A2 June 2022 Q5

EdexcelCurrent spec9 marksTransportation Problems

5. A standard transportation problem is described in the linear programming formulation below.

Let \(x_{ij}\) be the number of units transported from \(i\) to \(j\)

where \(i \in \{\text{A, B, C, D}\}\)
\(\qquad j \in \{\text{R, S, T}\}\) and \(x_{ij} \geqslant 0\)

Minimise \(P = 23x_{\text{AR}} + 17x_{\text{AS}} + 24x_{\text{AT}} + 15x_{\text{BR}} + 29x_{\text{BS}} + 32x_{\text{BT}}\)
\(\qquad\qquad + 25x_{\text{CR}} + 25x_{\text{CS}} + 27x_{\text{CT}} + 19x_{\text{DR}} + 20x_{\text{DS}} + 25x_{\text{DT}}\)

subject to

\[\begin{aligned} \sum x_{\text{A}j} &\leqslant 34 \\ \sum x_{\text{B}j} &\leqslant 27 \\ \sum x_{\text{C}j} &\leqslant 41 \\ \sum x_{\text{D}j} &\leqslant 18 \\ \sum x_{i\text{R}} &\geqslant 44 \\ \sum x_{i\text{S}} &\geqslant 37 \\ \sum x_{i\text{T}} &\geqslant k \end{aligned}\]

Given that the problem is balanced,

(a) state the value of \(k\). (1)
(b) Explain precisely what the constraint \(\sum x_{i\text{R}} \geqslant 44\) means in the transportation problem. (2)
(c) Use the north-west corner method to obtain the cost of an initial solution to this transportation problem. (2)
(d) Perform one iteration of the stepping-stone method to obtain an improved solution. You must make your method clear by showing the route and the
  • shadow costs
  • improvement indices
  • entering cell and exiting cell.
(4)
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