The Key Solution Algorithm of Linear Programming Model

. Linear programming problem is a common problem, and to solve the linear model is more plagued. The paper generating algorithm is based on mathematical theory and composition. The design of feasible solution algorithm illustrates key linear programming model, then we can find a better way to solve the linear programming model solutions


Introduction
According to mathematical theory of linear programming model [1][2] (1): S,t condition of the equation And x ୧ = 0(i = 1,2, ⋯ , n)m + n equations form an optimal solution m+n plane (side) closed set Ω, the linear programming model (1) on the closed set Ω border on the objective function s,t condition of the equation ‫ݖ‬ = ܿ ‫ݔ‬ ୀଵ From (0,0, ⋯, 0) (z = 0) moving along the surface normal direction, so that the value of z is gradually increased until the value of the maximum, then getting a linear programming model (1) of the optimal solution ‫ݔ(‬ ଵ , ‫ݔ‬ ଶ , ⋯ , ‫ݔ‬ ) and the corresponding z maximum.This method is theoretically established, but in practice it would be inconvenient to solve over three-dimensional linear programming model [3] .This paper designs the key feasible solution algorithmsof computer of graphic linear programming model.

The Combination Generation Algorithm
In the key feasible solution algorithms of computer of graphic linear programming model, it is supposed to discuss the combination generation algorithm of ‫ܥ‬ , ‫ܥ‬ linear equations are generated from m + n equations [4] .Let‫ܥ‬ = ∅ ‫ܥ‬ = 1,2,3, ⋯ , ݊.When generating the combination of ‫ܥ‬ , n numbers of the generated combination, each and all combinations are arranged by the dictionary order.

The Generation of Combination
The combination of and the combination of ),borrowing documents [2] to discuss the digital triangles and trigonometric operations and "⨁" and the triangular numbers counted by " °"): (To write conveniently, matrix form is used above.)

The Generation of Combination
The combination of ‫ܥ‬ ଶ ଶ is ‫ܥ,21‬  Theorem: the combination of ( 2, 2)

The Design of Key Feasible Solution Algorithm
From the foregoing, if the feasible solutions of linear programming model ( 1) is the closed setΩ, and the optimal solution is in the boundary of this closed set Ω, and the possible case is unique (a vertex of closed set Ω) or infinite (an edge or a surface of closed set Ω).So it only need to find each vertex of the closed set Ω when calculating.These vertices of the closed set Ω(‫ݔ‬ ଵ , ‫ݔ‬ ଶ , ⋯ , ‫ݔ‬ )are called key feasible solutions, then determining the optimal solution of the linear programming model 1 from key feasible solution [5][6][7][8][9][10] .
To find the key feasible solution on the closed set Ω, changing the restrictions of linear programming model to: . This is a linear equation containing n variables and m + n equations, the key feasible solution of the closed set Ω is the intersection from m + n equations obtained C ୫ା୬ ୬ equations.The intersection satisfies the points of restrictions in model ( 1),and it corresponds to the value of the objective function z.Finding the maximum value from these values, there may be three cases, one (unique), two (an edge) or three (a surface).
Computer algorithm as following: (1) Let max=0; (2) Generating m+n equations to ‫ܥ‬ ା equation groups by the algorithm of generated combination and solving it; (3) Substituting the linear programming model (1) restrictions, judging whether it is the key feasible solution on the closed set Ω, if so, calculating the value of objective function z; (4) Judging whether ‫ݖ‬ > max is established, if so, max = z returningthe key feasible solution of equation groups is ‫ݔ(‬ ଵ , ‫ݔ‬ ଶ , ⋯ , ‫ݔ‬ ) .If ‫ݖ‬ = max is satisfied, keeping the key feasible solution of equation groups as ‫ݔ(‬ ଵ , ‫ݔ‬ ଶ , ⋯ , ‫ݔ‬ ) (5) Returning to 1,until getting the solution of ‫ܥ‬ ା equation groups; (6) Outputting the value of z and the optimal solution ‫ݔ(‬ ଵ , ‫ݔ‬ ଶ , ⋯ , ‫ݔ‬ ); With this algorithm, we can easily calculate the optimal solution of linear programming model ( 1) ‫ݔ(‬ ଵ , ‫ݔ‬ ଶ , ⋯ , ‫ݔ‬ ).
existence of the optimal solution on the closed set Ω, directly calculating the intersection and determining whether it is the key feasible solution of the closed set Ω, and immediately seeking z, greatly reducing the number of calculations, and occupying less memory as a fast and effective way.Meanwhile, it will have a good reference for the algorithm of integer programming by improving this algorithm.This section will be discussed about the pull-off algorithm of the solutions of integer programming model in another text.