Optimization of Vegetable Planting and Allocation

In this paper, we aiming at question about vegetable planting. First we use the method of line segment superposition to calculate the minimum distance between 8 vegetable planting bases and 35 sales points’ and every route must through traffic junction. After we get the shortest distance, we change the conditions and establish the relationship between the planting base supply and the demand of each point of sale. According to the above relationship to write lingo program, and get optimal allocation scheme from direct allocation. Then we change the procedure and get the optimal allocation schemes between expanding the planting area and ensuring the vegetable species two circumstance.


Introduction
The data we use is from Jilin Province mathematical modeling contest. Contest provides JG City, vegetable growing base day vegetable supply, "Vegetable sales point daily demand for vegetables" and "daily shortage of compensation standards" and other information.
The 8 vegetable planting bases, through 15 major traffic junctions, the daily delivery of vegetables to the urban area of 35 vegetable sales point. If the vegetable sales point of demand cannot be met, the city government will give a certain shortage of compensation. At the same time, the city government will give the corresponding freight subsidies, based on the number of vegetables grown in the supply of vegetables and the distance, in order to improve the enthusiasm of vegetable cultivation, freight subsidy standard for 0.04 Yuan (ton/km).

Solution of the Shortest Distance Based on the Superposition of Line Segments
To design a transport plan, we must first solve the problem of distance, from planting to the point of sale need to go through 15 traffic junctions, so we need to calculate the shortest distance of each route according to different road conditions. Suppose our vegetable transport only considers the following three transport pathways: Base -point of sale, can be directly get through the information we have; Base -traffic junction -point of sale, calculate two segment distance; Base -traffic junction -point of sale -point of sale, calculate three segment distances.

Transport Scheme Design
The second part has calculated the shortest distance of each transport path, in order to not only can we transport vegetables to point of sales, but also to make the government's shortage of compensation and freight subsidies to the minimum, we consider the following two aspects.
The total cost is the sum of the cost of the freight and the shortage, that is . First consider the freight subsidies ( amount dis tan ce freight freight subsidy standard P = i i ) , and freight and distance is proportional to. Secondly considering the shortage of compensation Finally, according to the requirement of each point of sale, supply of each point of sale and the shortest distance between bases and point of sales, we can get the constrained condition and establish the objective function , use LINGO programming to find the minimum optimal solution.

Model of Freight Subsidy
Freight subsidies can be expressed by the following formula:

Model of Shortage Compensation Standard
The relationship between the actual receive amount and the demand is as follows: The relationship between the total supply and the actual supply of the base: Shortage compensation standard can be expressed by the following formula:

The Optimal Solution of the Total Cost
Using lingo13.0 software programming to solve the total cost, part of the running results are as follows (due to the program and the results are relatively long, so only with the final results): Global optimal solution found.

Based on the Expansion of the Planting Area of the Scheme Design
In order to meet the needs of the residents of the vegetable supply, we consider the situation of expanding the scale of vegetable planting base. Before the implementation of the expansion of planting area, the total supply of original eight bases are 210 (tons / day), but the total demand of all vegetable ales are 360.1(tons / day), so the difference between the total demand and the supply is 90.1 (tons / day), and each vegetable planting base needs add 90.1 (tons / day).
So we get the new constraint conditions of after expanding area, add the constraint conditions to lingo programming of the third part and after some fine-tuning will be able to find the solution, the constraint conditions are as follows: The constraint conditions are incorporated into the lingo13.0 software, and the results are as follows: Global optimal solution found.

Based on Given Vegetable Species of Scheme Design
In order to improve the quality of residents' life, vegetable planting base not only to ensure the total supply of vegetables, but also to meet the needs of the residents of vegetable species. Each vegetable planting base can plant 12 kinds of vegetables, and the demand for each type of vegetable to the point of sale is known. Then, we combine the relationship between the amount of planting base supply and the amount of sale demand, find out the supply source and quantity of each kind of vegetables.

The Sum of All Kinds of Vegetables Supplied by the Base Is Equal to the Amount of the Supply
The above constraints are incorporated into the lingo13.0 software to solve yi R , and the results are shown as follows:

Transport Scheme Design
Observing the supply of these 12 kinds of vegetables, we found that there is at least one source of supply for each. So we decided to according to the type of vegetables, will be the optimal solution to be divided into 12 decomposition (that is, the optimal solution of each vegetable), in order to obtain the 12 after the decomposition of the sum, so as to obtain the final minimum optimal solution.
Take vegetables 1 as an example, planting base can provide vegetables 1 to the various points of sale, set respectively express the vegetables amount provided by the planting base to the point of sales i , we can get the following constraints:  28

Shortage Compensation Standard Can Be Expressed by the Following Formula
We incorporated the above constraints conditions into programming, the minimum optimal value of the total cost of vegetables 1 was solved, and the results are as follows: Global optimal solution found. That, is . In the same way, we can obtain the minimum optimal value of the total cost of the remaining 12 kinds of vegetables.

Model Advantages
6.1.1 Be able to take into account the planting base, traffic junction, point of sale of three aspects of the analysis and calculation, and get the shortest distance; 6.1.2 Reasonable assumptions help us to better solve the problem, and make a lot of problems are better to start; 6.1.3 Through the lingo model to solve the linear programming problem, relatively simple and easy to operate, and the initial value is convenient to change, the model is more flexible.

6.2.1
The shortest distance calculation method has some limitations, may leak to calculate the distance between the two points, Or there is a very individual point is not the shortest distance, can seek a more appropriate way to calculate distance; 6.2.2 The model in the transport of the consideration is unilateral, some stiff, If we want to apply it in practice, the transport subsidies constraints need to become more flexible;