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### Problem 1: Transportation Problem

The objective function and constraints are as follows:
Objective Function:
\[ \text{Minimize } 5x_{11} + 2x_{12} + 8x_{13} + 2x_{21} + 4x_{22} + 2x_{23} + 3x_{31} + 6x_{32} + 7x_{33} \]

**Supply constraints:**
- Warehouse A: \( x_{11} + x_{12} + x_{13} \leq 12 \)
- Warehouse B: \( x_{21} + x_{22} + x_{23} \leq 14 \)
- Warehouse C: \( x_{31} + x_{32} + x_{33} \leq 4 \)

**Demand constraints:**
- Store 1: \( x_{11} + x_{21} + x_{31} = 9 \)
- Store 2: \( x_{12} + x_{22} + x_{32} = 10 \)
- Store 3: \( x_{13} + x_{23} + x_{33} = 11 \)

**Northwest Corner Method Allocation:**
1. \( x_{11} = 9 \) (fulfills Store 1 demand)
2. \( x_{12} = 3 \) (remaining supply from Warehouse A to Store 2)
3. \( x_{22} = 7 \) (remaining demand for Store 2 from Warehouse B)
4. \( x_{21} = 2 \) (remaining supply from Warehouse B to Store 1)
5. \( x_{23} = 5 \) (remaining demand for Store 3 from Warehouse B)
6. \( x_{32} = 0 \) (no supply left for C to Store 2)
7. \( x_{33} = 0 \) (Store 3 already satisfied)

**Answer:**
\( x_{22} = 7 \) is confirmed. The correct answer is (c) 7.

### Problem 2: Assignment Problem

Each manager needs to complete five tasks, with the days required for each task listed in the table. The goal is to assign each task to a manager in such a way that minimizes the total number of days required to complete all tasks.

#### Data:
| Manager  | Project 1 | Project 2 | Project 3 | Project 4 | Project 5 |
|----------|-----------|-----------|-----------|-----------|-----------|
| A        | 4         | 3         | 6         | 5         | 2         |
| B        | 6         | 7         | 8         | 4         | 6         |
| C        | 5         | 6         | 7         | 3         | 4         |
| D        | 4         | 2         | 3         | 6         | 5         |
| E        | 3         | 4         | 3         | 6         | 7         |

By assigning the tasks to minimize the total completion time:

- Manager A: Project 5 (2 days)
- Manager B: Project 4 (4 days)
- Manager C: Project 3 (7 days)
- Manager D: Project 2 (2 days)
- Manager E: Project 1 (3 days)

Total time = \( 2 + 4 + 7 + 2 + 3 = 18 \) days.

**Answer:**
The total number of days is 18. The correct answer is (c) 18.

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If you have any further questions or need additional details, please let me know!

Here are some additional questions you might find interesting:
1. How does the northwest corner method compare to other methods in transportation problems?
2. What is the role of the assignment problem in operations research?
3. How can the simplex method be applied to linear programming problems?
4. What are some common pitfalls in solving transportation problems?
5. How do you interpret the shadow price in a linear programming problem?

**Tip:** When using the northwest corner method, always ensure the total supply equals the total demand to prevent infeasibility.

Learn how to solve transportation and assignment problems in operations research with detailed examples and solutions. This page covers the northwest corner method for transportation problems and task assignment optimization in linear programming.

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