Formulate this problem as a PERTtype system by drawing the project network. [1 mark]
20240602 21:38:21
MAT2438 OPTIMISATION AND NETWORKS
Assignment Part 4
Due 11:59pm Sunday 26th of May 2024
Submission Guidelines:
 Be clear and concise. Show all your working.
 This is an individual assignment. The assignment solutions must be your own work.
 Your assignment submission can be typed or handwritten. You must submit via the Canvas submission link.
Question
1.Students in the MAT2438 class have a group project due at the end of the semester. They have a number of tasks that they need to perform in order to get the case study completed by the due date. One organised group has drawn up a list of tasks that need completing, the order they need to be completed in and the most optimistic, most pessimist and most likely time needed to complete each task. The tasks and estimated time are shown in the table below.
Activity

Description

Predecessor

o

p

m

A

Download the case study and read it.



0

7

2

B

Have a meeting to assign tasks.

A

1

5

3

C

Get a copy of the textbook.

B

0

4

2

D

Formulate the model.

C

2

6

4

E

Write back ground material for report.

C

2

5

2

F

Include model in report.

D, E

1

3

2

G

Run the model.

D

1

6

2

H

Obtain and validate results.

G

0

2

1

I

Include results in report.

H, F

1

3

2

J

Reread and copyedit report.

I

2

4

3

K

Print and submit the report

J

0

2

1

a. Formulate this problem as a PERTtype system by drawing the project network. [1 mark]
b. Determine the earliest and latest activity start and finish times based on the most likely time for each task.
c. Determine the critical path. How long is the project expected to take?[2 marks
d. Using a three point estimate, what is the probability they finish the project in less than 24 days? [4 marks]
2. A police station is responsible for minimising crime in 3 suburbs. The number of crimes in each suburb depends on the number of patrol cars assigned to each suburb as indicated in the table below.
Suburb

Number of patrol cars assigned to suburb

0

1

2

3

4

5

1

14

10

7

4

1

0

2

25

19

16

14

12

11

3

20

14

11

8

6

5

Five patrol cars are available to the police station. Use dynamic programming to determine how many patrol cars should be assigned to each suburb. [6 marks]
Overview
The course MAT2438, Optimisation and Networks, focuses on the mathematical and computational techniques used to solve optimization problems and analyze network structures. It covers a broad spectrum of topics, including linear programming, network flows, integer programming, and combinatorial optimization. The course is crucial for students in mathematics, computer science, engineering, and related fields, offering essential tools for solving realworld problems in logistics, telecommunications, transportation, and many other areas.Students in the MAT2438 class have a group project due at the end of the semester.
Conclusion
MAT2438, Optimisation and Networks, equips students with the fundamental concepts and techniques to tackle a wide range of optimization problems and analyze complex network structures. The skills acquired in this course are invaluable for addressing practical challenges in various professional fields, making it an essential component of the curriculum for students pursuing careers in mathematics, computer science, engineering, and beyond.a. Formulate this problem as a PERTtype system by drawing the project network.
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To address your problem systematically, we`ll follow the Project Evaluation and Review Technique (PERT) steps:
a. Formulate the Project Network
First, let`s list out the activities and their dependencies:
 A: Download the case study and read it. (No predecessor)
 B: Have a meeting to assign tasks. (Predecessor: A)
 C: Get a copy of the textbook. (Predecessor: B)
 D: Formulate the model. (Predecessor: C)
 E: Write background material for report. (Predecessor: C)
 F: Include model in report. (Predecessors: D, E)
 G: Run the model. (Predecessor: D)
 H: Obtain and validate results. (Predecessor: G)
 I: Include results in report. (Predecessors: H, F)
 J: Reread and copyedit report. (Predecessor: I)
 K: Print and submit the report. (Predecessor: J)
b. Calculate Earliest and Latest Start and Finish Times
We need to use the most likely time (m) for these calculations.
1. A:
 Earliest Start (ES): 0
 Earliest Finish (EF): 0 + 2 = 2
2. B:
 ES: 2
 EF: 2 + 3 = 5
3. C:
 ES: 5
 EF: 5 + 2 = 7
4. D:
 ES: 7
 EF: 7 + 4 = 11
5. E:
 ES: 7
 EF: 7 + 2 = 9
6. F:
 ES: max(11, 9) = 11
 EF: 11 + 2 = 13
7. G:
 ES: 11
 EF: 11 + 2 = 13
8. H:
 ES: 13
 EF: 13 + 1 = 14
9. I:
 ES: max(14, 13) = 14
 EF: 14 + 2 = 16
10. J:
 ES: 16
 EF: 16 + 3 = 19
11. K:
 ES: 19
 EF: 19 + 1 = 20
Next, calculate the Latest Start (LS) and Latest Finish (LF) times:
1. K:
 LF: 20
 LS: 20  1 = 19
2. J:
 LF: 19
 LS: 19  3 = 16
3. I:
 LF: 16
 LS: 16  2 = 14
4. H:
 LF: 14
 LS: 14  1 = 13
5. G:
 LF: 13
 LS: 13  2 = 11
6. F:
 LF: 14
 LS: 14  2 = 12
7. E:
 LF: 12
 LS: 12  2 = 10
8. D:
 LF: 11
 LS: 11  4 = 7
9. C:
 LF: 7
 LS: 7  2 = 5
10. B:
 LF: 5
 LS: 5  3 = 2
11. A:
 LF: 2
 LS: 2  2 = 0
c. Determine the Critical Path
The critical path is the longest path through the project network, which determines the shortest time possible to complete the project.
From the above calculations:
 Critical Path: A > B > C > D > G > H > I > J > K
 Project Duration: 20 days
d. Probability of Finishing in Less Than 24 Days
Using the threepoint estimate (PERT), we calculate the expected duration (Te) and standard deviation (σ) for each activity:
[ Te = frac{o + 4m + p}{6} ]
[ sigma = frac{p  o}{6} ]
We`ll calculate these for each activity and then sum them for the critical path.
For each critical path activity:
A: Te = (0 + 4*2 + 7) / 6 = 2.5, σ = (7  0) / 6 = 1.17
B: Te = (1 + 4*3 + 5) / 6 = 3, σ = (5  1) / 6 = 0.67
C: Te = (0 + 4*2 + 4) / 6 = 2, σ = (4  0) / 6 = 0.67
D: Te = (2 + 4*4 + 6) / 6 = 4, σ = (6  2) / 6 = 0.67
G: Te = (1 + 4*2 + 6) / 6 = 2.5, σ = (6  1) / 6 = 0.83
H: Te = (0 + 4*1 + 2) / 6 = 1, σ = (2  0) / 6 = 0.33
I: Te = (1 + 4*2 + 3) / 6 = 2, σ = (3  1) / 6 = 0.33
J: Te = (2 + 4*3 + 4) / 6 = 3, σ = (4  2) / 6 = 0.33
K: Te = (0 + 4*1 + 2) / 6 = 1, σ = (2  0) / 6 = 0.33
Summing Te for the critical path: 2.5 + 3 + 2 + 4 + 2.5 + 1 + 2 + 3 + 1 = 21
Summing σ for the critical path:
[ sqrt{1.17^2 + 0.67^2 + 0.67^2 + 0.67^2 + 0.83^2 + 0.33^2 + 0.33^2 + 0.33^2 + 0.33^2} approx 2.03 ]
We use the normal distribution to find the probability:
[ Z = frac{(24  21)}{2.03} = frac{3}{2.03} approx 1.48 ]
Using the Ztable, the probability of Z ≤ 1.48 is about 0.9306 or 93.06%.
Thus, there is a 93.06% probability that the project will be completed in less than 24 days.
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