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Chapter 4: Problem 13

Solve each linear programming problem by the simplex method. $$ \begin{array}{cc} \text { Maximize } & P=10 x+12 y \\ \text { subject to } & x+2 y \leq 12 \\ & 3 x+2 y \leq 24 \\ & x \geq 0, y \geq 0 \end{array} $$

Short Answer

Expert verified
The optimal solution for the given linear programming problem using the simplex method is \(x = 4\) and \(y = 6\), with a maximum value of the objective function \(P = 112\).

Step by step solution

01

Convert inequalities to equations by adding slack variables

Add slack variables \(s\) and \(t\) to the constraint inequalities to convert them into equations: 1. \(x + 2y + s = 12\) 2. \(3x + 2y + t = 24\) 3. \(x, y, s, t \geq 0\)
02

Set up the initial simplex tableau

Construct the initial simplex tableau with the coefficients of \(x\), \(y\), \(s\), \(t\), and the constants on the right-hand side: $$ \begin{array}{c|ccccc} & x & y & s & t & \text{Constants} \\ \hline s & 1 & 2 & 1 & 0 & 12 \\ t & 3 & 2 & 0 & 1 & 24 \\ \hline P & -10 & -12 & 0 & 0 & 0 \\ \end{array} $$
03

Perform row operations to pivot

Select the most negative value (-12) in the bottom row as the pivot column. The corresponding pivot row is the first one, as \(\frac{12}{2} \leq \frac{24}{2}\). Perform the following row operations: 1. Divide Row 1 by 2: \(\frac{1}{2}R_1 \rightarrow R_1\) 2. Add 2 times Row 1 to Row 2 to eliminate \(y\) in Row 2: \(R_2 + 2R_1 \rightarrow R_2\) 3. Add 12 times Row 1 to Row 3 to eliminate \(y\) in Row 3: \(R_3 + 12R_1 \rightarrow R_3\) The resulting tableau is: $$ \begin{array}{c|ccccc} & x & y & s & t & \text{Constants} \\ \hline s & 0.5 & 1 & 0.5 & 0 & 6 \\ t & 2 & 0 & -1 & 1 & 12 \\ \hline P & 4 & 0 & 6 & 0 & 72 \\ \end{array} $$ Now, all values in the bottom row, except for the last cell, are non-negative. Therefore, we have found an optimal solution.
04

Read the optimal solution and the maximum value of the objective function

From the final tableau: \(x = 4\), \(y = 6\), \(s = 0\), \(t = 12\) The optimal solution is \(x = 4\) and \(y = 6\), and the maximum value of the objective function \(P = 10x + 12y = 10(4) + 12(6) = 40 + 72 = 112\).

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Key Concepts

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

Linear Programming
Linear programming (LP) is a powerful mathematical technique used to find the best outcome, usually maximum profit or minimum cost, within a given mathematical model. Its application spans various fields such as business, economics, engineering, and military. LP problems involve an objective function, a set of linear inequalities or equations as constraints, and decision variables to be determined.

The LP problem presented here aims to maximize the profit function represented by the objective function, given the constraints that include the resources available. By utilizing the simplex method, a systematic process, we can navigate through potential solutions until we find the best one that maximizes the profits while adhering to the constraints.
Slack Variables
In linear programming, slack variables play a crucial role in transforming inequality constraints into equations, which can then be manipulated within the simplex method. They effectively measure the 'unused' portion of the resources, turning '<=' constraints into equations by adding a non-negative slack variable.

For instance, in the given exercise, the slack variables s and t are introduced to convert the two inequalities into equalities. This enables the construction of the initial simplex tableau, essential for proceeding with the simplex algorithm. The slack variables are treated as non-negative decision variables and become part of the LP's solution, indicating how much of the resources are left unused in the optimal solution.
Optimization
Optimization in the context of linear programming is the process of finding the most efficient solution to the problem鈥攎aximizing or minimizing the objective function, as determined by the problem's requirements. The simplex method is an iterative procedure used to carry out this optimization.

By selecting a pivot that results in the most increase in the objective function value if the goal is to maximize (or decrease if the goal is to minimize), the simplex method navigates through possible solutions within the feasible region. Through each iteration, it optimizes the objective value step by step until no further improvement is possible, leading to an optimal solution.
Objective Function
The objective function, in a linear programming problem, is the mathematical representation of the goal that needs to be achieved through optimization. It is a linear equation that includes all the decision variables, such as x and y in the given exercise, with their corresponding coefficients, which represent the contribution of each variable to the overall value.

In the example provided, the objective function is to maximize profit P, defined as \( P = 10x + 12y \). Upon finalizing the simplex method, the values obtained for x and y can be substituted into this function to find the maximum profit, which, in the given exercise, is \( P = 112 \). The objective function serves as the guiding beacon of the whole LP problem, providing a quantitative measure to define and evaluate the efficiency of potential solutions.

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Most popular questions from this chapter

Determine whether the given simplex tableau is in final form. If so, find the solution to the associated regular linear programming problem. If not, find the pivot element to be used in the next iteration of the simplex method. $$ \begin{array}{rrrrrrrr|r} x & y & z & s & t & u & v & P & \text { Constant } \\ \hline 1 & 0 & 0 & \frac{2}{5} & 0 & -\frac{6}{5} & -\frac{8}{5} & 0 & 4 \\ 0 & 0 & 0 & -\frac{2}{5} & 1 & \frac{6}{5} & \frac{8}{5} & 0 & 5 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 12 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 6 \\ \hline 0 & 0 & 0 & 72 & 0 & -16 & 12 & 1 & 4920 \end{array} $$

Natsano has at most \(\$ 50,000\) to invest in the common stocks of two companies. He estimates that an investment in company A will yield a return of \(10 \%\). whereas an investment in company \(\mathrm{B}\), which he feels is a riskier investment, will yield a return of \(20 \%\). If he decides that his investment in the stock of company \(\mathrm{A}\) is to exceed his investment in the stock of company B by at least \(\$ 20,000\), determine how much he should invest in the stock of each company in order to maximize the return on his investment.

Solve each linear programming problem by the simplex method. $$ \begin{array}{lr} \text { Maximize } & P=15 x+12 y \\ \text { subject to } & x+y \leq 12 \\ 3 x+y & \leq 30 \\ 10 x+7 y & \leq 70 \\ x & \geq 0, y \geq 0 \end{array} $$

Everest Deluxe World Travel has decided to advertise in the Sunday editions of two major newspapers in town. These advertisements are directed at three groups of potential customers. Fach advertisement in newspaper is seen by 70,000 group-A customers, 40,000 group-B customers, and 20,000 group-C customers. Fach advertisement in newspaper II is seen by 10,000 group-A, 20,000 group-B, and 40,000 group-C customers. Fach advertisement in ncwspaper I costs \(\$ 1000\), and cach advertiscment in newspaper II costs \(\$ 800\). Everest would like their advertiscments to be read by at least 2 million pcople from group A. \(1.4\) million people from group \(\mathrm{B}\), and 1 million people from group C. How many advertisements should Everest place in cach newspaper to achicve its advertising goals at a minimum cost? What is the minimum cost?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If a standard minimization linear programming problem has a unique solution, then so does the corresponding maximization problem with objective function \(P=-C\), where \(C=a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n x} x_{n}\) is the objective function for the minimization problem.
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