Problem 45 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If at least one of the coefficients \(a_{1}, a_{2}, \ldots, a_{n}\) of the objective function \(P=a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n}\) is positive, then \((0,0, \ldots, 0)\) cannot be the optimal solution of the standard (maximization) linear programming problem.

Step-by-Step Solution

Answer

The statement is true because in a standard maximization linear programming problem, having at least one positive coefficient in the objective function will cause the objective function to increase when the corresponding variable increases. This means that starting from the point \((0, 0, ..., 0)\), we can always find at least one direction (by increasing the variable with a positive coefficient) that will increase the objective function's value, given that this movement stays within the feasible region.

Step 1: Analyze the statement

We're given a standard maximization linear programming problem: Objective function: \(P = a_1x_1 + a_2x_2 + \cdots + a_nx_n\) We want to maximize P subject to some constraints (inequalities). The statement suggests that if at least one of the coefficients \((a_1, a_2, ..., a_n)\) is positive, then the optimal solution can't be at the point where all the variables are equal to zero \((0, 0, ..., 0)\). We will test this statement.

Step 2: Look for counterexamples or validate the statement

Let's suppose we have a linear programming problem with the following setup: Objective function: \(P = x_1 + x_2\) Constraints: \(x_1 \geq 1\) \(x_2 \geq 1\) We are supposed to find the solution that maximizes the value of P, and at least one of the coefficients is positive. If we look at these constraints, we can see that the point \((0, 0)\) is not a feasible solution, as it doesn't satisfy the constraints. We'll try another example to test the statement: Objective function: \(P = -x_1 + x_2\) Constraints: \(x_1 \leq 1\) \(x_2 \geq 1\) In this case, the all-zero point is not an optimal solution because the constraints do not allow it. Also, it's crucial to note that even at the point \((1,1)\), which lies within the feasible region, it is still not an optimal solution, as increasing \(x_2\) will increase the value of P. Since we haven't found any counterexamples yet, we can conclude the statement is true.

Step 3: Explanation of why the statement is true

The statement is true because, in a standard maximization linear programming problem, having at least one positive coefficient in the objective function will cause the objective function to increase when the corresponding variable increases. This means that starting from the point \((0, 0, ..., 0)\), we can always find at least one direction (by increasing the variable with a positive coefficient) that will increase the objective function's value, given that this movement stays within the feasible region. In conclusion, the statement is true: If at least one of the coefficients \(a_{1}, a_{2}, \ldots, a_{n}\) of the objective function \(P=a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n}\) is positive, then the all-zero solution \((0,0, \ldots, 0)\) cannot be the optimal solution of the standard (maximization) linear programming problem.

Key Concepts

Objective Function in Linear Programming

In the context of linear programming, the objective function is a mathematical expression that we aim to maximize or minimize based on a given set of variables and constraints. It represents the goal of the problem, such as maximizing profit or minimizing cost. For example, in a maximization problem, the objective function might be represented as a linear combination of variables:
\[\begin{equation} P = a_1x_1 + a_2x_2 + \/cdots + a_nx_n \end{equation}\]
where \( P \) is the value to be maximized, \( x_1, x_2, \ldots, x_n \) are the decision variables, and \( a_1, a_2, \ldots, a_n \) are the coefficients indicating the contribution of each variable to the objective function. The coefficients can be viewed as the 'weights' of the respective variables in achieving the goal. If any of these coefficients are positive, increasing the associated variable—while remaining within the feasible region—will increase the value of \( P \), contributing to the achievement of the maximization goal.

Maximization Problem

In a maximization problem, the primary goal is to find the highest possible value of the objective function within the realm of permissible solutions defined by the constraints. This typically involves resource allocation, scheduling, or financial decisions where the aim is to get the most benefit.
For instance, a business might want to maximize its profits subject to its available resources and market constraints. The mathematical characterization of this goal involves creating an objective function to represent profit and defining constraints that embody the limitations the business faces. It is essential to identify the optimal solution, which is the variable set that results in the highest value of the objective function while satisfying all of the constraints imposed on the problem.

Feasible Region

The feasible region consists of all possible points that satisfy the problem's constraints, and it is where the search for the optimal solution takes place in linear programming. This region is bounded by the constraints, which typically come in the form of linear inequalities or equations.

In graphical terms, if we are dealing with two variables, the feasible region can be plotted on a graph as a polygon or polyhedron (for higher dimensions), with edges defined by the constraints. The feasible region is key because it contains the possible solutions that meet the problem's requirements. The optimal solution—the one that maximizes or minimizes the objective function—will always be found at a vertex or an edge of the feasible region, according to the Fundamental Theorem of Linear Programming.

Optimization Constraints

The optimization constraints in linear programming are the set of restrictions or conditions that must be adhered to when solving an optimization problem. They define the boundaries of the feasible region and are expressed as linear inequalities or equations.

For example, a constraint might be in the form of \( x_1 \geq 0 \), indicating that the value of variable \( x_1 \) cannot be negative, or \( x_1 + 2x_2 \leq 5 \), meaning that the combined weighted sum of \( x_1 \) and \( x_2 \) must not exceed 5. Constraints often represent limitations such as available resources, legal restrictions, or physical capacities. The solution to a linear programming problem must satisfy all constraints; otherwise, it is considered infeasible and hence not a valid solution for the problem.