Integer programming, transportation problem - One Line Questions

1. Which of the following is a valid constraint for a binary integer programming problem? — x = 0 or x = 1
2. In a transportation problem, what does a 'source' represent? — A point where goods are produced or available
3. In a transportation problem, what does a 'destination' represent? — A point where goods are demanded or received
4. What is a 'transshipment problem'? — A transportation problem where goods can be shipped through intermediate points
5. In integer programming, a feasible solution that satisfies the integer requirements is called: — An integer feasible solution
6. A 'knapsack problem' is a type of integer programming problem that involves: — Selecting items with different values and weights to fit into a limited capacity
7. Which problem is a special case of the transportation problem where the objective is to minimize the total cost of shipping goods from a set of sources to a set of destinations? — Standard Transportation Problem
8. How is degeneracy typically handled in the transportation problem? — By making a small allocation (epsilon) to an empty cell
9. If total supply exceeds total demand in a transportation problem, how is it typically balanced? — By adding a dummy destination with zero demand
10. If total demand exceeds total supply in a transportation problem, how is it typically balanced? — By adding a dummy source with zero supply
11. Consider a fixed-charge problem in integer programming. What kind of variable is typically used to model the fixed cost? — Binary variable
12. The 'Big M' method in IP is used to handle: — Disjunctive constraints (either-or)
13. The transportation problem assumes that: — Shipments can be made in fractions
14. Which type of constraint is often introduced in IP to model 'either-or' situations? — Big M constraint
15. In the context of the transportation problem, what does the MODI (Modified Distribution) method aim to achieve? — Check for optimality and find improvement paths
16. The cutting-plane method for solving integer programs aims to: — Generate additional constraints (cuts) to tighten the LP relaxation
17. When is the Branch and Bound method guaranteed to find the optimal integer solution? — If all branches are explored or fathomed
18. What is 'LP relaxation' in the context of Branch and Bound for IP? — Ignoring the integer constraints and solving as an LP
19. What is the significance of the 'degeneracy' in a transportation problem? — It occurs when the number of independent basic variables is less than m + n - 1
20. Which method is specifically designed for finding the optimal solution to the transportation problem? — Stepping Stone Method
21. If a transportation problem has m sources and n destinations, how many basic variables are required for a non-degenerate basic feasible solution? — m + n - 1
22. The Stepping Stone Method improves an initial feasible solution by: — Identifying negative cost improvement cycles
23. Which type of integer programming problem requires all decision variables to be integers? — Pure Integer Programming (PIP)
24. What is a special case of Integer Programming where variables can only take values of 0 or 1? — Binary Integer Programming
25. The transportation problem formulation is a specific application of which broader class of optimization problems? — Network Flow Problems
26. The MODI method is an alternative to which other method for checking optimality in transportation problems? — Stepping Stone Method
27. Which of the following is NOT a standard method for finding an initial feasible solution to the transportation problem? — Stepping Stone Method
28. What condition must be met for a transportation problem to be 'balanced'? — Total supply equals total demand
29. The Assignment Problem can be viewed as a special case of the transportation problem where: — Supply at each source is 1 and demand at each destination is 1
30. In Mixed Integer Programming (MIP), what types of variables are allowed? — A mix of integer and continuous variables
31. Which of the following scenarios would most likely require Integer Programming? — Scheduling flights where the number of flights must be a whole number
32. In the context of solving the transportation problem using MODI, what are u_i and v_j typically used for? — Calculating opportunity costs or reduced costs
33. Which of the following is a common method for solving Integer Programming problems? — Branch and Bound Method
34. Which method is used to find an initial basic feasible solution for the transportation problem? — North-West Corner Rule
35. Which algorithm is commonly used to solve the Assignment Problem? — Hungarian Method
36. The Branch and Bound method for IP typically involves: — Solving a sequence of LP relaxations
37. The North-West Corner Rule for finding an initial solution starts allocation from which cell? — The top-left cell of the transportation tableau
38. What is the 'penalty cost' in Vogel's Approximation Method (VAM)? — The difference between the two lowest costs in a row or column
39. What is the 'gap' in the context of Branch and Bound for integer programming? — The difference between the objective function value of the LP relaxation and the best integer solution found so far
40. In integer programming, if the optimal solution to the LP relaxation is fractional, what action is typically taken in the Branch and Bound method? — A new branch is created based on the fractional variable
41. What is the main challenge in solving large-scale integer programming problems? — The solution space grows exponentially, leading to long computation times
42. Gomory's cutting-plane algorithm generates cuts from: — The tableau of the optimal LP relaxation solution
43. The Least Cost Method for initial solution allocation prioritizes: — The cell with the minimum transportation cost among available routes
44. In a transportation problem, if an optimal solution is not unique, it means: — There are alternative optimal solutions
45. What is the primary goal of Vogel's Approximation Method (VAM) for the transportation problem? — To obtain a better initial feasible solution with a lower cost
46. What is the primary objective of Integer Programming (IP) compared to Linear Programming (LP)? — To ensure all decision variables take on integer values
47. In the Branch and Bound method, when is a node considered 'fathomed'? — All of the above
48. Which of the following constraints is characteristic of integer programming and not standard linear programming? — xi is an integer
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