An industrial engineer is evaluating workflow in an online retail fulfillment center. Online fulfillment involves three tasks which must be performed in the following sequence: [1] pick the merchandise from inventory, [2] pack the merchandise in boxes, [3] ship the packed boxes. The amount of time it takes to complete each task is normally distributed with parameters given below: pick-(mean = 42 seconds; standard deviation=23 seconds) pack-(mean= 33 seconds; standard deviation = 12 seconds) ship-(mean = 28 seconds; standard deviation = 16 seconds) A)Assume that the time it takes to complete each task is independent of the other tasks. a. What is the probability that the pick task takes more than 60 seconds (a minute)? b. What is the probability that the entire process takes more than 120 seconds (two minutes), assuming that there is no time between tasks? Hint: the entire process requires completion of each task one after the other in the sequence shown. c. The industrial engineer times the entire process for a random sample of 20 orders (recall that there is no time between tasks). What is the probability that the entire process takes more than 120 seconds (two minutes) for at least 10 (half) of the orders in this sample? Hint: the entire process requires completion of each task one after the other in the sequence shown.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
An industrial engineer is evaluating workflow in an online retail fulfillment center. Online fulfillment involves three tasks which must be performed in the following sequence: [1] pick the merchandise from inventory, [2] pack the merchandise in boxes, [3] ship the packed boxes. The amount of time it takes to complete each task is
pick-(
pack-(mean= 33 seconds; standard deviation = 12 seconds)
ship-(mean = 28 seconds; standard deviation = 16 seconds)
A)Assume that the time it takes to complete each task is independent of the other tasks. a. What is the
b. What is the probability that the entire process takes more than 120 seconds (two minutes), assuming that there is no time between tasks? Hint: the entire process requires completion of each task one after the other in the sequence shown.
c. The industrial engineer times the entire process for a random sample of 20 orders (recall that there is no time between tasks). What is the probability that the entire process takes more than 120 seconds (two minutes) for at least 10 (half) of the orders in this sample? Hint: the entire process requires completion of each task one after the other in the sequence shown.
d. For the same sample of 20 orders, what is the probability that the sample mean for the entire process is greater than 120 seconds (two minutes)? Hint: the entire process requires completion of each task one after the other in the sequence shown.
e. By automating the ship task, the mean and standard deviation of this task’s time would be cut in half, so mean = 14 and standard deviation = 3. If workers are paid $18/hour, or $.005/second, how much would the online retailer expect to save on its 100,000 annual orders by automating
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