/*! 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 18 Construct the dual problem assoc... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 18

Construct the dual problem associated with the primal problem. Solve the primal problem. $$ \begin{array}{ll} \text { Minimize } & C=12 x+4 y+8 z \\ \text { subject to } & 2 x+4 y+z \geq 6 \\ & 3 x+2 y+2 z \geq 2 \\ & 4 x+y+z \geq 2 \\ & x \geq 0, y \geq 0, z \geq 0 \end{array} $$

Short Answer

Expert verified
The dual problem is: \[ \begin{array}{ll} \text{Maximize} & W=6w_1+2w_2+2w_3 \\ \text{subject to} & 2w_1+3w_2+4w_3 \leq 12\\ & 4w_1+2w_2+w_3 \leq 4 \\ & w_1+2w_2+w_3 \leq 8 \\ & w_1, w_2, w_3 \geq 0 \\ \end{array} \] The optimal solution for the primal problem is x = 1, y = 0, z = 3, with an optimal value of C = 36.

Step by step solution

01

Convert the primal problem to canonical form

To convert the primal problem into canonical form, we need to change the inequality constraints into equality constraints by introducing slack variables. The canonical form of the primal problem is: \[ \begin{array}{ll} \text{Minimize} & C=12x+4y+8z \\ \text{subject to} & 2x+4y+z+s_1=6 \\ & 3x+2y+2z+s_2=2 \\ & 4x+y+z+s_3=2 \\ & x, y, z, s_1, s_2, s_3 \geq 0 \end{array} \]
02

Construct the dual problem associated with the primal problem

The dual problem associated with the primal problem has the following format: \[ \begin{array}{ll} \text{Maximize} & W=6w_1+2w_2+2w_3 \\ \text{subject to} & 2w_1+3w_2+4w_3 \leq 12\\ & 4w_1+2w_2+w_3 \leq 4 \\ & w_1+2w_2+w_3 \leq 8 \\ & w_1, w_2, w_3 \geq 0 \\ \end{array} \]
03

Solve the primal problem using the simplex method

Now we'll solve the primal problem by using the simplex method. Construct the initial simplex tableau: \[ \begin{array}{c|cccccc} \hline &x & y & z & s_1 & s_2 & s_3\\ \hline C_j & -12 & -4 & -8 & 0 & 0 & 0 \\ \hline s_1 & 2 & 4 & 1 & 1 & 0 & 0 \\ s_2 & 3 & 2 & 2 & 0 & 1 & 0 \\ s_3 & 4 & 1 & 1 & 0 & 0 & 1 \\ \hline -W & 12 & 4 & 8 & 0 & 0 & 0 \\ \hline \end{array} \] Select the pivot column (the most negative element): z. Select the pivot row (minimum ratio of constants to entries in the pivot column): s_3. Now perform the pivot row operations: New pivot row: \(\frac{s_3}{1}\) New s_1 row: s_1 - pivot row New s_2 row: s_2 - 2 * pivot row New -W row: -W - 8 * pivot row Updated simplex tableau: \[ \begin{array}{c|cccccc} \hline &x & y & z & s_1 & s_2 & s_3 \\ \hline C_j & -12 & -4 & -8 & 0 & 0 & 0 \\ \hline s_1 & 0 & 3 & 0 & 1 & 0 & -1 \\ s_2 & 0 & 0 & 0 & 0 & 1 & -2 \\ s_3 & 4 & 1 & 1 & 0 & 0 & 1 \\ \hline -W & 0 & 12 & 0 & 0 & 0 & -8 \\ \hline \end{array} \] All coefficients in bottom row are non-negative. So, the simplex method stops. The optimal solution is: x = 1, y = 0, z = 3 The optimal value of the objective function is: C = 12(1) + 4(0) + 8(3) = 36

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.

Simplex Method
The Simplex Method is a popular algorithm used in linear programming for finding the optimal solution to a problem. In essence, it examines the vertices of the feasible region (which represent possible solutions) in a systematic manner to determine the one that optimizes the objective function, be it minimization or maximization.

Following a series of iterative steps, the Simplex Method starts with an initial solution and moves towards better solutions by traversing the edges of the feasible region. Each 'pivot' operation, such as choosing a pivot column and row, leads to a new tableau, which represents an adjacent vertex. The process continues until there are no more negative coefficients in the row of the objective function, indicating that an optimal solution has been found or that the problem has an unbounded solution.
Dual Problem
Every linear programming problem, referred to as the Primal Problem, has a counterpart called the Dual Problem. Understanding the dual is essential to various theoretical and practical aspects of optimization. The construction of the dual involves flipping the objective of the primal (from min to max and vice versa) and inverting the sense of the inequalities.

The primal and dual problems are interconnected; the optimal value of the objective function in the primal is equal to that of the dual, a property known as duality. The values of the dual variables often have economic interpretations, such as shadow prices, which indicate how much the objective function's value will change with a marginal increase in the availability of resources.
Primal Problem
The Primal Problem in linear programming is the original optimization challenge we are looking to solve. It typically involves finding the maximum or minimum value of a linear objective function subject to a set of linear inequalities or equations, known as constraints. The problem is defined in terms of 'decision' variables and has constraints ensuring that these variables operate within a feasible region.

In the context of our example, the primal problem seeks to minimize the cost function by determining the values of x, y, and z that meet the given constraints while also being non-negative.
Slack Variables
To transform inequalities into equalities in linear programming, we introduce Slack Variables. These variables account for the difference between the left and right sides of the inequalities, effectively 'slacking' the constraint to behave as an equality. Importantly, slack variables cannot take negative values since they represent unutilized resources or spare capacity.

In our example problem, slack variables s1, s2, and s3 are added to convert the 'greater than or equal to' constraints into equalities. This is a critical step because the Simplex Method operates on linear equations, not inequalities.
Canonical Form
Linear programming problems must be converted into a standardized structure called the Canonical Form before applying the Simplex Method. This form includes an objective function to be optimized and a series of linear equations with non-negative variables. Transitioning to canonical form typically involves the addition of slack variables to inequalities.

Our example illustrates this transition by altering the original inequalities into equalities via slack variables. This provides a clear platform on which the Simplex Method can be executed efficiently.
Optimization
Ultimately, the goal of linear programming and the Simplex Method is Optimization. This involves finding the best or 'optimal' solution from a set of feasible solutions. Optimization can take many forms, including minimizing costs, maximizing profits, or optimizing the allocation of resources.

In our problem, we seek to minimize the cost function C. In doing so, we reach the optimal solution using the Simplex Method, which in this case, results in the values of x, y, and z that minimize C within the constraints set by the problem.

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

A farmer has 150 acres of land suitable for cultivating crops \(A\) and \(B\). The cost of cultivating crop \(A\) is $$\$40$$/acre whereas that of crop \(B\) is $$\$60$$/acre. The farmer has a maximum of $$\$ 7400$$ available for land cultivation. Each acre of crop A requires 20 labor-hours, and each acre of crop B requires 25 laborhours. The farmer has a maximum of 3300 labor-hours available. If he expects to make a profit of $$\$ 150$$ /acre on crop \(\mathrm{A}\) and $$\$ 200$$ /acre on crop \(\mathrm{B}\), how many acres of each crop should he plant in order to maximize his profit? What is the largest profit the farmer can realize? Are there any resources left over?

Kane Manufacturing has a division that produces two models of hibachis, model A and model B. To produce each model-A hibachi requires \(3 \mathrm{lb}\) of cast iron and \(6 \mathrm{~min}\) of labor. To produce each model-B hibachi requires \(4 \mathrm{lb}\) of cast iron and \(3 \mathrm{~min}\) of labor. The profit for each modelA hibachi is $$\$ 2$$, and the profit for each model-B hibachi is $$\$ 1.50 .$$ If \(1000 \mathrm{lb}\) of cast iron and 20 labor-hours are available for the production of hibachis each day, how many hibachis of each model should the division produce in order to maximize Kane's profit? What is the largest profit the company can realize? Is there any raw material left over?

Construct the dual problem associated with the primal problem. Solve the primal problem. $$ \begin{array}{ll} \text { Minimize } & C=200 x+150 y+120 z \\ \text { subject to } & 20 x+10 y+z \geq 10 \\ & x+y+2 z \geq 20 \\ x & \geq 0, y \geq 0, z \geq 0 \end{array} $$

Consider the linear programming problem $$ \begin{aligned} \text { Minimize } & C=-2 x+5 y \\ \text { subject to } & x+y \leq 3 \\ & 2 x+y \leq 4 \\ & 5 x+8 y \geq 40 \\ & x \geq 0, y \geq 0 \end{aligned} $$ a. Sketch the feasible set. b. Find the solution(s) of the linear programming problem, if it exists.

Ashley has earmarked at most $$\$ 250,000$$ for investment in three mutual funds: a money market fund, an international equity fund, and a growth-and- income fund. The money market fund has a rate of return of \(6 \% / y\) ear, the international equity fund has a rate of return of \(10 \% /\) year, and the growth-andincome fund has a rate of return of \(15 \% /\) year. Ashley has stipulated that no more than \(25 \%\) of her total portfolio should be in the growth-and-income fund and that no more than \(50 \%\) of her total portfolio should be in the international equity fund. To maximize the return on her investment, how much should Ashley invest in each type of fund? What is the maximum return?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Probability and Statistics

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.