1. The following is a demonstration of the Central Limit Theorem.
(a) Using the inversion method for generating random numbers, generate a column of 100 random numbers from the exponential distribution with l = 0.5 (so that x = (-1/l)ln(u) with u ~ U(0,1)). Make a histogram of the resulting output data.
(b) Generate 2 more columns as before (so that now you have 3). Add corresponding elements together so that you have a column of sums. Make a histogram of the sums.
(c) Generate 7 more columns as before (so now you have 10). Once again add corresponding elements together so that you have a column of sums. Make another histogram.
What do you notice about the output? Fit the data using the Batch Fit from Crystal Ball.
2. A manufacturing process has 3 stages that must be completed to finish each product. The length of time for each stage is uncertain, but data has been collected so that we can build a static simulation of the process for the purpose of determining the distribution of the total time it takes to complete the manufacturing process. Each item is completed by a single worker, so there is no queueing between stages. Thus the total manufacturing time is just the sum of the times for the stages. The following describes the stages:
Stage 1: Time is distributed normally, with mean of 27 minutes, and standard deviation of 6 minutes.
Stage 2: There are three possibilities – with probability .5, the stage will be completed in 10 minutes; with probability .3, the stage will be completed in15 minutes; and with probability .2, the stage will be completed in 20 minutes.
Stage 3. The workers were asked how long it takes to perform stage three and they agreed that it was hard to tell. They did know that it never took less than 8 minutes, never took longer than 16 minutes, and usually took around 11 minutes.
Use this information to construct a static simulation of the system. Make 300 replications, and compute the summary statistics on the results. Also construct a histogram.
Repeat this problem using Crystal Ball.