Reliability of a manufacturing network. A team of industrial management university professors investigated the reliability of a manufacturing system that involves multiple production lines (Journal of Systems Sciences & Systems Engineering, March 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, then are processed by Line 2. The probability distribution of the maximum capacity level (x) of each line is shown below. Assume the lines operate independently.

a. Verify that the properties of discrete probability distributions are satisfied for each line in the system.

b. Find the probability that the maximum capacity level for Line 1 will exceed 30 items.

c. Repeat part b for Line 2.

d. Now consider the network of two production lines. What is the probability that a maximum capacity level exceeding 30 items is maintained throughout the network? [Hint: Apply the multiplicative law of probability for independent events.]

e. Find the mean maximum capacity for each line. Interpret the results practically.

f. Find the standard deviation of the maximum capacity for each line. Then, give an interval for each line that will contain the maximum capacity with probability of at least .75. Line Maximum Capacity, x p1x2 1 0 .01 12 .02 24 .02 36 .95 2 0 .002 35 .002 70 .996 Applying the Concepts—Intermediate