Find range for variable duration

5

1. The graphs below illustrate the precedences involved in running two projects, each consisting of the same activities \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E . \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Project 1} \includegraphics[alt={},max width=\textwidth]{8eba759f-38bc-4d14-ac65-9a0ee6c79741-6_280_385_429_495}
   \end{figure} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Project 2} \includegraphics[alt={},max width=\textwidth]{8eba759f-38bc-4d14-ac65-9a0ee6c79741-6_255_392_429_1187}
   \end{figure}
   1. For one activity the precedences in the two projects are different. State which activity and describe the difference.
   2. The table below shows the durations of the five activities.

      Activity	A	B	C	D	E
      Duration	2	1	\(x\)	3	2

      Give the total time for project 1 for all possible values of \(x\).
      Give the total time for project 2 for all possible values of \(x\).
2. The durations and precedences for the activities in a project are shown in the table.

   Activity	Duration	Immediate predecessors
   R	2	-
   S	1	-
   T	5	-
   w	3	R, S
   X	2	R, S, T
   Y	3	R
   Z	1	W, Y

   1. Draw an activity on arc network to represent this information.
   2. Find the early time and the late time for each event. Give the project duration and list the critical activities.

4 The table shows the activities involved in a project, their durations in hours and their immediate predecessors. The activities can be represented as an activity network.

1. Use standard algorithms to find the activities that form
   You must show working to demonstrate the use of the algorithms. Only one of the paths from part (a) has a practical interpretation.
2. What is the practical interpretation of the total weight of that path? The duration of activity E can be changed. No other durations change.
3. What is the smallest increase to the duration of E that will make activity E become part of a longest path through the network?

2 The activities involved in a project and their durations, in hours, are represented in the activity network below. \includegraphics[max width=\textwidth, alt={}, center]{74b6f747-7045-4902-8b21-0b59c007f7f6-3_446_1139_338_230}

1. Carry out a forward pass and a backward pass through the network.
2. Calculate the float for each activity. A delay means that activity B cannot finish until \(t\) hours have elapsed from the start of the project.
3. Determine the maximum value of \(t\) for which the project can be completed in 16 hours.

4. (a) Draw the activity network described in the precedence table below, using activity on arc and the minimum number of dummies. A project is modelled by the activity network drawn in (a). Each activity requires one worker. The project is to be completed in the shortest possible time. The table below gives the time, in days, to complete some of the activities. The critical activities for the project are B, F, I, J and L and the length of the critical path is 30 days.
(b) Calculate the duration of activity I.
(c) Find the range of possible values for the duration of activity K .

5.

1. Draw the activity network described in the precedence table above, using activity on arc and the minimum number of dummies. A project is modelled by the activity network drawn in (a). Each activity requires exactly one worker. The project is to be completed in the shortest possible time. The table below gives the time, in hours, to complete three of the activities.

   Activity	Duration (in hours)
   A	10
   E	7
   F	8

   The length of the critical path AEFK is 33 hours.
2. Determine the range of possible values for the duration of activity J. You must make your method and working clear.

4. (a) Draw an activity network described in this precedence table, using as few dummies as possible.

1. A different project is represented by the activity network shown in Fig. 3. The duration of each activity is shown in brackets. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{75ea31c7-11e7-4dd9-9312-4cf32bba622b-05_710_1580_1509_239}
   \end{figure} Find the range of values of \(x\) that will make \(D\) a critical activity.
   (2)

2. \begin{figure}[h] \end{figure} A project is modelled by the activity network shown in Figure 1. The activities are represented by the arcs. The number in brackets on each arc gives the time, in hours, to complete the corresponding activity. The exact duration, \(x\), of activity N is unknown, but it is given that \(5 < x < 10\) Each activity requires one worker. The project is to be completed in the shortest possible time.

1. Complete the precedence table in the answer book.
2. Complete Diagram 1 in the answer book to show the early event times and the late event times.
3. List the critical activities. It is given that activity J can be delayed by up to 4 hours without affecting the shortest possible completion time of the project.
4. Determine the value of \(x\). You must make the numbers used in your calculation clear.
5. Draw a cascade chart for this project on Grid 1 in the answer book.

3 The activities involved in a project and their durations are represented in the activity network below. \includegraphics[max width=\textwidth, alt={}, center]{a7bca340-6947-42b5-bc35-e6d429d6bed7-3_494_700_306_683}

1. Carry out a forward pass and a backward pass through the network.
2. Find the float for each activity. A delay means that the duration of activity E increases to \(x\).
3. Find the values of \(x\) for which activity E is not a critical activity.

1
Figure 1 below shows an activity diagram for a construction project. The time needed for each activity is given in days.

1. Find the earliest start time and the latest finish time for each activity and insert their values on Figure 1.
2. Find the critical paths and state the minimum time for completion of the project.
3. On Figure 2 opposite, draw a cascade diagram (Gantt chart) for the project, assuming that each activity starts as early as possible.
4. Activity \(J\) takes longer than expected so that its duration is \(x\) days, where \(x \geqslant 3\). Given that the minimum time for completion of the project is unchanged, find a further inequality relating to the maximum value of \(x\).
   1. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d0902228-7041-4449-9ccb-770352ce6bef-02_910_1355_1414_411}
      \end{figure}
   2. Critical paths are \(\_\_\_\_\) Minimum completion time is \(\_\_\_\_\) days. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{d0902228-7041-4449-9ccb-770352ce6bef-03_940_1160_390_520}
      \end{figure}
   3. \(\_\_\_\_\)

6 Answer parts (i), (ii) and (iii) of this question on the insert provided. The activity network for a project is shown below. The durations are in minutes. The events are numbered 1, 2, 3, etc. for reference. \includegraphics[max width=\textwidth, alt={}, center]{406831f5-74a3-415e-8849-2c381bfe47f4-06_747_1249_482_447}

1. Complete the table in the insert to show the immediate predecessors for each activity.
2. Explain why the dummy activity is needed between event 2 and event 3, and why the dummy activity is needed between event 4 and event 5 .
3. Carry out a forward pass to find the early event times and a backward pass to find the late event times. Record your early event times and late event times in the table in the insert. Write down the minimum project completion time and the critical activities. Suppose that the duration of activity \(K\) changes to \(x\) minutes.
4. Find, in terms of \(x\), expressions for the early event time and the late event time for event 9 .
5. Find the maximum duration of activity \(K\) that will not affect the minimum project completion time found in part (iii). \section*{ADVANCED GCE
   MATHEMATICS} Decision Mathematics 2
   INSERT for Questions 5 and 6
6. Dummy activity is needed between event 2 and event 3 because \(\_\_\_\_\) Dummy activity is needed between event 4 and event 5 because \(\_\_\_\_\)

7. Event	1	2	3	4	5	6	7	8	9	10
   Early event time
   Late event time

   Minimum project completion time = \(\_\_\_\_\) minutes Critical activities: \(\_\_\_\_\) \section*{Answer part (iv) and part (v) in your answer booklet.} OCR
   RECOGNISING ACHIEVEMENT

3. A project consists of 11 activities, some of which are dependent on others having been completed. The following precedence table summarises the relevant information.

1. Draw an activity network for the project.
2. Find the critical path and the minimum time in which the project can be completed. Activity \(F\) can be carried out more cheaply if it is allocated more time.
3. Find the maximum time that can be allocated to activity \(F\) without increasing the minimum time in which the project can be completed.