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Let's break down the tasks in your assignment one by one.

### 1. Linear Programming Problem

**Objective Function:**
\[ \text{Maximize } Z = 7x + 6y \]

**Subject to Constraints:**
\[
\begin{aligned}
2x + 4y & \leq 10 \\
3x + y & \leq 12 \\
x, y & \geq 0
\]

**Methods to Solve:**
- **Simplex Method**
- **Graphical Method**

### Solving the Linear Programming Problem

#### A. Graphical Method

1. **Plot the Constraints:**
   - Convert the inequalities into equalities and plot the lines:
     - \( 2x + 4y = 10 \) simplifies to \( y = \frac{5}{2} - \frac{1}{2}x \)
     - \( 3x + y = 12 \) simplifies to \( y = 12 - 3x \)

2. **Determine the Feasible Region:**
   - Identify the region that satisfies all the inequalities and the non-negative constraints \( x, y \geq 0 \).

3. **Find Corner Points:**
   - The feasible region's corner points are usually the intersections of the constraint lines with each other and the axes.

4. **Evaluate the Objective Function at Each Corner:**
   - Substitute the coordinates of each corner point into the objective function \( Z = 7x + 6y \).

5. **Determine the Maximum Value:**
   - The maximum value of \( Z \) among these points will be the solution.

#### B. Simplex Method

The Simplex method is more systematic and typically involves the following steps:
1. **Convert the inequalities into equations** by adding slack variables.
2. **Set up the initial simplex tableau.**
3. **Perform pivot operations** to maximize the objective function.
4. **Identify the optimal solution** from the final tableau.

### 2. Transportation Problem

**Objective:** Minimize the total transportation cost.

Given a table of costs, you're asked to:
- **Use the Northwest Corner Method** to allocate costs and find the total cost.
- **Use another method (e.g., Least Cost Method)** to allocate costs and find the total cost.

### Solving the Transportation Problem

#### A. Northwest Corner Method

1. **Start at the top-left corner** of the cost matrix and allocate as much as possible to minimize transportation cost.
2. **Move to the next cell** horizontally or vertically and continue the allocation.
3. **Calculate the total cost** based on the allocations.

#### B. Alternative Method (Least Cost Method)

1. **Identify the cell with the least cost** in the cost matrix and allocate as much as possible.
2. **Continue to the next lowest cost cell** and repeat the allocation.
3. **Calculate the total cost** based on the final allocations.

### Steps Forward
Would you like detailed step-by-step solutions for each of these problems, or is there a specific part you need help with?

### Suggested Questions:
1. Would you like to see a full solution using the Simplex method?
2. Should I graph the constraints and find the feasible region for you?
3. Do you need help setting up the initial simplex tableau?
4. Would you like a detailed walkthrough of the Northwest Corner method?
5. Is there a specific alternative method you'd prefer for the transportation problem?

### Tip:
When working on linear programming problems, always check the feasibility of the corner points before evaluating the objective function.

Explore detailed solutions for linear programming and transportation problems. Learn about formulating objective functions, applying the Simplex method, and optimizing transportation costs. Suitable for undergraduate students and learners interested in operational research.

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