that 5. (30 points) Suppose we have a miniload system with a storage rack is 60

Question

that 5. (30 points) Suppose we have a miniload system with a storage rack is 60 meters long and 20 meters high. There are two motors moving the s/r device from the bottom front of the rack to a random location to retrieve a tray. One motor moves the s/r machine horizontally at3 meters per second, while the other motor moves it vertically at 1 meter per second. The two motors can operate simultaneously so the time T from the origin to a random location (X, Y) is whichever time is longer of the time to move horizontally and to move vertically. Assume that 25% of the items cause 80% of the activity, and these items are stored in region closest to the bottom front of the rack. The travel time T is some function g(X, Y). (a) What is g(x, Y)? (b) What is P(T 5 seconds? (c) What is P[T 5 seconds |T

Explanation / Answer

a. The total travel time to (X,Y) is the maximum of the time to move the horizontal distance and vertical distance

So, g(X,Y) = max(X/3,Y)

b. For travel time to be less than 5 seconds, the maximum coordinate of the point to be retrieved can be (15,5)

So, the region of retrieval is the rectange made by (0,0), (0,5), (15,5), (15,0)

So, required probability = 15*5/(60*20) = 1/16

c. For P(T<=10), region is bounded by points (0,0), (0,10), (30,10), (30,0) and to get P(T<=5 | T<=10), we need the region as in the previous question

So, the required probability is 15*5/(30*10) = 1/4

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