Lesson 4D.1.3

4D.1.3 Transportation problems as linear programs Quiz: Pearson Edexcel Further Maths, Unit 40

20 questions

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Lesson 4D.1.3, Transportation problems as linear programs: 20 multiple choice questions for the Pearson Edexcel Further Maths (9FM0), Unit 40: Transportation problems, written with Revision Ninja.

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The 20 questions

  1. In the linear programming formulation of a transportation problem, what does the decision variable x_ij represent?

    • The number of units sent from source i to destination j.
    • The cost of sending one unit from source i to destination j.
    • The improvement index of cell (i, j).
    • The total supply available at source i.

  2. What is the objective of the transportation linear program with cost C_ij?

    • Maximise the sum of C_ij x_ij over all cells.
    • Maximise the total number of units shipped.
    • Minimise the number of occupied cells.
    • Minimise the sum of C_ij x_ij over all cells.

  3. Which statement describes a source constraint in the transportation linear program?

    • The units sent out of destination j equal its demand b_j.
    • The cost from source i equals its supply a_i.
    • The units sent out of source i equal its supply a_i, so the sum over j of x_ij = a_i.
    • The units received at source i equal its supply a_i.

  4. What condition must hold for a transportation problem to be balanced?

    • Total supply exceeds total demand by exactly one unit.
    • Total supply equals total demand.
    • Every cost is a whole number.
    • The number of sources equals the number of destinations.

  5. How is an unbalanced problem with supply exceeding demand made balanced?

    • Delete the cheapest source until the totals match.
    • Add a dummy destination with demand equal to the surplus and zero costs.
    • Add a dummy source with supply equal to the surplus and zero costs.
    • Double every demand value.

  6. For a problem with 2 sources and 3 destinations, how many decision variables does the linear program have?

    • 5
    • 9
    • 3
    • 6

  7. Why does a balanced transportation linear program have one redundant constraint?

    • The sum of the source constraints equals the sum of the destination constraints, so one constraint is implied by the others.
    • Each variable appears in exactly one constraint, so one of them is redundant.
    • The objective row can replace one of the constraints.
    • Slack variables always reduce the count of constraints by one.

  8. How many independent constraints does a balanced transportation problem with 2 sources and 3 destinations have?

    • 5
    • 4
    • 3
    • 6

  9. Supplies are 20 and 30, and demands are 10, 25 and 15. What is the common total for this balanced problem?

    • 45
    • 55
    • 50
    • 60

  10. Total supply is 60 and total demand is 50. What is the demand of the dummy destination needed to balance the problem?

    • 5
    • 10
    • 15
    • 60

  11. How many decision variables does a transportation problem with 4 sources and 4 destinations have?

    • 8
    • 20
    • 12
    • 16

  12. A source A has supply 15 and source B has supply 25; destinations X and Y each need 20. Costs: A to X 2, A to Y 5, B to X 4, B to Y 1. What is the cost of the allocation A to X 15, B to X 5, B to Y 20?

    • 75
    • 65
    • 70
    • 90

  13. In the example with supplies A = 15, B = 25 and demands X = 20, Y = 20, which constraint is x_BX + x_BY = 25?

    • The objective function.
    • The supply constraint for source B.
    • The demand constraint for destination X.
    • The demand constraint for destination Y.

  14. In the same example with supplies A = 15, B = 25 and demands X = 20, Y = 20, what is the demand constraint for destination X?

    • x_AY + x_BY = 20
    • x_AX + x_BX = 15
    • x_AX + x_BX = 20
    • x_AX + x_AY = 20

  15. In that same example, what is the total supply?

    • 40
    • 35
    • 50
    • 45

  16. Why are the optimal flows of a transportation problem with integer supplies and demands whole numbers?

    • The constraint matrix is totally unimodular, so every vertex of the feasible region has integer values.
    • The costs are integers, so the solution must be integer.
    • Every variable is restricted to the values 0 or 1.
    • The simplex method always rounds the answer to the nearest integer.

  17. A non-degenerate basic feasible solution of a balanced 3 by 3 transportation problem has how many basic (occupied) variables?

    • 9
    • 5
    • 6
    • 3

  18. Why is the cost of a dummy destination set to zero?

    • So that the dummy has the largest possible improvement index.
    • So that the dummy has no demand at all.
    • Units assigned to the dummy are not really shipped, so they should not add cost.
    • So that the objective row becomes all zeros.

  19. If every cost in one source's row is increased by 2, does the optimal allocation change?

    • Yes, the cost of every allocation doubles.
    • No, because every allocation from that source is increased by the same total, so the best allocation is unchanged.
    • Yes, the allocation always moves towards that source.
    • No, but the objective becomes a maximisation.

  20. Which statement about the non-negativity condition x_ij ≥ 0 in the transportation linear program is correct?

    • Each x_ij must be an integer between 0 and 1.
    • Units cannot be negative, so every variable must be at least zero.
    • Unoccupied cells must have negative values.
    • The variables must sum to zero across the table.

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