1. Excursions in Modern Mathematics Sixth Edition
Peter Tannenbaum

2. Chapter 8 The Mathematics of Scheduling
Directed Graphs and Critical Paths

3. The Mathematics of Scheduling Outline/learning Objectives
To understand and use digraph terminology.
To schedule a project on N processors using the priority-list model.
To apply the backflow algorithm to find the critical path of a project.
To implement the decreasing-time and critical- path algorithm.
To recognize optimal schedules and the difficulties faced in finding them.

4. The Mathematics of Scheduling
“Mythical Man-Month”
It takes one woman 9 months to make a baby so 9 women should be able to make a baby in 1 month. Right?

5. The Mathematics of Scheduling
8.1 The Basic Elements of Scheduling

6. The Basic Elements of Scheduling
Scheduling is part of project management within the planning phase of the System Development Life Cycle.

7. The Basic Elements of Scheduling
Overview of CPM
The purpose of Critical Path Method is to determine the sequencing and timing of events in order to complete a project.
CPM often is grouped with PERT (Program Evaluation Review Technique)
CPM activities are deterministic
PERT activities are probabilistic (probability of random events)
Deterministic because all the data is known.
Probabilistic because it deals in random events and the probability of them occurring.

8. The Basic Elements of Scheduling
History of CPM
The E.I. du Pont, de Nemours & Co. devised a technique (CPM) to control its large projects.
The technique saved the company $1 million in the first year of its implementation.
By 1959, Dr. Mauchly of the Du Pont effort established an organization to solve industrial problems using CPM.
Since that time, a great deal of work has been completed to improve upon all of these related techniques, much of it by people in the computer industry.
Retrieved from

9. The Basic Elements of Scheduling
CPM Applies to a variety of projects
NC State Building Program
Space Station Program
Polaris Ballistic Missile
Coordinating the Senior Prom
Building a House
Planning a Wedding
Apollo Space Program
Opening Ceremonies of the Olympics
Constructing the new Bay Bridge
Arranging a Basketball Game

10. The Basic Elements of Scheduling
The processors.
Every job requires workers. We will use the term processors to describe the “worker” who carry out the work. For the purposes of our discussion, we will use N (where N  2) to represent the number of processors and P1, P2, P3, …PN to denote the processors themselves.

11. The Basic Elements of Scheduling
The tasks.
In every complex project there are individual pieces of work, often called “jobs” or “tasks”. We will define a task as an indivisible unit of work that cannot be broken up into smaller units. Thus, it is always carried out by a single processor. We will use capital letters A, B, C…, to represent the tasks.

12. The Basic Elements of Scheduling
The tasks (continued).
At a particular moment in time a task can be in one of four possible states:
Completed
In Execution
Ready
Ineligible

13. The Basic Elements of Scheduling
The processing times.
The processing time for a given task X is the amount of time, without interruption, required by one processor to execute X. Thus, the notation X(5) tells us that the task called X has a processing time of 5 units (be it minutes, hours, days, or any other unit of time).

14. The Basic Elements of Scheduling
The precedence relations.
Precedence relations are formal restrictions on the order in which the tasks can be executed, much like those course prerequisites in the school catalog that tell you that you can’t take course Y until you have completed course X.

15. The Basic Elements of Scheduling
A precedence relation can be conveniently abbreviated by writing XY or described graphically in (a). Y is dependent of X.

16. The Basic Elements of Scheduling
When a pair of tasks X and Y have no precedence requirements between them, we say that the tasks are independent. Graphically, we can tell if there are no arrow connecting them (b).

17. The Basic Elements of Scheduling
Two final comments about precedence relations are in order. First, precedence relations are transitive: If XY and YZ , then it must be true that XZ as shown in (c).

18. The Basic Elements of Scheduling
The second observation is that we cannot have a set of precedence relations that form a cycle as shown in (d). Clearly, this is logically impossible!

19. The Basic Elements of Scheduling
Henry Laurence Gantt ( ) published the Gantt Chart in 1910 as a visual tool to display a project’s schedule and progress of the project.
“Gantt charts were used on large construction projects such as the Hoover Dam in 1931 and the interstate network started in 1956”

20. The Basic Elements of Scheduling
Task Schedules
Imagine you just wrecked your car. The garage is operated by two processors, P1 and P2. The repairs can be broken into four different tasks:

21. The Basic Elements of Scheduling
exterior body work (4 hours),
engine repairs (5 hours),
painting and exterior finish work (7 hours), and
repair transmission (3 hours).
The only precedence relation A C .

22. The Basic Elements of Scheduling
The schedule shown in (a) is very inefficient. All the short tasks are assigned one processor (P1) and all the long tasks to the other processor (P2).

23. The Basic Elements of Scheduling
Under this schedule, the project finishing time (the duration of the project from the start of the first task to the completion of the last task) is 12 hours. We will use Fin to denote the finishing time, so we can write Fin = 12 hours.

24. The Basic Elements of Scheduling
It looks better in schedule (b), but it violates the precedence relation A C (we cannot start task C until task A is completed).

25. The Basic Elements of Scheduling
On the other hand, if we force P2 to be idle for one hour, waiting for the green light to start task C, we get a perfectly good schedule as shown in (c).
Fin = 11 hours.

26. The Basic Elements of Scheduling
Can we do better? No! The precedence relation A(4) C(7) implies that the 11 hours is a minimum barrier that we cannot break. Thus, the schedule in (c) is an optimal schedule and Fin = 11 hours is the optimal finishing time.

27. The Basic Elements of Scheduling
From now on we will use Opt instead of Fin when we are referring to the optimal finishing time. Schedule (d) shows a different optimal schedule with finishing time Opt = 11 hours.

28. The Basic Elements of Scheduling
Minimum time for a project to be completed.
Does not take into consideration precedence relations.
Total processing time / N = Minimum Time Required

29. The Basic Elements of Scheduling
What happens if we start adding processors? (Opt = 11)
19/2 = 9.5 (3) 19/3 = 6.33 (14)
19/4 = 4.75 (25) 19/5 = 7.2 (36)
But what is the actual minimum time?

30. The Mathematics of Scheduling
8.2 Directed Graphs (DiGraphs)

31. Directed Graphs (DiGraphs)
PERT Diagrams
Program Evaluation and Review Technique
A network model that allows for randomness in activity times.
A tool used to control the length of projects
PERT was developed in the late 1950’s for the US Navy’s Polaris Project.
First used as a management tool for military projects
Adapted as an educational tool for business managers
It has the potential to reduce both the time and cost required to complete a project.

32. Directed Graphs (DiGraphs)
PERT and CRM Diagram: 3 B D A F 1 2 5 6 C E 4
Single start
node
Single finish
node

33. Directed Graphs (DiGraphs)
A directed graph or digraph for short, is a graph in which the edges have a direction associated with them, typically indicated by an arrowhead. Digraphs are particularly useful when we want to describe asymmetric relationships (A related to B does not imply that B must be related to A).

34. Directed Graphs (DiGraphs)
In a digraph, instead of talking about edges we talk about arcs. Every arc is defined by its starting and ending vertex. Thus, if we write XY, we are describing the arc in (b) as opposed to YX as shown in (c).

35. Directed Graphs (DiGraphs)
Arc-set: A list of all the arcs in a digraph
A = {CA, CB, DB, FD, FC, EC, EA}
Vertex-set: A list of all the vertices in a digraph.
V = {A, B, C, D, E, F}

36. Directed Graphs (DiGraphs)
Additional Terminology
If XY is an arc in the digraph, we say that vertex X is incident to vertex Y, or equivalently, that Y is incident from X.
The arc YZ is said to by adjacent to the arc XY if the starting point of YZ is the ending point of XY.

37. Directed Graphs (DiGraphs)
“Incident to” vs. “Incident from”

38. Directed Graphs (DiGraphs)
Additional Terminology
In a digraph, a path from vertex X to vertex W consists of a sequence of arcs XY, YZ, ZU, …, VW such that each are is adjacent to the one before it and no arc appears more than once in the sequence.
When the path starts and ends at the same vertex, we call it a cycle of the digraph.

39. Directed Graphs (DiGraphs)
Path vs. Circuit

40. Directed Graphs (DiGraphs)
Additional Terminology
In a digraph, the notion of the degree of a vertex is replaced by the concepts of indegree and outdegree. The outdegree of X is the number of arcs that have X as their starting points (outgoing arcs); the indegree of X is the number of arcs that have X as their ending point (incoming arcs).

41. Directed Graphs (DiGraphs)
Outdegree vs. Indegree

42. Directed Graphs (DiGraphs)

43. Directed Graphs (DiGraphs)
Project Digraph

44. The Mathematics of Scheduling
8.3 Scheduling with Priority Lists

45. Scheduling with Priority Lists
A priority list is a list of all the tasks prioritized in the order we prefer to execute them.
If task X comes before task Y in the priority list, then X gets priority over Y. This means that when it comes to a choice between the two, X is executed ahead of Y. However, if X is not yet ready for execution, we skip over it and move on to the first ready task after X in the priority list.
Priority List: E(5), D(2), C(7), B(5), A(6)

46. Scheduling with Priority Lists
Number of Priority Lists possible with M tasks: M!

47. Scheduling with Priority Lists
The process of scheduling tasks using a priority list and following these basic rules is known as the priority-list model for scheduling.

48. Scheduling with Priority Lists
At any particular moment in time throughout a project, a processor can be either busy or idle and a task can be ineligible, ready, in execution, or completed. At each stage of the schedule we need to keep track of the status of each task.

49. Scheduling with Priority Lists
One convenient record keeping strategy goes like this:
Task is ready to be processed.
Task is currently executing.
Task has completed executing.
Task is ineligible for execution.

50. Scheduling with Priority Lists
Priority List: A(6), B(5), C(7), D(2), E(5)
N=2

51. Scheduling with Priority Lists
Priority List: E(5), D(2), C(7), B(5), A(6)
N=2

52. Scheduling with Priority Lists
Priority List: E(5), D(2), C(7), B(5), A(6)
N=3

53. Scheduling with Priority Lists

54. Scheduling with Priority Lists

55. Scheduling with Priority Lists

56. Scheduling with Priority Lists

57. The Mathematics of Scheduling
8.4 The Decreasing-Time Algorithm

58. Decreasing-Time Algorithm
One first attempt to find a good priority list is to do the longer jobs first and leave the shorter jobs for last.
Decreasing-time priority list is the priority list listed in decreasing time order.
Decreasing-time algorithm is using that decreasing time list to create a schedule.

59. Decreasing-Time Algorithm
To use the decreasing-time algorithm we first prioritize the 15 tasks in a decreasing-time priority list: AD(8), AP(7), IW(7), AW(6), FW(6), AF(5), IF(5), ID(5), IP(4), HU(4), PU(3), PD(3), EU(2), IC (1)

60. Decreasing-Time Algorithm
Using the decreasing-time algorithm with N = 2 processors, we get the schedule shown below with project finishing time Fin = 42 hours.

61. Decreasing-Time Algorithm
Priority List:
N=2

62. Decreasing-Time Algorithm

63. Decreasing-Time Algorithm

64. The Mathematics of Scheduling
8.5 Critical Paths

65. Critical Paths and Times
In general, we would want to start the longer tasks first because they take the most amount of time.
Remember this example:
Fin = hours is the critical time.

66. Critical Paths and Times
Critical Paths and Critical Times
For a given vertex X of a project digraph, the critical path for X is the path from X to END with longest processing time. (The processing time of a path is defined to be the sum of the processing times of all the vertices in the path.) When we add the processing times of all the tasks along the critical path for a vertex X, we get the critical time for X. (By definition, the critical time of END is 0.)

67. Critical Paths and Times
Critical Paths and Critical Times
The path with longest processing time from START to END is called the critical path for the project, and the total processing time for this critical path is called the critical time for the project.

68. Critical Paths and Times
HU, IC, FW, END HU Critical Path/Time
START, AP, IF, IW, IP, HU, IC, FW, END Project Critical Path/Time

69. Critical Paths and Times
The Backflow Algorithm
Step 1. Find the critical time for every vertex of the project digraph. This is done by starting at END and working backward toward START according to the following rule: critical time for X = processing time of X plus largest critical time among the vertices incident from X.

70. Critical Paths and Times
The Backflow Algorithm
Step 1 (continued). The general idea is illustrated in the above figure. [To help with the recordkeeping, it is suggested that you write the critical time of the vertex in square brackets [ ] to distinguish it from the processing time in parentheses ( ).]

71. Critical Paths and Times
The Backflow Algorithm
Step 2. Once we have the critical time for every vertex in the project digraph, critical paths are found by just following the path along largest critical times. In other words, the critical path for any vertex X (and that includes START) is obtained by starting at X and moving to the adjacent vertex with largest critical time, and from there to the adjacent vertex with largest critical time, and so on.

72. Critical Paths and Times

73. Critical Paths and Times
Priority List:
N=2

74. Critical Paths and Times

75. Critical Paths and Times

76. The Mathematics of Scheduling
8.6 The Critical- Path Algorithm

77. The Critical-Path Algorithm
The concept of critical paths can be used to create very good schedules. The idea is to use critical times rather than processing times to prioritize the tasks. The priority list we obtain when we write the tasks in decreasing order of critical times (with ties broken randomly) is called the critical-time priority list, and the process of creating a schedule using the critical-time priority list is called the critical-path algorithm.

78. The Critical-Path Algorithm
Step 1 (Find critical times). Using the backflow algorithm, find the critical time for every task in the project. The critical times for each task are shown in red.

79. The Critical-Path Algorithm
Step 2 (Create priority list). Using the critical times obtained in Step 1, create the critical-time priority list.
The critical-time priority list for the project is AP[34], AF[32], AW[28], IF[27], IW[22], AD[18], IP[15], PL[11], HU[11], ID[10], IC[7], FW[6], PU[5], PD[3], EU[2].

80. The Critical-Path Algorithm
Step 3 (Create schedule). Using the critical-time priority list obtained in Step 2, create the schedule.
The timeline for the resulting schedule is given above. The project finishing time is Fin = 36 hours. This is a very good schedule, but it is not an optimal schedule.

81. The Critical-Path Algorithm
Step 3 (Continued). Using the critical-time priority list obtained in Step 2, create the schedule.
The above figure shows the timeline for an optimal schedule with finishing time Opt = 35 hours.

82. The Mathematics of Scheduling
8.7 Scheduling with Independent Tasks

83. Scheduling with Independent Tasks
In this section, we will briefly discuss what happens to scheduling problems in the special case when there are no precedence relations to worry about. This situation arises whenever we are scheduling tasks that are all independent. There are no efficient optimal algorithms known for scheduling, even when the tasks are all independent.

84. Scheduling with Independent Tasks
In this case, we just assign the tasks to the processors as they become free in exactly the order given by the priority list. Second, without precedence relations, the critical-path time of a task equals its processing time. This means that the critical-time list and decreasing-time list are exactly the same list, and, therefore, the decreasing-time algorithm and the critical-path algorithm become one and the same.

85. The Mathematics of Scheduling
The Relative Error
If we know the optimal finishing time Opt, we can measure how “close” a particular schedule is being optimal using the concept of relative error. For a project with finishing time Fin, the relative error (denoted by ) is given by

86. The Mathematics of Scheduling Conclusion
Precedence Relations
Project Digraph
Priority List Models
Independent Tasks