2023 Higher Education Society Cup National Mathematical Modeling Competition (Question D) In-depth Analysis | Mathematical Modeling Complete Code + Full Analysis of the Modeling Process

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Question one

In order to solve problem 1, we need to first establish a mathematical model to determine the reasonable number of breeding rams and foundation ewes on the farm, and estimate the range of annual slaughter sheep numbers. The following is the modeling idea for question 1:

Define variables:

1. Record the number of breeding rams as G.
2. Let the number of basic ewes be M.

Objective function: Our goal is to maximize the annual number of sheep slaughtered, that is, to maximize the number of lambs slaughtered each year.

Restrictions:

1. The ratio of breeding rams to basic ewes: According to the problem description, the ratio of breeding rams to basic ewes is generally not less than 1:50, so we can get the following constraints:

   G >= M / 50

2. Sheep pen requirements: Sheep at different stages only need different sheep pens. According to the problem description, we can list the following constraints:

   • Number of non-mating pens required: (M / 14) + (G / 4)
   • Number of sheep pens required during natural mating period: (M / 14)
   • Number of gestation pens required: (M / 8)
   • Number of lactation sheep pens required: (M / 6)
   • Number of sheep pens required during fattening period: (M / 14)
   • The number of sheep pens required for the rest period: (M / 14)

3. Time constraints: According to the problem description, we can calculate the time requirements of each stage:

   • Natural mating period: 20 days
   • Pregnancy period: 149 days
   • Lactation period: 40 days
   • Lamb fattening period: 210 days
   • Rest period of ewe during empty pregnancy: 20 days

4. Sheep life cycle constraints: We need to ensure that the state transition of each ewe within a life cycle is reasonable. The life cycle of a ewe includes: natural mating, pregnancy, childbirth, lactation, fattening, and rest during pregnancy. Therefore, we need to ensure that the status transition of the ewe is reasonable based on time constraints. For example, after natural mating, a ewe needs to go through various stages such as pregnancy, delivery, lactation, fattening, and rest.

5. Constraints on the number of sheep pens: The farm currently has 112 standard sheep pens.

6. Slaughter quantity constraint: According to the problem description, the farm hopes to sell no less than 1,500 sheep every year.

Synthesis of objectives and constraints: Based on the above objective function and constraints, we can use mathematical methods such as linear programming or integer programming to solve the problem of maximizing the annual number of sheep for slaughter. At the same time, we can also adjust the number of breeding rams and foundation ewes to meet the requirement of no less than 1,500 sheep for slaughter.

This mathematical model will allow you to determine the reasonable number of breeding rams and foundation ewes on a farm under given conditions and estimate the range of annualized sheep numbers, as well as the shortfall in existing standard pen numbers. You can solve this problem using linear programming software or a programming language such as Python's PuLP library.

Question 2

Modeling ideas:

1. We divided the problem into multiple time periods, each representing a different stage in the breeding process. Typical time periods include the natural mating period, pregnancy period, lactation period, fattening period, and empty pregnancy period.

2. We need to determine the number of breeding rams and foundation ewes for each time period and how to allocate them to different pens.

3. We need to consider factors such as the mating ratio of breeding rams to foundation ewes, pregnancy rate, number of lambs per litter, and lamb mortality rate.

4. Our goal is to maximize the annualized number of sheep slaughtered while ensuring compliance with pen capacity constraints.

Formulas and constraints:

The following is the specific formula and LaTeX code used to establish the linear programming model for question 2:

1. Define variables:

   • $G_t$: The number of breeding rams in the $t$th time period.
   • $M_t$: The base number of ewes in the $t$th time period.

   Among them, $t$ represents different time periods.

2. Objective function:

   Maximize the annual number of sheep slaughtered, that is, maximize the sum of the number of sheep slaughtered in all time periods:

   

3. Restrictions:

   • Ratio constraints of breeding rams and foundation ewes:

     

   • Sheep pen capacity constraints (demand for various types of sheep in each time period):

     • Non-mating period requirements:

       

     • Needs during the natural mating period:

       

     • Needs during pregnancy:

       

     • Lactation needs:

       

     • Needs during the fattening period:

       

     • Needs during the rest period:

       

   • Breeding sheep mating and breeding constraints:

     • Ewe conception rate: Assumed to be $0.85$.
     • Pregnancy period: Assumed to be $150$ days.
     • Number of lambs per litter: Assumed to be $2.2$.
     • Lamb mortality rate: assumed to be $3%$.

     With these constraints, the number of breeding rams and foundation ewes, as well as the number of lambs, can be determined for each time period.

Code:

Question three:

Modeling ideas:

1. Introducing random parameters: First, we need to introduce some random parameters, which represent uncertainty factors, such as the success rate of breeding sheep, the number of lambs delivered, lamb mortality, etc. These parameters should be modeled based on the probability distribution of the actual data.

2. Simulate multiple scenarios: We can use Monte Carlo simulation or random sampling methods to generate multiple possible farm scenarios, each with different parameter values. For each scenario, we can use the model in question one or two to calculate the corresponding production schedule and space requirements.

3. Statistical results: For each scenario, we can count key indicators such as the annualized number of sheep slaughtered and the demand for sheep pens. This will give us a distribution of results under uncertainty.

4. Risk analysis: Finally, we can analyze these statistical results to understand the risk and uncertainty levels of farm operations under different uncertainty conditions. This can include calculating risk measures such as value-at-risk (VaR) or conditional value-at-risk (CVaR) to assess possible losses.

Formulas and models:

In modeling, we need to consider the following uncertainties:

1. Breeding success rate: as a probability distribution parameter, may differ in each scenario.

2. Number of lambs delivered: The number of lambs per litter can be generated based on a distribution.

3. Lamb mortality: Consider generating the survival probability of each lamb.

4. Breastfeeding time: Consider changes in breastfeeding time under different scenarios.

5. Other factors: Other uncertainties can also be considered, such as disease outbreak rates, weather conditions, etc.

When simulating, we can use a random number generator in each scenario to simulate different values ​​for these parameters. For each scenario, we ran the model from question one or question two to calculate the corresponding production schedule and sheep pen requirements.

Finally, by analyzing the results of multiple scenarios, we can obtain information about the optimal production plan and corresponding risk assessment under uncertainty conditions.

Complete ideas + articles: In-depth analysis of the 2023 Higher Education Society Cup Mathematical Modeling National Competition (Question D)