Discussion
For a transportation problem, let \(c_{ij}\) denote the unit cost of the cell \((i, j)\). Assume that \(c_{11} = 10\), \(c_{12} = 12\), \(c_{21} = 11\) and \(c_{22} = 13\). Let \(\alpha_i\) and \(\beta_j\), \(i, j = 1, 2, 3\), represent the simplex multipliers associated with any basis corresponding to the unit cost \(c_{ij}\). Assume that \(\alpha_1 = x\), \(\alpha_2 = x + 1\), \(\beta_1 = y\) and \(\beta_2 = y + 2\). The relative cost coefficient \(d_{ij}\) is the difference between the current solution and the new improved solution.
If \(x = 4\), \(c_{13} = 19\) and \(\beta_3 = y + 5\), then which one of the following is TRUE?
Answer: B
The answer is \(y = 6\) and \(d_{13} = 4\).
In the u-v (MODI) method the simplex multipliers satisfy \(\alpha_i + \beta_j = c_{ij}\) on the basic cells, and the relative cost of a non-basic cell is \(d_{ij} = c_{ij} - \alpha_i - \beta_j\). With \(x = 4\), cell (1,1) gives
\[ \begin{aligned} \alpha_1 + \beta_1 &= 4 + y = 10 \\ y &= 6 \end{aligned} \]
and the other three given cells agree: \(4 + 8 = 12\), \(5 + 6 = 11\), \(5 + 8 = 13\). Then \(\beta_3 = y + 5 = 11\), and
\[ d_{13} = 19 - 4 - 11 = 4. \]
\(d_{13} = 7\) comes from using \(\beta_2 = 8\) in place of \(\beta_3\), and \(y = 5\) fails the condition on cell (1,1). A positive \(d_{13}\) means bringing cell (1,3) into the basis would not lower the cost.
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