Problem 81 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

At the end of Section \(6.3\), we mentioned that solving a linear programming problem in three variables and five constraints by the methods of corners requires that we solve \(563 \times 3\) systems of linear equations. Verify this assertion.

Step-by-Step Solution

Answer

The assertion is not correct, as solving the linear programming problem in the given exercise requires solving 30 systems of linear equations instead of 563 * 3. We found the number of corner points by calculating 5C3, which resulted in 10 corner points. Then, we multiplied it by 3 variables to get 30 systems of linear equations.

Step 1: Calculate number of corner points

We have five constraints, and we need to find the corner points of the feasible region. Since there are five constraints, the maximum number of combinations of these constraints is five choose 3 (5C3), which represents the number of ways to choose 3 constraints out of 5. The formula for calculating combinations is given by: \[nCr = \frac{n!}{r!(n-r)!}\] Computing the number of corner points: \[5C3 = \frac{5!}{3!(5-3)!} = \frac{120}{6 * 2} = 10\]

Step 2: Determine number of systems of linear equations

For each corner point, we have a system with three variables which need to be solved. As we have found in step 1, there are 10 corner points. Hence, we need to solve 10 systems with 3 variables each. Therefore, we must solve a total of: \[10 * 3 = 30\] systems of linear equations.

Step 3: Compare the calculated value with the given assertion

As per the assertion provided in the exercise, we needed to verify if solving a linear programming problem with 3 variables and 5 constraints by the methods of corners requires solving 563 * 3 systems of linear equations. Based on our calculations in the previous steps, we found that only 30 systems of linear equations are required to be solved. The assertion is not correct, as solving the linear programming problem in the given exercise requires solving 30 systems of linear equations instead of 563 * 3.

Key Concepts

Feasible Region

In linear programming, a feasible region refers to the area enclosed by the constraints on a graph where all the possible solutions to the problem exist. These constraints are usually defined by linear inequalities. When graphed, they form a polyhedron in space for problems involving more than two variables.

Understanding the feasible region is crucial, as it represents all the candidate solutions that satisfy all constraints at the same time. In simpler terms, any point within this region is a solution that does not violate any of the given constraints. Finding the feasible region helps to identify where the best (or optimal) solution to the linear programming problem could be.

To visualize this, imagine each constraint as a slice through the space that divides it into allowable and non-allowable areas. The intersection of all the allowable areas given by the constraints outlines the feasible region. Ensuring that the interpretation and plotting of this region are accurate is fundamental to successfully solving linear programming problems.

Corner Points

Corner points, also known as vertices, are key features in solving linear programming problems. These points occur at the intersections of the constraints and lie on the boundary of the feasible region.

The significance of corner points comes from the fundamental theorem of linear programming, which states that if an optimal solution exists, it will either be at one of the corner points or along the edge connecting two or more corner points of the feasible region.

In practical terms, this means that even though there may be infinitely many points within the feasible region, the solution-search can be limited to evaluating these corner points. This dramatically reduces the complexity of solving the linear problem, especially in models with higher numbers of constraints and variables. By systematically analyzing each corner point, decision-makers can efficiently determine the best possible outcome, such as maximizing profit or minimizing cost.

Systems of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. In linear programming, solving such systems is a crucial step for finding corner points, as each set of intersecting constraints forms a system.

When tackling a problem with three variables and five constraints, like the one given in the exercise, you look at different combinations of constraints to find points of intersection, or the corner points of the feasible region. Each combination is solved as a system to determine if a feasible solution (a point that satisfies all constraints) exists.

To solve a system, methods such as substitution, elimination, or matrix operations can be used. Solutions to these systems reveal where constraint boundaries intersect in the feasible region. Understanding these systems is vital not just for identifying potential solutions, but also for grasping how changes to constraints could alter the solution space.