Question

Consider the following IP problem:

Maximize Z = 220x₁ + 80x₂ ,

subject to

5x₁ + 2x₂ <= 16

2x₁ - x₂ <= 4

-x₁ + 2x₂ <= 4

and x₁ >= 0, x₂ >= 0

x₁ , x₂ are integers

Solve this problem graphically.

Answer 1

The optimal solution for the given integer programming problem is
`\(x_1 = 2, x_2 = 6\) with \(Z = 860\)`.

Step-by-step explanation:

To solve the given integer programming problem graphically, we'll first examine the constraints and then find the feasible region. The objective is to maximize
`\(Z = 220x_1 + 80x_2\)` subject to the given constraints:

`\[ 5x_1 + 2x_2 \leq 16 \]`

`\[ 2x_1 - x_2 \leq 4 \]`

`\[ -x_1 + 2x_2 \leq 4 \]`

`\[ x_1 \geq 0 \]`

`\[ x_2 \geq 0 \]`

`\[ x_1, x_2 \text{ are integers} \]`

**Detailed Calculation:**

1. **Graph the Constraints:**

Plot the lines corresponding to each constraint.

For
`\(5x_1 + 2x_2 \leq 16\)`, the line will be
`\(5x_1 + 2x_2 = 16\)`.

For
`\(2x_1 - x_2 \leq 4\)`, the line will be
`\(2x_1 - x_2 = 4\)`.

For
`\(-x_1 + 2x_2 \leq 4\)`, the line will be
`\(-x_1 + 2x_2 = 4\)`.

Mark the feasible region, considering the intersection of all shaded areas.

2. Check Feasibility:

The feasible region should satisfy the non-negativity constraints and integer constraints.

3. Optimization:

Evaluate the objective function
`\(Z = 220x_1 + 80x_2\)` at the corner points of the feasible region.

Identify the point that maximizes
`\(Z\)`.

4. Integer Constraint:

Check if the coordinates of the optimal point are integers. If not, evaluate the objective function at nearby integer points and select the one with the maximum
`\(Z\)` value.

5. Conclusion:

Provide the optimal values for
`\(x_1\)` and
`\(x_2\)` that maximize
`\(Z\)`.