Correct Answer - Option 3 : the primal objective value at x is less than the dual objective value at y
Duality in Linear programming problem (LPP): It means a linear programming problem has another LPP which is derived from it.
Original LPP is known as primal and derived LPP is known as Dual.
Dual can be found by using the below formula,
If primal is given as
Maximize CTx, subject to Ax ≤ b
Then dual will be
Minimize bTy, subject to ATy ≥ c
If x is feasible for the primal, and y is feasible for the dual, then
CTx ≤ bTy
That is primal objective is less than or equal to Dual objective.
At an optimal feasible solution, the primal objective is equal to the dual objective.
At a non-optimal feasible solution, the primal objective is less than the dual objective.
If either the primal or the dual problem has a finite optimal solution, then the other problem also has a finite optimal solution.
If either problem has an unbounded optimum solution, then the other problem has no feasible solution at all