Question 1: A dairy processing factory uses milk to produce two kinds of dairy products. 1 barrel of milk can be processed into 3kg in equipment A 12 hours, and 4kg in equipment B in 8 hours. According to market demand, all the products produced can be sold. And the profit per kilogram is 24 yuan, and the profit per kilogram is 16 yuan. At present, the processing plant supplies 50 barrels of milk every day, and the total labor time of regular workers is 480 hours every day. Equipment A can process 100 kg at most every day, and the processing capacity of equipment B is unlimited. Try to make a production plan for the site to maximize the daily profit, and further discuss the following three additional issues:

1) If 35 yuan can buy 1 barrel of milk, should this investment be made? How many barrels of milk can I buy every day if I invest?

2) If temporary workers can be hired to increase working hours, what is the maximum hourly wage paid to temporary workers?

3) Due to the change of market demand, the profit per kilogram has increased to 30 yuan. Should the production plan be changed

The goal of this optimization problem is to maximize the daily profit. The decision to be made is the production plan, that is, how many barrels of milk are used to produce A 1 and how many barrels of milk are used to produce A2 (or how many kilograms of A 1 and A2 are produced every day). Decision-making is limited by the supply of raw materials (milk), labor time and processing capacity of equipment A. According to the topic, the decision variables, objective functions and constraints are expressed by mathematical symbols and formulas, and the following model can be obtained.

basic model

Decision variables: suppose that x 1 barrel of milk is used to produce A 1 every day, and x2 barrel of milk is used to produce A2.

Objective function: let the daily profit be Z yuan. Barreled milk can produce 3 x 1 kg, with a profit of 24x3x 1, and x2 barreled milk can produce 4 x2 kg, with a profit of 16×4x2, so z=72x 1+64x2.

Constraints:

The total amount of raw materials (milk) produced by A 1 and A2 shall not exceed the daily supply, that is, x 1+x2≤50 barrels;

The total processing time of A 1 and A2 shall not exceed the total working time of regular workers every day, that is, 12x 1+8x2≤480 hours;

The output of equipment capacity shall not exceed the daily processing capacity of equipment A, namely 3x1≤100;

Non-negative constraints X 1 and X2 cannot be negative, that is, X 1 ≥ 0 and X2 ≥ 0.

To sum up, it can be concluded that

Max z=72x 1+64x2 (1)

South t. x 1+x2 ≤50 (2)

12x 1+8x2≤480 (3)

3x 1≤ 100 (4)

x 1≥0，x2≥0 (5)

This is the basic model of the problem. Because the objective function and constraints are linear with the decision variables, it is called linear programming (LP).