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The given problem involves a transportation model where three cement factories (P, Q, and R) transport cement to three states (A, B, and C). The task is to determine the optimal transportation plan to minimize costs while meeting the demands of the states and the supply from the factories.

### Key Information:
- **Factories:**
  - Factory P has 10 truckloads of cement.
  - Factory Q has 8 truckloads of cement.
  - Factory R has 11 truckloads of cement.

- **States' demand:**
  - State A requires 8 truckloads.
  - State B requires 9 truckloads.
  - State C requires 12 truckloads.

- **Cost matrix (in thousands of naira):**
  |      | A  | B  | C  |
  |------|----|----|----|
  | **P**| 6  | 4  | 2  |
  | **Q**| 9  | 7  | 5  |
  | **R**| 1  | 3  | 8  |

### Objective:
1. **Initial Feasible Transportation Plan:**
   Use a method such as the **North-West Corner Rule**, **Least Cost Method**, or **Vogel’s Approximation Method** to determine the initial allocation.
   
2. **Calculate Total Cost:**
   Based on the initial allocation, the total transportation cost is calculated using the cost matrix.

---

### Solution:

#### 1. **Initial Feasible Solution using North-West Corner Rule:**
   We start allocating from the top-left (north-west) corner of the cost matrix and proceed while respecting supply and demand limits.

1. **Allocate to (P, A)**: 
   - Supply from P = 10, Demand for A = 8.
   - Allocate 8 units to (P, A). Remaining supply for P = 10 - 8 = 2, and demand for A = 8 - 8 = 0.

2. **Allocate to (P, B)**: 
   - Supply from P = 2, Demand for B = 9.
   - Allocate 2 units to (P, B). Remaining supply for P = 0, and demand for B = 9 - 2 = 7.
   - P is now fully used.

3. **Allocate to (Q, B)**: 
   - Supply from Q = 8, Demand for B = 7.
   - Allocate 7 units to (Q, B). Remaining supply for Q = 8 - 7 = 1, and demand for B = 7 - 7 = 0.

4. **Allocate to (Q, C)**: 
   - Supply from Q = 1, Demand for C = 12.
   - Allocate 1 unit to (Q, C). Remaining supply for Q = 0, and demand for C = 12 - 1 = 11.
   - Q is now fully used.

5. **Allocate to (R, C)**: 
   - Supply from R = 11, Demand for C = 11.
   - Allocate 11 units to (R, C). Remaining supply for R = 11 - 11 = 0, and demand for C = 11 - 11 = 0.

#### Initial Feasible Plan:

|      | A  | B  | C  |
|------|----|----|----|
| **P**| 8  | 2  | 0  |
| **Q**| 0  | 7  | 1  |
| **R**| 0  | 0  | 11 |

#### 2. **Total Transportation Cost:**
To calculate the total cost, multiply the allocations by their respective costs in the matrix:

\[
\text{Total cost} = (8 \times 6) + (2 \times 4) + (7 \times 7) + (1 \times 5) + (11 \times 8)
\]
\[
= 48 + 8 + 49 + 5 + 88 = 198 \text{ thousand naira}.
\]

### Final Results:
- **Initial Feasible Transportation Plan**: Allocations provided in the table above.
- **Total Cost**: 198,000 Naira.

Would you like to explore an alternative method to improve the solution, such as the **Stepping Stone Method** or **MODI (Modified Distribution) Method** to further reduce the cost?

---

### Follow-up Questions:
1. How does the North-West Corner Rule help in solving transportation problems?
2. What is the next step to further optimize the transportation cost?
3. Can the total cost be minimized using other methods like the Least Cost Method or Vogel’s Approximation?
4. What are the advantages of using the Stepping Stone Method for transportation problems?
5. How can we handle cases where supply does not match demand in transportation problems?

#### Tip:
In transportation problems, if the total supply does not match the total demand, you introduce a **dummy row** (for supply) or **dummy column** (for demand) to balance the system without disrupting the overall calculations.

The cement factories P, Q, and R are to transport their products to states A, B, and C respectively. The numbers of truck loads of cement from factories P, Q, and R are 10, 8, and 11 respectively. The demand from states A, B, and C are 8, 9, and 12 respectively. The costs of transportation between the factories and the states (in thousands of naira) are shown in a table. (i) Obtain the initial feasible transportation plan. (ii) Calculate the least total cost of transportation.

Solve a transportation problem where factories P, Q, and R transport cement to states A, B, and C. Learn how to determine the initial feasible transportation plan and calculate the total cost using the North-West Corner method. Suitable for undergraduate operations research students.

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