Linear programming - simplex computational procedure, geometric interpretation, revised simplex method, duality, degeneracy, perturbation techniques - One Line Questions

1. If the primal problem is a minimization problem with 'greater than or equal to' constraints, its dual will be: — A maximization problem with 'less than or equal to' constraints
2. What is the 'dual' of a linear programming problem? — A problem derived from the original problem that provides information about the primal solution
3. The perturbation technique ensures that in each iteration of the simplex method: — A unique variable leaves the basis
4. In the context of the simplex method, what does a 'basic feasible solution' represent? — A vertex of the feasible region
5. What is a 'basic variable' in the context of the simplex method? — A variable that is part of the basis and has a non-zero value
6. Which technique is commonly used to resolve degeneracy and prevent cycling in the simplex method? — The perturbation technique (e.g., lexicographical method)
7. What is the 'optimality condition' for a minimization problem in the simplex method? — All coefficients in the objective row (Cj - Zj) must be non-negative
8. The duality theorem in linear programming states that: — If a primal problem has an optimal solution, so does its dual, and their optimal values are equal
9. If a linear programming problem has multiple optimal solutions, what is characteristic of its simplex tableau at optimality? — At least one non-basic variable has a zero coefficient in the objective row
10. In the revised simplex method, how are the reduced costs (or objective function coefficients in the non-basic variables) calculated? — Using the basis inverse and the original cost coefficients
11. The Big M method uses a large penalty (M) in the objective function to: — Discourage the use of artificial variables in the optimal solution
12. The revised simplex method is computationally more efficient for problems with: — A very large number of constraints
13. In the context of duality, what is the 'complementary slackness' condition? — Both (A) and (B)
14. The 'fundamental theorem of duality' states that for any primal-dual pair of linear programming problems: — If the primal is unbounded, the dual is unbounded
15. In the geometric interpretation, the simplex method moves from one vertex of the feasible region to an adjacent vertex by: — Changing one non-basic variable to basic and one basic variable to non-basic
16. If a primal linear programming problem has no feasible solution, what can be said about its dual problem? — It has an unbounded solution
17. Which of the following is a characteristic of the dual of a maximization problem? — It is a minimization problem
18. When using the perturbation technique, how is a variable with a zero value handled? — It is treated as having a very small positive value (epsilon)
19. What does the term 'degeneracy' imply about the basic feasible solution? — It is at the intersection of more than the usual number of constraint boundaries
20. Which of the following is NOT a characteristic of the simplex method's tableau? — It shows the current value of the objective function
21. If a primal problem has 'm' constraints and 'n' variables, its dual problem will have: — n variables and m constraints
22. The revised simplex method is an efficient alternative to the standard simplex method primarily because it: — Computes only the necessary elements of the tableau
23. What is the primary advantage of the revised simplex method over the standard simplex method for large-scale problems? — Reduced storage requirements by avoiding explicit tableau construction
24. What is a potential problem caused by degeneracy during the simplex computation? — The algorithm may cycle, returning to a previous basic feasible solution indefinitely
25. In the dual simplex method, which variable is selected to leave the basis? — The basic variable with the most negative value
26. In a maximization problem, if all coefficients in the objective function row (Cj - Zj) of the simplex tableau are non-negative, what can be concluded? — The current solution is optimal
27. What is the geometric interpretation of duality? — The dual problem provides lower bounds for the primal maximization problem
28. The dual simplex method is particularly useful when: — The initial solution is optimal but not feasible
29. What is the 'basis inverse' in the revised simplex method? — The inverse of the submatrix formed by the basic variables
30. What does the geometric interpretation of the simplex method illustrate? — The movement between vertices of the feasible region
31. The perturbation technique modifies the problem slightly to ensure that: — All basic variables are strictly positive
32. The perturbation technique can be viewed as introducing a small positive quantity 'epsilon' to: — The right-hand side values of the constraints
33. Degeneracy in linear programming occurs when: — At least one basic variable in a basic feasible solution is zero
34. What do the values of the dual variables at the optimal solution represent? — The shadow prices or marginal values of the resources
35. In the context of duality, if a dual variable is zero, what does it imply about the corresponding primal constraint? — The primal constraint is non-binding (has slack)
36. Which method is often used to find an initial basic feasible solution for problems with no obvious origin solution? — The Two-Phase Method or Big M Method
37. What does it mean for a constraint to be 'binding' at the optimal solution? — The constraint is satisfied as an equality
38. What is the primary difference in approach between the standard simplex method and the revised simplex method? — The standard method uses a tableau, while the revised method uses matrix inversions
39. What does the 'shadow price' of a resource represent in linear programming? — The optimal value of the dual variable associated with the resource constraint
40. The 'pricing out' operation in the revised simplex method involves calculating: — The reduced costs for the non-basic variables
41. What is the 'simplex criterion' for selecting the entering variable in a maximization problem? — The variable with the most negative coefficient in the objective row
42. What is the purpose of 'slack variables' in linear programming? — To convert inequality constraints into equality constraints
43. What is the role of the 'pivot element' in the simplex computational procedure? — To determine the direction of movement to a new basic feasible solution
44. What is the primary goal of the simplex method in linear programming? — To find the optimal solution to a linear programming problem
45. What is the purpose of 'artificial variables' in the simplex method? — Both (B) and (C)
46. If the primal problem is unbounded, what is the status of the dual problem? — Infeasible
47. When does a linear programming problem have an unbounded solution according to the simplex method? — When an entering variable can be increased indefinitely without violating constraints
48. The 'ratio test' in the simplex method is used to determine: — Which variable leaves the basis (the outgoing variable)
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