Without an objective, a vast number of solutions can be feasible, and therefore to find the "best" feasible solution, military-specified "ground rules" must be used that describe how goals can be achieved as opposed to specifying a goal itself.
Without an objective, a vast number of solutions can be feasible, and therefore to find the "best" feasible solution, military-specified "ground rules" must be used that describe how goals can be achieved as opposed to specifying a goal itself.Dantzig's core insight was to realize that most such ground rules can be translated into a linear objective function that needs to be maximized.
If the edge is finite, then the edge connects to another extreme point where the objective function has a greater value, otherwise the objective function is unbounded above on the edge and the linear program has no solution.
The simplex algorithm applies this insight by walking along edges of the polytope to extreme points with greater and greater objective values.
An extreme point or vertex of this polytope is known as basic feasible solution (BFS).
It can be shown that for a linear program in standard form, if the objective function has a maximum value on the feasible region, then it has this value on (at least) one of the extreme points.
This in itself reduces the problem to a finite computation since there is a finite number of extreme points, but the number of extreme points is unmanageably large for all but the smallest linear programs.
Simplex Method Of Solving Linear Programming Problem
It can also be shown that, if an extreme point is not a maximum point of the objective function, then there is an edge containing the point so that the objective function is strictly increasing on the edge moving away from the point.In the previous part we implemented and tested the simplex method on a simple example, and it has executed without any problems. In the first part, we have seen an example of the unbounded linear program.What will happen if we apply the simplex algorithm for it?In the latter case the linear program is called infeasible.In the second step, Phase II, the simplex algorithm is applied using the basic feasible solution found in Phase I as a starting point.On this example, we can see that on first iteration objective function value made no gains.In general, there might be longer runs of degenerate pivot steps.Let’s list some of the common pivot rules: It may happen that for some linear programs the simplex method cycles and theoretically, this is the only possibility of how it may fail.Such a situation is encountered very rarely in practice, if at all, and thus may implementations simply ignore the possibility of cycling.This continues until the maximum value is reached, or an unbounded edge is visited (concluding that the problem has no solution).The algorithm always terminates because the number of vertices in the polytope is finite; moreover since we jump between vertices always in the same direction (that of the objective function), we hope that the number of vertices visited will be small.