When in tasks is available N-unknown, the area of admissible decisions within the system of the limiting conditions is a convex polyhedron in N-dimensional space. Therefore, it is graphically impossible to solve such problem, here it is necessary to apply a simplex method of linear programming.
It is required to you
- - mathematical reference book
Instruction
1. Display the system of restrictions with a system of the linear equations which differs in the fact that the number of unknown in it are more than quantity of the equations. At a rank of system R choose R unknown. Lead a system by Gauss's method to a look: x1 = b1+a1r+1x r+1+ … + a1nx nx2 = b2+a2r+1x r+1+ … + a2nx n …………………… …. xr = br+ar, r+1x r+1+ … + amx n
2. Attach concrete significance to free variables then count base numbers which values are not negative. If base numbers are values from X1 to Xr, then the solution of the specified system from b1 to 0 will be basic provided that values from b1 to br ≥ 0.
3. At admissibility of the basic decision check it for optimality. If the decision is not that, pass to the following basic decision. At each new decision the linear form will approach an optimum.
4. Make a simplex the table. For this purpose members with variables in all equalities are had to the left part, and free members from variables are left in the right part. All this is displayed in a tabular form where basic variables are specified in columns, free members, X1 … by .Xr, Xr+1 … Xn, and in lines .Xr, Z are displayed by X1 ….
5. Look through the last line of the table and choose among coefficients or the minimum negative number by search of max, or maximum positive by search on min. If there are no similar values, so the found basic solution can be considered optimal.
6. Look through that column of the table which is identical to the chosen positive or negative value the last line. Choose in it positive sizes. If those are not found, then the task of decisions has no.
7. Among the remained coefficients of a column choose for what the ratio of the free member to this element is minimum. You receive the allowing coefficient, and a line at which it is present, will become key.
8. Transfer the basic variable corresponding to a line of the allowing element to the category free, and the free variable corresponding to a column of the allowing element – in category basic. Construct the new table with other names of basic variables.
9. Divide all elements of a key line, except a column of free members, into the allowing elements and again received values. Bring them in a line from the modified basic variable in the new table. The elements of a key column equal to zero are identical always to unit. The column where in a key line zero is found, and a line where in a key column there is zero, in the new table will remain. Write down results of transformation of elements from the old table in other columns of the new table.
10. Investigate options until you find an optimal solution.