Linear Programming and Simplex Methods - One Line Questions

1. What does degeneracy in the Simplex Method refer to? — A basic variable having a value of zero in a basic feasible solution.
2. What is the condition for multiple optimal solutions in a Linear Programming problem solved by the Simplex Method? — A non-basic variable has a zero coefficient in the objective function row of the optimal tableau.
3. If a maximization problem has an unbounded solution, what would be observed in the Simplex Tableau? — A pivot column can be selected, but all elements in that column are non-positive (≤ 0).
4. What is the 'dual problem' in Linear Programming? — A related LP problem derived from the primal problem, with constraints and objective function swapped.
5. When solving an LP problem using the Simplex Method, what indicates that the current basic feasible solution is optimal? — All coefficients in the objective function row (Cj - Zj row) are non-negative (for maximization).
6. The Simplex Method terminates when: — An optimal solution is found, or it's determined that the problem is infeasible or unbounded.
7. The 'Two-Phase Method' is an alternative to the Big M method for handling: — Artificial variables.
8. The graphical method for solving LP problems is only practical for problems with: — At most two decision variables.
9. Which of the following scenarios would require the use of artificial variables? — Constraints of the type '≥' or '=' when the origin is not a feasible starting point.
10. Bland's rule is a technique used to prevent: — Cycling in the Simplex Method.
11. Which of the following is a fundamental assumption of Linear Programming? — Divisibility of variables, certainty, and proportionality.
12. Which of the following describes the 'primal-dual relationship' in Linear Programming? — For every LP problem (primal), there exists a corresponding dual problem, and their optimal solutions are related.
13. What is the interpretation of a negative shadow price for a constraint in a maximization problem? — Increasing the RHS of that constraint would decrease the optimal objective value.
14. What happens to the 'basic variable' corresponding to the pivot row when a pivot operation is performed in the Simplex Method? — It becomes a non-basic variable.
15. What does it mean if a variable remains non-basic in the optimal solution of an LP problem? — Its optimal value is zero, and it does not contribute to the objective function at the optimum.
16. In Linear Programming, what are the constraints typically expressed as? — Linear equalities or inequalities.
17. In Phase I of the Two-Phase Method, the objective is to: — Minimize the sum of artificial variables.
18. If a problem involves minimizing a function, it can be converted to a maximization problem by: — Multiplying the objective function by -1.
19. Which of the following is NOT an assumption of Linear Programming? — Integer values for all variables.
20. The 'Simplex Tableau' is a tabular representation used in the Simplex Method to: — Organize the coefficients of the objective function and constraints for iterative calculations.
21. Which method is commonly used to solve Linear Programming problems graphically? — Plotting the constraints and identifying the corner points of the feasible region.
22. What is the Simplex Method primarily designed for? — Solving Linear Programming problems systematically.
23. Cycling in the Simplex Method occurs when: — The algorithm returns to a previous basic feasible solution, preventing convergence.
24. The 'reduced cost' of a non-basic variable in the optimal Simplex Tableau represents: — The amount by which the objective function would decrease (for maximization) if that variable were increased by one unit.
25. What does a 'shadow price' (or dual value) represent in Linear Programming? — The change in the optimal objective function value per unit increase in the RHS of a constraint, within the range of feasibility.
26. If a non-basic variable has a negative coefficient in the objective function row (Cj - Zj) of the optimal Simplex Tableau for a maximization problem, what does this imply? — The current solution is not optimal, and introducing this variable would improve the objective function.
27. According to the Duality Theorem, if the primal problem has an optimal solution, then: — The dual problem also has an optimal solution, and their optimal objective values are equal.
28. If the primal problem is infeasible, what can be said about the dual problem? — The dual problem is either unbounded or infeasible.
29. The Dual Simplex Method is particularly useful when: — The initial basic solution is feasible but not optimal, and we need to maintain feasibility.
30. The 'pivot row' in the Simplex Method is determined by: — The minimum ratio test (positive ratios of the right-hand side to the corresponding pivot column element).
31. In the Simplex Method, the 'pivot column' is typically chosen based on: — The most negative coefficient in the objective function row (for maximization).
32. Sensitivity analysis helps in understanding how changes in LP model parameters affect: — The optimal solution and the optimal objective value.
33. If the optimal solution to the primal problem is unique, what can be said about the optimal solution to the dual problem? — The optimal solution to the dual problem is also unique.
34. In the context of sensitivity analysis for an LP problem solved by the Simplex Method, what does changing the right-hand side (RHS) of a constraint affect? — The optimal value of the objective function and the values of the basic variables.
35. If Phase I of the Two-Phase Method results in an objective value greater than zero, it implies: — The original LP problem has no feasible solution.
36. The Simplex Method guarantees finding an optimal solution if one exists, provided that: — The problem is well-defined and does not exhibit cycling.
37. The 'range of feasibility' for a shadow price refers to: — The range of changes in the RHS of a constraint for which the current basis remains optimal.
38. What does the 'feasible region' represent in a Linear Programming problem? — The set of all possible solutions that satisfy all the constraints.
39. The 'Basis' in the Simplex Method refers to: — The set of basic variables in a particular iteration.
40. What is the fundamental difference between the Simplex Method and the graphical method for solving LP problems? — The Simplex Method can handle problems with many variables and constraints, while the graphical method is limited to two variables.
41. In the context of the Simplex Method, what are 'slack variables' used for? — To convert inequality constraints of the type '≤' into equalities.
42. What are 'surplus variables' used for in the Simplex Method? — To convert inequality constraints of the type '≥' into equalities.
43. What is the purpose of the 'replacement ratio' in the Simplex Method? — To determine which current basic variable will leave the basis.
44. What is the 'Big M' method used for in Linear Programming? — To handle problems with artificial variables in both maximization and minimization problems.
45. What is an 'artificial variable' introduced in the Simplex Method? — To help find an initial basic feasible solution when the origin is not feasible.
46. What is the primary objective of Linear Programming (LP)? — To maximize or minimize a linear objective function subject to linear constraints.
47. In a minimization problem using the Simplex Method, when is the optimal solution reached? — When all coefficients in the objective function row (Cj - Zj row) are non-positive (≤ 0).
48. Consider an LP problem: Maximize Z = 3x1 + 2x2, subject to x1 + x2 ≤ 4, 2x1 + x2 ≤ 6, x1, x2 ≥ 0. If we add slack variables s1 and s2, the constraints become x1 + x2 + s1 = 4 and 2x1 + x2 + s2 = 6. What is the initial basic feasible solution? — x1=0, x2=0, s1=4, s2=6, Z=0
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