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Operation Research – Transportation Problem : Balanced and Unbalanced

  1. What is the primary objective of a standard transportation problem?
    a) To determine the least-cost shipping plan from sources to destinations
    b) To maximize the number of transportation routes
    c) To eliminate all warehouses
    d) To ensure every source ships an equal quantity
    Answer: a) To determine the least-cost shipping plan from sources to destinations
  2. In a transportation model, an origin usually represents:
    a) A customer with a demand requirement
    b) A source having an available supply
    c) A route that cannot be used
    d) The total transportation cost
    Answer: b) A source having an available supply
  3. In a transportation model, a destination normally represents:
    a) A source of raw materials
    b) A transportation carrier
    c) A location having a demand requirement
    d) A dummy allocation only
    Answer: c) A location having a demand requirement
  4. The decision variable (x_{ij}) usually represents:
    a) The supply available at source (i)
    b) The demand required at destination (j)
    c) The cost of opening a warehouse
    d) The quantity transported from source (i) to destination (j)
    Answer: d) The quantity transported from source (i) to destination (j)
  5. In a cost-minimization transportation model, (c_{ij}) represents:
    a) The unit transportation cost from source (i) to destination (j)
    b) The total demand at destination (j)
    c) The amount allocated to a dummy destination
    d) The number of basic variables
    Answer: a) The unit transportation cost from source (i) to destination (j)
  6. Which expression represents the usual transportation-cost objective?
    a) Minimize (\sum a_i+\sum b_j)
    b) Minimize (\sum_i\sum_j c_{ij}x_{ij})
    c) Maximize (\sum_i\sum_j x_{ij}) only
    d) Minimize (\sum_i c_{ij}) without allocations
    Answer: b) Minimize (\sum_i\sum_j c_{ij}x_{ij})
  7. A supply constraint ensures that:
    a) Every transportation cost is positive
    b) Every destination receives the same quantity
    c) Shipments from a source respect its available supply
    d) All routes are used
    Answer: c) Shipments from a source respect its available supply
  8. A demand constraint ensures that:
    a) Every source uses the same carrier
    b) Every source has equal capacity
    c) Total transportation cost is zero
    d) Each destination receives its required quantity
    Answer: d) Each destination receives its required quantity
  9. The nonnegativity restriction requires:
    a) Every shipment quantity to be zero or positive
    b) Every cost to be strictly positive
    c) Supply to be larger than demand
    d) Every route to receive an allocation
    Answer: a) Every shipment quantity to be zero or positive
  10. A transportation problem is a special type of:
    a) Dynamic-programming problem only
    b) Linear-programming problem
    c) Queuing problem
    d) Game-theory problem
    Answer: b) Linear-programming problem
  11. A transportation table normally contains:
    a) Only the transportation costs
    b) Only supplies and demands
    c) Unit values, source supplies and destination demands
    d) Only the final optimal allocations
    Answer: c) Unit values, source supplies and destination demands
  12. The feasible region of a transportation problem is determined by:
    a) The highest unit cost only
    b) The lowest-cost route only
    c) The number of dummy cells
    d) Supply, demand and nonnegativity constraints
    Answer: d) Supply, demand and nonnegativity constraints
  13. A feasible transportation solution must:
    a) Satisfy all applicable supply and demand constraints
    b) Use only the cheapest cell in every row
    c) Have the minimum possible total cost
    d) Allocate to every cell
    Answer: a) Satisfy all applicable supply and demand constraints
  14. An optimal transportation solution is a feasible solution that:
    a) Contains the largest number of allocations
    b) Produces the best objective-function value
    c) Uses every available route
    d) Has no zero-cost cells
    Answer: b) Produces the best objective-function value
  15. An initial basic feasible solution is:
    a) Always the final optimal solution
    b) A solution containing only dummy allocations
    c) A starting feasible allocation used for further optimization
    d) A transportation table without costs
    Answer: c) A starting feasible allocation used for further optimization
  16. Which method is not primarily an initial-solution method?
    a) Northwest Corner Method
    b) Least-Cost Method
    c) Vogel’s Approximation Method
    d) MODI Method
    Answer: d) MODI Method
  17. The Northwest Corner Method begins allocation in:
    a) The top-left available cell
    b) The cell having the highest cost
    c) The bottom-right cell
    d) The row having the largest supply
    Answer: a) The top-left available cell
  18. The Least-Cost Method gives initial priority to:
    a) The cell with the largest demand
    b) The available cell with the lowest unit cost
    c) The source with the smallest supply
    d) The top-left available cell
    Answer: b) The available cell with the lowest unit cost
  19. Vogel’s Approximation Method uses penalties based on:
    a) The difference between total supply and demand
    b) The number of occupied cells
    c) The difference between the two smallest costs in each row or column
    d) The largest allocation in each row
    Answer: c) The difference between the two smallest costs in each row or column
  20. Which method is generally used to test and improve a transportation solution?
    a) Northwest Corner Method only
    b) Row-Minima Method only
    c) Column-Minima Method only
    d) MODI or stepping-stone method
    Answer: d) MODI or stepping-stone method

  1. A transportation problem is balanced when:
    a) Total supply equals total demand
    b) Every source has the same supply
    c) Every destination has the same demand
    d) All unit costs are equal
    Answer: a) Total supply equals total demand
  2. If supplies are 40, 60 and 50 units, total supply equals:
    a) 100 units
    b) 150 units
    c) 160 units
    d) 200 units
    Answer: b) 150 units
  3. Supplies are 30, 50 and 70, while demands are 40, 60 and 50. The problem is:
    a) Infeasible
    b) A maximization problem
    c) Balanced
    d) Degenerate automatically
    Answer: c) Balanced
  4. In a balanced transportation problem, total allocated quantity equals:
    a) The number of destinations
    b) The number of occupied cells
    c) The total transportation cost
    d) Both total supply and total demand
    Answer: d) Both total supply and total demand
  5. Which statement is true for a balanced transportation problem?
    a) No dummy source or destination is required solely for balancing
    b) Every cell must have a positive allocation
    c) Supply constraints can be ignored
    d) Demand may remain unsatisfied
    Answer: a) No dummy source or destination is required solely for balancing
  6. Sources have supplies of 25, 35 and 40 units. Destinations demand 20, 30 and 50 units. The model is:
    a) Unbalanced by 10 units
    b) Balanced at 100 units
    c) Balanced at 90 units
    d) Unbalanced by 20 units
    Answer: b) Balanced at 100 units
  7. In a balanced model with (m) sources and (n) destinations, a nondegenerate basic feasible solution normally has:
    a) (mn) occupied cells
    b) (m+n) occupied cells
    c) (m+n-1) occupied cells
    d) (m-n) occupied cells
    Answer: c) (m+n-1) occupied cells
  8. For a transportation problem with 3 sources and 4 destinations, the required number of basic cells for a nondegenerate solution is:
    a) 4
    b) 5
    c) 7
    d) 6
    Answer: d) 6
  9. A basic feasible solution is degenerate when the number of occupied independent cells is:
    a) Less than (m+n-1)
    b) Greater than (mn)
    c) Equal to (m+n)
    d) Equal to the total demand
    Answer: a) Less than (m+n-1)
  10. Degeneracy is often handled by placing:
    a) A large allocation in every empty cell
    b) A very small quantity, commonly denoted by epsilon, in an eligible empty cell
    c) A dummy destination in every model
    d) A negative allocation in an empty cell
    Answer: b) A very small quantity, commonly denoted by epsilon, in an eligible empty cell
  11. An epsilon allocation should be placed so that it:
    a) Changes total supply and demand
    b) Increases transportation cost significantly
    c) Maintains independent basic-cell structure without creating a closed loop
    d) Is assigned to the highest-cost route
    Answer: c) Maintains independent basic-cell structure without creating a closed loop
  12. In the Northwest Corner Method, the allocation to a selected cell equals:
    a) The larger of remaining supply and demand
    b) The average of remaining supply and demand
    c) The transportation cost in the cell
    d) The smaller of remaining supply and demand
    Answer: d) The smaller of remaining supply and demand
  13. If a source has 45 units and a destination needs 30 units, the first allowable allocation is:
    a) 30 units
    b) 45 units
    c) 15 units
    d) 75 units
    Answer: a) 30 units
  14. After allocating 30 units from a source having 45 units, the remaining source supply is:
    a) 10 units
    b) 15 units
    c) 30 units
    d) 45 units
    Answer: b) 15 units
  15. If both the remaining supply and demand become zero after one allocation, then:
    a) The problem becomes unbalanced
    b) The solution is automatically optimal
    c) Both a row and a column are satisfied simultaneously
    d) All other cells must be allocated zero permanently
    Answer: c) Both a row and a column are satisfied simultaneously
  16. Which initial method ignores transportation costs when choosing the first cell?
    a) Least-Cost Method
    b) Vogel’s Approximation Method
    c) Matrix-Minima Method
    d) Northwest Corner Method
    Answer: d) Northwest Corner Method
  17. A limitation of the Northwest Corner Method is that it:
    a) May produce an expensive initial solution because it ignores costs
    b) Cannot produce a feasible solution
    c) Applies only to maximization models
    d) Requires profit conversion first
    Answer: a) May produce an expensive initial solution because it ignores costs
  18. Compared with the Northwest Corner Method, the Least-Cost Method explicitly considers:
    a) Only source capacities
    b) Unit transportation costs
    c) Only destination demands
    d) Row and column totals without costs
    Answer: b) Unit transportation costs
  19. Vogel’s Approximation Method commonly gives:
    a) An infeasible solution
    b) A solution with no basic cells
    c) A relatively good initial solution near the optimum
    d) A solution that must contain a dummy source
    Answer: c) A relatively good initial solution near the optimum
  20. If two cells have the same lowest cost during the Least-Cost Method, the analyst may:
    a) Stop because no solution exists
    b) Select both without checking supply
    c) Convert the problem into maximization
    d) Use a consistent tie-breaking rule
    Answer: d) Use a consistent tie-breaking rule

  1. A transportation problem is unbalanced when:
    a) Total supply does not equal total demand
    b) Every transportation cost is different
    c) The number of sources differs from the number of destinations
    d) There is more than one optimal solution
    Answer: a) Total supply does not equal total demand
  2. If total supply is 500 units and total demand is 430 units, the excess supply is:
    a) 30 units
    b) 70 units
    c) 430 units
    d) 930 units
    Answer: b) 70 units
  3. When total supply exceeds total demand, balancing normally requires adding:
    a) A real source
    b) A dummy source
    c) A dummy destination
    d) A prohibited route
    Answer: c) A dummy destination
  4. When total demand exceeds total supply, balancing normally requires adding:
    a) A new cost row containing real production
    b) A dummy destination
    c) An additional real warehouse
    d) A dummy source
    Answer: d) A dummy source
  5. The demand assigned to a dummy destination equals:
    a) Excess total supply
    b) Total real demand
    c) The lowest destination demand
    d) The largest source supply
    Answer: a) Excess total supply
  6. The supply assigned to a dummy source equals:
    a) Total real supply
    b) The amount by which demand exceeds supply
    c) The largest destination demand
    d) The smallest source supply
    Answer: b) The amount by which demand exceeds supply
  7. Supplies are 60, 80 and 100 units, while total demand is 210 units. The dummy destination demand should be:
    a) 20 units
    b) 25 units
    c) 30 units
    d) 40 units
    Answer: c) 30 units
  8. Total demand is 350 units and total supply is 300 units. The dummy source supply should be:
    a) 25 units
    b) 30 units
    c) 40 units
    d) 50 units
    Answer: d) 50 units
  9. Dummy-cell transportation costs are ordinarily entered as:
    a) Zero, unless a penalty or opportunity cost is specified
    b) The highest real transportation cost
    c) Negative infinity
    d) The average of all real costs
    Answer: a) Zero, unless a penalty or opportunity cost is specified
  10. An allocation to a dummy destination usually represents:
    a) Unmet customer demand
    b) Unused or surplus supply
    c) Additional profit
    d) A prohibited shipment
    Answer: b) Unused or surplus supply
  11. An allocation from a dummy source normally represents:
    a) Surplus production
    b) Physical shipment from a new factory
    c) Unmet demand or externally supplied quantity
    d) Excess transportation capacity
    Answer: c) Unmet demand or externally supplied quantity
  12. Why is a dummy row or column added?
    a) To increase the number of profitable routes
    b) To reduce all real costs
    c) To create degeneracy deliberately
    d) To make total supply equal total demand
    Answer: d) To make total supply equal total demand
  13. After balancing an unbalanced model, it can generally be solved using:
    a) Standard transportation-solution methods
    b) Only the simplex method in tableau form
    c) Only assignment algorithms
    d) Only network diagrams
    Answer: a) Standard transportation-solution methods
  14. Supplies are 50 and 70 units. Demands are 30, 40 and 20 units. Which adjustment is needed?
    a) Add a dummy source of 30 units
    b) Add a dummy destination of 30 units
    c) Add a dummy destination of 90 units
    d) No adjustment is needed
    Answer: b) Add a dummy destination of 30 units
  15. Supplies are 40, 30 and 20. Demands are 50 and 60. Which adjustment is appropriate?
    a) Add a dummy destination of 20
    b) Add a dummy destination of 110
    c) Add a dummy source of 20
    d) Add a dummy source of 90
    Answer: c) Add a dummy source of 20
  16. A nonzero penalty in a dummy-source cell may represent:
    a) The cost of surplus capacity
    b) Normal internal shipping cost
    c) A discount for unmet demand
    d) The cost associated with shortage or external procurement
    Answer: d) The cost associated with shortage or external procurement
  17. If excess supply has a disposal cost, dummy-destination cells should contain:
    a) The relevant disposal or unused-capacity cost
    b) Negative allocations
    c) The destination demand
    d) No numerical value
    Answer: a) The relevant disposal or unused-capacity cost
  18. Which statement about an unbalanced problem is correct?
    a) It is always infeasible
    b) It can be balanced mathematically with a suitable dummy row or column
    c) It cannot be solved by Vogel’s Approximation Method
    d) It must always be converted into an assignment problem
    Answer: b) It can be balanced mathematically with a suitable dummy row or column
  19. In a balanced table created from an unbalanced problem, dummy allocations should:
    a) Always be ignored when evaluating the business meaning
    b) Be treated as real physical shipments
    c) Be interpreted according to the surplus or shortage they represent
    d) Always receive the highest transportation cost
    Answer: c) Be interpreted according to the surplus or shortage they represent
  20. If total supply and demand differ because demand may legally remain unmet, the dummy-source cost should ideally reflect:
    a) Zero under every circumstance
    b) The average real transportation cost
    c) The highest supply value
    d) The relevant shortage or lost-opportunity penalty
    Answer: d) The relevant shortage or lost-opportunity penalty

  1. A maximization transportation problem seeks to:
    a) Maximize total profit, revenue or benefit from allocations
    b) Minimize the number of sources
    c) Equalize all profit values
    d) Maximize unused supply
    Answer: a) Maximize total profit, revenue or benefit from allocations
  2. The entries in a maximization transportation table commonly represent:
    a) Supply capacities only
    b) Unit profits, returns or benefits
    c) Only fixed facility costs
    d) Dummy quantities
    Answer: b) Unit profits, returns or benefits
  3. A common way to solve a maximization transportation problem is to:
    a) Ignore the profit matrix
    b) Convert supply values into costs
    c) Transform the profit matrix into an equivalent loss or cost matrix
    d) Add all profit values together before allocation
    Answer: c) Transform the profit matrix into an equivalent loss or cost matrix
  4. A standard conversion from profit to opportunity-loss values is:
    a) Profit value minus minimum profit
    b) Maximum profit plus each profit
    c) Each profit divided by maximum profit
    d) Maximum table profit minus each cell’s profit
    Answer: d) Maximum table profit minus each cell’s profit
  5. If the largest unit profit is 50 and a cell profit is 32, its converted cost is:
    a) 18
    b) 32
    c) 50
    d) 82
    Answer: a) 18
  6. If the maximum profit is 70 and a cell profit is 70, the converted cost is:
    a) 70
    b) 0
    c) 1
    d) 140
    Answer: b) 0
  7. If the maximum profit is 60 and a route earns 25 per unit, its opportunity loss is:
    a) 25
    b) 30
    c) 35
    d) 85
    Answer: c) 35
  8. After converting profits into costs, the next step is generally to:
    a) Maximize the converted values
    b) Remove the supply constraints
    c) Ignore dummy requirements
    d) Solve the converted table as a minimization transportation problem
    Answer: d) Solve the converted table as a minimization transportation problem
  9. Why does subtracting each profit from the same maximum preserve the optimal allocation relationship?
    a) Higher original profits become lower converted opportunity costs
    b) It makes every route equally attractive
    c) It changes all supply quantities
    d) It eliminates destination demands
    Answer: a) Higher original profits become lower converted opportunity costs
  10. In a maximization model, the best original unit-profit cell becomes:
    a) The highest converted-cost cell
    b) A zero-cost cell after maximum-profit conversion
    c) A dummy cell automatically
    d) A prohibited route
    Answer: b) A zero-cost cell after maximum-profit conversion
  11. The final objective value of a maximization problem should be calculated using:
    a) Only the converted cost matrix
    b) Only dummy allocations
    c) The original profit values and final shipment quantities
    d) The largest profit value multiplied by total demand
    Answer: c) The original profit values and final shipment quantities
  12. Converted costs are mainly used to:
    a) Report the final total profit
    b) Replace the original supply values
    c) Change the number of destinations
    d) identify the optimal allocation using minimization procedures
    Answer: d) Identify the optimal allocation using minimization procedures
  13. A maximization problem with total supply equal to total demand is:
    a) A balanced maximization transportation problem
    b) An assignment problem automatically
    c) An infeasible problem
    d) A minimization model
    Answer: a) A balanced maximization transportation problem
  14. An unbalanced maximization transportation model should generally be:
    a) Solved without considering the imbalance
    b) Balanced using an appropriate dummy source or destination
    c) Converted into a queuing model
    d) Reduced to one destination
    Answer: b) Balanced using an appropriate dummy source or destination
  15. In a profit-maximization model with excess supply, a dummy destination normally represents:
    a) Additional customer demand
    b) Profit earned from all unused units
    c) Unused supply or unassigned capacity
    d) A required real shipment
    Answer: c) Unused supply or unassigned capacity
  16. If unused units earn no profit, original profit entries for a dummy destination should generally be:
    a) Equal to the maximum profit
    b) Equal to the minimum positive profit
    c) Negative infinity
    d) Zero
    Answer: d) Zero
  17. If unused supply incurs a loss, the dummy-destination value should reflect:
    a) The relevant negative profit or penalty
    b) The highest available route profit
    c) The total source supply
    d) No value under any circumstances
    Answer: a) The relevant negative profit or penalty
  18. A direct maximization approach would usually favor cells with:
    a) The lowest original profit
    b) High original unit profits, subject to feasibility
    c) The largest dummy allocation
    d) The smallest source supply only
    Answer: b) High original unit profits, subject to feasibility
  19. The optimal allocation in a profit model must still satisfy:
    a) Only the profit objective
    b) Only the source having the highest profit
    c) Supply, demand and nonnegativity requirements
    d) Equal allocations in all routes
    Answer: c) Supply, demand and nonnegativity requirements
  20. Maximizing unit profit route by route without considering all constraints may:
    a) Always produce the optimal solution
    b) Eliminate the need for balancing
    c) Guarantee feasibility
    d) Produce a feasible but nonoptimal or even infeasible plan
    Answer: d) Produce a feasible but nonoptimal or even infeasible plan

  1. The MODI method is also known as the:
    a) (u-v) method
    b) Northwest Corner Method
    c) Row-Minima Method
    d) Hungarian Method
    Answer: a) (u-v) method
  2. In the MODI method for a minimization problem, potentials satisfy which relation for occupied cells?
    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})
  3. For an unoccupied cell in a minimization problem, the MODI opportunity value is commonly calculated as:
    a) (u_i+v_j+c_{ij})
    b) (c_{ij}+x_{ij})
    c) (c_{ij}-(u_i+v_j))
    d) (u_i-v_j-c_{ij})
    Answer: c) (c_{ij}-(u_i+v_j))
  4. Under the convention (\Delta_{ij}=c_{ij}-(u_i+v_j)), a minimization solution is optimal when all unoccupied-cell values are:
    a) Negative
    b) Equal to transportation allocations
    c) Greater than each supply
    d) Zero or positive
    Answer: d) Zero or positive
  5. A negative opportunity value in a minimization model indicates that:
    a) Total cost may be reduced by introducing that route
    b) The route is prohibited
    c) The current solution is necessarily degenerate
    d) Supply exceeds demand
    Answer: a) Total cost may be reduced by introducing that route
  6. In the stepping-stone method, an unused cell is evaluated by constructing:
    a) A straight path through every cell
    b) A closed loop through occupied cells with alternating plus and minus signs
    c) A dummy row and column
    d) A new supply constraint
    Answer: b) A closed loop through occupied cells with alternating plus and minus signs
  7. The entering cell in an improvement loop is marked with:
    a) A minus sign only
    b) No sign
    c) A plus sign
    d) Both plus and minus signs
    Answer: c) A plus sign
  8. The amount shifted around an improvement loop is limited by:
    a) The largest allocation in a plus cell
    b) Total supply
    c) Total demand
    d) The smallest allocation in a minus-position cell
    Answer: d) The smallest allocation in a minus-position cell
  9. After adjusting an improvement loop:
    a) Row supplies and column demands remain satisfied
    b) Total supply is increased
    c) One destination demand must become unmet
    d) All occupied cells become empty
    Answer: a) Row supplies and column demands remain satisfied
  10. An alternate optimal solution may exist when an unoccupied cell has:
    a) A strictly positive opportunity value only
    b) A zero opportunity value at an optimal solution
    c) A negative allocation
    d) The highest unit cost
    Answer: b) A zero opportunity value at an optimal solution
  11. A prohibited transportation route may be represented by:
    a) A zero cost that encourages allocation
    b) A negative supply
    c) A very large cost or an explicit route restriction
    d) A dummy destination only
    Answer: c) A very large cost or an explicit route restriction
  12. A transportation problem may become infeasible when:
    a) It contains a zero-cost cell
    b) It has more destinations than sources
    c) It is initially unbalanced
    d) Route restrictions prevent required supply-demand connections
    Answer: d) Route restrictions prevent required supply-demand connections
  13. A transshipment model differs from a basic transportation model because it:
    a) Permits intermediate transfer points between origins and destinations
    b) Has no objective function
    c) Cannot contain supply constraints
    d) Applies only to maximization
    Answer: a) Permits intermediate transfer points between origins and destinations
  14. The assignment problem is a special transportation problem in which:
    a) Every cost is zero
    b) Each supply and each demand is usually one unit
    c) There is only one destination
    d) Total supply must exceed demand
    Answer: b) Each supply and each demand is usually one unit
  15. Which condition is necessary before using standard balanced transportation algorithms?
    a) Every route must have a unique cost
    b) The number of sources must equal the number of destinations
    c) Aggregate supply must equal aggregate demand
    d) Every source must have equal supply
    Answer: c) Aggregate supply must equal aggregate demand
  16. Which statement about the number of sources and destinations is correct?
    a) They must always be equal
    b) Sources must always exceed destinations
    c) Destinations must always exceed sources
    d) They need not be equal as long as total supply and demand are properly handled
    Answer: d) They need not be equal as long as total supply and demand are properly handled

  1. A company ships 20 units on a route costing $4 per unit and 30 units on a route costing $6 per unit. What is the total transportation cost?
    a) $260
    b) $180
    c) $300
    d) $500
    Answer: a) $260
  2. A maximization solution allocates 40 units to a route earning $8 each and 20 units to a route earning $5 each. What is the total profit?
    a) $320
    b) $420
    c) $500
    d) $520
    Answer: b) $420
  3. Supplies are 80, 70 and 50 units. Demands are 40, 60, 55 and 45 units. Which statement is correct?
    a) Demand exceeds supply by 20 units
    b) Supply exceeds demand by 10 units
    c) The transportation problem is balanced
    d) A dummy source of 200 units is required
    Answer: c) The transportation problem is balanced
  4. Which statement best summarizes transportation-problem analysis?
    a) The cheapest individual route always determines the full solution
    b) Balancing alone guarantees optimality
    c) Maximization models cannot use transportation methods
    d) A valid plan must satisfy supply and demand while optimizing total cost, profit or benefit
    Answer: d) A valid plan must satisfy supply and demand while optimizing total cost, profit or benefit
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