1、 The problem of minimum objective function value (min type)
There are three processing methods, and you can choose any one:
(1) In section 2.1.3, the problem of finding the minimum objective function value is transformed into the problem of finding the maximum objective function value. If the objective function is in the form of min z=∑ cjxj, let z=- z ', so the objective function can be transformed into max z'=- ∑ cjxj, and then solved by simplex method.
(2) Keep the min type of the objective function unchanged, and handle it by judging the test number. The judgment method of the optimal solution is opposite to that of the max type, that is, when all the test numbers cj zj ≥ 0, the optimal solution will be reached, otherwise continue to iterate; In addition, the method to determine the input variable is to select the non basic variable with the smallest test number as the input variable, and the method to determine the output variable is the same as that of the max type.
(3) Change the form of inspection number in the simplex table. That is to say, the test number cj zj in the simplex table is rewritten into the form of zj cj. The determination method of the optimal solution, the determination of the input variable and the output variable is the same as that of the max type.
The equation with two constraints is of "≥" type
The "≥" type constraint equation can be transformed into the "=" type constraint equation by subtracting the redundant variables from the left side of the constraint equation.
At the same time, it is known that the method to determine the initial basic feasible solution is to take the variable corresponding to the unit matrix as the basic variable, but the coefficient of the redundant variable is - 1, which cannot form the unit matrix, that is, the redundant variable cannot be taken as the initial basic variable. If the identity matrix cannot be found in the matrix of the changed constraint equations, the solution is:
The unit matrix is generated by artificially constructing variables, which are called artificial variables. In order not to make the artificial variable affect the value of the objective function, in the problem of finding the maximum or minimum of the objective function, the cj value of the artificial variable is set to a sufficiently small or sufficiently large number, namely - M or M, so that the possibility of the artificial variable entering the optimal solution is very small.
Dual simplex method
Three two stage method
(1) The first stage: construct a new objective function instead of the actual objective function, and solve it with the simplex method.
If the objective function is max type, let the objective function coefficient of the artificial variable cj=- 1, and the objective function coefficient of other variables cj=0;
If the objective function is min type, let the objective function coefficient of the artificial variable cj=1, and the objective function coefficient of other variables cj=0; (No optimal solution is calculated: cj zj>0, aij<0)
(2) The second stage: restore the original objective function and continue to use the simplex method to solve.
The steps to restore the original objective function are:
A、 Remove the columns of artificial variables from the optimal simplex table in the first stage;
B、 Restore the coefficients of the original objective function;
C、 Recalculate the test number and continue to use the simplex method to solve.
Four constraint equation is of "=" type
There are two processing methods, either of which can be selected:
(1) By adding artificial variables to the constraint equation, the coefficient matrix can form a unit matrix, which can be solved by the large M method or two-stage method.
(2) Change the equation into two non equations. For example, x1+2x2=10 can be replaced by two inequalities, x1+2x2 ≤ 10 and x1+2x2 ≥ 10. After this change, relaxation variables can be added, redundant variables can be subtracted, or artificial variables can be added to these two equations, and then they can be solved according to the previous method.
