Transportation and Assignment Problems - Question Bank

1. What is the main advantage of using the MODI method over the Stepping Stone method for transportation problems?
A) It is easier to understand conceptually.
B) It requires fewer iterations and is computationally more efficient.
C) It does not require a balanced problem.
D) It directly finds the optimal solution without an initial feasible solution.
2. When applying the Hungarian Method to a maximization problem, if the maximum profit is P, what is the resulting matrix for minimization?
A) A matrix where each element is P - original_profit.
B) A matrix where each element is P + original_profit.
C) A matrix where each element is original_profit / P.
D) A matrix where each element is P / original_profit.
3. The concept of 'loops' is fundamental to which methods for solving transportation problems?
A) North-West Corner Rule and Least Cost Method
B) Stepping Stone Method and MODI Method
C) Vogel's Approximation Method
D) Hungarian Method
4. Which of the following describes the 'Rim' in a transportation tableau?
A) The cells containing the shipping costs.
B) The cells representing supply and demand values.
C) The cells that are allocated in a feasible solution.
D) The cells that are not allocated.
5. In the Hungarian Method, if we have a 4x4 cost matrix and after line covering, we find 3 lines are needed, what is the implication?
A) The optimal assignment is found.
B) The problem is infeasible.
C) The matrix needs further adjustment by modifying the uncovered elements to find more zeros.
D) A dummy task must be added.
6. What is the primary difference between a transportation problem and a general linear programming problem?
A) Transportation problems involve only two variables.
B) Transportation problems have a specific structure (e.g., constraints for supply and demand) which allows for specialized solution methods.
C) Assignment problems are a type of transportation problem.
D) Transportation problems always have integer solutions.
7. The 'Stepping Stone' in the Stepping Stone Method refers to:
A) The initial feasible solution.
B) An unallocated cell that is part of a closed loop used to evaluate cost improvement.
C) The optimal solution.
D) A dummy cell.
8. If a transportation problem has m sources and n destinations, and it is balanced, how many basic variables are there in any basic feasible solution?
A) m + n
B) m * n
C) m + n - 1
D) m + n + 1
9. Which of the following is a key assumption in the standard transportation and assignment problems?
A) Costs are non-linear.
B) All supplies and demands are non-negative.
C) Routes are one-way only.
D) Capacity of routes is limited.
10. What is the 'cost matrix' in the context of assignment problems?
A) A matrix representing the supply at each source.
B) A matrix where rows represent agents and columns represent tasks, with entries being the cost or profit of assigning an agent to a task.
C) A matrix showing the demand at each destination.
D) A matrix representing the optimal solution.
11. If, during the Hungarian Method, we find that the minimum number of lines required to cover all zeros is less than n, it implies:
A) The problem is solved.
B) The current zeros are insufficient for an optimal assignment, and the matrix needs adjustment.
C) A degeneracy has occurred.
D) The problem is infeasible.
12. An unallocated cell (i, j) in a transportation problem is checked for optimality using the MODI method by calculating:
A) c_ij - (u_i + v_j)
B) u_i + v_j
C) (u_i + v_j) - c_ij
D) c_ij / (u_i + v_j)
13. When using the MODI method, the values of u_i and v_j are determined such that:
A) u_i + v_j = c_ij for all unallocated cells.
B) u_i + v_j = c_ij for all allocated cells.
C) u_i * v_j = c_ij for all allocated cells.
D) u_i - v_j = c_ij for all allocated cells.
14. The MODI method uses dual variables (u_i and v_j) to calculate:
A) Initial allocations.
B) Improvement indices (or opportunity costs) for unallocated cells.
C) The total supply.
D) The number of dummy sources.
15. What is the relationship between the number of allocations in a basic feasible solution of a transportation problem and the number of sources (m) and destinations (n)?
A) Exactly m + n
B) At least m + n - 1
C) Exactly m + n - 1
D) At most m + n
16. In the context of the Hungarian Algorithm, what does 'covering all zeros with minimum lines' attempt to achieve?
A) To find the minimum cost solution directly.
B) To determine if an optimal assignment can be made using the current zeros.
C) To balance the problem.
D) To calculate improvement indices.
17. The assignment problem can be viewed as a special case of the transportation problem where:
A) All supplies are greater than or equal to demands.
B) Supply from each source is 1 and demand at each destination is 1.
C) The number of sources is much larger than the number of destinations.
D) The transportation costs are extremely high.
18. Which of the following is NOT a method for finding an initial feasible solution for a transportation problem?
A) North-West Corner Rule
B) Least Cost Method
C) Vogel's Approximation Method
D) Hungarian Method
19. What is the minimum number of allocations required to represent a feasible solution in an assignment problem with n agents and n tasks?
A) n-1
B) n
C) n+1
D) 2n
20. For a maximization assignment problem, how is it typically converted to a minimization problem suitable for the Hungarian Method?
A) By adding a large constant to all elements.
B) By subtracting each element from the maximum element in the entire matrix.
C) By dividing all elements by the largest element.
D) By simply reversing the signs of all elements.
21. If the number of lines is less than n, what is done in the Hungarian Method?
A) The problem is solved.
B) The smallest uncovered element is subtracted from all uncovered elements, and added to elements at the intersection of two lines.
C) A dummy agent is added.
D) All costs are increased.
22. When covering zeros with minimum lines in the Hungarian Method, if the number of lines equals the number of agents/tasks (n), what does this indicate?
A) The problem is unbalanced.
B) An optimal assignment is possible with the current zeros.
C) Degeneracy is present.
D) The solution is not feasible.
23. If an optimal assignment cannot be made with the current zeros (i.e., fewer than n assignments are possible), what is the subsequent step in the Hungarian Method?
A) Introduce a dummy task.
B) Draw the minimum number of lines to cover all zeros.
C) Increase all costs by a small epsilon.
D) Re-apply the row and column reduction.
24. After performing row and column reductions in the Hungarian Method, the next step typically involves:
A) Finding the optimal assignment.
B) Making allocations to zero entries, aiming for one assignment per row and column.
C) Drawing minimum lines to cover all zeros.
D) Calculating improvement indices.
25. What is the first step in the Hungarian Method for a minimization problem?
A) Covering all zeros with a minimum number of lines.
B) Subtracting the smallest element from each row and then from each column.
C) Making allocations to zero entries.
D) Checking for optimality.
26. The Hungarian Method is an algorithm for:
A) Finding an initial feasible solution for transportation problems.
B) Solving the assignment problem.
C) Checking the optimality of transportation solutions.
D) Balancing transportation problems.
27. Which method is specifically designed to solve the assignment problem?
A) North-West Corner Rule
B) Least Cost Method
C) Hungarian Method
D) MODI Method
28. In an assignment problem, if the number of agents is not equal to the number of tasks, what is the usual procedure?
A) The problem becomes infeasible.
B) Dummy agents or dummy tasks are added with zero cost/profit.
C) The number of agents or tasks is reduced.
D) The problem is solved as a standard transportation problem.
29. What is the primary objective of an assignment problem?
A) To minimize the total cost or maximize the total profit of assigning tasks to agents.
B) To find the most efficient transportation route.
C) To balance supply and demand.
D) To determine the optimal production schedule.
30. What is an 'assignment problem'?
A) A problem of allocating resources from multiple sources to multiple destinations.
B) A special case of the transportation problem where supply and demand at each point are equal to 1.
C) A problem of finding the shortest path in a network.
D) A problem of maximizing production output.
31. The MODI method is an alternative to the Stepping Stone Method because it:
A) Is conceptually simpler and computationally faster.
B) Only works for balanced problems.
C) Focuses on the Northwest corner.
D) Does not require a feasible solution.
32. How is degeneracy typically resolved in a transportation problem?
A) By introducing a dummy source or destination.
B) By making a small positive allocation (epsilon, ε) to an empty cell in a way that does not violate feasibility.
C) By re-running the Northwest Corner Rule.
D) By increasing the total supply.
33. What is a 'degenerate solution' in a transportation problem?
A) A solution where the total cost is zero.
B) A solution where the number of allocated cells is less than (m + n - 1), where m is the number of sources and n is the number of destinations.
C) A solution where all supplies and demands are met.
D) A solution where the initial allocation is zero.
34. If all improvement indices are non-negative (for minimization problems), what can be concluded about the current solution?
A) It is not feasible.
B) It is optimal.
C) It needs further balancing.
D) It is a degenerate solution.
35. What is the significance of 'improvement indices' or 'opportunity costs' in the Stepping Stone/MODI method?
A) They indicate the per-unit cost of shipping.
B) They represent the change in total cost if one unit is shifted to an unallocated cell.
C) They determine the initial allocation.
D) They are used to balance the problem.
36. In the context of the Stepping Stone Method, what is a 'closed loop'?
A) A path that starts and ends at the same allocated cell, using only allocated and unallocated cells.
B) A path connecting two different destinations.
C) A route with zero shipping cost.
D) A path that covers all sources.
37. The Stepping Stone Method is used to:
A) Find the initial feasible solution.
B) Check for optimality and improve the current solution.
C) Balance the transportation problem.
D) Determine the number of sources and destinations.
38. After obtaining an initial basic feasible solution, how is optimality checked in a transportation problem?
A) By checking if all supplies and demands are met.
B) By using the Stepping Stone Method or MODI (Modified Distribution) Method.
C) By comparing the total cost with a benchmark cost.
D) By ensuring all allocated cells have positive values.
39. What is the core idea behind VAM for finding an initial solution?
A) To allocate to the cell with the absolute minimum cost.
B) To calculate penalties for not using a row or column and allocate to the lowest cost cell in the row/column with the highest penalty.
C) To allocate to the cell in the Northwest corner.
D) To allocate to the cell with the highest cost.
40. Vogel's Approximation Method (VAM) is generally preferred for finding an initial solution because it:
A) Is the quickest method to implement.
B) Often yields a solution closer to the optimal solution.
C) Guarantees the optimal solution directly.
D) Is only applicable to balanced transportation problems.
41. The Least Cost Method (or Matrix Minimum Method) aims to find an initial solution by:
A) Starting from the top-left cell and allocating units.
B) Prioritizing the cell with the lowest per-unit shipping cost.
C) Using a random allocation strategy.
D) Satisfying all demands first.
42. What is the objective of the Northwest Corner Rule in finding an initial solution?
A) To minimize the initial shipping cost.
B) To allocate as much as possible to the cell in the top-left corner and proceed systematically.
C) To prioritize routes with the lowest per-unit cost.
D) To ensure all demands are met first.
43. Which method is commonly used to find an initial basic feasible solution for a transportation problem?
A) Simplex Method
B) Hungarian Method
C) North-West Corner Rule
D) Branch and Bound Method
44. What is a 'balanced' transportation problem?
A) Where the cost of shipping is equal for all routes.
B) Where the total supply equals the total demand.
C) Where the number of sources equals the number of destinations.
D) Where all supplies and demands are non-negative.
45. If the total demand is greater than the total supply in a transportation problem, how is it typically handled?
A) The problem becomes infeasible.
B) A dummy destination with zero cost is introduced to absorb the excess supply.
C) A dummy source with zero cost is introduced to meet the excess demand.
D) The supply is increased to match the demand.
46. If the total supply is greater than the total demand in a transportation problem, how is it typically handled?
A) The problem becomes infeasible.
B) A dummy destination with zero cost is introduced to absorb the excess supply.
C) A dummy source with zero cost is introduced to meet the excess demand.
D) The demand is increased to match the supply.
47. What does 'demand' represent in the context of a transportation problem?
A) The capacity of a particular source to ship goods.
B) The quantity of goods required at a particular destination.
C) The maximum number of units that can be shipped.
D) The difference between supply and demand.
48. In a transportation problem, what does 'supply' refer to?
A) The demand at a particular destination.
B) The capacity of a particular source to ship goods.
C) The cost of shipping one unit from a source to a destination.
D) The total number of destinations.
49. What is the primary objective of a transportation problem?
A) To maximize profit by allocating resources to different production lines.
B) To minimize the total cost of shipping goods from various sources to different destinations.
C) To find the shortest path between two points in a network.
D) To determine the optimal production quantity to meet demand.
All Question Banks
E-Book Transportation and Assignment Problems One Line Questions Transportation and Assignment Problems Online Test Transportation and Assignment Problems