Problem 5 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use the technique developed in this section to solve the minimization problem. $$ \begin{aligned} \text { Minimize } & C=2 x-3 y-4 z \\ \text { subject to } &-x+2 y-z \leq 8 \\ & x-2 y+2 z \leq 10 \\ & 2 x+4 y-3 z \leq 12 \\ x & \geq 0, y \geq 0, z \geq 0 \end{aligned} $$

Step-by-Step Solution

Answer

In summary, to minimize the given objective function \(C = 2x - 3y - 4z\), subject to the given constraints, follow these steps: 1. Graph the inequalities and identify the feasible region in the first octant of the coordinate system (x ≥ 0, y ≥ 0, z ≥ 0). 2. Find the vertices of the feasible region by solving the system of equations formed by each pair of equations and the non-negativity constraints. 3. Compute the value of objective function C at each vertex: \(C = 2x - 3y - 4z\). 4. Choose the vertex with the minimum value of C, which is the solution to the minimization problem. Verify that the chosen vertex satisfies all the constraint inequalities.

Step 1: Graph the inequalities

Start by graphing each inequality on a coordinate system, where x represents the horizontal axis, y represents the vertical axis, and z represents the depth axis. To do this, we can first turn each inequality into an equation: 1. \(-x + 2y - z = 8\) 2. \(x - 2y + 2z = 10\) 3. \(2x + 4y -3z = 12\) With these equations, we can find intercepts and plot the planes to obtain the feasible region. Note that x, y, and z are all non-negative. Hence, you are working in the first octant of the coordinate system (x ≥ 0, y ≥ 0, z ≥ 0).

Step 2: Find the vertices of the feasible region

Now, solve the system of equations formed by taking each pair of equations and the non-negativity constraints to find the intersection points. These intersection points will be the vertices of the feasible region: 1. Intersection of equations 1 and 2: \(-x + 2y - z = 8\) \(x - 2y + 2z = 10\) 2. Intersection of equations 1 and 3: \(-x + 2y - z = 8\) \(2x + 4y -3z = 12\) 3. Intersection of equations 2 and 3: \(x - 2y + 2z = 10\) \(2x + 4y -3z = 12\) Also, find the intersections with the non-negativity constraints (x = 0, y = 0, z = 0).

Step 3: Compute the value of objective function C at each vertex

For each vertex obtained in the previous step, plug the coordinates (x, y, z) into the objective function: \(C=2x - 3y - 4z\) Compute the value of C at each vertex.

Step 4: Choose the vertex with the minimal value of C

Identify the vertex corresponding to the lowest value of C. This vertex will be the minimum point of the objective function C and will be the solution to the minimization problem. Make sure to verify that the chosen vertex satisfies all the constraint inequalities. If it does not, re-examine the previous steps to find the correct solution.

Key Concepts

Feasible Region

In linear programming, the **feasible region** is a key concept that helps us pinpoint the potential solutions to an optimization problem. It comprises all the points that satisfy a set of constraints, such as inequalities and equalities. Each point within this region is a potential solution to the problem. For the given problem:

• Inequalities in steps outline the boundaries of the region, indicating what combinations of variables are permissible.
• Since all variables must also be non-negative, we limit our feasible region to the first octant, where all variables are zero or positive.
• To visualize this region, we graph the inequalities and identify their intersections, creating a 3D region defined by the constraints.

Finding the **vertices** of this feasible region is crucial because in a linear programming problem with linear constraints and a linear objective function, the optimum value (either maximum or minimum) is always found at a vertex of the feasible region. Thus, identifying these vertices allows us to evaluate potential solutions for our objective function.

Objective Function

The **objective function** in linear programming is the formula we aim to minimize or maximize. It represents the goal of our problem, such as minimizing cost or maximizing profit. In our example:- The objective function is given by \(C = 2x - 3y - 4z\), representing a combination of the variables (x, y, z) with their respective coefficients.- Each coefficient indicates the contribution of that variable to the objective value, emphasizing higher/lower importance or cost.To solve the problem, we must compute the value of the objective function at each vertex of the feasible region. Evaluating each vertex provides the potential solutions within the constraints. By comparing the values of \(C\) at these vertices, we can determine which set of variables yields the minimal \(C\) value, representing the best solution to our minimization problem.In essence, the objective function determines where we should focus our search within the feasible region to achieve the optimal solution.

Non-Negativity Constraints

**Non-negativity constraints** are common in linear programming problems. They specify that the solution must have non-negative values for each of the variables involved. These constraints are essential for maintaining the context of real-world problems, where negative values are often meaningless, such as negative quantities or costs.In our exercise:

• We have the constraints \(x \geq 0\), \(y \geq 0\), and \(z \geq 0\).
• These ensure the variables remain within the practical confines of the problem — keeping quantities or values non-negative.
• In the graph of our feasible region, this constraint effectively limits the solutions to the first octant where all three axes (x, y, z) meet at the origin (0,0,0) and extend positively.

Non-negativity constraints simplify the solving process by providing a natural boundary that guides us in searching for the best solution. By removing the possibility of negative values, they narrow down the feasible region further towards practical and plausible results.