Operational research--linear programming and simplex method solving

1. The concept of linear programming

Linear programming is to study the extremum problem that makes a linear objective function take maximum (or minimum) under a set of linear inequalities or equality constraints .

2. Standard form of linear programming

features : The objective function is great , the equality constraint , and the variable is non-negative .

Make

The matrix expression of the linear programming standard form is:

Conventions:

How to make a standard shape:

(I) The objective function is greatly realized , that is, the order;

(II) Constraints are inequalities

The constraint is "" Inequality, then a non-negative relaxation variable is added to the left end of the constraint condition;

If the constraint is "" Inequality, a non-negative relaxation variable is subtracted from the left end of the constraint condition.

(III) If there is an unconstrained variable ,

3. simplex Method solution

(I) to the standard form (requirements), to determine the initial base, the establishment of an initial simplex table (assuming that a matrix exists in the unit matrix );

(II) If, then, the optimal solution has been obtained and stopped. Otherwise, move on to the next step;

(III) If in, existence, and, then, no optimal solution, stop. Otherwise, move on to the next step;

(IV) by, determined to swap in variables, by rule

can be determined to swap out variables;

(V) iterating over the main element

is about to iterate,

The new simplex table is obtained by replacing the simplex table column.

Repeat (Ⅱ) ~ (Ⅴ).

4. The simplex method solves the example

Two-stage method

In the first stage , we find the feasible solution of the initial base : Adding the artificial variable to the original linear programming problem, making the constraint matrix appear the unit sub-matrix, then using the sum of these artificial variables as the objective function, constructs the following model:

The solution to the above model (simplex method), if w=0, indicates that there is a basic feasible solution to the problem, the second stage can be carried out; otherwise , there is no feasible solution to the original problem, stop the operation.

Second Stage: in the final table of the first stage, the artificial variable is removed, and the coefficients of the objective function are replaced by the objective function coefficients of the original problem, as the initial table of the second stage calculation (calculated by Simplex method).

Cases:

First Stage

Phase II

the optimal solution of ∴ is ( 4 1 9) 0 0 ), the target function Z =–2

degradation: that is, the calculated θ(used to determine the swap out variable) has more than two of the same minimum ratio, resulting in the next iteration by one or several base variables equal to zero, which is degenerate (will produce degenerate solutions).

Although the variable is arbitrarily swapped out, the value of the target function is constant, but at this time the different bases are expressed as the same vertex, and the special case is never the optimal solution. This should be done as follows:

⑴: When there are more than two maximum values, the lowest non-base variable is chosen as the swap-in variable;

⑵. When there are more than two minimum values in θ , the base variable with the lowest subscript selected is the swap-out variable.

Reference documents:

[1] "Operations research" textbook writing group. Operations. Beijing: Tsinghua University Press.

Operational research--linear programming and simplex method solving