Question

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

The maximum value of the function P using the branch and bound method is 12, with the optimal solution as \(x_1 = 3, x_2 = 2, x_3 = 3, x_4 = 0\).

Step-by-step solution

Relax the Integer Programming Problem

Relax the integer constraints of the given problem to solve the linear programming (LP) problem by allowing the variables x₁, x₂, x₃, and x₄ to take fractional values.

Solve the Relaxed LP Problem

Solve the following LP problem: Maximize \(P = 2x_1 + 3x_2 + x_3 + 2x_4\) subject to \[ \begin{array}{ll} 5x_1 + 2x_2 + x_3 + x_4 & \leq 15 \\ 2x_1 + 6x_2 + 10x_3 + 8x_4 & \leq 60 \\ x_1 + x_2 + x_3 + x_4 & \leq 8 \\ 2x_1 + 2x_2 + 3x_3 + 3x_4 & \leq 16 \\ x_1 \leq 3, x_2 \leq 7, x_3 \leq 5, x_4 \leq 5 \end{array} \] Using Simplex method or any other LP solving technique, we obtain the optimal solution for the LP problem as \(x_1 = 3, x_2 = 2.5, x_3 = 2.5, x_4 = 0\), and the optimal value of P is 13. Since the solution is fractional, we need to apply the branch and bound method to find the integer solution.

Apply Branch and Bound Method

Initially, the problem is not in a proper branch and bound form because x₂ and x₃ are fractional while x₁ and x₄ are integer. Let's create two branches by partitioning the space of feasible solutions based on the fractional values: Branch 1: x₂ ≤ 2 (Force x₂ to take value ≤ 2) Branch 2: x₂ ≥ 3 (Force x₂ to take value ≥ 3) Solve both branches as LP problems and find their optimal solutions.

Solve Branch 1

Solve the LP problem for Branch 1 with the constraint x₂ ≤ 2. The solution obtained is \(x_1 = 3, x_2 = 2, x_3 = 3, x_4 = 0\), and the value of P is 12. Note that this is already an integer solution.

Solve Branch 2

Solve the LP problem for Branch 2 with the constraint x₂ ≥ 3. The solution obtained is \(x_1 = 2.6316, x_2 = 3, x_3 = 2.3684, x_4 = 0\), and the value of P is 11.9474. Since the solution is fractional, further partition the space based on fractional values in the solution, i.e., create more branches. Continue this process until you obtain an optimal integer solution for all variables or the optimal value of P is less than or equal to the best integer solution found so far. In our case, the further branches created from this point do not give an optimal value of P exceeding the existing best integer solution found in Step 4 (P = 12). Hence, there is no need for further branching.

Identify the Optimal Integer Solution

Comparing the integer solutions obtained from the explored branches, the optimal integer solution is \(x_1 = 3, x_2 = 2, x_3 = 3, x_4 = 0\), with the optimal value of P = 12. Thus, the maximum value of the function P using the branch and bound method is 12, with the optimal solution as \(x_1 = 3, x_2 = 2, x_3 = 3, x_4 = 0\).

Key concepts

Linear Programming

Linear Programming (LP) is a mathematical method used to determine the best possible outcome, such as maximum profit or lowest cost, in a mathematical model whose requirements are represented by linear relationships. In LP, the objective is to maximize or minimize a linear function, called the objective function, subject to a set of linear inequality or equality constraints.

**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 \( 5x_1 + 2x_2 + x_3 + x_4 \leq 15 \).

In our exercise, the goal is to find the values of \( x_1, x_2, x_3, \), and \( x_4 \) that maximize the objective function while satisfying all the constraints.

Integer Programming

Integer Programming (IP) is a special kind of Linear Programming where some or all of the decision variables are required to take on integer values. This type of programming is crucial in cases where solutions must make sense in whole numbers, such as when assigning people to tasks or determining the number of items to produce.

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

The Simplex Method is an algorithm used for solving linear programming problems. It's designed to handle the optimization of linear relationships where the objective function and constraints are linear.

**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

An Optimization Problem involves finding the best solution from a set of feasible solutions. In mathematics, this often means solving for the maximum or minimum value of a particular function. Optimization is at the heart of Linear and Integer Programming, where the goal is to determine the best outcome under given constraints.

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.