duality theory
associated with every linear program is another called its dual.the dual of this dual linear program is the original linear program (which is then referred to as the primal linear program).hence linear program come in primal/dual pairs.it turns out that every feasible solution for one of these two linear program gives a bound on the optimal objective function value for the other.these ideas are importan and form a subject called duality theory which is the topic of this chapter