Chapter 4.S, Problem 6P

One of the industrial robots designed by a leading producer of servomechanisms has four major components. Components’ reliabilities are .98, .95, .94 and .90 All of the components must function in order for the robot to operate effectively.

a. Compute the reliability of the robot.

b. Designers want to improve the reliability by adding a backup component. Due to spice limitations, only one backup can be added. The backup for any component will have the same reliability as the unit for which it is the backup. Which component should get the backup in order to achieve the highest reliability?

c. If one backup with a reliability of .92 can be added to any one of the main components, which component should get it to obtain the highest overall reliability?

Summary Introduction

To determine: The reliability of the robot.

Introduction:

Reliability:

Reliability can be termed as the measure of the capability of a product or any service to perform its intended job under a predetermined set of conditions. The product or service will yield the same results on the repeated trials. It is the degree of consistency of a measure.

Answer to Problem 6P

The reliability of the robot is 0.7876.

Explanation of Solution

Given information:

• There are 4 components.
• Reliabilities are 0.98, 0.95, 0.94, 0.90

Calculation of reliability of the robot:

The reliability is calculated by multiplying the probabilities of all the components.

$\begin{array}{c}\text{Reliability}=0.98\times 0.95\times 0.94\times 0.90\\ =0.7876\end{array}$

Hence, the reliability of the robot is 0.7876.

Summary Introduction

To determine: The component that should get the backup.

Introduction:

Reliability:

Reliability can be termed as the measure of the capability of a product or any service to perform its intended job under a predetermined set of conditions. The product or service will yield the same results on the repeated trials. It is the degree of consistency of a measure.

Answer to Problem 6P

The component that should get the backup is component 4.

Explanation of Solution

Given information:

• There are 4 components.
• Reliabilities are 0.98, 0.95, 0.94, 0.90
• Each backup will have the same reliability as the component.

Calculation of reliability of the robot:

The reliability is calculated by multiplying the probabilities of all the components.

$\begin{array}{c}\text{Reliability}=0.98\times 0.95\times 0.94\times 0.90\\ =0.7876\end{array}$

Calculation of reliability of the system:

Component 1 and backup for component 1:

$\begin{array}{c}\text{Reliability}=0.98+\left[\left(1-0.98\right)\times 0.98\right]\\ =0.98+\left[0.02\times 0.98\right]\\ =0.98+0.0196\\ =0.9996\end{array}$

$\begin{array}{c}\text{Reliability of robot}=0.9996\times 0.95\times 0.94\times 0.90\\ =0.8034\end{array}$

Component 2 and backup for component 2:

$\begin{array}{c}\text{Reliability}=0.95+\left[\left(1-0.95\right)\times 0.95\right]\\ =0.95+\left[0.05\times 0.95\right]\\ =0.95+0.0475\\ =0.9975\end{array}$

$\begin{array}{c}\text{Reliability of robot}=0.98\times 0.9975\times 0.94\times 0.90\\ =0.8270\end{array}$

Component 3 and backup for component 3:

$\begin{array}{c}\text{Reliability}=0.94+\left[\left(1-0.94\right)\times 0.94\right]\\ =0.94+\left[0.06\times 0.94\right]\\ =0.94+0.564\\ =0.9964\end{array}$

$\begin{array}{c}\text{Reliability of robot}=0.98\times 0.95\times 0.9964\times 0.90\\ =0.8349\end{array}$

Component 4 and backup for component 4:

$\begin{array}{c}\text{Reliability}=0.90+\left[\left(1-0.90\right)\times 0.90\right]\\ =0.90+\left[0.10\times 0.90\right]\\ =0.90+0.09\\ =0.99\end{array}$

$\begin{array}{c}\text{Reliability of robot}=0.98\times 0.95\times 0.94\times 0.99\\ =0.8664\end{array}$

The reliability of component 4 is more when compared with other components.

Hence, the component that should get the backup is component 4.

Summary Introduction

To determine: The component that should get the backup.

Introduction:

Reliability:

Reliability can be termed as the measure of the capability of a product or any service to perform its intended job under a predetermined set of conditions. The product or service will yield the same results on the repeated trials. It is the degree of consistency of a measure.

Answer to Problem 6P

The component that should get the backup is component 4.

Explanation of Solution

Given information:

• There are 4 components.
• Reliabilities are 0.98, 0.95, 0.94, 0.90
• Backup reliability is 0.92

Calculation of reliability of the robot:

The reliability is calculated by multiplying the probabilities of all the components.

$\begin{array}{c}\text{Reliability}=0.98\times 0.95\times 0.94\times 0.90\\ =0.7876\end{array}$

Calculation of reliability of the system:

Component 1 and backup for component 1:

$\begin{array}{c}\text{Reliability}=0.98+\left[\left(1-0.98\right)\times 0.92\right]\\ =0.98+\left[0.02\times 0.92\right]\\ =0.98+0.0184\\ =0.9984\end{array}$

$\begin{array}{c}\text{Reliability of robot}=0.9984\times 0.95\times 0.94\times 0.90\\ =0.8024\end{array}$

Component 2 and backup for component 2:

$\begin{array}{c}\text{Reliability}=0.95+\left[\left(1-0.95\right)\times 0.92\right]\\ =0.95+\left[0.05\times 0.92\right]\\ =0.95+0.0460\\ =0.9960\end{array}$

$\begin{array}{c}\text{Reliability of robot}=0.98\times 0.9960\times 0.94\times 0.90\\ =0.8258\end{array}$

Component 3 and backup for component 3:

$\begin{array}{c}\text{Reliability}=0.94+\left[\left(1-0.94\right)\times 0.92\right]\\ =0.94+\left[0.06\times 0.92\right]\\ =0.94+0.552\\ =0.9952\end{array}$

$\begin{array}{c}\text{Reliability of robot}=0.98\times 0.95\times 0.9952\times 0.90\\ =0.8339\end{array}$

Component 4 and backup for component 4:

$\begin{array}{c}\text{Reliability}=0.90+\left[\left(1-0.90\right)\times 0.92\right]\\ =0.90+\left[0.10\times 0.92\right]\\ =0.90+0.0920\\ =0.9920\end{array}$

$\begin{array}{c}\text{Reliability of robot}=0.98\times 0.95\times 0.94\times 0.9920\\ =0.8681\end{array}$

The reliability of component 4 is more when compared with other components.

Hence, the component that should get the backup is component 4.