Choose the most correct of the following statements relating to primal-dual linear programming problems. A. Shadow prices of the resources in the primal are optimal values of the dual variable. B. The optimal values of the objective functions of primal and dual are the same. C. If the primal problem has an unbounded solution, the dual problem would have infeasibility. D. All of the above.
Hint: For solving this question you should know about the primal-dual linear programming. In this problem we will compare our given statements with the standard statements which are given for the primal-dual linear programming and then we will find the wrong and right statements.
Complete step by step answer: According to our problem we have been asked to choose the most correct of the following statements relating to primal-dual linear programming problems. So, if we take a brief idea about the primal and dual, what are these and what are their linear programming problems, then in a simple way, let the primal be a maximization problem subject to some constraints. The dual would be a minimization problem, also subject to some constraints. As long as you construct the dual in the correct way (which is very straight forward for a linear program) then the dual arrives at the same optimal solution as a primal. So, the primal goes up until it hits its constraints. The dual goes down until it hits its constraints. In the end they should hit the same place and identify the same optimal point. In linear programming, working with the dual may save computation time/space. From the prime-dual relationship, the shadow prices of the resources in the primal are optimal values of the dual variable.
So, the correct answer is “Option A”.
Note: While solving these types of questions you have to keep in mind that if any one of the problems has an optimal feasible solution, then the other problem also has an optimal feasible solution. The optimal objective function value is the same for both primal and dual problems. If one problem has an unbounded solution then the other problem is infeasible.