Deterministic network planning CPM

Modelling a construction planning using LP software is in essence no more than translating the (graphical) relations between the different activities into mathematical equations. The relation between two activities in a construction planning can easily be translated into a mathematical formula. We will restrain ourselves to Finish-to-Start relations with a relation-duration of zero. In the critical path method the activity-duration is considered invariable. Consider the part of an AON* construction planning shown in Figure 3.3. This Finish-to-Start relationship means that activity B cannot start earlier than the earliest finish of activity A. In a mathematical equation:

where x1 represents the earliest start of activity B and x2 represents the earliest finish of activity A. We know the duration of activity A so the equation can be

*Of the two approaches for representing a network, Activity On Node (AON) and Activity On Arrow (AOA), the former is most commonly used because of its clarity and possibilities. The nodes are rectangles representing activities, the arrows represent the relationships.

Arrows show precedence relationships

Figure 3.3 Illustration of part of a construction planning rewritten as:

Arrows show precedence relationships

Figure 3.3 Illustration of part of a construction planning rewritten as:

where x1 represents the earliest start of activity B and x2 represents the earliest start of activity A and b1 represents the duration of activity A. This equation can be rewritten as:

This equation follows the standard form of a constraint from an LP model, as described in Chapter 1:

subject to:

In critical path planning we are interested in the shortest total project duration. So the objective function is to minimise the total project duration. We will explain this using the following example.

Consider a simple network planning with two parallel activities A and B. Activity A has a duration of 5 days and activity B has a duration of 3 days. The Start and Finish activities are artificial activities that define the project's start and the project's finish. We want to build a model to calculate the shortest project duration. This simple example is just to illustrate the modelling of a critical path planning.

We use the modelling ABC to build the LP model in Excel:

A

B

C

D

E

F

G

H

1

Endogenous variables

ES ST

ES A

ES B

ES FI

2

Outcome

0

0

0

0

Figure 3.4 Defining 'Adjustable' cells

A

B

C

D

E

F

G

H

1

Endogenous variables

ES ST

ES A

ES B

ES FI

2

Outcome

0

0

0

0

3

4

Objective function

-1

1

0

Figure 3.5 Defining 'Best' cell

A. Define the 'Adjustable' cells. In this case these are the earliest start dates of the different activities, following Figure 3.4. The cells B2 through E2 hold the adjustables.

B. Define the 'Best' cell. This would be the total project duration given the earliest start dates of the different activities. In Excel we create the entries following Figure 3.5, where cell F4 is the cell that needs to be minimised.

This might be confusing so we will explain this in more detail. The total project duration equals the earliest start of the artificial start-activity subtracted from the earliest start of the artificial finish-activity:

where xi represents the earliest start of the artificial start-activity and x4 represents the earliest start of the artificial finish-activity. Cell F4 must be the outcome of B4 times B2 added to the outcome of C4 times C2 added to the outcome of D4 times D2 added to the outcome of E4 times E2 or in mathematical terms:

In Excel, using the sumproduct function, we type in cell F4: =sumproduct(B$2:E$2;B4:E4)

C. Define the 'Constraints' that have to be met. These are the restrictions that represent the relationships between the different activities. In Excel we create the entries following Figure 3.6.

A

B

C

D

E

F

G

H

1

Endogenous variables

ES ST

ES A

ES B

ES FI

2

Outcome

0

0

0

0

3

4

Objective function

-1

1

0

5

required

available

6

St FS A

-1

1

0

> =

0

7

St FSB

-1

1

0

> =

0

8

A FS Fi

-1

1

0

> =

5

9

B FS Fi

-1

1

0

> =

3

Figure 3.6 Defining 'Constraints'

These are in fact representations of the following mathematical equation: anxi + ai2x2 +-----+ ainxn < bi (3.7)

Row 8 states for instance that the artificial finish-activity cannot start earlier than the earliest start of activity A added with the duration of activity A:

Where x4 represents the earliest start of the artificial finish-activity, x2 the earliest start of activity A and 5 equals the duration of activity A. We can rewrite this equation to meet the standard matrix layout:

Note that the formula created in cell F4 can be copied into cells F6 to F9. All other cells contain no formulas, just values entered.

The model is now ready to be solved. Figure 3.7 shows a screenshot of the actual solved model. As you can see the minimal project duration is 5 days. [O e_npra-1.xls]

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