Statistics Assignment Help >> Linear Programming Model Assignment Help, Standard Form

LINEAR PROGRAMMING

Linear programming can be said to be that mathematical method which is used for determining ways for achieving best outcomes while using a given mathematical model for a list of given requirements which are represented as linear relationships. Liner programming can also be said to be a specific case for mathematical programming or mathematical optimization.

If we define it in more formal terms, the linear programming can be said to be a technique used for optimization of linear objective function which is subject to linear equality and also to linear inequality constraints. The feasible region of the linear programming is a convex polyhedron which is a set defined as having intersection of many finite half spaces with each being defined by linear inequality. The objective function of it is the real valued affine function which is defined on this polyhedron. By using the linear programming polyhedron, we can find out the point on the polyhedron where the function attains the highest or smallest value, if the point exists.

The linear programs could also be expressed in canonical forms:-



Here x represents the variable vectors which are to be determined. C and b are known coefficients of vectors and A is the coefficient of known matrix. The objective function, c^Tx in this case, is the expression which has to be minimized or maximized. The equation Ax ≤ b is the constraint used for specifying the convex polytope, above which the optimization of the objective function is done.

The standard form

Standard form is most intuitive and the usual form which describes the linear programming problem. It has mainly following 4 parts:

A linear function which has to be maximized



Other forms like those forms of minimization problems are the problems with constraints on the alternative forms and also the problems involving negative variables that can always be rewritten into the equivalent problems, in the standard form.

The linear programming problems are also referred as primal problems. They can be converted into the dual problem that provides an upper bound to the optimal value of the primal problem. The primal problem can be expressed in the matrix form as

Maximizing the c^Tx subjects into Ax ≤ b, x ≥ 0;

Having the corresponding dual symmetric problem

Minimizing b^Ty subject into A^Ty ≥ c, y ≥ 0.

An alternate primal formulation can be:

Maximizing c^Tx subject into Ax ≤ b;

Having the corresponding dual asymmetric problem

Minimizing b^Ty subject into A^Ty = c, y ≥ 0.

The fundamental theory has two fundamental ideas. The first is that for the symmetric dual, the dual linear program’s dual is the original primal linear program. The second is that for every feasible solution of a linear program has a bound to the optimal value of the objective function of the dual. As per the weak duality theorem, the objective function value of any dual at any solution which is feasible is always equal to or greater than the objective function value of the primal at any of the primal solution.

Integral linear programs- the linear programs of the real variables having at least one integral optimal solution are termed as integral.

Hence, the polyhedron



Is integral when for all the feasible bounded functions c, the linear program


gets the optimum value x ^* having integer coordinates.

Help with Linear Programming Assignments - Home works

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