Chapter 17, Problem 13P

Rick Wing, salesperson for Wave Soldering Systems, Inc. (WSSI), has provided you with a proposal for improving the temperature control on your present machine. The machine uses a hot-air knife to cleanly remove excess solder from printed circuit boards; this is a great concept, but the hot-air temperature control lacks reliability. According to Wing, engineers at WSSI have improved the reliability of the critical temperature controls. The new system still has the four sensitive integrated circuits controlling the temperature, but the new machine has a backup for each. The four integrated circuits have reliabilities of .90, .92, .94, and .96. The four backup circuits all have a reliability of .90.

a) What is the reliability of the new temperature controller?

b) If you pay a premium, Wing says he can improve all four of the backup units to .93. What is the reliability of this option?

Summary Introduction

To determine: The reliability of the new temperature controller.

Explanation of Solution

Draw a schematic diagram as shown in figure of the reliabilities of the four integrated circuits along with their respective back ups for the wave soldering machine given that $R_1=0.90$ back up $R_{1\text{'}}=0.90$, $R_2=0.92$ back up $R_{2\text{'}}=0.90$, $R_3=0.94$ back up $R_{3\text{'}}=0.90$ and $R_4=0.96$ back up $R_{4\text{'}}=0.90$

The combined reliabilities R_C with backups for each integrated circuit can be expressed as follows

$\begin{array}{c}{R}_{\text{C}}=\left(R_1+R_1{}_{\text{'}}\times \left(1-R_1\right)\right)\times \left(R_2+R_2{}_{\text{'}}\times \left(1-R_2\right)\right)\times \left(R_3+R_3{}_{\text{'}}\times \left(1-R_3\right)\right)\times \left(R_4+R_4{}_{\text{'}}\times \left(1-R_4\right)\right)\\ =\left(0.9+0.9\times \left(1-0.9\right)\right)\times \left(0.92+0.90\times \left(1-0.92\right)\right)\\ \times \left(0.94+0.90\times \left(1-0.94\right)\right)\times \left(0.96+0.90\left(1-0.96\right)\right)\\ =0.99\times 0.992\times 0.994\times 0.996\\ R_{\text{C}}=0.9723\end{array}$

The reliability of the temperature controller is 0.9723.

Summary Introduction

To determine: The reliability of this option.

Explanation of Solution

The reliability of each of the backup integrated circuits has been improved from 0.9 to 0.93. To calculate the reliability of the temperature controller, again draw a schematic diagram as shown in figure of the reliabilities of the four integrated circuits along with their respective back ups for the wave soldering machine given that $R_1=0.90$ back up $R_{1\text{'}}=0.93$, $R_2=0.92$ back up $R_{2\text{'}}=0.93$, $R_3=0.94$ back up $R_{3\text{'}}=0.93$ and $R_4=0.96$ back up $R_{4\text{'}}=0.93$

The combined reliabilities R_C with backups for each integrated circuit can be expressed as follows

$\begin{array}{c}{R}_{\text{C}}=\left(R_1+R_1{}_{\text{'}}\times \left(1-R_1\right)\right)\times \left(R_2+R_2{}_{\text{'}}\times \left(1-R_2\right)\right)\times \left(R_3+R_3{}_{\text{'}}\times \left(1-R_3\right)\right)\times \left(R_4+R_4{}_{\text{'}}\times \left(1-R_4\right)\right)\\ =\left(0.9+0.93\times \left(1-0.9\right)\right)\times \left(0.92+0.93\times \left(1-0.92\right)\right)\\ \times \left(0.94+0.93\times \left(1-0.94\right)\right)\times \left(0.96+0.93\left(1-0.96\right)\right)\\ =0.993\times 0.9944\times 0.9958\times 0.9972\\ R_{\text{C}}=0.9805\end{array}$

The reliability of the temperature controller improves to 0.9805 from an earlier figure of 0.9723.