=2000
Simplification can get β=-1.
After calculation, β=0. 1 19 can be obtained.
Under the condition of t-day culture, the weight of each fish is mt =1/500 (1+β) t.
Let ct be the feed cost of each fish under the condition of t-day culture.
CT =/500( 1+β)t×0.05×0.2 =/500( 1+β)t
Let the total income for three years be m:
M= 10×5000×2×3-5000ct×3
Through calculation, it can be concluded that the maximum profit is:
m = 3×( 100000-3 19 18)= 19657.32
Therefore, under the state and conditions of this model, the income of 19657.32 yuan can be obtained in three years.