How to Identify and Solve Transportation Problems for Exams
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 Mathematics
1. What is a balanced transportation problem?
A balanced transportation problem occurs when the total supply from all sources exactly equals the total demand at all destinations. This equality ensures that all goods can be transported without any leftover supply or unmet demand, which simplifies the optimization process.
2. What is an unbalanced transportation problem?
An unbalanced transportation problem arises when the total supply does not equal the total demand. In such cases, to apply standard solving methods, a dummy row or dummy column is added to balance the problem by representing fictitious demand or supply, ensuring the total supply equals total demand.
3. How do you balance an unbalanced transportation problem?
To balance an unbalanced transportation problem, follow these steps:
1. Calculate the total supply and total demand.
2. If supply > demand, add a dummy destination (column) with demand equal to the surplus supply.
3. If demand > supply, add a dummy source (row) with supply equal to the surplus demand.
4. Assign zero transportation cost to all dummy cells. This balances the problem while allowing the use of standard methods to find feasible solutions.
4. What is the main difference between balanced and unbalanced transportation problems?
The key difference lies in the equality of total supply and total demand:
- In a balanced transportation problem, total supply equals total demand, making the model straightforward.
- In an unbalanced transportation problem, total supply and demand differ, requiring the addition of dummy rows or columns to balance the problem before solving.
5. Can you give an example of both balanced and unbalanced transportation problems?
Yes, here are simple examples:
- Balanced problem: Supply = 300 units from factories, Demand = 300 units at warehouses. No dummy additions required.
- Unbalanced problem: Supply = 350 units, Demand = 300 units. Add a dummy destination with demand = 50 units to balance before solving.
6. Why do we need to add a dummy row or column in unbalanced transportation problems?
Adding a dummy row or column ensures the total supply equals total demand, which is essential for applying standard transportation problem algorithms. These dummy entries represent fictitious supply or demand with zero transportation costs, allowing the problem to be solved as if it were balanced without affecting the real cost calculations.
7. What happens if you solve an unbalanced problem without balancing?
If you attempt to solve an unbalanced transportation problem without balancing, it can result in infeasible or incorrect solutions because standard algorithms assume total supply equals total demand. This can lead to unallocated supply or unmet demand, making the solution invalid.
8. How does a transportation problem differ from an assignment problem?
Both are types of linear programming problems but differ as follows:
- Transportation problems focus on minimizing the cost of shipping goods from multiple sources to multiple destinations considering supply and demand.
- Assignment problems allocate tasks to agents, with a one-to-one matching, aiming at minimizing total cost or time. The transportation problem allows multiple units per source-destination pair, while assignment problems are strictly one-to-one.
9. Why is balancing important for exam questions?
Balancing is crucial for exam problems because most solution techniques and formulas assume balanced conditions. Properly balancing the problem ensures students can apply methods like the NorthWest Corner method or Vogel’s Approximation Method correctly to arrive at feasible and accurate answers, aligning with standard exam requirements.
10. Can degenerate solutions appear in balanced problems?
Yes, degeneracy can occur in balanced transportation problems when the number of basic variables is less than required for a basic feasible solution. This causes multiple optimal solutions or cycling during iterations. Understanding degeneracy is important to troubleshoot and correctly interpret solution results in operations research.
11. What are some common methods to find an initial basic feasible solution in transportation problems?
The three most common methods to find the initial basic feasible solution are:
1. NorthWest Corner Method: Starts allocations from the top-left cell moving systematically.
2. Least Cost Method (LCM): Allocates supply based on the cheapest cost cell.
3. Vogel’s Approximation Method (VAM): Prioritizes allocations based on penalty costs to minimize total transportation cost effectively.
