Optimization problem is a common problem in engineering technology, economic management and scientific research, and plays an important role in solving extreme value problems. Zero-one programming is also a common mathematical tool, which can effectively represent the effectiveness of things. This paper is a problem of great practical significance. With the development of the information age, there are various ways for college students to accept knowledge. Newspapers, magazines and books have always won the favor of college students to varying degrees, and the products of this era of e-books came into being. For this problem with great practical significance, there should be an easy-to-understand model to make people look more acceptable.
Considering the establishment of a point of sale to maximize the number of people supplying books, it is necessary to establish an optimization model under constraints and choose whether there is a sales relationship between the two places as its decision-making variable, which will make people easy to understand and recognize. By establishing a linear programming model and applying Linggo software, the optimal solution is obtained. The relationship between B and E is to set up sales outlets in B(E) and sell books to E(B), and the relationship between D and G is to set up sales outlets in D(G) and sell books to G(D), but the number of college students is the largest,177,000. There are many ways to choose the optimal solution, which has great flexibility.
This model is suitable for only considering the choice of the address with the largest number of people, and has strong practicability and universality.
Optimization Model of Linggo for Zero-One Programming of Book Selling Problem
1. Restatement of the problem
A publishing house is going to supply books to college students in seven districts of a certain place. The number of college students in each district is as shown in the figure (unit: thousands). The publishing house is going to set up two book agency sales points in the city, and each agency can only sell books in the local area and an adjacent area. Publishers know that the larger the number of people covered by books, the greater the profits they will get, so they should choose two suitable sales agents to maximize the coverage. What needs to be solved now is to choose the appropriate agent sales point.
2. Problem analysis
Books are the ladder of human progress, and the problem of selling books is of general concern to people. In recent years, with the development of science and technology, the emergence of e-books and online bookstores, people read more and more ways, and the sale of books has attracted more and more attention from sellers. How to choose the point of sale, in order to make the book sell the most and the seller get the maximum benefit, this is the key to the problem.
In order to choose the best area among many candidate areas and formulate the best planning scheme, it is obviously necessary to establish an optimization model and the possibility of choosing or not choosing each area. This requires the use of 0- 1 planning mode to establish two sales agents. In order to get an optimal scheme, the publishing house needs to design a reasonable and effective investment scheme, provided that the following conditions are met:
1. Only two sales agents can be established.
2. Each sales agent can only sell books to college students in the local area and neighboring areas.
In the above requirements, the connection between every two adjacent areas indicates that the sales agency relationship has been established in this area. There are two options for this sales agency relationship: to establish or not to establish. Obviously, each region can only choose one sales or agent, and the optimal scheme is the connection of the maximum and the second largest option value. The above scheme constraints are transformed into constraints, the objective function and constraint decision scalar are transformed into mathematical symbols, and the optimal solution is obtained by LINGGO software.
Description of 3 symbols
Symbol representation symbol description
An area of 34,000 people.
B 290,000 population area
C 42 One 1000 people area
D 2 1 000 people area
E 560,000 people area
F 18 thousand population area
G 7 1 000 people area
X 1 AB establishes the entrustment relationship between the two regions.
Establish a consignment relationship between x2 AC and other regions.
X3 has established a consignment relationship between these two regions.
X4 BD has established a consignment relationship between the two regions.
X5 CD has established a consignment relationship between the two regions.
X6 DG has established a consignment relationship between the two regions.
X7 DF establishes a consignment relationship between two regions.
Establish x8 DE's consignment relationship with other regions.
X9 EF establishes a consignment relationship between two regions.
X 10 FG establishes a consignment relationship between the two regions.
X 1 1 BC has established a consignment relationship between the two regions.
The number of college students that Q can supply.
4. Question hypothesis
When choosing the agent sales point, only the total number of people in the region and surrounding areas is considered, and people's immigration, consumption power and people's needs are not considered;
1. There are only two sales agents, and each sales agent can only sell books to the region and its neighbors.
2. There is no turnover in seven sales areas.
The supply of books far meets the needs of students.
The sales agent sells books to students in two regions at the same price.
5. Do not consider reducing the purchase of books due to the problem of students' travel expenses in neighboring areas.
6. The sales of books are in direct proportion to the number of people.
7. Everyone's spending power is equal.
5. Establishment of the model
Decision variable: the entrustment relationship between ABCDEFG and the two places Xi (I = 1, 2,3 …10).
Xi= 1 indicates the establishment of entrustment relationship. Xi=0 indicates that no delegation relationship has been established.
Objective function: the number of college students available is Q thousand; Then q = 63 * x1+76 * x2+85 * x3+50 * x4+63 * X5+92 * X6+39 * x7+77 * x8+74 * x9+89 * x10+766.
Constraint condition
1. Only two sales agents can be established.
x 1+x2+x3+x4+X5+X6+x7+x8+x9+x 10 = 2；
2. There can only be one consignment relationship with.
x 1+x2 & lt； = 1;
There can only be one entrustment relationship with B.
x2+X5+x 1 1 & lt； = 1;
There can only be one entrustment relationship with C.
x 1+x3+x4+x 1 1 & lt； = 1;
There can only be one entrustment relationship with d.
x4+X5+X6+x7+x8 & lt； = 1;
There can only be one deposit relationship with E, namely
x3+x8+x9 & lt； = 1;
There can only be one entrustment relationship with F.
x7+x9+x 10 & lt； = 1;
There can only be one consignment relationship with G, namely
X6+x 10 & lt； = 1;
To sum up:
max Q = 63 * x 1+76 * x2+85 * x3+50 * x4+63 * X5+92 * X6+39 * x7+77 * x8+74 * x9+89 * x 10；
x 1+x2+x3+x4+X5+X6+x7+x8+x9+x 10 = 2；
x 1+x2 & lt； = 1;
x2+X5+x 1 1 & lt； = 1;
x 1+x3+x4+x 1 1 & lt； = 1;
x4+X5+X6+x7+x8 & lt； = 1;
x3+x8+x9 & lt； = 1;
x7+x9+x 10 & lt； = 1;
X6+x 10 & lt； = 1;
6. Solve the model
Enter the following code with lingo, as shown in appendix 1. Running LINDO teaching software can get the optimal solution of book sales problem, that is, the optimal scheme of establishing consignment relationship. The screenshot is as follows:
Target value: 177.0000
Variable value reduces cost
X 1 0.000000 22.00000
X2
X3 1.00000
X4 0.000000 38.00000
X5 0.000000 25.00000
X6 1.00000 0.000000
X7 0.000000 49.00000
X8 0.000000 1 1.00000
X9 0.000000 1 1.00000
x 10 0.00000 0.000000
x 1 1 0.000000 0.000000
It can be seen that the relationship between B and E is to set up a sales outlet in B(E) and sell books to E(B), and the relationship between D and G is to set up a sales outlet in D(G) and sell books to G(D), but the number of college students is the largest,177,000. (See Appendix 2 for detailed results)
However, considering the number of people in the region and the cost of buying books in reality, sales agents should be established in crowded areas. There are 56,000 people in Zone B and Zone E, and there are 7 1000 people in Zone D and Zone G, so it is better to set up two sales agents in Zone E and Zone G..
7. Evaluation and popularization of the model
Looking at the map of this area, you can roughly know that you should choose the area with the largest number of people as the sales point. Under the assumption in the question, choose the area with the largest number of people as the sales point, covering most of the population. The establishment of this model abstracts the problem of choosing a sales agent with good mathematical knowledge, which makes our choice more comprehensive, in-depth and orderly than subjective and blind. Choosing the least variables to consider the problem simplifies the analysis of model establishment. This is also the biggest drawback of the model, and the authenticity of the data is greatly limited, which is not conducive to practical application. Although there are many hypothetical variables, people can easily understand them.
There are too many assumptions in the question, some of which are divorced from reality, taking into account the transportation distance between sales points, the convenience of transportation, the book consumption of students during school, and the consumption of different groups of people.
8. Reference
1 Jiang Qiyuan Xie Jinxing Alfred Mathematical Modeling (Third Edition) Higher Education Press 2003
2/