Basic Feasible Solution in Transportation Problem

A transportation problem is a type of linear programming problem that involves determining the optimal allocation of goods from a set of sources to a set of destinations. The objective is to minimize the total transportation cost while satisfying supply and demand constraints.

A basic feasible solution (BFS) in a transportation problem refers to a feasible solution that satisfies the conservation of supply and demand. In other words, it is a solution where all the supply and demand quantities are allocated and there is no excess or shortage.

Characteristics of a Basic Feasible Solution
To understand why option 'B' is the correct answer, let's look at the characteristics of a basic feasible solution in a transportation problem:

1. Closed Loop: A closed loop in a transportation problem refers to a situation where a transportation route forms a closed loop, i.e., goods are transported from a source to a destination and then back to the same source. In a basic feasible solution, there should not be any closed loops.

2. No Excess or Shortage: A basic feasible solution ensures that all the supply and demand quantities are fully allocated without any excess or shortage. This means that the total supply equals the total demand.

3. Degeneracy: Degeneracy occurs in a transportation problem when the number of allocated cells is less than the sum of the number of sources and destinations. A basic feasible solution does not involve degeneracy.

4. Total Allocations: In a basic feasible solution, the total number of allocations is equal to the sum of the number of sources and destinations.

Why Option 'B' is the Correct Answer
Based on the characteristics mentioned above, we can conclude that option 'B' is the correct answer because:

- Option 'A' states that a basic feasible solution has a closed loop. However, a closed loop is not allowed in a basic feasible solution.

- Option 'C' states that a basic feasible solution involves degeneracy. However, degeneracy is not a characteristic of a basic feasible solution.

- Option 'D' states that the total number of allocations is one less than the sum of the number of sources and destinations. However, in a basic feasible solution, the total number of allocations should be equal to the sum of the number of sources and destinations.

Therefore, option 'B' is the correct answer because a basic feasible solution in a transportation problem does not have a closed loop and satisfies the conservation of supply and demand.