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Ross Hopkins, president of Hopkins Hospitality, has developed the tasks, durations, and predecessor relationships in the following table for building new motels.

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\multirow[b]{2}{}{Activity} & \multicolumn{3}{|c|}{Time (weeks)} & \multirow{2}{}{\begin{tabular}{l} Immediate \\ Predecessor(s) \end{tabular}} & \multirow[b]{2}{}{Activity} & \multicolumn{3}{|c|}{Time (weeks)} & \multirow{2}{}{\begin{tabular}{l} Immediate \\ Predecessor(s) \end{tabular}} \\
\hline
& a & [tex]$m$[/tex] & [tex]$b$[/tex] & & & a & [tex]$m$[/tex] & [tex]$b$[/tex] & \\
\hline
A & 4 & 9 & 10 & - & G & 3 & 3 & 4 & [tex]$\overline{C, E}$[/tex] \\
\hline
B & 2 & 9 & 24 & A & H & 2 & 2 & 2 & F \\
\hline
C & 9 & 12 & 18 & A & I & 5 & 5 & 5 & F \\
\hline
D & 4 & 7 & 10 & A & J & 6 & 7 & 14 & D, G, H \\
\hline
E & 1 & 3 & 4 & B & K & 1 & 1 & 4 & I, J \\
\hline
F & 5 & 8 & 20 & C, E & & & & & \\
\hline
\end{tabular}

a) The expected (estimated) time for activity [tex]$C$[/tex] is 12.5 weeks. (Round your response to two decimal places.)

b) The variance for activity [tex]$C$[/tex] is [tex]$\square$[/tex] weeks. (Round your response to two decimal places.)



Answer :

GinnyAnswer
To solve this problem using the PERT (Program Evaluation Review Technique) formulas, let's find the expected time and the variance for activity \( C \).

### Step-by-Step Solution

a) Expected (estimated) time \( t_e \) for activity \( C \)

The PERT formula for expected time \( t_e \) is:
[tex]\[ t_e = \frac{a + 4m + b}{6} \][/tex]

Given:
- Optimistic time (\( a \)) = 9 weeks
- Most likely time (\( m \)) = 12 weeks
- Pessimistic time (\( b \)) = 18 weeks

Substitute the values into the formula:
[tex]\[ t_e = \frac{9 + 4(12) + 18}{6} \][/tex]

Calculate inside the parentheses first:
[tex]\[ t_e = \frac{9 + 48 + 18}{6} \][/tex]
[tex]\[ t_e = \frac{75}{6} \approx 12.5 \][/tex]

So, the expected (estimated) time for activity \( C \) is 12.5 weeks.

b) Variance for activity \( C \)

The PERT formula for the variance \( \sigma^2 \) is:
[tex]\[ \text{Variance} = \left( \frac{b - a}{6} \right)^2 \][/tex]

Given:
- Optimistic time (\( a \)) = 9 weeks
- Pessimistic time (\( b \)) = 18 weeks

Substitute the values into the formula:
[tex]\[ \text{Variance} = \left( \frac{18 - 9}{6} \right)^2 \][/tex]

Calculate inside the parentheses first:
[tex]\[ \text{Variance} = \left( \frac{9}{6} \right)^2 \][/tex]
[tex]\[ \text{Variance} = \left( 1.5 \right)^2 \][/tex]
[tex]\[ \text{Variance} = 2.25 \][/tex]

So, the variance for activity [tex]\( C \)[/tex] is 2.25 weeks.

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