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Chapter 4: Problem 49

Describe at least one drawback to the method of finding the corner points of a feasible region by drawing its graph, when the feasible region arises from real-life constraints.

Short Answer

Expert verified
One drawback of finding the corner points of a feasible region by drawing its graph in real-life scenarios is the problem of scaling. The magnitude of constraints can vary significantly, making it difficult to fit them all on a single graph and leading to inaccuracies in representation. Additionally, graphical methods are less effective for high-dimensional or nonlinear problems, can be time-consuming, and may struggle with identifying multiple solutions.

Step by step solution

01

Problem of Scaling

When the feasible region arises from real-life constraints, the magnitude of the constraints can vary significantly, making it difficult to fit them all on a single graph. Scaling can become an issue, especially if the difference in magnitudes is large. This can lead to inaccuracies in the representation and make it challenging to identify the corner points of the feasible region correctly.
02

Difficulty in Drawing Accurate Graphs

In real-life scenarios, the constraints and objective functions may not be linear, making it difficult to draw accurate graphs for these functions. Moreover, complex or non-linear functions may require advanced graphical techniques or specialized software to be represented. This increases the complexity of finding corner points and might not be feasible in some scenarios.
03

Ineffectiveness with High-Dimensional Problems

Graphical representation of feasible regions works well for two-dimensional problems, where each variable can be plotted on an axis. However, it becomes ineffective when dealing with high-dimensional problems, which often occur in real-life situations. Visualizing and analyzing the corner points of a multidimensional feasible region is not possible using this method.
04

Efficiency and Time Consumption

The graphical method of finding the corner points of a feasible region may not be the most efficient or time-saving method, especially in real-life situations. This method involves manually drawing each constraint, analyzing the graph, and identifying the corner points. This can be time-consuming and error-prone, especially when there are multiple constraints or the constraints are not simple to represent graphically.
05

Ambiguity in Identifying Multiple Solutions

In some cases, the feasible region may have multiple optima or infinite corner points, making it difficult to identify all possible solutions using the graphical method. Real-life scenarios often involve multiple constraints that may result in such complexities, thus limiting the applicability of this method.

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Most popular questions from this chapter

Enormous State University's Business School is buying computers. The school has two models from which to choose, the Pomegranate and the iZac. Each Pomegranate comes with \(400 \mathrm{MB}\) of memory and \(80 \mathrm{~GB}\) of disk space; each iZac has \(300 \mathrm{MB}\) of memory and \(100 \mathrm{~GB}\) of disk space. For reasons related to its accreditation, the school would like to be able to say that it has a total of at least \(48,000 \mathrm{MB}\) of memory and at least \(12,800 \mathrm{~GB}\) of disk space. If the Pomegranate and the iZac cost \(\$ 2,000\) each, how many of each should the school buy to keep the cost as low as possible? HINT [See Example 4.]

Finance Senator Porkbarrel habitually overdraws his three bank accounts, at the Congressional Integrity Bank, Citizens' Trust, and Checks R Us. There are no penalties because the overdrafts are subsidized by the taxpayer. The Senate Ethics Committee tends to let slide irregular banking activities as long as they are not flagrant. At the moment (due to Congress" preoccupation with a Supreme Court nominee), a total overdraft of up to \(\$ 10,000\) will be overlooked. Porkbarrel's conscience makes him hesitate to overdraw accounts at banks whose names include expressions like "integrity" and "citizens' trust." The effect is that his overdrafts at the first two banks combined amount to no more than one-quarter of the total. On the other hand, the financial officers at Integrity Bank, aware that Senator Porkbarrel is a member of the Senate Banking Committee, "suggest" that he overdraw at least \(\$ 2,500\) from their bank. Find the amount he should overdraw from each bank in order to avoid investigation by the Ethics Committee and overdraw his account at Integrity by as much as his sense of guilt will allow.

$$ \begin{aligned} \text { Minimize } & c=6 s+6 t \\ \text { subject to } & s+2 t \geq 20 \\ & 2 s+t \geq 20 \\ & s \geq 0, t \geq 0 . \end{aligned} $$

$$ \begin{aligned} \text { Minimize } & c=s+t \\ \text { subject to } & s+2 t \geq 6 \\ & 2 s+t \geq 6 \\ & s \geq 0, t \geq 0 . \end{aligned} $$

Solve the LP problems. If no optimal solution exists, indicate whether the feasible region is empty or the objective function is unbounded. Minimize \(\quad \begin{aligned} c=2 x+4 y & \\ \text { subject to } & 0.1 x+0.1 y \geq 1 \\ & x+2 y \geq 14 \\ x \geq 0, y \geq 0 & \end{aligned}\)
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