Question

A project has three activities A, B, and C that must be carried out sequentially; that is, A -> -> C. The probability distributions of the times required to complete each of the activities A, B, and C are uniformly distributed in intervals (1,6], [2,3] and (3,6) respectively. Find the total project completion time and run 1000 simulation trials in Excel. The project completion time" is a(n)... Select one: O a. Input of the simulation b. Decision variable in the simulation O c. A fixed value in the simulation O d. Output of the simulation Which of the following Excel functions would properly generate a random number for the duration of activity A in the project described above? Select one: O a. 1+ 4* RAND() O b. 1+ 5 RANDO O c. NORM.INV(RAND(),5,1) O d. NORM.INV(RAND(),1,5) O e. 5* RANDO)

Answer 1

The project consists of three sequential activities. Therefore, A is carried out first. This is followed by B, which is then followed by C. Eac of the activities follows a uniform distribution in their respective intervals. A lies in the interval [1, 6]. B lies in the interval [2, 3], and C lies in the interval [3, 6]. We are to determine the total project completion time as would be predicted by running Excel 1000 trial simulations.

1. The simulation will involve creating 1000 trials each for activities A, B, and C. The project completion trail will be the sum of the times required to complete each of the three activities. Therefore, the project completion time will be the output of the simulation. Therefore, the answer is d.

2. Each of the activities are uniformly distributed. Hence, they have an equal probability to occur throughout their interval. Excel has a function called RAND() which can be used for this purpose. RAND() also draws values from a uniform distribution but of unit size in the range 0 to 1. In order to increase the width of the scale, we can simply multiply the output by the desired width. So say for instance, I want the size of the distribution to be of 100 units, I can simply use 100 * RAND() and then the output will lie in the uniform distribution of range [0, 100]. In order to change the lower limit of the range from 0 to some other value x, we can simply add x to the RAND() function. Therefore, in order to get a uniform distribution in the range [10, 110], the command would be 10 + 100 * RAND().

Now in the question, each of the activities follow a uniform distribution with the ranges mentioned. Therefore, a random number can be drawn from the desired uniform distribution by summing the lower limit of the the distribution with the product of the size of the distribution (the difference between the maximum and the minimum values) and the RAND() function.

Therefore, for acitivity A we have 1 + 5 * RAND(), for activity B we have 2 + 1 * RAND(), and for activity C we have 3 + 3 * RAND().

Hence, the answer is b.

3. The output of the simulation will be the sum of the times taken for each of the three activities. Therefore, it can be obtained using the SUM() function of excel. In order to perform 1000 simulations, the formula from the first row can simply be dragged down for 1000 cells. In order to compute the standard deviation of the output time, we can use the excel function STDEV() and provide it with the range of the output. In my case it came out to be 1.6879, which is very close to 1.70 given in the options. Repeating the simulation should give you similar results. Therefore, the answer is d.

4. RAND() is a pseudo-random number generator as mentioned above, and it gives values between 0 and 1. If 1000 cells have this function, then 1000 values will be drawn from this interval. Therefore, the probability of getting a value less than or equal to 0.2 will be:

0.2 x = 0.2 x1=0.2 1-0

For 1000 cells, the frequency will be 1000 * 0.2 = 200.

Therefore, the number of cells that will have value less than or equal to 0.2 will be close to 200.

Hence, the correct answer is b.

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