Worksheet

1

For each of the following networks and activity table pairs, find the earliest starting time (EST) for each vertex:

a

Activity | Dependencies | Duration |
---|---|---|

A | - | 2 |

B | A | 1 |

C | A | 5 |

D | B | 3 |

E | D | 7 |

F | C,E | 5 |

b

Activity | Dependencies | Duration |
---|---|---|

A | - | 5 |

B | - | 2 |

C | A,B | 3 |

D | A,B | 4 |

E | C | 5 |

F | C | 6 |

G | D,E,F | 7 |

H | G | 3 |

2

For each of the following networks and activity table pairs, find the earliest starting time (EST) and latest starting time (LST) for each vertex:

a

Activity | Dependencies | Duration |
---|---|---|

A | - | 5 |

B | - | 7 |

C | - | 4 |

D | A | 2 |

E | C | 3 |

F | C | 1 |

G | A | 9 |

H | B,D,E,F,G | 6 |

b

Activity | Dependencies | Duration |
---|---|---|

A | - | 5 |

B | - | 6 |

C | - | 4 |

D | A,B | 7 |

E | C | 2 |

F | C | 3 |

G | D | 1 |

H | E,F | 5 |

I | H | 8 |

J | G,I | 9 |

c

Activity | Predecessor | Duration |
---|---|---|

A | - | 5 |

B | - | 6 |

C | - | 9 |

D | - | 4 |

E | A,B | 2 |

F | D | 3 |

G | F | 1 |

H | C,E | 7 |

I | C,E,G | 8 |

J | H | 5 |

K | I | 3 |

L | J,K | 1 |

3

For each of the following networks, find the earliest starting time (EST) and latest starting time (LST) for each vertex:

a

b

c

d

e

4

The following networks have the earliest and latest starting times marked at each vertex. Determine the critical path through each network by listing the activities in order.

a

b

5

Determine whether the following would lead to a delay of the deadline of the whole project:

a

Delaying a critical activity.

b

Delaying a non-critical activity by a duration less than its float time.

c

Delaying a non-critical activity by a duration more than its float time.

6

Determine whether each of the following statements is true or false regarding critical paths in networks:

a

There may be multiple critical paths with the same duration through a network.

b

The average duration of all paths on the network is equal to the duration of the critical path.

c

Activities on the critical path may have a non-zero float time.

d

There is only one critical path in every network.

e

The critical path is the shortest path in the network.

7

Determine whether the following statements about the critical path are true or false:

a

The critical path is the set of activities that have a negative float.

b

There is no float for any activity along the critical path.

c

Not all networks have a critical path.

d

The critical path is the set of activities that have a positive float.

e

The earliest start and latest start of all activities on the critical path are equivalent.

8

For each of the following networks:

i

Find the earliest starting time (EST) and latest starting time (LST) for each vertex.

ii

Determine the critical path through the network by listing the activities in order.

iii

Calculate the duration of the critical path.

a

b

9

For each of the following activity tables:

i

Find the earliest starting time (EST) and latest starting time (LST) for each vertex.

ii

Determine the critical path through the network by listing the activities in order.

iii

Calculate the duration of the critical path.

a

Activity | Dependencies | Duration |
---|---|---|

A | - | 5 |

B | A | 1 |

C | A | 6 |

D | A | 2 |

E | B,C,D | 4 |

F | E | 8 |

b

Activity | Dependencies | Duration |
---|---|---|

A | - | 5 |

B | - | 4 |

C | - | 6 |

D | A | 7 |

E | B | 2 |

F | D | 1 |

G | C | 3 |

H | G | 5 |

I | E,F,H | 4 |

10

Consider the following activity table:

Activity | Predecessor | Duration |
---|---|---|

A | - | 5 |

B | - | 5 |

C | B | 8 |

D | C | 4 |

E | C | 3 |

F | A | 1 |

G | D,E,F | 7 |

H | G | 8 |

I | G | 5 |

J | H,I | 2 |

a

Construct the network which represents the information in the activity table.

b

Find the earliest starting time (EST) and latest starting time (LST) for each vertex.

c

Determine a critical path through the network by listing the activities in order.

d

Calculate the duration of the critical path.

11

Calculate the float time of the following activity:

12

For each of the following networks with the critical path highlighted in red, list the non-critical activities and the float time for each activity.

a

b

c

13

For each of the following networks:

i

Determine a critical path through the network by listing the activities in order.

ii

List the non-critical activities and float time for each activity.

a

b

c

14

For each of the following networks:

i

Find the earliest starting time (EST) and latest starting time (LST) for each vertex.

ii

Determine a critical path through the network by listing the activities in order.

iii

List the non-critical activities and float time for each activity.

a

b

c

d

15

In order to insert a window, several activities need to be performed to complete the project. The following table displays the project’s activities and their descriptions, dependencies and durations:

Activity | Description | Dependencies | Duration (hours) |
---|---|---|---|

A | \text{Buy handtools} | - | 2 |

B | \text{Buy raw material} | - | 4 |

C | \text{Cut a hole in the wall} | A | 3 |

D | \text{Mix up cement} | B | 2 |

E | \text{Lay thin film of cement on the hole's borders} | C, D | 3 |

F | \text{Insert window frame} | E | 1 |

G | \text{Insert window} | F | 1 |

H | \text{Insert sealant and clean smudges} | G | 2 |

a

Construct a network which represents the information in the activity table.

b

Find the earliest starting time (EST) and latest starting time (LST) for each vertex.

c

Determine a critical path through the network by listing the activities in order.

d

Calculate the duration of the critical path.

e

If the sealant arrived one hour after that activity's earliest start time, how would that affect the project finish time?

16

The following activity table details the steps required to bake a cake:

Activity | Description | Dependencies | Duration (minutes) |
---|---|---|---|

A | \text{Look up recipe} | - | 15 |

B | \text{Buy ingredients} | A | 30 |

C | \text{Prepare cooking utensils} | A | 10 |

D | \text{Preheat oven} | - | 5 |

E | \text{Mix dry ingredients (sugar, flour,}\\ \text{baking powder, etc)} | B, C | 4 |

F | \text{Mix wet ingredients (eggs, milk, oil, etc)} | B, C | 6 |

G | \text{Combine both mixtures} | E, F | 3 |

H | \text{Grease baking pan} | B, C | 1 |

I | \text{Pour mixture into baking pan} | G, H | 2 |

J | \text{Insert baking pan into oven} | D, I | 30 |

a

Construct a network which correctly represents the information in the activity table.

b

Find the earliest starting time (EST) and latest starting time (LST) for each vertex.

c

Determine a critical path through the network by listing the activities in order.

d

Calculate the duration of the critical path.

e

If you forgot to preheat the oven at the beginning and instead turned it on after you had poured the mixture into the pan, how would that affect the total time taken to bake the cake?

17

To manufacture a product, the following steps are to be taken:

Activity | Description | Dependencies | Duration (Days) |
---|---|---|---|

A | \text{Obtain workers} | - | 7 |

B | \text{Obtain raw materials} | - | 4 |

C | \text{Obtain design from engineers} | - | 3 |

D | \text{Train workers to use machines} | A | 9 |

E | \text{Produce part 1} | B, C, D | 2 |

F | \text{Produce part 2} | B, C, D | 1 |

G | \text{Produce part 3} | B, C, D | 3 |

H | \text{Test parts} | E, F, G | 4 |

I | \text{Assemble parts} | H | 2 |

J | \text{Test product} | I | 3 |

K | \text{Start mass production} | J | 3 |

a

Construct a network which correctly represents the information in the activity table.

b

Find the earliest starting time (EST) and latest starting time (LST) for each vertex.

c

Determine a critical path through the network by listing the activities in order.

d

Calculate the duration of the critical path.

e

If the production of Part 2 was delayed by 1 day, how would that affect the overall production time?

18

At a restaurant, taking the order of and preparing a steak with mushroom sauce and a side salad goes through several stages before reaching the customer. The following table describes these stages:

Activity | Description | Dependencies | Duration (mins) |
---|---|---|---|

A | \text{Take order} | - | 5 |

B | \text{Relay order to kitchen} | A | 2 |

C | \text{Chef 1 cooks steak} | B | 9 |

D | \text{Chef 2 chops vegetables} | B | 4 |

E | \text{Chef 3 prepares the sauce} | B | 4 |

F | \text{Chef 2 finishes salad} | D | 3 |

G | \text{Chef 1 prepares the plate} | C, E, F | 2 |

H | \text{Waiter takes order to the table} | G | 2 |

a

Construct a network which correctly represents the information in the activity table.

b

Find the earliest starting time (EST) and latest starting time (LST) for each vertex.

c

Determine a critical path through the network by listing the activities in order.

d

Calculate the duration of the critical path.

e

If Chef 2 takes an extra minute chopping up the vegetables, how will that affect the time taken to reach the customer?

19

The following table describes the steps involved in producing a movie:

Activity | Description | Dependencies | Duration (Days) |
---|---|---|---|

A | \text{Obtain script from writers} | - | 5 |

B | \text{Obtain equipment and set} | - | 10 |

C | \text{Cast and hire actors} | A | 13 |

D | \text{Hire employees} | A, B | 9 |

E | \text{Ready costumes and scenes} | D | 20 |

F | \text{Record scenes} | C, E | 32 |

G | \text{Edit recordings} | F | 11 |

H | \text{Combine scenes and finalise movie} | G | 6 |

a

Construct a network which correctly represents the information in the activity table.

b

Find the earliest starting time (EST) and latest starting time (LST) for each vertex.

c

Determine a critical path through the network by listing the activities in order.

d

Calculate the duration of the critical path.

e

If delivering the movie equipment took a few more days than planned, how would that affect the movie production time?

20

The following activity table describes the steps to build a dressing table:

Activity | Description | Dependencies | Duration (Days) |
---|---|---|---|

A | \text{Obtain wood, handles, junctions..} | - | 7 |

B | \text{Obtain mirror} | - | 5 |

C | \text{Obtain tools} | - | 3 |

D | \text{Build frame} | A, C | 4 |

E | \text{Build drawers} | D | 2 |

F | \text{Build top} | D | 2 |

G | \text{Fit top} | F | 2 |

H | \text{Fit drawers} | E | 1 |

I | \text{Sand surfaces then polish} | G, H | 2 |

J | \text{Insert drawer handles} | I | 2 |

K | \text{Insert mirror} | B, I | 1 |

a

Construct a network which correctly represents the information in the activity table.

b

Find the earliest starting time (EST) and latest starting time (LST) for each vertex.

c

Find the critical path by listing the activities in order.

d

Calculate the duration of the critical path.

e

If the delivery of the mirror took an extra 5 days, how would that affect the project completion time?

21

To start a fire while camping, the following steps should be taken:

Activity | Description | Dependencies | Duration (mins) |
---|---|---|---|

A | \text{Gather logs, twigs, and dried leaves} | - | 30 |

B | \text{Obtain lighter or matches} | - | 5 |

C | \text{Pile up the dried leaves at the bottom} | A | 7 |

D | \text{Cover dried leaves with twigs} | C | 3 |

E | \text{Place two logs at the side of the pile}\\ \text{and one log across} | D | 2 |

F | \text{Light up the pile from the bottom} | B, E | 2 |

a

Construct a network which correctly represents the information in the activity table.

b

Find the earliest starting time (EST) and latest starting time (LST) for each vertex.

c

Determine a critical path through the network by listing the activities in order.

d

Calculate the duration of the critical path.

e

If finding a lighter or matches took 15 minutes longer, how would that affect the time taken to start the fire?

22

A company that produces padlocks takes the following steps in their manufacturing process:

Activity | Description | Dependencies | Duration (minutes) |
---|---|---|---|

A | \text{Obtaining raw materials} | - | 2 |

B | \text{Bolt cutting} | A | 3 |

C | \text{Drilling and cutting the body} | B | 6 |

D | \text{Machining the barrel} | B | 5 |

E | \text{Pinning the barrel} | D | 3 |

F | \text{Groove cutting shackles} | B | 1 |

G | \text{Bending shackles} | F | 1 |

H | \text{Inserting shackle into body} | C, E, G | 1 |

I | \text{Inserting barrel into body} | H | 2 |

J | \text{Testing key set} | I | 1 |

K | \text{Packaging} | J | 3 |

a

Construct a network which correctly represents the information in the activity table.

b

Find the earliest starting time (EST) and latest starting time (LST) for each vertex.

c

Determine a critical path through the network by listing the activities in order.

d

Calculate the duration of the critical path.

e

If the machine that pins the barrel broke down and took 2 minutes to repair before being operational, how would that affect the time taken to produce a padlock?

23

A project consists of 11 activities, P to Z. The project network representing the scheduling of these activities is shown below:

\text{Activity} | P | Q | R | S | T | U | V | W | X | Y | Z |

\text{Time (days)} | 4 | 5 | 9 | 8 | 5 | 9 | 6 | 8 | 7 | 9 | 7 |

\text{Immediate}\\ \text{Predecessors} | - | - | - | Q | P | Q | Q | R | T, S | U, W | V, X |

a

Construct a project network.

b

Determine the minimum completion time for the kitchen project.

c

Complete a backward scan on your project network and annotate each vertex with the latest start time.

d

State the critical path by listing the task labels.

24

Zara has hired three builders to renovate her kitchen. The table below shows the required activities, together with the times taken (in hours) and the immediate predecessors for each activity:

\text{Activity} | A | B | C | D | E | F | G | H | I |

\text{Time (hours)} | 5 | 3 | 6 | 3 | 3 | 4 | 5 | 4 | 5 |

\text{Immediate Predecessors} | - | - | - | A | B, F | C | D, E, I | A | C |

a

Create a project network for this renovation.

b

Complete a forward scan on your project network and annotate each vertex with the earliest start time for each task.

c

Complete the backward scan on your project network and annotate each vertex with the latest start time.

d

State the critical path by listing the task labels.

e

Determine the minimum completion time for the kitchen project.

f

How much delay could be introduced to task D before it would cause a change to the critical path?

g

If task C is delayed by 2.5 hours, by how much will the minimum completion time be affected?

h

The builder decides that it is now possible to commence task F, without waiting for task C to be completed. What is the new minimum completion time?

25

The following table contains information for a project in a small landscaping company:

Activity | Immediate Predecessors | Time (hours) |
---|---|---|

P | - | 1 |

Q | - | 2 |

R | P | 5 |

S | P | 2 |

T | - | 2 |

U | - | 1 |

V | QR | 4 |

W | S | 2 |

X | TU | 3 |

Y | WX | 3 |

Z | VY | 2 |

a

Construct a project network.

b

Complete a forward scan on your project network and annotate each vertex with the earliest start time for each task.

c

State the minimum completion time for this project

d

Complete a backward scan on your project network and annotate each vertex with the latest start time.

e

State the critical path by listing the task labels.

f

The landscapers start work at 7:30 am.

i

At what time of day should they have commenced Activity Y?

ii

What is the earliest time of day that Activity V can be expected to start?

iii

Due to unforseen delays, Activity X is not started until 1 pm. At what time of day is the project now expected to be completed?

g

One of piece of machinery breaks and Activity S cannot start until R is completed. Calculate the new minimum completion time for this project.

26

A small swimming club is planning a competition. The various tasks that need to be completed in preparation for the competition, A–M, are described in the precedence table below:

Job | Time (days) | Prerequisites | EST | LST |
---|---|---|---|---|

A | 2 | - | 0 | 7 |

B | 3 | A | 2 | 9 |

C | 8 | - | 0 | 0 |

D | 7 | - | 0 | 1 |

E | 7 | A | 2 | 9 |

F | 5 | E | 9 | 16 |

G | 9 | B,H,J | 12 | 12 |

H | 4 | C | 8 | 8 |

I | 6 | F,G | 21 | 21 |

J | 4 | D | 7 | 8 |

K | 5 | H,J | 12 | 16 |

L | 2 | I | 27 | 27 |

M | 8 | K | 17 | 21 |

a

State the critical path by listing the task labels.

b

Calculate the minimum completion time for the competition preparation.

c

Task J is delayed by 3 days, state its effect the minimum completion time.

27

Consider the given precedence table for preparing a special exhibition at a gallery. The times needed to complete each activity are given in days:

a

Construct an activity network for this table.

b

Determine a critical path through the network by listing the activities in order.

c

Hence, state the minimum completion time for the exhibition to be ready.

d

Find the float time for activity J.

e

The curators want to have the exhibition ready early. If they can only reduce one of the following tasks: H, E or J, which one should they reduce?

f

Determine the most they can reduce task E by and still have an effect on the overall completion time.

Task | Time (days) | Prerequisites |
---|---|---|

A | 5 | - |

B | 8 | A |

C | 11 | A |

D | 3 | A |

E | 9 | B |

F | 2 | B |

G | 6 | F |

H | 10 | D |

J | 9 | F |

K | 4 | E, G |

L | 7 | C, F, H |

M | 4 | K, J |

28

The network shown below displays the estimated times for Gore-FX to create a mask for their online store. (times shown in hours) The forwards and backwards scan with critical path highlighted is shown:

a

If task K is delayed by 3 hours, find the new completion time for a mask.

b

If task F is delayed by 4 hours, state its impact on the completion time.

c

Determine the most that task K can be reduced by and still have an effect on the overall completion time.

d

If task D is delayed by 5 hours it becomes critical and the network will contain 2 critical paths. Write the second critical path formed.

29

The network shown below displays the major steps in the production process for a toy manufacturer’s new toy (times shown in hours). The forwards and backwards scan with critical path highlighted is shown:

a

If task M is delayed by 4 hours, find the new completion time for a toy.

b

If task H is delayed by 4 hours due to a malfunction, state its impact on the completion time.

c

Determine the most that task E can be reduced by and still have an effect on the overall completion time.

d

If task L is delayed by 2 hours it becomes critical and the network will contain 2 critical paths. Write the second critical path formed.

30

A pastry chef plans on baking an elaborate wedding cake. the various jobs, A-L, are described in the precedence table below:

Task | Description | Time (hours) | Prerequisites | EST | LST |
---|---|---|---|---|---|

A | \text{Design Cake} | 1 | \text{None} | 0 | 0 |

B | \text{Bake Cake} | 2 | A | 1 | 1 |

C | \text{Make Ganache} | 0.5 | A | 1 | 1 |

D | \text{Make Fondant} | 1 | A | 1 | 1.5 |

E | \text{Chill Fondant} | 8 | D | 2 | 2.5 |

F | \text{Make Buttercream} | 0.5 | G | 3 | 10.5 |

G | \text{Chill Cake} | 6 | B | 3 | 3 |

H | \text{Cut Cake} | 0.5 | G | 9 | 9 |

J | \text{Layer Cake} | 1 | C, H | 9.5 | 9.5 |

K | \text{Cover in Fondant} | 0.5 | E, J | 10.5 | 10.5 |

L | \text{Decorate Cake} | 2 | F, K | 11 | 11 |

a

How many critical tasks are there? Justify your answer.

b

Calculate the minimum completion time for the wedding cake.

c

Explain why there would be no advantage in reducing the time taken to chill the fondant.

d

Which task has the longest leeway and how much leeway does it have?

e

Draw a labelled network diagram using the information in the precedence table.

f

Discuss a limitation of using critical path analysis for this situation.

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use forward and backward scanning to determine the earliest starting time (EST) and latest starting times (LST) for each activity in the project

use ESTs and LSTs to locate the critical path(s) for the project

use the critical path to determine the minimum time for a project to be completed

calculate float times for non-critical activities