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Chapter 8: Problem 8

In the next two problems, convert each minimization problem into a maximization problem, the dual, and then solve by the simplex method. $$ \begin{aligned} &\begin{aligned} \text { Minimize } z=& 4 x_{1}+3 x_{2} \\ & x_{1}+x_{2} \geq 10 \end{aligned}\\\ &\text { subject to }\\\ &3 x_{1}+2 x_{2} \geq 24\\\ &x, x_{2} \geq 0 \end{aligned} $$

Short Answer

Expert verified
The optimal solution for the maximization (dual) problem is \( y_{1}=2, y_{2}=1 \) with a maximum value \( w=20 \). This infers that the minimal value \( z \) in original minimization problem is also \( 20 \).

Step by step solution

01

Convert Minimization to Maximization Problem

Minimization is the original problem given, and the dual problem is the maximization problem. In the dual problem, each variable and constraint of the primal problem gets reversed. So, the maximization problem becomes: \[ \begin{aligned} \text { Maximize } w=& 10y_{1}+24y_{2} \\ & y_{1}+3y_{2} \leq 4, \\ &y_{1}+2y_{2} \leq 3, \\ & y_{1}, y_{2} \geq 0 \end{aligned} \]
02

Setup Simplex Matrix

To solve the maximization problem using simplex method, we set up a simplex table. The objective function is \( w = 10y_{1} + 24y_{2} - w \) and the constraints form the initial equations. We have limited upper bounds hence: \[\begin{array} \hline y_{1} & y_{2} & s_{1} & s_{2} & w \end{array} \] \[\begin{array} {c|cccc} 1 & 3 & 1 & 0 & 4 \\ 1 & 2 & 0 & 1 & 3 \\ \hline -10 & -24 & 0 & 0 & 0 \\ \end{array}\] Where the \( s_{1} \) and \( s_{2} \) are slack variables added to make the constraints equals.
03

Apply Simplex Method

The simplex method is an algorithm for finding a maximum in the context of linear programming. Basically, we are looking for the pivot element and use it to make other elements in the pivot column zero. After several iterations, when all coefficients of the decision variables in the objective row (last row here) are negative, the process stops. The maximum value of the new table's last column (w column here) is the optimal solution value and corresponding assignments are the optimal solution: \[\begin{array} \hline y_{1} & y_{2} & s_{1} & s_{2} & w \\ \end{array} \] \[\begin{array} {c|cccc} 0 & 1 & 1/3 & -1/3 & 1 \\ 1 & 0 & -1/3 & 1/3 & 2 \\ \hline 0 & 0 & 10/3 & -8/3 & 20 \\ \end{array}\] Therefore optimal values are \( y_{1}=2,y_{2}=1,w=20 \).
04

Find Optimal Solution of Original Problem

Although the problem sought a dual solution, it's interesting to note that in linear programming, the optimal values of the primal and dual problems are equal. Hence, the optimal solution of the original minimization problem has \( z = 20 \).

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Key Concepts

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

Linear Programming
Linear programming is a mathematical technique used to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships. It's commonly used in various fields, such as business, economics, and engineering. In linear programming, we often deal with problems involving optimization, where we want to maximize or minimize a linear objective function, subject to a set of linear inequalities or equations called constraints.

Some key characteristics of linear programming include:
  • Objective Function: This is a linear function that you want to maximize or minimize. For example, in the original exercise, the objective function is to minimize 4\(x_1\) + 3\(x_2\).
  • Constraints: These are the limitations or requirements the solution must satisfy. Constraints are generally given as linear inequalities or equations. In our exercise, the constraints are \(x_1 + x_2 \geq 10\) and \(3x_1 + 2x_2 \geq 24\).
  • Decision Variables: These are the unknowns that we aim to solve for. In the exercise, \(x_1\) and \(x_2\) are our decision variables.
Linear programming problems are solved using various methods, with the simplex method being one of the most popular, as it iteratively approaches the optimal solution.
Duality in Optimization
Duality is a fundamental concept in optimization, especially in linear programming. It offers a way to derive a second related optimization problem from a given problem, providing deep insights and often simplifying the solution process. For any given linear programming problem, known as the 'primal' problem, there is a corresponding 'dual' problem.

Here are some important aspects of duality in optimization:
  • Relation between Primal and Dual Problems: Each constraint in the primal becomes a variable in the dual and vice versa. In our example, the original problem (primal) is a minimization problem, and its dual is a maximization problem.
  • Objective Functions: The objective of the primal problem, which in our case is to minimize \(z = 4x_1 + 3x_2\), becomes the function of the constraints in the dual problem, creating a new objective to maximize.
  • Optimal Values: One remarkable property of duality is that the optimal value of the objective function in the primal is equal to that of the dual, if they both have a feasible solution. In the provided solution, both primal and dual optimal values are 20.
Understanding duality not only provides alternative ways to solve problems but also helps in deriving bounds and ensuring the precision of solutions.
Minimization to Maximization
Converting a minimization problem into a maximization problem is a common step in applying the simplex method, particularly when dealing with dual problems. The dual problem often offers a more straightforward path for solving the optimization problem.

Here's how to approach this conversion:
  • Objective Function Switch: The conversion involves creating a maximization problem from a minimization problem by changing the roles of constraints and variables. In the example, the original minimization function \( z = 4x_1 + 3x_2 \) is transformed into a maximization problem with a new objective function \( w = 10y_1 + 24y_2 \).
  • Reversing Constraints: Inequalities can be flipped, and slack variables added where necessary, to change a '\geq' to a '\leq' format suitable for maximization (or vice versa). As shown in the problem, constraints are modified to fit the dual problem requirements.
  • Solving via Simplex Method: Once converted, the simplex method can be applied effectively to find the optimal solution to the new maximization problem. This involves setting up a tableau, identifying pivot elements, and iterating until an optimal solution is reached, as illustrated in the provided step-by-step solution.
This conversion plays a crucial role in optimization, as it allows the utilization of duality properties and simplifies complex problems for easier computation.

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Most popular questions from this chapter

A department store sells three different types of televisions: small, medium, and large. The store can sell up to 200 sets a month. The small, medium, and large televisions require, respectively, 3,6 , and 6 cubic feet of storage space, and a maximum of 1,020 cubic feet of storage space is available. The three types, small, medium, and large, take up, respectively, \(2,2,\) and 4 sales hours of labor, and a maximum of 600 hours of labor is available. If the profit made from each of these types is \(\$ 40, \$ 80,\) and \(\$ 100,\) respectively, how many of each type of television should be sold to maximize profit, and what is the maximum profit?

Solve the following linear programming problems using the simplex method. Maximize \(z=5 x_{1}+8 x_{2}\) subject to \(x_{1}+2 x_{2} \leq 30\) \(3 x_{1}+x_{2} \leq 30\) \(x_{1} \geq 0 ; x_{2} \geq 0\)

For his classes, Professor Wright gives three types of quizzes, objective, recall, and recall-plus. To keep his students on their toes, he has decided to give at least 20 quizzes next quarter. The three types, objective, recall, and recall-plus quizzes, require the students to spend, respectively, 10 minutes, 30 minutes, and 60 minutes for preparation, and Professor Wright would like them to spend at least 12 hours( 720 minutes) preparing for these quizzes above and beyond the normal study time. An average score on an objective quiz is \(5,\) on a recall type \(6,\) and on a recall-plus 7 , and Dr. Wright would like the students to score at least 130 points on all quizzes. It takes the professor one minute to grade an objective quiz, 2 minutes to grade a recall type quiz, and 3 minutes to grade a recall-plus quiz. How many of each type should he give in order to minimize his grading time?

A company produces three products, \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C},\) at its two factories, Factory I and Factory II. Daily production of each factory for each product is listed below. $$ \begin{array}{|l|l|l|} \hline & \text { Factory I } & \text { Factory II } \\ \hline \text { Product A } & 10 & 20 \\ \hline \text { Product B } & 20 & 20 \\ \hline \text { Product C } & 20 & 10 \\ \hline \end{array} $$ The company must produce at least 1000 units of product A, 1600 units of \(\mathrm{B},\) and 700 units of C. If the cost of operating Factory I is \(\$ 4,000\) per day and the cost of operating Factory II is \(\$ 5000\), how many days should each factory operate to complete the order at a minimum cost, and what is the minimum cost?

Solve the following linear programming problems using the simplex method. Minimize \(z=4 x_{1}+6 x_{2}+7 x_{3}\) $$ x_{1}+x_{2}+2 x_{3} \geq 20 $$ subject to $$ \begin{array}{r} x_{1}+2 x_{2}+x_{3} \geq 30 \\ x_{1}, x_{2}, x_{3} \geq 0 \end{array} $$
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