Chapter 11: Problem 784

Use the branch and bound method to solve the integer programming problem Maximize \(\mathrm{P}=2 \mathrm{x}_{1}+3 \mathrm{x}_{2}+\mathrm{x}_{3}+2 \mathrm{x}_{4}\) subject to $$ \begin{array}{ll} 5 \mathrm{x}_{1}+2 \mathrm{x}_{2}+\mathrm{x}_{3}+\mathrm{x}_{4} & \leq 15 \\ 2 \mathrm{x}_{1}+6 \mathrm{x}_{2}+10 \mathrm{x}_{3}+8 \mathrm{x}_{4} & \leq 60 \\\ \mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}+\mathrm{x}_{4} & \leq 8 \\ 2 \mathrm{x}_{1}+2 \mathrm{x}_{2}+3 \mathrm{x}_{3}+3 \mathrm{x}_{4} & \leq 16 \\\ \mathrm{x}_{1} \leq 3, \mathrm{x}_{2} \leq 7, \mathrm{x}_{3} \leq 5, \mathrm{x}_{4} \leq 5 . \end{array} $$

### Short Answer

## Step by step solution

## Relax the Integer Programming Problem

## Solve the Relaxed LP Problem

## Apply Branch and Bound Method

## Solve Branch 1

## Solve Branch 2

## Identify the Optimal Integer Solution

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### Linear Programming

**Key Components of Linear Programming:**

- Objective Function: A formula that needs to be maximized or minimized, such as \( P = 2x_1 + 3x_2 + x_3 + 2x_4 \).
- Variables: The unknowns whose values are to be determined, such as \( x_1, x_2, x_3, \), and \( x_4 \).
- Constraints: These are restrictions that limit the degree to which the objective function can be pursued, often expressed in the form of inequalities such as \( 5x_1 + 2x_2 + x_3 + x_4 \leq 15 \).

###### Integer Programming

In our exercise, we take a linear programming problem and add the constraint that variables \( x_1, x_2, x_3, \), and \( x_4 \) must be integers. This adds a layer of complexity because traditional simple techniques cannot be directly applied. Thus, methods such as Branch and Bound are employed for solving IP problems by systematically searching for possible integer solutions until all options are exhausted or the best solution is found.

###### Simplex Method

**How the Simplex Method Works:**

- It starts at a feasible corner point of the solution space and examines adjacent vertices for improvement.
- This method moves from one vertex to another, searching for one that improves the objective function, until no more improvements can be found.

For instance, in our exercise, the Simplex Method finds an optimal solution for the relaxed problem by allowing variables to take fractional values. Once an optimal solution is obtained for the LP problem, this solution is assessed to see if it satisfies the integer requirement. If it doesn’t, further actions such as Branch and Bound are needed to reach an integer solution.

###### Optimization Problem

In our exercise setup, the problem is to maximize the linear objective function \( P = 2x_1 + 3x_2 + x_3 + 2x_4 \) subject to a series of constraints. While Linear Programming can quickly handle scenarios when variables can take any value, Integer Programming and methods like Branch and Bound become crucial when these variables need to be non-fractional.

The calculation involves relaxing the integer constraints initially, solving the linear program, and then iteratively partitioning the solution space to find the best combination of integer values that yield the optimal objective value.