1. What is the primary purpose of an assignment problem?
   a) To assign resources to tasks on a one-to-one basis optimally
   b) To determine inventory reorder quantities
   c) To calculate project completion time
   d) To forecast future demand
   Answer: a) To assign resources to tasks on a one-to-one basis optimally
2. The assignment problem is a special case of the:
   a) Queuing model
   b) Transportation model
   c) Inventory model
   d) Replacement model
   Answer: b) Transportation model
3. In a standard assignment problem, each worker is assigned to:
   a) Several jobs simultaneously
   b) No job
   c) Exactly one job
   d) At least two jobs
   Answer: c) Exactly one job
4. In a standard assignment problem, each job is assigned to:
   a) Every worker
   b) Several workers
   c) No worker
   d) Exactly one worker
   Answer: d) Exactly one worker
5. Which is a common objective of an assignment problem?
   a) Minimize total assignment cost
   b) Maximize inventory levels
   c) Minimize project activities
   d) Maximize waiting time
   Answer: a) Minimize total assignment cost
6. An assignment problem may also be formulated to:
   a) Minimize demand
   b) Maximize total profit or effectiveness
   c) Increase idle time
   d) Eliminate all constraints
   Answer: b) Maximize total profit or effectiveness
7. Which of the following is an assignment-problem application?
   a) Determining economic order quantity
   b) Scheduling project activities
   c) Assigning salespeople to sales territories
   d) Calculating customer waiting time
   Answer: c) Assigning salespeople to sales territories
8. Which situation best represents an assignment problem?
   a) Shipping products from several factories to warehouses
   b) Deciding how much inventory to order
   c) Determining the shortest project duration
   d) Assigning four machines to four jobs
   Answer: d) Assigning four machines to four jobs
9. The rows of an assignment table usually represent:
   a) Agents, workers or machines
   b) Objective coefficients only
   c) Supply quantities greater than one
   d) Total costs only
   Answer: a) Agents, workers or machines
10. The columns of an assignment table usually represent:
    a) Decision rules
    b) Jobs, tasks or destinations
    c) Shadow prices
    d) Slack variables
    Answer: b) Jobs, tasks or destinations
11. The value in cell ((i,j)) of a cost matrix generally represents:
    a) The total number of assignments
    b) The supply available from agent (i)
    c) The cost of assigning agent (i) to job (j)
    d) The demand of job (j)
    Answer: c) The cost of assigning agent (i) to job (j)
12. A balanced assignment problem has:
    a) More jobs than agents
    b) More agents than jobs
    c) Unlimited agents
    d) An equal number of agents and jobs
    Answer: d) An equal number of agents and jobs
13. An unbalanced assignment problem occurs when:
    a) The number of agents and jobs is unequal
    b) Every assignment cost is zero
    c) The objective is maximization
    d) All costs are equal
    Answer: a) The number of agents and jobs is unequal
14. An unbalanced assignment problem is balanced by adding:
    a) A surplus constraint
    b) A dummy row or dummy column
    c) A new objective function
    d) An artificial objective value
    Answer: b) A dummy row or dummy column
15. If there are more jobs than workers, balancing requires:
    a) Deleting some jobs
    b) Adding a dummy job
    c) Adding one or more dummy workers
    d) Increasing every assignment cost
    Answer: c) Adding one or more dummy workers
16. If there are more workers than jobs, balancing requires:
    a) Removing workers
    b) Adding artificial costs only
    c) Adding a dummy worker
    d) Adding one or more dummy jobs
    Answer: d) Adding one or more dummy jobs
17. The cost of assigning an agent to a dummy job is normally:
    a) Zero
    b) One
    c) A very large number
    d) Equal to the highest cost
    Answer: a) Zero
18. Assignment to a dummy job generally means that the agent is:
    a) Assigned to two real jobs
    b) Left unassigned or idle
    c) Removed from the model
    d) Given maximum cost
    Answer: b) Left unassigned or idle
19. Assignment of a dummy worker to a job generally means that the job is:
    a) Completed twice
    b) Assigned at zero real cost automatically
    c) Left unassigned or unmet
    d) Removed from the matrix
    Answer: c) Left unassigned or unmet
20. The most widely used manual solution method for assignment problems is the:
    a) Simplex tableau method
    b) Northwest Corner Method
    c) Stepping-Stone Method
    d) Hungarian Method
    Answer: d) Hungarian Method
21. The Hungarian Method was developed primarily for solving:
    a) Assignment problems
    b) Queuing problems
    c) Inventory problems
    d) Replacement problems
    Answer: a) Assignment problems
22. The first major step in the Hungarian Method for minimization is:
    a) Column covering
    b) Row reduction
    c) Drawing loops
    d) Calculating shadow prices
    Answer: b) Row reduction
23. Row reduction involves:
    a) Adding the largest element to each row
    b) Dividing each row by its smallest value
    c) Subtracting the smallest row element from every element in that row
    d) Deleting the smallest element in each row
    Answer: c) Subtracting the smallest row element from every element in that row
24. After row reduction, the next standard step is:
    a) Assigning every zero immediately
    b) Adding dummy variables
    c) Applying MODI
    d) Column reduction
    Answer: d) Column reduction
25. Column reduction involves:
    a) Subtracting the smallest column element from every element in that column
    b) Dividing each column by the largest value
    c) Adding the smallest value to the column
    d) Removing all zeros
    Answer: a) Subtracting the smallest column element from every element in that column
26. In the Hungarian Method, an optimal assignment is sought among:
    a) The largest entries
    b) Independent zeros
    c) Negative entries only
    d) Dummy cells only
    Answer: b) Independent zeros
27. Two zeros are independent when they:
    a) Have equal numerical values
    b) Occur in adjacent cells
    c) Do not lie in the same row or column
    d) Appear in the same row
    Answer: c) Do not lie in the same row or column
28. A complete assignment in an (n\times n) problem requires:
    a) One zero
    b) (n-1) independent zeros
    c) (2n) zeros
    d) (n) independent zeros
    Answer: d) (n) independent zeros
29. If a row contains only one available zero, that zero should normally be:
    a) Selected for assignment
    b) Replaced with a dummy value
    c) Increased by the smallest uncovered number
    d) Deleted
    Answer: a) Selected for assignment
30. After selecting an assignment in a row, other zeros in the corresponding column are usually:
    a) Selected as well
    b) Crossed out from consideration
    c) Converted into costs
    d) Made negative
    Answer: b) Crossed out from consideration
31. If a complete set of independent zeros cannot be found, the next step is to:
    a) Stop and declare infeasibility
    b) Add another worker
    c) Cover all zeros using the minimum number of lines
    d) Return to the original cost matrix
    Answer: c) Cover all zeros using the minimum number of lines
32. If the minimum number of lines covering all zeros equals the order of the matrix, the current solution is:
    a) Unbalanced
    b) Infeasible
    c) Degenerate
    d) Capable of yielding an optimal assignment
    Answer: d) Capable of yielding an optimal assignment
33. When the number of covering lines is less than the matrix order, one should identify:
    a) The smallest uncovered element
    b) The largest covered element
    c) The average assignment cost
    d) The number of dummy jobs
    Answer: a) The smallest uncovered element
34. The smallest uncovered element is subtracted from:
    a) Every matrix element
    b) All uncovered elements
    c) All covered elements
    d) Only assigned cells
    Answer: b) All uncovered elements
35. The smallest uncovered element is added to elements located at:
    a) Uncovered positions
    b) Assigned positions only
    c) Intersections of two covering lines
    d) Every zero position
    Answer: c) Intersections of two covering lines

Section B: Mathematical Formulation of the Assignment Problem

36. In mathematical formulation, (x_{ij}=1) usually means:
    a) Agent (i) is not assigned to job (j)
    b) Job (j) has no demand
    c) The assignment cost is one
    d) Agent (i) is assigned to job (j)
    Answer: d) Agent (i) is assigned to job (j)
37. In a binary assignment model, (x_{ij}=0) means:
    a) Agent (i) is not assigned to job (j)
    b) Agent (i) is assigned to every job
    c) Job (j) has zero cost
    d) The model is infeasible
    Answer: a) Agent (i) is not assigned to job (j)
38. Assignment-problem decision variables are normally:
    a) Continuous without bounds
    b) Binary variables
    c) Negative variables
    d) Unrestricted variables
    Answer: b) Binary variables
39. Which restriction is appropriate for assignment variables?
    a) (x_{ij}\geq1)
    b) (x_{ij}\leq0)
    c) (x_{ij}\in{0,1})
    d) (x_{ij}) unrestricted
    Answer: c) (x_{ij}\in{0,1})
40. The objective function for a cost-minimization assignment problem is:
    a) Maximize (\sum_i\sum_j c_{ij}x_{ij})
    b) Minimize (\sum_i x_{ij}) only
    c) Maximize the number of assignments
    d) Minimize (\sum_i\sum_j c_{ij}x_{ij})
    Answer: d) Minimize (\sum_i\sum_j c_{ij}x_{ij})
41. In the objective function, (c_{ij}) represents:
    a) The cost of assigning agent (i) to job (j)
    b) The total number of agents
    c) The binary decision variable
    d) The number of dummy rows
    Answer: a) The cost of assigning agent (i) to job (j)
42. The constraint (\sum_j x_{ij}=1) means:
    a) Every job receives one agent
    b) Agent (i) is assigned to exactly one job
    c) Agent (i) performs every job
    d) Job (j) is left unassigned
    Answer: b) Agent (i) is assigned to exactly one job
43. The constraint (\sum_i x_{ij}=1) means:
    a) Agent (i) receives one job
    b) Every assignment cost equals one
    c) Job (j) is assigned to exactly one agent
    d) Every worker receives job (j)
    Answer: c) Job (j) is assigned to exactly one agent
44. For (n) agents and (n) jobs, the model contains how many binary variables?
    a) (n)
    b) (2n)
    c) (n+1)
    d) (n^2)
    Answer: d) (n^2)
45. A (4\times4) assignment problem contains:
    a) 16 decision variables
    b) 8 decision variables
    c) 4 decision variables
    d) 20 decision variables
    Answer: a) 16 decision variables
46. A balanced (5\times5) assignment model normally has:
    a) Five total constraints
    b) Ten assignment constraints before binary restrictions
    c) Twenty-five assignment constraints
    d) Fifty constraints
    Answer: b) Ten assignment constraints before binary restrictions
47. The row constraints ensure that:
    a) Every job has the same cost
    b) Every worker is assigned to all jobs
    c) Each agent receives exactly one assignment
    d) Every row contains zero cost
    Answer: c) Each agent receives exactly one assignment
48. The column constraints ensure that:
    a) Each agent receives several jobs
    b) Every cost is minimized independently
    c) Every column sum is zero
    d) Each job receives exactly one agent
    Answer: d) Each job receives exactly one agent
49. In a profit-maximization assignment problem, the objective is:
    a) Maximize (\sum_i\sum_j p_{ij}x_{ij})
    b) Minimize (\sum_i\sum_j p_{ij}x_{ij})
    c) Maximize (\sum_i x_{ij}) only
    d) Minimize the number of jobs
    Answer: a) Maximize (\sum_i\sum_j p_{ij}x_{ij})
50. To solve a maximization assignment problem using the minimization Hungarian Method, profits may be converted into:
    a) Supplies
    b) Opportunity-loss or equivalent cost values
    c) Demands
    d) Slack variables
    Answer: b) Opportunity-loss or equivalent cost values
51. A common conversion from profit to cost is to subtract each profit from:
    a) The smallest profit in the matrix
    b) The average profit
    c) The largest profit in the matrix
    d) The total profit
    Answer: c) The largest profit in the matrix
52. If the largest profit is 20 and a cell profit is 12, its converted cost is:
    a) 20
    b) 12
    c) 32
    d) 8
    Answer: d) 8
53. Why does subtracting each profit from the largest profit preserve the optimal assignment?
    a) It converts profit maximization into equivalent cost minimization
    b) It changes the number of agents
    c) It eliminates all constraints
    d) It makes every entry positive
    Answer: a) It converts profit maximization into equivalent cost minimization
54. A prohibited assignment may be represented by:
    a) A zero cost
    b) A very large cost (M)
    c) A negative cost
    d) A dummy column only
    Answer: b) A very large cost (M)
55. The purpose of assigning a very large cost to a prohibited cell is to:
    a) Encourage that assignment
    b) Make the problem balanced
    c) Prevent the assignment from being selected
    d) Increase the number of zeros
    Answer: c) Prevent the assignment from being selected
56. If a particular assignment is mandatory, the model may:
    a) Set its cost to infinity
    b) Delete the corresponding row
    c) Remove every other worker
    d) Fix the corresponding variable equal to one
    Answer: d) Fix the corresponding variable equal to one
57. If (x_{23}=1), this means:
    a) Agent 2 is assigned to job 3
    b) Agent 3 is assigned to job 2
    c) Job 2 has cost 3
    d) Agent 2 performs three jobs
    Answer: a) Agent 2 is assigned to job 3
58. If (\sum_{j=1}^{4}x_{2j}=1), the constraint applies to:
    a) Job 2
    b) Agent 2
    c) All jobs together only
    d) The objective function
    Answer: b) Agent 2
59. If (\sum_{i=1}^{4}x_{i3}=1), the constraint applies to:
    a) Agent 3
    b) The third cost coefficient
    c) Job 3
    d) All agents receiving job 3
    Answer: c) Job 3
60. The standard assignment formulation assumes that each agent has a capacity of:
    a) Zero
    b) Unlimited units
    c) Two units
    d) One unit
    Answer: d) One unit
61. The standard assignment formulation assumes that each job requires:
    a) One unit
    b) No agent
    c) Several units
    d) Unlimited capacity
    Answer: a) One unit
62. A solution assigning one worker to two jobs would violate:
    a) A column constraint only
    b) A row constraint
    c) The objective function
    d) A dummy constraint only
    Answer: b) A row constraint
63. A solution assigning two workers to the same job would violate:
    a) A row constraint only
    b) The binary restrictions only
    c) A column constraint
    d) The cost-minimization objective
    Answer: c) A column constraint
64. The total number of actual assignments in a balanced (n\times n) problem is:
    a) (n^2)
    b) (2n)
    c) (n-1)
    d) (n)
    Answer: d) (n)
65. In a (3\times3) balanced assignment problem, a feasible solution contains:
    a) Three selected assignments
    b) Nine selected assignments
    c) Six selected assignments
    d) One selected assignment
    Answer: a) Three selected assignments
66. Which condition defines a feasible assignment?
    a) Every row and column contains several assignments
    b) Exactly one selected cell appears in every row and column
    c) Only the lowest-cost cell is selected
    d) Every zero is selected
    Answer: b) Exactly one selected cell appears in every row and column
67. The linear-programming relaxation of an assignment problem often yields integer solutions because of:
    a) The use of Big M
    b) Artificial variables
    c) The special structure and total unimodularity of its constraint matrix
    d) The presence of negative costs
    Answer: c) The special structure and total unimodularity of its constraint matrix
68. Which statement about the assignment model is correct?
    a) Binary restrictions are never needed conceptually
    b) Every cost must be positive
    c) The number of jobs must always exceed agents
    d) Its feasible solutions have a one-to-one matching structure
    Answer: d) Its feasible solutions have a one-to-one matching structure
69. If all assignment costs are equal, then:
    a) Every feasible complete assignment has the same total cost
    b) No feasible assignment exists
    c) Only one assignment is optimal
    d) The model becomes unbalanced
    Answer: a) Every feasible complete assignment has the same total cost
70. Multiple optimal assignments occur when:
    a) No zero appears in the reduced matrix
    b) More than one complete assignment gives the same optimal value
    c) The matrix is rectangular
    d) All assignments are prohibited
    Answer: b) More than one complete assignment gives the same optimal value

Section C: Relationship Between Transportation and Assignment Models

71. The assignment problem is considered a special transportation problem because:
    a) It always has multiple sources
    b) It uses the same costs only
    c) Every source supply and destination demand equals one
    d) It requires the Northwest Corner Method
    Answer: c) Every source supply and destination demand equals one
72. In an assignment model, the supply of each agent is:
    a) Equal to the number of jobs
    b) Unlimited
    c) Zero
    d) One unit
    Answer: d) One unit
73. In an assignment model, the demand of each job is:
    a) One unit
    b) Equal to total supply
    c) Greater than one
    d) Unlimited
    Answer: a) One unit
74. A transportation model generally allows each source to ship:
    a) Exactly one unit only
    b) Multiple units
    c) No units
    d) One unit to every destination
    Answer: b) Multiple units
75. An assignment model restricts each source-agent to:
    a) Several destinations
    b) An unlimited number of jobs
    c) One destination or job
    d) No real job
    Answer: c) One destination or job
76. Which statement distinguishes a transportation model from an assignment model?
    a) Transportation models have no costs
    b) Assignment models have no constraints
    c) Transportation models cannot be balanced
    d) Transportation supplies and demands may exceed one
    Answer: d) Transportation supplies and demands may exceed one
77. In transportation terminology, agents in an assignment problem correspond to:
    a) Sources
    b) Destinations
    c) Routes
    d) Demands
    Answer: a) Sources
78. In transportation terminology, jobs in an assignment problem correspond to:
    a) Sources
    b) Destinations
    c) Supplies
    d) Basic variables
    Answer: b) Destinations
79. The assignment-cost matrix corresponds to the transportation model’s:
    a) Supply column
    b) Demand row
    c) Unit transportation-cost matrix
    d) Allocation table only
    Answer: c) Unit transportation-cost matrix
80. A balanced assignment problem is analogous to a balanced transportation problem because:
    a) All costs are equal
    b) Every cell is selected
    c) The number of variables equals the number of constraints
    d) Total supply equals total demand
    Answer: d) Total supply equals total demand
81. For an (n\times n) assignment problem, total supply equals:
    a) (n)
    b) (n^2)
    c) (2n)
    d) One
    Answer: a) (n)
82. For an (n\times n) assignment problem, total demand equals:
    a) (n^2)
    b) (n)
    c) One
    d) (2n)
    Answer: b) (n)
83. Which transportation constraint corresponds to assigning each agent once?
    a) Demand constraint
    b) Cost constraint
    c) Source-supply constraint with supply one
    d) Nonnegativity constraint only
    Answer: c) Source-supply constraint with supply one
84. Which transportation constraint corresponds to filling each job once?
    a) Source constraint
    b) Objective constraint
    c) Route constraint
    d) Destination-demand constraint with demand one
    Answer: d) Destination-demand constraint with demand one
85. Which method is especially efficient for assignment problems compared with general transportation methods?
    a) Hungarian Method
    b) Northwest Corner Method
    c) Least-Cost Method
    d) Vogel’s Approximation Method
    Answer: a) Hungarian Method
86. The Northwest Corner Method is mainly designed to obtain:
    a) An optimal assignment directly
    b) An initial feasible transportation solution
    c) Shadow prices
    d) A project schedule
    Answer: b) An initial feasible transportation solution
87. Why is the Hungarian Method preferred for many assignment problems?
    a) It permits multiple assignments in a row
    b) It ignores one-to-one restrictions
    c) It exploits the special structure of the assignment matrix
    d) It requires no cost data
    Answer: c) It exploits the special structure of the assignment matrix
88. A general transportation allocation (x_{ij}=5) would mean:
    a) Five workers are assigned to one job
    b) The assignment is prohibited
    c) The route cost is five
    d) Five units are shipped from source (i) to destination (j)
    Answer: d) Five units are shipped from source (i) to destination (j)
89. In a standard assignment problem, (x_{ij}=5) is invalid because:
    a) Assignment variables are binary
    b) Costs cannot equal five
    c) Every row must total five
    d) Jobs require five agents
    Answer: a) Assignment variables are binary
90. Which model permits splitting a source’s supply among multiple destinations?
    a) Standard assignment model
    b) General transportation model
    c) Binary matching model only
    d) Hungarian reduction model
    Answer: b) General transportation model
91. Which model normally prohibits splitting an agent among several jobs?
    a) General transportation model
    b) Transshipment model
    c) Assignment model
    d) Inventory model
    Answer: c) Assignment model
92. A transportation model becomes an assignment model when:
    a) All costs are zero
    b) There is only one source
    c) Supplies and demands are unrestricted
    d) Every supply and demand value equals one
    Answer: d) Every supply and demand value equals one
93. A dummy destination in a transportation model is similar to a dummy job in an assignment model because both:
    a) Balance excess supply or excess agents
    b) Increase demand
    c) Create real operating costs
    d) Make the model unbounded
    Answer: a) Balance excess supply or excess agents
94. A dummy source in a transportation model is similar to a dummy worker because both:
    a) represent excess supply
    b) Balance excess demand or excess jobs
    c) increase every cost
    d) remove real destinations
    Answer: b) Balance excess demand or excess jobs
95. In both assignment and transportation models, the objective often seeks to:
    a) Maximize the number of constraints
    b) Equalize all routes
    c) Minimize total cost
    d) Increase unused capacity
    Answer: c) Minimize total cost
96. Which statement is true of both assignment and transportation models?
    a) Every variable must be binary
    b) Every supply must equal one
    c) Every problem must be square
    d) Both use costs associated with source-destination combinations
    Answer: d) Both use costs associated with source-destination combinations
97. Which statement is unique to the standard assignment model?
    a) Every row and column must contain exactly one selected assignment
    b) Total supply must equal total demand
    c) Costs are associated with routes
    d) Dummy rows may be added
    Answer: a) Every row and column must contain exactly one selected assignment
98. Which statement best summarizes the relationship between the models?
    a) They are unrelated optimization models
    b) The assignment model is a unit-supply, unit-demand transportation model
    c) The transportation model is a special case of assignment
    d) Both require identical solution methods
    Answer: b) The assignment model is a unit-supply, unit-demand transportation model
99. If a problem requires assigning six employees to six jobs, it can be formulated as:
    a) An inventory model only
    b) A queuing model
    c) A (6\times6) assignment or unit transportation model
    d) A replacement model
    Answer: c) A (6\times6) assignment or unit transportation model
100. Which statement best describes an assignment problem?
     a) It distributes unlimited quantities across routes
     b) It determines order quantities under uncertain demand
     c) It schedules activities in a project network
     d) It finds the best one-to-one matching between agents and tasks
     Answer: d) It finds the best one-to-one matching between agents and tasks