How to Solve Balanced and Unbalanced Transportation Problems with Methods and Examples
The concept of Balanced and Unbalanced Transportation Problems is essential in mathematics, especially within linear programming and operations research. Understanding these problems is crucial for optimizing transport costs and solving practical distribution challenges in exams and real-life.
Understanding Balanced and Unbalanced Transportation Problems
A balanced transportation problem occurs when the total supply from all sources equals the total demand at all destinations. An unbalanced transportation problem arises when total supply and total demand are not equal. These concepts are widely used in linear programming, distribution logistics, and exam-level mathematics. They help set up the structure for finding the optimal way to transport goods, minimizing costs while meeting constraints.
Balanced Transportation Problem
Balanced transportation problem is when the sum of all supplies equals the sum of all demands. This condition makes it possible to solve the problem directly using transportation algorithms without adjustments. Supplies and demands are arranged in a matrix, and optimized routes and costs are calculated.
- Every supply gets distributed, and all demands are satisfied.
- No need to add artificial or dummy rows/columns.
- Common in textbook and exam questions to teach basic methods.
Unbalanced Transportation Problem
Unbalanced transportation problem occurs when the total supply is not equal to the total demand. This may happen if there is surplus production or insufficient requirement.
- The problem cannot be solved directly using standard transportation methods.
- Balancing is done by introducing a fake (dummy) source or destination with supply or demand equal to the surplus or deficit.
- Making it balanced is an essential exam and practical skill.
Here’s a helpful table to understand the differences more clearly:
Balanced vs Unbalanced Transportation Problems Table
| Aspect | Balanced | Unbalanced |
|---|---|---|
| Supply = Demand? | Yes | No |
| Need for Dummy Row/Column | No | Yes |
| Solved by Standard Algorithms Directly | Yes | Not until balanced |
| Example in Textbooks | Common | Often as advanced step |
This table shows how the pattern of balanced and unbalanced transportation problems appears regularly in mathematics and logistics.
How to Balance an Unbalanced Transportation Problem
To solve an unbalanced transportation problem, follow these steps:
2. If Supply > Demand, add a dummy column (destination) with demand = Supply – Demand.
3. If Demand > Supply, add a dummy row (source) with supply = Demand – Supply.
4. All costs in dummy row or column are set to zero.
5. The problem is now balanced, and standard methods such as NorthWest Corner, Least Cost, or Vogel’s Approximation can be used.
Worked Example – Solving a Transportation Problem
Let’s see examples of both balanced and unbalanced transportation problems, step by step:
Example 1: Balanced Transportation Problem
Supplies: O1=300, O2=400, O3=500; Demands: D1=250, D2=350, D3=400, D4=200
Total Supply = 1200, Total Demand = 1200.
Step 1: Start with NorthWest Corner Method at (O1, D1).
Step 2: Assign minimum of supply (O1=300) and demand (D1=250) to cell (O1, D1): allocate 250.
Supply left O1: 300–250=50; D1 demand met and column cancelled.
Step 3: Move to (O1, D2): min(50,350)=50 to (O1, D2). O1 done; D2 left=350–50=300.
Step 4: Next, (O2, D2): min(400,300)=300. O2 left=100, D2 done.
Step 5: (O2, D3): min(100,400)=100. O2 done; D3 left=300.
Step 6: (O3, D3): min(500,300)=300. O3 left=200, D3 done.
Step 7: (O3, D4): min(200,200)=200. Now all supplies and demands met.
Final allocation fills the table, confirming the problem is balanced.
Example 2: Unbalanced Transportation Problem
Total Supply = 117, Total Demand = 95.
Step 1: Since Supply > Demand, add dummy destination D6 with demand 22.
Step 2: Add a new column for D6 in matrix; costs in this column = 0.
Step 3: Now, total supply and demand both = 117; the new table is balanced.
Step 4: Solve by standard transportation methods (e.g., NorthWest Corner, Least Cost, or Vogel’s Approximation Method).
Common Mistakes to Avoid
- Forgetting to check if total supply equals total demand before starting calculations.
- Not adding a dummy row or column to balance the problem, leading to incorrect or unsolvable setups.
- Assigning costs in dummy row/column incorrectly (must always be zero for proper balancing).
Real-World Applications
Balanced and unbalanced transportation problems are used in logistics, supply chain management, factory production planning, and distribution networks to minimize transportation costs and resource use. Learning these concepts with Vedantu helps students apply maths skills in practical, job-related tasks.
Page Summary
We explored Balanced and Unbalanced Transportation Problems, their differences, stepwise methods to balance, and detailed examples. Mastering this topic prepares you for real-world analysis and competitive exams. Practice more with Vedantu for deeper understanding and exam confidence.
Related topics you may also like:
Linear Programming,
Types of Linear Programming,
Linear Equations in Two Variables,
Application of Linear Equations,
Sets,
Experiment Design
FAQs on Balanced and Unbalanced Transportation Problems in Operations Research
1. What is a balanced transportation problem?
A balanced transportation problem is a transportation model in which total supply equals total demand. In mathematical terms, if total supply = total demand, the problem is balanced.
- If total supply = 500 units and total demand = 500 units, the problem is balanced.
- No dummy row or column is required.
- Standard methods like North-West Corner, Least Cost, and Vogel’s Approximation Method (VAM) can be applied directly.
2. What is an unbalanced transportation problem?
An unbalanced transportation problem occurs when total supply is not equal to total demand. If total supply ≠ total demand, the model must be adjusted.
- If supply > demand, add a dummy demand column.
- If demand > supply, add a dummy supply row.
- The dummy cells usually have zero transportation cost.
3. How do you balance an unbalanced transportation problem?
You balance an unbalanced transportation problem by adding a dummy row or column so that total supply equals total demand.
- Step 1: Calculate total supply and total demand.
- Step 2: If supply > demand, add a dummy demand column with demand equal to the excess supply.
- Step 3: If demand > supply, add a dummy supply row with supply equal to the shortage.
- Step 4: Assign zero cost to all dummy cells.
4. What is the formula for the transportation problem?
The objective of a transportation problem is to minimize total transportation cost given by Z = ΣΣ cij xij.
- cij = cost of transporting one unit from source i to destination j
- xij = number of units transported
- Σ xij = supply of source i
- Σ xij = demand of destination j
- xij ≥ 0
5. What is the difference between balanced and unbalanced transportation problems?
The main difference is whether total supply equals total demand.
- Balanced transportation problem: Total supply = Total demand.
- Unbalanced transportation problem: Total supply ≠ Total demand.
- Balanced problems require no modification.
- Unbalanced problems require adding a dummy row or column.
6. Can you give an example of a balanced transportation problem?
A balanced transportation problem occurs when supply equals demand, such as when total supply and demand are both 100 units.
- Factory A supply = 40 units
- Factory B supply = 60 units
- Market X demand = 50 units
- Market Y demand = 50 units
Total demand = 50 + 50 = 100
Since both totals are equal, the problem is balanced and can be solved directly.
7. Why do we add a dummy row or column in an unbalanced transportation problem?
A dummy row or column is added to make total supply equal total demand in an unbalanced transportation problem.
- If supply exceeds demand, a dummy demand column absorbs excess supply.
- If demand exceeds supply, a dummy supply row meets unmet demand.
- Dummy transportation costs are usually zero.
8. What are the methods to find an initial basic feasible solution in a transportation problem?
The main methods to find an initial basic feasible solution are North-West Corner Method, Least Cost Method, and Vogel’s Approximation Method (VAM).
- North-West Corner: Allocate starting from the top-left cell.
- Least Cost Method: Allocate to the cell with the minimum cost first.
- VAM: Uses penalty values to get a better starting solution.
9. What is the condition for a basic feasible solution in a transportation problem?
A basic feasible solution must contain exactly m + n − 1 allocations, where m is the number of sources and n is the number of destinations.
- If allocations < m + n − 1, the solution is degenerate.
- If allocations = m + n − 1, it is non-degenerate.
10. What are the real-life applications of balanced and unbalanced transportation problems?
Balanced and unbalanced transportation problems are used to minimize distribution cost in logistics and supply chain management.
- Shipping goods from factories to warehouses
- Allocating products to retail stores
- Distributing raw materials to plants
- Optimizing fuel or resource allocation
