Deck 8: The Mathematics of Scheduling: Chasing the Critical Path

Question 1

Consider the digraph with vertex set V = {I, C, H, A, B, O, D} and arc set A = {IC, ID, CH, HA, HO, AI, AB, BO, BD, OD}. Draw the digraph.

Answer 1

Here is the digraph.

Question 2

Consider the digraph shown above ; which vertices, if any, are incident to E?

Answer 2

A and B

Question 3

Consider the digraph shown above ; what is the outdegree of vertex A?

Answer 3

2

Question 4

Consider the digraph shown above ; what is the indegree of vertex B?

Question 5

Consider the project digraph shown below; apply the decreasing-time algorithm to find the project finish time fin using N = 3 processors.

Question 6

Suppose that you work at a local convenience store. The store has just closed and you and two other co-workers must complete a list of tasks before any of you can leave for the night. Each task may only be worked on by one person. What is the optimal amount of time that it will take to complete the list of tasks knowing that all tasks are independent from one another?

Question 7

The catalog at your college gives the following information regarding math classes.
A prerequisite for Calculus III is Calculus II. A prerequisite for Calculus II is Calculus I. Linear Algebra can only be taken after both Calculus II and Fundamentals of Mathematical Proofs have been taken. A prerequisite for Fundamentals of Mathematical Proofs is Calculus I. In order to take Real Analysis I, a student must have completed Calculus III and Linear Algebra. A prerequisite for Real Analysis II is Real
Analysis I.
Draw a digraph which models the relationships between the math classes Calculus I, Calculus II,
Calculus III, Fundamentals of Mathematical Proofs, Linear Algebra, Real Analysis I, and Real
Analysis II.

Question 8

Consider the project digraph shown below; you plan on applying the critical-path algorithm to complete the project. However, you have a choice of using either N = 2 processors or N = 3 processors. Is it more efficient to use three processors as opposed to two? Explain.

Question 9

A project consists of six tasks labeled A through F. The table shown below gives the processing times and precedence relations for the tasks. Construct the project digraph.

Question 10

Consider the digraph shown below; give the arc set A.

Question 11

Consider the project digraph shown below; apply the decreasing-time algorithm to find the project finish time fin using N = 2 processors.

Question 12

Consider the digraph shown above ; which arcs, if any, are adjacent to BO?

Question 13

A project consists of six tasks labeled A through F . The table shown below gives the processing times and precedence relations for these tasks. Construct the project digraph.

Question 14

Consider the project digraph from problem 8; apply the critical-path algorithm to find the project finish time fin using N = 2 processors.

Question 15

A project consists of six tasks labeled A through F. The table shown below gives the processing times and precedence relations for the tasks. Construct the project digraph.

Consider the project shown above; use the priority list A, B, C, D, E, F to find the project finish time fin using N = 2 processors.

Question 16

Consider the digraph shown above ; which arcs, if any, are adjacent to AD?

Question 17

Consider the project from problem 6; use the priority list A, B, C, D, E, F to find the project finish time
fin using N = 2 processors.

Question 18

Suppose that you work at a local convenience store. The store has just closed and you and one other co-worker must complete a list of tasks before any of you can leave for the night. Each task may only be worked on by one person. What is the optimal amount of time that it will take to complete the list of tasks knowing that all tasks are independent from one another? Hint: You are guaranteed that a
project finish time is optimal if there is zero idle time.

Question 19

Consider the project digraph shown above; apply the critical-path algorithm to find the project finish time fin using N = 3 processors.

Question 20

Consider the digraph shown above ; which vertices, if any, are incident to I?

Question 21

Consider the project digraph shown below; apply the critical-path algorithm to find the project finish time f i n using N=2 processors.

A) 15
B) 16
C) 18
D) 20
E) None of the above.

Question 22

Which of the digraphs below have an arc set of A = {GH, GO, GT, HO, HT, TO, OS}?

A) figure 1
B) figure 2
C) figure 3
D) Both figure 2 and figure 3
E) None of the above.

Question 23

A project consists of seven tasks labeled A through G. The table shown below gives the processing times and precedence relations. Which project digraph represents this situation?

A) figure I
B) figure 2
C) figure 3
D) figure 4
E) None of the above.

Question 24

Consider the project digraph shown below; what is the critical time for vertex D?

A) 2
B) 5
C) 8
D) 11
E) None of the above.

Question 25

Consider the project digraph shown below; what is the critical time for vertex A?

A) 4
B) 7
C) 11
D) 12
E) None of the above.

Question 26

Consider the digraphs shown above ; which of the digraphs have an outdegree of 3 at vertex G?

A) Only figure 1
B) Only figure 2
C) Both figure 1 and figure 2
D) Both figure 1 and figure 3
E) None of the above.

Question 27

Suppose that you work at a local convenience store. The store has just closed and you and one other co-worker must complete a list of tasks before any of you can leave for the night. Each task may only be worked on by one person. What is the optimal amount of time that it will take to complete the list of tasks knowing that all tasks are independent from one another?

A) 28
B) 35
C) 37
D) 42
E) None of the above.

Question 28

Consider the digraphs shown above ; how many vertices from the digraph shown in figure 2 are incident to vertex O?

A) 0
B) 1
C) 2
D) 3
E) None of the above.

Question 29

Consider the project digraph shown below; apply the critical-path algorithm to find the project finish time fin using N = 2 processors.

A) 17
B) 18
C) 19
D) 21
E) None of the above.

Question 30

A project consists of seven tasks labeled A through G . The table shown below gives the processing times and precedence relations. Which project digraph represents this situation?

Consider the project above; use the priority list A, C, B, F, G, D, E to find the project finish time fin using N = 3 processors.

A) 12
B) 15
C) 14
D) 16
E) None of the above.

Question 31

A project consists of seven tasks labeled A through G. The table shown below gives the processing times and precedence relations. Which project digraph represents this situation?

Consider the project shown above ; use the priority list B, D, F, E, A, G, C to find the project finish time fin using N = 3 processors.

A) 12
B) 15
C) 19
D) 21
E) None of the above.

Question 32

Consider the project digraph shown below; apply the decreasing-time algorithm to find the project finish time fin using N=2 processors.

A) 16
B) 18
C) 19
D) 22
E) None of the above.

Question 33

Consider the digraphs shown above ; how many arcs from the digraph shown in figure 2 are adjacent to arc HT?

A) 0
B) 1
C) 2
D) 3
E) None of the above.

Question 34

In a specific project, the addition of more processors will always cause the optimal time to decrease.

A) True, as long as the number of processors is less than four.
B) True, as long as the number of processors is more than four.
C) True, no matter the number of processors.
D) False.
E) None of the above.

Question 35

A project consists of seven tasks labeled A through G . The table shown below gives the processing times and precedence relations. Which project digraph represents this situation?

A) figure 1
B) figure 2
C) figure 3
D) figure 4
E) None of the above.

Question 36

In a project, which of the following relations is true?

A) The Optimal Time is always greater than the Minimal Time.
B) The Optimal Time is always less than the Minimal Time.
C) The Optimal Time is always equal to the Minimal Time.
D) The Optimal Time is always greater than or equal to the Minimal Time.
E) The Optimal Time is always less than or equal to the Minimal Time.

Question 37

Which of the digraphs below have arc OS as part of its arc set?

A) figure 2
B) figure 3
C) Both figure 1 and figure 2
D) Both figure 1 and figure 3
E) None of the above.

Question 38

Suppose that you work at a local convenience store. The store has just closed and you and three other co-workers must complete a list of tasks before any of you can leave for the night. Each task may only be worked on by one person. What is the optimal amount of time that it will take to complete the list of tasks knowing that all tasks are independent from one another?


(a) 18
(b) 22
(c)25
(d)27
(e)None of the

Question 39

Consider the project digraph shown below; apply the decreasing-time algorithm to find the project finish time fin using N = 2 processors.

Question 40

Consider the digraphs shown above ; which of the digraphs have an outdegree of 3 at vertex O?

A) Only figure 1
B) Only figure 2
C) Only figure 3
D) Both figure 1 and figure 3
E) None of the above.