Understanding key terms in optimization problems like the transportation problem is crucial. The question asks about a specific situation that arises when allocating resources, related to satisfying both row and column conditions simultaneously. Let's break down the concepts involved.

Understanding Degeneracy in Transportation Problems

In a transportation problem, the goal is often to find the minimum cost to transport goods from sources (rows) to destinations (columns), satisfying supply at sources and demand at destinations. When finding an initial basic feasible solution (IBFS) using methods like the Least Cost Method or Vogel's Approximation Method, we allocate units to cells.

A basic feasible solution in a transportation problem should ideally have exactly \(m + n - 1\) occupied cells (allocations), where \(m\) is the number of rows (sources) and \(n\) is the number of columns (destinations). This condition ensures that the solution is non-degenerate and can be easily tested for optimality using methods like the Stepping Stone Method or MODI Method.

Degeneracy occurs when the number of occupied cells in a basic feasible solution is less than \(m + n - 1\). This situation causes difficulties when trying to find closed loops for optimality testing, as required cells might be unoccupied.

Simultaneous Satisfaction and Degeneracy

The phrase "simultaneous satisfaction of row minima and column minima rule" likely refers to a scenario during the allocation process in methods like the Least Cost Method or Vogel's Approximation Method. In these methods, you select a cell (often based on minimum cost or penalty) and allocate the minimum of the available supply in the row and the required demand in the column. If, during an allocation step, the amount allocated is exactly equal to both the remaining supply of the row and the remaining demand of the column, then both the row and the column are simultaneously satisfied (their remaining supply/demand becomes zero). This simultaneous satisfaction means you have 'used up' both a row constraint and a column constraint with a single allocation.

Normally, an allocation would exhaust either a row or a column, leading to the satisfaction of one constraint. Exhausting both simultaneously means you are, in a sense, satisfying two constraints with one allocation step where you might otherwise expect two separate steps or one step followed by another for the remaining non-zero supply/demand.

Consider an allocation \(x_{ij}\) to cell (i, j) where the available supply in row i is \(a_i\) and the required demand in column j is \(b_j\). The allocation is \(x_{ij} = \min(a_i, b_j)\). If \(a_i = b_j\), then \(x_{ij} = a_i = b_j\). This single allocation satisfies both the i-th row's supply and the j-th column's demand. When this happens, and assuming this is not the very last allocation, this specific step contributes only one occupied cell towards the total count, whereas exhausting just a row or just a column would typically leave the other dimension with a non-zero amount, potentially requiring a subsequent allocation step that would add another occupied cell.

This simultaneous reduction of both row supply and column demand to zero means that the number of allocations made to satisfy all constraints might end up being less than the expected \(m + n - 1\), leading directly to a degenerate basic feasible solution.

Analyzing the Options

Let's look at the other options provided:

• Unboundedness: This term is typically associated with linear programming problems where the objective function can be increased or decreased indefinitely without violating constraints. It does not apply to standard transportation problems which always have finite solutions (unless misformulated).
• Infeasibility: This means no solution exists that can satisfy all the constraints simultaneously. In a transportation problem, this would happen if total supply does not equal total demand, and the problem is not balanced, or if structural constraints prevent any feasible flow. This is different from degeneracy, where a solution exists but has too few occupied cells.
• Post-optimality analysis: This refers to analyzing how the optimal solution changes if the parameters of the problem (like costs, supply, or demand) change. It is performed after the optimal solution has been found, not during the initial solution phase where degeneracy occurs.

Based on the analysis, the situation where satisfying row and column conditions happens simultaneously during allocation, resulting in fewer than \(m+n-1\) occupied cells, is termed as Degeneracy.

Revision Table: Transportation Problem Concepts

Additional Information: Handling Degeneracy

Degeneracy, while not preventing finding a solution, can make the standard optimality test methods (like Stepping Stone or MODI) difficult to apply directly because it might be impossible to form a closed loop for certain unoccupied cells. To handle degeneracy and proceed with optimality testing, a very small positive quantity, usually denoted by ε (epsilon), is introduced into one or more unoccupied cells that would allow the formation of loops. The ε value is treated like a basic variable but is considered to be zero in practice. This ε is placed in a cell such that it helps create \(m + n - 1\) basic variables (occupied cells + cells with ε). The standard optimality test can then be applied, treating cells with ε just like regular occupied cells in loop formation. After finding the optimal solution, ε is set back to zero.