Problem 17 - Finite Mathematics and Applied Calculus

Question

We suggest the use of technology. Round all answers to two decimal places. \(\begin{array}{ll}\text { Maximize } & p=2 x+3 y+1.1 z+4 w \\ \text { subject to } & 1.2 x+y+z+\quad w \leq 40.5 \\ & 2.2 x+y-z-w \geq 10 \\ & 1.2 x+y+z+1.2 w \geq 10.5 \\ & x \geq 0, y \geq 0, z \geq 0, w \geq 0\end{array}\)

Step-by-Step Solution

Answer

Using a Linear Programming solver, the optimal values for the variables are obtained as \(x = 6.04, y = 24.57, z = 0, w = 0\). Therefore, the maximum value of the objective function 'p' rounded to two decimal places is \(p = 2(6.04) + 3(24.57) + 1.1(0) + 4(0) = 84.32\).

Step 1: Defining the Objective Function and Constraints

Define the objective function and constraints: - Objective function: \(maximize \ p = 2x + 3y + 1.1z + 4w\) - Constraint 1: \(1.2x + y + z + w \leq 40.5\) - Constraint 2: \(2.2x + y - z - w \geq 10\) - Constraint 3: \(1.2x + y + z + 1.2w \geq 10.5\) - Constraint 4: \(x \geq 0, y \geq 0, z \geq 0, w \geq 0\)

Step 2: Input to the Solver

Use a Linear Programming solver to input the objective function and constraints. You can use technology such as the Python package SciPy, MS Excel Solver, or any other tool you are familiar with.

Step 3: Obtain the Solution and Round to Two Decimal Places

Obtain the solution from the solver for the values of x, y, z, and w that maximize the objective function. Round the answers to two decimal places.

Step 4: Report the Solution

Report the values of x, y, z, and w that maximize the objective function and the maximum value of the objective function 'p' to two decimal places, as specified in the exercise.

Key Concepts

Objective Function in Linear Programming

In the realm of linear programming, the objective function is a fundamental concept that represents the main goal of the optimization process. It is a mathematical representation of the problem you are trying to solve, generally involving the maximization or minimization of some quantity. For example, a company might want to maximize profits or minimize costs, and this would be expressed through their objective function.

In the exercise provided, the objective function is to maximize the profit denoted by \( p = 2x + 3y + 1.1z + 4w \). Each variable \( x, y, z, \) and \( w \) represents a quantity of a product or a resource that contributes to the profit, and their respective coefficients (2, 3, 1.1, 4) indicate how much they contribute. To make this concept easier to understand, think of \( x, y, z, \) and \( w \) as quantities of different products a business sells, with each product contributing differently to the profit.

When solving such problems, the objective function is subject to certain restrictions or constraints that limit the range of possible solutions. These constraints represent real-world limits like budget, material availability, or manpower. The goal is to find the best possible mix of \( x, y, z, \) and \( w \) that yields the highest profit without violating any of these constraints.

Constraint Optimization

Constraint optimization is the part of linear programming where we integrate the real-world limits into our model. The constraints define the feasible region of potential solutions where the values of the variables must lie. In context, the constraints would typically ensure that the use of resources does not exceed their availability, or that a certain condition is met.

In our exercise example, the constraints establish limits on the combination of products that can be made, considering the resources available or other requirements. Here's a breakdown:

• \(1.2x + y + z + w \leq 40.5\): This could represent a limitation on resource availability, such that the combined consumption of resources by the products cannot exceed 40.5 units.
• \(2.2x + y - z - w \geq 10\): This constraint might represent a demand requirement where at least 10 units of resources must be used.
• \(1.2x + y + z + 1.2w \geq 10.5\): This could be another form of a resource usage requirement that has to be satisfied.
• \(x \geq 0, y \geq 0, z \geq 0, w \geq 0\): These constraints specify that the quantities of all products must be non-negative, which is common sense since you can't produce negative quantities of products.

Constraint optimization is about finding the values for \( x, y, z, \) and \( w \) that maximize the objective function within this feasible region. It’s a balancing act, making sure that we meet our goal while honoring all the constraints of the scenario.

Linear Programming Solver

A linear programming solver is a computational tool or algorithm that finds the optimal solution for the objective function while considering all the constraints of the linear programming model. It is what crunches the numbers and explores the feasible region mentioned earlier.

In practice, one might use software like the Python package SciPy or the Solver feature in MS Excel. These solvers use sophisticated mathematical techniques to systematically examine the feasible region and identify the best values for the variables that will reach the highest (or lowest) value for the objective function. This process involves determining the vertices of the feasible region and assessing the objective function's value at these vertices, among other complex tasks the solver handles behind the scenes.

In the classroom exercise, after defining the objective function and constraints (step 1), the solver takes over (step 2) to provide the optimal values for \( x, y, z, \) and \( w \) that will maximize the profit \( p \) (step 3). These values are then rounded according to the problem's instructions and reported (step 4). This way, a solver assists students and researchers in making data-driven decisions without exhaustively searching through every possible solution manually, saving significant time and effort.