/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Solve each linear programming pr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 8

Solve each linear programming problem by the method of corners. $$ \begin{array}{ll} \text { Maximize } & P=x+2 y \\ \text { subject to } & x+y \leq 4 \\ & 2 x+y \leq 5 \\ x & \geq 0, y \geq 0 \end{array} $$

Short Answer

Expert verified
The solution to the given linear programming problem using the method of corners is \(x = 0\) and \(y = 4\), with a maximum value of the objective function \(P = 8\).

Step by step solution

01

Graph the feasible region

First, let's graph the constraints to determine the feasible region: 1. \(x + y \leq 4\) 2. \(2x + y \leq 5\) 3. \(x \geq 0\) 4. \(y \geq 0\) The feasible region is the area that satisfies all four constraints. You can plot these inequalities on graph paper or an online graphing tool. The resulting graph will show the feasible region as a polygon enclosed by the intersecting lines representing the constraints.
02

Identify the corner points

The corner points are the vertices of the feasible region polygon. On the graph, there are four such points: 1. \((0, 0)\) - Origin, where both \(x\) and \(y\) are 0 2. \((0, 4)\) - Intersection of \(y \leq 4 - x\) and \(y \geq 0\) 3. \((2, 2)\) - Intersection of \(y \leq 4 - x\) and \(y \leq 5 - 2x\) 4. \((2.5, 0)\) - Intersection of \(y \leq 5 - 2x\) and \(x \geq 0\)
03

Calculate the objective function value at each corner

Now, we will calculate the value of the objective function \(P = x + 2y\) at each corner point: 1. \(P(0, 0) = 0 + 2 \cdot 0 = 0\) 2. \(P(0, 4) = 0 + 2 \cdot 4 = 8\) 3. \(P(2, 2) = 2 + 2 \cdot 2 = 6\) 4. \(P(2.5, 0) = 2.5 + 2 \cdot 0 = 2.5\)
04

Determine the maximum value and corresponding corner point

We now compare the values of P at each corner point: 1. \(P(0, 0) = 0\) 2. \(P(0, 4) = 8\) 3. \(P(2, 2) = 6\) 4. \(P(2.5, 0) = 2.5\) The maximum value of P is 8, and it occurs at the corner point (0, 4). Therefore, the solution to the linear programming problem is \(x = 0\) and \(y = 4\), which maximizes the value of P at \(P = 8\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Feasible Region
In linear programming, the feasible region is the set of all possible points that satisfy the linear inequalities representing the constraints of the optimization problem. Think of this region as the 'playing field' within which we must find the best solution.
To visualize this, imagine plotting the inequalities on a graph. Each constraint is represented by a line, and the area that satisfies all these conditions simultaneously is the feasible region. It is typically a polygon on the graph, and all its points are candidates for the optimal solution. In the linear programming problem above, constraints like (1. (2.(3.(4.(5.((ewline functions on these canonical 'corner' or 'extrem', 'corner', 'corner', 'corner', boundaries of the feasible region, where two or more constraints intersect.
  • The origin ((((((((((((((((((((((() represents the lowest value for both variables within the feasible space.
  • The intersection of inequalities (},(,}
    follows that the feasible region for our example is a quadrilateral with vertices at the corner points identified. The process of evaluating these vertices is pivotal as it leads us to the potential maximum (or minimum) values of our objective function.
Method of Corners
The method of corners, also known as the vertex method, is a way of finding the optimal solution to a linear programming problem by evaluating the objective function at each vertex of the feasible region. Because of the properties of linear functions, the optimal solution will always be found at one of these corner points when the problem is bounded and the feasible region is a convex set.
In our problem, as in most linear programming problems, we first graph the constraints to find the feasible region. Next, by identifying the corner points, which are the intersections of the boundaries of this region, we only need to calculate the value of the objective function at these points rather than across the entire region. These points are where the optimal solution may lie. We determine these points graphically or algebraically and then plug them into our objective function. The corner that gives us the highest (or lowest, depending on the problem) value tells us the optimal solution. The method of corners simplifies the problem significantly, allowing us to discard countless other points in the region that we now know will not yield the optimal solution.
Objective Function
An objective function is an equation that we aim to maximize or minimize in a linear programming problem. In the exercise provided, the objective function we want to maximize is P = x + 2y. It represents a measurable quantity we are trying to optimize 鈥� in this case, P, which could stand for profit, productivity, or any other metric of success in various contexts.
After establishing the feasible region and identifying the corner points, we evaluate the objective function at each corner. The maximum (or minimum for minimization problems) value of this function within the limits of the feasible region will be the optimal solution. This function is linear, which is crucial because it guarantees that the maximum or minimum value must occur at a vertex of the feasible region, thus marrying the concept to the method of corners. To find the optimal point, we simply look at which corner point gives us the highest value of the function, which in the given problem is at the point (0, 4) with a value of 8. Therefore, under given constraints, this is where we would see the highest attainable value of P, hence, our optimal solution.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{array}{r} x-y \leq 0 \\ 2 x+3 y \geq 10 \end{array} $$

Solve each linear programming problem by the method of corners. $$ \begin{array}{ll} \text { Maximize } & P=4 x+2 y \\ \text { subject to } & x+y \leq 8 \\ & 2 x+y \leq 10 \\ & x \geq 0, y \geq 0 \end{array} $$

MANUFACTURING-PRODUCTION SCHEDULING A division of the Winston Furniture Company manufactures dining tables and chairs. Each table requires 40 board feet of wood and 3 labor-hours. Each chair requires 16 board feet of wood and 4 labor-hours. The profit for each table is \(\$ 45\), and the profit for each chair is \(\$ 20 .\) In a certain week, the company has 3200 board feet of wood available, and 520 labor-hours. How many tables and chairs should Winston manufacture in order to maximize its profits?

A company manufactures two products, \(A\) and \(B\), on two machines, 1 and II. It has been determined that the company will realize a profit of \(\$ 3 /\) unit of product \(A\) and a profit of \(\$ 4 /\) unit of product B. To manufacture a unit of product A requires 6 min on machine I and 5 min on machine II. To manufacture a unit of product B requires 9 min on machine I and 4 min on machine II. There are \(5 \mathrm{hr}\) of machine time available on machine I and \(3 \mathrm{hr}\) of machine time available on machine II in each work shift. How many units of each product should be produced in each shift to maximize the company's profit? What is the optimal profit?

Kane Manufacturing has a division that produces two models of fireplace grates, model A and model B. To produce each model-A grate requires \(3 \mathrm{lb}\) of cast iron and 6 min of labor. To produce each model-B grate requires \(4 \mathrm{lb}\) of cast iron and \(3 \mathrm{~min}\) of labor. The profit for each model-A grate is \(\$ 2\), and the profit for each model-B grate is \(\$ 1.50 .1000 \mathrm{lb}\) of cast iron and 20 labor-hours are available for the production of grates each day. Because of an excess inventory of model-B grates, management has decided to limit the production of model-B grates to no more than 200 grates per day. How many grates of each model should the division produce daily to maximize Kane's profit? a. Use the method of corners to solve the problem. b. Find the range of values that the contribution to the profit of a model-A grate can assume without changing the optimal solution. c. Find the range of values that the resource for cast iron can assume without changing the optimal solution. d. Find the shadow price for the resource for cast iron. e. Identify the binding and nonbinding constraints.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.