- Which of the following is a condition for a linear program to be feasible?
- (A) All constraints must be satisfied.
- (B) The objective function must be maximized.
- (C) The feasible region must be unbounded.
- (D) The constraints must be non-linear.
Answer: (A) All constraints must be satisfied.
- In the simplex method, what happens when an artificial variable has a non-zero value in the optimal solution?
- (A) The solution is infeasible.
- (B) The solution is unbounded.
- (C) The solution is optimal but not feasible.
- (D) The solution is feasible.
Answer: (A) The solution is infeasible.
- In the duality theory of linear programming, what does the dual of a maximization problem represent?
- (A) A minimization problem.
- (B) Another maximization problem.
- (C) An infeasible solution.
- (D) An equivalent primal problem.
Answer: (A) A minimization problem.
- Which of the following statements is true about a degenerate basic feasible solution in the simplex method?
- (A) It is always an optimal solution.
- (B) It corresponds to a point where the feasible region is not unique.
- (C) It implies that the solution is unbounded.
- (D) It does not allow further iterations in the simplex method.
Answer: (B) It corresponds to a point where the feasible region is not unique.
- In the dual-simplex method, which of the following is required to obtain an optimal solution?
- (A) The objective function must be maximized.
- (B) All the constraints must be active.
- (C) The solution must be feasible and the optimal value must be reached.
- (D) The solution must satisfy the dual feasibility condition.
Answer: (D) The solution must satisfy the dual feasibility condition.
- Which of the following is the correct relationship between the primal and dual linear programs?
- (A) The optimal solution to the primal problem is always the optimal solution to the dual problem.
- (B) The objective values of the primal and dual problems are equal at optimality.
- (C) The dual problem cannot be solved if the primal problem has a solution.
- (D) The primal problem must have an optimal solution for the dual to be feasible.
Answer: (B) The objective values of the primal and dual problems are equal at optimality.
- What does the ‘dual gap’ refer to in linear programming?
- (A) The difference between the primal and dual objective function values.
- (B) The distance between the primal feasible region and the dual feasible region.
- (C) The sum of the slack variables in the primal problem.
- (D) The sum of the artificial variables in the dual problem.
Answer: (A) The difference between the primal and dual objective function values.
- Which of the following is true about the Simplex method?
- (A) It can only be used for maximization problems.
- (B) It always finds the global optimum for linear problems.
- (C) It is guaranteed to find an optimal solution in polynomial time.
- (D) It is not applicable for problems with integer constraints.
Answer: (B) It always finds the global optimum for linear problems.
- In linear programming, what is the significance of a slack variable in a constraint?
- (A) It represents a non-negativity condition for the variables.
- (B) It converts the inequality constraint into an equality.
- (C) It defines the objective function in a maximization problem.
- (D) It increases the value of the objective function.
Answer: (B) It converts the inequality constraint into an equality.
- If a linear programming problem has multiple optimal solutions, what happens in the Simplex method?
- (A) The Simplex method will stop at the first optimal solution found.
- (B) The Simplex method will keep iterating until all solutions are found.
- (C) The Simplex method will always find one solution, but other optimal solutions cannot be found.
- (D) The Simplex method will never stop, as there is an infinite number of optimal solutions.
Answer: (A) The Simplex method will stop at the first optimal solution found.