Correct Answer - Option 3 : the primal objective value at x is less than the dual objective value at y
Concept:
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:
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