Discuss the situation of a linear program that has one or mo

Discuss the situation of a linear program that has one or more columns of the A matrix equal to zero. Consider both the case where the corresponding variables are required to be nonnegative and the case where some are free.

Solution

If one or more columns of a matrix A has value zero then the solution is said to be a degenerate basic solution.If we consider Ax=b may not have any basic solutions we make certain assumptions that the rows of A are linearly independent and the rank of A is m.We also can have some components of a basic solution are zero.Every linear program either is infeasible,unbounded and has an unique optimal solution value.Every linear program has an extreme point that is an optimal solution.

a)corresponding variables are required to be nonnegative:

it can be written as F(x1,x2...)=b

which is a linear equality.here x1,x2 ... cannot be neagative if they are negative then solution does not exist or there is no feasible region.

b)corresponding variables are free

we can replace the free variables with non negative variables we use x=u-v where uo and v are greater then 0.we can write non negativity constraints as simple another inequality.

using transformations we can convert any any linear programme in 3 standard forms

a)max{cx\\Ax<-b}

b) max{cx\\Ax<-b,x>-0}

c)max{cx\\Ax=b,x>-0}