Abstract In this paper, a model is developed based on the conditions and requirements of the problem, this is a multivariate linear programming model, and by solving this model, the problem is solved in its entirety.
The objective function of this profitability is additive and the total sales remain constant if the supply of agricultural products exceeds the demand, i.e., if all the agricultural products can be sold. Therefore, the algorithm of minimum transportation cost can be directly applied to find the maximum profitability problem of this base.
The modeling process is generally divided into eight parts: 1. restatement of the problem; 2. model assumptions; 3. notation and textual description; 4. problem analysis; 5. modeling; 6. model solution; 7. analysis of the results of the model; and 8. evaluation of the model and further discussion.
In the modeling, the problem of preservation of agricultural products is not considered, and the problem of costs other than transportation costs is not considered.
To develop a produce transportation plan is to arrange for the sale of eight vegetables to six markets.
In a linear programming model, profit earned = total sales - total transportation costs. The total sales remain constant so that the total transportation cost is simply considered. Since the objective function of this model is multivariate linear, the resulting obtained profit = total sales - total transportation costs = $1505,885.
The model is analyzed in the paper with certain results and has a wide applicability.
Finally, the problem is y discussed,
Keywords Transportation scheme Model assumptions Problem analysis Establishment of the model Analysis of the results of the model Evaluation and discussion of the model
1. Restatement of the problem
Direct topic
2. Model assumptions
(1) In the formulation of the plan for transporting agricultural products, the data is accurate to 1 units, i.e., to the ton, this assumption guarantees a more accurate result in theory.
(2) Damage to vegetables during transportation is not considered.
(3) Costs other than freight during transportation of vegetables are not to be considered.
3. Symbols and text description
Y denotes the profit gained from the sale of vegetables at the base;
i=1,2..., 8 where 1 denotes cabbage, 2 denotes potato, 3 denotes tomato, 4 denotes bean curd, 5 denotes cucumber, 6 denotes pumpkin, 7 denotes eggplant, 8 denotes zucchini;
j=1,2 , ..., 6 where 1 denotes market A, 2 denotes market B, 3 denotes market C, 4 denotes market D, 5 denotes market E, and 6 denotes market F;
denotes the transportation of the ith (i=1, 2, ..., 8) vegetable to the jth (j=1, 2, ..., 6) market in the total amount.
4. Problem Analysis
Developing a produce transportation plan is a scheme for arranging the sale of vegetables in eight to six markets. The objective is to make the most profit. And from the data given in the question, it is clear that the supply of vegetables exceeds the demand, so the base of the vegetables can be sold out and profit, so the total income of the base is the sale of all the vegetables after the proceeds of the transportation program has nothing to do with. Therefore, to maximize profit, the transportation plan for the produce should be adjusted. In addition, the table shows that the supply of produce exceeds the demand, so the transportation plan is limited by the supply of the base and by the demand in the market,
Use the linear programming model.
5. Modeling
Basic model
Decision variables: let the total amount of the ith agricultural product transported to the jth market be .
Objective function: let the profit is Y dollars. From the title to get
MaxY=400x11+400x12+400x13+400x14+400x15+400x16-80x11-130x12-150x13-120x14-100x15-110x16+320x21+320x22+320x23+320x24+ 320x25+320x26-65x21-105x22-120x23-100x24-80x25-85x26+510x31+510x32+510x33+510x34+510x35+510x36-100x31-165x32-170x33-140x34-120x35- 130x36+300x41+300x42+300x43+300x44+300x45+300x46-70x41-110x42-125x43-105x44-85x45-90x46+230x51+230x52+230x53+230x54+230x55+230x56- 95x51-160x52-165x53-135x54-115x55-125x56+650x61+650x62+650x63+650x64+650x65+650x66-60x61-100x62-120x63-100x64-80x65-85x66+500x71+ 500x72+500x73+500x74+500x75+500x76-90x71-150x72-160x73-130x74-110x75-120x76+260x81+260x82+260x83+260x84+260x85+260x86-90x81-160x82 -165x83-130x84-115x85-120x86.
Constraints:
Supply of raw materials The total amount of various agricultural products transported must not exceed the supply, and by the supply exceeds the demand, so the supply of agricultural products can be sold in full, i.e.
x11+x12+x13+x14+x15+x16=826
x21+x22+x23+x24+x25+x26=594
x31+x32+x33+x34+x35+x36=600
x41+x42+x43+x44+x45+x46=356
x51+x52+x53+x54+x55+x56= 423
x61+x62+x63+x64+x65+x66=890
x71+x72+x73+x74+x75+x76=600
x81+x82+x83+x84+x85+x86=500
Market Demand The transportation of all kinds of agricultural commodities to each market shall not exceed the demand for the respective agricultural products in the respective markets, i.e.
x11<=160;x12<=130;x13<=200;x14<=150;x15<=140;x16<=180;
x21<=60;x22<=180. x23<=160;x24<=100;x25<=20;x26<=130;
x31<=100;x32<=140;x33<=200;x34<=60;x35<=80;x36<=90;
x41<=70;x42<=90;x43<=140;x44<=100;x45<=40;x46<=80;
x51<=50;x52<=100;x53<=130;x54<=90;x55<= 90;x56<=70;
x61<=200;x62<=210;x63<=130;x64<=100;x65<=240;x66<=150;
x71<=120;x72<=150;x73< ;=90;x74<=150;x75<=100;x76<=90;
x81<=60;x82<=90;x83<=150;x84<=140;x85<=100;x86<=80.
Non-negative constraints None of the constraints can be for negative values, that is, >=0.
6. Model solution
6.1 Algorithmic thinking
This topic is the use of linear programming to achieve, the algorithm is relatively simple and clear, by finding the total revenue from the sale of agricultural products minus the total cost of transportation is equal to the profit obtained, and through linear programming to take the optimal solution.
6.2 model solution
According to Exhibit 4, the transportation program is:
Total amount transported to a market (unit; tons)
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