Finding Mathematical Modeling Problem D Answers to Natural Casing Matching Problems

Question D: Natural casing matching problems.

The production and processing of natural casings (hereinafter referred to as casings) is a traditional industry in China, and its export volume ranks first in the world. After the casing is cleaned and organized, it is divided into small sections (raw materials) of varying lengths and then assembled into the process. The traditional production method relies on manual labor, while measuring the length of raw materials while mental arithmetic, and assembling the finished products (bundles) of raw materials according to the specified number and total length.

Raw materials are divided into bins according to length, usually in meters, such as: meters are calculated by 3 meters, meters are calculated by meters, and the rest are so on. Table 1 shows the specifications of several common finished products, the length is in meters, indicating that there is no upper limit, but the actual length is less than 26 meters.

Table 1 Specifications of finished products.

Minimum length Maximum length Number of roots Total length.

In order to improve production efficiency, the company plans to change the assembly process, first measure all raw materials, and establish a raw material list. Table 2 is a description of a batch of raw materials.

Table 2 Description of raw materials.

Length root number 43 59 39 41 27 28 34 21

Length root number 24 24 20 25 21 23 21 18

Length root number 31 23 22 59 18 25 35 29

Length root number 30 42 28 42 45 49 50 64

Length root number 52 63 49 35 27 16 12 2

The length of the root number 0 6 0 0 0 1

According to the above description of finished products and raw materials, a raw material matching scheme is designed, and workers produce according to this scheme "according to the prescription".

The company has the following specific requirements for the matching scheme:

1) For a given batch of raw materials, the more bundles of finished products loaded the better;

2) For the scheme with the same number of finished product bundles, the more finished products with the shortest length and longest length, the better the scheme;

3) In order to improve the utilization rate of raw materials, the total length is allowed to have an error of meters, and the total number of roots is allowed to be less than the standard by 1;

4) If there is a surplus of raw materials corresponding to a certain specification, it can be downgraded. For example, the raw material with a length of 14 meters can be bundled with a length between meters, and the finished product belongs to the specifications of rice;

5) In order to preserve the freshness of food, it is required to generate a plan within 30 minutes.

Please establish a mathematical model of the above problems, give a solution method, solve the actual data given in Table 1 and Table 2, and give a matching scheme.

Solve with an optimization model.

We do it too, hahahaha.