The production and processing of natural casing (hereinafter referred to as casing) is a traditional industry in China, and its export volume ranks first in the world. The cleaned casing is divided into small pieces (raw materials) with different lengths and enters the assembly process. The traditional production method relies on manpower, while measuring the length of raw materials and calculating in the mind, assembling raw materials into finished products (bundles) according to the specified quantity and total length.

Raw materials are graded according to length, usually 0.5m as a grade, for example, 3-3.4m is 3m, 3.5m-3.9m is 3m, and so on. Table 1 shows the specifications of several common finished products. The unit of length is meters. ∞ means there is no upper limit, but the actual length is less than 26 meters.

Table 1 finished product specification table

Minimum length, maximum length, number of elements and total length.

3 6.5 20 89

7 13.5 8 89

14 ∞ 5 89

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

Table 2 Description of raw materials

Length 3-3.4 3.5-3.9 4-4.4 4.5-4.9 5-5.4 5-5.9 6-6.4 6.5-6.9

Root number 43 59 39 465 438+0 27 28 34 265 438+0

Length 7-7.4 7.5-7.9 8-8.4 8.5-8.9 9-9.4 9.5-9.910-10.410.5-10.9.

Root number 24 24 20 25 2123 2118

Length11-1.41.5-1.912-12.4/.

Root number 3 1 23 22 59 18 25 35 29

Length15-15.415.5-16-16.5-16.9 65438+.

Number of roots 30 42 28 42 45 49 50 64

Length19-19.419.5-19.9 20-20.4 20.5-20.9 21.421.5-2/kloc-.

Number of roots 52 63 49 35 27 65 438+06 65 438+02 2

Length 23-23.4 23.5-23.9 24-24.4 24.5-24.9 25-25.4 25.5-25.9

Root number 0 6000 1

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

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, the better;

(2) For the scheme with the same number of bundles of finished products, the more finished products with the shortest 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 0.5m, and the total number of roots is allowed to be less than the standard1;

(4) Raw materials corresponding to a certain specification, if surplus, can be degraded and used. For example, raw materials with a length of 14m can be bundled with those with a length of 7- 13.5m, and the finished product belongs to the specification of 7- 13.5m;

(5) In order to keep the food fresh, it is required to come up with a plan within 30 minutes.

Please establish the mathematical model of the above problems, give the solution, and solve the actual data given in table 1 and table 2, and give the collocation scheme.