Mathematics for E&BI
Chapter 1: Project planning
1.1 From project to network
A network is constructed to get better insight into the organization of the project
A network consists of a set of events that are connected by arcs
An arc in such a network represents an activity of the project and an event represents the
start and/or the end of an activity
The start event of an activity is the event from which this activity starts and the end event of
an activity is the event from which this activity ends.
For the sake of convenience, we will assign a number to each event: the number 1 to the
start event of the project and the highest number to the end event of the project.
The other events will be numbered in an arbitrary way.
Event 1 is the start event of activity A
Plan van aanpak: A general approach to constructing a network is to start on the left-hand
side with those activities that do not depend on any other activity. Subsequently, add those
activities that depend on these first activities, then those that only depend on the latest
activities, and so on.
There are three rules:
1. each network has precisely one start event and one end event of the project
2. each activity is represented by precisely one arc
3. any two events are connected by at most one arc
Dummy: These dummy activities are needed to keep up our conditions for a network or to
preserve the order in which the activities have to be executed.
1) The first example illustrates that a dummy activity is needed in case two activities have the
same start and end event
2) The second example shows that a dummy activity could also be used to
preserve the order in which the activities have to be handled
, 1,2 The duration of a project: project analysis
In this section we will focus on the calculation of the minimum time needed to complete a
project.
We will assign a duration to each activity
Moreover, we have to investigate what activities are critical in a project, i.e. the activities
that cannot be delayed without delaying the completion of the project
For the calculation of the minimum project time and critical activities we will introduce the
notions of earliest event time, latest event time and total float of an activity
Notation: d (i,j) -> the arc that represents this activity is directed from i to j -> where i is the
number of the start event of this activity and j is the number of the end event of this activity.
Duration of activity: The time that is needed to execute an activity is called the duration of an
activity -> noted like d(I,j)
ET(i) : The earliest event time of event -> The earliest event time for an event is the earliest
time at which all the activities starting from that event can start -> (langste weg naar dat
punt) -> the earliest event time of the start event equal to 0, i.e. ET(1) = 0
Explanation:
Suppose activity A ends at time 3 in event i, activity B ends at time 6 in event i and activity C
ends at time 5 at event i. As activities D and E have
to wait until all the activities A,B and C are ended,
they can start earliest at time 6. Hence, the
earliest event time ET(i) = 6.
Minimum project time: The minimum project time
is equal to the earliest event time of the end event of the project. (laatste moment wanneer
hele project af is. (Aka langste tijd tot einde)
Critical activities: which activities cannot be delayed without increasing the project time ->
For this reason we introduce the latest event time
Latest event time: The latest event time for an event is the latest time at which at least one
activity starting from that event must start without causing a delay to the project -> notation
LT (i) -> we take the latest event time of the end event equal to the minimum project time
Total float of an activity: The total float of an activity is the amount of time for which the
starting time of the activity can be delayed without causing an increase in the completion
time of the project if it is assumed that no other activities are delayed -> T F(i, j) -> T F(i, j) =
LT(j) − ET(i) − d(i, j)
, Example:
Critical activity/critical: path A critical activity is an activity with total float equal to zero. A
path from the start event to the end event consisting only of critical activities is called a
critical path
1,3 Application: a production process
We illustrate by means of an example the use of project planning in a production process ->
we show how to proceed if the timing of the project does not meet a given deadline
1,4 Diagnostic Exercises
Just exercises
, Chapter 2: Linear Programming (LP)
2,1 Introduction to Linear Programming
Example: Top-Cycles wants to maximize its daily revenue. In order to do so Top Cycles has to
decide how many ATB and racing frames should be produced given its restrictions on the
supply of aluminum and steel. The following table summarizes all the features of the above
described production process of bike frames of Top-Cycles:
Let us start introducing the decision variables:
X1 = number of ATB frames produced each day
X2 = number of racing frames produced each day
The expression is called Top-Cycles’ objective function: The goal of Top-Cycles is to maximize
1960x1 + 1240x2
Steel and aluminum constraints:
4x1+ 5x2 ≤ 70
6x1 + 2x2 ≤ 72
Last but not least, producing a negative number of frames makes no sense, so Top-Cycles is
confronted with the following sign restrictions x1 ≥ 0, x2 ≥ 0
Objective function: The function in a LP problem that is to be maximized -> For convenience,
in a maximization LP problem the objective function is denoted by z
Feasible region: the set of all points satisfying the constraints and sign restrictions
An optimal solution is a point in the feasible region which has the largest objective
function value of all points in the feasible region
The optimal value is the value of the objective function in the optimal solution
Stappenplan:
1) Draw the lines:
4x1+ 5x2 ≤ 70
6x1 + 2x2 ≤ 72
2) Take all points that are below both lines, but still above the x1-axis and to the right of the x2-
axis
3) Solve the intersection point.
Chapter 1: Project planning
1.1 From project to network
A network is constructed to get better insight into the organization of the project
A network consists of a set of events that are connected by arcs
An arc in such a network represents an activity of the project and an event represents the
start and/or the end of an activity
The start event of an activity is the event from which this activity starts and the end event of
an activity is the event from which this activity ends.
For the sake of convenience, we will assign a number to each event: the number 1 to the
start event of the project and the highest number to the end event of the project.
The other events will be numbered in an arbitrary way.
Event 1 is the start event of activity A
Plan van aanpak: A general approach to constructing a network is to start on the left-hand
side with those activities that do not depend on any other activity. Subsequently, add those
activities that depend on these first activities, then those that only depend on the latest
activities, and so on.
There are three rules:
1. each network has precisely one start event and one end event of the project
2. each activity is represented by precisely one arc
3. any two events are connected by at most one arc
Dummy: These dummy activities are needed to keep up our conditions for a network or to
preserve the order in which the activities have to be executed.
1) The first example illustrates that a dummy activity is needed in case two activities have the
same start and end event
2) The second example shows that a dummy activity could also be used to
preserve the order in which the activities have to be handled
, 1,2 The duration of a project: project analysis
In this section we will focus on the calculation of the minimum time needed to complete a
project.
We will assign a duration to each activity
Moreover, we have to investigate what activities are critical in a project, i.e. the activities
that cannot be delayed without delaying the completion of the project
For the calculation of the minimum project time and critical activities we will introduce the
notions of earliest event time, latest event time and total float of an activity
Notation: d (i,j) -> the arc that represents this activity is directed from i to j -> where i is the
number of the start event of this activity and j is the number of the end event of this activity.
Duration of activity: The time that is needed to execute an activity is called the duration of an
activity -> noted like d(I,j)
ET(i) : The earliest event time of event -> The earliest event time for an event is the earliest
time at which all the activities starting from that event can start -> (langste weg naar dat
punt) -> the earliest event time of the start event equal to 0, i.e. ET(1) = 0
Explanation:
Suppose activity A ends at time 3 in event i, activity B ends at time 6 in event i and activity C
ends at time 5 at event i. As activities D and E have
to wait until all the activities A,B and C are ended,
they can start earliest at time 6. Hence, the
earliest event time ET(i) = 6.
Minimum project time: The minimum project time
is equal to the earliest event time of the end event of the project. (laatste moment wanneer
hele project af is. (Aka langste tijd tot einde)
Critical activities: which activities cannot be delayed without increasing the project time ->
For this reason we introduce the latest event time
Latest event time: The latest event time for an event is the latest time at which at least one
activity starting from that event must start without causing a delay to the project -> notation
LT (i) -> we take the latest event time of the end event equal to the minimum project time
Total float of an activity: The total float of an activity is the amount of time for which the
starting time of the activity can be delayed without causing an increase in the completion
time of the project if it is assumed that no other activities are delayed -> T F(i, j) -> T F(i, j) =
LT(j) − ET(i) − d(i, j)
, Example:
Critical activity/critical: path A critical activity is an activity with total float equal to zero. A
path from the start event to the end event consisting only of critical activities is called a
critical path
1,3 Application: a production process
We illustrate by means of an example the use of project planning in a production process ->
we show how to proceed if the timing of the project does not meet a given deadline
1,4 Diagnostic Exercises
Just exercises
, Chapter 2: Linear Programming (LP)
2,1 Introduction to Linear Programming
Example: Top-Cycles wants to maximize its daily revenue. In order to do so Top Cycles has to
decide how many ATB and racing frames should be produced given its restrictions on the
supply of aluminum and steel. The following table summarizes all the features of the above
described production process of bike frames of Top-Cycles:
Let us start introducing the decision variables:
X1 = number of ATB frames produced each day
X2 = number of racing frames produced each day
The expression is called Top-Cycles’ objective function: The goal of Top-Cycles is to maximize
1960x1 + 1240x2
Steel and aluminum constraints:
4x1+ 5x2 ≤ 70
6x1 + 2x2 ≤ 72
Last but not least, producing a negative number of frames makes no sense, so Top-Cycles is
confronted with the following sign restrictions x1 ≥ 0, x2 ≥ 0
Objective function: The function in a LP problem that is to be maximized -> For convenience,
in a maximization LP problem the objective function is denoted by z
Feasible region: the set of all points satisfying the constraints and sign restrictions
An optimal solution is a point in the feasible region which has the largest objective
function value of all points in the feasible region
The optimal value is the value of the objective function in the optimal solution
Stappenplan:
1) Draw the lines:
4x1+ 5x2 ≤ 70
6x1 + 2x2 ≤ 72
2) Take all points that are below both lines, but still above the x1-axis and to the right of the x2-
axis
3) Solve the intersection point.
