📌 What is a Transportation Problem?

A Transportation Problem (TP) is a type of Linear Programming Problem (LPP) that deals with optimally transporting goods from multiple origins (sources) to multiple destinations (sinks), minimizing cost or maximizing profit.

🎯 Objectives of a Transportation Problem:

• Minimize total transportation cost

• Satisfy supply at sources and demand at destinations

• Determine the optimal shipping schedule

🧱 Structure:

• Let’s say:

  • You have m origins (e.g., factories)

  • You have n destinations (e.g., warehouses or retail outlets)

You are given:

• Supply at each origin

• Demand at each destination

• Cost matrix: cost to transport from each origin to each destination

🧠 Solution Methods:

1. Initial Feasible Solution

   • North-West Corner Rule

   • Least Cost Method

   • Vogel’s Approximation Method (VAM)

2. Optimality Test

   • Modified Distribution Method (MODI)

   • Stepping Stone Method

🔄 Trans-shipment Problem:

📌 What is a Trans-shipment Problem?

It’s an extension of the transportation problem where:

• Goods can be routed through intermediate points (called trans-shipment nodes)

• These nodes can act as both origins and destinations

❓ So, What Happens to the Size of the Problem?

If you start with a transportation problem having:

• m origins

• n destinations

👉 When it becomes a trans-shipment problem, you:

• Treat each origin and destination as a trans-shipment point

• So, total nodes = m + n

“A transportation problem with m origins and n destinations becomes a trans-shipment problem with (m + n) sources and (m + n) destinations.”

Because:

• In trans-shipment, each node can send and receive

• So all m + n points are treated as both sources and destinations

🧠 Example:

• Original TP: 2 factories → 3 warehouses

• Trans-shipment version: 2 + 3 = 5 nodes

• Each of the 5 nodes is now both a potential sender and receiver