What are the conditions for solving a transportation problem?

Conditions for Solving a Transportation Problem

A transportation problem is a mathematical optimization problem that aims to determine the optimal distribution of goods from a set of origins (supply points) to a set of destinations (demand points) in a way that minimizes the total transportation cost. To solve a transportation problem, certain conditions must be met:

1. Balanced Model:

The total supply from all origins must equal the total demand at all destinations. This ensures that all goods are accounted for and that there are no shortages or surpluses.

M = N

Where M is the number of supply points and N is the number of demand points.

2. Non-negativity Constraint:

The decision variables (xij), which represent the quantity of goods shipped from origin i to destination j, must be non-negative. This means that goods cannot be shipped in negative quantities.

xij ≥ 0

3. Capacity Constraints:

The total quantity shipped from each origin cannot exceed its supply capacity. Similarly, the total quantity received at each destination cannot exceed its demand capacity.

∑j=1^N xij ≤ Si

∑i=1^M xij ≤ Dj

Where Si is the supply capacity at origin i and Dj is the demand capacity at destination j.

4. Matrix Solution:

The transportation problem can be formulated as a matrix (x) where the elements xij represent the quantity of goods shipped from origin i to destination j. The optimal solution is obtained by solving this matrix using techniques such as the northwest corner method, Vogel’s approximation method, or the stepping stone method.

5. Integer Solution:

In some cases, the optimal solution may require fractional values for xij. This indicates that the goods cannot be physically divided into smaller units. To ensure a feasible solution, the values of xij may need to be rounded to the nearest integer.

6. Feasible Solution:

A feasible solution to a transportation problem must satisfy all the constraints mentioned above. In other words, it must ensure that the total supply equals the total demand, that the shipping quantities are non-negative, and that the capacity constraints are met.

Meeting these conditions is essential for successfully solving a transportation problem and obtaining an optimal solution that minimizes the total transportation cost.

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