Question

The penalty of a row in a transportation problem is obtained by

• A. Deducting smallest element in the row from all other elements of the row
• B. Deducting smallest element in the row from the next highest element of the row
• C. Adding smallest element in the row to all other elements of the row
• D. None of the options are correct

Hint:

Vogel's Approximation Method (VAM) Penalty rule rule can be easily summarized as: \[ \text{Penalty} = \text{Second Minimum Element} - \text{First Minimum Element} \] This calculated value measures the financial risk of missing out on the absolute cheapest route.

Answer 1

The Correct Option is B

Concept: In optimization modeling, Vogel's Approximation Method (VAM) is a structured algorithm used to find an initial feasible solution for a transportation network problem. VAM relies on calculating opportunity cost penalties for each row and column. A row penalty represents the cost penalty incurred if an optimizer fails to allocate cargo to the absolute cheapest cell in that row, forcing an allocation to the next best alternative routing cell.

Step 1: Understanding the algorithmic formulation of a row penalty.
To calculate row and column penalties during VAM analysis, follow these operational steps: 1. Examine the cost elements within a specific row of the transportation matrix. 2. Identify the absolute minimum unit cost element present in that row. 3. Identify the next lowest unit cost element (the second smallest or next highest element) in the same row. 4. Subtract the smallest element from this next highest element. Mathematically, for any row $i$: \[ \text{Penalty}_i = (\text{Second Smallest Cost Value in Row } i) - (\text{Absolute Smallest Cost Value in Row } i) \]

Step 2: Evaluating the text options.
Reviewing our choices: * Option (1) describes a subtraction process similar to row reduction in the Hungarian Assignment method, not VAM. * Option (2) states: "Deducting smallest element in the row from the next highest element of the row." This perfectly matches the mathematical definition of calculating a VAM opportunity penalty. Hence, Option (2) is correct.