Schedule with limited workers - determine minimum time

5 An online shopping company selects some of its parcels to be checked before posting them. Each selected parcel must pass through three checks, which may be carried out in any order. One person must check the contents, another must check the postage and a third person must check the address. The parcels are classified according to the type of customer as 'new', 'occasional' or 'regular'. The table shows the time taken, in minutes, for each check on each type of parcel. The manager in charge of checking at the company has allocated each type of parcel a 'value' to represent how useful it is for generating additional income. In suitable units, these values are as follows. $$\text { new } = 8 \text { points } \quad \text { occasional } = 7 \text { points } \quad \text { regular } = 4 \text { points }$$ The manager wants to find out how many parcels of each type her department should check each hour, on average, to maximise the total value. She models this objective as $$\text { Maximise } P = 8 x + 7 y + 4 z .$$

1. What do the variables \(x , y\) and \(z\) represent?
2. Write down the constraints on the values of \(x , y\) and \(z\). The manager changes the value of parcels for regular customers to 0 points.
3. Explain what effect this has on the objective and simplify the constraints.
4. Use a graphical method to represent the feasible region for the manager's new problem. You should choose scales so that the feasible region can be clearly seen. Hence determine the optimal strategy. Now suppose that there is exactly one hour available for checking and the manager wants to find out how many parcels of each type her department should check in that hour to maximise the total value. The value of parcels for regular customers is still 0 points.
5. Find the optimal strategy in this situation.
6. Give a reason why, even if all the timings and values are correct, the total value may be less than this maximum. \section*{Question 6 is printed overleaf.}

1
Figure 1 below shows an activity diagram for a project. Each activity requires one worker. The duration required for each activity is given in hours.

1. Find the earliest start time and the latest finish time for each activity and insert their values on Figure 1.
2. On Figure 2 opposite, complete the precedence table.
3. Find the critical path.
4. Find the float time of activity \(E\).
5. Using Figure 3 on page 5, draw a resource histogram to illustrate how the project can be completed in the minimum time, assuming that each activity is to start as early as possible.
6. Given that there are two workers available for the project, find the minimum completion time for the project.
7. Given that there is only one worker available for the project, find the minimum completion time for the project. Figure 1 \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{(a)} \includegraphics[alt={},max width=\textwidth]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-02_629_1550_1818_292}
   \end{figure} (b) \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2}

   Activity	Immediate predecessor(s)
   A
   B
   C
   D
   E
   \(F\)
   G
   H
   I
   J
   \(K\)

   \end{table}
   \includegraphics[max width=\textwidth, alt={}]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-05_2486_1717_221_150}

1 Figure 2, on the page opposite, shows an activity diagram for a project. Each activity requires one worker. The duration required for each activity is given in hours.

1. On Figure 1 below, complete the precedence table.
2. Find the earliest start time and the latest finish time for each activity and insert their values on Figure 2.
3. List the critical paths.
4. Find the float time of activity \(E\).
5. Using Figure 3 opposite, draw a Gantt diagram to illustrate how the project can be completed in the minimum time, assuming that each activity is to start as early as possible.
6. Given that there is only one worker available for the project, find the minimum completion time for the project.
7. Given that there are two workers available for the project, find the minimum completion time for the project. Show a suitable allocation of tasks to the two workers.
   [0pt] [2 marks] \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1}

   Activity	Immediate predecessor(s)
   A
   B
   C
   D
   E
   \(F\)
   G
   \(H\)
   I
   J

   \end{table} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{b0f9523e-51dd-495f-99ec-4724243b5619-03_1071_1561_376_278}
   \end{figure} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{b0f9523e-51dd-495f-99ec-4724243b5619-03_801_1301_1644_420}
   \end{figure}

1
Figure 1 below shows an activity diagram for a project. Each activity requires one worker. The duration required for each activity is given in hours.

1. Find the earliest start time and the latest finish time for each activity and insert these values on Figure 1.
2. 1. Find the critical path.
   2. Find the float time of activity \(F\).
3. Using Figure 2 on page 3, draw a resource histogram to illustrate how the project can be completed in the minimum time, assuming that each activity is to start as early as possible.
4. 1. Given that there are two workers available for the project, find the minimum completion time for the project.
   2. Write down an allocation of tasks to the two workers that corresponds to your answer in part (d)(i). \section*{Answer space for question 1} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{34de3f03-a275-44fb-88b2-b88038bcec97-02_687_1655_1941_189}
      \end{figure} \section*{Answer space for question 1} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{34de3f03-a275-44fb-88b2-b88038bcec97-03_1115_1575_434_283}
      \end{figure}
      \includegraphics[max width=\textwidth, alt={}]{34de3f03-a275-44fb-88b2-b88038bcec97-03_1024_1593_1683_267}

\includegraphics{figure_3} A project is modelled by the activity network shown in Fig 3. The activities are represented by the edges. The number in brackets on each edge gives the time, in days, taken to complete the activity.

1. Calculate the early time and the late time for each event. Write these in the boxes on the answer sheet. [4]
2. Hence determine the critical activities and the length of the critical path. [2]
3. Obtain the total float for each of the non-critical activities. [3]
4. On the first grid on the answer sheet, draw a cascade (Gantt) chart showing the information obtained in parts (b) and (c). [4]

Each activity requires one worker. Only two workers are available.

1. On the second grid on the answer sheet, draw up a schedule and find the minimum time in which the 2 workers can complete the project. [4]

A project is undertaken by Higton Engineering Ltd. The project is broken down into 11 separate activities \(A\), \(B\), \(\ldots\), \(K\) Figure 3 below shows a completed activity network for the project, along with the earliest start time, duration, latest finish time and the number of workers required for each activity. All times and durations are given in days. \includegraphics{figure_3}

1. Write down the critical path. [1 mark]
2. Using Figure 4 below, draw a resource histogram for the project to show how the project can be completed in the minimum possible time. Assume that each activity is to start as early as possible. [3 marks] \includegraphics{figure_4}
3. Higton Engineering Ltd only has four workers available to work on the project. Find the minimum completion time for the project. Use Figure 5 below in your answer. [3 marks] \includegraphics{figure_5} Minimum completion time _____________________________________