1. What is the main objective of a standard transportation problem?
   a) To determine the least-cost plan for shipping goods from sources to destinations
   b) To maximize the number of transportation routes
   c) To ensure that every route receives an allocation
   d) To make all source capacities equal
   Answer: a) To determine the least-cost plan for shipping goods from sources to destinations
2. A transportation problem is a special type of:
   a) Queuing model
   b) Linear programming model
   c) Inventory model
   d) Game-theory model
   Answer: b) Linear programming model
3. In a transportation model, an origin usually represents:
   a) A customer location
   b) An unused route
   c) A source with an available supply
   d) A unit transportation cost
   Answer: c) A source with an available supply
4. In a transportation model, a destination normally represents:
   a) A production facility
   b) A source of raw materials
   c) A transportation carrier
   d) A location with a demand requirement
   Answer: d) A location with a demand requirement
5. Which of the following is a common application of the transportation model?
   a) Shipping products from factories to warehouses
   b) Determining employee salaries
   c) Selecting advertising messages
   d) Forecasting stock prices
   Answer: a) Shipping products from factories to warehouses
6. Which quantity generally moves from an origin to a destination?
   a) The destination demand
   b) A shipment or allocation
   c) The source label
   d) The objective-function sign
   Answer: b) A shipment or allocation
7. Which information is essential for setting up a transportation problem?
   a) Employee attendance data
   b) Product-advertising expenses
   c) Source supplies, destination demands and unit transportation costs
   d) Organizational reporting relationships
   Answer: c) Source supplies, destination demands and unit transportation costs
8. A transportation problem is balanced when:
   a) Every transportation cost is equal
   b) Every source has the same supply
   c) The number of sources equals the number of destinations
   d) Total supply equals total demand
   Answer: d) Total supply equals total demand
9. A transportation problem is unbalanced when:
   a) Total supply differs from total demand
   b) Some transportation costs are zero
   c) A route has no allocation
   d) The number of variables exceeds the number of constraints
   Answer: a) Total supply differs from total demand
10. When total supply exceeds total demand, the model is balanced by adding:
    a) A real source
    b) A dummy destination
    c) A prohibited route
    d) A second objective function
    Answer: b) A dummy destination
11. When total demand exceeds total supply, the model is balanced by adding:
    a) A dummy destination
    b) A new real warehouse
    c) A dummy source
    d) A negative transportation cost
    Answer: c) A dummy source
12. An allocation to a dummy destination usually represents:
    a) An additional customer order
    b) A real physical shipment
    c) A shortage at a destination
    d) Unused or surplus supply
    Answer: d) Unused or surplus supply
13. An allocation from a dummy source generally represents:
    a) Unmet demand or an externally supplied quantity
    b) Excess production capacity
    c) A prohibited route
    d) A transportation discount
    Answer: a) Unmet demand or an externally supplied quantity
14. Dummy-cell transportation costs are usually assigned a value of:
    a) The highest real transportation cost
    b) Zero, unless a penalty is specified
    c) Negative infinity
    d) The average transportation cost
    Answer: b) Zero, unless a penalty is specified
15. A feasible transportation solution must:
    a) Use every available route
    b) Have the lowest possible total cost
    c) Satisfy all supply and demand requirements
    d) Contain only positive costs
    Answer: c) Satisfy all supply and demand requirements

Section B: Mathematical Model Formulation

16. In a transportation model, (x_{ij}) normally represents:
    a) The supply at origin (i)
    b) The demand at destination (j)
    c) The cost of using origin (i)
    d) The quantity shipped from origin (i) to destination (j)
    Answer: d) The quantity shipped from origin (i) to destination (j)
17. In the mathematical model, (c_{ij}) usually represents:
    a) The unit transportation cost from origin (i) to destination (j)
    b) The amount shipped on a route
    c) The total supply available
    d) The number of destinations
    Answer: a) The unit transportation cost from origin (i) to destination (j)
18. If (a_i) denotes source supply, which expression represents its supply constraint in a balanced model?
    a) (\sum_i x_{ij}=a_i)
    b) (\sum_j x_{ij}=a_i)
    c) (\sum_j c_{ij}=a_i)
    d) (\sum_i c_{ij}x_{ij}=a_i)
    Answer: b) (\sum_j x_{ij}=a_i)
19. If (b_j) denotes destination demand, which expression represents its demand constraint?
    a) (\sum_j x_{ij}=b_j)
    b) (\sum_i c_{ij}=b_j)
    c) (\sum_i x_{ij}=b_j)
    d) (\sum_i a_i=b_j)
    Answer: c) (\sum_i x_{ij}=b_j)
20. The objective function of a cost-minimization transportation problem is:
    a) Maximize (\sum_i\sum_j x_{ij})
    b) Minimize (\sum_i a_i+\sum_j b_j)
    c) Maximize (\sum_i\sum_j c_{ij})
    d) Minimize (\sum_i\sum_j c_{ij}x_{ij})
    Answer: d) Minimize (\sum_i\sum_j c_{ij}x_{ij})
21. The restriction (x_{ij}\geq 0) means that:
    a) Shipment quantities cannot be negative
    b) Transportation costs cannot be negative
    c) Every route must receive a positive shipment
    d) Total supply must exceed total demand
    Answer: a) Shipment quantities cannot be negative
22. In a balanced transportation model, the sum of all source supplies must equal:
    a) The number of routes
    b) The sum of all destination demands
    c) The total transportation cost
    d) The number of basic variables
    Answer: b) The sum of all destination demands
23. If there are three sources and four destinations, how many decision variables are possible?
    a) 7
    b) 10
    c) 12
    d) 16
    Answer: c) 12
24. In a model with (m) sources and (n) destinations, the maximum number of shipment variables is:
    a) (m+n)
    b) (m-n)
    c) (m+n-1)
    d) (mn)
    Answer: d) (mn)
25. Which expression represents the balancing condition?
    a) (\sum_i a_i=\sum_j b_j)
    b) (\sum_i a_i>\sum_j b_j)
    c) (\sum_i a_i<\sum_j b_j)
    d) (\sum_i a_i+\sum_j b_j=0)
    Answer: a) (\sum_i a_i=\sum_j b_j)
26. If supplies are 30, 40 and 50 units, total supply is:
    a) 100 units
    b) 120 units
    c) 130 units
    d) 150 units
    Answer: b) 120 units
27. If demands are 25, 35, 30 and 30 units, total demand is:
    a) 90 units
    b) 110 units
    c) 120 units
    d) 130 units
    Answer: c) 120 units
28. Supplies total 180 units and demands total 160 units. What is required?
    a) A dummy source of 20 units
    b) A dummy destination of 160 units
    c) A dummy source of 180 units
    d) A dummy destination of 20 units
    Answer: d) A dummy destination of 20 units
29. Supplies total 200 units and demands total 240 units. What should be added?
    a) A dummy source of 40 units
    b) A dummy destination of 40 units
    c) A dummy source of 200 units
    d) A dummy destination of 240 units
    Answer: a) A dummy source of 40 units
30. Which assumption is commonly made in the basic transportation model?
    a) Shipment costs increase nonlinearly with volume
    b) Unit transportation cost on each route is known and constant
    c) Every route must carry the same quantity
    d) Sources and destinations must be equal in number
    Answer: b) Unit transportation cost on each route is known and constant

Section C: Setting Up a Transportation Table

31. In a transportation table, rows usually represent:
    a) Destination demands
    b) Transportation methods
    c) Origins or sources
    d) Opportunity costs
    Answer: c) Origins or sources
32. In a transportation table, columns usually represent:
    a) Source capacities
    b) Basic variables
    c) Dummy allocations only
    d) Destinations
    Answer: d) Destinations
33. Source supplies are generally displayed:
    a) At the right side of the corresponding rows
    b) At the top of the columns
    c) Inside the transportation-cost cells
    d) Under the destination labels
    Answer: a) At the right side of the corresponding rows
34. Destination demands are commonly displayed:
    a) Beside the source labels
    b) At the bottom of the corresponding columns
    c) In the objective-function row
    d) In a separate profit table only
    Answer: b) At the bottom of the corresponding columns
35. The unit transportation cost is normally written:
    a) Outside the table
    b) In the supply column
    c) Inside each route cell
    d) In the demand row only
    Answer: c) Inside each route cell
36. A shipment allocation is entered:
    a) In the row heading
    b) In the column heading
    c) Beside the source supply only
    d) In the appropriate origin-destination cell
    Answer: d) In the appropriate origin-destination cell
37. Before allocations are made, the first arithmetic check is to compare:
    a) Total supply with total demand
    b) The highest and lowest costs
    c) The number of rows with the number of columns
    d) The number of positive and zero costs
    Answer: a) Total supply with total demand
38. A prohibited route may be represented in the table by:
    a) A zero cost in all cases
    b) A very large cost or an explicit restriction
    c) A dummy destination
    d) A negative supply
    Answer: b) A very large cost or an explicit restriction
39. The total cost of a transportation plan is found by:
    a) Adding all table costs without using allocations
    b) Adding total supply and total demand
    c) Multiplying each allocation by its unit cost and summing the results
    d) Multiplying the largest cost by total supply
    Answer: c) Multiplying each allocation by its unit cost and summing the results
40. Which cell represents shipment from source 2 to destination 3?
    a) (x_{12})
    b) (x_{32})
    c) (x_{23})’s cost only
    d) (x_{23})
    Answer: d) (x_{23})
41. If cell ((2,3)) has a unit cost of $8 and an allocation of 15 units, its cost contribution is:
    a) $120
    b) $23
    c) $80
    d) $150
    Answer: a) $120
42. Which change converts an unbalanced transportation table into a balanced one?
    a) Removing its highest-cost route
    b) Adding a suitable dummy row or column
    c) Dividing every supply by total demand
    d) Replacing all costs with zeros
    Answer: b) Adding a suitable dummy row or column
43. If a dummy destination is added, its demand equals:
    a) Total destination demand
    b) The smallest source supply
    c) Excess supply
    d) The largest source supply
    Answer: c) Excess supply
44. If a dummy source is added, its supply equals:
    a) Total source supply
    b) The highest destination demand
    c) Excess supply
    d) Excess demand
    Answer: d) Excess demand
45. After completing an allocation table, each row total should equal:
    a) Its corresponding source supply
    b) The total transportation cost
    c) The number of destinations
    d) The highest allocation in the row
    Answer: a) Its corresponding source supply

Section D: Basic Feasible Solutions

46. A basic feasible solution is:
    a) A solution that is always optimal
    b) A feasible starting allocation with an appropriate set of basic cells
    c) A solution containing shipments in every cell
    d) A table containing only costs
    Answer: b) A feasible starting allocation with an appropriate set of basic cells
47. For a transportation problem with (m) sources and (n) destinations, a nondegenerate basic feasible solution contains:
    a) (mn) basic cells
    b) (m+n) basic cells
    c) (m+n-1) basic cells
    d) (m-n) basic cells
    Answer: c) (m+n-1) basic cells
48. In a 3-source, 5-destination problem, the required number of basic cells is:
    a) 5
    b) 6
    c) 8
    d) 7
    Answer: d) 7
49. In a 4-source, 4-destination problem, a nondegenerate solution should have:
    a) 7 basic cells
    b) 8 basic cells
    c) 12 basic cells
    d) 16 basic cells
    Answer: a) 7 basic cells
50. Which statement describes feasibility?
    a) The solution has the minimum total cost
    b) All supply and demand constraints are satisfied
    c) Every route has an allocation
    d) All opportunity costs equal zero
    Answer: b) All supply and demand constraints are satisfied
51. A basic cell is generally a cell that:
    a) Has the lowest cost in its row
    b) Is located on the table boundary
    c) Contains an allocation or an epsilon basic quantity
    d) Has zero demand
    Answer: c) Contains an allocation or an epsilon basic quantity
52. An unoccupied cell is normally called:
    a) A dummy cell
    b) A supply cell
    c) A prohibited cell
    d) A nonbasic cell
    Answer: d) A nonbasic cell
53. A feasible solution may fail to be optimal because:
    a) Another feasible allocation may have a lower total cost
    b) It satisfies all requirements
    c) It has (m+n-1) basic cells
    d) It includes positive allocations
    Answer: a) Another feasible allocation may have a lower total cost
54. Independence of basic cells means that:
    a) All basic cells have different costs
    b) The selected basic cells do not form a closed loop among themselves
    c) Every row has one allocation
    d) Every column has two allocations
    Answer: b) The selected basic cells do not form a closed loop among themselves
55. If a solution has more than (m+n-1) positive allocations, it is not:
    a) Feasible
    b) Balanced
    c) A basic solution in the usual transportation sense
    d) Cost minimizing
    Answer: c) A basic solution in the usual transportation sense
56. The main role of an initial basic feasible solution is to:
    a) Prove that every route is profitable
    b) Replace the mathematical formulation
    c) Eliminate all dummy variables
    d) Provide a starting point for optimality testing
    Answer: d) Provide a starting point for optimality testing
57. Which method always begins in the upper-left cell?
    a) Northwest Corner Method
    b) Vogel’s Approximation Method
    c) MODI Method
    d) Stepping-Stone Method
    Answer: a) Northwest Corner Method
58. Which method selects the currently available cell with the lowest unit cost?
    a) Northwest Corner Method
    b) Least-Cost Method
    c) MODI Method
    d) Row-Reduction Method
    Answer: b) Least-Cost Method
59. Which initial-solution method uses row and column penalties?
    a) Northwest Corner Method
    b) Stepping-Stone Method
    c) Vogel’s Approximation Method
    d) MODI Method
    Answer: c) Vogel’s Approximation Method
60. Which technique is primarily an optimality-testing method rather than an initial-allocation method?
    a) Northwest Corner Method
    b) Least-Cost Method
    c) Vogel’s Approximation Method
    d) MODI Method
    Answer: d) MODI Method

Section E: Developing an Initial Feasible Solution

61. Under the Northwest Corner Method, the allocation made in a selected cell is:
    a) The smaller of the remaining row supply and column demand
    b) The larger of the remaining supply and demand
    c) The unit transportation cost
    d) The average of supply and demand
    Answer: a) The smaller of the remaining row supply and column demand
62. If a source has 40 units remaining and a destination requires 25 units, the allocation is:
    a) 15 units
    b) 25 units
    c) 40 units
    d) 65 units
    Answer: b) 25 units
63. After allocating 25 units from a source with 40 units, the remaining supply is:
    a) 10 units
    b) 25 units
    c) 15 units
    d) 40 units
    Answer: c) 15 units
64. If a row’s supply becomes zero after an allocation, the next step is normally to:
    a) Increase the row supply
    b) Add a dummy destination
    c) Delete the entire table
    d) Cross out or leave the satisfied row
    Answer: d) Cross out or leave the satisfied row
65. If a column’s demand becomes zero, the column is:
    a) Satisfied and removed from further allocation consideration
    b) Converted into a source
    c) Assigned another shipment automatically
    d) Made into a dummy column
    Answer: a) Satisfied and removed from further allocation consideration
66. A major limitation of the Northwest Corner Method is that it:
    a) Cannot generate a feasible solution
    b) Ignores unit transportation costs
    c) Requires a maximization objective
    d) Always creates degeneracy
    Answer: b) Ignores unit transportation costs
67. The Least-Cost Method begins by identifying:
    a) The largest supply
    b) The smallest demand
    c) The lowest-cost available cell
    d) The northwest cell
    Answer: c) The lowest-cost available cell
68. If two cells have the same lowest cost, the analyst may:
    a) Stop the procedure
    b) Allocate to both without checking feasibility
    c) Convert the table into a maximization model
    d) Apply a consistent tie-breaking rule
    Answer: d) Apply a consistent tie-breaking rule
69. A common tie-breaking approach is to choose the cell that permits:
    a) The largest possible allocation
    b) The smallest possible allocation
    c) The highest total cost
    d) The addition of a dummy row
    Answer: a) The largest possible allocation
70. Vogel’s Approximation Method calculates a penalty as:
    a) The sum of the two lowest costs
    b) The difference between the two lowest costs
    c) The difference between supply and demand
    d) The highest cost minus the lowest allocation
    Answer: b) The difference between the two lowest costs
71. After computing penalties in VAM, the next allocation is made in:
    a) The row with the lowest supply
    b) The column with the greatest demand
    c) The least-cost cell in the row or column with the highest penalty
    d) The northwest available cell
    Answer: c) The least-cost cell in the row or column with the highest penalty
72. Why does VAM use penalties?
    a) To calculate dummy demand
    b) To identify degeneracy directly
    c) To determine the number of sources
    d) To estimate the opportunity lost by not using a low-cost route
    Answer: d) To estimate the opportunity lost by not using a low-cost route
73. Which initial-solution method often produces the best starting solution among the three standard methods?
    a) Vogel’s Approximation Method
    b) Northwest Corner Method
    c) Random allocation
    d) Row-maximum method
    Answer: a) Vogel’s Approximation Method
74. An initial feasible solution should be checked to ensure:
    a) All costs are different
    b) Row and column allocations match supply and demand
    c) Every cell has an allocation
    d) Every allocation is equal
    Answer: b) Row and column allocations match supply and demand
75. If the remaining supply equals the remaining demand in the final cell, the allocation should be:
    a) Zero
    b) Twice the remaining demand
    c) The common remaining quantity
    d) The unit transportation cost
    Answer: c) The common remaining quantity

Section F: Degeneracy in Transportation Problems

76. Degeneracy occurs when the number of basic occupied cells is:
    a) Greater than (mn)
    b) Equal to total supply
    c) Equal to (m+n)
    d) Less than (m+n-1)
    Answer: d) Less than (m+n-1)
77. For a 3-by-4 transportation table, a solution with only five basic cells is:
    a) Degenerate
    b) Nondegenerate
    c) Infeasible automatically
    d) Unbalanced automatically
    Answer: a) Degenerate
78. A 4-by-5 transportation problem requires how many independent basic cells?
    a) 7
    b) 8
    c) 9
    d) 20
    Answer: b) 8
79. Which symbol is commonly used to handle degeneracy?
    a) (\lambda)
    b) (\mu)
    c) (\epsilon)
    d) (\pi)
    Answer: c) (\epsilon)
80. Epsilon represents:
    a) A large penalty cost
    b) A negative shipment quantity
    c) Total excess supply
    d) A very small positive quantity treated as a basic allocation
    Answer: d) A very small positive quantity treated as a basic allocation
81. The purpose of adding epsilon is to:
    a) Increase the number of basic cells to (m+n-1)
    b) Change the total transportation cost significantly
    c) Make total supply exceed demand
    d) Remove a destination
    Answer: a) Increase the number of basic cells to (m+n-1)
82. An epsilon should be placed in an empty cell that:
    a) Has the highest transportation cost
    b) Does not create a closed loop with existing basic cells
    c) Is necessarily in the first row
    d) Belongs to a dummy destination
    Answer: b) Does not create a closed loop with existing basic cells
83. Why must an epsilon cell remain independent?
    a) To maximize the total transportation cost
    b) To create another dummy row
    c) To preserve a valid basic-variable structure
    d) To ensure that all rows have equal supply
    Answer: c) To preserve a valid basic-variable structure
84. Degeneracy may arise when:
    a) All costs are different
    b) Total supply exceeds demand
    c) The Northwest Corner Method is used
    d) A row and a column are satisfied simultaneously
    Answer: d) A row and a column are satisfied simultaneously
85. When a row and column are satisfied simultaneously, degeneracy may be prevented by:
    a) Retaining an appropriate zero or epsilon basic cell
    b) Increasing one demand value
    c) Deleting both the row and column without adjustment
    d) Changing all transportation costs
    Answer: a) Retaining an appropriate zero or epsilon basic cell
86. Does an epsilon allocation materially change the calculated transportation cost?
    a) Yes, it always doubles the cost
    b) No, it is treated as negligibly small
    c) Yes, it becomes the largest cost
    d) No, because its cost must always be negative
    Answer: b) No, it is treated as negligibly small
87. Degeneracy must be corrected before applying MODI because MODI requires:
    a) Equal row and column totals only
    b) Positive allocations in every cell
    c) (m+n-1) independent basic cells
    d) A maximization objective
    Answer: c) (m+n-1) independent basic cells
88. Which statement about degeneracy is correct?
    a) It means the model has no feasible solution
    b) It means total supply is unequal to total demand
    c) It means the current solution is always optimal
    d) It concerns an insufficient number of basic cells
    Answer: d) It concerns an insufficient number of basic cells

Section G: Optimality Check and Solution Improvement

89. The MODI method is also called the:
    a) (u-v) method
    b) Northwest Corner Method
    c) Matrix-Minima Method
    d) Row-Column Method
    Answer: a) (u-v) method
90. In MODI, for each basic cell, the potentials satisfy:
    a) (u_i-v_j=c_{ij})
    b) (u_i+v_j=c_{ij})
    c) (u_iv_j=c_{ij})
    d) (u_i/v_j=c_{ij})
    Answer: b) (u_i+v_j=c_{ij})
91. To begin finding MODI potentials, one potential is usually assigned:
    a) The largest transportation cost
    b) The total supply
    c) A value of zero
    d) A value of one
    Answer: c) A value of zero
92. For a nonbasic cell, the reduced or opportunity cost is commonly calculated as:
    a) (u_i+v_j+c_{ij})
    b) (u_i-v_j)
    c) (x_{ij}-c_{ij})
    d) (c_{ij}-(u_i+v_j))
    Answer: d) (c_{ij}-(u_i+v_j))
93. Under the convention (\Delta_{ij}=c_{ij}-(u_i+v_j)), a minimization solution is optimal when:
    a) Every nonbasic (\Delta_{ij}) is zero or positive
    b) Every nonbasic (\Delta_{ij}) is negative
    c) Every allocation is equal
    d) Every route is occupied
    Answer: a) Every nonbasic (\Delta_{ij}) is zero or positive
94. A negative reduced cost in a minimization problem indicates that:
    a) The current solution is infeasible
    b) Total cost may be reduced by introducing that cell
    c) The model is unbalanced
    d) The cell must remain empty
    Answer: b) Total cost may be reduced by introducing that cell
95. If several nonbasic cells have negative reduced costs, the entering cell is commonly chosen as the one with:
    a) The largest allocation
    b) The highest unit cost
    c) The most negative reduced cost
    d) The smallest destination demand
    Answer: c) The most negative reduced cost
96. After selecting an entering cell, the next step is to construct:
    a) A dummy row
    b) A new mathematical model
    c) A second transportation table
    d) A closed adjustment loop
    Answer: d) A closed adjustment loop
97. In an adjustment loop, signs are assigned:
    a) Alternately as plus and minus, beginning with plus at the entering cell
    b) As plus signs in every cell
    c) As minus signs in every occupied cell
    d) According to transportation costs
    Answer: a) Alternately as plus and minus, beginning with plus at the entering cell
98. The quantity transferred around the loop equals:
    a) The largest allocation in a plus cell
    b) The smallest allocation in a minus cell
    c) The average allocation in the loop
    d) The entering cell’s unit cost
    Answer: b) The smallest allocation in a minus cell
99. An alternate optimal solution may exist when, at optimality:
    a) A basic cell has a negative allocation
    b) Total supply exceeds total demand
    c) A nonbasic cell has zero reduced cost
    d) Every nonbasic cell has a positive reduced cost
    Answer: c) A nonbasic cell has zero reduced cost
100. Which statement best summarizes the transportation-solution procedure?
     a) Select the cheapest route and assign all supply to it
     b) Begin with MODI before finding a feasible solution
     c) Ignore degeneracy when testing optimality
     d) Formulate and balance the model, find an initial feasible solution, correct degeneracy and test for optimality
     Answer: d) Formulate and balance the model, find an initial feasible solution, correct degeneracy and test for optimality