[Management Operations Research] Chapter 6 | Transportation Problems (1, Mathematical Model and Properties of Transportation Problems)

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introduction

In the previous learning process, we were exposed to more general linear programming problems. However, as people's demand for transportation—moving people or objects from one spatial location to another spatial location becomes more and more complex, we often encounter this type of problem in real life, such as knowing the output of each place of origin 1. The sales volume of each sales place and the unit freight between the place of production and the place of sale, how to organize the transportation plan so that the transportation volume is as large as possible and the total freight is as small as possible.

For this kind of linear programming problem, due to its relatively fixed model structure, it will be cumbersome and complicated to solve it with the previous general method. Therefore, we need another feasible and simple method to solve.

The homework method on the table is a simple and effective method, and it is also the focus of our study.

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1. Mathematical model and characteristics of transportation problems

1.1 Mathematical model of transportation problem

Before learning how to apply the table method, let's understand the mathematical model of the transportation problem.

To give a typical example, suppose an item has $m$ production areas ($A_1,A_2,\dots ,A_m$), the yields of each origin are $a_1,a_2,\dots ,a_m$; have $n$ dots ($B_1,B_2,\dots ,B_n$), the sales capacity of each sales area is $b_1,b_2,\dots ,b_n$; Assume from $The\; i$ place of origin goes to thejjthThe freight rate of the unit item in the sales place of$j$$c_{ij}$, how to transport these items to minimize the total shipping cost?

任$x_{ij}$Real estate $A_i$Shipped to destination $B_j$If the total output of all production areas is equal to the total sales volume of all sales areas, this problem is called the production-sales balance transportation problem, otherwise, it is called the production-sales imbalance transportation problem.

For the production-sales balance problem, its mathematical model can be expressed as:

ex $The\; meaning\; of\; the\; n$ constraints is that the quantity of items shipped to a sales place is the same as the sales quantity of the sales place, and the middle$The\; meaning\; of\; the\; m$ constraints is that the quantity shipped from a production place to each sales place is the same as the output of the production place, and finally$m\times n$ constraints guarantee non-negative conditions.

If the simplex method is used to solve it, many artificial variables need to be introduced, which is very troublesome.

1.2 Characteristics of the transportation problem

Let the decision variable $$x_{ij}^0=\frac{a_i{b}_j}{Q}(i=1,2,\dots ,m;j=1,2,\dots ,n)$$ 其中 $Q={\sum }_{i=1}^m{a}_i={\sum }_{j=1}^n{b}_j$. Then $x_{ij}^0$is a feasible solution of the transportation problem; at the same time, the objective function of the model has a lower bound, so the transportation problem must have a finite optimal solution.

Analyzing the structure of the mathematical model in 1.1, and sorting out the constraint equations, the transportation problem has the following characteristics:

• The constraint coefficient matrix is ​​equal to 0 or 1;
• Each column of the constraint coefficient matrix has two non-zero elements, indicating that each variable is in the front nnOccurs once in$n$$also\; appears\; once\; in\; the\; m$ constraint equations.

For the production-sales balance transportation problem, it also has the following characteristics:

• All constraints are equality.
• The sum of the output of each production area is equal to the sum of the sales volume of each sales area.

1.3 Solution of transportation problem

The solution to the transportation problem represents a transportation scheme where each variable $x_{ij}$The value represented by $A_i$Shipping to $B_j$number of items.

It is conceivable to solve the transportation problem in the same way as the general linear programming problem. For example, iterative method is used, that is, a basic feasible solution is found first, and then the optimality of the solution is checked; if it is not the optimal solution, iterative adjustment is performed to obtain A better solution is obtained; continue to test and adjust improvements until an optimal solution is obtained.

In order to solve the transportation problem according to the above ideas, the solution XX obtained by each step of iteration is required$X$ must be a feasible solution, which means:

1. $X$ must satisfy all constraints;
2. The coefficient column vector of the constraint equation system corresponding to the basic variable is linearly independent;
3. The number of non-zero variables in the solution cannot be greater than $(m+n-1)$ because the transportation problem has$(m+n)$ constraint equations, but since total sales are equal to total output, there are only$(m+n-1)$ the constraint equations are linearly independent;
4. In order to make the iteration go smoothly, the number of basic variables is kept as ( m + n − 1 ) (m+n-1) during the iteration$(m+n-1)$ pcs.

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write at the end

As for the specific method introduction, I plan to talk about it separately later.