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To find the initial feasible solution using the **North-West Corner Method** from the given transportation matrix, we follow these steps:

### Step-by-step solution:

1. **Step 1 (Cell A1)**: Start at the top-left corner (Warehouse I to Store A). The supply is 10 and the demand is 60. Allocate **10 units** to this cell, reducing the demand for Store A to 50 (60 - 10) and the supply for Warehouse I to 0. Since Warehouse I's supply is now exhausted, move to the next row.

2. **Step 2 (Cell A2)**: Now move to Warehouse II and continue in Store A. The supply for Warehouse II is 20, and the remaining demand for Store A is 50. Allocate **20 units** to this cell, reducing the demand for Store A to 30 (50 - 20) and the supply for Warehouse II to 0. Since Warehouse II's supply is exhausted, move to Warehouse III.

3. **Step 3 (Cell A3)**: In Warehouse III, allocate the remaining demand for Store A (30). Warehouse III has a supply of 30, so allocate **30 units** to this cell, fully satisfying the demand for Store A (30 - 30 = 0) and reducing the supply for Warehouse III to 0. Now move to Store B.

4. **Step 4 (Cell B4)**: Now move to Store B and Warehouse IV. Warehouse IV has 40 units of supply, and Store B has 60 units of demand. Allocate **40 units** to this cell, reducing the demand for Store B to 20 (60 - 40), and exhaust the supply for Warehouse IV. Move to the next warehouse.

5. **Step 5 (Cell B5)**: Move to Warehouse V. The demand for Store B is 20, and Warehouse V has a supply of 50. Allocate **20 units** to this cell, fully satisfying Store B’s demand (20 - 20 = 0). This reduces Warehouse V's supply to 30 (50 - 20). Move to Store C.

6. **Step 6 (Cell C5)**: Store C needs 20 units, and Warehouse V has 30 units of supply remaining. Allocate **20 units** to this cell, fully satisfying Store C's demand (20 - 20 = 0). Warehouse V now has 10 units of supply left. Move to Store D.

7. **Step 7 (Cell D5)**: The final demand is for Store D, which needs 10 units. Warehouse V has exactly 10 units left. Allocate **10 units** to this cell, fully satisfying both the demand for Store D and the supply for Warehouse V.

### Initial Feasible Solution Table:

| Warehouse | A  | B  | C  | D  | Supply |
|-----------|----|----|----|----|--------|
| I         | 10 |  0 |  0 |  0 | 10     |
| II        | 20 |  0 |  0 |  0 | 20     |
| III       | 30 |  0 |  0 |  0 | 30     |
| IV        |  0 | 40 |  0 |  0 | 40     |
| V         |  0 | 20 | 20 | 10 | 50     |
| **Demand**| 60 | 60 | 20 | 10 |        |

The **initial feasible solution** using the **North-West Corner Method** allocates a total of 150 units, matching both the total supply and demand. 

Would you like a breakdown of the costs associated with this solution or have any further questions?

### 5 Follow-up Questions:
1. What are the limitations of the North-West Corner Method compared to other methods like Vogel’s approximation?
2. How can we check if this solution is optimal?
3. Could the same initial allocation be achieved if the demands and supplies were not balanced?
4. What would be the next step to move towards the optimal solution in transportation problems?
5. How does this method handle cases where the supply exceeds demand or vice versa?

### Tip:
When using the North-West Corner Method, always keep track of how the demand and supply are exhausted at each step to avoid errors in allocation.

Find the initial feasible solution using North-West Corner method for the given transportation matrix.

Solve a transportation problem using the North-West Corner Method. This step-by-step approach calculates the initial feasible solution by allocating supplies and demands in the matrix based on warehouse and store requirements.

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