The question asks for the mathematical formulation of a transportation problem. Here's the detailed explanation:

Mathematical Formulation of a Transportation Problem

The transportation problem involves determining the most cost-effective way to distribute a product from multiple sources (supply points) to multiple destinations (demand points) while satisfying supply and demand constraints. The mathematical formulation is as follows:

1. Decision Variables: Let $x_{ij}$ represent the amount of goods transported from source $i$ to destination $j$, where:

   • $i$ = 1, 2, ..., $m$ (number of sources),
   • $j$ = 1, 2, ..., $n$ (number of destinations).

2. Objective Function: Minimize the total transportation cost: $Z = \sum_{i=1}^m \sum_{j=1}^n c_{ij} x_{ij}$ where $c_{ij}$ is the cost of transporting one unit of goods from source $i$ to destination $j$.

3. Constraints:

   • Supply Constraints: Ensure that the total quantity transported from each source does not exceed its supply: $\sum_{j=1}^n x_{ij} \leq s_i, \quad \forall i = 1, 2, ..., m$ where $s_i$ is the supply available at source $i$.
   • Demand Constraints: Ensure that the total quantity transported to each destination meets its demand: $\sum_{i=1}^m x_{ij} \geq d_j, \quad \forall j = 1, 2, ..., n$ where $d_j$ is the demand at destination $j$.
   • Non-negativity Constraints: Transportation quantities cannot be negative: $x_{ij} \geq 0, \quad \forall i, j$

4. Balanced Transportation Problem: If the total supply equals total demand ($\sum_{i=1}^m s_i = \sum_{j=1}^n d_j$), the problem is balanced. Otherwise, a dummy source or destination is introduced to balance it.

This formulation provides a basis for solving the transportation problem using methods like the North-West Corner Rule, Least Cost Method, or Vogel's Approximation Method, followed by optimization techniques such as the Modified Distribution Method (MODI).

Would you like a more detailed breakdown or example of how this formulation works in practice?

Related Questions:

1. How is the transportation problem solved using the North-West Corner Rule?
2. What is the difference between a balanced and unbalanced transportation problem?
3. How does the Modified Distribution Method (MODI) optimize a transportation problem?
4. Can you explain the role of dummy sources or destinations in transportation problems?
5. What are some real-world applications of transportation problems in logistics?

Tip:

Ensure you balance the problem before solving it; unbalanced problems often lead to incorrect solutions without proper adjustments.