Section A: Hungarian Method

1. The Hungarian Method is primarily used to solve:
   a) Assignment problems
   b) Inventory problems
   c) Queuing problems
   d) Replacement problems
   Answer: a) Assignment problems
2. The Hungarian Method is designed mainly for problems involving:
   a) Multiple shipments from every source
   b) One-to-one assignment of agents to jobs
   c) Random customer arrivals
   d) Equipment replacement over time
   Answer: b) One-to-one assignment of agents to jobs
3. The first step in the Hungarian Method for a minimization problem is usually:
   a) Covering all zeros
   b) Selecting assignments
   c) Row reduction
   d) Adding a dummy row
   Answer: c) Row reduction
4. Row reduction is performed by:
   a) Adding the largest value to each row
   b) Dividing each row by its smallest value
   c) Replacing each row with zeros
   d) Subtracting the smallest row element from every element in that row
   Answer: d) Subtracting the smallest row element from every element in that row
5. The purpose of row reduction is to ensure that:
   a) Every row contains at least one zero
   b) Every row contains exactly one zero
   c) Every column contains equal values
   d) Every assignment cost becomes positive
   Answer: a) Every row contains at least one zero
6. After row reduction, the next standard step is:
   a) Selecting the largest elements
   b) Column reduction
   c) Calculating opportunity costs
   d) Applying the MODI Method
   Answer: b) Column reduction
7. Column reduction involves:
   a) Adding the smallest value to each column
   b) Dividing each column by its largest value
   c) Subtracting the smallest column value from every element in that column
   d) Deleting columns containing zeros
   Answer: c) Subtracting the smallest column value from every element in that column
8. After row and column reduction, the next objective is to:
   a) Restore the original costs
   b) Add artificial variables
   c) Calculate transportation penalties
   d) Find a set of independent zeros
   Answer: d) Find a set of independent zeros
9. Independent zeros are zeros that:
   a) Do not lie in the same row or column
   b) Have the same numerical value
   c) Appear only in dummy cells
   d) Are located on the main diagonal
   Answer: a) Do not lie in the same row or column
10. In an (n \times n) assignment problem, a complete assignment requires:
    a) (n-1) independent zeros
    b) (n) independent zeros
    c) (2n) independent zeros
    d) (n^2) independent zeros
    Answer: b) (n) independent zeros
11. If a row contains only one available zero, the usual action is to:
    a) Delete the row
    b) Add another zero
    c) Assign that zero
    d) Replace it with the smallest positive number
    Answer: c) Assign that zero
12. After selecting a zero for assignment, the other zeros in the same row and column are:
    a) Added to the solution
    b) Converted into costs
    c) Made negative
    d) Crossed out from further consideration
    Answer: d) Crossed out from further consideration
13. A complete optimal assignment is possible when:
    a) One independent zero can be selected in every row and column
    b) Every reduced cost is positive
    c) Only diagonal zeros remain
    d) The smallest cost is zero
    Answer: a) One independent zero can be selected in every row and column
14. If a complete set of independent zeros cannot be selected, the next step is to:
    a) Add a dummy job immediately
    b) Cover all zeros using the minimum number of lines
    c) Return to the original matrix
    d) Delete the row with the highest cost
    Answer: b) Cover all zeros using the minimum number of lines
15. The minimum-line test is used to determine whether:
    a) The problem is balanced
    b) Every cost is nonnegative
    c) Enough independent zeros exist for a complete assignment
    d) The original matrix is square
    Answer: c) Enough independent zeros exist for a complete assignment
16. If the minimum number of lines covering all zeros equals the order of the matrix, the solution is:
    a) Unbalanced
    b) Infeasible
    c) Necessarily unique
    d) Ready for an optimal assignment
    Answer: d) Ready for an optimal assignment
17. If the number of covering lines is less than the order of the matrix, the next step is to find:
    a) The smallest uncovered element
    b) The largest covered element
    c) The average reduced cost
    d) The number of dummy assignments
    Answer: a) The smallest uncovered element
18. The smallest uncovered element is subtracted from:
    a) Every element in the matrix
    b) All uncovered elements
    c) All zero elements
    d) All doubly covered elements
    Answer: b) All uncovered elements
19. The smallest uncovered element is added to:
    a) All uncovered elements
    b) Every row minimum
    c) Elements at intersections of two covering lines
    d) All selected zeros
    Answer: c) Elements at intersections of two covering lines
20. Elements covered by exactly one line are:
    a) Increased
    b) Decreased
    c) Replaced with zero
    d) Left unchanged
    Answer: d) Left unchanged
21. The adjustment step in the Hungarian Method is intended to:
    a) Create additional zeros without changing the optimal assignment structure
    b) Increase the total assignment cost
    c) Remove all dummy assignments
    d) Convert the problem into transportation form
    Answer: a) Create additional zeros without changing the optimal assignment structure
22. Which statement about zero selection is correct?
    a) Every zero must be selected
    b) Only independent zeros are selected for assignments
    c) Zeros in the same row should all be selected
    d) Dummy zeros are always excluded
    Answer: b) Only independent zeros are selected for assignments
23. If two selected zeros lie in the same column, the assignment is:
    a) Optimal
    b) Balanced
    c) Infeasible under one-to-one assignment rules
    d) A multiple optimum
    Answer: c) Infeasible under one-to-one assignment rules
24. If two selected zeros lie in the same row, the solution violates:
    a) The column-reduction rule
    b) The minimum-line test
    c) The cost objective only
    d) The one-job-per-agent condition
    Answer: d) The one-job-per-agent condition
25. In a (4\times4) assignment problem, the optimal solution must contain:
    a) Four assignments
    b) Eight assignments
    c) Sixteen assignments
    d) Three assignments
    Answer: a) Four assignments
26. In a (5\times5) assignment problem, the number of selected independent zeros must be:
    a) Four
    b) Five
    c) Ten
    d) Twenty-five
    Answer: b) Five
27. The Hungarian Method normally begins with a:
    a) Transportation tableau
    b) Project network
    c) Square cost or effectiveness matrix
    d) Probability distribution
    Answer: c) Square cost or effectiveness matrix
28. Which condition is essential before directly applying the standard Hungarian Method?
    a) All costs must be equal
    b) The problem must be a maximization model
    c) Every row must already contain a zero
    d) The assignment matrix must be square
    Answer: d) The assignment matrix must be square
29. A rectangular assignment matrix is converted into a square matrix by:
    a) Adding dummy rows or columns
    b) Removing real agents
    c) Multiplying every cost by zero
    d) Adding artificial objective coefficients
    Answer: a) Adding dummy rows or columns
30. The Hungarian Method solves a minimization assignment problem by manipulating:
    a) Assignment quantities
    b) The cost matrix
    c) Supply and demand values
    d) Shadow prices
    Answer: b) The cost matrix
31. A maximization assignment problem can be solved by first converting it into:
    a) A transportation maximization problem
    b) A queuing model
    c) An equivalent minimization problem
    d) A dynamic-programming model
    Answer: c) An equivalent minimization problem
32. A common method for converting profits into equivalent costs is to:
    a) Add every profit to the largest profit
    b) Divide each profit by the smallest profit
    c) Replace all profits with zeros
    d) Subtract each profit from the largest profit
    Answer: d) Subtract each profit from the largest profit
33. If the largest profit is 40 and a particular profit is 27, the converted cost is:
    a) 13
    b) 27
    c) 40
    d) 67
    Answer: a) 13
34. If the largest effectiveness value is 50 and a cell contains 35, its converted value is:
    a) 35
    b) 15
    c) 50
    d) 85
    Answer: b) 15
35. The conversion of a maximization matrix to a minimization matrix:
    a) Changes the feasible assignments
    b) Increases the number of jobs
    c) Preserves the relative optimal assignment structure
    d) Requires adding artificial variables
    Answer: c) Preserves the relative optimal assignment structure
36. A prohibited assignment is usually represented by:
    a) A zero cost
    b) A negative cost
    c) The smallest matrix value
    d) A very large cost (M)
    Answer: d) A very large cost (M)
37. The purpose of a very large cost in a prohibited cell is to:
    a) Prevent that assignment from being selected
    b) Encourage that assignment
    c) Balance the matrix
    d) Create more zeros
    Answer: a) Prevent that assignment from being selected
38. When a prohibited assignment is represented by (M), the analyst should ensure that:
    a) (M) is the smallest value
    b) (M) is sufficiently large relative to feasible costs
    c) (M) is equal to zero after reduction
    d) (M) is used in every dummy cell
    Answer: b) (M) is sufficiently large relative to feasible costs
39. A mandatory assignment can be handled by:
    a) Giving it the largest cost
    b) Deleting its row only
    c) Fixing that assignment and removing the corresponding row and column
    d) Adding a dummy job
    Answer: c) Fixing that assignment and removing the corresponding row and column
40. Which statement best describes the final step of the Hungarian Method?
    a) Select every zero in the matrix
    b) Add all reduced costs
    c) Use the reduced matrix total as the objective value
    d) Evaluate the selected assignments using the original cost matrix
    Answer: d) Evaluate the selected assignments using the original cost matrix

Section B: Multiple Optimal Solutions

41. Multiple optimal solutions occur when:
    a) More than one feasible assignment gives the same optimal total value
    b) The assignment matrix is rectangular
    c) Every cost is different
    d) No complete assignment exists
    Answer: a) More than one feasible assignment gives the same optimal total value
42. Another term for multiple optimal solutions is:
    a) Degenerate assignments
    b) Alternate optimal assignments
    c) Prohibited assignments
    d) Dummy assignments
    Answer: b) Alternate optimal assignments
43. Multiple optima can be detected when:
    a) Only one zero exists in each row
    b) Every row has a dummy assignment
    c) Different complete sets of independent zeros are available
    d) The matrix has no zeros
    Answer: c) Different complete sets of independent zeros are available
44. If two distinct complete assignments have the same minimum cost, the problem has:
    a) No optimal solution
    b) An unbounded solution
    c) An infeasible solution
    d) Multiple optimal solutions
    Answer: d) Multiple optimal solutions
45. Which condition is necessary for alternate optimal assignments?
    a) At least two feasible complete assignments must have the same best value
    b) Every matrix element must be zero
    c) The problem must be unbalanced
    d) A dummy row must be used
    Answer: a) At least two feasible complete assignments must have the same best value
46. Multiple optimal assignments are more likely when:
    a) All costs are distinct
    b) Several reduced zeros allow different independent selections
    c) Every job has only one feasible agent
    d) All prohibited cells are removed
    Answer: b) Several reduced zeros allow different independent selections
47. If all entries in a (3\times3) cost matrix are equal, then:
    a) No assignment is feasible
    b) Only the diagonal assignment is optimal
    c) Every complete assignment has the same total cost
    d) The problem is unbalanced
    Answer: c) Every complete assignment has the same total cost
48. The existence of multiple optimal assignments means that management:
    a) Must reject the model
    b) Has no feasible solution
    c) Must add another worker
    d) Can choose among equally optimal alternatives using qualitative criteria
    Answer: d) Can choose among equally optimal alternatives using qualitative criteria
49. When several optimal assignments exist, a secondary consideration may include:
    a) Employee preference
    b) Increasing total cost
    c) Removing constraints
    d) Ignoring feasibility
    Answer: a) Employee preference
50. Which secondary criterion may help select among alternate optima?
    a) The number of zeros only
    b) Skill compatibility or workload balance
    c) The matrix order
    d) The number of reduction steps
    Answer: b) Skill compatibility or workload balance
51. Multiple optimal solutions do not imply that the problem is:
    a) Balanced
    b) A minimization model
    c) Infeasible
    d) Solved by the Hungarian Method
    Answer: c) Infeasible
52. If two complete assignments have costs of 75 and 75, while all others cost more, then:
    a) The problem is unbounded
    b) The matrix is invalid
    c) No unique solution exists
    d) Both assignments are optimal
    Answer: d) Both assignments are optimal
53. A unique optimal assignment exists when:
    a) Only one feasible complete assignment attains the best objective value
    b) Every matrix entry is equal
    c) The problem contains a dummy row
    d) More than one zero appears in each row
    Answer: a) Only one feasible complete assignment attains the best objective value
54. The presence of several zeros in the reduced matrix:
    a) Always guarantees multiple optima
    b) May indicate alternate assignments but requires complete feasible matching checks
    c) Makes the problem infeasible
    d) Requires deleting rows
    Answer: b) May indicate alternate assignments but requires complete feasible matching checks
55. To confirm multiple optimal assignments, the analyst should:
    a) Compare only the reduced matrix values
    b) Count the number of zeros
    c) Construct and evaluate distinct complete assignments
    d) Add another dummy column
    Answer: c) Construct and evaluate distinct complete assignments
56. Which cost matrix is most likely to have many optimal assignments?
    a) A matrix with highly varied costs
    b) A matrix with many prohibited cells
    c) A matrix with one feasible assignment
    d) A matrix with identical costs
    Answer: d) A matrix with identical costs
57. If two workers can be exchanged between two jobs without changing total cost, this suggests:
    a) Alternate optimal assignments
    b) Infeasibility
    c) Unboundedness
    d) An incorrect row reduction
    Answer: a) Alternate optimal assignments
58. Suppose assignment A costs 60 and assignment B costs 60. If 60 is the minimum possible cost, then:
    a) Only A is optimal
    b) Both A and B are optimal
    c) Neither is optimal
    d) The problem is unbalanced
    Answer: b) Both A and B are optimal
59. An alternate optimum may be identified by changing:
    a) The original cost data
    b) The number of agents
    c) The selection of independent zeros while maintaining feasibility
    d) The objective from minimization to maximization
    Answer: c) The selection of independent zeros while maintaining feasibility
60. When reporting multiple optimal solutions, the analyst should:
    a) Show only one and conceal the others
    b) Declare the model infeasible
    c) Average the assignments
    d) Present the alternative assignments and their equal objective values
    Answer: d) Present the alternative assignments and their equal objective values
61. Which statement about alternate optimal assignments is correct?
    a) They may differ in assignments but have the same optimal objective value
    b) They must use the same worker-job pairs
    c) They occur only in unbalanced problems
    d) They occur only in maximization problems
    Answer: a) They may differ in assignments but have the same optimal objective value
62. Multiple optimal solutions can provide managers with:
    a) Less flexibility
    b) Greater implementation flexibility
    c) No useful information
    d) An invalid objective value
    Answer: b) Greater implementation flexibility
63. If one optimal assignment is operationally inconvenient, management may:
    a) Select a higher-cost assignment
    b) Change the original data
    c) Choose another equally optimal assignment
    d) Remove a job
    Answer: c) Choose another equally optimal assignment
64. Which statement is false about multiple optimal assignments?
    a) They have equal optimal objective values
    b) They may involve different worker-job combinations
    c) They can occur in balanced models
    d) They always require dummy assignments
    Answer: d) They always require dummy assignments
65. If every feasible assignment has the same total cost, then:
    a) Every feasible assignment is optimal
    b) No assignment is optimal
    c) The Hungarian Method cannot be used
    d) The model is infeasible
    Answer: a) Every feasible assignment is optimal
66. A secondary objective may be added when:
    a) There is no feasible solution
    b) Several primary-optimal assignments exist
    c) The matrix is not square
    d) All assignments are prohibited
    Answer: b) Several primary-optimal assignments exist
67. Which is an appropriate secondary objective?
    a) Increase the primary cost
    b) Violate one assignment constraint
    c) Minimize employee travel among equal-cost assignments
    d) Assign two jobs to one worker
    Answer: c) Minimize employee travel among equal-cost assignments
68. Alternate optimal solutions are identified after ensuring that:
    a) The number of jobs exceeds workers
    b) All costs are different
    c) Every zero is selected
    d) Each candidate solution satisfies one-to-one assignment rules
    Answer: d) Each candidate solution satisfies one-to-one assignment rules
69. Multiple optimal assignments are valuable because they allow:
    a) Consideration of qualitative factors without sacrificing the primary objective
    b) Violation of model constraints
    c) Elimination of all costs
    d) Unlimited assignments
    Answer: a) Consideration of qualitative factors without sacrificing the primary objective
70. Which statement best summarizes multiple optimal solutions?
    a) They indicate a computational error
    b) They represent different feasible matchings with the same best objective value
    c) They occur when no full assignment exists
    d) They require all matrix entries to be zero
    Answer: b) They represent different feasible matchings with the same best objective value

Section C: Unbalanced Assignment Problems

71. An unbalanced assignment problem occurs when:
    a) Every assignment cost is unequal
    b) The objective is maximization
    c) The number of agents and jobs is different
    d) The matrix contains prohibited cells
    Answer: c) The number of agents and jobs is different
72. Before applying the standard Hungarian Method, an unbalanced problem should be:
    a) Converted into a transportation problem
    b) Solved graphically
    c) Reduced to one row
    d) Balanced by adding dummy rows or columns
    Answer: d) Balanced by adding dummy rows or columns
73. If there are more workers than jobs, the model is balanced by adding:
    a) Dummy jobs
    b) Dummy workers
    c) Additional real jobs
    d) Artificial constraints
    Answer: a) Dummy jobs
74. If there are more jobs than workers, the model is balanced by adding:
    a) Dummy jobs
    b) Dummy workers
    c) More costs to each row
    d) A surplus variable
    Answer: b) Dummy workers
75. The number of dummy rows or columns added equals:
    a) The number of real assignments
    b) The matrix order
    c) The difference between the number of agents and jobs
    d) The number of zero costs
    Answer: c) The difference between the number of agents and jobs
76. If there are six workers and four jobs, the balanced matrix requires:
    a) Two dummy workers
    b) One dummy job
    c) Four dummy jobs
    d) Two dummy jobs
    Answer: d) Two dummy jobs
77. If there are three workers and five jobs, the balanced matrix requires:
    a) Two dummy workers
    b) Two dummy jobs
    c) Five dummy workers
    d) One dummy job
    Answer: a) Two dummy workers
78. The usual cost assigned to a dummy cell is:
    a) The largest real cost
    b) Zero
    c) One
    d) A negative number
    Answer: b) Zero
79. A zero dummy cost assumes that:
    a) The dummy assignment is highly profitable
    b) The job must be completed
    c) Leaving a worker or job unmatched has no direct penalty
    d) The problem is infeasible
    Answer: c) Leaving a worker or job unmatched has no direct penalty
80. If leaving a job unassigned incurs a penalty, the dummy-worker cells should contain:
    a) Zero only
    b) The smallest real cost
    c) A negative value
    d) The appropriate penalty cost
    Answer: d) The appropriate penalty cost
81. Assignment of a real worker to a dummy job means the worker is:
    a) Idle or unassigned
    b) Assigned to two jobs
    c) Removed permanently
    d) Given a prohibited assignment
    Answer: a) Idle or unassigned
82. Assignment of a dummy worker to a real job means the job is:
    a) Completed by the most efficient worker
    b) Left unassigned or externally unmet
    c) Assigned twice
    d) Removed from the objective
    Answer: b) Left unassigned or externally unmet
83. After balancing an unbalanced model, the resulting matrix is:
    a) Rectangular
    b) Triangular
    c) Square
    d) Diagonal
    Answer: c) Square
84. Once balanced, the problem can be solved using:
    a) Only the simplex method
    b) Only transportation MODI
    c) Only graphical analysis
    d) The standard Hungarian Method
    Answer: d) The standard Hungarian Method
85. Adding dummy rows or columns changes:
    a) The matrix dimensions but not the real assignment costs
    b) Every original assignment cost
    c) The number of real workers
    d) The objective from minimization to maximization
    Answer: a) The matrix dimensions but not the real assignment costs
86. A (4\times6) assignment matrix requires:
    a) Two dummy columns
    b) Two dummy rows
    c) Four dummy rows
    d) Six dummy columns
    Answer: b) Two dummy rows
87. A (7\times5) assignment matrix requires:
    a) Two dummy rows
    b) Seven dummy columns
    c) Two dummy columns
    d) Five dummy rows
    Answer: c) Two dummy columns
88. In a minimization model, a dummy assignment with zero cost may be selected because:
    a) It represents the most expensive option
    b) It violates feasibility
    c) It forces multiple assignments
    d) It balances unmatched capacity without adding cost
    Answer: d) It balances unmatched capacity without adding cost
89. If idle workers create a cost, dummy-job entries should represent:
    a) Idle-time costs
    b) Zero under all circumstances
    c) The highest production profit
    d) Transportation demand
    Answer: a) Idle-time costs
90. If an unfilled job creates a lost-sales penalty, dummy-worker entries should contain:
    a) Zero only
    b) The lost-sales penalty
    c) The lowest real assignment cost
    d) A negative value
    Answer: b) The lost-sales penalty
91. An unbalanced assignment problem is not necessarily:
    a) A minimization problem
    b) Rectangular initially
    c) Infeasible
    d) Solvable by the Hungarian Method
    Answer: c) Infeasible
92. The purpose of balancing is to:
    a) Make all costs equal
    b) Eliminate real assignments
    c) Create multiple optima
    d) Establish a square one-to-one assignment structure
    Answer: d) Establish a square one-to-one assignment structure
93. If there are eight machines and six jobs, the number of machines that may remain idle is:
    a) Two
    b) Six
    c) Eight
    d) Fourteen
    Answer: a) Two
94. If there are five workers and seven jobs, the number of jobs that may remain unmatched is:
    a) Five
    b) Two
    c) Seven
    d) Twelve
    Answer: b) Two
95. A dummy row corresponds to:
    a) An extra real worker
    b) An idle real worker
    c) A fictitious worker
    d) A prohibited worker
    Answer: c) A fictitious worker
96. A dummy column corresponds to:
    a) A fictitious worker
    b) An additional cost constraint
    c) A real job with no worker
    d) A fictitious job
    Answer: d) A fictitious job
97. Which statement about dummy assignments is correct?
    a) They represent unmatched agents or jobs in the balanced model
    b) They are always prohibited
    c) They must have the largest cost
    d) They create real production output
    Answer: a) They represent unmatched agents or jobs in the balanced model
98. If a dummy assignment has a nonzero penalty, it means:
    a) The model is incorrectly formulated
    b) Leaving an agent or job unmatched has an economic consequence
    c) The matrix cannot be balanced
    d) The Hungarian Method cannot be applied
    Answer: b) Leaving an agent or job unmatched has an economic consequence
99. Which statement best distinguishes balanced and unbalanced assignment problems?
    a) Balanced problems are always minimization problems
    b) Unbalanced problems cannot have optimal solutions
    c) Balanced problems have equal numbers of agents and jobs
    d) Unbalanced problems contain no cost matrix
    Answer: c) Balanced problems have equal numbers of agents and jobs
100. Which statement best summarizes the treatment of an unbalanced assignment problem?
     a) Delete the extra agents or jobs
     b) Solve the rectangular matrix without adjustment
     c) Convert it into a transportation model only
     d) Add dummy agents or jobs, assign suitable dummy costs, and apply the Hungarian Method
     Answer: d) Add dummy agents or jobs, assign suitable dummy costs, and apply the Hungarian Method