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Chapter 6: Problem 3

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. $$ \begin{array}{rrrrr|r} x & y & u & v & P & \text { Constant } \\ \hline 0 & \frac{1}{2} & 1 & -\frac{1}{2} & 0 & 2 \\ 1 & \frac{1}{2} & 0 & \frac{1}{2} & 0 & 4 \\ \hline 0 & -\frac{1}{2} & 0 & \frac{3}{2} & 1 & 12 \end{array} $$

Short Answer

Expert verified
The given simplex tableau is not in its final form since there is a negative entry \(-\frac{1}{2}\) in the bottom row. The pivot element to be used in the next iteration of the simplex method is \(\frac{1}{2}\), which is located in the first row and second column.

Step by step solution

01

Check if the tableau is in its final form

First, let's look at the bottom row of the tableau (excluding the last entry in the 'Constants' column): \(0, -\frac{1}{2}, 0, \frac{3}{2}, 1\) There is one negative element, \(-\frac{1}{2}\), which means this simplex tableau is not in its final form yet.
02

Locate the pivot element

Since there's a negative entry in the bottom row, we'll need to find a pivot element to perform the next iteration of the simplex method. To do this, following steps can be performed: 1. Choose the most negative entry in the bottom row (ignoring the 'Constant' column), which in our case is \(-\frac{1}{2}\). 2. Now, perform the minimum ratio test with the entries in the positive 'Constant' column. Divide each entry in the positive 'Constant' column by the corresponding entry in the pivot column (the column containing the pivot element), provided that the corresponding entry in the pivot column is positive. The ratios are: \[\frac{2}{\frac{1}{2}} = 4; \quad \frac{4}{\frac{1}{2}} = 8 \quad\text{(Ignoring the last row as it is irrelevant)}\] The minimum ratio is 4, which corresponds to the first row of the simplex tableau. Therefore, the pivot element is the entry in the first row and the second column, which is \(\frac{1}{2}\). To summarize, the tableau is not in its final form yet, and the pivot element to be used in the next iteration of the simplex method is \(\frac{1}{2}\).

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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 methodology used for optimizing a linear objective function, subject to a set of linear inequalities or equations called constraints. Such problems are ubiquitous in economics, engineering, and science, where resources must be allocated efficiently among competing activities.

The objective function represents what needs to be maximized or minimized, like profit, cost, or time. Constraints, on the other hand, represent the restrictions or limits within which the objective function needs to function. These could include material availability, budget caps, or staffing limitations.

In linear programming, all relationships are linear, which allows for clear and directed methods of finding the best possible solution, given the restrictions. One of the primary methods used to solve linear programming problems is the simplex method, which is often represented and worked through using a simplex tableau. The tableau is a tabular method that systematically examines vertices of the feasible region to find the optimal solution.
Pivot Element
In the simplex method, the pivot element plays a critical role in moving from one basic feasible solution to another, with the aim of improving the objective function. Identifying the correct pivot element ensures the method progresses correctly towards the optimum solution.

The pivot element is chosen by a specific set of rules: It must be in a column associated with a negative coefficient in the objective function row, known as the pivot column. The goal is to eliminate this negative coefficient through elementary row operations.

Next, within the pivot column, the pivot element is selected by taking ratios of each element in an associated 'Constants' column (excluding negative and zero entries) to the corresponding element in the pivot column. The smallest positive ratio determines the pivot row. The intersecting element of the pivot row and pivot column is the pivot element 鈥� an essential component to the next iteration of the simplex method that makes the tableau reflect a new and improved solution path.
Simplex Method
The simplex method is a procedure to solve linear programming problems. It is an iterative algorithm that starts at a basic feasible solution and walks along the edges of the feasible region to find the optimal solution. The journey from one vertex to the next is guided by the pivot element until it ends up at the optimal vertex that maximizes or minimizes the objective function.

The starting point often involves constructing a simplex tableau that organizes all the information from the constraints and objective function into a structured format. The tableau facilitates the implementation of the simplex algorithm by allowing the use of elementary row operations guided by the pivot element.

As demonstrated in the example given, the simplex method iterates through tableaus by pivoting on each tableau鈥檚 chosen element, refining the solution until the final tableau exhibits no further negative entries in the objective function row, signifying that the optimal solution has been reached.

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Most popular questions from this chapter

Solve each linear programming problem by the simplex method. $$ \begin{aligned} \text { Maximize } & P=2 x+6 y+6 z \\ \text { subject to } & 2 x+y+3 z \leq 10 \\ & 4 x+y+2 z \leq 56 \\ & 6 x+4 y+3 z \leq 126 \\ & 2 x+y+z \leq 32 \\ & x \geq 0, y \geq 0, z \geq 0 \end{aligned} $$

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. Choosing the pivot row by requiring that the ratio associated with that row be the smallest ensures that the iteration will not take us from a feasible point to a nonfeasible point.

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 available. How many tables and chairs should Winston manufacture in order to maximize its profit? What is the maximum profit?

Solve each linear programming problem by the method of corners. Find the maximum and minimum of \(P=10 x+12 y\) subject to $$ \begin{aligned} 5 x+2 y & \geq 63 \\ x+y & \geq 18 \\ 3 x+2 y & \leq 51 \\ x \geq 0, y & \geq 0 \end{aligned} $$

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?
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