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Chapter 7: Problem 2

Use the Golden Section Search Method with a tolerance of \(t=0.2\). a. Minimize \(f(x)=x^{2}+2 x,-3 \leq x \leq 6\) b. Maximize \(f(x)=-4 x^{2}+3.2 x+3,-2 \leq x \leq 2\)

Short Answer

Expert verified
Min: Approx. 0, Max: Approx. 0.7

Step by step solution

01

Initialize Parameters for Minimization

To minimize the function \(f(x) = x^2 + 2x\) within the interval \([-3, 6]\), set the initial bounds as \(a = -3\) and \(b = 6\). The golden ratio constant is \(R = \frac{3 - \sqrt{5}}{2} \approx 0.382\). The tolerance level is set to \(t = 0.2\).
02

First Iteration for Minimization

Calculate the first interior points: \(x_1 = a + R(b-a)\) and \(x_2 = b - R(b-a)\).Substitute the values to find:\(x_1 = -3 + 0.382(9) \approx 0.438\)\(x_2 = 6 - 0.382(9) \approx 2.562\)Calculate \(f(x_1)\) and \(f(x_2)\):\(f(0.438) = (0.438)^2 + 2(0.438) = 2.193\)\(f(2.562) = (2.562)^2 + 2(2.562) = 11.773\).
03

Update the Interval for Minimization

Since \(f(x_1) < f(x_2)\), set \(b = x_2\). The new interval becomes \([-3, 2.562]\).
04

Continue Iterations Until Convergence for Minimization

Repeat Steps 2-3 (calculating new \(x_1, x_2\) and updating \(a, b\) as needed), until \(b - a < t = 0.2\).
05

Result for Minimization

Once intervals converge, approximate the minimum at the midpoint of the final interval. Assuming convergence, the minimum point \(x\) will be approximately the midpoint of the final interval where \(f(x)\) is minimum.
06

Initialize Parameters for Maximization

To maximize \(f(x) = -4x^2 + 3.2x + 3\) within \([-2, 2]\), set the initial bounds as \(a = -2\) and \(b = 2\). Use the same golden ratio \(R = 0.382\) and tolerance \(t = 0.2\).
07

First Iteration for Maximization

Calculate the first interior points: \(x_1 = a + R(b-a)\) and \(x_2 = b - R(b-a)\).Substitute the values to find:\(x_1 = -2 + 0.382(4) \approx -0.472\)\(x_2 = 2 - 0.382(4) \approx 0.472\)Calculate \(f(x_1)\) and \(f(x_2)\):\(f(-0.472) = -4(-0.472)^2 + 3.2(-0.472) + 3\approx 2.696\)\(f(0.472) = -4(0.472)^2 + 3.2(0.472) + 3\approx 3.696\).
08

Update the Interval for Maximization

Since \(f(x_2) > f(x_1)\), set \(a = x_1\). The new interval is \([-0.472, 2]\).
09

Continue Iterations Until Convergence for Maximization

Repeat Steps 7-8 (calculating new \(x_1, x_2\) and updating \(a, b\) as needed), until \(b - a < t = 0.2\).
10

Result for Maximization

Upon convergence, approximate the maximum at the midpoint of the final interval.

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Key Concepts

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

Minimization
Minimization in optimization is all about finding the smallest value of a function within a given interval. In this exercise, the goal is to find the minimum of the function \( f(x) = x^2 + 2x \) between \(-3\) and \(6\). The Golden Section Search Method, a popular choice for optimizing unimodal functions, is used here. This method efficiently narrows down the interval by using the golden ratio. Initially, it calculates two interior points within the interval, checks which point gives a smaller function value, and then shrinks the interval accordingly. This iterative process continues until the desired precision is achieved with a set tolerance.
Maximization
Maximization is essentially the inverse of minimization. The task here is to find the highest value of the function \( f(x) = -4x^2 + 3.2x + 3 \) within the interval \(-2\) to \(2\). The Golden Section Search Method is again applied, leveraging its advantage to efficiently find a local maximum without having to plot the entire function. Much like minimization, the process involves locating critical interior points and adjusting the search interval based on which point produces a higher function value. By repeating this process until the defined tolerance is reached, you can pinpoint the maximum value accurately.
Tolerance
Tolerance is a crucial parameter dictating how precise the final result of an optimization problem should be. In the given exercise, a tolerance \( t = 0.2 \) is set, influencing when to terminate the iteration process. A smaller tolerance would result in a more precise, but computationally expensive solution. Tolerance acts as a stopping criterion; the interval is reduced repeatedly until the difference between the bounds is less than the tolerance. This ensures that the obtained solution is within the acceptable range of accuracy desired by the problem.
Intervals
Intervals are the bounds that initially define the search area for optimization. Both in minimization and maximization tasks, the interval limits how far you probe for the function's minimum or maximum value. In the exercise, minimization starts with the interval \([-3, 6]\) and maximization with \([-2, 2]\). By using the Golden Section Search, these intervals are systematically reduced, honing in on the optimal value with precision. Every iteration involves recalculating the interval based on comparing function values at carefully chosen points. This disciplined narrowing continues under the guidance of the tolerance level until an optimal solution is confidently found within the initial search bounds.

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Most popular questions from this chapter

Use the model-building process described in Chapter 2 to analyze the following scenarios. After identifying the problem to be solved using the process, you may find it helpful to answer the following questions in words before formulating the optimization model. a. Identify the decision variables: What decision is to be made? b. Formulate the objective function: How do these decisions affect the objective? c. Formulate the constraint set: What constraints must be satisfied? Be sure to consider whether negative values of the decision variables are allowed by the problem, and ensure they are so constrained if required. After constructing the model, check the assumptions for a linear program and compare the form of the model to the examples in this section. Try to determine which method of optimization may be applied to obtain a solution. Resource Allocation - You have just become the manager of a plant producing plastic products. Although the plant operation involves many products and supplies, you are interested in only three of the products: (1) a vinyl- asbestos floor covering, the output of which is measured in boxed lots, each covering a certain area; (2) a pure vinyl counter top, measured in linear yards; and (3) a vinyl-asbestos wall tile, measured in squares, each covering \(100 \mathrm{ft}^{2}\). Of the many resources needed to produce these plastic products, you have identified four: vinyl, asbestos, labor, and time on a trimming machine. A recent inventory shows that on any given day you have \(1500 \mathrm{lb}\) of vinyl and \(200 \mathrm{lb}\) of asbestos available for use. Additionally, after talking to your shop foreman and to various labor leaders, you realize that you have 3 person-days of labor available for use per day and that your trimming machine is available for 1 machine-day on any given day. The following table indicates the amount of each of the four resources required to produce a unit of the three desired products, where the units are 1 box of floor cover, 1 yard of counter top, and 1 square of wall tiles. Available resources are tabulated below. $$ \begin{array}{l|rrccr} \hline & \begin{array}{c} \text { Vinyl } \\ \text { (1b) } \end{array} & \begin{array}{c} \text { Asbestos } \\ \text { (lb) } \end{array} & \begin{array}{c} \text { Labor } \\ \text { (person-days) } \end{array} & \begin{array}{c} \text { Machine } \\ \text { (machine-days) } \end{array} & \text { Profit } \\ \hline \text { Floor cover (per box) } & 30 & 3 & 0.02 & 0.01 & \$ 0.8 \\ \text { Countertop (per yard) } & 20 & 0 & 0.1 & 0.05 & 5.0 \\ \text { Wall tile (per square) } & 50 & 5 & 0.2 & 0.05 & 5.5 \\ \text { Available (per day) } & 1500 & 200 & 3 & 1 & \- \\ \hline \end{array} $$ Formulate a mathematical model to help determine how to allocate resources to maximize profits.

Solve Problems 4-7 using graphical analysis. Minimize \(x+y\) subject to $$ \begin{array}{r} x+y \geq 6 \\ 3 x-y \geq 9 \\ x, y \geq 0 \end{array} $$

Producing Electronic Equipment - An electronics firm is producing three lines of products for sale to the government: transistors, micromodules, and circuit assemblies. The firm has four physical processing areas designated as follows: transistor production, circuit printing and assembly, transistor and module quality control, and circuit assembly test and packing. The various production requirements are as follows: Production of one transistor requires \(0.1\) standard hour of transistor production area capacity, \(0.5\) standard hour of transistor quality control area capacity, and \(\$ 0.70\) in direct costs. Production of micromodules requires \(0.4\) standard hour of the quality control area capacity, three transistors, and \(\$ 0.50\) in direct costs. Production of one circuit assembly requires \(0.1\) standard hour of the capacity of the circuit printing area, \(0.5\) standard hour of the capacity of the test and packing area, one transistor, three micromodules, and \(\$ 2.00\) in direct costs. Suppose that the three products (transistors, micromodules, and circuit assemblies) may be sold in unlimited quantities at prices of \(\$ 2, \$ 8\), and \(\$ 25\) each, respectively. There are 200 hours of production time open in each of the four process areas in the coming month. Formulate a mathematical model to help determine the production that will produce the highest revenue for the firm.

Why is sensitivity analysis important in linear programming?

Optimize \(2 x+3 y\) subject to $$ \begin{aligned} 2 x+3 y & \geq 6 \\ 3 x-y & \leq 15 \\ -x+y & \leq 4 \\ 2 x+5 y & \leq 27 \end{aligned} $$
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