## Related questions with answers

A tea.in of industrial management university professors investigated the reliability of a manufacturing system that involves multiple production lines (Journal of Systems Sciences & Systems Engineering, Mar. 2013). An example of such a network is a system for producing integrated circuit (IC) cards with two production lines set up in sequence. Items (IC cards) first pass through Line 1 and then are processed by Line 2. Toe probability distribution of the maximum capacity level of each line is shown in the next column. Assume the lines operate independently.

$\begin{matrix} \text{Line} & \text{Maximum Capacity, x} & \text{p(x)}\\ \text{1} & \text{0} & \text{.01}\\ \text{1} & \text{12} & \text{.02}\\ \text{1} & \text{24} & \text{.02}\\ \text{1} & \text{36} & \text{.95}\\ \text{2} & \text{0} & \text{.002}\\ \text{2} & \text{35} & \text{.002}\\ \text{2} & \text{70} & \text{.996}\\ \end{matrix}$

Now consider the network of two production lines. What is the probability that a maximum capacity level of 30 items is maintained throughout the network?

Solution

VerifiedGiven:

$\begin{array}{ l | c c c c } Line\:1\:x & 0 & 12 & 24 & 36 \\ \hline p(x) & 0.01 & 0.02 & 0.02 & 0.95 \\ \\ Line\:2\:x & 0 & 35 & 70 \\ \hline p(x) & 0.002 & 0.002 & 0.996 \end{array}$

Since we assume that the lines operate independently, it is appropriate to use the **Multiplication rule** for independent events: $P(A\cap B)=P(A\text{ and }B)=P(A)\times P(B)$

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