Question

Sketch the region determined by the constraints. Then find the minimum anc maximum values of the objective function and where they occur, subject to the indicated constraints. Objective function: $$ z=x $$ Constraints: $$ \begin{array}{r} x \geq 0 \\ y \geq 0 \\ 2 x+3 y \leq 60 \\ 2 x+y \leq 28 \\ 4 x+y \leq 48 \end{array} $$

Short answer

The region is the polygonal bounded by the lines \(2x+3y =60\), \(2x+y=28\), and \(4x+y=48\). The minimum and maximum values of the objective function occur at the vertices of the feasible region.

Step-by-step solution

Draw the Constraints

Represent the inequalities on a coordinate plane. The inequality \( x \geq 0 \) means that the region is to the right of the y-axis. Similarly, \( y \geq 0 \) means the region is above the x-axis. Sketch the other lines as follows: \( 2x+3y=60 \) and fill the area below (since it's \( \leq \)), \( 2x+y=28 \) and fill the area below, \( 4x+y=48 \) and fill the area below.

Identify the Feasible Region

The feasible region is where all the shaded regions from the constraints overlap. It is the polygonal region bounded by the lines \( 2x+3y=60 \), \( 2x+y = 28 \), and \( 4x+y = 48 \).

Find Vertices of the Feasible Region

The vertices of the feasible region are the intersection points of the bounding lines. Solve the equations to find these points.

Evaluate Objective Function

\( z \) takes its minimum and maximum values at the vertices of the feasible region. Substitute each pair of coordinates identified in the previous step into the objective function \( z=x \) and find out which gives the maximum and minimum values.

Key concepts

Objective Function

Linear programming is a method used to determine the best possible outcome of a mathematical model whose requirements are represented by linear relationships. A key component of linear programming is the objective function, which is the formula that needs to be maximized or minimized. In the given exercise, the objective function is surprisingly simple:

\[ z = x \]
This means that we are aiming to find the maximum or minimum value of 'x' while adhering to the given constraints. Despite its simplicity, the objective function directs the entire optimization process and is crucial for making business, engineering, economic decisions, or any other field where linear programming is applied. In the practical sense, it could represent costs, profits, or any other quantity that needs to be optimized.

Constraints

The side conditions of a linear programming problem, also known as constraints, are the restrictions or limitations on the values that the variables in the objective function can assume. They're typically represented by linear inequalities or equations. In the context of our exercise, the constraints that inform the feasible region are:

\[ \begin{align*} x &\text{ (must be greater than or equal to)}\text{ } 0, \ y &\text{ (must be greater than or equal to)}\text{ } 0, \ 2x + 3y &\text{ (must be less than or equal to)}\text{ } 60, \ 2x + y &\text{ (must be less than or equal to)}\text{ } 28, \ 4x + y &\text{ (must be less than or equal to)}\text{ } 48. \text{ } \text{ } \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ } \text{ } \text{ } \text{ } \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ } \text{ }\end{align*} \]
These set the playing field in which the objective function needs to operate. By graphing these constraints on a coordinate plane, one can visually comprehend the feasible region, to which solutions are confined.

Feasible Region

The feasible region represents all the possible points that satisfy the constraints of a linear programming problem. It is typically a polygonal shape on the graph where each edge is a boundary created by a constraint. In our case, the region is the overlap of all the areas shaded by our step 1 constraints. In other words, it's the 'playing field' where our solution can legally reside.

The significance of identifying the feasible region cannot be overstated. It gives a visual boundary of solutions and is essential for answering questions regarding optimization. Once the region is identified, we can determine the points that will give the best result for our objective function - usually at the corners or the 'vertices' of this region.

Vertices Identification

Identifying the vertices of the feasible region is a critical step in solving linear programming problems because the optimal solution occurs at one of these vertices. They represent the intersection points of the constraint lines. To find these, we set the equations equal to each other and solve for the values that satisfy both conditions. Once we find these points graphically or algebraically, we plug them back into the objective function to see which one gives us the maximum or minimum value.

For our provided exercise, we find the vertices by solving systems of equations generated by the intersecting lines of the constraints. If these points weren't clear from the graph, we'd have to calculate by setting the constraints equal to one another and finding their intersection points. Then, as our objective function is simply \( z = x \), evaluating it at each vertex gives us our minimum and maximum values straightforwardly.