"Mathematical Model (Fifth Edition)" Study Notes (2) Chapter 3 Simple Optimization Model Chapter 4 Mathematical Programming Model

Chapter 3 Simple Optimization Models

Key words: simple optimization differential method modeling idea

This chapter and the two consecutive chapters of Chapter 4 are all optimization and planning problems, which can be regarded as a kind of problems-the content is also from simple to complex. In Chapter 3, there are mainly a few simple optimization models, which can be solved by solving the function extreme value problem, directly using the differential method. Although models and mathematical calculations are not difficult, we still need to learn and introduce the idea of ​​modeling, result analysis and result interpretation after solving.

3.1 Storage model
Two kinds of discussions are allowed for shortage, and a minimum cost result is introduced in the middle - the economic order quantity formula. Graph the memory function, observe the rules, and interpret the results.

3.2 Timing of selling live pigs
The key points are sensitivity analysis and robustness analysis—this is very important for the optimization model to be practical and effective.

3.3 Fighting forest fires
Highlights are several assumptions made about the form of fire spread, and the actual interpretations that correspond to those assumptions. As long as it is reasonable and justified, it is a good simplified algorithm for practical problems.

3.4 Optimal price
Mainly eliciting marginal income, editorial expenditure, and a well-known law of economics - maximum profit is achieved when marginal income equals editorial support.

3.5 Vascular branches
is a very interesting section. We use mathematical models to study physiological problems. We still only focus on modeling and mathematics, and do not discuss too much physiological knowledge such as the geometric shape of the vascular system, and replace them with reasonable and powerful assumptions.

3.6 Consumer's choice
When a consumer buys two products, how should the money be allocated. The allocation ratio enables him to obtain the optimal ratio of flight attendant consumer equilibrium with the greatest satisfaction, and the key to establishing the consumer equilibrium model is to determine the utility function.

3.7 Iceberg transportation
is also a very interesting problem. Considering various factors and based on some assumptions, this section studies how to transport icebergs to minimize the cost. Among them, the empirical formula is established with actual data, and the second is to assume that the iceberg is spherical, which simplifies the calculation of melting laws and so on.

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Vascular model

Chapter 4 Mathematical Programming Models

Keywords: mathematical programming method lingo/lindo software results in-depth analysis of the number of variables

Constraints, feasible regions, and objective functions constitute what is often called a "mathematical programming" model. This chapter reveals the essence of mathematical programming and the difference between it and traditional optimization mathematical problems: common sense optimization models belong to the category of function extreme value problems, but in reality, more decision variables and constraints are larger, and the optimal solution Problems that are often achieved on the boundary and therefore cannot be solved by traditional "differentiation methods" - hence the introduction of "mathematical programming" methods.

This chapter has a lot of content, but they are all one type of problem. There are several main points:

1. The use of lingo and lindo solutions - there are some unnoticed information in the running results, which can be used as result analysis, and the first two sections describe more;
2. Some details: Express a sentence in a mathematical formula, which is often used as a constraint, such as the formula (19) of p102;
3. The processing of multi-objective planning, the "course selection strategy" of p109 - the basic idea is to form a new goal through weighted combination, so as to turn it into single-objective planning;
4. As in the previous chapters, after solving a problem, the problem is solved, but the analysis is not over - we have to learn the spirit of this further discussion and find that the result is "exactly the same as..." and so on, You might as well ask yourself one more question: "Is this accidental?" Then continue to analyze and draw general conclusions, so that you can often see more scenery and draw conclusions that are more valuable/inspiring, rather than just solving That's all. Such as p109 course selection strategy.
5. Reduce the number of variables, simplify the model and formula (for the sake of simplification, and lingo has restrictions on the number of variables), p115 sales examples.
6. When seeking the optimal solution, in order to reduce the search range and speed up, you can first go to a special case to find a feasible solution, and then let the optimal solution be at least better than it.
   

Production and sales of dairy products

The production and sale of dairy products is a complicated process, which needs to consider multiple factors for mathematical modeling. Here are some possible modeling elements:

1. Production costs: including raw material costs, labor costs, equipment costs, etc. These costs can be expressed mathematically and take into account the cost differences of different products.

2. Production capacity: According to the capacity limitation of the factory, determine the maximum production volume of each dairy product. This can be expressed by constraints such as production time, equipment capacity, etc.

3. Supply chain management: Considering suppliers and distribution channels, build a supply chain network model to determine the best raw material procurement and product distribution strategies.

4. Demand Forecasting: Forecast the demand for different dairy products through historical data or market research. This can be modeled using time series analysis, regression models, etc.

5. Inventory management: Determine reasonable inventory levels and replenishment strategies based on demand forecasts and production plans to avoid excess or stock-out situations.

6. Customer Satisfaction: Consider customer preference and satisfaction with different products, build mathematical models to optimize product mix and pricing strategy to maximize customer satisfaction and market share.

7. Sales forecast: Based on factors such as marketing activities and competition, forecast the sales volume of different dairy products and formulate corresponding sales strategies.

To sum up, the mathematical modeling of production and sales of dairy products involves many aspects such as cost, production capacity, supply chain, demand forecast, inventory management, customer satisfaction and sales forecast. Through reasonable modeling and optimization methods, production efficiency can be improved, costs can be reduced, and sustainable development and market competitive advantages can be achieved.

When it comes to mathematical modeling of production and sales of dairy products, the following factors can also be considered:

8. Quality control: Establish a quality control model to monitor and control quality problems in the production process of dairy products to ensure that products meet standards and customer requirements.

9. Seasonal factors: Consider the impact of different seasons on the demand for dairy products, and the impact of seasonal fluctuations on production planning and inventory management. For example, the demand for ice cream may be higher in summer.

10. Resource optimization: Consider resource utilization efficiency, such as energy and water resources. Mathematical models can be built to optimize the use of resources to reduce production costs and environmental impact.

11. Product mix optimization: Considering the complementarity and substitution between different dairy products, establish a mathematical model to determine the best product mix to maximize sales revenue.

12. Price strategy: By establishing a pricing model, considering market demand, cost and competition, determine the best product pricing strategy to maximize profit or market share.

13. Marketing strategy: Incorporate marketing, promotional activities and advertising into mathematical modeling to optimize marketing strategies and increase product exposure and sales.

14. Risk management: Consider uncertain factors in the production process, such as raw material price fluctuations, demand fluctuations, supply chain interruptions, etc., and establish a risk management model to minimize risks and losses.

Through mathematical modeling and optimization of the above factors, it can help dairy production and sales companies make more scientific and effective decisions, improve production efficiency, reduce costs, increase profits, and meet customer needs.

Unconstrained optimization refers to the problem of finding the maximum or minimum of a function without any constraints. In the production and sales of dairy products, unconstrained optimization can be applied to the following aspects:

1. Minimization of production costs: By optimizing various costs in the production process, including raw material costs, labor costs and equipment costs, etc., to minimize production costs.

2. Maximization of sales profit: Maximize sales profit by optimizing factors such as pricing strategy, product mix and marketing strategy.

3. Maximize production efficiency: optimize the allocation and utilization of production resources to increase production efficiency and maximize the use of available resources.

4. Product quality optimization: By optimizing the parameters and control variables in the production process, the product quality can be optimized.

5. Maximization of production capacity: By optimizing the factory's capacity planning and production scheduling, to maximize production capacity and meet market demand.

When performing unconstrained optimization, various mathematical methods and optimization algorithms can be used, such as gradient descent method, Newton method, genetic algorithm, etc. These methods can help find the maximum or minimum of a function and provide better production and sales decisions.

A Simple Classification of Constrained Optimization

MATLAB provides a powerful optimization toolbox for solving a variety of unconstrained and constrained optimization problems. The toolbox contains a variety of optimization algorithms and functions that can help users find the maximum or minimum of a function.

The following are commonly used functions and algorithms in MATLAB Optimization Toolbox:

1. fminunc: Function for unconstrained optimization problems. It uses gradient-based methods (such as quasi-Newton method, conjugate gradient method) to find the minimum of a function at a local minimum.

2. fminsearch: is also a function for unconstrained optimization, but it does not require gradient information. It uses a global search algorithm such as simulated annealing to find the minimum of the function.

3. fmincon: Function for constrained optimization problems. It can handle equality constraints, inequality constraints, and boundary constraints, and uses algorithms such as the interior point method to solve minimization problems.

4. lsqnonlin: Function for nonlinear least squares problems. It finds the optimal solution by minimizing the residual vector and can handle both equality constraints and inequality constraints.

In addition to the above functions, MATLAB Optimization Toolbox also provides many other functions and tools, such as fminimaxfor minimax problems, patternsearchfor pattern search, gafor genetic algorithms, etc.

Using the MATLAB optimization toolbox, you can solve various optimization problems in the production and sales of dairy products by writing appropriate objective functions and constraints, and selecting appropriate optimization algorithms, so as to improve efficiency, reduce costs or maximize profits.
Here is a simple example of solving an unconstrained optimization problem using the MATLAB Optimization Toolbox:

Suppose there is an objective function f(x) = x^2 + 3x + 2, and our goal is to find the value of x that makes the function obtain the minimum value.

% 定义目标函数
f = @(x) x^2 + 3*x + 2;

% 调用优化函数 fminunc
x0 = 0;  % 初始猜测值
x_opt = fminunc(f, x0);

% 输出最优解和最优值
fprintf('最优解 x = %f\n', x_opt);
fprintf('最优值 f(x) = %f\n', f(x_opt));

Running the above code, MATLAB will use the default quasi-Newton method (BFGS) to find the x value that minimizes the objective function. In this example, the minimum occurs at x ≈ -1.5 and the optimum is f(x) ≈ -0.25.

Please note that this is just a simple example, and optimization problems in real applications may be more complex, and may require the definition of more variables, constraints, and objective functions. You can choose the appropriate optimization function and algorithm to solve your problem according to the specific problem requirements and constraints.