Transportation problems Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Transportation problems questions. See exactly how to solve problems on transportation, tableau, degeneracy, balancing.

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Further Maths70 questionsStep-by-step solutions
Question 1
2 markseasy
The table below shows the cost (in pounds) of transporting one unit from each supply point SiS_i to each demand point DjD_j, together with the supplies and demands. D1D2D3SupplyS143910S27597S37585Demand1417\begin{array}{c|ccc|c} & D_1 & D_2 & D_3 & \text{Supply} \\ \hline S_1 & 4 & 3 & 9 & 10 \\ S_2 & 7 & 5 & 9 & 7 \\ S_3 & 7 & 5 & 8 & 5 \\ \hline \text{Demand} & 14 & 1 & 7 & \end{array} Find the total supply available.

Worked solution

  1. Read the supplies from the table

    s1=10, s2=7, s3=5s_1 = 10,\ s_2 = 7,\ s_3 = 5

    The supply of each source is the entry in the final column of its row.

  2. Add the supplies together

    10+7+5=2210 + 7 + 5 = 22

    The total supply is the sum of the individual supplies.

  3. State the total supply

    total supply=22\text{total supply} = 22

    This is how many units are available in total.

Answer
2222
Question 2
2 markseasy
The table below shows the cost (in pounds) of transporting one unit from each supply point SiS_i to each demand point DjD_j, together with the supplies and demands. D1D2D3SupplyS15374S227710S32646Demand3611\begin{array}{c|ccc|c} & D_1 & D_2 & D_3 & \text{Supply} \\ \hline S_1 & 5 & 3 & 7 & 4 \\ S_2 & 2 & 7 & 7 & 10 \\ S_3 & 2 & 6 & 4 & 6 \\ \hline \text{Demand} & 3 & 6 & 11 & \end{array} How many cost cells does this transportation tableau contain?

Worked solution

  1. Read the dimensions of the tableau

    m=3 sources,n=3 destinationsm = 3 \text{ sources}, \quad n = 3 \text{ destinations}

    Every source is paired with every destination in the tableau.

  2. Multiply the number of rows by the number of columns

    m×n=3×3=9m \times n = 3 \times 3 = 9

    There is one cell for each source-destination pair.

  3. Recall what a transportation problem asks

    min⁡∑i∑jcijxij\min \sum_i \sum_j c_{ij} x_{ij}

    The aim is to meet every demand from the available supplies at least total cost.

  4. State the number of cells

    cells=9\text{cells} = 9

    The tableau has one cost cell for every route.

Answer
99
Question 3
2 markseasy
The table below shows the cost (in pounds) of transporting one unit from each supply point SiS_i to each demand point DjD_j, together with the supplies and demands. D1D2D3SupplyS19396S24458S39439Demand31010\begin{array}{c|ccc|c} & D_1 & D_2 & D_3 & \text{Supply} \\ \hline S_1 & 9 & 3 & 9 & 6 \\ S_2 & 4 & 4 & 5 & 8 \\ S_3 & 9 & 4 & 3 & 9 \\ \hline \text{Demand} & 3 & 10 & 10 & \end{array} How many cells must be occupied in a non-degenerate basic feasible solution of this problem?

Worked solution

  1. Recall the rule for a non-degenerate solution

    occupied cells=m+n−1\text{occupied cells} = m + n - 1

    A basic feasible solution occupies exactly m+n−1m + n - 1 cells when it is non-degenerate.

  2. Substitute the dimensions

    m+n−1=3+3−1=5m + n - 1 = 3 + 3 - 1 = 5

    Here there are 33 sources and 33 destinations.

  3. Recall what a transportation problem asks

    min⁡∑i∑jcijxij\min \sum_i \sum_j c_{ij} x_{ij}

    The aim is to meet every demand from the available supplies at least total cost.

  4. State the number of occupied cells

    m+n−1=5m + n - 1 = 5

    A non-degenerate basic feasible solution occupies this many cells.

Answer
55
Question 4
2 markseasy
The table below shows the cost (in pounds) of transporting one unit from each supply point SiS_i to each demand point DjD_j, together with the supplies and demands. D1D2D3SupplyS13825S233712S34836Demand845\begin{array}{c|ccc|c} & D_1 & D_2 & D_3 & \text{Supply} \\ \hline S_1 & 3 & 8 & 2 & 5 \\ S_2 & 3 & 3 & 7 & 12 \\ S_3 & 4 & 8 & 3 & 6 \\ \hline \text{Demand} & 8 & 4 & 5 & \end{array} This problem is unbalanced. State the number of units that must be given to the dummy destination used to balance it.

Worked solution

  1. Compare total supply and total demand

    ∑si=23,∑dj=17\sum s_i = 23, \quad \sum d_j = 17

    The problem is unbalanced because these totals are not equal.

  2. Find the size of the imbalance

    ∣23−17∣=6\left|23 - 17\right| = 6

    This surplus must be absorbed to balance the problem.

  3. State the dummy demand

    demand = 6

    This balances total supply against total demand.

Answer
66
Question 5
2 markseasy
The table below shows the cost (in pounds) of transporting one unit from each supply point SiS_i to each demand point DjD_j, together with the supplies and demands. D1D2D3SupplyS145410S24688S36266Demand464\begin{array}{c|ccc|c} & D_1 & D_2 & D_3 & \text{Supply} \\ \hline S_1 & 4 & 5 & 4 & 10 \\ S_2 & 4 & 6 & 8 & 8 \\ S_3 & 6 & 2 & 6 & 6 \\ \hline \text{Demand} & 4 & 6 & 4 & \end{array} Is this transportation problem balanced, and if not how should it be balanced?

Worked solution

  1. Compute the total supply and the total demand

    ∑si=24,∑dj=14\sum s_i = 24, \quad \sum d_j = 14

    A problem is balanced only when these two totals are equal.

  2. Compare the two totals

    24≠1424 \ne 14

    They differ by 10, so the problem is unbalanced.

  3. Recall what a transportation problem asks

    min⁡∑i∑jcijxij\min \sum_i \sum_j c_{ij} x_{ij}

    The aim is to meet every demand from the available supplies at least total cost.

  4. State how to balance the problem

    add a dummy with zero costs carrying 10 units\text{add a dummy with zero costs carrying } 10 \text{ units}

    The dummy takes up the surplus without adding any real transport cost.

Answer
not balanced: add a dummy carrying 10 units\text{not balanced: add a dummy carrying } 10 \text{ units}

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