you may need to use the appropriate technology to answer this question. mueller & associates is a urban…

you may need to use the appropriate technology to answer this question. mueller & associates is a urban planning firm that is designing a new public park in an omaha suburb. coordination of the architect and subcontractors will require a major effort to meet the 42 - week completion date requested by the owner. the mueller project manager prepared the following project network. estimates of the optimistic, most probable, and pessimistic times (in weeks) for the activities are as follows. (a) find the critical path. (enter your answers as a comma - separated list.) (b) what is the expected project completion time (in weeks)? (c) based only on the critical path, what is the estimated probability the project can be completed in 42 weeks as requested by the owner? (round your answer to four decimal places.) (d) based only on the critical path, what is the estimated probability the building project could run more than 3 months late? use 55 weeks for this calculation. (round your answer to four decimal places.)

you may need to use the appropriate technology to answer this question. mueller & associates is a urban planning firm that is designing a new public park in an omaha suburb. coordination of the architect and subcontractors will require a major effort to meet the 42 - week completion date requested by the owner. the mueller project manager prepared the following project network. estimates of the optimistic, most probable, and pessimistic times (in weeks) for the activities are as follows. (a) find the critical path. (enter your answers as a comma - separated list.) (b) what is the expected project completion time (in weeks)? (c) based only on the critical path, what is the estimated probability the project can be completed in 42 weeks as requested by the owner? (round your answer to four decimal places.) (d) based only on the critical path, what is the estimated probability the building project could run more than 3 months late? use 55 weeks for this calculation. (round your answer to four decimal places.)

Answer

Explanation:

Step1: Calculate expected time for each activity

The formula for expected time $t_e$ of an activity is $t_e=\frac{t_o + 4t_m+t_p}{6}$, where $t_o$ is the optimistic time, $t_m$ is the most - probable time and $t_p$ is the pessimistic time. For activity A: $t_{eA}=\frac{4 + 4\times8+12}{6}=\frac{4 + 32+12}{6}=\frac{48}{6}=8$ For activity B: $t_{eB}=\frac{6 + 4\times7+8}{6}=\frac{6 + 28+8}{6}=\frac{42}{6}=7$ For activity C: $t_{eC}=\frac{6+4\times12 + 18}{6}=\frac{6+48+18}{6}=\frac{72}{6}=12$ For activity D: $t_{eD}=\frac{3+4\times5+7}{6}=\frac{3 + 20+7}{6}=\frac{30}{6}=5$ For activity E: $t_{eE}=\frac{6+4\times9+10}{6}=\frac{6+36+10}{6}=\frac{52}{6}\approx8.67$ For activity F: $t_{eF}=\frac{5+4\times8+17}{6}=\frac{5+32+17}{6}=\frac{54}{6}=9$ For activity G: $t_{eG}=\frac{10+4\times15+20}{6}=\frac{10+60+20}{6}=\frac{90}{6}=15$ For activity H: $t_{eH}=\frac{5+4\times6+13}{6}=\frac{5+24+13}{6}=\frac{42}{6}=7$

Step2: Identify possible paths and their lengths

Path 1: Start - A - B - F - H - Finish, length $L_1=t_{eA}+t_{eB}+t_{eF}+t_{eH}=8 + 7+9+7=31$ Path 2: Start - A - C - G - Finish, length $L_2=t_{eA}+t_{eC}+t_{eG}=8 + 12+15=35$ Path 3: Start - A - C - F - H - Finish, length $L_3=t_{eA}+t_{eC}+t_{eF}+t_{eH}=8 + 12+9+7=36$ Path 4: Start - A - D - E - G - Finish, length $L_4=t_{eA}+t_{eD}+t_{eE}+t_{eG}=8 + 5+8.67+15=36.67$

Step3: Determine the critical path

The critical path is the longest path. So the critical path is Start - A - D - E - G - Finish.

Step4: Calculate expected project completion time

The expected project completion time is the length of the critical path. So it is $36.67$ weeks.

Step5: Calculate variances for activities on critical path

The formula for variance $\sigma^2$ of an activity is $\sigma^2=\left(\frac{t_p - t_o}{6}\right)^2$ For activity A: $\sigma_{A}^2=\left(\frac{12 - 4}{6}\right)^2=\left(\frac{8}{6}\right)^2=\frac{64}{36}\approx1.78$ For activity D: $\sigma_{D}^2=\left(\frac{7 - 3}{6}\right)^2=\left(\frac{4}{6}\right)^2=\frac{16}{36}\approx0.44$ For activity E: $\sigma_{E}^2=\left(\frac{10 - 6}{6}\right)^2=\left(\frac{4}{6}\right)^2=\frac{16}{36}\approx0.44$ For activity G: $\sigma_{G}^2=\left(\frac{20 - 10}{6}\right)^2=\left(\frac{10}{6}\right)^2=\frac{100}{36}\approx2.78$ The total variance of the critical path $\sigma^2=\sigma_{A}^2+\sigma_{D}^2+\sigma_{E}^2+\sigma_{G}^2=1.78+0.44 + 0.44+2.78=5.44$ The standard deviation $\sigma=\sqrt{5.44}\approx2.33$

Step6: Calculate probability of completing in 42 weeks

We use the z - score formula $z=\frac{x-\mu}{\sigma}$, where $x = 42$, $\mu=36.67$ and $\sigma\approx2.33$. $z=\frac{42-36.67}{2.33}=\frac{5.33}{2.33}\approx2.29$ Looking up the z - value in the standard normal distribution table, the probability $P(Z\leq2.29)\approx0.9890$

Step7: Calculate probability of running more than 3 months (55 weeks) late

$z=\frac{55 - 36.67}{2.33}=\frac{18.33}{2.33}\approx7.87$ $P(Z>7.87)=1 - P(Z\leq7.87)\approx0$ (since for a standard normal distribution, values beyond $z = 3.49$ have a very small probability)

Answer:

(a) Start, A, D, E, G, Finish (b) $36.67$ (c) $0.9890$ (d) $0.0000$